LMFDB - Embedded newform 9576.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.a (trivial)

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:	yes
Analytic conductor:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.41421 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.41421 q^{5} -1.00000 q^{7} +2.82843 q^{11} +5.65685 q^{13} -2.24264 q^{17} +1.00000 q^{19} +8.48528 q^{23} +6.65685 q^{25} -10.2426 q^{29} +8.82843 q^{31} -3.41421 q^{35} -6.00000 q^{37} +8.82843 q^{41} -8.82843 q^{43} +10.5858 q^{47} +1.00000 q^{49} +5.07107 q^{53} +9.65685 q^{55} +2.34315 q^{59} +2.00000 q^{61} +19.3137 q^{65} -9.17157 q^{67} -12.7279 q^{71} -8.82843 q^{73} -2.82843 q^{77} +2.34315 q^{79} +12.2426 q^{83} -7.65685 q^{85} +16.8284 q^{89} -5.65685 q^{91} +3.41421 q^{95} -0.343146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^7	$ 2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.24264	−0.543920	−0.271960	−	0.962309i	$-0.587672\pi$
$17$	−2.24264	−0.543920	−0.271960	+	0.962309i	$0.587672\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.2426	−1.90201	−0.951005	−	0.309175i	$-0.899947\pi$
$29$	−10.2426	−1.90201	−0.951005	+	0.309175i	$0.899947\pi$
$30$	0	0
$31$	8.82843	1.58563	0.792816	−	0.609461i	$-0.208614\pi$
$31$	8.82843	1.58563	0.792816	+	0.609461i	$0.208614\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	−8.82843	−1.34632	−0.673161	−	0.739496i	$-0.735064\pi$
$43$	−8.82843	−1.34632	−0.673161	+	0.739496i	$0.735064\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5858	1.54410	0.772048	−	0.635564i	$-0.219233\pi$
$47$	10.5858	1.54410	0.772048	+	0.635564i	$0.219233\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.07107	0.696565	0.348282	−	0.937390i	$-0.386765\pi$
$53$	5.07107	0.696565	0.348282	+	0.937390i	$0.386765\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.34315	0.305052	0.152526	−	0.988299i	$-0.451259\pi$
$59$	2.34315	0.305052	0.152526	+	0.988299i	$0.451259\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	19.3137	2.39557
$66$	0	0
$67$	−9.17157	−1.12049	−0.560243	−	0.828328i	$-0.689292\pi$
$67$	−9.17157	−1.12049	−0.560243	+	0.828328i	$0.689292\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	−8.82843	−1.03329	−0.516645	−	0.856200i	$-0.672819\pi$
$73$	−8.82843	−1.03329	−0.516645	+	0.856200i	$0.672819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2426	1.34380	0.671902	−	0.740640i	$-0.265478\pi$
$83$	12.2426	1.34380	0.671902	+	0.740640i	$0.265478\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.8284	1.78381	0.891905	−	0.452223i	$-0.149369\pi$
$89$	16.8284	1.78381	0.891905	+	0.452223i	$0.149369\pi$
$90$	0	0
$91$	−5.65685	−0.592999
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.41421	0.350291
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.07107	−0.902605	−0.451302	−	0.892371i	$-0.649040\pi$
$101$	−9.07107	−0.902605	−0.451302	+	0.892371i	$0.649040\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.7279	1.61715	0.808575	−	0.588394i	$-0.200239\pi$
$107$	16.7279	1.61715	0.808575	+	0.588394i	$0.200239\pi$
$108$	0	0
$109$	−16.1421	−1.54614	−0.773068	−	0.634323i	$-0.781279\pi$
$109$	−16.1421	−1.54614	−0.773068	+	0.634323i	$0.781279\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.58579	−0.807683	−0.403841	−	0.914829i	$-0.632325\pi$
$113$	−8.58579	−0.807683	−0.403841	+	0.914829i	$0.632325\pi$
$114$	0	0
$115$	28.9706	2.70152
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.24264	0.205583
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−8.48528	−0.752947	−0.376473	−	0.926427i	$-0.622863\pi$
$127$	−8.48528	−0.752947	−0.376473	+	0.926427i	$0.622863\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.89949	−0.864923	−0.432461	−	0.901652i	$-0.642355\pi$
