LMFDB - Embedded newform 8464.2.a.ch.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.159078$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.159078 q^{3} +0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} +O(q^{10})$	$q+0.159078 q^{3} +0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} -2.40638 q^{11} -6.46974 q^{13} +0.00595930 q^{15} -1.41265 q^{17} -3.79359 q^{19} -0.505629 q^{21} -4.99860 q^{25} -0.950440 q^{27} -1.34251 q^{29} -5.26684 q^{31} -0.382801 q^{33} -0.119072 q^{35} -2.86429 q^{37} -1.02919 q^{39} +1.53077 q^{41} +3.15025 q^{43} -0.111437 q^{45} +7.00603 q^{47} +3.10291 q^{49} -0.224721 q^{51} -12.8957 q^{53} -0.0901467 q^{55} -0.603475 q^{57} +0.932170 q^{59} +4.33766 q^{61} +9.45509 q^{63} -0.242367 q^{65} +10.5734 q^{67} -12.6991 q^{71} -12.6489 q^{73} -0.795165 q^{75} +7.64869 q^{77} -7.50885 q^{79} +8.77289 q^{81} +12.5406 q^{83} -0.0529202 q^{85} -0.213563 q^{87} -7.79190 q^{89} +20.5641 q^{91} -0.837836 q^{93} -0.142114 q^{95} +2.28969 q^{97} +7.15824 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.159078	0.0918435	0.0459217	−	0.998945i	$-0.485378\pi$
$3$	0.159078	0.0918435	0.0459217	+	0.998945i	$0.485378\pi$
$4$	0	0
$5$	0.0374616	0.0167533	0.00837667	−	0.999965i	$-0.497334\pi$
$5$	0.0374616	0.0167533	0.00837667	+	0.999965i	$0.497334\pi$
$6$	0	0
$7$	−3.17851	−1.20136	−0.600682	−	0.799488i	$-0.705104\pi$
$7$	−3.17851	−1.20136	−0.600682	+	0.799488i	$0.705104\pi$
$8$	0	0
$9$	−2.97469	−0.991565
$10$	0	0
$11$	−2.40638	−0.725550	−0.362775	−	0.931877i	$-0.618171\pi$
$11$	−2.40638	−0.725550	−0.362775	+	0.931877i	$0.618171\pi$
$12$	0	0
$13$	−6.46974	−1.79438	−0.897191	−	0.441642i	$-0.854396\pi$
$13$	−6.46974	−1.79438	−0.897191	+	0.441642i	$0.854396\pi$
$14$	0	0
$15$	0.00595930	0.00153868
$16$	0	0
$17$	−1.41265	−0.342618	−0.171309	−	0.985217i	$-0.554800\pi$
$17$	−1.41265	−0.342618	−0.171309	+	0.985217i	$0.554800\pi$
$18$	0	0
$19$	−3.79359	−0.870310	−0.435155	−	0.900356i	$-0.643306\pi$
$19$	−3.79359	−0.870310	−0.435155	+	0.900356i	$0.643306\pi$
$20$	0	0
$21$	−0.505629	−0.110337
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.99860	−0.999719
$26$	0	0
$27$	−0.950440	−0.182912
$28$	0	0
$29$	−1.34251	−0.249298	−0.124649	−	0.992201i	$-0.539780\pi$
$29$	−1.34251	−0.249298	−0.124649	+	0.992201i	$0.539780\pi$
$30$	0	0
$31$	−5.26684	−0.945952	−0.472976	−	0.881075i	$-0.656820\pi$
$31$	−5.26684	−0.945952	−0.472976	+	0.881075i	$0.656820\pi$
$32$	0	0
$33$	−0.382801	−0.0666370
$34$	0	0
$35$	−0.119072	−0.0201268
$36$	0	0
$37$	−2.86429	−0.470886	−0.235443	−	0.971888i	$-0.575654\pi$
$37$	−2.86429	−0.470886	−0.235443	+	0.971888i	$0.575654\pi$
$38$	0	0
$39$	−1.02919	−0.164802
$40$	0	0
$41$	1.53077	0.239066	0.119533	−	0.992830i	$-0.461860\pi$
$41$	1.53077	0.239066	0.119533	+	0.992830i	$0.461860\pi$
$42$	0	0
$43$	3.15025	0.480408	0.240204	−	0.970722i	$-0.422786\pi$
$43$	3.15025	0.480408	0.240204	+	0.970722i	$0.422786\pi$
$44$	0	0
$45$	−0.111437	−0.0166120
$46$	0	0
$47$	7.00603	1.02193	0.510967	−	0.859600i	$-0.329287\pi$
$47$	7.00603	1.02193	0.510967	+	0.859600i	$0.329287\pi$
$48$	0	0
$49$	3.10291	0.443274
$50$	0	0
$51$	−0.224721	−0.0314673
$52$	0	0
$53$	−12.8957	−1.77136	−0.885681	−	0.464294i	$-0.846308\pi$
$53$	−12.8957	−1.77136	−0.885681	+	0.464294i	$0.846308\pi$
$54$	0	0
$55$	−0.0901467	−0.0121554
$56$	0	0
$57$	−0.603475	−0.0799323
$58$	0	0
$59$	0.932170	0.121358	0.0606791	−	0.998157i	$-0.480673\pi$
$59$	0.932170	0.121358	0.0606791	+	0.998157i	$0.480673\pi$
$60$	0	0
$61$	4.33766	0.555380	0.277690	−	0.960671i	$-0.410431\pi$
$61$	4.33766	0.555380	0.277690	+	0.960671i	$0.410431\pi$
$62$	0	0
$63$	9.45509	1.19123
$64$	0	0
$65$	−0.242367	−0.0300619
$66$	0	0
$67$	10.5734	1.29175	0.645874	−	0.763444i	$-0.276493\pi$
$67$	10.5734	1.29175	0.645874	+	0.763444i	$0.276493\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.6991	−1.50710	−0.753551	−	0.657390i	$-0.771660\pi$
$71$	−12.6991	−1.50710	−0.753551	+	0.657390i	$0.771660\pi$
$72$	0	0
$73$	−12.6489	−1.48044	−0.740218	−	0.672367i	$-0.765278\pi$
$73$	−12.6489	−1.48044	−0.740218	+	0.672367i	$0.765278\pi$
$74$	0	0
$75$	−0.795165	−0.0918177
$76$	0	0
$77$	7.64869	0.871649
$78$	0	0
$79$	−7.50885	−0.844812	−0.422406	−	0.906407i	$-0.638814\pi$
$79$	−7.50885	−0.844812	−0.422406	+	0.906407i	$0.638814\pi$
$80$	0	0
$81$	8.77289	0.974765
$82$	0	0
$83$	12.5406	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$83$	12.5406	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$84$	0	0
$85$	−0.0529202	−0.00574000
$86$	0	0
$87$	−0.213563	−0.0228964
$88$	0	0
