LMFDB - Embedded newform 4176.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, 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("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.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:	$33.3455278841$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +2.82843 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +2.82843 q^{7} -0.414214 q^{11} -3.82843 q^{13} -0.828427 q^{17} -6.00000 q^{19} +3.65685 q^{23} -4.00000 q^{25} -1.00000 q^{29} -10.0711 q^{31} +2.82843 q^{35} -4.00000 q^{37} +4.48528 q^{41} -3.58579 q^{43} -3.24264 q^{47} +1.00000 q^{49} -9.48528 q^{53} -0.414214 q^{55} -3.65685 q^{59} -4.82843 q^{61} -3.82843 q^{65} -5.65685 q^{67} -8.82843 q^{71} +4.00000 q^{73} -1.17157 q^{77} +2.41421 q^{79} +7.65685 q^{83} -0.828427 q^{85} +12.4853 q^{89} -10.8284 q^{91} -6.00000 q^{95} +4.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 12 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} - 6 q^{31} - 8 q^{37} - 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 2 q^{53} + 2 q^{55} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 12 q^{71} + 8 q^{73} - 8 q^{77} + 2 q^{79} + 4 q^{83} + 4 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 - 12 * q^19 - 4 * q^23 - 8 * q^25 - 2 * q^29 - 6 * q^31 - 8 * q^37 - 8 * q^41 - 10 * q^43 + 2 * q^47 + 2 * q^49 - 2 * q^53 + 2 * q^55 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 12 * q^71 + 8 * q^73 - 8 * q^77 + 2 * q^79 + 4 * q^83 + 4 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^95 - 8 * 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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.0711	−1.80882	−0.904409	−	0.426667i	$-0.859687\pi$
$31$	−10.0711	−1.80882	−0.904409	+	0.426667i	$0.859687\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.48528	0.700483	0.350242	−	0.936659i	$-0.386099\pi$
$41$	4.48528	0.700483	0.350242	+	0.936659i	$0.386099\pi$
$42$	0	0
$43$	−3.58579	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$43$	−3.58579	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.24264	−0.472988	−0.236494	−	0.971633i	$-0.575998\pi$
$47$	−3.24264	−0.472988	−0.236494	+	0.971633i	$0.575998\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.48528	−1.30290	−0.651452	−	0.758690i	$-0.725840\pi$
$53$	−9.48528	−1.30290	−0.651452	+	0.758690i	$0.725840\pi$
$54$	0	0
$55$	−0.414214	−0.0558525
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.82843	−0.474858
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.17157	−0.133513
$78$	0	0
$79$	2.41421	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$79$	2.41421	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.65685	0.840449	0.420224	−	0.907420i	$-0.361951\pi$
$83$	7.65685	0.840449	0.420224	+	0.907420i	$0.361951\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.4853	1.32344	0.661719	−	0.749752i	$-0.269827\pi$
$89$	12.4853	1.32344	0.661719	+	0.749752i	$0.269827\pi$
$90$	0	0
$91$	−10.8284	−1.13513
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	4.48528	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$97$	4.48528	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.34315	0.233152	0.116576	−	0.993182i	$-0.462808\pi$
$101$	2.34315	0.233152	0.116576	+	0.993182i	$0.462808\pi$
$102$	0	0
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.8284	−1.43352	−0.716759	−	0.697321i	$-0.754375\pi$
$107$	−14.8284	−1.43352	−0.716759	+	0.697321i	$0.754375\pi$
$108$	0	0
$109$	12.6569	1.21231	0.606153	−	0.795348i	$-0.292712\pi$
$109$	12.6569	1.21231	0.606153	+	0.795348i	$0.292712\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.3137	1.25245	0.626224	−	0.779643i	$-0.284599\pi$
$113$	13.3137	1.25245	0.626224	+	0.779643i	$0.284599\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.34315	−0.214796
$120$	0	0
$121$	−10.8284	−0.984402
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	4.34315	0.385392	0.192696	−	0.981259i	$-0.438277\pi$
$127$	4.34315	0.385392	0.192696	+	0.981259i	$0.438277\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.3137	1.86219	0.931094	−	0.364780i	$-0.118856\pi$
