LMFDB - Embedded newform 1440.2.k.f.721.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(721,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1440.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.k (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.6
Root			$1.40680 - 0.144584i$ of defining polynomial
Character	$\chi$	$=$	1440.721
Dual form			1440.2.k.f.721.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +3.62721 q^{7} +O(q^{10})$	$q+1.00000i q^{5} +3.62721 q^{7} +6.20555i q^{11} -0.578337i q^{13} -1.42166 q^{17} +5.62721i q^{19} -5.62721 q^{23} -1.00000 q^{25} -2.00000i q^{29} +2.57834 q^{31} +3.62721i q^{35} -7.83276i q^{37} -5.25443 q^{41} +7.25443i q^{43} +6.78389 q^{47} +6.15667 q^{49} +2.00000i q^{53} -6.20555 q^{55} -2.20555i q^{59} +12.4111i q^{61} +0.578337 q^{65} -4.00000i q^{67} +8.41110 q^{71} -6.00000 q^{73} +22.5089i q^{77} -5.42166 q^{79} +3.25443i q^{83} -1.42166i q^{85} +13.2544 q^{89} -2.09775i q^{91} -5.62721 q^{95} +4.84333 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{7}+O(q^{10}) $ 6 * q - 4 * q^7	$ 6 q - 4 q^{7} - 12 q^{17} - 8 q^{23} - 6 q^{25} + 12 q^{31} + 20 q^{41} + 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{71} - 36 q^{73} - 36 q^{79} + 28 q^{89} - 8 q^{95} + 36 q^{97}+O(q^{100}) $ 6 * q - 4 * q^7 - 12 * q^17 - 8 * q^23 - 6 * q^25 + 12 * q^31 + 20 * q^41 + 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^71 - 36 * q^73 - 36 * q^79 + 28 * q^89 - 8 * q^95 + 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.62721	1.37096	0.685479	−	0.728093i	$-0.259593\pi$
$7$	3.62721	1.37096	0.685479	+	0.728093i	$0.259593\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.20555i	1.87104i	0.353269	+	0.935522i	$0.385070\pi$
$11$	6.20555i	1.87104i	−0.353269	+	0.935522i	$0.614930\pi$
$12$	0	0
$13$	− 0.578337i	− 0.160402i	−0.996779	−	0.0802009i	$-0.974444\pi$
$13$	− 0.578337i	− 0.160402i	0.996779	−	0.0802009i	$-0.0255562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.42166	−0.344804	−0.172402	−	0.985027i	$-0.555153\pi$
$17$	−1.42166	−0.344804	−0.172402	+	0.985027i	$0.555153\pi$
$18$	0	0
$19$	5.62721i	1.29097i	0.763772	+	0.645486i	$0.223345\pi$
$19$	5.62721i	1.29097i	−0.763772	+	0.645486i	$0.776655\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.62721	−1.17336	−0.586678	−	0.809821i	$-0.699564\pi$
$23$	−5.62721	−1.17336	−0.586678	+	0.809821i	$0.699564\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	2.57834	0.463083	0.231542	−	0.972825i	$-0.425623\pi$
$31$	2.57834	0.463083	0.231542	+	0.972825i	$0.425623\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.62721i	0.613111i
$36$	0	0
$37$	− 7.83276i	− 1.28770i	−0.765152	−	0.643849i	$-0.777336\pi$
$37$	− 7.83276i	− 1.28770i	0.765152	−	0.643849i	$-0.222664\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.25443	−0.820603	−0.410302	−	0.911950i	$-0.634577\pi$
$41$	−5.25443	−0.820603	−0.410302	+	0.911950i	$0.634577\pi$
$42$	0	0
$43$	7.25443i	1.10629i	0.833085	+	0.553145i	$0.186572\pi$
$43$	7.25443i	1.10629i	−0.833085	+	0.553145i	$0.813428\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.78389	0.989532	0.494766	−	0.869026i	$-0.335254\pi$
$47$	6.78389	0.989532	0.494766	+	0.869026i	$0.335254\pi$
$48$	0	0
$49$	6.15667	0.879525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	−6.20555	−0.836756
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.20555i	− 0.287138i	−0.989640	−	0.143569i	$-0.954142\pi$
$59$	− 2.20555i	− 0.287138i	0.989640	−	0.143569i	$-0.0458579\pi$
$60$	0	0
$61$	12.4111i	1.58908i	0.607213	+	0.794539i	$0.292288\pi$
$61$	12.4111i	1.58908i	−0.607213	+	0.794539i	$0.707712\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.578337	0.0717339
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.41110	0.998214	0.499107	−	0.866540i	$-0.333661\pi$
$71$	8.41110	0.998214	0.499107	+	0.866540i	$0.333661\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	22.5089i	2.56512i
$78$	0	0
$79$	−5.42166	−0.609985	−0.304992	−	0.952355i	$-0.598654\pi$
$79$	−5.42166	−0.609985	−0.304992	+	0.952355i	$0.598654\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.25443i	0.357220i	0.983920	+	0.178610i	$0.0571600\pi$
$83$	3.25443i	0.357220i	−0.983920	+	0.178610i	$0.942840\pi$
$84$	0	0
$85$	− 1.42166i	− 0.154201i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.2544	1.40497	0.702483	−	0.711700i	$-0.252075\pi$
$89$	13.2544	1.40497	0.702483	+	0.711700i	$0.252075\pi$
$90$	0	0
$91$	− 2.09775i	− 0.219904i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.62721	−0.577340
$96$	0	0
$97$	4.84333	0.491765	0.245883	−	0.969300i	$-0.420922\pi$
