LMFDB - Embedded newform 8496.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.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:	$67.8409015573$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 354)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14637 q^{5} -1.19656 q^{7} +O(q^{10})$	$q-1.14637 q^{5} -1.19656 q^{7} -6.17513 q^{11} +2.34292 q^{13} -1.48929 q^{17} +4.97858 q^{19} +0.685846 q^{23} -3.68585 q^{25} -1.83221 q^{29} -3.83221 q^{31} +1.37169 q^{35} +6.92839 q^{37} +10.8610 q^{41} -4.17513 q^{43} -11.6644 q^{47} -5.56825 q^{49} -2.51806 q^{53} +7.07896 q^{55} +1.00000 q^{59} -6.41767 q^{61} -2.68585 q^{65} -11.6644 q^{67} -9.02877 q^{71} +13.7648 q^{73} +7.38890 q^{77} +9.78202 q^{79} -5.78202 q^{83} +1.70727 q^{85} -2.87819 q^{89} -2.80344 q^{91} -5.70727 q^{95} +5.31415 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} + q^{7}+O(q^{10}) $ 3 * q - 2 * q^5 + q^7	$ 3 q - 2 q^{5} + q^{7} + q^{11} + q^{13} + 3 q^{17} - 10 q^{23} + q^{25} + 8 q^{29} + 2 q^{31} - 20 q^{35} + 9 q^{37} + q^{41} + 7 q^{43} - 8 q^{47} + 12 q^{49} + 18 q^{53} + 3 q^{59} + 4 q^{65} - 8 q^{67} - 9 q^{71} + 8 q^{73} + 21 q^{77} + 19 q^{79} - 7 q^{83} + 8 q^{85} - 13 q^{91} - 20 q^{95} + 28 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 + q^7 + q^11 + q^13 + 3 * q^17 - 10 * q^23 + q^25 + 8 * q^29 + 2 * q^31 - 20 * q^35 + 9 * q^37 + q^41 + 7 * q^43 - 8 * q^47 + 12 * q^49 + 18 * q^53 + 3 * q^59 + 4 * q^65 - 8 * q^67 - 9 * q^71 + 8 * q^73 + 21 * q^77 + 19 * q^79 - 7 * q^83 + 8 * q^85 - 13 * q^91 - 20 * q^95 + 28 * 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
$5$	−1.14637	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$5$	−1.14637	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$6$	0	0
$7$	−1.19656	−0.452256	−0.226128	−	0.974098i	$-0.572607\pi$
$7$	−1.19656	−0.452256	−0.226128	+	0.974098i	$0.572607\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.17513	−1.86187	−0.930937	−	0.365181i	$-0.881007\pi$
$11$	−6.17513	−1.86187	−0.930937	+	0.365181i	$0.881007\pi$
$12$	0	0
$13$	2.34292	0.649810	0.324905	−	0.945747i	$-0.394668\pi$
$13$	2.34292	0.649810	0.324905	+	0.945747i	$0.394668\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.48929	−0.361206	−0.180603	−	0.983556i	$-0.557805\pi$
$17$	−1.48929	−0.361206	−0.180603	+	0.983556i	$0.557805\pi$
$18$	0	0
$19$	4.97858	1.14216	0.571082	−	0.820893i	$-0.306524\pi$
$19$	4.97858	1.14216	0.571082	+	0.820893i	$0.306524\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.685846	0.143009	0.0715044	−	0.997440i	$-0.477220\pi$
$23$	0.685846	0.143009	0.0715044	+	0.997440i	$0.477220\pi$
$24$	0	0
$25$	−3.68585	−0.737169
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.83221	−0.340233	−0.170117	−	0.985424i	$-0.554414\pi$
$29$	−1.83221	−0.340233	−0.170117	+	0.985424i	$0.554414\pi$
$30$	0	0
$31$	−3.83221	−0.688286	−0.344143	−	0.938917i	$-0.611830\pi$
$31$	−3.83221	−0.688286	−0.344143	+	0.938917i	$0.611830\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.37169	0.231858
$36$	0	0
$37$	6.92839	1.13902	0.569510	−	0.821985i	$-0.307133\pi$
$37$	6.92839	1.13902	0.569510	+	0.821985i	$0.307133\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.8610	1.69620	0.848100	−	0.529836i	$-0.177747\pi$
$41$	10.8610	1.69620	0.848100	+	0.529836i	$0.177747\pi$
$42$	0	0
$43$	−4.17513	−0.636702	−0.318351	−	0.947973i	$-0.603129\pi$
$43$	−4.17513	−0.636702	−0.318351	+	0.947973i	$0.603129\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.6644	−1.70143	−0.850716	−	0.525626i	$-0.823831\pi$
$47$	−11.6644	−1.70143	−0.850716	+	0.525626i	$0.823831\pi$
$48$	0	0
$49$	−5.56825	−0.795464
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.51806	−0.345882	−0.172941	−	0.984932i	$-0.555327\pi$
$53$	−2.51806	−0.345882	−0.172941	+	0.984932i	$0.555327\pi$
$54$	0	0
$55$	7.07896	0.954527
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−6.41767	−0.821699	−0.410849	−	0.911703i	$-0.634768\pi$
$61$	−6.41767	−0.821699	−0.410849	+	0.911703i	$0.634768\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.68585	−0.333138
$66$	0	0
$67$	−11.6644	−1.42504	−0.712518	−	0.701654i	$-0.752445\pi$
$67$	−11.6644	−1.42504	−0.712518	+	0.701654i	$0.752445\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.02877	−1.07152	−0.535759	−	0.844371i	$-0.679974\pi$
$71$	−9.02877	−1.07152	−0.535759	+	0.844371i	$0.679974\pi$
$72$	0	0
$73$	13.7648	1.61105	0.805524	−	0.592563i	$-0.201884\pi$
$73$	13.7648	1.61105	0.805524	+	0.592563i	$0.201884\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.38890	0.842044
