LMFDB - Embedded newform 8470.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +2.73205 q^{12} +4.00000 q^{13} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +0.535898 q^{17} -4.46410 q^{18} -0.732051 q^{19} -1.00000 q^{20} +2.73205 q^{21} +8.19615 q^{23} -2.73205 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -7.66025 q^{29} +2.73205 q^{30} +0.535898 q^{31} -1.00000 q^{32} -0.535898 q^{34} -1.00000 q^{35} +4.46410 q^{36} -2.73205 q^{37} +0.732051 q^{38} +10.9282 q^{39} +1.00000 q^{40} +2.19615 q^{41} -2.73205 q^{42} -4.92820 q^{43} -4.46410 q^{45} -8.19615 q^{46} +6.00000 q^{47} +2.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.46410 q^{51} +4.00000 q^{52} +5.26795 q^{53} -4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} +7.66025 q^{58} +8.39230 q^{59} -2.73205 q^{60} -8.00000 q^{61} -0.535898 q^{62} +4.46410 q^{63} +1.00000 q^{64} -4.00000 q^{65} +15.8564 q^{67} +0.535898 q^{68} +22.3923 q^{69} +1.00000 q^{70} +1.07180 q^{71} -4.46410 q^{72} -6.00000 q^{73} +2.73205 q^{74} +2.73205 q^{75} -0.732051 q^{76} -10.9282 q^{78} +7.66025 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} +14.9282 q^{83} +2.73205 q^{84} -0.535898 q^{85} +4.92820 q^{86} -20.9282 q^{87} -8.92820 q^{89} +4.46410 q^{90} +4.00000 q^{91} +8.19615 q^{92} +1.46410 q^{93} -6.00000 q^{94} +0.732051 q^{95} -2.73205 q^{96} +2.73205 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 8 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{21} + 6 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} + 8 q^{27} + 2 q^{28} + 2 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{45} - 6 q^{46} + 12 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{51} + 8 q^{52} + 14 q^{53} - 8 q^{54} - 2 q^{56} - 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} - 16 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{65} + 4 q^{67} + 8 q^{68} + 24 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} - 12 q^{73} + 2 q^{74} + 2 q^{75} + 2 q^{76} - 8 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 16 q^{83} + 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{87} - 4 q^{89} + 2 q^{90} + 8 q^{91} + 6 q^{92} - 4 q^{93} - 12 q^{94} - 2 q^{95} - 2 q^{96} + 2 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 8 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^21 + 6 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 + 8 * q^27 + 2 * q^28 + 2 * q^29 + 2 * q^30 + 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 - 2 * q^42 + 4 * q^43 - 2 * q^45 - 6 * q^46 + 12 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 4 * q^51 + 8 * q^52 + 14 * q^53 - 8 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 - 16 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^65 + 4 * q^67 + 8 * q^68 + 24 * q^69 + 2 * q^70 + 16 * q^71 - 2 * q^72 - 12 * q^73 + 2 * q^74 + 2 * q^75 + 2 * q^76 - 8 * q^78 - 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 16 * q^83 + 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^87 - 4 * q^89 + 2 * q^90 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 12 * q^94 - 2 * q^95 - 2 * q^96 + 2 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.73205	−1.11536
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	4.46410	1.48803
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	2.73205	0.788675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−0.267261
$15$	−2.73205	−0.705412
$16$	1.00000	0.250000
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	−4.46410	−1.05220
$19$	−0.732051	−0.167944	−0.0839720	−	0.996468i	$-0.526761\pi$
$19$	−0.732051	−0.167944	−0.0839720	+	0.996468i	$0.526761\pi$
$20$	−1.00000	−0.223607
$21$	2.73205	0.596182
$22$	0	0
$23$	8.19615	1.70902	0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615	1.70902	0.854508	+	0.519438i	$0.173859\pi$
$24$	−2.73205	−0.557678
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	4.00000	0.769800
$28$	1.00000	0.188982
$29$	−7.66025	−1.42247	−0.711237	−	0.702953i	$-0.751865\pi$
$29$	−7.66025	−1.42247	−0.711237	+	0.702953i	$0.751865\pi$
$30$	2.73205	0.498802
$31$	0.535898	0.0962502	0.0481251	−	0.998841i	$-0.484675\pi$
$31$	0.535898	0.0962502	0.0481251	+	0.998841i	$0.484675\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.535898	−0.0919058
$35$	−1.00000	−0.169031
$36$	4.46410	0.744017
$37$	−2.73205	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$38$	0.732051	0.118754
$39$	10.9282	1.74991
$40$	1.00000	0.158114
$41$	2.19615	0.342981	0.171491	−	0.985186i	$-0.445142\pi$
$41$	2.19615	0.342981	0.171491	+	0.985186i	$0.445142\pi$
$42$	−2.73205	−0.421565
$43$	−4.92820	−0.751544	−0.375772	−	0.926712i	$-0.622622\pi$
$43$	−4.92820	−0.751544	−0.375772	+	0.926712i	$0.622622\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	−8.19615	−1.20846
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	2.73205	0.394338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	1.46410	0.205015
$52$	4.00000	0.554700
$53$	5.26795	0.723608	0.361804	−	0.932254i	$-0.382161\pi$
$53$	5.26795	0.723608	0.361804	+	0.932254i	$0.382161\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−2.00000	−0.264906
$58$	7.66025	1.00584
$59$	8.39230	1.09259	0.546293	−	0.837594i	$-0.316039\pi$
$59$	8.39230	1.09259	0.546293	+	0.837594i	$0.316039\pi$
$60$	−2.73205	−0.352706
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−0.535898	−0.0680592
$63$	4.46410	0.562424
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	15.8564	1.93717	0.968584	−	0.248686i	$-0.0799987\pi$
