LMFDB - Embedded newform 378.4.g.b.109.2

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:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.11184604443.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 $ x^6 + 43x^4 - 210x^3 + 1849x^2 - 4515x + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$1.60692 + 2.78326i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.b.163.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-0.301070 - 0.521469i) q^{5} +(-17.6665 + 5.55825i) q^{7} -8.00000 q^{8} +(0.602141 - 1.04294i) q^{10} +(4.52503 - 7.83758i) q^{11} +5.60214 q^{13} +(-27.2937 - 25.0411i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-6.61646 + 11.4601i) q^{17} +(-24.8134 - 42.9780i) q^{19} +2.40856 q^{20} +18.1001 q^{22} +(-65.0049 - 112.592i) q^{23} +(62.3187 - 107.939i) q^{25} +(5.60214 + 9.70319i) q^{26} +(16.0787 - 72.3151i) q^{28} -50.2607 q^{29} +(124.287 - 215.272i) q^{31} +(16.0000 - 27.7128i) q^{32} -26.4659 q^{34} +(8.21732 + 7.53912i) q^{35} +(-15.1176 - 26.1845i) q^{37} +(49.6267 - 85.9560i) q^{38} +(2.40856 + 4.17175i) q^{40} -277.880 q^{41} -48.9644 q^{43} +(18.1001 + 31.3503i) q^{44} +(130.010 - 225.184i) q^{46} +(-121.813 - 210.986i) q^{47} +(281.212 - 196.390i) q^{49} +249.275 q^{50} +(-11.2043 + 19.4064i) q^{52} +(-243.524 + 421.796i) q^{53} -5.44941 q^{55} +(141.332 - 44.4660i) q^{56} +(-50.2607 - 87.0541i) q^{58} +(-44.7885 + 77.5759i) q^{59} +(-339.433 - 587.916i) q^{61} +497.150 q^{62} +64.0000 q^{64} +(-1.68664 - 2.92134i) q^{65} +(350.562 - 607.191i) q^{67} +(-26.4659 - 45.8402i) q^{68} +(-4.84082 + 21.7719i) q^{70} +376.105 q^{71} +(134.482 - 232.929i) q^{73} +(30.2352 - 52.3689i) q^{74} +198.507 q^{76} +(-36.3783 + 163.614i) q^{77} +(-19.6704 - 34.0701i) q^{79} +(-4.81712 + 8.34351i) q^{80} +(-277.880 - 481.303i) q^{82} -491.595 q^{83} +7.96808 q^{85} +(-48.9644 - 84.8089i) q^{86} +(-36.2002 + 62.7006i) q^{88} +(342.378 + 593.015i) q^{89} +(-98.9703 + 31.1381i) q^{91} +520.039 q^{92} +(243.626 - 421.972i) q^{94} +(-14.9411 + 25.8788i) q^{95} -328.527 q^{97} +(621.369 + 290.683i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

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
$5$	−0.301070	−	0.521469i	−0.0269285	−	0.0466416i	0.852247	−	0.523139i	$-0.175239\pi$
$5$	−0.301070	−	0.521469i	−0.0269285	−	0.0466416i	−0.879176	+	0.476498i	$0.841906\pi$
$6$		0			0
$7$	−17.6665	+	5.55825i	−0.953902	+	0.300117i
$7$	−17.6665	+	5.55825i	−0.953902	+	0.300117i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	0.602141	−	1.04294i	0.0190414	−	0.0329806i
$11$	4.52503	−	7.83758i	0.124032	−	0.214829i	−0.797322	−	0.603554i	$-0.793751\pi$
$11$	4.52503	−	7.83758i	0.124032	−	0.214829i	0.921354	+	0.388725i	$0.127084\pi$
$12$		0			0
$13$	5.60214			0.119520			0.0597598	−	0.998213i	$-0.480967\pi$
$13$	5.60214			0.119520			0.0597598	+	0.998213i	$0.480967\pi$
$14$	−27.2937	−	25.0411i	−0.521039	−	0.478036i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−6.61646	+	11.4601i	−0.0943958	+	0.163498i	−0.909356	−	0.416018i	$-0.863425\pi$
$17$	−6.61646	+	11.4601i	−0.0943958	+	0.163498i	0.814960	+	0.579517i	$0.196759\pi$
$18$		0			0
$19$	−24.8134	−	42.9780i	−0.299609	−	0.518938i	0.676437	−	0.736500i	$-0.263523\pi$
$19$	−24.8134	−	42.9780i	−0.299609	−	0.518938i	−0.976047	+	0.217562i	$0.930190\pi$
$20$	2.40856			0.0269285
$21$		0			0
$22$	18.1001			0.175407
$23$	−65.0049	−	112.592i	−0.589324	−	1.02074i	−0.994321	−	0.106422i	$-0.966061\pi$
$23$	−65.0049	−	112.592i	−0.589324	−	1.02074i	0.404997	−	0.914318i	$-0.367273\pi$
$24$		0			0
$25$	62.3187	−	107.939i	0.498550	−	0.863513i
$26$	5.60214	+	9.70319i	0.0422565	+	0.0731905i
$27$		0			0
$28$	16.0787	−	72.3151i	0.108521	−	0.488081i
$29$	−50.2607			−0.321834			−0.160917	−	0.986968i	$-0.551445\pi$
$29$	−50.2607			−0.321834			−0.160917	+	0.986968i	$0.551445\pi$
$30$		0			0
$31$	124.287	−	215.272i	0.720087	−	1.24723i	−0.240878	−	0.970555i	$-0.577435\pi$
$31$	124.287	−	215.272i	0.720087	−	1.24723i	0.960965	−	0.276671i	$-0.0892313\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−26.4659			−0.133496
$35$	8.21732	+	7.53912i	0.0396852	+	0.0364098i
$36$		0			0
$37$	−15.1176	−	26.1845i	−0.0671708	−	0.116343i	0.830484	−	0.557042i	$-0.188064\pi$
$37$	−15.1176	−	26.1845i	−0.0671708	−	0.116343i	−0.897655	+	0.440699i	$0.854731\pi$
$38$	49.6267	−	85.9560i	0.211856	−	0.366945i
$39$		0			0
$40$	2.40856	+	4.17175i	0.00952068	+	0.0164903i
$41$	−277.880			−1.05848			−0.529239	−	0.848473i	$-0.677522\pi$
$41$	−277.880			−1.05848			−0.529239	+	0.848473i	$0.677522\pi$
$42$		0			0
$43$	−48.9644			−0.173651			−0.0868256	−	0.996224i	$-0.527672\pi$
$43$	−48.9644			−0.173651			−0.0868256	+	0.996224i	$0.527672\pi$
$44$	18.1001	+	31.3503i	0.0620158	+	0.107414i
$45$		0			0
$46$	130.010	−	225.184i	0.416715	−	0.721772i
$47$	−121.813	−	210.986i	−0.378048	−	0.654798i	0.612730	−	0.790292i	$-0.290071\pi$
$47$	−121.813	−	210.986i	−0.378048	−	0.654798i	−0.990778	+	0.135494i	$0.956738\pi$
$48$		0			0
$49$	281.212	−	196.390i	0.819859	−	0.572565i
$50$	249.275			0.705056
$51$		0			0
$52$	−11.2043	+	19.4064i	−0.0298799	+	0.0517535i
$53$	−243.524	+	421.796i	−0.631143	+	1.09317i	0.356176	+	0.934419i	$0.384081\pi$
$53$	−243.524	+	421.796i	−0.631143	+	1.09317i	−0.987318	+	0.158752i	$0.949253\pi$
$54$		0			0
$55$	−5.44941			−0.0133600
$56$	141.332	−	44.4660i	0.337255	−	0.106107i
$57$		0			0
$58$	−50.2607	−	87.0541i	−0.113785	−	0.197082i
$59$	−44.7885	+	77.5759i	−0.0988299	+	0.171178i	−0.911201	−	0.411963i	$-0.864843\pi$
$59$	−44.7885	+	77.5759i	−0.0988299	+	0.171178i	0.812371	+	0.583141i	$0.198177\pi$
$60$		0			0
$61$	−339.433	−	587.916i	−0.712459	−	1.23402i	−0.963931	−	0.266151i	$-0.914248\pi$
$61$	−339.433	−	587.916i	−0.712459	−	1.23402i	0.251473	−	0.967864i	$-0.419085\pi$
$62$	497.150			1.01836
$63$		0			0
$64$	64.0000			0.125000
$65$	−1.68664	−	2.92134i	−0.00321849	−	0.00557458i
$66$		0			0
$67$	350.562	−	607.191i	0.639223	−	1.10717i	−0.346380	−	0.938094i	$-0.612589\pi$
$67$	350.562	−	607.191i	0.639223	−	1.10717i	0.985604	−	0.169073i	$-0.0540773\pi$
$68$	−26.4659	−	45.8402i	−0.0471979	−	0.0817491i
$69$		0			0
$70$	−4.84082	+	21.7719i	−0.00826555	+	0.0371749i
$71$	376.105			0.628669			0.314334	−	0.949312i	$-0.398219\pi$
$71$	376.105			0.628669			0.314334	+	0.949312i	$0.398219\pi$
$72$		0			0
$73$	134.482	−	232.929i	0.215615	−	0.373456i	−0.737848	−	0.674967i	$-0.764158\pi$
$73$	134.482	−	232.929i	0.215615	−	0.373456i	0.953463	+	0.301511i	$0.0974911\pi$
$74$	30.2352	−	52.3689i	0.0474969	−	0.0822671i
$75$		0			0
$76$	198.507			0.299609
$77$	−36.3783	+	163.614i	−0.0538401	+	0.242150i
$78$		0			0
$79$	−19.6704	−	34.0701i	−0.0280138	−	0.0485213i	0.851679	−	0.524064i	$-0.175585\pi$
$79$	−19.6704	−	34.0701i	−0.0280138	−	0.0485213i	−0.879692	+	0.475543i	$0.842252\pi$
$80$	−4.81712	+	8.34351i	−0.00673214	+	0.0116604i
$81$		0			0
$82$	−277.880	−	481.303i	−0.374228	−	0.648183i
$83$	−491.595			−0.650115			−0.325058	−	0.945694i	$-0.605384\pi$
