LMFDB - Embedded newform 4650.2.d.a.3349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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.a.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +6.00000i q^{22} +2.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +8.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} +2.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -8.00000i q^{58} +4.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +4.00000i q^{67} -4.00000i q^{68} +2.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +6.00000 q^{74} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +4.00000 q^{86} -8.00000i q^{87} -6.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -7.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{11} + 2 q^{16} + 2 q^{24} - 4 q^{26} + 16 q^{29} + 2 q^{31} + 8 q^{34} + 2 q^{36} - 4 q^{39} - 4 q^{41} + 12 q^{44} + 4 q^{46} + 14 q^{49} + 8 q^{51} + 2 q^{54} + 8 q^{61} - 2 q^{64} + 12 q^{66} + 4 q^{69} - 16 q^{71} + 12 q^{74} + 8 q^{79} + 2 q^{81} + 8 q^{86} + 4 q^{89} - 8 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 + 2 * q^16 + 2 * q^24 - 4 * q^26 + 16 * q^29 + 2 * q^31 + 8 * q^34 + 2 * q^36 - 4 * q^39 - 4 * q^41 + 12 * q^44 + 4 * q^46 + 14 * q^49 + 8 * q^51 + 2 * q^54 + 8 * q^61 - 2 * q^64 + 12 * q^66 + 4 * q^69 - 16 * q^71 + 12 * q^74 + 8 * q^79 + 2 * q^81 + 8 * q^86 + 4 * q^89 - 8 * q^94 - 2 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

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
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	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	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	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
$22$	6.00000i	1.27920i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	0	0
$29$	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
$33$	6.00000i	1.04447i
$34$	4.00000	0.685994
$35$	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
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	2.00000	0.294884
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	2.00000i	0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 8.00000i	− 1.05045i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	− 1.00000i	− 0.127000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 4.00000i	− 0.485071i
$69$	2.00000	0.240772
$70$	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	$-0.0752085\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	0	0
$78$	2.00000i	0.226455i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	− 8.00000i	− 0.857690i
$88$	− 6.00000i	− 0.639602i
$89$	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
$91$	0	0
$92$	− 2.00000i	− 0.208514i
$93$	− 1.00000i	− 0.103695i
$94$	−4.00000	−0.412568
$95$	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
$99$	6.00000	0.603023
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	− 4.00000i	− 0.396059i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	2.00000	0.196116
$105$	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
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	− 4.00000i	− 0.362143i
$123$	2.00000i	0.180334i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	0	0
$127$	− 14.0000i	− 1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$	− 14.0000i	− 1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 6.00000i	− 0.522233i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 8.00000i	− 0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	− 8.00000i	− 0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$	− 2.00000i	− 0.170251i
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	8.00000i	0.671345i
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	− 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
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 4.00000i	− 0.318223i
$159$	−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
$166$	0	0
$167$	10.0000i	0.773823i	0.922117	+	0.386912i	$0.126458\pi$
$167$	10.0000i	0.773823i	−0.922117	+	0.386912i	$0.873542\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−8.00000	−0.606478
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	−2.00000	−0.147442
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	− 24.0000i	− 1.75505i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	− 6.00000i	− 0.426401i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	16.0000	1.11477
$207$	− 2.00000i	− 0.139010i
$208$	− 2.00000i	− 0.138675i
$209$	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
$213$	8.00000i	0.548151i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−4.00000	−0.270295
$220$	0	0
$221$	8.00000	0.538138
$222$	− 6.00000i	− 0.402694i
$223$	18.0000i	1.20537i	0.797980	+	0.602685i	$0.205902\pi$
$223$	18.0000i	1.20537i	−0.797980	+	0.602685i	$0.794098\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	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
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 25.0000i	− 1.60706i
$243$	− 1.00000i	− 0.0641500i
$244$	−4.00000	−0.256074
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	1.00000i	0.0635001i
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	− 12.0000i	− 0.754434i
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.0000i	1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$	26.0000i	1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	−8.00000	−0.495188
