LMFDB - Embedded newform 8036.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.503623$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.503623 q^{3} +2.23729 q^{5} -2.74636 q^{9} +O(q^{10})$	$q+0.503623 q^{3} +2.23729 q^{5} -2.74636 q^{9} +0.0347289 q^{11} -4.89514 q^{13} +1.12675 q^{15} +3.49414 q^{17} -3.71793 q^{19} -1.75503 q^{23} +0.00545373 q^{25} -2.89400 q^{27} +6.79246 q^{29} +6.54224 q^{31} +0.0174903 q^{33} -6.20094 q^{37} -2.46531 q^{39} +1.00000 q^{41} +4.01766 q^{43} -6.14440 q^{45} -3.34655 q^{47} +1.75973 q^{51} +6.04352 q^{53} +0.0776986 q^{55} -1.87243 q^{57} +2.77858 q^{59} -7.57287 q^{61} -10.9518 q^{65} -5.47362 q^{67} -0.883871 q^{69} -8.89496 q^{71} +12.5510 q^{73} +0.00274662 q^{75} +11.0598 q^{79} +6.78161 q^{81} -16.7856 q^{83} +7.81740 q^{85} +3.42084 q^{87} -17.2525 q^{89} +3.29482 q^{93} -8.31807 q^{95} -4.79644 q^{97} -0.0953783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * 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
$3$	0.503623	0.290767	0.145383	−	0.989375i	$-0.453558\pi$
$3$	0.503623	0.290767	0.145383	+	0.989375i	$0.453558\pi$
$4$	0	0
$5$	2.23729	1.00055	0.500273	−	0.865868i	$-0.333233\pi$
$5$	2.23729	1.00055	0.500273	+	0.865868i	$0.333233\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.74636	−0.915455
$10$	0	0
$11$	0.0347289	0.0104712	0.00523558	−	0.999986i	$-0.498333\pi$
$11$	0.0347289	0.0104712	0.00523558	+	0.999986i	$0.498333\pi$
$12$	0	0
$13$	−4.89514	−1.35767	−0.678834	−	0.734292i	$-0.737514\pi$
$13$	−4.89514	−1.35767	−0.678834	+	0.734292i	$0.737514\pi$
$14$	0	0
$15$	1.12675	0.290925
$16$	0	0
$17$	3.49414	0.847454	0.423727	−	0.905790i	$-0.360722\pi$
$17$	3.49414	0.847454	0.423727	+	0.905790i	$0.360722\pi$
$18$	0	0
$19$	−3.71793	−0.852951	−0.426475	−	0.904499i	$-0.640245\pi$
$19$	−3.71793	−0.852951	−0.426475	+	0.904499i	$0.640245\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.75503	−0.365948	−0.182974	−	0.983118i	$-0.558572\pi$
$23$	−1.75503	−0.365948	−0.182974	+	0.983118i	$0.558572\pi$
$24$	0	0
$25$	0.00545373	0.00109075
$26$	0	0
$27$	−2.89400	−0.556951
$28$	0	0
$29$	6.79246	1.26133	0.630664	−	0.776056i	$-0.282783\pi$
$29$	6.79246	1.26133	0.630664	+	0.776056i	$0.282783\pi$
$30$	0	0
$31$	6.54224	1.17502	0.587510	−	0.809217i	$-0.300108\pi$
$31$	6.54224	1.17502	0.587510	+	0.809217i	$0.300108\pi$
$32$	0	0
$33$	0.0174903	0.00304467
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.20094	−1.01943	−0.509714	−	0.860344i	$-0.670249\pi$
$37$	−6.20094	−1.01943	−0.509714	+	0.860344i	$0.670249\pi$
$38$	0	0
$39$	−2.46531	−0.394765
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.01766	0.612687	0.306343	−	0.951921i	$-0.400894\pi$
$43$	4.01766	0.612687	0.306343	+	0.951921i	$0.400894\pi$
$44$	0	0
$45$	−6.14440	−0.915954
$46$	0	0
$47$	−3.34655	−0.488144	−0.244072	−	0.969757i	$-0.578483\pi$
$47$	−3.34655	−0.488144	−0.244072	+	0.969757i	$0.578483\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.75973	0.246412
$52$	0	0
$53$	6.04352	0.830141	0.415070	−	0.909789i	$-0.363757\pi$
$53$	6.04352	0.830141	0.415070	+	0.909789i	$0.363757\pi$
$54$	0	0
$55$	0.0776986	0.0104769
$56$	0	0
$57$	−1.87243	−0.248010
$58$	0	0
$59$	2.77858	0.361740	0.180870	−	0.983507i	$-0.442109\pi$
$59$	2.77858	0.361740	0.180870	+	0.983507i	$0.442109\pi$
$60$	0	0
$61$	−7.57287	−0.969607	−0.484803	−	0.874623i	$-0.661109\pi$
$61$	−7.57287	−0.969607	−0.484803	+	0.874623i	$0.661109\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.9518	−1.35841
$66$	0	0
$67$	−5.47362	−0.668709	−0.334355	−	0.942447i	$-0.608518\pi$
$67$	−5.47362	−0.668709	−0.334355	+	0.942447i	$0.608518\pi$
$68$	0	0
$69$	−0.883871	−0.106406
$70$	0	0
$71$	−8.89496	−1.05564	−0.527819	−	0.849357i	$-0.676990\pi$
$71$	−8.89496	−1.05564	−0.527819	+	0.849357i	$0.676990\pi$
$72$	0	0
$73$	12.5510	1.46899	0.734493	−	0.678616i	$-0.237420\pi$
$73$	12.5510	1.46899	0.734493	+	0.678616i	$0.237420\pi$
$74$	0	0
$75$	0.00274662	0.000317153 0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0598	1.24432	0.622161	−	0.782889i	$-0.286255\pi$
$79$	11.0598	1.24432	0.622161	+	0.782889i	$0.286255\pi$
$80$	0	0
$81$	6.78161	0.753512
$82$	0	0
$83$	−16.7856	−1.84246	−0.921228	−	0.389024i	$-0.872812\pi$
$83$	−16.7856	−1.84246	−0.921228	+	0.389024i	$0.872812\pi$
$84$	0	0
$85$	7.81740	0.847916
$86$	0	0
$87$	3.42084	0.366752
$88$	0	0
$89$	−17.2525	−1.82876	−0.914381	−	0.404856i	$-0.867322\pi$
