LMFDB - Embedded newform 1024.4.b.j.513.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.b (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:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 36x^{8} + 405x^{6} + 1380x^{4} + 420x^{2} + 32 $ x^10 + 36x^8 + 405x^6 + 1380x^4 + 420x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.6
Root			$-3.94652i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.4.b.j.513.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.07024i q^{3} +11.6331i q^{5} -2.67171 q^{7} +25.8546 q^{9} +O(q^{10})$	$q+1.07024i q^{3} +11.6331i q^{5} -2.67171 q^{7} +25.8546 q^{9} +63.9525i q^{11} -50.0586i q^{13} -12.4503 q^{15} +72.4991 q^{17} -27.4961i q^{19} -2.85937i q^{21} +139.462 q^{23} -10.3299 q^{25} +56.5672i q^{27} +93.3995i q^{29} +188.682 q^{31} -68.4447 q^{33} -31.0803i q^{35} -118.886i q^{37} +53.5748 q^{39} -104.629 q^{41} +44.5275i q^{43} +300.770i q^{45} +488.151 q^{47} -335.862 q^{49} +77.5916i q^{51} -211.510i q^{53} -743.968 q^{55} +29.4275 q^{57} +402.624i q^{59} -322.538i q^{61} -69.0758 q^{63} +582.338 q^{65} +196.789i q^{67} +149.258i q^{69} -453.655 q^{71} +259.747 q^{73} -11.0555i q^{75} -170.862i q^{77} -323.190 q^{79} +637.533 q^{81} +797.471i q^{83} +843.392i q^{85} -99.9602 q^{87} +866.853 q^{89} +133.742i q^{91} +201.935i q^{93} +319.866 q^{95} -936.077 q^{97} +1653.47i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) $ 10 * q - 28 * q^7 - 54 * q^9	$ 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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
$3$	1.07024i	0.205968i	0.994683	+	0.102984i	$0.0328391\pi$
$3$	1.07024i	0.205968i	−0.994683	+	0.102984i	$0.967161\pi$
$4$	0	0
$5$	11.6331i	1.04050i	0.854014	+	0.520250i	$0.174161\pi$
$5$	11.6331i	1.04050i	−0.854014	+	0.520250i	$0.825839\pi$
$6$	0	0
$7$	−2.67171	−0.144259	−0.0721293	−	0.997395i	$-0.522979\pi$
$7$	−2.67171	−0.144259	−0.0721293	+	0.997395i	$0.522979\pi$
$8$	0	0
$9$	25.8546	0.957577
$10$	0	0
$11$	63.9525i	1.75295i	0.481451	+	0.876473i	$0.340110\pi$
$11$	63.9525i	1.75295i	−0.481451	+	0.876473i	$0.659890\pi$
$12$	0	0
$13$	− 50.0586i	− 1.06798i	−0.845491	−	0.533990i	$-0.820692\pi$
$13$	− 50.0586i	− 1.06798i	0.845491	−	0.533990i	$-0.179308\pi$
$14$	0	0
$15$	−12.4503	−0.214310
$16$	0	0
$17$	72.4991	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$17$	72.4991	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$18$	0	0
$19$	− 27.4961i	− 0.332002i	−0.986126	−	0.166001i	$-0.946915\pi$
$19$	− 27.4961i	− 0.332002i	0.986126	−	0.166001i	$-0.0530855\pi$
$20$	0	0
$21$	− 2.85937i	− 0.0297127i
$22$	0	0
$23$	139.462	1.26434	0.632170	−	0.774830i	$-0.282165\pi$
$23$	139.462	1.26434	0.632170	+	0.774830i	$0.282165\pi$
$24$	0	0
$25$	−10.3299	−0.0826390
$26$	0	0
$27$	56.5672i	0.403199i
$28$	0	0
$29$	93.3995i	0.598064i	0.954243	+	0.299032i	$0.0966637\pi$
$29$	93.3995i	0.598064i	−0.954243	+	0.299032i	$0.903336\pi$
$30$	0	0
$31$	188.682	1.09317	0.546584	−	0.837404i	$-0.315928\pi$
$31$	188.682	1.09317	0.546584	+	0.837404i	$0.315928\pi$
$32$	0	0
$33$	−68.4447	−0.361051
$34$	0	0
$35$	− 31.0803i	− 0.150101i
$36$	0	0
$37$	− 118.886i	− 0.528238i	−0.964490	−	0.264119i	$-0.914919\pi$
$37$	− 118.886i	− 0.528238i	0.964490	−	0.264119i	$-0.0850811\pi$
$38$	0	0
$39$	53.5748	0.219970
$40$	0	0
$41$	−104.629	−0.398545	−0.199272	−	0.979944i	$-0.563858\pi$
$41$	−104.629	−0.398545	−0.199272	+	0.979944i	$0.563858\pi$
$42$	0	0
$43$	44.5275i	0.157916i	0.996878	+	0.0789579i	$0.0251593\pi$
$43$	44.5275i	0.157916i	−0.996878	+	0.0789579i	$0.974841\pi$
$44$	0	0
$45$	300.770i	0.996358i
$46$	0	0
$47$	488.151	1.51498	0.757491	−	0.652846i	$-0.226425\pi$
$47$	488.151	1.51498	0.757491	+	0.652846i	$0.226425\pi$
$48$	0	0
$49$	−335.862	−0.979189
$50$	0	0
$51$	77.5916i	0.213039i
$52$	0	0
$53$	− 211.510i	− 0.548173i	−0.961705	−	0.274087i	$-0.911624\pi$
$53$	− 211.510i	− 0.548173i	0.961705	−	0.274087i	$-0.0883755\pi$
$54$	0	0
$55$	−743.968	−1.82394
$56$	0	0
$57$	29.4275	0.0683819
$58$	0	0
$59$	402.624i	0.888426i	0.895921	+	0.444213i	$0.146517\pi$
$59$	402.624i	0.888426i	−0.895921	+	0.444213i	$0.853483\pi$
$60$	0	0
$61$	− 322.538i	− 0.676997i	−0.940967	−	0.338498i	$-0.890081\pi$
$61$	− 322.538i	− 0.676997i	0.940967	−	0.338498i	$-0.109919\pi$
$62$	0	0
$63$	−69.0758	−0.138139
$64$	0	0
$65$	582.338	1.11123
$66$	0	0
$67$	196.789i	0.358829i	0.983774	+	0.179415i	$0.0574203\pi$
$67$	196.789i	0.358829i	−0.983774	+	0.179415i	$0.942580\pi$
$68$	0	0
$69$	149.258i	0.260414i
$70$	0	0
$71$	−453.655	−0.758294	−0.379147	−	0.925336i	$-0.623783\pi$
$71$	−453.655	−0.758294	−0.379147	+	0.925336i	$0.623783\pi$
$72$	0	0
$73$	259.747	0.416454	0.208227	−	0.978081i	$-0.433231\pi$
$73$	259.747	0.416454	0.208227	+	0.978081i	$0.433231\pi$
$74$	0	0
$75$	− 11.0555i	− 0.0170210i
$76$	0	0
$77$	− 170.862i	− 0.252877i
$78$	0	0
$79$	−323.190	−0.460275	−0.230138	−	0.973158i	$-0.573918\pi$
