LMFDB - Embedded newform 378.4.k.b.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

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([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, 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:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 86 x^{14} + 5225 x^{12} - 158916 x^{10} + 3517046 x^{8} - 29955345 x^{6} + 190411550 x^{4} + \cdots + 1500625 $ x^16 - 86x^14 + 5225x^12 - 158916x^10 + 3517046x^8 - 29955345x^6 + 190411550x^4 - 17021375*x^2 + 1500625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.1
Root			$-4.99672 + 2.88485i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.b.215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.79998 - 8.31381i) q^{5} +(-15.4648 - 10.1902i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.79998 - 8.31381i) q^{5} +(-15.4648 - 10.1902i) q^{7} +8.00000i q^{8} +(16.6276 + 9.59997i) q^{10} +(44.8102 + 25.8712i) q^{11} -27.2029i q^{13} +(36.9760 + 2.18506i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(68.9704 - 119.460i) q^{17} +(-48.0551 + 27.7446i) q^{19} -38.3999 q^{20} -103.485 q^{22} +(-76.0479 + 43.9063i) q^{23} +(16.4203 - 28.4408i) q^{25} +(27.2029 + 47.1168i) q^{26} +(-66.2294 + 33.1914i) q^{28} +277.784i q^{29} +(-297.157 - 171.564i) q^{31} +(27.7128 + 16.0000i) q^{32} +275.881i q^{34} +(-10.4883 + 177.484i) q^{35} +(43.1604 + 74.7560i) q^{37} +(55.4892 - 96.1102i) q^{38} +(66.5105 - 38.3999i) q^{40} +87.1111 q^{41} -348.898 q^{43} +(179.241 - 103.485i) q^{44} +(87.8125 - 152.096i) q^{46} +(-17.7760 - 30.7890i) q^{47} +(135.321 + 315.178i) q^{49} +65.6813i q^{50} +(-94.2336 - 54.4058i) q^{52} +(-429.574 - 248.015i) q^{53} -496.725i q^{55} +(81.5213 - 123.719i) q^{56} +(-277.784 - 481.136i) q^{58} +(2.59089 - 4.48755i) q^{59} +(-689.977 + 398.358i) q^{61} +686.255 q^{62} -64.0000 q^{64} +(-226.160 + 130.573i) q^{65} +(-356.143 + 616.857i) q^{67} +(-275.881 - 477.841i) q^{68} +(-159.318 - 317.900i) q^{70} +571.676i q^{71} +(186.789 + 107.843i) q^{73} +(-149.512 - 86.3208i) q^{74} +221.957i q^{76} +(-429.350 - 856.716i) q^{77} +(-449.673 - 778.856i) q^{79} +(-76.7997 + 133.021i) q^{80} +(-150.881 + 87.1111i) q^{82} -720.873 q^{83} -1324.23 q^{85} +(604.310 - 348.898i) q^{86} +(-206.969 + 358.481i) q^{88} +(-414.054 - 717.163i) q^{89} +(-277.202 + 420.688i) q^{91} +351.250i q^{92} +(61.5780 + 35.5521i) q^{94} +(461.327 + 266.347i) q^{95} -255.034i q^{97} +(-549.561 - 410.583i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) $ 16 * q + 32 * q^4 - 52 * q^7	$ 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−4.79998	−	8.31381i	−0.429324	−	0.743610i	0.567490	−	0.823380i	$-0.307915\pi$
$5$	−4.79998	−	8.31381i	−0.429324	−	0.743610i	−0.996813	+	0.0797703i	$0.974581\pi$
$6$		0			0
$7$	−15.4648	−	10.1902i	−0.835022	−	0.550217i
$7$	−15.4648	−	10.1902i	−0.835022	−	0.550217i
$8$			8.00000i			0.353553i
$9$		0			0
$10$	16.6276	+	9.59997i	0.525812	+	0.303578i
$11$	44.8102	+	25.8712i	1.22825	+	0.709132i	0.966664	−	0.256048i	$-0.0824205\pi$
$11$	44.8102	+	25.8712i	1.22825	+	0.709132i	0.261588	+	0.965180i	$0.415754\pi$
$12$		0			0
$13$		−	27.2029i		−	0.580363i	−0.956972	−	0.290182i	$-0.906284\pi$
$13$		−	27.2029i		−	0.580363i	0.956972	−	0.290182i	$-0.0937157\pi$
$14$	36.9760	+	2.18506i	0.705875	+	0.0417130i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	68.9704	−	119.460i	0.983986	−	1.70431i	0.337629	−	0.941279i	$-0.390375\pi$
$17$	68.9704	−	119.460i	0.983986	−	1.70431i	0.646357	−	0.763035i	$-0.276292\pi$
$18$		0			0
$19$	−48.0551	+	27.7446i	−0.580242	+	0.335003i	−0.761230	−	0.648483i	$-0.775404\pi$
$19$	−48.0551	+	27.7446i	−0.580242	+	0.335003i	0.180988	+	0.983485i	$0.442071\pi$
$20$	−38.3999			−0.429324
$21$		0			0
$22$	−103.485			−1.00286
$23$	−76.0479	+	43.9063i	−0.689438	+	0.398047i	−0.803402	−	0.595438i	$-0.796979\pi$
$23$	−76.0479	+	43.9063i	−0.689438	+	0.398047i	0.113963	+	0.993485i	$0.463645\pi$
$24$		0			0
$25$	16.4203	−	28.4408i	0.131363	−	0.227527i
$26$	27.2029	+	47.1168i	0.205189	+	0.355399i
$27$		0			0
$28$	−66.2294	+	33.1914i	−0.447006	+	0.224021i
$29$			277.784i			1.77873i	0.457197	+	0.889366i	$0.348854\pi$
$29$			277.784i			1.77873i	−0.457197	+	0.889366i	$0.651146\pi$
$30$		0			0
$31$	−297.157	−	171.564i	−1.72165	−	0.993993i	−0.915546	−	0.402212i	$-0.868241\pi$
$31$	−297.157	−	171.564i	−1.72165	−	0.993993i	−0.806099	−	0.591780i	$-0.798425\pi$
$32$	27.7128	+	16.0000i	0.153093	+	0.0883883i
$33$		0			0
$34$			275.881i			1.39157i
$35$	−10.4883	+	177.484i	−0.0506525	+	0.857152i
$36$		0			0
$37$	43.1604	+	74.7560i	0.191771	+	0.332157i	0.945837	−	0.324641i	$-0.105244\pi$
$37$	43.1604	+	74.7560i	0.191771	+	0.332157i	−0.754066	+	0.656798i	$0.771910\pi$
$38$	55.4892	−	96.1102i	0.236883	−	0.410293i
$39$		0			0
$40$	66.5105	−	38.3999i	0.262906	−	0.151789i
$41$	87.1111			0.331816			0.165908	−	0.986141i	$-0.446944\pi$
$41$	87.1111			0.331816			0.165908	+	0.986141i	$0.446944\pi$
$42$		0			0
$43$	−348.898			−1.23736			−0.618680	−	0.785643i	$-0.712332\pi$
$43$	−348.898			−1.23736			−0.618680	+	0.785643i	$0.712332\pi$
$44$	179.241	−	103.485i	0.614126	−	0.354566i
$45$		0			0
$46$	87.8125	−	152.096i	0.281462	−	0.487506i
$47$	−17.7760	−	30.7890i	−0.0551681	−	0.0955540i	0.837122	−	0.547016i	$-0.184236\pi$
$47$	−17.7760	−	30.7890i	−0.0551681	−	0.0955540i	−0.892291	+	0.451462i	$0.850903\pi$
$48$		0			0
$49$	135.321	+	315.178i	0.394522	+	0.918886i
$50$			65.6813i			0.185775i
$51$		0			0
$52$	−94.2336	−	54.4058i	−0.251305	−	0.145091i
$53$	−429.574	−	248.015i	−1.11333	−	0.642782i	−0.173642	−	0.984809i	$-0.555553\pi$
$53$	−429.574	−	248.015i	−1.11333	−	0.642782i	−0.939690	+	0.342026i	$0.888887\pi$
$54$		0			0
$55$		−	496.725i		−	1.21779i
$56$	81.5213	−	123.719i	0.194531	−	0.295225i
$57$		0			0
$58$	−277.784	−	481.136i	−0.628876	−	1.08925i
$59$	2.59089	−	4.48755i	0.00571703	−	0.00990219i	−0.863153	−	0.504943i	$-0.831514\pi$
$59$	2.59089	−	4.48755i	0.00571703	−	0.00990219i	0.868870	+	0.495041i	$0.164847\pi$
$60$		0			0
$61$	−689.977	+	398.358i	−1.44824	+	0.836140i	−0.998377	−	0.0569583i	$-0.981860\pi$
$61$	−689.977	+	398.358i	−1.44824	+	0.836140i	−0.449861	+	0.893099i	$0.648526\pi$
$62$	686.255			1.40572
$63$		0			0
$64$	−64.0000			−0.125000
$65$	−226.160	+	130.573i	−0.431564	+	0.249164i
$66$		0			0
$67$	−356.143	+	616.857i	−0.649399	+	1.12479i	0.333868	+	0.942620i	$0.391646\pi$
$67$	−356.143	+	616.857i	−0.649399	+	1.12479i	−0.983267	+	0.182172i	$0.941687\pi$
$68$	−275.881	−	477.841i	−0.491993	−	0.852157i
$69$		0			0
$70$	−159.318	−	317.900i	−0.272031	−	0.542804i
$71$			571.676i			0.955570i	0.878477	+	0.477785i	$0.158560\pi$
$71$			571.676i			0.955570i	−0.878477	+	0.477785i	$0.841440\pi$
$72$		0			0
$73$	186.789	+	107.843i	0.299479	+	0.172904i	0.642209	−	0.766530i	$-0.278018\pi$
$73$	186.789	+	107.843i	0.299479	+	0.172904i	−0.342730	+	0.939434i	$0.611352\pi$
$74$	−149.512	−	86.3208i	−0.234871	−	0.135603i
$75$		0			0
$76$			221.957i			0.335003i
$77$	−429.350	−	856.716i	−0.635441	−	1.26795i
$78$		0			0