$78$	0	0
$79$	9.78202	1.10056	0.550282	−	0.834979i	$-0.314520\pi$
$79$	9.78202	1.10056	0.550282	+	0.834979i	$0.314520\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.78202	−0.634659	−0.317330	−	0.948315i	$-0.602786\pi$
$83$	−5.78202	−0.634659	−0.317330	+	0.948315i	$0.602786\pi$
$84$	0	0
$85$	1.70727	0.185179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.87819	−0.305088	−0.152544	−	0.988297i	$-0.548747\pi$
$89$	−2.87819	−0.305088	−0.152544	+	0.988297i	$0.548747\pi$
$90$	0	0
$91$	−2.80344	−0.293881
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.70727	−0.585553
$96$	0	0
$97$	5.31415	0.539571	0.269785	−	0.962921i	$-0.413047\pi$
$97$	5.31415	0.539571	0.269785	+	0.962921i	$0.413047\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.51071	−0.249825	−0.124913	−	0.992168i	$-0.539865\pi$
$101$	−2.51071	−0.249825	−0.124913	+	0.992168i	$0.539865\pi$
$102$	0	0
$103$	3.24675	0.319912	0.159956	−	0.987124i	$-0.448865\pi$
$103$	3.24675	0.319912	0.159956	+	0.987124i	$0.448865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.10038	−0.396399	−0.198200	−	0.980162i	$-0.563509\pi$
$107$	−4.10038	−0.396399	−0.198200	+	0.980162i	$0.563509\pi$
$108$	0	0
$109$	2.75325	0.263714	0.131857	−	0.991269i	$-0.457906\pi$
$109$	2.75325	0.263714	0.131857	+	0.991269i	$0.457906\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−0.786230	−0.0733164
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.78202	0.163357
$120$	0	0
$121$	27.1323	2.46657
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.95715	0.890595
$126$	0	0
$127$	−2.29273	−0.203447	−0.101723	−	0.994813i	$-0.532436\pi$
$127$	−2.29273	−0.203447	−0.101723	+	0.994813i	$0.532436\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.7648	1.20264	0.601318	−	0.799009i	$-0.294642\pi$
$131$	13.7648	1.20264	0.601318	+	0.799009i	$0.294642\pi$
$132$	0	0
$133$	−5.95715	−0.516551
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2969	−0.965163	−0.482581	−	0.875851i	$-0.660301\pi$
$137$	−11.2969	−0.965163	−0.482581	+	0.875851i	$0.660301\pi$
$138$	0	0
$139$	0.978577	0.0830018	0.0415009	−	0.999138i	$-0.486786\pi$
$139$	0.978577	0.0830018	0.0415009	+	0.999138i	$0.486786\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−14.4679	−1.20986
$144$	0	0
$145$	2.10038	0.174427
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.15371	0.749901	0.374951	−	0.927045i	$-0.377660\pi$
$149$	9.15371	0.749901	0.374951	+	0.927045i	$0.377660\pi$
$150$	0	0
$151$	−2.02456	−0.164756	−0.0823781	−	0.996601i	$-0.526252\pi$
$151$	−2.02456	−0.164756	−0.0823781	+	0.996601i	$0.526252\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.39312	0.352864
$156$	0	0
$157$	23.5395	1.87866	0.939328	−	0.343022i	$-0.111450\pi$
$157$	23.5395	1.87866	0.939328	+	0.343022i	$0.111450\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.820654	−0.0646766
$162$	0	0
$163$	13.3288	1.04400	0.521998	−	0.852947i	$-0.325187\pi$
$163$	13.3288	1.04400	0.521998	+	0.852947i	$0.325187\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.1249	1.09302	0.546510	−	0.837452i	$-0.315956\pi$
$167$	14.1249	1.09302	0.546510	+	0.837452i	$0.315956\pi$
$168$	0	0
$169$	−7.51071	−0.577747
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−25.4036	−1.93140	−0.965700	−	0.259661i	$-0.916389\pi$
$173$	−25.4036	−1.93140	−0.965700	+	0.259661i	$0.916389\pi$
$174$	0	0
$175$	4.41033	0.333389
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.5682	0.789908	0.394954	−	0.918701i	$-0.370761\pi$
$179$	10.5682	0.789908	0.394954	+	0.918701i	$0.370761\pi$
$180$	0	0
$181$	19.3717	1.43989	0.719943	−	0.694033i	$-0.244168\pi$
$181$	19.3717	1.43989	0.719943	+	0.694033i	$0.244168\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.94246	−0.583941
$186$	0	0
$187$	9.19656	0.672519
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.6644	1.42287	0.711434	−	0.702753i	$-0.248046\pi$
$191$	19.6644	1.42287	0.711434	+	0.702753i	$0.248046\pi$
$192$	0	0
$193$	1.82487	0.131357	0.0656783	−	0.997841i	$-0.479079\pi$
$193$	1.82487	0.131357	0.0656783	+	0.997841i	$0.479079\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.73183	0.693364	0.346682	−	0.937983i	$-0.387308\pi$
$197$	9.73183	0.693364	0.346682	+	0.937983i	$0.387308\pi$
$198$	0	0
$199$	−8.24989	−0.584819	−0.292409	−	0.956293i	$-0.594457\pi$
$199$	−8.24989	−0.584819	−0.292409	+	0.956293i	$0.594457\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.19235	0.153873
$204$	0	0
$205$	−12.4507	−0.869591
