LMFDB - Embedded newform 2850.2.d.x.799.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2850,2,Mod(799,2850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2850.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2850.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.7573645761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2850.799
Dual form			2850.2.d.x.799.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.73205i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.73205i q^{7} +1.00000i q^{8} -1.00000 q^{9} +0.267949 q^{11} -1.00000i q^{12} -0.732051i q^{13} -2.73205 q^{14} +1.00000 q^{16} +4.19615i q^{17} +1.00000i q^{18} +1.00000 q^{19} +2.73205 q^{21} -0.267949i q^{22} +7.92820i q^{23} -1.00000 q^{24} -0.732051 q^{26} -1.00000i q^{27} +2.73205i q^{28} -1.73205 q^{29} +4.46410 q^{31} -1.00000i q^{32} +0.267949i q^{33} +4.19615 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +0.732051 q^{39} +10.9282 q^{41} -2.73205i q^{42} -2.19615i q^{43} -0.267949 q^{44} +7.92820 q^{46} -3.46410i q^{47} +1.00000i q^{48} -0.464102 q^{49} -4.19615 q^{51} +0.732051i q^{52} -1.73205i q^{53} -1.00000 q^{54} +2.73205 q^{56} +1.00000i q^{57} +1.73205i q^{58} -2.19615 q^{59} -6.66025 q^{61} -4.46410i q^{62} +2.73205i q^{63} -1.00000 q^{64} +0.267949 q^{66} -3.73205i q^{67} -4.19615i q^{68} -7.92820 q^{69} +1.80385 q^{71} -1.00000i q^{72} -4.46410i q^{73} -2.00000 q^{74} -1.00000 q^{76} -0.732051i q^{77} -0.732051i q^{78} +12.4641 q^{79} +1.00000 q^{81} -10.9282i q^{82} -0.267949i q^{83} -2.73205 q^{84} -2.19615 q^{86} -1.73205i q^{87} +0.267949i q^{88} +16.8564 q^{89} -2.00000 q^{91} -7.92820i q^{92} +4.46410i q^{93} -3.46410 q^{94} +1.00000 q^{96} +9.12436i q^{97} +0.464102i q^{98} -0.267949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 8 q^{11} - 4 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{21} - 4 q^{24} + 4 q^{26} + 4 q^{31} - 4 q^{34} + 4 q^{36} - 4 q^{39} + 16 q^{41} - 8 q^{44} + 4 q^{46} + 12 q^{49} + 4 q^{51} - 4 q^{54} + 4 q^{56} + 12 q^{59} + 8 q^{61} - 4 q^{64} + 8 q^{66} - 4 q^{69} + 28 q^{71} - 8 q^{74} - 4 q^{76} + 36 q^{79} + 4 q^{81} - 4 q^{84} + 12 q^{86} + 12 q^{89} - 8 q^{91} + 4 q^{96} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 8 * q^11 - 4 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^21 - 4 * q^24 + 4 * q^26 + 4 * q^31 - 4 * q^34 + 4 * q^36 - 4 * q^39 + 16 * q^41 - 8 * q^44 + 4 * q^46 + 12 * q^49 + 4 * q^51 - 4 * q^54 + 4 * q^56 + 12 * q^59 + 8 * q^61 - 4 * q^64 + 8 * q^66 - 4 * q^69 + 28 * q^71 - 8 * q^74 - 4 * q^76 + 36 * q^79 + 4 * q^81 - 4 * q^84 + 12 * q^86 + 12 * q^89 - 8 * q^91 + 4 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$.

$n$	$1027$	$1351$	$1901$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 2.73205i	− 1.03262i	−0.856402	−	0.516309i	$-0.827306\pi$
$7$	− 2.73205i	− 1.03262i	0.856402	−	0.516309i	$-0.172694\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.267949	0.0807897	0.0403949	−	0.999184i	$-0.487138\pi$
$11$	0.267949	0.0807897	0.0403949	+	0.999184i	$0.487138\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 0.732051i	− 0.203034i	−0.994834	−	0.101517i	$-0.967630\pi$
$13$	− 0.732051i	− 0.203034i	0.994834	−	0.101517i	$-0.0323697\pi$
$14$	−2.73205	−0.730171
$15$	0	0
$16$	1.00000	0.250000
$17$	4.19615i	1.01772i	0.860850	+	0.508858i	$0.169932\pi$
$17$	4.19615i	1.01772i	−0.860850	+	0.508858i	$0.830068\pi$
$18$	1.00000i	0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.73205	0.596182
$22$	− 0.267949i	− 0.0571270i
$23$	7.92820i	1.65314i	0.562831	+	0.826572i	$0.309712\pi$
$23$	7.92820i	1.65314i	−0.562831	+	0.826572i	$0.690288\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.732051	−0.143567
$27$	− 1.00000i	− 0.192450i
$28$	2.73205i	0.516309i
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	4.46410	0.801776	0.400888	−	0.916127i	$-0.368702\pi$
$31$	4.46410	0.801776	0.400888	+	0.916127i	$0.368702\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0.267949i	0.0466440i
$34$	4.19615	0.719634
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 1.00000i	− 0.162221i
$39$	0.732051	0.117222
$40$	0	0
$41$	10.9282	1.70670	0.853349	−	0.521340i	$-0.174568\pi$
$41$	10.9282	1.70670	0.853349	+	0.521340i	$0.174568\pi$
$42$	− 2.73205i	− 0.421565i
$43$	− 2.19615i	− 0.334910i	−0.985880	−	0.167455i	$-0.946445\pi$
$43$	− 2.19615i	− 0.334910i	0.985880	−	0.167455i	$-0.0535549\pi$
$44$	−0.267949	−0.0403949
$45$	0	0
$46$	7.92820	1.16895
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	1.00000i	0.144338i
$49$	−0.464102	−0.0663002
$50$	0	0
$51$	−4.19615	−0.587579
$52$	0.732051i	0.101517i
$53$	− 1.73205i	− 0.237915i	−0.992899	−	0.118958i	$-0.962045\pi$
$53$	− 1.73205i	− 0.237915i	0.992899	−	0.118958i	$-0.0379553\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.73205	0.365086
$57$	1.00000i	0.132453i
$58$	1.73205i	0.227429i
$59$	−2.19615	−0.285915	−0.142957	−	0.989729i	$-0.545661\pi$
$59$	−2.19615	−0.285915	−0.142957	+	0.989729i	$0.545661\pi$
$60$	0	0
$61$	−6.66025	−0.852758	−0.426379	−	0.904545i	$-0.640211\pi$
$61$	−6.66025	−0.852758	−0.426379	+	0.904545i	$0.640211\pi$
$62$	− 4.46410i	− 0.566941i
$63$	2.73205i	0.344206i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.267949	0.0329823
$67$	− 3.73205i	− 0.455943i	−0.973668	−	0.227971i	$-0.926791\pi$
$67$	− 3.73205i	− 0.455943i	0.973668	−	0.227971i	$-0.0732092\pi$
$68$	− 4.19615i	− 0.508858i
$69$	−7.92820	−0.954444
$70$	0	0