$83$	−491.595			−0.650115			−0.325058	+	0.945694i	$0.605384\pi$
$84$		0			0
$85$	7.96808			0.0101678
$86$	−48.9644	−	84.8089i	−0.0613950	−	0.106339i
$87$		0			0
$88$	−36.2002	+	62.7006i	−0.0438518	+	0.0759535i
$89$	342.378	+	593.015i	0.407775	+	0.706286i	0.994640	−	0.103398i	$-0.0329716\pi$
$89$	342.378	+	593.015i	0.407775	+	0.706286i	−0.586865	+	0.809685i	$0.699638\pi$
$90$		0			0
$91$	−98.9703	+	31.1381i	−0.114010	+	0.0358699i
$92$	520.039			0.589324
$93$		0			0
$94$	243.626	−	421.972i	0.267320	−	0.463012i
$95$	−14.9411	+	25.8788i	−0.0161361	+	0.0279485i
$96$		0			0
$97$	−328.527			−0.343885			−0.171942	−	0.985107i	$-0.555004\pi$
$97$	−328.527			−0.343885			−0.171942	+	0.985107i	$0.555004\pi$
$98$	621.369	+	290.683i	0.640487	+	0.299627i
$99$		0			0
$100$	249.275	+	431.757i	0.249275	+	0.431757i
$101$	−189.567	+	328.339i	−0.186758	+	0.323475i	−0.944168	−	0.329466i	$-0.893131\pi$
$101$	−189.567	+	328.339i	−0.186758	+	0.323475i	0.757409	+	0.652940i	$0.226465\pi$
$102$		0			0
$103$	130.161	+	225.446i	0.124516	+	0.215669i	0.921544	−	0.388274i	$-0.126929\pi$
$103$	130.161	+	225.446i	0.124516	+	0.215669i	−0.797027	+	0.603943i	$0.793595\pi$
$104$	−44.8171			−0.0422565
$105$		0			0
$106$	−974.095			−0.892571
$107$	544.146	+	942.488i	0.491631	+	0.851530i	0.999954	−	0.00963652i	$-0.00306745\pi$
$107$	544.146	+	942.488i	0.491631	+	0.851530i	−0.508322	+	0.861167i	$0.669734\pi$
$108$		0			0
$109$	−251.781	+	436.097i	−0.221250	+	0.383216i	−0.955188	−	0.296001i	$-0.904347\pi$
$109$	−251.781	+	436.097i	−0.221250	+	0.383216i	0.733938	+	0.679217i	$0.237680\pi$
$110$	−5.44941	−	9.43865i	−0.00472346	−	0.00818127i
$111$		0			0
$112$	218.349	+	200.328i	0.184215	+	0.169011i
$113$	−1280.25			−1.06580			−0.532901	−	0.846178i	$-0.678898\pi$
$113$	−1280.25			−1.06580			−0.532901	+	0.846178i	$0.678898\pi$
$114$		0			0
$115$	−39.1421	+	67.7961i	−0.0317393	+	0.0549741i
$116$	100.521	−	174.108i	0.0804584	−	0.139358i
$117$		0			0
$118$	−179.154			−0.139767
$119$	53.1921	−	239.235i	0.0409757	−	0.184291i
$120$		0			0
$121$	624.548	+	1081.75i	0.469232	+	0.812734i
$122$	678.867	−	1175.83i	0.503785	−	0.872580i
$123$		0			0
$124$	497.150	+	861.089i	0.360043	+	0.623613i
$125$	−150.317			−0.107558
$126$		0			0
$127$	−920.069			−0.642857			−0.321429	−	0.946934i	$-0.604163\pi$
$127$	−920.069			−0.642857			−0.321429	+	0.946934i	$0.604163\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	3.37328	−	5.84269i	0.00227581	−	0.00394183i
$131$	−1392.23	−	2411.42i	−0.928551	−	1.60830i	−0.785749	−	0.618545i	$-0.787722\pi$
$131$	−1392.23	−	2411.42i	−0.928551	−	1.60830i	−0.142802	−	0.989751i	$-0.545611\pi$
$132$		0			0
$133$	677.248	+	621.353i	0.441540	+	0.405099i
$134$	1402.25			0.903998
$135$		0			0
$136$	52.9317	−	91.6804i	0.0333740	−	0.0578054i
$137$	−1207.94	+	2092.22i	−0.753295	+	1.30475i	0.192922	+	0.981214i	$0.438204\pi$
$137$	−1207.94	+	2092.22i	−0.753295	+	1.30475i	−0.946217	+	0.323532i	$0.895130\pi$
$138$		0			0
$139$	−94.5986			−0.0577248			−0.0288624	−	0.999583i	$-0.509188\pi$
$139$	−94.5986			−0.0577248			−0.0288624	+	0.999583i	$0.509188\pi$
$140$	−42.5509	+	13.3874i	−0.0256872	+	0.00808172i
$141$		0			0
$142$	376.105	+	651.433i	0.222268	+	0.384979i
$143$	25.3498	−	43.9072i	0.0148242	−	0.0256763i
$144$		0			0
$145$	15.1320	+	26.2094i	0.00866651	+	0.0150108i
$146$	537.926			0.304925
$147$		0			0
$148$	120.941			0.0671708
$149$	−1172.33	−	2030.53i	−0.644569	−	1.11643i	−0.984401	−	0.175941i	$-0.943703\pi$
$149$	−1172.33	−	2030.53i	−0.644569	−	1.11643i	0.339831	−	0.940486i	$-0.389630\pi$
$150$		0			0
$151$	113.816	−	197.135i	0.0613391	−	0.106242i	−0.833725	−	0.552180i	$-0.813796\pi$
$151$	113.816	−	197.135i	0.0613391	−	0.106242i	0.895064	+	0.445937i	$0.147130\pi$
$152$	198.507	+	343.824i	0.105928	+	0.183472i
$153$		0			0
$154$	−319.766	+	100.605i	−0.167321	+	0.0526427i
$155$	−149.677			−0.0775635
$156$		0			0
$157$	−172.605	+	298.961i	−0.0877414	+	0.151973i	−0.906556	−	0.422085i	$-0.861298\pi$
$157$	−172.605	+	298.961i	−0.0877414	+	0.151973i	0.818815	+	0.574058i	$0.194632\pi$
$158$	39.3408	−	68.1402i	0.0198088	−	0.0343098i
$159$		0			0
$160$	−19.2685			−0.00952068
$161$	1774.22	+	1627.79i	0.868499	+	0.796820i
$162$		0			0
$163$	1645.72	+	2850.47i	0.790813	+	1.36973i	0.925464	+	0.378836i	$0.123675\pi$
$163$	1645.72	+	2850.47i	0.790813	+	1.36973i	−0.134651	+	0.990893i	$0.542991\pi$
$164$	555.760	−	962.605i	0.264619	−	0.458334i
$165$		0			0
$166$	−491.595	−	851.468i	−0.229850	−	0.398113i
$167$	−3657.31			−1.69468			−0.847338	−	0.531054i	$-0.821796\pi$
$167$	−3657.31			−1.69468			−0.847338	+	0.531054i	$0.821796\pi$
$168$		0			0
$169$	−2165.62			−0.985715
$170$	7.96808	+	13.8011i	0.00359485	+	0.00622646i
$171$		0			0
$172$	97.9289	−	169.618i	0.0434128	−	0.0751932i
$173$	−1325.61	−	2296.03i	−0.582569	−	1.00904i	−0.995174	−	0.0981286i	$-0.968714\pi$
$173$	−1325.61	−	2296.03i	−0.582569	−	1.00904i	0.412605	−	0.910910i	$-0.364619\pi$
$174$		0			0
$175$	−501.002	+	2253.29i	−0.216413	+	0.973331i
$176$	−144.801			−0.0620158
$177$		0			0
$178$	−684.755	+	1186.03i	−0.288340	+	0.499420i
$179$	−700.967	+	1214.11i	−0.292697	+	0.506965i	−0.974446	−	0.224620i	$-0.927886\pi$
$179$	−700.967	+	1214.11i	−0.292697	+	0.506965i	0.681750	+	0.731585i	$0.261219\pi$
$180$		0			0
$181$	−2.91675			−0.00119779			−0.000598897	−	1.00000i	$-0.500191\pi$
$181$	−2.91675			−0.00119779			−0.000598897		1.00000i	$0.500191\pi$
$182$	−152.903	−	140.284i	−0.0622743	−	0.0571347i
$183$		0			0
$184$	520.039	+	900.734i	0.208358	+	0.360886i
$185$	−9.10293	+	15.7667i	−0.00361762	+	0.00626591i
$186$		0			0
$187$	59.8794	+	103.714i	0.0234161	+	0.0405579i
$188$	974.503			0.378048
$189$		0			0
$190$	−59.7645			−0.0228199
$191$	−126.055	−	218.334i	−0.0477540	−	0.0827124i	0.841160	−	0.540786i	$-0.181873\pi$
$191$	−126.055	−	218.334i	−0.0477540	−	0.0827124i	−0.888914	+	0.458073i	$0.848540\pi$
$192$		0			0
$193$	−986.702	+	1709.02i	−0.368002	+	0.637398i	−0.989253	−	0.146213i	$-0.953291\pi$
$193$	−986.702	+	1709.02i	−0.368002	+	0.637398i	0.621251	+	0.783612i	$0.286625\pi$
$194$	−328.527	−	569.025i	−0.121582	−	0.210586i
$195$		0			0
$196$	117.891	+	1366.93i	0.0429631	+	0.498151i
$197$	1788.56			0.646850			0.323425	−	0.946254i	$-0.395166\pi$
$197$	1788.56			0.646850			0.323425	+	0.946254i	$0.395166\pi$
$198$		0			0
$199$	−461.066	+	798.589i	−0.164242	+	0.284475i	−0.936386	−	0.350973i	$-0.885851\pi$
$199$	−461.066	+	798.589i	−0.164242	+	0.284475i	0.772144	+	0.635447i	$0.219184\pi$
$200$	−498.550	+	863.513i	−0.176264	+	0.305298i
$201$		0			0
$202$	−758.266			−0.264116
$203$	887.931	−	279.361i	0.306998	−	0.0965878i
$204$		0			0
$205$	83.6615	+	144.906i	0.0285033	+	0.0493691i
$206$	−260.323	+	450.892i	−0.0880464	+	0.152501i
$207$		0			0
$208$	−44.8171	−	77.6255i	−0.0149399	−	0.0258767i
$209$	−449.125			−0.148644
$210$		0			0
$211$	−5130.61			−1.67396			−0.836981	−	0.547232i	$-0.815682\pi$
$211$	−5130.61			−1.67396			−0.836981	+	0.547232i	$0.815682\pi$