$131$	−9.89949	−0.864923	−0.432461	+	0.901652i	$0.642355\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.6569	0.995912	0.497956	−	0.867202i	$-0.334084\pi$
$137$	11.6569	0.995912	0.497956	+	0.867202i	$0.334084\pi$
$138$	0	0
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.0000	1.33799
$144$	0	0
$145$	−34.9706	−2.90415
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	0	0
$151$	8.48528	0.690522	0.345261	−	0.938507i	$-0.387790\pi$
$151$	8.48528	0.690522	0.345261	+	0.938507i	$0.387790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	30.1421	2.42107
$156$	0	0
$157$	−7.65685	−0.611083	−0.305542	−	0.952179i	$-0.598838\pi$
$157$	−7.65685	−0.611083	−0.305542	+	0.952179i	$0.598838\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.48528	−0.668734
$162$	0	0
$163$	−1.51472	−0.118642	−0.0593210	−	0.998239i	$-0.518894\pi$
$163$	−1.51472	−0.118642	−0.0593210	+	0.998239i	$0.518894\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.8284	1.76652	0.883258	−	0.468887i	$-0.155345\pi$
$167$	22.8284	1.76652	0.883258	+	0.468887i	$0.155345\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.17157	0.241130	0.120565	−	0.992705i	$-0.461529\pi$
$173$	3.17157	0.241130	0.120565	+	0.992705i	$0.461529\pi$
$174$	0	0
$175$	−6.65685	−0.503211
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.07107	−0.229542	−0.114771	−	0.993392i	$-0.536613\pi$
$179$	−3.07107	−0.229542	−0.114771	+	0.993392i	$0.536613\pi$
$180$	0	0
$181$	−9.31371	−0.692283	−0.346141	−	0.938182i	$-0.612508\pi$
$181$	−9.31371	−0.692283	−0.346141	+	0.938182i	$0.612508\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.4853	−1.50611
$186$	0	0
$187$	−6.34315	−0.463857
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.8284	1.07295	0.536474	−	0.843917i	$-0.319756\pi$
$191$	14.8284	1.07295	0.536474	+	0.843917i	$0.319756\pi$
$192$	0	0
$193$	−0.343146	−0.0247002	−0.0123501	−	0.999924i	$-0.503931\pi$
$193$	−0.343146	−0.0247002	−0.0123501	+	0.999924i	$0.503931\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.9706	−1.06661	−0.533304	−	0.845924i	$-0.679050\pi$
$197$	−14.9706	−1.06661	−0.533304	+	0.845924i	$0.679050\pi$
$198$	0	0
$199$	−14.1421	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$199$	−14.1421	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.2426	0.718892
$204$	0	0
$205$	30.1421	2.10522
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.82843	0.195646
$210$	0	0
$211$	28.9706	1.99442	0.997208	−	0.0746754i	$-0.0237921\pi$
$211$	28.9706	1.99442	0.997208	+	0.0746754i	$0.0237921\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−30.1421	−2.05568
$216$	0	0
$217$	−8.82843	−0.599313
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.6863	−0.853372
$222$	0	0
$223$	−0.828427	−0.0554756	−0.0277378	−	0.999615i	$-0.508830\pi$
$223$	−0.828427	−0.0554756	−0.0277378	+	0.999615i	$0.508830\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.5147	−0.764259	−0.382129	−	0.924109i	$-0.624809\pi$
$227$	−11.5147	−0.764259	−0.382129	+	0.924109i	$0.624809\pi$
$228$	0	0
$229$	10.4853	0.692887	0.346443	−	0.938071i	$-0.387389\pi$
$229$	10.4853	0.692887	0.346443	+	0.938071i	$0.387389\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.4853	−0.686914	−0.343457	−	0.939168i	$-0.611598\pi$
$233$	−10.4853	−0.686914	−0.343457	+	0.939168i	$0.611598\pi$
$234$	0	0
$235$	36.1421	2.35765
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.34315	0.151565	0.0757827	−	0.997124i	$-0.475854\pi$
$239$	2.34315	0.151565	0.0757827	+	0.997124i	$0.475854\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.41421	0.218126
$246$	0	0
$247$	5.65685	0.359937
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.24264	0.267793	0.133897	−	0.990995i	$-0.457251\pi$
$251$	4.24264	0.267793	0.133897	+	0.990995i	$0.457251\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.65685	−0.102166	−0.0510830	−	0.998694i	$-0.516267\pi$
