LMFDB - Embedded newform 9576.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38197 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.38197 q^{5} +1.00000 q^{7} -0.854102 q^{11} +0.763932 q^{13} -6.47214 q^{17} -1.00000 q^{19} -7.70820 q^{23} -3.09017 q^{25} -5.85410 q^{29} -4.47214 q^{31} -1.38197 q^{35} +11.0902 q^{37} -5.09017 q^{41} +0.145898 q^{43} +5.14590 q^{47} +1.00000 q^{49} +5.61803 q^{53} +1.18034 q^{55} +7.09017 q^{59} -1.14590 q^{61} -1.05573 q^{65} +2.94427 q^{67} +6.56231 q^{71} +13.2361 q^{73} -0.854102 q^{77} -0.145898 q^{79} -4.00000 q^{83} +8.94427 q^{85} +10.0902 q^{89} +0.763932 q^{91} +1.38197 q^{95} +16.5623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 5 * q^5 + 2 * q^7	$ 2 q - 5 q^{5} + 2 q^{7} + 5 q^{11} + 6 q^{13} - 4 q^{17} - 2 q^{19} - 2 q^{23} + 5 q^{25} - 5 q^{29} - 5 q^{35} + 11 q^{37} + q^{41} + 7 q^{43} + 17 q^{47} + 2 q^{49} + 9 q^{53} - 20 q^{55} + 3 q^{59} - 9 q^{61} - 20 q^{65} - 12 q^{67} - 7 q^{71} + 22 q^{73} + 5 q^{77} - 7 q^{79} - 8 q^{83} + 9 q^{89} + 6 q^{91} + 5 q^{95} + 13 q^{97}+O(q^{100}) $ 2 * q - 5 * q^5 + 2 * q^7 + 5 * q^11 + 6 * q^13 - 4 * q^17 - 2 * q^19 - 2 * q^23 + 5 * q^25 - 5 * q^29 - 5 * q^35 + 11 * q^37 + q^41 + 7 * q^43 + 17 * q^47 + 2 * q^49 + 9 * q^53 - 20 * q^55 + 3 * q^59 - 9 * q^61 - 20 * q^65 - 12 * q^67 - 7 * q^71 + 22 * q^73 + 5 * q^77 - 7 * q^79 - 8 * q^83 + 9 * q^89 + 6 * q^91 + 5 * q^95 + 13 * 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
$5$	−1.38197	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−1.38197	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.854102	−0.257521	−0.128761	−	0.991676i	$-0.541100\pi$
$11$	−0.854102	−0.257521	−0.128761	+	0.991676i	$0.541100\pi$
$12$	0	0
$13$	0.763932	0.211877	0.105938	−	0.994373i	$-0.466215\pi$
$13$	0.763932	0.211877	0.105938	+	0.994373i	$0.466215\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.85410	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$29$	−5.85410	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$30$	0	0
$31$	−4.47214	−0.803219	−0.401610	−	0.915811i	$-0.631549\pi$
$31$	−4.47214	−0.803219	−0.401610	+	0.915811i	$0.631549\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.38197	−0.233595
$36$	0	0
$37$	11.0902	1.82321	0.911606	−	0.411064i	$-0.134843\pi$
$37$	11.0902	1.82321	0.911606	+	0.411064i	$0.134843\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.09017	−0.794951	−0.397475	−	0.917613i	$-0.630114\pi$
$41$	−5.09017	−0.794951	−0.397475	+	0.917613i	$0.630114\pi$
$42$	0	0
$43$	0.145898	0.0222492	0.0111246	−	0.999938i	$-0.496459\pi$
$43$	0.145898	0.0222492	0.0111246	+	0.999938i	$0.496459\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.14590	0.750606	0.375303	−	0.926902i	$-0.377539\pi$
$47$	5.14590	0.750606	0.375303	+	0.926902i	$0.377539\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.61803	0.771696	0.385848	−	0.922562i	$-0.373909\pi$
$53$	5.61803	0.771696	0.385848	+	0.922562i	$0.373909\pi$
$54$	0	0
$55$	1.18034	0.159157
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.09017	0.923062	0.461531	−	0.887124i	$-0.347300\pi$
$59$	7.09017	0.923062	0.461531	+	0.887124i	$0.347300\pi$
$60$	0	0
$61$	−1.14590	−0.146717	−0.0733586	−	0.997306i	$-0.523372\pi$
$61$	−1.14590	−0.146717	−0.0733586	+	0.997306i	$0.523372\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.05573	−0.130947
$66$	0	0
$67$	2.94427	0.359700	0.179850	−	0.983694i	$-0.442439\pi$
$67$	2.94427	0.359700	0.179850	+	0.983694i	$0.442439\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.56231	0.778802	0.389401	−	0.921068i	$-0.372682\pi$
$71$	6.56231	0.778802	0.389401	+	0.921068i	$0.372682\pi$
$72$	0	0
$73$	13.2361	1.54916	0.774582	−	0.632473i	$-0.217960\pi$
$73$	13.2361	1.54916	0.774582	+	0.632473i	$0.217960\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.854102	−0.0973340
$78$	0	0
$79$	−0.145898	−0.0164148	−0.00820741	−	0.999966i	$-0.502613\pi$
$79$	−0.145898	−0.0164148	−0.00820741	+	0.999966i	$0.502613\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0902	1.06956	0.534778	−	0.844993i	$-0.320395\pi$
$89$	10.0902	1.06956	0.534778	+	0.844993i	$0.320395\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.38197	0.141787
$96$	0	0
$97$	16.5623	1.68165	0.840824	−	0.541309i	$-0.182071\pi$
