LMFDB - Embedded newform 3850.2.c.h.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1849,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3850.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.c (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:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.h.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +3.00000i q^{18} -4.00000 q^{19} -1.00000i q^{22} +4.00000i q^{23} -2.00000 q^{26} -1.00000i q^{28} +2.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} -3.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{56} +2.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} +8.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -8.00000i q^{67} +6.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -10.0000 q^{74} +4.00000 q^{76} -1.00000i q^{77} +9.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} -12.0000 q^{86} +1.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} -4.00000i q^{92} +12.0000 q^{94} +14.0000i q^{97} -1.00000i q^{98} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^9	$ 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{26} + 4 q^{29} + 16 q^{31} + 12 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 2 q^{56} + 24 q^{59} + 12 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} + 8 q^{76} + 18 q^{81} - 24 q^{86} + 12 q^{89} - 4 q^{91} + 24 q^{94} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^26 + 4 * q^29 + 16 * q^31 + 12 * q^34 - 6 * q^36 - 12 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 2 * q^56 + 24 * q^59 + 12 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 + 8 * q^76 + 18 * q^81 - 24 * q^86 + 12 * q^89 - 4 * q^91 + 24 * q^94 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$.

$n$	$1751$	$2201$	$2927$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	3.00000	1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	3.00000i	0.707107i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−3.00000	−0.500000
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	2.00000i	0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000i	1.01600i
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 3.00000i	− 0.353553i
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	4.00000	0.458831
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 6.00000i	− 0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	1.00000i	0.106600i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 1.00000i	− 0.101015i
$99$	−3.00000	−0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	6.00000i	0.554700i
$118$	12.0000i	1.10469i
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.00000i	0.543214i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	−3.00000	−0.267261
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	8.00000	0.691095
$135$	0	0
$136$	−6.00000	−0.514496
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	− 2.00000i	− 0.167248i
$144$	3.00000	0.250000
$145$	0	0
$146$	−14.0000	−1.15865
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	4.00000i	0.324443i
$153$	− 18.0000i	− 1.45521i
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	9.00000i	0.707107i
$163$	24.0000i	1.87983i	0.341415	+	0.939913i	$0.389094\pi$
$163$	24.0000i	1.87983i	−0.341415	+	0.939913i	$0.610906\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−12.0000	−0.917663
$172$	− 12.0000i	− 0.914991i
$173$	26.0000i	1.97674i	0.152057	+	0.988372i	$0.451410\pi$
$173$	26.0000i	1.97674i	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.00000i	0.449719i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	− 2.00000i	− 0.148250i
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	6.00000i	0.438763i
$188$	12.0000i	0.875190i
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	− 3.00000i	− 0.213201i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	12.0000	0.836080
$207$	12.0000i	0.834058i
$208$	2.00000i	0.138675i
$209$	4.00000	0.276686
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 12.0000i	− 0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$	− 12.0000i	− 0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 2.00000i	− 0.131306i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	6.00000i	0.388922i
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	1.00000i	0.0642824i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 4.00000i	− 0.251478i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	− 4.00000i	− 0.247121i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	8.00000i	0.488678i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 4.00000i	− 0.239904i
$279$	24.0000	1.43684
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	− 6.00000i	− 0.354169i
$288$	3.00000i	0.176777i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	− 22.0000i	− 1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	− 22.0000i	− 1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	18.0000i	1.04271i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	−12.0000	−0.691669
$302$	16.0000i	0.920697i
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	18.0000	1.02899
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	1.00000i	0.0569803i
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	18.0000i	1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$	18.0000i	1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	− 4.00000i	− 0.222911i
$323$	24.0000i	1.33540i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−24.0000	−1.32924
$327$	0	0
$328$	6.00000i	0.331295i
$329$	12.0000	0.661581
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 4.00000i	− 0.219529i
$333$	30.0000i	1.64399i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	− 12.0000i	− 0.648886i
$343$	− 1.00000i	− 0.0539949i
$344$	12.0000	0.646997