$263$	−1.65685	−0.102166	−0.0510830	+	0.998694i	$0.516267\pi$
$264$	0	0
$265$	17.3137	1.06357
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.6569	0.710731	0.355365	−	0.934727i	$-0.384356\pi$
$269$	11.6569	0.710731	0.355365	+	0.934727i	$0.384356\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.8284	1.13540
$276$	0	0
$277$	−11.3137	−0.679775	−0.339887	−	0.940466i	$-0.610389\pi$
$277$	−11.3137	−0.679775	−0.339887	+	0.940466i	$0.610389\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.75736	−0.104835	−0.0524176	−	0.998625i	$-0.516693\pi$
$281$	−1.75736	−0.104835	−0.0524176	+	0.998625i	$0.516693\pi$
$282$	0	0
$283$	14.1421	0.840663	0.420331	−	0.907371i	$-0.361914\pi$
$283$	14.1421	0.840663	0.420331	+	0.907371i	$0.361914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.82843	−0.521126
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.51472	−0.0884908	−0.0442454	−	0.999021i	$-0.514088\pi$
$293$	−1.51472	−0.0884908	−0.0442454	+	0.999021i	$0.514088\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	8.82843	0.508862
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.82843	0.390995
$306$	0	0
$307$	−16.1421	−0.921280	−0.460640	−	0.887587i	$-0.652380\pi$
$307$	−16.1421	−0.921280	−0.460640	+	0.887587i	$0.652380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0416	1.59009	0.795047	−	0.606547i	$-0.207446\pi$
$311$	28.0416	1.59009	0.795047	+	0.606547i	$0.207446\pi$
$312$	0	0
$313$	2.97056	0.167906	0.0839531	−	0.996470i	$-0.473245\pi$
$313$	2.97056	0.167906	0.0839531	+	0.996470i	$0.473245\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.2132	−1.52845	−0.764223	−	0.644952i	$-0.776877\pi$
$317$	−27.2132	−1.52845	−0.764223	+	0.644952i	$0.776877\pi$
$318$	0	0
$319$	−28.9706	−1.62204
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.24264	−0.124784
$324$	0	0
$325$	37.6569	2.08883
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.5858	−0.583613
$330$	0	0
$331$	−14.1421	−0.777322	−0.388661	−	0.921381i	$-0.627062\pi$
$331$	−14.1421	−0.777322	−0.388661	+	0.921381i	$0.627062\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−31.3137	−1.71085
$336$	0	0
$337$	−8.82843	−0.480915	−0.240458	−	0.970660i	$-0.577297\pi$
$337$	−8.82843	−0.480915	−0.240458	+	0.970660i	$0.577297\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.9706	1.35223
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9706	−1.12576	−0.562879	−	0.826539i	$-0.690306\pi$
$347$	−20.9706	−1.12576	−0.562879	+	0.826539i	$0.690306\pi$
$348$	0	0
$349$	29.3137	1.56913	0.784563	−	0.620049i	$-0.212887\pi$
$349$	29.3137	1.56913	0.784563	+	0.620049i	$0.212887\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.100505	−0.00534934	−0.00267467	−	0.999996i	$-0.500851\pi$
$353$	−0.100505	−0.00534934	−0.00267467	+	0.999996i	$0.500851\pi$
$354$	0	0
$355$	−43.4558	−2.30640
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.48528	−0.236724	−0.118362	−	0.992971i	$-0.537764\pi$
$359$	−4.48528	−0.236724	−0.118362	+	0.992971i	$0.537764\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.1421	−1.57771
$366$	0	0
$367$	5.85786	0.305778	0.152889	−	0.988243i	$-0.451142\pi$
$367$	5.85786	0.305778	0.152889	+	0.988243i	$0.451142\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.07107	−0.263277
$372$	0	0
$373$	24.3431	1.26044	0.630220	−	0.776416i	$-0.282965\pi$
$373$	24.3431	1.26044	0.630220	+	0.776416i	$0.282965\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−57.9411	−2.98412
$378$	0	0
$379$	14.8284	0.761685	0.380843	−	0.924640i	$-0.375634\pi$
$379$	14.8284	0.761685	0.380843	+	0.924640i	$0.375634\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.4853	1.04675	0.523374	−	0.852103i	$-0.324673\pi$
$383$	20.4853	1.04675	0.523374	+	0.852103i	$0.324673\pi$
$384$	0	0
$385$	−9.65685	−0.492159
