LMFDB - Embedded newform 4680.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.00000 q^{7} -4.24264 q^{11} -1.00000 q^{13} +4.82843 q^{17} -8.24264 q^{19} -5.07107 q^{23} +1.00000 q^{25} +9.65685 q^{29} -1.41421 q^{31} +2.00000 q^{35} -1.17157 q^{37} +0.828427 q^{41} -1.75736 q^{43} -2.00000 q^{47} -3.00000 q^{49} +3.17157 q^{53} +4.24264 q^{55} +5.41421 q^{59} -7.31371 q^{61} +1.00000 q^{65} +0.828427 q^{67} +9.89949 q^{71} +6.82843 q^{73} +8.48528 q^{77} +6.82843 q^{79} +2.00000 q^{83} -4.82843 q^{85} +10.0000 q^{89} +2.00000 q^{91} +8.24264 q^{95} +0.343146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} - 2 q^{13} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} + 4 q^{35} - 8 q^{37} - 4 q^{41} - 12 q^{43} - 4 q^{47} - 6 q^{49} + 12 q^{53} + 8 q^{59} + 8 q^{61} + 2 q^{65} - 4 q^{67} + 8 q^{73} + 8 q^{79} + 4 q^{83} - 4 q^{85} + 20 q^{89} + 4 q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 - 2 * q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 + 4 * q^35 - 8 * q^37 - 4 * q^41 - 12 * q^43 - 4 * q^47 - 6 * q^49 + 12 * q^53 + 8 * q^59 + 8 * q^61 + 2 * q^65 - 4 * q^67 + 8 * q^73 + 8 * q^79 + 4 * q^83 - 4 * q^85 + 20 * q^89 + 4 * q^91 + 8 * q^95 + 12 * 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
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−8.24264	−1.89099	−0.945496	−	0.325634i	$-0.894422\pi$
$19$	−8.24264	−1.89099	−0.945496	+	0.325634i	$0.894422\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.07107	−1.05739	−0.528695	−	0.848812i	$-0.677319\pi$
$23$	−5.07107	−1.05739	−0.528695	+	0.848812i	$0.677319\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.65685	1.79323	0.896616	−	0.442808i	$-0.146018\pi$
$29$	9.65685	1.79323	0.896616	+	0.442808i	$0.146018\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−1.17157	−0.192605	−0.0963027	−	0.995352i	$-0.530702\pi$
$37$	−1.17157	−0.192605	−0.0963027	+	0.995352i	$0.530702\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	−1.75736	−0.267995	−0.133997	−	0.990982i	$-0.542781\pi$
$43$	−1.75736	−0.267995	−0.133997	+	0.990982i	$0.542781\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.17157	0.435649	0.217825	−	0.975988i	$-0.430104\pi$
$53$	3.17157	0.435649	0.217825	+	0.975988i	$0.430104\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.41421	0.704871	0.352435	−	0.935836i	$-0.385354\pi$
$59$	5.41421	0.704871	0.352435	+	0.935836i	$0.385354\pi$
$60$	0	0
$61$	−7.31371	−0.936424	−0.468212	−	0.883616i	$-0.655102\pi$
$61$	−7.31371	−0.936424	−0.468212	+	0.883616i	$0.655102\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	0.828427	0.101208	0.0506042	−	0.998719i	$-0.483885\pi$
$67$	0.828427	0.101208	0.0506042	+	0.998719i	$0.483885\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.89949	1.17485	0.587427	−	0.809277i	$-0.300141\pi$
$71$	9.89949	1.17485	0.587427	+	0.809277i	$0.300141\pi$
$72$	0	0
$73$	6.82843	0.799207	0.399603	−	0.916688i	$-0.369148\pi$
$73$	6.82843	0.799207	0.399603	+	0.916688i	$0.369148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.48528	0.966988
$78$	0	0
$79$	6.82843	0.768258	0.384129	−	0.923279i	$-0.374502\pi$
$79$	6.82843	0.768258	0.384129	+	0.923279i	$0.374502\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.24264	0.845677
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	8.58579	0.845983	0.422991	−	0.906134i	$-0.360980\pi$
$103$	8.58579	0.845983	0.422991	+	0.906134i	$0.360980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.07107	0.876933	0.438467	−	0.898747i	$-0.355522\pi$
$107$	9.07107	0.876933	0.438467	+	0.898747i	$0.355522\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.17157	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$113$	3.17157	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$114$	0	0
$115$	5.07107	0.472880
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.65685	−0.885242
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.89949	−0.346024	−0.173012	−	0.984920i	$-0.555350\pi$