$97$	4.84333	0.491765	0.245883	+	0.969300i	$0.420922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	−2.47054	−0.243429	−0.121715	−	0.992565i	$-0.538839\pi$
$103$	−2.47054	−0.243429	−0.121715	+	0.992565i	$0.538839\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0978i	1.36288i	0.731873	+	0.681441i	$0.238646\pi$
$107$	14.0978i	1.36288i	−0.731873	+	0.681441i	$0.761354\pi$
$108$	0	0
$109$	7.25443i	0.694848i	0.937708	+	0.347424i	$0.112944\pi$
$109$	7.25443i	0.694848i	−0.937708	+	0.347424i	$0.887056\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.08719	0.854851	0.427425	−	0.904051i	$-0.359421\pi$
$113$	9.08719	0.854851	0.427425	+	0.904051i	$0.359421\pi$
$114$	0	0
$115$	− 5.62721i	− 0.524740i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.15667	−0.472712
$120$	0	0
$121$	−27.5089	−2.50080
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	10.4705	0.929110	0.464555	−	0.885544i	$-0.346214\pi$
$127$	10.4705	0.929110	0.464555	+	0.885544i	$0.346214\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.4600i	− 1.17600i	−0.808860	−	0.588002i	$-0.799915\pi$
$131$	− 13.4600i	− 1.17600i	0.808860	−	0.588002i	$-0.200085\pi$
$132$	0	0
$133$	20.4111i	1.76987i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.5783	0.903768	0.451884	−	0.892077i	$-0.350752\pi$
$137$	10.5783	0.903768	0.451884	+	0.892077i	$0.350752\pi$
$138$	0	0
$139$	12.4705i	1.05774i	0.848704	+	0.528869i	$0.177384\pi$
$139$	12.4705i	1.05774i	−0.848704	+	0.528869i	$0.822616\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.58890	0.300119
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	0	0
$151$	−12.6761	−1.03157	−0.515783	−	0.856719i	$-0.672499\pi$
$151$	−12.6761	−1.03157	−0.515783	+	0.856719i	$0.672499\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.57834i	0.207097i
$156$	0	0
$157$	− 1.32391i	− 0.105660i	−0.998604	−	0.0528298i	$-0.983176\pi$
$157$	− 1.32391i	− 0.105660i	0.998604	−	0.0528298i	$-0.0168241\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.4111	−1.60862
$162$	0	0
$163$	− 15.2544i	− 1.19482i	−0.801936	−	0.597409i	$-0.796197\pi$
$163$	− 15.2544i	− 1.19482i	0.801936	−	0.597409i	$-0.203803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7839	0.834482	0.417241	−	0.908796i	$-0.362997\pi$
$167$	10.7839	0.834482	0.417241	+	0.908796i	$0.362997\pi$
$168$	0	0
$169$	12.6655	0.974271
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.6655i	− 1.03897i	−0.854479	−	0.519485i	$-0.826124\pi$
$173$	− 13.6655i	− 1.03897i	0.854479	−	0.519485i	$-0.173876\pi$
$174$	0	0
$175$	−3.62721	−0.274192
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.04888i	− 0.676345i	−0.941084	−	0.338172i	$-0.890191\pi$
$179$	− 9.04888i	− 0.676345i	0.941084	−	0.338172i	$-0.109809\pi$
$180$	0	0
$181$	− 23.2544i	− 1.72849i	−0.503073	−	0.864244i	$-0.667797\pi$
$181$	− 23.2544i	− 1.72849i	0.503073	−	0.864244i	$-0.332203\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.83276	0.575876
$186$	0	0
$187$	− 8.82220i	− 0.645143i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	25.6655	1.84745	0.923723	−	0.383062i	$-0.125131\pi$
$193$	25.6655	1.84745	0.923723	+	0.383062i	$0.125131\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.1567i	− 1.07987i	−0.841707	−	0.539934i	$-0.818449\pi$
$197$	− 15.1567i	− 1.07987i	0.841707	−	0.539934i	$-0.181551\pi$
$198$	0	0
$199$	20.6761	1.46569	0.732845	−	0.680396i	$-0.238192\pi$
$199$	20.6761	1.46569	0.732845	+	0.680396i	$0.238192\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.25443i	− 0.509161i
$204$	0	0
$205$	− 5.25443i	− 0.366985i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−34.9200	−2.41546
$210$	0	0
$211$	2.03831i	0.140323i	0.997536	+	0.0701616i	$0.0223515\pi$
$211$	2.03831i	0.140323i	−0.997536	+	0.0701616i	$0.977648\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.25443	−0.494748
$216$	0	0
$217$	9.35218	0.634867
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.822200i	0.0553072i
$222$	0	0
$223$	7.21611	0.483227	0.241613	−	0.970373i	$-0.422323\pi$
$223$	7.21611	0.483227	0.241613	+	0.970373i	$0.422323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.15667i	0.0767712i	0.999263	+	0.0383856i	$0.0122215\pi$
$227$	1.15667i	0.0767712i	−0.999263	+	0.0383856i	$0.987778\pi$
$228$	0	0
$229$	14.0978i	0.931606i	0.884889	+	0.465803i	$0.154234\pi$
$229$	14.0978i	0.931606i	−0.884889	+	0.465803i	$0.845766\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5783	0.955059	0.477529	−	0.878616i	$-0.341532\pi$
$233$	14.5783	0.955059	0.477529	+	0.878616i	$0.341532\pi$
$234$	0	0
$235$	6.78389i	0.442532i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.2544	1.24547	0.622733	−	0.782435i	$-0.286022\pi$