$79$	−323.190	−0.460275	−0.230138	+	0.973158i	$0.573918\pi$
$80$	0	0
$81$	637.533	0.874531
$82$	0	0
$83$	797.471i	1.05462i	0.849672	+	0.527312i	$0.176800\pi$
$83$	797.471i	1.05462i	−0.849672	+	0.527312i	$0.823200\pi$
$84$	0	0
$85$	843.392i	1.07622i
$86$	0	0
$87$	−99.9602	−0.123182
$88$	0	0
$89$	866.853	1.03243	0.516215	−	0.856459i	$-0.327341\pi$
$89$	866.853	1.03243	0.516215	+	0.856459i	$0.327341\pi$
$90$	0	0
$91$	133.742i	0.154065i
$92$	0	0
$93$	201.935i	0.225158i
$94$	0	0
$95$	319.866	0.345448
$96$	0	0
$97$	−936.077	−0.979837	−0.489919	−	0.871768i	$-0.662974\pi$
$97$	−936.077	−0.979837	−0.489919	+	0.871768i	$0.662974\pi$
$98$	0	0
$99$	1653.47i	1.67858i
$100$	0	0
$101$	− 2.24639i	− 0.00221311i	−0.999999	−	0.00110656i	$-0.999648\pi$
$101$	− 2.24639i	− 0.00221311i	0.999999	−	0.00110656i	$-0.000352228\pi$
$102$	0	0
$103$	−1388.28	−1.32807	−0.664036	−	0.747700i	$-0.731158\pi$
$103$	−1388.28	−1.32807	−0.664036	+	0.747700i	$0.731158\pi$
$104$	0	0
$105$	33.2635	0.0309160
$106$	0	0
$107$	1161.81i	1.04969i	0.851198	+	0.524845i	$0.175877\pi$
$107$	1161.81i	1.04969i	−0.851198	+	0.524845i	$0.824123\pi$
$108$	0	0
$109$	− 753.489i	− 0.662121i	−0.943610	−	0.331060i	$-0.892594\pi$
$109$	− 753.489i	− 0.662121i	0.943610	−	0.331060i	$-0.107406\pi$
$110$	0	0
$111$	127.237	0.108800
$112$	0	0
$113$	67.2680	0.0560003	0.0280002	−	0.999608i	$-0.491086\pi$
$113$	67.2680	0.0560003	0.0280002	+	0.999608i	$0.491086\pi$
$114$	0	0
$115$	1622.38i	1.31554i
$116$	0	0
$117$	− 1294.24i	− 1.02267i
$118$	0	0
$119$	−193.696	−0.149211
$120$	0	0
$121$	−2758.92	−2.07282
$122$	0	0
$123$	− 111.979i	− 0.0820876i
$124$	0	0
$125$	1333.97i	0.954514i
$126$	0	0
$127$	1903.59	1.33005	0.665026	−	0.746820i	$-0.268421\pi$
$127$	1903.59	1.33005	0.665026	+	0.746820i	$0.268421\pi$
$128$	0	0
$129$	−47.6552	−0.0325257
$130$	0	0
$131$	1298.86i	0.866271i	0.901329	+	0.433136i	$0.142593\pi$
$131$	1298.86i	0.866271i	−0.901329	+	0.433136i	$0.857407\pi$
$132$	0	0
$133$	73.4615i	0.0478941i
$134$	0	0
$135$	−658.054	−0.419528
$136$	0	0
$137$	−477.234	−0.297612	−0.148806	−	0.988866i	$-0.547543\pi$
$137$	−477.234	−0.297612	−0.148806	+	0.988866i	$0.547543\pi$
$138$	0	0
$139$	2140.97i	1.30643i	0.757171	+	0.653217i	$0.226581\pi$
$139$	2140.97i	1.30643i	−0.757171	+	0.653217i	$0.773419\pi$
$140$	0	0
$141$	522.440i	0.312038i
$142$	0	0
$143$	3201.37	1.87211
$144$	0	0
$145$	−1086.53	−0.622285
$146$	0	0
$147$	− 359.454i	− 0.201682i
$148$	0	0
$149$	− 530.829i	− 0.291861i	−0.989295	−	0.145930i	$-0.953382\pi$
$149$	− 530.829i	− 0.291861i	0.989295	−	0.145930i	$-0.0466175\pi$
$150$	0	0
$151$	−2997.52	−1.61546	−0.807730	−	0.589553i	$-0.799304\pi$
$151$	−2997.52	−1.61546	−0.807730	+	0.589553i	$0.799304\pi$
$152$	0	0
$153$	1874.43	0.990451
$154$	0	0
$155$	2194.96i	1.13744i
$156$	0	0
$157$	− 2134.06i	− 1.08482i	−0.840115	−	0.542409i	$-0.817512\pi$
$157$	− 2134.06i	− 1.08482i	0.840115	−	0.542409i	$-0.182488\pi$
$158$	0	0
$159$	226.368	0.112906
$160$	0	0
$161$	−372.601	−0.182392
$162$	0	0
$163$	− 2015.52i	− 0.968515i	−0.874925	−	0.484258i	$-0.839090\pi$
$163$	− 2015.52i	− 0.968515i	0.874925	−	0.484258i	$-0.160910\pi$
$164$	0	0
$165$	− 796.227i	− 0.375674i
$166$	0	0
$167$	−792.415	−0.367179	−0.183590	−	0.983003i	$-0.558772\pi$
$167$	−792.415	−0.367179	−0.183590	+	0.983003i	$0.558772\pi$
$168$	0	0
$169$	−308.861	−0.140583
$170$	0	0
$171$	− 710.900i	− 0.317917i
$172$	0	0
$173$	1094.03i	0.480794i	0.970675	+	0.240397i	$0.0772776\pi$
$173$	1094.03i	0.480794i	−0.970675	+	0.240397i	$0.922722\pi$
$174$	0	0
$175$	27.5984	0.0119214
$176$	0	0
$177$	−430.905	−0.182988
$178$	0	0
$179$	− 602.526i	− 0.251592i	−0.992056	−	0.125796i	$-0.959852\pi$
$179$	− 602.526i	− 0.251592i	0.992056	−	0.125796i	$-0.0401484\pi$
$180$	0	0
$181$	3702.50i	1.52047i	0.649649	+	0.760234i	$0.274916\pi$
$181$	3702.50i	1.52047i	−0.649649	+	0.760234i	$0.725084\pi$
$182$	0	0
$183$	345.194	0.139440
$184$	0	0
$185$	1383.02	0.549631
$186$	0	0
$187$	4636.50i	1.81312i
$188$	0	0
$189$	− 151.131i	− 0.0581649i
$190$	0	0
$191$	3216.39	1.21848	0.609240	−	0.792986i	$-0.291475\pi$
$191$	3216.39	1.21848	0.609240	+	0.792986i	$0.291475\pi$
$192$	0	0
$193$	2852.57	1.06390	0.531950	−	0.846776i	$-0.321459\pi$
$193$	2852.57	1.06390	0.531950	+	0.846776i	$0.321459\pi$
$194$	0	0
$195$	623.243i	0.228879i
$196$	0	0
$197$	2275.50i	0.822956i	0.911420	+	0.411478i	$0.134987\pi$
$197$	2275.50i	0.822956i	−0.911420	+	0.411478i	$0.865013\pi$
$198$	0	0
$199$	747.136	0.266146	0.133073	−	0.991106i	$-0.457516\pi$
$199$	747.136	0.266146	0.133073	+	0.991106i	$0.457516\pi$
$200$	0	0
$201$	−210.612	−0.0739074
$202$	0	0
$203$	− 249.536i	− 0.0862758i
$204$	0	0
$205$	− 1217.17i	− 0.414685i
$206$	0	0
$207$	3605.73	1.21070
$208$	0	0
$209$	1758.44	0.581981
$210$	0	0
$211$	3149.64i	1.02763i	0.857901	+	0.513816i	$0.171769\pi$
$211$	3149.64i	1.02763i	−0.857901	+	0.513816i	$0.828231\pi$
$212$	0	0
