LMFDB - Embedded newform 378.4.g.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 $ x^8 - x^7 + 18x^6 + 9x^5 + 283x^4 - 48x^3 + 186x^2 + 40x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.445324 - 0.771324i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.c.163.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-6.50453 - 11.2662i) q^{5} +(18.4787 - 1.24034i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-6.50453 - 11.2662i) q^{5} +(18.4787 - 1.24034i) q^{7} +8.00000 q^{8} +(-13.0091 + 22.5324i) q^{10} +(-13.6652 + 23.6688i) q^{11} +81.3058 q^{13} +(-20.6270 - 30.7657i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(3.11340 - 5.39257i) q^{17} +(29.0966 + 50.3968i) q^{19} +52.0363 q^{20} +54.6607 q^{22} +(-0.151573 - 0.262533i) q^{23} +(-22.1179 + 38.3094i) q^{25} +(-81.3058 - 140.826i) q^{26} +(-32.6607 + 66.4927i) q^{28} +48.9779 q^{29} +(-0.201323 + 0.348702i) q^{31} +(-16.0000 + 27.7128i) q^{32} -12.4536 q^{34} +(-134.169 - 200.116i) q^{35} +(-103.469 - 179.213i) q^{37} +(58.1932 - 100.794i) q^{38} +(-52.0363 - 90.1295i) q^{40} +322.582 q^{41} +13.5700 q^{43} +(-54.6607 - 94.6751i) q^{44} +(-0.303147 + 0.525066i) q^{46} +(-271.328 - 469.953i) q^{47} +(339.923 - 45.8396i) q^{49} +88.4717 q^{50} +(-162.612 + 281.652i) q^{52} +(326.119 - 564.854i) q^{53} +355.542 q^{55} +(147.829 - 9.92269i) q^{56} +(-48.9779 - 84.8323i) q^{58} +(141.268 - 244.683i) q^{59} +(97.9748 + 169.697i) q^{61} +0.805294 q^{62} +64.0000 q^{64} +(-528.857 - 916.007i) q^{65} +(-18.4344 + 31.9293i) q^{67} +(12.4536 + 21.5703i) q^{68} +(-212.443 + 432.504i) q^{70} +121.705 q^{71} +(-92.0556 + 159.445i) q^{73} +(-206.938 + 358.427i) q^{74} -232.773 q^{76} +(-223.157 + 454.317i) q^{77} +(-552.442 - 956.858i) q^{79} +(-104.073 + 180.259i) q^{80} +(-322.582 - 558.729i) q^{82} -993.883 q^{83} -81.0049 q^{85} +(-13.5700 - 23.5040i) q^{86} +(-109.321 + 189.350i) q^{88} +(795.853 + 1378.46i) q^{89} +(1502.42 - 100.847i) q^{91} +1.21259 q^{92} +(-542.655 + 939.907i) q^{94} +(378.520 - 655.616i) q^{95} -361.831 q^{97} +(-419.320 - 542.925i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	$ 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−6.50453	−	11.2662i	−0.581783	−	1.00768i	−0.995268	−	0.0971671i	$-0.969022\pi$
$5$	−6.50453	−	11.2662i	−0.581783	−	1.00768i	0.413485	−	0.910511i	$-0.364311\pi$
$6$		0			0
$7$	18.4787	−	1.24034i	0.997755	−	0.0669719i
$7$	18.4787	−	1.24034i	0.997755	−	0.0669719i
$8$	8.00000			0.353553
$9$		0			0
$10$	−13.0091	+	22.5324i	−0.411383	+	0.712536i
$11$	−13.6652	+	23.6688i	−0.374564	+	0.648764i	−0.990262	−	0.139218i	$-0.955541\pi$
$11$	−13.6652	+	23.6688i	−0.374564	+	0.648764i	0.615698	+	0.787983i	$0.288874\pi$
$12$		0			0
$13$	81.3058			1.73463			0.867315	−	0.497760i	$-0.165844\pi$
$13$	81.3058			1.73463			0.867315	+	0.497760i	$0.165844\pi$
$14$	−20.6270	−	30.7657i	−0.393771	−	0.587319i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	3.11340	−	5.39257i	0.0444183	−	0.0769347i	−0.842962	−	0.537974i	$-0.819190\pi$
$17$	3.11340	−	5.39257i	0.0444183	−	0.0769347i	0.887380	+	0.461039i	$0.152523\pi$
$18$		0			0
$19$	29.0966	+	50.3968i	0.351327	+	0.608517i	0.986482	−	0.163868i	$-0.0523971\pi$
$19$	29.0966	+	50.3968i	0.351327	+	0.608517i	−0.635155	+	0.772385i	$0.719064\pi$
$20$	52.0363			0.581783
$21$		0			0
$22$	54.6607			0.529714
$23$	−0.151573	−	0.262533i	−0.00137414	−	0.00238008i	0.865338	−	0.501190i	$-0.167104\pi$
$23$	−0.151573	−	0.262533i	−0.00137414	−	0.00238008i	−0.866712	+	0.498809i	$0.833771\pi$
$24$		0			0
$25$	−22.1179	+	38.3094i	−0.176943	+	0.306475i
$26$	−81.3058	−	140.826i	−0.613284	−	1.06224i
$27$		0			0
$28$	−32.6607	+	66.4927i	−0.220439	+	0.448783i
$29$	48.9779			0.313620			0.156810	−	0.987629i	$-0.449879\pi$
$29$	48.9779			0.313620			0.156810	+	0.987629i	$0.449879\pi$
$30$		0			0
$31$	−0.201323	+	0.348702i	−0.00116641	+	0.00202028i	−0.866608	−	0.498990i	$-0.833705\pi$
$31$	−0.201323	+	0.348702i	−0.00116641	+	0.00202028i	0.865442	+	0.501010i	$0.167038\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−12.4536			−0.0628169
$35$	−134.169	−	200.116i	−0.647963	−	0.966453i
$36$		0			0
$37$	−103.469	−	179.213i	−0.459735	−	0.796284i	0.539212	−	0.842170i	$-0.318722\pi$
$37$	−103.469	−	179.213i	−0.459735	−	0.796284i	−0.998947	+	0.0458863i	$0.985389\pi$
$38$	58.1932	−	100.794i	0.248426	−	0.430286i
$39$		0			0
$40$	−52.0363	−	90.1295i	−0.205691	−	0.356268i
$41$	322.582			1.22875			0.614377	−	0.789013i	$-0.289407\pi$
$41$	322.582			1.22875			0.614377	+	0.789013i	$0.289407\pi$
$42$		0			0
$43$	13.5700			0.0481258			0.0240629	−	0.999710i	$-0.492340\pi$
$43$	13.5700			0.0481258			0.0240629	+	0.999710i	$0.492340\pi$
$44$	−54.6607	−	94.6751i	−0.187282	−	0.324382i
$45$		0			0
$46$	−0.303147	+	0.525066i	−0.000971664	+	0.00168297i
$47$	−271.328	−	469.953i	−0.842068	−	1.45851i	−0.888143	−	0.459567i	$-0.848005\pi$
$47$	−271.328	−	469.953i	−0.842068	−	1.45851i	0.0460745	−	0.998938i	$-0.485329\pi$
$48$		0			0
$49$	339.923	−	45.8396i	0.991030	−	0.133643i
$50$	88.4717			0.250236
$51$		0			0
$52$	−162.612	+	281.652i	−0.433657	+	0.751117i
$53$	326.119	−	564.854i	0.845205	−	1.46394i	−0.0402385	−	0.999190i	$-0.512812\pi$
$53$	326.119	−	564.854i	0.845205	−	1.46394i	0.885443	−	0.464747i	$-0.153855\pi$
$54$		0			0
$55$	355.542			0.871661
$56$	147.829	−	9.92269i	0.352760	−	0.0236781i
$57$		0			0
$58$	−48.9779	−	84.8323i	−0.110881	−	0.192052i
$59$	141.268	−	244.683i	0.311720	−	0.539916i	−0.667015	−	0.745045i	$-0.732428\pi$
$59$	141.268	−	244.683i	0.311720	−	0.539916i	0.978735	+	0.205129i	$0.0657615\pi$
$60$		0			0
$61$	97.9748	+	169.697i	0.205646	+	0.356189i	0.950338	−	0.311219i	$-0.100737\pi$
$61$	97.9748	+	169.697i	0.205646	+	0.356189i	−0.744693	+	0.667408i	$0.767404\pi$
$62$	0.805294			0.00164956
$63$		0			0
$64$	64.0000			0.125000
$65$	−528.857	−	916.007i	−1.00918	−	1.74795i
$66$		0			0
$67$	−18.4344	+	31.9293i	−0.0336137	+	0.0582207i	−0.882343	−	0.470607i	$-0.844035\pi$
$67$	−18.4344	+	31.9293i	−0.0336137	+	0.0582207i	0.848729	+	0.528828i	$0.177368\pi$
$68$	12.4536	+	21.5703i	0.0222091	+	0.0384674i
$69$		0			0
$70$	−212.443	+	432.504i	−0.362739	+	0.738487i
$71$	121.705			0.203433			0.101716	−	0.994813i	$-0.467567\pi$
$71$	121.705			0.203433			0.101716	+	0.994813i	$0.467567\pi$
$72$		0			0
$73$	−92.0556	+	159.445i	−0.147593	+	0.255639i	−0.930337	−	0.366705i	$-0.880486\pi$
$73$	−92.0556	+	159.445i	−0.147593	+	0.255639i	0.782744	+	0.622343i	$0.213819\pi$
$74$	−206.938	+	358.427i	−0.325081	+	0.563058i
$75$		0			0
$76$	−232.773			−0.351327
$77$	−223.157	+	454.317i	−0.330274	+	0.672393i
$78$		0			0
$79$	−552.442	−	956.858i	−0.786767	−	1.36272i	−0.927938	−	0.372735i	$-0.878420\pi$
$79$	−552.442	−	956.858i	−0.786767	−	1.36272i	0.141171	−	0.989985i	$-0.454913\pi$
$80$	−104.073	+	180.259i	−0.145446	+	0.251920i
$81$		0			0
$82$	−322.582	−	558.729i	−0.434430	−	0.752455i
$83$	−993.883			−1.31437			−0.657186	−	0.753729i	$-0.728253\pi$
$83$	−993.883			−1.31437			−0.657186	+	0.753729i	$0.728253\pi$
$84$		0			0
$85$	−81.0049			−0.103367
$86$	−13.5700	−	23.5040i	−0.0170150	−	0.0294709i
$87$		0			0
$88$	−109.321	+	189.350i	−0.132428	+	0.229373i
