LMFDB - Embedded newform 828.4.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [828,4,Mod(1,828)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("828.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 828 = 2^{2} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	828.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.8535814848$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.59261$ of defining polynomial
Character	$\chi$	$=$	828.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+16.0718 q^{5} -35.2976 q^{7} -35.8117 q^{11} +26.4662 q^{13} +92.3131 q^{17} +71.9314 q^{19} +23.0000 q^{23} +133.304 q^{25} -131.757 q^{29} -330.509 q^{31} -567.297 q^{35} -55.6698 q^{37} -403.826 q^{41} -289.771 q^{43} +141.061 q^{47} +902.922 q^{49} -0.542497 q^{53} -575.560 q^{55} +241.813 q^{59} -779.976 q^{61} +425.360 q^{65} -324.411 q^{67} -408.286 q^{71} +124.854 q^{73} +1264.07 q^{77} -896.602 q^{79} -508.623 q^{83} +1483.64 q^{85} +908.860 q^{89} -934.194 q^{91} +1156.07 q^{95} -549.593 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 10 q^{5} - 46 q^{7} + 64 q^{11} - 44 q^{13} + 88 q^{17} - 94 q^{19} + 69 q^{23} + 181 q^{25} - 308 q^{29} - 140 q^{31} - 192 q^{35} + 26 q^{37} - 584 q^{41} - 478 q^{43} + 28 q^{47} + 695 q^{49} - 356 q^{53}+ \cdots + 736 q^{97}+O(q^{100}) $ 3 * q + 10 * q^5 - 46 * q^7 + 64 * q^11 - 44 * q^13 + 88 * q^17 - 94 * q^19 + 69 * q^23 + 181 * q^25 - 308 * q^29 - 140 * q^31 - 192 * q^35 + 26 * q^37 - 584 * q^41 - 478 * q^43 + 28 * q^47 + 695 * q^49 - 356 * q^53 - 1004 * q^55 - 144 * q^59 - 1052 * q^61 - 150 * q^65 - 2008 * q^67 - 360 * q^71 - 252 * q^73 + 576 * q^77 - 720 * q^79 + 1404 * q^83 - 1632 * q^85 - 534 * q^89 - 1526 * q^91 + 108 * q^95 + 736 * q^97

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
$5$	16.0718	1.43751	0.718754	−	0.695264i	$-0.244713\pi$
$5$	16.0718	1.43751	0.718754	+	0.695264i	$0.244713\pi$
$6$	0	0
$7$	−35.2976	−1.90589	−0.952946	−	0.303140i	$-0.901965\pi$
$7$	−35.2976	−1.90589	−0.952946	+	0.303140i	$0.901965\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−35.8117	−0.981604	−0.490802	−	0.871271i	$-0.663296\pi$
$11$	−35.8117	−0.981604	−0.490802	+	0.871271i	$0.663296\pi$
$12$	0	0
$13$	26.4662	0.564646	0.282323	−	0.959319i	$-0.408895\pi$
$13$	26.4662	0.564646	0.282323	+	0.959319i	$0.408895\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	92.3131	1.31701	0.658507	−	0.752575i	$-0.271188\pi$
$17$	92.3131	1.31701	0.658507	+	0.752575i	$0.271188\pi$
$18$	0	0
$19$	71.9314	0.868536	0.434268	−	0.900784i	$-0.357007\pi$
$19$	71.9314	0.868536	0.434268	+	0.900784i	$0.357007\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	133.304	1.06643
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−131.757	−0.843675	−0.421838	−	0.906671i	$-0.638615\pi$
$29$	−131.757	−0.843675	−0.421838	+	0.906671i	$0.638615\pi$
$30$	0	0
$31$	−330.509	−1.91488	−0.957439	−	0.288634i	$-0.906799\pi$
$31$	−330.509	−1.91488	−0.957439	+	0.288634i	$0.906799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−567.297	−2.73974
$36$	0	0
$37$	−55.6698	−0.247353	−0.123677	−	0.992323i	$-0.539469\pi$
$37$	−55.6698	−0.247353	−0.123677	+	0.992323i	$0.539469\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−403.826	−1.53822	−0.769110	−	0.639117i	$-0.779300\pi$
$41$	−403.826	−1.53822	−0.769110	+	0.639117i	$0.779300\pi$
$42$	0	0
$43$	−289.771	−1.02767	−0.513833	−	0.857890i	$-0.671775\pi$
$43$	−289.771	−1.02767	−0.513833	+	0.857890i	$0.671775\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	141.061	0.437785	0.218892	−	0.975749i	$-0.429756\pi$
$47$	141.061	0.437785	0.218892	+	0.975749i	$0.429756\pi$
$48$	0	0
$49$	902.922	2.63242
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.542497	−0.00140599	−0.000702997	−	1.00000i	$-0.500224\pi$
$53$	−0.542497	−0.00140599	−0.000702997		1.00000i	$0.500224\pi$
$54$	0	0
$55$	−575.560	−1.41106
$56$	0	0
$57$	0	0
$58$	0	0
$59$	241.813	0.533582	0.266791	−	0.963754i	$-0.414037\pi$
$59$	241.813	0.533582	0.266791	+	0.963754i	$0.414037\pi$
$60$	0	0
$61$	−779.976	−1.63714	−0.818571	−	0.574405i	$-0.805234\pi$
$61$	−779.976	−1.63714	−0.818571	+	0.574405i	$0.805234\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	425.360	0.811684
$66$	0	0
$67$	−324.411	−0.591538	−0.295769	−	0.955259i	$-0.595576\pi$
$67$	−324.411	−0.591538	−0.295769	+	0.955259i	$0.595576\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−408.286	−0.682460	−0.341230	−	0.939980i	$-0.610844\pi$
$71$	−408.286	−0.682460	−0.341230	+	0.939980i	$0.610844\pi$
$72$	0	0
$73$	124.854	0.200179	0.100090	−	0.994978i	$-0.468087\pi$
