LMFDB - Embedded newform 5175.2.a.bc.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5175,2,Mod(1,5175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,4,-6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.3225830460$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} +3.41421 q^{7} -4.41421 q^{8} -2.82843 q^{11} -8.24264 q^{14} +3.00000 q^{16} -7.41421 q^{17} -6.24264 q^{19} +6.82843 q^{22} -1.00000 q^{23} +13.0711 q^{28} +8.48528 q^{29} +8.48528 q^{31} +1.58579 q^{32} +17.8995 q^{34} +4.82843 q^{37} +15.0711 q^{38} -1.65685 q^{41} +1.75736 q^{43} -10.8284 q^{44} +2.41421 q^{46} +0.343146 q^{47} +4.65685 q^{49} +5.07107 q^{53} -15.0711 q^{56} -20.4853 q^{58} +7.65685 q^{59} -0.828427 q^{61} -20.4853 q^{62} -9.82843 q^{64} -8.58579 q^{67} -28.3848 q^{68} -13.6569 q^{71} -13.3137 q^{73} -11.6569 q^{74} -23.8995 q^{76} -9.65685 q^{77} +7.89949 q^{79} +4.00000 q^{82} -6.82843 q^{83} -4.24264 q^{86} +12.4853 q^{88} +13.0711 q^{89} -3.82843 q^{92} -0.828427 q^{94} +10.0000 q^{97} -11.2426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} - 8 q^{14} + 6 q^{16} - 12 q^{17} - 4 q^{19} + 8 q^{22} - 2 q^{23} + 12 q^{28} + 6 q^{32} + 16 q^{34} + 4 q^{37} + 16 q^{38} + 8 q^{41} + 12 q^{43} - 16 q^{44}+ \cdots - 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 - 8 * q^14 + 6 * q^16 - 12 * q^17 - 4 * q^19 + 8 * q^22 - 2 * q^23 + 12 * q^28 + 6 * q^32 + 16 * q^34 + 4 * q^37 + 16 * q^38 + 8 * q^41 + 12 * q^43 - 16 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^49 - 4 * q^53 - 16 * q^56 - 24 * q^58 + 4 * q^59 + 4 * q^61 - 24 * q^62 - 14 * q^64 - 20 * q^67 - 20 * q^68 - 16 * q^71 - 4 * q^73 - 12 * q^74 - 28 * q^76 - 8 * q^77 - 4 * q^79 + 8 * q^82 - 8 * q^83 + 8 * q^88 + 12 * q^89 - 2 * q^92 + 4 * q^94 + 20 * q^97 - 14 * q^98

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−8.24264	−2.20294
$15$	0	0
$16$	3.00000	0.750000
$17$	−7.41421	−1.79821	−0.899105	−	0.437732i	$-0.855782\pi$
$17$	−7.41421	−1.79821	−0.899105	+	0.437732i	$0.855782\pi$
$18$	0	0
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	0	0
$21$	0	0
$22$	6.82843	1.45583
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	13.0711	2.47020
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	17.8995	3.06974
$35$	0	0
$36$	0	0
$37$	4.82843	0.793789	0.396894	−	0.917864i	$-0.370088\pi$
$37$	4.82843	0.793789	0.396894	+	0.917864i	$0.370088\pi$
$38$	15.0711	2.44485
$39$	0	0
$40$	0	0
$41$	−1.65685	−0.258757	−0.129379	−	0.991595i	$-0.541298\pi$
$41$	−1.65685	−0.258757	−0.129379	+	0.991595i	$0.541298\pi$
$42$	0	0
$43$	1.75736	0.267995	0.133997	−	0.990982i	$-0.457219\pi$
$43$	1.75736	0.267995	0.133997	+	0.990982i	$0.457219\pi$
$44$	−10.8284	−1.63245
$45$	0	0
$46$	2.41421	0.355956
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.07107	0.696565	0.348282	−	0.937390i	$-0.386765\pi$
$53$	5.07107	0.696565	0.348282	+	0.937390i	$0.386765\pi$
$54$	0	0
$55$	0	0
$56$	−15.0711	−2.01396
$57$	0	0
$58$	−20.4853	−2.68985
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	−0.828427	−0.106069	−0.0530346	−	0.998593i	$-0.516889\pi$
$61$	−0.828427	−0.106069	−0.0530346	+	0.998593i	$0.516889\pi$
$62$	−20.4853	−2.60163
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	−8.58579	−1.04892	−0.524460	−	0.851435i	$-0.675733\pi$
$67$	−8.58579	−1.04892	−0.524460	+	0.851435i	$0.675733\pi$
$68$	−28.3848	−3.44216
$69$	0	0
$70$	0	0
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	−11.6569	−1.35508
$75$	0	0
$76$	−23.8995	−2.74146
$77$	−9.65685	−1.10050
$78$	0	0
$79$	7.89949	0.888763	0.444381	−	0.895838i	$-0.353424\pi$
$79$	7.89949	0.888763	0.444381	+	0.895838i	$0.353424\pi$
$80$	0	0
$81$	0	0
$82$	4.00000	0.441726
$83$	−6.82843	−0.749517	−0.374759	−	0.927122i	$-0.622274\pi$
$83$	−6.82843	−0.749517	−0.374759	+	0.927122i	$0.622274\pi$
$84$	0	0
$85$	0	0
$86$	−4.24264	−0.457496
$87$	0	0
$88$	12.4853	1.33094
$89$	13.0711	1.38553	0.692765	−	0.721163i	$-0.256392\pi$
$89$	13.0711	1.38553	0.692765	+	0.721163i	$0.256392\pi$
$90$	0	0
$91$	0	0
$92$	−3.82843	−0.399141
$93$	0	0
$94$	−0.828427	−0.0854457
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−11.2426	−1.13568
$99$	0	0
$100$	0	0
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	3.41421	0.336412	0.168206	−	0.985752i	$-0.446203\pi$
