LMFDB - Embedded newform 480.2.k.b.241.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(241,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, 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("480.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.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:	$3.83281929702$
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			241.6
Root			$1.40680 + 0.144584i$ of defining polynomial
Character	$\chi$	$=$	480.241
Dual form			480.2.k.b.241.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} +3.62721 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.00000i q^{5} +3.62721 q^{7} -1.00000 q^{9} -6.20555i q^{11} -0.578337i q^{13} +1.00000 q^{15} +1.42166 q^{17} +5.62721i q^{19} +3.62721i q^{21} +5.62721 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +2.57834 q^{31} +6.20555 q^{33} -3.62721i q^{35} -7.83276i q^{37} +0.578337 q^{39} +5.25443 q^{41} +7.25443i q^{43} +1.00000i q^{45} -6.78389 q^{47} +6.15667 q^{49} +1.42166i q^{51} -2.00000i q^{53} -6.20555 q^{55} -5.62721 q^{57} +2.20555i q^{59} +12.4111i q^{61} -3.62721 q^{63} -0.578337 q^{65} -4.00000i q^{67} +5.62721i q^{69} -8.41110 q^{71} -6.00000 q^{73} -1.00000i q^{75} -22.5089i q^{77} -5.42166 q^{79} +1.00000 q^{81} -3.25443i q^{83} -1.42166i q^{85} -2.00000 q^{87} -13.2544 q^{89} -2.09775i q^{91} +2.57834i q^{93} +5.62721 q^{95} +4.84333 q^{97} +6.20555i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{7} - 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^7 - 6 * q^9	$ 6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81} - 12 q^{87} - 28 q^{89} + 8 q^{95} + 36 q^{97}+O(q^{100}) $ 6 * q - 4 * q^7 - 6 * q^9 + 6 * q^15 + 12 * q^17 + 8 * q^23 - 6 * q^25 + 12 * q^31 + 8 * q^33 - 20 * q^41 - 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 - 36 * q^73 - 36 * q^79 + 6 * q^81 - 12 * q^87 - 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 $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	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
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 6.20555i	− 1.87104i	−0.353269	−	0.935522i	$-0.614930\pi$
$11$	− 6.20555i	− 1.87104i	0.353269	−	0.935522i	$-0.385070\pi$
$12$	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
$15$	1.00000	0.258199
$16$	0	0
$17$	1.42166	0.344804	0.172402	−	0.985027i	$-0.444847\pi$
$17$	1.42166	0.344804	0.172402	+	0.985027i	$0.444847\pi$
$18$	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
$21$	3.62721i	0.791523i
$22$	0	0
$23$	5.62721	1.17336	0.586678	−	0.809821i	$-0.300436\pi$
$23$	5.62721	1.17336	0.586678	+	0.809821i	$0.300436\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	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
$33$	6.20555	1.08025
$34$	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
$39$	0.578337	0.0926081
$40$	0	0
$41$	5.25443	0.820603	0.410302	−	0.911950i	$-0.365423\pi$
$41$	5.25443	0.820603	0.410302	+	0.911950i	$0.365423\pi$
$42$	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
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−6.78389	−0.989532	−0.494766	−	0.869026i	$-0.664746\pi$
$47$	−6.78389	−0.989532	−0.494766	+	0.869026i	$0.664746\pi$
$48$	0	0
$49$	6.15667	0.879525
$50$	0	0
$51$	1.42166i	0.199073i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	−6.20555	−0.836756
$56$	0	0
$57$	−5.62721	−0.745343
$58$	0	0
$59$	2.20555i	0.287138i	0.989640	+	0.143569i	$0.0458579\pi$
$59$	2.20555i	0.287138i	−0.989640	+	0.143569i	$0.954142\pi$
$60$	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
$63$	−3.62721	−0.456986
$64$	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
$69$	5.62721i	0.677437i
$70$	0	0
$71$	−8.41110	−0.998214	−0.499107	−	0.866540i	$-0.666339\pi$
