LMFDB - Embedded newform 4950.2.c.i.199.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(199,4950)] 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("4950.199"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.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,-2,0,0,0,0,2,0,0,2,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)

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

Self dual:	no
Analytic conductor:	$39.5259490005$
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 550)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.i.199.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -1.00000 q^{11} -3.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000 q^{19} +1.00000i q^{22} +3.00000i q^{23} -3.00000 q^{26} +5.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -12.0000i q^{37} -1.00000i q^{38} -8.00000 q^{41} +5.00000i q^{43} +1.00000 q^{44} +3.00000 q^{46} +8.00000i q^{47} +7.00000 q^{49} +3.00000i q^{52} -10.0000i q^{53} -5.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} +3.00000i q^{62} -1.00000 q^{64} +14.0000i q^{67} -4.00000i q^{68} +5.00000 q^{71} +4.00000i q^{73} -12.0000 q^{74} -1.00000 q^{76} +8.00000 q^{79} +8.00000i q^{82} -9.00000i q^{83} +5.00000 q^{86} -1.00000i q^{88} +3.00000 q^{89} -3.00000i q^{92} +8.00000 q^{94} +3.00000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{11} + 2 q^{16} + 2 q^{19} - 6 q^{26} + 10 q^{29} - 6 q^{31} + 8 q^{34} - 16 q^{41} + 2 q^{44} + 6 q^{46} + 14 q^{49} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 10 q^{71} - 24 q^{74} - 2 q^{76}+ \cdots + 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^11 + 2 * q^16 + 2 * q^19 - 6 * q^26 + 10 * q^29 - 6 * q^31 + 8 * q^34 - 16 * q^41 + 2 * q^44 + 6 * q^46 + 14 * q^49 + 16 * q^59 + 20 * q^61 - 2 * q^64 + 10 * q^71 - 24 * q^74 - 2 * q^76 + 16 * q^79 + 10 * q^86 + 6 * q^89 + 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$.

$n$	$551$	$2377$	$4501$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	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	$0.838794\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	0.213201i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	−3.00000	−0.588348
$27$	0	0
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	− 12.0000i	− 1.97279i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	− 12.0000i	− 1.97279i	0.164399	−	0.986394i	$-0.447432\pi$
$38$	− 1.00000i	− 0.162221i
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	3.00000	0.442326
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	3.00000i	0.416025i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 5.00000i	− 0.656532i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	3.00000i	0.381000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−12.0000	−1.39497
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	8.00000i	0.883452i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	5.00000	0.539164
$87$	0	0
$88$	− 1.00000i	− 0.106600i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	0	0
$92$	− 3.00000i	− 0.312772i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	3.00000i	0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$	3.00000i	0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	0	0
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	0	0
$103$	− 1.00000i	− 0.0985329i	−0.998786	−	0.0492665i	$-0.984312\pi$
$103$	− 1.00000i	− 0.0985329i	0.998786	−	0.0492665i	$-0.0156884\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−10.0000	−0.971286
$107$	− 17.0000i	− 1.64345i	−0.569883	−	0.821726i	$-0.693011\pi$
$107$	− 17.0000i	− 1.64345i	0.569883	−	0.821726i	$-0.306989\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	−5.00000	−0.464238
$117$	0	0
$118$	− 8.00000i	− 0.736460i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 10.0000i	− 0.905357i
$123$	0	0
$124$	3.00000	0.269408
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	0	0
$133$	0	0
$134$	14.0000	1.20942
$135$	0	0
$136$	−4.00000	−0.342997
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	0	0
$142$	− 5.00000i	− 0.419591i
$143$	3.00000i	0.250873i
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	12.0000i	0.986394i
$149$	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
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	1.00000i	0.0811107i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−9.00000	−0.698535
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	0	0
$172$	− 5.00000i	− 0.381246i