$103$	3.41421	0.336412	0.168206	+	0.985752i	$0.446203\pi$
$104$	0	0
$105$	0	0
$106$	−12.2426	−1.18911
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	0	0
$111$	0	0
$112$	10.2426	0.967839
$113$	−15.4142	−1.45005	−0.725024	−	0.688724i	$-0.758171\pi$
$113$	−15.4142	−1.45005	−0.725024	+	0.688724i	$0.758171\pi$
$114$	0	0
$115$	0	0
$116$	32.4853	3.01618
$117$	0	0
$118$	−18.4853	−1.70171
$119$	−25.3137	−2.32050
$120$	0	0
$121$	−3.00000	−0.272727
$122$	2.00000	0.181071
$123$	0	0
$124$	32.4853	2.91726
$125$	0	0
$126$	0	0
$127$	4.48528	0.398004	0.199002	−	0.979999i	$-0.436230\pi$
$127$	4.48528	0.398004	0.199002	+	0.979999i	$0.436230\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	−21.3137	−1.84813
$134$	20.7279	1.79062
$135$	0	0
$136$	32.7279	2.80640
$137$	−0.585786	−0.0500471	−0.0250236	−	0.999687i	$-0.507966\pi$
$137$	−0.585786	−0.0500471	−0.0250236	+	0.999687i	$0.507966\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	32.9706	2.76683
$143$	0	0
$144$	0	0
$145$	0	0
$146$	32.1421	2.66010
$147$	0	0
$148$	18.4853	1.51948
$149$	−16.5858	−1.35876	−0.679380	−	0.733786i	$-0.737751\pi$
$149$	−16.5858	−1.35876	−0.679380	+	0.733786i	$0.737751\pi$
$150$	0	0
$151$	−9.65685	−0.785864	−0.392932	−	0.919568i	$-0.628539\pi$
$151$	−9.65685	−0.785864	−0.392932	+	0.919568i	$0.628539\pi$
$152$	27.5563	2.23512
$153$	0	0
$154$	23.3137	1.87867
$155$	0	0
$156$	0	0
$157$	1.51472	0.120888	0.0604439	−	0.998172i	$-0.480748\pi$
$157$	1.51472	0.120888	0.0604439	+	0.998172i	$0.480748\pi$
$158$	−19.0711	−1.51721
$159$	0	0
$160$	0	0
$161$	−3.41421	−0.269078
$162$	0	0
$163$	−18.8284	−1.47476	−0.737378	−	0.675480i	$-0.763936\pi$
$163$	−18.8284	−1.47476	−0.737378	+	0.675480i	$0.763936\pi$
$164$	−6.34315	−0.495316
$165$	0	0
$166$	16.4853	1.27951
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	6.72792	0.512999
$173$	−10.8284	−0.823270	−0.411635	−	0.911349i	$-0.635042\pi$
$173$	−10.8284	−0.823270	−0.411635	+	0.911349i	$0.635042\pi$
$174$	0	0
$175$	0	0
$176$	−8.48528	−0.639602
$177$	0	0
$178$	−31.5563	−2.36525
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−11.1716	−0.830376	−0.415188	−	0.909736i	$-0.636284\pi$
$181$	−11.1716	−0.830376	−0.415188	+	0.909736i	$0.636284\pi$
$182$	0	0
$183$	0	0
$184$	4.41421	0.325420
$185$	0	0
$186$	0	0
$187$	20.9706	1.53352
$188$	1.31371	0.0958120
$189$	0	0
$190$	0	0
$191$	−0.686292	−0.0496583	−0.0248292	−	0.999692i	$-0.507904\pi$
$191$	−0.686292	−0.0496583	−0.0248292	+	0.999692i	$0.507904\pi$
$192$	0	0
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	−24.1421	−1.73330
$195$	0	0
$196$	17.8284	1.27346
$197$	9.17157	0.653448	0.326724	−	0.945120i	$-0.394055\pi$
$197$	9.17157	0.653448	0.326724	+	0.945120i	$0.394055\pi$
$198$	0	0
$199$	−2.24264	−0.158977	−0.0794883	−	0.996836i	$-0.525329\pi$
$199$	−2.24264	−0.158977	−0.0794883	+	0.996836i	$0.525329\pi$
$200$	0	0
$201$	0	0
$202$	27.3137	1.92179
$203$	28.9706	2.03333
$204$	0	0
$205$	0	0
$206$	−8.24264	−0.574292
$207$	0	0
$208$	0	0
$209$	17.6569	1.22135
$210$	0	0
$211$	−12.4853	−0.859522	−0.429761	−	0.902943i	$-0.641402\pi$
$211$	−12.4853	−0.859522	−0.429761	+	0.902943i	$0.641402\pi$
$212$	19.4142	1.33337
$213$	0	0
$214$	−20.4853	−1.40035
$215$	0	0
$216$	0	0
$217$	28.9706	1.96665
$218$	6.00000	0.406371
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.9706	0.868573	0.434287	−	0.900775i	$-0.357001\pi$
$223$	12.9706	0.868573	0.434287	+	0.900775i	$0.357001\pi$
$224$	5.41421	0.361752
$225$	0	0
$226$	37.2132	2.47539
$227$	6.14214	0.407668	0.203834	−	0.979005i	$-0.434660\pi$
$227$	6.14214	0.407668	0.203834	+	0.979005i	$0.434660\pi$
$228$	0	0
$229$	14.4853	0.957214	0.478607	−	0.878029i	$-0.341142\pi$
$229$	14.4853	0.957214	0.478607	+	0.878029i	$0.341142\pi$
$230$	0	0
$231$	0	0
$232$	−37.4558	−2.45910
$233$	−21.6569	−1.41879	−0.709394	−	0.704812i	$-0.751031\pi$
$233$	−21.6569	−1.41879	−0.709394	+	0.704812i	$0.751031\pi$
$234$	0	0
$235$	0	0
$236$	29.3137	1.90816
$237$	0	0
$238$	61.1127	3.96135
$239$	−19.3137	−1.24930	−0.624650	−	0.780905i	$-0.714758\pi$
$239$	−19.3137	−1.24930	−0.624650	+	0.780905i	$0.714758\pi$
$240$	0	0
$241$	−22.9706	−1.47966	−0.739832	−	0.672792i	$-0.765095\pi$
$241$	−22.9706	−1.47966	−0.739832	+	0.672792i	$0.765095\pi$
$242$	7.24264	0.465575
$243$	0	0