$127$	−3.89949	−0.346024	−0.173012	+	0.984920i	$0.555350\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.31371	−0.289520	−0.144760	−	0.989467i	$-0.546241\pi$
$131$	−3.31371	−0.289520	−0.144760	+	0.989467i	$0.546241\pi$
$132$	0	0
$133$	16.4853	1.42946
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	8.48528	0.719712	0.359856	−	0.933008i	$-0.382826\pi$
$139$	8.48528	0.719712	0.359856	+	0.933008i	$0.382826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.24264	0.354787
$144$	0	0
$145$	−9.65685	−0.801958
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.9706	−0.898744	−0.449372	−	0.893345i	$-0.648352\pi$
$149$	−10.9706	−0.898744	−0.449372	+	0.893345i	$0.648352\pi$
$150$	0	0
$151$	0.928932	0.0755954	0.0377977	−	0.999285i	$-0.487966\pi$
$151$	0.928932	0.0755954	0.0377977	+	0.999285i	$0.487966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.41421	0.113592
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.1421	0.799312
$162$	0	0
$163$	−22.4853	−1.76118	−0.880592	−	0.473876i	$-0.842854\pi$
$163$	−22.4853	−1.76118	−0.880592	+	0.473876i	$0.842854\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.3137	1.03025	0.515123	−	0.857116i	$-0.327746\pi$
$167$	13.3137	1.03025	0.515123	+	0.857116i	$0.327746\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.8284	1.58356	0.791778	−	0.610809i	$-0.209156\pi$
$173$	20.8284	1.58356	0.791778	+	0.610809i	$0.209156\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	23.3137	1.73289	0.866447	−	0.499269i	$-0.166398\pi$
$181$	23.3137	1.73289	0.866447	+	0.499269i	$0.166398\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.17157	0.0861358
$186$	0	0
$187$	−20.4853	−1.49803
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.6569	1.56703	0.783517	−	0.621370i	$-0.213423\pi$
$191$	21.6569	1.56703	0.783517	+	0.621370i	$0.213423\pi$
$192$	0	0
$193$	−8.34315	−0.600553	−0.300276	−	0.953852i	$-0.597079\pi$
$193$	−8.34315	−0.600553	−0.300276	+	0.953852i	$0.597079\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.343146	0.0244481	0.0122241	−	0.999925i	$-0.496109\pi$
$197$	0.343146	0.0244481	0.0122241	+	0.999925i	$0.496109\pi$
$198$	0	0
$199$	6.34315	0.449654	0.224827	−	0.974399i	$-0.427818\pi$
$199$	6.34315	0.449654	0.224827	+	0.974399i	$0.427818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−19.3137	−1.35556
$204$	0	0
$205$	−0.828427	−0.0578599
$206$	0	0
$207$	0	0
$208$	0	0
$209$	34.9706	2.41896
$210$	0	0
$211$	5.65685	0.389434	0.194717	−	0.980859i	$-0.437621\pi$
$211$	5.65685	0.389434	0.194717	+	0.980859i	$0.437621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.75736	0.119851
$216$	0	0
$217$	2.82843	0.192006
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.82843	−0.324795
$222$	0	0
$223$	25.3137	1.69513	0.847566	−	0.530691i	$-0.178067\pi$
$223$	25.3137	1.69513	0.847566	+	0.530691i	$0.178067\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.8284	−1.38243	−0.691216	−	0.722649i	$-0.742925\pi$
$227$	−20.8284	−1.38243	−0.691216	+	0.722649i	$0.742925\pi$
$228$	0	0
$229$	−13.7990	−0.911863	−0.455931	−	0.890015i	$-0.650694\pi$
$229$	−13.7990	−0.911863	−0.455931	+	0.890015i	$0.650694\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.72792	−0.564562	−0.282281	−	0.959332i	$-0.591091\pi$
$239$	−8.72792	−0.564562	−0.282281	+	0.959332i	$0.591091\pi$
$240$	0	0
$241$	7.17157	0.461962	0.230981	−	0.972958i	$-0.425807\pi$
$241$	7.17157	0.461962	0.230981	+	0.972958i	$0.425807\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	8.24264	0.524467
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.1421	−0.640166	−0.320083	−	0.947390i	$-0.603711\pi$
$251$	−10.1421	−0.640166	−0.320083	+	0.947390i	$0.603711\pi$
$252$	0	0
$253$	21.5147	1.35262
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.9706	−0.933838	−0.466919	−	0.884300i	$-0.654636\pi$
$257$	−14.9706	−0.933838	−0.466919	+	0.884300i	$0.654636\pi$
$258$	0	0
$259$	2.34315	0.145596
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.7574	−0.848315	−0.424158	−	0.905588i	$-0.639430\pi$