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−35.4558	−1.79768	−0.898841	−	0.438274i	$-0.855590\pi$
$389$	−35.4558	−1.79768	−0.898841	+	0.438274i	$0.855590\pi$
$390$	0	0
$391$	−19.0294	−0.962360
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.00000	0.402524
$396$	0	0
$397$	1.31371	0.0659331	0.0329666	−	0.999456i	$-0.489505\pi$
$397$	1.31371	0.0659331	0.0329666	+	0.999456i	$0.489505\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.3848	0.618466	0.309233	−	0.950986i	$-0.399928\pi$
$401$	12.3848	0.618466	0.309233	+	0.950986i	$0.399928\pi$
$402$	0	0
$403$	49.9411	2.48774
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−18.6274	−0.921066	−0.460533	−	0.887642i	$-0.652342\pi$
$409$	−18.6274	−0.921066	−0.460533	+	0.887642i	$0.652342\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.34315	−0.115299
$414$	0	0
$415$	41.7990	2.05183
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.72792	0.230974	0.115487	−	0.993309i	$-0.463157\pi$
$419$	4.72792	0.230974	0.115487	+	0.993309i	$0.463157\pi$
$420$	0	0
$421$	−1.51472	−0.0738229	−0.0369114	−	0.999319i	$-0.511752\pi$
$421$	−1.51472	−0.0738229	−0.0369114	+	0.999319i	$0.511752\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−14.9289	−0.724160
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.4142	1.60951	0.804753	−	0.593610i	$-0.202298\pi$
$431$	33.4142	1.60951	0.804753	+	0.593610i	$0.202298\pi$
$432$	0	0
$433$	4.62742	0.222379	0.111190	−	0.993799i	$-0.464534\pi$
$433$	4.62742	0.222379	0.111190	+	0.993799i	$0.464534\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	−41.7990	−1.99496	−0.997478	−	0.0709697i	$-0.977391\pi$
$439$	−41.7990	−1.99496	−0.997478	+	0.0709697i	$0.977391\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.686292	−0.0326067	−0.0163033	−	0.999867i	$-0.505190\pi$
$443$	−0.686292	−0.0326067	−0.0163033	+	0.999867i	$0.505190\pi$
$444$	0	0
$445$	57.4558	2.72367
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.8701	−0.796147	−0.398074	−	0.917353i	$-0.630321\pi$
$449$	−16.8701	−0.796147	−0.398074	+	0.917353i	$0.630321\pi$
$450$	0	0
$451$	24.9706	1.17582
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−19.3137	−0.905441
$456$	0	0
$457$	−27.3137	−1.27768	−0.638841	−	0.769339i	$-0.720586\pi$
$457$	−27.3137	−1.27768	−0.638841	+	0.769339i	$0.720586\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.5269	1.79438	0.897189	−	0.441648i	$-0.145606\pi$
$461$	38.5269	1.79438	0.897189	+	0.441648i	$0.145606\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.89949	−0.458094	−0.229047	−	0.973415i	$-0.573561\pi$
$467$	−9.89949	−0.458094	−0.229047	+	0.973415i	$0.573561\pi$
$468$	0	0
$469$	9.17157	0.423504
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.9706	−1.14815
$474$	0	0
$475$	6.65685	0.305437
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.727922	0.0332596	0.0166298	−	0.999862i	$-0.494706\pi$
$479$	0.727922	0.0332596	0.0166298	+	0.999862i	$0.494706\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.17157	−0.0531984
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.65685	0.435808	0.217904	−	0.975970i	$-0.430078\pi$
$491$	9.65685	0.435808	0.217904	+	0.975970i	$0.430078\pi$
$492$	0	0
$493$	22.9706	1.03454
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.7279	0.570925
$498$	0	0
$499$	43.5980	1.95171	0.975857	−	0.218411i	$-0.0700874\pi$
$499$	43.5980	1.95171	0.975857	+	0.218411i	$0.0700874\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.1838	−1.88088	−0.940441	−	0.339958i	$-0.889587\pi$
$503$	−42.1838	−1.88088	−0.940441	+	0.339958i	$0.889587\pi$
$504$	0	0
$505$	−30.9706	−1.37817
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.9706	0.663559	0.331779	−	0.943357i	$-0.392351\pi$
$509$	14.9706	0.663559	0.331779	+	0.943357i	$0.392351\pi$
$510$	0	0
$511$	8.82843	0.390547
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.3137	1.20359
