LMFDB - Embedded newform 4050.2.c.q.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(649,4050)] 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("4050.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,6,0,0,-8,0,2,0,0,8] 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:	$32.3394128186$
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 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.q.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} +4.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +4.00000 q^{19} -3.00000i q^{22} +6.00000i q^{23} +4.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +8.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +1.00000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} -9.00000 q^{49} -4.00000i q^{52} +4.00000 q^{56} -6.00000i q^{58} +9.00000 q^{59} +8.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.00000i q^{68} -6.00000 q^{71} -14.0000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +6.00000i q^{82} +9.00000i q^{83} +1.00000 q^{86} +3.00000i q^{88} +9.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} +12.0000 q^{94} -7.00000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{11} - 8 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{26} + 12 q^{29} + 16 q^{31} - 6 q^{34} - 12 q^{41} - 6 q^{44} + 12 q^{46} - 18 q^{49} + 8 q^{56} + 18 q^{59} + 16 q^{61} - 2 q^{64} - 12 q^{71}+ \cdots + 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^11 - 8 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^26 + 12 * q^29 + 16 * q^31 - 6 * q^34 - 12 * q^41 - 6 * q^44 + 12 * q^46 - 18 * q^49 + 8 * q^56 + 18 * q^59 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 16 * q^74 - 8 * q^76 - 16 * q^79 + 2 * q^86 + 18 * q^89 + 32 * q^91 + 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 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
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 3.00000i	− 0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	4.00000i	0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 8.00000i	− 1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	3.00000i	0.363803i
$69$	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	$0.615871\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	−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
$81$	0	0
$82$	6.00000i	0.662589i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	3.00000i	0.319801i
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−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
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	− 3.00000i	− 0.282216i	−0.989994	−	0.141108i	$-0.954933\pi$
$113$	− 3.00000i	− 0.282216i	0.989994	−	0.141108i	$-0.0450665\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 9.00000i	− 0.828517i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	3.00000	0.257248
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	−14.0000	−1.15865
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	24.0000	1.89146
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	9.00000	0.698535
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	− 1.00000i	− 0.0762493i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	− 9.00000i	− 0.674579i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 16.0000i	− 1.18600i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	9.00000	0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	− 12.0000i	− 0.844317i
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	4.00000i	0.277350i
$209$	12.0000	0.830057
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	− 32.0000i	− 2.17230i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−3.00000	−0.199557
$227$	− 27.0000i	− 1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$	− 27.0000i	− 1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 15.0000i	− 0.982683i	−0.870967	−	0.491341i	$-0.836507\pi$
$233$	− 15.0000i	− 0.982683i	0.870967	−	0.491341i	$-0.163493\pi$
$234$	0	0
$235$	0	0
$236$	−9.00000	−0.585850
$237$	0	0
$238$	12.0000i	0.777844i
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	8.00000i	0.508001i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	18.0000i	1.13165i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.00000i	0.187135i	0.995613	+	0.0935674i	$0.0298271\pi$
$257$	3.00000i	0.187135i	−0.995613	+	0.0935674i	$0.970173\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	−16.0000	−0.981023
$267$	0	0
$268$	4.00000i	0.244339i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	11.0000i	0.659736i
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	1.00000i	0.0594438i	0.999558	+	0.0297219i	$0.00946217\pi$
$283$	1.00000i	0.0594438i	−0.999558	+	0.0297219i	$0.990538\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	24.0000i	1.41668i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	14.0000i	0.819288i
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	4.00000	0.230556