$71$	−8.41110	−0.998214	−0.499107	+	0.866540i	$0.666339\pi$
$72$	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
$75$	− 1.00000i	− 0.115470i
$76$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	− 3.25443i	− 0.357220i	−0.983920	−	0.178610i	$-0.942840\pi$
$83$	− 3.25443i	− 0.357220i	0.983920	−	0.178610i	$-0.0571600\pi$
$84$	0	0
$85$	− 1.42166i	− 0.154201i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−13.2544	−1.40497	−0.702483	−	0.711700i	$-0.747925\pi$
$89$	−13.2544	−1.40497	−0.702483	+	0.711700i	$0.747925\pi$
$90$	0	0
$91$	− 2.09775i	− 0.219904i
$92$	0	0
$93$	2.57834i	0.267361i
$94$	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
$99$	6.20555i	0.623681i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	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
$105$	3.62721	0.353980
$106$	0	0
$107$	− 14.0978i	− 1.36288i	−0.731873	−	0.681441i	$-0.761354\pi$
$107$	− 14.0978i	− 1.36288i	0.731873	−	0.681441i	$-0.238646\pi$
$108$	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
$111$	7.83276	0.743453
$112$	0	0
$113$	−9.08719	−0.854851	−0.427425	−	0.904051i	$-0.640579\pi$
$113$	−9.08719	−0.854851	−0.427425	+	0.904051i	$0.640579\pi$
$114$	0	0
$115$	− 5.62721i	− 0.524740i
$116$	0	0
$117$	0.578337i	0.0534673i
$118$	0	0
$119$	5.15667	0.472712
$120$	0	0
$121$	−27.5089	−2.50080
$122$	0	0
$123$	5.25443i	0.473776i
$124$	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
$129$	−7.25443	−0.638717
$130$	0	0
$131$	13.4600i	1.17600i	0.808860	+	0.588002i	$0.200085\pi$
$131$	13.4600i	1.17600i	−0.808860	+	0.588002i	$0.799915\pi$
$132$	0	0
$133$	20.4111i	1.76987i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.5783	−0.903768	−0.451884	−	0.892077i	$-0.649248\pi$
$137$	−10.5783	−0.903768	−0.451884	+	0.892077i	$0.649248\pi$
$138$	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
$141$	− 6.78389i	− 0.571306i
$142$	0	0
$143$	−3.58890	−0.300119
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	6.15667i	0.507794i
$148$	0	0
$149$	2.00000i	0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$	2.00000i	0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$	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
$153$	−1.42166	−0.114935
$154$	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
$159$	2.00000	0.158610
$160$	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
$165$	− 6.20555i	− 0.483101i
$166$	0	0
$167$	−10.7839	−0.834482	−0.417241	−	0.908796i	$-0.637003\pi$
$167$	−10.7839	−0.834482	−0.417241	+	0.908796i	$0.637003\pi$
$168$	0	0
$169$	12.6655	0.974271
$170$	0	0
$171$	− 5.62721i	− 0.430324i
$172$	0	0
$173$	13.6655i	1.03897i	0.854479	+	0.519485i	$0.173876\pi$
$173$	13.6655i	1.03897i	−0.854479	+	0.519485i	$0.826124\pi$
$174$	0	0
$175$	−3.62721	−0.274192
$176$	0	0
$177$	−2.20555	−0.165779
$178$	0	0
$179$	9.04888i	0.676345i	0.941084	+	0.338172i	$0.109809\pi$
$179$	9.04888i	0.676345i	−0.941084	+	0.338172i	$0.890191\pi$
$180$	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
$183$	−12.4111	−0.917455
$184$	0	0
$185$	−7.83276	−0.575876
$186$	0	0
$187$	− 8.82220i	− 0.645143i
$188$	0	0
$189$	− 3.62721i	− 0.263841i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	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
$195$	− 0.578337i	− 0.0414156i
$196$	0	0
$197$	15.1567i	1.07987i	0.841707	+	0.539934i	$0.181551\pi$
$197$	15.1567i	1.07987i	−0.841707	+	0.539934i	$0.818449\pi$
$198$	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
$201$	4.00000	0.282138
$202$	0	0
$203$	7.25443i	0.509161i
$204$	0	0
$205$	− 5.25443i	− 0.366985i
$206$	0	0
$207$	−5.62721	−0.391118
$208$	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
$213$	− 8.41110i	− 0.576319i
$214$	0	0
$215$	7.25443	0.494748