$262$	0	0
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	− 4.00000i	− 0.244339i
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	−8.00000	−0.483298
$275$	0	0
$276$	−2.00000	−0.120386
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	− 2.00000i	− 0.119952i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	4.00000i	0.238197i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	12.0000	0.709575
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−14.0000	−0.820695
$292$	4.00000i	0.234082i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	− 6.00000i	− 0.348155i
$298$	− 14.0000i	− 0.810998i
$299$	4.00000	0.231326
$300$	0	0
$301$	0	0
$302$	− 12.0000i	− 0.690522i
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	−4.00000	−0.228665
$307$	− 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	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	− 2.00000i	− 0.113228i
$313$	− 12.0000i	− 0.678280i	−0.940736	−	0.339140i	$-0.889864\pi$
$313$	− 12.0000i	− 0.678280i	0.940736	−	0.339140i	$-0.110136\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	6.00000i	0.336463i
$319$	−48.0000	−2.68748
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	− 10.0000i	− 0.553001i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	10.0000	0.547176
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	− 9.00000i	− 0.489535i
$339$	6.00000	0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	8.00000i	0.428845i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	6.00000i	0.319801i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	− 2.00000i	− 0.105703i
$359$	−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
$362$	− 8.00000i	− 0.420471i
$363$	− 25.0000i	− 1.31216i
$364$	0	0
$365$	0	0
$366$	−4.00000	−0.209083
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	2.00000i	0.104257i
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	1.00000i	0.0518476i
$373$	18.0000i	0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$373$	18.0000i	0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	4.00000	0.206284
$377$	− 16.0000i	− 0.824042i
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	8.00000i	0.409316i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	− 4.00000i	− 0.203331i
$388$	14.0000i	0.710742i
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	7.00000i	0.353553i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 2.00000i	− 0.0996271i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	− 36.0000i	− 1.78445i
$408$	4.00000i	0.198030i
$409$	−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
$411$	−8.00000	−0.394611
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 2.00000i	− 0.0979404i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	4.00000i	0.194717i
$423$	4.00000i	0.194487i
$424$	6.00000	0.291386
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	12.0000i	0.580042i
$429$	12.0000	0.579365
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	1.00000i	0.0481125i
$433$	− 28.0000i	− 1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$	− 28.0000i	− 1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	4.00000i	0.191127i
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 8.00000i	− 0.380521i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	18.0000	0.852325
$447$	− 14.0000i	− 0.662177i
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	− 6.00000i	− 0.282216i
$453$	− 12.0000i	− 0.563809i
$454$	−12.0000	−0.563188
$455$	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	$0.940087\pi$
$458$	16.0000i	0.747631i
$459$	−4.00000	−0.186704
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	6.00000	0.277945
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	− 24.0000i	− 1.09773i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	4.00000i	0.181071i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−34.0000	−1.53440	−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000	−1.53440	−0.767199	+	0.641409i	$0.778350\pi$
$492$	− 2.00000i	− 0.0901670i
$493$	32.0000i	1.44121i
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	0	0
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	0	0
$501$	10.0000	0.446767
$502$	− 18.0000i	− 0.803379i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	− 9.00000i	− 0.399704i
$508$	14.0000i	0.621150i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	−4.00000	−0.176090
$517$	24.0000i	1.05552i
$518$	0	0
$519$	14.0000	0.614532
$520$	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
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	18.0000	0.784837
$527$	4.00000i	0.174243i
$528$	6.00000i	0.261116i
$529$	19.0000	0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.00000i	0.173259i
$534$	−2.00000	−0.0865485
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 2.00000i	− 0.0863064i
$538$	− 16.0000i	− 0.689809i
$539$	−42.0000	−1.80907
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	− 16.0000i	− 0.687259i
$543$	− 8.00000i	− 0.343313i
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	44.0000i	1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$	44.0000i	1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$	8.00000i	0.341743i
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	2.00000i	0.0851257i
$553$	0	0
$554$	−18.0000	−0.764747