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−30.7434	−2.12656
$210$	0	0
$211$	−2.31836	−0.159603	−0.0798014	−	0.996811i	$-0.525429\pi$
$211$	−2.31836	−0.159603	−0.0798014	+	0.996811i	$0.525429\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.78623	0.326418
$216$	0	0
$217$	4.58546	0.311281
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.48929	−0.234715
$222$	0	0
$223$	14.8034	0.991312	0.495656	−	0.868519i	$-0.334928\pi$
$223$	14.8034	0.991312	0.495656	+	0.868519i	$0.334928\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.5468	1.69560	0.847801	−	0.530314i	$-0.177926\pi$
$227$	25.5468	1.69560	0.847801	+	0.530314i	$0.177926\pi$
$228$	0	0
$229$	−0.0501921	−0.00331679	−0.00165839	−	0.999999i	$-0.500528\pi$
$229$	−0.0501921	−0.00331679	−0.00165839	+	0.999999i	$0.500528\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.4078	1.46798	0.733992	−	0.679158i	$-0.237655\pi$
$233$	22.4078	1.46798	0.733992	+	0.679158i	$0.237655\pi$
$234$	0	0
$235$	13.3717	0.872273
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.8255	−1.73520	−0.867598	−	0.497266i	$-0.834337\pi$
$239$	−26.8255	−1.73520	−0.867598	+	0.497266i	$0.834337\pi$
$240$	0	0
$241$	1.53213	0.0986934	0.0493467	−	0.998782i	$-0.484286\pi$
$241$	1.53213	0.0986934	0.0493467	+	0.998782i	$0.484286\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.38325	0.407811
$246$	0	0
$247$	11.6644	0.742189
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	−4.23519	−0.266264
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.7178	−1.04282	−0.521412	−	0.853305i	$-0.674595\pi$
$257$	−16.7178	−1.04282	−0.521412	+	0.853305i	$0.674595\pi$
$258$	0	0
$259$	−8.29021	−0.515129
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.0073	−0.987055	−0.493528	−	0.869730i	$-0.664293\pi$
$263$	−16.0073	−0.987055	−0.493528	+	0.869730i	$0.664293\pi$
$264$	0	0
$265$	2.88661	0.177323
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.7606	1.14385	0.571927	−	0.820305i	$-0.306196\pi$
$269$	18.7606	1.14385	0.571927	+	0.820305i	$0.306196\pi$
$270$	0	0
$271$	22.5682	1.37092	0.685462	−	0.728109i	$-0.259600\pi$
$271$	22.5682	1.37092	0.685462	+	0.728109i	$0.259600\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.7606	1.37252
$276$	0	0
$277$	20.2070	1.21412	0.607062	−	0.794655i	$-0.292348\pi$
$277$	20.2070	1.21412	0.607062	+	0.794655i	$0.292348\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.9828	−0.953453	−0.476727	−	0.879052i	$-0.658177\pi$
$281$	−15.9828	−0.953453	−0.476727	+	0.879052i	$0.658177\pi$
$282$	0	0
$283$	10.5682	0.628217	0.314109	−	0.949387i	$-0.398294\pi$
$283$	10.5682	0.628217	0.314109	+	0.949387i	$0.398294\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.9958	−0.767117
$288$	0	0
$289$	−14.7820	−0.869531
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.5181	−0.614471	−0.307236	−	0.951633i	$-0.599404\pi$
$293$	−10.5181	−0.614471	−0.307236	+	0.951633i	$0.599404\pi$
$294$	0	0
$295$	−1.14637	−0.0667440
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.60688	0.0929285
$300$	0	0
$301$	4.99579	0.287953
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.35700	0.421261
$306$	0	0
$307$	4.74338	0.270719	0.135360	−	0.990797i	$-0.456781\pi$
$307$	4.74338	0.270719	0.135360	+	0.990797i	$0.456781\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.9070	0.902001	0.451001	−	0.892524i	$-0.351067\pi$
$311$	15.9070	0.902001	0.451001	+	0.892524i	$0.351067\pi$
$312$	0	0
$313$	6.24989	0.353264	0.176632	−	0.984277i	$-0.443480\pi$
$313$	6.24989	0.353264	0.176632	+	0.984277i	$0.443480\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.7679	1.05411	0.527056	−	0.849830i	$-0.323296\pi$
$317$	18.7679	1.05411	0.527056	+	0.849830i	$0.323296\pi$
$318$	0	0
$319$	11.3142	0.633471
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.41454	−0.412556
$324$	0	0
$325$	−8.63565	−0.479020
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.9572	0.769483
$330$	0	0
$331$	27.3717	1.50448	0.752242	−	0.658887i	$-0.228972\pi$
$331$	27.3717	1.50448	0.752242	+	0.658887i	$0.228972\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.3717	0.730574
$336$	0	0
$337$	−13.3288	−0.726069	−0.363034	−	0.931776i	$-0.618259\pi$
$337$	−13.3288	−0.726069	−0.363034	+	0.931776i	$0.618259\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.6644	1.28150
$342$	0	0
$343$	15.0386	0.812010
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.3074	−1.51962	−0.759811	−	0.650144i	$-0.774709\pi$
$347$	−28.3074	−1.51962	−0.759811	+	0.650144i	$0.774709\pi$