$244$	−3.17157	−0.203039
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−37.4558	−2.37845
$249$	0	0
$250$	0	0
$251$	−20.4853	−1.29302	−0.646510	−	0.762906i	$-0.723772\pi$
$251$	−20.4853	−1.29302	−0.646510	+	0.762906i	$0.723772\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	−10.8284	−0.679436
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−6.34315	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$257$	−6.34315	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$258$	0	0
$259$	16.4853	1.02435
$260$	0	0
$261$	0	0
$262$	−40.9706	−2.53117
$263$	−23.3137	−1.43758	−0.718792	−	0.695225i	$-0.755305\pi$
$263$	−23.3137	−1.43758	−0.718792	+	0.695225i	$0.755305\pi$
$264$	0	0
$265$	0	0
$266$	51.4558	3.15496
$267$	0	0
$268$	−32.8701	−2.00786
$269$	18.1421	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$269$	18.1421	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$270$	0	0
$271$	−12.4853	−0.758427	−0.379213	−	0.925309i	$-0.623805\pi$
$271$	−12.4853	−0.758427	−0.379213	+	0.925309i	$0.623805\pi$
$272$	−22.2426	−1.34866
$273$	0	0
$274$	1.41421	0.0854358
$275$	0	0
$276$	0	0
$277$	−7.65685	−0.460056	−0.230028	−	0.973184i	$-0.573882\pi$
$277$	−7.65685	−0.460056	−0.230028	+	0.973184i	$0.573882\pi$
$278$	9.65685	0.579180
$279$	0	0
$280$	0	0
$281$	8.58579	0.512185	0.256093	−	0.966652i	$-0.417565\pi$
$281$	8.58579	0.512185	0.256093	+	0.966652i	$0.417565\pi$
$282$	0	0
$283$	−15.2132	−0.904331	−0.452166	−	0.891934i	$-0.649348\pi$
$283$	−15.2132	−0.904331	−0.452166	+	0.891934i	$0.649348\pi$
$284$	−52.2843	−3.10250
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	37.9706	2.23356
$290$	0	0
$291$	0	0
$292$	−50.9706	−2.98283
$293$	−7.41421	−0.433143	−0.216571	−	0.976267i	$-0.569487\pi$
$293$	−7.41421	−0.433143	−0.216571	+	0.976267i	$0.569487\pi$
$294$	0	0
$295$	0	0
$296$	−21.3137	−1.23883
$297$	0	0
$298$	40.0416	2.31955
$299$	0	0
$300$	0	0
$301$	6.00000	0.345834
$302$	23.3137	1.34155
$303$	0	0
$304$	−18.7279	−1.07412
$305$	0	0
$306$	0	0
$307$	−10.8284	−0.618011	−0.309005	−	0.951060i	$-0.599996\pi$
$307$	−10.8284	−0.618011	−0.309005	+	0.951060i	$0.599996\pi$
$308$	−36.9706	−2.10659
$309$	0	0
$310$	0	0
$311$	−9.31371	−0.528132	−0.264066	−	0.964505i	$-0.585064\pi$
$311$	−9.31371	−0.528132	−0.264066	+	0.964505i	$0.585064\pi$
$312$	0	0
$313$	−1.31371	−0.0742552	−0.0371276	−	0.999311i	$-0.511821\pi$
$313$	−1.31371	−0.0742552	−0.0371276	+	0.999311i	$0.511821\pi$
$314$	−3.65685	−0.206368
$315$	0	0
$316$	30.2426	1.70128
$317$	25.4558	1.42974	0.714871	−	0.699256i	$-0.246485\pi$
$317$	25.4558	1.42974	0.714871	+	0.699256i	$0.246485\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	8.24264	0.459344
$323$	46.2843	2.57533
$324$	0	0
$325$	0	0
$326$	45.4558	2.51757
$327$	0	0
$328$	7.31371	0.403832
$329$	1.17157	0.0645909
$330$	0	0
$331$	−6.82843	−0.375324	−0.187662	−	0.982234i	$-0.560091\pi$
$331$	−6.82843	−0.375324	−0.187662	+	0.982234i	$0.560091\pi$
$332$	−26.1421	−1.43474
$333$	0	0
$334$	−13.6569	−0.747270
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	31.3848	1.70711
$339$	0	0
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−7.75736	−0.418249
$345$	0	0
$346$	26.1421	1.40541
$347$	−25.3137	−1.35891	−0.679456	−	0.733717i	$-0.737784\pi$
$347$	−25.3137	−1.35891	−0.679456	+	0.733717i	$0.737784\pi$
$348$	0	0
$349$	−20.9706	−1.12253	−0.561264	−	0.827637i	$-0.689685\pi$
$349$	−20.9706	−1.12253	−0.561264	+	0.827637i	$0.689685\pi$
$350$	0	0
$351$	0	0
$352$	−4.48528	−0.239066
$353$	−5.85786	−0.311783	−0.155891	−	0.987774i	$-0.549825\pi$
$353$	−5.85786	−0.311783	−0.155891	+	0.987774i	$0.549825\pi$
$354$	0	0
$355$	0	0
$356$	50.0416	2.65220
$357$	0	0
$358$	43.4558	2.29671
$359$	32.2843	1.70390	0.851949	−	0.523624i	$-0.175420\pi$
$359$	32.2843	1.70390	0.851949	+	0.523624i	$0.175420\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	26.9706	1.41754
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.5563	−0.707636	−0.353818	−	0.935314i	$-0.615117\pi$
$367$	−13.5563	−0.707636	−0.353818	+	0.935314i	$0.615117\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	0	0
$371$	17.3137	0.898883
$372$	0	0
$373$	−6.48528	−0.335795	−0.167898	−	0.985804i	$-0.553698\pi$
$373$	−6.48528	−0.335795	−0.167898	+	0.985804i	$0.553698\pi$
$374$	−50.6274	−2.61788