$263$	−13.7574	−0.848315	−0.424158	+	0.905588i	$0.639430\pi$
$264$	0	0
$265$	−3.17157	−0.194828
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.6274	−1.01379	−0.506896	−	0.862007i	$-0.669207\pi$
$269$	−16.6274	−1.01379	−0.506896	+	0.862007i	$0.669207\pi$
$270$	0	0
$271$	9.89949	0.601351	0.300676	−	0.953726i	$-0.402788\pi$
$271$	9.89949	0.601351	0.300676	+	0.953726i	$0.402788\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	4.14214	0.248877	0.124438	−	0.992227i	$-0.460287\pi$
$277$	4.14214	0.248877	0.124438	+	0.992227i	$0.460287\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.4853	1.34136	0.670680	−	0.741747i	$-0.266003\pi$
$281$	22.4853	1.34136	0.670680	+	0.741747i	$0.266003\pi$
$282$	0	0
$283$	3.89949	0.231801	0.115900	−	0.993261i	$-0.463025\pi$
$283$	3.89949	0.231801	0.115900	+	0.993261i	$0.463025\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.65685	−0.0978010
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.82843	−0.165238	−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843	−0.165238	−0.0826192	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	−5.41421	−0.315228
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.07107	0.293267
$300$	0	0
$301$	3.51472	0.202585
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.31371	0.418782
$306$	0	0
$307$	15.6569	0.893584	0.446792	−	0.894638i	$-0.352566\pi$
$307$	15.6569	0.893584	0.446792	+	0.894638i	$0.352566\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.1421	0.575108	0.287554	−	0.957764i	$-0.407158\pi$
$311$	10.1421	0.575108	0.287554	+	0.957764i	$0.407158\pi$
$312$	0	0
$313$	−7.17157	−0.405361	−0.202681	−	0.979245i	$-0.564965\pi$
$313$	−7.17157	−0.405361	−0.202681	+	0.979245i	$0.564965\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.4853	−1.37523	−0.687615	−	0.726075i	$-0.741342\pi$
$317$	−24.4853	−1.37523	−0.687615	+	0.726075i	$0.741342\pi$
$318$	0	0
$319$	−40.9706	−2.29391
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−39.7990	−2.21448
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−0.928932	−0.0510587	−0.0255294	−	0.999674i	$-0.508127\pi$
$331$	−0.928932	−0.0510587	−0.0255294	+	0.999674i	$0.508127\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.828427	−0.0452618
$336$	0	0
$337$	−36.1421	−1.96879	−0.984394	−	0.175980i	$-0.943691\pi$
$337$	−36.1421	−1.96879	−0.984394	+	0.175980i	$0.943691\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.38478	−0.235387	−0.117694	−	0.993050i	$-0.537550\pi$
$347$	−4.38478	−0.235387	−0.117694	+	0.993050i	$0.537550\pi$
$348$	0	0
$349$	35.4558	1.89791	0.948954	−	0.315415i	$-0.102144\pi$
$349$	35.4558	1.89791	0.948954	+	0.315415i	$0.102144\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.17157	−0.275255	−0.137628	−	0.990484i	$-0.543948\pi$
$353$	−5.17157	−0.275255	−0.137628	+	0.990484i	$0.543948\pi$
$354$	0	0
$355$	−9.89949	−0.525411
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.5563	1.45437	0.727184	−	0.686442i	$-0.240829\pi$
$359$	27.5563	1.45437	0.727184	+	0.686442i	$0.240829\pi$
$360$	0	0
$361$	48.9411	2.57585
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.82843	−0.357416
$366$	0	0
$367$	−22.7279	−1.18639	−0.593194	−	0.805060i	$-0.702133\pi$
$367$	−22.7279	−1.18639	−0.593194	+	0.805060i	$0.702133\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.34315	−0.329320
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.65685	−0.497353
$378$	0	0
$379$	16.2426	0.834328	0.417164	−	0.908831i	$-0.363024\pi$
$379$	16.2426	0.834328	0.417164	+	0.908831i	$0.363024\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.0000	−1.12415	−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000	−1.12415	−0.562074	+	0.827087i	$0.689996\pi$
$384$	0	0
$385$	−8.48528	−0.432450
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.3137	0.675032	0.337516	−	0.941320i	$-0.390413\pi$
$389$	13.3137	0.675032	0.337516	+	0.941320i	$0.390413\pi$
$390$	0	0
