LMFDB - Embedded newform 4356.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4356,2,Mod(1,4356)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4356, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4356.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4356.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:	$34.7828351205$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 55 $ x^4 - 15*x^2 + 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 396)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.52626$ of defining polynomial
Character	$\chi$	$=$	4356.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52626 q^{5} -1.00000 q^{7} +0.236068 q^{13} +6.61382 q^{17} -3.85410 q^{19} +0.964944 q^{23} +1.38197 q^{25} -3.12262 q^{29} +6.85410 q^{31} +2.52626 q^{35} -6.70820 q^{37} +9.14008 q^{41} -3.00000 q^{43} +9.73645 q^{47} -6.00000 q^{49} -1.56131 q^{53} -10.7014 q^{59} -0.909830 q^{61} -0.596368 q^{65} +4.32624 q^{67} +10.7014 q^{71} -14.4721 q^{73} +0.708204 q^{79} -6.01745 q^{83} -16.7082 q^{85} +7.21019 q^{89} -0.236068 q^{91} +9.73645 q^{95} +11.5623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 8 q^{13} - 2 q^{19} + 10 q^{25} + 14 q^{31} - 12 q^{43} - 24 q^{49} - 26 q^{61} - 14 q^{67} - 40 q^{73} - 24 q^{79} - 40 q^{85} + 8 q^{91} + 6 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 8 * q^13 - 2 * q^19 + 10 * q^25 + 14 * q^31 - 12 * q^43 - 24 * q^49 - 26 * q^61 - 14 * q^67 - 40 * q^73 - 24 * q^79 - 40 * q^85 + 8 * q^91 + 6 * 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
$5$	−2.52626	−1.12978	−0.564888	−	0.825168i	$-0.691081\pi$
$5$	−2.52626	−1.12978	−0.564888	+	0.825168i	$0.691081\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.61382	1.60409	0.802044	−	0.597265i	$-0.203746\pi$
$17$	6.61382	1.60409	0.802044	+	0.597265i	$0.203746\pi$
$18$	0	0
$19$	−3.85410	−0.884192	−0.442096	−	0.896968i	$-0.645765\pi$
$19$	−3.85410	−0.884192	−0.442096	+	0.896968i	$0.645765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.964944	0.201205	0.100602	−	0.994927i	$-0.467923\pi$
$23$	0.964944	0.201205	0.100602	+	0.994927i	$0.467923\pi$
$24$	0	0
$25$	1.38197	0.276393
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.12262	−0.579857	−0.289928	−	0.957048i	$-0.593631\pi$
$29$	−3.12262	−0.579857	−0.289928	+	0.957048i	$0.593631\pi$
$30$	0	0
$31$	6.85410	1.23103	0.615517	−	0.788124i	$-0.288947\pi$
$31$	6.85410	1.23103	0.615517	+	0.788124i	$0.288947\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.52626	0.427015
$36$	0	0
$37$	−6.70820	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$37$	−6.70820	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.14008	1.42744	0.713720	−	0.700431i	$-0.247009\pi$
$41$	9.14008	1.42744	0.713720	+	0.700431i	$0.247009\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.73645	1.42021	0.710103	−	0.704098i	$-0.248648\pi$
$47$	9.73645	1.42021	0.710103	+	0.704098i	$0.248648\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.56131	−0.214463	−0.107231	−	0.994234i	$-0.534199\pi$
$53$	−1.56131	−0.214463	−0.107231	+	0.994234i	$0.534199\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.7014	−1.39320	−0.696601	−	0.717459i	$-0.745305\pi$
$59$	−10.7014	−1.39320	−0.696601	+	0.717459i	$0.745305\pi$
$60$	0	0
$61$	−0.909830	−0.116492	−0.0582459	−	0.998302i	$-0.518551\pi$
$61$	−0.909830	−0.116492	−0.0582459	+	0.998302i	$0.518551\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.596368	−0.0739703
$66$	0	0
$67$	4.32624	0.528534	0.264267	−	0.964450i	$-0.414870\pi$
$67$	4.32624	0.528534	0.264267	+	0.964450i	$0.414870\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.7014	1.27002	0.635011	−	0.772503i	$-0.280996\pi$
$71$	10.7014	1.27002	0.635011	+	0.772503i	$0.280996\pi$
$72$	0	0
$73$	−14.4721	−1.69384	−0.846918	−	0.531724i	$-0.821544\pi$
$73$	−14.4721	−1.69384	−0.846918	+	0.531724i	$0.821544\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.708204	0.0796792	0.0398396	−	0.999206i	$-0.487315\pi$
$79$	0.708204	0.0796792	0.0398396	+	0.999206i	$0.487315\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.01745	−0.660501	−0.330251	−	0.943893i	$-0.607133\pi$
$83$	−6.01745	−0.660501	−0.330251	+	0.943893i	$0.607133\pi$
$84$	0	0
$85$	−16.7082	−1.81226