$212$	−974.095	−	1687.18i	−0.315571	−	0.546586i
$213$		0			0
$214$	−1088.29	+	1884.98i	−0.347636	+	0.602123i
$215$	14.7417	+	25.5334i	0.00467618	+	0.00809938i
$216$		0			0
$217$	−999.190	+	4493.93i	−0.312578	+	1.40584i
$218$	−1007.12			−0.312895
$219$		0			0
$220$	10.8988	−	18.8773i	0.00333999	−	0.00578503i
$221$	−37.0664	+	64.2008i	−0.0112821	+	0.0195412i
$222$		0			0
$223$	3777.82			1.13445			0.567223	−	0.823564i	$-0.308018\pi$
$223$	3777.82			1.13445			0.567223	+	0.823564i	$0.308018\pi$
$224$	−128.630	+	578.521i	−0.0383680	+	0.172563i
$225$		0			0
$226$	−1280.25	−	2217.45i	−0.376818	−	0.652667i
$227$	1661.63	−	2878.02i	0.485842	−	0.841502i	−0.514026	−	0.857775i	$-0.671847\pi$
$227$	1661.63	−	2878.02i	0.485842	−	0.841502i	0.999868	+	0.0162724i	$0.00517988\pi$
$228$		0			0
$229$	477.230	+	826.587i	0.137713	+	0.238526i	0.926631	−	0.375973i	$-0.122692\pi$
$229$	477.230	+	826.587i	0.137713	+	0.238526i	−0.788918	+	0.614499i	$0.789358\pi$
$230$	−156.568			−0.0448861
$231$		0			0
$232$	402.086			0.113785
$233$	2002.14	+	3467.80i	0.562937	+	0.975035i	0.997238	+	0.0742678i	$0.0236619\pi$
$233$	2002.14	+	3467.80i	0.562937	+	0.975035i	−0.434301	+	0.900768i	$0.643005\pi$
$234$		0			0
$235$	−73.3485	+	127.043i	−0.0203605	+	0.0352655i
$236$	−179.154	−	310.304i	−0.0494150	−	0.0855892i
$237$		0			0
$238$	467.560	−	147.104i	0.127342	−	0.0400644i
$239$	2768.29			0.749230			0.374615	−	0.927181i	$-0.377775\pi$
$239$	2768.29			0.749230			0.374615	+	0.927181i	$0.377775\pi$
$240$		0			0
$241$	3568.69	−	6181.15i	0.953857	−	1.65213i	0.216896	−	0.976195i	$-0.430407\pi$
$241$	3568.69	−	6181.15i	0.953857	−	1.65213i	0.736962	−	0.675935i	$-0.236260\pi$
$242$	−1249.10	+	2163.50i	−0.331797	+	0.574690i
$243$		0			0
$244$	2715.47			0.712459
$245$	−187.076	−	87.5161i	−0.0487830	−	0.0228212i
$246$		0			0
$247$	−139.008	−	240.769i	−0.0358092	−	0.0620233i
$248$	−994.300	+	1722.18i	−0.254589	+	0.440961i
$249$		0			0
$250$	−150.317	−	260.356i	−0.0380275	−	0.0658655i
$251$	3925.58			0.987172			0.493586	−	0.869697i	$-0.335686\pi$
$251$	3925.58			0.987172			0.493586	+	0.869697i	$0.335686\pi$
$252$		0			0
$253$	−1176.60			−0.292379
$254$	−920.069	−	1593.61i	−0.227284	−	0.393668i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1123.14	+	1945.33i	0.272605	+	0.472165i	0.969528	−	0.244980i	$-0.0787815\pi$
$257$	1123.14	+	1945.33i	0.272605	+	0.472165i	−0.696923	+	0.717146i	$0.745448\pi$
$258$		0			0
$259$	412.615	+	378.561i	0.0989910	+	0.0908210i
$260$	13.4931			0.00321849
$261$		0			0
$262$	2784.47	−	4822.84i	0.656584	−	1.13724i
$263$	641.744	−	1111.53i	0.150463	−	0.260609i	−0.780935	−	0.624612i	$-0.785257\pi$
$263$	641.744	−	1111.53i	0.150463	−	0.260609i	0.931398	+	0.364003i	$0.118590\pi$
$264$		0			0
$265$	293.271			0.0679830
$266$	−398.967	+	1794.38i	−0.0919632	+	0.413611i
$267$		0			0
$268$	1402.25	+	2428.76i	0.319612	+	0.553583i
$269$	−2360.33	+	4088.22i	−0.534989	+	0.926629i	0.464174	+	0.885744i	$0.346351\pi$
$269$	−2360.33	+	4088.22i	−0.534989	+	0.926629i	−0.999164	+	0.0408850i	$0.986982\pi$
$270$		0			0
$271$	−2568.52	−	4448.81i	−0.575744	−	0.997217i	−0.995960	−	0.0897933i	$-0.971379\pi$
$271$	−2568.52	−	4448.81i	−0.575744	−	0.997217i	0.420217	−	0.907424i	$-0.361954\pi$
$272$	211.727			0.0471979
$273$		0			0
$274$	−4831.77			−1.06532
$275$	−563.988	−	976.855i	−0.123672	−	0.214206i
$276$		0			0
$277$	2885.47	−	4997.79i	0.625889	−	1.08407i	−0.362479	−	0.931992i	$-0.618070\pi$
$277$	2885.47	−	4997.79i	0.625889	−	1.08407i	0.988368	−	0.152080i	$-0.0485971\pi$
$278$	−94.5986	−	163.850i	−0.0204088	−	0.0353491i
$279$		0			0
$280$	−65.7385	−	60.3130i	−0.0140308	−	0.0128728i
$281$	3594.63			0.763123			0.381562	−	0.924343i	$-0.375386\pi$
$281$	3594.63			0.763123			0.381562	+	0.924343i	$0.375386\pi$
$282$		0			0
$283$	3765.14	−	6521.41i	0.790863	−	1.36982i	−0.134570	−	0.990904i	$-0.542965\pi$
$283$	3765.14	−	6521.41i	0.790863	−	1.36982i	0.925433	−	0.378911i	$-0.123701\pi$
$284$	−752.211	+	1302.87i	−0.157167	+	0.272222i
$285$		0			0
$286$	101.399			0.0209646
$287$	4909.18	−	1544.53i	1.00968	−	0.317667i
$288$		0			0
$289$	2368.94	+	4103.13i	0.482179	+	0.835158i
$290$	−30.2640	+	52.4188i	−0.00612815	+	0.0106143i
$291$		0			0
$292$	537.926	+	931.716i	0.107807	+	0.186728i
$293$	−4469.73			−0.891209			−0.445605	−	0.895230i	$-0.647011\pi$
$293$	−4469.73			−0.891209			−0.445605	+	0.895230i	$0.647011\pi$
$294$		0			0
$295$	53.9379			0.0106454
$296$	120.941	+	209.476i	0.0237485	+	0.0411335i
$297$		0			0
$298$	2344.66	−	4061.06i	0.455779	−	0.789433i
$299$	−364.167	−	630.755i	−0.0704358	−	0.121998i
$300$		0			0
$301$	865.031	−	272.156i	0.165646	−	0.0521157i
$302$	455.264			0.0867466
$303$		0			0
$304$	−397.014	+	687.648i	−0.0749023	+	0.129735i
$305$	−204.387	+	354.008i	−0.0383710	+	0.0664605i
$306$		0			0
$307$	2811.31			0.522638			0.261319	−	0.965252i	$-0.415843\pi$
$307$	2811.31			0.522638			0.261319	+	0.965252i	$0.415843\pi$
$308$	−494.019	−	453.246i	−0.0913939	−	0.0838509i
$309$		0			0
$310$	−149.677	−	259.248i	−0.0274229	−	0.0474978i
$311$	2580.95	−	4470.33i	0.470585	−	0.815077i	−0.528849	−	0.848716i	$-0.677376\pi$
$311$	2580.95	−	4470.33i	0.470585	−	0.815077i	0.999434	+	0.0336386i	$0.0107095\pi$
$312$		0			0
$313$	−3358.55	−	5817.18i	−0.606506	−	1.05050i	−0.991812	−	0.127710i	$-0.959237\pi$
$313$	−3358.55	−	5817.18i	−0.606506	−	1.05050i	0.385306	−	0.922789i	$-0.374096\pi$
$314$	−690.421			−0.124085
$315$		0			0
$316$	157.363			0.0280138
$317$	4091.97	+	7087.51i	0.725010	+	1.25575i	0.958970	+	0.283508i	$0.0914982\pi$
$317$	4091.97	+	7087.51i	0.725010	+	1.25575i	−0.233960	+	0.972246i	$0.575168\pi$
$318$		0			0
$319$	−227.431	+	393.922i	−0.0399175	+	0.0691392i
$320$	−19.2685	−	33.3740i	−0.00336607	−	0.00583020i
$321$		0			0
$322$	−1045.19	+	4700.84i	−0.180889	+	0.813563i
$323$	656.707			0.113127
$324$		0			0
$325$	349.118	−	604.690i	0.0595864	−	0.103207i
$326$	−3291.44	+	5700.93i	−0.559189	+	0.968545i
$327$		0			0
$328$	2223.04			0.374228
$329$	3324.72	+	3050.32i	0.557137	+	0.511155i
$330$		0			0
$331$	−1774.96	−	3074.32i	−0.294745	−	0.510514i	0.680180	−	0.733045i	$-0.261902\pi$
$331$	−1774.96	−	3074.32i	−0.294745	−	0.510514i	−0.974926	+	0.222531i	$0.928568\pi$
$332$	983.190	−	1702.94i	0.162529	−	0.281508i
$333$		0			0
$334$	−3657.31	−	6334.64i	−0.599158	−	1.03777i
$335$	−422.175			−0.0688534
$336$		0			0
$337$	−7152.10			−1.15608			−0.578041	−	0.816008i	$-0.696183\pi$
$337$	−7152.10			−1.15608			−0.578041	+	0.816008i	$0.696183\pi$
$338$	−2165.62	−	3750.96i	−0.348503	−	0.603625i
$339$		0			0
$340$	−15.9362	+	27.6023i	−0.00254194	+	0.00440277i
$341$	−1124.81	−	1948.22i	−0.178627	−	0.309391i
$342$		0			0
$343$	−3876.45	+	5032.57i	−0.610229	+	0.792225i
$344$	391.715			0.0613950
$345$		0			0
$346$	2651.22	−	4592.05i	0.411938	−	0.713498i
$347$	3138.58	−	5436.18i	0.485555	−	0.841007i	−0.514307	−	0.857606i	$-0.671951\pi$
$347$	3138.58	−	5436.18i	0.485555	−	0.841007i	0.999862	+	0.0165996i	$0.00528406\pi$