$89$	−17.2525	−1.82876	−0.914381	+	0.404856i	$0.867322\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.29482	0.341657
$94$	0	0
$95$	−8.31807	−0.853416
$96$	0	0
$97$	−4.79644	−0.487005	−0.243502	−	0.969900i	$-0.578296\pi$
$97$	−4.79644	−0.487005	−0.243502	+	0.969900i	$0.578296\pi$
$98$	0	0
$99$	−0.0953783	−0.00958588
$100$	0	0
$101$	−8.13968	−0.809928	−0.404964	−	0.914333i	$-0.632716\pi$
$101$	−8.13968	−0.809928	−0.404964	+	0.914333i	$0.632716\pi$
$102$	0	0
$103$	5.19146	0.511529	0.255765	−	0.966739i	$-0.417673\pi$
$103$	5.19146	0.511529	0.255765	+	0.966739i	$0.417673\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.53846	0.825445	0.412722	−	0.910857i	$-0.364578\pi$
$107$	8.53846	0.825445	0.412722	+	0.910857i	$0.364578\pi$
$108$	0	0
$109$	−12.4885	−1.19618	−0.598090	−	0.801429i	$-0.704073\pi$
$109$	−12.4885	−1.19618	−0.598090	+	0.801429i	$0.704073\pi$
$110$	0	0
$111$	−3.12293	−0.296416
$112$	0	0
$113$	−16.3562	−1.53867	−0.769333	−	0.638848i	$-0.779411\pi$
$113$	−16.3562	−1.53867	−0.769333	+	0.638848i	$0.779411\pi$
$114$	0	0
$115$	−3.92650	−0.366148
$116$	0	0
$117$	13.4438	1.24288
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9988	−0.999890
$122$	0	0
$123$	0.503623	0.0454102
$124$	0	0
$125$	−11.1742	−0.999454
$126$	0	0
$127$	−9.87504	−0.876268	−0.438134	−	0.898910i	$-0.644360\pi$
$127$	−9.87504	−0.876268	−0.438134	+	0.898910i	$0.644360\pi$
$128$	0	0
$129$	2.02338	0.178149
$130$	0	0
$131$	−1.93323	−0.168907	−0.0844533	−	0.996427i	$-0.526914\pi$
$131$	−1.93323	−0.168907	−0.0844533	+	0.996427i	$0.526914\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.47471	−0.557254
$136$	0	0
$137$	9.97925	0.852585	0.426292	−	0.904585i	$-0.359820\pi$
$137$	9.97925	0.852585	0.426292	+	0.904585i	$0.359820\pi$
$138$	0	0
$139$	−6.88510	−0.583986	−0.291993	−	0.956420i	$-0.594318\pi$
$139$	−6.88510	−0.583986	−0.291993	+	0.956420i	$0.594318\pi$
$140$	0	0
$141$	−1.68540	−0.141936
$142$	0	0
$143$	−0.170003	−0.0142164
$144$	0	0
$145$	15.1967	1.26202
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.5701	−0.947860	−0.473930	−	0.880563i	$-0.657165\pi$
$149$	−11.5701	−0.947860	−0.473930	+	0.880563i	$0.657165\pi$
$150$	0	0
$151$	−20.8805	−1.69923	−0.849616	−	0.527401i	$-0.823167\pi$
$151$	−20.8805	−1.69923	−0.849616	+	0.527401i	$0.823167\pi$
$152$	0	0
$153$	−9.59619	−0.775806
$154$	0	0
$155$	14.6369	1.17566
$156$	0	0
$157$	−20.5341	−1.63880	−0.819400	−	0.573222i	$-0.805693\pi$
$157$	−20.5341	−1.63880	−0.819400	+	0.573222i	$0.805693\pi$
$158$	0	0
$159$	3.04365	0.241377
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.1021	−1.49619	−0.748094	−	0.663592i	$-0.769031\pi$
$163$	−19.1021	−1.49619	−0.748094	+	0.663592i	$0.769031\pi$
$164$	0	0
$165$	0.0391308	0.00304633
$166$	0	0
$167$	−2.14270	−0.165807	−0.0829036	−	0.996558i	$-0.526419\pi$
$167$	−2.14270	−0.165807	−0.0829036	+	0.996558i	$0.526419\pi$
$168$	0	0
$169$	10.9624	0.843263
$170$	0	0
$171$	10.2108	0.780838
$172$	0	0
$173$	−22.1008	−1.68029	−0.840146	−	0.542360i	$-0.817531\pi$
$173$	−22.1008	−1.68029	−0.840146	+	0.542360i	$0.817531\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.39935	0.105182
$178$	0	0
$179$	−4.74876	−0.354939	−0.177469	−	0.984126i	$-0.556791\pi$
$179$	−4.74876	−0.354939	−0.177469	+	0.984126i	$0.556791\pi$
$180$	0	0
$181$	23.5181	1.74809	0.874044	−	0.485846i	$-0.161489\pi$
$181$	23.5181	1.74809	0.874044	+	0.485846i	$0.161489\pi$
$182$	0	0
$183$	−3.81387	−0.281929
$184$	0	0
$185$	−13.8733	−1.01998
$186$	0	0
$187$	0.121348	0.00887384
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.62903	−0.624375	−0.312187	−	0.950021i	$-0.601062\pi$
$191$	−8.62903	−0.624375	−0.312187	+	0.950021i	$0.601062\pi$
$192$	0	0
$193$	9.39697	0.676409	0.338204	−	0.941073i	$-0.390181\pi$
$193$	9.39697	0.676409	0.338204	+	0.941073i	$0.390181\pi$
$194$	0	0
$195$	−5.51560	−0.394980
$196$	0	0
$197$	−6.23988	−0.444573	−0.222286	−	0.974981i	$-0.571352\pi$
$197$	−6.23988	−0.444573	−0.222286	+	0.974981i	$0.571352\pi$
$198$	0	0
$199$	−0.445056	−0.0315492	−0.0157746	−	0.999876i	$-0.505021\pi$
$199$	−0.445056	−0.0315492	−0.0157746	+	0.999876i	$0.505021\pi$
$200$	0	0
$201$	−2.75664	−0.194438
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.23729	0.156259
$206$	0	0
$207$	4.81994	0.335009
$208$	0	0
$209$	−0.129120	−0.00893139
$210$	0	0
$211$	−1.08557	−0.0747339	−0.0373670	−	0.999302i	$-0.511897\pi$
$211$	−1.08557	−0.0747339	−0.0373670	+	0.999302i	$0.511897\pi$
$212$	0	0