$391$	−24.4853	−1.23827
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.82843	−0.343575
$396$	0	0
$397$	1.17157	0.0587996	0.0293998	−	0.999568i	$-0.490640\pi$
$397$	1.17157	0.0587996	0.0293998	+	0.999568i	$0.490640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.6274	−1.42958	−0.714792	−	0.699337i	$-0.753479\pi$
$401$	−28.6274	−1.42958	−0.714792	+	0.699337i	$0.753479\pi$
$402$	0	0
$403$	1.41421	0.0704470
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.97056	0.246382
$408$	0	0
$409$	4.82843	0.238750	0.119375	−	0.992849i	$-0.461911\pi$
$409$	4.82843	0.238750	0.119375	+	0.992849i	$0.461911\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.8284	−0.532832
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.1421	1.66795	0.833976	−	0.551800i	$-0.186059\pi$
$419$	34.1421	1.66795	0.833976	+	0.551800i	$0.186059\pi$
$420$	0	0
$421$	35.6569	1.73781	0.868904	−	0.494980i	$-0.164825\pi$
$421$	35.6569	1.73781	0.868904	+	0.494980i	$0.164825\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.82843	0.234213
$426$	0	0
$427$	14.6274	0.707870
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.2132	−0.636458	−0.318229	−	0.948014i	$-0.603088\pi$
$431$	−13.2132	−0.636458	−0.318229	+	0.948014i	$0.603088\pi$
$432$	0	0
$433$	−6.97056	−0.334984	−0.167492	−	0.985873i	$-0.553567\pi$
$433$	−6.97056	−0.334984	−0.167492	+	0.985873i	$0.553567\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.7990	1.99952
$438$	0	0
$439$	28.2843	1.34993	0.674967	−	0.737848i	$-0.264158\pi$
$439$	28.2843	1.34993	0.674967	+	0.737848i	$0.264158\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.41421	0.162214	0.0811071	−	0.996705i	$-0.474154\pi$
$443$	3.41421	0.162214	0.0811071	+	0.996705i	$0.474154\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.85786	−0.370836	−0.185418	−	0.982660i	$-0.559364\pi$
$449$	−7.85786	−0.370836	−0.185418	+	0.982660i	$0.559364\pi$
$450$	0	0
$451$	−3.51472	−0.165502
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−32.6274	−1.52625	−0.763123	−	0.646253i	$-0.776335\pi$
$457$	−32.6274	−1.52625	−0.763123	+	0.646253i	$0.776335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.1421	0.938113	0.469056	−	0.883168i	$-0.344594\pi$
$461$	20.1421	0.938113	0.469056	+	0.883168i	$0.344594\pi$
$462$	0	0
$463$	22.4853	1.04498	0.522490	−	0.852646i	$-0.325003\pi$
$463$	22.4853	1.04498	0.522490	+	0.852646i	$0.325003\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.2426	−0.659071	−0.329535	−	0.944143i	$-0.606892\pi$
$467$	−14.2426	−0.659071	−0.329535	+	0.944143i	$0.606892\pi$
$468$	0	0
$469$	−1.65685	−0.0765064
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.45584	0.342820
$474$	0	0
$475$	−8.24264	−0.378198
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.2426	1.47320	0.736602	−	0.676327i	$-0.236429\pi$
$479$	32.2426	1.47320	0.736602	+	0.676327i	$0.236429\pi$
$480$	0	0
$481$	1.17157	0.0534191
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.343146	−0.0155814
$486$	0	0
$487$	6.48528	0.293876	0.146938	−	0.989146i	$-0.453058\pi$
$487$	6.48528	0.293876	0.146938	+	0.989146i	$0.453058\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.1127	0.862544	0.431272	−	0.902222i	$-0.358065\pi$
$491$	19.1127	0.862544	0.431272	+	0.902222i	$0.358065\pi$
$492$	0	0
$493$	46.6274	2.09999
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.7990	−0.888106
$498$	0	0
$499$	−26.5858	−1.19014	−0.595072	−	0.803673i	$-0.702876\pi$
$499$	−26.5858	−1.19014	−0.595072	+	0.803673i	$0.702876\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.4142	1.22234	0.611170	−	0.791500i	$-0.290699\pi$
$503$	27.4142	1.22234	0.611170	+	0.791500i	$0.290699\pi$
$504$	0	0
$505$	7.65685	0.340726
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.5147	−0.776326	−0.388163	−	0.921591i	$-0.626890\pi$
$509$	−17.5147	−0.776326	−0.388163	+	0.921591i	$0.626890\pi$
$510$	0	0
$511$	−13.6569	−0.604144
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.58579	−0.378335
$516$	0	0