$97$	16.5623	1.68165	0.840824	+	0.541309i	$0.182071\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.94427	−0.690981	−0.345490	−	0.938422i	$-0.612287\pi$
$101$	−6.94427	−0.690981	−0.345490	+	0.938422i	$0.612287\pi$
$102$	0	0
$103$	0.291796	0.0287515	0.0143758	−	0.999897i	$-0.495424\pi$
$103$	0.291796	0.0287515	0.0143758	+	0.999897i	$0.495424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1803	0.984171	0.492085	−	0.870547i	$-0.336235\pi$
$107$	10.1803	0.984171	0.492085	+	0.870547i	$0.336235\pi$
$108$	0	0
$109$	−10.5623	−1.01169	−0.505843	−	0.862626i	$-0.668818\pi$
$109$	−10.5623	−1.01169	−0.505843	+	0.862626i	$0.668818\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.944272	0.0888296	0.0444148	−	0.999013i	$-0.485858\pi$
$113$	0.944272	0.0888296	0.0444148	+	0.999013i	$0.485858\pi$
$114$	0	0
$115$	10.6525	0.993348
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.47214	−0.593300
$120$	0	0
$121$	−10.2705	−0.933683
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	8.85410	0.785675	0.392837	−	0.919608i	$-0.371494\pi$
$127$	8.85410	0.785675	0.392837	+	0.919608i	$0.371494\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.180340	−0.0157564	−0.00787818	−	0.999969i	$-0.502508\pi$
$131$	−0.180340	−0.0157564	−0.00787818	+	0.999969i	$0.502508\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.56231	0.133477	0.0667384	−	0.997771i	$-0.478741\pi$
$137$	1.56231	0.133477	0.0667384	+	0.997771i	$0.478741\pi$
$138$	0	0
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.652476	−0.0545628
$144$	0	0
$145$	8.09017	0.671852
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.94427	−0.732743	−0.366372	−	0.930469i	$-0.619400\pi$
$149$	−8.94427	−0.732743	−0.366372	+	0.930469i	$0.619400\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.18034	0.496417
$156$	0	0
$157$	16.3820	1.30742	0.653712	−	0.756744i	$-0.273211\pi$
$157$	16.3820	1.30742	0.653712	+	0.756744i	$0.273211\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.70820	−0.607492
$162$	0	0
$163$	−18.5623	−1.45391	−0.726956	−	0.686684i	$-0.759066\pi$
$163$	−18.5623	−1.45391	−0.726956	+	0.686684i	$0.759066\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.1803	1.68634	0.843170	−	0.537647i	$-0.180687\pi$
$173$	22.1803	1.68634	0.843170	+	0.537647i	$0.180687\pi$
$174$	0	0
$175$	−3.09017	−0.233595
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.1246	1.42944	0.714720	−	0.699410i	$-0.246554\pi$
$179$	19.1246	1.42944	0.714720	+	0.699410i	$0.246554\pi$
$180$	0	0
$181$	−2.76393	−0.205441	−0.102721	−	0.994710i	$-0.532755\pi$
$181$	−2.76393	−0.205441	−0.102721	+	0.994710i	$0.532755\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−15.3262	−1.12681
$186$	0	0
$187$	5.52786	0.404237
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7082	0.847176	0.423588	−	0.905855i	$-0.360770\pi$
$191$	11.7082	0.847176	0.423588	+	0.905855i	$0.360770\pi$
$192$	0	0
$193$	2.76393	0.198952	0.0994761	−	0.995040i	$-0.468283\pi$
$193$	2.76393	0.198952	0.0994761	+	0.995040i	$0.468283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−28.0344	−1.98731	−0.993654	−	0.112476i	$-0.964122\pi$
$199$	−28.0344	−1.98731	−0.993654	+	0.112476i	$0.964122\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.85410	−0.410877
$204$	0	0
$205$	7.03444	0.491307
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.854102	0.0590795
$210$	0	0
$211$	21.8885	1.50687	0.753435	−	0.657523i	$-0.228396\pi$
$211$	21.8885	1.50687	0.753435	+	0.657523i	$0.228396\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.201626	−0.0137508
$216$	0	0
$217$	−4.47214	−0.303588
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.94427	−0.332588
$222$	0	0
$223$	−20.1803	−1.35138	−0.675688	−	0.737188i	$-0.736153\pi$
$223$	−20.1803	−1.35138	−0.675688	+	0.737188i	$0.736153\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.94427	−0.328163	−0.164081	−	0.986447i	$-0.552466\pi$
$227$	−4.94427	−0.328163	−0.164081	+	0.986447i	$0.552466\pi$
$228$	0	0
$229$	6.32624	0.418050	0.209025	−	0.977910i	$-0.432971\pi$
$229$	6.32624	0.418050	0.209025	+	0.977910i	$0.432971\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.8541	1.10415	0.552074	−	0.833795i	$-0.313836\pi$
$233$	16.8541	1.10415	0.552074	+	0.833795i	$0.313836\pi$
$234$	0	0