$216$	0	0
$217$	9.35218	0.634867
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	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
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 1.15667i	− 0.0767712i	−0.999263	−	0.0383856i	$-0.987778\pi$
$227$	− 1.15667i	− 0.0767712i	0.999263	−	0.0383856i	$-0.0122215\pi$
$228$	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
$231$	22.5089	1.48097
$232$	0	0
$233$	−14.5783	−0.955059	−0.477529	−	0.878616i	$-0.658468\pi$
$233$	−14.5783	−0.955059	−0.477529	+	0.878616i	$0.658468\pi$
$234$	0	0
$235$	6.78389i	0.442532i
$236$	0	0
$237$	− 5.42166i	− 0.352175i
$238$	0	0
$239$	−19.2544	−1.24547	−0.622733	−	0.782435i	$-0.713978\pi$
$239$	−19.2544	−1.24547	−0.622733	+	0.782435i	$0.713978\pi$
$240$	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
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 6.15667i	− 0.393335i
$246$	0	0
$247$	3.25443	0.207074
$248$	0	0
$249$	3.25443	0.206241
$250$	0	0
$251$	7.14663i	0.451091i	0.974233	+	0.225546i	$0.0724165\pi$
$251$	7.14663i	0.451091i	−0.974233	+	0.225546i	$0.927584\pi$
$252$	0	0
$253$	− 34.9200i	− 2.19540i
$254$	0	0
$255$	1.42166	0.0890280
$256$	0	0
$257$	7.73501	0.482497	0.241248	−	0.970463i	$-0.422443\pi$
$257$	7.73501	0.482497	0.241248	+	0.970463i	$0.422443\pi$
$258$	0	0
$259$	− 28.4111i	− 1.76538i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	18.7839	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$263$	18.7839	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	− 13.2544i	− 0.811158i
$268$	0	0
$269$	8.50885i	0.518794i	0.965771	+	0.259397i	$0.0835238\pi$
$269$	8.50885i	0.518794i	−0.965771	+	0.259397i	$0.916476\pi$
$270$	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
$273$	2.09775	0.126962
$274$	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
$279$	−2.57834	−0.154361
$280$	0	0
$281$	13.6655	0.815217	0.407608	−	0.913157i	$-0.366363\pi$
$281$	13.6655	0.815217	0.407608	+	0.913157i	$0.366363\pi$
$282$	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
$285$	5.62721i	0.333327i
$286$	0	0
$287$	19.0589	1.12501
$288$	0	0
$289$	−14.9789	−0.881110
$290$	0	0
$291$	4.84333i	0.283921i
$292$	0	0
$293$	− 4.31335i	− 0.251989i	−0.992031	−	0.125994i	$-0.959788\pi$
$293$	− 4.31335i	− 0.251989i	0.992031	−	0.125994i	$-0.0402121\pi$
$294$	0	0
$295$	2.20555	0.128412
$296$	0	0
$297$	−6.20555	−0.360083
$298$	0	0
$299$	− 3.25443i	− 0.188208i
$300$	0	0
$301$	26.3133i	1.51668i
$302$	0	0
$303$	2.00000	0.114897
$304$	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
$309$	− 2.47054i	− 0.140544i
$310$	0	0
$311$	20.0766	1.13844	0.569221	−	0.822185i	$-0.307245\pi$
$311$	20.0766	1.13844	0.569221	+	0.822185i	$0.307245\pi$
$312$	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
$315$	3.62721i	0.204370i
$316$	0	0
$317$	24.1744i	1.35777i	0.734245	+	0.678884i	$0.237536\pi$
$317$	24.1744i	1.35777i	−0.734245	+	0.678884i	$0.762464\pi$
$318$	0	0
$319$	12.4111	0.694888
$320$	0	0
$321$	14.0978	0.786860
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0.578337i	0.0320804i
$326$	0	0
$327$	−7.25443	−0.401171
$328$	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
$333$	7.83276i	0.429233i
$334$	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
$339$	− 9.08719i	− 0.493548i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	−3.05892	−0.165166
$344$	0	0
$345$	5.62721	0.302959
$346$	0	0
$347$	23.6655i	1.27043i	0.772335	+	0.635216i	$0.219089\pi$
$347$	23.6655i	1.27043i	−0.772335	+	0.635216i	$0.780911\pi$
$348$	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
$351$	−0.578337	−0.0308694
$352$	0	0
$353$	−15.9305	−0.847896	−0.423948	−	0.905687i	$-0.639356\pi$
$353$	−15.9305	−0.847896	−0.423948	+	0.905687i	$0.639356\pi$