$89$	795.853	+	1378.46i	0.947868	+	1.64176i	0.749904	+	0.661547i	$0.230100\pi$
$89$	795.853	+	1378.46i	0.947868	+	1.64176i	0.197965	+	0.980209i	$0.436567\pi$
$90$		0			0
$91$	1502.42	−	100.847i	1.73074	−	0.116171i
$92$	1.21259			0.00137414
$93$		0			0
$94$	−542.655	+	939.907i	−0.595432	+	1.03132i
$95$	378.520	−	655.616i	0.408793	−	0.708050i
$96$		0			0
$97$	−361.831			−0.378746			−0.189373	−	0.981905i	$-0.560646\pi$
$97$	−361.831			−0.378746			−0.189373	+	0.981905i	$0.560646\pi$
$98$	−419.320	−	542.925i	−0.432221	−	0.559629i
$99$		0			0
$100$	−88.4717	−	153.238i	−0.0884717	−	0.153238i
$101$	292.761	−	507.078i	0.288424	−	0.499566i	−0.685010	−	0.728534i	$-0.740202\pi$
$101$	292.761	−	507.078i	0.288424	−	0.499566i	0.973434	+	0.228969i	$0.0735353\pi$
$102$		0			0
$103$	−818.558	−	1417.78i	−0.783057	−	1.35629i	−0.930153	−	0.367172i	$-0.880326\pi$
$103$	−818.558	−	1417.78i	−0.783057	−	1.35629i	0.147096	−	0.989122i	$-0.453007\pi$
$104$	650.447			0.613284
$105$		0			0
$106$	−1304.47			−1.19530
$107$	−743.870	−	1288.42i	−0.672081	−	1.16408i	−0.977313	−	0.211800i	$-0.932067\pi$
$107$	−743.870	−	1288.42i	−0.672081	−	1.16408i	0.305232	−	0.952278i	$-0.401266\pi$
$108$		0			0
$109$	661.260	−	1145.34i	0.581075	−	1.00645i	−0.414277	−	0.910151i	$-0.635965\pi$
$109$	661.260	−	1145.34i	0.581075	−	1.00645i	0.995352	−	0.0963011i	$-0.0307012\pi$
$110$	−355.542	−	615.818i	−0.308179	−	0.533781i
$111$		0			0
$112$	−165.016	−	246.125i	−0.139219	−	0.207649i
$113$	1342.87			1.11794			0.558968	−	0.829189i	$-0.311197\pi$
$113$	1342.87			1.11794			0.558968	+	0.829189i	$0.311197\pi$
$114$		0			0
$115$	−1.97183	+	3.41531i	−0.00159890	+	0.00276938i
$116$	−97.9559	+	169.665i	−0.0784050	+	0.135801i
$117$		0			0
$118$	−565.071			−0.440839
$119$	50.8429	−	103.509i	0.0391661	−	0.0797368i
$120$		0			0
$121$	292.026	+	505.804i	0.219403	+	0.380018i
$122$	195.950	−	339.395i	0.145413	−	0.251863i
$123$		0			0
$124$	−0.805294	−	1.39481i	−0.000583206	−	0.00101014i
$125$	−1050.67			−0.751795
$126$		0			0
$127$	579.894			0.405175			0.202588	−	0.979264i	$-0.435065\pi$
$127$	579.894			0.405175			0.202588	+	0.979264i	$0.435065\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1057.71	+	1832.01i	−0.713597	+	1.23599i
$131$	1422.68	+	2464.15i	0.948853	+	1.64346i	0.747846	+	0.663873i	$0.231088\pi$
$131$	1422.68	+	2464.15i	0.948853	+	1.64346i	0.201008	+	0.979590i	$0.435578\pi$
$132$		0			0
$133$	600.176	+	895.177i	0.391292	+	0.583622i
$134$	73.7375			0.0475370
$135$		0			0
$136$	24.9072	−	43.1405i	0.0157042	−	0.0272005i
$137$	−1264.00	+	2189.32i	−0.788255	+	1.36530i	0.138780	+	0.990323i	$0.455682\pi$
$137$	−1264.00	+	2189.32i	−0.788255	+	1.36530i	−0.927035	+	0.374975i	$0.877651\pi$
$138$		0			0
$139$	1466.86			0.895092			0.447546	−	0.894261i	$-0.352298\pi$
$139$	1466.86			0.895092			0.447546	+	0.894261i	$0.352298\pi$
$140$	961.562	−	64.5425i	0.580477	−	0.0389631i
$141$		0			0
$142$	−121.705	−	210.799i	−0.0719244	−	0.124577i
$143$	−1111.06	+	1924.41i	−0.649730	+	1.12537i
$144$		0			0
$145$	−318.579	−	551.794i	−0.182459	−	0.316028i
$146$	368.222			0.208728
$147$		0			0
$148$	827.751			0.459735
$149$	1104.10	+	1912.37i	0.607059	+	1.05146i	0.991722	+	0.128400i	$0.0409842\pi$
$149$	1104.10	+	1912.37i	0.607059	+	1.05146i	−0.384663	+	0.923057i	$0.625683\pi$
$150$		0			0
$151$	−1224.97	+	2121.71i	−0.660175	+	1.14346i	0.320394	+	0.947284i	$0.396185\pi$
$151$	−1224.97	+	2121.71i	−0.660175	+	1.14346i	−0.980569	+	0.196172i	$0.937149\pi$
$152$	232.773	+	403.175i	0.124213	+	0.215143i
$153$		0			0
$154$	1010.06	−	67.7977i	0.528524	−	0.0354759i
$155$	5.23806			0.00271439
$156$		0			0
$157$	1565.68	−	2711.84i	0.795893	−	1.37853i	−0.126378	−	0.991982i	$-0.540335\pi$
$157$	1565.68	−	2711.84i	0.795893	−	1.37853i	0.922271	−	0.386544i	$-0.126331\pi$
$158$	−1104.88	+	1913.72i	−0.556328	+	0.963589i
$159$		0			0
$160$	416.290			0.205691
$161$	−3.12651	−	4.66326i	−0.00153045	−	0.00228271i
$162$		0			0
$163$	1810.22	+	3135.39i	0.869860	+	1.50664i	0.862139	+	0.506672i	$0.169125\pi$
$163$	1810.22	+	3135.39i	0.869860	+	1.50664i	0.00772179	+	0.999970i	$0.497542\pi$
$164$	−645.165	+	1117.46i	−0.307188	+	0.532066i
$165$		0			0
$166$	993.883	+	1721.46i	0.464700	+	0.804885i
$167$	1156.26			0.535775			0.267887	−	0.963450i	$-0.413674\pi$
$167$	1156.26			0.535775			0.267887	+	0.963450i	$0.413674\pi$
$168$		0			0
$169$	4413.64			2.00894
$170$	81.0049	+	140.305i	0.0365458	+	0.0632992i
$171$		0			0
$172$	−27.1400	+	47.0079i	−0.0120314	+	0.0208391i
$173$	−531.941	−	921.349i	−0.233773	−	0.404907i	0.725142	−	0.688599i	$-0.241774\pi$
$173$	−531.941	−	921.349i	−0.233773	−	0.404907i	−0.958915	+	0.283692i	$0.908441\pi$
$174$		0			0
$175$	−361.194	+	735.341i	−0.156021	+	0.317637i
$176$	437.286			0.187282
$177$		0			0
$178$	1591.71	−	2756.92i	0.670244	−	1.16090i
$179$	−875.381	+	1516.20i	−0.365526	+	0.633109i	−0.988860	−	0.148846i	$-0.952444\pi$
$179$	−875.381	+	1516.20i	−0.365526	+	0.633109i	0.623335	+	0.781955i	$0.285777\pi$
$180$		0			0
$181$	−3313.36			−1.36066			−0.680332	−	0.732904i	$-0.738164\pi$
$181$	−3313.36			−1.36066			−0.680332	+	0.732904i	$0.738164\pi$
$182$	−1677.10	−	2501.43i	−0.683047	−	1.01878i
$183$		0			0
$184$	−1.21259	−	2.10026i	−0.000485832	−	0.000841486i
$185$	−1346.03	+	2331.40i	−0.534932	+	0.926529i
$186$		0			0
$187$	85.0903	+	147.381i	0.0332750	+	0.0576340i
$188$	2170.62			0.842068
$189$		0			0
$190$	−1514.08			−0.578120
$191$	−265.789	−	460.360i	−0.100690	−	0.174400i	0.811279	−	0.584659i	$-0.198772\pi$
$191$	−265.789	−	460.360i	−0.100690	−	0.174400i	−0.911969	+	0.410259i	$0.865438\pi$
$192$		0			0
$193$	1145.72	−	1984.44i	0.427309	−	0.740121i	−0.569324	−	0.822113i	$-0.692795\pi$
$193$	1145.72	−	1984.44i	0.427309	−	0.740121i	0.996633	+	0.0819920i	$0.0261282\pi$
$194$	361.831	+	626.710i	0.133907	+	0.231934i
$195$		0			0
$196$	−521.053	+	1269.21i	−0.189888	+	0.462539i
$197$	2967.84			1.07335			0.536674	−	0.843789i	$-0.319680\pi$
$197$	2967.84			1.07335			0.536674	+	0.843789i	$0.319680\pi$
$198$		0			0
$199$	−1568.96	+	2717.52i	−0.558899	+	0.968041i	0.438690	+	0.898638i	$0.355443\pi$
$199$	−1568.96	+	2717.52i	−0.558899	+	0.968041i	−0.997589	+	0.0694024i	$0.977891\pi$
$200$	−176.943	+	306.475i	−0.0625590	+	0.108355i
$201$		0			0
$202$	−1171.05			−0.407894
$203$	905.048	−	60.7491i	0.312916	−	0.0210037i
$204$		0			0
$205$	−2098.25	−	3634.27i	−0.714868	−	1.23819i
$206$	−1637.12	+	2835.57i	−0.553705	+	0.959045i
$207$		0			0
$208$	−650.447	−	1126.61i	−0.216829	−	0.375558i
$209$	−1590.44			−0.526379
$210$		0			0
$211$	1306.36			0.426226			0.213113	−	0.977028i	$-0.431640\pi$
$211$	1306.36			0.426226			0.213113	+	0.977028i	$0.431640\pi$
$212$	1304.47	+	2259.42i	0.422602	+	0.731969i
$213$		0			0
$214$	−1487.74	+	2576.84i	−0.475233	+	0.823127i
$215$	−88.2667	−	152.882i	−0.0279988	−	0.0484953i
$216$		0			0
$217$	−3.28768	+	6.69327i	−0.00102849	+	0.00209386i
$218$	−2645.04			−0.821765
$219$		0			0
$220$	−711.085	+	1231.64i	−0.217915	+	0.377440i
$221$	253.138	−	438.447i	0.0770493	−	0.133453i
$222$		0			0
$223$	2155.86			0.647387			0.323693	−	0.946162i	$-0.395075\pi$
$223$	2155.86			0.647387			0.323693	+	0.946162i	$0.395075\pi$