$235$	−7.11146	−0.463900
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.7082	1.40419	0.702093	−	0.712085i	$-0.252249\pi$
$239$	21.7082	1.40419	0.702093	+	0.712085i	$0.252249\pi$
$240$	0	0
$241$	−15.5623	−1.00246	−0.501228	−	0.865315i	$-0.667118\pi$
$241$	−15.5623	−1.00246	−0.501228	+	0.865315i	$0.667118\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.38197	−0.0882906
$246$	0	0
$247$	−0.763932	−0.0486078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.7639	−1.56309	−0.781543	−	0.623852i	$-0.785567\pi$
$251$	−24.7639	−1.56309	−0.781543	+	0.623852i	$0.785567\pi$
$252$	0	0
$253$	6.58359	0.413907
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.3262	1.14316	0.571580	−	0.820547i	$-0.306331\pi$
$257$	18.3262	1.14316	0.571580	+	0.820547i	$0.306331\pi$
$258$	0	0
$259$	11.0902	0.689110
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.7639	1.28036	0.640179	−	0.768225i	$-0.278860\pi$
$263$	20.7639	1.28036	0.640179	+	0.768225i	$0.278860\pi$
$264$	0	0
$265$	−7.76393	−0.476935
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.58359	0.157524	0.0787622	−	0.996893i	$-0.474903\pi$
$269$	2.58359	0.157524	0.0787622	+	0.996893i	$0.474903\pi$
$270$	0	0
$271$	10.1459	0.616319	0.308160	−	0.951335i	$-0.400287\pi$
$271$	10.1459	0.616319	0.308160	+	0.951335i	$0.400287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.63932	0.159157
$276$	0	0
$277$	−31.7082	−1.90516	−0.952581	−	0.304286i	$-0.901582\pi$
$277$	−31.7082	−1.90516	−0.952581	+	0.304286i	$0.901582\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.8885	−1.06714	−0.533571	−	0.845756i	$-0.679150\pi$
$281$	−17.8885	−1.06714	−0.533571	+	0.845756i	$0.679150\pi$
$282$	0	0
$283$	14.4721	0.860279	0.430140	−	0.902762i	$-0.358464\pi$
$283$	14.4721	0.860279	0.430140	+	0.902762i	$0.358464\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.09017	−0.300463
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−0.763932	−0.0446294	−0.0223147	−	0.999751i	$-0.507104\pi$
$293$	−0.763932	−0.0446294	−0.0223147	+	0.999751i	$0.507104\pi$
$294$	0	0
$295$	−9.79837	−0.570483
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.88854	−0.340543
$300$	0	0
$301$	0.145898	0.00840942
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.58359	0.0906762
$306$	0	0
$307$	15.0902	0.861241	0.430621	−	0.902533i	$-0.358295\pi$
$307$	15.0902	0.861241	0.430621	+	0.902533i	$0.358295\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.4508	1.72671	0.863355	−	0.504598i	$-0.168359\pi$
$311$	30.4508	1.72671	0.863355	+	0.504598i	$0.168359\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.3262	1.31013	0.655066	−	0.755572i	$-0.272641\pi$
$317$	23.3262	1.31013	0.655066	+	0.755572i	$0.272641\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.47214	0.360119
$324$	0	0
$325$	−2.36068	−0.130947
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.14590	0.283703
$330$	0	0
$331$	35.3050	1.94054	0.970268	−	0.242034i	$-0.0778145\pi$
$331$	35.3050	1.94054	0.970268	+	0.242034i	$0.0778145\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.06888	−0.222307
$336$	0	0
$337$	−4.76393	−0.259508	−0.129754	−	0.991546i	$-0.541419\pi$
$337$	−4.76393	−0.259508	−0.129754	+	0.991546i	$0.541419\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.81966	0.206846
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.47214	−0.347442	−0.173721	−	0.984795i	$-0.555579\pi$
$347$	−6.47214	−0.347442	−0.173721	+	0.984795i	$0.555579\pi$
$348$	0	0
$349$	19.8885	1.06461	0.532305	−	0.846553i	$-0.321326\pi$
$349$	19.8885	1.06461	0.532305	+	0.846553i	$0.321326\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.3607	−1.08369	−0.541845	−	0.840479i	$-0.682274\pi$
$353$	−20.3607	−1.08369	−0.541845	+	0.840479i	$0.682274\pi$
$354$	0	0
$355$	−9.06888	−0.481326
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.2361	0.909685	0.454842	−	0.890572i	$-0.349696\pi$
$359$	17.2361	0.909685	0.454842	+	0.890572i	$0.349696\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.2918	−0.957436
$366$	0	0
$367$	21.3262	1.11322	0.556610	−	0.830774i	$-0.312102\pi$
$367$	21.3262	1.11322	0.556610	+	0.830774i	$0.312102\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.61803	0.291674
$372$	0	0
$373$	−20.0344	−1.03734	−0.518672	−	0.854973i	$-0.673573\pi$