$131$	21.3137	1.86219	0.931094	+	0.364780i	$0.118856\pi$
$132$	0	0
$133$	−16.9706	−1.47153
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.58579	0.132610
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.82843	0.641330	0.320665	−	0.947193i	$-0.396094\pi$
$149$	7.82843	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.0711	−0.808928
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.3431	0.815154
$162$	0	0
$163$	−3.92893	−0.307738	−0.153869	−	0.988091i	$-0.549173\pi$
$163$	−3.92893	−0.307738	−0.153869	+	0.988091i	$0.549173\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.17157	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$167$	−3.17157	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.3431	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$173$	−12.3431	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$174$	0	0
$175$	−11.3137	−0.855236
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	8.31371	0.617953	0.308977	−	0.951070i	$-0.400014\pi$
$181$	8.31371	0.617953	0.308977	+	0.951070i	$0.400014\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0.343146	0.0250933
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.3137	1.83164	0.915818	−	0.401594i	$-0.131544\pi$
$191$	25.3137	1.83164	0.915818	+	0.401594i	$0.131544\pi$
$192$	0	0
$193$	−5.17157	−0.372258	−0.186129	−	0.982525i	$-0.559594\pi$
$193$	−5.17157	−0.372258	−0.186129	+	0.982525i	$0.559594\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	4.48528	0.313266
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.48528	0.171911
$210$	0	0
$211$	19.3848	1.33450	0.667252	−	0.744832i	$-0.267471\pi$
$211$	19.3848	1.33450	0.667252	+	0.744832i	$0.267471\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.58579	−0.244549
$216$	0	0
$217$	−28.4853	−1.93371
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.17157	0.213343
$222$	0	0
$223$	3.17157	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$223$	3.17157	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.14214	−0.540413	−0.270206	−	0.962802i	$-0.587092\pi$
$227$	−8.14214	−0.540413	−0.270206	+	0.962802i	$0.587092\pi$
$228$	0	0
$229$	−3.51472	−0.232259	−0.116130	−	0.993234i	$-0.537049\pi$
$229$	−3.51472	−0.232259	−0.116130	+	0.993234i	$0.537049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.3137	−1.19977	−0.599885	−	0.800086i	$-0.704787\pi$
$233$	−18.3137	−1.19977	−0.599885	+	0.800086i	$0.704787\pi$
$234$	0	0
$235$	−3.24264	−0.211527
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.6569	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$239$	−19.6569	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$240$	0	0
$241$	−18.3137	−1.17969	−0.589845	−	0.807517i	$-0.700811\pi$
$241$	−18.3137	−1.17969	−0.589845	+	0.807517i	$0.700811\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	22.9706	1.46158
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.0711	1.26687	0.633437	−	0.773794i	$-0.281643\pi$
$251$	20.0711	1.26687	0.633437	+	0.773794i	$0.281643\pi$
$252$	0	0
$253$	−1.51472	−0.0952295
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1716	1.13351	0.566756	−	0.823886i	$-0.308198\pi$
$257$	18.1716	1.13351	0.566756	+	0.823886i	$0.308198\pi$
$258$	0	0
$259$	−11.3137	−0.703000
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.75736	0.170026	0.0850130	−	0.996380i	$-0.472907\pi$
$263$	2.75736	0.170026	0.0850130	+	0.996380i	$0.472907\pi$
$264$	0	0
$265$	−9.48528	−0.582676
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.4558	−1.91790	−0.958948	−	0.283581i	$-0.908478\pi$
$269$	−31.4558	−1.91790	−0.958948	+	0.283581i	$0.908478\pi$
$270$	0	0
$271$	−16.5563	−1.00573	−0.502863	−	0.864366i	$-0.667720\pi$
$271$	−16.5563	−1.00573	−0.502863	+	0.864366i	$0.667720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.65685	0.0999121
$276$	0	0
$277$	−17.3137	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$277$	−17.3137	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.9706	−1.90720	−0.953602	−	0.301070i	$-0.902656\pi$
$281$	−31.9706	−1.90720	−0.953602	+	0.301070i	$0.902656\pi$
$282$	0	0
$283$	−11.6569	−0.692928	−0.346464	−	0.938063i	$-0.612618\pi$