$89$	−7.79190	−0.825940	−0.412970	−	0.910745i	$-0.635509\pi$
$89$	−7.79190	−0.825940	−0.412970	+	0.910745i	$0.635509\pi$
$90$	0	0
$91$	20.5641	2.15571
$92$	0	0
$93$	−0.837836	−0.0868795
$94$	0	0
$95$	−0.142114	−0.0145806
$96$	0	0
$97$	2.28969	0.232482	0.116241	−	0.993221i	$-0.462915\pi$
$97$	2.28969	0.232482	0.116241	+	0.993221i	$0.462915\pi$
$98$	0	0
$99$	7.15824	0.719430
$100$	0	0
$101$	−7.83402	−0.779514	−0.389757	−	0.920918i	$-0.627441\pi$
$101$	−7.83402	−0.779514	−0.389757	+	0.920918i	$0.627441\pi$
$102$	0	0
$103$	15.9171	1.56836	0.784178	−	0.620536i	$-0.213085\pi$
$103$	15.9171	1.56836	0.784178	+	0.620536i	$0.213085\pi$
$104$	0	0
$105$	−0.0189417	−0.00184852
$106$	0	0
$107$	−11.8774	−1.14823	−0.574117	−	0.818773i	$-0.694655\pi$
$107$	−11.8774	−1.14823	−0.574117	+	0.818773i	$0.694655\pi$
$108$	0	0
$109$	9.79620	0.938306	0.469153	−	0.883117i	$-0.344559\pi$
$109$	9.79620	0.938306	0.469153	+	0.883117i	$0.344559\pi$
$110$	0	0
$111$	−0.455644	−0.0432478
$112$	0	0
$113$	12.6578	1.19074	0.595371	−	0.803451i	$-0.297005\pi$
$113$	12.6578	1.19074	0.595371	+	0.803451i	$0.297005\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19.2455	1.77925
$118$	0	0
$119$	4.49013	0.411609
$120$	0	0
$121$	−5.20935	−0.473577
$122$	0	0
$123$	0.243511	0.0219567
$124$	0	0
$125$	−0.374563	−0.0335020
$126$	0	0
$127$	9.00093	0.798703	0.399352	−	0.916798i	$-0.369235\pi$
$127$	9.00093	0.798703	0.399352	+	0.916798i	$0.369235\pi$
$128$	0	0
$129$	0.501134	0.0441224
$130$	0	0
$131$	−17.0958	−1.49367	−0.746833	−	0.665012i	$-0.768426\pi$
$131$	−17.0958	−1.49367	−0.746833	+	0.665012i	$0.768426\pi$
$132$	0	0
$133$	12.0580	1.04556
$134$	0	0
$135$	−0.0356050	−0.00306439
$136$	0	0
$137$	−16.9732	−1.45012	−0.725061	−	0.688685i	$-0.758188\pi$
$137$	−16.9732	−1.45012	−0.725061	+	0.688685i	$0.758188\pi$
$138$	0	0
$139$	−9.48689	−0.804667	−0.402334	−	0.915493i	$-0.631801\pi$
$139$	−9.48689	−0.804667	−0.402334	+	0.915493i	$0.631801\pi$
$140$	0	0
$141$	1.11450	0.0938580
$142$	0	0
$143$	15.5686	1.30191
$144$	0	0
$145$	−0.0502926	−0.00417657
$146$	0	0
$147$	0.493604	0.0407118
$148$	0	0
$149$	−5.27184	−0.431886	−0.215943	−	0.976406i	$-0.569283\pi$
$149$	−5.27184	−0.431886	−0.215943	+	0.976406i	$0.569283\pi$
$150$	0	0
$151$	10.2191	0.831621	0.415810	−	0.909451i	$-0.363498\pi$
$151$	10.2191	0.831621	0.415810	+	0.909451i	$0.363498\pi$
$152$	0	0
$153$	4.20221	0.339728
$154$	0	0
$155$	−0.197304	−0.0158478
$156$	0	0
$157$	19.8369	1.58316	0.791580	−	0.611065i	$-0.209259\pi$
$157$	19.8369	1.58316	0.791580	+	0.611065i	$0.209259\pi$
$158$	0	0
$159$	−2.05142	−0.162688
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.2507	1.50783	0.753917	−	0.656969i	$-0.228162\pi$
$163$	19.2507	1.50783	0.753917	+	0.656969i	$0.228162\pi$
$164$	0	0
$165$	−0.0143403	−0.00111639
$166$	0	0
$167$	0.977670	0.0756544	0.0378272	−	0.999284i	$-0.487956\pi$
$167$	0.977670	0.0756544	0.0378272	+	0.999284i	$0.487956\pi$
$168$	0	0
$169$	28.8575	2.21981
$170$	0	0
$171$	11.2848	0.862968
$172$	0	0
$173$	−10.9967	−0.836062	−0.418031	−	0.908433i	$-0.637280\pi$
$173$	−10.9967	−0.836062	−0.418031	+	0.908433i	$0.637280\pi$
$174$	0	0
$175$	15.8881	1.20103
$176$	0	0
$177$	0.148287	0.0111460
$178$	0	0
$179$	−5.44344	−0.406862	−0.203431	−	0.979089i	$-0.565209\pi$
$179$	−5.44344	−0.406862	−0.203431	+	0.979089i	$0.565209\pi$
$180$	0	0
$181$	−15.7379	−1.16979	−0.584894	−	0.811110i	$-0.698864\pi$
$181$	−15.7379	−1.16979	−0.584894	+	0.811110i	$0.698864\pi$
$182$	0	0
$183$	0.690024	0.0510081
$184$	0	0
$185$	−0.107301	−0.00788892
$186$	0	0
$187$	3.39937	0.248587
$188$	0	0
$189$	3.02098	0.219744
$190$	0	0
$191$	0.930577	0.0673342	0.0336671	−	0.999433i	$-0.489281\pi$
$191$	0.930577	0.0673342	0.0336671	+	0.999433i	$0.489281\pi$
$192$	0	0
$193$	−15.2524	−1.09789	−0.548947	−	0.835857i	$-0.684971\pi$
$193$	−15.2524	−1.09789	−0.548947	+	0.835857i	$0.684971\pi$
$194$	0	0
$195$	−0.0385551	−0.00276099
$196$	0	0
$197$	−20.5486	−1.46402	−0.732012	−	0.681292i	$-0.761418\pi$
$197$	−20.5486	−1.46402	−0.732012	+	0.681292i	$0.761418\pi$
$198$	0	0
$199$	−17.7683	−1.25956	−0.629781	−	0.776773i	$-0.716855\pi$
$199$	−17.7683	−1.25956	−0.629781	+	0.776773i	$0.716855\pi$
$200$	0	0
$201$	1.68199	0.118639
$202$	0	0
$203$	4.26718	0.299497
$204$	0	0
$205$	0.0573451	0.00400516
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.12881	0.631453
$210$	0	0
$211$	5.40387	0.372018	0.186009	−	0.982548i	$-0.440445\pi$
$211$	5.40387	0.372018	0.186009	+	0.982548i	$0.440445\pi$
$212$	0	0
$213$	−2.02014	−0.138417
$214$	0	0
$215$	0.118013	0.00804844
$216$	0	0