$517$	8.48528	0.373182
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.6863	0.555797	0.277898	−	0.960611i	$-0.410362\pi$
$521$	12.6863	0.555797	0.277898	+	0.960611i	$0.410362\pi$
$522$	0	0
$523$	−23.6985	−1.03626	−0.518131	−	0.855301i	$-0.673372\pi$
$523$	−23.6985	−1.03626	−0.518131	+	0.855301i	$0.673372\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.82843	−0.297451
$528$	0	0
$529$	2.71573	0.118075
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.828427	−0.0358832
$534$	0	0
$535$	−9.07107	−0.392176
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.7279	0.548230
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.31371	0.227614
$546$	0	0
$547$	−33.0711	−1.41402	−0.707008	−	0.707205i	$-0.749956\pi$
$547$	−33.0711	−1.41402	−0.707008	+	0.707205i	$0.749956\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−79.5980	−3.39099
$552$	0	0
$553$	−13.6569	−0.580749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.1421	−1.78562	−0.892810	−	0.450434i	$-0.851269\pi$
$557$	−42.1421	−1.78562	−0.892810	+	0.450434i	$0.851269\pi$
$558$	0	0
$559$	1.75736	0.0743284
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.5563	−0.739912	−0.369956	−	0.929049i	$-0.620627\pi$
$563$	−17.5563	−0.739912	−0.369956	+	0.929049i	$0.620627\pi$
$564$	0	0
$565$	−3.17157	−0.133429
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.3137	1.31274	0.656369	−	0.754440i	$-0.272091\pi$
$569$	31.3137	1.31274	0.656369	+	0.754440i	$0.272091\pi$
$570$	0	0
$571$	−34.8284	−1.45752	−0.728762	−	0.684767i	$-0.759904\pi$
$571$	−34.8284	−1.45752	−0.728762	+	0.684767i	$0.759904\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.07107	−0.211478
$576$	0	0
$577$	34.1421	1.42136	0.710678	−	0.703518i	$-0.248388\pi$
$577$	34.1421	1.42136	0.710678	+	0.703518i	$0.248388\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−13.4558	−0.557284
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.4853	0.597872	0.298936	−	0.954273i	$-0.403368\pi$
$587$	14.4853	0.597872	0.298936	+	0.954273i	$0.403368\pi$
$588$	0	0
$589$	11.6569	0.480312
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	9.65685	0.395892
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.4853	1.16388	0.581939	−	0.813233i	$-0.302294\pi$
$599$	28.4853	1.16388	0.581939	+	0.813233i	$0.302294\pi$
$600$	0	0
$601$	−31.9411	−1.30291	−0.651453	−	0.758689i	$-0.725840\pi$
$601$	−31.9411	−1.30291	−0.651453	+	0.758689i	$0.725840\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−37.5563	−1.52437	−0.762183	−	0.647362i	$-0.775872\pi$
$607$	−37.5563	−1.52437	−0.762183	+	0.647362i	$0.775872\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.65685	−0.308253	−0.154127	−	0.988051i	$-0.549256\pi$
$617$	−7.65685	−0.308253	−0.154127	+	0.988051i	$0.549256\pi$
$618$	0	0
$619$	−12.0416	−0.483994	−0.241997	−	0.970277i	$-0.577802\pi$
$619$	−12.0416	−0.483994	−0.241997	+	0.970277i	$0.577802\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	12.7279	0.506691	0.253345	−	0.967376i	$-0.418469\pi$
$631$	12.7279	0.506691	0.253345	+	0.967376i	$0.418469\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.89949	0.154747
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.6569	−1.56635	−0.783176	−	0.621800i	$-0.786402\pi$
$641$	−39.6569	−1.56635	−0.783176	+	0.621800i	$0.786402\pi$
$642$	0	0
$643$	9.31371	0.367297	0.183648	−	0.982992i	$-0.441209\pi$
$643$	9.31371	0.367297	0.183648	+	0.982992i	$0.441209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.3553	−1.31133	−0.655667	−	0.755050i	$-0.727612\pi$
$647$	−33.3553	−1.31133	−0.655667	+	0.755050i	$0.727612\pi$
$648$	0	0
$649$	−22.9706	−0.901673
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.9706	1.68157	0.840784	−	0.541371i	$-0.182094\pi$
$653$	42.9706	1.68157	0.840784	+	0.541371i	$0.182094\pi$
$654$	0	0
$655$	3.31371	0.129477
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.4853	0.486358	0.243179	−	0.969981i	$-0.421810\pi$
$659$	12.4853	0.486358	0.243179	+	0.969981i	$0.421810\pi$
$660$	0	0