$79$	−449.673	−	778.856i	−0.640407	−	1.10922i	−0.985342	−	0.170591i	$-0.945432\pi$
$79$	−449.673	−	778.856i	−0.640407	−	1.10922i	0.344935	−	0.938627i	$-0.387901\pi$
$80$	−76.7997	+	133.021i	−0.107331	+	0.185903i
$81$		0			0
$82$	−150.881	+	87.1111i	−0.203195	+	0.117315i
$83$	−720.873			−0.953326			−0.476663	−	0.879086i	$-0.658154\pi$
$83$	−720.873			−0.953326			−0.476663	+	0.879086i	$0.658154\pi$
$84$		0			0
$85$	−1324.23			−1.68979
$86$	604.310	−	348.898i	0.757725	−	0.437473i
$87$		0			0
$88$	−206.969	+	358.481i	−0.250716	+	0.434253i
$89$	−414.054	−	717.163i	−0.493142	−	0.854148i	0.506826	−	0.862048i	$-0.330818\pi$
$89$	−414.054	−	717.163i	−0.493142	−	0.854148i	−0.999969	+	0.00790053i	$0.997485\pi$
$90$		0			0
$91$	−277.202	+	420.688i	−0.319326	+	0.484616i
$92$			351.250i			0.398047i
$93$		0			0
$94$	61.5780	+	35.5521i	0.0675669	+	0.0390097i
$95$	461.327	+	266.347i	0.498223	+	0.287649i
$96$		0			0
$97$		−	255.034i		−	0.266957i	−0.991052	−	0.133478i	$-0.957385\pi$
$97$		−	255.034i		−	0.266957i	0.991052	−	0.133478i	$-0.0426147\pi$
$98$	−549.561	−	410.583i	−0.566470	−	0.423216i
$99$		0			0
$100$	−65.6813	−	113.763i	−0.0656813	−	0.113763i
$101$	−587.991	+	1018.43i	−0.579281	+	1.00334i	0.416282	+	0.909236i	$0.363333\pi$
$101$	−587.991	+	1018.43i	−0.579281	+	1.00334i	−0.995562	+	0.0941075i	$0.970000\pi$
$102$		0			0
$103$	1443.43	−	833.365i	1.38083	−	0.797222i	0.388572	−	0.921418i	$-0.372968\pi$
$103$	1443.43	−	833.365i	1.38083	−	0.797222i	0.992258	+	0.124196i	$0.0396352\pi$
$104$	217.623			0.205189
$105$		0			0
$106$	992.060			0.909032
$107$	332.343	−	191.878i	0.300269	−	0.173360i	−0.342295	−	0.939593i	$-0.611204\pi$
$107$	332.343	−	191.878i	0.300269	−	0.173360i	0.642564	+	0.766232i	$0.277871\pi$
$108$		0			0
$109$	889.896	−	1541.35i	0.781987	−	1.35444i	−0.148795	−	0.988868i	$-0.547540\pi$
$109$	889.896	−	1541.35i	0.781987	−	1.35444i	0.930783	−	0.365573i	$-0.119127\pi$
$110$	496.725	+	860.352i	0.430553	+	0.745740i
$111$		0			0
$112$	−17.4805	+	295.808i	−0.0147478	+	0.249565i
$113$			210.382i			0.175142i	0.996158	+	0.0875712i	$0.0279105\pi$
$113$			210.382i			0.175142i	−0.996158	+	0.0875712i	$0.972089\pi$
$114$		0			0
$115$	730.057	+	421.499i	0.591984	+	0.341782i
$116$	962.272	+	555.568i	0.770213	+	0.444683i
$117$		0			0
$118$			10.3636i			0.00808510i
$119$	−2283.93	+	1144.61i	−1.75939	+	0.881733i
$120$		0			0
$121$	673.134	+	1165.90i	0.505736	+	0.875960i
$122$	796.717	−	1379.95i	0.591240	−	1.02406i
$123$		0			0
$124$	−1188.63	+	686.255i	−0.860823	+	0.496996i
$125$	−1515.26			−1.08424
$126$		0			0
$127$	−355.794			−0.248595			−0.124298	−	0.992245i	$-0.539668\pi$
$127$	−355.794			−0.248595			−0.124298	+	0.992245i	$0.539668\pi$
$128$	110.851	−	64.0000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	261.147	−	452.320i	0.176185	−	0.305162i
$131$	26.1971	+	45.3747i	0.0174722	+	0.0302627i	0.874629	−	0.484792i	$-0.161105\pi$
$131$	26.1971	+	45.3747i	0.0174722	+	0.0302627i	−0.857157	+	0.515055i	$0.827771\pi$
$132$		0			0
$133$	1025.89	+	60.6237i	0.668839	+	0.0395243i
$134$		−	1424.57i		−	0.918389i
$135$		0			0
$136$	955.681	+	551.763i	0.602566	+	0.347892i
$137$	258.810	+	149.424i	0.161399	+	0.0931836i	0.578524	−	0.815666i	$-0.303629\pi$
$137$	258.810	+	149.424i	0.161399	+	0.0931836i	−0.417125	+	0.908849i	$0.636962\pi$
$138$		0			0
$139$			1416.82i			0.864553i	0.901741	+	0.432276i	$0.142289\pi$
$139$			1416.82i			0.864553i	−0.901741	+	0.432276i	$0.857711\pi$
$140$	593.847	+	391.301i	0.358494	+	0.236221i
$141$		0			0
$142$	−571.676	−	990.172i	−0.337845	−	0.585165i
$143$	703.770	−	1218.97i	0.411554	−	0.712833i
$144$		0			0
$145$	2309.45	−	1333.36i	1.32268	−	0.763651i
$146$	−431.370			−0.244524
$147$		0			0
$148$	345.283			0.191771
$149$	1519.29	−	877.165i	0.835338	−	0.482283i	−0.0203385	−	0.999793i	$-0.506474\pi$
$149$	1519.29	−	877.165i	0.835338	−	0.482283i	0.855677	+	0.517510i	$0.173141\pi$
$150$		0			0
$151$	−1564.69	+	2710.12i	−0.843262	+	1.46057i	0.0438605	+	0.999038i	$0.486034\pi$
$151$	−1564.69	+	2710.12i	−0.843262	+	1.46057i	−0.887122	+	0.461535i	$0.847299\pi$
$152$	−221.957	−	384.441i	−0.118441	−	0.205146i
$153$		0			0
$154$	1600.37	+	1054.53i	0.837413	+	0.551793i
$155$			3294.01i			1.70698i
$156$		0			0
$157$	75.3277	+	43.4905i	0.0382918	+	0.0221078i	0.519024	−	0.854760i	$-0.326296\pi$
$157$	75.3277	+	43.4905i	0.0382918	+	0.0221078i	−0.480732	+	0.876868i	$0.659629\pi$
$158$	1557.71	+	899.346i	0.784335	+	0.452836i
$159$		0			0
$160$		−	307.199i		−	0.151789i
$161$	1623.48	+	95.9378i	0.794708	+	0.0469625i
$162$		0			0
$163$	−728.072	−	1261.06i	−0.349859	−	0.605973i	0.636365	−	0.771388i	$-0.280437\pi$
$163$	−728.072	−	1261.06i	−0.349859	−	0.605973i	−0.986224	+	0.165415i	$0.947104\pi$
$164$	174.222	−	301.762i	0.0829540	−	0.143681i
$165$		0			0
$166$	1248.59	−	720.873i	0.583791	−	0.337052i
$167$	−350.270			−0.162303			−0.0811517	−	0.996702i	$-0.525860\pi$
$167$	−350.270			−0.162303			−0.0811517	+	0.996702i	$0.525860\pi$
$168$		0			0
$169$	1457.00			0.663178
$170$	2293.63	−	1324.23i	1.03478	−	0.597432i
$171$		0			0
$172$	−697.797	+	1208.62i	−0.309340	+	0.535793i
$173$	927.796	+	1606.99i	0.407740	+	0.706227i	0.994636	−	0.103436i	$-0.0329836\pi$
$173$	927.796	+	1606.99i	0.407740	+	0.706227i	−0.586896	+	0.809662i	$0.699650\pi$
$174$		0			0
$175$	−543.754	+	272.507i	−0.234880	+	0.117712i
$176$		−	827.877i		−	0.354566i
$177$		0			0
$178$	1434.33	+	828.109i	0.603974	+	0.348704i
$179$	−448.009	−	258.658i	−0.187071	−	0.108006i	0.403539	−	0.914962i	$-0.367780\pi$
$179$	−448.009	−	258.658i	−0.187071	−	0.108006i	−0.590611	+	0.806957i	$0.701113\pi$
$180$		0			0
$181$			593.488i			0.243722i	0.992547	+	0.121861i	$0.0388861\pi$
$181$			593.488i			0.243722i	−0.992547	+	0.121861i	$0.961114\pi$
$182$	59.4399	−	1005.85i	0.0242087	−	0.409664i
$183$		0			0
$184$	−351.250	−	608.383i	−0.140731	−	0.243753i
$185$	414.338	−	717.655i	0.164664	−	0.285206i
$186$		0			0
$187$	6181.15	−	3568.69i	2.41717	−	1.39555i
$188$	−142.208			−0.0551681
$189$		0			0
$190$	−1065.39			−0.406797
$191$	−2959.65	+	1708.76i	−1.12122	+	0.647336i	−0.941712	−	0.336420i	$-0.890784\pi$
$191$	−2959.65	+	1708.76i	−1.12122	+	0.647336i	−0.179508	+	0.983757i	$0.557450\pi$
$192$		0			0
$193$	951.974	−	1648.87i	0.355050	−	0.614964i	−0.632077	−	0.774906i	$-0.717797\pi$
$193$	951.974	−	1648.87i	0.355050	−	0.614964i	0.987126	+	0.159942i	$0.0511306\pi$
$194$	255.034	+	441.733i	0.0943836	+	0.163477i
$195$		0			0
$196$	1362.45	+	161.590i	0.496520	+	0.0588883i
$197$			1294.73i			0.468254i	0.972206	+	0.234127i	$0.0752231\pi$
$197$			1294.73i			0.468254i	−0.972206	+	0.234127i	$0.924777\pi$
$198$		0			0
$199$	−265.433	−	153.248i	−0.0945531	−	0.0545902i	0.451978	−	0.892029i	$-0.350719\pi$
$199$	−265.433	−	153.248i	−0.0945531	−	0.0545902i	−0.546531	+	0.837439i	$0.684052\pi$
$200$	227.527	+	131.363i	0.0804428	+	0.0464437i
$201$		0			0
$202$		−	2351.97i		−	0.819226i
$203$	2830.66	−	4295.88i	0.978688	−	1.48528i
$204$		0			0
$205$	−418.132	−	724.225i	−0.142457	−	0.246742i
$206$	−1666.73	+	2886.86i	−0.563721	+	0.976394i