$67$	15.8564	1.93717	0.968584	+	0.248686i	$0.0799987\pi$
$68$	0.535898	0.0649872
$69$	22.3923	2.69572
$70$	1.00000	0.119523
$71$	1.07180	0.127199	0.0635994	−	0.997976i	$-0.479742\pi$
$71$	1.07180	0.127199	0.0635994	+	0.997976i	$0.479742\pi$
$72$	−4.46410	−0.526099
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.73205	0.317594
$75$	2.73205	0.315470
$76$	−0.732051	−0.0839720
$77$	0	0
$78$	−10.9282	−1.23738
$79$	7.66025	0.861846	0.430923	−	0.902389i	$-0.358188\pi$
$79$	7.66025	0.861846	0.430923	+	0.902389i	$0.358188\pi$
$80$	−1.00000	−0.111803
$81$	−2.46410	−0.273789
$82$	−2.19615	−0.242524
$83$	14.9282	1.63858	0.819292	−	0.573377i	$-0.194367\pi$
$83$	14.9282	1.63858	0.819292	+	0.573377i	$0.194367\pi$
$84$	2.73205	0.298091
$85$	−0.535898	−0.0581263
$86$	4.92820	0.531422
$87$	−20.9282	−2.24374
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	4.46410	0.470558
$91$	4.00000	0.419314
$92$	8.19615	0.854508
$93$	1.46410	0.151820
$94$	−6.00000	−0.618853
$95$	0.732051	0.0751068
$96$	−2.73205	−0.278839
$97$	2.73205	0.277398	0.138699	−	0.990335i	$-0.455708\pi$
$97$	2.73205	0.277398	0.138699	+	0.990335i	$0.455708\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	−1.46410	−0.144968
$103$	2.39230	0.235721	0.117860	−	0.993030i	$-0.462396\pi$
$103$	2.39230	0.235721	0.117860	+	0.993030i	$0.462396\pi$
$104$	−4.00000	−0.392232
$105$	−2.73205	−0.266621
$106$	−5.26795	−0.511668
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	4.00000	0.384900
$109$	−16.7321	−1.60264	−0.801320	−	0.598236i	$-0.795868\pi$
$109$	−16.7321	−1.60264	−0.801320	+	0.598236i	$0.795868\pi$
$110$	0	0
$111$	−7.46410	−0.708461
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	2.00000	0.187317
$115$	−8.19615	−0.764295
$116$	−7.66025	−0.711237
$117$	17.8564	1.65083
$118$	−8.39230	−0.772574
$119$	0.535898	0.0491257
$120$	2.73205	0.249401
$121$	0	0
$122$	8.00000	0.724286
$123$	6.00000	0.541002
$124$	0.535898	0.0481251
$125$	−1.00000	−0.0894427
$126$	−4.46410	−0.397694
$127$	8.39230	0.744697	0.372348	−	0.928093i	$-0.378553\pi$
$127$	8.39230	0.744697	0.372348	+	0.928093i	$0.378553\pi$
$128$	−1.00000	−0.0883883
$129$	−13.4641	−1.18545
$130$	4.00000	0.350823
$131$	10.1962	0.890842	0.445421	−	0.895321i	$-0.353054\pi$
$131$	10.1962	0.890842	0.445421	+	0.895321i	$0.353054\pi$
$132$	0	0
$133$	−0.732051	−0.0634769
$134$	−15.8564	−1.36978
$135$	−4.00000	−0.344265
$136$	−0.535898	−0.0459529
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	−22.3923	−1.90616
$139$	−3.66025	−0.310459	−0.155229	−	0.987878i	$-0.549612\pi$
$139$	−3.66025	−0.310459	−0.155229	+	0.987878i	$0.549612\pi$
$140$	−1.00000	−0.0845154
$141$	16.3923	1.38048
$142$	−1.07180	−0.0899432
$143$	0	0
$144$	4.46410	0.372008
$145$	7.66025	0.636149
$146$	6.00000	0.496564
$147$	2.73205	0.225336
$148$	−2.73205	−0.224573
$149$	1.80385	0.147777	0.0738885	−	0.997267i	$-0.476459\pi$
$149$	1.80385	0.147777	0.0738885	+	0.997267i	$0.476459\pi$
$150$	−2.73205	−0.223071
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	0.732051	0.0593772
$153$	2.39230	0.193406
$154$	0	0
$155$	−0.535898	−0.0430444
$156$	10.9282	0.874957
$157$	−16.5359	−1.31971	−0.659854	−	0.751394i	$-0.729382\pi$
$157$	−16.5359	−1.31971	−0.659854	+	0.751394i	$0.729382\pi$
$158$	−7.66025	−0.609417
$159$	14.3923	1.14138
$160$	1.00000	0.0790569
$161$	8.19615	0.645947
$162$	2.46410	0.193598
$163$	12.9282	1.01262	0.506308	−	0.862353i	$-0.331010\pi$
$163$	12.9282	1.01262	0.506308	+	0.862353i	$0.331010\pi$
$164$	2.19615	0.171491
$165$	0	0
$166$	−14.9282	−1.15865
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	−2.73205	−0.210782
$169$	3.00000	0.230769
$170$	0.535898	0.0411015
$171$	−3.26795	−0.249906
$172$	−4.92820	−0.375772
$173$	4.92820	0.374684	0.187342	−	0.982295i	$-0.440013\pi$
$173$	4.92820	0.374684	0.187342	+	0.982295i	$0.440013\pi$
$174$	20.9282	1.58656
$175$	1.00000	0.0755929
$176$	0	0
$177$	22.9282	1.72339
$178$	8.92820	0.669197
$179$	6.39230	0.477783	0.238892	−	0.971046i	$-0.423216\pi$
$179$	6.39230	0.477783	0.238892	+	0.971046i	$0.423216\pi$
$180$	−4.46410	−0.332734
$181$	−4.92820	−0.366310	−0.183155	−	0.983084i	$-0.558631\pi$
$181$	−4.92820	−0.366310	−0.183155	+	0.983084i	$0.558631\pi$
$182$	−4.00000	−0.296500
$183$	−21.8564	−1.61567
$184$	−8.19615	−0.604228
$185$	2.73205	0.200864
$186$	−1.46410	−0.107353
$187$	0	0
$188$	6.00000	0.437595
$189$	4.00000	0.290957
$190$	−0.732051	−0.0531085
$191$	−20.3923	−1.47554	−0.737768	−	0.675055i	$-0.764120\pi$
$191$	−20.3923	−1.47554	−0.737768	+	0.675055i	$0.764120\pi$
$192$	2.73205	0.197169
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−2.73205	−0.196150
$195$	−10.9282	−0.782585
$196$	1.00000	0.0714286
$197$	−3.46410	−0.246807	−0.123404	−	0.992357i	$-0.539381\pi$
$197$	−3.46410	−0.246807	−0.123404	+	0.992357i	$0.539381\pi$
$198$	0	0
$199$	−1.85641	−0.131597	−0.0657986	−	0.997833i	$-0.520959\pi$
$199$	−1.85641	−0.131597	−0.0657986	+	0.997833i	$0.520959\pi$
$200$	−1.00000	−0.0707107
$201$	43.3205	3.05559
$202$	−8.00000	−0.562878
$203$	−7.66025	−0.537644
$204$	1.46410	0.102508
$205$	−2.19615	−0.153386
$206$	−2.39230	−0.166680
$207$	36.5885	2.54307
$208$	4.00000	0.277350
$209$	0	0
$210$	2.73205	0.188529
$211$	−13.8564	−0.953914	−0.476957	−	0.878927i	$-0.658260\pi$
$211$	−13.8564	−0.953914	−0.476957	+	0.878927i	$0.658260\pi$
$212$	5.26795	0.361804
$213$	2.92820	0.200637
$214$	2.00000	0.136717
$215$	4.92820	0.336101
$216$	−4.00000	−0.272166
$217$	0.535898	0.0363792
$218$	16.7321	1.13324
$219$	−16.3923	−1.10769
$220$	0	0