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.21019	0.764279	0.382139	−	0.924105i	$-0.375187\pi$
$89$	7.21019	0.764279	0.382139	+	0.924105i	$0.375187\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.73645	0.998938
$96$	0	0
$97$	11.5623	1.17397	0.586987	−	0.809596i	$-0.300314\pi$
$97$	11.5623	1.17397	0.586987	+	0.809596i	$0.300314\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.21019	0.717441	0.358720	−	0.933445i	$-0.383213\pi$
$101$	7.21019	0.717441	0.358720	+	0.933445i	$0.383213\pi$
$102$	0	0
$103$	−13.4164	−1.32196	−0.660979	−	0.750404i	$-0.729859\pi$
$103$	−13.4164	−1.32196	−0.660979	+	0.750404i	$0.729859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.2627	−1.18548	−0.592740	−	0.805394i	$-0.701954\pi$
$107$	−12.2627	−1.18548	−0.592740	+	0.805394i	$0.701954\pi$
$108$	0	0
$109$	−18.1803	−1.74136	−0.870680	−	0.491849i	$-0.836321\pi$
$109$	−18.1803	−1.74136	−0.870680	+	0.491849i	$0.836321\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.2627	−1.15358	−0.576789	−	0.816893i	$-0.695694\pi$
$113$	−12.2627	−1.15358	−0.576789	+	0.816893i	$0.695694\pi$
$114$	0	0
$115$	−2.43769	−0.227316
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.61382	−0.606288
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.14008	0.817513
$126$	0	0
$127$	16.1803	1.43577	0.717886	−	0.696160i	$-0.245110\pi$
$127$	16.1803	1.43577	0.717886	+	0.696160i	$0.245110\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.54371	−0.746467	−0.373234	−	0.927737i	$-0.621751\pi$
$131$	−8.54371	−0.746467	−0.373234	+	0.927737i	$0.621751\pi$
$132$	0	0
$133$	3.85410	0.334193
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.3853	−1.31446	−0.657228	−	0.753691i	$-0.728271\pi$
$137$	−15.3853	−1.31446	−0.657228	+	0.753691i	$0.728271\pi$
$138$	0	0
$139$	−13.1459	−1.11502	−0.557510	−	0.830170i	$-0.688243\pi$
$139$	−13.1459	−1.11502	−0.557510	+	0.830170i	$0.688243\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.88854	0.655108
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.3328	−0.846497	−0.423249	−	0.906014i	$-0.639110\pi$
$149$	−10.3328	−0.846497	−0.423249	+	0.906014i	$0.639110\pi$
$150$	0	0
$151$	−21.1246	−1.71910	−0.859548	−	0.511055i	$-0.829255\pi$
$151$	−21.1246	−1.71910	−0.859548	+	0.511055i	$0.829255\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−17.3152	−1.39079
$156$	0	0
$157$	2.29180	0.182905	0.0914526	−	0.995809i	$-0.470849\pi$
$157$	2.29180	0.182905	0.0914526	+	0.995809i	$0.470849\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.964944	−0.0760482
$162$	0	0
$163$	14.3820	1.12648	0.563241	−	0.826292i	$-0.309554\pi$
$163$	14.3820	1.12648	0.563241	+	0.826292i	$0.309554\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.9991	−1.70235	−0.851173	−	0.524886i	$-0.824108\pi$
$167$	−21.9991	−1.70235	−0.851173	+	0.524886i	$0.824108\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.50865	−0.722930	−0.361465	−	0.932386i	$-0.617723\pi$
$173$	−9.50865	−0.722930	−0.361465	+	0.932386i	$0.617723\pi$
$174$	0	0
$175$	−1.38197	−0.104467
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.3853	−1.14995	−0.574977	−	0.818170i	$-0.694989\pi$
$179$	−15.3853	−1.14995	−0.574977	+	0.818170i	$0.694989\pi$
$180$	0	0
$181$	−16.2361	−1.20682	−0.603409	−	0.797432i	$-0.706191\pi$
$181$	−16.2361	−1.20682	−0.603409	+	0.797432i	$0.706191\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9466	1.24594
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.5605	1.70477	0.852387	−	0.522911i	$-0.175154\pi$
$191$	23.5605	1.70477	0.852387	+	0.522911i	$0.175154\pi$
$192$	0	0
$193$	−11.1459	−0.802299	−0.401150	−	0.916013i	$-0.631389\pi$
$193$	−11.1459	−0.802299	−0.401150	+	0.916013i	$0.631389\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.0867	1.85860	0.929301	−	0.369324i	$-0.120411\pi$
$197$	26.0867	1.85860	0.929301	+	0.369324i	$0.120411\pi$
$198$	0	0
$199$	4.23607	0.300287	0.150143	−	0.988664i	$-0.452026\pi$
$199$	4.23607	0.300287	0.150143	+	0.988664i	$0.452026\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.12262	0.219165
$204$	0	0
$205$	−23.0902	−1.61269
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.3820	−0.714724	−0.357362	−	0.933966i	$-0.616324\pi$