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	1.00000i	0.0423334i
$559$	8.00000	0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	26.0000i	1.09674i
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	− 12.0000i	− 0.501745i
$573$	8.00000i	0.334205i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	0	0
$582$	14.0000i	0.580319i
$583$	36.0000i	1.49097i
$584$	4.00000	0.165521
$585$	0	0
$586$	−30.0000	−1.23929
$587$	− 32.0000i	− 1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$	− 32.0000i	− 1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$	7.00000i	0.288675i
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	6.00000i	0.246598i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−14.0000	−0.573462
$597$	− 20.0000i	− 0.818546i
$598$	− 4.00000i	− 0.163572i
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	−12.0000	−0.488273
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	4.00000i	0.161690i
$613$	− 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
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	− 16.0000i	− 0.643614i
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	24.0000i	0.962312i
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−12.0000	−0.479616
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	−24.0000	−0.956943
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	4.00000i	0.159111i
$633$	4.00000i	0.158986i
$634$	26.0000	1.03259
$635$	0	0
$636$	6.00000	0.237915
$637$	− 14.0000i	− 0.554700i
$638$	48.0000i	1.90034i
$639$	8.00000	0.316475
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	12.0000i	0.473602i
$643$	− 44.0000i	− 1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$	− 44.0000i	− 1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	4.00000i	0.156055i
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	18.0000i	0.699590i
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	16.0000i	0.619522i
$668$	− 10.0000i	− 0.386912i
$669$	18.0000	0.695920
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	32.0000i	1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$	32.0000i	1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	− 6.00000i	− 0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	6.00000i	0.229752i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.0000i	0.610438i
$688$	4.00000i	0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−16.0000	−0.607352
$695$	0	0
$696$	8.00000	0.303239
$697$	− 8.00000i	− 0.303022i
$698$	− 30.0000i	− 1.13552i
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	0	0
$704$	6.00000	0.226134
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	2.00000i	0.0749532i
$713$	2.00000i	0.0749006i
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	− 24.0000i	− 0.896296i
$718$	24.0000i	0.895672i
$719$	−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
$721$	0	0
$722$	19.0000i	0.707107i
$723$	− 18.0000i	− 0.669427i
$724$	−8.00000	−0.297318
$725$	0	0
$726$	−25.0000	−0.927837
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	4.00000i	0.147844i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	2.00000	0.0737210
$737$	− 24.0000i	− 0.884051i
$738$	− 2.00000i	− 0.0736210i
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 22.0000i	− 0.807102i	−0.914957	−	0.403551i	$-0.867776\pi$
$743$	− 22.0000i	− 0.807102i	0.914957	−	0.403551i	$-0.132224\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	18.0000	0.659027
$747$	0	0
$748$	24.0000i	0.877527i
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	− 4.00000i	− 0.145865i
$753$	− 18.0000i	− 0.655956i
$754$	−16.0000	−0.582686
$755$	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
$759$	−12.0000	−0.435572
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	14.0000i	0.507166i
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	18.0000i	0.647834i
$773$	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
$778$	− 20.0000i	− 0.717035i
$779$	0	0
$780$	0	0
$781$	48.0000	1.71758
$782$	8.00000i	0.286079i
$783$	8.00000i	0.285897i
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	− 18.0000i	− 0.641223i
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	6.00000i	0.213201i
$793$	− 8.00000i	− 0.284088i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−20.0000	−0.708881
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	26.0000i	0.918092i
$803$	24.0000i	0.846942i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	− 16.0000i	− 0.563227i
$808$	2.00000i	0.0703598i
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	−36.0000	−1.26180
$815$	0	0
$816$	4.00000	0.140028
$817$	0	0
$818$	14.0000i	0.489499i
$819$	0	0
$820$	0	0
$821$	16.0000	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$821$	16.0000	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$822$	8.00000i	0.279032i
$823$	6.00000i	0.209147i	0.994517	+	0.104573i	$0.0333477\pi$
$823$	6.00000i	0.209147i	−0.994517	+	0.104573i	$0.966652\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 32.0000i	− 1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$	− 32.0000i	− 1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$	2.00000i	0.0695048i
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	2.00000i	0.0693375i
$833$	28.0000i	0.970143i
$834$	−2.00000	−0.0692543
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	− 26.0000i	− 0.896019i
$843$	26.0000i	0.895488i
$844$	4.00000	0.137686