$373$	−20.0344	−1.03734	−0.518672	+	0.854973i	$0.673573\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.47214	−0.230327
$378$	0	0
$379$	−5.12461	−0.263234	−0.131617	−	0.991301i	$-0.542017\pi$
$379$	−5.12461	−0.263234	−0.131617	+	0.991301i	$0.542017\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.0689	1.63864	0.819322	−	0.573334i	$-0.194350\pi$
$383$	32.0689	1.63864	0.819322	+	0.573334i	$0.194350\pi$
$384$	0	0
$385$	1.18034	0.0601557
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.3607	0.728115	0.364058	−	0.931376i	$-0.381391\pi$
$389$	14.3607	0.728115	0.364058	+	0.931376i	$0.381391\pi$
$390$	0	0
$391$	49.8885	2.52297
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.201626	0.0101449
$396$	0	0
$397$	−1.50658	−0.0756130	−0.0378065	−	0.999285i	$-0.512037\pi$
$397$	−1.50658	−0.0756130	−0.0378065	+	0.999285i	$0.512037\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.8885	1.29281	0.646406	−	0.762994i	$-0.276271\pi$
$401$	25.8885	1.29281	0.646406	+	0.762994i	$0.276271\pi$
$402$	0	0
$403$	−3.41641	−0.170183
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.47214	−0.469516
$408$	0	0
$409$	2.09017	0.103352	0.0516761	−	0.998664i	$-0.483544\pi$
$409$	2.09017	0.103352	0.0516761	+	0.998664i	$0.483544\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.09017	0.348884
$414$	0	0
$415$	5.52786	0.271352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.5279	−0.953999	−0.476999	−	0.878904i	$-0.658276\pi$
$419$	−19.5279	−0.953999	−0.476999	+	0.878904i	$0.658276\pi$
$420$	0	0
$421$	−11.5279	−0.561834	−0.280917	−	0.959732i	$-0.590639\pi$
$421$	−11.5279	−0.561834	−0.280917	+	0.959732i	$0.590639\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.0000	0.970143
$426$	0	0
$427$	−1.14590	−0.0554539
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.9098	0.573676	0.286838	−	0.957979i	$-0.407396\pi$
$431$	11.9098	0.573676	0.286838	+	0.957979i	$0.407396\pi$
$432$	0	0
$433$	15.2705	0.733854	0.366927	−	0.930250i	$-0.380410\pi$
$433$	15.2705	0.733854	0.366927	+	0.930250i	$0.380410\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.70820	0.368733
$438$	0	0
$439$	39.3050	1.87592	0.937961	−	0.346739i	$-0.112711\pi$
$439$	39.3050	1.87592	0.937961	+	0.346739i	$0.112711\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.2705	−1.29566	−0.647831	−	0.761785i	$-0.724324\pi$
$443$	−27.2705	−1.29566	−0.647831	+	0.761785i	$0.724324\pi$
$444$	0	0
$445$	−13.9443	−0.661022
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.81966	−0.180261	−0.0901305	−	0.995930i	$-0.528728\pi$
$449$	−3.81966	−0.180261	−0.0901305	+	0.995930i	$0.528728\pi$
$450$	0	0
$451$	4.34752	0.204717
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.05573	−0.0494933
$456$	0	0
$457$	−4.27051	−0.199766	−0.0998830	−	0.994999i	$-0.531847\pi$
$457$	−4.27051	−0.199766	−0.0998830	+	0.994999i	$0.531847\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.03444	0.141328	0.0706640	−	0.997500i	$-0.477488\pi$
$461$	3.03444	0.141328	0.0706640	+	0.997500i	$0.477488\pi$
$462$	0	0
$463$	0.763932	0.0355029	0.0177515	−	0.999842i	$-0.494349\pi$
$463$	0.763932	0.0355029	0.0177515	+	0.999842i	$0.494349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.1803	−0.841286	−0.420643	−	0.907226i	$-0.638195\pi$
$467$	−18.1803	−0.841286	−0.420643	+	0.907226i	$0.638195\pi$
$468$	0	0
$469$	2.94427	0.135954
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.124612	−0.00572966
$474$	0	0
$475$	3.09017	0.141787
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.3951	−1.66294	−0.831468	−	0.555573i	$-0.812499\pi$
$479$	−36.3951	−1.66294	−0.831468	+	0.555573i	$0.812499\pi$
$480$	0	0
$481$	8.47214	0.386296
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.8885	−1.03932
$486$	0	0
$487$	23.8541	1.08093	0.540466	−	0.841366i	$-0.318248\pi$
$487$	23.8541	1.08093	0.540466	+	0.841366i	$0.318248\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.8885	−1.34885	−0.674426	−	0.738343i	$-0.735609\pi$
$491$	−29.8885	−1.34885	−0.674426	+	0.738343i	$0.735609\pi$
$492$	0	0
$493$	37.8885	1.70641
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.56231	0.294360
$498$	0	0
$499$	1.96556	0.0879905	0.0439952	−	0.999032i	$-0.485991\pi$
$499$	1.96556	0.0879905	0.0439952	+	0.999032i	$0.485991\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13.8541	0.617724	0.308862	−	0.951107i	$-0.400052\pi$