$211$	−10.3820	−0.714724	−0.357362	+	0.933966i	$0.616324\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.57877	0.516868
$216$	0	0
$217$	−6.85410	−0.465287
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.56131	0.105025
$222$	0	0
$223$	−6.41641	−0.429674	−0.214837	−	0.976650i	$-0.568922\pi$
$223$	−6.41641	−0.429674	−0.214837	+	0.976650i	$0.568922\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.2976	−1.61269	−0.806344	−	0.591446i	$-0.798557\pi$
$227$	−24.2976	−1.61269	−0.806344	+	0.591446i	$0.798557\pi$
$228$	0	0
$229$	11.4164	0.754417	0.377209	−	0.926128i	$-0.376884\pi$
$229$	11.4164	0.754417	0.377209	+	0.926128i	$0.376884\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.64888	−0.370070	−0.185035	−	0.982732i	$-0.559240\pi$
$233$	−5.64888	−0.370070	−0.185035	+	0.982732i	$0.559240\pi$
$234$	0	0
$235$	−24.5967	−1.60451
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.68393	0.302979	0.151489	−	0.988459i	$-0.451593\pi$
$239$	4.68393	0.302979	0.151489	+	0.988459i	$0.451593\pi$
$240$	0	0
$241$	−22.9443	−1.47797	−0.738985	−	0.673722i	$-0.764695\pi$
$241$	−22.9443	−1.47797	−0.738985	+	0.673722i	$0.764695\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.1575	0.968379
$246$	0	0
$247$	−0.909830	−0.0578911
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.7188	1.05528	0.527642	−	0.849467i	$-0.323076\pi$
$251$	16.7188	1.05528	0.527642	+	0.849467i	$0.323076\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.45614	0.277967	0.138983	−	0.990295i	$-0.455617\pi$
$257$	4.45614	0.277967	0.138983	+	0.990295i	$0.455617\pi$
$258$	0	0
$259$	6.70820	0.416828
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.7014	−0.659876	−0.329938	−	0.944003i	$-0.607028\pi$
$263$	−10.7014	−0.659876	−0.329938	+	0.944003i	$0.607028\pi$
$264$	0	0
$265$	3.94427	0.242295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.2978	0.688836	0.344418	−	0.938816i	$-0.388076\pi$
$269$	11.2978	0.688836	0.344418	+	0.938816i	$0.388076\pi$
$270$	0	0
$271$	2.14590	0.130354	0.0651770	−	0.997874i	$-0.479239\pi$
$271$	2.14590	0.130354	0.0651770	+	0.997874i	$0.479239\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.27051	0.496927	0.248463	−	0.968641i	$-0.420074\pi$
$277$	8.27051	0.496927	0.248463	+	0.968641i	$0.420074\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.2802	−1.09050	−0.545251	−	0.838273i	$-0.683566\pi$
$281$	−18.2802	−1.09050	−0.545251	+	0.838273i	$0.683566\pi$
$282$	0	0
$283$	−25.4164	−1.51085	−0.755424	−	0.655236i	$-0.772569\pi$
$283$	−25.4164	−1.51085	−0.755424	+	0.655236i	$0.772569\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.14008	−0.539522
$288$	0	0
$289$	26.7426	1.57310
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.14008	0.533969	0.266985	−	0.963701i	$-0.413973\pi$
$293$	9.14008	0.533969	0.266985	+	0.963701i	$0.413973\pi$
$294$	0	0
$295$	27.0344	1.57401
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.227792	0.0131736
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.29846	0.131610
$306$	0	0
$307$	14.0902	0.804168	0.402084	−	0.915603i	$-0.368286\pi$
$307$	14.0902	0.804168	0.402084	+	0.915603i	$0.368286\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.01745	−0.341219	−0.170609	−	0.985339i	$-0.554574\pi$
$311$	−6.01745	−0.341219	−0.170609	+	0.985339i	$0.554574\pi$
$312$	0	0
$313$	−13.1803	−0.744997	−0.372498	−	0.928033i	$-0.621499\pi$
$313$	−13.1803	−0.744997	−0.372498	+	0.928033i	$0.621499\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.4202	−1.54007	−0.770037	−	0.637999i	$-0.779762\pi$
$317$	−27.4202	−1.54007	−0.770037	+	0.637999i	$0.779762\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−25.4903	−1.41832
$324$	0	0
$325$	0.326238	0.0180964
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.73645	−0.536788
$330$	0	0
$331$	10.4164	0.572538	0.286269	−	0.958149i	$-0.407585\pi$
$331$	10.4164	0.572538	0.286269	+	0.958149i	$0.407585\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.9292	−0.597125
$336$	0	0
$337$	−13.2361	−0.721015	−0.360507	−	0.932756i	$-0.617396\pi$
$337$	−13.2361	−0.721015	−0.360507	+	0.932756i	$0.617396\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.4204	0.774127	0.387063	−	0.922053i	$-0.373489\pi$
$347$	14.4204	0.774127	0.387063	+	0.922053i	$0.373489\pi$
$348$	0	0