$217$	16.7407	1.13643
$218$	0	0
$219$	−2.01215	−0.135968
$220$	0	0
$221$	9.13949	0.614789
$222$	0	0
$223$	−23.5755	−1.57873	−0.789367	−	0.613921i	$-0.789591\pi$
$223$	−23.5755	−1.57873	−0.789367	+	0.613921i	$0.789591\pi$
$224$	0	0
$225$	14.8693	0.991286
$226$	0	0
$227$	14.8000	0.982313	0.491157	−	0.871071i	$-0.336574\pi$
$227$	14.8000	0.982313	0.491157	+	0.871071i	$0.336574\pi$
$228$	0	0
$229$	−0.758365	−0.0501142	−0.0250571	−	0.999686i	$-0.507977\pi$
$229$	−0.758365	−0.0501142	−0.0250571	+	0.999686i	$0.507977\pi$
$230$	0	0
$231$	1.21673	0.0800553
$232$	0	0
$233$	18.6348	1.22081	0.610404	−	0.792091i	$-0.291007\pi$
$233$	18.6348	1.22081	0.610404	+	0.792091i	$0.291007\pi$
$234$	0	0
$235$	0.262457	0.0171208
$236$	0	0
$237$	−1.19449	−0.0775904
$238$	0	0
$239$	12.1829	0.788045	0.394023	−	0.919101i	$-0.371083\pi$
$239$	12.1829	0.788045	0.394023	+	0.919101i	$0.371083\pi$
$240$	0	0
$241$	9.42155	0.606896	0.303448	−	0.952848i	$-0.401862\pi$
$241$	9.42155	0.606896	0.303448	+	0.952848i	$0.401862\pi$
$242$	0	0
$243$	4.24689	0.272438
$244$	0	0
$245$	0.116240	0.00742631
$246$	0	0
$247$	24.5435	1.56167
$248$	0	0
$249$	1.99493	0.126424
$250$	0	0
$251$	28.9432	1.82688	0.913440	−	0.406973i	$-0.133416\pi$
$251$	28.9432	1.82688	0.913440	+	0.406973i	$0.133416\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.00841842	−0.000527182 0
$256$	0	0
$257$	−4.50409	−0.280957	−0.140479	−	0.990084i	$-0.544864\pi$
$257$	−4.50409	−0.280957	−0.140479	+	0.990084i	$0.544864\pi$
$258$	0	0
$259$	9.10417	0.565705
$260$	0	0
$261$	3.99356	0.247195
$262$	0	0
$263$	−4.56784	−0.281665	−0.140832	−	0.990033i	$-0.544978\pi$
$263$	−4.56784	−0.281665	−0.140832	+	0.990033i	$0.544978\pi$
$264$	0	0
$265$	−0.483094	−0.0296762
$266$	0	0
$267$	−1.23952	−0.0758572
$268$	0	0
$269$	−22.9431	−1.39887	−0.699433	−	0.714698i	$-0.746564\pi$
$269$	−22.9431	−1.39887	−0.699433	+	0.714698i	$0.746564\pi$
$270$	0	0
$271$	24.2580	1.47357	0.736783	−	0.676129i	$-0.236344\pi$
$271$	24.2580	1.47357	0.736783	+	0.676129i	$0.236344\pi$
$272$	0	0
$273$	3.27129	0.197987
$274$	0	0
$275$	12.0285	0.725346
$276$	0	0
$277$	−16.3549	−0.982668	−0.491334	−	0.870971i	$-0.663491\pi$
$277$	−16.3549	−0.982668	−0.491334	+	0.870971i	$0.663491\pi$
$278$	0	0
$279$	15.6672	0.937973
$280$	0	0
$281$	31.0273	1.85094	0.925468	−	0.378826i	$-0.123672\pi$
$281$	31.0273	1.85094	0.925468	+	0.378826i	$0.123672\pi$
$282$	0	0
$283$	7.83374	0.465668	0.232834	−	0.972517i	$-0.425200\pi$
$283$	7.83374	0.465668	0.232834	+	0.972517i	$0.425200\pi$
$284$	0	0
$285$	−0.0226071	−0.00133913
$286$	0	0
$287$	−4.86557	−0.287205
$288$	0	0
$289$	−15.0044	−0.882613
$290$	0	0
$291$	0.364238	0.0213520
$292$	0	0
$293$	−18.8084	−1.09880	−0.549399	−	0.835560i	$-0.685143\pi$
$293$	−18.8084	−1.09880	−0.549399	+	0.835560i	$0.685143\pi$
$294$	0	0
$295$	0.0349206	0.00203316
$296$	0	0
$297$	2.28712	0.132712
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−10.0131	−0.577145
$302$	0	0
$303$	−1.24622	−0.0715932
$304$	0	0
$305$	0.162496	0.00930447
$306$	0	0
$307$	−7.07318	−0.403688	−0.201844	−	0.979418i	$-0.564693\pi$
$307$	−7.07318	−0.403688	−0.201844	+	0.979418i	$0.564693\pi$
$308$	0	0
$309$	2.53205	0.144043
$310$	0	0
$311$	−34.5175	−1.95731	−0.978653	−	0.205519i	$-0.934112\pi$
$311$	−34.5175	−1.95731	−0.978653	+	0.205519i	$0.934112\pi$
$312$	0	0
$313$	−21.6164	−1.22183	−0.610916	−	0.791696i	$-0.709199\pi$
$313$	−21.6164	−1.22183	−0.610916	+	0.791696i	$0.709199\pi$
$314$	0	0
$315$	0.354203	0.0199571
$316$	0	0
$317$	5.48160	0.307878	0.153939	−	0.988080i	$-0.450804\pi$
$317$	5.48160	0.307878	0.153939	+	0.988080i	$0.450804\pi$
$318$	0	0
$319$	3.23059	0.180878
$320$	0	0
$321$	−1.88943	−0.105458
$322$	0	0
$323$	5.35902	0.298184
$324$	0	0
$325$	32.3396	1.79388
$326$	0	0
$327$	1.55836	0.0861772
$328$	0	0
$329$	−22.2687	−1.22771
$330$	0	0
$331$	5.18383	0.284929	0.142464	−	0.989800i	$-0.454497\pi$
$331$	5.18383	0.284929	0.142464	+	0.989800i	$0.454497\pi$
$332$	0	0
$333$	8.52039	0.466914
$334$	0	0
$335$	0.396097	0.0216411
$336$	0	0
$337$	−30.1142	−1.64042	−0.820212	−	0.572059i	$-0.806145\pi$
$337$	−30.1142	−1.64042	−0.820212	+	0.572059i	$0.806145\pi$
$338$	0	0
$339$	2.01357	0.109362
$340$	0	0
$341$	12.6740	0.686335
$342$	0	0
$343$	12.3869	0.668831
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.0197	−1.02103	−0.510516	−	0.859868i	$-0.670546\pi$
$347$	−19.0197	−1.02103	−0.510516	+	0.859868i	$0.670546\pi$
$348$	0	0
$349$	−10.1168	−0.541538	−0.270769	−	0.962644i	$-0.587278\pi$
$349$	−10.1168	−0.541538	−0.270769	+	0.962644i	$0.587278\pi$
$350$	0	0
