LMFDB - Embedded newform 6032.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
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} - 2x^{9} - 10x^{8} + 21x^{7} + 28x^{6} - 67x^{5} - 20x^{4} + 76x^{3} - 8x^{2} - 26x + 8 $ x^10 - 2x^9 - 10x^8 + 21x^7 + 28x^6 - 67x^5 - 20x^4 + 76x^3 - 8x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.57583$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0834797 q^{3} +3.65561 q^{5} +2.67943 q^{7} -2.99303 q^{9} +O(q^{10})$	$q-0.0834797 q^{3} +3.65561 q^{5} +2.67943 q^{7} -2.99303 q^{9} +0.842216 q^{11} -1.00000 q^{13} -0.305170 q^{15} +2.23422 q^{17} +5.38243 q^{19} -0.223678 q^{21} +2.82814 q^{23} +8.36352 q^{25} +0.500297 q^{27} +1.00000 q^{29} +0.255843 q^{31} -0.0703080 q^{33} +9.79498 q^{35} +1.03111 q^{37} +0.0834797 q^{39} -3.34405 q^{41} +1.27024 q^{43} -10.9414 q^{45} +0.0511321 q^{47} +0.179368 q^{49} -0.186512 q^{51} -5.17237 q^{53} +3.07882 q^{55} -0.449324 q^{57} -0.825413 q^{59} -10.8922 q^{61} -8.01963 q^{63} -3.65561 q^{65} +7.97332 q^{67} -0.236092 q^{69} +8.85519 q^{71} -6.79784 q^{73} -0.698185 q^{75} +2.25666 q^{77} +10.5740 q^{79} +8.93733 q^{81} +4.36892 q^{83} +8.16745 q^{85} -0.0834797 q^{87} +5.48790 q^{89} -2.67943 q^{91} -0.0213577 q^{93} +19.6761 q^{95} -10.5659 q^{97} -2.52078 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	$ 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) $ 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * 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
$3$	−0.0834797	−0.0481971	−0.0240985	−	0.999710i	$-0.507672\pi$
$3$	−0.0834797	−0.0481971	−0.0240985	+	0.999710i	$0.507672\pi$
$4$	0	0
$5$	3.65561	1.63484	0.817420	−	0.576042i	$-0.195403\pi$
$5$	3.65561	1.63484	0.817420	+	0.576042i	$0.195403\pi$
$6$	0	0
$7$	2.67943	1.01273	0.506365	−	0.862319i	$-0.330989\pi$
$7$	2.67943	1.01273	0.506365	+	0.862319i	$0.330989\pi$
$8$	0	0
$9$	−2.99303	−0.997677
$10$	0	0
$11$	0.842216	0.253938	0.126969	−	0.991907i	$-0.459475\pi$
$11$	0.842216	0.253938	0.126969	+	0.991907i	$0.459475\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.305170	−0.0787945
$16$	0	0
$17$	2.23422	0.541878	0.270939	−	0.962597i	$-0.412666\pi$
$17$	2.23422	0.541878	0.270939	+	0.962597i	$0.412666\pi$
$18$	0	0
$19$	5.38243	1.23481	0.617407	−	0.786644i	$-0.288183\pi$
$19$	5.38243	1.23481	0.617407	+	0.786644i	$0.288183\pi$
$20$	0	0
$21$	−0.223678	−0.0488106
$22$	0	0
$23$	2.82814	0.589707	0.294854	−	0.955542i	$-0.404729\pi$
$23$	2.82814	0.589707	0.294854	+	0.955542i	$0.404729\pi$
$24$	0	0
$25$	8.36352	1.67270
$26$	0	0
$27$	0.500297	0.0962821
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.255843	0.0459507	0.0229753	−	0.999736i	$-0.492686\pi$
$31$	0.255843	0.0459507	0.0229753	+	0.999736i	$0.492686\pi$
$32$	0	0
$33$	−0.0703080	−0.0122391
$34$	0	0
$35$	9.79498	1.65565
$36$	0	0
$37$	1.03111	0.169514	0.0847569	−	0.996402i	$-0.472989\pi$
$37$	1.03111	0.169514	0.0847569	+	0.996402i	$0.472989\pi$
$38$	0	0
$39$	0.0834797	0.0133675
$40$	0	0
$41$	−3.34405	−0.522253	−0.261126	−	0.965305i	$-0.584094\pi$
$41$	−3.34405	−0.522253	−0.261126	+	0.965305i	$0.584094\pi$
$42$	0	0
$43$	1.27024	0.193711	0.0968553	−	0.995298i	$-0.469122\pi$
$43$	1.27024	0.193711	0.0968553	+	0.995298i	$0.469122\pi$
$44$	0	0
$45$	−10.9414	−1.63104
$46$	0	0
$47$	0.0511321	0.00745838	0.00372919	−	0.999993i	$-0.498813\pi$
$47$	0.0511321	0.00745838	0.00372919	+	0.999993i	$0.498813\pi$
$48$	0	0
$49$	0.179368	0.0256240
$50$	0	0
$51$	−0.186512	−0.0261169
$52$	0	0
$53$	−5.17237	−0.710479	−0.355240	−	0.934775i	$-0.615601\pi$
$53$	−5.17237	−0.710479	−0.355240	+	0.934775i	$0.615601\pi$
$54$	0	0
$55$	3.07882	0.415148
$56$	0	0
$57$	−0.449324	−0.0595144
$58$	0	0
$59$	−0.825413	−0.107460	−0.0537298	−	0.998556i	$-0.517111\pi$
$59$	−0.825413	−0.107460	−0.0537298	+	0.998556i	$0.517111\pi$
$60$	0	0
$61$	−10.8922	−1.39460	−0.697302	−	0.716777i	$-0.745617\pi$
$61$	−10.8922	−1.39460	−0.697302	+	0.716777i	$0.745617\pi$
$62$	0	0
$63$	−8.01963	−1.01038
$64$	0	0
$65$	−3.65561	−0.453423
$66$	0	0
$67$	7.97332	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$67$	7.97332	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$68$	0	0
$69$	−0.236092	−0.0284221
$70$	0	0
$71$	8.85519	1.05092	0.525459	−	0.850819i	$-0.323894\pi$
$71$	8.85519	1.05092	0.525459	+	0.850819i	$0.323894\pi$
$72$	0	0
$73$	−6.79784	−0.795626	−0.397813	−	0.917466i	$-0.630231\pi$
$73$	−6.79784	−0.795626	−0.397813	+	0.917466i	$0.630231\pi$
$74$	0	0
$75$	−0.698185	−0.0806194
$76$	0	0
$77$	2.25666	0.257171
$78$	0	0
$79$	10.5740	1.18967	0.594833	−	0.803849i	$-0.297218\pi$
$79$	10.5740	1.18967	0.594833	+	0.803849i	$0.297218\pi$
$80$	0	0
$81$	8.93733	0.993037
