LMFDB - Embedded newform 3850.2.c.j.1849.1

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 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 3.00000i	− 0.707107i
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	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	$0.692861\pi$
$38$	− 6.00000i	− 0.973329i
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	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	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	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
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	2.00000i	0.254000i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	4.00000i	0.485071i
$69$	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
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	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
$82$	− 4.00000i	− 0.441726i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	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
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	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
$103$	− 18.0000i	− 1.77359i	−0.462160	−	0.886796i	$-0.652926\pi$
$103$	− 18.0000i	− 1.77359i	0.462160	−	0.886796i	$-0.347074\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 6.00000i	− 0.554700i
$118$	− 12.0000i	− 1.10469i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000i	1.26750i
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	−3.00000	−0.267261
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	−12.0000	−1.03664
$135$	0	0
$136$	4.00000	0.342997
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	8.00000i	0.671345i
$143$	2.00000i	0.167248i
$144$	3.00000	0.250000
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	6.00000i	0.486664i
$153$	− 12.0000i	− 0.970143i
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	− 8.00000i	− 0.638470i	−0.947676	−	0.319235i	$-0.896574\pi$
$157$	− 8.00000i	− 0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	− 9.00000i	− 0.707107i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	6.00000	0.465690
$167$	4.00000i	0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$	4.00000i	0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	18.0000	1.37649
$172$	− 8.00000i	− 0.609994i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	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
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	− 2.00000i	− 0.145865i
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	3.00000i	0.213201i
$199$	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
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	−18.0000	−1.25412
$207$	− 12.0000i	− 0.834058i
$208$	− 2.00000i	− 0.138675i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−16.0000	−1.09374
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	− 14.0000i	− 0.948200i
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 2.00000i	− 0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$	− 2.00000i	− 0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 30.0000i	− 1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	− 30.0000i	− 1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	4.00000i	0.259281i
$239$	−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
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	− 12.0000i	− 0.763542i
$248$	− 2.00000i	− 0.127000i
$249$	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	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	6.00000	0.371391
$262$	− 6.00000i	− 0.370681i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	12.0000i	0.733017i
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	− 4.00000i	− 0.242536i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	14.0000i	0.839664i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 6.00000i	− 0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$	− 6.00000i	− 0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	− 4.00000i	− 0.236113i
$288$	− 3.00000i	− 0.176777i
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	2.00000i	0.115857i
$299$	−8.00000	−0.462652
$300$	0	0
$301$	8.00000	0.461112
$302$	24.0000i	1.38104i
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	−12.0000	−0.685994
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	0	0
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	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
$326$	−4.00000	−0.221540
$327$	0	0
$328$	4.00000i	0.220863i
$329$	2.00000	0.110264
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	− 6.00000i	− 0.329293i
$333$	30.0000i	1.64399i
$334$	4.00000	0.218870
$335$	0	0
$336$	0	0
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	− 18.0000i	− 0.973329i
$343$	1.00000i	0.0539949i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	14.0000	0.752645
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	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	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	4.00000i	0.211407i
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 20.0000i	− 1.05118i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	− 4.00000i	− 0.208514i
$369$	12.0000	0.624695
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−2.00000	−0.103142
$377$	− 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
$382$	4.00000i	0.204658i
$383$	10.0000i	0.510976i	0.966812	+	0.255488i	$0.0822362\pi$
$383$	10.0000i	0.510976i	−0.966812	+	0.255488i	$0.917764\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	24.0000i	1.21999i
$388$	14.0000i	0.710742i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	3.00000	0.150756
$397$	24.0000i	1.20453i	0.798298	+	0.602263i	$0.205734\pi$