$516$	0	0
$517$	29.9411	1.31681
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.9706	1.70733	0.853666	−	0.520821i	$-0.174374\pi$
$521$	38.9706	1.70733	0.853666	+	0.520821i	$0.174374\pi$
$522$	0	0
$523$	−0.142136	−0.00621516	−0.00310758	−	0.999995i	$-0.500989\pi$
$523$	−0.142136	−0.00621516	−0.00310758	+	0.999995i	$0.500989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.7990	−0.862458
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	49.9411	2.16319
$534$	0	0
$535$	57.1127	2.46920
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−55.1127	−2.36077
$546$	0	0
$547$	−35.5980	−1.52206	−0.761030	−	0.648717i	$-0.775306\pi$
$547$	−35.5980	−1.52206	−0.761030	+	0.648717i	$0.775306\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.2426	−0.436351
$552$	0	0
$553$	−2.34315	−0.0996407
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.17157	−0.134384	−0.0671919	−	0.997740i	$-0.521404\pi$
$557$	−3.17157	−0.134384	−0.0671919	+	0.997740i	$0.521404\pi$
$558$	0	0
$559$	−49.9411	−2.11228
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.4853	1.03193	0.515966	−	0.856609i	$-0.327433\pi$
$563$	24.4853	1.03193	0.515966	+	0.856609i	$0.327433\pi$
$564$	0	0
$565$	−29.3137	−1.23324
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.5269	0.776689	0.388344	−	0.921514i	$-0.373047\pi$
$569$	18.5269	0.776689	0.388344	+	0.921514i	$0.373047\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	56.4853	2.35560
$576$	0	0
$577$	4.34315	0.180808	0.0904038	−	0.995905i	$-0.471184\pi$
$577$	4.34315	0.180808	0.0904038	+	0.995905i	$0.471184\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.2426	−0.507910
$582$	0	0
$583$	14.3431	0.594032
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.928932	0.0383411	0.0191706	−	0.999816i	$-0.493897\pi$
$587$	0.928932	0.0383411	0.0191706	+	0.999816i	$0.493897\pi$
$588$	0	0
$589$	8.82843	0.363769
$590$	0	0
$591$	0	0
$592$	0	0
$593$	48.3848	1.98692	0.993462	−	0.114161i	$-0.0364178\pi$
$593$	48.3848	1.98692	0.993462	+	0.114161i	$0.0364178\pi$
$594$	0	0
$595$	7.65685	0.313900
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.1005	−0.903002	−0.451501	−	0.892271i	$-0.649111\pi$
$599$	−22.1005	−0.903002	−0.451501	+	0.892271i	$0.649111\pi$
$600$	0	0
$601$	−9.31371	−0.379914	−0.189957	−	0.981792i	$-0.560835\pi$
$601$	−9.31371	−0.379914	−0.189957	+	0.981792i	$0.560835\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.2426	−0.416423
$606$	0	0
$607$	25.9411	1.05292	0.526459	−	0.850201i	$-0.323519\pi$
$607$	25.9411	1.05292	0.526459	+	0.850201i	$0.323519\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	59.8823	2.42258
$612$	0	0
$613$	−12.2843	−0.496157	−0.248079	−	0.968740i	$-0.579799\pi$
$613$	−12.2843	−0.496157	−0.248079	+	0.968740i	$0.579799\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.9411	1.44694	0.723468	−	0.690358i	$-0.242547\pi$
$617$	35.9411	1.44694	0.723468	+	0.690358i	$0.242547\pi$
$618$	0	0
$619$	11.5147	0.462816	0.231408	−	0.972857i	$-0.425667\pi$
$619$	11.5147	0.462816	0.231408	+	0.972857i	$0.425667\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.8284	−0.674217
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.4558	0.536520
$630$	0	0
$631$	15.4558	0.615287	0.307644	−	0.951502i	$-0.400460\pi$
$631$	15.4558	0.615287	0.307644	+	0.951502i	$0.400460\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−28.9706	−1.14966
$636$	0	0
$637$	5.65685	0.224133
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−47.6985	−1.88398	−0.941988	−	0.335645i	$-0.891046\pi$
$641$	−47.6985	−1.88398	−0.941988	+	0.335645i	$0.891046\pi$
$642$	0	0
$643$	21.1716	0.834925	0.417463	−	0.908694i	$-0.362919\pi$
$643$	21.1716	0.834925	0.417463	+	0.908694i	$0.362919\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.727922	−0.0286176	−0.0143088	−	0.999898i	$-0.504555\pi$