$71$	1.80385	0.214077	0.107039	−	0.994255i	$-0.465863\pi$
$71$	1.80385	0.214077	0.107039	+	0.994255i	$0.465863\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.46410i	− 0.522484i	−0.965273	−	0.261242i	$-0.915868\pi$
$73$	− 4.46410i	− 0.522484i	0.965273	−	0.261242i	$-0.0841320\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	− 0.732051i	− 0.0834249i
$78$	− 0.732051i	− 0.0828884i
$79$	12.4641	1.40232	0.701160	−	0.713003i	$-0.252666\pi$
$79$	12.4641	1.40232	0.701160	+	0.713003i	$0.252666\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.9282i	− 1.20682i
$83$	− 0.267949i	− 0.0294112i	−0.999892	−	0.0147056i	$-0.995319\pi$
$83$	− 0.267949i	− 0.0294112i	0.999892	−	0.0147056i	$-0.00468111\pi$
$84$	−2.73205	−0.298091
$85$	0	0
$86$	−2.19615	−0.236817
$87$	− 1.73205i	− 0.185695i
$88$	0.267949i	0.0285635i
$89$	16.8564	1.78678	0.893388	−	0.449286i	$-0.148322\pi$
$89$	16.8564	1.78678	0.893388	+	0.449286i	$0.148322\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 7.92820i	− 0.826572i
$93$	4.46410i	0.462906i
$94$	−3.46410	−0.357295
$95$	0	0
$96$	1.00000	0.102062
$97$	9.12436i	0.926438i	0.886244	+	0.463219i	$0.153306\pi$
$97$	9.12436i	0.926438i	−0.886244	+	0.463219i	$0.846694\pi$
$98$	0.464102i	0.0468813i
$99$	−0.267949	−0.0269299
$100$	0	0
$101$	12.7321	1.26689	0.633443	−	0.773789i	$-0.281641\pi$
$101$	12.7321	1.26689	0.633443	+	0.773789i	$0.281641\pi$
$102$	4.19615i	0.415481i
$103$	10.4641i	1.03106i	0.856872	+	0.515529i	$0.172405\pi$
$103$	10.4641i	1.03106i	−0.856872	+	0.515529i	$0.827595\pi$
$104$	0.732051	0.0717835
$105$	0	0
$106$	−1.73205	−0.168232
$107$	− 8.19615i	− 0.792352i	−0.918175	−	0.396176i	$-0.870337\pi$
$107$	− 8.19615i	− 0.792352i	0.918175	−	0.396176i	$-0.129663\pi$
$108$	1.00000i	0.0962250i
$109$	18.3923	1.76166	0.880832	−	0.473430i	$-0.156984\pi$
$109$	18.3923	1.76166	0.880832	+	0.473430i	$0.156984\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 2.73205i	− 0.258155i
$113$	− 7.39230i	− 0.695410i	−0.937604	−	0.347705i	$-0.886961\pi$
$113$	− 7.39230i	− 0.695410i	0.937604	−	0.347705i	$-0.113039\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	1.73205	0.160817
$117$	0.732051i	0.0676781i
$118$	2.19615i	0.202172i
$119$	11.4641	1.05091
$120$	0	0
$121$	−10.9282	−0.993473
$122$	6.66025i	0.602991i
$123$	10.9282i	0.985363i
$124$	−4.46410	−0.400888
$125$	0	0
$126$	2.73205	0.243390
$127$	− 5.00000i	− 0.443678i	−0.975083	−	0.221839i	$-0.928794\pi$
$127$	− 5.00000i	− 0.443678i	0.975083	−	0.221839i	$-0.0712060\pi$
$128$	1.00000i	0.0883883i
$129$	2.19615	0.193360
$130$	0	0
$131$	13.7321	1.19977	0.599887	−	0.800084i	$-0.295212\pi$
$131$	13.7321	1.19977	0.599887	+	0.800084i	$0.295212\pi$
$132$	− 0.267949i	− 0.0233220i
$133$	− 2.73205i	− 0.236899i
$134$	−3.73205	−0.322400
$135$	0	0
$136$	−4.19615	−0.359817
$137$	8.39230i	0.717003i	0.933529	+	0.358501i	$0.116712\pi$
$137$	8.39230i	0.717003i	−0.933529	+	0.358501i	$0.883288\pi$
$138$	7.92820i	0.674893i
$139$	−3.12436	−0.265004	−0.132502	−	0.991183i	$-0.542301\pi$
$139$	−3.12436	−0.265004	−0.132502	+	0.991183i	$0.542301\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	− 1.80385i	− 0.151376i
$143$	− 0.196152i	− 0.0164031i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.46410	−0.369452
$147$	− 0.464102i	− 0.0382785i
$148$	2.00000i	0.164399i
$149$	−5.85641	−0.479776	−0.239888	−	0.970801i	$-0.577111\pi$
$149$	−5.85641	−0.479776	−0.239888	+	0.970801i	$0.577111\pi$
$150$	0	0
$151$	−4.53590	−0.369126	−0.184563	−	0.982821i	$-0.559087\pi$
$151$	−4.53590	−0.369126	−0.184563	+	0.982821i	$0.559087\pi$
$152$	1.00000i	0.0811107i
$153$	− 4.19615i	− 0.339239i
$154$	−0.732051	−0.0589903
$155$	0	0
$156$	−0.732051	−0.0586110
$157$	− 13.8564i	− 1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$	− 13.8564i	− 1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$	− 12.4641i	− 0.991591i
$159$	1.73205	0.137361
$160$	0	0
$161$	21.6603	1.70707
$162$	− 1.00000i	− 0.0785674i
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	−10.9282	−0.853349
$165$	0	0
$166$	−0.267949	−0.0207969
$167$	4.92820i	0.381356i	0.981653	+	0.190678i	$0.0610686\pi$
$167$	4.92820i	0.381356i	−0.981653	+	0.190678i	$0.938931\pi$
$168$	2.73205i	0.210782i
$169$	12.4641	0.958777
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	2.19615i	0.167455i
$173$	17.5885i	1.33723i	0.743611	+	0.668613i	$0.233112\pi$
$173$	17.5885i	1.33723i	−0.743611	+	0.668613i	$0.766888\pi$
$174$	−1.73205	−0.131306
$175$	0	0
$176$	0.267949	0.0201974
$177$	− 2.19615i	− 0.165073i
$178$	− 16.8564i	− 1.26344i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.5885	1.08435	0.542176	−	0.840265i	$-0.317601\pi$
$181$	14.5885	1.08435	0.542176	+	0.840265i	$0.317601\pi$
$182$	2.00000i	0.148250i
$183$	− 6.66025i	− 0.492340i
$184$	−7.92820	−0.584475
$185$	0	0
$186$	4.46410	0.327324
$187$	1.12436i	0.0822210i
$188$	3.46410i	0.252646i
$189$	−2.73205	−0.198727
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	16.0526i	1.15549i	0.816218	+	0.577744i	$0.196067\pi$
$193$	16.0526i	1.15549i	−0.816218	+	0.577744i	$0.803933\pi$
$194$	9.12436	0.655091
$195$	0	0
$196$	0.464102	0.0331501
$197$	− 6.58846i	− 0.469408i	−0.972067	−	0.234704i	$-0.924588\pi$
$197$	− 6.58846i	− 0.469408i	0.972067	−	0.234704i	$-0.0754121\pi$
$198$	0.267949i	0.0190423i
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	3.73205	0.263239
$202$	− 12.7321i	− 0.895824i
$203$	4.73205i	0.332125i
$204$	4.19615	0.293789
$205$	0	0
$206$	10.4641	0.729069
$207$	− 7.92820i	− 0.551048i