$239$	19.2544	1.24547	0.622733	+	0.782435i	$0.286022\pi$
$240$	0	0
$241$	−13.6655	−0.880274	−0.440137	−	0.897931i	$-0.645070\pi$
$241$	−13.6655	−0.880274	−0.440137	+	0.897931i	$0.645070\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.15667i	0.393335i
$246$	0	0
$247$	3.25443	0.207074
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.14663i	− 0.451091i	−0.974233	−	0.225546i	$-0.927584\pi$
$251$	− 7.14663i	− 0.451091i	0.974233	−	0.225546i	$-0.0724165\pi$
$252$	0	0
$253$	− 34.9200i	− 2.19540i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.73501	−0.482497	−0.241248	−	0.970463i	$-0.577557\pi$
$257$	−7.73501	−0.482497	−0.241248	+	0.970463i	$0.577557\pi$
$258$	0	0
$259$	− 28.4111i	− 1.76538i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.7839	−1.15826	−0.579132	−	0.815234i	$-0.696608\pi$
$263$	−18.7839	−1.15826	−0.579132	+	0.815234i	$0.696608\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 8.50885i	− 0.518794i	−0.965771	−	0.259397i	$-0.916476\pi$
$269$	− 8.50885i	− 0.518794i	0.965771	−	0.259397i	$-0.0835238\pi$
$270$	0	0
$271$	−30.9894	−1.88247	−0.941237	−	0.337746i	$-0.890335\pi$
$271$	−30.9894	−1.88247	−0.941237	+	0.337746i	$0.890335\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.20555i	− 0.374209i
$276$	0	0
$277$	9.51941i	0.571966i	0.958235	+	0.285983i	$0.0923201\pi$
$277$	9.51941i	0.571966i	−0.958235	+	0.285983i	$0.907680\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.6655	−0.815217	−0.407608	−	0.913157i	$-0.633637\pi$
$281$	−13.6655	−0.815217	−0.407608	+	0.913157i	$0.633637\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.0589	−1.12501
$288$	0	0
$289$	−14.9789	−0.881110
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.31335i	0.251989i	0.992031	+	0.125994i	$0.0402121\pi$
$293$	4.31335i	0.251989i	−0.992031	+	0.125994i	$0.959788\pi$
$294$	0	0
$295$	2.20555	0.128412
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.25443i	0.188208i
$300$	0	0
$301$	26.3133i	1.51668i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.4111	−0.710658
$306$	0	0
$307$	− 25.5678i	− 1.45923i	−0.683858	−	0.729615i	$-0.739699\pi$
$307$	− 25.5678i	− 1.45923i	0.683858	−	0.729615i	$-0.260301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0766	−1.13844	−0.569221	−	0.822185i	$-0.692755\pi$
$311$	−20.0766	−1.13844	−0.569221	+	0.822185i	$0.692755\pi$
$312$	0	0
$313$	7.15667	0.404519	0.202260	−	0.979332i	$-0.435172\pi$
$313$	7.15667	0.404519	0.202260	+	0.979332i	$0.435172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.1744i	− 1.35777i	−0.734245	−	0.678884i	$-0.762464\pi$
$317$	− 24.1744i	− 1.35777i	0.734245	−	0.678884i	$-0.237536\pi$
$318$	0	0
$319$	12.4111	0.694888
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0.578337i	0.0320804i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.6066	1.35661
$330$	0	0
$331$	27.1950i	1.49477i	0.664390	+	0.747386i	$0.268691\pi$
$331$	27.1950i	1.49477i	−0.664390	+	0.747386i	$0.731309\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	22.8222	1.24320	0.621602	−	0.783333i	$-0.286482\pi$
$337$	22.8222	1.24320	0.621602	+	0.783333i	$0.286482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	−3.05892	−0.165166
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.6655i	− 1.27043i	−0.772335	−	0.635216i	$-0.780911\pi$
$347$	− 23.6655i	− 1.27043i	0.772335	−	0.635216i	$-0.219089\pi$
$348$	0	0
$349$	− 34.9200i	− 1.86922i	−0.355671	−	0.934611i	$-0.615748\pi$
$349$	− 34.9200i	− 1.86922i	0.355671	−	0.934611i	$-0.384252\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.9305	0.847896	0.423948	−	0.905687i	$-0.360644\pi$
$353$	15.9305	0.847896	0.423948	+	0.905687i	$0.360644\pi$
$354$	0	0
$355$	8.41110i	0.446415i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.41110	0.443921	0.221960	−	0.975056i	$-0.428754\pi$
$359$	8.41110	0.443921	0.221960	+	0.975056i	$0.428754\pi$
$360$	0	0
$361$	−12.6655	−0.666607
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.00000i	− 0.314054i
$366$	0	0
$367$	24.4494	1.27625	0.638124	−	0.769933i	$-0.279711\pi$
$367$	24.4494	1.27625	0.638124	+	0.769933i	$0.279711\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.25443i	0.376631i
$372$	0	0
$373$	− 0.167237i	− 0.00865920i	−0.999991	−	0.00432960i	$-0.998622\pi$
$373$	− 0.167237i	− 0.00865920i	0.999991	−	0.00432960i	$-0.00137816\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.15667	−0.0595718
$378$	0	0
$379$	− 7.72496i	− 0.396805i	−0.980121	−	0.198402i	$-0.936425\pi$
$379$	− 7.72496i	− 0.396805i	0.980121	−	0.198402i	$-0.0635753\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.62721	−0.0831467	−0.0415734	−	0.999135i	$-0.513237\pi$