$283$	−11.6569	−0.692928	−0.346464	+	0.938063i	$0.612618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.6863	0.748848
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	0	0
$295$	−3.65685	−0.212910
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.0000	−0.809641
$300$	0	0
$301$	−10.1421	−0.584583
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.82843	−0.276475
$306$	0	0
$307$	−2.89949	−0.165483	−0.0827415	−	0.996571i	$-0.526368\pi$
$307$	−2.89949	−0.165483	−0.0827415	+	0.996571i	$0.526368\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.68629	0.152326	0.0761628	−	0.997095i	$-0.475733\pi$
$311$	2.68629	0.152326	0.0761628	+	0.997095i	$0.475733\pi$
$312$	0	0
$313$	9.82843	0.555536	0.277768	−	0.960648i	$-0.410405\pi$
$313$	9.82843	0.555536	0.277768	+	0.960648i	$0.410405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.4558	1.76674	0.883368	−	0.468680i	$-0.155270\pi$
$317$	31.4558	1.76674	0.883368	+	0.468680i	$0.155270\pi$
$318$	0	0
$319$	0.414214	0.0231915
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.97056	0.276570
$324$	0	0
$325$	15.3137	0.849452
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.17157	−0.505645
$330$	0	0
$331$	2.41421	0.132697	0.0663486	−	0.997797i	$-0.478865\pi$
$331$	2.41421	0.132697	0.0663486	+	0.997797i	$0.478865\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	21.7990	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$337$	21.7990	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.17157	0.225903
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.48528	0.133417	0.0667084	−	0.997773i	$-0.478750\pi$
$347$	2.48528	0.133417	0.0667084	+	0.997773i	$0.478750\pi$
$348$	0	0
$349$	−5.14214	−0.275252	−0.137626	−	0.990484i	$-0.543947\pi$
$349$	−5.14214	−0.275252	−0.137626	+	0.990484i	$0.543947\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.9706	−1.43550	−0.717749	−	0.696302i	$-0.754828\pi$
$353$	−26.9706	−1.43550	−0.717749	+	0.696302i	$0.754828\pi$
$354$	0	0
$355$	−8.82843	−0.468564
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.92893	0.207361	0.103681	−	0.994611i	$-0.466938\pi$
$359$	3.92893	0.207361	0.103681	+	0.994611i	$0.466938\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26.8284	−1.39286
$372$	0	0
$373$	−26.3137	−1.36247	−0.681236	−	0.732064i	$-0.738557\pi$
$373$	−26.3137	−1.36247	−0.681236	+	0.732064i	$0.738557\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.82843	0.197174
$378$	0	0
$379$	6.97056	0.358054	0.179027	−	0.983844i	$-0.442705\pi$
$379$	6.97056	0.358054	0.179027	+	0.983844i	$0.442705\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.51472	−0.179594	−0.0897969	−	0.995960i	$-0.528622\pi$
$383$	−3.51472	−0.179594	−0.0897969	+	0.995960i	$0.528622\pi$
$384$	0	0
$385$	−1.17157	−0.0597089
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.02944	−0.153599	−0.0767993	−	0.997047i	$-0.524470\pi$
$389$	−3.02944	−0.153599	−0.0767993	+	0.997047i	$0.524470\pi$
$390$	0	0
$391$	−3.02944	−0.153205
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.41421	0.121472
$396$	0	0
$397$	19.3431	0.970805	0.485402	−	0.874291i	$-0.338673\pi$
$397$	19.3431	0.970805	0.485402	+	0.874291i	$0.338673\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.6569	0.931679	0.465839	−	0.884869i	$-0.345752\pi$
$401$	18.6569	0.931679	0.465839	+	0.884869i	$0.345752\pi$
$402$	0	0
$403$	38.5563	1.92063
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.65685	0.0821272
$408$	0	0
$409$	−18.9706	−0.938034	−0.469017	−	0.883189i	$-0.655392\pi$
$409$	−18.9706	−0.938034	−0.469017	+	0.883189i	$0.655392\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.3431	−0.508953
$414$	0	0
$415$	7.65685	0.375860
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.51472	−0.464824	−0.232412	−	0.972617i	$-0.574662\pi$
$419$	−9.51472	−0.464824	−0.232412	+	0.972617i	$0.574662\pi$
$420$	0	0
$421$	37.1127	1.80876	0.904381	−	0.426726i	$-0.140333\pi$
$421$	37.1127	1.80876	0.904381	+	0.426726i	$0.140333\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.31371	0.160738
$426$	0	0
$427$	−13.6569	−0.660901