$213$	−4.47971	−0.306945
$214$	0	0
$215$	8.98865	0.613021
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.32099	0.427133
$220$	0	0
$221$	−17.1043	−1.15056
$222$	0	0
$223$	18.1979	1.21862	0.609310	−	0.792932i	$-0.291447\pi$
$223$	18.1979	1.21862	0.609310	+	0.792932i	$0.291447\pi$
$224$	0	0
$225$	−0.0149779	−0.000998528 0
$226$	0	0
$227$	−4.72895	−0.313872	−0.156936	−	0.987609i	$-0.550162\pi$
$227$	−4.72895	−0.313872	−0.156936	+	0.987609i	$0.550162\pi$
$228$	0	0
$229$	21.4742	1.41905	0.709527	−	0.704678i	$-0.248909\pi$
$229$	21.4742	1.41905	0.709527	+	0.704678i	$0.248909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.4815	−1.34179	−0.670894	−	0.741553i	$-0.734089\pi$
$233$	−20.4815	−1.34179	−0.670894	+	0.741553i	$0.734089\pi$
$234$	0	0
$235$	−7.48718	−0.488410
$236$	0	0
$237$	5.56996	0.361808
$238$	0	0
$239$	−25.1167	−1.62466	−0.812331	−	0.583197i	$-0.801802\pi$
$239$	−25.1167	−1.62466	−0.812331	+	0.583197i	$0.801802\pi$
$240$	0	0
$241$	2.82214	0.181790	0.0908951	−	0.995860i	$-0.471027\pi$
$241$	2.82214	0.181790	0.0908951	+	0.995860i	$0.471027\pi$
$242$	0	0
$243$	12.0974	0.776047
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.1998	1.15802
$248$	0	0
$249$	−8.45360	−0.535725
$250$	0	0
$251$	−15.6780	−0.989589	−0.494794	−	0.869010i	$-0.664757\pi$
$251$	−15.6780	−0.989589	−0.494794	+	0.869010i	$0.664757\pi$
$252$	0	0
$253$	−0.0609502	−0.00383191
$254$	0	0
$255$	3.93702	0.246546
$256$	0	0
$257$	26.8874	1.67719	0.838596	−	0.544754i	$-0.183377\pi$
$257$	26.8874	1.67719	0.838596	+	0.544754i	$0.183377\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−18.6546	−1.15469
$262$	0	0
$263$	0.259170	0.0159811	0.00799055	−	0.999968i	$-0.497457\pi$
$263$	0.259170	0.0159811	0.00799055	+	0.999968i	$0.497457\pi$
$264$	0	0
$265$	13.5211	0.830593
$266$	0	0
$267$	−8.68875	−0.531743
$268$	0	0
$269$	6.46974	0.394467	0.197234	−	0.980357i	$-0.436804\pi$
$269$	6.46974	0.394467	0.197234	+	0.980357i	$0.436804\pi$
$270$	0	0
$271$	22.0403	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$271$	22.0403	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.000189402 0	1.14214e−5 0
$276$	0	0
$277$	−15.9441	−0.957986	−0.478993	−	0.877819i	$-0.658998\pi$
$277$	−15.9441	−0.957986	−0.478993	+	0.877819i	$0.658998\pi$
$278$	0	0
$279$	−17.9674	−1.07568
$280$	0	0
$281$	0.325569	0.0194218	0.00971090	−	0.999953i	$-0.496909\pi$
$281$	0.325569	0.0194218	0.00971090	+	0.999953i	$0.496909\pi$
$282$	0	0
$283$	1.76090	0.104675	0.0523373	−	0.998629i	$-0.483333\pi$
$283$	1.76090	0.104675	0.0523373	+	0.998629i	$0.483333\pi$
$284$	0	0
$285$	−4.18917	−0.248145
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4.79096	−0.281821
$290$	0	0
$291$	−2.41560	−0.141605
$292$	0	0
$293$	−10.5794	−0.618058	−0.309029	−	0.951053i	$-0.600004\pi$
$293$	−10.5794	−0.618058	−0.309029	+	0.951053i	$0.600004\pi$
$294$	0	0
$295$	6.21647	0.361937
$296$	0	0
$297$	−0.100506	−0.00583193
$298$	0	0
$299$	8.59110	0.496836
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.09933	−0.235500
$304$	0	0
$305$	−16.9427	−0.970135
$306$	0	0
$307$	15.3000	0.873215	0.436608	−	0.899652i	$-0.356180\pi$
$307$	15.3000	0.873215	0.436608	+	0.899652i	$0.356180\pi$
$308$	0	0
$309$	2.61454	0.148736
$310$	0	0
$311$	6.59334	0.373874	0.186937	−	0.982372i	$-0.440144\pi$
$311$	6.59334	0.373874	0.186937	+	0.982372i	$0.440144\pi$
$312$	0	0
$313$	13.4815	0.762020	0.381010	−	0.924571i	$-0.375576\pi$
$313$	13.4815	0.762020	0.381010	+	0.924571i	$0.375576\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.2309	−0.686958	−0.343479	−	0.939160i	$-0.611605\pi$
$317$	−12.2309	−0.686958	−0.343479	+	0.939160i	$0.611605\pi$
$318$	0	0
$319$	0.235895	0.0132076
$320$	0	0
$321$	4.30017	0.240012
$322$	0	0
$323$	−12.9910	−0.722837
$324$	0	0
$325$	−0.0266968	−0.00148087
$326$	0	0
$327$	−6.28949	−0.347809
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.24358	−0.508073	−0.254037	−	0.967195i	$-0.581758\pi$
$331$	−9.24358	−0.508073	−0.254037	+	0.967195i	$0.581758\pi$
$332$	0	0
$333$	17.0300	0.933240
$334$	0	0
$335$	−12.2461	−0.669074
$336$	0	0
$337$	22.7806	1.24094	0.620470	−	0.784230i	$-0.286942\pi$
$337$	22.7806	1.24094	0.620470	+	0.784230i	$0.286942\pi$
$338$	0	0
$339$	−8.23738	−0.447393
$340$	0	0
$341$	0.227205	0.0123038
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.97747	−0.106464
$346$	0	0
$347$	13.4907	0.724217	0.362109	−	0.932136i	$-0.382057\pi$
$347$	13.4907	0.724217	0.362109	+	0.932136i	$0.382057\pi$