$82$	0	0
$83$	4.36892	0.479551	0.239776	−	0.970828i	$-0.422926\pi$
$83$	4.36892	0.479551	0.239776	+	0.970828i	$0.422926\pi$
$84$	0	0
$85$	8.16745	0.885884
$86$	0	0
$87$	−0.0834797	−0.00894997
$88$	0	0
$89$	5.48790	0.581716	0.290858	−	0.956766i	$-0.406059\pi$
$89$	5.48790	0.581716	0.290858	+	0.956766i	$0.406059\pi$
$90$	0	0
$91$	−2.67943	−0.280881
$92$	0	0
$93$	−0.0213577	−0.00221469
$94$	0	0
$95$	19.6761	2.01872
$96$	0	0
$97$	−10.5659	−1.07281	−0.536404	−	0.843961i	$-0.680218\pi$
$97$	−10.5659	−1.07281	−0.536404	+	0.843961i	$0.680218\pi$
$98$	0	0
$99$	−2.52078	−0.253348
$100$	0	0
$101$	17.5444	1.74574	0.872869	−	0.487955i	$-0.162257\pi$
$101$	17.5444	1.74574	0.872869	+	0.487955i	$0.162257\pi$
$102$	0	0
$103$	12.8165	1.26284	0.631422	−	0.775439i	$-0.282471\pi$
$103$	12.8165	1.26284	0.631422	+	0.775439i	$0.282471\pi$
$104$	0	0
$105$	−0.817682	−0.0797976
$106$	0	0
$107$	−11.7216	−1.13317	−0.566586	−	0.824002i	$-0.691736\pi$
$107$	−11.7216	−1.13317	−0.566586	+	0.824002i	$0.691736\pi$
$108$	0	0
$109$	9.67559	0.926753	0.463377	−	0.886161i	$-0.346638\pi$
$109$	9.67559	0.926753	0.463377	+	0.886161i	$0.346638\pi$
$110$	0	0
$111$	−0.0860769	−0.00817006
$112$	0	0
$113$	10.5356	0.991104	0.495552	−	0.868578i	$-0.334966\pi$
$113$	10.5356	0.991104	0.495552	+	0.868578i	$0.334966\pi$
$114$	0	0
$115$	10.3386	0.964077
$116$	0	0
$117$	2.99303	0.276706
$118$	0	0
$119$	5.98645	0.548777
$120$	0	0
$121$	−10.2907	−0.935516
$122$	0	0
$123$	0.279160	0.0251710
$124$	0	0
$125$	12.2957	1.09976
$126$	0	0
$127$	11.2503	0.998299	0.499150	−	0.866516i	$-0.333646\pi$
$127$	11.2503	0.998299	0.499150	+	0.866516i	$0.333646\pi$
$128$	0	0
$129$	−0.106040	−0.00933628
$130$	0	0
$131$	−16.8651	−1.47351	−0.736756	−	0.676159i	$-0.763643\pi$
$131$	−16.8651	−1.47351	−0.736756	+	0.676159i	$0.763643\pi$
$132$	0	0
$133$	14.4219	1.25053
$134$	0	0
$135$	1.82889	0.157406
$136$	0	0
$137$	−12.7913	−1.09284	−0.546418	−	0.837512i	$-0.684009\pi$
$137$	−12.7913	−1.09284	−0.546418	+	0.837512i	$0.684009\pi$
$138$	0	0
$139$	−6.30863	−0.535091	−0.267546	−	0.963545i	$-0.586213\pi$
$139$	−6.30863	−0.535091	−0.267546	+	0.963545i	$0.586213\pi$
$140$	0	0
$141$	−0.00426849	−0.000359472 0
$142$	0	0
$143$	−0.842216	−0.0704297
$144$	0	0
$145$	3.65561	0.303582
$146$	0	0
$147$	−0.0149736	−0.00123500
$148$	0	0
$149$	−10.8086	−0.885473	−0.442736	−	0.896652i	$-0.645992\pi$
$149$	−10.8086	−0.885473	−0.442736	+	0.896652i	$0.645992\pi$
$150$	0	0
$151$	3.07756	0.250448	0.125224	−	0.992128i	$-0.460035\pi$
$151$	3.07756	0.250448	0.125224	+	0.992128i	$0.460035\pi$
$152$	0	0
$153$	−6.68709	−0.540619
$154$	0	0
$155$	0.935262	0.0751221
$156$	0	0
$157$	4.83232	0.385661	0.192831	−	0.981232i	$-0.438233\pi$
$157$	4.83232	0.385661	0.192831	+	0.981232i	$0.438233\pi$
$158$	0	0
$159$	0.431788	0.0342430
$160$	0	0
$161$	7.57780	0.597215
$162$	0	0
$163$	14.6157	1.14479	0.572394	−	0.819979i	$-0.306015\pi$
$163$	14.6157	1.14479	0.572394	+	0.819979i	$0.306015\pi$
$164$	0	0
$165$	−0.257019	−0.0200089
$166$	0	0
$167$	−13.3082	−1.02982	−0.514909	−	0.857245i	$-0.672174\pi$
$167$	−13.3082	−1.02982	−0.514909	+	0.857245i	$0.672174\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−16.1098	−1.23194
$172$	0	0
$173$	−4.43125	−0.336901	−0.168451	−	0.985710i	$-0.553876\pi$
$173$	−4.43125	−0.336901	−0.168451	+	0.985710i	$0.553876\pi$
$174$	0	0
$175$	22.4095	1.69400
$176$	0	0
$177$	0.0689053	0.00517924
$178$	0	0
$179$	3.13050	0.233984	0.116992	−	0.993133i	$-0.462675\pi$
$179$	3.13050	0.233984	0.116992	+	0.993133i	$0.462675\pi$
$180$	0	0
$181$	−18.7057	−1.39039	−0.695193	−	0.718823i	$-0.744681\pi$
$181$	−18.7057	−1.39039	−0.695193	+	0.718823i	$0.744681\pi$
$182$	0	0
$183$	0.909279	0.0672158
$184$	0	0
$185$	3.76935	0.277128
$186$	0	0
$187$	1.88170	0.137603
$188$	0	0
$189$	1.34051	0.0975079
$190$	0	0
$191$	−13.7365	−0.993939	−0.496969	−	0.867768i	$-0.665554\pi$
$191$	−13.7365	−0.993939	−0.496969	+	0.867768i	$0.665554\pi$
$192$	0	0
$193$	−1.51428	−0.109000	−0.0545001	−	0.998514i	$-0.517357\pi$
$193$	−1.51428	−0.109000	−0.0545001	+	0.998514i	$0.517357\pi$
$194$	0	0
$195$	0.305170	0.0218537
$196$	0	0
$197$	19.6888	1.40277	0.701383	−	0.712784i	$-0.252566\pi$
$197$	19.6888	1.40277	0.701383	+	0.712784i	$0.252566\pi$
$198$	0	0
$199$	−13.7540	−0.974996	−0.487498	−	0.873124i	$-0.662090\pi$
$199$	−13.7540	−0.974996	−0.487498	+	0.873124i	$0.662090\pi$
$200$	0	0
$201$	−0.665611	−0.0469485
$202$	0	0
$203$	2.67943	0.188059
$204$	0	0
$205$	−12.2246	−0.853800
$206$	0	0
$207$	−8.46470	−0.588337
$208$	0	0
$209$	4.53317	0.313566
$210$	0	0
$211$	−0.364088	−0.0250649	−0.0125324	−	0.999921i	$-0.503989\pi$
$211$	−0.364088	−0.0250649	−0.0125324	+	0.999921i	$0.503989\pi$