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.6569	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$431$	19.6569	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$432$	0	0
$433$	30.6274	1.47186	0.735930	−	0.677058i	$-0.236745\pi$
$433$	30.6274	1.47186	0.735930	+	0.677058i	$0.236745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.9411	−1.04959
$438$	0	0
$439$	0.343146	0.0163775	0.00818873	−	0.999966i	$-0.497393\pi$
$439$	0.343146	0.0163775	0.00818873	+	0.999966i	$0.497393\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.3431	−1.15658	−0.578289	−	0.815832i	$-0.696279\pi$
$443$	−24.3431	−1.15658	−0.578289	+	0.815832i	$0.696279\pi$
$444$	0	0
$445$	12.4853	0.591859
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.9706	1.65036	0.825181	−	0.564868i	$-0.191073\pi$
$449$	34.9706	1.65036	0.825181	+	0.564868i	$0.191073\pi$
$450$	0	0
$451$	−1.85786	−0.0874834
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.8284	−0.507644
$456$	0	0
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−38.3553	−1.77487	−0.887437	−	0.460930i	$-0.847516\pi$
$467$	−38.3553	−1.77487	−0.887437	+	0.460930i	$0.847516\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.48528	0.0682933
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.89949	0.315246	0.157623	−	0.987499i	$-0.449617\pi$
$479$	6.89949	0.315246	0.157623	+	0.987499i	$0.449617\pi$
$480$	0	0
$481$	15.3137	0.698245
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.48528	0.203666
$486$	0	0
$487$	11.5147	0.521782	0.260891	−	0.965368i	$-0.415984\pi$
$487$	11.5147	0.521782	0.260891	+	0.965368i	$0.415984\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.2426	−0.958667	−0.479333	−	0.877633i	$-0.659122\pi$
$491$	−21.2426	−0.958667	−0.479333	+	0.877633i	$0.659122\pi$
$492$	0	0
$493$	0.828427	0.0373105
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.9706	−1.12008
$498$	0	0
$499$	−18.9706	−0.849239	−0.424620	−	0.905372i	$-0.639592\pi$
$499$	−18.9706	−0.849239	−0.424620	+	0.905372i	$0.639592\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.272078	0.0121314	0.00606568	−	0.999982i	$-0.498069\pi$
$503$	0.272078	0.0121314	0.00606568	+	0.999982i	$0.498069\pi$
$504$	0	0
$505$	2.34315	0.104269
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.5147	0.466057	0.233028	−	0.972470i	$-0.425137\pi$
$509$	10.5147	0.466057	0.233028	+	0.972470i	$0.425137\pi$
$510$	0	0
$511$	11.3137	0.500489
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.82843	0.212766
$516$	0	0
$517$	1.34315	0.0590715
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.1421	1.27674	0.638370	−	0.769730i	$-0.279609\pi$
$521$	29.1421	1.27674	0.638370	+	0.769730i	$0.279609\pi$
$522$	0	0
$523$	−4.68629	−0.204917	−0.102459	−	0.994737i	$-0.532671\pi$
$523$	−4.68629	−0.204917	−0.102459	+	0.994737i	$0.532671\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.34315	0.363433
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.1716	−0.743783
$534$	0	0
$535$	−14.8284	−0.641089
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.414214	−0.0178414
$540$	0	0
$541$	−10.3431	−0.444687	−0.222343	−	0.974968i	$-0.571371\pi$
$541$	−10.3431	−0.444687	−0.222343	+	0.974968i	$0.571371\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.6569	0.542160
$546$	0	0
$547$	−35.7990	−1.53065	−0.765327	−	0.643641i	$-0.777423\pi$
$547$	−35.7990	−1.53065	−0.765327	+	0.643641i	$0.777423\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	6.82843	0.290374
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.3137	0.733605	0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137	0.733605	0.366803	+	0.930299i	$0.380452\pi$
$558$	0	0
$559$	13.7279	0.580629
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.757359	−0.0319189	−0.0159594	−	0.999873i	$-0.505080\pi$
$563$	−0.757359	−0.0319189	−0.0159594	+	0.999873i	$0.505080\pi$
$564$	0	0
$565$	13.3137	0.560112
$566$	0	0
$567$	0	0
$568$	0	0
$569$	39.6569	1.66250	0.831251	−	0.555897i	$-0.187625\pi$
$569$	39.6569	1.66250	0.831251	+	0.555897i	$0.187625\pi$
$570$	0	0
$571$	−14.6274	−0.612138	−0.306069	−	0.952009i	$-0.599014\pi$
$571$	−14.6274	−0.612138	−0.306069	+	0.952009i	$0.599014\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.6274	−0.610005