$348$		0			0
$349$	5735.43			0.879686			0.439843	−	0.898075i	$-0.355034\pi$
$349$	5735.43			0.879686			0.439843	+	0.898075i	$0.355034\pi$
$350$	−4403.82	+	1385.53i	−0.672554	+	0.211599i
$351$		0			0
$352$	−144.801	−	250.802i	−0.0219259	−	0.0379767i
$353$	−1308.40	+	2266.21i	−0.197278	+	0.341695i	−0.947645	−	0.319326i	$-0.896543\pi$
$353$	−1308.40	+	2266.21i	−0.197278	+	0.341695i	0.750367	+	0.661021i	$0.229877\pi$
$354$		0			0
$355$	−113.234	−	196.127i	−0.0169291	−	0.0293221i
$356$	−2739.02			−0.407775
$357$		0			0
$358$	−2803.87			−0.413936
$359$	−5826.07	−	10091.0i	−0.856513	−	1.48352i	−0.875234	−	0.483699i	$-0.839293\pi$
$359$	−5826.07	−	10091.0i	−0.856513	−	1.48352i	0.0187217	−	0.999825i	$-0.494040\pi$
$360$		0			0
$361$	2198.09	−	3807.21i	0.320469	−	0.555068i
$362$	−2.91675	−	5.05197i	−0.000423484	−	0.000733496i
$363$		0			0
$364$	90.0751	−	405.119i	0.0129704	−	0.0583352i
$365$	−161.954			−0.0232248
$366$		0			0
$367$	6237.16	−	10803.1i	0.887132	−	1.53656i	0.0438807	−	0.999037i	$-0.486028\pi$
$367$	6237.16	−	10803.1i	0.887132	−	1.53656i	0.843251	−	0.537520i	$-0.180639\pi$
$368$	−1040.08	+	1801.47i	−0.147331	+	0.255185i
$369$		0			0
$370$	−36.4117			−0.00511609
$371$	1957.77	−	8805.23i	0.273969	−	1.23220i
$372$		0			0
$373$	−963.950	−	1669.61i	−0.133811	−	0.231767i	0.791332	−	0.611387i	$-0.209388\pi$
$373$	−963.950	−	1669.61i	−0.133811	−	0.231767i	−0.925143	+	0.379620i	$0.876055\pi$
$374$	−119.759	+	207.428i	−0.0165577	+	0.0286788i
$375$		0			0
$376$	974.503	+	1687.89i	0.133660	+	0.231506i
$377$	−281.567			−0.0384654
$378$		0			0
$379$	−2946.20			−0.399304			−0.199652	−	0.979867i	$-0.563981\pi$
$379$	−2946.20			−0.399304			−0.199652	+	0.979867i	$0.563981\pi$
$380$	−59.7645	−	103.515i	−0.00806804	−	0.0139743i
$381$		0			0
$382$	252.110	−	436.667i	0.0337672	−	0.0584865i
$383$	−1558.73	−	2699.80i	−0.207956	−	0.360191i	0.743114	−	0.669164i	$-0.233348\pi$
$383$	−1558.73	−	2699.80i	−0.207956	−	0.360191i	−0.951071	+	0.308974i	$0.900015\pi$
$384$		0			0
$385$	96.2720	−	30.2891i	0.0127441	−	0.00400955i
$386$	−3946.81			−0.520433
$387$		0			0
$388$	657.053	−	1138.05i	0.0859712	−	0.148906i
$389$	−918.419	+	1590.75i	−0.119706	+	0.207337i	−0.919651	−	0.392736i	$-0.871529\pi$
$389$	−918.419	+	1590.75i	−0.119706	+	0.207337i	0.799945	+	0.600073i	$0.204862\pi$
$390$		0			0
$391$	1720.41			0.222519
$392$	−2249.69	+	1571.12i	−0.289864	+	0.202432i
$393$		0			0
$394$	1788.56	+	3097.87i	0.228696	+	0.396113i
$395$	−11.8443	+	20.5150i	−0.00150874	+	0.00261322i
$396$		0			0
$397$	−677.402	−	1173.30i	−0.0856369	−	0.148327i	0.820026	−	0.572327i	$-0.193959\pi$
$397$	−677.402	−	1173.30i	−0.0856369	−	0.148327i	−0.905662	+	0.423999i	$0.860626\pi$
$398$	−1844.26			−0.232273
$399$		0			0
$400$	−1994.20			−0.249275
$401$	6128.83	+	10615.4i	0.763240	+	1.32197i	0.941172	+	0.337928i	$0.109726\pi$
$401$	6128.83	+	10615.4i	0.763240	+	1.32197i	−0.177932	+	0.984043i	$0.556941\pi$
$402$		0			0
$403$	696.276	−	1205.98i	0.0860644	−	0.149068i
$404$	−758.266	−	1313.36i	−0.0933791	−	0.161737i
$405$		0			0
$406$	1371.80	+	1258.58i	0.167688	+	0.153848i
$407$	−273.630			−0.0333252
$408$		0			0
$409$	3807.82	−	6595.34i	0.460354	−	0.797356i	−0.538625	−	0.842546i	$-0.681056\pi$
$409$	3807.82	−	6595.34i	0.460354	−	0.797356i	0.998978	+	0.0451896i	$0.0143892\pi$
$410$	−167.323	+	289.812i	−0.0201549	+	0.0349092i
$411$		0			0
$412$	−1041.29			−0.124516
$413$	360.070	−	1619.44i	0.0429005	−	0.192948i
$414$		0			0
$415$	148.005	+	256.352i	0.0175067	+	0.0303224i
$416$	89.6342	−	155.251i	0.0105641	−	0.0182976i
$417$		0			0
$418$	−449.125	−	777.907i	−0.0525536	−	0.0910255i
$419$	8575.60			0.999870			0.499935	−	0.866063i	$-0.333357\pi$
$419$	8575.60			0.999870			0.499935	+	0.866063i	$0.333357\pi$
$420$		0			0
$421$	−14776.6			−1.71061			−0.855303	−	0.518128i	$-0.826629\pi$
$421$	−14776.6			−1.71061			−0.855303	+	0.518128i	$0.826629\pi$
$422$	−5130.61	−	8886.48i	−0.591835	−	1.02509i
$423$		0			0
$424$	1948.19	−	3374.37i	0.223143	−	0.386494i
$425$	824.659	+	1428.35i	0.0941220	+	0.163024i
$426$		0			0
$427$	9264.39	+	8499.77i	1.04997	+	0.963309i
$428$	−4353.17			−0.491631
$429$		0			0
$430$	−29.4835	+	51.0669i	−0.00330656	+	0.00572712i
$431$	8168.58	−	14148.4i	0.912916	−	1.58122i	0.102992	−	0.994682i	$-0.467159\pi$
$431$	8168.58	−	14148.4i	0.912916	−	1.58122i	0.809924	−	0.586534i	$-0.199508\pi$
$432$		0			0
$433$	6150.96			0.682670			0.341335	−	0.939942i	$-0.389121\pi$
$433$	6150.96			0.682670			0.341335	+	0.939942i	$0.389121\pi$
$434$	−8782.91	+	2763.28i	−0.971412	+	0.305626i
$435$		0			0
$436$	−1007.12	−	1744.39i	−0.110625	−	0.191608i
$437$	−3225.98	+	5587.56i	−0.353134	+	0.611646i
$438$		0			0
$439$	−1299.24	−	2250.35i	−0.141252	−	0.244655i	0.786717	−	0.617314i	$-0.211779\pi$
$439$	−1299.24	−	2250.35i	−0.141252	−	0.244655i	−0.927968	+	0.372659i	$0.878446\pi$
$440$	43.5952			0.00472346
$441$		0			0
$442$	−148.265			−0.0159554
$443$	4915.60	+	8514.08i	0.527195	+	0.913129i	0.999498	+	0.0316922i	$0.0100896\pi$
$443$	4915.60	+	8514.08i	0.527195	+	0.913129i	−0.472303	+	0.881436i	$0.656577\pi$
$444$		0			0
$445$	206.159	−	357.079i	0.0219616	−	0.0380385i
$446$	3777.82	+	6543.38i	0.401088	+	0.694704i
$447$		0			0
$448$	−1130.66	+	355.728i	−0.119238	+	0.0375146i
$449$	−13649.3			−1.43463			−0.717317	−	0.696747i	$-0.754630\pi$
$449$	−13649.3			−1.43463			−0.717317	+	0.696747i	$0.754630\pi$
$450$		0			0
$451$	−1257.42	+	2177.91i	−0.131285	+	0.227392i
$452$	2560.49	−	4434.91i	0.266450	−	0.461505i
$453$		0			0
$454$	6646.51			0.687084
$455$	46.0346	+	42.2352i	0.00474315	+	0.00435169i
$456$		0			0
$457$	6155.95	+	10662.4i	0.630117	+	1.09139i	0.987527	+	0.157447i	$0.0503263\pi$
$457$	6155.95	+	10662.4i	0.630117	+	1.09139i	−0.357411	+	0.933947i	$0.616340\pi$
$458$	−954.461	+	1653.17i	−0.0973778	+	0.168663i
$459$		0			0
$460$	−156.568	−	271.184i	−0.0158696	−	0.0274870i
$461$	−6291.03			−0.635580			−0.317790	−	0.948161i	$-0.602941\pi$
$461$	−6291.03			−0.635580			−0.317790	+	0.948161i	$0.602941\pi$
$462$		0			0
$463$	−5964.10			−0.598651			−0.299326	−	0.954151i	$-0.596762\pi$
$463$	−5964.10			−0.598651			−0.299326	+	0.954151i	$0.596762\pi$
$464$	402.086	+	696.433i	0.0402292	+	0.0696790i
$465$		0			0
$466$	−4004.27	+	6935.60i	−0.398057	+	0.689454i
$467$	−4290.20	−	7430.85i	−0.425111	−	0.736314i	0.571320	−	0.820728i	$-0.306432\pi$
$467$	−4290.20	−	7430.85i	−0.425111	−	0.736314i	−0.996431	+	0.0844134i	$0.973098\pi$
$468$		0			0
$469$	−2818.29	+	12675.5i	−0.277477	+	1.24797i
$470$	−293.394			−0.0287942
$471$		0			0
$472$	358.308	−	620.608i	0.0349417	−	0.0605207i
$473$	−221.565	+	383.762i	−0.0215382	+	0.0373053i
$474$		0			0
$475$	−6185.35			−0.597480
$476$	722.351	+	662.733i	0.0695565	+	0.0638158i
$477$		0			0
$478$	2768.29	+	4794.82i	0.264893	+	0.458808i
$479$	7089.88	−	12280.0i	0.676294	−	1.17138i	−0.299794	−	0.954004i	$-0.596918\pi$
$479$	7089.88	−	12280.0i	0.676294	−	1.17138i	0.976089	−	0.217372i	$-0.0697486\pi$