$212$	0	0
$213$	−0.739229	−0.0506512
$214$	0	0
$215$	4.64353	0.316686
$216$	0	0
$217$	0.685513	0.0465357
$218$	0	0
$219$	0.567482	0.0383468
$220$	0	0
$221$	−2.23422	−0.150290
$222$	0	0
$223$	−5.08948	−0.340817	−0.170408	−	0.985374i	$-0.554509\pi$
$223$	−5.08948	−0.340817	−0.170408	+	0.985374i	$0.554509\pi$
$224$	0	0
$225$	−25.0323	−1.66882
$226$	0	0
$227$	−9.24954	−0.613913	−0.306957	−	0.951723i	$-0.599311\pi$
$227$	−9.24954	−0.613913	−0.306957	+	0.951723i	$0.599311\pi$
$228$	0	0
$229$	8.84455	0.584465	0.292232	−	0.956347i	$-0.405602\pi$
$229$	8.84455	0.584465	0.292232	+	0.956347i	$0.405602\pi$
$230$	0	0
$231$	−0.188386	−0.0123949
$232$	0	0
$233$	9.09107	0.595576	0.297788	−	0.954632i	$-0.403751\pi$
$233$	9.09107	0.595576	0.297788	+	0.954632i	$0.403751\pi$
$234$	0	0
$235$	0.186919	0.0121933
$236$	0	0
$237$	−0.882714	−0.0573384
$238$	0	0
$239$	11.1128	0.718824	0.359412	−	0.933179i	$-0.382977\pi$
$239$	11.1128	0.718824	0.359412	+	0.933179i	$0.382977\pi$
$240$	0	0
$241$	16.7352	1.07801	0.539005	−	0.842302i	$-0.318800\pi$
$241$	16.7352	1.07801	0.539005	+	0.842302i	$0.318800\pi$
$242$	0	0
$243$	−2.24698	−0.144144
$244$	0	0
$245$	0.655699	0.0418911
$246$	0	0
$247$	−5.38243	−0.342476
$248$	0	0
$249$	−0.364716	−0.0231130
$250$	0	0
$251$	−10.5557	−0.666271	−0.333136	−	0.942879i	$-0.608107\pi$
$251$	−10.5557	−0.666271	−0.333136	+	0.942879i	$0.608107\pi$
$252$	0	0
$253$	2.38190	0.149749
$254$	0	0
$255$	−0.681817	−0.0426970
$256$	0	0
$257$	−7.74463	−0.483097	−0.241549	−	0.970389i	$-0.577655\pi$
$257$	−7.74463	−0.483097	−0.241549	+	0.970389i	$0.577655\pi$
$258$	0	0
$259$	2.76280	0.171672
$260$	0	0
$261$	−2.99303	−0.185264
$262$	0	0
$263$	15.5276	0.957474	0.478737	−	0.877958i	$-0.341095\pi$
$263$	15.5276	0.957474	0.478737	+	0.877958i	$0.341095\pi$
$264$	0	0
$265$	−18.9082	−1.16152
$266$	0	0
$267$	−0.458128	−0.0280370
$268$	0	0
$269$	−19.0261	−1.16004	−0.580020	−	0.814602i	$-0.696955\pi$
$269$	−19.0261	−1.16004	−0.580020	+	0.814602i	$0.696955\pi$
$270$	0	0
$271$	13.3992	0.813943	0.406971	−	0.913441i	$-0.366585\pi$
$271$	13.3992	0.813943	0.406971	+	0.913441i	$0.366585\pi$
$272$	0	0
$273$	0.223678	0.0135376
$274$	0	0
$275$	7.04389	0.424763
$276$	0	0
$277$	−11.4909	−0.690420	−0.345210	−	0.938525i	$-0.612192\pi$
$277$	−11.4909	−0.690420	−0.345210	+	0.938525i	$0.612192\pi$
$278$	0	0
$279$	−0.765745	−0.0458439
$280$	0	0
$281$	11.2339	0.670157	0.335078	−	0.942190i	$-0.391237\pi$
$281$	11.2339	0.670157	0.335078	+	0.942190i	$0.391237\pi$
$282$	0	0
$283$	16.2069	0.963402	0.481701	−	0.876336i	$-0.340019\pi$
$283$	16.2069	0.963402	0.481701	+	0.876336i	$0.340019\pi$
$284$	0	0
$285$	−1.64255	−0.0972965
$286$	0	0
$287$	−8.96016	−0.528902
$288$	0	0
$289$	−12.0083	−0.706368
$290$	0	0
$291$	0.882042	0.0517062
$292$	0	0
$293$	4.81900	0.281529	0.140765	−	0.990043i	$-0.455044\pi$
$293$	4.81900	0.281529	0.140765	+	0.990043i	$0.455044\pi$
$294$	0	0
$295$	−3.01739	−0.175679
$296$	0	0
$297$	0.421358	0.0244497
$298$	0	0
$299$	−2.82814	−0.163555
$300$	0	0
$301$	3.40354	0.196177
$302$	0	0
$303$	−1.46461	−0.0841394
$304$	0	0
$305$	−39.8177	−2.27996
$306$	0	0
$307$	−20.6734	−1.17989	−0.589946	−	0.807443i	$-0.700851\pi$
$307$	−20.6734	−1.17989	−0.589946	+	0.807443i	$0.700851\pi$
$308$	0	0
$309$	−1.06992	−0.0608654
$310$	0	0
$311$	33.4315	1.89573	0.947863	−	0.318677i	$-0.103238\pi$
$311$	33.4315	1.89573	0.947863	+	0.318677i	$0.103238\pi$
$312$	0	0
$313$	−19.2947	−1.09060	−0.545299	−	0.838241i	$-0.683584\pi$
$313$	−19.2947	−1.09060	−0.545299	+	0.838241i	$0.683584\pi$
$314$	0	0
$315$	−29.3167	−1.65181
$316$	0	0
$317$	17.2901	0.971110	0.485555	−	0.874206i	$-0.338618\pi$
$317$	17.2901	0.971110	0.485555	+	0.874206i	$0.338618\pi$
$318$	0	0
$319$	0.842216	0.0471551
$320$	0	0
$321$	0.978519	0.0546156
$322$	0	0
$323$	12.0255	0.669118
$324$	0	0
$325$	−8.36352	−0.463925
$326$	0	0
$327$	−0.807716	−0.0446668
$328$	0	0
$329$	0.137005	0.00755333
$330$	0	0
$331$	25.0185	1.37514	0.687570	−	0.726118i	$-0.258677\pi$
$331$	25.0185	1.37514	0.687570	+	0.726118i	$0.258677\pi$
$332$	0	0
$333$	−3.08615	−0.169120
$334$	0	0
$335$	29.1474	1.59249
$336$	0	0
$337$	10.3493	0.563764	0.281882	−	0.959449i	$-0.409041\pi$
$337$	10.3493	0.563764	0.281882	+	0.959449i	$0.409041\pi$
$338$	0	0
$339$	−0.879507	−0.0477683
$340$	0	0
$341$	0.215475	0.0116686
$342$	0	0
$343$	−18.2754	−0.986781
$344$	0	0
$345$	−0.863062	−0.0464657
$346$	0	0
$347$	3.50532	0.188176	0.0940878	−	0.995564i	$-0.470007\pi$
$347$	3.50532	0.188176	0.0940878	+	0.995564i	$0.470007\pi$
$348$	0	0
$349$	−11.3063	−0.605215	−0.302607	−	0.953115i	$-0.597857\pi$
$349$	−11.3063	−0.605215	−0.302607	+	0.953115i	$0.597857\pi$