$345$	0	0
$346$	−26.0000	−1.39777
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 10.0000i	− 0.525588i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	4.00000i	0.208514i
$369$	−18.0000	−0.937043
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	−12.0000	−0.618853
$377$	4.00000i	0.206010i
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	− 24.0000i	− 1.22795i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	0	0
$385$	0	0
$386$	−22.0000	−1.11977
$387$	36.0000i	1.82998i
$388$	− 14.0000i	− 0.710742i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	−14.0000	−0.705310
$395$	0	0
$396$	3.00000	0.150756
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	16.0000i	0.797017i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	− 10.0000i	− 0.495682i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	12.0000i	0.591198i
$413$	12.0000i	0.590481i
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	4.00000i	0.195646i
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	− 28.0000i	− 1.36302i
$423$	− 36.0000i	− 1.75038i
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	6.00000	0.287348
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	12.0000i	0.570782i
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	0	0
$445$	0	0
$446$	12.0000	0.568216
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	6.00000i	0.282216i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	10.0000i	0.467269i
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	28.0000i	1.30127i	0.759390	+	0.650635i	$0.225497\pi$
$463$	28.0000i	1.30127i	−0.759390	+	0.650635i	$0.774503\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 6.00000i	− 0.277350i
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	− 12.0000i	− 0.552345i
$473$	− 12.0000i	− 0.551761i
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	18.0000i	0.824163i
$478$	− 16.0000i	− 0.731823i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	8.00000	0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	4.00000	0.177822
$507$	0	0
$508$	8.00000i	0.354943i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	− 10.0000i	− 0.439375i
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	6.00000i	0.262613i
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	36.0000	1.56227
$532$	4.00000i	0.173422i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	−8.00000	−0.345547
$537$	0	0
$538$	18.0000i	0.776035i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	0	0
$543$	0	0
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	− 14.0000i	− 0.598050i
$549$	18.0000	0.768221
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	24.0000i	1.01600i
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	− 30.0000i	− 1.26547i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	9.00000i	0.377964i
$568$	8.00000i	0.335673i
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	2.00000i	0.0836242i
$573$	0	0
$574$	6.00000	0.250435
$575$	0	0
$576$	−3.00000	−0.125000
$577$	46.0000i	1.91501i	0.288425	+	0.957503i	$0.406868\pi$
$577$	46.0000i	1.91501i	−0.288425	+	0.957503i	$0.593132\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	− 6.00000i	− 0.248495i
$584$	14.0000	0.579324
$585$	0	0
$586$	22.0000	0.908812
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	10.0000i	0.410997i
$593$	− 42.0000i	− 1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$	− 42.0000i	− 1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	− 12.0000i	− 0.489083i
$603$	− 24.0000i	− 0.977356i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	18.0000i	0.727607i
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	−18.0000	−0.719425
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	60.0000	2.39236
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	− 2.00000i	− 0.0791808i
$639$	−24.0000	−0.949425
$640$	0	0
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	4.00000i	0.157256i	0.996904	+	0.0786281i	$0.0250540\pi$
$647$	4.00000i	0.157256i	−0.996904	+	0.0786281i	$0.974946\pi$
$648$	− 9.00000i	− 0.353553i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	− 24.0000i	− 0.939913i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	42.0000i	1.63858i
$658$	12.0000i	0.467809i
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	−30.0000	−1.16248
$667$	8.00000i	0.309761i
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	22.0000i	0.848038i	0.905653	+	0.424019i	$0.139381\pi$
$673$	22.0000i	0.848038i	−0.905653	+	0.424019i	$0.860619\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	12.0000i	0.457496i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	− 26.0000i	− 0.988372i
$693$	− 3.00000i	− 0.113961i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	10.0000i	0.378506i
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	− 40.0000i	− 1.50863i
$704$	1.00000	0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	20.0000i	0.741759i	0.928681	+	0.370879i	$0.120944\pi$
$727$	20.0000i	0.741759i	−0.928681	+	0.370879i	$0.879056\pi$
$728$	2.00000i	0.0741249i
$729$	27.0000	1.00000
$730$	0	0
$731$	72.0000	2.66302
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	−4.00000	−0.147442
$737$	8.00000i	0.294684i
$738$	− 18.0000i	− 0.662589i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	− 6.00000i	− 0.220267i
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	30.0000	1.09838
$747$	12.0000i	0.439057i
$748$	− 6.00000i	− 0.219382i
$749$	4.00000	0.146157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 12.0000i	− 0.437595i
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	− 46.0000i	− 1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$	− 46.0000i	− 1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$	− 4.00000i	− 0.145287i
$759$	0	0