$73$	124.854	0.200179	0.100090	+	0.994978i	$0.468087\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1264.07	1.87083
$78$	0	0
$79$	−896.602	−1.27691	−0.638453	−	0.769661i	$-0.720425\pi$
$79$	−896.602	−1.27691	−0.638453	+	0.769661i	$0.720425\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−508.623	−0.672634	−0.336317	−	0.941749i	$-0.609181\pi$
$83$	−508.623	−0.672634	−0.336317	+	0.941749i	$0.609181\pi$
$84$	0	0
$85$	1483.64	1.89322
$86$	0	0
$87$	0	0
$88$	0	0
$89$	908.860	1.08246	0.541230	−	0.840875i	$-0.317959\pi$
$89$	908.860	1.08246	0.541230	+	0.840875i	$0.317959\pi$
$90$	0	0
$91$	−934.194	−1.07615
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1156.07	1.24853
$96$	0	0
$97$	−549.593	−0.575286	−0.287643	−	0.957738i	$-0.592872\pi$
$97$	−549.593	−0.575286	−0.287643	+	0.957738i	$0.592872\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1828.65	−1.80156	−0.900782	−	0.434272i	$-0.857006\pi$
$101$	−1828.65	−1.80156	−0.900782	+	0.434272i	$0.857006\pi$
$102$	0	0
$103$	−747.379	−0.714965	−0.357483	−	0.933920i	$-0.616365\pi$
$103$	−747.379	−0.714965	−0.357483	+	0.933920i	$0.616365\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−43.4383	−0.0392462	−0.0196231	−	0.999807i	$-0.506247\pi$
$107$	−43.4383	−0.0392462	−0.0196231	+	0.999807i	$0.506247\pi$
$108$	0	0
$109$	534.245	0.469462	0.234731	−	0.972060i	$-0.424579\pi$
$109$	534.245	0.469462	0.234731	+	0.972060i	$0.424579\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−506.919	−0.422008	−0.211004	−	0.977485i	$-0.567673\pi$
$113$	−506.919	−0.422008	−0.211004	+	0.977485i	$0.567673\pi$
$114$	0	0
$115$	369.652	0.299741
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3258.43	−2.51008
$120$	0	0
$121$	−48.5191	−0.0364531
$122$	0	0
$123$	0	0
$124$	0	0
$125$	133.458	0.0954949
$126$	0	0
$127$	58.5859	0.0409343	0.0204672	−	0.999791i	$-0.493485\pi$
$127$	58.5859	0.0409343	0.0204672	+	0.999791i	$0.493485\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2337.89	1.55926	0.779629	−	0.626242i	$-0.215408\pi$
$131$	2337.89	1.55926	0.779629	+	0.626242i	$0.215408\pi$
$132$	0	0
$133$	−2539.01	−1.65534
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1212.36	−0.756050	−0.378025	−	0.925795i	$-0.623397\pi$
$137$	−1212.36	−0.756050	−0.378025	+	0.925795i	$0.623397\pi$
$138$	0	0
$139$	−1153.23	−0.703710	−0.351855	−	0.936055i	$-0.614449\pi$
$139$	−1153.23	−0.703710	−0.351855	+	0.936055i	$0.614449\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−947.801	−0.554259
$144$	0	0
$145$	−2117.57	−1.21279
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−252.502	−0.138830	−0.0694152	−	0.997588i	$-0.522113\pi$
$149$	−252.502	−0.138830	−0.0694152	+	0.997588i	$0.522113\pi$
$150$	0	0
$151$	−557.908	−0.300675	−0.150337	−	0.988635i	$-0.548036\pi$
$151$	−557.908	−0.300675	−0.150337	+	0.988635i	$0.548036\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5311.89	−2.75265
$156$	0	0
$157$	1012.24	0.514555	0.257278	−	0.966338i	$-0.417174\pi$
$157$	1012.24	0.514555	0.257278	+	0.966338i	$0.417174\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−811.845	−0.397406
$162$	0	0
$163$	3203.54	1.53939	0.769695	−	0.638411i	$-0.220408\pi$
$163$	3203.54	1.53939	0.769695	+	0.638411i	$0.220408\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	355.080	0.164533	0.0822663	−	0.996610i	$-0.473784\pi$
$167$	355.080	0.164533	0.0822663	+	0.996610i	$0.473784\pi$
$168$	0	0
$169$	−1496.54	−0.681175
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3926.70	1.72567	0.862836	−	0.505483i	$-0.168686\pi$
$173$	3926.70	1.72567	0.862836	+	0.505483i	$0.168686\pi$
$174$	0	0
$175$	−4705.31	−2.03250
$176$	0	0
$177$	0	0
$178$	0	0
$179$	808.736	0.337697	0.168849	−	0.985642i	$-0.445995\pi$
$179$	808.736	0.337697	0.168849	+	0.985642i	$0.445995\pi$
$180$	0	0
$181$	−2181.75	−0.895957	−0.447979	−	0.894044i	$-0.647856\pi$
$181$	−2181.75	−0.895957	−0.447979	+	0.894044i	$0.647856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−894.716	−0.355572
$186$	0	0
$187$	−3305.89	−1.29279
$188$	0	0
$189$	0	0
$190$	0	0
$191$	171.454	0.0649528	0.0324764	−	0.999473i	$-0.489661\pi$
$191$	171.454	0.0649528	0.0324764	+	0.999473i	$0.489661\pi$
$192$	0	0
$193$	−4290.56	−1.60021	−0.800106	−	0.599858i	$-0.795224\pi$
$193$	−4290.56	−1.60021	−0.800106	+	0.599858i	$0.795224\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1228.86	−0.444431	−0.222216	−	0.974998i	$-0.571329\pi$
$197$	−1228.86	−0.444431	−0.222216	+	0.974998i	$0.571329\pi$
$198$	0	0
$199$	3093.78	1.10207	0.551036	−	0.834481i	$-0.314233\pi$