$351$	6.14910	0.328215
$352$	0	0
$353$	5.48359	0.291862	0.145931	−	0.989295i	$-0.453382\pi$
$353$	5.48359	0.291862	0.145931	+	0.989295i	$0.453382\pi$
$354$	0	0
$355$	−0.475727	−0.0252490
$356$	0	0
$357$	0.714278	0.0378036
$358$	0	0
$359$	18.8674	0.995785	0.497893	−	0.867239i	$-0.334107\pi$
$359$	18.8674	0.995785	0.497893	+	0.867239i	$0.334107\pi$
$360$	0	0
$361$	−4.60866	−0.242561
$362$	0	0
$363$	−0.828691	−0.0434950
$364$	0	0
$365$	−0.473846	−0.0248022
$366$	0	0
$367$	−3.02519	−0.157914	−0.0789568	−	0.996878i	$-0.525159\pi$
$367$	−3.02519	−0.157914	−0.0789568	+	0.996878i	$0.525159\pi$
$368$	0	0
$369$	−4.55357	−0.237050
$370$	0	0
$371$	40.9891	2.12805
$372$	0	0
$373$	−16.8982	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$373$	−16.8982	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$374$	0	0
$375$	−0.0595846	−0.00307694
$376$	0	0
$377$	8.68569	0.447336
$378$	0	0
$379$	12.6158	0.648030	0.324015	−	0.946052i	$-0.394967\pi$
$379$	12.6158	0.648030	0.324015	+	0.946052i	$0.394967\pi$
$380$	0	0
$381$	1.43185	0.0733557
$382$	0	0
$383$	−15.8612	−0.810471	−0.405236	−	0.914212i	$-0.632811\pi$
$383$	−15.8612	−0.810471	−0.405236	+	0.914212i	$0.632811\pi$
$384$	0	0
$385$	0.286532	0.0146030
$386$	0	0
$387$	−9.37102	−0.476356
$388$	0	0
$389$	14.4267	0.731464	0.365732	−	0.930720i	$-0.380819\pi$
$389$	14.4267	0.731464	0.365732	+	0.930720i	$0.380819\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.71956	−0.137183
$394$	0	0
$395$	−0.281293	−0.0141534
$396$	0	0
$397$	10.2008	0.511963	0.255982	−	0.966682i	$-0.417601\pi$
$397$	10.2008	0.511963	0.255982	+	0.966682i	$0.417601\pi$
$398$	0	0
$399$	1.91815	0.0960277
$400$	0	0
$401$	−24.5279	−1.22486	−0.612432	−	0.790524i	$-0.709808\pi$
$401$	−24.5279	−1.22486	−0.612432	+	0.790524i	$0.709808\pi$
$402$	0	0
$403$	34.0751	1.69740
$404$	0	0
$405$	0.328646	0.0163306
$406$	0	0
$407$	6.89256	0.341652
$408$	0	0
$409$	14.4370	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$409$	14.4370	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$410$	0	0
$411$	−2.70006	−0.133184
$412$	0	0
$413$	−2.96291	−0.145795
$414$	0	0
$415$	0.469791	0.0230611
$416$	0	0
$417$	−1.50915	−0.0739034
$418$	0	0
$419$	9.51133	0.464659	0.232329	−	0.972637i	$-0.425365\pi$
$419$	9.51133	0.464659	0.232329	+	0.972637i	$0.425365\pi$
$420$	0	0
$421$	−15.8981	−0.774824	−0.387412	−	0.921907i	$-0.626631\pi$
$421$	−15.8981	−0.774824	−0.387412	+	0.921907i	$0.626631\pi$
$422$	0	0
$423$	−20.8408	−1.01331
$424$	0	0
$425$	7.06128	0.342522
$426$	0	0
$427$	−13.7873	−0.667214
$428$	0	0
$429$	2.47662	0.119572
$430$	0	0
$431$	−23.5296	−1.13338	−0.566690	−	0.823931i	$-0.691776\pi$
$431$	−23.5296	−1.13338	−0.566690	+	0.823931i	$0.691776\pi$
$432$	0	0
$433$	−20.8487	−1.00192	−0.500962	−	0.865469i	$-0.667020\pi$
$433$	−20.8487	−1.00192	−0.500962	+	0.865469i	$0.667020\pi$
$434$	0	0
$435$	−0.00800042	−0.000383591 0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	3.78161	0.180486	0.0902432	−	0.995920i	$-0.471236\pi$
$439$	3.78161	0.180486	0.0902432	+	0.995920i	$0.471236\pi$
$440$	0	0
$441$	−9.23022	−0.439534
$442$	0	0
$443$	−34.3887	−1.63386	−0.816928	−	0.576740i	$-0.804325\pi$
$443$	−34.3887	−1.63386	−0.816928	+	0.576740i	$0.804325\pi$
$444$	0	0
$445$	−0.291897	−0.0138372
$446$	0	0
$447$	−0.838632	−0.0396659
$448$	0	0
$449$	13.2765	0.626557	0.313279	−	0.949661i	$-0.398573\pi$
$449$	13.2765	0.626557	0.313279	+	0.949661i	$0.398573\pi$
$450$	0	0
$451$	−3.68361	−0.173454
$452$	0	0
$453$	1.62563	0.0763789
$454$	0	0
$455$	0.770365	0.0361153
$456$	0	0
$457$	12.1984	0.570615	0.285307	−	0.958436i	$-0.407904\pi$
$457$	12.1984	0.570615	0.285307	+	0.958436i	$0.407904\pi$
$458$	0	0
$459$	1.34264	0.0626691
$460$	0	0
$461$	14.4173	0.671481	0.335741	−	0.941955i	$-0.391013\pi$
$461$	14.4173	0.671481	0.335741	+	0.941955i	$0.391013\pi$
$462$	0	0
$463$	−14.6488	−0.680789	−0.340395	−	0.940283i	$-0.610561\pi$
$463$	−14.6488	−0.680789	−0.340395	+	0.940283i	$0.610561\pi$
$464$	0	0
$465$	−0.0313867	−0.00145552
$466$	0	0
$467$	−42.1973	−1.95266	−0.976328	−	0.216295i	$-0.930603\pi$
$467$	−42.1973	−1.95266	−0.976328	+	0.216295i	$0.930603\pi$
$468$	0	0
$469$	−33.6077	−1.55186
$470$	0	0
$471$	3.15561	0.145403
$472$	0	0
$473$	−7.58068	−0.348560
$474$	0	0
$475$	18.9626	0.870065
$476$	0	0
$477$	38.3608	1.75642
$478$	0	0
$479$	−9.64609	−0.440741	−0.220371	−	0.975416i	$-0.570727\pi$
$479$	−9.64609	−0.440741	−0.220371	+	0.975416i	$0.570727\pi$
$480$	0	0
$481$	18.5312	0.844950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.0857753	0.00389486
$486$	0	0