$354$	0	0
$355$	8.41110i	0.446415i
$356$	0	0
$357$	5.15667i	0.272920i
$358$	0	0
$359$	−8.41110	−0.443921	−0.221960	−	0.975056i	$-0.571246\pi$
$359$	−8.41110	−0.443921	−0.221960	+	0.975056i	$0.571246\pi$
$360$	0	0
$361$	−12.6655	−0.666607
$362$	0	0
$363$	− 27.5089i	− 1.44384i
$364$	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
$369$	−5.25443	−0.273534
$370$	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
$375$	−1.00000	−0.0516398
$376$	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
$381$	10.4705i	0.536422i
$382$	0	0
$383$	1.62721	0.0831467	0.0415734	−	0.999135i	$-0.486763\pi$
$383$	1.62721	0.0831467	0.0415734	+	0.999135i	$0.486763\pi$
$384$	0	0
$385$	−22.5089	−1.14716
$386$	0	0
$387$	− 7.25443i	− 0.368763i
$388$	0	0
$389$	12.3133i	0.624312i	0.950031	+	0.312156i	$0.101051\pi$
$389$	12.3133i	0.624312i	−0.950031	+	0.312156i	$0.898949\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−13.4600	−0.678966
$394$	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
$399$	−20.4111	−1.02183
$400$	0	0
$401$	−14.4111	−0.719656	−0.359828	−	0.933019i	$-0.617165\pi$
$401$	−14.4111	−0.719656	−0.359828	+	0.933019i	$0.617165\pi$
$402$	0	0
$403$	− 1.49115i	− 0.0742794i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	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
$411$	− 10.5783i	− 0.521791i
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	−3.25443	−0.159753
$416$	0	0
$417$	−12.4705	−0.610685
$418$	0	0
$419$	7.36222i	0.359668i	0.983697	+	0.179834i	$0.0575561\pi$
$419$	7.36222i	0.359668i	−0.983697	+	0.179834i	$0.942444\pi$
$420$	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
$423$	6.78389	0.329844
$424$	0	0
$425$	−1.42166	−0.0689608
$426$	0	0
$427$	45.0177i	2.17856i
$428$	0	0
$429$	− 3.58890i	− 0.173274i
$430$	0	0
$431$	−8.41110	−0.405148	−0.202574	−	0.979267i	$-0.564931\pi$
$431$	−8.41110	−0.405148	−0.202574	+	0.979267i	$0.564931\pi$
$432$	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
$435$	2.00000i	0.0958927i
$436$	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
$441$	−6.15667	−0.293175
$442$	0	0
$443$	− 21.3522i	− 1.01447i	−0.861807	−	0.507236i	$-0.830667\pi$
$443$	− 21.3522i	− 1.01447i	0.861807	−	0.507236i	$-0.169333\pi$
$444$	0	0
$445$	13.2544i	0.628320i
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	−20.3133	−0.958646	−0.479323	−	0.877639i	$-0.659118\pi$
$449$	−20.3133	−0.958646	−0.479323	+	0.877639i	$0.659118\pi$
$450$	0	0
$451$	− 32.6066i	− 1.53539i
$452$	0	0
$453$	− 12.6761i	− 0.595575i
$454$	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
$459$	− 1.42166i	− 0.0663575i
$460$	0	0
$461$	28.5089i	1.32779i	0.747826	+	0.663895i	$0.231098\pi$
$461$	28.5089i	1.32779i	−0.747826	+	0.663895i	$0.768902\pi$
$462$	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
$465$	2.57834	0.119568
$466$	0	0
$467$	29.5678i	1.36823i	0.729373	+	0.684117i	$0.239812\pi$
$467$	29.5678i	1.36823i	−0.729373	+	0.684117i	$0.760188\pi$
$468$	0	0
$469$	− 14.5089i	− 0.669957i
$470$	0	0
$471$	1.32391	0.0610026
$472$	0	0
$473$	45.0177	2.06992
$474$	0	0
$475$	− 5.62721i	− 0.258194i
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	22.0978	1.00967	0.504836	−	0.863215i	$-0.331553\pi$
$479$	22.0978	1.00967	0.504836	+	0.863215i	$0.331553\pi$
$480$	0	0
$481$	−4.52998	−0.206549
$482$	0	0
$483$	20.4111i	0.928737i
$484$	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
$489$	15.2544	0.689829
$490$	0	0
$491$	− 18.2056i	− 0.821605i	−0.911724	−	0.410802i	$-0.865249\pi$
$491$	− 18.2056i	− 0.821605i	0.911724	−	0.410802i	$-0.134751\pi$
$492$	0	0
$493$	2.84333i	0.128057i
$494$	0	0
$495$	6.20555	0.278919
$496$	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
$501$	− 10.7839i	− 0.481789i