$208$	− 0.732051i	− 0.0507586i
$209$	0.267949	0.0185344
$210$	0	0
$211$	−13.7321	−0.945353	−0.472677	−	0.881236i	$-0.656712\pi$
$211$	−13.7321	−0.945353	−0.472677	+	0.881236i	$0.656712\pi$
$212$	1.73205i	0.118958i
$213$	1.80385i	0.123598i
$214$	−8.19615	−0.560277
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 12.1962i	− 0.827929i
$218$	− 18.3923i	− 1.24568i
$219$	4.46410	0.301656
$220$	0	0
$221$	3.07180	0.206631
$222$	− 2.00000i	− 0.134231i
$223$	10.4641i	0.700728i	0.936614	+	0.350364i	$0.113942\pi$
$223$	10.4641i	0.700728i	−0.936614	+	0.350364i	$0.886058\pi$
$224$	−2.73205	−0.182543
$225$	0	0
$226$	−7.39230	−0.491729
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	9.73205	0.643112	0.321556	−	0.946891i	$-0.395794\pi$
$229$	9.73205	0.643112	0.321556	+	0.946891i	$0.395794\pi$
$230$	0	0
$231$	0.732051	0.0481654
$232$	− 1.73205i	− 0.113715i
$233$	− 3.12436i	− 0.204683i	−0.994749	−	0.102342i	$-0.967366\pi$
$233$	− 3.12436i	− 0.204683i	0.994749	−	0.102342i	$-0.0326335\pi$
$234$	0.732051	0.0478557
$235$	0	0
$236$	2.19615	0.142957
$237$	12.4641i	0.809630i
$238$	− 11.4641i	− 0.743107i
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−19.8038	−1.27568	−0.637839	−	0.770170i	$-0.720171\pi$
$241$	−19.8038	−1.27568	−0.637839	+	0.770170i	$0.720171\pi$
$242$	10.9282i	0.702492i
$243$	1.00000i	0.0641500i
$244$	6.66025	0.426379
$245$	0	0
$246$	10.9282	0.696757
$247$	− 0.732051i	− 0.0465793i
$248$	4.46410i	0.283471i
$249$	0.267949	0.0169806
$250$	0	0
$251$	4.53590	0.286303	0.143152	−	0.989701i	$-0.454276\pi$
$251$	4.53590	0.286303	0.143152	+	0.989701i	$0.454276\pi$
$252$	− 2.73205i	− 0.172103i
$253$	2.12436i	0.133557i
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000i	0.561405i	0.959795	+	0.280702i	$0.0905674\pi$
$257$	9.00000i	0.561405i	−0.959795	+	0.280702i	$0.909433\pi$
$258$	− 2.19615i	− 0.136726i
$259$	−5.46410	−0.339523
$260$	0	0
$261$	1.73205	0.107211
$262$	− 13.7321i	− 0.848369i
$263$	− 15.0000i	− 0.924940i	−0.886635	−	0.462470i	$-0.846963\pi$
$263$	− 15.0000i	− 0.924940i	0.886635	−	0.462470i	$-0.153037\pi$
$264$	−0.267949	−0.0164911
$265$	0	0
$266$	−2.73205	−0.167513
$267$	16.8564i	1.03160i
$268$	3.73205i	0.227971i
$269$	27.4641	1.67452	0.837258	−	0.546808i	$-0.184157\pi$
$269$	27.4641	1.67452	0.837258	+	0.546808i	$0.184157\pi$
$270$	0	0
$271$	−26.1962	−1.59130	−0.795651	−	0.605755i	$-0.792871\pi$
$271$	−26.1962	−1.59130	−0.795651	+	0.605755i	$0.792871\pi$
$272$	4.19615i	0.254429i
$273$	− 2.00000i	− 0.121046i
$274$	8.39230	0.506998
$275$	0	0
$276$	7.92820	0.477222
$277$	0.803848i	0.0482985i	0.999708	+	0.0241493i	$0.00768770\pi$
$277$	0.803848i	0.0482985i	−0.999708	+	0.0241493i	$0.992312\pi$
$278$	3.12436i	0.187386i
$279$	−4.46410	−0.267259
$280$	0	0
$281$	−23.7846	−1.41887	−0.709435	−	0.704770i	$-0.751050\pi$
$281$	−23.7846	−1.41887	−0.709435	+	0.704770i	$0.751050\pi$
$282$	− 3.46410i	− 0.206284i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−1.80385	−0.107039
$285$	0	0
$286$	−0.196152	−0.0115987
$287$	− 29.8564i	− 1.76237i
$288$	1.00000i	0.0589256i
$289$	−0.607695	−0.0357468
$290$	0	0
$291$	−9.12436	−0.534879
$292$	4.46410i	0.261242i
$293$	24.2679i	1.41775i	0.705335	+	0.708874i	$0.250797\pi$
$293$	24.2679i	1.41775i	−0.705335	+	0.708874i	$0.749203\pi$
$294$	−0.464102	−0.0270670
$295$	0	0
$296$	2.00000	0.116248
$297$	− 0.267949i	− 0.0155480i
$298$	5.85641i	0.339253i
$299$	5.80385	0.335645
$300$	0	0
$301$	−6.00000	−0.345834
$302$	4.53590i	0.261012i
$303$	12.7321i	0.731437i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−4.19615	−0.239878
$307$	− 23.7321i	− 1.35446i	−0.735772	−	0.677230i	$-0.763180\pi$
$307$	− 23.7321i	− 1.35446i	0.735772	−	0.677230i	$-0.236820\pi$
$308$	0.732051i	0.0417125i
$309$	−10.4641	−0.595282
$310$	0	0
$311$	5.32051	0.301698	0.150849	−	0.988557i	$-0.451799\pi$
$311$	5.32051	0.301698	0.150849	+	0.988557i	$0.451799\pi$
$312$	0.732051i	0.0414442i
$313$	− 9.92820i	− 0.561175i	−0.959828	−	0.280588i	$-0.909471\pi$
$313$	− 9.92820i	− 0.561175i	0.959828	−	0.280588i	$-0.0905293\pi$
$314$	−13.8564	−0.781962
$315$	0	0
$316$	−12.4641	−0.701160
$317$	1.73205i	0.0972817i	0.998816	+	0.0486408i	$0.0154890\pi$
$317$	1.73205i	0.0972817i	−0.998816	+	0.0486408i	$0.984511\pi$
$318$	− 1.73205i	− 0.0971286i
$319$	−0.464102	−0.0259847
$320$	0	0
$321$	8.19615	0.457465
$322$	− 21.6603i	− 1.20708i
$323$	4.19615i	0.233480i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.00000	0.110770
$327$	18.3923i	1.01710i
$328$	10.9282i	0.603409i
$329$	−9.46410	−0.521773
$330$	0	0
$331$	−25.7321	−1.41436	−0.707181	−	0.707033i	$-0.750033\pi$
$331$	−25.7321	−1.41436	−0.707181	+	0.707033i	$0.750033\pi$
$332$	0.267949i	0.0147056i
$333$	2.00000i	0.109599i
$334$	4.92820	0.269659
$335$	0	0
$336$	2.73205	0.149046
$337$	17.4641i	0.951330i	0.879626	+	0.475665i	$0.157793\pi$
$337$	17.4641i	0.951330i	−0.879626	+	0.475665i	$0.842207\pi$
$338$	− 12.4641i	− 0.677958i
$339$	7.39230	0.401495
$340$	0	0
$341$	1.19615	0.0647753
$342$	1.00000i	0.0540738i
$343$	− 17.8564i	− 0.964155i
$344$	2.19615	0.118409
$345$	0	0
$346$	17.5885	0.945561
$347$	20.7846i	1.11578i	0.829916	+	0.557888i	$0.188388\pi$
$347$	20.7846i	1.11578i	−0.829916	+	0.557888i	$0.811612\pi$
$348$	1.73205i	0.0928477i
$349$	−21.0526	−1.12692	−0.563459	−	0.826144i	$-0.690530\pi$
$349$	−21.0526	−1.12692	−0.563459	+	0.826144i	$0.690530\pi$
$350$	0	0
$351$	−0.732051	−0.0390740
$352$	− 0.267949i	− 0.0142817i
$353$	22.0526i	1.17374i	0.809681	+	0.586870i	$0.199640\pi$