$487$	−19.0439	−0.862961	−0.431480	−	0.902122i	$-0.642009\pi$
$487$	−19.0439	−0.862961	−0.431480	+	0.902122i	$0.642009\pi$
$488$	0	0
$489$	3.06236	0.138485
$490$	0	0
$491$	8.20761	0.370404	0.185202	−	0.982700i	$-0.440706\pi$
$491$	8.20761	0.370404	0.185202	+	0.982700i	$0.440706\pi$
$492$	0	0
$493$	1.89650	0.0854140
$494$	0	0
$495$	0.268159	0.0120528
$496$	0	0
$497$	40.3641	1.81058
$498$	0	0
$499$	−25.5214	−1.14250	−0.571248	−	0.820777i	$-0.693541\pi$
$499$	−25.5214	−1.14250	−0.571248	+	0.820777i	$0.693541\pi$
$500$	0	0
$501$	0.155525	0.00694836
$502$	0	0
$503$	−27.9623	−1.24678	−0.623388	−	0.781912i	$-0.714244\pi$
$503$	−27.9623	−1.24678	−0.623388	+	0.781912i	$0.714244\pi$
$504$	0	0
$505$	−0.293475	−0.0130595
$506$	0	0
$507$	4.59058	0.203875
$508$	0	0
$509$	−16.2940	−0.722221	−0.361110	−	0.932523i	$-0.617602\pi$
$509$	−16.2940	−0.722221	−0.361110	+	0.932523i	$0.617602\pi$
$510$	0	0
$511$	40.2045	1.77854
$512$	0	0
$513$	3.60558	0.159190
$514$	0	0
$515$	0.596279	0.0262752
$516$	0	0
$517$	−16.8591	−0.741464
$518$	0	0
$519$	−1.74932	−0.0767868
$520$	0	0
$521$	26.1985	1.14778	0.573888	−	0.818934i	$-0.305434\pi$
$521$	26.1985	1.14778	0.573888	+	0.818934i	$0.305434\pi$
$522$	0	0
$523$	30.8177	1.34756	0.673782	−	0.738930i	$-0.264669\pi$
$523$	30.8177	1.34756	0.673782	+	0.738930i	$0.264669\pi$
$524$	0	0
$525$	2.52744	0.110306
$526$	0	0
$527$	7.44021	0.324101
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−2.77292	−0.120335
$532$	0	0
$533$	−9.90368	−0.428976
$534$	0	0
$535$	−0.444948	−0.0192368
$536$	0	0
$537$	−0.865929	−0.0373676
$538$	0	0
$539$	−7.46678	−0.321617
$540$	0	0
$541$	9.87335	0.424489	0.212244	−	0.977217i	$-0.431923\pi$
$541$	9.87335	0.424489	0.212244	+	0.977217i	$0.431923\pi$
$542$	0	0
$543$	−2.50354	−0.107437
$544$	0	0
$545$	0.366981	0.0157197
$546$	0	0
$547$	−17.2917	−0.739339	−0.369669	−	0.929163i	$-0.620529\pi$
$547$	−17.2917	−0.739339	−0.369669	+	0.929163i	$0.620529\pi$
$548$	0	0
$549$	−12.9032	−0.550696
$550$	0	0
$551$	5.09293	0.216966
$552$	0	0
$553$	23.8669	1.01493
$554$	0	0
$555$	−0.0170692	−0.000724545 0
$556$	0	0
$557$	23.5446	0.997618	0.498809	−	0.866712i	$-0.333771\pi$
$557$	23.5446	0.997618	0.498809	+	0.866712i	$0.333771\pi$
$558$	0	0
$559$	−20.3813	−0.862036
$560$	0	0
$561$	0.540764	0.0228311
$562$	0	0
$563$	22.0864	0.930830	0.465415	−	0.885093i	$-0.345905\pi$
$563$	22.0864	0.930830	0.465415	+	0.885093i	$0.345905\pi$
$564$	0	0
$565$	0.474180	0.0199489
$566$	0	0
$567$	−27.8847	−1.17105
$568$	0	0
$569$	30.6044	1.28300	0.641501	−	0.767123i	$-0.278312\pi$
$569$	30.6044	1.28300	0.641501	+	0.767123i	$0.278312\pi$
$570$	0	0
$571$	37.3214	1.56185	0.780926	−	0.624624i	$-0.214748\pi$
$571$	37.3214	1.56185	0.780926	+	0.624624i	$0.214748\pi$
$572$	0	0
$573$	0.148034	0.00618420
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.1654	0.506454	0.253227	−	0.967407i	$-0.418508\pi$
$577$	12.1654	0.506454	0.253227	+	0.967407i	$0.418508\pi$
$578$	0	0
$579$	−2.42632	−0.100834
$580$	0	0
$581$	−39.8604	−1.65369
$582$	0	0
$583$	31.0319	1.28521
$584$	0	0
$585$	0.720967	0.0298083
$586$	0	0
$587$	41.0071	1.69254	0.846272	−	0.532752i	$-0.178842\pi$
$587$	41.0071	1.69254	0.846272	+	0.532752i	$0.178842\pi$
$588$	0	0
$589$	19.9802	0.823271
$590$	0	0
$591$	−3.26881	−0.134461
$592$	0	0
$593$	−28.5318	−1.17166	−0.585831	−	0.810433i	$-0.699232\pi$
$593$	−28.5318	−1.17166	−0.585831	+	0.810433i	$0.699232\pi$
$594$	0	0
$595$	0.168207	0.00689583
$596$	0	0
$597$	−2.82654	−0.115682
$598$	0	0
$599$	−12.7872	−0.522470	−0.261235	−	0.965275i	$-0.584130\pi$
$599$	−12.7872	−0.522470	−0.261235	+	0.965275i	$0.584130\pi$
$600$	0	0
$601$	10.6783	0.435575	0.217788	−	0.975996i	$-0.430116\pi$
$601$	10.6783	0.435575	0.217788	+	0.975996i	$0.430116\pi$
$602$	0	0
$603$	−31.4527	−1.28085
$604$	0	0
$605$	−0.195151	−0.00793400
$606$	0	0
$607$	−36.4595	−1.47985	−0.739923	−	0.672691i	$-0.765138\pi$
$607$	−36.4595	−1.47985	−0.739923	+	0.672691i	$0.765138\pi$
$608$	0	0
$609$	0.678812	0.0275069
$610$	0	0
$611$	−45.3272	−1.83374
$612$	0	0
$613$	−19.3819	−0.782826	−0.391413	−	0.920215i	$-0.628014\pi$
$613$	−19.3819	−0.782826	−0.391413	+	0.920215i	$0.628014\pi$
$614$	0	0
$615$	0.00912232	0.000367847 0
$616$	0	0
$617$	17.7144	0.713157	0.356578	−	0.934265i	$-0.383943\pi$
$617$	17.7144	0.713157	0.356578	+	0.934265i	$0.383943\pi$
$618$	0	0
$619$	−32.5171	−1.30697	−0.653487	−	0.756938i	$-0.726694\pi$
$619$	−32.5171	−1.30697	−0.653487	+	0.756938i	$0.726694\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.7666	0.992254
$624$	0	0
$625$	24.9790	0.999158
$626$	0	0
$627$	1.45219	0.0579949