$224$	−261.286	+	531.942i	−0.0779370	+	0.158669i
$225$		0			0
$226$	−1342.87	−	2325.92i	−0.395250	−	0.684594i
$227$	−2564.22	+	4441.37i	−0.749751	+	1.29861i	0.198191	+	0.980163i	$0.436493\pi$
$227$	−2564.22	+	4441.37i	−0.749751	+	1.29861i	−0.947942	+	0.318444i	$0.896840\pi$
$228$		0			0
$229$	−2973.64	−	5150.49i	−0.858094	−	1.48626i	−0.873744	−	0.486386i	$-0.838315\pi$
$229$	−2973.64	−	5150.49i	−0.858094	−	1.48626i	0.0156498	−	0.999878i	$-0.495018\pi$
$230$	7.88732			0.00226119
$231$		0			0
$232$	391.824			0.110881
$233$	−2567.16	−	4446.46i	−0.721804	−	1.25020i	−0.960276	−	0.279052i	$-0.909980\pi$
$233$	−2567.16	−	4446.46i	−0.721804	−	1.25020i	0.238471	−	0.971150i	$-0.423354\pi$
$234$		0			0
$235$	−3529.72	+	6113.66i	−0.979803	+	1.69707i
$236$	565.071	+	978.732i	0.155860	+	0.269958i
$237$		0			0
$238$	−230.126	+	15.4467i	−0.0626759	+	0.00420697i
$239$	7241.83			1.95998			0.979989	−	0.199050i	$-0.0637858\pi$
$239$	7241.83			1.95998			0.979989	+	0.199050i	$0.0637858\pi$
$240$		0			0
$241$	3332.02	−	5771.23i	0.890599	−	1.54256i	0.0514393	−	0.998676i	$-0.483619\pi$
$241$	3332.02	−	5771.23i	0.890599	−	1.54256i	0.839159	−	0.543886i	$-0.183048\pi$
$242$	584.052	−	1011.61i	0.155142	−	0.268713i
$243$		0			0
$244$	−783.798			−0.205646
$245$	−2727.48	−	3531.47i	−0.711234	−	0.920888i
$246$		0			0
$247$	2365.72	+	4097.56i	0.609423	+	1.05555i
$248$	−1.61059	+	2.78962i	−0.000412389	+	0.000714278i
$249$		0			0
$250$	1050.67	+	1819.81i	0.265800	+	0.460379i
$251$	−6024.90			−1.51509			−0.757546	−	0.652781i	$-0.773602\pi$
$251$	−6024.90			−1.51509			−0.757546	+	0.652781i	$0.773602\pi$
$252$		0			0
$253$	8.28511			0.00205882
$254$	−579.894	−	1004.41i	−0.143251	−	0.248118i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1936.59	+	3354.27i	0.470043	+	0.814138i	0.999413	−	0.0342531i	$-0.0109052\pi$
$257$	1936.59	+	3354.27i	0.470043	+	0.814138i	−0.529371	+	0.848391i	$0.677572\pi$
$258$		0			0
$259$	−2134.25	−	3183.29i	−0.512031	−	0.763707i
$260$	4230.85			1.00918
$261$		0			0
$262$	2845.35	−	4928.30i	0.670941	−	1.16210i
$263$	2416.95	−	4186.28i	0.566675	−	0.981511i	−0.430216	−	0.902726i	$-0.641563\pi$
$263$	2416.95	−	4186.28i	0.566675	−	0.981511i	0.996892	−	0.0787847i	$-0.0251040\pi$
$264$		0			0
$265$	−8485.00			−1.96690
$266$	950.316	−	1934.71i	0.219051	−	0.445958i
$267$		0			0
$268$	−73.7375	−	127.717i	−0.0168069	−	0.0291103i
$269$	1026.81	−	1778.49i	0.232736	−	0.403110i	−0.725876	−	0.687825i	$-0.758566\pi$
$269$	1026.81	−	1778.49i	0.232736	−	0.403110i	0.958612	+	0.284715i	$0.0918989\pi$
$270$		0			0
$271$	−82.2034	−	142.380i	−0.0184262	−	0.0319151i	0.856665	−	0.515873i	$-0.172532\pi$
$271$	−82.2034	−	142.380i	−0.0184262	−	0.0319151i	−0.875091	+	0.483958i	$0.839199\pi$
$272$	−99.6288			−0.0222091
$273$		0			0
$274$	5056.01			1.11476
$275$	−604.491	−	1047.01i	−0.132553	−	0.229589i
$276$		0			0
$277$	−1454.42	+	2519.13i	−0.315479	+	0.546425i	−0.979539	−	0.201254i	$-0.935498\pi$
$277$	−1454.42	+	2519.13i	−0.315479	+	0.546425i	0.664061	+	0.747679i	$0.268832\pi$
$278$	−1466.86	−	2540.68i	−0.316463	−	0.548130i
$279$		0			0
$280$	−1073.35	−	1600.93i	−0.229090	−	0.341693i
$281$	4676.92			0.992888			0.496444	−	0.868069i	$-0.334639\pi$
$281$	4676.92			0.992888			0.496444	+	0.868069i	$0.334639\pi$
$282$		0			0
$283$	−843.271	+	1460.59i	−0.177128	+	0.306795i	−0.940896	−	0.338696i	$-0.890014\pi$
$283$	−843.271	+	1460.59i	−0.177128	+	0.306795i	0.763768	+	0.645491i	$0.223347\pi$
$284$	−243.410	+	421.599i	−0.0508582	+	0.0880890i
$285$		0			0
$286$	4444.24			0.918857
$287$	5960.90	−	400.111i	1.22600	−	0.0822919i
$288$		0			0
$289$	2437.11	+	4221.20i	0.496054	+	0.859191i
$290$	−637.157	+	1103.59i	−0.129018	+	0.223465i
$291$		0			0
$292$	−368.222	−	637.780i	−0.0737965	−	0.127819i
$293$	−6140.31			−1.22430			−0.612151	−	0.790741i	$-0.709696\pi$
$293$	−6140.31			−1.22430			−0.612151	+	0.790741i	$0.709696\pi$
$294$		0			0
$295$	−3675.53			−0.725415
$296$	−827.751	−	1433.71i	−0.162541	−	0.281529i
$297$		0			0
$298$	2208.21	−	3824.73i	0.429256	−	0.743492i
$299$	−12.3238	−	21.3455i	−0.00238363	−	0.00412856i
$300$		0			0
$301$	250.756	−	16.8314i	0.0480177	−	0.00322307i
$302$	4899.87			0.933628
$303$		0			0
$304$	465.546	−	806.349i	0.0878319	−	0.152129i
$305$	1274.56	−	2207.60i	0.239282	−	0.414449i
$306$		0			0
$307$	−7548.41			−1.40329			−0.701645	−	0.712526i	$-0.747551\pi$
$307$	−7548.41			−1.40329			−0.701645	+	0.712526i	$0.747551\pi$
$308$	−1127.49	−	1681.67i	−0.208586	−	0.311111i
$309$		0			0
$310$	−5.23806	−	9.07259i	−0.000959683	−	0.00166222i
$311$	−850.079	+	1472.38i	−0.154995	+	0.268460i	−0.933057	−	0.359728i	$-0.882870\pi$
$311$	−850.079	+	1472.38i	−0.154995	+	0.268460i	0.778062	+	0.628188i	$0.216203\pi$
$312$		0			0
$313$	1467.61	+	2541.98i	0.265030	+	0.459045i	0.967571	−	0.252597i	$-0.0812848\pi$
$313$	1467.61	+	2541.98i	0.265030	+	0.459045i	−0.702542	+	0.711643i	$0.747951\pi$
$314$	−6262.73			−1.12556
$315$		0			0
$316$	4419.54			0.786767
$317$	−1817.34	−	3147.72i	−0.321993	−	0.557709i	0.658906	−	0.752225i	$-0.271020\pi$
$317$	−1817.34	−	3147.72i	−0.321993	−	0.557709i	−0.980899	+	0.194517i	$0.937686\pi$
$318$		0			0
$319$	−669.292	+	1159.25i	−0.117471	+	0.203465i
$320$	−416.290	−	721.036i	−0.0727229	−	0.125960i
$321$		0			0
$322$	−4.95049	+	10.0785i	−0.000856771	+	0.00174427i
$323$	362.358			0.0624214
$324$		0			0
$325$	−1798.32	+	3114.78i	−0.306931	+	0.531621i
$326$	3620.44	−	6270.78i	0.615084	−	1.06536i
$327$		0			0
$328$	2580.66			0.434430
$329$	−5596.68	−	8347.58i	−0.937857	−	1.39884i
$330$		0			0
$331$	199.480	+	345.509i	0.0331251	+	0.0573743i	0.882113	−	0.471038i	$-0.156121\pi$
$331$	199.480	+	345.509i	0.0331251	+	0.0573743i	−0.848988	+	0.528413i	$0.822787\pi$
$332$	1987.77	−	3442.91i	0.328593	−	0.569140i
$333$		0			0
$334$	−1156.26	−	2002.71i	−0.189425	−	0.328094i
$335$	479.628			0.0782236
$336$		0			0
$337$	−1094.51			−0.176918			−0.0884592	−	0.996080i	$-0.528194\pi$
$337$	−1094.51			−0.176918			−0.0884592	+	0.996080i	$0.528194\pi$
$338$	−4413.64	−	7644.65i	−0.710268	−	1.23022i
$339$		0			0
$340$	162.010	−	280.609i	0.0258418	−	0.0447593i
$341$	−5.50224	−	9.53016i	−0.000873792	−	0.00151345i
$342$		0			0
$343$	6224.47	−	1268.67i	0.979854	−	0.199714i
$344$	108.560			0.0170150
$345$		0			0
$346$	−1063.88	+	1842.70i	−0.165302	+	0.286312i
$347$	1373.78	−	2379.45i	0.212531	−	0.368115i	−0.739975	−	0.672634i	$-0.765163\pi$
$347$	1373.78	−	2379.45i	0.212531	−	0.368115i	0.952506	+	0.304520i	$0.0984960\pi$
$348$		0			0
$349$	−2780.82			−0.426516			−0.213258	−	0.976996i	$-0.568407\pi$
$349$	−2780.82			−0.426516			−0.213258	+	0.976996i	$0.568407\pi$
$350$	1634.84	−	109.735i	0.249674	−	0.0167588i
$351$		0			0
$352$	−437.286	−	757.401i	−0.0662142	−	0.114686i
$353$	−510.735	+	884.620i	−0.0770076	+	0.133381i	−0.901958	−	0.431825i	$-0.857870\pi$
$353$	−510.735	+	884.620i	−0.0770076	+	0.133381i	0.824950	+	0.565206i	$0.191203\pi$
$354$		0			0
$355$	−791.635	−	1371.15i	−0.118354	−	0.204995i
$356$	−6366.83			−0.947868
$357$		0			0
$358$	3501.53			0.516931
$359$	−1956.68	−	3389.06i	−0.287659	−	0.498239i	0.685592	−	0.727986i	$-0.259544\pi$