$348$	0	0
$349$	−32.4005	−1.73436	−0.867178	−	0.497997i	$-0.834069\pi$
$349$	−32.4005	−1.73436	−0.867178	+	0.497997i	$0.834069\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.60688	0.404874	0.202437	−	0.979295i	$-0.435114\pi$
$353$	7.60688	0.404874	0.202437	+	0.979295i	$0.435114\pi$
$354$	0	0
$355$	10.3503	0.549335
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.1867	1.01263	0.506317	−	0.862347i	$-0.331006\pi$
$359$	19.1867	1.01263	0.506317	+	0.862347i	$0.331006\pi$
$360$	0	0
$361$	5.78623	0.304538
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.7795	−0.825937
$366$	0	0
$367$	34.2400	1.78731	0.893657	−	0.448750i	$-0.148131\pi$
$367$	34.2400	1.78731	0.893657	+	0.448750i	$0.148131\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.01300	0.156427
$372$	0	0
$373$	23.5212	1.21788	0.608941	−	0.793216i	$-0.291595\pi$
$373$	23.5212	1.21788	0.608941	+	0.793216i	$0.291595\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.29273	−0.221087
$378$	0	0
$379$	−2.30742	−0.118524	−0.0592622	−	0.998242i	$-0.518875\pi$
$379$	−2.30742	−0.118524	−0.0592622	+	0.998242i	$0.518875\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.35762	0.222664	0.111332	−	0.993783i	$-0.464488\pi$
$383$	4.35762	0.222664	0.111332	+	0.993783i	$0.464488\pi$
$384$	0	0
$385$	−8.47038	−0.431691
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.4177	1.33943	0.669715	−	0.742619i	$-0.266416\pi$
$389$	26.4177	1.33943	0.669715	+	0.742619i	$0.266416\pi$
$390$	0	0
$391$	−1.02142	−0.0516556
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.2138	−0.564226
$396$	0	0
$397$	14.0330	0.704295	0.352148	−	0.935945i	$-0.385452\pi$
$397$	14.0330	0.704295	0.352148	+	0.935945i	$0.385452\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.7795	−1.58699	−0.793496	−	0.608575i	$-0.791741\pi$
$401$	−31.7795	−1.58699	−0.793496	+	0.608575i	$0.791741\pi$
$402$	0	0
$403$	−8.97858	−0.447255
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−42.7837	−2.12071
$408$	0	0
$409$	−20.4078	−1.00910	−0.504551	−	0.863382i	$-0.668342\pi$
$409$	−20.4078	−1.00910	−0.504551	+	0.863382i	$0.668342\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.19656	−0.0588788
$414$	0	0
$415$	6.62831	0.325371
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.50398	−0.366593	−0.183297	−	0.983058i	$-0.558677\pi$
$419$	−7.50398	−0.366593	−0.183297	+	0.983058i	$0.558677\pi$
$420$	0	0
$421$	2.67850	0.130542	0.0652710	−	0.997868i	$-0.479209\pi$
$421$	2.67850	0.130542	0.0652710	+	0.997868i	$0.479209\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.48929	0.266270
$426$	0	0
$427$	7.67912	0.371618
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.8139	−0.954403	−0.477202	−	0.878794i	$-0.658349\pi$
$431$	−19.8139	−0.954403	−0.477202	+	0.878794i	$0.658349\pi$
$432$	0	0
$433$	−24.8610	−1.19474	−0.597371	−	0.801965i	$-0.703788\pi$
$433$	−24.8610	−1.19474	−0.597371	+	0.801965i	$0.703788\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.41454	0.163340
$438$	0	0
$439$	7.48929	0.357444	0.178722	−	0.983900i	$-0.442804\pi$
$439$	7.48929	0.357444	0.178722	+	0.983900i	$0.442804\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.7178	−0.509216	−0.254608	−	0.967044i	$-0.581946\pi$
$443$	−10.7178	−0.509216	−0.254608	+	0.967044i	$0.581946\pi$
$444$	0	0
$445$	3.29946	0.156409
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.0319048	−0.00150568	−0.000752841	−	1.00000i	$-0.500240\pi$
$449$	−0.0319048	−0.00150568	−0.000752841		1.00000i	$0.500240\pi$
$450$	0	0
$451$	−67.0680	−3.15811
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.21377	0.150664
$456$	0	0
$457$	−9.89962	−0.463084	−0.231542	−	0.972825i	$-0.574377\pi$
$457$	−9.89962	−0.463084	−0.231542	+	0.972825i	$0.574377\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.94560	0.230339	0.115170	−	0.993346i	$-0.463259\pi$
$461$	4.94560	0.230339	0.115170	+	0.993346i	$0.463259\pi$
$462$	0	0
$463$	−12.0821	−0.561503	−0.280751	−	0.959781i	$-0.590584\pi$
$463$	−12.0821	−0.561503	−0.280751	+	0.959781i	$0.590584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.76060	0.127745	0.0638726	−	0.997958i	$-0.479655\pi$
$467$	2.76060	0.127745	0.0638726	+	0.997958i	$0.479655\pi$
$468$	0	0
$469$	13.9572	0.644482
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25.7820	1.18546
$474$	0	0
$475$	−18.3503	−0.841968
$476$	0	0
$477$	0	0
$478$	0	0
$479$	28.0821	1.28310	0.641552	−	0.767080i	$-0.278291\pi$
$479$	28.0821	1.28310	0.641552	+	0.767080i	$0.278291\pi$
$480$	0	0