$213$	− 485.520i	− 0.156185i
$214$	0	0
$215$	−517.995	−0.164311
$216$	0	0
$217$	−504.102	−0.157699
$218$	0	0
$219$	277.993i	0.0857763i
$220$	0	0
$221$	− 3629.20i	− 1.10464i
$222$	0	0
$223$	−358.053	−0.107520	−0.0537601	−	0.998554i	$-0.517121\pi$
$223$	−358.053	−0.107520	−0.0537601	+	0.998554i	$0.517121\pi$
$224$	0	0
$225$	−267.075	−0.0791332
$226$	0	0
$227$	4886.67i	1.42881i	0.699733	+	0.714404i	$0.253302\pi$
$227$	4886.67i	1.42881i	−0.699733	+	0.714404i	$0.746698\pi$
$228$	0	0
$229$	− 2022.37i	− 0.583589i	−0.956481	−	0.291794i	$-0.905748\pi$
$229$	− 2022.37i	− 0.583589i	0.956481	−	0.291794i	$-0.0942523\pi$
$230$	0	0
$231$	182.864	0.0520847
$232$	0	0
$233$	926.479	0.260496	0.130248	−	0.991481i	$-0.458423\pi$
$233$	926.479	0.260496	0.130248	+	0.991481i	$0.458423\pi$
$234$	0	0
$235$	5678.73i	1.57634i
$236$	0	0
$237$	− 345.892i	− 0.0948021i
$238$	0	0
$239$	792.472	0.214480	0.107240	−	0.994233i	$-0.465799\pi$
$239$	792.472	0.214480	0.107240	+	0.994233i	$0.465799\pi$
$240$	0	0
$241$	−1449.01	−0.387299	−0.193650	−	0.981071i	$-0.562033\pi$
$241$	−1449.01	−0.387299	−0.193650	+	0.981071i	$0.562033\pi$
$242$	0	0
$243$	2209.63i	0.583325i
$244$	0	0
$245$	− 3907.13i	− 1.01885i
$246$	0	0
$247$	−1376.42	−0.354572
$248$	0	0
$249$	−853.487	−0.217219
$250$	0	0
$251$	− 5062.94i	− 1.27319i	−0.771199	−	0.636594i	$-0.780343\pi$
$251$	− 5062.94i	− 1.27319i	0.771199	−	0.636594i	$-0.219657\pi$
$252$	0	0
$253$	8918.93i	2.21632i
$254$	0	0
$255$	−902.634	−0.221667
$256$	0	0
$257$	−4708.87	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$257$	−4708.87	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$258$	0	0
$259$	317.629i	0.0762028i
$260$	0	0
$261$	2414.81i	0.572692i
$262$	0	0
$263$	−2967.82	−0.695830	−0.347915	−	0.937526i	$-0.613110\pi$
$263$	−2967.82	−0.695830	−0.347915	+	0.937526i	$0.613110\pi$
$264$	0	0
$265$	2460.53	0.570374
$266$	0	0
$267$	927.743i	0.212648i
$268$	0	0
$269$	938.519i	0.212723i	0.994328	+	0.106362i	$0.0339201\pi$
$269$	938.519i	0.212723i	−0.994328	+	0.106362i	$0.966080\pi$
$270$	0	0
$271$	−8058.74	−1.80640	−0.903199	−	0.429223i	$-0.858788\pi$
$271$	−8058.74	−1.80640	−0.903199	+	0.429223i	$0.858788\pi$
$272$	0	0
$273$	−143.136	−0.0317326
$274$	0	0
$275$	− 660.622i	− 0.144862i
$276$	0	0
$277$	− 682.325i	− 0.148003i	−0.997258	−	0.0740017i	$-0.976423\pi$
$277$	− 682.325i	− 0.148003i	0.997258	−	0.0740017i	$-0.0235770\pi$
$278$	0	0
$279$	4878.29	1.04679
$280$	0	0
$281$	−5899.10	−1.25235	−0.626175	−	0.779682i	$-0.715381\pi$
$281$	−5899.10	−1.25235	−0.626175	+	0.779682i	$0.715381\pi$
$282$	0	0
$283$	− 961.519i	− 0.201966i	−0.994888	−	0.100983i	$-0.967801\pi$
$283$	− 961.519i	− 0.201966i	0.994888	−	0.100983i	$-0.0321988\pi$
$284$	0	0
$285$	342.334i	0.0711513i
$286$	0	0
$287$	279.538	0.0574935
$288$	0	0
$289$	343.118	0.0698388
$290$	0	0
$291$	− 1001.83i	− 0.201815i
$292$	0	0
$293$	− 5024.51i	− 1.00183i	−0.865498	−	0.500913i	$-0.832998\pi$
$293$	− 5024.51i	− 1.00183i	0.865498	−	0.500913i	$-0.167002\pi$
$294$	0	0
$295$	−4683.78	−0.924407
$296$	0	0
$297$	−3617.62	−0.706786
$298$	0	0
$299$	− 6981.26i	− 1.35029i
$300$	0	0
$301$	− 118.964i	− 0.0227807i
$302$	0	0
$303$	2.40419	0.000455831 0
$304$	0	0
$305$	3752.13	0.704415
$306$	0	0
$307$	− 3868.67i	− 0.719207i	−0.933105	−	0.359603i	$-0.882912\pi$
$307$	− 3868.67i	− 0.719207i	0.933105	−	0.359603i	$-0.117088\pi$
$308$	0	0
$309$	− 1485.80i	− 0.273541i
$310$	0	0
$311$	−5796.70	−1.05692	−0.528458	−	0.848960i	$-0.677229\pi$
$311$	−5796.70	−1.05692	−0.528458	+	0.848960i	$0.677229\pi$
$312$	0	0
$313$	8362.62	1.51017	0.755085	−	0.655627i	$-0.227596\pi$
$313$	8362.62	1.51017	0.755085	+	0.655627i	$0.227596\pi$
$314$	0	0
$315$	− 803.569i	− 0.143733i
$316$	0	0
$317$	− 487.064i	− 0.0862973i	−0.999069	−	0.0431487i	$-0.986261\pi$
$317$	− 487.064i	− 0.0862973i	0.999069	−	0.0431487i	$-0.0137389\pi$
$318$	0	0
$319$	−5973.13	−1.04837
$320$	0	0
$321$	−1243.42	−0.216203
$322$	0	0
$323$	− 1993.44i	− 0.343400i
$324$	0	0
$325$	517.099i	0.0882569i
$326$	0	0
$327$	806.416	0.136376
$328$	0	0
$329$	−1304.20	−0.218549
$330$	0	0
$331$	− 3801.21i	− 0.631218i	−0.948889	−	0.315609i	$-0.897791\pi$
$331$	− 3801.21i	− 0.631218i	0.948889	−	0.315609i	$-0.102209\pi$
$332$	0	0
$333$	− 3073.76i	− 0.505828i
$334$	0	0
$335$	−2289.27	−0.373361
$336$	0	0
$337$	1795.31	0.290199	0.145099	−	0.989417i	$-0.453650\pi$
$337$	1795.31	0.290199	0.145099	+	0.989417i	$0.453650\pi$
$338$	0	0
$339$	71.9931i	0.0115343i
$340$	0	0
$341$	12066.7i	1.91627i
$342$	0	0
$343$	1813.72	0.285515
$344$	0	0
$345$	−1736.34	−0.270960
$346$	0	0
$347$	− 2782.23i	− 0.430426i	−0.976567	−	0.215213i	$-0.930955\pi$
$347$	− 2782.23i	− 0.430426i	0.976567	−	0.215213i	$-0.0690445\pi$
$348$	0	0
$349$	− 10413.4i	− 1.59718i	−0.601876	−	0.798589i	$-0.705580\pi$
$349$	− 10413.4i	− 1.59718i	0.601876	−	0.798589i	$-0.294420\pi$
$350$	0	0
$351$	2831.68	0.430609
$352$	0	0