$359$	−1956.68	−	3389.06i	−0.287659	−	0.498239i	−0.973250	+	0.229747i	$0.926210\pi$
$360$		0			0
$361$	1736.27	−	3007.31i	0.253138	−	0.438448i
$362$	3313.36	+	5738.91i	0.481067	+	0.833233i
$363$		0			0
$364$	−2655.51	+	5406.24i	−0.382380	+	0.778473i
$365$	2395.12			0.343469
$366$		0			0
$367$	−2359.67	+	4087.06i	−0.335623	+	0.581316i	−0.983604	−	0.180340i	$-0.942280\pi$
$367$	−2359.67	+	4087.06i	−0.335623	+	0.581316i	0.647981	+	0.761656i	$0.275613\pi$
$368$	−2.42517	+	4.20053i	−0.000343535	+	0.000595020i
$369$		0			0
$370$	5384.14			0.756508
$371$	5325.63	−	10842.3i	0.745265	−	1.51726i
$372$		0			0
$373$	−3950.30	−	6842.11i	−0.548361	−	0.949789i	−0.998387	−	0.0567733i	$-0.981919\pi$
$373$	−3950.30	−	6842.11i	−0.548361	−	0.949789i	0.450026	−	0.893015i	$-0.351415\pi$
$374$	170.181	−	294.762i	0.0235290	−	0.0407534i
$375$		0			0
$376$	−2170.62	−	3759.63i	−0.297716	−	0.515659i
$377$	3982.19			0.544014
$378$		0			0
$379$	−6062.83			−0.821706			−0.410853	−	0.911702i	$-0.634769\pi$
$379$	−6062.83			−0.821706			−0.410853	+	0.911702i	$0.634769\pi$
$380$	1514.08	+	2622.46i	0.204396	+	0.354025i
$381$		0			0
$382$	−531.577	+	920.719i	−0.0711986	+	0.123320i
$383$	−1373.25	−	2378.53i	−0.183210	−	0.317330i	0.759762	−	0.650202i	$-0.225316\pi$
$383$	−1373.25	−	2378.53i	−0.183210	−	0.317330i	−0.942972	+	0.332872i	$0.891982\pi$
$384$		0			0
$385$	6569.95	−	440.992i	0.869704	−	0.0583767i
$386$	−4582.88			−0.604307
$387$		0			0
$388$	723.662	−	1253.42i	0.0946865	−	0.164002i
$389$	−6931.88	+	12006.4i	−0.903497	+	1.56490i	−0.0805753	+	0.996749i	$0.525676\pi$
$389$	−6931.88	+	12006.4i	−0.903497	+	1.56490i	−0.822922	+	0.568155i	$0.807658\pi$
$390$		0			0
$391$	−1.88764			−0.000244148
$392$	2719.39	−	366.716i	0.350382	−	0.0472499i
$393$		0			0
$394$	−2967.84	−	5140.45i	−0.379486	−	0.657289i
$395$	−7186.76	+	12447.8i	−0.915456	+	1.58562i
$396$		0			0
$397$	2267.06	+	3926.67i	0.286601	+	0.496408i	0.972996	−	0.230821i	$-0.0741412\pi$
$397$	2267.06	+	3926.67i	0.286601	+	0.496408i	−0.686395	+	0.727229i	$0.740808\pi$
$398$	6275.85			0.790402
$399$		0			0
$400$	707.774			0.0884717
$401$	−3616.09	−	6263.25i	−0.450322	−	0.779980i	0.548084	−	0.836423i	$-0.315357\pi$
$401$	−3616.09	−	6263.25i	−0.450322	−	0.779980i	−0.998406	+	0.0564433i	$0.982024\pi$
$402$		0			0
$403$	−16.3688	+	28.3515i	−0.00202329	+	0.00350444i
$404$	1171.05	+	2028.31i	0.144212	+	0.249783i
$405$		0			0
$406$	−1010.27	−	1506.84i	−0.123495	−	0.184195i
$407$	5655.68			0.688800
$408$		0			0
$409$	−2831.73	+	4904.70i	−0.342348	+	0.592963i	−0.984868	−	0.173305i	$-0.944555\pi$
$409$	−2831.73	+	4904.70i	−0.342348	+	0.592963i	0.642521	+	0.766268i	$0.277889\pi$
$410$	−4196.50	+	7268.55i	−0.505488	+	0.875531i
$411$		0			0
$412$	6548.46			0.783057
$413$	2306.95	−	4696.64i	0.274861	−	0.559580i
$414$		0			0
$415$	6464.75	+	11197.3i	0.764679	+	1.32446i
$416$	−1300.89	+	2253.21i	−0.153321	+	0.265560i
$417$		0			0
$418$	1590.44	+	2754.73i	0.186103	+	0.322340i
$419$	1506.26			0.175622			0.0878109	−	0.996137i	$-0.472013\pi$
$419$	1506.26			0.175622			0.0878109	+	0.996137i	$0.472013\pi$
$420$		0			0
$421$	−12171.3			−1.40901			−0.704506	−	0.709698i	$-0.748831\pi$
$421$	−12171.3			−1.40901			−0.704506	+	0.709698i	$0.748831\pi$
$422$	−1306.36	−	2262.69i	−0.150694	−	0.261009i
$423$		0			0
$424$	2608.95	−	4518.83i	0.298825	−	0.517580i
$425$	137.724	+	238.545i	0.0157190	+	0.0272262i
$426$		0			0
$427$	2020.93	+	3014.26i	0.229039	+	0.341617i
$428$	5950.96			0.672081
$429$		0			0
$430$	−176.533	+	305.765i	−0.0197981	+	0.0342914i
$431$	1010.84	−	1750.82i	0.112971	−	0.195671i	−0.803996	−	0.594635i	$-0.797297\pi$
$431$	1010.84	−	1750.82i	0.112971	−	0.195671i	0.916967	+	0.398964i	$0.130630\pi$
$432$		0			0
$433$	5605.18			0.622096			0.311048	−	0.950394i	$-0.399320\pi$
$433$	5605.18			0.622096			0.311048	+	0.950394i	$0.399320\pi$
$434$	14.8808	−	0.998835i	0.00164585	−	0.000110474i
$435$		0			0
$436$	2645.04	+	4581.34i	0.290538	+	0.503226i
$437$	8.82055	−	15.2776i	0.000965547	−	0.00167238i
$438$		0			0
$439$	6372.97	+	11038.3i	0.692859	+	1.20007i	0.970897	+	0.239497i	$0.0769825\pi$
$439$	6372.97	+	11038.3i	0.692859	+	1.20007i	−0.278038	+	0.960570i	$0.589684\pi$
$440$	2844.34			0.308179
$441$		0			0
$442$	−1012.55			−0.108964
$443$	−1303.21	−	2257.23i	−0.139768	−	0.242086i	0.787640	−	0.616135i	$-0.211302\pi$
$443$	−1303.21	−	2257.23i	−0.139768	−	0.242086i	−0.927409	+	0.374049i	$0.877969\pi$
$444$		0			0
$445$	10353.3	−	17932.5i	1.10291	−	1.91029i
$446$	−2155.86	−	3734.06i	−0.228886	−	0.396442i
$447$		0			0
$448$	1182.64	−	79.3815i	0.124719	−	0.00837148i
$449$	7512.78			0.789644			0.394822	−	0.918758i	$-0.370806\pi$
$449$	7512.78			0.789644			0.394822	+	0.918758i	$0.370806\pi$
$450$		0			0
$451$	−4408.15	+	7635.13i	−0.460247	+	0.797171i
$452$	−2685.75	+	4651.85i	−0.279484	+	0.484081i
$453$		0			0
$454$	10256.9			1.06031
$455$	−10908.7	−	16270.6i	−1.12398	−	1.67644i
$456$		0			0
$457$	−978.068	−	1694.06i	−0.100114	−	0.173403i	0.811617	−	0.584189i	$-0.198587\pi$
$457$	−978.068	−	1694.06i	−0.100114	−	0.173403i	−0.911731	+	0.410787i	$0.865254\pi$
$458$	−5947.28	+	10301.0i	−0.606764	+	1.05095i
$459$		0			0
$460$	−7.88732	−	13.6612i	−0.000799452	−	0.00138469i
$461$	4344.87			0.438961			0.219480	−	0.975617i	$-0.429564\pi$
$461$	4344.87			0.438961			0.219480	+	0.975617i	$0.429564\pi$
$462$		0			0
$463$	12010.6			1.20557			0.602786	−	0.797903i	$-0.294057\pi$
$463$	12010.6			1.20557			0.602786	+	0.797903i	$0.294057\pi$
$464$	−391.824	−	678.658i	−0.0392025	−	0.0679007i
$465$		0			0
$466$	−5134.33	+	8892.91i	−0.510393	+	0.884026i
$467$	2711.13	+	4695.82i	0.268643	+	0.465303i	0.968512	−	0.248968i	$-0.0800915\pi$
$467$	2711.13	+	4695.82i	0.268643	+	0.465303i	−0.699869	+	0.714271i	$0.746758\pi$
$468$		0			0
$469$	−301.040	+	612.876i	−0.0296391	+	0.0603411i
$470$	14118.9			1.38565
$471$		0			0
$472$	1130.14	−	1957.46i	0.110210	−	0.190889i
$473$	−185.437	+	321.186i	−0.0180262	+	0.0312223i
$474$		0			0
$475$	−2574.23			−0.248660
$476$	256.881	+	383.144i	0.0247355	+	0.0368936i
$477$		0			0
$478$	−7241.83	−	12543.2i	−0.692957	−	1.20024i
$479$	−1821.01	+	3154.09i	−0.173704	+	0.300864i	−0.939712	−	0.341967i	$-0.888907\pi$
$479$	−1821.01	+	3154.09i	−0.173704	+	0.300864i	0.766008	+	0.642831i	$0.222240\pi$
$480$		0			0
$481$	−8412.63	−	14571.1i	−0.797469	−	1.38126i
$482$	−13328.1			−1.25950
$483$		0			0
$484$	−2336.21			−0.219403
$485$	2353.54	+	4076.45i	0.220348	+	0.381654i
$486$		0			0
$487$	−2332.62	+	4040.21i	−0.217045	+	0.375933i	−0.953903	−	0.300114i	$-0.902975\pi$
$487$	−2332.62	+	4040.21i	−0.217045	+	0.375933i	0.736858	+	0.676047i	$0.236308\pi$
$488$	783.798	+	1357.58i	0.0727067	+	0.125932i
$489$		0			0
$490$	−3389.21	+	8255.60i	−0.312467	+	0.761123i
$491$	−8983.95			−0.825744			−0.412872	−	0.910789i	$-0.635474\pi$
$491$	−8983.95			−0.825744			−0.412872	+	0.910789i	$0.635474\pi$
$492$		0			0
$493$	152.488	−	264.117i	0.0139305	−	0.0241282i
$494$	4731.45	−	8195.11i	0.430927	−	0.746388i
$495$		0			0
$496$	6.44235			0.000583206
$497$	2248.95	−	150.955i	0.202976	−	0.0136243i
$498$		0			0
$499$	−91.1916	−	157.948i	−0.00818095	−	0.0141698i	0.861906	−	0.507068i	$-0.169271\pi$