$302$	10.0000i	0.575435i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 1.00000i	− 0.0570730i	−0.999593	−	0.0285365i	$-0.990915\pi$
$307$	− 1.00000i	− 0.0570730i	0.999593	−	0.0285365i	$-0.00908469\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	− 17.0000i	− 0.960897i	−0.877023	−	0.480448i	$-0.840474\pi$
$313$	− 17.0000i	− 0.960897i	0.877023	−	0.480448i	$-0.159526\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	8.00000	0.450035
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	− 24.0000i	− 1.33747i
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	0	0
$326$	−11.0000	−0.609234
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	48.0000	2.64633
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	− 9.00000i	− 0.493939i
$333$	0	0
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	8.00000i	0.431959i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	6.00000	0.322562
$347$	27.0000i	1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$	27.0000i	1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	− 3.00000i	− 0.159901i
$353$	9.00000i	0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$	9.00000i	0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$	0	0
$355$	0	0
$356$	−9.00000	−0.476999
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	−16.0000	−0.838628
$365$	0	0
$366$	0	0
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 8.00000i	− 0.414224i	−0.978317	−	0.207112i	$-0.933593\pi$
$373$	− 8.00000i	− 0.414224i	0.978317	−	0.207112i	$-0.0664065\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−12.0000	−0.618853
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	0	0
$382$	− 6.00000i	− 0.306987i
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	0	0
$386$	−5.00000	−0.254493
$387$	0	0
$388$	7.00000i	0.355371i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	2.00000i	0.100251i
$399$	0	0
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	32.0000i	1.59403i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	−24.0000	−1.19110
$407$	24.0000i	1.18964i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	− 36.0000i	− 1.77144i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	− 12.0000i	− 0.586939i
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	− 23.0000i	− 1.11962i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 32.0000i	− 1.54859i
$428$	− 12.0000i	− 0.580042i
$429$	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	$0.453841\pi$
$432$	0	0
$433$	7.00000i	0.336399i	0.985753	+	0.168199i	$0.0537952\pi$
$433$	7.00000i	0.336399i	−0.985753	+	0.168199i	$0.946205\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	2.00000	0.0957826
$437$	24.0000i	1.14808i
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	4.00000i	0.188982i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	3.00000i	0.141108i
$453$	0	0
$454$	−27.0000	−1.26717
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	8.00000i	0.373815i
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−15.0000	−0.694862
$467$	9.00000i	0.416470i	0.978079	+	0.208235i	$0.0667719\pi$
$467$	9.00000i	0.416470i	−0.978079	+	0.208235i	$0.933228\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	9.00000i	0.414259i
$473$	3.00000i	0.137940i
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	18.0000i	0.823301i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	− 26.0000i	− 1.18427i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	16.0000	0.719874
$495$	0	0
$496$	8.00000	0.359211
$497$	24.0000i	1.07655i
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	0	0
$505$	0	0
$506$	18.0000	0.800198
$507$	0	0
$508$	16.0000i	0.709885i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	3.00000	0.132324
$515$	0	0
$516$	0	0
$517$	36.0000i	1.58328i
$518$	− 32.0000i	− 1.40600i
$519$	0	0
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	− 29.0000i	− 1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$	− 29.0000i	− 1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	16.0000i	0.693688i
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	− 12.0000i	− 0.517357i
$539$	−27.0000	−1.16297
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	4.00000i	0.171815i
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	32.0000i	1.36078i
$554$	2.00000	0.0849719
$555$	0	0
$556$	11.0000	0.466504
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	3.00000i	0.126435i	0.998000	+	0.0632175i	$0.0201362\pi$
$563$	3.00000i	0.126435i	−0.998000	+	0.0632175i	$0.979864\pi$
$564$	0	0
$565$	0	0
$566$	1.00000	0.0420331
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	− 12.0000i	− 0.501745i
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	0	0
$584$	14.0000	0.579324
$585$	0	0
$586$	12.0000	0.495715
$587$	− 3.00000i	− 0.123823i	−0.998082	−	0.0619116i	$-0.980280\pi$
$587$	− 3.00000i	− 0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	45.0000i	1.84793i	0.382479	+	0.923964i	$0.375070\pi$
$593$	45.0000i	1.84793i	−0.382479	+	0.923964i	$0.624930\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	24.0000i	0.981433i