$199$	3093.78	1.10207	0.551036	+	0.834481i	$0.314233\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4650.69	1.60795
$204$	0	0
$205$	−6490.22	−2.21120
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2575.99	−0.852559
$210$	0	0
$211$	1834.96	0.598692	0.299346	−	0.954145i	$-0.403232\pi$
$211$	1834.96	0.598692	0.299346	+	0.954145i	$0.403232\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4657.15	−1.47728
$216$	0	0
$217$	11666.2	3.64955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2443.18	0.743647
$222$	0	0
$223$	−1002.48	−0.301035	−0.150518	−	0.988607i	$-0.548094\pi$
$223$	−1002.48	−0.301035	−0.150518	+	0.988607i	$0.548094\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4895.79	1.43148	0.715738	−	0.698369i	$-0.246091\pi$
$227$	4895.79	1.43148	0.715738	+	0.698369i	$0.246091\pi$
$228$	0	0
$229$	3751.95	1.08269	0.541345	−	0.840801i	$-0.317915\pi$
$229$	3751.95	1.08269	0.541345	+	0.840801i	$0.317915\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1757.11	−0.494043	−0.247021	−	0.969010i	$-0.579452\pi$
$233$	−1757.11	−0.494043	−0.247021	+	0.969010i	$0.579452\pi$
$234$	0	0
$235$	2267.11	0.629319
$236$	0	0
$237$	0	0
$238$	0	0
$239$	121.152	0.0327895	0.0163947	−	0.999866i	$-0.494781\pi$
$239$	121.152	0.0327895	0.0163947	+	0.999866i	$0.494781\pi$
$240$	0	0
$241$	5902.90	1.57776	0.788878	−	0.614550i	$-0.210663\pi$
$241$	5902.90	1.57776	0.788878	+	0.614550i	$0.210663\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14511.6	3.78413
$246$	0	0
$247$	1903.75	0.490416
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2001.32	−0.503276	−0.251638	−	0.967821i	$-0.580969\pi$
$251$	−2001.32	−0.503276	−0.251638	+	0.967821i	$0.580969\pi$
$252$	0	0
$253$	−823.670	−0.204679
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2287.08	−0.555114	−0.277557	−	0.960709i	$-0.589525\pi$
$257$	−2287.08	−0.555114	−0.277557	+	0.960709i	$0.589525\pi$
$258$	0	0
$259$	1965.01	0.471428
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2996.57	−0.702572	−0.351286	−	0.936268i	$-0.614255\pi$
$263$	−2996.57	−0.702572	−0.351286	+	0.936268i	$0.614255\pi$
$264$	0	0
$265$	−8.71892	−0.00202113
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4012.29	−0.909419	−0.454710	−	0.890640i	$-0.650257\pi$
$269$	−4012.29	−0.909419	−0.454710	+	0.890640i	$0.650257\pi$
$270$	0	0
$271$	2747.49	0.615860	0.307930	−	0.951409i	$-0.400364\pi$
$271$	2747.49	0.615860	0.307930	+	0.951409i	$0.400364\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4773.84	−1.04681
$276$	0	0
$277$	−1108.74	−0.240497	−0.120248	−	0.992744i	$-0.538369\pi$
$277$	−1108.74	−0.240497	−0.120248	+	0.992744i	$0.538369\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7189.03	−1.52620	−0.763098	−	0.646282i	$-0.776323\pi$
$281$	−7189.03	−1.52620	−0.763098	+	0.646282i	$0.776323\pi$
$282$	0	0
$283$	544.421	0.114355	0.0571775	−	0.998364i	$-0.481790\pi$
$283$	544.421	0.114355	0.0571775	+	0.998364i	$0.481790\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14254.1	2.93168
$288$	0	0
$289$	3608.71	0.734523
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3092.72	0.616651	0.308326	−	0.951281i	$-0.400231\pi$
$293$	3092.72	0.616651	0.308326	+	0.951281i	$0.400231\pi$
$294$	0	0
$295$	3886.37	0.767028
$296$	0	0
$297$	0	0
$298$	0	0
$299$	608.722	0.117737
$300$	0	0
$301$	10228.2	1.95862
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12535.6	−2.35341
$306$	0	0
$307$	−4592.21	−0.853717	−0.426859	−	0.904318i	$-0.640380\pi$
$307$	−4592.21	−0.853717	−0.426859	+	0.904318i	$0.640380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6451.07	1.17623	0.588114	−	0.808778i	$-0.299871\pi$
$311$	6451.07	1.17623	0.588114	+	0.808778i	$0.299871\pi$
$312$	0	0
$313$	−5187.10	−0.936716	−0.468358	−	0.883539i	$-0.655154\pi$
$313$	−5187.10	−0.936716	−0.468358	+	0.883539i	$0.655154\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6807.19	−1.20609	−0.603044	−	0.797708i	$-0.706046\pi$
$317$	−6807.19	−1.20609	−0.603044	+	0.797708i	$0.706046\pi$
$318$	0	0
$319$	4718.43	0.828155
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6640.21	1.14387
$324$	0	0
$325$	3528.05	0.602156
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4979.12	−0.834370
$330$	0	0
$331$	−6121.87	−1.01658	−0.508291	−	0.861186i	$-0.669722\pi$
$331$	−6121.87	−1.01658	−0.508291	+	0.861186i	$0.669722\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5213.87	−0.850341
$336$	0	0
$337$	−6467.75	−1.04546	−0.522731	−	0.852498i	$-0.675087\pi$
$337$	−6467.75	−1.04546	−0.522731	+	0.852498i	$0.675087\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11836.1	1.87965
$342$	0	0
$343$	−19763.9	−3.11123