$350$	0	0
$351$	−0.500297	−0.0267039
$352$	0	0
$353$	2.66529	0.141859	0.0709296	−	0.997481i	$-0.477403\pi$
$353$	2.66529	0.141859	0.0709296	+	0.997481i	$0.477403\pi$
$354$	0	0
$355$	32.3712	1.71808
$356$	0	0
$357$	−0.499747	−0.0264494
$358$	0	0
$359$	10.8751	0.573967	0.286984	−	0.957935i	$-0.407347\pi$
$359$	10.8751	0.573967	0.286984	+	0.957935i	$0.407347\pi$
$360$	0	0
$361$	9.97052	0.524764
$362$	0	0
$363$	0.859063	0.0450891
$364$	0	0
$365$	−24.8503	−1.30072
$366$	0	0
$367$	−23.7088	−1.23759	−0.618794	−	0.785553i	$-0.712379\pi$
$367$	−23.7088	−1.23759	−0.618794	+	0.785553i	$0.712379\pi$
$368$	0	0
$369$	10.0088	0.521040
$370$	0	0
$371$	−13.8590	−0.719525
$372$	0	0
$373$	1.64225	0.0850323	0.0425162	−	0.999096i	$-0.486463\pi$
$373$	1.64225	0.0850323	0.0425162	+	0.999096i	$0.486463\pi$
$374$	0	0
$375$	−1.02644	−0.0530054
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	2.95611	0.151845	0.0759225	−	0.997114i	$-0.475810\pi$
$379$	2.95611	0.151845	0.0759225	+	0.997114i	$0.475810\pi$
$380$	0	0
$381$	−0.939169	−0.0481151
$382$	0	0
$383$	23.7477	1.21345	0.606725	−	0.794912i	$-0.292483\pi$
$383$	23.7477	1.21345	0.606725	+	0.794912i	$0.292483\pi$
$384$	0	0
$385$	8.24949	0.420433
$386$	0	0
$387$	−3.80188	−0.193261
$388$	0	0
$389$	24.1105	1.22245	0.611224	−	0.791457i	$-0.290677\pi$
$389$	24.1105	1.22245	0.611224	+	0.791457i	$0.290677\pi$
$390$	0	0
$391$	6.31868	0.319549
$392$	0	0
$393$	1.40789	0.0710189
$394$	0	0
$395$	38.6544	1.94491
$396$	0	0
$397$	−19.0750	−0.957348	−0.478674	−	0.877993i	$-0.658882\pi$
$397$	−19.0750	−0.957348	−0.478674	+	0.877993i	$0.658882\pi$
$398$	0	0
$399$	−1.20393	−0.0602720
$400$	0	0
$401$	−15.5884	−0.778446	−0.389223	−	0.921144i	$-0.627256\pi$
$401$	−15.5884	−0.778446	−0.389223	+	0.921144i	$0.627256\pi$
$402$	0	0
$403$	−0.255843	−0.0127444
$404$	0	0
$405$	32.6714	1.62346
$406$	0	0
$407$	0.868419	0.0430459
$408$	0	0
$409$	5.73325	0.283491	0.141745	−	0.989903i	$-0.454729\pi$
$409$	5.73325	0.283491	0.141745	+	0.989903i	$0.454729\pi$
$410$	0	0
$411$	1.06782	0.0526715
$412$	0	0
$413$	−2.21164	−0.108828
$414$	0	0
$415$	15.9711	0.783990
$416$	0	0
$417$	0.526643	0.0257898
$418$	0	0
$419$	14.0133	0.684597	0.342298	−	0.939591i	$-0.388795\pi$
$419$	14.0133	0.684597	0.342298	+	0.939591i	$0.388795\pi$
$420$	0	0
$421$	28.2011	1.37444	0.687220	−	0.726450i	$-0.258831\pi$
$421$	28.2011	1.37444	0.687220	+	0.726450i	$0.258831\pi$
$422$	0	0
$423$	−0.153040	−0.00744105
$424$	0	0
$425$	18.6859	0.906401
$426$	0	0
$427$	−29.1850	−1.41236
$428$	0	0
$429$	0.0703080	0.00339450
$430$	0	0
$431$	−34.1518	−1.64503	−0.822517	−	0.568741i	$-0.807431\pi$
$431$	−34.1518	−1.64503	−0.822517	+	0.568741i	$0.807431\pi$
$432$	0	0
$433$	−27.4212	−1.31778	−0.658889	−	0.752240i	$-0.728973\pi$
$433$	−27.4212	−1.31778	−0.658889	+	0.752240i	$0.728973\pi$
$434$	0	0
$435$	−0.305170	−0.0146318
$436$	0	0
$437$	15.2222	0.728178
$438$	0	0
$439$	−33.5764	−1.60251	−0.801256	−	0.598322i	$-0.795835\pi$
$439$	−33.5764	−1.60251	−0.801256	+	0.598322i	$0.795835\pi$
$440$	0	0
$441$	−0.536853	−0.0255644
$442$	0	0
$443$	−27.6040	−1.31150	−0.655752	−	0.754976i	$-0.727648\pi$
$443$	−27.6040	−1.31150	−0.655752	+	0.754976i	$0.727648\pi$
$444$	0	0
$445$	20.0616	0.951013
$446$	0	0
$447$	0.902297	0.0426772
$448$	0	0
$449$	40.7723	1.92416	0.962082	−	0.272759i	$-0.0879363\pi$
$449$	40.7723	1.92416	0.962082	+	0.272759i	$0.0879363\pi$
$450$	0	0
$451$	−2.81641	−0.132620
$452$	0	0
$453$	−0.256914	−0.0120709
$454$	0	0
$455$	−9.79498	−0.459196
$456$	0	0
$457$	28.1218	1.31548	0.657741	−	0.753244i	$-0.271512\pi$
$457$	28.1218	1.31548	0.657741	+	0.753244i	$0.271512\pi$
$458$	0	0
$459$	1.11777	0.0521732
$460$	0	0
$461$	−8.95736	−0.417186	−0.208593	−	0.978003i	$-0.566888\pi$
$461$	−8.95736	−0.417186	−0.208593	+	0.978003i	$0.566888\pi$
$462$	0	0
$463$	−20.5942	−0.957093	−0.478546	−	0.878062i	$-0.658836\pi$
$463$	−20.5942	−0.957093	−0.478546	+	0.878062i	$0.658836\pi$
$464$	0	0
$465$	−0.0780754	−0.00362066
$466$	0	0
$467$	−4.91859	−0.227605	−0.113803	−	0.993503i	$-0.536303\pi$
$467$	−4.91859	−0.227605	−0.113803	+	0.993503i	$0.536303\pi$
$468$	0	0
$469$	21.3640	0.986497
$470$	0	0
$471$	−0.403401	−0.0185877
$472$	0	0
$473$	1.06982	0.0491904
$474$	0	0
$475$	45.0160	2.06548
$476$	0	0
$477$	15.4811	0.708829
$478$	0	0
$479$	19.6227	0.896586	0.448293	−	0.893887i	$-0.352032\pi$
$479$	19.6227	0.896586	0.448293	+	0.893887i	$0.352032\pi$
$480$	0	0
$481$	−1.03111	−0.0470146
$482$	0	0
$483$	−0.632593	−0.0287840
$484$	0	0
$485$	−38.6250	−1.75387
$486$	0	0
$487$	−6.98071	−0.316326	−0.158163	−	0.987413i	$-0.550557\pi$
$487$	−6.98071	−0.316326	−0.158163	+	0.987413i	$0.550557\pi$
$488$	0	0