$349$	−11.7639	−0.629709	−0.314854	−	0.949140i	$-0.601956\pi$
$349$	−11.7639	−0.629709	−0.314854	+	0.949140i	$0.601956\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.2802	0.972954	0.486477	−	0.873693i	$-0.338282\pi$
$353$	18.2802	0.972954	0.486477	+	0.873693i	$0.338282\pi$
$354$	0	0
$355$	−27.0344	−1.43484
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.3151	1.59997	0.799984	−	0.600021i	$-0.204841\pi$
$359$	30.3151	1.59997	0.799984	+	0.600021i	$0.204841\pi$
$360$	0	0
$361$	−4.14590	−0.218205
$362$	0	0
$363$	0	0
$364$	0	0
$365$	36.5603	1.91365
$366$	0	0
$367$	−1.56231	−0.0815517	−0.0407759	−	0.999168i	$-0.512983\pi$
$367$	−1.56231	−0.0815517	−0.0407759	+	0.999168i	$0.512983\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.56131	0.0810593
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.737151	−0.0379652
$378$	0	0
$379$	19.8328	1.01874	0.509372	−	0.860547i	$-0.329878\pi$
$379$	19.8328	1.01874	0.509372	+	0.860547i	$0.329878\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.5079	−0.945712	−0.472856	−	0.881140i	$-0.656777\pi$
$383$	−18.5079	−0.945712	−0.472856	+	0.881140i	$0.656777\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.8940	−1.26217	−0.631087	−	0.775712i	$-0.717391\pi$
$389$	−24.8940	−1.26217	−0.631087	+	0.775712i	$0.717391\pi$
$390$	0	0
$391$	6.38197	0.322750
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.78910	−0.0900196
$396$	0	0
$397$	21.0000	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.7890	0.738525	0.369263	−	0.929325i	$-0.379610\pi$
$401$	14.7890	0.738525	0.369263	+	0.929325i	$0.379610\pi$
$402$	0	0
$403$	1.61803	0.0806000
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.7014	0.526581
$414$	0	0
$415$	15.2016	0.746218
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.1042	1.56839	0.784196	−	0.620514i	$-0.213076\pi$
$419$	32.1042	1.56839	0.784196	+	0.620514i	$0.213076\pi$
$420$	0	0
$421$	13.8541	0.675208	0.337604	−	0.941288i	$-0.390384\pi$
$421$	13.8541	0.675208	0.337604	+	0.941288i	$0.390384\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.14008	0.443359
$426$	0	0
$427$	0.909830	0.0440298
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.54371	0.411536	0.205768	−	0.978601i	$-0.434031\pi$
$431$	8.54371	0.411536	0.205768	+	0.978601i	$0.434031\pi$
$432$	0	0
$433$	−10.7639	−0.517282	−0.258641	−	0.965974i	$-0.583275\pi$
$433$	−10.7639	−0.517282	−0.258641	+	0.965974i	$0.583275\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.71899	−0.177903
$438$	0	0
$439$	14.7082	0.701984	0.350992	−	0.936378i	$-0.385844\pi$
$439$	14.7082	0.701984	0.350992	+	0.936378i	$0.385844\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.5254	1.16524	0.582619	−	0.812746i	$-0.302028\pi$
$443$	24.5254	1.16524	0.582619	+	0.812746i	$0.302028\pi$
$444$	0	0
$445$	−18.2148	−0.863463
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.36787	−0.442097	−0.221049	−	0.975263i	$-0.570948\pi$
$449$	−9.36787	−0.442097	−0.221049	+	0.975263i	$0.570948\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.596368	0.0279582
$456$	0	0
$457$	−24.2705	−1.13533	−0.567663	−	0.823261i	$-0.692152\pi$
$457$	−24.2705	−1.13533	−0.567663	+	0.823261i	$0.692152\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−27.0517	−1.25992	−0.629961	−	0.776627i	$-0.716929\pi$
$461$	−27.0517	−1.25992	−0.629961	+	0.776627i	$0.716929\pi$
$462$	0	0
$463$	−32.5623	−1.51330	−0.756649	−	0.653821i	$-0.773165\pi$
$463$	−32.5623	−1.51330	−0.756649	+	0.653821i	$0.773165\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.9999	−0.601561	−0.300781	−	0.953693i	$-0.597247\pi$
$467$	−12.9999	−0.601561	−0.300781	+	0.953693i	$0.597247\pi$
$468$	0	0
$469$	−4.32624	−0.199767
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.32624	−0.244385
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.56131	−0.0713382	−0.0356691	−	0.999364i	$-0.511356\pi$
$479$	−1.56131	−0.0713382	−0.0356691	+	0.999364i	$0.511356\pi$
$480$	0	0
$481$	−1.58359	−0.0722056
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−29.2093	−1.32633
$486$	0	0
$487$	1.76393	0.0799314	0.0399657	−	0.999201i	$-0.487275\pi$
$487$	1.76393	0.0799314	0.0399657	+	0.999201i	$0.487275\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.2093	−1.31820	−0.659099	−	0.752056i	$-0.729062\pi$