$481$	16.2327	0.740146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.09196	−0.276622
$486$	0	0
$487$	−39.3692	−1.78399	−0.891994	−	0.452048i	$-0.850694\pi$
$487$	−39.3692	−1.78399	−0.891994	+	0.452048i	$0.850694\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.1004	1.08764	0.543818	−	0.839203i	$-0.316978\pi$
$491$	24.1004	1.08764	0.543818	+	0.839203i	$0.316978\pi$
$492$	0	0
$493$	2.72869	0.122894
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.8034	0.484601
$498$	0	0
$499$	−42.6577	−1.90962	−0.954810	−	0.297216i	$-0.903942\pi$
$499$	−42.6577	−1.90962	−0.954810	+	0.297216i	$0.903942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.72869	−0.121666	−0.0608332	−	0.998148i	$-0.519376\pi$
$503$	−2.72869	−0.121666	−0.0608332	+	0.998148i	$0.519376\pi$
$504$	0	0
$505$	2.87819	0.128078
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.6644	0.605665	0.302832	−	0.953044i	$-0.402068\pi$
$509$	13.6644	0.605665	0.302832	+	0.953044i	$0.402068\pi$
$510$	0	0
$511$	−16.4704	−0.728607
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.72196	−0.164009
$516$	0	0
$517$	72.0294	3.16785
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.2797	−0.581796	−0.290898	−	0.956754i	$-0.593954\pi$
$521$	−13.2797	−0.581796	−0.290898	+	0.956754i	$0.593954\pi$
$522$	0	0
$523$	39.9143	1.74533	0.872665	−	0.488319i	$-0.162390\pi$
$523$	39.9143	1.74533	0.872665	+	0.488319i	$0.162390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.70727	0.248613
$528$	0	0
$529$	−22.5296	−0.979548
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.4464	1.10221
$534$	0	0
$535$	4.70054	0.203222
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34.3847	1.48105
$540$	0	0
$541$	−25.7722	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$541$	−25.7722	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.15623	−0.135198
$546$	0	0
$547$	−15.9656	−0.682639	−0.341319	−	0.939947i	$-0.610874\pi$
$547$	−15.9656	−0.682639	−0.341319	+	0.939947i	$0.610874\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.12181	−0.388602
$552$	0	0
$553$	−11.7047	−0.497737
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.11025	0.174157	0.0870784	−	0.996201i	$-0.472247\pi$
$557$	4.11025	0.174157	0.0870784	+	0.996201i	$0.472247\pi$
$558$	0	0
$559$	−9.78202	−0.413735
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.3931	1.19663	0.598314	−	0.801262i	$-0.295838\pi$
$563$	28.3931	1.19663	0.598314	+	0.801262i	$0.295838\pi$
$564$	0	0
$565$	−2.29273	−0.0964559
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.3288	0.558774	0.279387	−	0.960179i	$-0.409869\pi$
$569$	13.3288	0.558774	0.279387	+	0.960179i	$0.409869\pi$
$570$	0	0
$571$	−34.1579	−1.42946	−0.714732	−	0.699398i	$-0.753451\pi$
$571$	−34.1579	−1.42946	−0.714732	+	0.699398i	$0.753451\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.52792	−0.105422
$576$	0	0
$577$	−19.6044	−0.816140	−0.408070	−	0.912951i	$-0.633798\pi$
$577$	−19.6044	−0.816140	−0.408070	+	0.912951i	$0.633798\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.91852	0.287029
$582$	0	0
$583$	15.5493	0.643988
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.6321	1.67706	0.838532	−	0.544852i	$-0.183414\pi$
$587$	40.6321	1.67706	0.838532	+	0.544852i	$0.183414\pi$
$588$	0	0
$589$	−19.0790	−0.786135
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.5426	0.679324	0.339662	−	0.940548i	$-0.389687\pi$
$593$	16.5426	0.679324	0.339662	+	0.940548i	$0.389687\pi$
$594$	0	0
$595$	−2.04285	−0.0837485
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.8940	1.50745	0.753723	−	0.657192i	$-0.228256\pi$
$599$	36.8940	1.50745	0.753723	+	0.657192i	$0.228256\pi$
$600$	0	0
$601$	16.1923	0.660500	0.330250	−	0.943894i	$-0.392867\pi$
$601$	16.1923	0.660500	0.330250	+	0.943894i	$0.392867\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−31.1035	−1.26454
$606$	0	0
$607$	2.11760	0.0859506	0.0429753	−	0.999076i	$-0.486316\pi$
$607$	2.11760	0.0859506	0.0429753	+	0.999076i	$0.486316\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27.3288	−1.10561
$612$	0	0
$613$	−34.2719	−1.38423	−0.692115	−	0.721787i	$-0.743321\pi$
$613$	−34.2719	−1.38423	−0.692115	+	0.721787i	$0.743321\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.6963	1.43708	0.718540	−	0.695486i	$-0.244811\pi$
$617$	35.6963	1.43708	0.718540	+	0.695486i	$0.244811\pi$
$618$	0	0
$619$	2.15792	0.0867342	0.0433671	−	0.999059i	$-0.486191\pi$
$619$	2.15792	0.0867342	0.0433671	+	0.999059i	$0.486191\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.44392	0.137978