$353$	10644.3	1.60493	0.802466	−	0.596698i	$-0.203521\pi$
$353$	10644.3	1.60493	0.802466	+	0.596698i	$0.203521\pi$
$354$	0	0
$355$	− 5277.43i	− 0.789005i
$356$	0	0
$357$	− 207.302i	− 0.0307327i
$358$	0	0
$359$	7459.42	1.09664	0.548319	−	0.836269i	$-0.315268\pi$
$359$	7459.42	1.09664	0.548319	+	0.836269i	$0.315268\pi$
$360$	0	0
$361$	6102.96	0.889775
$362$	0	0
$363$	− 2952.72i	− 0.426935i
$364$	0	0
$365$	3021.68i	0.433320i
$366$	0	0
$367$	6251.35	0.889149	0.444574	−	0.895742i	$-0.353355\pi$
$367$	6251.35	0.889149	0.444574	+	0.895742i	$0.353355\pi$
$368$	0	0
$369$	−2705.14	−0.381637
$370$	0	0
$371$	565.094i	0.0790787i
$372$	0	0
$373$	− 12603.3i	− 1.74953i	−0.484552	−	0.874763i	$-0.661017\pi$
$373$	− 12603.3i	− 1.74953i	0.484552	−	0.874763i	$-0.338983\pi$
$374$	0	0
$375$	−1427.68	−0.196600
$376$	0	0
$377$	4675.45	0.638721
$378$	0	0
$379$	1674.47i	0.226943i	0.993541	+	0.113472i	$0.0361971\pi$
$379$	1674.47i	0.226943i	−0.993541	+	0.113472i	$0.963803\pi$
$380$	0	0
$381$	2037.31i	0.273949i
$382$	0	0
$383$	−2880.38	−0.384283	−0.192142	−	0.981367i	$-0.561543\pi$
$383$	−2880.38	−0.384283	−0.192142	+	0.981367i	$0.561543\pi$
$384$	0	0
$385$	1987.66	0.263119
$386$	0	0
$387$	1151.24i	0.151217i
$388$	0	0
$389$	13073.3i	1.70397i	0.523567	+	0.851984i	$0.324601\pi$
$389$	13073.3i	1.70397i	−0.523567	+	0.851984i	$0.675399\pi$
$390$	0	0
$391$	10110.9	1.30774
$392$	0	0
$393$	−1390.09	−0.178424
$394$	0	0
$395$	− 3759.72i	− 0.478916i
$396$	0	0
$397$	6021.44i	0.761228i	0.924734	+	0.380614i	$0.124287\pi$
$397$	6021.44i	0.761228i	−0.924734	+	0.380614i	$0.875713\pi$
$398$	0	0
$399$	−78.6216	−0.00986467
$400$	0	0
$401$	−12722.6	−1.58437	−0.792187	−	0.610278i	$-0.791058\pi$
$401$	−12722.6	−1.58437	−0.792187	+	0.610278i	$0.791058\pi$
$402$	0	0
$403$	− 9445.14i	− 1.16748i
$404$	0	0
$405$	7416.51i	0.909949i
$406$	0	0
$407$	7603.08	0.925972
$408$	0	0
$409$	232.991	0.0281678	0.0140839	−	0.999901i	$-0.495517\pi$
$409$	232.991	0.0281678	0.0140839	+	0.999901i	$0.495517\pi$
$410$	0	0
$411$	− 510.756i	− 0.0612987i
$412$	0	0
$413$	− 1075.69i	− 0.128163i
$414$	0	0
$415$	−9277.08	−1.09734
$416$	0	0
$417$	−2291.35	−0.269084
$418$	0	0
$419$	− 8663.03i	− 1.01006i	−0.863101	−	0.505032i	$-0.831481\pi$
$419$	− 8663.03i	− 1.01006i	0.863101	−	0.505032i	$-0.168519\pi$
$420$	0	0
$421$	− 11749.9i	− 1.36023i	−0.733107	−	0.680113i	$-0.761931\pi$
$421$	− 11749.9i	− 1.36023i	0.733107	−	0.680113i	$-0.238069\pi$
$422$	0	0
$423$	12620.9	1.45071
$424$	0	0
$425$	−748.907	−0.0854760
$426$	0	0
$427$	861.727i	0.0976626i
$428$	0	0
$429$	3426.24i	0.385596i
$430$	0	0
$431$	−8737.57	−0.976506	−0.488253	−	0.872702i	$-0.662366\pi$
$431$	−8737.57	−0.976506	−0.488253	+	0.872702i	$0.662366\pi$
$432$	0	0
$433$	11627.5	1.29049	0.645247	−	0.763974i	$-0.276755\pi$
$433$	11627.5	1.29049	0.645247	+	0.763974i	$0.276755\pi$
$434$	0	0
$435$	− 1162.85i	− 0.128171i
$436$	0	0
$437$	− 3834.66i	− 0.419763i
$438$	0	0
$439$	17631.8	1.91690	0.958450	−	0.285261i	$-0.0920804\pi$
$439$	17631.8	1.91690	0.958450	+	0.285261i	$0.0920804\pi$
$440$	0	0
$441$	−8683.57	−0.937649
$442$	0	0
$443$	− 6434.41i	− 0.690085i	−0.938587	−	0.345043i	$-0.887864\pi$
$443$	− 6434.41i	− 0.690085i	0.938587	−	0.345043i	$-0.112136\pi$
$444$	0	0
$445$	10084.2i	1.07424i
$446$	0	0
$447$	568.116	0.0601140
$448$	0	0
$449$	−12926.5	−1.35867	−0.679334	−	0.733830i	$-0.737731\pi$
$449$	−12926.5	−1.35867	−0.679334	+	0.733830i	$0.737731\pi$
$450$	0	0
$451$	− 6691.30i	− 0.698627i
$452$	0	0
$453$	− 3208.07i	− 0.332734i
$454$	0	0
$455$	−1555.84	−0.160305
$456$	0	0
$457$	−9320.32	−0.954018	−0.477009	−	0.878898i	$-0.658279\pi$
$457$	−9320.32	−0.954018	−0.477009	+	0.878898i	$0.658279\pi$
$458$	0	0
$459$	4101.07i	0.417041i
$460$	0	0
$461$	− 18222.1i	− 1.84098i	−0.390771	−	0.920488i	$-0.627792\pi$
$461$	− 18222.1i	− 1.84098i	0.390771	−	0.920488i	$-0.372208\pi$
$462$	0	0
$463$	−7038.37	−0.706482	−0.353241	−	0.935532i	$-0.614920\pi$
$463$	−7038.37	−0.706482	−0.353241	+	0.935532i	$0.614920\pi$
$464$	0	0
$465$	−2349.14	−0.234277
$466$	0	0
$467$	− 8487.77i	− 0.841043i	−0.907283	−	0.420522i	$-0.861847\pi$
$467$	− 8487.77i	− 0.841043i	0.907283	−	0.420522i	$-0.138153\pi$
$468$	0	0
$469$	− 525.761i	− 0.0517642i
$470$	0	0
$471$	2283.96	0.223438
$472$	0	0
$473$	−2847.65	−0.276818
$474$	0	0
$475$	284.031i	0.0274363i
$476$	0	0
$477$	− 5468.51i	− 0.524918i
$478$	0	0
$479$	−587.317	−0.0560234	−0.0280117	−	0.999608i	$-0.508918\pi$
$479$	−587.317	−0.0560234	−0.0280117	+	0.999608i	$0.508918\pi$
$480$	0	0
$481$	−5951.28	−0.564148
$482$	0	0
$483$	− 398.773i	− 0.0375669i
$484$	0	0
$485$	− 10889.5i	− 1.01952i
$486$	0	0
$487$	−8366.45	−0.778481	−0.389240	−	0.921136i	$-0.627262\pi$
$487$	−8366.45	−0.778481	−0.389240	+	0.921136i	$0.627262\pi$
$488$	0	0
$489$	2157.10	0.199483
$490$	0	0
$491$	− 2162.76i	− 0.198786i	−0.995048	−	0.0993929i	$-0.968310\pi$
$491$	− 2162.76i	− 0.198786i	0.995048	−	0.0993929i	$-0.0316901\pi$