$647$	−0.727922	−0.0286176	−0.0143088	+	0.999898i	$0.504555\pi$
$648$	0	0
$649$	6.62742	0.260149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.5980	−1.47132	−0.735661	−	0.677350i	$-0.763128\pi$
$653$	−37.5980	−1.47132	−0.735661	+	0.677350i	$0.763128\pi$
$654$	0	0
$655$	−33.7990	−1.32064
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.2426	−0.632723	−0.316362	−	0.948639i	$-0.602461\pi$
$659$	−16.2426	−0.632723	−0.316362	+	0.948639i	$0.602461\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.41421	−0.132398
$666$	0	0
$667$	−86.9117	−3.36523
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.65685	0.218380
$672$	0	0
$673$	21.3137	0.821583	0.410792	−	0.911729i	$-0.365252\pi$
$673$	21.3137	0.821583	0.410792	+	0.911729i	$0.365252\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.9411	−1.53506	−0.767531	−	0.641012i	$-0.778515\pi$
$677$	−39.9411	−1.53506	−0.767531	+	0.641012i	$0.778515\pi$
$678$	0	0
$679$	0.343146	0.0131687
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.8995	0.837961	0.418980	−	0.907995i	$-0.362388\pi$
$683$	21.8995	0.837961	0.418980	+	0.907995i	$0.362388\pi$
$684$	0	0
$685$	39.7990	1.52064
$686$	0	0
$687$	0	0
$688$	0	0
$689$	28.6863	1.09286
$690$	0	0
$691$	28.9706	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$691$	28.9706	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−42.6274	−1.61695
$696$	0	0
$697$	−19.7990	−0.749940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.07107	0.341153
$708$	0	0
$709$	−1.65685	−0.0622245	−0.0311122	−	0.999516i	$-0.509905\pi$
$709$	−1.65685	−0.0622245	−0.0311122	+	0.999516i	$0.509905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	74.9117	2.80546
$714$	0	0
$715$	54.6274	2.04295
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.38478	−0.238112	−0.119056	−	0.992888i	$-0.537987\pi$
$719$	−6.38478	−0.238112	−0.119056	+	0.992888i	$0.537987\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−68.1838	−2.53228
$726$	0	0
$727$	−26.6274	−0.987556	−0.493778	−	0.869588i	$-0.664385\pi$
$727$	−26.6274	−0.987556	−0.493778	+	0.869588i	$0.664385\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.7990	0.732292
$732$	0	0
$733$	−20.8284	−0.769316	−0.384658	−	0.923059i	$-0.625681\pi$
$733$	−20.8284	−0.769316	−0.384658	+	0.923059i	$0.625681\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.9411	−0.955554
$738$	0	0
$739$	−37.7990	−1.39046	−0.695229	−	0.718788i	$-0.744697\pi$
$739$	−37.7990	−1.39046	−0.695229	+	0.718788i	$0.744697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.0711	1.43338	0.716689	−	0.697393i	$-0.245657\pi$
$743$	39.0711	1.43338	0.716689	+	0.697393i	$0.245657\pi$
$744$	0	0
$745$	−63.1127	−2.31227
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−16.7279	−0.611225
$750$	0	0
$751$	12.2010	0.445221	0.222611	−	0.974907i	$-0.428542\pi$
$751$	12.2010	0.445221	0.222611	+	0.974907i	$0.428542\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.9706	1.05435
$756$	0	0
$757$	4.34315	0.157854	0.0789272	−	0.996880i	$-0.474851\pi$
$757$	4.34315	0.157854	0.0789272	+	0.996880i	$0.474851\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−47.2132	−1.71148	−0.855739	−	0.517408i	$-0.826897\pi$
$761$	−47.2132	−1.71148	−0.855739	+	0.517408i	$0.826897\pi$
$762$	0	0
$763$	16.1421	0.584385
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.2548	0.478604
$768$	0	0
$769$	1.02944	0.0371225	0.0185612	−	0.999828i	$-0.494091\pi$
$769$	1.02944	0.0371225	0.0185612	+	0.999828i	$0.494091\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.5147	0.486091	0.243045	−	0.970015i	$-0.421854\pi$
$773$	13.5147	0.486091	0.243045	+	0.970015i	$0.421854\pi$
$774$	0	0
$775$	58.7696	2.11106
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.82843	0.316311
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−26.1421	−0.933053