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5815.67	0.899716	0.449858	−	0.893100i	$-0.351475\pi$
$347$	5815.67	0.899716	0.449858	+	0.893100i	$0.351475\pi$
$348$	0	0
$349$	10877.0	1.66830	0.834148	−	0.551541i	$-0.185960\pi$
$349$	10877.0	1.66830	0.834148	+	0.551541i	$0.185960\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3507.96	0.528923	0.264462	−	0.964396i	$-0.414806\pi$
$353$	3507.96	0.528923	0.264462	+	0.964396i	$0.414806\pi$
$354$	0	0
$355$	−6561.91	−0.981042
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3256.28	−0.478718	−0.239359	−	0.970931i	$-0.576937\pi$
$359$	−3256.28	−0.478718	−0.239359	+	0.970931i	$0.576937\pi$
$360$	0	0
$361$	−1684.88	−0.245645
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2006.64	0.287760
$366$	0	0
$367$	−8590.93	−1.22192	−0.610958	−	0.791663i	$-0.709215\pi$
$367$	−8590.93	−1.22192	−0.610958	+	0.791663i	$0.709215\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.1489	0.00267967
$372$	0	0
$373$	−9049.15	−1.25616	−0.628079	−	0.778149i	$-0.716159\pi$
$373$	−9049.15	−1.25616	−0.628079	+	0.778149i	$0.716159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3487.09	−0.476378
$378$	0	0
$379$	10260.9	1.39068	0.695342	−	0.718679i	$-0.255253\pi$
$379$	10260.9	1.39068	0.695342	+	0.718679i	$0.255253\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1185.33	−0.158140	−0.0790702	−	0.996869i	$-0.525195\pi$
$383$	−1185.33	−0.158140	−0.0790702	+	0.996869i	$0.525195\pi$
$384$	0	0
$385$	20315.9	2.68934
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8067.50	1.05151	0.525757	−	0.850635i	$-0.323782\pi$
$389$	8067.50	1.05151	0.525757	+	0.850635i	$0.323782\pi$
$390$	0	0
$391$	2123.20	0.274616
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14410.0	−1.83556
$396$	0	0
$397$	1783.75	0.225501	0.112750	−	0.993623i	$-0.464034\pi$
$397$	1783.75	0.225501	0.112750	+	0.993623i	$0.464034\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4678.99	0.582687	0.291344	−	0.956618i	$-0.405898\pi$
$401$	4678.99	0.582687	0.291344	+	0.956618i	$0.405898\pi$
$402$	0	0
$403$	−8747.33	−1.08123
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1993.63	0.242803
$408$	0	0
$409$	−9444.19	−1.14177	−0.570887	−	0.821029i	$-0.693400\pi$
$409$	−9444.19	−1.14177	−0.570887	+	0.821029i	$0.693400\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8535.41	−1.01695
$414$	0	0
$415$	−8174.50	−0.966917
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2572.71	0.299965	0.149982	−	0.988689i	$-0.452078\pi$
$419$	2572.71	0.299965	0.149982	+	0.988689i	$0.452078\pi$
$420$	0	0
$421$	6620.54	0.766426	0.383213	−	0.923660i	$-0.374818\pi$
$421$	6620.54	0.766426	0.383213	+	0.923660i	$0.374818\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12305.7	1.40450
$426$	0	0
$427$	27531.3	3.12022
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13881.0	−1.55133	−0.775667	−	0.631142i	$-0.782586\pi$
$431$	−13881.0	−1.55133	−0.775667	+	0.631142i	$0.782586\pi$
$432$	0	0
$433$	14558.4	1.61578	0.807889	−	0.589335i	$-0.200610\pi$
$433$	14558.4	1.61578	0.807889	+	0.589335i	$0.200610\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1654.42	0.181102
$438$	0	0
$439$	3763.83	0.409198	0.204599	−	0.978846i	$-0.434411\pi$
$439$	3763.83	0.409198	0.204599	+	0.978846i	$0.434411\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2355.23	0.252597	0.126299	−	0.991992i	$-0.459690\pi$
$443$	2355.23	0.252597	0.126299	+	0.991992i	$0.459690\pi$
$444$	0	0
$445$	14607.0	1.55605
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9439.98	0.992206	0.496103	−	0.868264i	$-0.334764\pi$
$449$	9439.98	0.992206	0.496103	+	0.868264i	$0.334764\pi$
$450$	0	0
$451$	14461.7	1.50992
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15014.2	−1.54698
$456$	0	0
$457$	17258.1	1.76652	0.883262	−	0.468881i	$-0.155343\pi$
$457$	17258.1	1.76652	0.883262	+	0.468881i	$0.155343\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18097.6	1.82839	0.914194	−	0.405276i	$-0.132825\pi$
$461$	18097.6	1.82839	0.914194	+	0.405276i	$0.132825\pi$
$462$	0	0
$463$	−15791.5	−1.58509	−0.792543	−	0.609816i	$-0.791243\pi$
$463$	−15791.5	−1.58509	−0.792543	+	0.609816i	$0.791243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16881.1	1.67273	0.836363	−	0.548176i	$-0.184678\pi$
$467$	16881.1	1.67273	0.836363	+	0.548176i	$0.184678\pi$
$468$	0	0
$469$	11450.9	1.12741
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10377.2	1.00876
$474$	0	0
$475$	9588.73	0.926234
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15989.4	1.52521	0.762604	−	0.646866i	$-0.223921\pi$
$479$	15989.4	1.52521	0.762604	+	0.646866i	$0.223921\pi$