$492$	0	0
$493$	6771.38i	0.618596i
$494$	0	0
$495$	−19235.0	−1.74656
$496$	0	0
$497$	1212.03	0.109390
$498$	0	0
$499$	− 16071.8i	− 1.44183i	−0.693026	−	0.720913i	$-0.743723\pi$
$499$	− 16071.8i	− 1.44183i	0.693026	−	0.720913i	$-0.256277\pi$
$500$	0	0
$501$	− 848.077i	− 0.0756273i
$502$	0	0
$503$	12570.2	1.11427	0.557137	−	0.830421i	$-0.311900\pi$
$503$	12570.2	1.11427	0.557137	+	0.830421i	$0.311900\pi$
$504$	0	0
$505$	26.1326	0.00230274
$506$	0	0
$507$	− 330.556i	− 0.0289556i
$508$	0	0
$509$	16801.4i	1.46308i	0.681797	+	0.731542i	$0.261199\pi$
$509$	16801.4i	1.46308i	−0.681797	+	0.731542i	$0.738801\pi$
$510$	0	0
$511$	−693.968	−0.0600770
$512$	0	0
$513$	1555.38	0.133863
$514$	0	0
$515$	− 16150.1i	− 1.38186i
$516$	0	0
$517$	31218.5i	2.65568i
$518$	0	0
$519$	−1170.87	−0.0990283
$520$	0	0
$521$	6612.98	0.556085	0.278042	−	0.960569i	$-0.410314\pi$
$521$	6612.98	0.556085	0.278042	+	0.960569i	$0.410314\pi$
$522$	0	0
$523$	− 7253.92i	− 0.606485i	−0.952913	−	0.303243i	$-0.901931\pi$
$523$	− 7253.92i	− 0.606485i	0.952913	−	0.303243i	$-0.0980693\pi$
$524$	0	0
$525$	29.5370i	0.00245543i
$526$	0	0
$527$	13679.3	1.13070
$528$	0	0
$529$	7282.60	0.598553
$530$	0	0
$531$	10409.7i	0.850737i
$532$	0	0
$533$	5237.59i	0.425638i
$534$	0	0
$535$	−13515.5	−1.09220
$536$	0	0
$537$	644.849	0.0518199
$538$	0	0
$539$	− 21479.2i	− 1.71647i
$540$	0	0
$541$	− 15511.9i	− 1.23273i	−0.787461	−	0.616365i	$-0.788605\pi$
$541$	− 15511.9i	− 1.23273i	0.787461	−	0.616365i	$-0.211395\pi$
$542$	0	0
$543$	−3962.57	−0.313168
$544$	0	0
$545$	8765.44	0.688936
$546$	0	0
$547$	18510.4i	1.44689i	0.690382	+	0.723445i	$0.257442\pi$
$547$	18510.4i	1.44689i	−0.690382	+	0.723445i	$0.742558\pi$
$548$	0	0
$549$	− 8339.09i	− 0.648277i
$550$	0	0
$551$	2568.12	0.198558
$552$	0	0
$553$	863.469	0.0663986
$554$	0	0
$555$	1480.17i	0.113207i
$556$	0	0
$557$	− 7141.59i	− 0.543266i	−0.962401	−	0.271633i	$-0.912436\pi$
$557$	− 7141.59i	− 0.543266i	0.962401	−	0.271633i	$-0.0875637\pi$
$558$	0	0
$559$	2228.98	0.168651
$560$	0	0
$561$	−4962.18	−0.373446
$562$	0	0
$563$	4594.86i	0.343961i	0.985100	+	0.171981i	$0.0550167\pi$
$563$	4594.86i	0.343961i	−0.985100	+	0.171981i	$0.944983\pi$
$564$	0	0
$565$	782.538i	0.0582683i
$566$	0	0
$567$	−1703.30	−0.126159
$568$	0	0
$569$	2806.05	0.206741	0.103371	−	0.994643i	$-0.467037\pi$
$569$	2806.05	0.206741	0.103371	+	0.994643i	$0.467037\pi$
$570$	0	0
$571$	− 17025.4i	− 1.24779i	−0.781506	−	0.623897i	$-0.785548\pi$
$571$	− 17025.4i	− 1.24779i	0.781506	−	0.623897i	$-0.214452\pi$
$572$	0	0
$573$	3442.31i	0.250968i
$574$	0	0
$575$	−1440.62	−0.104484
$576$	0	0
$577$	7206.84	0.519973	0.259987	−	0.965612i	$-0.416282\pi$
$577$	7206.84	0.519973	0.259987	+	0.965612i	$0.416282\pi$
$578$	0	0
$579$	3052.95i	0.219130i
$580$	0	0
$581$	− 2130.61i	− 0.152138i
$582$	0	0
$583$	13526.6	0.960918
$584$	0	0
$585$	15056.1	1.06409
$586$	0	0
$587$	− 14675.2i	− 1.03188i	−0.856626	−	0.515938i	$-0.827443\pi$
$587$	− 14675.2i	− 1.03188i	0.856626	−	0.515938i	$-0.172557\pi$
$588$	0	0
$589$	− 5188.01i	− 0.362934i
$590$	0	0
$591$	−2435.33	−0.169503
$592$	0	0
$593$	4758.60	0.329531	0.164766	−	0.986333i	$-0.447313\pi$
$593$	4758.60	0.329531	0.164766	+	0.986333i	$0.447313\pi$
$594$	0	0
$595$	− 2253.29i	− 0.155254i
$596$	0	0
$597$	799.617i	0.0548176i
$598$	0	0
$599$	−14256.4	−0.972455	−0.486227	−	0.873832i	$-0.661627\pi$
$599$	−14256.4	−0.972455	−0.486227	+	0.873832i	$0.661627\pi$
$600$	0	0
$601$	10385.2	0.704862	0.352431	−	0.935838i	$-0.385355\pi$
$601$	10385.2	0.704862	0.352431	+	0.935838i	$0.385355\pi$
$602$	0	0
$603$	5087.89i	0.343606i
$604$	0	0
$605$	− 32094.9i	− 2.15677i
$606$	0	0
$607$	−16243.6	−1.08618	−0.543088	−	0.839676i	$-0.682745\pi$
$607$	−16243.6	−1.08618	−0.543088	+	0.839676i	$0.682745\pi$
$608$	0	0
$609$	267.064	0.0177701
$610$	0	0
$611$	− 24436.1i	− 1.61797i
$612$	0	0
$613$	− 707.817i	− 0.0466369i	−0.999728	−	0.0233185i	$-0.992577\pi$
$613$	− 707.817i	− 0.0466369i	0.999728	−	0.0233185i	$-0.00742317\pi$
$614$	0	0
$615$	1302.66	0.0854121
$616$	0	0
$617$	−11575.9	−0.755316	−0.377658	−	0.925945i	$-0.623271\pi$
$617$	−11575.9	−0.755316	−0.377658	+	0.925945i	$0.623271\pi$
$618$	0	0
$619$	− 25959.4i	− 1.68562i	−0.538214	−	0.842808i	$-0.680901\pi$
$619$	− 25959.4i	− 1.68562i	0.538214	−	0.842808i	$-0.319099\pi$
$620$	0	0
$621$	7888.97i	0.509780i
$622$	0	0
$623$	−2315.98	−0.148937
$624$	0	0
$625$	−16809.5	−1.07581
$626$	0	0
$627$	1881.96i	0.119870i
$628$	0	0
$629$	− 8619.15i	− 0.546372i
$630$	0	0
$631$	10224.8	0.645079	0.322539	−	0.946556i	$-0.395463\pi$
$631$	10224.8	0.645079	0.322539	+	0.946556i	$0.395463\pi$
$632$	0	0
$633$	−3370.88	−0.211660
$634$	0	0
$635$	22144.8i	1.38392i
$636$	0	0
$637$	16812.8i	1.04576i
$638$	0	0
$639$	−11729.0	−0.726125
$640$	0	0
$641$	19804.4	1.22032	0.610162	−	0.792277i	$-0.291104\pi$
$641$	19804.4	1.22032	0.610162	+	0.792277i	$0.291104\pi$