$375$	0	0
$376$	−1.51472	−0.0781156
$377$	0	0
$378$	0	0
$379$	0.585786	0.0300898	0.0150449	−	0.999887i	$-0.495211\pi$
$379$	0.585786	0.0300898	0.0150449	+	0.999887i	$0.495211\pi$
$380$	0	0
$381$	0	0
$382$	1.65685	0.0847720
$383$	−21.6569	−1.10661	−0.553307	−	0.832978i	$-0.686634\pi$
$383$	−21.6569	−1.10661	−0.553307	+	0.832978i	$0.686634\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	38.2843	1.94359
$389$	−3.89949	−0.197712	−0.0988561	−	0.995102i	$-0.531518\pi$
$389$	−3.89949	−0.197712	−0.0988561	+	0.995102i	$0.531518\pi$
$390$	0	0
$391$	7.41421	0.374953
$392$	−20.5563	−1.03825
$393$	0	0
$394$	−22.1421	−1.11550
$395$	0	0
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	5.41421	0.271390
$399$	0	0
$400$	0	0
$401$	30.0416	1.50021	0.750104	−	0.661320i	$-0.230004\pi$
$401$	30.0416	1.50021	0.750104	+	0.661320i	$0.230004\pi$
$402$	0	0
$403$	0	0
$404$	−43.3137	−2.15494
$405$	0	0
$406$	−69.9411	−3.47112
$407$	−13.6569	−0.676945
$408$	0	0
$409$	14.3431	0.709223	0.354611	−	0.935014i	$-0.384613\pi$
$409$	14.3431	0.709223	0.354611	+	0.935014i	$0.384613\pi$
$410$	0	0
$411$	0	0
$412$	13.0711	0.643965
$413$	26.1421	1.28637
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−42.6274	−2.08498
$419$	20.4853	1.00077	0.500386	−	0.865803i	$-0.333192\pi$
$419$	20.4853	1.00077	0.500386	+	0.865803i	$0.333192\pi$
$420$	0	0
$421$	−39.4558	−1.92296	−0.961480	−	0.274875i	$-0.911364\pi$
$421$	−39.4558	−1.92296	−0.961480	+	0.274875i	$0.911364\pi$
$422$	30.1421	1.46730
$423$	0	0
$424$	−22.3848	−1.08710
$425$	0	0
$426$	0	0
$427$	−2.82843	−0.136877
$428$	32.4853	1.57024
$429$	0	0
$430$	0	0
$431$	8.68629	0.418404	0.209202	−	0.977872i	$-0.432913\pi$
$431$	8.68629	0.418404	0.209202	+	0.977872i	$0.432913\pi$
$432$	0	0
$433$	3.65685	0.175737	0.0878686	−	0.996132i	$-0.471994\pi$
$433$	3.65685	0.175737	0.0878686	+	0.996132i	$0.471994\pi$
$434$	−69.9411	−3.35728
$435$	0	0
$436$	−9.51472	−0.455672
$437$	6.24264	0.298626
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	−31.3137	−1.48275
$447$	0	0
$448$	−33.5563	−1.58539
$449$	10.6274	0.501539	0.250769	−	0.968047i	$-0.419316\pi$
$449$	10.6274	0.501539	0.250769	+	0.968047i	$0.419316\pi$
$450$	0	0
$451$	4.68629	0.220669
$452$	−59.0122	−2.77570
$453$	0	0
$454$	−14.8284	−0.695933
$455$	0	0
$456$	0	0
$457$	−34.9706	−1.63585	−0.817927	−	0.575322i	$-0.804877\pi$
$457$	−34.9706	−1.63585	−0.817927	+	0.575322i	$0.804877\pi$
$458$	−34.9706	−1.63407
$459$	0	0
$460$	0	0
$461$	−9.17157	−0.427163	−0.213581	−	0.976925i	$-0.568513\pi$
$461$	−9.17157	−0.427163	−0.213581	+	0.976925i	$0.568513\pi$
$462$	0	0
$463$	−8.97056	−0.416897	−0.208449	−	0.978033i	$-0.566841\pi$
$463$	−8.97056	−0.416897	−0.208449	+	0.978033i	$0.566841\pi$
$464$	25.4558	1.18176
$465$	0	0
$466$	52.2843	2.42202
$467$	−5.85786	−0.271070	−0.135535	−	0.990773i	$-0.543275\pi$
$467$	−5.85786	−0.271070	−0.135535	+	0.990773i	$0.543275\pi$
$468$	0	0
$469$	−29.3137	−1.35358
$470$	0	0
$471$	0	0
$472$	−33.7990	−1.55572
$473$	−4.97056	−0.228547
$474$	0	0
$475$	0	0
$476$	−96.9117	−4.44194
$477$	0	0
$478$	46.6274	2.13269
$479$	16.6863	0.762416	0.381208	−	0.924489i	$-0.375508\pi$
$479$	16.6863	0.762416	0.381208	+	0.924489i	$0.375508\pi$
$480$	0	0
$481$	0	0
$482$	55.4558	2.52594
$483$	0	0
$484$	−11.4853	−0.522058
$485$	0	0
$486$	0	0
$487$	−8.97056	−0.406495	−0.203247	−	0.979127i	$-0.565150\pi$
$487$	−8.97056	−0.406495	−0.203247	+	0.979127i	$0.565150\pi$
$488$	3.65685	0.165538
$489$	0	0
$490$	0	0
$491$	6.68629	0.301748	0.150874	−	0.988553i	$-0.451791\pi$
$491$	6.68629	0.301748	0.150874	+	0.988553i	$0.451791\pi$
$492$	0	0
$493$	−62.9117	−2.83340
$494$	0	0
$495$	0	0
$496$	25.4558	1.14300
$497$	−46.6274	−2.09153
$498$	0	0
$499$	−24.4853	−1.09611	−0.548056	−	0.836442i	$-0.684632\pi$
$499$	−24.4853	−1.09611	−0.548056	+	0.836442i	$0.684632\pi$
$500$	0	0
$501$	0	0
$502$	49.4558	2.20732
$503$	−17.6569	−0.787280	−0.393640	−	0.919265i	$-0.628784\pi$
$503$	−17.6569	−0.787280	−0.393640	+	0.919265i	$0.628784\pi$
$504$	0	0
$505$	0	0
$506$	−6.82843	−0.303561
$507$	0	0
$508$	17.1716	0.761865
$509$	−38.1421	−1.69062	−0.845310	−	0.534276i	$-0.820584\pi$
$509$	−38.1421	−1.69062	−0.845310	+	0.534276i	$0.820584\pi$
$510$	0	0
$511$	−45.4558	−2.01085
$512$	31.2426	1.38074
$513$	0	0
$514$	15.3137	0.675459