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	− 4.00000i	− 0.163028i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	−1.00000	−0.0403567
$615$	0	0
$616$	12.0000	0.483494
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	0	0
$622$	− 6.00000i	− 0.240578i
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	−17.0000	−0.679457
$627$	0	0
$628$	4.00000i	0.159617i
$629$	24.0000	0.956943
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 36.0000i	− 1.42637i
$638$	− 18.0000i	− 0.712627i
$639$	0	0
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	−12.0000	−0.472134
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	0	0
$652$	11.0000i	0.430793i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	− 48.0000i	− 1.87123i
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 17.0000i	− 0.660724i
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	− 18.0000i	− 0.696441i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	3.00000	0.115385
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	0	0
$682$	− 24.0000i	− 0.919007i
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	1.00000i	0.0381246i
$689$	0	0
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	14.0000i	0.529908i
$699$	0	0
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	32.0000i	1.20690i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	9.00000	0.338719
$707$	− 48.0000i	− 1.80523i
$708$	0	0
$709$	52.0000	1.95290	0.976450	−	0.215742i	$-0.0692169\pi$
$709$	52.0000	1.95290	0.976450	+	0.215742i	$0.0692169\pi$
$710$	0	0
$711$	0	0
$712$	9.00000i	0.337289i
$713$	48.0000i	1.79761i
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	0	0
$719$	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
$721$	−32.0000	−1.19174
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	16.0000i	0.592999i
$729$	0	0
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	6.00000	0.221163
$737$	− 12.0000i	− 0.442026i
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−8.00000	−0.292901
$747$	0	0
$748$	9.00000i	0.329073i
$749$	48.0000	1.75388
$750$	0	0
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	24.0000	0.874028
$755$	0	0
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	29.0000i	1.05333i
$759$	0	0
$760$	0	0
$761$	−39.0000	−1.41375	−0.706874	−	0.707339i	$-0.749895\pi$
$761$	−39.0000	−1.41375	−0.706874	+	0.707339i	$0.749895\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−18.0000	−0.650366
$767$	36.0000i	1.29988i
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	5.00000i	0.179954i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	0	0
$775$	0	0
$776$	7.00000	0.251285
$777$	0	0
$778$	− 6.00000i	− 0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−18.0000	−0.644091
$782$	− 18.0000i	− 0.643679i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	32.0000i	1.13635i
$794$	14.0000	0.496841
$795$	0	0
$796$	2.00000	0.0708881
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	0	0
$802$	15.0000i	0.529668i
$803$	− 42.0000i	− 1.48215i
$804$	0	0
$805$	0	0
$806$	32.0000	1.12715
$807$	0	0
$808$	12.0000i	0.422159i
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	24.0000i	0.842235i
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	− 22.0000i	− 0.769212i
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−36.0000	−1.25260
$827$	15.0000i	0.521601i	0.965393	+	0.260801i	$0.0839865\pi$
$827$	15.0000i	0.521601i	−0.965393	+	0.260801i	$0.916014\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	− 4.00000i	− 0.138675i
$833$	27.0000i	0.935495i
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	9.00000i	0.310900i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 32.0000i	− 1.10279i
$843$	0	0
$844$	−23.0000	−0.791693
$845$	0	0
$846$	0	0
$847$	8.00000i	0.274883i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 27.0000i	− 0.922302i	−0.887322	−	0.461151i	$-0.847437\pi$
$857$	− 27.0000i	− 0.922302i	0.887322	−	0.461151i	$-0.152563\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	0	0
$862$	− 6.00000i	− 0.204361i
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	7.00000	0.237870
$867$	0	0
$868$	32.0000i	1.08615i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	− 16.0000i	− 0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$	− 16.0000i	− 0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$	14.0000i	0.472477i
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	13.0000i	0.437485i	0.975783	+	0.218742i	$0.0701954\pi$
$883$	13.0000i	0.437485i	−0.975783	+	0.218742i	$0.929805\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−9.00000	−0.302361
$887$	30.0000i	1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$887$	30.0000i	1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$888$	0	0
$889$	−64.0000	−2.14649
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	18.0000i	0.600668i
$899$	48.0000	1.60089
$900$	0	0
$901$	0	0
$902$	18.0000i	0.599334i
$903$	0	0
$904$	3.00000	0.0997785
$905$	0	0
$906$	0	0