$642$	0	0
$643$	22175.9i	1.36008i	0.733174	+	0.680041i	$0.238038\pi$
$643$	22175.9i	1.36008i	−0.733174	+	0.680041i	$0.761962\pi$
$644$	0	0
$645$	− 554.380i	− 0.0338429i
$646$	0	0
$647$	−9232.26	−0.560985	−0.280493	−	0.959856i	$-0.590498\pi$
$647$	−9232.26	−0.560985	−0.280493	+	0.959856i	$0.590498\pi$
$648$	0	0
$649$	−25748.8	−1.55736
$650$	0	0
$651$	− 539.511i	− 0.0324810i
$652$	0	0
$653$	27857.1i	1.66942i	0.550690	+	0.834710i	$0.314365\pi$
$653$	27857.1i	1.66942i	−0.550690	+	0.834710i	$0.685635\pi$
$654$	0	0
$655$	−15109.8	−0.901355
$656$	0	0
$657$	6715.66	0.398786
$658$	0	0
$659$	− 5498.55i	− 0.325028i	−0.986706	−	0.162514i	$-0.948040\pi$
$659$	− 5498.55i	− 0.325028i	0.986706	−	0.162514i	$-0.0519602\pi$
$660$	0	0
$661$	11469.6i	0.674908i	0.941342	+	0.337454i	$0.109566\pi$
$661$	11469.6i	0.674908i	−0.941342	+	0.337454i	$0.890434\pi$
$662$	0	0
$663$	3884.13	0.227522
$664$	0	0
$665$	−854.587	−0.0498338
$666$	0	0
$667$	13025.7i	0.756156i
$668$	0	0
$669$	− 383.204i	− 0.0221458i
$670$	0	0
$671$	20627.1	1.18674
$672$	0	0
$673$	−28428.2	−1.62827	−0.814135	−	0.580676i	$-0.802788\pi$
$673$	−28428.2	−1.62827	−0.814135	+	0.580676i	$0.802788\pi$
$674$	0	0
$675$	− 584.333i	− 0.0333200i
$676$	0	0
$677$	− 23995.5i	− 1.36222i	−0.732181	−	0.681110i	$-0.761498\pi$
$677$	− 23995.5i	− 1.36222i	0.732181	−	0.681110i	$-0.238502\pi$
$678$	0	0
$679$	2500.92	0.141350
$680$	0	0
$681$	−5229.92	−0.294289
$682$	0	0
$683$	13506.0i	0.756650i	0.925673	+	0.378325i	$0.123500\pi$
$683$	13506.0i	0.756650i	−0.925673	+	0.378325i	$0.876500\pi$
$684$	0	0
$685$	− 5551.73i	− 0.309665i
$686$	0	0
$687$	2164.42	0.120201
$688$	0	0
$689$	−10587.9	−0.585439
$690$	0	0
$691$	29500.1i	1.62408i	0.583605	+	0.812038i	$0.301642\pi$
$691$	29500.1i	1.62408i	−0.583605	+	0.812038i	$0.698358\pi$
$692$	0	0
$693$	− 4417.57i	− 0.242150i
$694$	0	0
$695$	−24906.1	−1.35934
$696$	0	0
$697$	−7585.52	−0.412227
$698$	0	0
$699$	991.557i	0.0536540i
$700$	0	0
$701$	33227.5i	1.79028i	0.445787	+	0.895139i	$0.352924\pi$
$701$	33227.5i	1.79028i	−0.445787	+	0.895139i	$0.647076\pi$
$702$	0	0
$703$	−3268.91	−0.175376
$704$	0	0
$705$	−6077.62	−0.324676
$706$	0	0
$707$	6.00170i	0 0.000319261i
$708$	0	0
$709$	6448.04i	0.341553i	0.985310	+	0.170777i	$0.0546277\pi$
$709$	6448.04i	0.341553i	−0.985310	+	0.170777i	$0.945372\pi$
$710$	0	0
$711$	−8355.95	−0.440749
$712$	0	0
$713$	26313.9	1.38214
$714$	0	0
$715$	37242.0i	1.94793i
$716$	0	0
$717$	848.138i	0.0441761i
$718$	0	0
$719$	6494.67	0.336871	0.168436	−	0.985713i	$-0.446128\pi$
$719$	6494.67	0.336871	0.168436	+	0.985713i	$0.446128\pi$
$720$	0	0
$721$	3709.08	0.191586
$722$	0	0
$723$	− 1550.80i	− 0.0797714i
$724$	0	0
$725$	− 964.806i	− 0.0494234i
$726$	0	0
$727$	24866.4	1.26856	0.634280	−	0.773103i	$-0.281296\pi$
$727$	24866.4	1.26856	0.634280	+	0.773103i	$0.281296\pi$
$728$	0	0
$729$	14848.5	0.754384
$730$	0	0
$731$	3228.20i	0.163337i
$732$	0	0
$733$	− 21092.1i	− 1.06283i	−0.847112	−	0.531414i	$-0.821661\pi$
$733$	− 21092.1i	− 1.06283i	0.847112	−	0.531414i	$-0.178339\pi$
$734$	0	0
$735$	4181.58	0.209850
$736$	0	0
$737$	−12585.1	−0.629008
$738$	0	0
$739$	11952.7i	0.594977i	0.954725	+	0.297489i	$0.0961491\pi$
$739$	11952.7i	0.594977i	−0.954725	+	0.297489i	$0.903851\pi$
$740$	0	0
$741$	− 1473.10i	− 0.0730305i
$742$	0	0
$743$	−5622.43	−0.277614	−0.138807	−	0.990319i	$-0.544327\pi$
$743$	−5622.43	−0.277614	−0.138807	+	0.990319i	$0.544327\pi$
$744$	0	0
$745$	6175.21	0.303681
$746$	0	0
$747$	20618.3i	1.00988i
$748$	0	0
$749$	− 3104.02i	− 0.151427i
$750$	0	0
$751$	32314.9	1.57016	0.785079	−	0.619396i	$-0.212622\pi$
$751$	32314.9	1.57016	0.785079	+	0.619396i	$0.212622\pi$
$752$	0	0
$753$	5418.58	0.262236
$754$	0	0
$755$	− 34870.5i	− 1.68089i
$756$	0	0
$757$	− 17950.4i	− 0.861847i	−0.902389	−	0.430923i	$-0.858188\pi$
$757$	− 17950.4i	− 0.861847i	0.902389	−	0.430923i	$-0.141812\pi$
$758$	0	0
$759$	−9545.42	−0.456491
$760$	0	0
$761$	−13108.2	−0.624404	−0.312202	−	0.950016i	$-0.601067\pi$
$761$	−13108.2	−0.624404	−0.312202	+	0.950016i	$0.601067\pi$
$762$	0	0
$763$	2013.10i	0.0955166i
$764$	0	0
$765$	21805.5i	1.03056i
$766$	0	0
$767$	20154.8	0.948822
$768$	0	0
$769$	23661.2	1.10955	0.554776	−	0.832000i	$-0.312804\pi$
$769$	23661.2	1.10955	0.554776	+	0.832000i	$0.312804\pi$
$770$	0	0
$771$	− 5039.63i	− 0.235406i
$772$	0	0
$773$	30222.4i	1.40624i	0.711071	+	0.703120i	$0.248210\pi$
$773$	30222.4i	1.40624i	−0.711071	+	0.703120i	$0.751790\pi$
$774$	0	0
$775$	−1949.06	−0.0903384
$776$	0	0
$777$	−339.940	−0.0156954
$778$	0	0
$779$	2876.89i	0.132318i
$780$	0	0
$781$	− 29012.3i	− 1.32925i
$782$	0	0
$783$	−5283.35	−0.241139
$784$	0	0
$785$	24825.8	1.12875
$786$	0	0
$787$	29544.2i	1.33817i	0.743188	+	0.669083i	$0.233313\pi$
$787$	29544.2i	1.33817i	−0.743188	+	0.669083i	$0.766687\pi$
$788$	0	0
$789$	− 3176.28i	− 0.143319i
$790$	0	0
$791$	−179.720	−0.00807853
$792$	0	0