$173$	− 13.0000i	− 0.988372i	−0.869356	−	0.494186i	$-0.835466\pi$
$173$	− 13.0000i	− 0.988372i	0.869356	−	0.494186i	$-0.164534\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	− 3.00000i	− 0.224860i
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	3.00000	0.215387
$195$	0	0
$196$	−7.00000	−0.500000
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	13.0000	0.921546	0.460773	−	0.887518i	$-0.347572\pi$
$199$	13.0000	0.921546	0.460773	+	0.887518i	$0.347572\pi$
$200$	0	0
$201$	0	0
$202$	17.0000i	1.19612i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	− 3.00000i	− 0.208013i
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	10.0000i	0.686803i
$213$	0	0
$214$	−17.0000	−1.16210
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 13.0000i	− 0.880471i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	− 19.0000i	− 1.26107i	−0.776159	−	0.630537i	$-0.782835\pi$
$227$	− 19.0000i	− 1.26107i	0.776159	−	0.630537i	$-0.217165\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	5.00000i	0.328266i
$233$	30.0000i	1.96537i	0.185296	+	0.982683i	$0.440675\pi$
$233$	30.0000i	1.96537i	−0.185296	+	0.982683i	$0.559325\pi$
$234$	0	0
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	0	0
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	− 3.00000i	− 0.190885i
$248$	− 3.00000i	− 0.190500i
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	− 3.00000i	− 0.188608i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 7.00000i	− 0.436648i	−0.975876	−	0.218324i	$-0.929941\pi$
$257$	− 7.00000i	− 0.436648i	0.975876	−	0.218324i	$-0.0700590\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 13.0000i	− 0.803143i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 14.0000i	− 0.855186i
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	13.0000i	0.779688i
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	−5.00000	−0.296695
$285$	0	0
$286$	3.00000	0.177394
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	12.0000	0.697486
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	9.00000	0.520483
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−8.00000	−0.450035
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	− 8.00000i	− 0.441726i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	9.00000i	0.493939i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.0000i	0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$	10.0000i	0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$	− 4.00000i	− 0.217571i
$339$	0	0
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	0	0
$344$	−5.00000	−0.269582
$345$	0	0
$346$	−13.0000	−0.698884
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	0	0
$352$	1.00000i	0.0533002i
$353$	− 17.0000i	− 0.904819i	−0.891810	−	0.452409i	$-0.850565\pi$
$353$	− 17.0000i	− 0.904819i	0.891810	−	0.452409i	$-0.149435\pi$
$354$	0	0
$355$	0	0
$356$	−3.00000	−0.159000
$357$	0	0
$358$	− 24.0000i	− 1.26844i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	16.0000i	0.840941i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.00000i	0.0521996i	0.999659	+	0.0260998i	$0.00830876\pi$
$367$	1.00000i	0.0521996i	−0.999659	+	0.0260998i	$0.991691\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−8.00000	−0.412568
$377$	− 15.0000i	− 0.772539i
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	0	0
$382$	− 9.00000i	− 0.460480i
$383$	13.0000i	0.664269i	0.943232	+	0.332134i	$0.107769\pi$
$383$	13.0000i	0.664269i	−0.943232	+	0.332134i	$0.892231\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	− 3.00000i	− 0.152302i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	7.00000i	0.353553i
$393$	0	0
$394$	3.00000	0.151138
$395$	0	0
$396$	0	0
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	− 13.0000i	− 0.651631i
$399$	0	0
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	0	0
$403$	9.00000i	0.448322i
$404$	17.0000	0.845782
$405$	0	0
$406$	0	0
$407$	12.0000i	0.594818i
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	0	0
$412$	1.00000i	0.0492665i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−3.00000	−0.147087
$417$	0	0
$418$	1.00000i	0.0489116i
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 28.0000i	− 1.36302i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	17.0000i	0.821726i
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	0	0
$435$	0	0
$436$	−13.0000	−0.622587
$437$	3.00000i	0.143509i
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	0	0
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	−19.0000	−0.891714
$455$	0	0
$456$	0	0
