LMFDB - Embedded newform 4650.2.d.v.3349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(3349,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,2,0,0,-2,0,4,0,0,-8,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$37.1304369399$
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 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.v.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{21} -2.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +8.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} -8.00000i q^{58} +12.0000 q^{59} -8.00000 q^{61} -1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -8.00000i q^{77} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +16.0000i q^{83} -4.00000 q^{84} -4.00000 q^{86} +8.00000i q^{87} +2.00000i q^{88} +2.00000 q^{89} -8.00000 q^{91} -6.00000i q^{92} +1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} - 8 q^{14} + 2 q^{16} + 8 q^{21} - 2 q^{24} - 4 q^{26} + 16 q^{29} + 2 q^{31} + 2 q^{36} + 4 q^{39} + 12 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} - 2 q^{54}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 - 8 * q^14 + 2 * q^16 + 8 * q^21 - 2 * q^24 - 4 * q^26 + 16 * q^29 + 2 * q^31 + 2 * q^36 + 4 * q^39 + 12 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 - 2 * q^54 + 8 * q^56 + 24 * q^59 - 16 * q^61 - 2 * q^64 + 4 * q^66 - 12 * q^69 - 16 * q^71 - 4 * q^74 - 8 * q^79 + 2 * q^81 - 8 * q^84 - 8 * q^86 + 4 * q^89 - 16 * q^91 - 24 * q^94 + 2 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	− 2.00000i	− 0.426401i
$23$	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
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	− 1.00000i	− 0.176777i
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	− 8.00000i	− 1.05045i
$59$	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
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	− 1.00000i	− 0.127000i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 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
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	− 2.00000i	− 0.226455i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−4.00000	−0.431331
$87$	8.00000i	0.857690i
$88$	2.00000i	0.213201i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 6.00000i	− 0.625543i
$93$	1.00000i	0.103695i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	9.00000i	0.909137i
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	1.00000i	0.0962250i
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 4.00000i	− 0.377964i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	2.00000i	0.184900i
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	8.00000i	0.724286i
$123$	6.00000i	0.541002i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	4.00000	0.356348
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	− 2.00000i	− 0.174078i
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	6.00000i	0.510754i
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	8.00000i	0.671345i
$143$	− 4.00000i	− 0.334497i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 9.00000i	− 0.742307i
$148$	2.00000i	0.164399i
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	−8.00000	−0.644658
$155$	0	0
$156$	−2.00000	−0.160128
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	4.00000i	0.318223i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	24.0000	1.89146
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	16.0000	1.24184
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	4.00000i	0.308607i
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	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	$-0.239873\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	2.00000	0.150756
$177$	12.0000i	0.901975i
$178$	− 2.00000i	− 0.149906i
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	8.00000i	0.592999i
$183$	− 8.00000i	− 0.591377i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	12.0000i	0.875190i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	2.00000i	0.142134i
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	10.0000i	0.703598i
$203$	− 32.0000i	− 2.24596i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	− 6.00000i	− 0.417029i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	− 2.00000i	− 0.137361i
$213$	− 8.00000i	− 0.548151i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 4.00000i	− 0.271538i
$218$	− 18.0000i	− 1.21911i
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	− 2.00000i	− 0.134231i