$793$	−16145.8	−0.723020
$794$	0	0
$795$	2633.36i	0.117479i
$796$	0	0
$797$	− 9665.91i	− 0.429591i	−0.976659	−	0.214795i	$-0.931092\pi$
$797$	− 9665.91i	− 0.429591i	0.976659	−	0.214795i	$-0.0689085\pi$
$798$	0	0
$799$	35390.5	1.56699
$800$	0	0
$801$	22412.1	0.988631
$802$	0	0
$803$	16611.5i	0.730021i
$804$	0	0
$805$	− 4334.52i	− 0.189778i
$806$	0	0
$807$	−1004.44	−0.0438142
$808$	0	0
$809$	15807.0	0.686952	0.343476	−	0.939162i	$-0.388396\pi$
$809$	15807.0	0.686952	0.343476	+	0.939162i	$0.388396\pi$
$810$	0	0
$811$	16295.5i	0.705565i	0.935705	+	0.352783i	$0.114764\pi$
$811$	16295.5i	0.705565i	−0.935705	+	0.352783i	$0.885236\pi$
$812$	0	0
$813$	− 8624.81i	− 0.372061i
$814$	0	0
$815$	23446.9	1.00774
$816$	0	0
$817$	1224.33	0.0524284
$818$	0	0
$819$	3457.84i	0.147529i
$820$	0	0
$821$	− 436.743i	− 0.0185657i	−0.999957	−	0.00928286i	$-0.997045\pi$
$821$	− 436.743i	− 0.0185657i	0.999957	−	0.00928286i	$-0.00295487\pi$
$822$	0	0
$823$	5633.49	0.238604	0.119302	−	0.992858i	$-0.461934\pi$
$823$	5633.49	0.238604	0.119302	+	0.992858i	$0.461934\pi$
$824$	0	0
$825$	707.026	0.0298369
$826$	0	0
$827$	− 22394.7i	− 0.941645i	−0.882228	−	0.470822i	$-0.843957\pi$
$827$	− 22394.7i	− 0.941645i	0.882228	−	0.470822i	$-0.156043\pi$
$828$	0	0
$829$	− 8768.39i	− 0.367357i	−0.982986	−	0.183678i	$-0.941199\pi$
$829$	− 8768.39i	− 0.367357i	0.982986	−	0.183678i	$-0.0588005\pi$
$830$	0	0
$831$	730.253	0.0304840
$832$	0	0
$833$	−24349.7	−1.01281
$834$	0	0
$835$	− 9218.27i	− 0.382050i
$836$	0	0
$837$	10673.2i	0.440764i
$838$	0	0
$839$	−30644.3	−1.26098	−0.630488	−	0.776199i	$-0.717145\pi$
$839$	−30644.3	−1.26098	−0.630488	+	0.776199i	$0.717145\pi$
$840$	0	0
$841$	15665.5	0.642319
$842$	0	0
$843$	− 6313.47i	− 0.257945i
$844$	0	0
$845$	− 3593.02i	− 0.146277i
$846$	0	0
$847$	7371.03	0.299022
$848$	0	0
$849$	1029.06	0.0415986
$850$	0	0
$851$	− 16580.1i	− 0.667871i
$852$	0	0
$853$	43559.2i	1.74846i	0.485508	+	0.874232i	$0.338635\pi$
$853$	43559.2i	1.74846i	−0.485508	+	0.874232i	$0.661365\pi$
$854$	0	0
$855$	8270.00	0.330793
$856$	0	0
$857$	41788.5	1.66566	0.832828	−	0.553533i	$-0.186721\pi$
$857$	41788.5	1.66566	0.832828	+	0.553533i	$0.186721\pi$
$858$	0	0
$859$	16849.6i	0.669267i	0.942348	+	0.334633i	$0.108612\pi$
$859$	16849.6i	0.669267i	−0.942348	+	0.334633i	$0.891388\pi$
$860$	0	0
$861$	299.174i	0.0118418i
$862$	0	0
$863$	27636.7	1.09011	0.545054	−	0.838401i	$-0.316509\pi$
$863$	27636.7	1.09011	0.545054	+	0.838401i	$0.316509\pi$
$864$	0	0
$865$	−12727.0	−0.500266
$866$	0	0
$867$	367.220i	0.0143846i
$868$	0	0
$869$	− 20668.8i	− 0.806837i
$870$	0	0
$871$	9850.95	0.383223
$872$	0	0
$873$	−24201.9	−0.938270
$874$	0	0
$875$	− 3563.98i	− 0.137697i
$876$	0	0
$877$	16855.0i	0.648977i	0.945890	+	0.324488i	$0.105192\pi$
$877$	16855.0i	0.648977i	−0.945890	+	0.324488i	$0.894808\pi$
$878$	0	0
$879$	5377.45	0.206344
$880$	0	0
$881$	13330.0	0.509759	0.254880	−	0.966973i	$-0.417964\pi$
$881$	13330.0	0.509759	0.254880	+	0.966973i	$0.417964\pi$
$882$	0	0
$883$	35598.7i	1.35673i	0.734725	+	0.678365i	$0.237311\pi$
$883$	35598.7i	1.35673i	−0.734725	+	0.678365i	$0.762689\pi$
$884$	0	0
$885$	− 5012.78i	− 0.190399i
$886$	0	0
$887$	−48821.3	−1.84810	−0.924048	−	0.382278i	$-0.875140\pi$
$887$	−48821.3	−1.84810	−0.924048	+	0.382278i	$0.875140\pi$
$888$	0	0
$889$	−5085.84	−0.191871
$890$	0	0
$891$	40771.8i	1.53301i
$892$	0	0
$893$	− 13422.2i	− 0.502977i
$894$	0	0
$895$	7009.27	0.261781
$896$	0	0
$897$	7471.64	0.278117
$898$	0	0
$899$	17622.8i	0.653785i
$900$	0	0
$901$	− 15334.3i	− 0.566992i
$902$	0	0
$903$	127.321	0.00469210
$904$	0	0
$905$	−43071.7	−1.58205
$906$	0	0
$907$	− 7523.90i	− 0.275443i	−0.990471	−	0.137722i	$-0.956022\pi$
$907$	− 7523.90i	− 0.275443i	0.990471	−	0.137722i	$-0.0439779\pi$
$908$	0	0
$909$	− 58.0796i	− 0.00211923i
$910$	0	0
$911$	26016.0	0.946155	0.473077	−	0.881021i	$-0.343143\pi$
$911$	26016.0	0.946155	0.473077	+	0.881021i	$0.343143\pi$
$912$	0	0
$913$	−51000.2	−1.84870
$914$	0	0
$915$	4015.69i	0.145087i
$916$	0	0
$917$	− 3470.16i	− 0.124967i
$918$	0	0
$919$	−24082.1	−0.864413	−0.432206	−	0.901775i	$-0.642265\pi$
$919$	−24082.1	−0.864413	−0.432206	+	0.901775i	$0.642265\pi$
$920$	0	0
$921$	4140.41	0.148134
$922$	0	0
$923$	22709.3i	0.809844i
$924$	0	0
$925$	1228.08i	0.0436530i
$926$	0	0
$927$	−35893.5	−1.27173
$928$	0	0
$929$	9324.93	0.329323	0.164661	−	0.986350i	$-0.447347\pi$
$929$	9324.93	0.329323	0.164661	+	0.986350i	$0.447347\pi$
$930$	0	0
$931$	9234.89i	0.325093i
$932$	0	0
$933$	− 6203.87i	− 0.217691i
$934$	0	0
$935$	−53937.0	−1.88656
$936$	0	0
$937$	15535.2	0.541636	0.270818	−	0.962631i	$-0.412706\pi$
$937$	15535.2	0.541636	0.270818	+	0.962631i	$0.412706\pi$
$938$	0	0
$939$	8950.03i	0.311047i
$940$	0	0
$941$	48118.3i	1.66696i	0.552547	+	0.833482i	$0.313656\pi$
$941$	48118.3i	1.66696i	−0.552547	+	0.833482i	$0.686344\pi$
$942$	0	0
$943$	−14591.8	−0.503896