$457$	− 12.0000i	− 0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$	− 12.0000i	− 0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$	− 26.0000i	− 1.21490i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	29.0000i	1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$	29.0000i	1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	30.0000	1.38972
$467$	− 14.0000i	− 0.647843i	−0.946084	−	0.323921i	$-0.894999\pi$
$467$	− 14.0000i	− 0.647843i	0.946084	−	0.323921i	$-0.105001\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	8.00000i	0.368230i
$473$	− 5.00000i	− 0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 22.0000i	− 1.00626i
$479$	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
$481$	−36.0000	−1.64146
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 3.00000i	− 0.135943i	−0.997687	−	0.0679715i	$-0.978347\pi$
$487$	− 3.00000i	− 0.135943i	0.997687	−	0.0679715i	$-0.0216527\pi$
$488$	10.0000i	0.452679i
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	20.0000i	0.900755i
$494$	−3.00000	−0.134976
$495$	0	0
$496$	−3.00000	−0.134704
$497$	0	0
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	0	0
$502$	20.0000i	0.892644i
$503$	30.0000i	1.33763i	0.743427	+	0.668817i	$0.233199\pi$
$503$	30.0000i	1.33763i	−0.743427	+	0.668817i	$0.766801\pi$
$504$	0	0
$505$	0	0
$506$	−3.00000	−0.133366
$507$	0	0
$508$	4.00000i	0.177471i
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−7.00000	−0.308757
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.00000	−0.306676	−0.153338	−	0.988174i	$-0.549002\pi$
$521$	−7.00000	−0.306676	−0.153338	+	0.988174i	$0.549002\pi$
$522$	0	0
$523$	19.0000i	0.830812i	0.909636	+	0.415406i	$0.136360\pi$
$523$	19.0000i	0.830812i	−0.909636	+	0.415406i	$0.863640\pi$
$524$	−13.0000	−0.567908
$525$	0	0
$526$	−12.0000	−0.523225
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.0000i	1.03956i
$534$	0	0
$535$	0	0
$536$	−14.0000	−0.604708
$537$	0	0
$538$	2.00000i	0.0862261i
$539$	−7.00000	−0.301511
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	− 22.0000i	− 0.944981i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	− 31.0000i	− 1.32546i	−0.748857	−	0.662732i	$-0.769397\pi$
$547$	− 31.0000i	− 1.32546i	0.748857	−	0.662732i	$-0.230603\pi$
$548$	3.00000i	0.128154i
$549$	0	0
$550$	0	0
$551$	5.00000	0.213007
$552$	0	0
$553$	0	0
$554$	−18.0000	−0.764747
$555$	0	0
$556$	13.0000	0.551323
$557$	3.00000i	0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$	3.00000i	0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$	0	0
$559$	15.0000	0.634432
$560$	0	0
$561$	0	0
$562$	12.0000i	0.506189i
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	5.00000i	0.209795i
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	− 3.00000i	− 0.125436i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.0000i	1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$	26.0000i	1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.0000i	0.414158i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	0	0
$591$	0	0
$592$	− 12.0000i	− 0.493197i
$593$	2.00000i	0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$	2.00000i	0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	− 9.00000i	− 0.368037i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	36.0000	1.46847	0.734235	−	0.678895i	$-0.237541\pi$
$601$	36.0000	1.46847	0.734235	+	0.678895i	$0.237541\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	− 1.00000i	− 0.0405554i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	19.0000i	0.764911i	0.923974	+	0.382456i	$0.124922\pi$
$617$	19.0000i	0.764911i	−0.923974	+	0.382456i	$0.875078\pi$
$618$	0	0
$619$	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
$622$	27.0000i	1.08260i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	6.00000	0.239808
$627$	0	0
$628$	6.00000i	0.239426i
$629$	48.0000	1.91389
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	−32.0000	−1.27088
$635$	0	0
$636$	0	0
$637$	− 21.0000i	− 0.832050i
$638$	5.00000i	0.197952i
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	44.0000i	1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$	44.0000i	1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	−8.00000	−0.312348
$657$	0	0
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	28.0000i	1.08825i
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	0	0
$667$	15.0000i	0.580802i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	− 20.0000i	− 0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$	− 20.0000i	− 0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−4.00000	−0.153846