$207$		0			0
$208$	−376.934	+	217.623i	−0.125652	+	0.0725454i
$209$	−2871.14			−0.950244
$210$		0			0
$211$	3414.52			1.11405			0.557026	−	0.830495i	$-0.311942\pi$
$211$	3414.52			1.11405			0.557026	+	0.830495i	$0.311942\pi$
$212$	−1718.30	+	992.060i	−0.556666	+	0.321391i
$213$		0			0
$214$	−383.757	+	664.686i	−0.122584	+	0.212322i
$215$	1674.71	+	2900.68i	0.531228	+	0.920114i
$216$		0			0
$217$	2847.22	+	5681.28i	0.890700	+	1.77728i
$218$			3559.58i			1.10590i
$219$		0			0
$220$	−1720.70	−	993.449i	−0.527318	−	0.304447i
$221$	−3249.66	−	1876.19i	−0.989122	−	0.571070i
$222$		0			0
$223$			331.944i			0.0996799i	0.998757	+	0.0498400i	$0.0158711\pi$
$223$			331.944i			0.0996799i	−0.998757	+	0.0498400i	$0.984129\pi$
$224$	−265.531	−	529.835i	−0.0792033	−	0.158041i
$225$		0			0
$226$	−210.382	−	364.393i	−0.0619222	−	0.107252i
$227$	−946.926	+	1640.12i	−0.276871	+	0.479555i	−0.970605	−	0.240676i	$-0.922631\pi$
$227$	−946.926	+	1640.12i	−0.276871	+	0.479555i	0.693734	+	0.720231i	$0.255964\pi$
$228$		0			0
$229$	−2460.07	+	1420.32i	−0.709896	+	0.409858i	−0.811022	−	0.585015i	$-0.801089\pi$
$229$	−2460.07	+	1420.32i	−0.709896	+	0.409858i	0.101127	+	0.994874i	$0.467755\pi$
$230$	−1685.99			−0.483353
$231$		0			0
$232$	−2222.27			−0.628876
$233$	3958.46	−	2285.42i	1.11299	−	0.642587i	0.173390	−	0.984853i	$-0.444528\pi$
$233$	3958.46	−	2285.42i	1.11299	−	0.642587i	0.939603	+	0.342267i	$0.111195\pi$
$234$		0			0
$235$	−170.649	+	295.573i	−0.0473699	+	0.0820472i
$236$	−10.3636	−	17.9502i	−0.00285852	−	0.00495110i
$237$		0			0
$238$	2811.28	−	4266.46i	0.765664	−	1.16199i
$239$		−	357.209i		−	0.0966775i	−0.998831	−	0.0483387i	$-0.984607\pi$
$239$		−	357.209i		−	0.0966775i	0.998831	−	0.0483387i	$-0.0153927\pi$
$240$		0			0
$241$	−4420.93	−	2552.42i	−1.18165	−	0.682224i	−0.225252	−	0.974301i	$-0.572320\pi$
$241$	−4420.93	−	2552.42i	−1.18165	−	0.682224i	−0.956395	+	0.292077i	$0.905654\pi$
$242$	−2331.81	−	1346.27i	−0.619397	−	0.357609i
$243$		0			0
$244$			3186.87i			0.836140i
$245$	1970.79	−	2637.88i	0.513915	−	0.687870i
$246$		0			0
$247$	754.734	+	1307.24i	0.194423	+	0.336751i
$248$	1372.51	−	2377.26i	0.351429	−	0.608694i
$249$		0			0
$250$	2624.52	−	1515.26i	0.663956	−	0.383335i
$251$	−3617.28			−0.909645			−0.454822	−	0.890582i	$-0.650297\pi$
$251$	−3617.28			−0.909645			−0.454822	+	0.890582i	$0.650297\pi$
$252$		0			0
$253$	−4543.62			−1.12907
$254$	616.253	−	355.794i	0.152233	−	0.0878918i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	835.142	+	1446.51i	0.202703	+	0.351092i	0.949399	−	0.314074i	$-0.101694\pi$
$257$	835.142	+	1446.51i	0.202703	+	0.351092i	−0.746695	+	0.665166i	$0.768361\pi$
$258$		0			0
$259$	94.3081	−	1595.90i	0.0226255	−	0.382874i
$260$			1044.59i			0.249164i
$261$		0			0
$262$	−90.7494	−	52.3942i	−0.0213989	−	0.0123547i
$263$	−29.5214	−	17.0442i	−0.00692155	−	0.00399616i	0.496535	−	0.868017i	$-0.334605\pi$
$263$	−29.5214	−	17.0442i	−0.00692155	−	0.00399616i	−0.503457	+	0.864020i	$0.667939\pi$
$264$		0			0
$265$			4761.87i			1.10385i
$266$	−1837.51	+	920.882i	−0.423552	+	0.212267i
$267$		0			0
$268$	1424.57	+	2467.43i	0.324700	+	0.562396i
$269$	3092.36	−	5356.13i	0.700909	−	1.21401i	−0.267238	−	0.963630i	$-0.586111\pi$
$269$	3092.36	−	5356.13i	0.700909	−	1.21401i	0.968148	−	0.250380i	$-0.0805556\pi$
$270$		0			0
$271$	−1423.47	+	821.841i	−0.319076	+	0.184219i	−0.650981	−	0.759094i	$-0.725642\pi$
$271$	−1423.47	+	821.841i	−0.319076	+	0.184219i	0.331904	+	0.943313i	$0.392309\pi$
$272$	−2207.05			−0.491993
$273$		0			0
$274$	−597.696			−0.131781
$275$	1471.59	−	849.626i	0.322693	−	0.186307i
$276$		0			0
$277$	−2428.41	+	4206.12i	−0.526747	+	0.912352i	0.472768	+	0.881187i	$0.343255\pi$
$277$	−2428.41	+	4206.12i	−0.526747	+	0.912352i	−0.999514	+	0.0311647i	$0.990078\pi$
$278$	−1416.82	−	2454.00i	−0.305665	−	0.529428i
$279$		0			0
$280$	−1419.87	−	83.9060i	−0.303049	−	0.0179084i
$281$		−	5754.68i		−	1.22169i	−0.791750	−	0.610846i	$-0.790830\pi$
$281$		−	5754.68i		−	1.22169i	0.791750	−	0.610846i	$-0.209170\pi$
$282$		0			0
$283$	−2181.75	−	1259.63i	−0.458274	−	0.264584i	0.253044	−	0.967455i	$-0.418568\pi$
$283$	−2181.75	−	1259.63i	−0.458274	−	0.264584i	−0.711318	+	0.702870i	$0.751901\pi$
$284$	1980.34	+	1143.35i	0.413774	+	0.238893i
$285$		0			0
$286$			2815.08i			0.582025i
$287$	−1347.16	−	887.676i	−0.277074	−	0.182571i
$288$		0			0
$289$	−7057.32	−	12223.6i	−1.43646	−	2.48802i
$290$	−2666.72	+	4618.89i	−0.539983	+	0.935278i
$291$		0			0
$292$	747.155	−	431.370i	0.149740	−	0.0864522i
$293$	−4257.92			−0.848976			−0.424488	−	0.905433i	$-0.639546\pi$
$293$	−4257.92			−0.848976			−0.424488	+	0.905433i	$0.639546\pi$
$294$		0			0
$295$	−49.7449			−0.00981783
$296$	−598.048	+	345.283i	−0.117435	+	0.0678013i
$297$		0			0
$298$	−1754.33	+	3038.59i	−0.341026	+	0.590674i
$299$	1194.38	+	2068.72i	0.231012	+	0.400125i
$300$		0			0
$301$	5395.65	+	3555.33i	1.03322	+	0.680817i
$302$		−	6258.75i		−	1.19255i
$303$		0			0
$304$	768.882	+	443.914i	0.145060	+	0.0837507i
$305$	6623.76	+	3824.23i	1.24352	+	0.717949i
$306$		0			0
$307$		−	1312.64i		−	0.244028i	−0.992528	−	0.122014i	$-0.961065\pi$
$307$		−	1312.64i		−	0.244028i	0.992528	−	0.122014i	$-0.0389353\pi$
$308$	−3826.45	−	226.120i	−0.707897	−	0.0418324i
$309$		0			0
$310$	−3294.01	−	5705.40i	−0.603508	−	1.04531i
$311$	−72.2772	+	125.188i	−0.0131783	+	0.0228255i	−0.872539	−	0.488544i	$-0.837528\pi$
$311$	−72.2772	+	125.188i	−0.0131783	+	0.0228255i	0.859361	+	0.511369i	$0.170862\pi$
$312$		0			0
$313$	6793.99	−	3922.51i	1.22690	−	0.708350i	0.260519	−	0.965469i	$-0.416106\pi$
$313$	6793.99	−	3922.51i	1.22690	−	0.708350i	0.966380	+	0.257119i	$0.0827731\pi$
$314$	−173.962			−0.0312651
$315$		0			0
$316$	−3597.38			−0.640407
$317$	−4032.21	+	2328.00i	−0.714422	+	0.412472i	−0.812696	−	0.582688i	$-0.802001\pi$
$317$	−4032.21	+	2328.00i	−0.714422	+	0.412472i	0.0982743	+	0.995159i	$0.468668\pi$
$318$		0			0
$319$	−7186.60	+	12447.6i	−1.26135	+	2.18473i
$320$	307.199	+	532.084i	0.0536654	+	0.0929513i
$321$		0			0
$322$	−2907.89	+	1457.31i	−0.503261	+	0.252213i
$323$			7654.23i			1.31855i
$324$		0			0
$325$	−773.673	−	446.680i	−0.132048	−	0.0762380i
$326$	2522.12	+	1456.14i	0.428488	+	0.247388i
$327$		0			0
$328$			696.889i			0.117315i
$329$	−38.8417	+	657.287i	−0.00650885	+	0.110144i
$330$		0			0
$331$	−3330.90	−	5769.30i	−0.553121	−	0.958034i	−0.998047	−	0.0624665i	$-0.980103\pi$
$331$	−3330.90	−	5769.30i	−0.553121	−	0.958034i	0.444926	−	0.895567i	$-0.353230\pi$
$332$	−1441.75	+	2497.18i	−0.238331	+	0.412802i
$333$		0			0
$334$	606.685	−	350.270i	0.0993902	−	0.0573829i
$335$	6837.91			1.11521
$336$		0			0
$337$	−5828.97			−0.942209			−0.471105	−	0.882077i	$-0.656145\pi$
$337$	−5828.97			−0.942209			−0.471105	+	0.882077i	$0.656145\pi$
$338$	−2523.60	+	1457.00i	−0.406112	+	0.234469i
$339$		0			0
$340$	−2648.45	+	4587.25i	−0.422449	+	0.731702i
$341$	−8877.11	−	15375.6i	−1.40974	−	2.44175i
$342$		0			0
$343$	1119.00	−	6253.12i	0.176152	−	0.984363i
$344$		−	2791.19i		−	0.437473i