$502$	0	0
$503$	−2.03831	−0.0908839	−0.0454419	−	0.998967i	$-0.514470\pi$
$503$	−2.03831	−0.0908839	−0.0454419	+	0.998967i	$0.514470\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	12.6655i	0.562496i
$508$	0	0
$509$	− 40.7044i	− 1.80419i	−0.431539	−	0.902094i	$-0.642029\pi$
$509$	− 40.7044i	− 1.80419i	0.431539	−	0.902094i	$-0.357971\pi$
$510$	0	0
$511$	−21.7633	−0.962751
$512$	0	0
$513$	5.62721	0.248448
$514$	0	0
$515$	2.47054i	0.108865i
$516$	0	0
$517$	42.0978i	1.85146i
$518$	0	0
$519$	−13.6655	−0.599850
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	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
$525$	− 3.62721i	− 0.158305i
$526$	0	0
$527$	3.66553	0.159673
$528$	0	0
$529$	8.66553	0.376762
$530$	0	0
$531$	− 2.20555i	− 0.0957127i
$532$	0	0
$533$	− 3.03883i	− 0.131626i
$534$	0	0
$535$	−14.0978	−0.609499
$536$	0	0
$537$	−9.04888	−0.390488
$538$	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
$543$	23.2544	0.997943
$544$	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
$549$	− 12.4111i	− 0.529693i
$550$	0	0
$551$	−11.2544	−0.479455
$552$	0	0
$553$	−19.6655	−0.836263
$554$	0	0
$555$	− 7.83276i	− 0.332482i
$556$	0	0
$557$	− 33.6655i	− 1.42645i	−0.700933	−	0.713227i	$-0.747233\pi$
$557$	− 33.6655i	− 1.42645i	0.700933	−	0.713227i	$-0.252767\pi$
$558$	0	0
$559$	4.19550	0.177451
$560$	0	0
$561$	8.82220	0.372474
$562$	0	0
$563$	5.35218i	0.225567i	0.993620	+	0.112784i	$0.0359767\pi$
$563$	5.35218i	0.225567i	−0.993620	+	0.112784i	$0.964023\pi$
$564$	0	0
$565$	9.08719i	0.382301i
$566$	0	0
$567$	3.62721	0.152329
$568$	0	0
$569$	5.58890	0.234299	0.117149	−	0.993114i	$-0.462624\pi$
$569$	5.58890	0.234299	0.117149	+	0.993114i	$0.462624\pi$
$570$	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
$573$	8.00000i	0.334205i
$574$	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
$579$	25.6655i	1.06662i
$580$	0	0
$581$	− 11.8045i	− 0.489733i
$582$	0	0
$583$	−12.4111	−0.514015
$584$	0	0
$585$	0.578337	0.0239113
$586$	0	0
$587$	− 1.90225i	− 0.0785142i	−0.999229	−	0.0392571i	$-0.987501\pi$
$587$	− 1.90225i	− 0.0785142i	0.999229	−	0.0392571i	$-0.0124991\pi$
$588$	0	0
$589$	14.5089i	0.597827i
$590$	0	0
$591$	−15.1567	−0.623462
$592$	0	0
$593$	−2.57834	−0.105880	−0.0529398	−	0.998598i	$-0.516859\pi$
$593$	−2.57834	−0.105880	−0.0529398	+	0.998598i	$0.516859\pi$
$594$	0	0
$595$	− 5.15667i	− 0.211403i
$596$	0	0
$597$	20.6761i	0.846216i
$598$	0	0
$599$	−26.7244	−1.09193	−0.545966	−	0.837808i	$-0.683837\pi$
$599$	−26.7244	−1.09193	−0.545966	+	0.837808i	$0.683837\pi$
$600$	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
$603$	4.00000i	0.162893i
$604$	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
$609$	−7.25443	−0.293964
$610$	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
$615$	5.25443	0.211879
$616$	0	0
$617$	19.7350	0.794502	0.397251	−	0.917710i	$-0.369964\pi$
$617$	19.7350	0.794502	0.397251	+	0.917710i	$0.369964\pi$
$618$	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
$621$	− 5.62721i	− 0.225812i
$622$	0	0
$623$	−48.0766	−1.92615
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	34.9200i	1.39457i
$628$	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
$633$	−2.03831	−0.0810157
$634$	0	0
$635$	− 10.4705i	− 0.415511i
$636$	0	0
$637$	− 3.56063i	− 0.141077i
$638$	0	0
$639$	8.41110	0.332738
$640$	0	0
$641$	−27.9789	−1.10510	−0.552550	−	0.833480i	$-0.686345\pi$
$641$	−27.9789	−1.10510	−0.552550	+	0.833480i	$0.686345\pi$
$642$	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
$645$	7.25443i	0.285643i
$646$	0	0
$647$	49.3694	1.94091	0.970455	−	0.241282i	$-0.0775679\pi$
$647$	49.3694	1.94091	0.970455	+	0.241282i	$0.0775679\pi$
$648$	0	0
$649$	13.6867	0.537248
$650$	0	0