$576$	0	0
$577$	−29.7990	−1.24055	−0.620274	−	0.784385i	$-0.712979\pi$
$577$	−29.7990	−1.24055	−0.620274	+	0.784385i	$0.712979\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.6569	0.898478
$582$	0	0
$583$	3.92893	0.162720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.65685	0.316032	0.158016	−	0.987437i	$-0.449490\pi$
$587$	7.65685	0.316032	0.158016	+	0.987437i	$0.449490\pi$
$588$	0	0
$589$	60.4264	2.48983
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.4853	0.800165	0.400082	−	0.916479i	$-0.368982\pi$
$593$	19.4853	0.800165	0.400082	+	0.916479i	$0.368982\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.87006	0.403280	0.201640	−	0.979460i	$-0.435373\pi$
$599$	9.87006	0.403280	0.201640	+	0.979460i	$0.435373\pi$
$600$	0	0
$601$	−17.1716	−0.700443	−0.350222	−	0.936667i	$-0.613894\pi$
$601$	−17.1716	−0.700443	−0.350222	+	0.936667i	$0.613894\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.8284	−0.440238
$606$	0	0
$607$	7.72792	0.313667	0.156833	−	0.987625i	$-0.449871\pi$
$607$	7.72792	0.313667	0.156833	+	0.987625i	$0.449871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.4142	0.502225
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.686292	−0.0276291	−0.0138145	−	0.999905i	$-0.504397\pi$
$617$	−0.686292	−0.0276291	−0.0138145	+	0.999905i	$0.504397\pi$
$618$	0	0
$619$	−33.5858	−1.34993	−0.674963	−	0.737851i	$-0.735841\pi$
$619$	−33.5858	−1.34993	−0.674963	+	0.737851i	$0.735841\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	35.3137	1.41481
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.31371	0.132126
$630$	0	0
$631$	36.8284	1.46612	0.733058	−	0.680166i	$-0.238092\pi$
$631$	36.8284	1.46612	0.733058	+	0.680166i	$0.238092\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.34315	0.172352
$636$	0	0
$637$	−3.82843	−0.151688
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.7990	−0.703018	−0.351509	−	0.936185i	$-0.614331\pi$
$641$	−17.7990	−0.703018	−0.351509	+	0.936185i	$0.614331\pi$
$642$	0	0
$643$	−32.4853	−1.28109	−0.640547	−	0.767919i	$-0.721292\pi$
$643$	−32.4853	−1.28109	−0.640547	+	0.767919i	$0.721292\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	39.6569	1.55907	0.779536	−	0.626358i	$-0.215455\pi$
$647$	39.6569	1.55907	0.779536	+	0.626358i	$0.215455\pi$
$648$	0	0
$649$	1.51472	0.0594579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.1421	1.17955	0.589776	−	0.807567i	$-0.299216\pi$
$653$	30.1421	1.17955	0.589776	+	0.807567i	$0.299216\pi$
$654$	0	0
$655$	21.3137	0.832796
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.4142	0.561498	0.280749	−	0.959781i	$-0.409417\pi$
$659$	14.4142	0.561498	0.280749	+	0.959781i	$0.409417\pi$
$660$	0	0
$661$	33.3137	1.29575	0.647877	−	0.761745i	$-0.275657\pi$
$661$	33.3137	1.29575	0.647877	+	0.761745i	$0.275657\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.9706	−0.658090
$666$	0	0
$667$	−3.65685	−0.141594
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−21.6274	−0.833676	−0.416838	−	0.908981i	$-0.636862\pi$
$673$	−21.6274	−0.833676	−0.416838	+	0.908981i	$0.636862\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	12.6863	0.486855
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.9706	0.802416	0.401208	−	0.915987i	$-0.368590\pi$
$683$	20.9706	0.802416	0.401208	+	0.915987i	$0.368590\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36.3137	1.38344
$690$	0	0
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−3.71573	−0.140743
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.1127	1.51504	0.757518	−	0.652814i	$-0.226412\pi$
$701$	40.1127	1.51504	0.757518	+	0.652814i	$0.226412\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.62742	0.249250
$708$	0	0
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.8284	−1.37924
$714$	0	0
$715$	1.58579	0.0593051
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.1421	−0.751175	−0.375587	−	0.926787i	$-0.622559\pi$
$719$	−20.1421	−0.751175	−0.375587	+	0.926787i	$0.622559\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−1.31371	−0.0487228	−0.0243614	−	0.999703i	$-0.507755\pi$