$480$	0	0
$481$	−1473.37	−0.139667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8832.97	−0.826978
$486$	0	0
$487$	9349.64	0.869964	0.434982	−	0.900439i	$-0.356755\pi$
$487$	9349.64	0.869964	0.434982	+	0.900439i	$0.356755\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2846.67	0.261646	0.130823	−	0.991406i	$-0.458238\pi$
$491$	2846.67	0.261646	0.130823	+	0.991406i	$0.458238\pi$
$492$	0	0
$493$	−12162.9	−1.11113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14411.5	1.30070
$498$	0	0
$499$	5936.20	0.532547	0.266274	−	0.963898i	$-0.414208\pi$
$499$	5936.20	0.532547	0.266274	+	0.963898i	$0.414208\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15918.1	1.41104	0.705522	−	0.708688i	$-0.250713\pi$
$503$	15918.1	1.41104	0.705522	+	0.708688i	$0.250713\pi$
$504$	0	0
$505$	−29389.8	−2.58976
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4051.52	−0.352810	−0.176405	−	0.984318i	$-0.556447\pi$
$509$	−4051.52	−0.352810	−0.176405	+	0.984318i	$0.556447\pi$
$510$	0	0
$511$	−4407.06	−0.381520
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12011.7	−1.02777
$516$	0	0
$517$	−5051.64	−0.429731
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16599.5	1.39585	0.697925	−	0.716171i	$-0.254107\pi$
$521$	16599.5	1.39585	0.697925	+	0.716171i	$0.254107\pi$
$522$	0	0
$523$	−8884.68	−0.742830	−0.371415	−	0.928467i	$-0.621127\pi$
$523$	−8884.68	−0.742830	−0.371415	+	0.928467i	$0.621127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30510.4	−2.52192
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10687.7	−0.868550
$534$	0	0
$535$	−698.134	−0.0564167
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−32335.2	−2.58400
$540$	0	0
$541$	5434.91	0.431913	0.215957	−	0.976403i	$-0.430713\pi$
$541$	5434.91	0.431913	0.215957	+	0.976403i	$0.430713\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8586.29	0.674856
$546$	0	0
$547$	−23125.1	−1.80760	−0.903801	−	0.427953i	$-0.859235\pi$
$547$	−23125.1	−1.80760	−0.903801	+	0.427953i	$0.859235\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9477.43	−0.732763
$552$	0	0
$553$	31647.9	2.43364
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−326.574	−0.0248427	−0.0124213	−	0.999923i	$-0.503954\pi$
$557$	−326.574	−0.0248427	−0.0124213	+	0.999923i	$0.503954\pi$
$558$	0	0
$559$	−7669.13	−0.580268
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4445.50	−0.332780	−0.166390	−	0.986060i	$-0.553211\pi$
$563$	−4445.50	−0.332780	−0.166390	+	0.986060i	$0.553211\pi$
$564$	0	0
$565$	−8147.12	−0.606640
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5704.62	−0.420299	−0.210150	−	0.977669i	$-0.567395\pi$
$569$	−5704.62	−0.420299	−0.210150	+	0.977669i	$0.567395\pi$
$570$	0	0
$571$	−686.306	−0.0502995	−0.0251497	−	0.999684i	$-0.508006\pi$
$571$	−686.306	−0.0502995	−0.0251497	+	0.999684i	$0.508006\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3065.99	0.222366
$576$	0	0
$577$	6.09356	0.000439651 0	0.000219825	−	1.00000i	$-0.499930\pi$
$577$	6.09356	0.000439651 0	0.000219825		1.00000i	$0.499930\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	17953.2	1.28197
$582$	0	0
$583$	19.4278	0.00138013
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21294.9	1.49734	0.748668	−	0.662946i	$-0.230694\pi$
$587$	21294.9	1.49734	0.748668	+	0.662946i	$0.230694\pi$
$588$	0	0
$589$	−23774.0	−1.66314
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3829.95	−0.265223	−0.132612	−	0.991168i	$-0.542336\pi$
$593$	−3829.95	−0.265223	−0.132612	+	0.991168i	$0.542336\pi$
$594$	0	0
$595$	−52369.0	−3.60827
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20004.6	−1.36455	−0.682277	−	0.731094i	$-0.739010\pi$
$599$	−20004.6	−1.36455	−0.682277	+	0.731094i	$0.739010\pi$
$600$	0	0
$601$	−10778.4	−0.731546	−0.365773	−	0.930704i	$-0.619195\pi$
$601$	−10778.4	−0.731546	−0.365773	+	0.930704i	$0.619195\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−779.790	−0.0524016
$606$	0	0
$607$	−13666.1	−0.913823	−0.456911	−	0.889512i	$-0.651044\pi$
$607$	−13666.1	−0.913823	−0.456911	+	0.889512i	$0.651044\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3733.35	0.247193
$612$	0	0
$613$	4694.25	0.309297	0.154648	−	0.987970i	$-0.450576\pi$
$613$	4694.25	0.309297	0.154648	+	0.987970i	$0.450576\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25810.5	−1.68411	−0.842053	−	0.539395i	$-0.818653\pi$
$617$	−25810.5	−1.68411	−0.842053	+	0.539395i	$0.818653\pi$
$618$	0	0
$619$	−4358.87	−0.283034	−0.141517	−	0.989936i	$-0.545198\pi$
$619$	−4358.87	−0.283034	−0.141517	+	0.989936i	$0.545198\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32080.6	−2.06305