$677$	41.0000i	1.57576i	0.615830	+	0.787879i	$0.288821\pi$
$677$	41.0000i	1.57576i	−0.615830	+	0.787879i	$0.711179\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 3.00000i	− 0.114876i
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	5.00000i	0.190623i
$689$	−30.0000	−1.14291
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	13.0000i	0.494186i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	− 32.0000i	− 1.21209i
$698$	21.0000i	0.794862i
$699$	0	0
$700$	0	0
$701$	29.0000	1.09531	0.547657	−	0.836703i	$-0.315520\pi$
$701$	29.0000	1.09531	0.547657	+	0.836703i	$0.315520\pi$
$702$	0	0
$703$	− 12.0000i	− 0.452589i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−17.0000	−0.639803
$707$	0	0
$708$	0	0
$709$	−44.0000	−1.65245	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000	−1.65245	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	0	0
$712$	3.00000i	0.112430i
$713$	− 9.00000i	− 0.337053i
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	8.00000i	0.298557i
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	0	0
$722$	18.0000i	0.669891i
$723$	0	0
$724$	16.0000	0.594635
$725$	0	0
$726$	0	0
$727$	5.00000i	0.185440i	0.995692	+	0.0927199i	$0.0295561\pi$
$727$	5.00000i	0.185440i	−0.995692	+	0.0927199i	$0.970444\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	45.0000i	1.66211i	0.556188	+	0.831056i	$0.312263\pi$
$733$	45.0000i	1.66211i	−0.556188	+	0.831056i	$0.687737\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	3.00000	0.110581
$737$	− 14.0000i	− 0.515697i
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	0	0
$745$	0	0
$746$	−6.00000	−0.219676
$747$	0	0
$748$	4.00000i	0.146254i
$749$	0	0
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	−15.0000	−0.546268
$755$	0	0
$756$	0	0
$757$	− 52.0000i	− 1.88997i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	− 52.0000i	− 1.88997i	0.327111	−	0.944986i	$-0.393925\pi$
$758$	6.00000i	0.217930i
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	−9.00000	−0.325609
$765$	0	0
$766$	13.0000	0.469709
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0000i	− 0.719816i
$773$	28.0000i	1.00709i	0.863969	+	0.503545i	$0.167971\pi$
$773$	28.0000i	1.00709i	−0.863969	+	0.503545i	$0.832029\pi$
$774$	0	0
$775$	0	0
$776$	−3.00000	−0.107694
$777$	0	0
$778$	− 24.0000i	− 0.860442i
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−5.00000	−0.178914
$782$	12.0000i	0.429119i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 30.0000i	− 1.06533i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	−13.0000	−0.460773
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	0	0
$802$	− 25.0000i	− 0.882781i
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	9.00000	0.317011
$807$	0	0
$808$	− 17.0000i	− 0.598058i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	12.0000	0.420600
$815$	0	0
$816$	0	0
$817$	5.00000i	0.174928i
$818$	− 8.00000i	− 0.279713i
$819$	0	0
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	0	0
$827$	− 11.0000i	− 0.382507i	−0.981541	−	0.191254i	$-0.938745\pi$
$827$	− 11.0000i	− 0.382507i	0.981541	−	0.191254i	$-0.0612553\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	0	0
$832$	3.00000i	0.104006i
$833$	28.0000i	0.970143i
$834$	0	0
$835$	0	0
$836$	1.00000	0.0345857
$837$	0	0
$838$	− 30.0000i	− 1.03633i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	− 2.00000i	− 0.0689246i
$843$	0	0
$844$	−28.0000	−0.963800
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	0	0
$850$	0	0
$851$	36.0000	1.23406
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	0	0
$855$	0	0
$856$	17.0000	0.581048
$857$	4.00000i	0.136637i	0.997664	+	0.0683187i	$0.0217635\pi$
$857$	4.00000i	0.136637i	−0.997664	+	0.0683187i	$0.978237\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	0	0
$861$	0	0
$862$	− 8.00000i	− 0.272481i
$863$	9.00000i	0.306364i	0.988198	+	0.153182i	$0.0489520\pi$
$863$	9.00000i	0.306364i	−0.988198	+	0.153182i	$0.951048\pi$
$864$	0	0
$865$	0	0
$866$	−11.0000	−0.373795
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	42.0000	1.42312
$872$	13.0000i	0.440236i
$873$	0	0
$874$	3.00000	0.101477
$875$	0	0
$876$	0	0
$877$	17.0000i	0.574049i	0.957923	+	0.287025i	$0.0926662\pi$
$877$	17.0000i	0.574049i	−0.957923	+	0.287025i	$0.907334\pi$
$878$	− 4.00000i	− 0.134993i
$879$	0	0
$880$	0	0
$881$	−53.0000	−1.78562	−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000	−1.78562	−0.892808	+	0.450438i	$0.851268\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	16.0000	0.537531