$383$	−1.62721	−0.0831467	−0.0415734	+	0.999135i	$0.513237\pi$
$384$	0	0
$385$	−22.5089	−1.14716
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 12.3133i	− 0.624312i	−0.950031	−	0.312156i	$-0.898949\pi$
$389$	− 12.3133i	− 0.624312i	0.950031	−	0.312156i	$-0.101051\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 5.42166i	− 0.272793i
$396$	0	0
$397$	− 19.0872i	− 0.957959i	−0.877826	−	0.478979i	$-0.841007\pi$
$397$	− 19.0872i	− 0.957959i	0.877826	−	0.478979i	$-0.158993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4111	0.719656	0.359828	−	0.933019i	$-0.382835\pi$
$401$	14.4111	0.719656	0.359828	+	0.933019i	$0.382835\pi$
$402$	0	0
$403$	− 1.49115i	− 0.0742794i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.6066	2.40934
$408$	0	0
$409$	−8.31335	−0.411069	−0.205534	−	0.978650i	$-0.565893\pi$
$409$	−8.31335	−0.411069	−0.205534	+	0.978650i	$0.565893\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	−3.25443	−0.159753
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 7.36222i	− 0.359668i	−0.983697	−	0.179834i	$-0.942444\pi$
$419$	− 7.36222i	− 0.359668i	0.983697	−	0.179834i	$-0.0575561\pi$
$420$	0	0
$421$	− 30.0978i	− 1.46687i	−0.679757	−	0.733437i	$-0.737915\pi$
$421$	− 30.0978i	− 1.46687i	0.679757	−	0.733437i	$-0.262085\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.42166	0.0689608
$426$	0	0
$427$	45.0177i	2.17856i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.41110	0.405148	0.202574	−	0.979267i	$-0.435069\pi$
$431$	8.41110	0.405148	0.202574	+	0.979267i	$0.435069\pi$
$432$	0	0
$433$	−4.31335	−0.207286	−0.103643	−	0.994615i	$-0.533050\pi$
$433$	−4.31335	−0.207286	−0.103643	+	0.994615i	$0.533050\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 31.6655i	− 1.51477i
$438$	0	0
$439$	−9.83276	−0.469292	−0.234646	−	0.972081i	$-0.575393\pi$
$439$	−9.83276	−0.469292	−0.234646	+	0.972081i	$0.575393\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.3522i	1.01447i	0.861807	+	0.507236i	$0.169333\pi$
$443$	21.3522i	1.01447i	−0.861807	+	0.507236i	$0.830667\pi$
$444$	0	0
$445$	13.2544i	0.628320i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.3133	0.958646	0.479323	−	0.877639i	$-0.340882\pi$
$449$	20.3133	0.958646	0.479323	+	0.877639i	$0.340882\pi$
$450$	0	0
$451$	− 32.6066i	− 1.53539i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.09775	0.0983441
$456$	0	0
$457$	−3.35218	−0.156808	−0.0784041	−	0.996922i	$-0.524982\pi$
$457$	−3.35218	−0.156808	−0.0784041	+	0.996922i	$0.524982\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 28.5089i	− 1.32779i	−0.747826	−	0.663895i	$-0.768902\pi$
$461$	− 28.5089i	− 1.32779i	0.747826	−	0.663895i	$-0.231098\pi$
$462$	0	0
$463$	−23.6272	−1.09805	−0.549025	−	0.835806i	$-0.685001\pi$
$463$	−23.6272	−1.09805	−0.549025	+	0.835806i	$0.685001\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.5678i	− 1.36823i	−0.729373	−	0.684117i	$-0.760188\pi$
$467$	− 29.5678i	− 1.36823i	0.729373	−	0.684117i	$-0.239812\pi$
$468$	0	0
$469$	− 14.5089i	− 0.669957i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−45.0177	−2.06992
$474$	0	0
$475$	− 5.62721i	− 0.258194i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.0978	−1.00967	−0.504836	−	0.863215i	$-0.668447\pi$
$479$	−22.0978	−1.00967	−0.504836	+	0.863215i	$0.668447\pi$
$480$	0	0
$481$	−4.52998	−0.206549
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.84333i	0.219924i
$486$	0	0
$487$	−4.03831	−0.182993	−0.0914967	−	0.995805i	$-0.529165\pi$
$487$	−4.03831	−0.182993	−0.0914967	+	0.995805i	$0.529165\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.2056i	0.821605i	0.911724	+	0.410802i	$0.134751\pi$
$491$	18.2056i	0.821605i	−0.911724	+	0.410802i	$0.865249\pi$
$492$	0	0
$493$	2.84333i	0.128057i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.5089	1.36851
$498$	0	0
$499$	− 0.0594386i	− 0.00266084i	−0.999999	−	0.00133042i	$-0.999577\pi$
$499$	− 0.0594386i	− 0.00266084i	0.999999	−	0.00133042i	$-0.000423486\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.03831	0.0908839	0.0454419	−	0.998967i	$-0.485530\pi$
$503$	2.03831	0.0908839	0.0454419	+	0.998967i	$0.485530\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.7044i	1.80419i	0.431539	+	0.902094i	$0.357971\pi$
$509$	40.7044i	1.80419i	−0.431539	+	0.902094i	$0.642029\pi$
$510$	0	0
$511$	−21.7633	−0.962751
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.47054i	− 0.108865i
$516$	0	0
$517$	42.0978i	1.85146i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	− 35.3311i	− 1.54492i	−0.635064	−	0.772460i	$-0.719026\pi$