$221$	2.14359	0.144194
$222$	7.46410	0.500958
$223$	−11.4641	−0.767693	−0.383847	−	0.923397i	$-0.625401\pi$
$223$	−11.4641	−0.767693	−0.383847	+	0.923397i	$0.625401\pi$
$224$	−1.00000	−0.0668153
$225$	4.46410	0.297607
$226$	−18.0000	−1.19734
$227$	−23.3205	−1.54784	−0.773918	−	0.633286i	$-0.781706\pi$
$227$	−23.3205	−1.54784	−0.773918	+	0.633286i	$0.781706\pi$
$228$	−2.00000	−0.132453
$229$	23.3205	1.54106	0.770531	−	0.637402i	$-0.219991\pi$
$229$	23.3205	1.54106	0.770531	+	0.637402i	$0.219991\pi$
$230$	8.19615	0.540438
$231$	0	0
$232$	7.66025	0.502920
$233$	23.8564	1.56289	0.781443	−	0.623977i	$-0.214484\pi$
$233$	23.8564	1.56289	0.781443	+	0.623977i	$0.214484\pi$
$234$	−17.8564	−1.16731
$235$	−6.00000	−0.391397
$236$	8.39230	0.546293
$237$	20.9282	1.35943
$238$	−0.535898	−0.0347371
$239$	20.0526	1.29709	0.648546	−	0.761175i	$-0.275377\pi$
$239$	20.0526	1.29709	0.648546	+	0.761175i	$0.275377\pi$
$240$	−2.73205	−0.176353
$241$	−22.9808	−1.48032	−0.740161	−	0.672430i	$-0.765251\pi$
$241$	−22.9808	−1.48032	−0.740161	+	0.672430i	$0.765251\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	−8.00000	−0.512148
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	−2.92820	−0.186317
$248$	−0.535898	−0.0340296
$249$	40.7846	2.58462
$250$	1.00000	0.0632456
$251$	−24.7846	−1.56439	−0.782195	−	0.623033i	$-0.785900\pi$
$251$	−24.7846	−1.56439	−0.782195	+	0.623033i	$0.785900\pi$
$252$	4.46410	0.281212
$253$	0	0
$254$	−8.39230	−0.526580
$255$	−1.46410	−0.0916856
$256$	1.00000	0.0625000
$257$	−7.80385	−0.486791	−0.243395	−	0.969927i	$-0.578261\pi$
$257$	−7.80385	−0.486791	−0.243395	+	0.969927i	$0.578261\pi$
$258$	13.4641	0.838238
$259$	−2.73205	−0.169761
$260$	−4.00000	−0.248069
$261$	−34.1962	−2.11669
$262$	−10.1962	−0.629920
$263$	28.3923	1.75074	0.875372	−	0.483449i	$-0.160616\pi$
$263$	28.3923	1.75074	0.875372	+	0.483449i	$0.160616\pi$
$264$	0	0
$265$	−5.26795	−0.323608
$266$	0.732051	0.0448849
$267$	−24.3923	−1.49278
$268$	15.8564	0.968584
$269$	−4.39230	−0.267804	−0.133902	−	0.990995i	$-0.542751\pi$
$269$	−4.39230	−0.267804	−0.133902	+	0.990995i	$0.542751\pi$
$270$	4.00000	0.243432
$271$	32.3923	1.96769	0.983846	−	0.179016i	$-0.0572913\pi$
$271$	32.3923	1.96769	0.983846	+	0.179016i	$0.0572913\pi$
$272$	0.535898	0.0324936
$273$	10.9282	0.661405
$274$	7.85641	0.474623
$275$	0	0
$276$	22.3923	1.34786
$277$	−17.3205	−1.04069	−0.520344	−	0.853957i	$-0.674196\pi$
$277$	−17.3205	−1.04069	−0.520344	+	0.853957i	$0.674196\pi$
$278$	3.66025	0.219527
$279$	2.39230	0.143224
$280$	1.00000	0.0597614
$281$	−23.3205	−1.39118	−0.695592	−	0.718437i	$-0.744858\pi$
$281$	−23.3205	−1.39118	−0.695592	+	0.718437i	$0.744858\pi$
$282$	−16.3923	−0.976148
$283$	−9.46410	−0.562582	−0.281291	−	0.959622i	$-0.590763\pi$
$283$	−9.46410	−0.562582	−0.281291	+	0.959622i	$0.590763\pi$
$284$	1.07180	0.0635994
$285$	2.00000	0.118470
$286$	0	0
$287$	2.19615	0.129635
$288$	−4.46410	−0.263050
$289$	−16.7128	−0.983107
$290$	−7.66025	−0.449826
$291$	7.46410	0.437553
$292$	−6.00000	−0.351123
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	−2.73205	−0.159336
$295$	−8.39230	−0.488619
$296$	2.73205	0.158797
$297$	0	0
$298$	−1.80385	−0.104494
$299$	32.7846	1.89598
$300$	2.73205	0.157735
$301$	−4.92820	−0.284057
$302$	1.80385	0.103800
$303$	21.8564	1.25562
$304$	−0.732051	−0.0419860
$305$	8.00000	0.458079
$306$	−2.39230	−0.136759
$307$	13.4641	0.768437	0.384218	−	0.923242i	$-0.374471\pi$
$307$	13.4641	0.768437	0.384218	+	0.923242i	$0.374471\pi$
$308$	0	0
$309$	6.53590	0.371814
$310$	0.535898	0.0304370
$311$	17.3205	0.982156	0.491078	−	0.871116i	$-0.336603\pi$
$311$	17.3205	0.982156	0.491078	+	0.871116i	$0.336603\pi$
$312$	−10.9282	−0.618688
$313$	−2.73205	−0.154425	−0.0772123	−	0.997015i	$-0.524602\pi$
$313$	−2.73205	−0.154425	−0.0772123	+	0.997015i	$0.524602\pi$
$314$	16.5359	0.933175
$315$	−4.46410	−0.251524
$316$	7.66025	0.430923
$317$	−4.58846	−0.257713	−0.128857	−	0.991663i	$-0.541131\pi$
$317$	−4.58846	−0.257713	−0.128857	+	0.991663i	$0.541131\pi$
$318$	−14.3923	−0.807080
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−5.46410	−0.304976
$322$	−8.19615	−0.456754
$323$	−0.392305	−0.0218284
$324$	−2.46410	−0.136895
$325$	4.00000	0.221880
$326$	−12.9282	−0.716027
$327$	−45.7128	−2.52792
$328$	−2.19615	−0.121262
$329$	6.00000	0.330791
$330$	0	0
$331$	5.60770	0.308227	0.154113	−	0.988053i	$-0.450748\pi$
$331$	5.60770	0.308227	0.154113	+	0.988053i	$0.450748\pi$
$332$	14.9282	0.819292
$333$	−12.1962	−0.668345
$334$	−5.07180	−0.277516
$335$	−15.8564	−0.866328
$336$	2.73205	0.149046
$337$	27.8564	1.51744	0.758718	−	0.651420i	$-0.225826\pi$
$337$	27.8564	1.51744	0.758718	+	0.651420i	$0.225826\pi$
$338$	−3.00000	−0.163178
$339$	49.1769	2.67092
$340$	−0.535898	−0.0290632
$341$	0	0
$342$	3.26795	0.176710
$343$	1.00000	0.0539949
$344$	4.92820	0.265711
$345$	−22.3923	−1.20556
$346$	−4.92820	−0.264942
$347$	4.92820	0.264560	0.132280	−	0.991212i	$-0.457770\pi$
$347$	4.92820	0.264560	0.132280	+	0.991212i	$0.457770\pi$
$348$	−20.9282	−1.12187
$349$	5.46410	0.292487	0.146243	−	0.989249i	$-0.453282\pi$
$349$	5.46410	0.292487	0.146243	+	0.989249i	$0.453282\pi$
$350$	−1.00000	−0.0534522
$351$	16.0000	0.854017
$352$	0	0
$353$	24.9808	1.32959	0.664796	−	0.747025i	$-0.268519\pi$
$353$	24.9808	1.32959	0.664796	+	0.747025i	$0.268519\pi$
$354$	−22.9282	−1.21862
$355$	−1.07180	−0.0568851
$356$	−8.92820	−0.473194
$357$	1.46410	0.0774885
$358$	−6.39230	−0.337844
$359$	24.4449	1.29015	0.645075	−	0.764119i	$-0.276826\pi$