$491$	−29.2093	−1.31820	−0.659099	+	0.752056i	$0.729062\pi$
$492$	0	0
$493$	−20.6525	−0.930141
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.7014	−0.480023
$498$	0	0
$499$	9.56231	0.428068	0.214034	−	0.976826i	$-0.431340\pi$
$499$	9.56231	0.428068	0.214034	+	0.976826i	$0.431340\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	44.7354	1.99465	0.997327	−	0.0730614i	$-0.0232769\pi$
$503$	44.7354	1.99465	0.997327	+	0.0730614i	$0.0232769\pi$
$504$	0	0
$505$	−18.2148	−0.810547
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.8415	0.879458	0.439729	−	0.898131i	$-0.355075\pi$
$509$	19.8415	0.879458	0.439729	+	0.898131i	$0.355075\pi$
$510$	0	0
$511$	14.4721	0.640210
$512$	0	0
$513$	0	0
$514$	0	0
$515$	33.8933	1.49352
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.2978	−0.494964	−0.247482	−	0.968893i	$-0.579603\pi$
$521$	−11.2978	−0.494964	−0.247482	+	0.968893i	$0.579603\pi$
$522$	0	0
$523$	29.0344	1.26959	0.634794	−	0.772682i	$-0.281085\pi$
$523$	29.0344	1.26959	0.634794	+	0.772682i	$0.281085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	45.3318	1.97468
$528$	0	0
$529$	−22.0689	−0.959517
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.15768	0.0934595
$534$	0	0
$535$	30.9787	1.33933
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	11.8541	0.509648	0.254824	−	0.966987i	$-0.417983\pi$
$541$	11.8541	0.509648	0.254824	+	0.966987i	$0.417983\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	45.9282	1.96735
$546$	0	0
$547$	25.3262	1.08287	0.541436	−	0.840742i	$-0.317881\pi$
$547$	25.3262	1.08287	0.541436	+	0.840742i	$0.317881\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0349	0.512704
$552$	0	0
$553$	−0.708204	−0.0301159
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.9815	−1.22799	−0.613994	−	0.789311i	$-0.710438\pi$
$557$	−28.9815	−1.22799	−0.613994	+	0.789311i	$0.710438\pi$
$558$	0	0
$559$	−0.708204	−0.0299538
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.227792	0.00960030	0.00480015	−	0.999988i	$-0.498472\pi$
$563$	0.227792	0.00960030	0.00480015	+	0.999988i	$0.498472\pi$
$564$	0	0
$565$	30.9787	1.30328
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.6656	0.866348	0.433174	−	0.901310i	$-0.357394\pi$
$569$	20.6656	0.866348	0.433174	+	0.901310i	$0.357394\pi$
$570$	0	0
$571$	25.9230	1.08484	0.542422	−	0.840106i	$-0.317508\pi$
$571$	25.9230	1.08484	0.542422	+	0.840106i	$0.317508\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.33352	0.0556116
$576$	0	0
$577$	−22.1803	−0.923380	−0.461690	−	0.887041i	$-0.652757\pi$
$577$	−22.1803	−0.923380	−0.461690	+	0.887041i	$0.652757\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.01745	0.249646
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0524	0.745101	0.372550	−	0.928012i	$-0.378483\pi$
$587$	18.0524	0.745101	0.372550	+	0.928012i	$0.378483\pi$
$588$	0	0
$589$	−26.4164	−1.08847
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.78910	0.0734697	0.0367348	−	0.999325i	$-0.488304\pi$
$593$	1.78910	0.0734697	0.0367348	+	0.999325i	$0.488304\pi$
$594$	0	0
$595$	16.7082	0.684970
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	12.4164	0.506476	0.253238	−	0.967404i	$-0.418504\pi$
$601$	12.4164	0.506476	0.253238	+	0.967404i	$0.418504\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.72949	0.232553	0.116276	−	0.993217i	$-0.462904\pi$
$607$	5.72949	0.232553	0.116276	+	0.993217i	$0.462904\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.29846	0.0929859
$612$	0	0
$613$	36.8328	1.48766	0.743832	−	0.668367i	$-0.233006\pi$
$613$	36.8328	1.48766	0.743832	+	0.668367i	$0.233006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.6655	−1.35532	−0.677661	−	0.735375i	$-0.737006\pi$
$617$	−33.6655	−1.35532	−0.677661	+	0.735375i	$0.737006\pi$
$618$	0	0
$619$	−35.7639	−1.43747	−0.718737	−	0.695282i	$-0.755279\pi$
$619$	−35.7639	−1.43747	−0.718737	+	0.695282i	$0.755279\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.21019	−0.288870
$624$	0	0
$625$	−30.0000	−1.20000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44.3669	−1.76902
$630$	0	0
$631$	−13.4377	−0.534946	−0.267473	−	0.963565i	$-0.586189\pi$
$631$	−13.4377	−0.534946	−0.267473	+	0.963565i	$0.586189\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−40.8757	−1.62210