$624$	0	0
$625$	−14518.1	−0.929156
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5139.06	−0.325767
$630$	0	0
$631$	12601.2	0.795004	0.397502	−	0.917601i	$-0.369877\pi$
$631$	12601.2	0.795004	0.397502	+	0.917601i	$0.369877\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	941.584	0.0588435
$636$	0	0
$637$	23896.9	1.48639
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18975.9	1.16927	0.584637	−	0.811295i	$-0.301237\pi$
$641$	18975.9	1.16927	0.584637	+	0.811295i	$0.301237\pi$
$642$	0	0
$643$	3625.90	0.222382	0.111191	−	0.993799i	$-0.464533\pi$
$643$	3625.90	0.222382	0.111191	+	0.993799i	$0.464533\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18092.6	−1.09937	−0.549685	−	0.835372i	$-0.685252\pi$
$647$	−18092.6	−1.09937	−0.549685	+	0.835372i	$0.685252\pi$
$648$	0	0
$649$	−8659.73	−0.523766
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3510.61	−0.210384	−0.105192	−	0.994452i	$-0.533546\pi$
$653$	−3510.61	−0.210384	−0.105192	+	0.994452i	$0.533546\pi$
$654$	0	0
$655$	37574.2	2.24145
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4632.23	−0.273818	−0.136909	−	0.990584i	$-0.543717\pi$
$659$	−4632.23	−0.273818	−0.136909	+	0.990584i	$0.543717\pi$
$660$	0	0
$661$	15948.2	0.938446	0.469223	−	0.883080i	$-0.344534\pi$
$661$	15948.2	0.938446	0.469223	+	0.883080i	$0.344534\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−40806.5	−2.37956
$666$	0	0
$667$	−3030.40	−0.175918
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27932.3	1.60703
$672$	0	0
$673$	7107.81	0.407111	0.203556	−	0.979063i	$-0.434750\pi$
$673$	7107.81	0.407111	0.203556	+	0.979063i	$0.434750\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23843.1	1.35357	0.676784	−	0.736182i	$-0.263373\pi$
$677$	23843.1	1.35357	0.676784	+	0.736182i	$0.263373\pi$
$678$	0	0
$679$	19399.3	1.09643
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11190.3	−0.626919	−0.313460	−	0.949602i	$-0.601488\pi$
$683$	−11190.3	−0.626919	−0.313460	+	0.949602i	$0.601488\pi$
$684$	0	0
$685$	−19484.8	−1.08683
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.3578	−0.000793889 0
$690$	0	0
$691$	−3665.88	−0.201818	−0.100909	−	0.994896i	$-0.532175\pi$
$691$	−3665.88	−0.201818	−0.100909	+	0.994896i	$0.532175\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18534.5	−1.01159
$696$	0	0
$697$	−37278.4	−2.02586
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21583.4	−1.16290	−0.581450	−	0.813582i	$-0.697514\pi$
$701$	−21583.4	−1.16290	−0.581450	+	0.813582i	$0.697514\pi$
$702$	0	0
$703$	−4004.41	−0.214835
$704$	0	0
$705$	0	0
$706$	0	0
$707$	64547.2	3.43359
$708$	0	0
$709$	−245.437	−0.0130008	−0.00650042	−	0.999979i	$-0.502069\pi$
$709$	−245.437	−0.0130008	−0.00650042	+	0.999979i	$0.502069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7601.72	−0.399280
$714$	0	0
$715$	−15232.9	−0.796752
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16969.1	0.880166	0.440083	−	0.897957i	$-0.354949\pi$
$719$	16969.1	0.880166	0.440083	+	0.897957i	$0.354949\pi$
$720$	0	0
$721$	26380.7	1.36265
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−17563.7	−0.899721
$726$	0	0
$727$	18831.4	0.960686	0.480343	−	0.877081i	$-0.340512\pi$
$727$	18831.4	0.960686	0.480343	+	0.877081i	$0.340512\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−26749.7	−1.35345
$732$	0	0
$733$	19019.5	0.958391	0.479195	−	0.877708i	$-0.340928\pi$
$733$	19019.5	0.958391	0.479195	+	0.877708i	$0.340928\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11617.7	0.580657
$738$	0	0
$739$	4165.01	0.207324	0.103662	−	0.994613i	$-0.466944\pi$
$739$	4165.01	0.207324	0.103662	+	0.994613i	$0.466944\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−462.458	−0.0228344	−0.0114172	−	0.999935i	$-0.503634\pi$
$743$	−462.458	−0.0228344	−0.0114172	+	0.999935i	$0.503634\pi$
$744$	0	0
$745$	−4058.16	−0.199570
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1533.27	0.0747990
$750$	0	0
$751$	−4450.92	−0.216267	−0.108133	−	0.994136i	$-0.534487\pi$
$751$	−4450.92	−0.216267	−0.108133	+	0.994136i	$0.534487\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8966.60	−0.432223
$756$	0	0
$757$	−14260.9	−0.684703	−0.342351	−	0.939572i	$-0.611223\pi$
$757$	−14260.9	−0.684703	−0.342351	+	0.939572i	$0.611223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38730.7	1.84493	0.922463	−	0.386086i	$-0.126173\pi$
$761$	38730.7	1.84493	0.922463	+	0.386086i	$0.126173\pi$
$762$	0	0
$763$	−18857.6	−0.894744
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6399.86	0.301285
$768$	0	0