$348$	0	0
$349$	−11.9983	−0.642256	−0.321128	−	0.947036i	$-0.604062\pi$
$349$	−11.9983	−0.642256	−0.321128	+	0.947036i	$0.604062\pi$
$350$	0	0
$351$	14.1665	0.756154
$352$	0	0
$353$	−0.965277	−0.0513765	−0.0256883	−	0.999670i	$-0.508178\pi$
$353$	−0.965277	−0.0513765	−0.0256883	+	0.999670i	$0.508178\pi$
$354$	0	0
$355$	−19.9006	−1.05621
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.6388	1.61705	0.808526	−	0.588460i	$-0.200266\pi$
$359$	30.6388	1.61705	0.808526	+	0.588460i	$0.200266\pi$
$360$	0	0
$361$	−5.17702	−0.272475
$362$	0	0
$363$	−5.53925	−0.290735
$364$	0	0
$365$	28.0803	1.46979
$366$	0	0
$367$	25.6550	1.33918	0.669591	−	0.742730i	$-0.266469\pi$
$367$	25.6550	1.33918	0.669591	+	0.742730i	$0.266469\pi$
$368$	0	0
$369$	−2.74636	−0.142970
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.96781	−0.309001	−0.154501	−	0.987993i	$-0.549377\pi$
$373$	−5.96781	−0.309001	−0.154501	+	0.987993i	$0.549377\pi$
$374$	0	0
$375$	−5.62760	−0.290608
$376$	0	0
$377$	−33.2500	−1.71246
$378$	0	0
$379$	−17.5808	−0.903068	−0.451534	−	0.892254i	$-0.649123\pi$
$379$	−17.5808	−0.903068	−0.451534	+	0.892254i	$0.649123\pi$
$380$	0	0
$381$	−4.97330	−0.254790
$382$	0	0
$383$	15.2734	0.780436	0.390218	−	0.920723i	$-0.372400\pi$
$383$	15.2734	0.780436	0.390218	+	0.920723i	$0.372400\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.0339	−0.560887
$388$	0	0
$389$	30.0106	1.52160	0.760799	−	0.648987i	$-0.224807\pi$
$389$	30.0106	1.52160	0.760799	+	0.648987i	$0.224807\pi$
$390$	0	0
$391$	−6.13231	−0.310124
$392$	0	0
$393$	−0.973617	−0.0491125
$394$	0	0
$395$	24.7439	1.24500
$396$	0	0
$397$	26.1206	1.31095	0.655477	−	0.755215i	$-0.272467\pi$
$397$	26.1206	1.31095	0.655477	+	0.755215i	$0.272467\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0174	−1.19937	−0.599686	−	0.800235i	$-0.704708\pi$
$401$	−24.0174	−1.19937	−0.599686	+	0.800235i	$0.704708\pi$
$402$	0	0
$403$	−32.0252	−1.59529
$404$	0	0
$405$	15.1724	0.753923
$406$	0	0
$407$	−0.215352	−0.0106746
$408$	0	0
$409$	17.5462	0.867602	0.433801	−	0.901009i	$-0.357172\pi$
$409$	17.5462	0.867602	0.433801	+	0.901009i	$0.357172\pi$
$410$	0	0
$411$	5.02578	0.247903
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−37.5541	−1.84346
$416$	0	0
$417$	−3.46749	−0.169804
$418$	0	0
$419$	12.0459	0.588482	0.294241	−	0.955731i	$-0.404933\pi$
$419$	12.0459	0.588482	0.294241	+	0.955731i	$0.404933\pi$
$420$	0	0
$421$	−8.45948	−0.412290	−0.206145	−	0.978521i	$-0.566092\pi$
$421$	−8.45948	−0.412290	−0.206145	+	0.978521i	$0.566092\pi$
$422$	0	0
$423$	9.19083	0.446874
$424$	0	0
$425$	0.0190561	0.000924357 0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.0856175	−0.00413365
$430$	0	0
$431$	12.7468	0.613993	0.306997	−	0.951711i	$-0.400676\pi$
$431$	12.7468	0.613993	0.306997	+	0.951711i	$0.400676\pi$
$432$	0	0
$433$	−0.834608	−0.0401087	−0.0200544	−	0.999799i	$-0.506384\pi$
$433$	−0.834608	−0.0401087	−0.0200544	+	0.999799i	$0.506384\pi$
$434$	0	0
$435$	7.65340	0.366952
$436$	0	0
$437$	6.52506	0.312136
$438$	0	0
$439$	−33.0626	−1.57799	−0.788996	−	0.614399i	$-0.789399\pi$
$439$	−33.0626	−1.57799	−0.788996	+	0.614399i	$0.789399\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.9251	1.51681	0.758403	−	0.651786i	$-0.225980\pi$
$443$	31.9251	1.51681	0.758403	+	0.651786i	$0.225980\pi$
$444$	0	0
$445$	−38.5988	−1.82976
$446$	0	0
$447$	−5.82697	−0.275606
$448$	0	0
$449$	−16.7227	−0.789193	−0.394597	−	0.918854i	$-0.629116\pi$
$449$	−16.7227	−0.789193	−0.394597	+	0.918854i	$0.629116\pi$
$450$	0	0
$451$	0.0347289	0.00163532
$452$	0	0
$453$	−10.5159	−0.494081
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.13120	−0.286806	−0.143403	−	0.989664i	$-0.545804\pi$
$457$	−6.13120	−0.286806	−0.143403	+	0.989664i	$0.545804\pi$
$458$	0	0
$459$	−10.1121	−0.471990
$460$	0	0
$461$	12.3826	0.576715	0.288358	−	0.957523i	$-0.406891\pi$
$461$	12.3826	0.576715	0.288358	+	0.957523i	$0.406891\pi$
$462$	0	0
$463$	−15.1953	−0.706187	−0.353094	−	0.935588i	$-0.614870\pi$
$463$	−15.1953	−0.706187	−0.353094	+	0.935588i	$0.614870\pi$
$464$	0	0
$465$	7.37146	0.341843
$466$	0	0
$467$	17.6623	0.817312	0.408656	−	0.912689i	$-0.365998\pi$
$467$	17.6623	0.817312	0.408656	+	0.912689i	$0.365998\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.3414	−0.476509
$472$	0	0
$473$	0.139529	0.00641555
$474$	0	0
$475$	−0.0202766	−0.000930352 0
$476$	0	0
$477$	−16.5977	−0.759956
$478$	0	0