$489$	−1.22011	−0.0551754
$490$	0	0
$491$	0.121771	0.00549543	0.00274771	−	0.999996i	$-0.499125\pi$
$491$	0.121771	0.00549543	0.00274771	+	0.999996i	$0.499125\pi$
$492$	0	0
$493$	2.23422	0.100624
$494$	0	0
$495$	−9.21500	−0.414183
$496$	0	0
$497$	23.7269	1.06430
$498$	0	0
$499$	30.2595	1.35460	0.677301	−	0.735706i	$-0.263149\pi$
$499$	30.2595	1.35460	0.677301	+	0.735706i	$0.263149\pi$
$500$	0	0
$501$	1.11096	0.0496342
$502$	0	0
$503$	−8.39254	−0.374205	−0.187102	−	0.982340i	$-0.559910\pi$
$503$	−8.39254	−0.374205	−0.187102	+	0.982340i	$0.559910\pi$
$504$	0	0
$505$	64.1357	2.85400
$506$	0	0
$507$	−0.0834797	−0.00370747
$508$	0	0
$509$	−1.67698	−0.0743308	−0.0371654	−	0.999309i	$-0.511833\pi$
$509$	−1.67698	−0.0743308	−0.0371654	+	0.999309i	$0.511833\pi$
$510$	0	0
$511$	−18.2144	−0.805755
$512$	0	0
$513$	2.69281	0.118890
$514$	0	0
$515$	46.8521	2.06455
$516$	0	0
$517$	0.0430643	0.00189396
$518$	0	0
$519$	0.369919	0.0162377
$520$	0	0
$521$	−22.3057	−0.977232	−0.488616	−	0.872499i	$-0.662498\pi$
$521$	−22.3057	−0.977232	−0.488616	+	0.872499i	$0.662498\pi$
$522$	0	0
$523$	11.7876	0.515436	0.257718	−	0.966220i	$-0.417030\pi$
$523$	11.7876	0.515436	0.257718	+	0.966220i	$0.417030\pi$
$524$	0	0
$525$	−1.87074	−0.0816458
$526$	0	0
$527$	0.571609	0.0248997
$528$	0	0
$529$	−15.0016	−0.652246
$530$	0	0
$531$	2.47049	0.107210
$532$	0	0
$533$	3.34405	0.144847
$534$	0	0
$535$	−42.8498	−1.85256
$536$	0	0
$537$	−0.261333	−0.0112774
$538$	0	0
$539$	0.151066	0.00650689
$540$	0	0
$541$	31.7095	1.36330	0.681649	−	0.731680i	$-0.261263\pi$
$541$	31.7095	1.36330	0.681649	+	0.731680i	$0.261263\pi$
$542$	0	0
$543$	1.56155	0.0670125
$544$	0	0
$545$	35.3702	1.51509
$546$	0	0
$547$	−41.8879	−1.79100	−0.895498	−	0.445065i	$-0.853180\pi$
$547$	−41.8879	−1.79100	−0.895498	+	0.445065i	$0.853180\pi$
$548$	0	0
$549$	32.6007	1.39137
$550$	0	0
$551$	5.38243	0.229299
$552$	0	0
$553$	28.3323	1.20481
$554$	0	0
$555$	−0.314664	−0.0133567
$556$	0	0
$557$	−13.2371	−0.560875	−0.280438	−	0.959872i	$-0.590480\pi$
$557$	−13.2371	−0.560875	−0.280438	+	0.959872i	$0.590480\pi$
$558$	0	0
$559$	−1.27024	−0.0537256
$560$	0	0
$561$	−0.157084	−0.00663207
$562$	0	0
$563$	−23.0283	−0.970526	−0.485263	−	0.874368i	$-0.661276\pi$
$563$	−23.0283	−0.970526	−0.485263	+	0.874368i	$0.661276\pi$
$564$	0	0
$565$	38.5140	1.62030
$566$	0	0
$567$	23.9470	1.00568
$568$	0	0
$569$	−27.3213	−1.14537	−0.572685	−	0.819776i	$-0.694098\pi$
$569$	−27.3213	−1.14537	−0.572685	+	0.819776i	$0.694098\pi$
$570$	0	0
$571$	30.3443	1.26987	0.634935	−	0.772565i	$-0.281027\pi$
$571$	30.3443	1.26987	0.634935	+	0.772565i	$0.281027\pi$
$572$	0	0
$573$	1.14672	0.0479049
$574$	0	0
$575$	23.6532	0.986405
$576$	0	0
$577$	9.39824	0.391254	0.195627	−	0.980678i	$-0.437326\pi$
$577$	9.39824	0.391254	0.195627	+	0.980678i	$0.437326\pi$
$578$	0	0
$579$	0.126412	0.00525349
$580$	0	0
$581$	11.7062	0.485656
$582$	0	0
$583$	−4.35625	−0.180418
$584$	0	0
$585$	10.9414	0.452370
$586$	0	0
$587$	−1.74220	−0.0719083	−0.0359542	−	0.999353i	$-0.511447\pi$
$587$	−1.74220	−0.0719083	−0.0359542	+	0.999353i	$0.511447\pi$
$588$	0	0
$589$	1.37705	0.0567405
$590$	0	0
$591$	−1.64361	−0.0676092
$592$	0	0
$593$	35.3442	1.45141	0.725706	−	0.688005i	$-0.241513\pi$
$593$	35.3442	1.45141	0.725706	+	0.688005i	$0.241513\pi$
$594$	0	0
$595$	21.8841	0.897162
$596$	0	0
$597$	1.14818	0.0469919
$598$	0	0
$599$	−42.9256	−1.75389	−0.876946	−	0.480589i	$-0.840423\pi$
$599$	−42.9256	−1.75389	−0.876946	+	0.480589i	$0.840423\pi$
$600$	0	0
$601$	12.2493	0.499660	0.249830	−	0.968290i	$-0.419625\pi$
$601$	12.2493	0.499660	0.249830	+	0.968290i	$0.419625\pi$
$602$	0	0
$603$	−23.8644	−0.971833
$604$	0	0
$605$	−37.6187	−1.52942
$606$	0	0
$607$	4.33847	0.176093	0.0880465	−	0.996116i	$-0.471938\pi$
$607$	4.33847	0.176093	0.0880465	+	0.996116i	$0.471938\pi$
$608$	0	0
$609$	−0.223678	−0.00906391
$610$	0	0
$611$	−0.0511321	−0.00206858
$612$	0	0
$613$	−12.1903	−0.492361	−0.246180	−	0.969224i	$-0.579176\pi$
$613$	−12.1903	−0.492361	−0.246180	+	0.969224i	$0.579176\pi$
$614$	0	0
$615$	1.02050	0.0411507
$616$	0	0
$617$	−44.1031	−1.77552	−0.887762	−	0.460303i	$-0.847741\pi$
$617$	−44.1031	−1.77552	−0.887762	+	0.460303i	$0.847741\pi$
$618$	0	0
$619$	−0.0417533	−0.00167821	−0.000839104	−	1.00000i	$-0.500267\pi$
$619$	−0.0417533	−0.00167821	−0.000839104		1.00000i	$0.500267\pi$
$620$	0	0
$621$	1.41491	0.0567783
$622$	0	0
$623$	14.7045	0.589122
$624$	0	0
$625$	3.13087	0.125235
$626$	0	0
$627$	−0.378428	−0.0151129
$628$	0	0
$629$	2.30373	0.0918557
$630$	0	0
$631$	9.71227	0.386639	0.193320	−	0.981136i	$-0.438075\pi$
$631$	9.71227	0.386639	0.193320	+	0.981136i	$0.438075\pi$
$632$	0	0