$523$	− 35.3311i	− 1.54492i	0.635064	−	0.772460i	$-0.280974\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.66553	−0.159673
$528$	0	0
$529$	8.66553	0.376762
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.03883i	0.131626i
$534$	0	0
$535$	−14.0978	−0.609499
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38.2056i	1.64563i
$540$	0	0
$541$	3.05892i	0.131513i	0.997836	+	0.0657567i	$0.0209461\pi$
$541$	3.05892i	0.131513i	−0.997836	+	0.0657567i	$0.979054\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.25443	−0.310745
$546$	0	0
$547$	32.0766i	1.37150i	0.727838	+	0.685749i	$0.240525\pi$
$547$	32.0766i	1.37150i	−0.727838	+	0.685749i	$0.759475\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.2544	0.479455
$552$	0	0
$553$	−19.6655	−0.836263
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.6655i	1.42645i	0.700933	+	0.713227i	$0.252767\pi$
$557$	33.6655i	1.42645i	−0.700933	+	0.713227i	$0.747233\pi$
$558$	0	0
$559$	4.19550	0.177451
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 5.35218i	− 0.225567i	−0.993620	−	0.112784i	$-0.964023\pi$
$563$	− 5.35218i	− 0.225567i	0.993620	−	0.112784i	$-0.0359767\pi$
$564$	0	0
$565$	9.08719i	0.382301i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.58890	−0.234299	−0.117149	−	0.993114i	$-0.537376\pi$
$569$	−5.58890	−0.234299	−0.117149	+	0.993114i	$0.537376\pi$
$570$	0	0
$571$	10.3728i	0.434088i	0.976162	+	0.217044i	$0.0696414\pi$
$571$	10.3728i	0.434088i	−0.976162	+	0.217044i	$0.930359\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.62721	0.234671
$576$	0	0
$577$	21.6655	0.901948	0.450974	−	0.892537i	$-0.351077\pi$
$577$	21.6655	0.901948	0.450974	+	0.892537i	$0.351077\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8045i	0.489733i
$582$	0	0
$583$	−12.4111	−0.514015
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.90225i	0.0785142i	0.999229	+	0.0392571i	$0.0124991\pi$
$587$	1.90225i	0.0785142i	−0.999229	+	0.0392571i	$0.987501\pi$
$588$	0	0
$589$	14.5089i	0.597827i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.57834	0.105880	0.0529398	−	0.998598i	$-0.483141\pi$
$593$	2.57834	0.105880	0.0529398	+	0.998598i	$0.483141\pi$
$594$	0	0
$595$	− 5.15667i	− 0.211403i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.7244	1.09193	0.545966	−	0.837808i	$-0.316163\pi$
$599$	26.7244	1.09193	0.545966	+	0.837808i	$0.316163\pi$
$600$	0	0
$601$	33.3311	1.35960	0.679801	−	0.733397i	$-0.262066\pi$
$601$	33.3311	1.35960	0.679801	+	0.733397i	$0.262066\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 27.5089i	− 1.11839i
$606$	0	0
$607$	21.9406	0.890540	0.445270	−	0.895396i	$-0.353108\pi$
$607$	21.9406	0.890540	0.445270	+	0.895396i	$0.353108\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.92337i	− 0.158723i
$612$	0	0
$613$	− 3.42166i	− 0.138200i	−0.997610	−	0.0690998i	$-0.977987\pi$
$613$	− 3.42166i	− 0.138200i	0.997610	−	0.0690998i	$-0.0220127\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.7350	−0.794502	−0.397251	−	0.917710i	$-0.630036\pi$
$617$	−19.7350	−0.794502	−0.397251	+	0.917710i	$0.630036\pi$
$618$	0	0
$619$	20.4705i	0.822780i	0.911459	+	0.411390i	$0.134957\pi$
$619$	20.4705i	0.822780i	−0.911459	+	0.411390i	$0.865043\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	48.0766	1.92615
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.1355i	0.444003i
$630$	0	0
$631$	−1.08719	−0.0432803	−0.0216402	−	0.999766i	$-0.506889\pi$
$631$	−1.08719	−0.0432803	−0.0216402	+	0.999766i	$0.506889\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.4705i	0.415511i
$636$	0	0
$637$	− 3.56063i	− 0.141077i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.9789	1.10510	0.552550	−	0.833480i	$-0.313655\pi$
$641$	27.9789	1.10510	0.552550	+	0.833480i	$0.313655\pi$
$642$	0	0
$643$	4.94108i	0.194857i	0.995243	+	0.0974285i	$0.0310617\pi$
$643$	4.94108i	0.194857i	−0.995243	+	0.0974285i	$0.968938\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−49.3694	−1.94091	−0.970455	−	0.241282i	$-0.922432\pi$
$647$	−49.3694	−1.94091	−0.970455	+	0.241282i	$0.922432\pi$
$648$	0	0
$649$	13.6867	0.537248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.1744i	1.57214i	0.618134	+	0.786072i	$0.287889\pi$
$653$	40.1744i	1.57214i	−0.618134	+	0.786072i	$0.712111\pi$
$654$	0	0
$655$	13.4600	0.525925
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.1255i	0.822933i	0.911425	+	0.411466i	$0.134983\pi$
$659$	21.1255i	0.822933i	−0.911425	+	0.411466i	$0.865017\pi$
$660$	0	0
$661$	10.9200i	0.424737i	0.977190	+	0.212368i	$0.0681177\pi$