$515$	0	0
$516$	0	0
$517$	−0.970563	−0.0426853
$518$	−39.7990	−1.74867
$519$	0	0
$520$	0	0
$521$	18.7279	0.820485	0.410243	−	0.911976i	$-0.365444\pi$
$521$	18.7279	0.820485	0.410243	+	0.911976i	$0.365444\pi$
$522$	0	0
$523$	35.6985	1.56099	0.780493	−	0.625165i	$-0.214968\pi$
$523$	35.6985	1.56099	0.780493	+	0.625165i	$0.214968\pi$
$524$	64.9706	2.83825
$525$	0	0
$526$	56.2843	2.45411
$527$	−62.9117	−2.74048
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−81.5980	−3.53772
$533$	0	0
$534$	0	0
$535$	0	0
$536$	37.8995	1.63701
$537$	0	0
$538$	−43.7990	−1.88831
$539$	−13.1716	−0.567340
$540$	0	0
$541$	4.97056	0.213701	0.106851	−	0.994275i	$-0.465923\pi$
$541$	4.97056	0.213701	0.106851	+	0.994275i	$0.465923\pi$
$542$	30.1421	1.29472
$543$	0	0
$544$	−11.7574	−0.504093
$545$	0	0
$546$	0	0
$547$	−12.4853	−0.533832	−0.266916	−	0.963720i	$-0.586005\pi$
$547$	−12.4853	−0.533832	−0.266916	+	0.963720i	$0.586005\pi$
$548$	−2.24264	−0.0958009
$549$	0	0
$550$	0	0
$551$	−52.9706	−2.25662
$552$	0	0
$553$	26.9706	1.14690
$554$	18.4853	0.785364
$555$	0	0
$556$	−15.3137	−0.649446
$557$	22.2426	0.942451	0.471225	−	0.882013i	$-0.343812\pi$
$557$	22.2426	0.942451	0.471225	+	0.882013i	$0.343812\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−20.7279	−0.874355
$563$	−21.1716	−0.892275	−0.446138	−	0.894964i	$-0.647201\pi$
$563$	−21.1716	−0.892275	−0.446138	+	0.894964i	$0.647201\pi$
$564$	0	0
$565$	0	0
$566$	36.7279	1.54379
$567$	0	0
$568$	60.2843	2.52947
$569$	8.58579	0.359935	0.179967	−	0.983673i	$-0.442401\pi$
$569$	8.58579	0.359935	0.179967	+	0.983673i	$0.442401\pi$
$570$	0	0
$571$	1.75736	0.0735432	0.0367716	−	0.999324i	$-0.488293\pi$
$571$	1.75736	0.0735432	0.0367716	+	0.999324i	$0.488293\pi$
$572$	0	0
$573$	0	0
$574$	13.6569	0.570026
$575$	0	0
$576$	0	0
$577$	32.2843	1.34401	0.672006	−	0.740546i	$-0.265433\pi$
$577$	32.2843	1.34401	0.672006	+	0.740546i	$0.265433\pi$
$578$	−91.6690	−3.81293
$579$	0	0
$580$	0	0
$581$	−23.3137	−0.967216
$582$	0	0
$583$	−14.3431	−0.594032
$584$	58.7696	2.43190
$585$	0	0
$586$	17.8995	0.739421
$587$	−7.02944	−0.290136	−0.145068	−	0.989422i	$-0.546340\pi$
$587$	−7.02944	−0.290136	−0.145068	+	0.989422i	$0.546340\pi$
$588$	0	0
$589$	−52.9706	−2.18261
$590$	0	0
$591$	0	0
$592$	14.4853	0.595341
$593$	−0.686292	−0.0281826	−0.0140913	−	0.999901i	$-0.504486\pi$
$593$	−0.686292	−0.0281826	−0.0140913	+	0.999901i	$0.504486\pi$
$594$	0	0
$595$	0	0
$596$	−63.4975	−2.60096
$597$	0	0
$598$	0	0
$599$	4.68629	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$599$	4.68629	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$600$	0	0
$601$	1.65685	0.0675845	0.0337922	−	0.999429i	$-0.489242\pi$
$601$	1.65685	0.0675845	0.0337922	+	0.999429i	$0.489242\pi$
$602$	−14.4853	−0.590376
$603$	0	0
$604$	−36.9706	−1.50431
$605$	0	0
$606$	0	0
$607$	14.3431	0.582170	0.291085	−	0.956697i	$-0.405984\pi$
$607$	14.3431	0.582170	0.291085	+	0.956697i	$0.405984\pi$
$608$	−9.89949	−0.401478
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−39.4558	−1.59361	−0.796803	−	0.604239i	$-0.793477\pi$
$613$	−39.4558	−1.59361	−0.796803	+	0.604239i	$0.793477\pi$
$614$	26.1421	1.05501
$615$	0	0
$616$	42.6274	1.71751
$617$	33.5563	1.35093	0.675464	−	0.737393i	$-0.263943\pi$
$617$	33.5563	1.35093	0.675464	+	0.737393i	$0.263943\pi$
$618$	0	0
$619$	−22.2426	−0.894007	−0.447004	−	0.894532i	$-0.647509\pi$
$619$	−22.2426	−0.894007	−0.447004	+	0.894532i	$0.647509\pi$
$620$	0	0
$621$	0	0
$622$	22.4853	0.901578
$623$	44.6274	1.78796
$624$	0	0
$625$	0	0
$626$	3.17157	0.126762
$627$	0	0
$628$	5.79899	0.231405
$629$	−35.7990	−1.42740
$630$	0	0
$631$	−24.8701	−0.990061	−0.495031	−	0.868875i	$-0.664843\pi$
$631$	−24.8701	−0.990061	−0.495031	+	0.868875i	$0.664843\pi$
$632$	−34.8701	−1.38706
$633$	0	0
$634$	−61.4558	−2.44072
$635$	0	0
$636$	0	0
$637$	0	0
$638$	57.9411	2.29391
$639$	0	0
$640$	0	0
$641$	−2.92893	−0.115686	−0.0578429	−	0.998326i	$-0.518422\pi$
$641$	−2.92893	−0.115686	−0.0578429	+	0.998326i	$0.518422\pi$
$642$	0	0
$643$	8.38478	0.330663	0.165332	−	0.986238i	$-0.447131\pi$
$643$	8.38478	0.330663	0.165332	+	0.986238i	$0.447131\pi$
$644$	−13.0711	−0.515072
$645$	0	0
$646$	−111.740	−4.39636
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−21.6569	−0.850106