$480$		0			0
$481$	−84.6910	−	146.689i	−0.00802822	−	0.0139053i
$482$	14274.8			1.34896
$483$		0			0
$484$	−4996.39			−0.469232
$485$	98.9096	+	171.316i	0.00926031	+	0.0160393i
$486$		0			0
$487$	−2794.43	+	4840.09i	−0.260016	+	0.450360i	−0.966246	−	0.257622i	$-0.917061\pi$
$487$	−2794.43	+	4840.09i	−0.260016	+	0.450360i	0.706230	+	0.707982i	$0.250394\pi$
$488$	2715.47	+	4703.33i	0.251892	+	0.436290i
$489$		0			0
$490$	−35.4934	−	411.541i	−0.00327230	−	0.0379419i
$491$	9149.11			0.840924			0.420462	−	0.907310i	$-0.361868\pi$
$491$	9149.11			0.840924			0.420462	+	0.907310i	$0.361868\pi$
$492$		0			0
$493$	332.548	−	575.990i	0.0303797	−	0.0526193i
$494$	278.016	−	481.538i	0.0253209	−	0.0438571i
$495$		0			0
$496$	−3977.20			−0.360043
$497$	−6644.47	+	2090.49i	−0.599689	+	0.188674i
$498$		0			0
$499$	10709.1	+	18548.8i	0.960735	+	1.66404i	0.720661	+	0.693288i	$0.243839\pi$
$499$	10709.1	+	18548.8i	0.960735	+	1.66404i	0.240075	+	0.970754i	$0.422828\pi$
$500$	300.634	−	520.713i	0.0268895	−	0.0465740i
$501$		0			0
$502$	3925.58	+	6799.30i	0.349018	+	0.604517i
$503$	−11552.6			−1.02407			−0.512034	−	0.858965i	$-0.671108\pi$
$503$	−11552.6			−1.02407			−0.512034	+	0.858965i	$0.671108\pi$
$504$		0			0
$505$	228.292			0.0201165
$506$	−1176.60	−	2037.92i	−0.103372	−	0.179045i
$507$		0			0
$508$	1840.14	−	3187.21i	0.160714	−	0.278365i
$509$	9713.19	+	16823.7i	0.845834	+	1.46503i	0.884895	+	0.465791i	$0.154230\pi$
$509$	9713.19	+	16823.7i	0.845834	+	1.46503i	−0.0390601	+	0.999237i	$0.512436\pi$
$510$		0			0
$511$	−1081.14	+	4862.53i	−0.0935950	+	0.420950i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−2246.28	+	3890.66i	−0.192761	+	0.333871i
$515$	78.3755	−	135.750i	0.00670609	−	0.0116153i
$516$		0			0
$517$	−2204.83			−0.187559
$518$	−243.072	+	1093.23i	−0.0206177	+	0.0927294i
$519$		0			0
$520$	13.4931	+	23.3707i	0.00113791	+	0.00197091i
$521$	−3458.28	+	5989.91i	−0.290806	+	0.503691i	−0.974001	−	0.226546i	$-0.927257\pi$
$521$	−3458.28	+	5989.91i	−0.290806	+	0.503691i	0.683195	+	0.730236i	$0.260590\pi$
$522$		0			0
$523$	−7971.04	−	13806.2i	−0.666442	−	1.15431i	−0.978892	−	0.204377i	$-0.934483\pi$
$523$	−7971.04	−	13806.2i	−0.666442	−	1.15431i	0.312450	−	0.949934i	$-0.398850\pi$
$524$	11137.9			0.928551
$525$		0			0
$526$	2566.98			0.212786
$527$	1644.69	+	2848.68i	0.135946	+	0.235466i
$528$		0			0
$529$	−2367.78	+	4101.11i	−0.194606	+	0.337068i
$530$	293.271	+	507.961i	0.0240356	+	0.0416309i
$531$		0			0
$532$	−3506.93	+	1103.35i	−0.285798	+	0.0899179i
$533$	−1556.72			−0.126509
$534$		0			0
$535$	327.652	−	567.510i	0.0264778	−	0.0458610i
$536$	−2804.49	+	4857.53i	−0.225999	+	0.391443i
$537$		0			0
$538$	−9441.34			−0.756589
$539$	−266.729	−	3092.69i	−0.0213151	−	0.247146i
$540$		0			0
$541$	8881.02	+	15382.4i	0.705776	+	1.22244i	0.966411	+	0.257003i	$0.0827350\pi$
$541$	8881.02	+	15382.4i	0.705776	+	1.22244i	−0.260634	+	0.965438i	$0.583932\pi$
$542$	5137.04	−	8897.62i	0.407112	−	0.705139i
$543$		0			0
$544$	211.727	+	366.722i	0.0166870	+	0.0289027i
$545$	303.215			0.0238317
$546$		0			0
$547$	−9998.28			−0.781527			−0.390764	−	0.920491i	$-0.627789\pi$
$547$	−9998.28			−0.781527			−0.390764	+	0.920491i	$0.627789\pi$
$548$	−4831.77	−	8368.87i	−0.376648	−	0.652373i
$549$		0			0
$550$	1127.98	−	1953.71i	0.0874492	−	0.151466i
$551$	1247.14	+	2160.10i	0.0964243	+	0.167012i
$552$		0			0
$553$	536.877	+	492.567i	0.0412845	+	0.0378772i
$554$	11541.9			0.885141
$555$		0			0
$556$	189.197	−	327.699i	0.0144312	−	0.0249956i
$557$	1558.09	−	2698.69i	0.118525	−	0.205291i	−0.800658	−	0.599121i	$-0.795517\pi$
$557$	1558.09	−	2698.69i	0.118525	−	0.205291i	0.919183	+	0.393830i	$0.128850\pi$
$558$		0			0
$559$	−274.306			−0.0207547
$560$	38.7266	−	174.175i	0.00292231	−	0.0131433i
$561$		0			0
$562$	3594.63	+	6226.08i	0.269805	+	0.467316i
$563$	−11072.6	+	19178.4i	−0.828875	+	1.43565i	0.0700470	+	0.997544i	$0.477685\pi$
$563$	−11072.6	+	19178.4i	−0.828875	+	1.43565i	−0.898922	+	0.438109i	$0.855648\pi$
$564$		0			0
$565$	385.444	+	667.609i	0.0287005	+	0.0497107i
$566$	15060.6			1.11845
$567$		0			0
$568$	−3008.84			−0.222268
$569$	10360.5	+	17944.9i	0.763331	+	1.32213i	0.941125	+	0.338060i	$0.109771\pi$
$569$	10360.5	+	17944.9i	0.763331	+	1.32213i	−0.177794	+	0.984068i	$0.556896\pi$
$570$		0			0
$571$	1524.28	−	2640.12i	0.111715	−	0.193495i	−0.804747	−	0.593618i	$-0.797699\pi$
$571$	1524.28	−	2640.12i	0.111715	−	0.193495i	0.916462	+	0.400123i	$0.131032\pi$
$572$	101.399	+	175.629i	0.00741210	+	0.0128381i
$573$		0			0
$574$	7584.37	+	6958.41i	0.551508	+	0.505991i
$575$	−16204.1			−1.17523
$576$		0			0
$577$	−5648.14	+	9782.86i	−0.407513	+	0.705834i	−0.994610	−	0.103683i	$-0.966937\pi$
$577$	−5648.14	+	9782.86i	−0.407513	+	0.705834i	0.587097	+	0.809516i	$0.300271\pi$
$578$	−4737.89	+	8206.27i	−0.340952	+	0.590546i
$579$		0			0
$580$	−121.056			−0.00866651
$581$	8684.77	−	2732.41i	0.620147	−	0.195111i
$582$		0			0
$583$	2203.90	+	3817.27i	0.156563	+	0.271175i
$584$	−1075.85	+	1863.43i	−0.0762314	+	0.132037i
$585$		0			0
$586$	−4469.73	−	7741.79i	−0.315090	−	0.545752i
$587$	−5531.46			−0.388940			−0.194470	−	0.980908i	$-0.562299\pi$
$587$	−5531.46			−0.388940			−0.194470	+	0.980908i	$0.562299\pi$
$588$		0			0
$589$	−12336.0			−0.862978
$590$	53.9379	+	93.4233i	0.00376371	+	0.00651894i
$591$		0			0
$592$	−241.882	+	418.951i	−0.0167927	+	0.0290858i
$593$	−2880.05	−	4988.40i	−0.199443	−	0.345445i	0.748905	−	0.662677i	$-0.230580\pi$
$593$	−2880.05	−	4988.40i	−0.199443	−	0.345445i	−0.948348	+	0.317232i	$0.897247\pi$
$594$		0			0
$595$	−140.768	+	44.2886i	−0.00969906	+	0.00305152i
$596$	9378.62			0.644569
$597$		0			0
$598$	728.333	−	1261.51i	0.0498056	−	0.0862659i
$599$	8560.31	−	14826.9i	0.583915	−	1.01137i	−0.411095	−	0.911592i	$-0.634854\pi$
$599$	8560.31	−	14826.9i	0.583915	−	1.01137i	0.995010	−	0.0997775i	$-0.0318131\pi$
$600$		0			0
$601$	−2611.63			−0.177255			−0.0886277	−	0.996065i	$-0.528248\pi$
$601$	−2611.63			−0.177255			−0.0886277	+	0.996065i	$0.528248\pi$
$602$	1336.42	+	1226.12i	0.0904791	+	0.0830116i
$603$		0			0
$604$	455.264	+	788.540i	0.0306696	+	0.0531212i
$605$	376.066	−	651.365i	0.0252715	−	0.0437715i
$606$		0			0
$607$	13815.8	+	23929.7i	0.923831	+	1.60012i	0.793430	+	0.608662i	$0.208293\pi$
$607$	13815.8	+	23929.7i	0.923831	+	1.60012i	0.130402	+	0.991461i	$0.458373\pi$
$608$	−1588.06			−0.105928
$609$		0			0
$610$	−817.547			−0.0542647
$611$	−682.413	−	1181.97i	−0.0451841	−	0.0782611i
$612$		0			0
$613$	−823.643	+	1426.59i	−0.0542686	+	0.0939959i	−0.891883	−	0.452265i	$-0.850616\pi$
$613$	−823.643	+	1426.59i	−0.0542686	+	0.0939959i	0.837615	+	0.546261i	$0.183949\pi$
$614$	2811.31	+	4869.33i	0.184780	+	0.320049i
$615$		0			0
$616$	291.026	−	1308.91i	0.0190354	−	0.0856129i
$617$	−15808.0			−1.03145			−0.515727	−	0.856753i	$-0.672478\pi$
$617$	−15808.0			−1.03145			−0.515727	+	0.856753i	$0.672478\pi$
$618$		0			0