$503$	13.8541	0.617724	0.308862	+	0.951107i	$0.400052\pi$
$504$	0	0
$505$	9.59675	0.427050
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.23607	−0.409382	−0.204691	−	0.978827i	$-0.565619\pi$
$509$	−9.23607	−0.409382	−0.204691	+	0.978827i	$0.565619\pi$
$510$	0	0
$511$	13.2361	0.585529
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.403252	−0.0177694
$516$	0	0
$517$	−4.39512	−0.193297
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.3050	1.72198	0.860991	−	0.508621i	$-0.169845\pi$
$521$	39.3050	1.72198	0.860991	+	0.508621i	$0.169845\pi$
$522$	0	0
$523$	−1.88854	−0.0825803	−0.0412901	−	0.999147i	$-0.513147\pi$
$523$	−1.88854	−0.0825803	−0.0412901	+	0.999147i	$0.513147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.9443	1.26083
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.88854	−0.168432
$534$	0	0
$535$	−14.0689	−0.608251
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.854102	−0.0367888
$540$	0	0
$541$	0.291796	0.0125453	0.00627265	−	0.999980i	$-0.498003\pi$
$541$	0.291796	0.0125453	0.00627265	+	0.999980i	$0.498003\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.5967	0.625256
$546$	0	0
$547$	−12.2918	−0.525559	−0.262780	−	0.964856i	$-0.584639\pi$
$547$	−12.2918	−0.525559	−0.262780	+	0.964856i	$0.584639\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.85410	0.249393
$552$	0	0
$553$	−0.145898	−0.00620422
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.1803	−1.02455	−0.512277	−	0.858820i	$-0.671198\pi$
$557$	−24.1803	−1.02455	−0.512277	+	0.858820i	$0.671198\pi$
$558$	0	0
$559$	0.111456	0.00471409
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.79837	0.160082	0.0800412	−	0.996792i	$-0.474495\pi$
$563$	3.79837	0.160082	0.0800412	+	0.996792i	$0.474495\pi$
$564$	0	0
$565$	−1.30495	−0.0548997
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.291796	−0.0122327	−0.00611636	−	0.999981i	$-0.501947\pi$
$569$	−0.291796	−0.0122327	−0.00611636	+	0.999981i	$0.501947\pi$
$570$	0	0
$571$	−17.2148	−0.720416	−0.360208	−	0.932872i	$-0.617294\pi$
$571$	−17.2148	−0.720416	−0.360208	+	0.932872i	$0.617294\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	23.8197	0.993348
$576$	0	0
$577$	−1.05573	−0.0439505	−0.0219753	−	0.999759i	$-0.506996\pi$
$577$	−1.05573	−0.0439505	−0.0219753	+	0.999759i	$0.506996\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−4.79837	−0.198728
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1246	0.789357	0.394679	−	0.918819i	$-0.370856\pi$
$587$	19.1246	0.789357	0.394679	+	0.918819i	$0.370856\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.1246	−1.11387	−0.556937	−	0.830555i	$-0.688024\pi$
$593$	−27.1246	−1.11387	−0.556937	+	0.830555i	$0.688024\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.2016	0.743698	0.371849	−	0.928293i	$-0.378724\pi$
$599$	18.2016	0.743698	0.371849	+	0.928293i	$0.378724\pi$
$600$	0	0
$601$	−27.8885	−1.13760	−0.568799	−	0.822477i	$-0.692592\pi$
$601$	−27.8885	−1.13760	−0.568799	+	0.822477i	$0.692592\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.1935	0.577048
$606$	0	0
$607$	4.76393	0.193362	0.0966810	−	0.995315i	$-0.469177\pi$
$607$	4.76393	0.193362	0.0966810	+	0.995315i	$0.469177\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.93112	0.159036
$612$	0	0
$613$	−0.944272	−0.0381388	−0.0190694	−	0.999818i	$-0.506070\pi$
$613$	−0.944272	−0.0381388	−0.0190694	+	0.999818i	$0.506070\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.0902	−1.73474	−0.867372	−	0.497660i	$-0.834193\pi$
$617$	−43.0902	−1.73474	−0.867372	+	0.497660i	$0.834193\pi$
$618$	0	0
$619$	31.7082	1.27446	0.637230	−	0.770674i	$-0.280080\pi$
$619$	31.7082	1.27446	0.637230	+	0.770674i	$0.280080\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.0902	0.404254
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−71.7771	−2.86194
$630$	0	0
$631$	−4.36068	−0.173596	−0.0867980	−	0.996226i	$-0.527663\pi$
$631$	−4.36068	−0.173596	−0.0867980	+	0.996226i	$0.527663\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.2361	−0.485574
$636$	0	0
$637$	0.763932	0.0302681
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.65248	−0.341752	−0.170876	−	0.985293i	$-0.554660\pi$