$353$	22.0526i	1.17374i	−0.809681	+	0.586870i	$0.800360\pi$
$354$	−2.19615	−0.116724
$355$	0	0
$356$	−16.8564	−0.893388
$357$	11.4641i	0.606745i
$358$	− 12.0000i	− 0.634220i
$359$	3.07180	0.162123	0.0810616	−	0.996709i	$-0.474169\pi$
$359$	3.07180	0.162123	0.0810616	+	0.996709i	$0.474169\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 14.5885i	− 0.766752i
$363$	− 10.9282i	− 0.573582i
$364$	2.00000	0.104828
$365$	0	0
$366$	−6.66025	−0.348137
$367$	− 16.7321i	− 0.873406i	−0.899606	−	0.436703i	$-0.856146\pi$
$367$	− 16.7321i	− 0.873406i	0.899606	−	0.436703i	$-0.143854\pi$
$368$	7.92820i	0.413286i
$369$	−10.9282	−0.568900
$370$	0	0
$371$	−4.73205	−0.245676
$372$	− 4.46410i	− 0.231453i
$373$	− 13.5167i	− 0.699866i	−0.936775	−	0.349933i	$-0.886204\pi$
$373$	− 13.5167i	− 0.699866i	0.936775	−	0.349933i	$-0.113796\pi$
$374$	1.12436	0.0581390
$375$	0	0
$376$	3.46410	0.178647
$377$	1.26795i	0.0653027i
$378$	2.73205i	0.140522i
$379$	−20.3923	−1.04748	−0.523741	−	0.851877i	$-0.675464\pi$
$379$	−20.3923	−1.04748	−0.523741	+	0.851877i	$0.675464\pi$
$380$	0	0
$381$	5.00000	0.256158
$382$	− 3.00000i	− 0.153493i
$383$	− 25.5167i	− 1.30384i	−0.758288	−	0.651920i	$-0.773964\pi$
$383$	− 25.5167i	− 1.30384i	0.758288	−	0.651920i	$-0.226036\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	16.0526	0.817054
$387$	2.19615i	0.111637i
$388$	− 9.12436i	− 0.463219i
$389$	33.7128	1.70931	0.854654	−	0.519198i	$-0.173769\pi$
$389$	33.7128	1.70931	0.854654	+	0.519198i	$0.173769\pi$
$390$	0	0
$391$	−33.2679	−1.68243
$392$	− 0.464102i	− 0.0234407i
$393$	13.7321i	0.692690i
$394$	−6.58846	−0.331922
$395$	0	0
$396$	0.267949	0.0134650
$397$	8.26795i	0.414956i	0.978240	+	0.207478i	$0.0665256\pi$
$397$	8.26795i	0.414956i	−0.978240	+	0.207478i	$0.933474\pi$
$398$	− 14.0000i	− 0.701757i
$399$	2.73205	0.136774
$400$	0	0
$401$	−10.8564	−0.542143	−0.271072	−	0.962559i	$-0.587378\pi$
$401$	−10.8564	−0.542143	−0.271072	+	0.962559i	$0.587378\pi$
$402$	− 3.73205i	− 0.186138i
$403$	− 3.26795i	− 0.162788i
$404$	−12.7321	−0.633443
$405$	0	0
$406$	4.73205	0.234848
$407$	− 0.535898i	− 0.0265635i
$408$	− 4.19615i	− 0.207741i
$409$	−22.3923	−1.10723	−0.553614	−	0.832773i	$-0.686752\pi$
$409$	−22.3923	−1.10723	−0.553614	+	0.832773i	$0.686752\pi$
$410$	0	0
$411$	−8.39230	−0.413962
$412$	− 10.4641i	− 0.515529i
$413$	6.00000i	0.295241i
$414$	−7.92820	−0.389650
$415$	0	0
$416$	−0.732051	−0.0358917
$417$	− 3.12436i	− 0.153000i
$418$	− 0.267949i	− 0.0131058i
$419$	13.8564	0.676930	0.338465	−	0.940979i	$-0.390092\pi$
$419$	13.8564	0.676930	0.338465	+	0.940979i	$0.390092\pi$
$420$	0	0
$421$	−34.7846	−1.69530	−0.847649	−	0.530557i	$-0.821983\pi$
$421$	−34.7846	−1.69530	−0.847649	+	0.530557i	$0.821983\pi$
$422$	13.7321i	0.668466i
$423$	3.46410i	0.168430i
$424$	1.73205	0.0841158
$425$	0	0
$426$	1.80385	0.0873967
$427$	18.1962i	0.880574i
$428$	8.19615i	0.396176i
$429$	0.196152	0.00947033
$430$	0	0
$431$	5.32051	0.256280	0.128140	−	0.991756i	$-0.459099\pi$
$431$	5.32051	0.256280	0.128140	+	0.991756i	$0.459099\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 13.8038i	− 0.663371i	−0.943390	−	0.331685i	$-0.892383\pi$
$433$	− 13.8038i	− 0.663371i	0.943390	−	0.331685i	$-0.107617\pi$
$434$	−12.1962	−0.585434
$435$	0	0
$436$	−18.3923	−0.880832
$437$	7.92820i	0.379257i
$438$	− 4.46410i	− 0.213303i
$439$	15.7846	0.753358	0.376679	−	0.926344i	$-0.377066\pi$
$439$	15.7846	0.753358	0.376679	+	0.926344i	$0.377066\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	− 3.07180i	− 0.146110i
$443$	− 20.6603i	− 0.981598i	−0.871273	−	0.490799i	$-0.836705\pi$
$443$	− 20.6603i	− 0.981598i	0.871273	−	0.490799i	$-0.163295\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	10.4641	0.495490
$447$	− 5.85641i	− 0.276999i
$448$	2.73205i	0.129077i
$449$	−17.5359	−0.827570	−0.413785	−	0.910375i	$-0.635794\pi$
$449$	−17.5359	−0.827570	−0.413785	+	0.910375i	$0.635794\pi$
$450$	0	0
$451$	2.92820	0.137884
$452$	7.39230i	0.347705i
$453$	− 4.53590i	− 0.213115i
$454$	−6.00000	−0.281594
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	18.9282i	0.885424i	0.896664	+	0.442712i	$0.145984\pi$
$457$	18.9282i	0.885424i	−0.896664	+	0.442712i	$0.854016\pi$
$458$	− 9.73205i	− 0.454749i
$459$	4.19615	0.195860
$460$	0	0
$461$	9.85641	0.459059	0.229529	−	0.973302i	$-0.426281\pi$
$461$	9.85641	0.459059	0.229529	+	0.973302i	$0.426281\pi$
$462$	− 0.732051i	− 0.0340581i
$463$	− 0.679492i	− 0.0315787i	−0.999875	−	0.0157893i	$-0.994974\pi$
$463$	− 0.679492i	− 0.0315787i	0.999875	−	0.0157893i	$-0.00502611\pi$
$464$	−1.73205	−0.0804084
$465$	0	0
$466$	−3.12436	−0.144733
$467$	3.19615i	0.147900i	0.997262	+	0.0739501i	$0.0235606\pi$
$467$	3.19615i	0.147900i	−0.997262	+	0.0739501i	$0.976439\pi$
$468$	− 0.732051i	− 0.0338391i
$469$	−10.1962	−0.470815
$470$	0	0
$471$	13.8564	0.638470
$472$	− 2.19615i	− 0.101086i
$473$	− 0.588457i	− 0.0270573i
$474$	12.4641	0.572495
$475$	0	0
$476$	−11.4641	−0.525456
$477$	1.73205i	0.0793052i
$478$	− 14.0000i	− 0.640345i
$479$	−30.8564	−1.40987	−0.704933	−	0.709274i	$-0.749023\pi$
$479$	−30.8564	−1.40987	−0.704933	+	0.709274i	$0.749023\pi$
$480$	0	0
$481$	−1.46410	−0.0667573
$482$	19.8038i	0.902041i
$483$	21.6603i	0.985576i
$484$	10.9282	0.496737
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 29.8564i	− 1.35292i	−0.736478	−	0.676461i	$-0.763513\pi$
$487$	− 29.8564i	− 1.35292i	0.736478	−	0.676461i	$-0.236487\pi$
$488$	− 6.66025i	− 0.301496i