$359$	24.4449	1.29015	0.645075	+	0.764119i	$0.276826\pi$
$360$	4.46410	0.235279
$361$	−18.4641	−0.971795
$362$	4.92820	0.259021
$363$	0	0
$364$	4.00000	0.209657
$365$	6.00000	0.314054
$366$	21.8564	1.14245
$367$	−14.3923	−0.751272	−0.375636	−	0.926767i	$-0.622576\pi$
$367$	−14.3923	−0.751272	−0.375636	+	0.926767i	$0.622576\pi$
$368$	8.19615	0.427254
$369$	9.80385	0.510368
$370$	−2.73205	−0.142033
$371$	5.26795	0.273498
$372$	1.46410	0.0759101
$373$	33.3205	1.72527	0.862635	−	0.505826i	$-0.168812\pi$
$373$	33.3205	1.72527	0.862635	+	0.505826i	$0.168812\pi$
$374$	0	0
$375$	−2.73205	−0.141082
$376$	−6.00000	−0.309426
$377$	−30.6410	−1.57809
$378$	−4.00000	−0.205738
$379$	29.8564	1.53362	0.766810	−	0.641874i	$-0.221843\pi$
$379$	29.8564	1.53362	0.766810	+	0.641874i	$0.221843\pi$
$380$	0.732051	0.0375534
$381$	22.9282	1.17465
$382$	20.3923	1.04336
$383$	3.46410	0.177007	0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410	0.177007	0.0885037	+	0.996076i	$0.471792\pi$
$384$	−2.73205	−0.139419
$385$	0	0
$386$	16.0000	0.814379
$387$	−22.0000	−1.11832
$388$	2.73205	0.138699
$389$	26.3923	1.33814	0.669071	−	0.743198i	$-0.266692\pi$
$389$	26.3923	1.33814	0.669071	+	0.743198i	$0.266692\pi$
$390$	10.9282	0.553371
$391$	4.39230	0.222128
$392$	−1.00000	−0.0505076
$393$	27.8564	1.40517
$394$	3.46410	0.174519
$395$	−7.66025	−0.385429
$396$	0	0
$397$	7.46410	0.374613	0.187306	−	0.982302i	$-0.440024\pi$
$397$	7.46410	0.374613	0.187306	+	0.982302i	$0.440024\pi$
$398$	1.85641	0.0930532
$399$	−2.00000	−0.100125
$400$	1.00000	0.0500000
$401$	20.3923	1.01834	0.509172	−	0.860665i	$-0.329952\pi$
$401$	20.3923	1.01834	0.509172	+	0.860665i	$0.329952\pi$
$402$	−43.3205	−2.16063
$403$	2.14359	0.106780
$404$	8.00000	0.398015
$405$	2.46410	0.122442
$406$	7.66025	0.380172
$407$	0	0
$408$	−1.46410	−0.0724838
$409$	30.1962	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$409$	30.1962	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$410$	2.19615	0.108460
$411$	−21.4641	−1.05875
$412$	2.39230	0.117860
$413$	8.39230	0.412958
$414$	−36.5885	−1.79822
$415$	−14.9282	−0.732797
$416$	−4.00000	−0.196116
$417$	−10.0000	−0.489702
$418$	0	0
$419$	37.8564	1.84941	0.924703	−	0.380689i	$-0.124313\pi$
$419$	37.8564	1.84941	0.924703	+	0.380689i	$0.124313\pi$
$420$	−2.73205	−0.133310
$421$	24.2487	1.18181	0.590905	−	0.806741i	$-0.298771\pi$
$421$	24.2487	1.18181	0.590905	+	0.806741i	$0.298771\pi$
$422$	13.8564	0.674519
$423$	26.7846	1.30231
$424$	−5.26795	−0.255834
$425$	0.535898	0.0259949
$426$	−2.92820	−0.141872
$427$	−8.00000	−0.387147
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−4.92820	−0.237659
$431$	−16.7321	−0.805955	−0.402977	−	0.915210i	$-0.632025\pi$
$431$	−16.7321	−0.805955	−0.402977	+	0.915210i	$0.632025\pi$
$432$	4.00000	0.192450
$433$	−29.2679	−1.40653	−0.703264	−	0.710929i	$-0.748275\pi$
$433$	−29.2679	−1.40653	−0.703264	+	0.710929i	$0.748275\pi$
$434$	−0.535898	−0.0257239
$435$	20.9282	1.00343
$436$	−16.7321	−0.801320
$437$	−6.00000	−0.287019
$438$	16.3923	0.783255
$439$	−29.4641	−1.40624	−0.703122	−	0.711069i	$-0.748211\pi$
$439$	−29.4641	−1.40624	−0.703122	+	0.711069i	$0.748211\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	−2.14359	−0.101960
$443$	−28.9282	−1.37442	−0.687210	−	0.726459i	$-0.741165\pi$
$443$	−28.9282	−1.37442	−0.687210	+	0.726459i	$0.741165\pi$
$444$	−7.46410	−0.354231
$445$	8.92820	0.423237
$446$	11.4641	0.542841
$447$	4.92820	0.233096
$448$	1.00000	0.0472456
$449$	−10.2487	−0.483667	−0.241833	−	0.970318i	$-0.577749\pi$
$449$	−10.2487	−0.483667	−0.241833	+	0.970318i	$0.577749\pi$
$450$	−4.46410	−0.210440
$451$	0	0
$452$	18.0000	0.846649
$453$	−4.92820	−0.231547
$454$	23.3205	1.09449
$455$	−4.00000	−0.187523
$456$	2.00000	0.0936586
$457$	−15.8564	−0.741731	−0.370866	−	0.928687i	$-0.620939\pi$
$457$	−15.8564	−0.741731	−0.370866	+	0.928687i	$0.620939\pi$
$458$	−23.3205	−1.08970
$459$	2.14359	0.100054
$460$	−8.19615	−0.382148
$461$	−4.67949	−0.217946	−0.108973	−	0.994045i	$-0.534756\pi$
$461$	−4.67949	−0.217946	−0.108973	+	0.994045i	$0.534756\pi$
$462$	0	0
$463$	7.41154	0.344444	0.172222	−	0.985058i	$-0.444905\pi$
$463$	7.41154	0.344444	0.172222	+	0.985058i	$0.444905\pi$
$464$	−7.66025	−0.355618
$465$	−1.46410	−0.0678961
$466$	−23.8564	−1.10513
$467$	40.5885	1.87821	0.939105	−	0.343631i	$-0.111657\pi$
$467$	40.5885	1.87821	0.939105	+	0.343631i	$0.111657\pi$
$468$	17.8564	0.825413
$469$	15.8564	0.732181
$470$	6.00000	0.276759
$471$	−45.1769	−2.08164
$472$	−8.39230	−0.386287
$473$	0	0
$474$	−20.9282	−0.961264
$475$	−0.732051	−0.0335888
$476$	0.535898	0.0245629
$477$	23.5167	1.07675
$478$	−20.0526	−0.917183
$479$	−19.7128	−0.900701	−0.450351	−	0.892852i	$-0.648701\pi$
$479$	−19.7128	−0.900701	−0.450351	+	0.892852i	$0.648701\pi$
$480$	2.73205	0.124700
$481$	−10.9282	−0.498283
$482$	22.9808	1.04675
$483$	22.3923	1.01889
$484$	0	0
$485$	−2.73205	−0.124056
$486$	18.7321	0.849703
$487$	−30.0526	−1.36181	−0.680906	−	0.732371i	$-0.738414\pi$
$487$	−30.0526	−1.36181	−0.680906	+	0.732371i	$0.738414\pi$
$488$	8.00000	0.362143
$489$	35.3205	1.59725
$490$	1.00000	0.0451754
$491$	−9.07180	−0.409404	−0.204702	−	0.978824i	$-0.565623\pi$
$491$	−9.07180	−0.409404	−0.204702	+	0.978824i	$0.565623\pi$
$492$	6.00000	0.270501
$493$	−4.10512	−0.184885
$494$	2.92820	0.131746
$495$	0	0
$496$	0.535898	0.0240625
$497$	1.07180	0.0480767
$498$	−40.7846	−1.82760
$499$	−22.3923	−1.00242	−0.501209	−	0.865326i	$-0.667111\pi$
$499$	−22.3923	−1.00242	−0.501209	+	0.865326i	$0.667111\pi$