$651$	9.35218i	0.366541i
$652$	0	0
$653$	− 40.1744i	− 1.57214i	−0.618134	−	0.786072i	$-0.712111\pi$
$653$	− 40.1744i	− 1.57214i	0.618134	−	0.786072i	$-0.287889\pi$
$654$	0	0
$655$	13.4600	0.525925
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	− 21.1255i	− 0.822933i	−0.911425	−	0.411466i	$-0.865017\pi$
$659$	− 21.1255i	− 0.822933i	0.911425	−	0.411466i	$-0.134983\pi$
$660$	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
$663$	0.822200	0.0319316
$664$	0	0
$665$	20.4111	0.791509
$666$	0	0
$667$	11.2544i	0.435773i
$668$	0	0
$669$	7.21611i	0.278991i
$670$	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
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	− 30.4877i	− 1.17174i	−0.810406	−	0.585869i	$-0.800753\pi$
$677$	− 30.4877i	− 1.17174i	0.810406	−	0.585869i	$-0.199247\pi$
$678$	0	0
$679$	17.5678	0.674189
$680$	0	0
$681$	1.15667	0.0443239
$682$	0	0
$683$	− 35.2544i	− 1.34897i	−0.738287	−	0.674487i	$-0.764365\pi$
$683$	− 35.2544i	− 1.34897i	0.738287	−	0.674487i	$-0.235635\pi$
$684$	0	0
$685$	10.5783i	0.404177i
$686$	0	0
$687$	−14.0978	−0.537863
$688$	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
$693$	22.5089i	0.855041i
$694$	0	0
$695$	12.4705	0.473034
$696$	0	0
$697$	7.47002	0.282947
$698$	0	0
$699$	− 14.5783i	− 0.551403i
$700$	0	0
$701$	34.8222i	1.31522i	0.753360	+	0.657608i	$0.228432\pi$
$701$	34.8222i	1.31522i	−0.753360	+	0.657608i	$0.771568\pi$
$702$	0	0
$703$	44.0766	1.66238
$704$	0	0
$705$	−6.78389	−0.255496
$706$	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
$711$	5.42166	0.203328
$712$	0	0
$713$	14.5089	0.543361
$714$	0	0
$715$	3.58890i	0.134217i
$716$	0	0
$717$	− 19.2544i	− 0.719070i
$718$	0	0
$719$	3.66553	0.136701	0.0683505	−	0.997661i	$-0.478226\pi$
$719$	3.66553	0.136701	0.0683505	+	0.997661i	$0.478226\pi$
$720$	0	0
$721$	−8.96117	−0.333731
$722$	0	0
$723$	− 13.6655i	− 0.508226i
$724$	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
$729$	−1.00000	−0.0370370
$730$	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
$735$	6.15667	0.227092
$736$	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
$741$	3.25443i	0.119554i
$742$	0	0
$743$	23.3139	0.855303	0.427651	−	0.903944i	$-0.359341\pi$
$743$	23.3139	0.855303	0.427651	+	0.903944i	$0.359341\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	3.25443i	0.119073i
$748$	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
$753$	−7.14663	−0.260438
$754$	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
$759$	34.9200	1.26751
$760$	0	0
$761$	−17.1355	−0.621163	−0.310582	−	0.950547i	$-0.600524\pi$
$761$	−17.1355	−0.621163	−0.310582	+	0.950547i	$0.600524\pi$
$762$	0	0
$763$	26.3133i	0.952607i
$764$	0	0
$765$	1.42166i	0.0514003i
$766$	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
$771$	7.73501i	0.278570i
$772$	0	0
$773$	− 3.15667i	− 0.113538i	−0.998387	−	0.0567688i	$-0.981920\pi$
$773$	− 3.15667i	− 0.113538i	0.998387	−	0.0567688i	$-0.0180798\pi$
$774$	0	0
$775$	−2.57834	−0.0926166
$776$	0	0
$777$	28.4111	1.01924
$778$	0	0
$779$	29.5678i	1.05938i
$780$	0	0
$781$	52.1955i	1.86770i
$782$	0	0
$783$	2.00000	0.0714742
$784$	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
$789$	18.7839i	0.668724i
$790$	0	0
$791$	−32.9612	−1.17196
$792$	0	0
$793$	7.17780	0.254891
$794$	0	0
$795$	− 2.00000i	− 0.0709327i
$796$	0	0
$797$	− 10.0000i	− 0.354218i	−0.984191	−	0.177109i	$-0.943325\pi$
$797$	− 10.0000i	− 0.354218i	0.984191	−	0.177109i	$-0.0566745\pi$
$798$	0	0
$799$	−9.64440	−0.341194
$800$	0	0
$801$	13.2544	0.468322
$802$	0	0
$803$	37.2333i	1.31393i
$804$	0	0
$805$	− 20.4111i	− 0.719397i
$806$	0	0
$807$	−8.50885	−0.299526
$808$	0	0
$809$	29.0388	1.02095	0.510475	−	0.859892i	$-0.329469\pi$