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	8.00000i	0.525226i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000i	0.449977i
$243$	1.00000i	0.0641500i
$244$	8.00000	0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	0	0
$248$	1.00000i	0.0635001i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	− 4.00000i	− 0.251976i
$253$	12.0000i	0.754434i
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	− 4.00000i	− 0.249029i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−8.00000	−0.495188
$262$	4.00000i	0.247121i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	−2.00000	−0.123091
$265$	0	0
$266$	0	0
$267$	2.00000i	0.122398i
$268$	4.00000i	0.244339i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	− 8.00000i	− 0.484182i
$274$	−12.0000	−0.724947
$275$	0	0
$276$	6.00000	0.361158
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	18.0000i	1.07957i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−4.00000	−0.236525
$287$	− 24.0000i	− 1.41668i
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	2.00000	0.117242
$292$	4.00000i	0.234082i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	2.00000	0.116248
$297$	− 2.00000i	− 0.116052i
$298$	14.0000i	0.810998i
$299$	12.0000	0.693978
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 20.0000i	− 1.15087i
$303$	− 10.0000i	− 0.574485i
$304$	0	0
$305$	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	$-0.0364132\pi$
$308$	8.00000i	0.455842i
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	2.00000i	0.113228i
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	4.00000	0.225018
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	2.00000i	0.112154i
$319$	16.0000	0.895828
$320$	0	0
$321$	4.00000	0.223258
$322$	− 24.0000i	− 1.33747i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	18.0000i	0.995402i
$328$	6.00000i	0.331295i
$329$	−48.0000	−2.64633
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	− 16.0000i	− 0.878114i
$333$	2.00000i	0.109599i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	4.00000	0.218218
$337$	− 16.0000i	− 0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$	− 16.0000i	− 0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	− 9.00000i	− 0.489535i
$339$	18.0000	0.977626
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	8.00000i	0.431959i
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	− 8.00000i	− 0.428845i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 2.00000i	− 0.106600i
$353$	− 4.00000i	− 0.212899i	−0.994318	−	0.106449i	$-0.966052\pi$
$353$	− 4.00000i	− 0.212899i	0.994318	−	0.106449i	$-0.0339482\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	14.0000i	0.739923i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	12.0000i	0.630706i
$363$	− 7.00000i	− 0.367405i
$364$	8.00000	0.419314
$365$	0	0
$366$	−8.00000	−0.418167
$367$	18.0000i	0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$	18.0000i	0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$	6.00000i	0.312772i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	8.00000	0.415339
$372$	− 1.00000i	− 0.0518476i
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	− 16.0000i	− 0.824042i
$378$	4.00000i	0.205738i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	− 8.00000i	− 0.409316i
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	4.00000i	0.203331i
$388$	2.00000i	0.101535i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	− 4.00000i	− 0.201773i
$394$	2.00000	0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	− 12.0000i	− 0.601506i
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 2.00000i	− 0.0996271i
$404$	10.0000	0.497519
$405$	0	0
$406$	−32.0000	−1.58813
$407$	− 4.00000i	− 0.198273i
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	− 4.00000i	− 0.197066i
$413$	− 48.0000i	− 2.36193i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 18.0000i	− 0.881464i
$418$	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
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	20.0000i	0.973585i
$423$	12.0000i	0.583460i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−8.00000	−0.387601
$427$	32.0000i	1.54859i
$428$	4.00000i	0.193347i
$429$	4.00000	0.193122
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	4.00000i	0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$	4.00000i	0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	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
$441$	9.00000	0.428571
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	2.00000	0.0947027
$447$	− 14.0000i	− 0.662177i
$448$	4.00000i	0.188982i