$479$	−10.0089	−0.457318	−0.228659	−	0.973507i	$-0.573434\pi$
$479$	−10.0089	−0.457318	−0.228659	+	0.973507i	$0.573434\pi$
$480$	0	0
$481$	30.3545	1.38404
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.7310	−0.487270
$486$	0	0
$487$	12.9803	0.588195	0.294098	−	0.955775i	$-0.404981\pi$
$487$	12.9803	0.588195	0.294098	+	0.955775i	$0.404981\pi$
$488$	0	0
$489$	−9.62023	−0.435042
$490$	0	0
$491$	19.0806	0.861094	0.430547	−	0.902568i	$-0.358321\pi$
$491$	19.0806	0.861094	0.430547	+	0.902568i	$0.358321\pi$
$492$	0	0
$493$	23.7338	1.06892
$494$	0	0
$495$	−0.213389	−0.00959111
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−36.0503	−1.61383	−0.806916	−	0.590667i	$-0.798865\pi$
$499$	−36.0503	−1.61383	−0.806916	+	0.590667i	$0.798865\pi$
$500$	0	0
$501$	−1.07911	−0.0482112
$502$	0	0
$503$	28.3359	1.26344	0.631718	−	0.775198i	$-0.282350\pi$
$503$	28.3359	1.26344	0.631718	+	0.775198i	$0.282350\pi$
$504$	0	0
$505$	−18.2108	−0.810370
$506$	0	0
$507$	5.52093	0.245193
$508$	0	0
$509$	−19.6986	−0.873125	−0.436563	−	0.899674i	$-0.643804\pi$
$509$	−19.6986	−0.873125	−0.436563	+	0.899674i	$0.643804\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.7597	0.475052
$514$	0	0
$515$	11.6148	0.511808
$516$	0	0
$517$	−0.116222	−0.00511144
$518$	0	0
$519$	−11.1305	−0.488573
$520$	0	0
$521$	−3.39751	−0.148848	−0.0744238	−	0.997227i	$-0.523712\pi$
$521$	−3.39751	−0.148848	−0.0744238	+	0.997227i	$0.523712\pi$
$522$	0	0
$523$	−21.4286	−0.937006	−0.468503	−	0.883462i	$-0.655206\pi$
$523$	−21.4286	−0.937006	−0.468503	+	0.883462i	$0.655206\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.8595	0.995777
$528$	0	0
$529$	−19.9199	−0.866082
$530$	0	0
$531$	−7.63098	−0.331156
$532$	0	0
$533$	−4.89514	−0.212032
$534$	0	0
$535$	19.1030	0.825895
$536$	0	0
$537$	−2.39158	−0.103204
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23.4976	1.01024	0.505120	−	0.863049i	$-0.331448\pi$
$541$	23.4976	1.01024	0.505120	+	0.863049i	$0.331448\pi$
$542$	0	0
$543$	11.8443	0.508286
$544$	0	0
$545$	−27.9403	−1.19683
$546$	0	0
$547$	37.0416	1.58378	0.791891	−	0.610662i	$-0.209097\pi$
$547$	37.0416	1.58378	0.791891	+	0.610662i	$0.209097\pi$
$548$	0	0
$549$	20.7979	0.887631
$550$	0	0
$551$	−25.2539	−1.07585
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.98690	−0.296577
$556$	0	0
$557$	−9.58290	−0.406040	−0.203020	−	0.979175i	$-0.565076\pi$
$557$	−9.58290	−0.406040	−0.203020	+	0.979175i	$0.565076\pi$
$558$	0	0
$559$	−19.6670	−0.831825
$560$	0	0
$561$	0.0611136	0.00258022
$562$	0	0
$563$	2.63712	0.111141	0.0555706	−	0.998455i	$-0.482302\pi$
$563$	2.63712	0.111141	0.0555706	+	0.998455i	$0.482302\pi$
$564$	0	0
$565$	−36.5936	−1.53951
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.64222	0.110768	0.0553838	−	0.998465i	$-0.482362\pi$
$569$	2.64222	0.110768	0.0553838	+	0.998465i	$0.482362\pi$
$570$	0	0
$571$	46.5030	1.94609	0.973045	−	0.230616i	$-0.0740741\pi$
$571$	46.5030	1.94609	0.973045	+	0.230616i	$0.0740741\pi$
$572$	0	0
$573$	−4.34578	−0.181547
$574$	0	0
$575$	−0.00957143	−0.000399156 0
$576$	0	0
$577$	17.9120	0.745688	0.372844	−	0.927894i	$-0.378383\pi$
$577$	17.9120	0.745688	0.372844	+	0.927894i	$0.378383\pi$
$578$	0	0
$579$	4.73253	0.196677
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.209885	0.00869255
$584$	0	0
$585$	30.0777	1.24356
$586$	0	0
$587$	25.1374	1.03753	0.518766	−	0.854917i	$-0.326392\pi$
$587$	25.1374	1.03753	0.518766	+	0.854917i	$0.326392\pi$
$588$	0	0
$589$	−24.3236	−1.00224
$590$	0	0
$591$	−3.14254	−0.129267
$592$	0	0
$593$	6.48493	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$593$	6.48493	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.224140	−0.00917346
$598$	0	0
$599$	39.6276	1.61914	0.809569	−	0.587025i	$-0.199701\pi$
$599$	39.6276	1.61914	0.809569	+	0.587025i	$0.199701\pi$
$600$	0	0
$601$	17.5441	0.715638	0.357819	−	0.933791i	$-0.383521\pi$
$601$	17.5441	0.715638	0.357819	+	0.933791i	$0.383521\pi$
$602$	0	0
$603$	15.0326	0.612173
$604$	0	0
$605$	−24.6075	−1.00044
$606$	0	0
$607$	−16.6556	−0.676029	−0.338014	−	0.941141i	$-0.609755\pi$
$607$	−16.6556	−0.676029	−0.338014	+	0.941141i	$0.609755\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.3818	0.662737
$612$	0	0
$613$	−30.6412	−1.23759	−0.618793	−	0.785554i	$-0.712378\pi$
$613$	−30.6412	−1.23759	−0.618793	+	0.785554i	$0.712378\pi$
$614$	0	0
$615$	1.12675	0.0454349
$616$	0	0
$617$	−25.2343	−1.01589	−0.507947	−	0.861388i	$-0.669595\pi$