$619$	−2005.62	+	3473.83i	−0.130230	+	0.225566i	−0.923765	−	0.382959i	$-0.874905\pi$
$619$	−2005.62	+	3473.83i	−0.130230	+	0.225566i	0.793535	+	0.608525i	$0.208238\pi$
$620$	299.354	−	518.496i	0.0193909	−	0.0335860i
$621$		0			0
$622$	10323.8			0.665508
$623$	−9344.74	−	8573.50i	−0.600946	−	0.551348i
$624$		0			0
$625$	−7744.58	−	13414.0i	−0.495653	−	0.858497i
$626$	6717.10	−	11634.4i	0.428864	−	0.742815i
$627$		0			0
$628$	−690.421	−	1195.84i	−0.0438707	−	0.0759863i
$629$	400.100			0.0253626
$630$		0			0
$631$	−11682.4			−0.737036			−0.368518	−	0.929621i	$-0.620135\pi$
$631$	−11682.4			−0.737036			−0.368518	+	0.929621i	$0.620135\pi$
$632$	157.363	+	272.561i	0.00990438	+	0.0171549i
$633$		0			0
$634$	−8183.95	+	14175.0i	−0.512659	+	0.887952i
$635$	277.005	+	479.787i	0.0173112	+	0.0299839i
$636$		0			0
$637$	1575.39	−	1100.20i	0.0979892	−	0.0684327i
$638$	−909.724			−0.0564519
$639$		0			0
$640$	38.5370	−	66.7480i	0.00238017	−	0.00412257i
$641$	5405.96	−	9363.40i	0.333109	−	0.576961i	−0.650011	−	0.759925i	$-0.725236\pi$
$641$	5405.96	−	9363.40i	0.333109	−	0.576961i	0.983120	+	0.182964i	$0.0585690\pi$
$642$		0			0
$643$	−2335.44			−0.143236			−0.0716180	−	0.997432i	$-0.522816\pi$
$643$	−2335.44			−0.143236			−0.0716180	+	0.997432i	$0.522816\pi$
$644$	−9187.28	+	2890.51i	−0.562158	+	0.176866i
$645$		0			0
$646$	656.707	+	1137.45i	0.0399966	+	0.0692761i
$647$	−12753.8	+	22090.2i	−0.774965	+	1.34228i	0.159850	+	0.987141i	$0.448899\pi$
$647$	−12753.8	+	22090.2i	−0.774965	+	1.34228i	−0.934814	+	0.355137i	$0.884434\pi$
$648$		0			0
$649$	405.338	+	702.067i	0.0245161	+	0.0424631i
$650$	1396.47			0.0842680
$651$		0			0
$652$	−13165.7			−0.790813
$653$	2171.15	+	3760.54i	0.130113	+	0.225362i	0.923720	−	0.383069i	$-0.125133\pi$
$653$	2171.15	+	3760.54i	0.130113	+	0.225362i	−0.793607	+	0.608431i	$0.791799\pi$
$654$		0			0
$655$	−838.321	+	1452.01i	−0.0500090	+	0.0866182i
$656$	2223.04	+	3850.42i	0.132310	+	0.229167i
$657$		0			0
$658$	−1958.59	+	8808.91i	−0.116039	+	0.521896i
$659$	4148.85			0.245245			0.122622	−	0.992453i	$-0.460870\pi$
$659$	4148.85			0.245245			0.122622	+	0.992453i	$0.460870\pi$
$660$		0			0
$661$	8975.78	−	15546.5i	0.528165	−	0.914809i	−0.471296	−	0.881975i	$-0.656213\pi$
$661$	8975.78	−	15546.5i	0.528165	−	0.914809i	0.999461	−	0.0328337i	$-0.0104532\pi$
$662$	3549.92	−	6148.65i	0.208417	−	0.360988i
$663$		0			0
$664$	3932.76			0.229850
$665$	120.117	−	540.235i	0.00700442	−	0.0315029i
$666$		0			0
$667$	3267.19	+	5658.94i	0.189664	+	0.328508i
$668$	7314.61	−	12669.3i	0.423669	−	0.733816i
$669$		0			0
$670$	−422.175	−	731.229i	−0.0243434	−	0.0421639i
$671$	−6143.78			−0.353470
$672$		0			0
$673$	13667.9			0.782850			0.391425	−	0.920210i	$-0.371982\pi$
$673$	13667.9			0.782850			0.391425	+	0.920210i	$0.371982\pi$
$674$	−7152.10	−	12387.8i	−0.408737	−	0.707953i
$675$		0			0
$676$	4331.23	−	7501.91i	0.246429	−	0.426827i
$677$	8632.32	+	14951.6i	0.490054	+	0.848799i	0.999934	−	0.0114463i	$-0.00364355\pi$
$677$	8632.32	+	14951.6i	0.490054	+	0.848799i	−0.509880	+	0.860246i	$0.670310\pi$
$678$		0			0
$679$	5803.92	−	1826.03i	0.328032	−	0.103206i
$680$	−63.7447			−0.00359485
$681$		0			0
$682$	2249.62	−	3896.45i	0.126308	−	0.218772i
$683$	9857.82	−	17074.3i	0.552268	−	0.956556i	−0.445842	−	0.895112i	$-0.647096\pi$
$683$	9857.82	−	17074.3i	0.552268	−	0.956556i	0.998110	−	0.0614450i	$-0.0195709\pi$
$684$		0			0
$685$	1454.70			0.0811406
$686$	−12593.1	−	1681.64i	−0.700885	−	0.0935937i
$687$		0			0
$688$	391.715	+	678.471i	0.0217064	+	0.0375966i
$689$	−1364.25	+	2362.96i	−0.0754339	+	0.130655i
$690$		0			0
$691$	−5806.90	−	10057.8i	−0.319689	−	0.553717i	0.660734	−	0.750620i	$-0.270245\pi$
$691$	−5806.90	−	10057.8i	−0.319689	−	0.553717i	−0.980423	+	0.196903i	$0.936912\pi$
$692$	10604.9			0.582569
$693$		0			0
$694$	12554.3			0.686679
$695$	28.4808	+	49.3302i	0.00155444	+	0.00269238i
$696$		0			0
$697$	1838.58	−	3184.52i	0.0999158	−	0.173059i
$698$	5735.43	+	9934.05i	0.311016	+	0.538695i
$699$		0			0
$700$	−6803.63	−	6242.11i	−0.367361	−	0.337042i
$701$	−16100.4			−0.867481			−0.433740	−	0.901038i	$-0.642806\pi$
$701$	−16100.4			−0.867481			−0.433740	+	0.901038i	$0.642806\pi$
$702$		0			0
$703$	−750.237	+	1299.45i	−0.0402500	+	0.0697150i
$704$	289.602	−	501.605i	0.0155039	−	0.0268536i
$705$		0			0
$706$	−5233.59			−0.278993
$707$	1523.99	−	6854.27i	0.0810688	−	0.364613i
$708$		0			0
$709$	−1098.83	−	1903.22i	−0.0582049	−	0.100814i	0.835455	−	0.549559i	$-0.185204\pi$
$709$	−1098.83	−	1903.22i	−0.0582049	−	0.100814i	−0.893660	+	0.448745i	$0.851871\pi$
$710$	226.468	−	392.255i	0.0119707	−	0.0207339i
$711$		0			0
$712$	−2739.02	−	4744.12i	−0.144170	−	0.249710i
$713$	−32317.2			−1.69746
$714$		0			0
$715$	−30.5283			−0.00159678
$716$	−2803.87	−	4856.44i	−0.146348	−	0.253483i
$717$		0			0
$718$	11652.1	−	20182.1i	0.605646	−	1.04901i
$719$	−4950.87	−	8575.16i	−0.256796	−	0.444784i	0.708586	−	0.705625i	$-0.249333\pi$
$719$	−4950.87	−	8575.16i	−0.256796	−	0.444784i	−0.965382	+	0.260841i	$0.916000\pi$
$720$		0			0
$721$	−3552.58	−	3259.38i	−0.183502	−	0.168357i
$722$	8792.38			0.453211
$723$		0			0
$724$	5.83351	−	10.1039i	0.000299448	−	0.000518660i
$725$	−3132.18	+	5425.10i	−0.160450	+	0.277908i
$726$		0			0
$727$	24524.8			1.25114			0.625568	−	0.780170i	$-0.284867\pi$
$727$	24524.8			1.25114			0.625568	+	0.780170i	$0.284867\pi$
$728$	791.763	−	249.105i	0.0403086	−	0.0126819i
$729$		0			0
$730$	−161.954	−	280.512i	−0.00821120	−	0.0142222i
$731$	323.971	−	561.135i	0.0163920	−	0.0283917i
$732$		0			0
$733$	−4870.58	−	8436.09i	−0.245428	−	0.425094i	0.716824	−	0.697255i	$-0.245595\pi$
$733$	−4870.58	−	8436.09i	−0.245428	−	0.425094i	−0.962252	+	0.272160i	$0.912262\pi$
$734$	24948.7			1.25459
$735$		0			0
$736$	−4160.31			−0.208358
$737$	−3172.60	−	5495.11i	−0.158568	−	0.274647i
$738$		0			0
$739$	16049.5	−	27798.6i	0.798905	−	1.38374i	−0.121426	−	0.992601i	$-0.538747\pi$
$739$	16049.5	−	27798.6i	0.798905	−	1.38374i	0.920330	−	0.391143i	$-0.127920\pi$
$740$	−36.4117	−	63.0669i	−0.00180881	−	0.00313295i
$741$		0			0
$742$	17208.9	−	5414.26i	0.851425	−	0.267876i
$743$	−21404.8			−1.05688			−0.528442	−	0.848970i	$-0.677223\pi$
$743$	−21404.8			−1.05688			−0.528442	+	0.848970i	$0.677223\pi$
$744$		0			0
$745$	−705.906	+	1222.67i	−0.0347146	+	0.0601275i
$746$	1927.90	−	3339.22i	0.0946186	−	0.163884i
$747$		0			0
$748$	−479.035			−0.0234161
$749$	−14851.7	−	13626.0i	−0.724527	−	0.664730i
$750$		0			0
$751$	−12218.6	−	21163.2i	−0.593692	−	1.02830i	−0.993730	−	0.111806i	$-0.964337\pi$
$751$	−12218.6	−	21163.2i	−0.593692	−	1.02830i	0.400038	−	0.916498i	$-0.368997\pi$
$752$	−1949.01	+	3375.78i	−0.0945119	+	0.163699i
$753$		0			0
$754$	−281.567	−	487.689i	−0.0135996	−	0.0235552i
$755$	−137.066			−0.00660709
$756$		0			0
$757$	35339.6			1.69675			0.848375	−	0.529395i	$-0.177581\pi$
$757$	35339.6			1.69675			0.848375	+	0.529395i	$0.177581\pi$
$758$	−2946.20	−	5102.97i	−0.141175	−	0.244523i
$759$		0			0