$633$	0.0303940	0.00120805
$634$	0	0
$635$	41.1266	1.63206
$636$	0	0
$637$	−0.179368	−0.00710681
$638$	0	0
$639$	−26.5039	−1.04848
$640$	0	0
$641$	−3.98695	−0.157475	−0.0787374	−	0.996895i	$-0.525089\pi$
$641$	−3.98695	−0.157475	−0.0787374	+	0.996895i	$0.525089\pi$
$642$	0	0
$643$	12.1024	0.477273	0.238636	−	0.971109i	$-0.423300\pi$
$643$	12.1024	0.477273	0.238636	+	0.971109i	$0.423300\pi$
$644$	0	0
$645$	−0.387640	−0.0152633
$646$	0	0
$647$	−49.8136	−1.95838	−0.979188	−	0.202955i	$-0.934946\pi$
$647$	−49.8136	−1.95838	−0.979188	+	0.202955i	$0.934946\pi$
$648$	0	0
$649$	−0.695176	−0.0272881
$650$	0	0
$651$	−0.0572265	−0.00224288
$652$	0	0
$653$	−1.74183	−0.0681632	−0.0340816	−	0.999419i	$-0.510851\pi$
$653$	−1.74183	−0.0681632	−0.0340816	+	0.999419i	$0.510851\pi$
$654$	0	0
$655$	−61.6523	−2.40896
$656$	0	0
$657$	20.3461	0.793778
$658$	0	0
$659$	−3.59254	−0.139945	−0.0699727	−	0.997549i	$-0.522291\pi$
$659$	−3.59254	−0.139945	−0.0699727	+	0.997549i	$0.522291\pi$
$660$	0	0
$661$	6.56296	0.255270	0.127635	−	0.991821i	$-0.459261\pi$
$661$	6.56296	0.255270	0.127635	+	0.991821i	$0.459261\pi$
$662$	0	0
$663$	0.186512	0.00724353
$664$	0	0
$665$	52.7208	2.04442
$666$	0	0
$667$	2.82814	0.109506
$668$	0	0
$669$	0.424868	0.0164264
$670$	0	0
$671$	−9.17360	−0.354143
$672$	0	0
$673$	7.41055	0.285656	0.142828	−	0.989748i	$-0.454380\pi$
$673$	7.41055	0.285656	0.142828	+	0.989748i	$0.454380\pi$
$674$	0	0
$675$	4.18424	0.161052
$676$	0	0
$677$	23.9856	0.921840	0.460920	−	0.887442i	$-0.347520\pi$
$677$	23.9856	0.921840	0.460920	+	0.887442i	$0.347520\pi$
$678$	0	0
$679$	−28.3107	−1.08647
$680$	0	0
$681$	0.772149	0.0295888
$682$	0	0
$683$	−12.8176	−0.490450	−0.245225	−	0.969466i	$-0.578862\pi$
$683$	−12.8176	−0.490450	−0.245225	+	0.969466i	$0.578862\pi$
$684$	0	0
$685$	−46.7602	−1.78661
$686$	0	0
$687$	−0.738341	−0.0281695
$688$	0	0
$689$	5.17237	0.197052
$690$	0	0
$691$	41.8661	1.59266	0.796331	−	0.604861i	$-0.206771\pi$
$691$	41.8661	1.59266	0.796331	+	0.604861i	$0.206771\pi$
$692$	0	0
$693$	−6.75426	−0.256573
$694$	0	0
$695$	−23.0619	−0.874789
$696$	0	0
$697$	−7.47134	−0.282997
$698$	0	0
$699$	−0.758920	−0.0287050
$700$	0	0
$701$	33.4931	1.26502	0.632509	−	0.774553i	$-0.282025\pi$
$701$	33.4931	1.26502	0.632509	+	0.774553i	$0.282025\pi$
$702$	0	0
$703$	5.54988	0.209318
$704$	0	0
$705$	−0.0156040	−0.000587679 0
$706$	0	0
$707$	47.0092	1.76796
$708$	0	0
$709$	37.3001	1.40084	0.700418	−	0.713733i	$-0.252997\pi$
$709$	37.3001	1.40084	0.700418	+	0.713733i	$0.252997\pi$
$710$	0	0
$711$	−31.6483	−1.18690
$712$	0	0
$713$	0.723558	0.0270974
$714$	0	0
$715$	−3.07882	−0.115141
$716$	0	0
$717$	−0.927690	−0.0346452
$718$	0	0
$719$	−44.3033	−1.65223	−0.826117	−	0.563498i	$-0.809455\pi$
$719$	−44.3033	−1.65223	−0.826117	+	0.563498i	$0.809455\pi$
$720$	0	0
$721$	34.3409	1.27892
$722$	0	0
$723$	−1.39705	−0.0519569
$724$	0	0
$725$	8.36352	0.310613
$726$	0	0
$727$	−7.97391	−0.295736	−0.147868	−	0.989007i	$-0.547241\pi$
$727$	−7.97391	−0.295736	−0.147868	+	0.989007i	$0.547241\pi$
$728$	0	0
$729$	−26.6244	−0.986089
$730$	0	0
$731$	2.83801	0.104967
$732$	0	0
$733$	−19.9817	−0.738042	−0.369021	−	0.929421i	$-0.620307\pi$
$733$	−19.9817	−0.738042	−0.369021	+	0.929421i	$0.620307\pi$
$734$	0	0
$735$	−0.0547376	−0.00201903
$736$	0	0
$737$	6.71526	0.247360
$738$	0	0
$739$	−23.8624	−0.877794	−0.438897	−	0.898537i	$-0.644631\pi$
$739$	−23.8624	−0.877794	−0.438897	+	0.898537i	$0.644631\pi$
$740$	0	0
$741$	0.449324	0.0165063
$742$	0	0
$743$	−8.98839	−0.329752	−0.164876	−	0.986314i	$-0.552722\pi$
$743$	−8.98839	−0.329752	−0.164876	+	0.986314i	$0.552722\pi$
$744$	0	0
$745$	−39.5120	−1.44761
$746$	0	0
$747$	−13.0763	−0.478437
$748$	0	0
$749$	−31.4073	−1.14760
$750$	0	0
$751$	−18.2780	−0.666972	−0.333486	−	0.942755i	$-0.608225\pi$
$751$	−18.2780	−0.666972	−0.333486	+	0.942755i	$0.608225\pi$
$752$	0	0
$753$	0.881189	0.0321123
$754$	0	0
$755$	11.2504	0.409443
$756$	0	0
$757$	32.4430	1.17916	0.589581	−	0.807709i	$-0.299293\pi$
$757$	32.4430	1.17916	0.589581	+	0.807709i	$0.299293\pi$
$758$	0	0
$759$	−0.198841	−0.00721746
$760$	0	0
$761$	2.00998	0.0728618	0.0364309	−	0.999336i	$-0.488401\pi$
$761$	2.00998	0.0728618	0.0364309	+	0.999336i	$0.488401\pi$
$762$	0	0
$763$	25.9251	0.938552
$764$	0	0
$765$	−24.4454	−0.883826
$766$	0	0
$767$	0.825413	0.0298039
$768$	0	0
$769$	−50.1467	−1.80834	−0.904168	−	0.427177i	$-0.859508\pi$
$769$	−50.1467	−1.80834	−0.904168	+	0.427177i	$0.859508\pi$
$770$	0	0
$771$	0.646520	0.0232839
$772$	0	0
$773$	19.4122	0.698208	0.349104	−	0.937084i	$-0.386486\pi$
$773$	19.4122	0.698208	0.349104	+	0.937084i	$0.386486\pi$
$774$	0	0
$775$	2.13974	0.0768619
$776$	0	0