$650$	0	0
$651$	0	0
$652$	−72.0833	−2.82300
$653$	21.6569	0.847498	0.423749	−	0.905780i	$-0.360714\pi$
$653$	21.6569	0.847498	0.423749	+	0.905780i	$0.360714\pi$
$654$	0	0
$655$	0	0
$656$	−4.97056	−0.194068
$657$	0	0
$658$	−2.82843	−0.110264
$659$	34.1421	1.32999	0.664994	−	0.746848i	$-0.268434\pi$
$659$	34.1421	1.32999	0.664994	+	0.746848i	$0.268434\pi$
$660$	0	0
$661$	−11.8579	−0.461217	−0.230609	−	0.973047i	$-0.574072\pi$
$661$	−11.8579	−0.461217	−0.230609	+	0.973047i	$0.574072\pi$
$662$	16.4853	0.640719
$663$	0	0
$664$	30.1421	1.16974
$665$	0	0
$666$	0	0
$667$	−8.48528	−0.328551
$668$	21.6569	0.837929
$669$	0	0
$670$	0	0
$671$	2.34315	0.0904561
$672$	0	0
$673$	40.9706	1.57930	0.789650	−	0.613558i	$-0.210262\pi$
$673$	40.9706	1.57930	0.789650	+	0.613558i	$0.210262\pi$
$674$	53.1127	2.04582
$675$	0	0
$676$	−49.7696	−1.91421
$677$	−35.6985	−1.37200	−0.686002	−	0.727600i	$-0.740636\pi$
$677$	−35.6985	−1.37200	−0.686002	+	0.727600i	$0.740636\pi$
$678$	0	0
$679$	34.1421	1.31025
$680$	0	0
$681$	0	0
$682$	57.9411	2.21868
$683$	10.3431	0.395769	0.197885	−	0.980225i	$-0.436593\pi$
$683$	10.3431	0.395769	0.197885	+	0.980225i	$0.436593\pi$
$684$	0	0
$685$	0	0
$686$	19.3137	0.737401
$687$	0	0
$688$	5.27208	0.200996
$689$	0	0
$690$	0	0
$691$	−22.8284	−0.868434	−0.434217	−	0.900808i	$-0.642975\pi$
$691$	−22.8284	−0.868434	−0.434217	+	0.900808i	$0.642975\pi$
$692$	−41.4558	−1.57591
$693$	0	0
$694$	61.1127	2.31981
$695$	0	0
$696$	0	0
$697$	12.2843	0.465300
$698$	50.6274	1.91628
$699$	0	0
$700$	0	0
$701$	3.89949	0.147282	0.0736409	−	0.997285i	$-0.476538\pi$
$701$	3.89949	0.147282	0.0736409	+	0.997285i	$0.476538\pi$
$702$	0	0
$703$	−30.1421	−1.13683
$704$	27.7990	1.04771
$705$	0	0
$706$	14.1421	0.532246
$707$	−38.6274	−1.45273
$708$	0	0
$709$	−29.7990	−1.11912	−0.559562	−	0.828788i	$-0.689031\pi$
$709$	−29.7990	−1.11912	−0.559562	+	0.828788i	$0.689031\pi$
$710$	0	0
$711$	0	0
$712$	−57.6985	−2.16234
$713$	−8.48528	−0.317776
$714$	0	0
$715$	0	0
$716$	−68.9117	−2.57535
$717$	0	0
$718$	−77.9411	−2.90874
$719$	41.3137	1.54074	0.770371	−	0.637596i	$-0.220071\pi$
$719$	41.3137	1.54074	0.770371	+	0.637596i	$0.220071\pi$
$720$	0	0
$721$	11.6569	0.434124
$722$	−48.2132	−1.79431
$723$	0	0
$724$	−42.7696	−1.58952
$725$	0	0
$726$	0	0
$727$	−37.7574	−1.40034	−0.700171	−	0.713975i	$-0.746893\pi$
$727$	−37.7574	−1.40034	−0.700171	+	0.713975i	$0.746893\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.0294	−0.481911
$732$	0	0
$733$	9.79899	0.361934	0.180967	−	0.983489i	$-0.442077\pi$
$733$	9.79899	0.361934	0.180967	+	0.983489i	$0.442077\pi$
$734$	32.7279	1.20801
$735$	0	0
$736$	−1.58579	−0.0584529
$737$	24.2843	0.894523
$738$	0	0
$739$	43.3137	1.59332	0.796660	−	0.604427i	$-0.206598\pi$
$739$	43.3137	1.59332	0.796660	+	0.604427i	$0.206598\pi$
$740$	0	0
$741$	0	0
$742$	−41.7990	−1.53449
$743$	7.02944	0.257885	0.128943	−	0.991652i	$-0.458842\pi$
$743$	7.02944	0.257885	0.128943	+	0.991652i	$0.458842\pi$
$744$	0	0
$745$	0	0
$746$	15.6569	0.573238
$747$	0	0
$748$	80.2843	2.93548
$749$	28.9706	1.05856
$750$	0	0
$751$	−21.3553	−0.779267	−0.389634	−	0.920970i	$-0.627398\pi$
$751$	−21.3553	−0.779267	−0.389634	+	0.920970i	$0.627398\pi$
$752$	1.02944	0.0375397
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−24.1421	−0.877461	−0.438730	−	0.898619i	$-0.644572\pi$
$757$	−24.1421	−0.877461	−0.438730	+	0.898619i	$0.644572\pi$
$758$	−1.41421	−0.0513665
$759$	0	0
$760$	0	0
$761$	−31.7990	−1.15271	−0.576356	−	0.817199i	$-0.695526\pi$
$761$	−31.7990	−1.15271	−0.576356	+	0.817199i	$0.695526\pi$
$762$	0	0
$763$	−8.48528	−0.307188
$764$	−2.62742	−0.0950566
$765$	0	0
$766$	52.2843	1.88911
$767$	0	0
$768$	0	0
$769$	−34.9706	−1.26107	−0.630535	−	0.776161i	$-0.717165\pi$
$769$	−34.9706	−1.26107	−0.630535	+	0.776161i	$0.717165\pi$
$770$	0	0
$771$	0	0
$772$	6.34315	0.228295
$773$	−8.38478	−0.301579	−0.150790	−	0.988566i	$-0.548182\pi$
$773$	−8.38478	−0.301579	−0.150790	+	0.988566i	$0.548182\pi$
$774$	0	0
$775$	0	0
$776$	−44.1421	−1.58461
$777$	0	0
$778$	9.41421	0.337516
$779$	10.3431	0.370582
$780$	0	0
$781$	38.6274	1.38220
$782$	−17.8995	−0.640085
$783$	0	0
$784$	13.9706	0.498949
$785$	0	0
$786$	0	0
$787$	27.6985	0.987344	0.493672	−	0.869648i	$-0.335654\pi$