$661$	−42.9706	−1.67136	−0.835681	−	0.549216i	$-0.814927\pi$
$661$	−42.9706	−1.67136	−0.835681	+	0.549216i	$0.814927\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.4853	−0.639272
$666$	0	0
$667$	−48.9706	−1.89615
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.0294	1.19788
$672$	0	0
$673$	−5.79899	−0.223535	−0.111767	−	0.993734i	$-0.535651\pi$
$673$	−5.79899	−0.223535	−0.111767	+	0.993734i	$0.535651\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.14214	−0.312928	−0.156464	−	0.987684i	$-0.550009\pi$
$677$	−8.14214	−0.312928	−0.156464	+	0.987684i	$0.550009\pi$
$678$	0	0
$679$	−0.686292	−0.0263375
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.1127	0.960911	0.480455	−	0.877019i	$-0.340471\pi$
$683$	25.1127	0.960911	0.480455	+	0.877019i	$0.340471\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.17157	−0.120827
$690$	0	0
$691$	−46.5858	−1.77221	−0.886103	−	0.463488i	$-0.846598\pi$
$691$	−46.5858	−1.77221	−0.886103	+	0.463488i	$0.846598\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.48528	−0.321865
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.6274	−0.930165	−0.465082	−	0.885267i	$-0.653975\pi$
$701$	−24.6274	−0.930165	−0.465082	+	0.885267i	$0.653975\pi$
$702$	0	0
$703$	9.65685	0.364215
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.3137	0.575931
$708$	0	0
$709$	31.4558	1.18135	0.590675	−	0.806910i	$-0.298862\pi$
$709$	31.4558	1.18135	0.590675	+	0.806910i	$0.298862\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.17157	0.268578
$714$	0	0
$715$	−4.24264	−0.158666
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.6274	−0.993035	−0.496518	−	0.868027i	$-0.665388\pi$
$719$	−26.6274	−0.993035	−0.496518	+	0.868027i	$0.665388\pi$
$720$	0	0
$721$	−17.1716	−0.639503
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.65685	0.358647
$726$	0	0
$727$	6.24264	0.231527	0.115763	−	0.993277i	$-0.463069\pi$
$727$	6.24264	0.231527	0.115763	+	0.993277i	$0.463069\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.48528	−0.313839
$732$	0	0
$733$	−12.6274	−0.466404	−0.233202	−	0.972428i	$-0.574920\pi$
$733$	−12.6274	−0.466404	−0.233202	+	0.972428i	$0.574920\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.51472	−0.129466
$738$	0	0
$739$	21.4142	0.787735	0.393867	−	0.919167i	$-0.371137\pi$
$739$	21.4142	0.787735	0.393867	+	0.919167i	$0.371137\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.97056	0.108979	0.0544897	−	0.998514i	$-0.482647\pi$
$743$	2.97056	0.108979	0.0544897	+	0.998514i	$0.482647\pi$
$744$	0	0
$745$	10.9706	0.401930
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18.1421	−0.662899
$750$	0	0
$751$	−8.48528	−0.309632	−0.154816	−	0.987943i	$-0.549479\pi$
$751$	−8.48528	−0.309632	−0.154816	+	0.987943i	$0.549479\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.928932	−0.0338073
$756$	0	0
$757$	20.1421	0.732078	0.366039	−	0.930599i	$-0.380714\pi$
$757$	20.1421	0.732078	0.366039	+	0.930599i	$0.380714\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.6863	0.677378	0.338689	−	0.940898i	$-0.390017\pi$
$761$	18.6863	0.677378	0.338689	+	0.940898i	$0.390017\pi$
$762$	0	0
$763$	10.6274	0.384738
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.41421	−0.195496
$768$	0	0
$769$	−8.34315	−0.300862	−0.150431	−	0.988621i	$-0.548066\pi$
$769$	−8.34315	−0.300862	−0.150431	+	0.988621i	$0.548066\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.1421	0.364787	0.182394	−	0.983226i	$-0.441615\pi$
$773$	10.1421	0.364787	0.182394	+	0.983226i	$0.441615\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.82843	−0.244654
$780$	0	0
$781$	−42.0000	−1.50288
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.31371	−0.189654
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.34315	−0.225536
$792$	0	0
$793$	7.31371	0.259717
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.5980	1.04841	0.524207	−	0.851591i	$-0.324362\pi$
$797$	29.5980	1.04841	0.524207	+	0.851591i	$0.324362\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.9706	−1.02235