$641$	−8.65248	−0.341752	−0.170876	+	0.985293i	$0.554660\pi$
$642$	0	0
$643$	46.4721	1.83268	0.916341	−	0.400399i	$-0.131128\pi$
$643$	46.4721	1.83268	0.916341	+	0.400399i	$0.131128\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.2705	−0.521718	−0.260859	−	0.965377i	$-0.584006\pi$
$647$	−13.2705	−0.521718	−0.260859	+	0.965377i	$0.584006\pi$
$648$	0	0
$649$	−6.05573	−0.237708
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.34752	0.0527327	0.0263663	−	0.999652i	$-0.491606\pi$
$653$	1.34752	0.0527327	0.0263663	+	0.999652i	$0.491606\pi$
$654$	0	0
$655$	0.249224	0.00973797
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.94427	−0.348419	−0.174210	−	0.984709i	$-0.555737\pi$
$659$	−8.94427	−0.348419	−0.174210	+	0.984709i	$0.555737\pi$
$660$	0	0
$661$	36.8328	1.43263	0.716315	−	0.697777i	$-0.245827\pi$
$661$	36.8328	1.43263	0.716315	+	0.697777i	$0.245827\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.38197	0.0535903
$666$	0	0
$667$	45.1246	1.74723
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.978714	0.0377828
$672$	0	0
$673$	−20.7639	−0.800391	−0.400195	−	0.916430i	$-0.631058\pi$
$673$	−20.7639	−0.800391	−0.400195	+	0.916430i	$0.631058\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	16.5623	0.635603
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.8885	0.454902	0.227451	−	0.973789i	$-0.426961\pi$
$683$	11.8885	0.454902	0.227451	+	0.973789i	$0.426961\pi$
$684$	0	0
$685$	−2.15905	−0.0824932
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.29180	0.163504
$690$	0	0
$691$	−7.34752	−0.279513	−0.139756	−	0.990186i	$-0.544632\pi$
$691$	−7.34752	−0.279513	−0.139756	+	0.990186i	$0.544632\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.94427	−0.339276
$696$	0	0
$697$	32.9443	1.24785
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.7639	−1.31302	−0.656508	−	0.754319i	$-0.727967\pi$
$701$	−34.7639	−1.31302	−0.656508	+	0.754319i	$0.727967\pi$
$702$	0	0
$703$	−11.0902	−0.418274
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.94427	−0.261166
$708$	0	0
$709$	−4.18034	−0.156996	−0.0784980	−	0.996914i	$-0.525012\pi$
$709$	−4.18034	−0.156996	−0.0784980	+	0.996914i	$0.525012\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34.4721	1.29099
$714$	0	0
$715$	0.901699	0.0337216
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.88854	0.368780	0.184390	−	0.982853i	$-0.440969\pi$
$719$	9.88854	0.368780	0.184390	+	0.982853i	$0.440969\pi$
$720$	0	0
$721$	0.291796	0.0108671
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.0902	0.671852
$726$	0	0
$727$	−4.56231	−0.169207	−0.0846033	−	0.996415i	$-0.526962\pi$
$727$	−4.56231	−0.169207	−0.0846033	+	0.996415i	$0.526962\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.944272	−0.0349252
$732$	0	0
$733$	−47.5623	−1.75675	−0.878377	−	0.477969i	$-0.841373\pi$
$733$	−47.5623	−1.75675	−0.878377	+	0.477969i	$0.841373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.51471	−0.0926305
$738$	0	0
$739$	−12.4377	−0.457528	−0.228764	−	0.973482i	$-0.573468\pi$
$739$	−12.4377	−0.457528	−0.228764	+	0.973482i	$0.573468\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.7984	0.763018	0.381509	−	0.924365i	$-0.375404\pi$
$743$	20.7984	0.763018	0.381509	+	0.924365i	$0.375404\pi$
$744$	0	0
$745$	12.3607	0.452860
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.1803	0.371982
$750$	0	0
$751$	25.0902	0.915553	0.457777	−	0.889067i	$-0.348646\pi$
$751$	25.0902	0.915553	0.457777	+	0.889067i	$0.348646\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	27.6393	1.00590
$756$	0	0
$757$	30.4721	1.10753	0.553764	−	0.832673i	$-0.313191\pi$
$757$	30.4721	1.10753	0.553764	+	0.832673i	$0.313191\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.94427	−0.251730	−0.125865	−	0.992047i	$-0.540171\pi$
$761$	−6.94427	−0.251730	−0.125865	+	0.992047i	$0.540171\pi$
$762$	0	0
$763$	−10.5623	−0.382381
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.41641	0.195575
$768$	0	0
$769$	−0.652476	−0.0235289	−0.0117644	−	0.999931i	$-0.503745\pi$
$769$	−0.652476	−0.0235289	−0.0117644	+	0.999931i	$0.503745\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.2918	0.873715	0.436858	−	0.899531i	$-0.356091\pi$
$773$	24.2918	0.873715	0.436858	+	0.899531i	$0.356091\pi$
$774$	0	0
$775$	13.8197	0.496417