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	− 6.00000i	− 0.217215i
$764$	24.0000	0.868290
$765$	0	0
$766$	20.0000	0.722629
$767$	24.0000i	0.866590i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 22.0000i	− 0.791797i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	−36.0000	−1.29399
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	10.0000i	0.358517i
$779$	24.0000	0.859889
$780$	0	0
$781$	8.00000	0.286263
$782$	24.0000i	0.858238i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	− 14.0000i	− 0.498729i
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	3.00000i	0.106600i
$793$	12.0000i	0.426132i
$794$	14.0000	0.496841
$795$	0	0
$796$	16.0000	0.567105
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	0	0
$801$	18.0000	0.635999
$802$	2.00000i	0.0706225i
$803$	− 14.0000i	− 0.494049i
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	0	0
$814$	10.0000	0.350500
$815$	0	0
$816$	0	0
$817$	− 48.0000i	− 1.67931i
$818$	14.0000i	0.489499i
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	−12.0000	−0.417533
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	− 12.0000i	− 0.417029i
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	12.0000i	0.414533i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	38.0000i	1.30957i
$843$	0	0
$844$	28.0000	0.963800
$845$	0	0
$846$	36.0000	1.23771
$847$	1.00000i	0.0343604i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−40.0000	−1.37118
$852$	0	0
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−4.00000	−0.136717
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	16.0000i	0.544962i
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 8.00000i	− 0.271538i
$869$	0	0
$870$	0	0
$871$	16.0000	0.542139
$872$	6.00000i	0.203186i
$873$	42.0000i	1.42148i
$874$	16.0000	0.541208
$875$	0	0
$876$	0	0
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	− 8.00000i	− 0.269987i
$879$	0	0
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	− 3.00000i	− 0.101015i
$883$	− 24.0000i	− 0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$	− 24.0000i	− 0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	16.0000	0.537531
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−9.00000	−0.301511
$892$	12.0000i	0.401790i
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	− 2.00000i	− 0.0667409i
$899$	16.0000	0.533630
$900$	0	0
$901$	36.0000	1.19933
$902$	6.00000i	0.199778i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	− 12.0000i	− 0.398234i
$909$	18.0000	0.597022
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	− 4.00000i	− 0.132381i
$914$	22.0000	0.727695
$915$	0	0
$916$	−10.0000	−0.330409
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	−28.0000	−0.920137
$927$	− 36.0000i	− 1.18240i
$928$	2.00000i	0.0656532i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	10.0000i	0.327561i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	6.00000	0.196116
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	12.0000	0.390567
$945$	0	0
$946$	12.0000	0.390154
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 6.00000i	− 0.194461i
$953$	38.0000i	1.23094i	0.788160	+	0.615470i	$0.211034\pi$
$953$	38.0000i	1.23094i	−0.788160	+	0.615470i	$0.788966\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	−14.0000	−0.452084
$960$	0	0
$961$	33.0000	1.06452
$962$	− 20.0000i	− 0.644826i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	12.0000	0.384505
$975$	0	0
$976$	6.00000	0.192055
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−18.0000	−0.574696
$982$	4.00000i	0.127645i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	8.00000i	0.254514i
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	36.0000i	1.13956i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.h.1849.2		2
5.2	odd	4		770.2.a.c.1.1	✓	1
5.3	odd	4		3850.2.a.t.1.1		1
5.4	even	2	inner	3850.2.c.h.1849.1		2
15.2	even	4		6930.2.a.u.1.1		1
20.7	even	4		6160.2.a.i.1.1		1
35.27	even	4		5390.2.a.i.1.1		1
55.32	even	4		8470.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.c.1.1	✓	1	5.2	odd	4
3850.2.a.t.1.1		1	5.3	odd	4
3850.2.c.h.1849.1		2	5.4	even	2	inner
3850.2.c.h.1849.2		2	1.1	even	1	trivial
5390.2.a.i.1.1		1	35.27	even	4
6160.2.a.i.1.1		1	20.7	even	4
6930.2.a.u.1.1		1	15.2	even	4
8470.2.a.ba.1.1		1	55.32	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:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +3.00000i q^{18} -4.00000 q^{19} -1.00000i q^{22} +4.00000i q^{23} -2.00000 q^{26} -1.00000i q^{28} +2.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} -3.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{56} +2.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} +8.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -8.00000i q^{67} +6.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +14.0000i q^{73} -10.0000 q^{74} +4.00000 q^{76} -1.00000i q^{77} +9.00000 q^{81} -6.00000i q^{82} +4.00000i q^{83} -12.0000 q^{86} +1.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} -4.00000i q^{92} +12.0000 q^{94} +14.0000i q^{97} -1.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^9	\( 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{26} + 4 q^{29} + 16 q^{31} + 12 q^{34} - 6 q^{36} - 12 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 2 q^{56} + 24 q^{59} + 12 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} + 8 q^{76} + 18 q^{81} - 24 q^{86} + 12 q^{89} - 4 q^{91} + 24 q^{94} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^26 + 4 * q^29 + 16 * q^31 + 12 * q^34 - 6 * q^36 - 12 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 2 * q^56 + 24 * q^59 + 12 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 + 8 * q^76 + 18 * q^81 - 24 * q^86 + 12 * q^89 - 4 * q^91 + 24 * q^94 - 6 * q^99

Introduction

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