$804$	0	0
$805$	−10.1421	−0.357463
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.3431	1.06681	0.533404	−	0.845861i	$-0.320913\pi$
$809$	30.3431	1.06681	0.533404	+	0.845861i	$0.320913\pi$
$810$	0	0
$811$	−9.89949	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$811$	−9.89949	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.4853	0.787625
$816$	0	0
$817$	14.4853	0.506776
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.3137	−1.02306	−0.511528	−	0.859267i	$-0.670920\pi$
$821$	−29.3137	−1.02306	−0.511528	+	0.859267i	$0.670920\pi$
$822$	0	0
$823$	36.8701	1.28521	0.642605	−	0.766198i	$-0.277854\pi$
$823$	36.8701	1.28521	0.642605	+	0.766198i	$0.277854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.6274	0.856379	0.428190	−	0.903689i	$-0.359152\pi$
$827$	24.6274	0.856379	0.428190	+	0.903689i	$0.359152\pi$
$828$	0	0
$829$	25.6569	0.891099	0.445550	−	0.895257i	$-0.353008\pi$
$829$	25.6569	0.891099	0.445550	+	0.895257i	$0.353008\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.4853	−0.501885
$834$	0	0
$835$	−13.3137	−0.460740
$836$	0	0
$837$	0	0
$838$	0	0
$839$	18.1005	0.624899	0.312449	−	0.949934i	$-0.398851\pi$
$839$	18.1005	0.624899	0.312449	+	0.949934i	$0.398851\pi$
$840$	0	0
$841$	64.2548	2.21568
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.94113	0.203659
$852$	0	0
$853$	15.1127	0.517449	0.258724	−	0.965951i	$-0.416698\pi$
$853$	15.1127	0.517449	0.258724	+	0.965951i	$0.416698\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.9117	−1.67079	−0.835396	−	0.549649i	$-0.814761\pi$
$857$	−48.9117	−1.67079	−0.835396	+	0.549649i	$0.814761\pi$
$858$	0	0
$859$	49.1716	1.67771	0.838856	−	0.544353i	$-0.183225\pi$
$859$	49.1716	1.67771	0.838856	+	0.544353i	$0.183225\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.5980	0.871365	0.435683	−	0.900100i	$-0.356507\pi$
$863$	25.5980	0.871365	0.435683	+	0.900100i	$0.356507\pi$
$864$	0	0
$865$	−20.8284	−0.708188
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28.9706	−0.982759
$870$	0	0
$871$	−0.828427	−0.0280702
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−6.68629	−0.225780	−0.112890	−	0.993607i	$-0.536011\pi$
$877$	−6.68629	−0.225780	−0.112890	+	0.993607i	$0.536011\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.9411	1.41303	0.706516	−	0.707697i	$-0.250266\pi$
$881$	41.9411	1.41303	0.706516	+	0.707697i	$0.250266\pi$
$882$	0	0
$883$	−9.75736	−0.328361	−0.164181	−	0.986430i	$-0.552498\pi$
$883$	−9.75736	−0.328361	−0.164181	+	0.986430i	$0.552498\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.0416	0.605779	0.302889	−	0.953026i	$-0.402049\pi$
$887$	18.0416	0.605779	0.302889	+	0.953026i	$0.402049\pi$
$888$	0	0
$889$	7.79899	0.261570
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.4853	0.551659
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.6569	−0.455482
$900$	0	0
$901$	15.3137	0.510174
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.3137	−0.774974
$906$	0	0
$907$	−13.0711	−0.434018	−0.217009	−	0.976170i	$-0.569630\pi$
$907$	−13.0711	−0.434018	−0.217009	+	0.976170i	$0.569630\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.5980	0.914362	0.457181	−	0.889374i	$-0.348859\pi$
$911$	27.5980	0.914362	0.457181	+	0.889374i	$0.348859\pi$
$912$	0	0
$913$	−8.48528	−0.280822
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.62742	0.218857
$918$	0	0
$919$	59.1127	1.94995	0.974974	−	0.222319i	$-0.0713626\pi$
$919$	59.1127	1.94995	0.974974	+	0.222319i	$0.0713626\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.89949	−0.325846
$924$	0	0
$925$	−1.17157	−0.0385211
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−21.1127	−0.692685	−0.346343	−	0.938108i	$-0.612577\pi$
$929$	−21.1127	−0.692685	−0.346343	+	0.938108i	$0.612577\pi$
$930$	0	0
$931$	24.7279	0.810425
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.4853	0.669940
$936$	0	0
$937$	−10.2843	−0.335972	−0.167986	−	0.985789i	$-0.553726\pi$