$345$		0			0
$346$	−3213.98	−	1855.59i	−0.499378	−	0.288316i
$347$	6309.41	+	3642.74i	0.976100	+	0.563552i	0.901091	−	0.433631i	$-0.142768\pi$
$347$	6309.41	+	3642.74i	0.976100	+	0.563552i	0.0750099	+	0.997183i	$0.476101\pi$
$348$		0			0
$349$		−	8093.19i		−	1.24131i	−0.784082	−	0.620657i	$-0.786866\pi$
$349$		−	8093.19i		−	1.24131i	0.784082	−	0.620657i	$-0.213134\pi$
$350$	669.303	−	1015.75i	0.102216	−	0.155126i
$351$		0			0
$352$	827.877	+	1433.93i	0.125358	+	0.217126i
$353$	6294.32	−	10902.1i	0.949045	−	1.64379i	0.201602	−	0.979468i	$-0.435385\pi$
$353$	6294.32	−	10902.1i	0.949045	−	1.64379i	0.747443	−	0.664326i	$-0.231281\pi$
$354$		0			0
$355$	4752.81	−	2744.04i	0.710572	−	0.410249i
$356$	−3312.43			−0.493142
$357$		0			0
$358$	1034.63			0.152743
$359$	838.790	−	484.276i	0.123314	−	0.0711953i	−0.437074	−	0.899425i	$-0.643985\pi$
$359$	838.790	−	484.276i	0.123314	−	0.0711953i	0.560388	+	0.828230i	$0.310652\pi$
$360$		0			0
$361$	−1889.97	+	3273.53i	−0.275546	+	0.477260i
$362$	−593.488	−	1027.95i	−0.0861686	−	0.149248i
$363$		0			0
$364$	902.901	+	1801.63i	0.130013	+	0.259426i
$365$		−	2070.57i		−	0.296928i
$366$		0			0
$367$	−6313.17	−	3644.91i	−0.897942	−	0.518427i	−0.0214098	−	0.999771i	$-0.506815\pi$
$367$	−6313.17	−	3644.91i	−0.897942	−	0.518427i	−0.876532	+	0.481344i	$0.840149\pi$
$368$	1216.77	+	702.500i	0.172360	+	0.0995118i
$369$		0			0
$370$			1657.35i			0.232869i
$371$	4115.98	+	8212.94i	0.575986	+	1.14931i
$372$		0			0
$373$	−5332.76	−	9236.61i	−0.740267	−	1.28218i	−0.952373	−	0.304934i	$-0.901366\pi$
$373$	−5332.76	−	9236.61i	−0.740267	−	1.28218i	0.212106	−	0.977247i	$-0.431968\pi$
$374$	−7137.37	+	12362.3i	−0.986804	+	1.70919i
$375$		0			0
$376$	246.312	−	142.208i	0.0337834	−	0.0195049i
$377$	7556.53			1.03231
$378$		0			0
$379$	−7342.28			−0.995113			−0.497556	−	0.867432i	$-0.665769\pi$
$379$	−7342.28			−0.995113			−0.497556	+	0.867432i	$0.665769\pi$
$380$	1845.31	−	1065.39i	0.249112	−	0.143825i
$381$		0			0
$382$	3417.51	−	5919.31i	0.457736	−	0.792822i
$383$	6748.09	+	11688.0i	0.900290	+	1.55935i	0.827117	+	0.562029i	$0.189979\pi$
$383$	6748.09	+	11688.0i	0.900290	+	1.55935i	0.0731732	+	0.997319i	$0.476687\pi$
$384$		0			0
$385$	−5061.70	+	7681.76i	−0.670048	+	1.01688i
$386$			3807.90i			0.502116i
$387$		0			0
$388$	−883.465	−	510.069i	−0.115596	−	0.0667392i
$389$	−2204.30	−	1272.65i	−0.287307	−	0.165877i	0.349420	−	0.936966i	$-0.386379\pi$
$389$	−2204.30	−	1272.65i	−0.287307	−	0.165877i	−0.636727	+	0.771089i	$0.719712\pi$
$390$		0			0
$391$			12112.9i			1.56669i
$392$	−2521.42	+	1082.57i	−0.324875	+	0.139485i
$393$		0			0
$394$	−1294.73	−	2242.55i	−0.165553	−	0.286746i
$395$	−4316.85	+	7477.00i	−0.549884	+	0.952427i
$396$		0			0
$397$	2299.34	−	1327.52i	0.290681	−	0.167825i	−0.347568	−	0.937655i	$-0.612992\pi$
$397$	2299.34	−	1327.52i	0.290681	−	0.167825i	0.638249	+	0.769830i	$0.279659\pi$
$398$	612.992			0.0772023
$399$		0			0
$400$	−525.450			−0.0656813
$401$	7365.70	−	4252.59i	0.917270	−	0.529586i	0.0345071	−	0.999404i	$-0.489014\pi$
$401$	7365.70	−	4252.59i	0.917270	−	0.529586i	0.882763	+	0.469818i	$0.155681\pi$
$402$		0			0
$403$	−4667.03	+	8083.54i	−0.576877	+	0.999180i
$404$	2351.97	+	4073.72i	0.289640	+	0.501672i
$405$		0			0
$406$	−606.975	+	10271.3i	−0.0741962	+	1.25556i
$407$			4466.44i			0.543964i
$408$		0			0
$409$	−3071.59	−	1773.39i	−0.371346	−	0.214397i	0.302700	−	0.953086i	$-0.402112\pi$
$409$	−3071.59	−	1773.39i	−0.371346	−	0.214397i	−0.674046	+	0.738689i	$0.735445\pi$
$410$	1448.45	+	836.263i	0.174473	+	0.100732i
$411$		0			0
$412$		−	6666.92i		−	0.797222i
$413$	−85.7965	+	42.9976i	−0.0102222	+	0.00512294i
$414$		0			0
$415$	3460.18	+	5993.20i	0.409285	+	0.708903i
$416$	435.246	−	753.869i	0.0512974	−	0.0888496i
$417$		0			0
$418$	4972.97	−	2871.14i	0.581904	−	0.335962i
$419$	−1430.85			−0.166829			−0.0834146	−	0.996515i	$-0.526583\pi$
$419$	−1430.85			−0.166829			−0.0834146	+	0.996515i	$0.526583\pi$
$420$		0			0
$421$	−1900.59			−0.220021			−0.110011	−	0.993930i	$-0.535088\pi$
$421$	−1900.59			−0.220021			−0.110011	+	0.993930i	$0.535088\pi$
$422$	−5914.12	+	3414.52i	−0.682215	+	0.393877i
$423$		0			0
$424$	1984.12	−	3436.60i	0.227258	−	0.393622i
$425$	−2265.03	−	3923.15i	−0.258518	−	0.447766i
$426$		0			0
$427$	14729.7	+	870.437i	1.66937	+	0.0986496i
$428$		−	1535.03i		−	0.173360i
$429$		0			0
$430$	−5801.35	−	3349.41i	−0.650619	−	0.375635i
$431$	−11415.9	−	6590.96i	−1.27583	−	0.736602i	−0.299753	−	0.954017i	$-0.596904\pi$
$431$	−11415.9	−	6590.96i	−1.27583	−	0.736602i	−0.976079	+	0.217415i	$0.930237\pi$
$432$		0			0
$433$			1346.27i			0.149417i	0.997205	+	0.0747084i	$0.0238026\pi$
$433$			1346.27i			0.149417i	−0.997205	+	0.0747084i	$0.976197\pi$
$434$	−10612.8	−	6993.05i	−1.17380	−	0.773450i
$435$		0			0
$436$	−3559.58	−	6165.38i	−0.390994	−	0.677221i
$437$	2436.33	−	4219.84i	0.266694	−	0.461927i
$438$		0			0
$439$	4374.65	−	2525.70i	0.475605	−	0.274591i	−0.242978	−	0.970032i	$-0.578124\pi$
$439$	4374.65	−	2525.70i	0.475605	−	0.274591i	0.718583	+	0.695441i	$0.244791\pi$
$440$	3973.80			0.430553
$441$		0			0
$442$	7504.77			0.807614
$443$	9483.44	−	5475.27i	1.01709	−	0.587219i	0.103832	−	0.994595i	$-0.466890\pi$
$443$	9483.44	−	5475.27i	1.01709	−	0.587219i	0.913260	+	0.407376i	$0.133556\pi$
$444$		0			0
$445$	−3974.91	+	6884.74i	−0.423435	+	0.733411i
$446$	−331.944	−	574.944i	−0.0352422	−	0.0610412i
$447$		0			0
$448$	989.748	+	652.170i	0.104378	+	0.0687771i
$449$			2738.96i			0.287883i	0.989586	+	0.143942i	$0.0459777\pi$
$449$			2738.96i			0.287883i	−0.989586	+	0.143942i	$0.954022\pi$
$450$		0			0
$451$	3903.46	+	2253.66i	0.407554	+	0.235301i
$452$	728.785	+	420.764i	0.0758389	+	0.0437856i
$453$		0			0
$454$		−	3787.71i		−	0.391555i
$455$	4828.08	+	285.311i	0.497459	+	0.0293969i
$456$		0			0
$457$	348.716	+	603.994i	0.0356942	+	0.0618242i	0.883321	−	0.468769i	$-0.155302\pi$
$457$	348.716	+	603.994i	0.0356942	+	0.0618242i	−0.847626	+	0.530593i	$0.821969\pi$
$458$	2840.65	−	4920.14i	0.289814	−	0.501972i
$459$		0			0
$460$	2920.23	−	1685.99i	0.295992	−	0.170891i
$461$	16648.3			1.68197			0.840987	−	0.541056i	$-0.181975\pi$
$461$	16648.3			1.68197			0.840987	+	0.541056i	$0.181975\pi$
$462$		0			0
$463$	−10230.3			−1.02688			−0.513438	−	0.858127i	$-0.671628\pi$
$463$	−10230.3			−1.02688			−0.513438	+	0.858127i	$0.671628\pi$
$464$	3849.09	−	2222.27i	0.385107	−	0.222341i
$465$		0			0
$466$	−4570.83	+	7916.92i	−0.454377	+	0.787005i
$467$	4546.02	+	7873.94i	0.450460	+	0.780219i	0.998415	−	0.0562887i	$-0.0179267\pi$
$467$	4546.02	+	7873.94i	0.450460	+	0.780219i	−0.547955	+	0.836508i	$0.684593\pi$
$468$		0			0
$469$	11793.6	−	5910.43i	1.16114	−	0.581915i
$470$		−	682.597i		−	0.0669912i
$471$		0			0
$472$	35.9004	+	20.7271i	0.00350095	+	0.00202128i
$473$	−15634.2	−	9026.40i	−1.51979	−	0.877451i
$474$		0			0
$475$			1822.30i			0.176027i
$476$	−602.817	+	10201.0i	−0.0580464	+	0.982273i
$477$		0			0
$478$	357.209	+	618.704i	0.0341806	+	0.0592026i
$479$	7430.27	−	12869.6i	0.708763	−	1.22761i	−0.256553	−	0.966530i	$-0.582587\pi$