$499$	−91.1916	−	157.948i	−0.00818095	−	0.0141698i	−0.870087	+	0.492898i	$0.835937\pi$
$500$	2101.33	−	3639.61i	0.187949	−	0.325537i
$501$		0			0
$502$	6024.90	+	10435.4i	0.535666	+	0.927801i
$503$	16507.0			1.46325			0.731623	−	0.681709i	$-0.238763\pi$
$503$	16507.0			1.46325			0.731623	+	0.681709i	$0.238763\pi$
$504$		0			0
$505$	−7617.11			−0.671202
$506$	−8.28511	−	14.3502i	−0.000727901	−	0.00126076i
$507$		0			0
$508$	−1159.79	+	2008.81i	−0.101294	+	0.175446i
$509$	8221.25	+	14239.6i	0.715914	+	1.24000i	0.962606	+	0.270906i	$0.0873231\pi$
$509$	8221.25	+	14239.6i	0.715914	+	1.24000i	−0.246692	+	0.969094i	$0.579344\pi$
$510$		0			0
$511$	−1503.30	+	3060.51i	−0.130141	+	0.264949i
$512$	512.000			0.0441942
$513$		0			0
$514$	3873.17	−	6708.53i	0.332370	−	0.575682i
$515$	−10648.7	+	18444.0i	−0.911139	+	1.57814i
$516$		0			0
$517$	14831.0			1.26163
$518$	−3379.37	+	6879.93i	−0.286643	+	0.583565i
$519$		0			0
$520$	−4230.85	−	7328.05i	−0.356798	−	0.617993i
$521$	−8460.51	+	14654.0i	−0.711443	+	1.23226i	0.252873	+	0.967500i	$0.418625\pi$
$521$	−8460.51	+	14654.0i	−0.711443	+	1.23226i	−0.964316	+	0.264756i	$0.914709\pi$
$522$		0			0
$523$	401.147	+	694.806i	0.0335390	+	0.0580913i	0.882308	−	0.470673i	$-0.155989\pi$
$523$	401.147	+	694.806i	0.0335390	+	0.0580913i	−0.848769	+	0.528764i	$0.822656\pi$
$524$	−11381.4			−0.948853
$525$		0			0
$526$	−9667.81			−0.801400
$527$	1.25360	+	2.17130i	0.000103620	+	0.000179475i
$528$		0			0
$529$	6083.45	−	10536.9i	0.499996	−	0.866019i
$530$	8485.00	+	14696.5i	0.695406	+	1.20448i
$531$		0			0
$532$	−4301.34	+	288.717i	−0.350539	+	0.0235291i
$533$	26227.8			2.13143
$534$		0			0
$535$	−9677.06	+	16761.2i	−0.782011	+	1.35448i
$536$	−147.475	+	255.434i	−0.0118842	+	0.0205841i
$537$		0			0
$538$	−4107.26			−0.329138
$539$	−3560.14	+	8671.97i	−0.284501	+	0.693002i
$540$		0			0
$541$	−3844.50	−	6658.87i	−0.305523	−	0.529181i	0.671855	−	0.740683i	$-0.265498\pi$
$541$	−3844.50	−	6658.87i	−0.305523	−	0.529181i	−0.977378	+	0.211502i	$0.932165\pi$
$542$	−164.407	+	284.761i	−0.0130293	+	0.0225674i
$543$		0			0
$544$	99.6288	+	172.562i	0.00785212	+	0.0136003i
$545$	−17204.8			−1.35224
$546$		0			0
$547$	−22735.9			−1.77718			−0.888590	−	0.458702i	$-0.848314\pi$
$547$	−22735.9			−1.77718			−0.888590	+	0.458702i	$0.848314\pi$
$548$	−5056.01	−	8757.26i	−0.394128	−	0.682649i
$549$		0			0
$550$	−1208.98	+	2094.02i	−0.0937294	+	0.162344i
$551$	1425.09	+	2468.33i	0.110183	+	0.190843i
$552$		0			0
$553$	−11395.2	−	16996.3i	−0.876264	−	1.30697i
$554$	5817.67			0.446154
$555$		0			0
$556$	−2933.73	+	5081.37i	−0.223773	+	0.387586i
$557$	−2610.78	+	4522.01i	−0.198604	+	0.343992i	−0.948076	−	0.318044i	$-0.896974\pi$
$557$	−2610.78	+	4522.01i	−0.198604	+	0.343992i	0.749472	+	0.662036i	$0.230307\pi$
$558$		0			0
$559$	1103.32			0.0834804
$560$	−1699.54	+	3460.03i	−0.128248	+	0.261095i
$561$		0			0
$562$	−4676.92	−	8100.66i	−0.351039	−	0.608017i
$563$	−10098.1	+	17490.5i	−0.755926	+	1.30930i	0.188987	+	0.981980i	$0.439480\pi$
$563$	−10098.1	+	17490.5i	−0.755926	+	1.30930i	−0.944913	+	0.327322i	$0.893854\pi$
$564$		0			0
$565$	−8734.76	−	15129.1i	−0.650397	−	1.12652i
$566$	3373.08			0.250497
$567$		0			0
$568$	973.641			0.0719244
$569$	12415.9	+	21504.9i	0.914763	+	1.58442i	0.807248	+	0.590212i	$0.200956\pi$
$569$	12415.9	+	21504.9i	0.914763	+	1.58442i	0.107515	+	0.994203i	$0.465711\pi$
$570$		0			0
$571$	3951.24	−	6843.75i	0.289587	−	0.501580i	−0.684124	−	0.729366i	$-0.739815\pi$
$571$	3951.24	−	6843.75i	0.289587	−	0.501580i	0.973711	+	0.227786i	$0.0731486\pi$
$572$	−4444.24	−	7697.64i	−0.324865	−	0.562683i
$573$		0			0
$574$	−6653.91	−	9924.46i	−0.483848	−	0.721671i
$575$	13.4100			0.000972581
$576$		0			0
$577$	6371.20	−	11035.2i	0.459682	−	0.796193i	−0.539262	−	0.842138i	$-0.681297\pi$
$577$	6371.20	−	11035.2i	0.459682	−	0.796193i	0.998944	+	0.0459452i	$0.0146300\pi$
$578$	4874.23	−	8442.41i	0.350763	−	0.607540i
$579$		0			0
$580$	2548.63			0.182459
$581$	−18365.6	+	1232.75i	−1.31142	+	0.0880259i
$582$		0			0
$583$	8912.94	+	15437.7i	0.633167	+	1.09668i
$584$	−736.445	+	1275.56i	−0.0521820	+	0.0903819i
$585$		0			0
$586$	6140.31	+	10635.3i	0.432856	+	0.749729i
$587$	−3400.10			−0.239075			−0.119537	−	0.992830i	$-0.538141\pi$
$587$	−3400.10			−0.239075			−0.119537	+	0.992830i	$0.538141\pi$
$588$		0			0
$589$	−23.4313			−0.00163917
$590$	3675.53	+	6366.20i	0.256473	+	0.444224i
$591$		0			0
$592$	−1655.50	+	2867.41i	−0.114934	+	0.199071i
$593$	1714.44	+	2969.49i	0.118724	+	0.205636i	0.919262	−	0.393645i	$-0.128786\pi$
$593$	1714.44	+	2969.49i	0.118724	+	0.205636i	−0.800538	+	0.599282i	$0.795453\pi$
$594$		0			0
$595$	−1496.86	+	100.473i	−0.103135	+	0.00692270i
$596$	−8832.84			−0.607059
$597$		0			0
$598$	−24.6476	+	42.6909i	−0.00168548	+	0.00291933i
$599$	−4138.34	+	7167.82i	−0.282284	+	0.488930i	−0.971947	−	0.235200i	$-0.924425\pi$
$599$	−4138.34	+	7167.82i	−0.282284	+	0.488930i	0.689663	+	0.724130i	$0.257759\pi$
$600$		0			0
$601$	−17691.1			−1.20072			−0.600362	−	0.799728i	$-0.704977\pi$
$601$	−17691.1			−1.20072			−0.600362	+	0.799728i	$0.704977\pi$
$602$	−279.909	−	417.491i	−0.0189506	−	0.0282652i
$603$		0			0
$604$	−4899.87	−	8486.82i	−0.330087	−	0.571728i
$605$	3798.98	−	6580.04i	0.255290	−	0.442176i
$606$		0			0
$607$	1300.25	+	2252.09i	0.0869445	+	0.150592i	0.906218	−	0.422810i	$-0.138956\pi$
$607$	1300.25	+	2252.09i	0.0869445	+	0.150592i	−0.819274	+	0.573403i	$0.805623\pi$
$608$	−1862.18			−0.124213
$609$		0			0
$610$	−5098.24			−0.338396
$611$	−22060.5	−	38210.0i	−1.46068	−	2.52997i
$612$		0			0
$613$	8333.19	−	14433.5i	0.549061	−	0.951001i	−0.449278	−	0.893392i	$-0.648319\pi$
$613$	8333.19	−	14433.5i	0.549061	−	0.951001i	0.998339	−	0.0576096i	$-0.0183479\pi$
$614$	7548.41	+	13074.2i	0.496138	+	0.859337i
$615$		0			0
$616$	−1785.26	+	3634.54i	−0.116770	+	0.237727i
$617$	7263.76			0.473951			0.236976	−	0.971516i	$-0.423844\pi$
$617$	7263.76			0.473951			0.236976	+	0.971516i	$0.423844\pi$
$618$		0			0
$619$	−1890.69	+	3274.77i	−0.122768	+	0.212640i	−0.920858	−	0.389898i	$-0.872510\pi$
$619$	−1890.69	+	3274.77i	−0.122768	+	0.212640i	0.798091	+	0.602538i	$0.205844\pi$
$620$	−10.4761	+	18.1452i	−0.000678599	+	0.00117537i
$621$		0			0
$622$	3400.32			0.219197
$623$	16416.1	+	24485.0i	1.05569	+	1.57459i
$624$		0			0
$625$	9598.84	+	16625.7i	0.614325	+	1.06404i
$626$	2935.23	−	5083.96i	0.187404	−	0.324594i
$627$		0			0
$628$	6262.73	+	10847.4i	0.397946	+	0.689263i
$629$	−1288.56			−0.0816825
$630$		0			0
$631$	22105.4			1.39462			0.697309	−	0.716771i	$-0.254381\pi$
$631$	22105.4			1.39462			0.697309	+	0.716771i	$0.254381\pi$
$632$	−4419.54	−	7654.86i	−0.278164	−	0.481794i
$633$		0			0
$634$	−3634.68	+	6295.45i	−0.227684	+	0.394360i
$635$	−3771.94	−	6533.19i	−0.235724	−	0.408286i
$636$		0			0
$637$	27637.7	−	3727.02i	1.71907	−	0.231821i
$638$	2677.17			0.166129
$639$		0			0
$640$	−832.580	+	1442.07i	−0.0514229	+	0.0890670i
$641$	−1414.80	+	2450.50i	−0.0871779	+	0.150997i	−0.906317	−	0.422598i	$-0.861118\pi$
$641$	−1414.80	+	2450.50i	−0.0871779	+	0.150997i	0.819139	+	0.573595i	$0.194452\pi$
$642$		0			0
$643$	−3281.09			−0.201234			−0.100617	−	0.994925i	$-0.532082\pi$
