LMFDB - Embedded newform 4650.2.d.u.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,-2,0,0,0,0,-4,0,2,0,0,-16] 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:	$1$
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.u.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +2.00000 q^{21} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +1.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +8.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -8.00000i q^{57} +6.00000 q^{59} +2.00000 q^{61} -1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +6.00000i q^{68} -6.00000 q^{71} -1.00000i q^{72} +8.00000i q^{73} +4.00000 q^{74} +8.00000 q^{76} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +8.00000 q^{86} -8.00000 q^{91} +1.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} + 12 q^{51} - 2 q^{54} + 4 q^{56}+ \cdots + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 16 * q^19 + 4 * q^21 - 2 * q^24 - 8 * q^26 + 2 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 12 * q^41 + 6 * q^49 + 12 * q^51 - 2 * q^54 + 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 - 12 * q^71 + 8 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 16 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96

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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	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
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000i	0.235702i
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	− 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
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	8.00000i	1.29777i
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 2.00000i	− 0.308607i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	6.00000	0.840168
$52$	4.00000i	0.554700i
$53$	− 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
$56$	2.00000	0.267261
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 1.00000i	− 0.127000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	− 1.00000i	− 0.117851i
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	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
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	1.00000i	0.103695i
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	1.00000i	0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 2.00000i	− 0.188982i
$113$	− 18.0000i	− 1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$	− 18.0000i	− 1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	− 6.00000i	− 0.552345i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	− 6.00000i	− 0.541002i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	2.00000	0.178174
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	6.00000i	0.503509i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	3.00000i	0.247436i
$148$	− 4.00000i	− 0.328798i
$149$	−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	$0.394462\pi$
$152$	− 8.00000i	− 0.648886i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	8.00000i	0.636446i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	2.00000i	0.154303i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	8.00000	0.611775
$172$	− 8.00000i	− 0.609994i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	8.00000i	0.592999i
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	− 12.0000i	− 0.875190i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−10.0000	−0.696733
$207$	0	0
$208$	− 4.00000i	− 0.277350i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000i	0.412082i
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 2.00000i	− 0.135769i
$218$	2.00000i	0.135457i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	−24.0000	−1.61441
$222$	4.00000i	0.268462i
$223$	− 28.0000i	− 1.87502i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	− 28.0000i	− 1.87502i	0.347960	−	0.937509i	$-0.386874\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−18.0000	−1.19734
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	8.00000i	0.529813i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−6.00000	−0.390567
$237$	− 8.00000i	− 0.519656i
$238$	12.0000i	0.777844i
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	11.0000i	0.707107i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	32.0000i	2.03611i
$248$	1.00000i	0.0635001i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	8.00000i	0.498058i
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	16.0000	0.981023
$267$	0	0
$268$	2.00000i	0.122169i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	− 6.00000i	− 0.363803i
$273$	− 8.00000i	− 0.484182i
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	12.0000i	0.714590i
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	1.00000i	0.0589256i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 8.00000i	− 0.468165i
$293$	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
$296$	−4.00000	−0.232495
$297$	0	0
$298$	18.0000i	1.04271i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	− 8.00000i	− 0.460348i
$303$	− 6.00000i	− 0.344691i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	6.00000	0.342997
$307$	34.0000i	1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$	34.0000i	1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	4.00000i	0.226455i
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.0000i	2.67079i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.00000	0.110770
$327$	− 2.00000i	− 0.110600i
$328$	− 6.00000i	− 0.331295i
$329$	24.0000	1.32316
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 4.00000i	− 0.219199i
$334$	24.0000	1.31322
$335$	0	0
$336$	2.00000	0.109109
$337$	− 20.0000i	− 1.08947i	−0.838608	−	0.544735i	$-0.816630\pi$
$337$	− 20.0000i	− 1.08947i	0.838608	−	0.544735i	$-0.183370\pi$
$338$	3.00000i	0.163178i
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	− 8.00000i	− 0.432590i
$343$	− 20.0000i	− 1.07990i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	36.0000i	1.93258i	0.257454	+	0.966291i	$0.417117\pi$
$347$	36.0000i	1.93258i	−0.257454	+	0.966291i	$0.582883\pi$
$348$	0	0