$661$	10.9200i	0.424737i	−0.977190	+	0.212368i	$0.931882\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.4111	−0.791509
$666$	0	0
$667$	11.2544i	0.435773i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−77.0177	−2.97324
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.4877i	1.17174i	0.810406	+	0.585869i	$0.199247\pi$
$677$	30.4877i	1.17174i	−0.810406	+	0.585869i	$0.800753\pi$
$678$	0	0
$679$	17.5678	0.674189
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.2544i	1.34897i	0.738287	+	0.674487i	$0.235635\pi$
$683$	35.2544i	1.34897i	−0.738287	+	0.674487i	$0.764365\pi$
$684$	0	0
$685$	10.5783i	0.404177i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.15667	0.0440658
$690$	0	0
$691$	− 28.1361i	− 1.07035i	−0.844742	−	0.535173i	$-0.820246\pi$
$691$	− 28.1361i	− 1.07035i	0.844742	−	0.535173i	$-0.179754\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.4705	−0.473034
$696$	0	0
$697$	7.47002	0.282947
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 34.8222i	− 1.31522i	−0.753360	−	0.657608i	$-0.771568\pi$
$701$	− 34.8222i	− 1.31522i	0.753360	−	0.657608i	$-0.228432\pi$
$702$	0	0
$703$	44.0766	1.66238
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.25443i	0.272831i
$708$	0	0
$709$	7.58890i	0.285007i	0.989794	+	0.142504i	$0.0455152\pi$
$709$	7.58890i	0.285007i	−0.989794	+	0.142504i	$0.954485\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14.5089	−0.543361
$714$	0	0
$715$	3.58890i	0.134217i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.66553	−0.136701	−0.0683505	−	0.997661i	$-0.521774\pi$
$719$	−3.66553	−0.136701	−0.0683505	+	0.997661i	$0.521774\pi$
$720$	0	0
$721$	−8.96117	−0.333731
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	36.1149	1.33943	0.669714	−	0.742619i	$-0.266416\pi$
$727$	36.1149	1.33943	0.669714	+	0.742619i	$0.266416\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 10.3133i	− 0.381453i
$732$	0	0
$733$	34.0071i	1.25608i	0.778180	+	0.628041i	$0.216143\pi$
$733$	34.0071i	1.25608i	−0.778180	+	0.628041i	$0.783857\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.8222	0.914338
$738$	0	0
$739$	− 52.0172i	− 1.91348i	−0.290939	−	0.956742i	$-0.593968\pi$
$739$	− 52.0172i	− 1.91348i	0.290939	−	0.956742i	$-0.406032\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.3139	−0.855303	−0.427651	−	0.903944i	$-0.640659\pi$
$743$	−23.3139	−0.855303	−0.427651	+	0.903944i	$0.640659\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	51.1355i	1.86845i
$750$	0	0
$751$	−11.1083	−0.405348	−0.202674	−	0.979246i	$-0.564963\pi$
$751$	−11.1083	−0.405348	−0.202674	+	0.979246i	$0.564963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 12.6761i	− 0.461330i
$756$	0	0
$757$	13.3239i	0.484266i	0.970243	+	0.242133i	$0.0778470\pi$
$757$	13.3239i	0.484266i	−0.970243	+	0.242133i	$0.922153\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.1355	0.621163	0.310582	−	0.950547i	$-0.399476\pi$
$761$	17.1355	0.621163	0.310582	+	0.950547i	$0.399476\pi$
$762$	0	0
$763$	26.3133i	0.952607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.27555	−0.0460575
$768$	0	0
$769$	5.47002	0.197254	0.0986270	−	0.995124i	$-0.468555\pi$
$769$	5.47002	0.197254	0.0986270	+	0.995124i	$0.468555\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.15667i	0.113538i	0.998387	+	0.0567688i	$0.0180798\pi$
$773$	3.15667i	0.113538i	−0.998387	+	0.0567688i	$0.981920\pi$
$774$	0	0
$775$	−2.57834	−0.0926166
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 29.5678i	− 1.05938i
$780$	0	0
$781$	52.1955i	1.86770i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.32391	0.0472524
$786$	0	0
$787$	− 31.1355i	− 1.10986i	−0.831896	−	0.554931i	$-0.812745\pi$
$787$	− 31.1355i	− 1.10986i	0.831896	−	0.554931i	$-0.187255\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.9612	1.17196
$792$	0	0
$793$	7.17780	0.254891
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000i	0.354218i	0.984191	+	0.177109i	$0.0566745\pi$
$797$	10.0000i	0.354218i	−0.984191	+	0.177109i	$0.943325\pi$
$798$	0	0
$799$	−9.64440	−0.341194
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 37.2333i	− 1.31393i
$804$	0	0
$805$	− 20.4111i	− 0.719397i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29.0388	−1.02095	−0.510475	−	0.859892i	$-0.670531\pi$
$809$	−29.0388	−1.02095	−0.510475	+	0.859892i	$0.670531\pi$
$810$	0	0
$811$	− 2.58838i	− 0.0908904i	−0.998967	−	0.0454452i	$-0.985529\pi$
$811$	− 2.58838i	− 0.0908904i	0.998967	−	0.0454452i	$-0.0144706\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	15.2544	0.534339
$816$	0	0
$817$	−40.8222	−1.42819
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.1955i	1.33303i	0.745491	+	0.666516i	$0.232215\pi$