$643$	−3281.09			−0.201234			−0.100617	+	0.994925i	$0.532082\pi$
$644$	22.4070	−	1.50402i	0.00137106	−	9.20288e-5i
$645$		0			0
$646$	−362.358	−	627.622i	−0.0220693	−	0.0382252i
$647$	−10054.5	+	17414.8i	−0.610945	+	1.05819i	0.380137	+	0.924930i	$0.375877\pi$
$647$	−10054.5	+	17414.8i	−0.610945	+	1.05819i	−0.991082	+	0.133257i	$0.957456\pi$
$648$		0			0
$649$	3860.90	+	6687.27i	0.233519	+	0.404466i
$650$	7193.27			0.434067
$651$		0			0
$652$	−14481.7			−0.869860
$653$	4343.67	+	7523.46i	0.260308	+	0.450866i	0.966324	−	0.257330i	$-0.0828427\pi$
$653$	4343.67	+	7523.46i	0.260308	+	0.450866i	−0.706016	+	0.708196i	$0.749509\pi$
$654$		0			0
$655$	18507.7	−	32056.3i	1.10405	−	1.91228i
$656$	−2580.66	−	4469.83i	−0.153594	−	0.266033i
$657$		0			0
$658$	−8861.76	+	18041.3i	−0.525026	+	1.06888i
$659$	24386.8			1.44154			0.720771	−	0.693174i	$-0.243788\pi$
$659$	24386.8			1.44154			0.720771	+	0.693174i	$0.243788\pi$
$660$		0			0
$661$	−7767.45	+	13453.6i	−0.457063	+	0.791657i	−0.998804	−	0.0488887i	$-0.984432\pi$
$661$	−7767.45	+	13453.6i	−0.457063	+	0.791657i	0.541741	+	0.840545i	$0.317765\pi$
$662$	398.960	−	691.018i	0.0234230	−	0.0405698i
$663$		0			0
$664$	−7951.06			−0.464700
$665$	6181.36	−	12584.4i	0.360456	−	0.733838i
$666$		0			0
$667$	−7.42375	−	12.8583i	−0.000430958	−	0.000746441i
$668$	−2312.53	+	4005.42i	−0.133944	+	0.231997i
$669$		0			0
$670$	−479.628	−	830.741i	−0.0276562	−	0.0479020i
$671$	−5355.37			−0.308110
$672$		0			0
$673$	14978.7			0.857930			0.428965	−	0.903321i	$-0.358878\pi$
$673$	14978.7			0.857930			0.428965	+	0.903321i	$0.358878\pi$
$674$	1094.51	+	1895.74i	0.0625501	+	0.108340i
$675$		0			0
$676$	−8827.28	+	15289.3i	−0.502235	+	0.869896i
$677$	1096.32	+	1898.87i	0.0622376	+	0.107799i	0.895465	−	0.445131i	$-0.146843\pi$
$677$	1096.32	+	1898.87i	0.0622376	+	0.107799i	−0.833228	+	0.552930i	$0.813510\pi$
$678$		0			0
$679$	−6686.16	+	448.792i	−0.377896	+	0.0253653i
$680$	−648.039			−0.0365458
$681$		0			0
$682$	−11.0045	+	19.0603i	−0.000617864	+	0.00107017i
$683$	9228.24	−	15983.8i	0.516997	−	0.895465i	−0.482808	−	0.875726i	$-0.660383\pi$
$683$	9228.24	−	15983.8i	0.516997	−	0.895465i	0.999805	−	0.0197389i	$-0.00628351\pi$
$684$		0			0
$685$	32887.0			1.83437
$686$	−8421.88	−	9512.43i	−0.468730	−	0.529426i
$687$		0			0
$688$	−108.560	−	188.032i	−0.00601572	−	0.0104195i
$689$	26515.4	−	45925.9i	1.46612	−	2.53939i
$690$		0			0
$691$	15365.7	+	26614.2i	0.845933	+	1.46520i	0.884809	+	0.465955i	$0.154289\pi$
$691$	15365.7	+	26614.2i	0.845933	+	1.46520i	−0.0388756	+	0.999244i	$0.512378\pi$
$692$	4255.53			0.233773
$693$		0			0
$694$	−5495.11			−0.300564
$695$	−9541.27	−	16526.0i	−0.520749	−	0.901964i
$696$		0			0
$697$	1004.33	−	1739.55i	0.0545791	−	0.0945338i
$698$	2780.82	+	4816.53i	0.150796	+	0.261187i
$699$		0			0
$700$	−1824.91	−	2721.89i	−0.0985357	−	0.146968i
$701$	−10749.1			−0.579153			−0.289577	−	0.957155i	$-0.593514\pi$
$701$	−10749.1			−0.579153			−0.289577	+	0.957155i	$0.593514\pi$
$702$		0			0
$703$	6021.19	−	10429.0i	0.323035	−	0.559513i
$704$	−874.571	+	1514.80i	−0.0468205	+	0.0810955i
$705$		0			0
$706$	2042.94			0.108905
$707$	4780.90	−	9733.25i	0.254320	−	0.517760i
$708$		0			0
$709$	16684.6	+	28898.6i	0.883785	+	1.53076i	0.847100	+	0.531433i	$0.178346\pi$
$709$	16684.6	+	28898.6i	0.883785	+	1.53076i	0.0366848	+	0.999327i	$0.488320\pi$
$710$	−1583.27	+	2742.30i	−0.0836888	+	0.144953i
$711$		0			0
$712$	6366.83	+	11027.7i	0.335122	+	0.580448i
$713$	0.122061			6.41125e−6
$714$		0			0
$715$	28907.7			1.51201
$716$	−3501.53	−	6064.82i	−0.182763	−	0.316554i
$717$		0			0
$718$	−3913.35	+	6778.13i	−0.203405	+	0.352308i
$719$	−3355.34	−	5811.63i	−0.174038	−	0.301442i	0.765790	−	0.643091i	$-0.222348\pi$
$719$	−3355.34	−	5811.63i	−0.174038	−	0.301442i	−0.939828	+	0.341648i	$0.889015\pi$
$720$		0			0
$721$	−16884.4	−	25183.5i	−0.872132	−	1.30081i
$722$	−6945.10			−0.357991
$723$		0			0
$724$	6626.72	−	11477.8i	0.340166	−	0.589185i
$725$	−1083.29	+	1876.31i	−0.0554930	+	0.0961167i
$726$		0			0
$727$	−35427.5			−1.80733			−0.903667	−	0.428236i	$-0.859135\pi$
$727$	−35427.5			−1.80733			−0.903667	+	0.428236i	$0.859135\pi$
$728$	12019.4	−	806.773i	0.611907	−	0.0410728i
$729$		0			0
$730$	−2395.12	−	4148.46i	−0.121435	−	0.210331i
$731$	42.2489	−	73.1773i	0.00213766	−	0.00370254i
$732$		0			0
$733$	−7700.32	−	13337.4i	−0.388019	−	0.672069i	0.604164	−	0.796860i	$-0.293507\pi$
$733$	−7700.32	−	13337.4i	−0.388019	−	0.672069i	−0.992183	+	0.124791i	$0.960174\pi$
$734$	9438.67			0.474642
$735$		0			0
$736$	9.70070			0.000485832
$737$	−503.818	−	872.639i	−0.0251810	−	0.0436147i
$738$		0			0
$739$	42.2335	−	73.1506i	0.00210228	−	0.00364126i	−0.864972	−	0.501820i	$-0.832664\pi$
$739$	42.2335	−	73.1506i	0.00210228	−	0.00364126i	0.867075	+	0.498178i	$0.165997\pi$
$740$	−5384.14	−	9325.60i	−0.267466	−	0.463265i
$741$		0			0
$742$	−24105.0	+	1617.99i	−1.19262	+	0.0800515i
$743$	−23071.8			−1.13919			−0.569597	−	0.821924i	$-0.692901\pi$
$743$	−23071.8			−1.13919			−0.569597	+	0.821924i	$0.692901\pi$
$744$		0			0
$745$	14363.4	−	24878.1i	0.706353	−	1.22344i
$746$	−7900.59	+	13684.2i	−0.387750	+	0.671602i
$747$		0			0
$748$	−680.723			−0.0332750
$749$	−15343.8	−	22885.7i	−0.748532	−	1.11645i
$750$		0			0
$751$	16010.6	+	27731.1i	0.777940	+	1.34743i	0.933127	+	0.359547i	$0.117069\pi$
$751$	16010.6	+	27731.1i	0.777940	+	1.34743i	−0.155186	+	0.987885i	$0.549598\pi$
$752$	−4341.24	+	7519.25i	−0.210517	+	0.364626i
$753$		0			0
$754$	−3982.19	−	6897.36i	−0.192338	−	0.333139i
$755$	31871.4			1.53631
$756$		0			0
$757$	−26055.9			−1.25101			−0.625507	−	0.780218i	$-0.715108\pi$
$757$	−26055.9			−1.25101			−0.625507	+	0.780218i	$0.715108\pi$
$758$	6062.83	+	10501.1i	0.290517	+	0.503190i
$759$		0			0
$760$	3028.16	−	5244.92i	0.144530	−	0.250333i
$761$	−2459.81	−	4260.51i	−0.117172	−	0.202948i	0.801474	−	0.598030i	$-0.204050\pi$
$761$	−2459.81	−	4260.51i	−0.117172	−	0.202948i	−0.918646	+	0.395082i	$0.870716\pi$
$762$		0			0
$763$	10798.6	−	21984.5i	0.512367	−	1.04311i
$764$	2126.31			0.100690
$765$		0			0
$766$	−2746.49	+	4757.06i	−0.129549	+	0.224386i
$767$	11485.9	−	19894.2i	0.540719	−	0.936553i
$768$		0			0
$769$	−19360.7			−0.907887			−0.453943	−	0.891031i	$-0.649983\pi$
$769$	−19360.7			−0.907887			−0.453943	+	0.891031i	$0.649983\pi$
$770$	−7333.78	−	10938.5i	−0.343235	−	0.511943i
$771$		0			0
$772$	4582.88	+	7937.78i	0.213655	+	0.370061i
$773$	−12460.2	+	21581.7i	−0.579769	+	1.00419i	0.415737	+	0.909485i	$0.363524\pi$
$773$	−12460.2	+	21581.7i	−0.579769	+	1.00419i	−0.995505	+	0.0947040i	$0.969810\pi$
$774$		0			0
$775$	−8.90572	−	15.4252i	−0.000412778	−	0.000714952i
$776$	−2894.65			−0.133907
$777$		0			0
$778$	27727.5			1.27774
$779$	9386.06	+	16257.1i	0.431695	+	0.747717i
$780$		0			0
$781$	−1663.12	+	2880.61i	−0.0761987	+	0.131980i
$782$	1.88764	+	3.26948i	8.63193e−5	+	0.000149509i
$783$		0			0
$784$	−3354.56	−	4343.40i	−0.152813	−	0.197859i
$785$	−40736.2			−1.85215
$786$		0			0
$787$	3806.48	−	6593.01i	0.172409	−	0.298622i	−0.766852	−	0.641824i	$-0.778178\pi$
$787$	3806.48	−	6593.01i	0.172409	−	0.298622i	0.939262	+	0.343202i	$0.111511\pi$