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	10.9282	0.493183	0.246591	−	0.969120i	$-0.420689\pi$
$491$	10.9282	0.493183	0.246591	+	0.969120i	$0.420689\pi$
$492$	− 10.9282i	− 0.492681i
$493$	− 7.26795i	− 0.327332i
$494$	−0.732051	−0.0329365
$495$	0	0
$496$	4.46410	0.200444
$497$	− 4.92820i	− 0.221060i
$498$	− 0.267949i	− 0.0120071i
$499$	31.3731	1.40445	0.702226	−	0.711954i	$-0.252190\pi$
$499$	31.3731	1.40445	0.702226	+	0.711954i	$0.252190\pi$
$500$	0	0
$501$	−4.92820	−0.220176
$502$	− 4.53590i	− 0.202447i
$503$	9.85641i	0.439475i	0.975559	+	0.219738i	$0.0705202\pi$
$503$	9.85641i	0.439475i	−0.975559	+	0.219738i	$0.929480\pi$
$504$	−2.73205	−0.121695
$505$	0	0
$506$	2.12436	0.0944391
$507$	12.4641i	0.553550i
$508$	5.00000i	0.221839i
$509$	−38.1244	−1.68983	−0.844916	−	0.534899i	$-0.820350\pi$
$509$	−38.1244	−1.68983	−0.844916	+	0.534899i	$0.820350\pi$
$510$	0	0
$511$	−12.1962	−0.539526
$512$	− 1.00000i	− 0.0441942i
$513$	− 1.00000i	− 0.0441511i
$514$	9.00000	0.396973
$515$	0	0
$516$	−2.19615	−0.0966802
$517$	− 0.928203i	− 0.0408223i
$518$	5.46410i	0.240079i
$519$	−17.5885	−0.772048
$520$	0	0
$521$	−38.1769	−1.67256	−0.836280	−	0.548302i	$-0.815274\pi$
$521$	−38.1769	−1.67256	−0.836280	+	0.548302i	$0.815274\pi$
$522$	− 1.73205i	− 0.0758098i
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−13.7321	−0.599887
$525$	0	0
$526$	−15.0000	−0.654031
$527$	18.7321i	0.815981i
$528$	0.267949i	0.0116610i
$529$	−39.8564	−1.73289
$530$	0	0
$531$	2.19615	0.0953049
$532$	2.73205i	0.118449i
$533$	− 8.00000i	− 0.346518i
$534$	16.8564	0.729448
$535$	0	0
$536$	3.73205	0.161200
$537$	12.0000i	0.517838i
$538$	− 27.4641i	− 1.18406i
$539$	−0.124356	−0.00535638
$540$	0	0
$541$	6.80385	0.292520	0.146260	−	0.989246i	$-0.453276\pi$
$541$	6.80385	0.292520	0.146260	+	0.989246i	$0.453276\pi$
$542$	26.1962i	1.12522i
$543$	14.5885i	0.626051i
$544$	4.19615	0.179909
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	− 15.4449i	− 0.660375i	−0.943915	−	0.330187i	$-0.892888\pi$
$547$	− 15.4449i	− 0.660375i	0.943915	−	0.330187i	$-0.107112\pi$
$548$	− 8.39230i	− 0.358501i
$549$	6.66025	0.284253
$550$	0	0
$551$	−1.73205	−0.0737878
$552$	− 7.92820i	− 0.337447i
$553$	− 34.0526i	− 1.44806i
$554$	0.803848	0.0341522
$555$	0	0
$556$	3.12436	0.132502
$557$	14.1962i	0.601510i	0.953701	+	0.300755i	$0.0972387\pi$
$557$	14.1962i	0.601510i	−0.953701	+	0.300755i	$0.902761\pi$
$558$	4.46410i	0.188980i
$559$	−1.60770	−0.0679983
$560$	0	0
$561$	−1.12436	−0.0474703
$562$	23.7846i	1.00329i
$563$	9.66025i	0.407131i	0.979061	+	0.203566i	$0.0652530\pi$
$563$	9.66025i	0.407131i	−0.979061	+	0.203566i	$0.934747\pi$
$564$	−3.46410	−0.145865
$565$	0	0
$566$	−4.00000	−0.168133
$567$	− 2.73205i	− 0.114735i
$568$	1.80385i	0.0756878i
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	4.19615	0.175604	0.0878018	−	0.996138i	$-0.472016\pi$
$571$	4.19615	0.175604	0.0878018	+	0.996138i	$0.472016\pi$
$572$	0.196152i	0.00820154i
$573$	3.00000i	0.125327i
$574$	−29.8564	−1.24618
$575$	0	0
$576$	1.00000	0.0416667
$577$	31.2487i	1.30090i	0.759549	+	0.650450i	$0.225420\pi$
$577$	31.2487i	1.30090i	−0.759549	+	0.650450i	$0.774580\pi$
$578$	0.607695i	0.0252768i
$579$	−16.0526	−0.667122
$580$	0	0
$581$	−0.732051	−0.0303706
$582$	9.12436i	0.378217i
$583$	− 0.464102i	− 0.0192211i
$584$	4.46410	0.184726
$585$	0	0
$586$	24.2679	1.00250
$587$	− 15.7321i	− 0.649331i	−0.945829	−	0.324666i	$-0.894748\pi$
$587$	− 15.7321i	− 0.649331i	0.945829	−	0.324666i	$-0.105252\pi$
$588$	0.464102i	0.0191392i
$589$	4.46410	0.183940
$590$	0	0
$591$	6.58846	0.271013
$592$	− 2.00000i	− 0.0821995i
$593$	48.3923i	1.98723i	0.112807	+	0.993617i	$0.464016\pi$
$593$	48.3923i	1.98723i	−0.112807	+	0.993617i	$0.535984\pi$
$594$	−0.267949	−0.0109941
$595$	0	0
$596$	5.85641	0.239888
$597$	14.0000i	0.572982i
$598$	− 5.80385i	− 0.237337i
$599$	−13.6603	−0.558143	−0.279071	−	0.960270i	$-0.590027\pi$
$599$	−13.6603	−0.558143	−0.279071	+	0.960270i	$0.590027\pi$
$600$	0	0
$601$	28.0526	1.14429	0.572144	−	0.820153i	$-0.306112\pi$
$601$	28.0526	1.14429	0.572144	+	0.820153i	$0.306112\pi$
$602$	6.00000i	0.244542i
$603$	3.73205i	0.151981i
$604$	4.53590	0.184563
$605$	0	0
$606$	12.7321	0.517204
$607$	29.7846i	1.20892i	0.796635	+	0.604460i	$0.206611\pi$
$607$	29.7846i	1.20892i	−0.796635	+	0.604460i	$0.793389\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	−4.73205	−0.191752
$610$	0	0
$611$	−2.53590	−0.102591
$612$	4.19615i	0.169619i
$613$	2.78461i	0.112469i	0.998418	+	0.0562347i	$0.0179095\pi$
$613$	2.78461i	0.112469i	−0.998418	+	0.0562347i	$0.982091\pi$
$614$	−23.7321	−0.957748
$615$	0	0
$616$	0.732051	0.0294952
$617$	25.6077i	1.03093i	0.856912	+	0.515463i	$0.172380\pi$
$617$	25.6077i	1.03093i	−0.856912	+	0.515463i	$0.827620\pi$
$618$	10.4641i	0.420928i
$619$	−10.1962	−0.409818	−0.204909	−	0.978781i	$-0.565690\pi$
$619$	−10.1962	−0.409818	−0.204909	+	0.978781i	$0.565690\pi$
$620$	0	0
$621$	7.92820	0.318148
$622$	− 5.32051i	− 0.213333i
$623$	− 46.0526i	− 1.84506i
$624$	0.732051	0.0293055
$625$	0	0
$626$	−9.92820	−0.396811
$627$	0.267949i	0.0107009i
$628$	13.8564i	0.552931i
$629$	8.39230	0.334623
$630$	0	0
$631$	−10.9282	−0.435045	−0.217522	−	0.976055i	$-0.569798\pi$
$631$	−10.9282	−0.435045	−0.217522	+	0.976055i	$0.569798\pi$
$632$	12.4641i	0.495795i
$633$	− 13.7321i	− 0.545800i
$634$	1.73205	0.0687885
$635$	0	0
$636$	−1.73205	−0.0686803
$637$	0.339746i	0.0134612i
$638$	0.464102i	0.0183740i
$639$	−1.80385	−0.0713591