$624$	0	0
$625$	7.01469	0.280588
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.3184	−0.411420
$630$	0	0
$631$	47.7686	1.90164	0.950818	−	0.309750i	$-0.100245\pi$
$631$	47.7686	1.90164	0.950818	+	0.309750i	$0.100245\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.62831	0.104301
$636$	0	0
$637$	−13.0460	−0.516901
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.4637	−0.531782	−0.265891	−	0.964003i	$-0.585666\pi$
$641$	−13.4637	−0.531782	−0.265891	+	0.964003i	$0.585666\pi$
$642$	0	0
$643$	−28.4998	−1.12392	−0.561961	−	0.827164i	$-0.689953\pi$
$643$	−28.4998	−1.12392	−0.561961	+	0.827164i	$0.689953\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.7465	0.933572	0.466786	−	0.884370i	$-0.345412\pi$
$647$	23.7465	0.933572	0.466786	+	0.884370i	$0.345412\pi$
$648$	0	0
$649$	−6.17513	−0.242395
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.6399	0.768567	0.384284	−	0.923215i	$-0.374448\pi$
$653$	19.6399	0.768567	0.384284	+	0.923215i	$0.374448\pi$
$654$	0	0
$655$	−15.7795	−0.616556
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.5855	−0.646078	−0.323039	−	0.946386i	$-0.604704\pi$
$659$	−16.5855	−0.646078	−0.323039	+	0.946386i	$0.604704\pi$
$660$	0	0
$661$	50.3158	1.95706	0.978530	−	0.206105i	$-0.0660790\pi$
$661$	50.3158	1.95706	0.978530	+	0.206105i	$0.0660790\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.82908	0.264820
$666$	0	0
$667$	−1.25662	−0.0486563
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.6300	1.52990
$672$	0	0
$673$	−8.96388	−0.345532	−0.172766	−	0.984963i	$-0.555270\pi$
$673$	−8.96388	−0.345532	−0.172766	+	0.984963i	$0.555270\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.43910	0.132175	0.0660876	−	0.997814i	$-0.478948\pi$
$677$	3.43910	0.132175	0.0660876	+	0.997814i	$0.478948\pi$
$678$	0	0
$679$	−6.35869	−0.244024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.44017	−0.208162	−0.104081	−	0.994569i	$-0.533190\pi$
$683$	−5.44017	−0.208162	−0.104081	+	0.994569i	$0.533190\pi$
$684$	0	0
$685$	12.9504	0.494810
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.89962	−0.224757
$690$	0	0
$691$	4.42502	0.168336	0.0841678	−	0.996452i	$-0.473177\pi$
$691$	4.42502	0.168336	0.0841678	+	0.996452i	$0.473177\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.12181	−0.0425526
$696$	0	0
$697$	−16.1751	−0.612677
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.1176	0.910909	0.455455	−	0.890259i	$-0.349477\pi$
$701$	24.1176	0.910909	0.455455	+	0.890259i	$0.349477\pi$
$702$	0	0
$703$	34.4935	1.30095
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.00421	0.112985
$708$	0	0
$709$	9.52962	0.357892	0.178946	−	0.983859i	$-0.442731\pi$
$709$	9.52962	0.357892	0.178946	+	0.983859i	$0.442731\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.62831	−0.0984309
$714$	0	0
$715$	16.5855	0.620261
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.3221	−1.57835	−0.789174	−	0.614169i	$-0.789491\pi$
$719$	−42.3221	−1.57835	−0.789174	+	0.614169i	$0.789491\pi$
$720$	0	0
$721$	−3.88492	−0.144682
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.75325	0.250809
$726$	0	0
$727$	29.9975	1.11254	0.556272	−	0.831000i	$-0.312231\pi$
$727$	29.9975	1.11254	0.556272	+	0.831000i	$0.312231\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.21798	0.229980
$732$	0	0
$733$	−10.5363	−0.389169	−0.194584	−	0.980886i	$-0.562336\pi$
$733$	−10.5363	−0.389169	−0.194584	+	0.980886i	$0.562336\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	72.0294	2.65324
$738$	0	0
$739$	50.4481	1.85576	0.927882	−	0.372873i	$-0.121627\pi$
$739$	50.4481	1.85576	0.927882	+	0.372873i	$0.121627\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.1611	−1.58343	−0.791713	−	0.610893i	$-0.790810\pi$
$743$	−43.1611	−1.58343	−0.791713	+	0.610893i	$0.790810\pi$
$744$	0	0
$745$	−10.4935	−0.384452
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.90635	0.179274
$750$	0	0
$751$	2.13964	0.0780764	0.0390382	−	0.999238i	$-0.487571\pi$
$751$	2.13964	0.0780764	0.0390382	+	0.999238i	$0.487571\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.32088	0.0844656
$756$	0	0
$757$	32.3074	1.17423	0.587117	−	0.809502i	$-0.300263\pi$
$757$	32.3074	1.17423	0.587117	+	0.809502i	$0.300263\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.7925	−0.753728	−0.376864	−	0.926269i	$-0.622998\pi$
$761$	−20.7925	−0.753728	−0.376864	+	0.926269i	$0.622998\pi$
$762$	0	0
$763$	−3.29442	−0.119266
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.34292	0.0845980
$768$	0	0