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.09017	0.182374
$780$	0	0
$781$	−5.60488	−0.200558
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.6393	−0.808032
$786$	0	0
$787$	−44.3262	−1.58006	−0.790030	−	0.613068i	$-0.789935\pi$
$787$	−44.3262	−1.58006	−0.790030	+	0.613068i	$0.789935\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.944272	0.0335744
$792$	0	0
$793$	−0.875388	−0.0310859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.2918	−1.28552	−0.642761	−	0.766067i	$-0.722211\pi$
$797$	−36.2918	−1.28552	−0.642761	+	0.766067i	$0.722211\pi$
$798$	0	0
$799$	−33.3050	−1.17824
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.3050	−0.398943
$804$	0	0
$805$	10.6525	0.375450
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.0344	0.704373	0.352187	−	0.935930i	$-0.385438\pi$
$809$	20.0344	0.704373	0.352187	+	0.935930i	$0.385438\pi$
$810$	0	0
$811$	50.4508	1.77157	0.885784	−	0.464097i	$-0.153621\pi$
$811$	50.4508	1.77157	0.885784	+	0.464097i	$0.153621\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	25.6525	0.898567
$816$	0	0
$817$	−0.145898	−0.00510433
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.58359	−0.229769	−0.114884	−	0.993379i	$-0.536650\pi$
$821$	−6.58359	−0.229769	−0.114884	+	0.993379i	$0.536650\pi$
$822$	0	0
$823$	−2.58359	−0.0900584	−0.0450292	−	0.998986i	$-0.514338\pi$
$823$	−2.58359	−0.0900584	−0.0450292	+	0.998986i	$0.514338\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.0689	1.39333	0.696666	−	0.717396i	$-0.254666\pi$
$827$	40.0689	1.39333	0.696666	+	0.717396i	$0.254666\pi$
$828$	0	0
$829$	−4.58359	−0.159195	−0.0795974	−	0.996827i	$-0.525363\pi$
$829$	−4.58359	−0.159195	−0.0795974	+	0.996827i	$0.525363\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.47214	−0.224246
$834$	0	0
$835$	−2.76393	−0.0956498
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.9443	1.34451	0.672253	−	0.740322i	$-0.265327\pi$
$839$	38.9443	1.34451	0.672253	+	0.740322i	$0.265327\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.1591	0.590289
$846$	0	0
$847$	−10.2705	−0.352899
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−85.4853	−2.93040
$852$	0	0
$853$	13.0902	0.448199	0.224099	−	0.974566i	$-0.428056\pi$
$853$	13.0902	0.448199	0.224099	+	0.974566i	$0.428056\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−50.0000	−1.70797	−0.853984	−	0.520300i	$-0.825820\pi$
$857$	−50.0000	−1.70797	−0.853984	+	0.520300i	$0.825820\pi$
$858$	0	0
$859$	41.5967	1.41926	0.709631	−	0.704573i	$-0.248862\pi$
$859$	41.5967	1.41926	0.709631	+	0.704573i	$0.248862\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.2705	0.928299	0.464149	−	0.885757i	$-0.346360\pi$
$863$	27.2705	0.928299	0.464149	+	0.885757i	$0.346360\pi$
$864$	0	0
$865$	−30.6525	−1.04222
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.124612	0.00422717
$870$	0	0
$871$	2.24922	0.0762120
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	−54.5623	−1.84244	−0.921219	−	0.389044i	$-0.872805\pi$
$877$	−54.5623	−1.84244	−0.921219	+	0.389044i	$0.872805\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.5967	1.06452	0.532261	−	0.846580i	$-0.321343\pi$
$881$	31.5967	1.06452	0.532261	+	0.846580i	$0.321343\pi$
$882$	0	0
$883$	−49.0344	−1.65014	−0.825070	−	0.565030i	$-0.808864\pi$
$883$	−49.0344	−1.65014	−0.825070	+	0.565030i	$0.808864\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.41641	−0.316172	−0.158086	−	0.987425i	$-0.550532\pi$
$887$	−9.41641	−0.316172	−0.158086	+	0.987425i	$0.550532\pi$
$888$	0	0
$889$	8.85410	0.296957
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.14590	−0.172201
$894$	0	0
$895$	−26.4296	−0.883443
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.1803	0.873163
$900$	0	0
$901$	−36.3607	−1.21135
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.81966	0.126970
$906$	0	0
$907$	−47.0132	−1.56105	−0.780523	−	0.625127i	$-0.785047\pi$
$907$	−47.0132	−1.56105	−0.780523	+	0.625127i	$0.785047\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.97871	−0.0986892	−0.0493446	−	0.998782i	$-0.515713\pi$
$911$	−2.97871	−0.0986892	−0.0493446	+	0.998782i	$0.515713\pi$
$912$	0	0
$913$	3.41641	0.113067
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.180340	−0.00595535
$918$	0	0
$919$	−9.59675	−0.316567	−0.158284	−	0.987394i	$-0.550596\pi$