$809$	29.0388	1.02095	0.510475	+	0.859892i	$0.329469\pi$
$810$	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
$813$	− 30.9894i	− 1.08685i
$814$	0	0
$815$	−15.2544	−0.534339
$816$	0	0
$817$	−40.8222	−1.42819
$818$	0	0
$819$	2.09775i	0.0733014i
$820$	0	0
$821$	− 38.1955i	− 1.33303i	−0.745491	−	0.666516i	$-0.767785\pi$
$821$	− 38.1955i	− 1.33303i	0.745491	−	0.666516i	$-0.232215\pi$
$822$	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
$825$	−6.20555	−0.216050
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	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
$831$	−9.51941	−0.330225
$832$	0	0
$833$	8.75272	0.303264
$834$	0	0
$835$	10.7839i	0.373192i
$836$	0	0
$837$	− 2.57834i	− 0.0891204i
$838$	0	0
$839$	53.4288	1.84457	0.922284	−	0.386514i	$-0.126321\pi$
$839$	53.4288	1.84457	0.922284	+	0.386514i	$0.126321\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	13.6655i	0.470666i
$844$	0	0
$845$	− 12.6655i	− 0.435707i
$846$	0	0
$847$	−99.7805	−3.42850
$848$	0	0
$849$	−20.0000	−0.686398
$850$	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
$855$	−5.62721	−0.192447
$856$	0	0
$857$	−10.2439	−0.349924	−0.174962	−	0.984575i	$-0.555980\pi$
$857$	−10.2439	−0.349924	−0.174962	+	0.984575i	$0.555980\pi$
$858$	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
$861$	19.0589i	0.649526i
$862$	0	0
$863$	−38.8605	−1.32283	−0.661414	−	0.750021i	$-0.730043\pi$
$863$	−38.8605	−1.32283	−0.661414	+	0.750021i	$0.730043\pi$
$864$	0	0
$865$	13.6655	0.464642
$866$	0	0
$867$	− 14.9789i	− 0.508709i
$868$	0	0
$869$	33.6444i	1.14131i
$870$	0	0
$871$	−2.31335	−0.0783848
$872$	0	0
$873$	−4.84333	−0.163922
$874$	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
$879$	4.31335	0.145486
$880$	0	0
$881$	9.88112	0.332903	0.166452	−	0.986050i	$-0.446769\pi$
$881$	9.88112	0.332903	0.166452	+	0.986050i	$0.446769\pi$
$882$	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
$885$	2.20555i	0.0741388i
$886$	0	0
$887$	−11.6061	−0.389694	−0.194847	−	0.980834i	$-0.562421\pi$
$887$	−11.6061	−0.389694	−0.194847	+	0.980834i	$0.562421\pi$
$888$	0	0
$889$	37.9789	1.27377
$890$	0	0
$891$	− 6.20555i	− 0.207894i
$892$	0	0
$893$	− 38.1744i	− 1.27746i
$894$	0	0
$895$	9.04888	0.302471
$896$	0	0
$897$	3.25443	0.108662
$898$	0	0
$899$	5.15667i	0.171985i
$900$	0	0
$901$	− 2.84333i	− 0.0947249i
$902$	0	0
$903$	−26.3133	−0.875653
$904$	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
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	7.88112	0.261113	0.130557	−	0.991441i	$-0.458324\pi$
$911$	7.88112	0.261113	0.130557	+	0.991441i	$0.458324\pi$
$912$	0	0
$913$	−20.1955	−0.668374
$914$	0	0
$915$	12.4111i	0.410298i
$916$	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
$921$	25.5678	0.842487
$922$	0	0
$923$	4.86445i	0.160115i
$924$	0	0
$925$	7.83276i	0.257540i
$926$	0	0
$927$	2.47054	0.0811431
$928$	0	0
$929$	6.82220	0.223829	0.111915	−	0.993718i	$-0.464302\pi$
$929$	6.82220	0.223829	0.111915	+	0.993718i	$0.464302\pi$
$930$	0	0
$931$	34.6449i	1.13544i
$932$	0	0
$933$	20.0766i	0.657279i
$934$	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
$939$	7.15667i	0.233549i
$940$	0	0
$941$	− 0.508852i	− 0.0165881i	−0.999966	−	0.00829405i	$-0.997360\pi$
$941$	− 0.508852i	− 0.0165881i	0.999966	−	0.00829405i	$-0.00264011\pi$
$942$	0	0
$943$	29.5678	0.962859
$944$	0	0
$945$	−3.62721	−0.117993
$946$	0	0
$947$	− 1.68665i	− 0.0548088i	−0.999624	−	0.0274044i	$-0.991276\pi$
$947$	− 1.68665i	− 0.0548088i	0.999624	−	0.0274044i	$-0.00872419\pi$
$948$	0	0
$949$	3.47002i	0.112642i
$950$	0	0
$951$	−24.1744	−0.783908
$952$	0	0
$953$	9.22616	0.298865	0.149432	−	0.988772i	$-0.452255\pi$
$953$	9.22616	0.298865	0.149432	+	0.988772i	$0.452255\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	12.4111i	0.401194i