$760$	119.529	−	207.030i	0.00570497	−	0.00988129i
$761$	−12995.4	−	22508.7i	−0.619033	−	1.07220i	−0.989663	−	0.143415i	$-0.954192\pi$
$761$	−12995.4	−	22508.7i	−0.619033	−	1.07220i	0.370630	−	0.928781i	$-0.379142\pi$
$762$		0			0
$763$	2024.15	−	9103.78i	0.0960410	−	0.431951i
$764$	1008.44			0.0477540
$765$		0			0
$766$	3117.46	−	5399.59i	0.147047	−	0.254693i
$767$	−250.911	+	434.591i	−0.0118121	+	0.0204592i
$768$		0			0
$769$	−19140.3			−0.897551			−0.448776	−	0.893645i	$-0.648140\pi$
$769$	−19140.3			−0.897551			−0.448776	+	0.893645i	$0.648140\pi$
$770$	148.734	+	136.459i	0.00696106	+	0.00638654i
$771$		0			0
$772$	−3946.81	−	6836.07i	−0.184001	−	0.318699i
$773$	−14103.5	+	24428.0i	−0.656232	+	1.13663i	0.325351	+	0.945593i	$0.394517\pi$
$773$	−14103.5	+	24428.0i	−0.656232	+	1.13663i	−0.981583	+	0.191034i	$0.938816\pi$
$774$		0			0
$775$	−15490.9	−	26831.0i	−0.717998	−	1.24361i
$776$	2628.21			0.121582
$777$		0			0
$778$	−3673.68			−0.169290
$779$	6895.14	+	11942.7i	0.317130	+	0.549285i
$780$		0			0
$781$	1701.89	−	2947.75i	0.0779748	−	0.135056i
$782$	1720.41	+	2979.84i	0.0786723	+	0.136264i
$783$		0			0
$784$	−4970.95	−	2325.47i	−0.226446	−	0.105934i
$785$	207.865			0.00945099
$786$		0			0
$787$	2497.97	−	4326.62i	0.113142	−	0.195969i	−0.803893	−	0.594774i	$-0.797242\pi$
$787$	2497.97	−	4326.62i	0.113142	−	0.195969i	0.917036	+	0.398805i	$0.130575\pi$
$788$	−3577.12	+	6195.74i	−0.161712	+	0.280094i
$789$		0			0
$790$	−47.3773			−0.00213368
$791$	22617.5	−	7115.93i	1.01667	−	0.319865i
$792$		0			0
$793$	−1901.55	−	3293.59i	−0.0851528	−	0.147489i
$794$	1354.80	−	2346.59i	0.0605544	−	0.104883i
$795$		0			0
$796$	−1844.26	−	3194.36i	−0.0821208	−	0.142237i
$797$	−895.767			−0.0398114			−0.0199057	−	0.999802i	$-0.506337\pi$
$797$	−895.767			−0.0398114			−0.0199057	+	0.999802i	$0.506337\pi$
$798$		0			0
$799$	3223.88			0.142744
$800$	−1994.20	−	3454.05i	−0.0881320	−	0.152649i
$801$		0			0
$802$	−12257.7	+	21230.9i	−0.539692	+	0.934774i
$803$	−1217.07	−	2108.02i	−0.0534861	−	0.0926406i
$804$		0			0
$805$	314.677	−	1415.28i	0.0137775	−	0.0619654i
$806$	2785.10			0.121713
$807$		0			0
$808$	1516.53	−	2626.71i	0.0660290	−	0.114366i
$809$	467.063	−	808.978i	0.0202980	−	0.0351572i	−0.855698	−	0.517476i	$-0.826872\pi$
$809$	467.063	−	808.978i	0.0202980	−	0.0351572i	0.875996	+	0.482318i	$0.160205\pi$
$810$		0			0
$811$	−8341.06			−0.361152			−0.180576	−	0.983561i	$-0.557796\pi$
$811$	−8341.06			−0.361152			−0.180576	+	0.983561i	$0.557796\pi$
$812$	−808.127	+	3634.61i	−0.0349257	+	0.157081i
$813$		0			0
$814$	−273.630	−	473.942i	−0.0117822	−	0.0204074i
$815$	990.954	−	1716.38i	0.0425909	−	0.0737696i
$816$		0			0
$817$	1214.97	+	2104.39i	0.0520275	+	0.0901143i
$818$	15231.3			0.651039
$819$		0			0
$820$	−669.292			−0.0285033
$821$	15014.9	+	26006.5i	0.638273	+	1.10552i	0.985812	+	0.167856i	$0.0536842\pi$
$821$	15014.9	+	26006.5i	0.638273	+	1.10552i	−0.347539	+	0.937666i	$0.612982\pi$
$822$		0			0
$823$	−1378.74	+	2388.04i	−0.0583958	+	0.101144i	−0.893745	−	0.448575i	$-0.851932\pi$
$823$	−1378.74	+	2388.04i	−0.0583958	+	0.101144i	0.835350	+	0.549719i	$0.185265\pi$
$824$	−1041.29	−	1803.57i	−0.0440232	−	0.0762504i
$825$		0			0
$826$	3165.03	−	995.782i	0.133324	−	0.0419464i
$827$	21128.3			0.888394			0.444197	−	0.895929i	$-0.353489\pi$
$827$	21128.3			0.888394			0.444197	+	0.895929i	$0.353489\pi$
$828$		0			0
$829$	7810.43	−	13528.1i	0.327223	−	0.566766i	−0.654737	−	0.755857i	$-0.727221\pi$
$829$	7810.43	−	13528.1i	0.327223	−	0.566766i	0.981960	+	0.189091i	$0.0605540\pi$
$830$	−296.009	+	512.703i	−0.0123791	+	0.0214412i
$831$		0			0
$832$	358.537			0.0149399
$833$	390.010	+	4522.11i	0.0162221	+	0.188093i
$834$		0			0
$835$	1101.11	+	1907.17i	0.0456351	+	0.0790424i
$836$	898.249	−	1555.81i	0.0371610	−	0.0643647i
$837$		0			0
$838$	8575.60	+	14853.4i	0.353507	+	0.612293i
$839$	−45714.6			−1.88110			−0.940550	−	0.339655i	$-0.889690\pi$
$839$	−45714.6			−1.88110			−0.940550	+	0.339655i	$0.889690\pi$
$840$		0			0
$841$	−21862.9			−0.896423
$842$	−14776.6	−	25593.7i	−0.604791	−	1.04753i
$843$		0			0
$844$	10261.2	−	17773.0i	0.418490	−	0.724847i
$845$	652.003	+	1129.30i	0.0265439	+	0.0459753i
$846$		0			0
$847$	−17046.2	−	15639.4i	−0.691517	−	0.634444i
$848$	7792.76			0.315571
$849$		0			0
$850$	−1649.32	+	2856.70i	−0.0665543	+	0.115275i
$851$	−1965.44	+	3404.24i	−0.0791708	+	0.137128i
$852$		0			0
$853$	−4539.60			−0.182219			−0.0911096	−	0.995841i	$-0.529041\pi$
$853$	−4539.60			−0.182219			−0.0911096	+	0.995841i	$0.529041\pi$
$854$	−5457.65	+	24546.2i	−0.218685	+	0.983551i
$855$		0			0
$856$	−4353.17	−	7539.90i	−0.173818	−	0.301061i
$857$	19837.5	−	34359.6i	0.790707	−	1.36955i	−0.134822	−	0.990870i	$-0.543046\pi$
$857$	19837.5	−	34359.6i	0.790707	−	1.36955i	0.925530	−	0.378675i	$-0.123620\pi$
$858$		0			0
$859$	16090.6	+	27869.7i	0.639120	+	1.10699i	0.985626	+	0.168941i	$0.0540347\pi$
$859$	16090.6	+	27869.7i	0.639120	+	1.10699i	−0.346506	+	0.938048i	$0.612632\pi$
$860$	−117.934			−0.00467618
$861$		0			0
$862$	32674.3			1.29106
$863$	−1201.72	−	2081.44i	−0.0474009	−	0.0821007i	0.841352	−	0.540488i	$-0.181760\pi$
$863$	−1201.72	−	2081.44i	−0.0474009	−	0.0821007i	−0.888752	+	0.458388i	$0.848427\pi$
$864$		0			0
$865$	−798.205	+	1382.53i	−0.0313755	+	0.0543439i
$866$	6150.96	+	10653.8i	0.241360	+	0.418049i
$867$		0			0
$868$	−13569.1	−	12449.2i	−0.530603	−	0.486811i
$869$	−356.036			−0.0138984
$870$		0			0
$871$	1963.90	−	3401.57i	0.0763997	−	0.132328i
$872$	2014.25	−	3488.78i	0.0782236	−	0.135487i
$873$		0			0
$874$	−12903.9			−0.499407
$875$	2655.58	−	835.498i	0.102600	−	0.0322800i
$876$		0			0
$877$	−6688.04	−	11584.0i	−0.257513	−	0.446026i	0.708062	−	0.706150i	$-0.249570\pi$
$877$	−6688.04	−	11584.0i	−0.257513	−	0.446026i	−0.965575	+	0.260125i	$0.916236\pi$
$878$	2598.48	−	4500.71i	0.0998799	−	0.172997i
$879$		0			0
$880$	43.5952	+	75.5092i	0.00166999	+	0.00289252i
$881$	37240.1			1.42412			0.712060	−	0.702119i	$-0.247762\pi$
$881$	37240.1			1.42412			0.712060	+	0.702119i	$0.247762\pi$
$882$		0			0
$883$	15721.8			0.599186			0.299593	−	0.954067i	$-0.403149\pi$
$883$	15721.8			0.599186			0.299593	+	0.954067i	$0.403149\pi$
$884$	−148.265	−	256.803i	−0.00564107	−	0.00977062i
$885$		0			0
$886$	−9831.21	+	17028.2i	−0.372783	+	0.645680i
$887$	−6729.50	−	11655.8i	−0.254740	−	0.441223i	0.710085	−	0.704116i	$-0.248657\pi$
$887$	−6729.50	−	11655.8i	−0.254740	−	0.441223i	−0.964825	+	0.262893i	$0.915323\pi$
$888$		0			0
$889$	16254.4	−	5113.97i	0.613223	−	0.192933i
$890$	824.638			0.0310583
$891$		0			0
$892$	−7555.64	+	13086.8i	−0.283612	+	0.491230i
$893$	−6045.18	+	10470.6i	−0.226533	+	0.392367i
$894$		0			0
$895$	844.161			0.0315276
$896$	−1746.80	−	1602.63i	−0.0651299	−	0.0597545i
$897$		0			0
$898$	−13649.3	−	23641.3i	−0.507220	−	0.878531i
$899$	−6246.77	+	10819.7i	−0.231748	+	0.401399i
$900$		0			0
$901$	−3222.53	−	5581.59i	−0.119154	−	0.206382i
$902$	−5029.66			−0.185665