$500$	−1.00000	−0.0447214
$501$	13.8564	0.619059
$502$	24.7846	1.10619
$503$	−10.1436	−0.452280	−0.226140	−	0.974095i	$-0.572611\pi$
$503$	−10.1436	−0.452280	−0.226140	+	0.974095i	$0.572611\pi$
$504$	−4.46410	−0.198847
$505$	−8.00000	−0.355995
$506$	0	0
$507$	8.19615	0.364004
$508$	8.39230	0.372348
$509$	−0.928203	−0.0411419	−0.0205709	−	0.999788i	$-0.506548\pi$
$509$	−0.928203	−0.0411419	−0.0205709	+	0.999788i	$0.506548\pi$
$510$	1.46410	0.0648315
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	−2.92820	−0.129283
$514$	7.80385	0.344213
$515$	−2.39230	−0.105418
$516$	−13.4641	−0.592724
$517$	0	0
$518$	2.73205	0.120039
$519$	13.4641	0.591008
$520$	4.00000	0.175412
$521$	−24.2487	−1.06236	−0.531178	−	0.847260i	$-0.678250\pi$
$521$	−24.2487	−1.06236	−0.531178	+	0.847260i	$0.678250\pi$
$522$	34.1962	1.49672
$523$	−14.5359	−0.635610	−0.317805	−	0.948156i	$-0.602946\pi$
$523$	−14.5359	−0.635610	−0.317805	+	0.948156i	$0.602946\pi$
$524$	10.1962	0.445421
$525$	2.73205	0.119236
$526$	−28.3923	−1.23796
$527$	0.287187	0.0125101
$528$	0	0
$529$	44.1769	1.92074
$530$	5.26795	0.228825
$531$	37.4641	1.62580
$532$	−0.732051	−0.0317384
$533$	8.78461	0.380504
$534$	24.3923	1.05556
$535$	2.00000	0.0864675
$536$	−15.8564	−0.684892
$537$	17.4641	0.753632
$538$	4.39230	0.189366
$539$	0	0
$540$	−4.00000	−0.172133
$541$	−12.7321	−0.547394	−0.273697	−	0.961816i	$-0.588246\pi$
$541$	−12.7321	−0.547394	−0.273697	+	0.961816i	$0.588246\pi$
$542$	−32.3923	−1.39137
$543$	−13.4641	−0.577800
$544$	−0.535898	−0.0229765
$545$	16.7321	0.716722
$546$	−10.9282	−0.467684
$547$	23.7128	1.01389	0.506943	−	0.861979i	$-0.330775\pi$
$547$	23.7128	1.01389	0.506943	+	0.861979i	$0.330775\pi$
$548$	−7.85641	−0.335609
$549$	−35.7128	−1.52419
$550$	0	0
$551$	5.60770	0.238896
$552$	−22.3923	−0.953080
$553$	7.66025	0.325747
$554$	17.3205	0.735878
$555$	7.46410	0.316833
$556$	−3.66025	−0.155229
$557$	3.07180	0.130156	0.0650781	−	0.997880i	$-0.479270\pi$
$557$	3.07180	0.130156	0.0650781	+	0.997880i	$0.479270\pi$
$558$	−2.39230	−0.101274
$559$	−19.7128	−0.833763
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	23.3205	0.983716
$563$	39.7128	1.67370	0.836848	−	0.547436i	$-0.184396\pi$
$563$	39.7128	1.67370	0.836848	+	0.547436i	$0.184396\pi$
$564$	16.3923	0.690241
$565$	−18.0000	−0.757266
$566$	9.46410	0.397806
$567$	−2.46410	−0.103483
$568$	−1.07180	−0.0449716
$569$	−33.8564	−1.41933	−0.709667	−	0.704537i	$-0.751155\pi$
$569$	−33.8564	−1.41933	−0.709667	+	0.704537i	$0.751155\pi$
$570$	−2.00000	−0.0837708
$571$	19.3205	0.808538	0.404269	−	0.914640i	$-0.367526\pi$
$571$	19.3205	0.808538	0.404269	+	0.914640i	$0.367526\pi$
$572$	0	0
$573$	−55.7128	−2.32744
$574$	−2.19615	−0.0916656
$575$	8.19615	0.341803
$576$	4.46410	0.186004
$577$	1.26795	0.0527854	0.0263927	−	0.999652i	$-0.491598\pi$
$577$	1.26795	0.0527854	0.0263927	+	0.999652i	$0.491598\pi$
$578$	16.7128	0.695161
$579$	−43.7128	−1.81664
$580$	7.66025	0.318075
$581$	14.9282	0.619326
$582$	−7.46410	−0.309397
$583$	0	0
$584$	6.00000	0.248282
$585$	−17.8564	−0.738272
$586$	−27.7128	−1.14481
$587$	26.7321	1.10335	0.551675	−	0.834059i	$-0.313989\pi$
$587$	26.7321	1.10335	0.551675	+	0.834059i	$0.313989\pi$
$588$	2.73205	0.112668
$589$	−0.392305	−0.0161646
$590$	8.39230	0.345506
$591$	−9.46410	−0.389301
$592$	−2.73205	−0.112287
$593$	−28.6410	−1.17615	−0.588073	−	0.808808i	$-0.700113\pi$
$593$	−28.6410	−1.17615	−0.588073	+	0.808808i	$0.700113\pi$
$594$	0	0
$595$	−0.535898	−0.0219697
$596$	1.80385	0.0738885
$597$	−5.07180	−0.207575
$598$	−32.7846	−1.34066
$599$	−46.9282	−1.91743	−0.958717	−	0.284361i	$-0.908218\pi$
$599$	−46.9282	−1.91743	−0.958717	+	0.284361i	$0.908218\pi$
$600$	−2.73205	−0.111536
$601$	2.87564	0.117300	0.0586500	−	0.998279i	$-0.481320\pi$
$601$	2.87564	0.117300	0.0586500	+	0.998279i	$0.481320\pi$
$602$	4.92820	0.200859
$603$	70.7846	2.88257
$604$	−1.80385	−0.0733975
$605$	0	0
$606$	−21.8564	−0.887856
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	0.732051	0.0296886
$609$	−20.9282	−0.848054
$610$	−8.00000	−0.323911
$611$	24.0000	0.970936
$612$	2.39230	0.0967032
$613$	41.7128	1.68476	0.842382	−	0.538880i	$-0.181153\pi$
$613$	41.7128	1.68476	0.842382	+	0.538880i	$0.181153\pi$
$614$	−13.4641	−0.543367
$615$	−6.00000	−0.241943
$616$	0	0
$617$	38.3923	1.54562	0.772808	−	0.634640i	$-0.218852\pi$
$617$	38.3923	1.54562	0.772808	+	0.634640i	$0.218852\pi$
$618$	−6.53590	−0.262912
$619$	5.07180	0.203853	0.101926	−	0.994792i	$-0.467499\pi$
$619$	5.07180	0.203853	0.101926	+	0.994792i	$0.467499\pi$
$620$	−0.535898	−0.0215222
$621$	32.7846	1.31560
$622$	−17.3205	−0.694489
$623$	−8.92820	−0.357701
$624$	10.9282	0.437478
$625$	1.00000	0.0400000
$626$	2.73205	0.109195
$627$	0	0
$628$	−16.5359	−0.659854
$629$	−1.46410	−0.0583776
$630$	4.46410	0.177854
$631$	−25.8564	−1.02933	−0.514664	−	0.857392i	$-0.672083\pi$
$631$	−25.8564	−1.02933	−0.514664	+	0.857392i	$0.672083\pi$
$632$	−7.66025	−0.304709
$633$	−37.8564	−1.50466
$634$	4.58846	0.182231
$635$	−8.39230	−0.333038
$636$	14.3923	0.570692
$637$	4.00000	0.158486
$638$	0	0
$639$	4.78461	0.189276
$640$	1.00000	0.0395285
$641$	−23.8564	−0.942271	−0.471136	−	0.882061i	$-0.656156\pi$
$641$	−23.8564	−0.942271	−0.471136	+	0.882061i	$0.656156\pi$
$642$	5.46410	0.215651
$643$	4.19615	0.165480	0.0827400	−	0.996571i	$-0.473633\pi$
$643$	4.19615	0.165480	0.0827400	+	0.996571i	$0.473633\pi$
$644$	8.19615	0.322974
$645$	13.4641	0.530148
$646$	0.392305	0.0154350