$349$	−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
$351$	−4.00000	−0.213504
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	0	0
$357$	− 12.0000i	− 0.635107i
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 2.00000i	− 0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	8.00000	0.419314
$365$	0	0
$366$	2.00000	0.104542
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	− 1.00000i	− 0.0518476i
$373$	38.0000i	1.96757i	0.179364	+	0.983783i	$0.442596\pi$
$373$	38.0000i	1.96757i	−0.179364	+	0.983783i	$0.557404\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	2.00000i	0.102869i
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	18.0000i	0.920960i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 8.00000i	− 0.406663i
$388$	− 10.0000i	− 0.507673i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	− 6.00000i	− 0.302660i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	8.00000i	0.401004i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	− 2.00000i	− 0.0997509i
$403$	− 4.00000i	− 0.199254i
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	6.00000i	0.297044i
$409$	−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
$412$	10.0000i	0.492665i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	4.00000i	0.195881i
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	− 8.00000i	− 0.389434i
$423$	− 12.0000i	− 0.583460i
$424$	6.00000	0.291386
$425$	0	0
$426$	−6.00000	−0.290701
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 16.0000i	− 0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	− 16.0000i	− 0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	8.00000i	0.382255i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	24.0000i	1.14156i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−28.0000	−1.32584
$447$	− 18.0000i	− 0.851371i
$448$	2.00000i	0.0944911i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	18.0000i	0.846649i
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	8.00000	0.374634
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	14.0000i	0.654177i
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	18.0000	0.833834
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	− 4.00000i	− 0.184900i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	14.0000	0.645086
$472$	6.00000i	0.276172i
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	12.0000	0.550019
$477$	6.00000i	0.274721i
$478$	12.0000i	0.548867i
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	2.00000i	0.0905357i
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	6.00000i	0.270501i
$493$	0	0
$494$	32.0000	1.43975
$495$	0	0
$496$	1.00000	0.0449013
$497$	12.0000i	0.538274i
$498$	12.0000i	0.537733i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	− 16.0000i	− 0.709885i
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	30.0000	1.32324
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	6.00000	0.263371
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 40.0000i	− 1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$	− 40.0000i	− 1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	0	0
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	− 16.0000i	− 0.693688i
$533$	24.0000i	1.03956i
$534$	0	0
$535$	0	0
$536$	2.00000	0.0863868
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 32.0000i	− 1.37452i
$543$	2.00000i	0.0858282i
$544$	−6.00000	−0.257248
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000i	0.680389i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 42.0000i	− 1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$	− 42.0000i	− 1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	1.00000i	0.0423334i
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−22.0000	−0.924729
$567$	− 2.00000i	− 0.0839921i
$568$	− 6.00000i	− 0.251754i
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	19.0000i	0.790296i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	24.0000	0.995688
$582$	10.0000i	0.414513i
$583$	0	0
$584$	−8.00000	−0.331042
$585$	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
$589$	−8.00000	−0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	4.00000i	0.164399i
$593$	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
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 16.0000i	− 0.652111i
$603$	2.00000i	0.0814463i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	8.00000i	0.324443i
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	− 6.00000i	− 0.242536i
$613$	− 40.0000i	− 1.61558i	−0.589467	−	0.807792i	$-0.700662\pi$
$613$	− 40.0000i	− 1.61558i	0.589467	−	0.807792i	$-0.299338\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 10.0000i	− 0.402259i
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−16.0000	−0.639489
$627$	0	0
$628$	14.0000i	0.558661i
$629$	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	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	8.00000i	0.317971i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	48.0000	1.88853
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	2.00000	0.0783862
$652$	− 2.00000i	− 0.0783260i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	− 8.00000i	− 0.312110i
$658$	− 24.0000i	− 0.935617i
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000i	0.155464i
$663$	− 24.0000i	− 0.932083i
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	− 24.0000i	− 0.928588i
$669$	28.0000	1.08254
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	44.0000i	1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$	44.0000i	1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	3.00000	0.115385
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	− 18.0000i	− 0.691286i
$679$	20.0000	0.767530
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	−20.0000	−0.763604
$687$	− 14.0000i	− 0.534133i
$688$	8.00000i	0.304997i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	36.0000	1.36654
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	2.00000i	0.0757011i
$699$	−18.0000	−0.680823
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	4.00000i	0.150970i
$703$	− 32.0000i	− 1.20690i
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	12.0000i	0.451306i