$636$	0	0
$637$	−1.41641	−0.0561201
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.8765	−0.745578	−0.372789	−	0.927916i	$-0.621598\pi$
$641$	−18.8765	−0.745578	−0.372789	+	0.927916i	$0.621598\pi$
$642$	0	0
$643$	45.9787	1.81322	0.906612	−	0.421966i	$-0.138660\pi$
$643$	45.9787	1.81322	0.906612	+	0.421966i	$0.138660\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.368576	−0.0144902	−0.00724510	−	0.999974i	$-0.502306\pi$
$647$	−0.368576	−0.0144902	−0.00724510	+	0.999974i	$0.502306\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.2794	1.06753	0.533764	−	0.845634i	$-0.320777\pi$
$653$	27.2794	1.06753	0.533764	+	0.845634i	$0.320777\pi$
$654$	0	0
$655$	21.5836	0.843341
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0349	−0.468813	−0.234407	−	0.972139i	$-0.575315\pi$
$659$	−12.0349	−0.468813	−0.234407	+	0.972139i	$0.575315\pi$
$660$	0	0
$661$	−3.90983	−0.152075	−0.0760374	−	0.997105i	$-0.524227\pi$
$661$	−3.90983	−0.152075	−0.0760374	+	0.997105i	$0.524227\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.73645	−0.377563
$666$	0	0
$667$	−3.01316	−0.116670
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.1246	−0.390275	−0.195138	−	0.980776i	$-0.562515\pi$
$673$	−10.1246	−0.390275	−0.195138	+	0.980776i	$0.562515\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.1050	−0.388368	−0.194184	−	0.980965i	$-0.562206\pi$
$677$	−10.1050	−0.388368	−0.194184	+	0.980965i	$0.562206\pi$
$678$	0	0
$679$	−11.5623	−0.443721
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.5781	0.634342	0.317171	−	0.948368i	$-0.397267\pi$
$683$	16.5781	0.634342	0.317171	+	0.948368i	$0.397267\pi$
$684$	0	0
$685$	38.8673	1.48504
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.368576	−0.0140416
$690$	0	0
$691$	−33.3050	−1.26698	−0.633490	−	0.773751i	$-0.718378\pi$
$691$	−33.3050	−1.26698	−0.633490	+	0.773751i	$0.718378\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.2099	1.25972
$696$	0	0
$697$	60.4508	2.28974
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.7363	−0.858738	−0.429369	−	0.903129i	$-0.641264\pi$
$701$	−22.7363	−0.858738	−0.429369	+	0.903129i	$0.641264\pi$
$702$	0	0
$703$	25.8541	0.975106
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.21019	−0.271167
$708$	0	0
$709$	22.6738	0.851531	0.425766	−	0.904834i	$-0.360005\pi$
$709$	22.6738	0.851531	0.425766	+	0.904834i	$0.360005\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.61382	0.247690
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.2627	0.457322	0.228661	−	0.973506i	$-0.426565\pi$
$719$	12.2627	0.457322	0.228661	+	0.973506i	$0.426565\pi$
$720$	0	0
$721$	13.4164	0.499653
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.31536	−0.160268
$726$	0	0
$727$	−13.4377	−0.498376	−0.249188	−	0.968455i	$-0.580164\pi$
$727$	−13.4377	−0.498376	−0.249188	+	0.968455i	$0.580164\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19.8415	−0.733863
$732$	0	0
$733$	11.7082	0.432452	0.216226	−	0.976343i	$-0.430625\pi$
$733$	11.7082	0.432452	0.216226	+	0.976343i	$0.430625\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−15.8328	−0.582420	−0.291210	−	0.956659i	$-0.594058\pi$
$739$	−15.8328	−0.582420	−0.291210	+	0.956659i	$0.594058\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−51.2085	−1.87866	−0.939329	−	0.343019i	$-0.888551\pi$
$743$	−51.2085	−1.87866	−0.939329	+	0.343019i	$0.888551\pi$
$744$	0	0
$745$	26.1033	0.956352
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.2627	0.448069
$750$	0	0
$751$	−35.7426	−1.30427	−0.652134	−	0.758104i	$-0.726126\pi$
$751$	−35.7426	−1.30427	−0.652134	+	0.758104i	$0.726126\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	53.3662	1.94219
$756$	0	0
$757$	−33.4721	−1.21657	−0.608283	−	0.793720i	$-0.708141\pi$
$757$	−33.4721	−1.21657	−0.608283	+	0.793720i	$0.708141\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.1750	−0.767593	−0.383796	−	0.923418i	$-0.625384\pi$
$761$	−21.1750	−0.767593	−0.383796	+	0.923418i	$0.625384\pi$
$762$	0	0
$763$	18.1803	0.658172
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.52626	−0.0912178
$768$	0	0
$769$	−7.85410	−0.283226	−0.141613	−	0.989922i	$-0.545229\pi$