$821$	38.1955i	1.33303i	−0.745491	+	0.666516i	$0.767785\pi$
$822$	0	0
$823$	18.3517	0.639699	0.319849	−	0.947468i	$-0.396368\pi$
$823$	18.3517	0.639699	0.319849	+	0.947468i	$0.396368\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	− 24.7456i	− 0.859449i	−0.902960	−	0.429725i	$-0.858611\pi$
$829$	− 24.7456i	− 0.859449i	0.902960	−	0.429725i	$-0.141389\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.75272	−0.303264
$834$	0	0
$835$	10.7839i	0.373192i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−53.4288	−1.84457	−0.922284	−	0.386514i	$-0.873679\pi$
$839$	−53.4288	−1.84457	−0.922284	+	0.386514i	$0.873679\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.6655i	0.435707i
$846$	0	0
$847$	−99.7805	−3.42850
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.0766i	1.51093i
$852$	0	0
$853$	− 29.0661i	− 0.995203i	−0.867406	−	0.497602i	$-0.834214\pi$
$853$	− 29.0661i	− 0.995203i	0.867406	−	0.497602i	$-0.165786\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.2439	0.349924	0.174962	−	0.984575i	$-0.444020\pi$
$857$	10.2439	0.349924	0.174962	+	0.984575i	$0.444020\pi$
$858$	0	0
$859$	10.9794i	0.374612i	0.982302	+	0.187306i	$0.0599756\pi$
$859$	10.9794i	0.374612i	−0.982302	+	0.187306i	$0.940024\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.8605	1.32283	0.661414	−	0.750021i	$-0.269957\pi$
$863$	38.8605	1.32283	0.661414	+	0.750021i	$0.269957\pi$
$864$	0	0
$865$	13.6655	0.464642
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 33.6444i	− 1.14131i
$870$	0	0
$871$	−2.31335	−0.0783848
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.62721i	− 0.122622i
$876$	0	0
$877$	− 22.3416i	− 0.754423i	−0.926127	−	0.377211i	$-0.876883\pi$
$877$	− 22.3416i	− 0.754423i	0.926127	−	0.377211i	$-0.123117\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.88112	−0.332903	−0.166452	−	0.986050i	$-0.553231\pi$
$881$	−9.88112	−0.332903	−0.166452	+	0.986050i	$0.553231\pi$
$882$	0	0
$883$	− 10.6277i	− 0.357652i	−0.983881	−	0.178826i	$-0.942770\pi$
$883$	− 10.6277i	− 0.357652i	0.983881	−	0.178826i	$-0.0572298\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.6061	0.389694	0.194847	−	0.980834i	$-0.437579\pi$
$887$	11.6061	0.389694	0.194847	+	0.980834i	$0.437579\pi$
$888$	0	0
$889$	37.9789	1.27377
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38.1744i	1.27746i
$894$	0	0
$895$	9.04888	0.302471
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.15667i	− 0.171985i
$900$	0	0
$901$	− 2.84333i	− 0.0947249i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.2544	0.773003
$906$	0	0
$907$	− 0.195504i	− 0.00649159i	−0.999995	−	0.00324580i	$-0.998967\pi$
$907$	− 0.195504i	− 0.00649159i	0.999995	−	0.00324580i	$-0.00103317\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.88112	−0.261113	−0.130557	−	0.991441i	$-0.541676\pi$
$911$	−7.88112	−0.261113	−0.130557	+	0.991441i	$0.541676\pi$
$912$	0	0
$913$	−20.1955	−0.668374
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.8222i	− 1.61225i
$918$	0	0
$919$	−9.75614	−0.321825	−0.160913	−	0.986969i	$-0.551444\pi$
$919$	−9.75614	−0.321825	−0.160913	+	0.986969i	$0.551444\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.86445i	− 0.160115i
$924$	0	0
$925$	7.83276i	0.257540i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.82220	−0.223829	−0.111915	−	0.993718i	$-0.535698\pi$
$929$	−6.82220	−0.223829	−0.111915	+	0.993718i	$0.535698\pi$
$930$	0	0
$931$	34.6449i	1.13544i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.82220	0.288517
$936$	0	0
$937$	57.5266	1.87931	0.939655	−	0.342123i	$-0.111146\pi$
$937$	57.5266	1.87931	0.939655	+	0.342123i	$0.111146\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.508852i	0.0165881i	0.999966	+	0.00829405i	$0.00264011\pi$
$941$	0.508852i	0.0165881i	−0.999966	+	0.00829405i	$0.997360\pi$
$942$	0	0
$943$	29.5678	0.962859
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.68665i	0.0548088i	0.999624	+	0.0274044i	$0.00872419\pi$
$947$	1.68665i	0.0548088i	−0.999624	+	0.0274044i	$0.991276\pi$
$948$	0	0
$949$	3.47002i	0.112642i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.22616	−0.298865	−0.149432	−	0.988772i	$-0.547745\pi$
$953$	−9.22616	−0.298865	−0.149432	+	0.988772i	$0.547745\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	38.3699	1.23903
$960$	0	0
$961$	−24.3522	−0.785554
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.6655i	0.826203i
$966$	0	0
$967$	−12.2338	−0.393413	−0.196707	−	0.980462i	$-0.563025\pi$
$967$	−12.2338	−0.393413	−0.196707	+	0.980462i	$0.563025\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.2444i	1.06686i	0.845843	+	0.533431i	$0.179098\pi$