$640$	0	0
$641$	17.4641	0.689791	0.344895	−	0.938641i	$-0.387914\pi$
$641$	17.4641	0.689791	0.344895	+	0.938641i	$0.387914\pi$
$642$	− 8.19615i	− 0.323476i
$643$	− 40.9282i	− 1.61405i	−0.590517	−	0.807025i	$-0.701076\pi$
$643$	− 40.9282i	− 1.61405i	0.590517	−	0.807025i	$-0.298924\pi$
$644$	−21.6603	−0.853534
$645$	0	0
$646$	4.19615	0.165095
$647$	− 11.3923i	− 0.447878i	−0.974603	−	0.223939i	$-0.928108\pi$
$647$	− 11.3923i	− 0.447878i	0.974603	−	0.223939i	$-0.0718916\pi$
$648$	1.00000i	0.0392837i
$649$	−0.588457	−0.0230990
$650$	0	0
$651$	12.1962	0.478005
$652$	− 2.00000i	− 0.0783260i
$653$	10.5359i	0.412302i	0.978520	+	0.206151i	$0.0660937\pi$
$653$	10.5359i	0.412302i	−0.978520	+	0.206151i	$0.933906\pi$
$654$	18.3923	0.719196
$655$	0	0
$656$	10.9282	0.426675
$657$	4.46410i	0.174161i
$658$	9.46410i	0.368949i
$659$	−38.5885	−1.50319	−0.751596	−	0.659623i	$-0.770716\pi$
$659$	−38.5885	−1.50319	−0.751596	+	0.659623i	$0.770716\pi$
$660$	0	0
$661$	46.6410	1.81413	0.907063	−	0.420996i	$-0.138319\pi$
$661$	46.6410	1.81413	0.907063	+	0.420996i	$0.138319\pi$
$662$	25.7321i	1.00010i
$663$	3.07180i	0.119299i
$664$	0.267949	0.0103984
$665$	0	0
$666$	2.00000	0.0774984
$667$	− 13.7321i	− 0.531707i
$668$	− 4.92820i	− 0.190678i
$669$	−10.4641	−0.404566
$670$	0	0
$671$	−1.78461	−0.0688941
$672$	− 2.73205i	− 0.105391i
$673$	44.9808i	1.73388i	0.498412	+	0.866940i	$0.333917\pi$
$673$	44.9808i	1.73388i	−0.498412	+	0.866940i	$0.666083\pi$
$674$	17.4641	0.672692
$675$	0	0
$676$	−12.4641	−0.479389
$677$	8.66025i	0.332841i	0.986055	+	0.166420i	$0.0532208\pi$
$677$	8.66025i	0.332841i	−0.986055	+	0.166420i	$0.946779\pi$
$678$	− 7.39230i	− 0.283900i
$679$	24.9282	0.956657
$680$	0	0
$681$	6.00000	0.229920
$682$	− 1.19615i	− 0.0458030i
$683$	4.98076i	0.190584i	0.995449	+	0.0952918i	$0.0303784\pi$
$683$	4.98076i	0.190584i	−0.995449	+	0.0952918i	$0.969622\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−17.8564	−0.681761
$687$	9.73205i	0.371301i
$688$	− 2.19615i	− 0.0837275i
$689$	−1.26795	−0.0483050
$690$	0	0
$691$	−11.8038	−0.449040	−0.224520	−	0.974470i	$-0.572081\pi$
$691$	−11.8038	−0.449040	−0.224520	+	0.974470i	$0.572081\pi$
$692$	− 17.5885i	− 0.668613i
$693$	0.732051i	0.0278083i
$694$	20.7846	0.788973
$695$	0	0
$696$	1.73205	0.0656532
$697$	45.8564i	1.73694i
$698$	21.0526i	0.796851i
$699$	3.12436	0.118174
$700$	0	0
$701$	24.3923	0.921285	0.460642	−	0.887586i	$-0.347619\pi$
$701$	24.3923	0.921285	0.460642	+	0.887586i	$0.347619\pi$
$702$	0.732051i	0.0276295i
$703$	− 2.00000i	− 0.0754314i
$704$	−0.267949	−0.0100987
$705$	0	0
$706$	22.0526	0.829959
$707$	− 34.7846i	− 1.30821i
$708$	2.19615i	0.0825365i
$709$	39.3013	1.47599	0.737995	−	0.674806i	$-0.235773\pi$
$709$	39.3013	1.47599	0.737995	+	0.674806i	$0.235773\pi$
$710$	0	0
$711$	−12.4641	−0.467440
$712$	16.8564i	0.631721i
$713$	35.3923i	1.32545i
$714$	11.4641	0.429033
$715$	0	0
$716$	−12.0000	−0.448461
$717$	14.0000i	0.522840i
$718$	− 3.07180i	− 0.114638i
$719$	50.1769	1.87128	0.935642	−	0.352952i	$-0.114822\pi$
$719$	50.1769	1.87128	0.935642	+	0.352952i	$0.114822\pi$
$720$	0	0
$721$	28.5885	1.06469
$722$	− 1.00000i	− 0.0372161i
$723$	− 19.8038i	− 0.736513i
$724$	−14.5885	−0.542176
$725$	0	0
$726$	−10.9282	−0.405584
$727$	38.1051i	1.41324i	0.707593	+	0.706620i	$0.249781\pi$
$727$	38.1051i	1.41324i	−0.707593	+	0.706620i	$0.750219\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	9.21539	0.340844
$732$	6.66025i	0.246170i
$733$	− 24.5167i	− 0.905544i	−0.891626	−	0.452772i	$-0.850435\pi$
$733$	− 24.5167i	− 0.905544i	0.891626	−	0.452772i	$-0.149565\pi$
$734$	−16.7321	−0.617591
$735$	0	0
$736$	7.92820	0.292237
$737$	− 1.00000i	− 0.0368355i
$738$	10.9282i	0.402273i
$739$	−9.17691	−0.337578	−0.168789	−	0.985652i	$-0.553986\pi$
$739$	−9.17691	−0.337578	−0.168789	+	0.985652i	$0.553986\pi$
$740$	0	0
$741$	0.732051	0.0268926
$742$	4.73205i	0.173719i
$743$	18.9282i	0.694408i	0.937790	+	0.347204i	$0.112869\pi$
$743$	18.9282i	0.694408i	−0.937790	+	0.347204i	$0.887131\pi$
$744$	−4.46410	−0.163662
$745$	0	0
$746$	−13.5167	−0.494880
$747$	0.267949i	0.00980375i
$748$	− 1.12436i	− 0.0411105i
$749$	−22.3923	−0.818197
$750$	0	0
$751$	0.535898	0.0195552	0.00977760	−	0.999952i	$-0.496888\pi$
$751$	0.535898	0.0195552	0.00977760	+	0.999952i	$0.496888\pi$
$752$	− 3.46410i	− 0.126323i
$753$	4.53590i	0.165297i
$754$	1.26795	0.0461760
$755$	0	0
$756$	2.73205	0.0993637
$757$	17.5885i	0.639263i	0.947542	+	0.319632i	$0.103559\pi$
$757$	17.5885i	0.639263i	−0.947542	+	0.319632i	$0.896441\pi$
$758$	20.3923i	0.740682i
$759$	−2.12436	−0.0771092
$760$	0	0
$761$	4.73205	0.171537	0.0857684	−	0.996315i	$-0.472666\pi$
$761$	4.73205	0.171537	0.0857684	+	0.996315i	$0.472666\pi$
$762$	− 5.00000i	− 0.181131i
$763$	− 50.2487i	− 1.81913i
$764$	−3.00000	−0.108536
$765$	0	0
$766$	−25.5167	−0.921954
$767$	1.60770i	0.0580505i
$768$	1.00000i	0.0360844i
$769$	−20.7128	−0.746923	−0.373462	−	0.927646i	$-0.621829\pi$
$769$	−20.7128	−0.746923	−0.373462	+	0.927646i	$0.621829\pi$
$770$	0	0
$771$	−9.00000	−0.324127
$772$	− 16.0526i	− 0.577744i
$773$	10.6795i	0.384115i	0.981384	+	0.192057i	$0.0615160\pi$
$773$	10.6795i	0.384115i	−0.981384	+	0.192057i	$0.938484\pi$
$774$	2.19615	0.0789391
$775$	0	0
$776$	−9.12436	−0.327545
$777$	− 5.46410i	− 0.196024i
$778$	− 33.7128i	− 1.20866i
$779$	10.9282	0.391544
$780$	0	0
$781$	0.483340	0.0172952
$782$	33.2679i	1.18966i
$783$	1.73205i	0.0618984i
$784$	−0.464102	−0.0165751
$785$	0	0