$397$	24.0000i	1.20453i	−0.798298	+	0.602263i	$0.794266\pi$
$398$	− 14.0000i	− 0.701757i
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	− 10.0000i	− 0.495682i
$408$	0	0
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	0	0
$412$	18.0000i	0.886796i
$413$	− 12.0000i	− 0.590481i
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	6.00000i	0.293470i
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	8.00000i	0.389434i
$423$	6.00000i	0.291730i
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	14.0000i	0.677507i
$428$	16.0000i	0.773389i
$429$	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	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	−14.0000	−0.670478
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	8.00000i	0.380521i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	2.00000	0.0947027
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	14.0000i	0.658505i
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	20.0000i	0.934539i
$459$	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
$463$	32.0000i	1.48717i	0.668644	+	0.743583i	$0.266875\pi$
$463$	32.0000i	1.48717i	−0.668644	+	0.743583i	$0.733125\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−30.0000	−1.38972
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	6.00000i	0.277350i
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	− 18.0000i	− 0.824163i
$478$	16.0000i	0.731823i
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	− 12.0000i	− 0.546585i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	− 14.0000i	− 0.633750i
$489$	0	0
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	4.00000	0.177822
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	0	0
$517$	− 2.00000i	− 0.0879599i
$518$	− 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
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	36.0000	1.56227
$532$	6.00000i	0.260133i
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	12.0000i	0.517357i
$539$	1.00000	0.0430730
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	20.0000i	0.859074i
$543$	0	0
$544$	−4.00000	−0.171499
$545$	0	0
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−42.0000	−1.79252
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	−30.0000	−1.27458
$555$	0	0
$556$	14.0000	0.593732
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	6.00000i	0.254000i
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	10.0000i	0.421825i
$563$	34.0000i	1.43293i	0.697623	+	0.716465i	$0.254241\pi$
$563$	34.0000i	1.43293i	−0.697623	+	0.716465i	$0.745759\pi$
$564$	0	0
$565$	0	0
$566$	−6.00000	−0.252199
$567$	− 9.00000i	− 0.377964i
$568$	− 8.00000i	− 0.335673i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	−3.00000	−0.125000
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	6.00000i	0.248495i
$584$	4.00000	0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	10.0000i	0.410997i
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	2.00000	0.0819232
$597$	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
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	− 8.00000i	− 0.326056i
$603$	− 36.0000i	− 1.46603i
$604$	24.0000	0.976546
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	− 6.00000i	− 0.243332i
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	12.0000i	0.485071i
$613$	− 46.0000i	− 1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$	− 46.0000i	− 1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	− 14.0000i	− 0.561349i
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	8.00000i	0.319235i
$629$	40.0000	1.59490
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	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
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	9.00000i	0.353553i
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	0	0
$655$	0	0
$656$	4.00000	0.156174
$657$	− 12.0000i	− 0.468165i
$658$	− 2.00000i	− 0.0779681i
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	20.0000i	0.777322i
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	30.0000	1.16248
$667$	− 8.00000i	− 0.309761i
$668$	− 4.00000i	− 0.154765i
$669$	0	0
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−9.00000	−0.346154
$677$	26.0000i	0.999261i	0.866239	+	0.499631i	$0.166531\pi$
$677$	26.0000i	0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	− 2.00000i	− 0.0765840i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−18.0000	−0.688247
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	8.00000i	0.304997i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	− 14.0000i	− 0.532200i
$693$	3.00000i	0.113961i
$694$	8.00000	0.303676
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	− 10.0000i	− 0.378506i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	60.0000i	2.26294i
$704$	1.00000	0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	− 16.0000i	− 0.597115i
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	−20.0000	−0.743294
$725$	0	0
$726$	0	0
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	27.0000	1.00000
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−4.00000	−0.147442
$737$	12.0000i	0.442026i
$738$	− 12.0000i	− 0.441726i
$739$	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
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	18.0000i	0.658586i
$748$	− 4.00000i	− 0.146254i
$749$	−16.0000	−0.584627
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	2.00000i	0.0729325i
$753$	0	0
$754$	−4.00000	−0.145671
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	4.00000i	0.145287i
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	− 14.0000i	− 0.506834i
$764$	4.00000	0.144715
$765$	0	0