$887$	46.0000i	1.54453i	0.635301	+	0.772264i	$0.280876\pi$
$887$	46.0000i	1.54453i	−0.635301	+	0.772264i	$0.719124\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	24.0000i	0.803579i
$893$	8.00000i	0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	27.0000i	0.901002i
$899$	−15.0000	−0.500278
$900$	0	0
$901$	40.0000	1.33259
$902$	− 8.00000i	− 0.266371i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	0	0
$907$	30.0000i	0.996134i	0.867139	+	0.498067i	$0.165957\pi$
$907$	30.0000i	0.996134i	−0.867139	+	0.498067i	$0.834043\pi$
$908$	19.0000i	0.630537i
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	9.00000i	0.297857i
$914$	−12.0000	−0.396925
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	− 15.0000i	− 0.493731i
$924$	0	0
$925$	0	0
$926$	29.0000	0.952999
$927$	0	0
$928$	− 5.00000i	− 0.164133i
$929$	−25.0000	−0.820223	−0.410112	−	0.912035i	$-0.634510\pi$
$929$	−25.0000	−0.820223	−0.410112	+	0.912035i	$0.634510\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	− 30.0000i	− 0.982683i
$933$	0	0
$934$	−14.0000	−0.458094
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	8.00000	0.260378
$945$	0	0
$946$	−5.00000	−0.162564
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	0	0
$955$	0	0
$956$	−22.0000	−0.711531
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	36.0000i	1.16069i
$963$	0	0
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	54.0000i	1.73652i	0.496107	+	0.868261i	$0.334762\pi$
$967$	54.0000i	1.73652i	−0.496107	+	0.868261i	$0.665238\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	0	0
$974$	−3.00000	−0.0961262
$975$	0	0
$976$	10.0000	0.320092
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	0	0
$982$	− 9.00000i	− 0.287202i
$983$	− 57.0000i	− 1.81802i	−0.416777	−	0.909009i	$-0.636840\pi$
$983$	− 57.0000i	− 1.81802i	0.416777	−	0.909009i	$-0.363160\pi$
$984$	0	0
$985$	0	0
$986$	20.0000	0.636930
$987$	0	0
$988$	3.00000i	0.0954427i
$989$	−15.0000	−0.476972
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	3.00000i	0.0952501i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	− 10.0000i	− 0.316544i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.i.199.1		2
3.2	odd	2		550.2.b.e.199.2		2
5.2	odd	4		4950.2.a.bh.1.1		1
5.3	odd	4		4950.2.a.l.1.1		1
5.4	even	2	inner	4950.2.c.i.199.2		2
12.11	even	2		4400.2.b.d.4049.2		2
15.2	even	4		550.2.a.b.1.1	✓	1
15.8	even	4		550.2.a.l.1.1	yes	1
15.14	odd	2		550.2.b.e.199.1		2
60.23	odd	4		4400.2.a.g.1.1		1
60.47	odd	4		4400.2.a.y.1.1		1
60.59	even	2		4400.2.b.d.4049.1		2
165.32	odd	4		6050.2.a.y.1.1		1
165.98	odd	4		6050.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
550.2.a.b.1.1	✓	1	15.2	even	4
550.2.a.l.1.1	yes	1	15.8	even	4
550.2.b.e.199.1		2	15.14	odd	2
550.2.b.e.199.2		2	3.2	odd	2
4400.2.a.g.1.1		1	60.23	odd	4
4400.2.a.y.1.1		1	60.47	odd	4
4400.2.b.d.4049.1		2	60.59	even	2
4400.2.b.d.4049.2		2	12.11	even	2
4950.2.a.l.1.1		1	5.3	odd	4
4950.2.a.bh.1.1		1	5.2	odd	4
4950.2.c.i.199.1		2	1.1	even	1	trivial
4950.2.c.i.199.2		2	5.4	even	2	inner
6050.2.a.p.1.1		1	165.98	odd	4
6050.2.a.y.1.1		1	165.32	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -1.00000 q^{11} -3.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000 q^{19} +1.00000i q^{22} +3.00000i q^{23} -3.00000 q^{26} +5.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -12.0000i q^{37} -1.00000i q^{38} -8.00000 q^{41} +5.00000i q^{43} +1.00000 q^{44} +3.00000 q^{46} +8.00000i q^{47} +7.00000 q^{49} +3.00000i q^{52} -10.0000i q^{53} -5.00000i q^{58} +8.00000 q^{59} +10.0000 q^{61} +3.00000i q^{62} -1.00000 q^{64} +14.0000i q^{67} -4.00000i q^{68} +5.00000 q^{71} +4.00000i q^{73} -12.0000 q^{74} -1.00000 q^{76} +8.00000 q^{79} +8.00000i q^{82} -9.00000i q^{83} +5.00000 q^{86} -1.00000i q^{88} +3.00000 q^{89} -3.00000i q^{92} +8.00000 q^{94} +3.00000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{11} + 2 q^{16} + 2 q^{19} - 6 q^{26} + 10 q^{29} - 6 q^{31} + 8 q^{34} - 16 q^{41} + 2 q^{44} + 6 q^{46} + 14 q^{49} + 16 q^{59} + 20 q^{61} - 2 q^{64} + 10 q^{71} - 24 q^{74} - 2 q^{76}+ \cdots + 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^11 + 2 * q^16 + 2 * q^19 - 6 * q^26 + 10 * q^29 - 6 * q^31 + 8 * q^34 - 16 * q^41 + 2 * q^44 + 6 * q^46 + 14 * q^49 + 16 * q^59 + 20 * q^61 - 2 * q^64 + 10 * q^71 - 24 * q^74 - 2 * q^76 + 16 * q^79 + 10 * q^86 + 6 * q^89 + 16 * q^94

Introduction

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