$786$	0	0
$787$	16.1421	0.575405	0.287702	−	0.957720i	$-0.407109\pi$
$787$	16.1421	0.575405	0.287702	+	0.957720i	$0.407109\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.58579	0.305275
$792$	0	0
$793$	11.3137	0.401762
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.48528	−0.229720	−0.114860	−	0.993382i	$-0.536642\pi$
$797$	−6.48528	−0.229720	−0.114860	+	0.993382i	$0.536642\pi$
$798$	0	0
$799$	−23.7401	−0.839865
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−24.9706	−0.881192
$804$	0	0
$805$	−28.9706	−1.02108
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.9706	0.948234	0.474117	−	0.880462i	$-0.342767\pi$
$809$	26.9706	0.948234	0.474117	+	0.880462i	$0.342767\pi$
$810$	0	0
$811$	−44.9706	−1.57913	−0.789565	−	0.613667i	$-0.789694\pi$
$811$	−44.9706	−1.57913	−0.789565	+	0.613667i	$0.789694\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.17157	−0.181152
$816$	0	0
$817$	−8.82843	−0.308868
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.8284	0.587316	0.293658	−	0.955911i	$-0.405127\pi$
$821$	16.8284	0.587316	0.293658	+	0.955911i	$0.405127\pi$
$822$	0	0
$823$	44.1421	1.53870	0.769349	−	0.638829i	$-0.220581\pi$
$823$	44.1421	1.53870	0.769349	+	0.638829i	$0.220581\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.8995	0.761520	0.380760	−	0.924674i	$-0.375662\pi$
$827$	21.8995	0.761520	0.380760	+	0.924674i	$0.375662\pi$
$828$	0	0
$829$	50.2843	1.74644	0.873222	−	0.487322i	$-0.162026\pi$
$829$	50.2843	1.74644	0.873222	+	0.487322i	$0.162026\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.24264	−0.0777029
$834$	0	0
$835$	77.9411	2.69726
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.2548	−1.42428	−0.712138	−	0.702040i	$-0.752273\pi$
$839$	−41.2548	−1.42428	−0.712138	+	0.702040i	$0.752273\pi$
$840$	0	0
$841$	75.9117	2.61764
$842$	0	0
$843$	0	0
$844$	0	0
$845$	64.8701	2.23160
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−50.9117	−1.74523
$852$	0	0
$853$	1.02944	0.0352473	0.0176236	−	0.999845i	$-0.494390\pi$
$853$	1.02944	0.0352473	0.0176236	+	0.999845i	$0.494390\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.4853	0.904720	0.452360	−	0.891835i	$-0.350582\pi$
$857$	26.4853	0.904720	0.452360	+	0.891835i	$0.350582\pi$
$858$	0	0
$859$	53.6569	1.83075	0.915374	−	0.402604i	$-0.131895\pi$
$859$	53.6569	1.83075	0.915374	+	0.402604i	$0.131895\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.7574	−0.944871	−0.472436	−	0.881365i	$-0.656625\pi$
$863$	−27.7574	−0.944871	−0.472436	+	0.881365i	$0.656625\pi$
$864$	0	0
$865$	10.8284	0.368178
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.62742	0.224820
$870$	0	0
$871$	−51.8823	−1.75796
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	18.6863	0.630991	0.315496	−	0.948927i	$-0.397829\pi$
$877$	18.6863	0.630991	0.315496	+	0.948927i	$0.397829\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.3848	−0.686781	−0.343390	−	0.939193i	$-0.611575\pi$
$881$	−20.3848	−0.686781	−0.343390	+	0.939193i	$0.611575\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	8.48528	0.284587
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.5858	0.354240
$894$	0	0
$895$	−10.4853	−0.350484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−90.4264	−3.01589
$900$	0	0
$901$	−11.3726	−0.378876
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.7990	−1.05703
$906$	0	0
$907$	−31.5980	−1.04919	−0.524597	−	0.851351i	$-0.675784\pi$
$907$	−31.5980	−1.04919	−0.524597	+	0.851351i	$0.675784\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.4437	0.544802	0.272401	−	0.962184i	$-0.412182\pi$
$911$	16.4437	0.544802	0.272401	+	0.962184i	$0.412182\pi$
$912$	0	0
$913$	34.6274	1.14600
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.89949	0.326910
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−39.9411	−1.31326