$786$	13.7321	0.489806
$787$	− 22.1244i	− 0.788648i	−0.918971	−	0.394324i	$-0.870979\pi$
$787$	− 22.1244i	− 0.788648i	0.918971	−	0.394324i	$-0.129021\pi$
$788$	6.58846i	0.234704i
$789$	15.0000	0.534014
$790$	0	0
$791$	−20.1962	−0.718093
$792$	− 0.267949i	− 0.00952116i
$793$	4.87564i	0.173139i
$794$	8.26795	0.293419
$795$	0	0
$796$	−14.0000	−0.496217
$797$	20.0000i	0.708436i	0.935163	+	0.354218i	$0.115253\pi$
$797$	20.0000i	0.708436i	−0.935163	+	0.354218i	$0.884747\pi$
$798$	− 2.73205i	− 0.0967136i
$799$	14.5359	0.514243
$800$	0	0
$801$	−16.8564	−0.595592
$802$	10.8564i	0.383353i
$803$	− 1.19615i	− 0.0422113i
$804$	−3.73205	−0.131619
$805$	0	0
$806$	−3.26795	−0.115109
$807$	27.4641i	0.966782i
$808$	12.7321i	0.447912i
$809$	7.51666	0.264272	0.132136	−	0.991232i	$-0.457816\pi$
$809$	7.51666	0.264272	0.132136	+	0.991232i	$0.457816\pi$
$810$	0	0
$811$	−54.3731	−1.90930	−0.954648	−	0.297736i	$-0.903769\pi$
$811$	−54.3731	−1.90930	−0.954648	+	0.297736i	$0.903769\pi$
$812$	− 4.73205i	− 0.166062i
$813$	− 26.1962i	− 0.918739i
$814$	−0.535898	−0.0187832
$815$	0	0
$816$	−4.19615	−0.146895
$817$	− 2.19615i	− 0.0768336i
$818$	22.3923i	0.782929i
$819$	2.00000	0.0698857
$820$	0	0
$821$	47.5167	1.65834	0.829171	−	0.558994i	$-0.188813\pi$
$821$	47.5167	1.65834	0.829171	+	0.558994i	$0.188813\pi$
$822$	8.39230i	0.292715i
$823$	− 34.7846i	− 1.21252i	−0.795268	−	0.606258i	$-0.792670\pi$
$823$	− 34.7846i	− 1.21252i	0.795268	−	0.606258i	$-0.207330\pi$
$824$	−10.4641	−0.364534
$825$	0	0
$826$	6.00000	0.208767
$827$	3.26795i	0.113638i	0.998385	+	0.0568189i	$0.0180958\pi$
$827$	3.26795i	0.113638i	−0.998385	+	0.0568189i	$0.981904\pi$
$828$	7.92820i	0.275524i
$829$	4.19615	0.145738	0.0728692	−	0.997342i	$-0.476784\pi$
$829$	4.19615	0.145738	0.0728692	+	0.997342i	$0.476784\pi$
$830$	0	0
$831$	−0.803848	−0.0278852
$832$	0.732051i	0.0253793i
$833$	− 1.94744i	− 0.0674748i
$834$	−3.12436	−0.108188
$835$	0	0
$836$	−0.267949	−0.00926722
$837$	− 4.46410i	− 0.154302i
$838$	− 13.8564i	− 0.478662i
$839$	15.4115	0.532066	0.266033	−	0.963964i	$-0.414287\pi$
$839$	15.4115	0.532066	0.266033	+	0.963964i	$0.414287\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	34.7846i	1.19876i
$843$	− 23.7846i	− 0.819185i
$844$	13.7321	0.472677
$845$	0	0
$846$	3.46410	0.119098
$847$	29.8564i	1.02588i
$848$	− 1.73205i	− 0.0594789i
$849$	4.00000	0.137280
$850$	0	0
$851$	15.8564	0.543551
$852$	− 1.80385i	− 0.0617988i
$853$	11.4641i	0.392523i	0.980552	+	0.196262i	$0.0628802\pi$
$853$	11.4641i	0.392523i	−0.980552	+	0.196262i	$0.937120\pi$
$854$	18.1962	0.622660
$855$	0	0
$856$	8.19615	0.280139
$857$	− 29.0718i	− 0.993074i	−0.868016	−	0.496537i	$-0.834605\pi$
$857$	− 29.0718i	− 0.993074i	0.868016	−	0.496537i	$-0.165395\pi$
$858$	− 0.196152i	− 0.00669653i
$859$	−43.0718	−1.46959	−0.734795	−	0.678289i	$-0.762722\pi$
$859$	−43.0718	−1.46959	−0.734795	+	0.678289i	$0.762722\pi$
$860$	0	0
$861$	29.8564	1.01750
$862$	− 5.32051i	− 0.181217i
$863$	20.6795i	0.703938i	0.936012	+	0.351969i	$0.114488\pi$
$863$	20.6795i	0.703938i	−0.936012	+	0.351969i	$0.885512\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−13.8038	−0.469074
$867$	− 0.607695i	− 0.0206384i
$868$	12.1962i	0.413964i
$869$	3.33975	0.113293
$870$	0	0
$871$	−2.73205	−0.0925720
$872$	18.3923i	0.622842i
$873$	− 9.12436i	− 0.308813i
$874$	7.92820	0.268175
$875$	0	0
$876$	−4.46410	−0.150828
$877$	− 31.1769i	− 1.05277i	−0.850246	−	0.526385i	$-0.823547\pi$
$877$	− 31.1769i	− 1.05277i	0.850246	−	0.526385i	$-0.176453\pi$
$878$	− 15.7846i	− 0.532705i
$879$	−24.2679	−0.818538
$880$	0	0
$881$	−33.4641	−1.12743	−0.563717	−	0.825968i	$-0.690629\pi$
$881$	−33.4641	−1.12743	−0.563717	+	0.825968i	$0.690629\pi$
$882$	− 0.464102i	− 0.0156271i
$883$	− 30.7321i	− 1.03422i	−0.855920	−	0.517108i	$-0.827009\pi$
$883$	− 30.7321i	− 1.03422i	0.855920	−	0.517108i	$-0.172991\pi$
$884$	−3.07180	−0.103316
$885$	0	0
$886$	−20.6603	−0.694095
$887$	− 28.3923i	− 0.953320i	−0.879088	−	0.476660i	$-0.841847\pi$
$887$	− 28.3923i	− 0.953320i	0.879088	−	0.476660i	$-0.158153\pi$
$888$	2.00000i	0.0671156i
$889$	−13.6603	−0.458150
$890$	0	0
$891$	0.267949	0.00897664
$892$	− 10.4641i	− 0.350364i
$893$	− 3.46410i	− 0.115922i
$894$	−5.85641	−0.195868
$895$	0	0
$896$	2.73205	0.0912714
$897$	5.80385i	0.193785i
$898$	17.5359i	0.585181i
$899$	−7.73205	−0.257878
$900$	0	0
$901$	7.26795	0.242130
$902$	− 2.92820i	− 0.0974985i
$903$	− 6.00000i	− 0.199667i
$904$	7.39230	0.245864
$905$	0	0
$906$	−4.53590	−0.150695
$907$	14.6795i	0.487425i	0.969848	+	0.243712i	$0.0783653\pi$
$907$	14.6795i	0.487425i	−0.969848	+	0.243712i	$0.921635\pi$
$908$	6.00000i	0.199117i
$909$	−12.7321	−0.422295
$910$	0	0
$911$	−42.1051	−1.39500	−0.697502	−	0.716582i	$-0.745705\pi$
$911$	−42.1051	−1.39500	−0.697502	+	0.716582i	$0.745705\pi$
$912$	1.00000i	0.0331133i
$913$	− 0.0717968i	− 0.00237613i
$914$	18.9282	0.626089
$915$	0	0
$916$	−9.73205	−0.321556
$917$	− 37.5167i	− 1.23891i
$918$	− 4.19615i	− 0.138494i
$919$	49.5167	1.63340	0.816702	−	0.577060i	$-0.195800\pi$
$919$	49.5167	1.63340	0.816702	+	0.577060i	$0.195800\pi$
$920$	0	0
$921$	23.7321	0.781998
$922$	− 9.85641i	− 0.324603i
$923$	− 1.32051i	− 0.0434651i
$924$	−0.732051	−0.0240827
$925$	0	0
$926$	−0.679492	−0.0223295
$927$	− 10.4641i	− 0.343686i
$928$	1.73205i	0.0568574i
$929$	−29.0718	−0.953815	−0.476907	−	0.878954i	$-0.658242\pi$
$929$	−29.0718	−0.953815	−0.476907	+	0.878954i	$0.658242\pi$
$930$	0	0
$931$	−0.464102	−0.0152103