$769$	−7.85410	−0.283226	−0.141613	+	0.989922i	$0.545229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.7180	−1.39259	−0.696295	−	0.717756i	$-0.745169\pi$
$773$	−38.7180	−1.39259	−0.696295	+	0.717756i	$0.745169\pi$
$774$	0	0
$775$	9.47214	0.340249
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−35.2268	−1.26213
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.78966	−0.206642
$786$	0	0
$787$	−40.7214	−1.45156	−0.725780	−	0.687927i	$-0.758521\pi$
$787$	−40.7214	−1.45156	−0.725780	+	0.687927i	$0.758521\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.2627	0.436011
$792$	0	0
$793$	−0.214782	−0.00762712
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.66704	0.0944714	0.0472357	−	0.998884i	$-0.484959\pi$
$797$	2.66704	0.0944714	0.0472357	+	0.998884i	$0.484959\pi$
$798$	0	0
$799$	64.3951	2.27814
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	2.43769	0.0859174
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41.0165	−1.44206	−0.721031	−	0.692903i	$-0.756331\pi$
$809$	−41.0165	−1.44206	−0.721031	+	0.692903i	$0.756331\pi$
$810$	0	0
$811$	53.7984	1.88912	0.944558	−	0.328344i	$-0.106491\pi$
$811$	53.7984	1.88912	0.944558	+	0.328344i	$0.106491\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−36.3325	−1.27267
$816$	0	0
$817$	11.5623	0.404514
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.7881	1.28391	0.641957	−	0.766741i	$-0.278123\pi$
$821$	36.7881	1.28391	0.641957	+	0.766741i	$0.278123\pi$
$822$	0	0
$823$	23.6525	0.824473	0.412237	−	0.911077i	$-0.364748\pi$
$823$	23.6525	0.824473	0.412237	+	0.911077i	$0.364748\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.6480	0.961416	0.480708	−	0.876881i	$-0.340380\pi$
$827$	27.6480	0.961416	0.480708	+	0.876881i	$0.340380\pi$
$828$	0	0
$829$	−6.20163	−0.215391	−0.107696	−	0.994184i	$-0.534347\pi$
$829$	−6.20163	−0.215391	−0.107696	+	0.994184i	$0.534347\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−39.6829	−1.37493
$834$	0	0
$835$	55.5755	1.92327
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.7004	−1.57775	−0.788876	−	0.614552i	$-0.789337\pi$
$839$	−45.7004	−1.57775	−0.788876	+	0.614552i	$0.789337\pi$
$840$	0	0
$841$	−19.2492	−0.663766
$842$	0	0
$843$	0	0
$844$	0	0
$845$	32.7005	1.12493
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.47304	−0.221893
$852$	0	0
$853$	−33.1803	−1.13607	−0.568037	−	0.823003i	$-0.692297\pi$
$853$	−33.1803	−1.13607	−0.568037	+	0.823003i	$0.692297\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.2976	−0.829991	−0.414995	−	0.909824i	$-0.636217\pi$
$857$	−24.2976	−0.829991	−0.414995	+	0.909824i	$0.636217\pi$
$858$	0	0
$859$	−19.6525	−0.670534	−0.335267	−	0.942123i	$-0.608826\pi$
$859$	−19.6525	−0.670534	−0.335267	+	0.942123i	$0.608826\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.4201	1.37592	0.687958	−	0.725751i	$-0.258507\pi$
$863$	40.4201	1.37592	0.687958	+	0.725751i	$0.258507\pi$
$864$	0	0
$865$	24.0213	0.816748
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.02129	0.0346050
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.14008	−0.308991
$876$	0	0
$877$	34.7771	1.17434	0.587169	−	0.809464i	$-0.300242\pi$
$877$	34.7771	1.17434	0.587169	+	0.809464i	$0.300242\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.9817	0.538437	0.269218	−	0.963079i	$-0.413235\pi$
$881$	15.9817	0.538437	0.269218	+	0.963079i	$0.413235\pi$
$882$	0	0
$883$	14.8328	0.499164	0.249582	−	0.968354i	$-0.419707\pi$
$883$	14.8328	0.499164	0.249582	+	0.968354i	$0.419707\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.1926	−0.476540	−0.238270	−	0.971199i	$-0.576580\pi$
$887$	−14.1926	−0.476540	−0.238270	+	0.971199i	$0.576580\pi$
$888$	0	0
$889$	−16.1803	−0.542671
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−37.5253	−1.25573
$894$	0	0
$895$	38.8673	1.29919
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−21.4028	−0.713823
$900$	0	0
$901$	−10.3262	−0.344017
$902$	0	0
$903$	0	0
$904$	0	0
$905$	41.0165	1.36343
$906$	0	0
$907$	0.729490	0.0242223	0.0121112	−	0.999927i	$-0.496145\pi$
$907$	0.729490	0.0242223	0.0121112	+	0.999927i	$0.496145\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.6479	−1.34672	−0.673362	−	0.739313i	$-0.735150\pi$
$911$	−40.6479	−1.34672	−0.673362	+	0.739313i	$0.735150\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.54371	0.282138
