LMFDB - Embedded newform 4950.2.c.p.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(199,4950)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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 66)
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.p.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{11} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -4.00000 q^{19} -1.00000i q^{22} +4.00000i q^{23} +6.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} -1.00000i q^{32} -2.00000 q^{34} +6.00000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} -9.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} +4.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} -14.0000 q^{61} -1.00000 q^{64} +4.00000i q^{67} +2.00000i q^{68} +12.0000 q^{71} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000i q^{77} +4.00000 q^{79} -6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +10.0000 q^{89} +24.0000 q^{91} -4.00000i q^{92} +12.0000 q^{94} -14.0000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{11} - 8 q^{14} + 2 q^{16} - 8 q^{19} + 12 q^{26} + 12 q^{29} - 4 q^{34} + 12 q^{41} - 2 q^{44} + 8 q^{46} - 18 q^{49} + 8 q^{56} + 24 q^{59} - 28 q^{61} - 2 q^{64} + 24 q^{71} + 12 q^{74} + 8 q^{76} + 8 q^{79} - 8 q^{86} + 20 q^{89} + 48 q^{91} + 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^11 - 8 * q^14 + 2 * q^16 - 8 * q^19 + 12 * q^26 + 12 * q^29 - 4 * q^34 + 12 * q^41 - 2 * q^44 + 8 * q^46 - 18 * q^49 + 8 * q^56 + 24 * q^59 - 28 * q^61 - 2 * q^64 + 24 * q^71 + 12 * q^74 + 8 * q^76 + 8 * q^79 - 8 * q^86 + 20 * q^89 + 48 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

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
$7$	− 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
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	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
$52$	− 6.00000i	− 0.832050i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	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
$116$	−6.00000	−0.557086
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0000i	1.26750i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	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
$141$	0	0
$142$	− 12.0000i	− 1.00702i
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	− 24.0000i	− 1.77900i
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	14.0000i	0.985037i
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	6.00000i	0.416025i
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 2.00000i	− 0.137361i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−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
$232$	6.00000i	0.393919i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	8.00000i	0.518563i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	− 24.0000i	− 1.52708i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	− 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
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	6.00000	0.354787
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 6.00000i	− 0.351123i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	10.0000i	0.579284i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 4.00000i	− 0.230174i
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 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
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	− 16.0000i	− 0.891645i
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	6.00000i	0.331295i
$329$	48.0000	2.64633
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	− 4.00000i	− 0.219529i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	23.0000i	1.25104i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000i	0.431959i
$344$	4.00000	0.215666
$345$	0	0
$346$	−10.0000	−0.537603
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	− 1.00000i	− 0.0533002i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	− 20.0000i	− 1.05703i
$359$	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
$364$	−24.0000	−1.25794
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	4.00000i	0.208514i
$369$	0	0
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	36.0000i	1.85409i
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	14.0000i	0.710742i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−24.0000	−1.19110
$407$	6.00000i	0.297409i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 48.0000i	− 2.36193i
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	4.00000i	0.195646i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	0	0
$427$	56.0000i	2.71003i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	28.0000i	1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$	28.0000i	1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	4.00000i	0.188982i
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	0	0
$478$	− 8.00000i	− 0.365911i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 14.0000i	− 0.633750i
$489$	0	0
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	−24.0000	−1.07981
$495$	0	0
$496$	0	0
$497$	− 48.0000i	− 2.15309i
$498$	0	0
$499$	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
