LMFDB - Embedded newform 1950.2.e.l.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.e (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:	$15.5708283941$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.l.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +1.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +10.0000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -6.00000 q^{74} -1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +6.00000 q^{89} +4.00000i q^{92} +1.00000 q^{96} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{16} - 2 q^{24} + 2 q^{26} + 20 q^{29} - 12 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 8 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{69} + 32 q^{71} - 12 q^{74} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^16 - 2 * q^24 + 2 * q^26 + 20 * q^29 - 12 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 8 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 + 12 * q^61 - 2 * q^64 - 8 * q^69 + 32 * q^71 - 12 * q^74 + 2 * q^81 + 8 * q^86 + 12 * q^89 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000i	0.192450i
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	1.00000i	0.138675i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000i	1.31306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	− 1.00000i	− 0.113228i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	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
$86$	4.00000	0.431331
$87$	− 10.0000i	− 1.07211i
$88$	0	0
$89$	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
$91$	0	0
$92$	4.00000i	0.417029i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	7.00000i	0.707107i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000i	0.594089i
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	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
$111$	6.00000	0.569495
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	−10.0000	−0.928477
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000i	0.543214i
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−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
$142$	16.0000i	1.34269i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 7.00000i	− 0.577350i
$148$	− 6.00000i	− 0.493197i
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	0	0
$177$	0	0
$178$	6.00000i	0.449719i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000i	0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−12.0000	−0.836080
$207$	4.00000i	0.278019i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	6.00000i	0.412082i
$213$	− 16.0000i	− 1.09630i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.0000i	0.948200i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	6.00000i	0.402694i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	− 10.0000i	− 0.656532i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−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
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−6.00000	−0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	−12.0000	−0.752947
$255$	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	$0.940080\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	12.0000i	0.741362i
$263$	− 28.0000i	− 1.72655i	−0.504730	−	0.863277i	$-0.668408\pi$
$263$	− 28.0000i	− 1.72655i	0.504730	−	0.863277i	$-0.331592\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	4.00000	0.240772
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−14.0000	−0.820695
$292$	2.00000i	0.117041i
$293$	26.0000i	1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$	26.0000i	1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	14.0000i	0.810998i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	0	0
$302$	− 16.0000i	− 0.920697i
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	6.00000	0.342997
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	1.00000i	0.0566139i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 14.0000i	− 0.774202i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	− 4.00000i	− 0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	24.0000	1.31322
$335$	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	$-0.163099\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	10.0000	0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	10.0000i	0.536056i
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	20.0000i	1.05703i
$359$	−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
$361$	−19.0000	−1.00000
$362$	− 10.0000i	− 0.525588i
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.0000i	− 0.515026i
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	24.0000i	1.22795i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	4.00000i	0.203331i
$388$	14.0000i	0.710742i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	− 7.00000i	− 0.353553i
$393$	− 12.0000i	− 0.605320i
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 4.00000i	− 0.199502i
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	− 12.0000i	− 0.591198i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	1.00000	0.0490290
$417$	4.00000i	0.195881i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	16.0000	0.775203
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000i	0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	0	0
$438$	− 2.00000i	− 0.0955637i
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	6.00000i	0.285391i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	−16.0000	−0.757622
$447$	− 14.0000i	− 0.662177i
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	0	0
$452$	− 10.0000i	− 0.470360i
$453$	16.0000i	0.751746i
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	−18.0000	−0.833834
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 16.0000i	− 0.731823i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	− 14.0000i	− 0.637683i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 6.00000i	− 0.271607i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	2.00000i	0.0901670i
$493$	60.0000i	2.70226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	4.00000i	0.179244i
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	− 4.00000i	− 0.178529i
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	− 12.0000i	− 0.532414i
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	− 10.0000i	− 0.437688i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	28.0000	1.22086
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2.00000i	− 0.0866296i
$534$	6.00000	0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 20.0000i	− 0.863064i
$538$	− 14.0000i	− 0.603583i
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	− 24.0000i	− 1.03089i
$543$	10.0000i	0.429141i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	6.00000i	0.256307i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	4.00000i	0.170251i
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	4.00000	0.169638
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	0	0
$568$	− 16.0000i	− 0.671345i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	− 19.0000i	− 0.790296i
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	− 14.0000i	− 0.580319i
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	7.00000i	0.288675i
$589$	0	0
$590$	0	0
$591$	22.0000	0.904959
$592$	6.00000i	0.246598i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	−14.0000	−0.573462
$597$	− 16.0000i	− 0.654836i
$598$	− 4.00000i	− 0.163572i
$599$	−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
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	16.0000	0.651031
$605$	0	0
$606$	6.00000	0.243733
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.00000i	0.242536i