$479$	7430.27	−	12869.6i	0.708763	−	1.22761i	0.965316	−	0.261083i	$-0.0840798\pi$
$480$		0			0
$481$	2033.58	−	1174.09i	0.192772	−	0.111297i
$482$	10209.7			0.964811
$483$		0			0
$484$	5385.07			0.505736
$485$	−2120.31	+	1224.16i	−0.198512	+	0.114611i
$486$		0			0
$487$	8066.16	−	13971.0i	0.750539	−	1.29997i	−0.197023	−	0.980399i	$-0.563127\pi$
$487$	8066.16	−	13971.0i	0.750539	−	1.29997i	0.947562	−	0.319573i	$-0.103539\pi$
$488$	−3186.87	−	5519.82i	−0.295620	−	0.512029i
$489$		0			0
$490$	−775.627	+	6539.74i	−0.0715087	+	0.602929i
$491$			8536.33i			0.784601i	0.919837	+	0.392300i	$0.128321\pi$
$491$			8536.33i			0.784601i	−0.919837	+	0.392300i	$0.871679\pi$
$492$		0			0
$493$	33184.1	+	19158.9i	3.03152	+	1.75025i
$494$	−2614.47	−	1509.47i	−0.238119	−	0.137478i
$495$		0			0
$496$			5490.04i			0.496996i
$497$	5825.47	−	8840.87i	0.525771	−	0.797922i
$498$		0			0
$499$	−1994.38	−	3454.36i	−0.178919	−	0.309897i	0.762592	−	0.646880i	$-0.223927\pi$
$499$	−1994.38	−	3454.36i	−0.178919	−	0.309897i	−0.941511	+	0.336983i	$0.890593\pi$
$500$	−3030.53	+	5249.03i	−0.271059	+	0.469488i
$501$		0			0
$502$	6265.32	−	3617.28i	0.557041	−	0.321608i
$503$	2447.58			0.216963			0.108481	−	0.994098i	$-0.465401\pi$
$503$	2447.58			0.216963			0.108481	+	0.994098i	$0.465401\pi$
$504$		0			0
$505$	11289.4			0.994795
$506$	7869.79	−	4543.62i	0.691413	−	0.399187i
$507$		0			0
$508$	−711.588	+	1232.51i	−0.0621489	+	0.107645i
$509$	2538.65	+	4397.07i	0.221068	+	0.382901i	0.955133	−	0.296178i	$-0.0957123\pi$
$509$	2538.65	+	4397.07i	0.221068	+	0.382901i	−0.734064	+	0.679080i	$0.762379\pi$
$510$		0			0
$511$	−1789.72	−	3571.17i	−0.154937	−	0.309157i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	−2893.02	−	1670.28i	−0.248260	−	0.143333i
$515$	−13856.9	−	8000.28i	−1.18565	−	0.684533i
$516$		0			0
$517$		−	1839.55i		−	0.156486i
$518$	1432.55	+	2858.49i	0.121511	+	0.242461i
$519$		0			0
$520$	−1044.59	−	1809.28i	−0.0880927	−	0.152581i
$521$	−1220.65	+	2114.23i	−0.102644	+	0.177785i	−0.912773	−	0.408466i	$-0.866064\pi$
$521$	−1220.65	+	2114.23i	−0.102644	+	0.177785i	0.810129	+	0.586252i	$0.199397\pi$
$522$		0			0
$523$	15338.0	−	8855.38i	1.28238	−	0.740380i	0.305094	−	0.952322i	$-0.401312\pi$
$523$	15338.0	−	8855.38i	1.28238	−	0.740380i	0.977282	+	0.211942i	$0.0679787\pi$
$524$	209.577			0.0174722
$525$		0			0
$526$	68.1767			0.00565142
$527$	−40990.1	+	23665.6i	−3.38815	+	1.95615i
$528$		0			0
$529$	−2227.98	+	3858.97i	−0.183117	+	0.317167i
$530$	−4761.87	−	8247.80i	−0.390269	−	0.675965i
$531$		0			0
$532$	2261.78	−	3432.52i	0.184324	−	0.279735i
$533$		−	2369.67i		−	0.192574i
$534$		0			0
$535$	−3190.48	−	1842.02i	−0.257825	−	0.148855i
$536$	−4934.86	−	2849.14i	−0.397674	−	0.229597i
$537$		0			0
$538$			12369.4i			0.991236i
$539$	−2090.26	+	17624.1i	−0.167038	+	1.40839i
$540$		0			0
$541$	−5468.89	−	9472.40i	−0.434614	−	0.752773i	0.562650	−	0.826695i	$-0.309782\pi$
$541$	−5468.89	−	9472.40i	−0.434614	−	0.752773i	−0.997264	+	0.0739217i	$0.976449\pi$
$542$	1643.68	−	2846.94i	0.130262	−	0.225621i
$543$		0			0
$544$	3822.72	−	2207.05i	0.301283	−	0.173946i
$545$	−17085.9			−1.34290
$546$		0			0
$547$	−16232.7			−1.26885			−0.634426	−	0.772984i	$-0.718763\pi$
$547$	−16232.7			−1.26885			−0.634426	+	0.772984i	$0.718763\pi$
$548$	1035.24	−	597.696i	0.0806993	−	0.0465918i
$549$		0			0
$550$	−1699.25	+	2943.19i	−0.131739	+	0.228178i
$551$	−7707.01	−	13348.9i	−0.595880	−	1.03209i
$552$		0			0
$553$	−982.562	+	16627.1i	−0.0755566	+	1.27858i
$554$		−	9713.63i		−	0.744932i
$555$		0			0
$556$	4908.00	+	2833.63i	0.374362	+	0.216138i
$557$	−9250.07	−	5340.53i	−0.703659	−	0.406258i	0.105050	−	0.994467i	$-0.466500\pi$
$557$	−9250.07	−	5340.53i	−0.703659	−	0.406258i	−0.808709	+	0.588209i	$0.799833\pi$
$558$		0			0
$559$			9491.04i			0.718119i
$560$	2543.20	−	1274.54i	0.191910	−	0.0961774i
$561$		0			0
$562$	5754.68	+	9967.39i	0.431933	+	0.748130i
$563$	203.514	−	352.496i	0.0152346	−	0.0263871i	−0.858308	−	0.513136i	$-0.828484\pi$
$563$	203.514	−	352.496i	0.0152346	−	0.0263871i	0.873542	+	0.486748i	$0.161817\pi$
$564$		0			0
$565$	1749.08	−	1009.83i	0.130238	−	0.0751928i
$566$	5038.53			0.374179
$567$		0			0
$568$	−4573.41			−0.337845
$569$	−12670.1	+	7315.07i	−0.933493	+	0.538952i	−0.887914	−	0.460009i	$-0.847846\pi$
$569$	−12670.1	+	7315.07i	−0.933493	+	0.538952i	−0.0455781	+	0.998961i	$0.514513\pi$
$570$		0			0
$571$	10415.5	−	18040.2i	0.763354	−	1.32217i	−0.177758	−	0.984074i	$-0.556884\pi$
$571$	10415.5	−	18040.2i	0.763354	−	1.32217i	0.941112	−	0.338094i	$-0.109782\pi$
$572$	−2815.08	−	4875.86i	−0.205777	−	0.356416i
$573$		0			0
$574$	3221.02	+	190.343i	0.234221	+	0.0138410i
$575$			2883.82i			0.209154i
$576$		0			0
$577$	−11232.8	−	6485.25i	−0.810445	−	0.467910i	0.0366656	−	0.999328i	$-0.488326\pi$
$577$	−11232.8	−	6485.25i	−0.810445	−	0.467910i	−0.847110	+	0.531417i	$0.821660\pi$
$578$	24447.3	+	14114.6i	1.75929	+	1.01573i
$579$		0			0
$580$		−	10666.9i		−	0.763651i
$581$	11148.2	+	7345.81i	0.796048	+	0.524536i
$582$		0			0
$583$	−12832.9	−	22227.2i	−0.911635	−	1.57900i
$584$	−862.740	+	1494.31i	−0.0611309	+	0.105882i
$585$		0			0
$586$	7374.93	−	4257.92i	0.519890	−	0.300159i
$587$	5834.17			0.410225			0.205113	−	0.978738i	$-0.434244\pi$
$587$	5834.17			0.410225			0.205113	+	0.978738i	$0.434244\pi$
$588$		0			0
$589$	19039.9			1.33196
$590$	86.1607	−	49.7449i	0.00601217	−	0.00347113i
$591$		0			0
$592$	690.566	−	1196.10i	0.0479427	−	0.0830393i
$593$	−2466.66	−	4272.38i	−0.170815	−	0.295861i	0.767890	−	0.640582i	$-0.221307\pi$
$593$	−2466.66	−	4272.38i	−0.170815	−	0.295861i	−0.938705	+	0.344721i	$0.887973\pi$
$594$		0			0
$595$	20478.9	+	13494.1i	1.41101	+	0.929753i
$596$		−	7017.32i		−	0.482283i
$597$		0			0
$598$	−4137.44	−	2388.75i	−0.282931	−	0.163350i
$599$	14631.5	+	8447.53i	0.998045	+	0.576221i	0.907669	−	0.419686i	$-0.137860\pi$
$599$	14631.5	+	8447.53i	0.998045	+	0.576221i	0.0903756	+	0.995908i	$0.471193\pi$
$600$		0			0
$601$		−	4874.25i		−	0.330823i	−0.986225	−	0.165412i	$-0.947105\pi$
$601$		−	4874.25i		−	0.330823i	0.986225	−	0.165412i	$-0.0528953\pi$
$602$	−12900.9	−	762.364i	−0.873422	−	0.0516140i
$603$		0			0
$604$	6258.75	+	10840.5i	0.421631	+	0.730286i
$605$	6462.07	−	11192.6i	0.434248	−	0.752140i
$606$		0			0
$607$	390.353	−	225.370i	0.0261020	−	0.0150700i	−0.486892	−	0.873462i	$-0.661870\pi$
$607$	390.353	−	225.370i	0.0261020	−	0.0150700i	0.512994	+	0.858392i	$0.328536\pi$
$608$	−1775.66			−0.118441
$609$		0			0
$610$	−15296.9			−1.01533
$611$	−837.550	+	483.560i	−0.0554560	+	0.0320176i
$612$		0			0
$613$	−9461.80	+	16388.3i	−0.623423	+	1.07980i	0.365420	+	0.930843i	$0.380925\pi$
$613$	−9461.80	+	16388.3i	−0.623423	+	1.07980i	−0.988843	+	0.148958i	$0.952408\pi$
$614$	1312.64	+	2273.57i	0.0862769	+	0.149436i
$615$		0			0
$616$	6853.73	−	3434.80i	0.448286	−	0.224662i
$617$		−	8086.23i		−	0.527616i	−0.964575	−	0.263808i	$-0.915021\pi$
$617$		−	8086.23i		−	0.527616i	0.964575	−	0.263808i	$-0.0849786\pi$
$618$		0			0