$628$	0	0
$629$	4.04624	0.161334
$630$	0	0
$631$	2.62778	0.104610	0.0523052	−	0.998631i	$-0.483343\pi$
$631$	2.62778	0.104610	0.0523052	+	0.998631i	$0.483343\pi$
$632$	0	0
$633$	0.859634	0.0341674
$634$	0	0
$635$	0.337189	0.0133809
$636$	0	0
$637$	−20.0750	−0.795402
$638$	0	0
$639$	37.7758	1.49439
$640$	0	0
$641$	−30.9101	−1.22088	−0.610438	−	0.792064i	$-0.709007\pi$
$641$	−30.9101	−1.22088	−0.610438	+	0.792064i	$0.709007\pi$
$642$	0	0
$643$	−12.9499	−0.510696	−0.255348	−	0.966849i	$-0.582190\pi$
$643$	−12.9499	−0.510696	−0.255348	+	0.966849i	$0.582190\pi$
$644$	0	0
$645$	0.0187733	0.000739197 0
$646$	0	0
$647$	−24.9647	−0.981465	−0.490732	−	0.871310i	$-0.663271\pi$
$647$	−24.9647	−0.981465	−0.490732	+	0.871310i	$0.663271\pi$
$648$	0	0
$649$	−2.24315	−0.0880515
$650$	0	0
$651$	2.66307	0.104374
$652$	0	0
$653$	31.0751	1.21606	0.608031	−	0.793913i	$-0.291960\pi$
$653$	31.0751	1.21606	0.608031	+	0.793913i	$0.291960\pi$
$654$	0	0
$655$	−0.640435	−0.0250239
$656$	0	0
$657$	37.6265	1.46795
$658$	0	0
$659$	49.7960	1.93978	0.969888	−	0.243551i	$-0.0783124\pi$
$659$	49.7960	1.93978	0.969888	+	0.243551i	$0.0783124\pi$
$660$	0	0
$661$	−8.71606	−0.339015	−0.169508	−	0.985529i	$-0.554218\pi$
$661$	−8.71606	−0.339015	−0.169508	+	0.985529i	$0.554218\pi$
$662$	0	0
$663$	1.45389	0.0564643
$664$	0	0
$665$	0.451711	0.0175166
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−3.75034	−0.144996
$670$	0	0
$671$	−10.4380	−0.402956
$672$	0	0
$673$	−8.83627	−0.340613	−0.170307	−	0.985391i	$-0.554476\pi$
$673$	−8.83627	−0.340613	−0.170307	+	0.985391i	$0.554476\pi$
$674$	0	0
$675$	4.75087	0.182861
$676$	0	0
$677$	8.47603	0.325760	0.162880	−	0.986646i	$-0.447922\pi$
$677$	8.47603	0.325760	0.162880	+	0.986646i	$0.447922\pi$
$678$	0	0
$679$	−7.27779	−0.279296
$680$	0	0
$681$	2.35435	0.0902190
$682$	0	0
$683$	−7.90172	−0.302351	−0.151176	−	0.988507i	$-0.548306\pi$
$683$	−7.90172	−0.302351	−0.151176	+	0.988507i	$0.548306\pi$
$684$	0	0
$685$	−0.635844	−0.0242944
$686$	0	0
$687$	−0.120639	−0.00460266
$688$	0	0
$689$	83.4319	3.17850
$690$	0	0
$691$	0.908007	0.0345422	0.0172711	−	0.999851i	$-0.494502\pi$
$691$	0.908007	0.0345422	0.0172711	+	0.999851i	$0.494502\pi$
$692$	0	0
$693$	−22.7525	−0.864297
$694$	0	0
$695$	−0.355394	−0.0134809
$696$	0	0
$697$	−2.16245	−0.0819085
$698$	0	0
$699$	2.96438	0.112123
$700$	0	0
$701$	42.5456	1.60693	0.803463	−	0.595354i	$-0.202988\pi$
$701$	42.5456	1.60693	0.803463	+	0.595354i	$0.202988\pi$
$702$	0	0
$703$	10.8659	0.409817
$704$	0	0
$705$	0.0417510	0.00157243
$706$	0	0
$707$	24.9005	0.936479
$708$	0	0
$709$	−47.9843	−1.80209	−0.901045	−	0.433726i	$-0.857198\pi$
$709$	−47.9843	−1.80209	−0.901045	+	0.433726i	$0.857198\pi$
$710$	0	0
$711$	22.3365	0.837685
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0.583226	0.0218114
$716$	0	0
$717$	1.93802	0.0723768
$718$	0	0
$719$	30.6562	1.14328	0.571641	−	0.820504i	$-0.306307\pi$
$719$	30.6562	1.14328	0.571641	+	0.820504i	$0.306307\pi$
$720$	0	0
$721$	−50.5926	−1.88417
$722$	0	0
$723$	1.49876	0.0557394
$724$	0	0
$725$	6.71067	0.249228
$726$	0	0
$727$	30.5225	1.13202	0.566008	−	0.824400i	$-0.308487\pi$
$727$	30.5225	1.13202	0.566008	+	0.824400i	$0.308487\pi$
$728$	0	0
$729$	−25.6431	−0.949744
$730$	0	0
$731$	−4.45020	−0.164597
$732$	0	0
$733$	32.2009	1.18937	0.594684	−	0.803959i	$-0.297277\pi$
$733$	32.2009	1.18937	0.594684	+	0.803959i	$0.297277\pi$
$734$	0	0
$735$	0.0184912	0.000682058 0
$736$	0	0
$737$	−25.4436	−0.937228
$738$	0	0
$739$	−11.9938	−0.441200	−0.220600	−	0.975364i	$-0.570801\pi$
$739$	−11.9938	−0.441200	−0.220600	+	0.975364i	$0.570801\pi$
$740$	0	0
$741$	3.90433	0.143429
$742$	0	0
$743$	1.81638	0.0666366	0.0333183	−	0.999445i	$-0.489392\pi$
$743$	1.81638	0.0666366	0.0333183	+	0.999445i	$0.489392\pi$
$744$	0	0
$745$	−0.197492	−0.00723553
$746$	0	0
$747$	−37.3045	−1.36490
$748$	0	0
$749$	37.7525	1.37945
$750$	0	0
$751$	−18.4934	−0.674833	−0.337416	−	0.941356i	$-0.609553\pi$
$751$	−18.4934	−0.674833	−0.337416	+	0.941356i	$0.609553\pi$
$752$	0	0
$753$	4.60422	0.167787
$754$	0	0
$755$	0.382825	0.0139324
$756$	0	0
$757$	28.0298	1.01876	0.509380	−	0.860542i	$-0.329875\pi$
$757$	28.0298	1.01876	0.509380	+	0.860542i	$0.329875\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17.9096	−0.649221	−0.324611	−	0.945848i	$-0.605233\pi$
$761$	−17.9096	−0.649221	−0.324611	+	0.945848i	$0.605233\pi$
$762$	0	0
$763$	−31.1373	−1.12725
$764$	0	0
$765$	0.157421	0.00569158
$766$	0	0
$767$	−6.03090	−0.217763
$768$	0	0
$769$	6.23294	0.224766	0.112383	−	0.993665i	$-0.464152\pi$