$769$	−1.75011	−0.0631108	−0.0315554	−	0.999502i	$-0.510046\pi$
$769$	−1.75011	−0.0631108	−0.0315554	+	0.999502i	$0.510046\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.0491	−0.361442	−0.180721	−	0.983534i	$-0.557843\pi$
$773$	−10.0491	−0.361442	−0.180721	+	0.983534i	$0.557843\pi$
$774$	0	0
$775$	14.1249	0.507383
$776$	0	0
$777$	0	0
$778$	0	0
$779$	54.0722	1.93734
$780$	0	0
$781$	55.7539	1.99503
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−26.9848	−0.963131
$786$	0	0
$787$	−25.7135	−0.916589	−0.458294	−	0.888800i	$-0.651539\pi$
$787$	−25.7135	−0.916589	−0.458294	+	0.888800i	$0.651539\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.39312	−0.0850894
$792$	0	0
$793$	−15.0361	−0.533948
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.2180	−0.432783	−0.216392	−	0.976307i	$-0.569429\pi$
$797$	−12.2180	−0.432783	−0.216392	+	0.976307i	$0.569429\pi$
$798$	0	0
$799$	17.3717	0.614566
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−84.9995	−2.99957
$804$	0	0
$805$	0.940770	0.0331578
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−52.9504	−1.86164	−0.930819	−	0.365481i	$-0.880905\pi$
$809$	−52.9504	−1.86164	−0.930819	+	0.365481i	$0.880905\pi$
$810$	0	0
$811$	−0.425020	−0.0149245	−0.00746224	−	0.999972i	$-0.502375\pi$
$811$	−0.425020	−0.0149245	−0.00746224	+	0.999972i	$0.502375\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.2797	−0.535226
$816$	0	0
$817$	−20.7862	−0.727218
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.6749	1.07056	0.535281	−	0.844674i	$-0.320206\pi$
$821$	30.6749	1.07056	0.535281	+	0.844674i	$0.320206\pi$
$822$	0	0
$823$	43.6608	1.52192	0.760960	−	0.648798i	$-0.224728\pi$
$823$	43.6608	1.52192	0.760960	+	0.648798i	$0.224728\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.72196	0.198972	0.0994861	−	0.995039i	$-0.468280\pi$
$827$	5.72196	0.198972	0.0994861	+	0.995039i	$0.468280\pi$
$828$	0	0
$829$	23.8077	0.826874	0.413437	−	0.910533i	$-0.364328\pi$
$829$	23.8077	0.826874	0.413437	+	0.910533i	$0.364328\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.29273	0.287326
$834$	0	0
$835$	−16.1923	−0.560359
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.8996	1.10130	0.550649	−	0.834737i	$-0.314380\pi$
$839$	31.8996	1.10130	0.550649	+	0.834737i	$0.314380\pi$
$840$	0	0
$841$	−25.6430	−0.884241
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.61002	0.296194
$846$	0	0
$847$	−32.4653	−1.11552
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.75181	0.162890
$852$	0	0
$853$	−15.1856	−0.519946	−0.259973	−	0.965616i	$-0.583714\pi$
$853$	−15.1856	−0.519946	−0.259973	+	0.965616i	$0.583714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.51492	−0.120067	−0.0600337	−	0.998196i	$-0.519121\pi$
$857$	−3.51492	−0.120067	−0.0600337	+	0.998196i	$0.519121\pi$
$858$	0	0
$859$	22.7925	0.777670	0.388835	−	0.921307i	$-0.372878\pi$
$859$	22.7925	0.777670	0.388835	+	0.921307i	$0.372878\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.0790	0.513294	0.256647	−	0.966505i	$-0.417382\pi$
$863$	15.0790	0.513294	0.256647	+	0.966505i	$0.417382\pi$
$864$	0	0
$865$	29.1218	0.990171
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.4053	−2.04911
$870$	0	0
$871$	−27.3288	−0.926003
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.9143	−0.402777
$876$	0	0
$877$	−57.2285	−1.93247	−0.966234	−	0.257666i	$-0.917046\pi$
$877$	−57.2285	−1.93247	−0.966234	+	0.257666i	$0.917046\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.9718	0.605487	0.302743	−	0.953072i	$-0.402097\pi$
$881$	17.9718	0.605487	0.302743	+	0.953072i	$0.402097\pi$
$882$	0	0
$883$	8.98700	0.302437	0.151218	−	0.988500i	$-0.451680\pi$
$883$	8.98700	0.302437	0.151218	+	0.988500i	$0.451680\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.820654	0.0275549	0.0137774	−	0.999905i	$-0.495614\pi$
$887$	0.820654	0.0275549	0.0137774	+	0.999905i	$0.495614\pi$
$888$	0	0
$889$	2.74338	0.0920102
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−58.0722	−1.94331
$894$	0	0
$895$	−12.1151	−0.404962
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.02142	0.234178
$900$	0	0
$901$	3.75011	0.124934
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.2070	−0.738187
$906$	0	0
$907$	−58.2646	−1.93464	−0.967322	−	0.253552i	$-0.918401\pi$
$907$	−58.2646	−1.93464	−0.967322	+	0.253552i	$0.918401\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.1867	−0.768209	−0.384105	−	0.923290i	$-0.625490\pi$
$911$	−23.1867	−0.768209	−0.384105	+	0.923290i	$0.625490\pi$