$919$	−9.59675	−0.316567	−0.158284	+	0.987394i	$0.550596\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.01316	0.165010
$924$	0	0
$925$	−34.2705	−1.12681
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.23607	0.237408	0.118704	−	0.992930i	$-0.462126\pi$
$929$	7.23607	0.237408	0.118704	+	0.992930i	$0.462126\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.63932	−0.249832
$936$	0	0
$937$	9.88854	0.323045	0.161522	−	0.986869i	$-0.448360\pi$
$937$	9.88854	0.323045	0.161522	+	0.986869i	$0.448360\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.54102	−0.278429	−0.139215	−	0.990262i	$-0.544458\pi$
$941$	−8.54102	−0.278429	−0.139215	+	0.990262i	$0.544458\pi$
$942$	0	0
$943$	39.2361	1.27770
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.6869	1.54962	0.774808	−	0.632196i	$-0.217846\pi$
$947$	47.6869	1.54962	0.774808	+	0.632196i	$0.217846\pi$
$948$	0	0
$949$	10.1115	0.328232
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.5279	0.891715	0.445857	−	0.895104i	$-0.352899\pi$
$953$	27.5279	0.891715	0.445857	+	0.895104i	$0.352899\pi$
$954$	0	0
$955$	−16.1803	−0.523584
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.56231	0.0504495
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.81966	−0.122959
$966$	0	0
$967$	11.4164	0.367127	0.183563	−	0.983008i	$-0.441237\pi$
$967$	11.4164	0.367127	0.183563	+	0.983008i	$0.441237\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.5623	0.788242	0.394121	−	0.919059i	$-0.371049\pi$
$971$	24.5623	0.788242	0.394121	+	0.919059i	$0.371049\pi$
$972$	0	0
$973$	6.47214	0.207487
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56.0689	−1.79380	−0.896901	−	0.442231i	$-0.854187\pi$
$977$	−56.0689	−1.79380	−0.896901	+	0.442231i	$0.854187\pi$
$978$	0	0
$979$	−8.61803	−0.275434
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.9443	1.56108	0.780540	−	0.625106i	$-0.214944\pi$
$983$	48.9443	1.56108	0.780540	+	0.625106i	$0.214944\pi$
$984$	0	0
$985$	24.8754	0.792596
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.12461	−0.0357606
$990$	0	0
$991$	39.3262	1.24924	0.624620	−	0.780929i	$-0.285254\pi$
$991$	39.3262	1.24924	0.624620	+	0.780929i	$0.285254\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38.7426	1.22822
$996$	0	0
$997$	41.4508	1.31276	0.656381	−	0.754430i	$-0.272086\pi$
$997$	41.4508	1.31276	0.656381	+	0.754430i	$0.272086\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bc.1.2		2
3.2	odd	2		1064.2.a.e.1.1	✓	2
12.11	even	2		2128.2.a.d.1.2		2
21.20	even	2		7448.2.a.y.1.2		2
24.5	odd	2		8512.2.a.e.1.2		2
24.11	even	2		8512.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.e.1.1	✓	2	3.2	odd	2
2128.2.a.d.1.2		2	12.11	even	2
7448.2.a.y.1.2		2	21.20	even	2
8512.2.a.e.1.2		2	24.5	odd	2
8512.2.a.ba.1.1		2	24.11	even	2
9576.2.a.bc.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.38197 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.38197 q^{5} +1.00000 q^{7} -0.854102 q^{11} +0.763932 q^{13} -6.47214 q^{17} -1.00000 q^{19} -7.70820 q^{23} -3.09017 q^{25} -5.85410 q^{29} -4.47214 q^{31} -1.38197 q^{35} +11.0902 q^{37} -5.09017 q^{41} +0.145898 q^{43} +5.14590 q^{47} +1.00000 q^{49} +5.61803 q^{53} +1.18034 q^{55} +7.09017 q^{59} -1.14590 q^{61} -1.05573 q^{65} +2.94427 q^{67} +6.56231 q^{71} +13.2361 q^{73} -0.854102 q^{77} -0.145898 q^{79} -4.00000 q^{83} +8.94427 q^{85} +10.0902 q^{89} +0.763932 q^{91} +1.38197 q^{95} +16.5623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 5 * q^5 + 2 * q^7	\( 2 q - 5 q^{5} + 2 q^{7} + 5 q^{11} + 6 q^{13} - 4 q^{17} - 2 q^{19} - 2 q^{23} + 5 q^{25} - 5 q^{29} - 5 q^{35} + 11 q^{37} + q^{41} + 7 q^{43} + 17 q^{47} + 2 q^{49} + 9 q^{53} - 20 q^{55} + 3 q^{59} - 9 q^{61} - 20 q^{65} - 12 q^{67} - 7 q^{71} + 22 q^{73} + 5 q^{77} - 7 q^{79} - 8 q^{83} + 9 q^{89} + 6 q^{91} + 5 q^{95} + 13 q^{97}+O(q^{100}) \) 2 * q - 5 * q^5 + 2 * q^7 + 5 * q^11 + 6 * q^13 - 4 * q^17 - 2 * q^19 - 2 * q^23 + 5 * q^25 - 5 * q^29 - 5 * q^35 + 11 * q^37 + q^41 + 7 * q^43 + 17 * q^47 + 2 * q^49 + 9 * q^53 - 20 * q^55 + 3 * q^59 - 9 * q^61 - 20 * q^65 - 12 * q^67 - 7 * q^71 + 22 * q^73 + 5 * q^77 - 7 * q^79 - 8 * q^83 + 9 * q^89 + 6 * q^91 + 5 * q^95 + 13 * q^97

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