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	32.0000i	1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$	32.0000i	1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	0	0
$505$	0	0
$506$	4.00000	0.177822
$507$	0	0
$508$	− 12.0000i	− 0.532414i
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	− 24.0000i	− 1.05450i
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 16.0000i	− 0.693688i
$533$	36.0000i	1.55933i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	− 26.0000i	− 1.12094i
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	− 20.0000i	− 0.859074i
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	2.00000i	0.0854358i
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	26.0000	1.10463
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	− 22.0000i	− 0.928014i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	12.0000i	0.503509i
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	− 46.0000i	− 1.91501i	−0.288425	−	0.957503i	$-0.593132\pi$
$577$	− 46.0000i	− 1.91501i	0.288425	−	0.957503i	$-0.406868\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	2.00000i	0.0828315i
$584$	−6.00000	−0.248282
$585$	0	0
$586$	22.0000	0.908812
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	6.00000i	0.246598i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	24.0000i	0.981433i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000i	0.652111i
$603$	0	0
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	−72.0000	−2.91281
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	4.00000	0.161165
$617$	22.0000i	0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$	22.0000i	0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$	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
$622$	4.00000i	0.160385i
$623$	− 40.0000i	− 1.60257i
$624$	0	0
$625$	0	0
$626$	−26.0000	−1.03917
$627$	0	0
$628$	10.0000i	0.399043i
$629$	12.0000	0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 54.0000i	− 2.13956i
$638$	− 6.00000i	− 0.237542i
$639$	0	0
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	8.00000	0.314756
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	− 48.0000i	− 1.87123i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	− 20.0000i	− 0.777322i
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	− 46.0000i	− 1.76792i	−0.467559	−	0.883962i	$-0.654866\pi$
$677$	− 46.0000i	− 1.76792i	0.467559	−	0.883962i	$-0.345134\pi$
$678$	0	0
$679$	−56.0000	−2.14908
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	10.0000i	0.380143i
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	18.0000	0.677439
$707$	56.0000i	2.10610i
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	0	0
$718$	0	0
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	24.0000i	0.889499i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	4.00000	0.147442
$737$	4.00000i	0.147342i
$738$	0	0
$739$	−44.0000	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$740$	0	0
$741$	0	0
$742$	− 8.00000i	− 0.293689i
$743$	− 32.0000i	− 1.17397i	−0.809599	−	0.586983i	$-0.800316\pi$
$743$	− 32.0000i	− 1.17397i	0.809599	−	0.586983i	$-0.199684\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	−16.0000	−0.584627
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 28.0000i	− 1.01701i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	− 24.0000i	− 0.868858i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	72.0000i	2.59977i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	30.0000i	1.07555i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	12.0000	0.429394
$782$	− 8.00000i	− 0.286079i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	− 20.0000i	− 0.712923i	−0.934310	−	0.356462i	$-0.883983\pi$
$787$	− 20.0000i	− 0.712923i	0.934310	−	0.356462i	$-0.116017\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	0	0
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	− 84.0000i	− 2.98293i
$794$	22.0000	0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	54.0000i	1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$	54.0000i	1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	− 38.0000i	− 1.34183i
$803$	6.00000i	0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 14.0000i	− 0.492518i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	24.0000i	0.842235i
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	52.0000i	1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$	52.0000i	1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	− 6.00000i	− 0.208013i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	12.0000i	0.414533i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000i	0.344623i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	2.00000i	0.0686803i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	6.00000i	0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$	6.00000i	0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	56.0000	1.91628
$855$	0	0
$856$	4.00000	0.136717
$857$	− 26.0000i	− 0.888143i	−0.895991	−	0.444072i	$-0.853534\pi$
$857$	− 26.0000i	− 0.888143i	0.895991	−	0.444072i	$-0.146466\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.0000i	0.680808i	0.940279	+	0.340404i	$0.110564\pi$
