LMFDB - Embedded newform 1950.2.e.d.1249.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +2.00000i q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} +14.0000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +1.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 16 q^{31} + 2 q^{36} - 2 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 4 q^{74} + 4 q^{76} + 32 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 16 * q^31 + 2 * q^36 - 2 * q^39 + 12 * q^41 + 12 * q^46 + 6 * q^49 + 2 * q^54 - 4 * q^56 + 28 * q^61 - 2 * q^64 + 12 * q^69 + 4 * q^74 + 4 * q^76 + 32 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 12 * q^89 + 4 * q^91 - 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
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	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
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000i	0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	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
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	2.00000i	0.324443i
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 2.00000i	− 0.308607i
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	− 8.00000i	− 1.01600i
$63$	− 2.00000i	− 0.251976i
$64$	−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	$-0.0785709\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	1.00000i	0.113228i
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	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
$91$	2.00000	0.209657
$92$	− 6.00000i	− 0.625543i
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000i	0.188982i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 14.0000i	− 1.26750i
$123$	− 6.00000i	− 0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	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
$133$	− 4.00000i	− 0.346844i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 6.00000i	− 0.510754i
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	− 3.00000i	− 0.247436i
$148$	− 2.00000i	− 0.164399i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	1.00000	0.0800641
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 16.0000i	− 1.27289i
$159$	6.00000	0.475831
$160$	0	0
$161$	−12.0000	−0.945732
$162$	− 1.00000i	− 0.0785674i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	2.00000i	0.154303i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	− 4.00000i	− 0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 2.00000i	− 0.148250i
$183$	− 14.0000i	− 1.03491i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	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
$202$	12.0000i	0.844317i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	− 6.00000i	− 0.417029i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	16.0000i	1.08615i
$218$	− 4.00000i	− 0.270914i
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	− 2.00000i	− 0.134231i
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−12.0000	−0.798228
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 2.00000i	− 0.132453i
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−14.0000	−0.896258
$245$	0	0
$246$	−6.00000	−0.382546
$247$	2.00000i	0.127257i
$248$	8.00000i	0.508001i
$249$	12.0000	0.760469
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	− 4.00000i	− 0.249029i
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	−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
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	− 2.00000i	− 0.121046i
$274$	6.00000	0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	8.00000i	0.479808i
$279$	−8.00000	−0.478947
$280$	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
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	−4.00000	−0.234484
$292$	− 4.00000i	− 0.234082i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	18.0000i	1.04271i
$299$	6.00000	0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 8.00000i	− 0.460348i
$303$	12.0000i	0.689382i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	12.0000i	0.668734i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	− 4.00000i	− 0.221201i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	12.0000	0.656611
$335$	0	0
$336$	2.00000	0.109109
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	1.00000i	0.0543928i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	− 2.00000i	− 0.108148i
$343$	20.0000i	1.07990i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	− 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
$358$	− 6.00000i	− 0.317110i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 2.00000i	− 0.105118i
$363$	11.0000i	0.577350i
$364$	−2.00000	−0.104828
$365$	0	0
$366$	−14.0000	−0.731792
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	6.00000i	0.312772i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	8.00000i	0.414781i
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	2.00000i	0.102869i
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−8.00000	−0.407189
$387$	− 4.00000i	− 0.203331i
$388$	4.00000i	0.203069i
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	− 6.00000i	− 0.302660i
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 16.0000i	− 0.802008i
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	4.00000i	0.199502i
$403$	− 8.00000i	− 0.398508i
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	8.00000i	0.391762i
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000i	0.0481125i
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−4.00000	−0.191565
$437$	− 12.0000i	− 0.574038i
$438$	− 4.00000i	− 0.191127i
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	10.0000	0.473514
$447$	18.0000i	0.851371i
$448$	− 2.00000i	− 0.0944911i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	12.0000i	0.564433i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	− 4.00000i	− 0.186908i
$459$	0	0
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	8.00000	0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	−16.0000	−0.734904
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 26.0000i	− 1.18427i
$483$	12.0000i	0.546019i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	14.0000i	0.633750i
$489$	16.0000	0.723545
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	6.00000i	0.270501i
$493$	0	0
$494$	2.00000	0.0899843
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	30.0000i	1.33897i
$503$	18.0000i	0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$	18.0000i	0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	16.0000i	0.709885i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	4.00000i	0.175750i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−30.0000	−1.30806
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 6.00000i	− 0.259889i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	4.00000	0.172774
$537$	− 6.00000i	− 0.258919i
$538$	12.0000i	0.517357i
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	− 8.00000i	− 0.343629i
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	0	0
$552$	6.00000i	0.255377i
$553$	32.0000i	1.36078i
$554$	26.0000	1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	8.00000i	0.338667i
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	28.0000	1.17693
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−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
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	− 17.0000i	− 0.707107i
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−24.0000	−0.995688
$582$	4.00000i	0.165805i
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	3.00000i	0.123718i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	18.0000	0.740421
$592$	2.00000i	0.0821995i
$593$	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
$596$	18.0000	0.737309