$788$	−5935.68	+	10280.9i	−0.268337	+	0.464774i
$789$		0			0
$790$	28747.0			1.29465
$791$	24814.5	−	1665.61i	1.11543	−	0.0748703i
$792$		0			0
$793$	7965.92	+	13797.4i	0.356719	+	0.617856i
$794$	4534.13	−	7853.34i	0.202658	−	0.351013i
$795$		0			0
$796$	−6275.85	−	10870.1i	−0.279449	−	0.484020i
$797$	−4029.41			−0.179083			−0.0895414	−	0.995983i	$-0.528540\pi$
$797$	−4029.41			−0.179083			−0.0895414	+	0.995983i	$0.528540\pi$
$798$		0			0
$799$	−3379.01			−0.149613
$800$	−707.774	−	1225.90i	−0.0312795	−	0.0541777i
$801$		0			0
$802$	−7232.18	+	12526.5i	−0.318425	+	0.551529i
$803$	−2515.91	−	4357.69i	−0.110566	−	0.191506i
$804$		0			0
$805$	−32.2007	+	65.5561i	−0.00140984	+	0.00287025i
$806$	65.4751			0.00286137
$807$		0			0
$808$	2342.09	−	4056.62i	0.101973	−	0.176623i
$809$	−6913.15	+	11973.9i	−0.300437	+	0.520372i	−0.976235	−	0.216715i	$-0.930466\pi$
$809$	−6913.15	+	11973.9i	−0.300437	+	0.520372i	0.675798	+	0.737087i	$0.263799\pi$
$810$		0			0
$811$	−35205.7			−1.52434			−0.762170	−	0.647377i	$-0.775866\pi$
$811$	−35205.7			−1.52434			−0.762170	+	0.647377i	$0.775866\pi$
$812$	−1599.65	+	3256.68i	−0.0691341	+	0.140747i
$813$		0			0
$814$	−5655.68	−	9795.93i	−0.243528	−	0.421802i
$815$	23549.3	−	40788.5i	1.01214	−	1.75308i
$816$		0			0
$817$	394.842	+	683.886i	0.0169079	+	0.0292854i
$818$	11326.9			0.484153
$819$		0			0
$820$	16786.0			0.714868
$821$	−14420.9	−	24977.8i	−0.613025	−	1.06179i	−0.990728	−	0.135864i	$-0.956619\pi$
$821$	−14420.9	−	24977.8i	−0.613025	−	1.06179i	0.377702	−	0.925927i	$-0.376714\pi$
$822$		0			0
$823$	−1780.74	+	3084.33i	−0.0754224	+	0.130635i	−0.901270	−	0.433258i	$-0.857364\pi$
$823$	−1780.74	+	3084.33i	−0.0754224	+	0.130635i	0.825847	+	0.563894i	$0.190697\pi$
$824$	−6548.46	−	11342.3i	−0.276852	−	0.479523i
$825$		0			0
$826$	−10441.8	+	700.878i	−0.439849	+	0.0295238i
$827$	13329.6			0.560477			0.280238	−	0.959930i	$-0.409586\pi$
$827$	13329.6			0.560477			0.280238	+	0.959930i	$0.409586\pi$
$828$		0			0
$829$	−5982.51	+	10362.0i	−0.250641	+	0.434123i	−0.963702	−	0.266979i	$-0.913975\pi$
$829$	−5982.51	+	10362.0i	−0.250641	+	0.434123i	0.713062	+	0.701101i	$0.247308\pi$
$830$	12929.5	−	22394.5i	0.540710	−	0.936537i
$831$		0			0
$832$	5203.57			0.216829
$833$	811.124	−	1975.78i	0.0337380	−	0.0821808i
$834$		0			0
$835$	−7520.96	−	13026.7i	−0.311705	−	0.539889i
$836$	3180.88	−	5509.45i	0.131595	−	0.227929i
$837$		0			0
$838$	−1506.26	−	2608.92i	−0.0620917	−	0.107546i
$839$	−24647.0			−1.01420			−0.507098	−	0.861888i	$-0.669282\pi$
$839$	−24647.0			−1.01420			−0.507098	+	0.861888i	$0.669282\pi$
$840$		0			0
$841$	−21990.2			−0.901643
$842$	12171.3	+	21081.3i	0.498161	+	0.862840i
$843$		0			0
$844$	−2612.73	+	4525.37i	−0.106557	+	0.184561i
$845$	−28708.7	−	49724.9i	−1.16877	−	2.02436i
$846$		0			0
$847$	6023.62	+	8984.37i	0.244361	+	0.364471i
$848$	−10435.8			−0.422602
$849$		0			0
$850$	275.448	−	477.090i	0.0111150	−	0.0192518i
$851$	−31.3663	+	54.3280i	−0.00126348	+	0.00218841i
$852$		0			0
$853$	4460.76			0.179054			0.0895272	−	0.995984i	$-0.471464\pi$
$853$	4460.76			0.179054			0.0895272	+	0.995984i	$0.471464\pi$
$854$	3199.93	−	6514.61i	0.128219	−	0.261037i
$855$		0			0
$856$	−5950.96	−	10307.4i	−0.237616	−	0.411564i
$857$	15452.5	−	26764.5i	0.615924	−	1.06681i	−0.374297	−	0.927309i	$-0.622116\pi$
$857$	15452.5	−	26764.5i	0.615924	−	1.06681i	0.990222	−	0.139504i	$-0.0445507\pi$
$858$		0			0
$859$	11451.7	+	19834.9i	0.454863	+	0.787846i	0.998680	−	0.0513578i	$-0.0163549\pi$
$859$	11451.7	+	19834.9i	0.454863	+	0.787846i	−0.543817	+	0.839204i	$0.683022\pi$
$860$	706.133			0.0279988
$861$		0			0
$862$	−4043.35			−0.159765
$863$	−2023.53	−	3504.86i	−0.0798168	−	0.138247i	0.823354	−	0.567528i	$-0.192100\pi$
$863$	−2023.53	−	3504.86i	−0.0798168	−	0.138247i	−0.903171	+	0.429281i	$0.858767\pi$
$864$		0			0
$865$	−6920.06	+	11985.9i	−0.272010	+	0.471136i
$866$	−5605.18	−	9708.45i	−0.219944	−	0.380954i
$867$		0			0
$868$	−16.6108	−	24.7754i	−0.000649547	−	0.000968816i
$869$	30196.9			1.17878
$870$		0			0
$871$	−1498.82	+	2596.04i	−0.0583073	+	0.100991i
$872$	5290.08	−	9162.69i	0.205441	−	0.355835i
$873$		0			0
$874$	−35.2822			−0.00136549
$875$	−19414.9	+	1303.18i	−0.750108	+	0.0503491i
$876$		0			0
$877$	1451.75	+	2514.50i	0.0558973	+	0.0968170i	0.892620	−	0.450810i	$-0.148865\pi$
$877$	1451.75	+	2514.50i	0.0558973	+	0.0968170i	−0.836723	+	0.547627i	$0.815531\pi$
$878$	12745.9	−	22076.6i	0.489925	−	0.848576i
$879$		0			0
$880$	−2844.34	−	4926.54i	−0.108958	−	0.188720i
$881$	−23241.2			−0.888782			−0.444391	−	0.895833i	$-0.646580\pi$
$881$	−23241.2			−0.888782			−0.444391	+	0.895833i	$0.646580\pi$
$882$		0			0
$883$	−23984.8			−0.914105			−0.457052	−	0.889440i	$-0.651095\pi$
$883$	−23984.8			−0.914105			−0.457052	+	0.889440i	$0.651095\pi$
$884$	1012.55	+	1753.79i	0.0385246	+	0.0667266i
$885$		0			0
$886$	−2606.42	+	4514.46i	−0.0988312	+	0.171181i
$887$	18183.3	+	31494.4i	0.688316	+	1.19220i	0.972382	+	0.233393i	$0.0749830\pi$
$887$	18183.3	+	31494.4i	0.688316	+	1.19220i	−0.284067	+	0.958805i	$0.591684\pi$
$888$		0			0
$889$	10715.7	−	719.264i	0.404266	−	0.0271354i
$890$	−41413.2			−1.55975
$891$		0			0
$892$	−4311.72	+	7468.12i	−0.161847	+	0.280327i
$893$	15789.4	−	27348.1i	0.591683	−	1.02483i
$894$		0			0
$895$	22775.8			0.850627
$896$	−1320.13	−	1969.00i	−0.0492214	−	0.0734149i
$897$		0			0
$898$	−7512.78	−	13012.5i	−0.279181	−	0.483556i
$899$	−9.86041	+	17.0787i	−0.000365810	+	0.000633601i
$900$		0			0
$901$	−2030.68	−	3517.24i	−0.0750851	−	0.130051i
$902$	17632.6			0.650888
$903$		0			0
$904$	10743.0			0.395250
$905$	21551.9	+	37328.9i	0.791612	+	1.37111i
$906$		0			0
$907$	15682.0	−	27162.0i	0.574103	−	0.994375i	−0.422036	−	0.906579i	$-0.638684\pi$
$907$	15682.0	−	27162.0i	0.574103	−	0.994375i	0.996139	−	0.0877957i	$-0.0279823\pi$
$908$	−10256.9	−	17765.5i	−0.374876	−	0.649304i
$909$		0			0
$910$	−17272.8	+	35165.1i	−0.629219	+	1.28100i
$911$	−42177.1			−1.53391			−0.766954	−	0.641702i	$-0.778229\pi$
$911$	−42177.1			−1.53391			−0.766954	+	0.641702i	$0.778229\pi$
$912$		0			0
$913$	13581.6	−	23524.0i	0.492316	−	0.852717i
$914$	−1956.14	+	3388.13i	−0.0707913	+	0.122614i
$915$		0			0
$916$	23789.1			0.858094
$917$	29345.5	+	43769.6i	1.05679	+	1.57623i
$918$		0			0
$919$	−21419.0	−	37098.8i	−0.768823	−	1.33164i	−0.938201	−	0.346090i	$-0.887509\pi$
$919$	−21419.0	−	37098.8i	−0.768823	−	1.33164i	0.169378	−	0.985551i	$-0.445824\pi$
$920$	−15.7746	+	27.3225i	−0.000565298	+	0.000979125i
$921$		0			0
$922$	−4344.87	−	7525.54i	−0.155196	−	0.268807i
$923$	9895.33			0.352881
$924$		0			0
$925$	9154.07			0.325388
$926$	−12010.6	−	20803.0i	−0.426234	−	0.738259i
$927$		0			0
$928$	−783.647	+	1357.32i	−0.0277203	+	0.0480130i
$929$	12138.9	+	21025.2i	0.428703	+	0.742536i	0.996758	−	0.0804548i	$-0.0256373\pi$
$929$	12138.9	+	21025.2i	0.428703	+	0.742536i	−0.568055	+	0.822991i	$0.692304\pi$
$930$		0			0
$931$	12200.8	+	15797.3i	0.429500	+	0.556106i
$932$	20537.3			0.721804
$933$		0			0
$934$	5422.26	−	9391.64i	0.189959	−	0.329019i
$935$	1106.95	−	1917.29i	0.0387177	−	0.0670610i
$936$		0			0