$766$	10.0000	0.361315
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	48.0000i	1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$773$	48.0000i	1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	24.0000	0.859889
$780$	0	0
$781$	8.00000	0.286263
$782$	16.0000i	0.572159i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	− 3.00000i	− 0.106600i
$793$	28.0000i	0.994309i
$794$	24.0000	0.851728
$795$	0	0
$796$	−14.0000	−0.496217
$797$	− 16.0000i	− 0.566749i	−0.959009	−	0.283375i	$-0.908546\pi$
$797$	− 16.0000i	− 0.566749i	0.959009	−	0.283375i	$-0.0914540\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	18.0000	0.635999
$802$	18.0000i	0.635602i
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	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
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	2.00000i	0.0701862i
$813$	0	0
$814$	−10.0000	−0.350500
$815$	0	0
$816$	0	0
$817$	48.0000i	1.67931i
$818$	16.0000i	0.559427i
$819$	−6.00000	−0.209657
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	−12.0000	−0.417533
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	12.0000i	0.417029i
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	0	0
$832$	2.00000i	0.0693375i
$833$	4.00000i	0.138592i
$834$	0	0
$835$	0	0
$836$	6.00000	0.207514
$837$	0	0
$838$	− 32.0000i	− 1.10542i
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	2.00000i	0.0689246i
$843$	0	0
$844$	8.00000	0.275371
$845$	0	0
$846$	6.00000	0.206284
$847$	− 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	$-0.989099\pi$
$853$	− 2.00000i	− 0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	16.0000	0.546869
$857$	32.0000i	1.09310i	0.837427	+	0.546550i	$0.184059\pi$
$857$	32.0000i	1.09310i	−0.837427	+	0.546550i	$0.815941\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	− 16.0000i	− 0.544962i
$863$	44.0000i	1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$	44.0000i	1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	− 2.00000i	− 0.0678844i
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	14.0000i	0.474100i
$873$	− 42.0000i	− 1.42148i
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	28.0000i	0.944954i
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	3.00000i	0.101015i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	36.0000	1.20944
$887$	16.0000i	0.537227i	0.963248	+	0.268614i	$0.0865655\pi$
$887$	16.0000i	0.537227i	−0.963248	+	0.268614i	$0.913434\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−9.00000	−0.301511
$892$	− 2.00000i	− 0.0669650i
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−24.0000	−0.799556
$902$	4.00000i	0.133185i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	2.00000i	0.0663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	2.00000	0.0661541
$915$	0	0
$916$	20.0000	0.660819
$917$	− 6.00000i	− 0.198137i
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	− 30.0000i	− 0.987997i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	− 54.0000i	− 1.77359i
$928$	− 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
$931$	−6.00000	−0.196642
$932$	30.0000i	0.982683i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	6.00000	0.196116
$937$	12.0000i	0.392023i	0.980602	+	0.196011i	$0.0627990\pi$
$937$	12.0000i	0.392023i	−0.980602	+	0.196011i	$0.937201\pi$
$938$	12.0000i	0.391814i
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	12.0000	0.390567
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	− 4.00000i	− 0.129641i
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	6.00000	0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 20.0000i	− 0.644826i
$963$	− 48.0000i	− 1.54678i
$964$	−12.0000	−0.386494
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	56.0000	1.79713	0.898563	−	0.438845i	$-0.144612\pi$
$971$	56.0000	1.79713	0.898563	+	0.438845i	$0.144612\pi$
$972$	0	0
$973$	14.0000i	0.448819i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	−14.0000	−0.448129
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	42.0000	1.34096
$982$	36.0000i	1.14881i
$983$	− 18.0000i	− 0.574111i	−0.957914	−	0.287055i	$-0.907324\pi$
$983$	− 18.0000i	− 0.574111i	0.957914	−	0.287055i	$-0.0926764\pi$
$984$	0	0
$985$	0	0
$986$	−8.00000	−0.254772
$987$	0	0
$988$	12.0000i	0.381771i
$989$	32.0000	1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	2.00000i	0.0635001i
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	44.0000i	1.39280i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.j.1849.1		2
5.2	odd	4		3850.2.a.u.1.1		1
5.3	odd	4		154.2.a.a.1.1	✓	1
5.4	even	2	inner	3850.2.c.j.1849.2		2
15.8	even	4		1386.2.a.l.1.1		1
20.3	even	4		1232.2.a.e.1.1		1
35.3	even	12		1078.2.e.i.177.1		2
35.13	even	4		1078.2.a.d.1.1		1
35.18	odd	12		1078.2.e.j.177.1		2
35.23	odd	12		1078.2.e.j.67.1		2
35.33	even	12		1078.2.e.i.67.1		2
40.3	even	4		4928.2.a.w.1.1		1
40.13	odd	4		4928.2.a.v.1.1		1
55.43	even	4		1694.2.a.g.1.1		1
105.83	odd	4		9702.2.a.ba.1.1		1
140.83	odd	4		8624.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.a.1.1	✓	1	5.3	odd	4
1078.2.a.d.1.1		1	35.13	even	4
1078.2.e.i.67.1		2	35.33	even	12
1078.2.e.i.177.1		2	35.3	even	12
1078.2.e.j.67.1		2	35.23	odd	12
1078.2.e.j.177.1		2	35.18	odd	12
1232.2.a.e.1.1		1	20.3	even	4
1386.2.a.l.1.1		1	15.8	even	4
1694.2.a.g.1.1		1	55.43	even	4
3850.2.a.u.1.1		1	5.2	odd	4
3850.2.c.j.1849.1		2	1.1	even	1	trivial
3850.2.c.j.1849.2		2	5.4	even	2	inner
4928.2.a.v.1.1		1	40.13	odd	4
4928.2.a.w.1.1		1	40.3	even	4
8624.2.a.r.1.1		1	140.83	odd	4
9702.2.a.ba.1.1		1	105.83	odd	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)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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