$958$	0	0
$959$	−38.3699	−1.23903
$960$	0	0
$961$	−24.3522	−0.785554
$962$	0	0
$963$	14.0978i	0.454294i
$964$	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
$969$	−8.00000	−0.256997
$970$	0	0
$971$	− 33.2444i	− 1.06686i	−0.845843	−	0.533431i	$-0.820902\pi$
$971$	− 33.2444i	− 1.06686i	0.845843	−	0.533431i	$-0.179098\pi$
$972$	0	0
$973$	45.2333i	1.45011i
$974$	0	0
$975$	−0.578337	−0.0185216
$976$	0	0
$977$	7.93051	0.253720	0.126860	−	0.991921i	$-0.459510\pi$
$977$	7.93051	0.253720	0.126860	+	0.991921i	$0.459510\pi$
$978$	0	0
$979$	82.2510i	2.62875i
$980$	0	0
$981$	− 7.25443i	− 0.231616i
$982$	0	0
$983$	−41.8993	−1.33638	−0.668191	−	0.743990i	$-0.732931\pi$
$983$	−41.8993	−1.33638	−0.668191	+	0.743990i	$0.732931\pi$
$984$	0	0
$985$	15.1567	0.482932
$986$	0	0
$987$	− 24.6066i	− 0.783237i
$988$	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
$993$	−27.1950	−0.863007
$994$	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
$999$	−7.83276	−0.247818

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.5	✓	6	8.3	odd	2
120.2.k.b.61.6	yes	6	4.3	odd	2
360.2.k.f.181.1		6	12.11	even	2
360.2.k.f.181.2		6	24.11	even	2
480.2.k.b.241.3		6	8.5	even	2	inner
480.2.k.b.241.6		6	1.1	even	1	trivial
600.2.d.e.349.3		6	40.3	even	4
600.2.d.e.349.4		6	20.7	even	4
600.2.d.f.349.3		6	20.3	even	4
600.2.d.f.349.4		6	40.27	even	4
600.2.k.c.301.1		6	20.19	odd	2
600.2.k.c.301.2		6	40.19	odd	2
1440.2.k.f.721.3		6	24.5	odd	2
1440.2.k.f.721.6		6	3.2	odd	2
1800.2.d.q.1549.3		6	60.47	odd	4
1800.2.d.q.1549.4		6	120.83	odd	4
1800.2.d.r.1549.3		6	120.107	odd	4
1800.2.d.r.1549.4		6	60.23	odd	4
1800.2.k.p.901.5		6	120.59	even	2
1800.2.k.p.901.6		6	60.59	even	2
2400.2.d.e.49.2		6	5.3	odd	4
2400.2.d.e.49.5		6	40.37	odd	4
2400.2.d.f.49.2		6	40.13	odd	4
2400.2.d.f.49.5		6	5.2	odd	4
2400.2.k.c.1201.1		6	5.4	even	2
2400.2.k.c.1201.4		6	40.29	even	2
3840.2.a.bo.1.1		3	16.13	even	4
3840.2.a.bp.1.3		3	16.11	odd	4
3840.2.a.bq.1.3		3	16.3	odd	4
3840.2.a.br.1.1		3	16.5	even	4
7200.2.d.q.2449.2		6	120.53	even	4
7200.2.d.q.2449.5		6	15.2	even	4
7200.2.d.r.2449.2		6	15.8	even	4
7200.2.d.r.2449.5		6	120.77	even	4
7200.2.k.p.3601.1		6	120.29	odd	2
7200.2.k.p.3601.2		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} +3.62721 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.00000i q^{5} +3.62721 q^{7} -1.00000 q^{9} -6.20555i q^{11} -0.578337i q^{13} +1.00000 q^{15} +1.42166 q^{17} +5.62721i q^{19} +3.62721i q^{21} +5.62721 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +2.57834 q^{31} +6.20555 q^{33} -3.62721i q^{35} -7.83276i q^{37} +0.578337 q^{39} +5.25443 q^{41} +7.25443i q^{43} +1.00000i q^{45} -6.78389 q^{47} +6.15667 q^{49} +1.42166i q^{51} -2.00000i q^{53} -6.20555 q^{55} -5.62721 q^{57} +2.20555i q^{59} +12.4111i q^{61} -3.62721 q^{63} -0.578337 q^{65} -4.00000i q^{67} +5.62721i q^{69} -8.41110 q^{71} -6.00000 q^{73} -1.00000i q^{75} -22.5089i q^{77} -5.42166 q^{79} +1.00000 q^{81} -3.25443i q^{83} -1.42166i q^{85} -2.00000 q^{87} -13.2544 q^{89} -2.09775i q^{91} +2.57834i q^{93} +5.62721 q^{95} +4.84333 q^{97} +6.20555i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{7} - 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^7 - 6 * q^9	\( 6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81} - 12 q^{87} - 28 q^{89} + 8 q^{95} + 36 q^{97}+O(q^{100}) \) 6 * q - 4 * q^7 - 6 * q^9 + 6 * q^15 + 12 * q^17 + 8 * q^23 - 6 * q^25 + 12 * q^31 + 8 * q^33 - 20 * q^41 - 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 - 36 * q^73 - 36 * q^79 + 6 * q^81 - 12 * q^87 - 28 * q^89 + 8 * q^95 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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