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	18.0000i	0.846649i
$453$	20.0000i	0.939682i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	4.00000i	0.186908i
$459$	0	0
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	− 8.00000i	− 0.372194i
$463$	42.0000i	1.95191i	0.217982	+	0.975953i	$0.430053\pi$
$463$	42.0000i	1.95191i	−0.217982	+	0.975953i	$0.569947\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	6.00000	0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−18.0000	−0.829396
$472$	12.0000i	0.552345i
$473$	− 8.00000i	− 0.367840i
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	8.00000i	0.365911i
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	− 10.0000i	− 0.455488i
$483$	24.0000i	1.09204i
$484$	7.00000	0.318182
$485$	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	$0.985571\pi$
$488$	− 8.00000i	− 0.362143i
$489$	12.0000	0.542659
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	32.0000i	1.43540i
$498$	16.0000i	0.716977i
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	− 2.00000i	− 0.0892644i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	12.0000	0.533465
$507$	9.00000i	0.399704i
$508$	6.00000i	0.266207i
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−14.0000	−0.617514
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 24.0000i	− 1.05552i
$518$	8.00000i	0.351500i
$519$	18.0000	0.790112
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	8.00000i	0.350150i
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	2.00000	0.0865485
$535$	0	0
$536$	4.00000	0.172774
$537$	− 14.0000i	− 0.604145i
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	− 12.0000i	− 0.514969i
$544$	0	0
$545$	0	0
$546$	−8.00000	−0.342368
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	12.0000i	0.512615i
$549$	8.00000	0.341432
$550$	0	0
$551$	0	0
$552$	− 6.00000i	− 0.255377i
$553$	16.0000i	0.680389i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	18.0000	0.763370
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	1.00000i	0.0423334i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−20.0000	−0.840663
$567$	− 4.00000i	− 0.167984i
$568$	− 8.00000i	− 0.335673i
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	4.00000i	0.167248i
$573$	8.00000i	0.334205i
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	− 17.0000i	− 0.707107i
$579$	22.0000	0.914289
$580$	0	0
$581$	64.0000	2.65517
$582$	− 2.00000i	− 0.0829027i
$583$	4.00000i	0.165663i
$584$	4.00000	0.165521
$585$	0	0
$586$	−14.0000	−0.578335
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	− 2.00000i	− 0.0821995i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	14.0000	0.573462
$597$	12.0000i	0.491127i
$598$	− 12.0000i	− 0.490716i
$599$	−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
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	16.0000i	0.652111i
$603$	4.00000i	0.162893i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−10.0000	−0.406222
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	32.0000	1.29671
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	8.00000	0.322329
$617$	− 34.0000i	− 1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$	− 34.0000i	− 1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$	4.00000i	0.160904i
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	− 24.0000i	− 0.962312i
$623$	− 8.00000i	− 0.320513i
$624$	2.00000	0.0800641
$625$	0	0
$626$	4.00000	0.159872
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	0	0
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	− 4.00000i	− 0.159111i
$633$	− 20.0000i	− 0.794929i
$634$	−22.0000	−0.873732
$635$	0	0
$636$	2.00000	0.0793052
$637$	18.0000i	0.713186i
$638$	− 16.0000i	− 0.633446i
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 4.00000i	− 0.157867i
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	0	0
$647$	14.0000i	0.550397i	0.961387	+	0.275198i	$0.0887435\pi$
$647$	14.0000i	0.550397i	−0.961387	+	0.275198i	$0.911256\pi$
$648$	1.00000i	0.0392837i
$649$	24.0000	0.942082
$650$	0	0
$651$	4.00000	0.156772
$652$	12.0000i	0.469956i
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	6.00000	0.234261
$657$	4.00000i	0.156055i
$658$	48.0000i	1.87123i
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	− 26.0000i	− 1.01052i
$663$	0	0
$664$	−16.0000	−0.620920
$665$	0	0
$666$	2.00000	0.0774984
$667$	48.0000i	1.85857i
$668$	18.0000i	0.696441i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−16.0000	−0.617673
$672$	− 4.00000i	− 0.154303i
$673$	− 8.00000i	− 0.308377i	−0.988041	−	0.154189i	$-0.950724\pi$