$863$	20.0000i	0.680808i	−0.940279	+	0.340404i	$0.889436\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−24.0000	−0.813209
$872$	6.00000i	0.203186i
$873$	0	0
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	42.0000i	1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$	42.0000i	1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$	4.00000i	0.134993i
$879$	0	0
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	28.0000	0.940678
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	22.0000i	0.734150i
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	− 6.00000i	− 0.199778i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	4.00000i	0.132381i
$914$	34.0000	1.12462
$915$	0	0
$916$	14.0000	0.462573
$917$	16.0000i	0.528367i
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	14.0000i	0.461065i
$923$	72.0000i	2.36991i
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	− 10.0000i	− 0.327561i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	12.0000	0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	0	0
$951$	0	0
$952$	− 8.00000i	− 0.259281i
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	36.0000i	1.16069i
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−14.0000	−0.448129
$977$	− 26.0000i	− 0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$	− 26.0000i	− 0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	− 28.0000i	− 0.893516i
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	24.0000i	0.763542i
$989$	16.0000	0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	0	0
$994$	−48.0000	−1.52247
$995$	0	0
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.p.199.1		2
3.2	odd	2		1650.2.c.e.199.2		2
5.2	odd	4		4950.2.a.bu.1.1		1
5.3	odd	4		198.2.a.a.1.1		1
5.4	even	2	inner	4950.2.c.p.199.2		2
15.2	even	4		1650.2.a.k.1.1		1
15.8	even	4		66.2.a.b.1.1	✓	1
15.14	odd	2		1650.2.c.e.199.1		2
20.3	even	4		1584.2.a.f.1.1		1
35.13	even	4		9702.2.a.x.1.1		1
40.3	even	4		6336.2.a.cj.1.1		1
40.13	odd	4		6336.2.a.bw.1.1		1
45.13	odd	12		1782.2.e.v.1189.1		2
45.23	even	12		1782.2.e.e.1189.1		2
45.38	even	12		1782.2.e.e.595.1		2
45.43	odd	12		1782.2.e.v.595.1		2
55.43	even	4		2178.2.a.g.1.1		1
60.23	odd	4		528.2.a.j.1.1		1
105.83	odd	4		3234.2.a.t.1.1		1
120.53	even	4		2112.2.a.r.1.1		1
120.83	odd	4		2112.2.a.e.1.1		1
165.8	odd	20		726.2.e.o.493.1		4
165.38	even	20		726.2.e.g.487.1		4
165.53	even	20		726.2.e.g.565.1		4
165.68	odd	20		726.2.e.o.565.1		4
165.83	odd	20		726.2.e.o.487.1		4
165.98	odd	4		726.2.a.c.1.1		1
165.113	even	20		726.2.e.g.493.1		4
165.128	odd	20		726.2.e.o.511.1		4
165.158	even	20		726.2.e.g.511.1		4
660.263	even	4		5808.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	15.8	even	4
198.2.a.a.1.1		1	5.3	odd	4
528.2.a.j.1.1		1	60.23	odd	4
726.2.a.c.1.1		1	165.98	odd	4
726.2.e.g.487.1		4	165.38	even	20
726.2.e.g.493.1		4	165.113	even	20
726.2.e.g.511.1		4	165.158	even	20
726.2.e.g.565.1		4	165.53	even	20
726.2.e.o.487.1		4	165.83	odd	20
726.2.e.o.493.1		4	165.8	odd	20
726.2.e.o.511.1		4	165.128	odd	20
726.2.e.o.565.1		4	165.68	odd	20
1584.2.a.f.1.1		1	20.3	even	4
1650.2.a.k.1.1		1	15.2	even	4
1650.2.c.e.199.1		2	15.14	odd	2
1650.2.c.e.199.2		2	3.2	odd	2
1782.2.e.e.595.1		2	45.38	even	12
1782.2.e.e.1189.1		2	45.23	even	12
1782.2.e.v.595.1		2	45.43	odd	12
1782.2.e.v.1189.1		2	45.13	odd	12
2112.2.a.e.1.1		1	120.83	odd	4
2112.2.a.r.1.1		1	120.53	even	4
2178.2.a.g.1.1		1	55.43	even	4
3234.2.a.t.1.1		1	105.83	odd	4
4950.2.a.bu.1.1		1	5.2	odd	4
4950.2.c.p.199.1		2	1.1	even	1	trivial
4950.2.c.p.199.2		2	5.4	even	2	inner
5808.2.a.bc.1.1		1	660.263	even	4
6336.2.a.bw.1.1		1	40.13	odd	4
6336.2.a.cj.1.1		1	40.3	even	4
9702.2.a.x.1.1		1	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{11} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -4.00000 q^{19} -1.00000i q^{22} +4.00000i q^{23} +6.00000 q^{26} +4.00000i q^{28} +6.00000 q^{29} -1.00000i q^{32} -2.00000 q^{34} +6.00000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} -9.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} +4.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} -14.0000 q^{61} -1.00000 q^{64} +4.00000i q^{67} +2.00000i q^{68} +12.0000 q^{71} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000i q^{77} +4.00000 q^{79} -6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +10.0000 q^{89} +24.0000 q^{91} -4.00000i q^{92} +12.0000 q^{94} -14.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{11} - 8 q^{14} + 2 q^{16} - 8 q^{19} + 12 q^{26} + 12 q^{29} - 4 q^{34} + 12 q^{41} - 2 q^{44} + 8 q^{46} - 18 q^{49} + 8 q^{56} + 24 q^{59} - 28 q^{61} - 2 q^{64} + 24 q^{71} + 12 q^{74} + 8 q^{76} + 8 q^{79} - 8 q^{86} + 20 q^{89} + 48 q^{91} + 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^11 - 8 * q^14 + 2 * q^16 - 8 * q^19 + 12 * q^26 + 12 * q^29 - 4 * q^34 + 12 * q^41 - 2 * q^44 + 8 * q^46 - 18 * q^49 + 8 * q^56 + 24 * q^59 - 28 * q^61 - 2 * q^64 + 24 * q^71 + 12 * q^74 + 8 * q^76 + 8 * q^79 - 8 * q^86 + 20 * q^89 + 48 * q^91 + 24 * q^94

Introduction

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