$971$	33.2444i	1.06686i	−0.845843	+	0.533431i	$0.820902\pi$
$972$	0	0
$973$	45.2333i	1.45011i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.93051	−0.253720	−0.126860	−	0.991921i	$-0.540490\pi$
$977$	−7.93051	−0.253720	−0.126860	+	0.991921i	$0.540490\pi$
$978$	0	0
$979$	82.2510i	2.62875i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.8993	1.33638	0.668191	−	0.743990i	$-0.267069\pi$
$983$	41.8993	1.33638	0.668191	+	0.743990i	$0.267069\pi$
$984$	0	0
$985$	15.1567	0.482932
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 40.8222i	− 1.29807i
$990$	0	0
$991$	35.1849	1.11769	0.558843	−	0.829273i	$-0.311245\pi$
$991$	35.1849	1.11769	0.558843	+	0.829273i	$0.311245\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.6761i	0.655476i
$996$	0	0
$997$	8.04836i	0.254894i	0.991845	+	0.127447i	$0.0406783\pi$
$997$	8.04836i	0.254894i	−0.991845	+	0.127447i	$0.959322\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.k.f.721.6		6
3.2	odd	2		480.2.k.b.241.6		6
4.3	odd	2		360.2.k.f.181.1		6
5.2	odd	4		7200.2.d.q.2449.5		6
5.3	odd	4		7200.2.d.r.2449.2		6
5.4	even	2		7200.2.k.p.3601.2		6
8.3	odd	2		360.2.k.f.181.2		6
8.5	even	2	inner	1440.2.k.f.721.3		6
12.11	even	2		120.2.k.b.61.6	yes	6
15.2	even	4		2400.2.d.f.49.5		6
15.8	even	4		2400.2.d.e.49.2		6
15.14	odd	2		2400.2.k.c.1201.1		6
20.3	even	4		1800.2.d.r.1549.4		6
20.7	even	4		1800.2.d.q.1549.3		6
20.19	odd	2		1800.2.k.p.901.6		6
24.5	odd	2		480.2.k.b.241.3		6
24.11	even	2		120.2.k.b.61.5	✓	6
40.3	even	4		1800.2.d.q.1549.4		6
40.13	odd	4		7200.2.d.q.2449.2		6
40.19	odd	2		1800.2.k.p.901.5		6
40.27	even	4		1800.2.d.r.1549.3		6
40.29	even	2		7200.2.k.p.3601.1		6
40.37	odd	4		7200.2.d.r.2449.5		6
48.5	odd	4		3840.2.a.br.1.1		3
48.11	even	4		3840.2.a.bp.1.3		3
48.29	odd	4		3840.2.a.bo.1.1		3
48.35	even	4		3840.2.a.bq.1.3		3
60.23	odd	4		600.2.d.f.349.3		6
60.47	odd	4		600.2.d.e.349.4		6
60.59	even	2		600.2.k.c.301.1		6
120.29	odd	2		2400.2.k.c.1201.4		6
120.53	even	4		2400.2.d.f.49.2		6
120.59	even	2		600.2.k.c.301.2		6
120.77	even	4		2400.2.d.e.49.5		6
120.83	odd	4		600.2.d.e.349.3		6
120.107	odd	4		600.2.d.f.349.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.5	✓	6	24.11	even	2
120.2.k.b.61.6	yes	6	12.11	even	2
360.2.k.f.181.1		6	4.3	odd	2
360.2.k.f.181.2		6	8.3	odd	2
480.2.k.b.241.3		6	24.5	odd	2
480.2.k.b.241.6		6	3.2	odd	2
600.2.d.e.349.3		6	120.83	odd	4
600.2.d.e.349.4		6	60.47	odd	4
600.2.d.f.349.3		6	60.23	odd	4
600.2.d.f.349.4		6	120.107	odd	4
600.2.k.c.301.1		6	60.59	even	2
600.2.k.c.301.2		6	120.59	even	2
1440.2.k.f.721.3		6	8.5	even	2	inner
1440.2.k.f.721.6		6	1.1	even	1	trivial
1800.2.d.q.1549.3		6	20.7	even	4
1800.2.d.q.1549.4		6	40.3	even	4
1800.2.d.r.1549.3		6	40.27	even	4
1800.2.d.r.1549.4		6	20.3	even	4
1800.2.k.p.901.5		6	40.19	odd	2
1800.2.k.p.901.6		6	20.19	odd	2
2400.2.d.e.49.2		6	15.8	even	4
2400.2.d.e.49.5		6	120.77	even	4
2400.2.d.f.49.2		6	120.53	even	4
2400.2.d.f.49.5		6	15.2	even	4
2400.2.k.c.1201.1		6	15.14	odd	2
2400.2.k.c.1201.4		6	120.29	odd	2
3840.2.a.bo.1.1		3	48.29	odd	4
3840.2.a.bp.1.3		3	48.11	even	4
3840.2.a.bq.1.3		3	48.35	even	4
3840.2.a.br.1.1		3	48.5	odd	4
7200.2.d.q.2449.2		6	40.13	odd	4
7200.2.d.q.2449.5		6	5.2	odd	4
7200.2.d.r.2449.2		6	5.3	odd	4
7200.2.d.r.2449.5		6	40.37	odd	4
7200.2.k.p.3601.1		6	40.29	even	2
7200.2.k.p.3601.2		6	5.4	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +3.62721 q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +3.62721 q^{7} +6.20555i q^{11} -0.578337i q^{13} -1.42166 q^{17} +5.62721i q^{19} -5.62721 q^{23} -1.00000 q^{25} -2.00000i q^{29} +2.57834 q^{31} +3.62721i q^{35} -7.83276i q^{37} -5.25443 q^{41} +7.25443i q^{43} +6.78389 q^{47} +6.15667 q^{49} +2.00000i q^{53} -6.20555 q^{55} -2.20555i q^{59} +12.4111i q^{61} +0.578337 q^{65} -4.00000i q^{67} +8.41110 q^{71} -6.00000 q^{73} +22.5089i q^{77} -5.42166 q^{79} +3.25443i q^{83} -1.42166i q^{85} +13.2544 q^{89} -2.09775i q^{91} -5.62721 q^{95} +4.84333 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{7}+O(q^{10}) \) 6 * q - 4 * q^7	\( 6 q - 4 q^{7} - 12 q^{17} - 8 q^{23} - 6 q^{25} + 12 q^{31} + 20 q^{41} + 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{71} - 36 q^{73} - 36 q^{79} + 28 q^{89} - 8 q^{95} + 36 q^{97}+O(q^{100}) \) 6 * q - 4 * q^7 - 12 * q^17 - 8 * q^23 - 6 * q^25 + 12 * q^31 + 20 * q^41 + 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^71 - 36 * q^73 - 36 * q^79 + 28 * q^89 - 8 * q^95 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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