$673$	− 8.00000i	− 0.308377i	0.988041	−	0.154189i	$-0.0492764\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−9.00000	−0.346154
$677$	50.0000i	1.92166i	0.277145	+	0.960828i	$0.410612\pi$
$677$	50.0000i	1.92166i	−0.277145	+	0.960828i	$0.589388\pi$
$678$	− 18.0000i	− 0.691286i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	4.00000	0.153280
$682$	− 2.00000i	− 0.0765840i
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	− 4.00000i	− 0.152610i
$688$	− 4.00000i	− 0.152499i
$689$	4.00000	0.152388
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000i	0.684257i
$693$	8.00000i	0.303895i
$694$	16.0000	0.607352
$695$	0	0
$696$	−8.00000	−0.303239
$697$	0	0
$698$	2.00000i	0.0757011i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	2.00000i	0.0754851i
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−4.00000	−0.150542
$707$	40.0000i	1.50435i
$708$	− 12.0000i	− 0.450988i
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	2.00000i	0.0749532i
$713$	6.00000i	0.224702i
$714$	0	0
$715$	0	0
$716$	14.0000	0.523205
$717$	− 8.00000i	− 0.298765i
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	19.0000i	0.707107i
$723$	10.0000i	0.371904i
$724$	12.0000	0.445976
$725$	0	0
$726$	−7.00000	−0.259794
$727$	24.0000i	0.890111i	0.895503	+	0.445055i	$0.146816\pi$
$727$	24.0000i	0.890111i	−0.895503	+	0.445055i	$0.853184\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	8.00000i	0.295689i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	6.00000	0.221163
$737$	− 8.00000i	− 0.294684i
$738$	6.00000i	0.220863i
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	0	0
$742$	− 8.00000i	− 0.293689i
$743$	− 26.0000i	− 0.953847i	−0.878945	−	0.476924i	$-0.841752\pi$
$743$	− 26.0000i	− 0.953847i	0.878945	−	0.476924i	$-0.158248\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−6.00000	−0.219676
$747$	− 16.0000i	− 0.585409i
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	− 12.0000i	− 0.437595i
$753$	2.00000i	0.0728841i
$754$	−16.0000	−0.582686
$755$	0	0
$756$	4.00000	0.145479
$757$	− 18.0000i	− 0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	− 18.0000i	− 0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$	16.0000i	0.581146i
$759$	−12.0000	−0.435572
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	− 6.00000i	− 0.217357i
$763$	− 72.0000i	− 2.60658i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−6.00000	−0.216789
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	22.0000i	0.791797i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	− 8.00000i	− 0.286998i
$778$	− 36.0000i	− 1.29066i
$779$	0	0
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−72.0000	−2.56003
$792$	− 2.00000i	− 0.0710669i
$793$	16.0000i	0.568177i
$794$	18.0000	0.638796
$795$	0	0
$796$	−12.0000	−0.425329
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	10.0000i	0.353112i
$803$	− 8.00000i	− 0.282314i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	− 10.0000i	− 0.351799i
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	32.0000i	1.12298i
$813$	0	0
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	30.0000i	1.04893i
$819$	8.00000	0.279543
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 12.0000i	− 0.418548i
$823$	22.0000i	0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$	22.0000i	0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−48.0000	−1.67013
$827$	− 16.0000i	− 0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$	− 16.0000i	− 0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$	6.00000i	0.208514i
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	−18.0000	−0.623289
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	12.0000i	0.414533i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 18.0000i	− 0.620321i
$843$	6.00000i	0.206651i
$844$	20.0000	0.688428
$845$	0	0
$846$	12.0000	0.412568
$847$	28.0000i	0.962091i
$848$	2.00000i	0.0686803i
$849$	20.0000	0.686398
$850$	0	0
$851$	12.0000	0.411355
$852$	8.00000i	0.274075i
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	4.00000	0.136717
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	16.0000i	0.544962i
$863$	18.0000i	0.612727i	0.951915	+	0.306364i	$0.0991123\pi$
$863$	18.0000i	0.612727i	−0.951915	+	0.306364i	$0.900888\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	4.00000	0.135926
$867$	17.0000i	0.577350i
$868$	4.00000i	0.135769i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−8.00000	−0.271070
$872$	18.0000i	0.609557i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	− 24.0000i	− 0.809961i