$907$	17.0000i	0.564476i	0.959344	+	0.282238i	$0.0910767\pi$
$907$	17.0000i	0.564476i	−0.959344	+	0.282238i	$0.908923\pi$
$908$	27.0000i	0.896026i
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	27.0000i	0.893570i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	8.00000	0.264327
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	−14.0000	−0.460069
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	15.0000i	0.491341i
$933$	0	0
$934$	9.00000	0.294489
$935$	0	0
$936$	0	0
$937$	− 37.0000i	− 1.20874i	−0.796705	−	0.604369i	$-0.793425\pi$
$937$	− 37.0000i	− 1.20874i	0.796705	−	0.604369i	$-0.206575\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	− 36.0000i	− 1.17232i
$944$	9.00000	0.292925
$945$	0	0
$946$	3.00000	0.0975384
$947$	− 9.00000i	− 0.292461i	−0.989251	−	0.146230i	$-0.953286\pi$
$947$	− 9.00000i	− 0.292461i	0.989251	−	0.146230i	$-0.0467141\pi$
$948$	0	0
$949$	56.0000	1.81784
$950$	0	0
$951$	0	0
$952$	− 12.0000i	− 0.388922i
$953$	27.0000i	0.874616i	0.899312	+	0.437308i	$0.144068\pi$
$953$	27.0000i	0.874616i	−0.899312	+	0.437308i	$0.855932\pi$
$954$	0	0
$955$	0	0
$956$	18.0000	0.582162
$957$	0	0
$958$	− 12.0000i	− 0.387702i
$959$	72.0000	2.32500
$960$	0	0
$961$	33.0000	1.06452
$962$	32.0000i	1.03172i
$963$	0	0
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	− 28.0000i	− 0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$	− 28.0000i	− 0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	44.0000i	1.41058i
$974$	2.00000	0.0640841
$975$	0	0
$976$	8.00000	0.256074
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	27.0000	0.862924
$980$	0	0
$981$	0	0
$982$	15.0000i	0.478669i
$983$	− 54.0000i	− 1.72233i	−0.508323	−	0.861166i	$-0.669735\pi$
$983$	− 54.0000i	− 1.72233i	0.508323	−	0.861166i	$-0.330265\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 8.00000i	− 0.254000i
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	11.0000i	0.348199i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.q.649.1		2
3.2	odd	2		4050.2.c.e.649.2		2
5.2	odd	4		4050.2.a.bj.1.1		1
5.3	odd	4		4050.2.a.b.1.1		1
5.4	even	2	inner	4050.2.c.q.649.2		2
9.2	odd	6		1350.2.j.d.199.2		4
9.4	even	3		450.2.j.d.349.2		4
9.5	odd	6		1350.2.j.d.1099.1		4
9.7	even	3		450.2.j.d.49.1		4
15.2	even	4		4050.2.a.p.1.1		1
15.8	even	4		4050.2.a.t.1.1		1
15.14	odd	2		4050.2.c.e.649.1		2
45.2	even	12		1350.2.e.f.901.1		2
45.4	even	6		450.2.j.d.349.1		4
45.7	odd	12		450.2.e.a.301.1	yes	2
45.13	odd	12		450.2.e.h.151.1	yes	2
45.14	odd	6		1350.2.j.d.1099.2		4
45.22	odd	12		450.2.e.a.151.1	✓	2
45.23	even	12		1350.2.e.e.451.1		2
45.29	odd	6		1350.2.j.d.199.1		4
45.32	even	12		1350.2.e.f.451.1		2
45.34	even	6		450.2.j.d.49.2		4
45.38	even	12		1350.2.e.e.901.1		2
45.43	odd	12		450.2.e.h.301.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.e.a.151.1	✓	2	45.22	odd	12
450.2.e.a.301.1	yes	2	45.7	odd	12
450.2.e.h.151.1	yes	2	45.13	odd	12
450.2.e.h.301.1	yes	2	45.43	odd	12
450.2.j.d.49.1		4	9.7	even	3
450.2.j.d.49.2		4	45.34	even	6
450.2.j.d.349.1		4	45.4	even	6
450.2.j.d.349.2		4	9.4	even	3
1350.2.e.e.451.1		2	45.23	even	12
1350.2.e.e.901.1		2	45.38	even	12
1350.2.e.f.451.1		2	45.32	even	12
1350.2.e.f.901.1		2	45.2	even	12
1350.2.j.d.199.1		4	45.29	odd	6
1350.2.j.d.199.2		4	9.2	odd	6
1350.2.j.d.1099.1		4	9.5	odd	6
1350.2.j.d.1099.2		4	45.14	odd	6
4050.2.a.b.1.1		1	5.3	odd	4
4050.2.a.p.1.1		1	15.2	even	4
4050.2.a.t.1.1		1	15.8	even	4
4050.2.a.bj.1.1		1	5.2	odd	4
4050.2.c.e.649.1		2	15.14	odd	2
4050.2.c.e.649.2		2	3.2	odd	2
4050.2.c.q.649.1		2	1.1	even	1	trivial
4050.2.c.q.649.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +3.00000 q^{11} +4.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +4.00000 q^{19} -3.00000i q^{22} +6.00000i q^{23} +4.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +8.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} +1.00000i q^{43} -3.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} -9.00000 q^{49} -4.00000i q^{52} +4.00000 q^{56} -6.00000i q^{58} +9.00000 q^{59} +8.00000 q^{61} -8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +3.00000i q^{68} -6.00000 q^{71} -14.0000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +6.00000i q^{82} +9.00000i q^{83} +1.00000 q^{86} +3.00000i q^{88} +9.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} +12.0000 q^{94} -7.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{11} - 8 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{26} + 12 q^{29} + 16 q^{31} - 6 q^{34} - 12 q^{41} - 6 q^{44} + 12 q^{46} - 18 q^{49} + 8 q^{56} + 18 q^{59} + 16 q^{61} - 2 q^{64} - 12 q^{71}+ \cdots + 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^11 - 8 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^26 + 12 * q^29 + 16 * q^31 - 6 * q^34 - 12 * q^41 - 6 * q^44 + 12 * q^46 - 18 * q^49 + 8 * q^56 + 18 * q^59 + 16 * q^61 - 2 * q^64 - 12 * q^71 + 16 * q^74 - 8 * q^76 - 16 * q^79 + 2 * q^86 + 18 * q^89 + 32 * q^91 + 24 * q^94

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