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.2132	−1.41778	−0.708890	−	0.705319i	$-0.750804\pi$
$929$	−43.2132	−1.41778	−0.708890	+	0.705319i	$0.750804\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−21.6569	−0.708255
$936$	0	0
$937$	25.0294	0.817676	0.408838	−	0.912607i	$-0.365934\pi$
$937$	25.0294	0.817676	0.408838	+	0.912607i	$0.365934\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.2843	0.987239	0.493620	−	0.869678i	$-0.335674\pi$
$941$	30.2843	0.987239	0.493620	+	0.869678i	$0.335674\pi$
$942$	0	0
$943$	74.9117	2.43946
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.48528	−0.145752	−0.0728760	−	0.997341i	$-0.523218\pi$
$947$	−4.48528	−0.145752	−0.0728760	+	0.997341i	$0.523218\pi$
$948$	0	0
$949$	−49.9411	−1.62116
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50.0416	1.62101	0.810504	−	0.585734i	$-0.199193\pi$
$953$	50.0416	1.62101	0.810504	+	0.585734i	$0.199193\pi$
$954$	0	0
$955$	50.6274	1.63826
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.6569	−0.376419
$960$	0	0
$961$	46.9411	1.51423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.17157	−0.0377143
$966$	0	0
$967$	−24.8284	−0.798428	−0.399214	−	0.916858i	$-0.630717\pi$
$967$	−24.8284	−0.798428	−0.399214	+	0.916858i	$0.630717\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.6863	0.407122	0.203561	−	0.979062i	$-0.434748\pi$
$971$	12.6863	0.407122	0.203561	+	0.979062i	$0.434748\pi$
$972$	0	0
$973$	12.4853	0.400260
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−61.1543	−1.95650	−0.978250	−	0.207429i	$-0.933491\pi$
$977$	−61.1543	−1.95650	−0.978250	+	0.207429i	$0.933491\pi$
$978$	0	0
$979$	47.5980	1.52124
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.2843	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$984$	0	0
$985$	−51.1127	−1.62859
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−74.9117	−2.38205
$990$	0	0
$991$	24.2843	0.771415	0.385708	−	0.922621i	$-0.373957\pi$
$991$	24.2843	0.771415	0.385708	+	0.922621i	$0.373957\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−48.2843	−1.53071
$996$	0	0
$997$	19.6569	0.622539	0.311269	−	0.950322i	$-0.399246\pi$
$997$	19.6569	0.622539	0.311269	+	0.950322i	$0.399246\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.by.1.2		2
3.2	odd	2		3192.2.a.r.1.1	✓	2
12.11	even	2		6384.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.r.1.1	✓	2	3.2	odd	2
6384.2.a.bh.1.1		2	12.11	even	2
9576.2.a.by.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41421 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.41421 q^{5} -1.00000 q^{7} +2.82843 q^{11} +5.65685 q^{13} -2.24264 q^{17} +1.00000 q^{19} +8.48528 q^{23} +6.65685 q^{25} -10.2426 q^{29} +8.82843 q^{31} -3.41421 q^{35} -6.00000 q^{37} +8.82843 q^{41} -8.82843 q^{43} +10.5858 q^{47} +1.00000 q^{49} +5.07107 q^{53} +9.65685 q^{55} +2.34315 q^{59} +2.00000 q^{61} +19.3137 q^{65} -9.17157 q^{67} -12.7279 q^{71} -8.82843 q^{73} -2.82843 q^{77} +2.34315 q^{79} +12.2426 q^{83} -7.65685 q^{85} +16.8284 q^{89} -5.65685 q^{91} +3.41421 q^{95} -0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^7	\( 2 q + 4 q^{5} - 2 q^{7} + 4 q^{17} + 2 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 4 q^{35} - 12 q^{37} + 12 q^{41} - 12 q^{43} + 24 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{55} + 16 q^{59} + 4 q^{61} + 16 q^{65} - 24 q^{67} - 12 q^{73} + 16 q^{79} + 16 q^{83} - 4 q^{85} + 28 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^7 + 4 * q^17 + 2 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 4 * q^35 - 12 * q^37 + 12 * q^41 - 12 * q^43 + 24 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^55 + 16 * q^59 + 4 * q^61 + 16 * q^65 - 24 * q^67 - 12 * q^73 + 16 * q^79 + 16 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more