$619$	5242.84	+	3026.96i	0.340432	+	0.196549i	0.660463	−	0.750858i	$-0.270360\pi$
$619$	5242.84	+	3026.96i	0.340432	+	0.196549i	−0.320031	+	0.947407i	$0.603693\pi$
$620$	11410.8	+	6588.03i	0.739143	+	0.426744i
$621$		0			0
$622$		−	289.109i		−	0.0186370i
$623$	−904.733	+	15310.1i	−0.0581820	+	0.984567i
$624$		0			0
$625$	5220.71	+	9042.53i	0.334125	+	0.578722i
$626$	−7845.03	+	13588.0i	−0.500879	+	0.867548i
$627$		0			0
$628$	301.311	−	173.962i	0.0191459	−	0.0110539i
$629$	11907.2			0.754800
$630$		0			0
$631$	4505.89			0.284273			0.142137	−	0.989847i	$-0.454603\pi$
$631$	4505.89			0.284273			0.142137	+	0.989847i	$0.454603\pi$
$632$	6230.85	−	3597.38i	0.392168	−	0.226418i
$633$		0			0
$634$	4656.00	−	8064.43i	0.291661	−	0.505173i
$635$	1707.81	+	2958.01i	0.106728	+	0.184858i
$636$		0			0
$637$	8573.75	−	3681.13i	0.533288	−	0.228966i
$638$		−	28746.4i		−	1.78383i
$639$		0			0
$640$	−1064.17	−	614.398i	−0.0657265	−	0.0379472i
$641$	−17661.2	−	10196.7i	−1.08826	−	0.628306i	−0.155146	−	0.987892i	$-0.549585\pi$
$641$	−17661.2	−	10196.7i	−1.08826	−	0.628306i	−0.933112	+	0.359585i	$0.882918\pi$
$642$		0			0
$643$			17679.2i			1.08429i	0.840284	+	0.542147i	$0.182388\pi$
$643$			17679.2i			1.08429i	−0.840284	+	0.542147i	$0.817612\pi$
$644$	3579.30	−	5432.02i	0.219012	−	0.332378i
$645$		0			0
$646$	−7654.23	−	13257.5i	−0.466179	−	0.807445i
$647$	4256.01	−	7371.63i	0.258611	−	0.447927i	−0.707259	−	0.706954i	$-0.750069\pi$
$647$	4256.01	−	7371.63i	0.258611	−	0.447927i	0.965870	+	0.259028i	$0.0834020\pi$
$648$		0			0
$649$	232.196	−	134.059i	0.0140439	−	0.00810826i
$650$	1786.72			0.107817
$651$		0			0
$652$	−5824.58			−0.349859
$653$	7712.69	−	4452.93i	0.462207	−	0.266855i	−0.250765	−	0.968048i	$-0.580682\pi$
$653$	7712.69	−	4452.93i	0.462207	−	0.266855i	0.712972	+	0.701193i	$0.247349\pi$
$654$		0			0
$655$	251.491	−	435.596i	0.0150024	−	0.0259849i
$656$	−696.889	−	1207.05i	−0.0414770	−	0.0718403i
$657$		0			0
$658$	−590.011	−	1177.30i	−0.0349560	−	0.0697504i
$659$		−	5704.29i		−	0.337189i	−0.985685	−	0.168594i	$-0.946077\pi$
$659$		−	5704.29i		−	0.337189i	0.985685	−	0.168594i	$-0.0539228\pi$
$660$		0			0
$661$	−4146.15	−	2393.78i	−0.243974	−	0.140858i	0.373028	−	0.927820i	$-0.378319\pi$
$661$	−4146.15	−	2393.78i	−0.243974	−	0.140858i	−0.617002	+	0.786962i	$0.711653\pi$
$662$	11538.6	+	6661.81i	0.677432	+	0.391116i
$663$		0			0
$664$		−	5766.98i		−	0.337052i
$665$	−4420.22	−	8820.02i	−0.257758	−	0.514324i
$666$		0			0
$667$	−12196.5	−	21124.9i	−0.708019	−	1.22633i
$668$	−700.539	+	1213.37i	−0.0405759	+	0.0702795i
$669$		0			0
$670$	−11843.6	+	6837.91i	−0.682923	+	0.394286i
$671$	−41224.0			−2.37173
$672$		0			0
$673$	21739.0			1.24514			0.622568	−	0.782566i	$-0.286089\pi$
$673$	21739.0			1.24514			0.622568	+	0.782566i	$0.286089\pi$
$674$	10096.1	−	5828.97i	0.576983	−	0.333121i
$675$		0			0
$676$	2914.01	−	5047.21i	0.165795	−	0.287165i
$677$	424.391	+	735.067i	0.0240926	+	0.0417296i	0.877820	−	0.478990i	$-0.158997\pi$
$677$	424.391	+	735.067i	0.0240926	+	0.0417296i	−0.853728	+	0.520720i	$0.825664\pi$
$678$		0			0
$679$	−2598.84	+	3944.06i	−0.146884	+	0.222915i
$680$		−	10593.8i		−	0.597432i
$681$		0			0
$682$	30751.2	+	17754.2i	1.72658	+	0.996839i
$683$	−8593.02	−	4961.18i	−0.481410	−	0.277942i	0.239594	−	0.970873i	$-0.422986\pi$
$683$	−8593.02	−	4961.18i	−0.481410	−	0.277942i	−0.721004	+	0.692931i	$0.756319\pi$
$684$		0			0
$685$		−	2868.93i		−	0.160024i
$686$	4314.96	+	11949.7i	0.240154	+	0.665076i
$687$		0			0
$688$	2791.19	+	4834.48i	0.154670	+	0.267896i
$689$	−6746.72	+	11685.7i	−0.373047	+	0.646137i
$690$		0			0
$691$	−8175.18	+	4719.94i	−0.450070	+	0.259848i	−0.707860	−	0.706353i	$-0.750339\pi$
$691$	−8175.18	+	4719.94i	−0.450070	+	0.259848i	0.257790	+	0.966201i	$0.417006\pi$
$692$	7422.37			0.407740
$693$		0			0
$694$	−14571.0			−0.796983
$695$	11779.1	−	6800.70i	0.642890	−	0.371173i
$696$		0			0
$697$	6008.08	−	10406.3i	0.326503	−	0.565519i
$698$	8093.19	+	14017.8i	0.438871	+	0.760146i
$699$		0			0
$700$	−143.518	+	2428.63i	−0.00774922	+	0.131134i
$701$			24385.5i			1.31388i	0.753944	+	0.656938i	$0.228149\pi$
$701$			24385.5i			1.31388i	−0.753944	+	0.656938i	$0.771851\pi$
$702$		0			0
$703$	−4148.15	−	2394.94i	−0.222547	−	0.128488i
$704$	−2867.85	−	1655.75i	−0.153532	−	0.0886415i
$705$		0			0
$706$			25177.3i			1.34215i
$707$	19471.2	−	9758.12i	1.03577	−	0.519083i
$708$		0			0
$709$	−2430.55	−	4209.83i	−0.128746	−	0.222995i	0.794445	−	0.607336i	$-0.207762\pi$
$709$	−2430.55	−	4209.83i	−0.128746	−	0.222995i	−0.923191	+	0.384341i	$0.874429\pi$
$710$	−5488.07	+	9505.62i	−0.290090	+	0.502450i
$711$		0			0
$712$	5737.30	−	3312.43i	0.301987	−	0.174352i
$713$	30130.9			1.58262
$714$		0			0
$715$	−13512.3			−0.706759
$716$	−1792.04	+	1034.63i	−0.0935357	+	0.0540029i
$717$		0			0
$718$	−968.552	+	1677.58i	−0.0503427	+	0.0871960i
$719$	7739.73	+	13405.6i	0.401451	+	0.695333i	0.993901	−	0.110274i	$-0.0351727\pi$
$719$	7739.73	+	13405.6i	0.401451	+	0.695333i	−0.592450	+	0.805607i	$0.701839\pi$
$720$		0			0
$721$	−30814.5	−	1820.95i	−1.59167	−	0.0940580i
$722$		−	7559.89i		−	0.389681i
$723$		0			0
$724$	2055.90	+	1186.98i	0.105535	+	0.0609304i
$725$	7900.41	+	4561.30i	0.404709	+	0.233659i
$726$		0			0
$727$		−	11891.8i		−	0.606660i	−0.952886	−	0.303330i	$-0.901902\pi$
$727$		−	11891.8i		−	0.606660i	0.952886	−	0.303330i	$-0.0980984\pi$
$728$	−3365.50	−	2217.61i	−0.171338	−	0.112899i
$729$		0			0
$730$	2070.57	+	3586.33i	0.104980	+	0.181830i
$731$	−24063.6	+	41679.4i	−1.21755	+	2.10885i
$732$		0			0
$733$	−15111.5	+	8724.62i	−0.761467	+	0.439633i	−0.829822	−	0.558028i	$-0.811558\pi$
$733$	−15111.5	+	8724.62i	−0.761467	+	0.439633i	0.0683554	+	0.997661i	$0.478225\pi$
$734$	14579.6			0.733166
$735$		0			0
$736$	−2810.00			−0.140731
$737$	−31917.6	+	18427.6i	−1.59525	+	0.921019i
$738$		0			0
$739$	−15927.1	+	27586.6i	−0.792813	+	1.37319i	0.131406	+	0.991329i	$0.458051\pi$
$739$	−15927.1	+	27586.6i	−0.792813	+	1.37319i	−0.924219	+	0.381863i	$0.875283\pi$
$740$	−1657.35	−	2870.62i	−0.0823318	−	0.142603i
$741$		0			0
$742$	−15342.0	−	10109.2i	−0.759061	−	0.500165i
$743$		−	28287.9i		−	1.39674i	−0.715735	−	0.698372i	$-0.753908\pi$
$743$		−	28287.9i		−	1.39674i	0.715735	−	0.698372i	$-0.246092\pi$
$744$		0			0
$745$	−14585.2	−	8420.75i	−0.717261	−	0.414111i
$746$	18473.2	+	10665.5i	0.906639	+	0.523448i
$747$		0			0
$748$		−	28549.5i		−	1.39555i
$749$	−7094.89	−	419.265i	−0.346117	−	0.0204534i
$750$		0			0
$751$	15036.1	+	26043.3i	0.730592	+	1.26542i	0.956630	+	0.291305i	$0.0940894\pi$
$751$	15036.1	+	26043.3i	0.730592	+	1.26542i	−0.226038	+	0.974119i	$0.572577\pi$
$752$	−284.417	+	492.624i	−0.0137920	+	0.0238885i
$753$		0			0
$754$	−13088.3	+	7556.53i	−0.632158	+	0.364977i
$755$	30041.9			1.44813
$756$		0			0
$757$	1532.83			0.0735952			0.0367976	−	0.999323i	$-0.488284\pi$
$757$	1532.83			0.0735952			0.0367976	+	0.999323i	$0.488284\pi$
$758$	12717.2	−	7342.28i	0.609380	−	0.351826i
$759$		0			0
$760$	−2130.78	+	3690.62i	−0.101699	+	0.176148i