$787$	27.6985	0.987344	0.493672	+	0.869648i	$0.335654\pi$
$788$	35.1127	1.25084
$789$	0	0
$790$	0	0
$791$	−52.6274	−1.87122
$792$	0	0
$793$	0	0
$794$	−62.7696	−2.22761
$795$	0	0
$796$	−8.58579	−0.304315
$797$	37.0711	1.31312	0.656562	−	0.754272i	$-0.272010\pi$
$797$	37.0711	1.31312	0.656562	+	0.754272i	$0.272010\pi$
$798$	0	0
$799$	−2.54416	−0.0900058
$800$	0	0
$801$	0	0
$802$	−72.5269	−2.56101
$803$	37.6569	1.32888
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	49.9411	1.75692
$809$	−19.1127	−0.671967	−0.335983	−	0.941868i	$-0.609069\pi$
$809$	−19.1127	−0.671967	−0.335983	+	0.941868i	$0.609069\pi$
$810$	0	0
$811$	56.7696	1.99345	0.996724	−	0.0808743i	$-0.0257712\pi$
$811$	56.7696	1.99345	0.996724	+	0.0808743i	$0.0257712\pi$
$812$	110.912	3.89224
$813$	0	0
$814$	32.9706	1.15562
$815$	0	0
$816$	0	0
$817$	−10.9706	−0.383811
$818$	−34.6274	−1.21072
$819$	0	0
$820$	0	0
$821$	−11.3137	−0.394851	−0.197426	−	0.980318i	$-0.563258\pi$
$821$	−11.3137	−0.394851	−0.197426	+	0.980318i	$0.563258\pi$
$822$	0	0
$823$	−15.5147	−0.540809	−0.270405	−	0.962747i	$-0.587157\pi$
$823$	−15.5147	−0.540809	−0.270405	+	0.962747i	$0.587157\pi$
$824$	−15.0711	−0.525026
$825$	0	0
$826$	−63.1127	−2.19597
$827$	−7.79899	−0.271197	−0.135599	−	0.990764i	$-0.543296\pi$
$827$	−7.79899	−0.271197	−0.135599	+	0.990764i	$0.543296\pi$
$828$	0	0
$829$	52.9706	1.83974	0.919872	−	0.392219i	$-0.128292\pi$
$829$	52.9706	1.83974	0.919872	+	0.392219i	$0.128292\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−34.5269	−1.19629
$834$	0	0
$835$	0	0
$836$	67.5980	2.33793
$837$	0	0
$838$	−49.4558	−1.70842
$839$	−52.9706	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$839$	−52.9706	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	95.2548	3.28270
$843$	0	0
$844$	−47.7990	−1.64531
$845$	0	0
$846$	0	0
$847$	−10.2426	−0.351941
$848$	15.2132	0.522424
$849$	0	0
$850$	0	0
$851$	−4.82843	−0.165516
$852$	0	0
$853$	−51.9411	−1.77843	−0.889215	−	0.457489i	$-0.848749\pi$
$853$	−51.9411	−1.77843	−0.889215	+	0.457489i	$0.848749\pi$
$854$	6.82843	0.233664
$855$	0	0
$856$	−37.4558	−1.28021
$857$	−29.6569	−1.01306	−0.506529	−	0.862223i	$-0.669072\pi$
$857$	−29.6569	−1.01306	−0.506529	+	0.862223i	$0.669072\pi$
$858$	0	0
$859$	15.0294	0.512798	0.256399	−	0.966571i	$-0.417464\pi$
$859$	15.0294	0.512798	0.256399	+	0.966571i	$0.417464\pi$
$860$	0	0
$861$	0	0
$862$	−20.9706	−0.714260
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	0	0
$866$	−8.82843	−0.300002
$867$	0	0
$868$	110.912	3.76459
$869$	−22.3431	−0.757939
$870$	0	0
$871$	0	0
$872$	10.9706	0.371510
$873$	0	0
$874$	−15.0711	−0.509786
$875$	0	0
$876$	0	0
$877$	43.6569	1.47419	0.737094	−	0.675791i	$-0.236198\pi$
$877$	43.6569	1.47419	0.737094	+	0.675791i	$0.236198\pi$
$878$	−57.9411	−1.95542
$879$	0	0
$880$	0	0
$881$	−13.0711	−0.440375	−0.220188	−	0.975458i	$-0.570667\pi$
$881$	−13.0711	−0.440375	−0.220188	+	0.975458i	$0.570667\pi$
$882$	0	0
$883$	11.5147	0.387501	0.193751	−	0.981051i	$-0.437935\pi$
$883$	11.5147	0.387501	0.193751	+	0.981051i	$0.437935\pi$
$884$	0	0
$885$	0	0
$886$	−57.9411	−1.94657
$887$	−0.343146	−0.0115217	−0.00576085	−	0.999983i	$-0.501834\pi$
$887$	−0.343146	−0.0115217	−0.00576085	+	0.999983i	$0.501834\pi$
$888$	0	0
$889$	15.3137	0.513605
$890$	0	0
$891$	0	0
$892$	49.6569	1.66263
$893$	−2.14214	−0.0716838
$894$	0	0
$895$	0	0
$896$	70.1838	2.34468
$897$	0	0
$898$	−25.6569	−0.856180
$899$	72.0000	2.40133
$900$	0	0
$901$	−37.5980	−1.25257
$902$	−11.3137	−0.376705
$903$	0	0
$904$	68.0416	2.26303
$905$	0	0
$906$	0	0
$907$	35.6985	1.18535	0.592674	−	0.805442i	$-0.298072\pi$
$907$	35.6985	1.18535	0.592674	+	0.805442i	$0.298072\pi$
$908$	23.5147	0.780363
$909$	0	0
$910$	0	0
$911$	−13.3726	−0.443053	−0.221527	−	0.975154i	$-0.571104\pi$
$911$	−13.3726	−0.443053	−0.221527	+	0.975154i	$0.571104\pi$
$912$	0	0
$913$	19.3137	0.639190
$914$	84.4264	2.79258
$915$	0	0
$916$	55.4558	1.83231
$917$	57.9411	1.91338
$918$	0	0
$919$	−3.21320	−0.105994	−0.0529969	−	0.998595i	$-0.516877\pi$
$919$	−3.21320	−0.105994	−0.0529969	+	0.998595i	$0.516877\pi$
$920$	0	0
$921$	0	0
$922$	22.1421	0.729212
$923$	0	0
$924$	0	0
$925$	0	0
$926$	21.6569	0.711688
$927$	0	0
$928$	13.4558	0.441710
$929$	23.3137	0.764898	0.382449	−	0.923977i	$-0.375081\pi$