$727$	−1.31371	−0.0487228	−0.0243614	+	0.999703i	$0.507755\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.97056	0.109870
$732$	0	0
$733$	−41.2548	−1.52378	−0.761891	−	0.647705i	$-0.775729\pi$
$733$	−41.2548	−1.52378	−0.761891	+	0.647705i	$0.775729\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.34315	0.0863109
$738$	0	0
$739$	−4.07107	−0.149757	−0.0748783	−	0.997193i	$-0.523857\pi$
$739$	−4.07107	−0.149757	−0.0748783	+	0.997193i	$0.523857\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.6569	0.867886	0.433943	−	0.900940i	$-0.357122\pi$
$743$	23.6569	0.867886	0.433943	+	0.900940i	$0.357122\pi$
$744$	0	0
$745$	7.82843	0.286811
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−41.9411	−1.53250
$750$	0	0
$751$	−25.3137	−0.923710	−0.461855	−	0.886955i	$-0.652816\pi$
$751$	−25.3137	−0.923710	−0.461855	+	0.886955i	$0.652816\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.1421	0.514685
$756$	0	0
$757$	25.5147	0.927348	0.463674	−	0.886006i	$-0.346531\pi$
$757$	25.5147	0.927348	0.463674	+	0.886006i	$0.346531\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.5980	−1.65293	−0.826463	−	0.562991i	$-0.809650\pi$
$761$	−45.5980	−1.65293	−0.826463	+	0.562991i	$0.809650\pi$
$762$	0	0
$763$	35.7990	1.29601
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.0000	0.505511
$768$	0	0
$769$	−49.1127	−1.77105	−0.885525	−	0.464592i	$-0.846201\pi$
$769$	−49.1127	−1.77105	−0.885525	+	0.464592i	$0.846201\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.5147	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$773$	19.5147	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$774$	0	0
$775$	40.2843	1.44705
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.9117	−0.964211
$780$	0	0
$781$	3.65685	0.130853
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.48528	0.302853
$786$	0	0
$787$	54.0833	1.92786	0.963930	−	0.266156i	$-0.0857536\pi$
$787$	54.0833	1.92786	0.963930	+	0.266156i	$0.0857536\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37.6569	1.33892
$792$	0	0
$793$	18.4853	0.656432
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−51.7401	−1.83273	−0.916364	−	0.400345i	$-0.868890\pi$
$797$	−51.7401	−1.83273	−0.916364	+	0.400345i	$0.868890\pi$
$798$	0	0
$799$	2.68629	0.0950342
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.65685	−0.0584691
$804$	0	0
$805$	10.3431	0.364548
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.2843	−1.27569	−0.637844	−	0.770166i	$-0.720173\pi$
$809$	−36.2843	−1.27569	−0.637844	+	0.770166i	$0.720173\pi$
$810$	0	0
$811$	−10.8284	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$811$	−10.8284	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.92893	−0.137624
$816$	0	0
$817$	21.5147	0.752705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.48528	0.0518367	0.0259183	−	0.999664i	$-0.491749\pi$
$821$	1.48528	0.0518367	0.0259183	+	0.999664i	$0.491749\pi$
$822$	0	0
$823$	54.2843	1.89223	0.946115	−	0.323830i	$-0.104971\pi$
$823$	54.2843	1.89223	0.946115	+	0.323830i	$0.104971\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.8995	1.14403	0.572014	−	0.820244i	$-0.306162\pi$
$827$	32.8995	1.14403	0.572014	+	0.820244i	$0.306162\pi$
$828$	0	0
$829$	−29.7990	−1.03496	−0.517481	−	0.855695i	$-0.673130\pi$
$829$	−29.7990	−1.03496	−0.517481	+	0.855695i	$0.673130\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	−3.17157	−0.109757
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.92893	−0.273737	−0.136869	−	0.990589i	$-0.543704\pi$
$839$	−7.92893	−0.273737	−0.136869	+	0.990589i	$0.543704\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.65685	0.0569975
$846$	0	0
$847$	−30.6274	−1.05237
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.6274	−0.501421
$852$	0	0
$853$	−22.9706	−0.786497	−0.393249	−	0.919432i	$-0.628649\pi$
$853$	−22.9706	−0.786497	−0.393249	+	0.919432i	$0.628649\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.17157	0.210817	0.105408	−	0.994429i	$-0.466385\pi$
$857$	6.17157	0.210817	0.105408	+	0.994429i	$0.466385\pi$
$858$	0	0
$859$	−19.7279	−0.673108	−0.336554	−	0.941664i	$-0.609261\pi$