$761$	8760.38	+	15173.4i	0.417298	+	0.722781i	0.995667	−	0.0929947i	$-0.0296440\pi$
$761$	8760.38	+	15173.4i	0.417298	+	0.722781i	−0.578369	+	0.815775i	$0.696311\pi$
$762$		0			0
$763$	−29468.6	+	14768.4i	−1.39821	+	0.700725i
$764$			13670.1i			0.647336i
$765$		0			0
$766$	−23376.1	−	13496.2i	−1.10263	−	0.636601i
$767$	−122.074	−	70.4796i	−0.00574687	−	0.00331796i
$768$		0			0
$769$		−	37012.1i		−	1.73562i	−0.496896	−	0.867810i	$-0.665527\pi$
$769$		−	37012.1i		−	1.73562i	0.496896	−	0.867810i	$-0.334473\pi$
$770$	1085.37	−	18366.9i	0.0507976	−	0.859606i
$771$		0			0
$772$	−3807.90	−	6595.47i	−0.177525	−	0.307482i
$773$	5746.59	−	9953.39i	0.267388	−	0.463129i	−0.700799	−	0.713359i	$-0.747173\pi$
$773$	5746.59	−	9953.39i	0.267388	−	0.463129i	0.968186	+	0.250230i	$0.0805063\pi$
$774$		0			0
$775$	−9758.84	+	5634.27i	−0.452320	+	0.261147i
$776$	2040.28			0.0943836
$777$		0			0
$778$	5090.62			0.234585
$779$	−4186.13	+	2416.86i	−0.192534	+	0.111159i
$780$		0			0
$781$	−14789.9	+	25616.9i	−0.677625	+	1.17368i
$782$	−12112.9	−	20980.2i	−0.553909	−	0.959399i
$783$		0			0
$784$	3284.66	−	4396.49i	0.149629	−	0.200277i
$785$		−	835.015i		−	0.0379655i
$786$		0			0
$787$	−1310.44	−	756.583i	−0.0593547	−	0.0342685i	0.470029	−	0.882651i	$-0.344243\pi$
$787$	−1310.44	−	756.583i	−0.0593547	−	0.0342685i	−0.529384	+	0.848383i	$0.677577\pi$
$788$	4485.09	+	2589.47i	0.202760	+	0.117063i
$789$		0			0
$790$		−	17267.4i		−	0.777653i
$791$	2143.83	−	3253.52i	0.0963663	−	0.146248i
$792$		0			0
$793$	10836.5	+	18769.4i	0.485265	+	0.840504i
$794$	−2655.05	+	4598.67i	−0.118670	+	0.205543i
$795$		0			0
$796$	−1061.73	+	612.992i	−0.0472765	+	0.0272951i
$797$	−23237.8			−1.03278			−0.516391	−	0.856353i	$-0.672725\pi$
$797$	−23237.8			−1.03278			−0.516391	+	0.856353i	$0.672725\pi$
$798$		0			0
$799$	−4904.08			−0.217139
$800$	910.107	−	525.450i	0.0402214	−	0.0232218i
$801$		0			0
$802$	−8505.18	+	14731.4i	−0.374474	+	0.648608i
$803$	5580.02	+	9664.88i	0.245224	+	0.424740i
$804$		0			0
$805$	−6995.06	−	13957.8i	−0.306265	−	0.611115i
$806$		−	18668.1i		−	0.815827i
$807$		0			0
$808$	−8147.45	−	4703.93i	−0.354735	−	0.204807i
$809$	−14550.2	−	8400.57i	−0.632334	−	0.365078i	0.149321	−	0.988789i	$-0.452291\pi$
$809$	−14550.2	−	8400.57i	−0.632334	−	0.365078i	−0.781656	+	0.623710i	$0.785624\pi$
$810$		0			0
$811$			35748.4i			1.54784i	0.633284	+	0.773919i	$0.281706\pi$
$811$			35748.4i			1.54784i	−0.633284	+	0.773919i	$0.718294\pi$
$812$	−9220.04	−	18397.5i	−0.398473	−	0.795104i
$813$		0			0
$814$	−4466.44	−	7736.10i	−0.192320	−	0.333108i
$815$	−6989.47	+	12106.1i	−0.300405	+	0.520317i
$816$		0			0
$817$	16766.3	−	9680.05i	0.717968	−	0.414519i
$818$	7093.54			0.303203
$819$		0			0
$820$	−3345.05			−0.142457
$821$	−26979.1	+	15576.4i	−1.14687	+	0.662144i	−0.948122	−	0.317907i	$-0.897020\pi$
$821$	−26979.1	+	15576.4i	−1.14687	+	0.662144i	−0.198746	+	0.980051i	$0.563687\pi$
$822$		0			0
$823$	17122.2	−	29656.5i	0.725202	−	1.25609i	−0.233688	−	0.972312i	$-0.575080\pi$
$823$	17122.2	−	29656.5i	0.725202	−	1.25609i	0.958891	−	0.283776i	$-0.0915871\pi$
$824$	6666.92	+	11547.4i	0.281861	+	0.488197i
$825$		0			0
$826$	105.606	−	160.270i	0.00444856	−	0.00675124i
$827$			38454.9i			1.61694i	0.588540	+	0.808468i	$0.299703\pi$
$827$			38454.9i			1.61694i	−0.588540	+	0.808468i	$0.700297\pi$
$828$		0			0
$829$	−24682.3	−	14250.3i	−1.03408	−	0.597026i	−0.115929	−	0.993258i	$-0.536984\pi$
$829$	−24682.3	−	14250.3i	−1.03408	−	0.597026i	−0.918151	+	0.396232i	$0.870318\pi$
$830$	−11986.4	−	6920.35i	−0.501270	−	0.289408i
$831$		0			0
$832$			1740.98i			0.0725454i
$833$	46984.4	+	5572.45i	1.95428	+	0.231781i
$834$		0			0
$835$	1681.29	+	2912.08i	0.0696807	+	0.120691i
$836$	−5742.29	+	9945.93i	−0.237561	+	0.411468i
$837$		0			0
$838$	2478.30	−	1430.85i	0.102162	−	0.0589830i
$839$	−47336.4			−1.94784			−0.973919	−	0.226898i	$-0.927142\pi$
$839$	−47336.4			−1.94784			−0.973919	+	0.226898i	$0.927142\pi$
$840$		0			0
$841$	−52775.0			−2.16389
$842$	3291.91	−	1900.59i	0.134735	−	0.0777893i
$843$		0			0
$844$	6829.04	−	11828.2i	0.278513	−	0.482399i
$845$	−6993.59	−	12113.3i	−0.284718	−	0.493146i
$846$		0			0
$847$	1470.84	−	24889.8i	0.0596678	−	1.00971i
$848$			7936.48i			0.321391i
$849$		0			0
$850$	7846.30	+	4530.06i	0.316619	+	0.182800i
$851$	−6564.51	−	3790.02i	−0.264428	−	0.152668i
$852$		0			0
$853$		−	2542.72i		−	0.102065i	−0.998697	−	0.0510323i	$-0.983749\pi$
$853$		−	2542.72i		−	0.102065i	0.998697	−	0.0510323i	$-0.0162511\pi$
$854$	−26383.0	+	13222.1i	−1.05715	+	0.529801i
$855$		0			0
$856$	1535.03	+	2658.74i	0.0612922	+	0.106161i
$857$	12220.9	−	21167.2i	0.487116	−	0.843709i	−0.512774	−	0.858523i	$-0.671382\pi$
$857$	12220.9	−	21167.2i	0.487116	−	0.843709i	0.999890	+	0.0148139i	$0.00471559\pi$
$858$		0			0
$859$	32353.2	−	18679.1i	1.28507	−	0.741936i	0.307301	−	0.951612i	$-0.400574\pi$
$859$	32353.2	−	18679.1i	1.28507	−	0.741936i	0.977771	+	0.209676i	$0.0672409\pi$
$860$	13397.6			0.531228
$861$		0			0
$862$	26363.8			1.04171
$863$	8309.60	−	4797.55i	0.327766	−	0.189236i	−0.327083	−	0.944996i	$-0.606066\pi$
$863$	8309.60	−	4797.55i	0.327766	−	0.189236i	0.654849	+	0.755760i	$0.272732\pi$
$864$		0			0
$865$	8906.81	−	15427.1i	0.350105	−	0.606399i
$866$	−1346.27	−	2331.80i	−0.0528268	−	0.0914987i
$867$		0			0
$868$	25375.0	+	1499.51i	0.992262	+	0.0586367i
$869$		−	46534.3i		−	1.81653i
$870$		0			0
$871$	16780.3	+	9688.11i	0.652788	+	0.376887i
$872$	12330.8	+	7119.17i	0.478867	+	0.276474i
$873$		0			0
$874$			9745.30i			0.377162i
$875$	23433.3	+	15440.8i	0.905360	+	0.596565i
$876$		0			0
$877$	−21258.0	−	36820.0i	−0.818510	−	1.41770i	−0.906780	−	0.421604i	$-0.861467\pi$
$877$	−21258.0	−	36820.0i	−0.818510	−	1.41770i	0.0882706	−	0.996097i	$-0.471866\pi$
$878$	−5051.41	+	8749.30i	−0.194165	+	0.336303i
$879$		0			0
$880$	−6882.82	+	3973.80i	−0.263659	+	0.152223i
$881$	−28763.4			−1.09996			−0.549978	−	0.835179i	$-0.685364\pi$
$881$	−28763.4			−1.09996			−0.549978	+	0.835179i	$0.685364\pi$
$882$		0			0
$883$	−13750.7			−0.524062			−0.262031	−	0.965059i	$-0.584392\pi$
$883$	−13750.7			−0.524062			−0.262031	+	0.965059i	$0.584392\pi$
$884$	−12998.6	+	7504.77i	−0.494561	+	0.285535i
$885$		0			0
$886$	−10950.5	+	18966.9i	−0.415226	+	0.719193i
$887$	−2715.07	−	4702.64i	−0.102777	−	0.178015i	0.810051	−	0.586360i	$-0.199439\pi$
$887$	−2715.07	−	4702.64i	−0.102777	−	0.178015i	−0.912828	+	0.408345i	$0.866106\pi$
$888$		0			0
$889$	5502.29	+	3625.60i	0.207583	+	0.136781i
$890$		−	15899.6i		−	0.598828i
$891$		0			0
$892$	1149.89	+	663.888i	0.0431627	+	0.0249200i
$893$	1708.46	+	986.379i	0.0640217	+	0.0369629i
$894$		0			0
$895$			4966.22i			0.185478i
$896$	−2366.46	−	139.844i	−0.0882344	−	0.00521412i
$897$		0			0
$898$	−2738.96	−	4744.02i	−0.101782	−	0.176292i
$899$	47657.7	−	82545.6i	1.76805	−	3.06235i
$900$		0			0
$901$	−59255.8	+	34211.4i	−2.19101	+	1.26498i
$902$	−9014.66			−0.332766
$903$		0			0
$904$	−1683.06			−0.0619222
$905$	4934.15	−	2848.73i	0.181234	−	0.104635i
$906$		0			0