$708$	− 6.00000i	− 0.225494i
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	−12.0000	−0.449089
$715$	0	0
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	30.0000i	1.11959i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	− 45.0000i	− 1.67473i
$723$	2.00000i	0.0743808i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000	1.77534
$732$	− 2.00000i	− 0.0739221i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−20.0000	−0.738213
$735$	0	0
$736$	0	0
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	12.0000i	0.440534i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	38.0000	1.39128
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	4.00000i	0.145382i	0.997354	+	0.0726912i	$0.0231588\pi$
$757$	4.00000i	0.145382i	−0.997354	+	0.0726912i	$0.976841\pi$
$758$	32.0000i	1.16229i
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	16.0000i	0.579619i
$763$	4.00000i	0.144810i
$764$	18.0000	0.651217
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	1.00000i	0.0360844i
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	− 14.0000i	− 0.503871i
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−10.0000	−0.358979
$777$	8.00000i	0.286998i
$778$	24.0000i	0.860442i
$779$	48.0000	1.71978
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	16.0000i	0.566394i
$799$	72.0000	2.54718
$800$	0	0
$801$	0	0
$802$	− 36.0000i	− 1.27120i
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	− 64.0000i	− 2.23908i
$818$	14.0000i	0.489499i
$819$	8.00000	0.279543
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	6.00000i	0.209274i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−12.0000	−0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	4.00000i	0.138675i
$833$	− 18.0000i	− 0.623663i
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	30.0000i	1.03633i
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 26.0000i	− 0.896019i
$843$	− 18.0000i	− 0.619953i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−12.0000	−0.412568
$847$	22.0000i	0.755929i
$848$	− 6.00000i	− 0.206041i
$849$	22.0000	0.755038
$850$	0	0
$851$	0	0
$852$	6.00000i	0.205557i
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	0	0
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	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
$862$	− 6.00000i	− 0.204361i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.0000	−0.543702
$867$	− 19.0000i	− 0.645274i
$868$	2.00000i	0.0678844i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	− 2.00000i	− 0.0677285i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	8.00000i	0.269987i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	3.00000i	0.101015i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	− 4.00000i	− 0.134231i
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	28.0000i	0.937509i
$893$	− 96.0000i	− 3.21252i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	12.0000i	0.400445i
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	16.0000i	0.532447i
$904$	18.0000	0.598671
$905$	0	0
$906$	8.00000	0.265782
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	12.0000i	0.398234i
$909$	6.00000	0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	− 8.00000i	− 0.264906i
$913$	0	0
$914$	−8.00000	−0.264616
$915$	0	0
$916$	14.0000	0.462573
$917$	12.0000i	0.396275i
$918$	6.00000i	0.198030i
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	−34.0000	−1.12034
$922$	36.0000i	1.18560i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	10.0000i	0.328443i
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	− 18.0000i	− 0.589610i
$933$	− 18.0000i	− 0.589294i
$934$	−24.0000	−0.785304
$935$	0	0
$936$	−4.00000	−0.130744
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	4.00000i	0.130605i
$939$	16.0000	0.522140
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	− 14.0000i	− 0.456145i
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	8.00000i	0.259828i
$949$	32.0000	1.03876
$950$	0	0
$951$	6.00000	0.194563
$952$	− 12.0000i	− 0.388922i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	12.0000	0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 16.0000i	− 0.515861i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−48.0000	−1.54198
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 8.00000i	− 0.256468i
$974$	40.0000	1.28168
$975$	0	0
$976$	2.00000	0.0640184
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	2.00000i	0.0639529i
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 12.0000i	− 0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	− 32.0000i	− 1.01806i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 4.00000i	− 0.126936i
$994$	12.0000	0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	− 4.00000i	− 0.126618i
$999$	4.00000	0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.n.1.1	✓	1	5.2	odd	4
2790.2.a.k.1.1		1	15.2	even	4
4650.2.a.d.1.1		1	5.3	odd	4
4650.2.d.u.3349.1		2	1.1	even	1	trivial
4650.2.d.u.3349.2		2	5.4	even	2	inner
7440.2.a.b.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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +2.00000 q^{21} -1.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} +1.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +8.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -8.00000i q^{57} +6.00000 q^{59} +2.00000 q^{61} -1.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +6.00000i q^{68} -6.00000 q^{71} -1.00000i q^{72} +8.00000i q^{73} +4.00000 q^{74} +8.00000 q^{76} -4.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +8.00000 q^{86} -8.00000 q^{91} +1.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} + 12 q^{51} - 2 q^{54} + 4 q^{56}+ \cdots + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 16 * q^19 + 4 * q^21 - 2 * q^24 - 8 * q^26 + 2 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 12 * q^41 + 6 * q^49 + 12 * q^51 - 2 * q^54 + 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 - 12 * q^71 + 8 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 16 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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