$932$	3.12436i	0.102342i
$933$	5.32051i	0.174186i
$934$	3.19615	0.104581
$935$	0	0
$936$	−0.732051	−0.0239278
$937$	− 54.7846i	− 1.78974i	−0.446332	−	0.894868i	$-0.647270\pi$
$937$	− 54.7846i	− 1.78974i	0.446332	−	0.894868i	$-0.352730\pi$
$938$	10.1962i	0.332916i
$939$	9.92820	0.323995
$940$	0	0
$941$	−15.7321	−0.512850	−0.256425	−	0.966564i	$-0.582545\pi$
$941$	−15.7321	−0.512850	−0.256425	+	0.966564i	$0.582545\pi$
$942$	− 13.8564i	− 0.451466i
$943$	86.6410i	2.82142i
$944$	−2.19615	−0.0714787
$945$	0	0
$946$	−0.588457	−0.0191324
$947$	− 21.8564i	− 0.710238i	−0.934821	−	0.355119i	$-0.884440\pi$
$947$	− 21.8564i	− 0.710238i	0.934821	−	0.355119i	$-0.115560\pi$
$948$	− 12.4641i	− 0.404815i
$949$	−3.26795	−0.106082
$950$	0	0
$951$	−1.73205	−0.0561656
$952$	11.4641i	0.371554i
$953$	43.2487i	1.40096i	0.713670	+	0.700482i	$0.247031\pi$
$953$	43.2487i	1.40096i	−0.713670	+	0.700482i	$0.752969\pi$
$954$	1.73205	0.0560772
$955$	0	0
$956$	−14.0000	−0.452792
$957$	− 0.464102i	− 0.0150023i
$958$	30.8564i	0.996925i
$959$	22.9282	0.740390
$960$	0	0
$961$	−11.0718	−0.357155
$962$	1.46410i	0.0472045i
$963$	8.19615i	0.264117i
$964$	19.8038	0.637839
$965$	0	0
$966$	21.6603	0.696907
$967$	58.1051i	1.86853i	0.356573	+	0.934267i	$0.383945\pi$
$967$	58.1051i	1.86853i	−0.356573	+	0.934267i	$0.616055\pi$
$968$	− 10.9282i	− 0.351246i
$969$	−4.19615	−0.134800
$970$	0	0
$971$	55.7654	1.78960	0.894798	−	0.446471i	$-0.147319\pi$
$971$	55.7654	1.78960	0.894798	+	0.446471i	$0.147319\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	8.53590i	0.273648i
$974$	−29.8564	−0.956661
$975$	0	0
$976$	−6.66025	−0.213190
$977$	− 5.46410i	− 0.174812i	−0.996173	−	0.0874060i	$-0.972142\pi$
$977$	− 5.46410i	− 0.174812i	0.996173	−	0.0874060i	$-0.0278578\pi$
$978$	2.00000i	0.0639529i
$979$	4.51666	0.144353
$980$	0	0
$981$	−18.3923	−0.587221
$982$	− 10.9282i	− 0.348733i
$983$	− 47.6603i	− 1.52013i	−0.649849	−	0.760063i	$-0.725168\pi$
$983$	− 47.6603i	− 1.52013i	0.649849	−	0.760063i	$-0.274832\pi$
$984$	−10.9282	−0.348378
$985$	0	0
$986$	−7.26795	−0.231459
$987$	− 9.46410i	− 0.301246i
$988$	0.732051i	0.0232896i
$989$	17.4115	0.553655
$990$	0	0
$991$	−5.67949	−0.180415	−0.0902075	−	0.995923i	$-0.528753\pi$
$991$	−5.67949	−0.180415	−0.0902075	+	0.995923i	$0.528753\pi$
$992$	− 4.46410i	− 0.141735i
$993$	− 25.7321i	− 0.816582i
$994$	−4.92820	−0.156313
$995$	0	0
$996$	−0.267949	−0.00849030
$997$	24.9474i	0.790093i	0.918661	+	0.395047i	$0.129272\pi$
$997$	24.9474i	0.790093i	−0.918661	+	0.395047i	$0.870728\pi$
$998$	− 31.3731i	− 0.993097i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2850.2.d.x.799.1		4
5.2	odd	4		2850.2.a.bi.1.2	yes	2
5.3	odd	4		2850.2.a.bd.1.1	✓	2
5.4	even	2	inner	2850.2.d.x.799.4		4
15.2	even	4		8550.2.a.bs.1.2		2
15.8	even	4		8550.2.a.by.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.bd.1.1	✓	2	5.3	odd	4
2850.2.a.bi.1.2	yes	2	5.2	odd	4
2850.2.d.x.799.1		4	1.1	even	1	trivial
2850.2.d.x.799.4		4	5.4	even	2	inner
8550.2.a.bs.1.2		2	15.2	even	4
8550.2.a.by.1.1		2	15.8	even	4

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:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.73205i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.73205i q^{7} +1.00000i q^{8} -1.00000 q^{9} +0.267949 q^{11} -1.00000i q^{12} -0.732051i q^{13} -2.73205 q^{14} +1.00000 q^{16} +4.19615i q^{17} +1.00000i q^{18} +1.00000 q^{19} +2.73205 q^{21} -0.267949i q^{22} +7.92820i q^{23} -1.00000 q^{24} -0.732051 q^{26} -1.00000i q^{27} +2.73205i q^{28} -1.73205 q^{29} +4.46410 q^{31} -1.00000i q^{32} +0.267949i q^{33} +4.19615 q^{34} +1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +0.732051 q^{39} +10.9282 q^{41} -2.73205i q^{42} -2.19615i q^{43} -0.267949 q^{44} +7.92820 q^{46} -3.46410i q^{47} +1.00000i q^{48} -0.464102 q^{49} -4.19615 q^{51} +0.732051i q^{52} -1.73205i q^{53} -1.00000 q^{54} +2.73205 q^{56} +1.00000i q^{57} +1.73205i q^{58} -2.19615 q^{59} -6.66025 q^{61} -4.46410i q^{62} +2.73205i q^{63} -1.00000 q^{64} +0.267949 q^{66} -3.73205i q^{67} -4.19615i q^{68} -7.92820 q^{69} +1.80385 q^{71} -1.00000i q^{72} -4.46410i q^{73} -2.00000 q^{74} -1.00000 q^{76} -0.732051i q^{77} -0.732051i q^{78} +12.4641 q^{79} +1.00000 q^{81} -10.9282i q^{82} -0.267949i q^{83} -2.73205 q^{84} -2.19615 q^{86} -1.73205i q^{87} +0.267949i q^{88} +16.8564 q^{89} -2.00000 q^{91} -7.92820i q^{92} +4.46410i q^{93} -3.46410 q^{94} +1.00000 q^{96} +9.12436i q^{97} +0.464102i q^{98} -0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 8 q^{11} - 4 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{21} - 4 q^{24} + 4 q^{26} + 4 q^{31} - 4 q^{34} + 4 q^{36} - 4 q^{39} + 16 q^{41} - 8 q^{44} + 4 q^{46} + 12 q^{49} + 4 q^{51} - 4 q^{54} + 4 q^{56} + 12 q^{59} + 8 q^{61} - 4 q^{64} + 8 q^{66} - 4 q^{69} + 28 q^{71} - 8 q^{74} - 4 q^{76} + 36 q^{79} + 4 q^{81} - 4 q^{84} + 12 q^{86} + 12 q^{89} - 8 q^{91} + 4 q^{96} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 8 * q^11 - 4 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^21 - 4 * q^24 + 4 * q^26 + 4 * q^31 - 4 * q^34 + 4 * q^36 - 4 * q^39 + 16 * q^41 - 8 * q^44 + 4 * q^46 + 12 * q^49 + 4 * q^51 - 4 * q^54 + 4 * q^56 + 12 * q^59 + 8 * q^61 - 4 * q^64 + 8 * q^66 - 4 * q^69 + 28 * q^71 - 8 * q^74 - 4 * q^76 + 36 * q^79 + 4 * q^81 - 4 * q^84 + 12 * q^86 + 12 * q^89 - 8 * q^91 + 4 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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