$617$	−25.2343	−1.01589	−0.507947	+	0.861388i	$0.669595\pi$
$618$	0	0
$619$	−13.9528	−0.560811	−0.280405	−	0.959882i	$-0.590469\pi$
$619$	−13.9528	−0.560811	−0.280405	+	0.959882i	$0.590469\pi$
$620$	0	0
$621$	5.07905	0.203815
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−25.0272	−1.00109
$626$	0	0
$627$	−0.0650276	−0.00259695
$628$	0	0
$629$	−21.6670	−0.863918
$630$	0	0
$631$	24.2451	0.965182	0.482591	−	0.875846i	$-0.339696\pi$
$631$	24.2451	0.965182	0.482591	+	0.875846i	$0.339696\pi$
$632$	0	0
$633$	−0.546719	−0.0217301
$634$	0	0
$635$	−22.0933	−0.876746
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.4288	0.966389
$640$	0	0
$641$	17.0715	0.674285	0.337143	−	0.941454i	$-0.390540\pi$
$641$	17.0715	0.674285	0.337143	+	0.941454i	$0.390540\pi$
$642$	0	0
$643$	23.5024	0.926843	0.463422	−	0.886138i	$-0.346622\pi$
$643$	23.5024	0.926843	0.463422	+	0.886138i	$0.346622\pi$
$644$	0	0
$645$	4.52689	0.178246
$646$	0	0
$647$	43.1874	1.69787	0.848936	−	0.528495i	$-0.177244\pi$
$647$	43.1874	1.69787	0.848936	+	0.528495i	$0.177244\pi$
$648$	0	0
$649$	0.0964970	0.00378784
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.7968	−1.08777	−0.543886	−	0.839159i	$-0.683048\pi$
$653$	−27.7968	−1.08777	−0.543886	+	0.839159i	$0.683048\pi$
$654$	0	0
$655$	−4.32518	−0.168999
$656$	0	0
$657$	−34.4697	−1.34479
$658$	0	0
$659$	18.3588	0.715157	0.357578	−	0.933883i	$-0.383602\pi$
$659$	18.3588	0.715157	0.357578	+	0.933883i	$0.383602\pi$
$660$	0	0
$661$	−14.3458	−0.557986	−0.278993	−	0.960293i	$-0.590001\pi$
$661$	−14.3458	−0.557986	−0.278993	+	0.960293i	$0.590001\pi$
$662$	0	0
$663$	−8.61413	−0.334545
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.9209	−0.461581
$668$	0	0
$669$	9.16487	0.354334
$670$	0	0
$671$	−0.262998	−0.0101529
$672$	0	0
$673$	−13.8763	−0.534891	−0.267445	−	0.963573i	$-0.586180\pi$
$673$	−13.8763	−0.534891	−0.267445	+	0.963573i	$0.586180\pi$
$674$	0	0
$675$	−0.0157831	−0.000607491 0
$676$	0	0
$677$	9.06289	0.348315	0.174158	−	0.984718i	$-0.444280\pi$
$677$	9.06289	0.348315	0.174158	+	0.984718i	$0.444280\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.38161	−0.0912635
$682$	0	0
$683$	−9.03714	−0.345797	−0.172898	−	0.984940i	$-0.555313\pi$
$683$	−9.03714	−0.345797	−0.172898	+	0.984940i	$0.555313\pi$
$684$	0	0
$685$	22.3264	0.853049
$686$	0	0
$687$	10.8149	0.412614
$688$	0	0
$689$	−29.5839	−1.12706
$690$	0	0
$691$	−2.61647	−0.0995352	−0.0497676	−	0.998761i	$-0.515848\pi$
$691$	−2.61647	−0.0995352	−0.0497676	+	0.998761i	$0.515848\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.4039	−0.584305
$696$	0	0
$697$	3.49414	0.132350
$698$	0	0
$699$	−10.3150	−0.390147
$700$	0	0
$701$	−17.1379	−0.647288	−0.323644	−	0.946179i	$-0.604908\pi$
$701$	−17.1379	−0.647288	−0.323644	+	0.946179i	$0.604908\pi$
$702$	0	0
$703$	23.0546	0.869522
$704$	0	0
$705$	−3.77072	−0.142013
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−20.1513	−0.756799	−0.378399	−	0.925642i	$-0.623525\pi$
$709$	−20.1513	−0.756799	−0.378399	+	0.925642i	$0.623525\pi$
$710$	0	0
$711$	−30.3742	−1.13912
$712$	0	0
$713$	−11.4818	−0.429997
$714$	0	0
$715$	−0.380346	−0.0142241
$716$	0	0
$717$	−12.6493	−0.472398
$718$	0	0
$719$	19.5824	0.730301	0.365150	−	0.930949i	$-0.381018\pi$
$719$	19.5824	0.730301	0.365150	+	0.930949i	$0.381018\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.42130	0.0528585
$724$	0	0
$725$	0.0370442	0.00137579
$726$	0	0
$727$	−22.5529	−0.836440	−0.418220	−	0.908346i	$-0.637346\pi$
$727$	−22.5529	−0.836440	−0.418220	+	0.908346i	$0.637346\pi$
$728$	0	0
$729$	−14.2523	−0.527863
$730$	0	0
$731$	14.0383	0.519224
$732$	0	0
$733$	−13.8629	−0.512036	−0.256018	−	0.966672i	$-0.582411\pi$
$733$	−13.8629	−0.512036	−0.256018	+	0.966672i	$0.582411\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.190093	−0.00700217
$738$	0	0
$739$	−27.8669	−1.02510	−0.512550	−	0.858657i	$-0.671299\pi$
$739$	−27.8669	−1.02510	−0.512550	+	0.858657i	$0.671299\pi$
$740$	0	0
$741$	9.16583	0.336715
$742$	0	0
$743$	−24.6049	−0.902667	−0.451334	−	0.892355i	$-0.649052\pi$
$743$	−24.6049	−0.902667	−0.451334	+	0.892355i	$0.649052\pi$
$744$	0	0
$745$	−25.8856	−0.948377
$746$	0	0
$747$	46.0993	1.68668
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.557235	0.0203338	0.0101669	−	0.999948i	$-0.496764\pi$
$751$	0.557235	0.0203338	0.0101669	+	0.999948i	$0.496764\pi$
$752$	0	0
$753$	−7.89582	−0.287740
$754$	0	0