$769$	6.23294	0.224766	0.112383	+	0.993665i	$0.464152\pi$
$770$	0	0
$771$	−0.716499	−0.0258041
$772$	0	0
$773$	5.85310	0.210522	0.105261	−	0.994445i	$-0.466432\pi$
$773$	5.85310	0.210522	0.105261	+	0.994445i	$0.466432\pi$
$774$	0	0
$775$	26.3268	0.945686
$776$	0	0
$777$	1.44827	0.0519564
$778$	0	0
$779$	−5.80712	−0.208062
$780$	0	0
$781$	30.5587	1.09348
$782$	0	0
$783$	1.27597	0.0455996
$784$	0	0
$785$	0.743124	0.0265232
$786$	0	0
$787$	−17.5673	−0.626207	−0.313104	−	0.949719i	$-0.601369\pi$
$787$	−17.5673	−0.626207	−0.313104	+	0.949719i	$0.601369\pi$
$788$	0	0
$789$	−0.726640	−0.0258691
$790$	0	0
$791$	−40.2328	−1.43051
$792$	0	0
$793$	−28.0635	−0.996565
$794$	0	0
$795$	−0.0768494	−0.00272557
$796$	0	0
$797$	−18.6553	−0.660803	−0.330402	−	0.943840i	$-0.607184\pi$
$797$	−18.6553	−0.660803	−0.330402	+	0.943840i	$0.607184\pi$
$798$	0	0
$799$	−9.89708	−0.350133
$800$	0	0
$801$	23.1785	0.818973
$802$	0	0
$803$	30.4379	1.07413
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.64974	−0.128477
$808$	0	0
$809$	3.09727	0.108894	0.0544471	−	0.998517i	$-0.482660\pi$
$809$	3.09727	0.108894	0.0544471	+	0.998517i	$0.482660\pi$
$810$	0	0
$811$	−40.3170	−1.41572	−0.707860	−	0.706353i	$-0.750339\pi$
$811$	−40.3170	−1.41572	−0.707860	+	0.706353i	$0.750339\pi$
$812$	0	0
$813$	3.85890	0.135337
$814$	0	0
$815$	0.721164	0.0252613
$816$	0	0
$817$	−11.9508	−0.418104
$818$	0	0
$819$	−61.1720	−2.13752
$820$	0	0
$821$	8.71210	0.304054	0.152027	−	0.988376i	$-0.451420\pi$
$821$	8.71210	0.304054	0.152027	+	0.988376i	$0.451420\pi$
$822$	0	0
$823$	15.5972	0.543684	0.271842	−	0.962342i	$-0.412367\pi$
$823$	15.5972	0.543684	0.271842	+	0.962342i	$0.412367\pi$
$824$	0	0
$825$	1.91347	0.0666183
$826$	0	0
$827$	−2.31512	−0.0805045	−0.0402522	−	0.999190i	$-0.512816\pi$
$827$	−2.31512	−0.0805045	−0.0402522	+	0.999190i	$0.512816\pi$
$828$	0	0
$829$	8.11849	0.281967	0.140983	−	0.990012i	$-0.454974\pi$
$829$	8.11849	0.281967	0.140983	+	0.990012i	$0.454974\pi$
$830$	0	0
$831$	−2.60169	−0.0902517
$832$	0	0
$833$	−4.38334	−0.151874
$834$	0	0
$835$	0.0366251	0.00126746
$836$	0	0
$837$	5.00581	0.173026
$838$	0	0
$839$	−21.5246	−0.743111	−0.371555	−	0.928411i	$-0.621175\pi$
$839$	−21.5246	−0.743111	−0.371555	+	0.928411i	$0.621175\pi$
$840$	0	0
$841$	−27.1977	−0.937851
$842$	0	0
$843$	4.93575	0.169996
$844$	0	0
$845$	1.08105	0.0371892
$846$	0	0
$847$	16.5580	0.568938
$848$	0	0
$849$	1.24617	0.0427685
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−16.7675	−0.574109	−0.287054	−	0.957914i	$-0.592676\pi$
$853$	−16.7675	−0.574109	−0.287054	+	0.957914i	$0.592676\pi$
$854$	0	0
$855$	0.422746	0.0144576
$856$	0	0
$857$	23.8287	0.813972	0.406986	−	0.913434i	$-0.366580\pi$
$857$	23.8287	0.813972	0.406986	+	0.913434i	$0.366580\pi$
$858$	0	0
$859$	53.5155	1.82592	0.912962	−	0.408044i	$-0.133789\pi$
$859$	53.5155	1.82592	0.912962	+	0.408044i	$0.133789\pi$
$860$	0	0
$861$	−0.774003	−0.0263779
$862$	0	0
$863$	−51.5565	−1.75500	−0.877501	−	0.479574i	$-0.840791\pi$
$863$	−51.5565	−1.75500	−0.877501	+	0.479574i	$0.840791\pi$
$864$	0	0
$865$	−0.411953	−0.0140068
$866$	0	0
$867$	−2.38687	−0.0810622
$868$	0	0
$869$	18.0691	0.612953
$870$	0	0
$871$	−68.4073	−2.31789
$872$	0	0
$873$	−6.81112	−0.230521
$874$	0	0
$875$	1.19055	0.0402480
$876$	0	0
$877$	9.28442	0.313513	0.156756	−	0.987637i	$-0.449896\pi$
$877$	9.28442	0.313513	0.156756	+	0.987637i	$0.449896\pi$
$878$	0	0
$879$	−2.99199	−0.100917
$880$	0	0
$881$	41.5919	1.40127	0.700634	−	0.713521i	$-0.252901\pi$
$881$	41.5919	1.40127	0.700634	+	0.713521i	$0.252901\pi$
$882$	0	0
$883$	29.1045	0.979445	0.489723	−	0.871878i	$-0.337098\pi$
$883$	29.1045	0.979445	0.489723	+	0.871878i	$0.337098\pi$
$884$	0	0
$885$	0.00555508	0.000186732 0
$886$	0	0
$887$	−13.0977	−0.439779	−0.219889	−	0.975525i	$-0.570570\pi$
$887$	−13.0977	−0.439779	−0.219889	+	0.975525i	$0.570570\pi$
$888$	0	0
$889$	−28.6095	−0.959532
$890$	0	0
$891$	−21.1109	−0.707241
$892$	0	0
$893$	−26.5780	−0.889399
$894$	0	0
$895$	−0.203920	−0.00681629
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.07078	0.235824
$900$	0	0
$901$	18.2172	0.606901
$902$	0	0
$903$	−1.59286	−0.0530070
$904$	0	0
$905$	−0.589566	−0.0195978
$906$	0	0
$907$	41.4787	1.37728	0.688639	−	0.725104i	$-0.258209\pi$
$907$	41.4787	1.37728	0.688639	+	0.725104i	$0.258209\pi$
$908$	0	0
$909$	23.3038	0.772938
$910$	0	0
$911$	−28.8411	−0.955549	−0.477775	−	0.878482i	$-0.658556\pi$
$911$	−28.8411	−0.955549	−0.477775	+	0.878482i	$0.658556\pi$
$912$	0	0
$913$	−30.1774	−0.998727
$914$	0	0
$915$	0.0258494	0.000854555 0