$918$	0	0
$919$	53.8328	1.77578	0.887890	−	0.460055i	$-0.152170\pi$
$919$	53.8328	1.77578	0.887890	+	0.460055i	$0.152170\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.52626	0.0831527
$924$	0	0
$925$	−9.27051	−0.304812
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39.9107	1.30943	0.654714	−	0.755876i	$-0.272789\pi$
$929$	39.9107	1.30943	0.654714	+	0.755876i	$0.272789\pi$
$930$	0	0
$931$	23.1246	0.757879
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.1246	1.44149	0.720744	−	0.693201i	$-0.243800\pi$
$937$	44.1246	1.44149	0.720744	+	0.693201i	$0.243800\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.4035	−0.404342	−0.202171	−	0.979350i	$-0.564800\pi$
$941$	−12.4035	−0.404342	−0.202171	+	0.979350i	$0.564800\pi$
$942$	0	0
$943$	8.81966	0.287208
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.7014	0.347748	0.173874	−	0.984768i	$-0.444371\pi$
$947$	10.7014	0.347748	0.173874	+	0.984768i	$0.444371\pi$
$948$	0	0
$949$	−3.41641	−0.110901
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.5779	−0.958123	−0.479061	−	0.877781i	$-0.659023\pi$
$953$	−29.5779	−0.958123	−0.479061	+	0.877781i	$0.659023\pi$
$954$	0	0
$955$	−59.5197	−1.92601
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.3853	0.496818
$960$	0	0
$961$	15.9787	0.515442
$962$	0	0
$963$	0	0
$964$	0	0
$965$	28.1574	0.906418
$966$	0	0
$967$	57.5623	1.85108	0.925539	−	0.378651i	$-0.123612\pi$
$967$	57.5623	1.85108	0.925539	+	0.378651i	$0.123612\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.2626	0.810714	0.405357	−	0.914158i	$-0.367147\pi$
$971$	25.2626	0.810714	0.405357	+	0.914158i	$0.367147\pi$
$972$	0	0
$973$	13.1459	0.421438
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.2976	0.777349	0.388675	−	0.921375i	$-0.372933\pi$
$977$	24.2976	0.777349	0.388675	+	0.921375i	$0.372933\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.36787	−0.298789	−0.149394	−	0.988778i	$-0.547732\pi$
$983$	−9.36787	−0.298789	−0.149394	+	0.988778i	$0.547732\pi$
$984$	0	0
$985$	−65.9017	−2.09980
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.89483	−0.0920503
$990$	0	0
$991$	−20.3951	−0.647872	−0.323936	−	0.946079i	$-0.605006\pi$
$991$	−20.3951	−0.647872	−0.323936	+	0.946079i	$0.605006\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.7014	−0.339257
$996$	0	0
$997$	−12.0344	−0.381135	−0.190567	−	0.981674i	$-0.561033\pi$
$997$	−12.0344	−0.381135	−0.190567	+	0.981674i	$0.561033\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.2.a.x.1.2		4
3.2	odd	2	inner	4356.2.a.x.1.3		4
11.7	odd	10		396.2.j.d.181.1	✓	8
11.8	odd	10		396.2.j.d.361.1	yes	8
11.10	odd	2		4356.2.a.z.1.2		4
33.8	even	10		396.2.j.d.361.2	yes	8
33.29	even	10		396.2.j.d.181.2	yes	8
33.32	even	2		4356.2.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.2.j.d.181.1	✓	8	11.7	odd	10
396.2.j.d.181.2	yes	8	33.29	even	10
396.2.j.d.361.1	yes	8	11.8	odd	10
396.2.j.d.361.2	yes	8	33.8	even	10
4356.2.a.x.1.2		4	1.1	even	1	trivial
4356.2.a.x.1.3		4	3.2	odd	2	inner
4356.2.a.z.1.2		4	11.10	odd	2
4356.2.a.z.1.3		4	33.32	even	2

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:	\( N \)	\(=\)	\( 4356 = 2^{2} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4356.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52626 q^{5} -1.00000 q^{7} +0.236068 q^{13} +6.61382 q^{17} -3.85410 q^{19} +0.964944 q^{23} +1.38197 q^{25} -3.12262 q^{29} +6.85410 q^{31} +2.52626 q^{35} -6.70820 q^{37} +9.14008 q^{41} -3.00000 q^{43} +9.73645 q^{47} -6.00000 q^{49} -1.56131 q^{53} -10.7014 q^{59} -0.909830 q^{61} -0.596368 q^{65} +4.32624 q^{67} +10.7014 q^{71} -14.4721 q^{73} +0.708204 q^{79} -6.01745 q^{83} -16.7082 q^{85} +7.21019 q^{89} -0.236068 q^{91} +9.73645 q^{95} +11.5623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 8 q^{13} - 2 q^{19} + 10 q^{25} + 14 q^{31} - 12 q^{43} - 24 q^{49} - 26 q^{61} - 14 q^{67} - 40 q^{73} - 24 q^{79} - 40 q^{85} + 8 q^{91} + 6 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 8 * q^13 - 2 * q^19 + 10 * q^25 + 14 * q^31 - 12 * q^43 - 24 * q^49 - 26 * q^61 - 14 * q^67 - 40 * q^73 - 24 * q^79 - 40 * q^85 + 8 * q^91 + 6 * q^97

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