$777$	−0.230637	−0.00827407
$778$	0	0
$779$	−17.9991	−0.644885
$780$	0	0
$781$	7.45799	0.266868
$782$	0	0
$783$	0.500297	0.0178791
$784$	0	0
$785$	17.6651	0.630495
$786$	0	0
$787$	−8.60073	−0.306583	−0.153292	−	0.988181i	$-0.548987\pi$
$787$	−8.60073	−0.306583	−0.153292	+	0.988181i	$0.548987\pi$
$788$	0	0
$789$	−1.29624	−0.0461474
$790$	0	0
$791$	28.2294	1.00372
$792$	0	0
$793$	10.8922	0.386794
$794$	0	0
$795$	1.57845	0.0559819
$796$	0	0
$797$	−8.45555	−0.299511	−0.149755	−	0.988723i	$-0.547849\pi$
$797$	−8.45555	−0.299511	−0.149755	+	0.988723i	$0.547849\pi$
$798$	0	0
$799$	0.114240	0.00404153
$800$	0	0
$801$	−16.4254	−0.580365
$802$	0	0
$803$	−5.72525	−0.202040
$804$	0	0
$805$	27.7015	0.976351
$806$	0	0
$807$	1.58829	0.0559105
$808$	0	0
$809$	14.9649	0.526136	0.263068	−	0.964777i	$-0.415266\pi$
$809$	14.9649	0.526136	0.263068	+	0.964777i	$0.415266\pi$
$810$	0	0
$811$	49.7584	1.74725	0.873627	−	0.486596i	$-0.161762\pi$
$811$	49.7584	1.74725	0.873627	+	0.486596i	$0.161762\pi$
$812$	0	0
$813$	−1.11856	−0.0392296
$814$	0	0
$815$	53.4292	1.87154
$816$	0	0
$817$	6.83700	0.239196
$818$	0	0
$819$	8.01963	0.280229
$820$	0	0
$821$	−46.0843	−1.60835	−0.804176	−	0.594391i	$-0.797393\pi$
$821$	−46.0843	−1.60835	−0.804176	+	0.594391i	$0.797393\pi$
$822$	0	0
$823$	−2.49033	−0.0868075	−0.0434038	−	0.999058i	$-0.513820\pi$
$823$	−2.49033	−0.0868075	−0.0434038	+	0.999058i	$0.513820\pi$
$824$	0	0
$825$	−0.588022	−0.0204723
$826$	0	0
$827$	35.1800	1.22333	0.611665	−	0.791117i	$-0.290500\pi$
$827$	35.1800	1.22333	0.611665	+	0.791117i	$0.290500\pi$
$828$	0	0
$829$	−29.0783	−1.00993	−0.504965	−	0.863140i	$-0.668495\pi$
$829$	−29.0783	−1.00993	−0.504965	+	0.863140i	$0.668495\pi$
$830$	0	0
$831$	0.959256	0.0332762
$832$	0	0
$833$	0.400747	0.0138851
$834$	0	0
$835$	−48.6496	−1.68359
$836$	0	0
$837$	0.127997	0.00442423
$838$	0	0
$839$	26.1281	0.902041	0.451021	−	0.892513i	$-0.351060\pi$
$839$	26.1281	0.902041	0.451021	+	0.892513i	$0.351060\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.937801	−0.0322996
$844$	0	0
$845$	3.65561	0.125757
$846$	0	0
$847$	−27.5732	−0.947426
$848$	0	0
$849$	−1.35295	−0.0464331
$850$	0	0
$851$	2.91612	0.0999634
$852$	0	0
$853$	−11.5324	−0.394863	−0.197432	−	0.980317i	$-0.563260\pi$
$853$	−11.5324	−0.394863	−0.197432	+	0.980317i	$0.563260\pi$
$854$	0	0
$855$	−58.8911	−2.01403
$856$	0	0
$857$	15.1015	0.515858	0.257929	−	0.966164i	$-0.416960\pi$
$857$	15.1015	0.515858	0.257929	+	0.966164i	$0.416960\pi$
$858$	0	0
$859$	−4.28944	−0.146354	−0.0731769	−	0.997319i	$-0.523314\pi$
$859$	−4.28944	−0.146354	−0.0731769	+	0.997319i	$0.523314\pi$
$860$	0	0
$861$	0.747992	0.0254915
$862$	0	0
$863$	14.3539	0.488611	0.244305	−	0.969698i	$-0.421440\pi$
$863$	14.3539	0.488611	0.244305	+	0.969698i	$0.421440\pi$
$864$	0	0
$865$	−16.1989	−0.550780
$866$	0	0
$867$	1.00245	0.0340449
$868$	0	0
$869$	8.90558	0.302101
$870$	0	0
$871$	−7.97332	−0.270166
$872$	0	0
$873$	31.6242	1.07032
$874$	0	0
$875$	32.9456	1.11376
$876$	0	0
$877$	−26.9566	−0.910260	−0.455130	−	0.890425i	$-0.650407\pi$
$877$	−26.9566	−0.910260	−0.455130	+	0.890425i	$0.650407\pi$
$878$	0	0
$879$	−0.402289	−0.0135689
$880$	0	0
$881$	−36.5909	−1.23278	−0.616390	−	0.787441i	$-0.711405\pi$
$881$	−36.5909	−1.23278	−0.616390	+	0.787441i	$0.711405\pi$
$882$	0	0
$883$	27.7724	0.934615	0.467308	−	0.884095i	$-0.345224\pi$
$883$	27.7724	0.934615	0.467308	+	0.884095i	$0.345224\pi$
$884$	0	0
$885$	0.251891	0.00846723
$886$	0	0
$887$	−23.3333	−0.783456	−0.391728	−	0.920081i	$-0.628123\pi$
$887$	−23.3333	−0.783456	−0.391728	+	0.920081i	$0.628123\pi$
$888$	0	0
$889$	30.1443	1.01101
$890$	0	0
$891$	7.52716	0.252169
$892$	0	0
$893$	0.275215	0.00920970
$894$	0	0
$895$	11.4439	0.382527
$896$	0	0
$897$	0.236092	0.00788288
$898$	0	0
$899$	0.255843	0.00853283
$900$	0	0
$901$	−11.5562	−0.384993
$902$	0	0
$903$	−0.284126	−0.00945514
$904$	0	0
$905$	−68.3809	−2.27306
$906$	0	0
$907$	−20.0967	−0.667301	−0.333650	−	0.942697i	$-0.608280\pi$
$907$	−20.0967	−0.667301	−0.333650	+	0.942697i	$0.608280\pi$
$908$	0	0
$909$	−52.5111	−1.74168
$910$	0	0
$911$	41.8525	1.38663	0.693317	−	0.720633i	$-0.256149\pi$
$911$	41.8525	1.38663	0.693317	+	0.720633i	$0.256149\pi$
$912$	0	0
$913$	3.67958	0.121776
$914$	0	0
$915$	3.32397	0.109887
$916$	0	0
$917$	−45.1889	−1.49227
$918$	0	0
$919$	−25.3085	−0.834850	−0.417425	−	0.908711i	$-0.637067\pi$
$919$	−25.3085	−0.834850	−0.417425	+	0.908711i	$0.637067\pi$
$920$	0	0
$921$	1.72581	0.0568673
$922$	0	0
$923$	−8.85519	−0.291472
$924$	0	0
$925$	8.62372	0.283546
$926$	0	0
$927$	−38.3601	−1.25991
$928$	0	0
$929$	−4.21092	−0.138156	−0.0690779	−	0.997611i	$-0.522006\pi$