$929$	23.3137	0.764898	0.382449	+	0.923977i	$0.375081\pi$
$930$	0	0
$931$	−29.0711	−0.952766
$932$	−82.9117	−2.71586
$933$	0	0
$934$	14.1421	0.462745
$935$	0	0
$936$	0	0
$937$	−31.6569	−1.03418	−0.517092	−	0.855930i	$-0.672986\pi$
$937$	−31.6569	−1.03418	−0.517092	+	0.855930i	$0.672986\pi$
$938$	70.7696	2.31071
$939$	0	0
$940$	0	0
$941$	−24.3848	−0.794921	−0.397460	−	0.917619i	$-0.630108\pi$
$941$	−24.3848	−0.794921	−0.397460	+	0.917619i	$0.630108\pi$
$942$	0	0
$943$	1.65685	0.0539546
$944$	22.9706	0.747628
$945$	0	0
$946$	12.0000	0.390154
$947$	−2.68629	−0.0872927	−0.0436464	−	0.999047i	$-0.513897\pi$
$947$	−2.68629	−0.0872927	−0.0436464	+	0.999047i	$0.513897\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	111.740	3.62152
$953$	31.0122	1.00458	0.502292	−	0.864698i	$-0.332490\pi$
$953$	31.0122	1.00458	0.502292	+	0.864698i	$0.332490\pi$
$954$	0	0
$955$	0	0
$956$	−73.9411	−2.39143
$957$	0	0
$958$	−40.2843	−1.30153
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	−87.9411	−2.83239
$965$	0	0
$966$	0	0
$967$	15.5147	0.498920	0.249460	−	0.968385i	$-0.419747\pi$
$967$	15.5147	0.498920	0.249460	+	0.968385i	$0.419747\pi$
$968$	13.2426	0.425635
$969$	0	0
$970$	0	0
$971$	7.11270	0.228257	0.114129	−	0.993466i	$-0.463592\pi$
$971$	7.11270	0.228257	0.114129	+	0.993466i	$0.463592\pi$
$972$	0	0
$973$	−13.6569	−0.437819
$974$	21.6569	0.693930
$975$	0	0
$976$	−2.48528	−0.0795519
$977$	−30.0416	−0.961117	−0.480558	−	0.876963i	$-0.659566\pi$
$977$	−30.0416	−0.961117	−0.480558	+	0.876963i	$0.659566\pi$
$978$	0	0
$979$	−36.9706	−1.18158
$980$	0	0
$981$	0	0
$982$	−16.1421	−0.515116
$983$	−23.3137	−0.743592	−0.371796	−	0.928314i	$-0.621258\pi$
$983$	−23.3137	−0.743592	−0.371796	+	0.928314i	$0.621258\pi$
$984$	0	0
$985$	0	0
$986$	151.882	4.83692
$987$	0	0
$988$	0	0
$989$	−1.75736	−0.0558808
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	13.4558	0.427223
$993$	0	0
$994$	112.569	3.57046
$995$	0	0
$996$	0	0
$997$	−3.02944	−0.0959432	−0.0479716	−	0.998849i	$-0.515276\pi$
$997$	−3.02944	−0.0959432	−0.0479716	+	0.998849i	$0.515276\pi$
$998$	59.1127	1.87118
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bc.1.1		2
3.2	odd	2		5175.2.a.bo.1.2		2
5.4	even	2		207.2.a.e.1.2	yes	2
15.14	odd	2		207.2.a.b.1.1	✓	2
20.19	odd	2		3312.2.a.be.1.1		2
60.59	even	2		3312.2.a.u.1.2		2
115.114	odd	2		4761.2.a.z.1.2		2
345.344	even	2		4761.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.a.b.1.1	✓	2	15.14	odd	2
207.2.a.e.1.2	yes	2	5.4	even	2
3312.2.a.u.1.2		2	60.59	even	2
3312.2.a.be.1.1		2	20.19	odd	2
4761.2.a.k.1.1		2	345.344	even	2
4761.2.a.z.1.2		2	115.114	odd	2
5175.2.a.bc.1.1		2	1.1	even	1	trivial
5175.2.a.bo.1.2		2	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} +3.41421 q^{7} -4.41421 q^{8} -2.82843 q^{11} -8.24264 q^{14} +3.00000 q^{16} -7.41421 q^{17} -6.24264 q^{19} +6.82843 q^{22} -1.00000 q^{23} +13.0711 q^{28} +8.48528 q^{29} +8.48528 q^{31} +1.58579 q^{32} +17.8995 q^{34} +4.82843 q^{37} +15.0711 q^{38} -1.65685 q^{41} +1.75736 q^{43} -10.8284 q^{44} +2.41421 q^{46} +0.343146 q^{47} +4.65685 q^{49} +5.07107 q^{53} -15.0711 q^{56} -20.4853 q^{58} +7.65685 q^{59} -0.828427 q^{61} -20.4853 q^{62} -9.82843 q^{64} -8.58579 q^{67} -28.3848 q^{68} -13.6569 q^{71} -13.3137 q^{73} -11.6569 q^{74} -23.8995 q^{76} -9.65685 q^{77} +7.89949 q^{79} +4.00000 q^{82} -6.82843 q^{83} -4.24264 q^{86} +12.4853 q^{88} +13.0711 q^{89} -3.82843 q^{92} -0.828427 q^{94} +10.0000 q^{97} -11.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} - 8 q^{14} + 6 q^{16} - 12 q^{17} - 4 q^{19} + 8 q^{22} - 2 q^{23} + 12 q^{28} + 6 q^{32} + 16 q^{34} + 4 q^{37} + 16 q^{38} + 8 q^{41} + 12 q^{43} - 16 q^{44}+ \cdots - 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 - 8 * q^14 + 6 * q^16 - 12 * q^17 - 4 * q^19 + 8 * q^22 - 2 * q^23 + 12 * q^28 + 6 * q^32 + 16 * q^34 + 4 * q^37 + 16 * q^38 + 8 * q^41 + 12 * q^43 - 16 * q^44 + 2 * q^46 + 12 * q^47 - 2 * q^49 - 4 * q^53 - 16 * q^56 - 24 * q^58 + 4 * q^59 + 4 * q^61 - 24 * q^62 - 14 * q^64 - 20 * q^67 - 20 * q^68 - 16 * q^71 - 4 * q^73 - 12 * q^74 - 28 * q^76 - 8 * q^77 - 4 * q^79 + 8 * q^82 - 8 * q^83 + 8 * q^88 + 12 * q^89 - 2 * q^92 + 4 * q^94 + 20 * q^97 - 14 * q^98

Introduction

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