$859$	−19.7279	−0.673108	−0.336554	+	0.941664i	$0.609261\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.1127	0.582523	0.291262	−	0.956643i	$-0.405925\pi$
$863$	17.1127	0.582523	0.291262	+	0.956643i	$0.405925\pi$
$864$	0	0
$865$	−12.3431	−0.419680
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	21.6569	0.733815
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−25.4558	−0.860565
$876$	0	0
$877$	−37.1421	−1.25420	−0.627100	−	0.778938i	$-0.715758\pi$
$877$	−37.1421	−1.25420	−0.627100	+	0.778938i	$0.715758\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	−38.4264	−1.29315	−0.646576	−	0.762850i	$-0.723800\pi$
$883$	−38.4264	−1.29315	−0.646576	+	0.762850i	$0.723800\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.1005	0.574179	0.287089	−	0.957904i	$-0.407312\pi$
$887$	17.1005	0.574179	0.287089	+	0.957904i	$0.407312\pi$
$888$	0	0
$889$	12.2843	0.412001
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19.4558	0.651065
$894$	0	0
$895$	−6.48528	−0.216779
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.0711	0.335889
$900$	0	0
$901$	7.85786	0.261783
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.31371	0.276357
$906$	0	0
$907$	−22.2843	−0.739937	−0.369969	−	0.929044i	$-0.620632\pi$
$907$	−22.2843	−0.739937	−0.369969	+	0.929044i	$0.620632\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−15.4437	−0.511671	−0.255835	−	0.966720i	$-0.582351\pi$
$911$	−15.4437	−0.511671	−0.255835	+	0.966720i	$0.582351\pi$
$912$	0	0
$913$	−3.17157	−0.104964
$914$	0	0
$915$	0	0
$916$	0	0
$917$	60.2843	1.99076
$918$	0	0
$919$	−8.14214	−0.268584	−0.134292	−	0.990942i	$-0.542876\pi$
$919$	−8.14214	−0.268584	−0.134292	+	0.990942i	$0.542876\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	33.7990	1.11251
$924$	0	0
$925$	16.0000	0.526077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.6863	−0.613077	−0.306539	−	0.951858i	$-0.599171\pi$
$929$	−18.6863	−0.613077	−0.306539	+	0.951858i	$0.599171\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.343146	0.0112221
$936$	0	0
$937$	−16.6274	−0.543194	−0.271597	−	0.962411i	$-0.587552\pi$
$937$	−16.6274	−0.543194	−0.271597	+	0.962411i	$0.587552\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	56.5980	1.84504	0.922521	−	0.385948i	$-0.126125\pi$
$941$	56.5980	1.84504	0.922521	+	0.385948i	$0.126125\pi$
$942$	0	0
$943$	16.4020	0.534123
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.61522	0.0849834	0.0424917	−	0.999097i	$-0.486470\pi$
$947$	2.61522	0.0849834	0.0424917	+	0.999097i	$0.486470\pi$
$948$	0	0
$949$	−15.3137	−0.497104
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.6274	1.15409	0.577043	−	0.816714i	$-0.304207\pi$
$953$	35.6274	1.15409	0.577043	+	0.816714i	$0.304207\pi$
$954$	0	0
$955$	25.3137	0.819132
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.9411	−1.09602
$960$	0	0
$961$	70.4264	2.27182
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.17157	−0.166479
$966$	0	0
$967$	35.2426	1.13333	0.566663	−	0.823949i	$-0.308234\pi$
$967$	35.2426	1.13333	0.566663	+	0.823949i	$0.308234\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.6569	−0.502452	−0.251226	−	0.967928i	$-0.580834\pi$
$971$	−15.6569	−0.502452	−0.251226	+	0.967928i	$0.580834\pi$
$972$	0	0
$973$	−39.5980	−1.26945
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.1716	−1.15723	−0.578616	−	0.815600i	$-0.696407\pi$
$977$	−36.1716	−1.15723	−0.578616	+	0.815600i	$0.696407\pi$
$978$	0	0
$979$	−5.17157	−0.165284
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.8701	−0.697547	−0.348773	−	0.937207i	$-0.613402\pi$
$983$	−21.8701	−0.697547	−0.348773	+	0.937207i	$0.613402\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.1127	−0.416960
$990$	0	0
$991$	12.8284	0.407508	0.203754	−	0.979022i	$-0.434686\pi$
$991$	12.8284	0.407508	0.203754	+	0.979022i	$0.434686\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.485281	0.0153845
$996$	0	0
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bq.1.2		2
3.2	odd	2		464.2.a.h.1.1		2
4.3	odd	2		261.2.a.d.1.2		2
12.11	even	2		29.2.a.a.1.1	✓	2
20.19	odd	2		6525.2.a.o.1.1		2
24.5	odd	2		1856.2.a.w.1.2		2