$879$	14.0000	0.472208
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	− 9.00000i	− 0.303046i
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	2.00000i	0.0671156i
$889$	−24.0000	−0.804934
$890$	0	0
$891$	2.00000	0.0670025
$892$	− 2.00000i	− 0.0669650i
$893$	0	0
$894$	−14.0000	−0.468230
$895$	0	0
$896$	4.00000	0.133631
$897$	12.0000i	0.400668i
$898$	− 34.0000i	− 1.13459i
$899$	8.00000	0.266815
$900$	0	0
$901$	0	0
$902$	− 12.0000i	− 0.399556i
$903$	− 16.0000i	− 0.532447i
$904$	18.0000	0.598671
$905$	0	0
$906$	20.0000	0.664455
$907$	52.0000i	1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$	52.0000i	1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$	4.00000i	0.132745i
$909$	10.0000	0.331679
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	32.0000i	1.05905i
$914$	8.00000	0.264616
$915$	0	0
$916$	4.00000	0.132164
$917$	16.0000i	0.528367i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	− 24.0000i	− 0.790398i
$923$	16.0000i	0.526646i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	42.0000	1.38021
$927$	− 4.00000i	− 0.131377i
$928$	− 8.00000i	− 0.262613i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	0	0
$932$	− 6.00000i	− 0.196537i
$933$	24.0000i	0.785725i
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	16.0000i	0.522419i
$939$	−4.00000	−0.130535
$940$	0	0
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	18.0000i	0.586472i
$943$	36.0000i	1.17232i
$944$	12.0000	0.390567
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 28.0000i	− 0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$	− 28.0000i	− 0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$	4.00000i	0.129914i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	− 4.00000i	− 0.129573i	−0.997899	−	0.0647864i	$-0.979363\pi$
$953$	− 4.00000i	− 0.129573i	0.997899	−	0.0647864i	$-0.0206366\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	8.00000	0.258738
$957$	16.0000i	0.517207i
$958$	− 8.00000i	− 0.258468i
$959$	−48.0000	−1.55000
$960$	0	0
$961$	1.00000	0.0322581
$962$	4.00000i	0.128965i
$963$	4.00000i	0.128898i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	24.0000	0.772187
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	72.0000i	2.30821i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	− 12.0000i	− 0.383718i
$979$	4.00000	0.127841
$980$	0	0
$981$	−18.0000	−0.574696
$982$	42.0000i	1.34027i
$983$	54.0000i	1.72233i	0.508323	+	0.861166i	$0.330265\pi$
$983$	54.0000i	1.72233i	−0.508323	+	0.861166i	$0.669735\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	− 48.0000i	− 1.52786i
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	26.0000i	0.825085i
$994$	32.0000	1.01498
$995$	0	0
$996$	16.0000	0.506979
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	− 34.0000i	− 1.07625i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.v.3349.1		2
5.2	odd	4		4650.2.a.bw.1.1		1
5.3	odd	4		930.2.a.c.1.1	✓	1
5.4	even	2	inner	4650.2.d.v.3349.2		2
15.8	even	4		2790.2.a.o.1.1		1
20.3	even	4		7440.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.c.1.1	✓	1	5.3	odd	4
2790.2.a.o.1.1		1	15.8	even	4
4650.2.a.bw.1.1		1	5.2	odd	4
4650.2.d.v.3349.1		2	1.1	even	1	trivial
4650.2.d.v.3349.2		2	5.4	even	2	inner
7440.2.a.bb.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{21} -2.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +8.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +4.00000 q^{56} -8.00000i q^{58} +12.0000 q^{59} -8.00000 q^{61} -1.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -8.00000i q^{77} -2.00000i q^{78} -4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +16.0000i q^{83} -4.00000 q^{84} -4.00000 q^{86} +8.00000i q^{87} +2.00000i q^{88} +2.00000 q^{89} -8.00000 q^{91} -6.00000i q^{92} +1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} +9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} - 8 q^{14} + 2 q^{16} + 8 q^{21} - 2 q^{24} - 4 q^{26} + 16 q^{29} + 2 q^{31} + 2 q^{36} + 4 q^{39} + 12 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} - 2 q^{54}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 - 8 * q^14 + 2 * q^16 + 8 * q^21 - 2 * q^24 - 4 * q^26 + 16 * q^29 + 2 * q^31 + 2 * q^36 + 4 * q^39 + 12 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 - 2 * q^54 + 8 * q^56 + 24 * q^59 - 16 * q^61 - 2 * q^64 + 4 * q^66 - 12 * q^69 - 16 * q^71 - 4 * q^74 - 8 * q^79 + 2 * q^81 - 8 * q^84 - 8 * q^86 + 4 * q^89 - 16 * q^91 - 24 * q^94 + 2 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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