$597$	− 16.0000i	− 0.654836i
$598$	− 6.00000i	− 0.245358i
$599$	−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
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	8.00000i	0.326056i
$603$	4.00000i	0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	12.0000	0.487467
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 16.0000i	− 0.643614i
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	− 24.0000i	− 0.962312i
$623$	12.0000i	0.480770i
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	16.0000i	0.636446i
$633$	4.00000i	0.158986i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	12.0000i	0.473602i
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	0	0
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	− 16.0000i	− 0.626608i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	6.00000	0.234261
$657$	− 4.00000i	− 0.156055i
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 2.00000i	− 0.0777322i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	10.0000	0.386622
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	12.0000i	0.460857i
$679$	8.00000	0.307012
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	− 4.00000i	− 0.152610i
$688$	4.00000i	0.152499i
$689$	6.00000	0.228582
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	20.0000i	0.757011i
$699$	0	0
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	6.00000i	0.224860i
$713$	48.0000i	1.79761i
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	15.0000i	0.558242i
$723$	− 26.0000i	− 0.966950i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	11.0000	0.408248
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	2.00000i	0.0741249i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	14.0000i	0.517455i
$733$	− 50.0000i	− 1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	− 50.0000i	− 1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	6.00000i	0.220863i
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	12.0000i	0.440534i
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−26.0000	−0.951928
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	30.0000i	1.09326i
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	− 34.0000i	− 1.23494i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	16.0000i	0.579619i
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	8.00000i	0.287926i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	4.00000	0.143592
$777$	4.00000i	0.143499i
$778$	36.0000i	1.29066i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−30.0000	−1.06803
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	− 14.0000i	− 0.497155i
$794$	26.0000	0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	4.00000i	0.141598i
$799$	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
$806$	−8.00000	−0.281788
$807$	12.0000i	0.422420i
$808$	− 12.0000i	− 0.422159i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	14.0000i	0.489499i
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	6.00000i	0.208514i
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	1.00000i	0.0346688i
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	6.00000i	0.207267i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 32.0000i	− 1.10279i
$843$	30.0000i	1.03325i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	6.00000i	0.206041i
$849$	28.0000	0.960958
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	12.0000	0.410152
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	− 24.0000i	− 0.817443i
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	− 17.0000i	− 0.577350i
$868$	− 16.0000i	− 0.543075i
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	4.00000i	0.135457i
$873$	4.00000i	0.135379i
$874$	−12.0000	−0.405906
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 58.0000i	− 1.95852i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	− 58.0000i	− 1.95852i	0.202606	−	0.979260i	$-0.435059\pi$
$878$	32.0000i	1.07995i
$879$	6.00000	0.202375
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	3.00000i	0.101015i
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 30.0000i	− 1.00730i	−0.863907	−	0.503651i	$-0.831990\pi$
$887$	− 30.0000i	− 1.00730i	0.863907	−	0.503651i	$-0.168010\pi$
$888$	2.00000i	0.0671156i
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	− 10.0000i	− 0.334825i
$893$	0	0
$894$	18.0000	0.602010
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 6.00000i	− 0.200334i
$898$	− 30.0000i	− 1.00111i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.00000i	0.266223i
$904$	12.0000	0.399114
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 12.0000i	− 0.398234i
$909$	12.0000	0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	2.00000i	0.0662266i
$913$	0	0
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	12.0000i	0.396275i
$918$	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
$921$	−4.00000	−0.131804
$922$	42.0000i	1.38320i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	10.0000	0.328620
$927$	− 16.0000i	− 0.525509i
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	− 24.0000i	− 0.785725i
$934$	36.0000	1.17796
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	− 8.00000i	− 0.261209i
$939$	10.0000	0.326338
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	36.0000i	1.17232i
$944$	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	$0.801125\pi$
$948$	16.0000i	0.519656i
$949$	4.00000	0.129845
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	− 2.00000i	− 0.0644826i
$963$	12.0000i	0.386695i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	12.0000	0.386094
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	1.00000i	0.0320750i
$973$	− 16.0000i	− 0.512936i
$974$	2.00000	0.0640841
$975$	0	0
$976$	14.0000	0.448129
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	− 16.0000i	− 0.511624i
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	− 6.00000i	− 0.191468i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 2.00000i	− 0.0634681i
$994$	0	0
$995$	0	0
$996$	−12.0000	−0.380235
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	14.0000i	0.443162i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.d.1249.1		2
3.2	odd	2		5850.2.e.o.5149.2		2
5.2	odd	4		1950.2.a.o.1.1		1
5.3	odd	4		390.2.a.d.1.1	✓	1
5.4	even	2	inner	1950.2.e.d.1249.2		2
15.2	even	4		5850.2.a.g.1.1		1
15.8	even	4		1170.2.a.k.1.1		1
15.14	odd	2		5850.2.e.o.5149.1		2
20.3	even	4		3120.2.a.j.1.1		1
60.23	odd	4		9360.2.a.g.1.1		1
65.8	even	4		5070.2.b.m.1351.1		2
65.18	even	4		5070.2.b.m.1351.2		2
65.38	odd	4		5070.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.d.1.1	✓	1	5.3	odd	4
1170.2.a.k.1.1		1	15.8	even	4
1950.2.a.o.1.1		1	5.2	odd	4
1950.2.e.d.1249.1		2	1.1	even	1	trivial
1950.2.e.d.1249.2		2	5.4	even	2	inner
3120.2.a.j.1.1		1	20.3	even	4
5070.2.a.t.1.1		1	65.38	odd	4
5070.2.b.m.1351.1		2	65.8	even	4
5070.2.b.m.1351.2		2	65.18	even	4
5850.2.a.g.1.1		1	15.2	even	4
5850.2.e.o.5149.1		2	15.14	odd	2
5850.2.e.o.5149.2		2	3.2	odd	2
9360.2.a.g.1.1		1	60.23	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +2.00000i q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} +14.0000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +1.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 16 q^{31} + 2 q^{36} - 2 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 4 q^{74} + 4 q^{76} + 32 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^24 - 2 * q^26 + 16 * q^31 + 2 * q^36 - 2 * q^39 + 12 * q^41 + 12 * q^46 + 6 * q^49 + 2 * q^54 - 4 * q^56 + 28 * q^61 - 2 * q^64 + 12 * q^69 + 4 * q^74 + 4 * q^76 + 32 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 12 * q^89 + 4 * q^91 - 2 * q^96

Introduction

L-functions

Modular forms

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Database

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