$845$	0	0
$846$	4.00000	0.137523
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−12.0000	−0.411355
$852$	− 8.00000i	− 0.274075i
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	− 12.0000i	− 0.409673i
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	0	0
$862$	− 40.0000i	− 1.36241i
$863$	22.0000i	0.748889i	0.927249	+	0.374444i	$0.122167\pi$
$863$	22.0000i	0.748889i	−0.927249	+	0.374444i	$0.877833\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−28.0000	−0.951479
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	8.00000	0.271070
$872$	10.0000i	0.338643i
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	50.0000i	1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$	50.0000i	1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$	0	0
$879$	−30.0000	−1.01187
$880$	0	0
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	7.00000i	0.235702i
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	12.0000	0.403148
$887$	4.00000i	0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$	4.00000i	0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$	6.00000i	0.201347i
$889$	0	0
$890$	0	0
$891$	−6.00000	−0.201008
$892$	− 18.0000i	− 0.602685i
$893$	0	0
$894$	−14.0000	−0.468230
$895$	0	0
$896$	0	0
$897$	− 4.00000i	− 0.133556i
$898$	14.0000i	0.467186i
$899$	8.00000	0.266815
$900$	0	0
$901$	24.0000	0.799556
$902$	− 12.0000i	− 0.399556i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−12.0000	−0.398673
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	12.0000i	0.398234i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	8.00000	0.264616
$915$	0	0
$916$	16.0000	0.528655
$917$	0	0
$918$	4.00000i	0.132020i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	−14.0000	−0.460069
$927$	− 16.0000i	− 0.525509i
$928$	− 8.00000i	− 0.262613i
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	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
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	52.0000	1.69515	0.847576	−	0.530674i	$-0.178061\pi$
$941$	52.0000	1.69515	0.847576	+	0.530674i	$0.178061\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	− 4.00000i	− 0.130258i
$944$	0	0
$945$	0	0
$946$	−24.0000	−0.780307
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	4.00000i	0.129914i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	8.00000i	0.259145i	0.991570	+	0.129573i	$0.0413606\pi$
$953$	8.00000i	0.259145i	−0.991570	+	0.129573i	$0.958639\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	48.0000i	1.55162i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 12.0000i	− 0.386896i
$963$	12.0000i	0.386695i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	− 6.00000i	− 0.192947i	−0.995336	−	0.0964735i	$-0.969244\pi$
$967$	− 6.00000i	− 0.192947i	0.995336	−	0.0964735i	$-0.0307563\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	34.0000	1.08943
$975$	0	0
$976$	4.00000	0.128037
$977$	− 26.0000i	− 0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$	− 26.0000i	− 0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$	4.00000i	0.127906i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−10.0000	−0.319275
$982$	34.0000i	1.08498i
$983$	10.0000i	0.318950i	0.987202	+	0.159475i	$0.0509802\pi$
$983$	10.0000i	0.318950i	−0.987202	+	0.159475i	$0.949020\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	32.0000	1.01909
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	18.0000i	0.571213i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	− 38.0000i	− 1.20287i
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.d.a.3349.1		2
5.2	odd	4		930.2.a.l.1.1	✓	1
5.3	odd	4		4650.2.a.r.1.1		1
5.4	even	2	inner	4650.2.d.a.3349.2		2
15.2	even	4		2790.2.a.j.1.1		1
20.7	even	4		7440.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.l.1.1	✓	1	5.2	odd	4
2790.2.a.j.1.1		1	15.2	even	4
4650.2.a.r.1.1		1	5.3	odd	4
4650.2.d.a.3349.1		2	1.1	even	1	trivial
4650.2.d.a.3349.2		2	5.4	even	2	inner
7440.2.a.r.1.1		1	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +6.00000i q^{22} +2.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +8.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +6.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} +2.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -8.00000i q^{58} +4.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} +6.00000 q^{66} +4.00000i q^{67} -4.00000i q^{68} +2.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +6.00000 q^{74} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +4.00000 q^{86} -8.00000i q^{87} -6.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -7.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} + 2 q^{16} + 2 q^{24} - 4 q^{26} + 16 q^{29} + 2 q^{31} + 8 q^{34} + 2 q^{36} - 4 q^{39} - 4 q^{41} + 12 q^{44} + 4 q^{46} + 14 q^{49} + 8 q^{51} + 2 q^{54} + 8 q^{61} - 2 q^{64} + 12 q^{66} + 4 q^{69} - 16 q^{71} + 12 q^{74} + 8 q^{79} + 2 q^{81} + 8 q^{86} + 4 q^{89} - 8 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 + 2 * q^16 + 2 * q^24 - 4 * q^26 + 16 * q^29 + 2 * q^31 + 8 * q^34 + 2 * q^36 - 4 * q^39 - 4 * q^41 + 12 * q^44 + 4 * q^46 + 14 * q^49 + 8 * q^51 + 2 * q^54 + 8 * q^61 - 2 * q^64 + 12 * q^66 + 4 * q^69 - 16 * q^71 + 12 * q^74 + 8 * q^79 + 2 * q^81 + 8 * q^86 + 4 * q^89 - 8 * q^94 - 2 * q^96 + 12 * q^99

Introduction

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