$755$	−46.7157	−1.70016
$756$	0	0
$757$	0.923650	0.0335706	0.0167853	−	0.999859i	$-0.494657\pi$
$757$	0.923650	0.0335706	0.0167853	+	0.999859i	$0.494657\pi$
$758$	0	0
$759$	−0.0306959	−0.00111419
$760$	0	0
$761$	−24.0807	−0.872924	−0.436462	−	0.899723i	$-0.643769\pi$
$761$	−24.0807	−0.872924	−0.436462	+	0.899723i	$0.643769\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−21.4694	−0.776229
$766$	0	0
$767$	−13.6015	−0.491123
$768$	0	0
$769$	−12.5643	−0.453081	−0.226541	−	0.974002i	$-0.572742\pi$
$769$	−12.5643	−0.453081	−0.226541	+	0.974002i	$0.572742\pi$
$770$	0	0
$771$	13.5411	0.487672
$772$	0	0
$773$	−37.5637	−1.35107	−0.675536	−	0.737327i	$-0.736088\pi$
$773$	−37.5637	−1.35107	−0.675536	+	0.737327i	$0.736088\pi$
$774$	0	0
$775$	0.0356796	0.00128165
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.71793	−0.133209
$780$	0	0
$781$	−0.308913	−0.0110538
$782$	0	0
$783$	−19.6574	−0.702497
$784$	0	0
$785$	−45.9407	−1.63969
$786$	0	0
$787$	−10.0718	−0.359020	−0.179510	−	0.983756i	$-0.557451\pi$
$787$	−10.0718	−0.359020	−0.179510	+	0.983756i	$0.557451\pi$
$788$	0	0
$789$	0.130524	0.00464677
$790$	0	0
$791$	0	0
$792$	0	0
$793$	37.0703	1.31640
$794$	0	0
$795$	6.80953	0.241509
$796$	0	0
$797$	−9.25256	−0.327743	−0.163871	−	0.986482i	$-0.552398\pi$
$797$	−9.25256	−0.327743	−0.163871	+	0.986482i	$0.552398\pi$
$798$	0	0
$799$	−11.6933	−0.413680
$800$	0	0
$801$	47.3816	1.67415
$802$	0	0
$803$	0.435884	0.0153820
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.25831	0.114698
$808$	0	0
$809$	−38.3361	−1.34782	−0.673912	−	0.738811i	$-0.735387\pi$
$809$	−38.3361	−1.34782	−0.673912	+	0.738811i	$0.735387\pi$
$810$	0	0
$811$	23.2094	0.814992	0.407496	−	0.913207i	$-0.366402\pi$
$811$	23.2094	0.814992	0.407496	+	0.913207i	$0.366402\pi$
$812$	0	0
$813$	11.1000	0.389295
$814$	0	0
$815$	−42.7368	−1.49700
$816$	0	0
$817$	−14.9373	−0.522592
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.4485	−1.02776	−0.513880	−	0.857862i	$-0.671792\pi$
$821$	−29.4485	−1.02776	−0.513880	+	0.857862i	$0.671792\pi$
$822$	0	0
$823$	32.1900	1.12207	0.561036	−	0.827791i	$-0.310403\pi$
$823$	32.1900	1.12207	0.561036	+	0.827791i	$0.310403\pi$
$824$	0	0
$825$	9.53873e−5 0	3.32096e−6 0
$826$	0	0
$827$	−42.2487	−1.46913	−0.734566	−	0.678537i	$-0.762614\pi$
$827$	−42.2487	−1.46913	−0.734566	+	0.678537i	$0.762614\pi$
$828$	0	0
$829$	−36.7460	−1.27624	−0.638121	−	0.769936i	$-0.720288\pi$
$829$	−36.7460	−1.27624	−0.638121	+	0.769936i	$0.720288\pi$
$830$	0	0
$831$	−8.02980	−0.278551
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.79384	−0.165898
$836$	0	0
$837$	−18.9332	−0.654429
$838$	0	0
$839$	−45.6966	−1.57762	−0.788811	−	0.614636i	$-0.789303\pi$
$839$	−45.6966	−1.57762	−0.788811	+	0.614636i	$0.789303\pi$
$840$	0	0
$841$	17.1375	0.590947
$842$	0	0
$843$	0.163964	0.00564722
$844$	0	0
$845$	24.5261	0.843723
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.886830	0.0304359
$850$	0	0
$851$	10.8828	0.373058
$852$	0	0
$853$	31.6441	1.08347	0.541736	−	0.840549i	$-0.317767\pi$
$853$	31.6441	1.08347	0.541736	+	0.840549i	$0.317767\pi$
$854$	0	0
$855$	22.8444	0.781264
$856$	0	0
$857$	5.24557	0.179185	0.0895927	−	0.995978i	$-0.471443\pi$
$857$	5.24557	0.179185	0.0895927	+	0.995978i	$0.471443\pi$
$858$	0	0
$859$	10.0109	0.341569	0.170785	−	0.985308i	$-0.445370\pi$
$859$	10.0109	0.341569	0.170785	+	0.985308i	$0.445370\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.8606	−1.76535	−0.882677	−	0.469980i	$-0.844261\pi$
$863$	−51.8606	−1.76535	−0.882677	+	0.469980i	$0.844261\pi$
$864$	0	0
$865$	−49.4458	−1.68121
$866$	0	0
$867$	−2.41284	−0.0819443
$868$	0	0
$869$	0.384095	0.0130295
$870$	0	0
$871$	26.7941	0.907885
$872$	0	0
$873$	13.1728	0.445831
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.91274	0.267194	0.133597	−	0.991036i	$-0.457347\pi$
$877$	7.91274	0.267194	0.133597	+	0.991036i	$0.457347\pi$
$878$	0	0
$879$	−5.32805	−0.179711
$880$	0	0
$881$	2.15112	0.0724730	0.0362365	−	0.999343i	$-0.488463\pi$
$881$	2.15112	0.0724730	0.0362365	+	0.999343i	$0.488463\pi$
$882$	0	0
$883$	55.7464	1.87602	0.938008	−	0.346614i	$-0.112668\pi$
$883$	55.7464	1.87602	0.938008	+	0.346614i	$0.112668\pi$
$884$	0	0
$885$	3.13076	0.105239
$886$	0	0
$887$	−0.323643	−0.0108669	−0.00543344	−	0.999985i	$-0.501730\pi$
$887$	−0.323643	−0.0108669	−0.00543344	+	0.999985i	$0.501730\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0.235518	0.00789015