$647$	−5.60770	−0.220461	−0.110231	−	0.993906i	$-0.535159\pi$
$647$	−5.60770	−0.220461	−0.110231	+	0.993906i	$0.535159\pi$
$648$	2.46410	0.0967991
$649$	0	0
$650$	−4.00000	−0.156893
$651$	1.46410	0.0573827
$652$	12.9282	0.506308
$653$	−34.7321	−1.35917	−0.679585	−	0.733597i	$-0.737840\pi$
$653$	−34.7321	−1.35917	−0.679585	+	0.733597i	$0.737840\pi$
$654$	45.7128	1.78751
$655$	−10.1962	−0.398397
$656$	2.19615	0.0857453
$657$	−26.7846	−1.04497
$658$	−6.00000	−0.233904
$659$	40.3923	1.57346	0.786730	−	0.617297i	$-0.211772\pi$
$659$	40.3923	1.57346	0.786730	+	0.617297i	$0.211772\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−5.60770	−0.217949
$663$	5.85641	0.227444
$664$	−14.9282	−0.579327
$665$	0.732051	0.0283877
$666$	12.1962	0.472591
$667$	−62.7846	−2.43103
$668$	5.07180	0.196234
$669$	−31.3205	−1.21092
$670$	15.8564	0.612586
$671$	0	0
$672$	−2.73205	−0.105391
$673$	−43.8564	−1.69054	−0.845270	−	0.534339i	$-0.820560\pi$
$673$	−43.8564	−1.69054	−0.845270	+	0.534339i	$0.820560\pi$
$674$	−27.8564	−1.07299
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	−12.9282	−0.496871	−0.248436	−	0.968648i	$-0.579916\pi$
$677$	−12.9282	−0.496871	−0.248436	+	0.968648i	$0.579916\pi$
$678$	−49.1769	−1.88863
$679$	2.73205	0.104846
$680$	0.535898	0.0205508
$681$	−63.7128	−2.44148
$682$	0	0
$683$	1.60770	0.0615167	0.0307584	−	0.999527i	$-0.490208\pi$
$683$	1.60770	0.0615167	0.0307584	+	0.999527i	$0.490208\pi$
$684$	−3.26795	−0.124953
$685$	7.85641	0.300178
$686$	−1.00000	−0.0381802
$687$	63.7128	2.43080
$688$	−4.92820	−0.187886
$689$	21.0718	0.802772
$690$	22.3923	0.852460
$691$	−14.9282	−0.567896	−0.283948	−	0.958840i	$-0.591644\pi$
$691$	−14.9282	−0.567896	−0.283948	+	0.958840i	$0.591644\pi$
$692$	4.92820	0.187342
$693$	0	0
$694$	−4.92820	−0.187072
$695$	3.66025	0.138841
$696$	20.9282	0.793281
$697$	1.17691	0.0445788
$698$	−5.46410	−0.206819
$699$	65.1769	2.46522
$700$	1.00000	0.0377964
$701$	31.2679	1.18097	0.590487	−	0.807047i	$-0.298936\pi$
$701$	31.2679	1.18097	0.590487	+	0.807047i	$0.298936\pi$
$702$	−16.0000	−0.603881
$703$	2.00000	0.0754314
$704$	0	0
$705$	−16.3923	−0.617370
$706$	−24.9808	−0.940163
$707$	8.00000	0.300871
$708$	22.9282	0.861695
$709$	−3.85641	−0.144830	−0.0724152	−	0.997375i	$-0.523071\pi$
$709$	−3.85641	−0.144830	−0.0724152	+	0.997375i	$0.523071\pi$
$710$	1.07180	0.0402238
$711$	34.1962	1.28246
$712$	8.92820	0.334599
$713$	4.39230	0.164493
$714$	−1.46410	−0.0547926
$715$	0	0
$716$	6.39230	0.238892
$717$	54.7846	2.04597
$718$	−24.4449	−0.912274
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	−4.46410	−0.166367
$721$	2.39230	0.0890941
$722$	18.4641	0.687163
$723$	−62.7846	−2.33498
$724$	−4.92820	−0.183155
$725$	−7.66025	−0.284495
$726$	0	0
$727$	−45.7128	−1.69539	−0.847697	−	0.530480i	$-0.822012\pi$
$727$	−45.7128	−1.69539	−0.847697	+	0.530480i	$0.822012\pi$
$728$	−4.00000	−0.148250
$729$	−43.7846	−1.62165
$730$	−6.00000	−0.222070
$731$	−2.64102	−0.0976815
$732$	−21.8564	−0.807836
$733$	51.8564	1.91536	0.957680	−	0.287835i	$-0.0929354\pi$
$733$	51.8564	1.91536	0.957680	+	0.287835i	$0.0929354\pi$
$734$	14.3923	0.531230
$735$	−2.73205	−0.100773
$736$	−8.19615	−0.302114
$737$	0	0
$738$	−9.80385	−0.360885
$739$	−9.85641	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$739$	−9.85641	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$740$	2.73205	0.100432
$741$	−8.00000	−0.293887
$742$	−5.26795	−0.193392
$743$	−3.60770	−0.132353	−0.0661767	−	0.997808i	$-0.521080\pi$
$743$	−3.60770	−0.132353	−0.0661767	+	0.997808i	$0.521080\pi$
$744$	−1.46410	−0.0536766
$745$	−1.80385	−0.0660879
$746$	−33.3205	−1.21995
$747$	66.6410	2.43827
$748$	0	0
$749$	−2.00000	−0.0730784
$750$	2.73205	0.0997604
$751$	22.1436	0.808031	0.404016	−	0.914752i	$-0.367614\pi$
$751$	22.1436	0.808031	0.404016	+	0.914752i	$0.367614\pi$
$752$	6.00000	0.218797
$753$	−67.7128	−2.46759
$754$	30.6410	1.11588
$755$	1.80385	0.0656487
$756$	4.00000	0.145479
$757$	−10.3397	−0.375804	−0.187902	−	0.982188i	$-0.560169\pi$
$757$	−10.3397	−0.375804	−0.187902	+	0.982188i	$0.560169\pi$
$758$	−29.8564	−1.08443
$759$	0	0
$760$	−0.732051	−0.0265543
$761$	−34.5885	−1.25383	−0.626915	−	0.779087i	$-0.715683\pi$
$761$	−34.5885	−1.25383	−0.626915	+	0.779087i	$0.715683\pi$
$762$	−22.9282	−0.830601
$763$	−16.7321	−0.605741
$764$	−20.3923	−0.737768
$765$	−2.39230	−0.0864940
$766$	−3.46410	−0.125163
$767$	33.5692	1.21211
$768$	2.73205	0.0985844
$769$	3.26795	0.117845	0.0589226	−	0.998263i	$-0.481233\pi$
$769$	3.26795	0.117845	0.0589226	+	0.998263i	$0.481233\pi$
$770$	0	0
$771$	−21.3205	−0.767839
$772$	−16.0000	−0.575853
$773$	−45.7128	−1.64418	−0.822088	−	0.569361i	$-0.807191\pi$
$773$	−45.7128	−1.64418	−0.822088	+	0.569361i	$0.807191\pi$
$774$	22.0000	0.790774
$775$	0.535898	0.0192500
$776$	−2.73205	−0.0980749
$777$	−7.46410	−0.267773
$778$	−26.3923	−0.946210
$779$	−1.60770	−0.0576017
$780$	−10.9282	−0.391292
$781$	0	0
$782$	−4.39230	−0.157069
$783$	−30.6410	−1.09502
$784$	1.00000	0.0357143
$785$	16.5359	0.590192
$786$	−27.8564	−0.993605
$787$	4.78461	0.170553	0.0852765	−	0.996357i	$-0.472823\pi$
$787$	4.78461	0.170553	0.0852765	+	0.996357i	$0.472823\pi$
$788$	−3.46410	−0.123404
$789$	77.5692	2.76154
$790$	7.66025	0.272540
$791$	18.0000	0.640006
$792$	0	0
$793$	−32.0000	−1.13635
$794$	−7.46410	−0.264891
$795$	−14.3923	−0.510442
$796$	−1.85641	−0.0657986
$797$	18.3923	0.651489	0.325744	−	0.945458i	$-0.394385\pi$
$797$	18.3923	0.651489	0.325744	+	0.945458i	$0.394385\pi$
$798$	2.00000	0.0707992
$799$	3.21539	0.113752