$916$	0	0
$917$	54.3391	1.79443
$918$	0	0
$919$	−53.2249	−1.75573	−0.877864	−	0.478910i	$-0.841032\pi$
$919$	−53.2249	−1.75573	−0.877864	+	0.478910i	$0.841032\pi$
$920$	0	0
$921$	−1.12518	−0.0370761
$922$	0	0
$923$	82.1596	2.70432
$924$	0	0
$925$	14.3174	0.470754
$926$	0	0
$927$	−47.3484	−1.55513
$928$	0	0
$929$	1.06999	0.0351052	0.0175526	−	0.999846i	$-0.494413\pi$
$929$	1.06999	0.0351052	0.0175526	+	0.999846i	$0.494413\pi$
$930$	0	0
$931$	−11.7712	−0.385785
$932$	0	0
$933$	−5.49096	−0.179766
$934$	0	0
$935$	0.127346	0.00416466
$936$	0	0
$937$	−26.3024	−0.859261	−0.429631	−	0.903005i	$-0.641356\pi$
$937$	−26.3024	−0.859261	−0.429631	+	0.903005i	$0.641356\pi$
$938$	0	0
$939$	−3.43868	−0.112217
$940$	0	0
$941$	−0.382060	−0.0124548	−0.00622740	−	0.999981i	$-0.501982\pi$
$941$	−0.382060	−0.0124548	−0.00622740	+	0.999981i	$0.501982\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0.113171	0.00368145
$946$	0	0
$947$	−9.68922	−0.314857	−0.157429	−	0.987530i	$-0.550320\pi$
$947$	−9.68922	−0.314857	−0.157429	+	0.987530i	$0.550320\pi$
$948$	0	0
$949$	81.8348	2.65647
$950$	0	0
$951$	0.872000	0.0282765
$952$	0	0
$953$	−10.1573	−0.329027	−0.164513	−	0.986375i	$-0.552605\pi$
$953$	−10.1573	−0.329027	−0.164513	+	0.986375i	$0.552605\pi$
$954$	0	0
$955$	0.0348609	0.00112807
$956$	0	0
$957$	0.513914	0.0166125
$958$	0	0
$959$	53.9496	1.74212
$960$	0	0
$961$	−3.26043	−0.105175
$962$	0	0
$963$	35.3317	1.13855
$964$	0	0
$965$	−0.571381	−0.0183934
$966$	0	0
$967$	−15.2492	−0.490382	−0.245191	−	0.969475i	$-0.578851\pi$
$967$	−15.2492	−0.490382	−0.245191	+	0.969475i	$0.578851\pi$
$968$	0	0
$969$	0.852501	0.0273863
$970$	0	0
$971$	38.9528	1.25006	0.625028	−	0.780602i	$-0.285088\pi$
$971$	38.9528	1.25006	0.625028	+	0.780602i	$0.285088\pi$
$972$	0	0
$973$	30.1542	0.966698
$974$	0	0
$975$	5.14451	0.164756
$976$	0	0
$977$	21.8477	0.698971	0.349486	−	0.936942i	$-0.386356\pi$
$977$	21.8477	0.698971	0.349486	+	0.936942i	$0.386356\pi$
$978$	0	0
$979$	18.7503	0.599261
$980$	0	0
$981$	−29.1407	−0.930391
$982$	0	0
$983$	−54.5463	−1.73976	−0.869878	−	0.493267i	$-0.835803\pi$
$983$	−54.5463	−1.73976	−0.869878	+	0.493267i	$0.835803\pi$
$984$	0	0
$985$	−0.769782	−0.0245273
$986$	0	0
$987$	−3.54245	−0.112758
$988$	0	0
$989$	0	0
$990$	0	0
$991$	33.0686	1.05046	0.525230	−	0.850961i	$-0.323979\pi$
$991$	33.0686	1.05046	0.525230	+	0.850961i	$0.323979\pi$
$992$	0	0
$993$	0.824631	0.0261689
$994$	0	0
$995$	−0.665629	−0.0211019
$996$	0	0
$997$	−10.5509	−0.334150	−0.167075	−	0.985944i	$-0.553432\pi$
$997$	−10.5509	−0.334150	−0.167075	+	0.985944i	$0.553432\pi$
$998$	0	0
$999$	2.72233	0.0861309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.8		15
4.3	odd	2		4232.2.a.ba.1.8		15
23.5	odd	22		368.2.m.e.209.2		30
23.14	odd	22		368.2.m.e.81.2		30
23.22	odd	2		8464.2.a.cg.1.8		15
92.51	even	22		184.2.i.b.25.2	✓	30
92.83	even	22		184.2.i.b.81.2	yes	30
92.91	even	2		4232.2.a.bb.1.8		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.25.2	✓	30	92.51	even	22
184.2.i.b.81.2	yes	30	92.83	even	22
368.2.m.e.81.2		30	23.14	odd	22
368.2.m.e.209.2		30	23.5	odd	22
4232.2.a.ba.1.8		15	4.3	odd	2
4232.2.a.bb.1.8		15	92.91	even	2
8464.2.a.cg.1.8		15	23.22	odd	2
8464.2.a.ch.1.8		15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.159078 q^{3} +0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} +O(q^{10})\)	\(q+0.159078 q^{3} +0.0374616 q^{5} -3.17851 q^{7} -2.97469 q^{9} -2.40638 q^{11} -6.46974 q^{13} +0.00595930 q^{15} -1.41265 q^{17} -3.79359 q^{19} -0.505629 q^{21} -4.99860 q^{25} -0.950440 q^{27} -1.34251 q^{29} -5.26684 q^{31} -0.382801 q^{33} -0.119072 q^{35} -2.86429 q^{37} -1.02919 q^{39} +1.53077 q^{41} +3.15025 q^{43} -0.111437 q^{45} +7.00603 q^{47} +3.10291 q^{49} -0.224721 q^{51} -12.8957 q^{53} -0.0901467 q^{55} -0.603475 q^{57} +0.932170 q^{59} +4.33766 q^{61} +9.45509 q^{63} -0.242367 q^{65} +10.5734 q^{67} -12.6991 q^{71} -12.6489 q^{73} -0.795165 q^{75} +7.64869 q^{77} -7.50885 q^{79} +8.77289 q^{81} +12.5406 q^{83} -0.0529202 q^{85} -0.213563 q^{87} -7.79190 q^{89} +20.5641 q^{91} -0.837836 q^{93} -0.142114 q^{95} +2.28969 q^{97} +7.15824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45} + 36 q^{47} + 25 q^{49} + 35 q^{51} + 5 q^{53} + 10 q^{55} - 23 q^{57} + 22 q^{59} - 12 q^{61} + 35 q^{63} + 26 q^{65} + 58 q^{67} - 2 q^{71} + 5 q^{73} + 17 q^{75} + 26 q^{77} + 26 q^{79} - 21 q^{81} + 68 q^{83} - 72 q^{85} - 19 q^{87} + 6 q^{89} + 71 q^{91} - 55 q^{93} + 12 q^{95} - 40 q^{97} + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more