$937$	−4321.15			−0.150657			−0.0753286	−	0.997159i	$-0.524001\pi$
$937$	−4321.15			−0.150657			−0.0753286	+	0.997159i	$0.524001\pi$
$938$	1362.57	−	91.4594i	0.0474302	−	0.00318364i
$939$		0			0
$940$	−14118.9	−	24454.6i	−0.489901	−	0.848534i
$941$	−3091.03	+	5353.82i	−0.107083	+	0.185473i	−0.914587	−	0.404389i	$-0.867484\pi$
$941$	−3091.03	+	5353.82i	−0.107083	+	0.185473i	0.807505	+	0.589861i	$0.200818\pi$
$942$		0			0
$943$	−48.8949	−	84.6885i	−0.00168848	−	0.00292453i
$944$	−4520.57			−0.155860
$945$		0			0
$946$	741.747			0.0254929
$947$	19501.3	+	33777.3i	0.669174	+	1.15904i	0.978136	+	0.207969i	$0.0666852\pi$
$947$	19501.3	+	33777.3i	0.669174	+	1.15904i	−0.308962	+	0.951074i	$0.599981\pi$
$948$		0			0
$949$	−7484.66	+	12963.8i	−0.256019	+	0.443438i
$950$	2574.23	+	4458.69i	0.0879147	+	0.152273i
$951$		0			0
$952$	406.744	−	828.074i	0.0138473	−	0.0281912i
$953$	−33368.1			−1.13421			−0.567103	−	0.823647i	$-0.691936\pi$
$953$	−33368.1			−1.13421			−0.567103	+	0.823647i	$0.691936\pi$
$954$		0			0
$955$	−3457.66	+	5988.85i	−0.117160	+	0.202926i
$956$	−14483.7	+	25086.4i	−0.489995	+	0.848696i
$957$		0			0
$958$	7284.05			0.245654
$959$	−20641.6	+	42023.4i	−0.695049	+	1.41502i
$960$		0			0
$961$	14895.4	+	25799.6i	0.499997	+	0.866021i
$962$	−16825.3	+	29142.2i	−0.563896	+	0.976696i
$963$		0			0
$964$	13328.1	+	23084.9i	0.445299	+	0.771281i
$965$	−29809.5			−0.994406
$966$		0			0
$967$	4454.30			0.148129			0.0740645	−	0.997253i	$-0.476403\pi$
$967$	4454.30			0.148129			0.0740645	+	0.997253i	$0.476403\pi$
$968$	2336.21	+	4046.43i	0.0775708	+	0.134357i
$969$		0			0
$970$	4707.08	−	8152.91i	0.155810	−	0.269870i
$971$	−3588.58	−	6215.60i	−0.118602	−	0.205426i	0.800612	−	0.599184i	$-0.204508\pi$
$971$	−3588.58	−	6215.60i	−0.118602	−	0.205426i	−0.919214	+	0.393758i	$0.871175\pi$
$972$		0			0
$973$	27105.7	−	1819.40i	0.893082	−	0.0599460i
$974$	9330.46			0.306948
$975$		0			0
$976$	1567.60	−	2715.16i	0.0514114	−	0.0890472i
$977$	15698.2	−	27190.0i	0.514052	−	0.890363i	−0.485816	−	0.874061i	$-0.661477\pi$
$977$	15698.2	−	27190.0i	0.514052	−	0.890363i	0.999867	−	0.0163021i	$-0.00518934\pi$
$978$		0			0
$979$	−43501.9			−1.42015
$980$	17688.3	−	2385.32i	0.576564	−	0.0777513i
$981$		0			0
$982$	8983.95	+	15560.7i	0.291944	+	0.505663i
$983$	−4655.00	+	8062.70i	−0.151039	+	0.261608i	−0.931610	−	0.363460i	$-0.881595\pi$
$983$	−4655.00	+	8062.70i	−0.151039	+	0.261608i	0.780571	+	0.625068i	$0.214929\pi$
$984$		0			0
$985$	−19304.4	−	33436.2i	−0.624456	−	1.08159i
$986$	−609.952			−0.0197006
$987$		0			0
$988$	−18925.8			−0.609423
$989$	−2.05685	−	3.56258i	−6.61316e−5	−	0.000114543i
$990$		0			0
$991$	−2332.61	+	4040.20i	−0.0747707	+	0.129507i	−0.900987	−	0.433847i	$-0.857156\pi$
$991$	−2332.61	+	4040.20i	−0.0747707	+	0.129507i	0.826216	+	0.563354i	$0.190489\pi$
$992$	−6.44235	−	11.1585i	−0.000206194	−	0.000357139i
$993$		0			0
$994$	−2510.41	−	3744.34i	−0.0801060	−	0.119480i
$995$	40821.5			1.30063
$996$		0			0
$997$	−15569.9	+	26967.8i	−0.494587	+	0.856649i	−0.999981	−	0.00623944i	$-0.998014\pi$
$997$	−15569.9	+	26967.8i	−0.494587	+	0.856649i	0.505394	+	0.862889i	$0.331347\pi$
$998$	−182.383	+	315.897i	−0.00578481	+	0.0100196i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.c.109.1	✓	8
3.2	odd	2		378.4.g.f.109.4	yes	8
7.2	even	3	inner	378.4.g.c.163.1	yes	8
21.2	odd	6		378.4.g.f.163.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.c.109.1	✓	8	1.1	even	1	trivial
378.4.g.c.163.1	yes	8	7.2	even	3	inner
378.4.g.f.109.4	yes	8	3.2	odd	2
378.4.g.f.163.4	yes	8	21.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-6.50453 - 11.2662i) q^{5} +(18.4787 - 1.24034i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-6.50453 - 11.2662i) q^{5} +(18.4787 - 1.24034i) q^{7} +8.00000 q^{8} +(-13.0091 + 22.5324i) q^{10} +(-13.6652 + 23.6688i) q^{11} +81.3058 q^{13} +(-20.6270 - 30.7657i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(3.11340 - 5.39257i) q^{17} +(29.0966 + 50.3968i) q^{19} +52.0363 q^{20} +54.6607 q^{22} +(-0.151573 - 0.262533i) q^{23} +(-22.1179 + 38.3094i) q^{25} +(-81.3058 - 140.826i) q^{26} +(-32.6607 + 66.4927i) q^{28} +48.9779 q^{29} +(-0.201323 + 0.348702i) q^{31} +(-16.0000 + 27.7128i) q^{32} -12.4536 q^{34} +(-134.169 - 200.116i) q^{35} +(-103.469 - 179.213i) q^{37} +(58.1932 - 100.794i) q^{38} +(-52.0363 - 90.1295i) q^{40} +322.582 q^{41} +13.5700 q^{43} +(-54.6607 - 94.6751i) q^{44} +(-0.303147 + 0.525066i) q^{46} +(-271.328 - 469.953i) q^{47} +(339.923 - 45.8396i) q^{49} +88.4717 q^{50} +(-162.612 + 281.652i) q^{52} +(326.119 - 564.854i) q^{53} +355.542 q^{55} +(147.829 - 9.92269i) q^{56} +(-48.9779 - 84.8323i) q^{58} +(141.268 - 244.683i) q^{59} +(97.9748 + 169.697i) q^{61} +0.805294 q^{62} +64.0000 q^{64} +(-528.857 - 916.007i) q^{65} +(-18.4344 + 31.9293i) q^{67} +(12.4536 + 21.5703i) q^{68} +(-212.443 + 432.504i) q^{70} +121.705 q^{71} +(-92.0556 + 159.445i) q^{73} +(-206.938 + 358.427i) q^{74} -232.773 q^{76} +(-223.157 + 454.317i) q^{77} +(-552.442 - 956.858i) q^{79} +(-104.073 + 180.259i) q^{80} +(-322.582 - 558.729i) q^{82} -993.883 q^{83} -81.0049 q^{85} +(-13.5700 - 23.5040i) q^{86} +(-109.321 + 189.350i) q^{88} +(795.853 + 1378.46i) q^{89} +(1502.42 - 100.847i) q^{91} +1.21259 q^{92} +(-542.655 + 939.907i) q^{94} +(378.520 - 655.616i) q^{95} -361.831 q^{97} +(-419.320 - 542.925i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8	\( 8 q - 8 q^{2} - 16 q^{4} + 2 q^{5} + 6 q^{7} + 64 q^{8} + 4 q^{10} - 32 q^{11} - 4 q^{13} - 36 q^{14} - 64 q^{16} + 58 q^{17} + 70 q^{19} - 16 q^{20} + 128 q^{22} - 86 q^{23} - 156 q^{25} + 4 q^{26} + 48 q^{28} + 212 q^{29} - 64 q^{31} - 128 q^{32} - 232 q^{34} - 8 q^{35} - 146 q^{37} + 140 q^{38} + 16 q^{40} + 780 q^{41} + 880 q^{43} - 128 q^{44} - 172 q^{46} + 306 q^{47} + 50 q^{49} + 624 q^{50} + 8 q^{52} + 90 q^{53} - 64 q^{55} + 48 q^{56} - 212 q^{58} - 148 q^{59} - 364 q^{61} + 256 q^{62} + 512 q^{64} - 1296 q^{65} - 954 q^{67} + 232 q^{68} + 20 q^{70} + 1360 q^{71} - 54 q^{73} - 292 q^{74} - 560 q^{76} - 2224 q^{77} - 226 q^{79} + 32 q^{80} - 780 q^{82} + 3136 q^{83} + 3920 q^{85} - 880 q^{86} - 256 q^{88} + 1458 q^{89} + 3836 q^{91} + 688 q^{92} + 612 q^{94} - 1310 q^{95} - 4344 q^{97} - 776 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 16 * q^4 + 2 * q^5 + 6 * q^7 + 64 * q^8 + 4 * q^10 - 32 * q^11 - 4 * q^13 - 36 * q^14 - 64 * q^16 + 58 * q^17 + 70 * q^19 - 16 * q^20 + 128 * q^22 - 86 * q^23 - 156 * q^25 + 4 * q^26 + 48 * q^28 + 212 * q^29 - 64 * q^31 - 128 * q^32 - 232 * q^34 - 8 * q^35 - 146 * q^37 + 140 * q^38 + 16 * q^40 + 780 * q^41 + 880 * q^43 - 128 * q^44 - 172 * q^46 + 306 * q^47 + 50 * q^49 + 624 * q^50 + 8 * q^52 + 90 * q^53 - 64 * q^55 + 48 * q^56 - 212 * q^58 - 148 * q^59 - 364 * q^61 + 256 * q^62 + 512 * q^64 - 1296 * q^65 - 954 * q^67 + 232 * q^68 + 20 * q^70 + 1360 * q^71 - 54 * q^73 - 292 * q^74 - 560 * q^76 - 2224 * q^77 - 226 * q^79 + 32 * q^80 - 780 * q^82 + 3136 * q^83 + 3920 * q^85 - 880 * q^86 - 256 * q^88 + 1458 * q^89 + 3836 * q^91 + 688 * q^92 + 612 * q^94 - 1310 * q^95 - 4344 * q^97 - 776 * q^98

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