$800$	−1.00000	−0.0353553
$801$	−39.8564	−1.40826
$802$	−20.3923	−0.720077
$803$	0	0
$804$	43.3205	1.52780
$805$	−8.19615	−0.288876
$806$	−2.14359	−0.0755049
$807$	−12.0000	−0.422420
$808$	−8.00000	−0.281439
$809$	−44.1051	−1.55065	−0.775327	−	0.631560i	$-0.782415\pi$
$809$	−44.1051	−1.55065	−0.775327	+	0.631560i	$0.782415\pi$
$810$	−2.46410	−0.0865797
$811$	3.26795	0.114753	0.0573766	−	0.998353i	$-0.481726\pi$
$811$	3.26795	0.114753	0.0573766	+	0.998353i	$0.481726\pi$
$812$	−7.66025	−0.268822
$813$	88.4974	3.10374
$814$	0	0
$815$	−12.9282	−0.452855
$816$	1.46410	0.0512538
$817$	3.60770	0.126217
$818$	−30.1962	−1.05578
$819$	17.8564	0.623953
$820$	−2.19615	−0.0766930
$821$	−23.3731	−0.815726	−0.407863	−	0.913043i	$-0.633726\pi$
$821$	−23.3731	−0.815726	−0.407863	+	0.913043i	$0.633726\pi$
$822$	21.4641	0.748647
$823$	1.94744	0.0678835	0.0339418	−	0.999424i	$-0.489194\pi$
$823$	1.94744	0.0678835	0.0339418	+	0.999424i	$0.489194\pi$
$824$	−2.39230	−0.0833399
$825$	0	0
$826$	−8.39230	−0.292006
$827$	−20.7846	−0.722752	−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846	−0.722752	−0.361376	+	0.932420i	$0.617693\pi$
$828$	36.5885	1.27154
$829$	3.60770	0.125300	0.0626502	−	0.998036i	$-0.480045\pi$
$829$	3.60770	0.125300	0.0626502	+	0.998036i	$0.480045\pi$
$830$	14.9282	0.518165
$831$	−47.3205	−1.64153
$832$	4.00000	0.138675
$833$	0.535898	0.0185678
$834$	10.0000	0.346272
$835$	−5.07180	−0.175517
$836$	0	0
$837$	2.14359	0.0740934
$838$	−37.8564	−1.30773
$839$	−28.2487	−0.975254	−0.487627	−	0.873052i	$-0.662137\pi$
$839$	−28.2487	−0.975254	−0.487627	+	0.873052i	$0.662137\pi$
$840$	2.73205	0.0942647
$841$	29.6795	1.02343
$842$	−24.2487	−0.835666
$843$	−63.7128	−2.19439
$844$	−13.8564	−0.476957
$845$	−3.00000	−0.103203
$846$	−26.7846	−0.920874
$847$	0	0
$848$	5.26795	0.180902
$849$	−25.8564	−0.887390
$850$	−0.535898	−0.0183812
$851$	−22.3923	−0.767598
$852$	2.92820	0.100319
$853$	−24.6410	−0.843692	−0.421846	−	0.906667i	$-0.638618\pi$
$853$	−24.6410	−0.843692	−0.421846	+	0.906667i	$0.638618\pi$
$854$	8.00000	0.273754
$855$	3.26795	0.111762
$856$	2.00000	0.0683586
$857$	−24.9282	−0.851531	−0.425766	−	0.904833i	$-0.639995\pi$
$857$	−24.9282	−0.851531	−0.425766	+	0.904833i	$0.639995\pi$
$858$	0	0
$859$	13.0718	0.446004	0.223002	−	0.974818i	$-0.428414\pi$
$859$	13.0718	0.446004	0.223002	+	0.974818i	$0.428414\pi$
$860$	4.92820	0.168050
$861$	6.00000	0.204479
$862$	16.7321	0.569896
$863$	5.66025	0.192677	0.0963386	−	0.995349i	$-0.469287\pi$
$863$	5.66025	0.192677	0.0963386	+	0.995349i	$0.469287\pi$
$864$	−4.00000	−0.136083
$865$	−4.92820	−0.167564
$866$	29.2679	0.994565
$867$	−45.6603	−1.55070
$868$	0.535898	0.0181896
$869$	0	0
$870$	−20.9282	−0.709533
$871$	63.4256	2.14910
$872$	16.7321	0.566619
$873$	12.1962	0.412777
$874$	6.00000	0.202953
$875$	−1.00000	−0.0338062
$876$	−16.3923	−0.553845
$877$	−38.3923	−1.29642	−0.648208	−	0.761463i	$-0.724481\pi$
$877$	−38.3923	−1.29642	−0.648208	+	0.761463i	$0.724481\pi$
$878$	29.4641	0.994365
$879$	75.7128	2.55373
$880$	0	0
$881$	2.28719	0.0770573	0.0385286	−	0.999257i	$-0.487733\pi$
$881$	2.28719	0.0770573	0.0385286	+	0.999257i	$0.487733\pi$
$882$	−4.46410	−0.150314
$883$	−57.3205	−1.92899	−0.964494	−	0.264104i	$-0.914924\pi$
$883$	−57.3205	−1.92899	−0.964494	+	0.264104i	$0.914924\pi$
$884$	2.14359	0.0720969
$885$	−22.9282	−0.770723
$886$	28.9282	0.971862
$887$	−46.6410	−1.56605	−0.783026	−	0.621989i	$-0.786325\pi$
$887$	−46.6410	−1.56605	−0.783026	+	0.621989i	$0.786325\pi$
$888$	7.46410	0.250479
$889$	8.39230	0.281469
$890$	−8.92820	−0.299274
$891$	0	0
$892$	−11.4641	−0.383847
$893$	−4.39230	−0.146983
$894$	−4.92820	−0.164824
$895$	−6.39230	−0.213671
$896$	−1.00000	−0.0334077
$897$	89.5692	2.99063
$898$	10.2487	0.342004
$899$	−4.10512	−0.136913
$900$	4.46410	0.148803
$901$	2.82309	0.0940506
$902$	0	0
$903$	−13.4641	−0.448057
$904$	−18.0000	−0.598671
$905$	4.92820	0.163819
$906$	4.92820	0.163729
$907$	−9.32051	−0.309482	−0.154741	−	0.987955i	$-0.549454\pi$
$907$	−9.32051	−0.309482	−0.154741	+	0.987955i	$0.549454\pi$
$908$	−23.3205	−0.773918
$909$	35.7128	1.18452
$910$	4.00000	0.132599
$911$	43.0333	1.42576	0.712879	−	0.701287i	$-0.247391\pi$
$911$	43.0333	1.42576	0.712879	+	0.701287i	$0.247391\pi$
$912$	−2.00000	−0.0662266
$913$	0	0
$914$	15.8564	0.524483
$915$	21.8564	0.722551
$916$	23.3205	0.770531
$917$	10.1962	0.336707
$918$	−2.14359	−0.0707491
$919$	−35.3731	−1.16685	−0.583425	−	0.812167i	$-0.698288\pi$
$919$	−35.3731	−1.16685	−0.583425	+	0.812167i	$0.698288\pi$
$920$	8.19615	0.270219
$921$	36.7846	1.21209
$922$	4.67949	0.154111
$923$	4.28719	0.141114
$924$	0	0
$925$	−2.73205	−0.0898293
$926$	−7.41154	−0.243558
$927$	10.6795	0.350761
$928$	7.66025	0.251460
$929$	−36.2487	−1.18928	−0.594641	−	0.803991i	$-0.702706\pi$
$929$	−36.2487	−1.18928	−0.594641	+	0.803991i	$0.702706\pi$
$930$	1.46410	0.0480098
$931$	−0.732051	−0.0239920
$932$	23.8564	0.781443
$933$	47.3205	1.54920
$934$	−40.5885	−1.32809
$935$	0	0
$936$	−17.8564	−0.583655
$937$	−6.67949	−0.218209	−0.109105	−	0.994030i	$-0.534798\pi$
$937$	−6.67949	−0.218209	−0.109105	+	0.994030i	$0.534798\pi$
$938$	−15.8564	−0.517730
$939$	−7.46410	−0.243582
$940$	−6.00000	−0.195698
$941$	−6.92820	−0.225853	−0.112926	−	0.993603i	$-0.536022\pi$
$941$	−6.92820	−0.225853	−0.112926	+	0.993603i	$0.536022\pi$
$942$	45.1769	1.47194
$943$	18.0000	0.586161
$944$	8.39230	0.273146
$945$	−4.00000	−0.130120
$946$	0	0
$947$	2.39230	0.0777395	0.0388697	−	0.999244i	$-0.487624\pi$