$907$	16067.2	−	27829.1i	0.588204	−	1.01880i	−0.406263	−	0.913756i	$-0.633168\pi$
$907$	16067.2	−	27829.1i	0.588204	−	1.01880i	0.994468	−	0.105044i	$-0.0334983\pi$
$908$	3787.71	+	6560.50i	0.138435	+	0.239777i
$909$		0			0
$910$	−8647.80	+	4333.91i	−0.315024	+	0.157877i
$911$			35948.6i			1.30739i	0.756760	+	0.653693i	$0.226781\pi$
$911$			35948.6i			1.30739i	−0.756760	+	0.653693i	$0.773219\pi$
$912$		0			0
$913$	−32302.4	−	18649.8i	−1.17092	−	0.676034i
$914$	−1207.99	−	697.433i	−0.0437163	−	0.0252396i
$915$		0			0
$916$			11362.6i			0.409858i
$917$	57.2423	−	968.665i	0.00206140	−	0.0348835i
$918$		0			0
$919$	18145.6	+	31429.2i	0.651327	+	1.12813i	0.982801	+	0.184667i	$0.0591208\pi$
$919$	18145.6	+	31429.2i	0.651327	+	1.12813i	−0.331474	+	0.943464i	$0.607546\pi$
$920$	−3371.99	+	5840.46i	−0.120838	+	0.209298i
$921$		0			0
$922$	−28835.7	+	16648.3i	−1.02999	+	0.594667i
$923$	15551.2			0.554578
$924$		0			0
$925$	2834.83			0.100766
$926$	17719.4	−	10230.3i	0.628830	−	0.363055i
$927$		0			0
$928$	−4444.55	+	7698.18i	−0.157219	+	0.272312i
$929$	895.812	+	1551.59i	0.0316369	+	0.0547967i	0.881410	−	0.472351i	$-0.156595\pi$
$929$	895.812	+	1551.59i	0.0316369	+	0.0547967i	−0.849773	+	0.527148i	$0.823261\pi$
$930$		0			0
$931$	−15247.4	−	11391.5i	−0.536748	−	0.401010i
$932$		−	18283.3i		−	0.642587i
$933$		0			0
$934$	−15747.9	−	9092.04i	−0.551698	−	0.318523i
$935$	−59338.8	−	34259.3i	−2.07549	−	1.19829i
$936$		0			0
$937$			9949.70i			0.346897i	0.984843	+	0.173449i	$0.0554910\pi$
$937$			9949.70i			0.346897i	−0.984843	+	0.173449i	$0.944509\pi$
$938$	−14516.6	+	22030.7i	−0.505313	+	0.766875i
$939$		0			0
$940$	682.597	+	1182.29i	0.0236850	+	0.0410236i
$941$	20171.1	−	34937.3i	0.698787	−	1.21033i	−0.270100	−	0.962832i	$-0.587057\pi$
$941$	20171.1	−	34937.3i	0.698787	−	1.21033i	0.968887	−	0.247502i	$-0.0796098\pi$
$942$		0			0
$943$	−6624.61	+	3824.72i	−0.228767	+	0.132079i
$944$	−82.9084			−0.00285852
$945$		0			0
$946$	36105.6			1.24090
$947$	7541.67	−	4354.18i	0.258787	−	0.149411i	−0.364994	−	0.931010i	$-0.618929\pi$
$947$	7541.67	−	4354.18i	0.258787	−	0.149411i	0.623781	+	0.781599i	$0.285596\pi$
$948$		0			0
$949$	2933.63	−	5081.19i	0.100347	−	0.173807i
$950$	−1822.30	−	3156.32i	−0.0622351	−	0.107794i
$951$		0			0
$952$	−9156.88	−	18271.5i	−0.311740	−	0.622039i
$953$			2413.68i			0.0820428i	0.999158	+	0.0410214i	$0.0130612\pi$
$953$			2413.68i			0.0820428i	−0.999158	+	0.0410214i	$0.986939\pi$
$954$		0			0
$955$	28412.6	+	16404.0i	0.962732	+	0.555834i
$956$	−1237.41	−	714.417i	−0.0418626	−	0.0241694i
$957$		0			0
$958$			29721.1i			1.00234i
$959$	−2479.79	−	4948.13i	−0.0835002	−	0.166615i
$960$		0			0
$961$	43972.8	+	76163.1i	1.47604	+	2.55658i
$962$	−2348.17	+	4067.16i	−0.0786987	+	0.136310i
$963$		0			0
$964$	−17683.7	+	10209.7i	−0.590823	+	0.341112i
$965$	−18277.8			−0.609725
$966$		0			0
$967$	40922.6			1.36089			0.680446	−	0.732798i	$-0.261786\pi$
$967$	40922.6			1.36089			0.680446	+	0.732798i	$0.261786\pi$
$968$	−9327.22	+	5385.07i	−0.309699	+	0.178805i
$969$		0			0
$970$	2448.32	−	4240.62i	0.0810422	−	0.140369i
$971$	6941.97	+	12023.8i	0.229432	+	0.397388i	0.957640	−	0.287968i	$-0.0929798\pi$
$971$	6941.97	+	12023.8i	0.229432	+	0.397388i	−0.728208	+	0.685356i	$0.759646\pi$
$972$		0			0
$973$	14437.6	−	21910.8i	0.475692	−	0.721920i
$974$			32264.6i			1.06142i
$975$		0			0
$976$	11039.6	+	6373.73i	0.362059	+	0.209035i
$977$	26543.9	+	15325.1i	0.869206	+	0.501836i	0.867084	−	0.498162i	$-0.165991\pi$
$977$	26543.9	+	15325.1i	0.869206	+	0.501836i	0.00212159	+	0.999998i	$0.499325\pi$
$978$		0			0
$979$		−	42848.3i		−	1.39881i
$980$	−5196.32	−	12102.8i	−0.169378	−	0.394500i
$981$		0			0
$982$	−8536.33	−	14785.4i	−0.277398	−	0.480468i
$983$	4880.79	−	8453.78i	0.158365	−	0.274297i	−0.775914	−	0.630839i	$-0.782711\pi$
$983$	4880.79	−	8453.78i	0.158365	−	0.274297i	0.934279	+	0.356542i	$0.116044\pi$
$984$		0			0
$985$	10764.2	−	6214.70i	0.348198	−	0.201032i
$986$	−76635.5			−2.47522
$987$		0			0
$988$	6037.87			0.194423
$989$	26533.0	−	15318.8i	0.853083	−	0.492528i
$990$		0			0
$991$	1399.13	−	2423.36i	0.0448483	−	0.0776796i	−0.842730	−	0.538337i	$-0.819053\pi$
$991$	1399.13	−	2423.36i	0.0448483	−	0.0776796i	0.887578	+	0.460657i	$0.152386\pi$
$992$	−5490.04	−	9509.03i	−0.175715	−	0.304347i
$993$		0			0
$994$	−1249.15	+	21138.3i	−0.0398597	+	0.674513i
$995$			2942.35i			0.0937475i
$996$		0			0
$997$	9165.99	+	5291.99i	0.291163	+	0.168103i	0.638466	−	0.769650i	$-0.279569\pi$
$997$	9165.99	+	5291.99i	0.291163	+	0.168103i	−0.347303	+	0.937753i	$0.612902\pi$
$998$	6908.73	+	3988.75i	0.219130	+	0.126515i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.b.269.1	yes	16
3.2	odd	2	inner	378.4.k.b.269.8	yes	16
7.5	odd	6	inner	378.4.k.b.215.8	yes	16
21.5	even	6	inner	378.4.k.b.215.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.b.215.1	✓	16	21.5	even	6	inner
378.4.k.b.215.8	yes	16	7.5	odd	6	inner
378.4.k.b.269.1	yes	16	1.1	even	1	trivial
378.4.k.b.269.8	yes	16	3.2	odd	2	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.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.79998 - 8.31381i) q^{5} +(-15.4648 - 10.1902i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.79998 - 8.31381i) q^{5} +(-15.4648 - 10.1902i) q^{7} +8.00000i q^{8} +(16.6276 + 9.59997i) q^{10} +(44.8102 + 25.8712i) q^{11} -27.2029i q^{13} +(36.9760 + 2.18506i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(68.9704 - 119.460i) q^{17} +(-48.0551 + 27.7446i) q^{19} -38.3999 q^{20} -103.485 q^{22} +(-76.0479 + 43.9063i) q^{23} +(16.4203 - 28.4408i) q^{25} +(27.2029 + 47.1168i) q^{26} +(-66.2294 + 33.1914i) q^{28} +277.784i q^{29} +(-297.157 - 171.564i) q^{31} +(27.7128 + 16.0000i) q^{32} +275.881i q^{34} +(-10.4883 + 177.484i) q^{35} +(43.1604 + 74.7560i) q^{37} +(55.4892 - 96.1102i) q^{38} +(66.5105 - 38.3999i) q^{40} +87.1111 q^{41} -348.898 q^{43} +(179.241 - 103.485i) q^{44} +(87.8125 - 152.096i) q^{46} +(-17.7760 - 30.7890i) q^{47} +(135.321 + 315.178i) q^{49} +65.6813i q^{50} +(-94.2336 - 54.4058i) q^{52} +(-429.574 - 248.015i) q^{53} -496.725i q^{55} +(81.5213 - 123.719i) q^{56} +(-277.784 - 481.136i) q^{58} +(2.59089 - 4.48755i) q^{59} +(-689.977 + 398.358i) q^{61} +686.255 q^{62} -64.0000 q^{64} +(-226.160 + 130.573i) q^{65} +(-356.143 + 616.857i) q^{67} +(-275.881 - 477.841i) q^{68} +(-159.318 - 317.900i) q^{70} +571.676i q^{71} +(186.789 + 107.843i) q^{73} +(-149.512 - 86.3208i) q^{74} +221.957i q^{76} +(-429.350 - 856.716i) q^{77} +(-449.673 - 778.856i) q^{79} +(-76.7997 + 133.021i) q^{80} +(-150.881 + 87.1111i) q^{82} -720.873 q^{83} -1324.23 q^{85} +(604.310 - 348.898i) q^{86} +(-206.969 + 358.481i) q^{88} +(-414.054 - 717.163i) q^{89} +(-277.202 + 420.688i) q^{91} +351.250i q^{92} +(61.5780 + 35.5521i) q^{94} +(461.327 + 266.347i) q^{95} -255.034i q^{97} +(-549.561 - 410.583i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) \) 16 * q + 32 * q^4 - 52 * q^7	\( 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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