$944$	0	0
$945$	1758.13	0.0605205
$946$	0	0
$947$	5396.67i	0.185183i	0.995704	+	0.0925915i	$0.0295151\pi$
$947$	5396.67i	0.185183i	−0.995704	+	0.0925915i	$0.970485\pi$
$948$	0	0
$949$	− 13002.6i	− 0.444765i
$950$	0	0
$951$	521.277	0.0177745
$952$	0	0
$953$	−21759.7	−0.739629	−0.369815	−	0.929106i	$-0.620579\pi$
$953$	−21759.7	−0.739629	−0.369815	+	0.929106i	$0.620579\pi$
$954$	0	0
$955$	37416.7i	1.26783i
$956$	0	0
$957$	− 6392.70i	− 0.215932i
$958$	0	0
$959$	1275.03	0.0429331
$960$	0	0
$961$	5809.78	0.195018
$962$	0	0
$963$	30038.2i	1.00516i
$964$	0	0
$965$	33184.4i	1.10699i
$966$	0	0
$967$	19338.5	0.643108	0.321554	−	0.946891i	$-0.395795\pi$
$967$	19338.5	0.643108	0.321554	+	0.946891i	$0.395795\pi$
$968$	0	0
$969$	2133.47	0.0707294
$970$	0	0
$971$	− 5918.37i	− 0.195602i	−0.995206	−	0.0978009i	$-0.968819\pi$
$971$	− 5918.37i	− 0.195602i	0.995206	−	0.0978009i	$-0.0311809\pi$
$972$	0	0
$973$	− 5720.03i	− 0.188464i
$974$	0	0
$975$	−553.421	−0.0181781
$976$	0	0
$977$	−17841.9	−0.584249	−0.292125	−	0.956380i	$-0.594362\pi$
$977$	−17841.9	−0.584249	−0.292125	+	0.956380i	$0.594362\pi$
$978$	0	0
$979$	55437.4i	1.80979i
$980$	0	0
$981$	− 19481.1i	− 0.634032i
$982$	0	0
$983$	−61512.0	−1.99586	−0.997929	−	0.0643304i	$-0.979509\pi$
$983$	−61512.0	−1.99586	−0.997929	+	0.0643304i	$0.979509\pi$
$984$	0	0
$985$	−26471.2	−0.856286
$986$	0	0
$987$	− 1395.81i	− 0.0450142i
$988$	0	0
$989$	6209.89i	0.199659i
$990$	0	0
$991$	1827.73	0.0585870	0.0292935	−	0.999571i	$-0.490674\pi$
$991$	1827.73	0.0585870	0.0292935	+	0.999571i	$0.490674\pi$
$992$	0	0
$993$	4068.21	0.130011
$994$	0	0
$995$	8691.53i	0.276925i
$996$	0	0
$997$	− 35379.4i	− 1.12385i	−0.827188	−	0.561925i	$-0.810061\pi$
$997$	− 35379.4i	− 1.12385i	0.827188	−	0.561925i	$-0.189939\pi$
$998$	0	0
$999$	6725.07	0.212985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.b.j.513.6		10
4.3	odd	2		1024.4.b.k.513.5		10
8.3	odd	2		1024.4.b.k.513.6		10
8.5	even	2	inner	1024.4.b.j.513.5		10
16.3	odd	4		1024.4.a.m.1.6		10
16.5	even	4		1024.4.a.n.1.6		10
16.11	odd	4		1024.4.a.m.1.5		10
16.13	even	4		1024.4.a.n.1.5		10
32.3	odd	8		128.4.e.a.97.3		10
32.5	even	8		128.4.e.b.33.3		10
32.11	odd	8		64.4.e.a.17.3		10
32.13	even	8		16.4.e.a.5.3	✓	10
32.19	odd	8		64.4.e.a.49.3		10
32.21	even	8		16.4.e.a.13.3	yes	10
32.27	odd	8		128.4.e.a.33.3		10
32.29	even	8		128.4.e.b.97.3		10
96.11	even	8		576.4.k.a.145.2		10
96.53	odd	8		144.4.k.a.109.3		10
96.77	odd	8		144.4.k.a.37.3		10
96.83	even	8		576.4.k.a.433.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.3	✓	10	32.13	even	8
16.4.e.a.13.3	yes	10	32.21	even	8
64.4.e.a.17.3		10	32.11	odd	8
64.4.e.a.49.3		10	32.19	odd	8
128.4.e.a.33.3		10	32.27	odd	8
128.4.e.a.97.3		10	32.3	odd	8
128.4.e.b.33.3		10	32.5	even	8
128.4.e.b.97.3		10	32.29	even	8
144.4.k.a.37.3		10	96.77	odd	8
144.4.k.a.109.3		10	96.53	odd	8
576.4.k.a.145.2		10	96.11	even	8
576.4.k.a.433.2		10	96.83	even	8
1024.4.a.m.1.5		10	16.11	odd	4
1024.4.a.m.1.6		10	16.3	odd	4
1024.4.a.n.1.5		10	16.13	even	4
1024.4.a.n.1.6		10	16.5	even	4
1024.4.b.j.513.5		10	8.5	even	2	inner
1024.4.b.j.513.6		10	1.1	even	1	trivial
1024.4.b.k.513.5		10	4.3	odd	2
1024.4.b.k.513.6		10	8.3	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.07024i q^{3} +11.6331i q^{5} -2.67171 q^{7} +25.8546 q^{9} +O(q^{10})\)	\(q+1.07024i q^{3} +11.6331i q^{5} -2.67171 q^{7} +25.8546 q^{9} +63.9525i q^{11} -50.0586i q^{13} -12.4503 q^{15} +72.4991 q^{17} -27.4961i q^{19} -2.85937i q^{21} +139.462 q^{23} -10.3299 q^{25} +56.5672i q^{27} +93.3995i q^{29} +188.682 q^{31} -68.4447 q^{33} -31.0803i q^{35} -118.886i q^{37} +53.5748 q^{39} -104.629 q^{41} +44.5275i q^{43} +300.770i q^{45} +488.151 q^{47} -335.862 q^{49} +77.5916i q^{51} -211.510i q^{53} -743.968 q^{55} +29.4275 q^{57} +402.624i q^{59} -322.538i q^{61} -69.0758 q^{63} +582.338 q^{65} +196.789i q^{67} +149.258i q^{69} -453.655 q^{71} +259.747 q^{73} -11.0555i q^{75} -170.862i q^{77} -323.190 q^{79} +637.533 q^{81} +797.471i q^{83} +843.392i q^{85} -99.9602 q^{87} +866.853 q^{89} +133.742i q^{91} +201.935i q^{93} +319.866 q^{95} -936.077 q^{97} +1653.47i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{7} - 54 q^{9}+O(q^{10}) \) 10 * q - 28 * q^7 - 54 * q^9	\( 10 q - 28 q^{7} - 54 q^{9} + 124 q^{15} + 4 q^{17} - 276 q^{23} - 50 q^{25} + 368 q^{31} - 4 q^{33} - 732 q^{39} + 944 q^{47} - 94 q^{49} - 1380 q^{55} - 108 q^{57} + 2628 q^{63} - 492 q^{65} - 3468 q^{71} + 296 q^{73} + 4416 q^{79} - 482 q^{81} - 6036 q^{87} - 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q - 28 * q^7 - 54 * q^9 + 124 * q^15 + 4 * q^17 - 276 * q^23 - 50 * q^25 + 368 * q^31 - 4 * q^33 - 732 * q^39 + 944 * q^47 - 94 * q^49 - 1380 * q^55 - 108 * q^57 + 2628 * q^63 - 492 * q^65 - 3468 * q^71 + 296 * q^73 + 4416 * q^79 - 482 * q^81 - 6036 * q^87 - 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more