$912$	0	0
$913$	35.7047	1.18165
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.4704	−0.543900
$918$	0	0
$919$	−34.0539	−1.12334	−0.561668	−	0.827363i	$-0.689840\pi$
$919$	−34.0539	−1.12334	−0.561668	+	0.827363i	$0.689840\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.1537	−0.696283
$924$	0	0
$925$	−25.5370	−0.839650
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.335577	−0.0110099	−0.00550496	−	0.999985i	$-0.501752\pi$
$929$	−0.335577	−0.0110099	−0.00550496	+	0.999985i	$0.501752\pi$
$930$	0	0
$931$	−27.7220	−0.908551
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.5426	−0.344780
$936$	0	0
$937$	−54.4141	−1.77763	−0.888815	−	0.458266i	$-0.848471\pi$
$937$	−54.4141	−1.77763	−0.888815	+	0.458266i	$0.848471\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	18.4507	0.601474	0.300737	−	0.953707i	$-0.402767\pi$
$941$	18.4507	0.601474	0.300737	+	0.953707i	$0.402767\pi$
$942$	0	0
$943$	7.44896	0.242572
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.6938	1.28988	0.644938	−	0.764235i	$-0.276883\pi$
$947$	39.6938	1.28988	0.644938	+	0.764235i	$0.276883\pi$
$948$	0	0
$949$	32.2499	1.04688
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.1046	−0.683645	−0.341822	−	0.939765i	$-0.611044\pi$
$953$	−21.1046	−0.683645	−0.341822	+	0.939765i	$0.611044\pi$
$954$	0	0
$955$	−22.5426	−0.729462
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.5174	0.436501
$960$	0	0
$961$	−16.3142	−0.526263
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.09196	−0.0673427
$966$	0	0
$967$	−2.12494	−0.0683335	−0.0341668	−	0.999416i	$-0.510878\pi$
$967$	−2.12494	−0.0683335	−0.0341668	+	0.999416i	$0.510878\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.3435	0.877496	0.438748	−	0.898610i	$-0.355422\pi$
$971$	27.3435	0.877496	0.438748	+	0.898610i	$0.355422\pi$
$972$	0	0
$973$	−1.17092	−0.0375381
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.3074	1.16158	0.580789	−	0.814054i	$-0.302744\pi$
$977$	36.3074	1.16158	0.580789	+	0.814054i	$0.302744\pi$
$978$	0	0
$979$	17.7732	0.568035
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.14323	0.0683584	0.0341792	−	0.999416i	$-0.489118\pi$
$983$	2.14323	0.0683584	0.0341792	+	0.999416i	$0.489118\pi$
$984$	0	0
$985$	−11.1562	−0.355467
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.86350	−0.0910540
$990$	0	0
$991$	−46.9603	−1.49174	−0.745871	−	0.666090i	$-0.767967\pi$
$991$	−46.9603	−1.49174	−0.745871	+	0.666090i	$0.767967\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.45738	0.299819
$996$	0	0
$997$	16.1432	0.511261	0.255631	−	0.966775i	$-0.417717\pi$
$997$	16.1432	0.511261	0.255631	+	0.966775i	$0.417717\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bi.1.2		3
3.2	odd	2		2832.2.a.r.1.2		3
4.3	odd	2		1062.2.a.n.1.2		3
12.11	even	2		354.2.a.h.1.2	✓	3
60.59	even	2		8850.2.a.bu.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.a.h.1.2	✓	3	12.11	even	2
1062.2.a.n.1.2		3	4.3	odd	2
2832.2.a.r.1.2		3	3.2	odd	2
8496.2.a.bi.1.2		3	1.1	even	1	trivial
8850.2.a.bu.1.2		3	60.59	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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14637 q^{5} -1.19656 q^{7} +O(q^{10})\)	\(q-1.14637 q^{5} -1.19656 q^{7} -6.17513 q^{11} +2.34292 q^{13} -1.48929 q^{17} +4.97858 q^{19} +0.685846 q^{23} -3.68585 q^{25} -1.83221 q^{29} -3.83221 q^{31} +1.37169 q^{35} +6.92839 q^{37} +10.8610 q^{41} -4.17513 q^{43} -11.6644 q^{47} -5.56825 q^{49} -2.51806 q^{53} +7.07896 q^{55} +1.00000 q^{59} -6.41767 q^{61} -2.68585 q^{65} -11.6644 q^{67} -9.02877 q^{71} +13.7648 q^{73} +7.38890 q^{77} +9.78202 q^{79} -5.78202 q^{83} +1.70727 q^{85} -2.87819 q^{89} -2.80344 q^{91} -5.70727 q^{95} +5.31415 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} + q^{7}+O(q^{10}) \) 3 * q - 2 * q^5 + q^7	\( 3 q - 2 q^{5} + q^{7} + q^{11} + q^{13} + 3 q^{17} - 10 q^{23} + q^{25} + 8 q^{29} + 2 q^{31} - 20 q^{35} + 9 q^{37} + q^{41} + 7 q^{43} - 8 q^{47} + 12 q^{49} + 18 q^{53} + 3 q^{59} + 4 q^{65} - 8 q^{67} - 9 q^{71} + 8 q^{73} + 21 q^{77} + 19 q^{79} - 7 q^{83} + 8 q^{85} - 13 q^{91} - 20 q^{95} + 28 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 + q^7 + q^11 + q^13 + 3 * q^17 - 10 * q^23 + q^25 + 8 * q^29 + 2 * q^31 - 20 * q^35 + 9 * q^37 + q^41 + 7 * q^43 - 8 * q^47 + 12 * q^49 + 18 * q^53 + 3 * q^59 + 4 * q^65 - 8 * q^67 - 9 * q^71 + 8 * q^73 + 21 * q^77 + 19 * q^79 - 7 * q^83 + 8 * q^85 - 13 * q^91 - 20 * q^95 + 28 * q^97

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