$769$	−8841.70	−0.414616	−0.207308	−	0.978276i	$-0.566470\pi$
$769$	−8841.70	−0.414616	−0.207308	+	0.978276i	$0.566470\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−37453.4	−1.74270	−0.871348	−	0.490666i	$-0.836754\pi$
$773$	−37453.4	−1.74270	−0.871348	+	0.490666i	$0.836754\pi$
$774$	0	0
$775$	−44058.2	−2.04209
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−29047.7	−1.33600
$780$	0	0
$781$	14621.4	0.669906
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16268.5	0.739678
$786$	0	0
$787$	−24653.1	−1.11663	−0.558315	−	0.829629i	$-0.688552\pi$
$787$	−24653.1	−1.11663	−0.558315	+	0.829629i	$0.688552\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17893.0	0.804302
$792$	0	0
$793$	−20643.0	−0.924406
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13427.3	−0.596760	−0.298380	−	0.954447i	$-0.596446\pi$
$797$	−13427.3	−0.596760	−0.298380	+	0.954447i	$0.596446\pi$
$798$	0	0
$799$	13021.8	0.576568
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4471.25	−0.196497
$804$	0	0
$805$	−13047.8	−0.571274
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5683.88	0.247014	0.123507	−	0.992344i	$-0.460586\pi$
$809$	5683.88	0.247014	0.123507	+	0.992344i	$0.460586\pi$
$810$	0	0
$811$	−34247.3	−1.48284	−0.741421	−	0.671041i	$-0.765848\pi$
$811$	−34247.3	−1.48284	−0.741421	+	0.671041i	$0.765848\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	51486.8	2.21289
$816$	0	0
$817$	−20843.6	−0.892566
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7867.77	−0.334455	−0.167227	−	0.985918i	$-0.553481\pi$
$821$	−7867.77	−0.334455	−0.167227	+	0.985918i	$0.553481\pi$
$822$	0	0
$823$	2728.12	0.115548	0.0577741	−	0.998330i	$-0.481600\pi$
$823$	2728.12	0.115548	0.0577741	+	0.998330i	$0.481600\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11135.2	−0.468209	−0.234105	−	0.972211i	$-0.575216\pi$
$827$	−11135.2	−0.468209	−0.234105	+	0.972211i	$0.575216\pi$
$828$	0	0
$829$	41966.8	1.75822	0.879111	−	0.476618i	$-0.158137\pi$
$829$	41966.8	1.75822	0.879111	+	0.476618i	$0.158137\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	83351.5	3.46694
$834$	0	0
$835$	5706.79	0.236517
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44244.9	1.82062	0.910312	−	0.413922i	$-0.135841\pi$
$839$	44244.9	1.82062	0.910312	+	0.413922i	$0.135841\pi$
$840$	0	0
$841$	−7029.21	−0.288212
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24052.2	−0.979194
$846$	0	0
$847$	1712.61	0.0694756
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1280.41	−0.0515767
$852$	0	0
$853$	−36560.4	−1.46753	−0.733765	−	0.679403i	$-0.762239\pi$
$853$	−36560.4	−1.46753	−0.733765	+	0.679403i	$0.762239\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36088.2	−1.43845	−0.719223	−	0.694779i	$-0.755502\pi$
$857$	−36088.2	−1.43845	−0.719223	+	0.694779i	$0.755502\pi$
$858$	0	0
$859$	−43437.9	−1.72536	−0.862679	−	0.505751i	$-0.831215\pi$
$859$	−43437.9	−1.72536	−0.862679	+	0.505751i	$0.831215\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36323.1	−1.43274	−0.716369	−	0.697722i	$-0.754197\pi$
$863$	−36323.1	−1.43274	−0.716369	+	0.697722i	$0.754197\pi$
$864$	0	0
$865$	63109.3	2.48067
$866$	0	0
$867$	0	0
$868$	0	0
$869$	32108.9	1.25342
$870$	0	0
$871$	−8585.92	−0.334010
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4710.76	−0.182003
$876$	0	0
$877$	−39908.0	−1.53660	−0.768299	−	0.640092i	$-0.778896\pi$
$877$	−39908.0	−1.53660	−0.768299	+	0.640092i	$0.778896\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33462.4	−1.27966	−0.639829	−	0.768518i	$-0.720995\pi$
$881$	−33462.4	−1.27966	−0.639829	+	0.768518i	$0.720995\pi$
$882$	0	0
$883$	5773.57	0.220041	0.110021	−	0.993929i	$-0.464908\pi$
$883$	5773.57	0.220041	0.110021	+	0.993929i	$0.464908\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28559.1	1.08108	0.540542	−	0.841317i	$-0.318219\pi$
$887$	28559.1	1.08108	0.540542	+	0.841317i	$0.318219\pi$
$888$	0	0
$889$	−2067.94	−0.0780164
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10146.7	0.380232
$894$	0	0
$895$	12997.9	0.485442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43546.8	1.61554
$900$	0	0
$901$	−50.0796	−0.00185171
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−35064.7	−1.28795
$906$	0	0
$907$	9165.89	0.335555	0.167777	−	0.985825i	$-0.446341\pi$
$907$	9165.89	0.335555	0.167777	+	0.985825i	$0.446341\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11322.8	−0.411790	−0.205895	−	0.978574i	$-0.566010\pi$
$911$	−11322.8	−0.411790	−0.205895	+	0.978574i	$0.566010\pi$
$912$	0	0
$913$	18214.7	0.660260
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−82522.1	−2.97178