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	12.0000i	0.482711i
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	− 16.0000i	− 0.641542i
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	−36.0000	−1.43541
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	0	0
$633$	12.0000i	0.476957i
$634$	2.00000	0.0794301
$635$	0	0
$636$	6.00000	0.237915
$637$	− 7.00000i	− 0.277350i
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	4.00000i	0.157867i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	2.00000	0.0780869
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	8.00000i	0.310929i
$663$	− 6.00000i	− 0.233021i
$664$	4.00000	0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	− 40.0000i	− 1.54881i
$668$	24.0000i	0.928588i
$669$	16.0000	0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	42.0000i	1.61898i	0.587133	+	0.809491i	$0.300257\pi$
$673$	42.0000i	1.61898i	−0.587133	+	0.809491i	$0.699743\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	1.00000	0.0384615
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	10.0000i	0.384048i
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	− 4.00000i	− 0.152499i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	14.0000i	0.532200i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	−10.0000	−0.379049
$697$	12.0000i	0.454532i
$698$	− 34.0000i	− 1.28692i
$699$	18.0000	0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	1.00000i	0.0377426i
$703$	0	0
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	0	0
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	16.0000i	0.597531i
$718$	− 24.0000i	− 0.895672i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	− 19.0000i	− 0.707107i
$723$	14.0000i	0.520666i
$724$	10.0000	0.371647
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	6.00000i	0.221766i
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	− 2.00000i	− 0.0736210i
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	4.00000i	0.145768i
$754$	10.0000	0.364179
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	− 24.0000i	− 0.871719i
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	12.0000i	0.434714i
$763$	0	0
$764$	−24.0000	−0.868290
$765$	0	0
$766$	8.00000	0.289052
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	− 14.0000i	− 0.503871i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	18.0000i	0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	24.0000i	0.858238i
$783$	10.0000i	0.357371i
$784$	7.00000	0.250000
$785$	0	0
$786$	12.0000	0.428026
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	− 22.0000i	− 0.783718i
$789$	−28.0000	−0.996826
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 6.00000i	− 0.213066i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	18.0000i	0.635602i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	14.0000i	0.492823i
$808$	− 6.00000i	− 0.211079i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	24.0000i	0.841717i
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	0	0
$818$	− 2.00000i	− 0.0699284i
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	1.00000i	0.0346688i
$833$	42.0000i	1.45521i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	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
$841$	71.0000	2.44828
$842$	− 22.0000i	− 0.758170i
$843$	6.00000i	0.206651i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	24.0000	0.822709
$852$	16.0000i	0.548151i
$853$	− 38.0000i	− 1.30110i	−0.759465	−	0.650548i	$-0.774539\pi$
$853$	− 38.0000i	− 1.30110i	0.759465	−	0.650548i	$-0.225461\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	38.0000i	1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$	38.0000i	1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	− 14.0000i	− 0.474100i
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	24.0000i	0.809961i
$879$	26.0000	0.876958
$880$	0	0
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	− 7.00000i	− 0.235702i
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	4.00000	0.134383
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	14.0000	0.468230
$895$	0	0
$896$	0	0
$897$	4.00000i	0.133556i
$898$	14.0000i	0.467186i
$899$	0	0
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	10.0000	0.332595
$905$	0	0
$906$	−16.0000	−0.531564
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	20.0000i	0.663723i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	0	0
$914$	6.00000	0.198462
$915$	0	0
$916$	2.00000	0.0660819
$917$	0	0
$918$	− 6.00000i	− 0.198030i
$919$	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
$921$	4.00000	0.131804
$922$	− 14.0000i	− 0.461065i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	− 12.0000i	− 0.394132i
$928$	10.0000i	0.328266i
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	0	0
$932$	− 18.0000i	− 0.589610i
$933$	16.0000i	0.523816i
$934$	−28.0000	−0.916188
$935$	0	0
$936$	1.00000	0.0326860
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	14.0000i	0.456145i
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	10.0000i	0.323932i	0.986796	+	0.161966i	$0.0517835\pi$
$953$	10.0000i	0.323932i	−0.986796	+	0.161966i	$0.948217\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	6.00000i	0.193448i
$963$	− 4.00000i	− 0.128898i
$964$	14.0000	0.450910
$965$	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	$0.827968\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	− 4.00000i	− 0.127906i
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	20.0000i	0.638226i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−60.0000	−1.91079
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	40.0000i	1.26618i
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.l.1249.2		2
3.2	odd	2		5850.2.e.p.5149.1		2
5.2	odd	4		390.2.a.a.1.1	✓	1
5.3	odd	4		1950.2.a.y.1.1		1
5.4	even	2	inner	1950.2.e.l.1249.1		2
15.2	even	4		1170.2.a.m.1.1		1
15.8	even	4		5850.2.a.m.1.1		1
15.14	odd	2		5850.2.e.p.5149.2		2
20.7	even	4		3120.2.a.q.1.1		1
60.47	odd	4		9360.2.a.bn.1.1		1
65.12	odd	4		5070.2.a.s.1.1		1
65.47	even	4		5070.2.b.c.1351.1		2
65.57	even	4		5070.2.b.c.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.a.1.1	✓	1	5.2	odd	4
1170.2.a.m.1.1		1	15.2	even	4
1950.2.a.y.1.1		1	5.3	odd	4
1950.2.e.l.1249.1		2	5.4	even	2	inner
1950.2.e.l.1249.2		2	1.1	even	1	trivial
3120.2.a.q.1.1		1	20.7	even	4
5070.2.a.s.1.1		1	65.12	odd	4
5070.2.b.c.1351.1		2	65.47	even	4
5070.2.b.c.1351.2		2	65.57	even	4
5850.2.a.m.1.1		1	15.8	even	4
5850.2.e.p.5149.1		2	3.2	odd	2
5850.2.e.p.5149.2		2	15.14	odd	2
9360.2.a.bn.1.1		1	60.47	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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} +1.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +10.0000i q^{58} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +16.0000 q^{71} +1.00000i q^{72} -2.00000i q^{73} -6.00000 q^{74} -1.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +6.00000 q^{89} +4.00000i q^{92} +1.00000 q^{96} -14.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 2 q^{16} - 2 q^{24} + 2 q^{26} + 20 q^{29} - 12 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 8 q^{46} + 14 q^{49} + 12 q^{51} - 2 q^{54} + 12 q^{61} - 2 q^{64} - 8 q^{69} + 32 q^{71} - 12 q^{74} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 2 * q^16 - 2 * q^24 + 2 * q^26 + 20 * q^29 - 12 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 8 * q^46 + 14 * q^49 + 12 * q^51 - 2 * q^54 + 12 * q^61 - 2 * q^64 - 8 * q^69 + 32 * q^71 - 12 * q^74 + 2 * q^81 + 8 * q^86 + 12 * q^89 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Database

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