$903$		0			0
$904$	10242.0			0.376818
$905$	0.878148	+	1.52100i	3.22548e−5	+	5.58670e-5i
$906$		0			0
$907$	−13519.1	+	23415.7i	−0.494921	+	0.857229i	−0.999983	−	0.00585452i	$-0.998136\pi$
$907$	−13519.1	+	23415.7i	−0.494921	+	0.857229i	0.505062	+	0.863083i	$0.331470\pi$
$908$	6646.51	+	11512.1i	0.242921	+	0.420751i
$909$		0			0
$910$	−27.1190	+	121.969i	−0.000987895	+	0.00444313i
$911$	19883.2			0.723116			0.361558	−	0.932350i	$-0.382245\pi$
$911$	19883.2			0.723116			0.361558	+	0.932350i	$0.382245\pi$
$912$		0			0
$913$	−2224.48	+	3852.91i	−0.0806348	+	0.139664i
$914$	−12311.9	+	21324.8i	−0.445560	+	0.771732i
$915$		0			0
$916$	−3817.84			−0.137713
$917$	37999.2	+	34863.0i	1.36842	+	1.25548i
$918$		0			0
$919$	−16676.1	−	28883.9i	−0.598579	−	1.03677i	−0.993031	−	0.117853i	$-0.962399\pi$
$919$	−16676.1	−	28883.9i	−0.598579	−	1.03677i	0.394452	−	0.918916i	$-0.370934\pi$
$920$	313.137	−	542.369i	0.0112215	−	0.0194363i
$921$		0			0
$922$	−6291.03	−	10896.4i	−0.224711	−	0.389212i
$923$	2106.99			0.0751382
$924$		0			0
$925$	−3768.44			−0.133952
$926$	−5964.10	−	10330.1i	−0.211655	−	0.366597i
$927$		0			0
$928$	−804.171	+	1392.87i	−0.0284463	+	0.0492705i
$929$	494.177	+	855.939i	0.0174525	+	0.0302287i	0.874620	−	0.484809i	$-0.161111\pi$
$929$	494.177	+	855.939i	0.0174525	+	0.0302287i	−0.857167	+	0.515038i	$0.827778\pi$
$930$		0			0
$931$	−15418.2	−	7212.83i	−0.542763	−	0.253911i
$932$	−16017.1			−0.562937
$933$		0			0
$934$	8580.41	−	14861.7i	0.300599	−	0.520653i
$935$	36.0558	−	62.4505i	0.00126112	−	0.00218433i
$936$		0			0
$937$	40549.5			1.41376			0.706880	−	0.707333i	$-0.250102\pi$
$937$	40549.5			1.41376			0.706880	+	0.707333i	$0.250102\pi$
$938$	−24772.8	+	7794.04i	−0.862326	+	0.271305i
$939$		0			0
$940$	−293.394	−	508.173i	−0.0101803	−	0.0176328i
$941$	−5412.12	+	9374.07i	−0.187492	+	0.324746i	−0.944414	−	0.328760i	$-0.893369\pi$
$941$	−5412.12	+	9374.07i	−0.187492	+	0.324746i	0.756921	+	0.653506i	$0.226703\pi$
$942$		0			0
$943$	18063.6	+	31287.0i	0.623787	+	1.08043i
$944$	1433.23			0.0494150
$945$		0			0
$946$	−886.261			−0.0304597
$947$	28695.5	+	49702.1i	0.984667	+	1.70549i	0.643406	+	0.765525i	$0.277521\pi$
$947$	28695.5	+	49702.1i	0.984667	+	1.70549i	0.341261	+	0.939969i	$0.389146\pi$
$948$		0			0
$949$	753.385	−	1304.90i	0.0257702	−	0.0446353i
$950$	−6185.35	−	10713.3i	−0.211241	−	0.365881i
$951$		0			0
$952$	−425.537	+	1913.88i	−0.0144871	+	0.0651568i
$953$	−1316.06			−0.0447338			−0.0223669	−	0.999750i	$-0.507120\pi$
$953$	−1316.06			−0.0447338			−0.0223669	+	0.999750i	$0.507120\pi$
$954$		0			0
$955$	−75.9028	+	131.467i	−0.00257189	+	0.00445465i
$956$	−5536.58	+	9589.64i	−0.187307	+	0.324426i
$957$		0			0
$958$	28359.5			0.956425
$959$	9711.07	−	43676.2i	0.326994	−	1.47068i
$960$		0			0
$961$	−15999.2	−	27711.5i	−0.537049	−	0.930197i
$962$	169.382	−	293.378i	0.00567681	−	0.00983253i
$963$		0			0
$964$	14274.8	+	24724.6i	0.476929	+	0.826065i
$965$	1188.27			0.0396390
$966$		0			0
$967$	49471.0			1.64517			0.822585	−	0.568642i	$-0.192531\pi$
$967$	49471.0			1.64517			0.822585	+	0.568642i	$0.192531\pi$
$968$	−4996.39	−	8653.99i	−0.165899	−	0.287345i
$969$		0			0
$970$	−197.819	+	342.633i	−0.00654803	+	0.0113415i
$971$	25767.5	+	44630.6i	0.851615	+	1.47504i	0.879750	+	0.475437i	$0.157710\pi$
$971$	25767.5	+	44630.6i	0.851615	+	1.47504i	−0.0281347	+	0.999604i	$0.508957\pi$
$972$		0			0
$973$	1671.23	−	525.802i	0.0550638	−	0.0173242i
$974$	−11177.7			−0.367718
$975$		0			0
$976$	−5430.93	+	9406.65i	−0.178115	+	0.308504i
$977$	11810.8	−	20456.9i	0.386755	−	0.669880i	−0.605256	−	0.796031i	$-0.706929\pi$
$977$	11810.8	−	20456.9i	0.386755	−	0.669880i	0.992011	+	0.126151i	$0.0402625\pi$
$978$		0			0
$979$	6197.07			0.202308
$980$	677.316	−	473.017i	0.0220776	−	0.0154183i
$981$		0			0
$982$	9149.11	+	15846.7i	0.297311	+	0.514959i
$983$	19303.4	−	33434.5i	0.626330	−	1.08484i	−0.361952	−	0.932197i	$-0.617889\pi$
$983$	19303.4	−	33434.5i	0.626330	−	1.08484i	0.988282	−	0.152639i	$-0.0487772\pi$
$984$		0			0
$985$	−538.482	−	932.677i	−0.0174187	−	0.0301701i
$986$	1330.19			0.0429634
$987$		0			0
$988$	1112.06			0.0358092
$989$	3182.93	+	5512.99i	0.102337	+	0.177253i
$990$		0			0
$991$	4966.28	−	8601.86i	0.159192	−	0.275729i	−0.775386	−	0.631488i	$-0.782444\pi$
$991$	4966.28	−	8601.86i	0.159192	−	0.275729i	0.934578	+	0.355760i	$0.115778\pi$
$992$	−3977.20	−	6888.71i	−0.127295	−	0.220481i
$993$		0			0
$994$	−10265.3	−	9418.08i	−0.327561	−	0.300526i
$995$	555.253			0.0176912
$996$		0			0
$997$	−23598.0	+	40872.9i	−0.749604	+	1.29835i	0.198408	+	0.980119i	$0.436423\pi$
$997$	−23598.0	+	40872.9i	−0.749604	+	1.29835i	−0.948012	+	0.318233i	$0.896910\pi$
$998$	−21418.3	+	37097.5i	−0.679342	+	1.17666i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.b.109.2	yes	6
3.2	odd	2		378.4.g.a.109.2	✓	6
7.2	even	3	inner	378.4.g.b.163.2	yes	6
21.2	odd	6		378.4.g.a.163.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.a.109.2	✓	6	3.2	odd	2
378.4.g.a.163.2	yes	6	21.2	odd	6
378.4.g.b.109.2	yes	6	1.1	even	1	trivial
378.4.g.b.163.2	yes	6	7.2	even	3	inner

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)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-0.301070 - 0.521469i) q^{5} +(-17.6665 + 5.55825i) q^{7} -8.00000 q^{8} +(0.602141 - 1.04294i) q^{10} +(4.52503 - 7.83758i) q^{11} +5.60214 q^{13} +(-27.2937 - 25.0411i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(-6.61646 + 11.4601i) q^{17} +(-24.8134 - 42.9780i) q^{19} +2.40856 q^{20} +18.1001 q^{22} +(-65.0049 - 112.592i) q^{23} +(62.3187 - 107.939i) q^{25} +(5.60214 + 9.70319i) q^{26} +(16.0787 - 72.3151i) q^{28} -50.2607 q^{29} +(124.287 - 215.272i) q^{31} +(16.0000 - 27.7128i) q^{32} -26.4659 q^{34} +(8.21732 + 7.53912i) q^{35} +(-15.1176 - 26.1845i) q^{37} +(49.6267 - 85.9560i) q^{38} +(2.40856 + 4.17175i) q^{40} -277.880 q^{41} -48.9644 q^{43} +(18.1001 + 31.3503i) q^{44} +(130.010 - 225.184i) q^{46} +(-121.813 - 210.986i) q^{47} +(281.212 - 196.390i) q^{49} +249.275 q^{50} +(-11.2043 + 19.4064i) q^{52} +(-243.524 + 421.796i) q^{53} -5.44941 q^{55} +(141.332 - 44.4660i) q^{56} +(-50.2607 - 87.0541i) q^{58} +(-44.7885 + 77.5759i) q^{59} +(-339.433 - 587.916i) q^{61} +497.150 q^{62} +64.0000 q^{64} +(-1.68664 - 2.92134i) q^{65} +(350.562 - 607.191i) q^{67} +(-26.4659 - 45.8402i) q^{68} +(-4.84082 + 21.7719i) q^{70} +376.105 q^{71} +(134.482 - 232.929i) q^{73} +(30.2352 - 52.3689i) q^{74} +198.507 q^{76} +(-36.3783 + 163.614i) q^{77} +(-19.6704 - 34.0701i) q^{79} +(-4.81712 + 8.34351i) q^{80} +(-277.880 - 481.303i) q^{82} -491.595 q^{83} +7.96808 q^{85} +(-48.9644 - 84.8089i) q^{86} +(-36.2002 + 62.7006i) q^{88} +(342.378 + 593.015i) q^{89} +(-98.9703 + 31.1381i) q^{91} +520.039 q^{92} +(243.626 - 421.972i) q^{94} +(-14.9411 + 25.8788i) q^{95} -328.527 q^{97} +(621.369 + 290.683i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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