$929$	−4.21092	−0.138156	−0.0690779	+	0.997611i	$0.522006\pi$
$930$	0	0
$931$	0.965434	0.0316408
$932$	0	0
$933$	−2.79085	−0.0913685
$934$	0	0
$935$	6.87876	0.224959
$936$	0	0
$937$	39.6338	1.29478	0.647391	−	0.762158i	$-0.275860\pi$
$937$	39.6338	1.29478	0.647391	+	0.762158i	$0.275860\pi$
$938$	0	0
$939$	1.61071	0.0525637
$940$	0	0
$941$	−36.8245	−1.20045	−0.600223	−	0.799833i	$-0.704921\pi$
$941$	−36.8245	−1.20045	−0.600223	+	0.799833i	$0.704921\pi$
$942$	0	0
$943$	−9.45743	−0.307976
$944$	0	0
$945$	4.90040	0.159410
$946$	0	0
$947$	46.0369	1.49600	0.747999	−	0.663700i	$-0.231015\pi$
$947$	46.0369	1.49600	0.747999	+	0.663700i	$0.231015\pi$
$948$	0	0
$949$	6.79784	0.220667
$950$	0	0
$951$	−1.44337	−0.0468046
$952$	0	0
$953$	−34.5174	−1.11813	−0.559064	−	0.829125i	$-0.688839\pi$
$953$	−34.5174	−1.11813	−0.559064	+	0.829125i	$0.688839\pi$
$954$	0	0
$955$	−50.2154	−1.62493
$956$	0	0
$957$	−0.0703080	−0.00227273
$958$	0	0
$959$	−34.2735	−1.10675
$960$	0	0
$961$	−30.9345	−0.997889
$962$	0	0
$963$	35.0832	1.13054
$964$	0	0
$965$	−5.53562	−0.178198
$966$	0	0
$967$	−40.4423	−1.30054	−0.650268	−	0.759705i	$-0.725343\pi$
$967$	−40.4423	−1.30054	−0.650268	+	0.759705i	$0.725343\pi$
$968$	0	0
$969$	−1.00389	−0.0322495
$970$	0	0
$971$	−6.92660	−0.222285	−0.111143	−	0.993804i	$-0.535451\pi$
$971$	−6.92660	−0.222285	−0.111143	+	0.993804i	$0.535451\pi$
$972$	0	0
$973$	−16.9036	−0.541903
$974$	0	0
$975$	0.698185	0.0223598
$976$	0	0
$977$	−18.7859	−0.601015	−0.300508	−	0.953779i	$-0.597156\pi$
$977$	−18.7859	−0.601015	−0.300508	+	0.953779i	$0.597156\pi$
$978$	0	0
$979$	4.62200	0.147720
$980$	0	0
$981$	−28.9593	−0.924601
$982$	0	0
$983$	17.8834	0.570393	0.285196	−	0.958469i	$-0.407941\pi$
$983$	17.8834	0.570393	0.285196	+	0.958469i	$0.407941\pi$
$984$	0	0
$985$	71.9745	2.29330
$986$	0	0
$987$	−0.0114371	−0.000364048 0
$988$	0	0
$989$	3.59243	0.114232
$990$	0	0
$991$	−8.01853	−0.254717	−0.127358	−	0.991857i	$-0.540650\pi$
$991$	−8.01853	−0.254717	−0.127358	+	0.991857i	$0.540650\pi$
$992$	0	0
$993$	−2.08854	−0.0662777
$994$	0	0
$995$	−50.2794	−1.59396
$996$	0	0
$997$	−1.13077	−0.0358118	−0.0179059	−	0.999840i	$-0.505700\pi$
$997$	−1.13077	−0.0358118	−0.0179059	+	0.999840i	$0.505700\pi$
$998$	0	0
$999$	0.515862	0.0163211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bb.1.4		10
4.3	odd	2		3016.2.a.f.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.f.1.7	✓	10	4.3	odd	2
6032.2.a.bb.1.4		10	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0834797 q^{3} +3.65561 q^{5} +2.67943 q^{7} -2.99303 q^{9} +O(q^{10})\)	\(q-0.0834797 q^{3} +3.65561 q^{5} +2.67943 q^{7} -2.99303 q^{9} +0.842216 q^{11} -1.00000 q^{13} -0.305170 q^{15} +2.23422 q^{17} +5.38243 q^{19} -0.223678 q^{21} +2.82814 q^{23} +8.36352 q^{25} +0.500297 q^{27} +1.00000 q^{29} +0.255843 q^{31} -0.0703080 q^{33} +9.79498 q^{35} +1.03111 q^{37} +0.0834797 q^{39} -3.34405 q^{41} +1.27024 q^{43} -10.9414 q^{45} +0.0511321 q^{47} +0.179368 q^{49} -0.186512 q^{51} -5.17237 q^{53} +3.07882 q^{55} -0.449324 q^{57} -0.825413 q^{59} -10.8922 q^{61} -8.01963 q^{63} -3.65561 q^{65} +7.97332 q^{67} -0.236092 q^{69} +8.85519 q^{71} -6.79784 q^{73} -0.698185 q^{75} +2.25666 q^{77} +10.5740 q^{79} +8.93733 q^{81} +4.36892 q^{83} +8.16745 q^{85} -0.0834797 q^{87} +5.48790 q^{89} -2.67943 q^{91} -0.0213577 q^{93} +19.6761 q^{95} -10.5659 q^{97} -2.52078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9}+O(q^{10}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9	\( 10 q + 3 q^{3} - 5 q^{5} + 9 q^{9} + 13 q^{11} - 10 q^{13} - 3 q^{15} + 9 q^{17} + 12 q^{19} - 10 q^{21} + 5 q^{23} + 7 q^{25} + 3 q^{27} + 10 q^{29} - 5 q^{31} - 9 q^{33} + 23 q^{35} - 4 q^{37} - 3 q^{39} - 3 q^{41} + 27 q^{43} - 20 q^{45} + 16 q^{47} + 6 q^{49} + 34 q^{51} + 11 q^{53} + q^{55} + 27 q^{59} - 7 q^{61} + 6 q^{63} + 5 q^{65} + 35 q^{67} - 22 q^{69} + 21 q^{71} + 7 q^{75} - 18 q^{77} + 12 q^{79} + 6 q^{81} + 24 q^{83} - 2 q^{85} + 3 q^{87} - 23 q^{89} - 3 q^{93} + 17 q^{95} + 2 q^{97} + 65 q^{99}+O(q^{100}) \) 10 * q + 3 * q^3 - 5 * q^5 + 9 * q^9 + 13 * q^11 - 10 * q^13 - 3 * q^15 + 9 * q^17 + 12 * q^19 - 10 * q^21 + 5 * q^23 + 7 * q^25 + 3 * q^27 + 10 * q^29 - 5 * q^31 - 9 * q^33 + 23 * q^35 - 4 * q^37 - 3 * q^39 - 3 * q^41 + 27 * q^43 - 20 * q^45 + 16 * q^47 + 6 * q^49 + 34 * q^51 + 11 * q^53 + q^55 + 27 * q^59 - 7 * q^61 + 6 * q^63 + 5 * q^65 + 35 * q^67 - 22 * q^69 + 21 * q^71 + 7 * q^75 - 18 * q^77 + 12 * q^79 + 6 * q^81 + 24 * q^83 - 2 * q^85 + 3 * q^87 - 23 * q^89 - 3 * q^93 + 17 * q^95 + 2 * q^97 + 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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