$918$	0	0
$919$	−40564.8	−1.45605	−0.728025	−	0.685551i	$-0.759562\pi$
$919$	−40564.8	−1.45605	−0.728025	+	0.685551i	$0.759562\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10805.8	−0.385349
$924$	0	0
$925$	−7421.00	−0.263785
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27249.4	0.962352	0.481176	−	0.876624i	$-0.340210\pi$
$929$	27249.4	0.962352	0.481176	+	0.876624i	$0.340210\pi$
$930$	0	0
$931$	64948.4	2.28636
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−53131.8	−1.85839
$936$	0	0
$937$	−42035.0	−1.46555	−0.732777	−	0.680469i	$-0.761776\pi$
$937$	−42035.0	−1.46555	−0.732777	+	0.680469i	$0.761776\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30484.4	1.05607	0.528036	−	0.849222i	$-0.322929\pi$
$941$	30484.4	1.05607	0.528036	+	0.849222i	$0.322929\pi$
$942$	0	0
$943$	−9287.99	−0.320741
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29556.7	−1.01422	−0.507109	−	0.861882i	$-0.669286\pi$
$947$	−29556.7	−1.01422	−0.507109	+	0.861882i	$0.669286\pi$
$948$	0	0
$949$	3304.42	0.113031
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7163.52	0.243493	0.121747	−	0.992561i	$-0.461150\pi$
$953$	7163.52	0.243493	0.121747	+	0.992561i	$0.461150\pi$
$954$	0	0
$955$	2755.58	0.0933701
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42793.4	1.44095
$960$	0	0
$961$	79445.5	2.66676
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−68957.1	−2.30032
$966$	0	0
$967$	44103.3	1.46667	0.733333	−	0.679869i	$-0.237963\pi$
$967$	44103.3	1.46667	0.733333	+	0.679869i	$0.237963\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1860.63	0.0614937	0.0307469	−	0.999527i	$-0.490211\pi$
$971$	1860.63	0.0614937	0.0307469	+	0.999527i	$0.490211\pi$
$972$	0	0
$973$	40706.3	1.34120
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2851.14	−0.0933634	−0.0466817	−	0.998910i	$-0.514865\pi$
$977$	−2851.14	−0.0933634	−0.0466817	+	0.998910i	$0.514865\pi$
$978$	0	0
$979$	−32547.9	−1.06255
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3126.22	−0.101435	−0.0507176	−	0.998713i	$-0.516151\pi$
$983$	−3126.22	−0.101435	−0.0507176	+	0.998713i	$0.516151\pi$
$984$	0	0
$985$	−19750.1	−0.638874
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6664.73	−0.214283
$990$	0	0
$991$	−27509.6	−0.881808	−0.440904	−	0.897554i	$-0.645342\pi$
$991$	−27509.6	−0.881808	−0.440904	+	0.897554i	$0.645342\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	49722.8	1.58424
$996$	0	0
$997$	−9754.21	−0.309848	−0.154924	−	0.987926i	$-0.549513\pi$
$997$	−9754.21	−0.309848	−0.154924	+	0.987926i	$0.549513\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	828.4.a.f.1.3		3
3.2	odd	2		92.4.a.a.1.3	✓	3
12.11	even	2		368.4.a.k.1.1		3
15.2	even	4		2300.4.c.b.1749.2		6
15.8	even	4		2300.4.c.b.1749.5		6
15.14	odd	2		2300.4.a.b.1.1		3
24.5	odd	2		1472.4.a.w.1.1		3
24.11	even	2		1472.4.a.p.1.3		3
69.68	even	2		2116.4.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.a.a.1.3	✓	3	3.2	odd	2
368.4.a.k.1.1		3	12.11	even	2
828.4.a.f.1.3		3	1.1	even	1	trivial
1472.4.a.p.1.3		3	24.11	even	2
1472.4.a.w.1.1		3	24.5	odd	2
2116.4.a.a.1.3		3	69.68	even	2
2300.4.a.b.1.1		3	15.14	odd	2
2300.4.c.b.1749.2		6	15.2	even	4
2300.4.c.b.1749.5		6	15.8	even	4

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	828.a (trivial)

\(f(q)\)	\(=\)	\(q+16.0718 q^{5} -35.2976 q^{7} -35.8117 q^{11} +26.4662 q^{13} +92.3131 q^{17} +71.9314 q^{19} +23.0000 q^{23} +133.304 q^{25} -131.757 q^{29} -330.509 q^{31} -567.297 q^{35} -55.6698 q^{37} -403.826 q^{41} -289.771 q^{43} +141.061 q^{47} +902.922 q^{49} -0.542497 q^{53} -575.560 q^{55} +241.813 q^{59} -779.976 q^{61} +425.360 q^{65} -324.411 q^{67} -408.286 q^{71} +124.854 q^{73} +1264.07 q^{77} -896.602 q^{79} -508.623 q^{83} +1483.64 q^{85} +908.860 q^{89} -934.194 q^{91} +1156.07 q^{95} -549.593 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 10 q^{5} - 46 q^{7} + 64 q^{11} - 44 q^{13} + 88 q^{17} - 94 q^{19} + 69 q^{23} + 181 q^{25} - 308 q^{29} - 140 q^{31} - 192 q^{35} + 26 q^{37} - 584 q^{41} - 478 q^{43} + 28 q^{47} + 695 q^{49} - 356 q^{53}+ \cdots + 736 q^{97}+O(q^{100}) \) 3 * q + 10 * q^5 - 46 * q^7 + 64 * q^11 - 44 * q^13 + 88 * q^17 - 94 * q^19 + 69 * q^23 + 181 * q^25 - 308 * q^29 - 140 * q^31 - 192 * q^35 + 26 * q^37 - 584 * q^41 - 478 * q^43 + 28 * q^47 + 695 * q^49 - 356 * q^53 - 1004 * q^55 - 144 * q^59 - 1052 * q^61 - 150 * q^65 - 2008 * q^67 - 360 * q^71 - 252 * q^73 + 576 * q^77 - 720 * q^79 + 1404 * q^83 - 1632 * q^85 - 534 * q^89 - 1526 * q^91 + 108 * q^95 + 736 * q^97

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