$947$	2.39230	0.0777395	0.0388697	+	0.999244i	$0.487624\pi$
$948$	20.9282	0.679716
$949$	−24.0000	−0.779073
$950$	0.732051	0.0237509
$951$	−12.5359	−0.406504
$952$	−0.535898	−0.0173686
$953$	22.6410	0.733414	0.366707	−	0.930336i	$-0.380485\pi$
$953$	22.6410	0.733414	0.366707	+	0.930336i	$0.380485\pi$
$954$	−23.5167	−0.761380
$955$	20.3923	0.659879
$956$	20.0526	0.648546
$957$	0	0
$958$	19.7128	0.636892
$959$	−7.85641	−0.253697
$960$	−2.73205	−0.0881766
$961$	−30.7128	−0.990736
$962$	10.9282	0.352339
$963$	−8.92820	−0.287707
$964$	−22.9808	−0.740161
$965$	16.0000	0.515058
$966$	−22.3923	−0.720461
$967$	36.3923	1.17030	0.585149	−	0.810926i	$-0.301036\pi$
$967$	36.3923	1.17030	0.585149	+	0.810926i	$0.301036\pi$
$968$	0	0
$969$	−1.07180	−0.0344311
$970$	2.73205	0.0877209
$971$	3.60770	0.115776	0.0578882	−	0.998323i	$-0.481563\pi$
$971$	3.60770	0.115776	0.0578882	+	0.998323i	$0.481563\pi$
$972$	−18.7321	−0.600831
$973$	−3.66025	−0.117342
$974$	30.0526	0.962946
$975$	10.9282	0.349983
$976$	−8.00000	−0.256074
$977$	31.8564	1.01918	0.509588	−	0.860418i	$-0.329798\pi$
$977$	31.8564	1.01918	0.509588	+	0.860418i	$0.329798\pi$
$978$	−35.3205	−1.12943
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−74.6936	−2.38478
$982$	9.07180	0.289493
$983$	27.4641	0.875969	0.437984	−	0.898983i	$-0.355693\pi$
$983$	27.4641	0.875969	0.437984	+	0.898983i	$0.355693\pi$
$984$	−6.00000	−0.191273
$985$	3.46410	0.110375
$986$	4.10512	0.130734
$987$	16.3923	0.521773
$988$	−2.92820	−0.0931586
$989$	−40.3923	−1.28440
$990$	0	0
$991$	−3.60770	−0.114602	−0.0573011	−	0.998357i	$-0.518250\pi$
$991$	−3.60770	−0.114602	−0.0573011	+	0.998357i	$0.518250\pi$
$992$	−0.535898	−0.0170148
$993$	15.3205	0.486182
$994$	−1.07180	−0.0339953
$995$	1.85641	0.0588520
$996$	40.7846	1.29231
$997$	23.8564	0.755540	0.377770	−	0.925899i	$-0.376691\pi$
$997$	23.8564	0.755540	0.377770	+	0.925899i	$0.376691\pi$
$998$	22.3923	0.708816
$999$	−10.9282	−0.345753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bs.1.2	✓	2
11.10	odd	2		8470.2.a.cd.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bs.1.2	✓	2	1.1	even	1	trivial
8470.2.a.cd.1.2	yes	2	11.10	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} +1.00000 q^{7} -1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +2.73205 q^{12} +4.00000 q^{13} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +0.535898 q^{17} -4.46410 q^{18} -0.732051 q^{19} -1.00000 q^{20} +2.73205 q^{21} +8.19615 q^{23} -2.73205 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -7.66025 q^{29} +2.73205 q^{30} +0.535898 q^{31} -1.00000 q^{32} -0.535898 q^{34} -1.00000 q^{35} +4.46410 q^{36} -2.73205 q^{37} +0.732051 q^{38} +10.9282 q^{39} +1.00000 q^{40} +2.19615 q^{41} -2.73205 q^{42} -4.92820 q^{43} -4.46410 q^{45} -8.19615 q^{46} +6.00000 q^{47} +2.73205 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.46410 q^{51} +4.00000 q^{52} +5.26795 q^{53} -4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} +7.66025 q^{58} +8.39230 q^{59} -2.73205 q^{60} -8.00000 q^{61} -0.535898 q^{62} +4.46410 q^{63} +1.00000 q^{64} -4.00000 q^{65} +15.8564 q^{67} +0.535898 q^{68} +22.3923 q^{69} +1.00000 q^{70} +1.07180 q^{71} -4.46410 q^{72} -6.00000 q^{73} +2.73205 q^{74} +2.73205 q^{75} -0.732051 q^{76} -10.9282 q^{78} +7.66025 q^{79} -1.00000 q^{80} -2.46410 q^{81} -2.19615 q^{82} +14.9282 q^{83} +2.73205 q^{84} -0.535898 q^{85} +4.92820 q^{86} -20.9282 q^{87} -8.92820 q^{89} +4.46410 q^{90} +4.00000 q^{91} +8.19615 q^{92} +1.46410 q^{93} -6.00000 q^{94} +0.732051 q^{95} -2.73205 q^{96} +2.73205 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 8 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{21} + 6 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{26} + 8 q^{27} + 2 q^{28} + 2 q^{29} + 2 q^{30} + 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} - 2 q^{42} + 4 q^{43} - 2 q^{45} - 6 q^{46} + 12 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{51} + 8 q^{52} + 14 q^{53} - 8 q^{54} - 2 q^{56} - 4 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} - 16 q^{61} - 8 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{65} + 4 q^{67} + 8 q^{68} + 24 q^{69} + 2 q^{70} + 16 q^{71} - 2 q^{72} - 12 q^{73} + 2 q^{74} + 2 q^{75} + 2 q^{76} - 8 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 16 q^{83} + 2 q^{84} - 8 q^{85} - 4 q^{86} - 28 q^{87} - 4 q^{89} + 2 q^{90} + 8 q^{91} + 6 q^{92} - 4 q^{93} - 12 q^{94} - 2 q^{95} - 2 q^{96} + 2 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 8 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 + 2 * q^19 - 2 * q^20 + 2 * q^21 + 6 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^26 + 8 * q^27 + 2 * q^28 + 2 * q^29 + 2 * q^30 + 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 - 2 * q^42 + 4 * q^43 - 2 * q^45 - 6 * q^46 + 12 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 4 * q^51 + 8 * q^52 + 14 * q^53 - 8 * q^54 - 2 * q^56 - 4 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 - 16 * q^61 - 8 * q^62 + 2 * q^63 + 2 * q^64 - 8 * q^65 + 4 * q^67 + 8 * q^68 + 24 * q^69 + 2 * q^70 + 16 * q^71 - 2 * q^72 - 12 * q^73 + 2 * q^74 + 2 * q^75 + 2 * q^76 - 8 * q^78 - 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 16 * q^83 + 2 * q^84 - 8 * q^85 - 4 * q^86 - 28 * q^87 - 4 * q^89 + 2 * q^90 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 12 * q^94 - 2 * q^95 - 2 * q^96 + 2 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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