$937$	−10.2843	−0.335972	−0.167986	+	0.985789i	$0.553726\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.51472	0.0493784	0.0246892	−	0.999695i	$-0.492140\pi$
$941$	1.51472	0.0493784	0.0246892	+	0.999695i	$0.492140\pi$
$942$	0	0
$943$	−4.20101	−0.136804
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.0294	−0.423400	−0.211700	−	0.977335i	$-0.567900\pi$
$947$	−13.0294	−0.423400	−0.211700	+	0.977335i	$0.567900\pi$
$948$	0	0
$949$	−6.82843	−0.221660
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−47.9411	−1.55297	−0.776483	−	0.630139i	$-0.782998\pi$
$953$	−47.9411	−1.55297	−0.776483	+	0.630139i	$0.782998\pi$
$954$	0	0
$955$	−21.6569	−0.700799
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.34315	0.268575
$966$	0	0
$967$	−39.1716	−1.25967	−0.629836	−	0.776728i	$-0.716878\pi$
$967$	−39.1716	−1.25967	−0.629836	+	0.776728i	$0.716878\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.6274	−1.23961	−0.619806	−	0.784755i	$-0.712789\pi$
$971$	−38.6274	−1.23961	−0.619806	+	0.784755i	$0.712789\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.4264	0.717484	0.358742	−	0.933437i	$-0.383206\pi$
$977$	22.4264	0.717484	0.358742	+	0.933437i	$0.383206\pi$
$978$	0	0
$979$	−42.4264	−1.35595
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.5147	0.558633	0.279316	−	0.960199i	$-0.409892\pi$
$983$	17.5147	0.558633	0.279316	+	0.960199i	$0.409892\pi$
$984$	0	0
$985$	−0.343146	−0.0109335
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.91169	0.283375
$990$	0	0
$991$	−23.5980	−0.749615	−0.374807	−	0.927103i	$-0.622291\pi$
$991$	−23.5980	−0.749615	−0.374807	+	0.927103i	$0.622291\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.34315	−0.201091
$996$	0	0
$997$	−8.82843	−0.279599	−0.139800	−	0.990180i	$-0.544646\pi$
$997$	−8.82843	−0.279599	−0.139800	+	0.990180i	$0.544646\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.w.1.1		2
3.2	odd	2		520.2.a.c.1.1	✓	2
4.3	odd	2		9360.2.a.ck.1.2		2
12.11	even	2		1040.2.a.n.1.2		2
15.2	even	4		2600.2.d.i.1249.4		4
15.8	even	4		2600.2.d.i.1249.1		4
15.14	odd	2		2600.2.a.w.1.2		2
24.5	odd	2		4160.2.a.bn.1.2		2
24.11	even	2		4160.2.a.u.1.1		2
39.38	odd	2		6760.2.a.n.1.1		2
60.59	even	2		5200.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.c.1.1	✓	2	3.2	odd	2
1040.2.a.n.1.2		2	12.11	even	2
2600.2.a.w.1.2		2	15.14	odd	2
2600.2.d.i.1249.1		4	15.8	even	4
2600.2.d.i.1249.4		4	15.2	even	4
4160.2.a.u.1.1		2	24.11	even	2
4160.2.a.bn.1.2		2	24.5	odd	2
4680.2.a.w.1.1		2	1.1	even	1	trivial
5200.2.a.bl.1.1		2	60.59	even	2
6760.2.a.n.1.1		2	39.38	odd	2
9360.2.a.ck.1.2		2	4.3	odd	2

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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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.00000 q^{7} -4.24264 q^{11} -1.00000 q^{13} +4.82843 q^{17} -8.24264 q^{19} -5.07107 q^{23} +1.00000 q^{25} +9.65685 q^{29} -1.41421 q^{31} +2.00000 q^{35} -1.17157 q^{37} +0.828427 q^{41} -1.75736 q^{43} -2.00000 q^{47} -3.00000 q^{49} +3.17157 q^{53} +4.24264 q^{55} +5.41421 q^{59} -7.31371 q^{61} +1.00000 q^{65} +0.828427 q^{67} +9.89949 q^{71} +6.82843 q^{73} +8.48528 q^{77} +6.82843 q^{79} +2.00000 q^{83} -4.82843 q^{85} +10.0000 q^{89} +2.00000 q^{91} +8.24264 q^{95} +0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} - 2 q^{13} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} + 4 q^{35} - 8 q^{37} - 4 q^{41} - 12 q^{43} - 4 q^{47} - 6 q^{49} + 12 q^{53} + 8 q^{59} + 8 q^{61} + 2 q^{65} - 4 q^{67} + 8 q^{73} + 8 q^{79} + 4 q^{83} - 4 q^{85} + 20 q^{89} + 4 q^{91} + 8 q^{95} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 - 2 * q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 + 4 * q^35 - 8 * q^37 - 4 * q^41 - 12 * q^43 - 4 * q^47 - 6 * q^49 + 12 * q^53 + 8 * q^59 + 8 * q^61 + 2 * q^65 - 4 * q^67 + 8 * q^73 + 8 * q^79 + 4 * q^83 - 4 * q^85 + 20 * q^89 + 4 * q^91 + 8 * q^95 + 12 * q^97

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