24.11	even	2		1856.2.a.r.1.1		2
60.23	odd	4		725.2.b.b.349.4		4
60.47	odd	4		725.2.b.b.349.1		4
60.59	even	2		725.2.a.b.1.2		2
84.83	odd	2		1421.2.a.j.1.1		2
116.115	odd	2		7569.2.a.c.1.1		2
132.131	odd	2		3509.2.a.j.1.2		2
156.155	even	2		4901.2.a.g.1.2		2
204.203	even	2		8381.2.a.e.1.1		2
348.11	odd	28		841.2.e.k.63.1		24
348.23	even	14		841.2.d.j.645.2		12
348.35	even	14		841.2.d.f.645.1		12
348.47	odd	28		841.2.e.k.63.4		24
348.71	even	14		841.2.d.f.778.2		12
348.83	even	14		841.2.d.j.190.2		12
348.95	odd	28		841.2.e.k.267.1		24
348.107	even	14		841.2.d.j.574.1		12
348.119	odd	28		841.2.e.k.270.1		24
348.131	odd	28		841.2.e.k.196.1		24
348.143	odd	28		841.2.e.k.236.4		24
348.155	odd	28		841.2.e.k.651.1		24
348.167	even	14		841.2.d.f.571.1		12
348.179	even	14		841.2.d.f.605.1		12
348.191	odd	4		841.2.b.a.840.4		4
348.215	odd	4		841.2.b.a.840.1		4
348.227	even	14		841.2.d.j.605.2		12
348.239	even	14		841.2.d.j.571.2		12
348.251	odd	28		841.2.e.k.651.4		24
348.263	odd	28		841.2.e.k.236.1		24
348.275	odd	28		841.2.e.k.196.4		24
348.287	odd	28		841.2.e.k.270.4		24
348.299	even	14		841.2.d.f.574.2		12
348.311	odd	28		841.2.e.k.267.4		24
348.323	even	14		841.2.d.f.190.1		12
348.335	even	14		841.2.d.j.778.1		12
348.347	even	2		841.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.1	✓	2	12.11	even	2
261.2.a.d.1.2		2	4.3	odd	2
464.2.a.h.1.1		2	3.2	odd	2
725.2.a.b.1.2		2	60.59	even	2
725.2.b.b.349.1		4	60.47	odd	4
725.2.b.b.349.4		4	60.23	odd	4
841.2.a.d.1.2		2	348.347	even	2
841.2.b.a.840.1		4	348.215	odd	4
841.2.b.a.840.4		4	348.191	odd	4
841.2.d.f.190.1		12	348.323	even	14
841.2.d.f.571.1		12	348.167	even	14
841.2.d.f.574.2		12	348.299	even	14
841.2.d.f.605.1		12	348.179	even	14
841.2.d.f.645.1		12	348.35	even	14
841.2.d.f.778.2		12	348.71	even	14
841.2.d.j.190.2		12	348.83	even	14
841.2.d.j.571.2		12	348.239	even	14
841.2.d.j.574.1		12	348.107	even	14
841.2.d.j.605.2		12	348.227	even	14
841.2.d.j.645.2		12	348.23	even	14
841.2.d.j.778.1		12	348.335	even	14
841.2.e.k.63.1		24	348.11	odd	28
841.2.e.k.63.4		24	348.47	odd	28
841.2.e.k.196.1		24	348.131	odd	28
841.2.e.k.196.4		24	348.275	odd	28
841.2.e.k.236.1		24	348.263	odd	28
841.2.e.k.236.4		24	348.143	odd	28
841.2.e.k.267.1		24	348.95	odd	28
841.2.e.k.267.4		24	348.311	odd	28
841.2.e.k.270.1		24	348.119	odd	28
841.2.e.k.270.4		24	348.287	odd	28
841.2.e.k.651.1		24	348.155	odd	28
841.2.e.k.651.4		24	348.251	odd	28
1421.2.a.j.1.1		2	84.83	odd	2
1856.2.a.r.1.1		2	24.11	even	2
1856.2.a.w.1.2		2	24.5	odd	2
3509.2.a.j.1.2		2	132.131	odd	2
4176.2.a.bq.1.2		2	1.1	even	1	trivial
4901.2.a.g.1.2		2	156.155	even	2
6525.2.a.o.1.1		2	20.19	odd	2
7569.2.a.c.1.1		2	116.115	odd	2
8381.2.a.e.1.1		2	204.203	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:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +2.82843 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +2.82843 q^{7} -0.414214 q^{11} -3.82843 q^{13} -0.828427 q^{17} -6.00000 q^{19} +3.65685 q^{23} -4.00000 q^{25} -1.00000 q^{29} -10.0711 q^{31} +2.82843 q^{35} -4.00000 q^{37} +4.48528 q^{41} -3.58579 q^{43} -3.24264 q^{47} +1.00000 q^{49} -9.48528 q^{53} -0.414214 q^{55} -3.65685 q^{59} -4.82843 q^{61} -3.82843 q^{65} -5.65685 q^{67} -8.82843 q^{71} +4.00000 q^{73} -1.17157 q^{77} +2.41421 q^{79} +7.65685 q^{83} -0.828427 q^{85} +12.4853 q^{89} -10.8284 q^{91} -6.00000 q^{95} +4.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 12 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} - 6 q^{31} - 8 q^{37} - 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 2 q^{53} + 2 q^{55} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 12 q^{71} + 8 q^{73} - 8 q^{77} + 2 q^{79} + 4 q^{83} + 4 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 - 12 * q^19 - 4 * q^23 - 8 * q^25 - 2 * q^29 - 6 * q^31 - 8 * q^37 - 8 * q^41 - 10 * q^43 + 2 * q^47 + 2 * q^49 - 2 * q^53 + 2 * q^55 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 12 * q^71 + 8 * q^73 - 8 * q^77 + 2 * q^79 + 4 * q^83 + 4 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^95 - 8 * q^97

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