$892$	0	0
$893$	12.4422	0.416363
$894$	0	0
$895$	−10.6243	−0.355132
$896$	0	0
$897$	4.32668	0.144464
$898$	0	0
$899$	44.4379	1.48209
$900$	0	0
$901$	21.1169	0.703506
$902$	0	0
$903$	0	0
$904$	0	0
$905$	52.6168	1.74904
$906$	0	0
$907$	−18.7725	−0.623329	−0.311665	−	0.950192i	$-0.600887\pi$
$907$	−18.7725	−0.623329	−0.311665	+	0.950192i	$0.600887\pi$
$908$	0	0
$909$	22.3545	0.741452
$910$	0	0
$911$	23.2737	0.771093	0.385546	−	0.922688i	$-0.374013\pi$
$911$	23.2737	0.771093	0.385546	+	0.922688i	$0.374013\pi$
$912$	0	0
$913$	−0.582945	−0.0192927
$914$	0	0
$915$	−8.53273	−0.282083
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7.05597	0.232755	0.116377	−	0.993205i	$-0.462872\pi$
$919$	7.05597	0.232755	0.116377	+	0.993205i	$0.462872\pi$
$920$	0	0
$921$	7.70541	0.253902
$922$	0	0
$923$	43.5421	1.43321
$924$	0	0
$925$	−0.0338182	−0.00111194
$926$	0	0
$927$	−14.2576	−0.468282
$928$	0	0
$929$	37.4705	1.22937	0.614683	−	0.788774i	$-0.289284\pi$
$929$	37.4705	1.22937	0.614683	+	0.788774i	$0.289284\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.32056	0.108710
$934$	0	0
$935$	0.271490	0.00887868
$936$	0	0
$937$	−11.8703	−0.387786	−0.193893	−	0.981023i	$-0.562112\pi$
$937$	−11.8703	−0.387786	−0.193893	+	0.981023i	$0.562112\pi$
$938$	0	0
$939$	6.78960	0.221570
$940$	0	0
$941$	28.5706	0.931376	0.465688	−	0.884949i	$-0.345807\pi$
$941$	28.5706	0.931376	0.465688	+	0.884949i	$0.345807\pi$
$942$	0	0
$943$	−1.75503	−0.0571515
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54.7008	1.77754	0.888769	−	0.458356i	$-0.151562\pi$
$947$	54.7008	1.77754	0.888769	+	0.458356i	$0.151562\pi$
$948$	0	0
$949$	−61.4391	−1.99440
$950$	0	0
$951$	−6.15978	−0.199745
$952$	0	0
$953$	39.8735	1.29163	0.645814	−	0.763495i	$-0.276518\pi$
$953$	39.8735	1.29163	0.645814	+	0.763495i	$0.276518\pi$
$954$	0	0
$955$	−19.3056	−0.624715
$956$	0	0
$957$	0.118802	0.00384033
$958$	0	0
$959$	0	0
$960$	0	0
$961$	11.8009	0.380674
$962$	0	0
$963$	−23.4497	−0.755657
$964$	0	0
$965$	21.0237	0.676777
$966$	0	0
$967$	−3.68499	−0.118501	−0.0592507	−	0.998243i	$-0.518871\pi$
$967$	−3.68499	−0.118501	−0.0592507	+	0.998243i	$0.518871\pi$
$968$	0	0
$969$	−6.54255	−0.210177
$970$	0	0
$971$	12.5399	0.402423	0.201212	−	0.979548i	$-0.435512\pi$
$971$	12.5399	0.402423	0.201212	+	0.979548i	$0.435512\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.0134451	−0.000430588 0
$976$	0	0
$977$	19.8910	0.636369	0.318184	−	0.948029i	$-0.396927\pi$
$977$	19.8910	0.636369	0.318184	+	0.948029i	$0.396927\pi$
$978$	0	0
$979$	−0.599161	−0.0191493
$980$	0	0
$981$	34.2979	1.09505
$982$	0	0
$983$	−5.26889	−0.168052	−0.0840258	−	0.996464i	$-0.526778\pi$
$983$	−5.26889	−0.168052	−0.0840258	+	0.996464i	$0.526778\pi$
$984$	0	0
$985$	−13.9604	−0.444815
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7.05109	−0.224212
$990$	0	0
$991$	55.4163	1.76036	0.880178	−	0.474644i	$-0.157423\pi$
$991$	55.4163	1.76036	0.880178	+	0.474644i	$0.157423\pi$
$992$	0	0
$993$	−4.65528	−0.147731
$994$	0	0
$995$	−0.995718	−0.0315664
$996$	0	0
$997$	0.479233	0.0151775	0.00758873	−	0.999971i	$-0.497584\pi$
$997$	0.479233	0.0151775	0.00758873	+	0.999971i	$0.497584\pi$
$998$	0	0
$999$	17.9455	0.567771

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.5		8
7.2	even	3		1148.2.i.d.165.4	✓	16
7.4	even	3		1148.2.i.d.821.4	yes	16
7.6	odd	2		8036.2.a.n.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.4	✓	16	7.2	even	3
1148.2.i.d.821.4	yes	16	7.4	even	3
8036.2.a.m.1.5		8	1.1	even	1	trivial
8036.2.a.n.1.4		8	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+0.503623 q^{3} +2.23729 q^{5} -2.74636 q^{9} +O(q^{10})\)	\(q+0.503623 q^{3} +2.23729 q^{5} -2.74636 q^{9} +0.0347289 q^{11} -4.89514 q^{13} +1.12675 q^{15} +3.49414 q^{17} -3.71793 q^{19} -1.75503 q^{23} +0.00545373 q^{25} -2.89400 q^{27} +6.79246 q^{29} +6.54224 q^{31} +0.0174903 q^{33} -6.20094 q^{37} -2.46531 q^{39} +1.00000 q^{41} +4.01766 q^{43} -6.14440 q^{45} -3.34655 q^{47} +1.75973 q^{51} +6.04352 q^{53} +0.0776986 q^{55} -1.87243 q^{57} +2.77858 q^{59} -7.57287 q^{61} -10.9518 q^{65} -5.47362 q^{67} -0.883871 q^{69} -8.89496 q^{71} +12.5510 q^{73} +0.00274662 q^{75} +11.0598 q^{79} +6.78161 q^{81} -16.7856 q^{83} +7.81740 q^{85} +3.42084 q^{87} -17.2525 q^{89} +3.29482 q^{93} -8.31807 q^{95} -4.79644 q^{97} -0.0953783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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