LMFDB - Embedded newform 4950.2.c.bd.199.3

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 = [6,0,0,-6,0,0,0,0,0,0,-6,0,0,2,0,6,0,0,-6,0,0,0,0,0,0,4] 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:	$6$
Coefficient field:	6.0.270273600.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 66x^{3} - 15x^{2} - 172x + 712 $ x^6 - 2x^5 - 6x^4 + 66x^3 - 15x^2 - 172*x + 712
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$1.36268 - 1.85361i$ of defining polynomial
Character	$\chi$	$=$	4950.199
Dual form			4950.2.c.bd.199.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +4.70722i q^{7} +1.00000i q^{8} -1.00000 q^{11} +2.72536i q^{13} +4.70722 q^{14} +1.00000 q^{16} +2.70722i q^{17} +5.43259 q^{19} +1.00000i q^{22} +7.43259i q^{23} +2.72536 q^{26} -4.70722i q^{28} -5.70722 q^{29} +7.70722 q^{31} -1.00000i q^{32} +2.70722 q^{34} +4.70722i q^{37} -5.43259i q^{38} -12.1579 q^{41} -0.725363i q^{43} +1.00000 q^{44} +7.43259 q^{46} -8.70722i q^{47} -15.1579 q^{49} -2.72536i q^{52} -3.45073i q^{53} -4.70722 q^{56} +5.70722i q^{58} -8.70722 q^{59} +13.4144 q^{61} -7.70722i q^{62} -1.00000 q^{64} +2.00000i q^{67} -2.70722i q^{68} +12.8470 q^{71} -10.8652i q^{73} +4.70722 q^{74} -5.43259 q^{76} -4.70722i q^{77} -5.29278 q^{79} +12.1579i q^{82} -2.25650i q^{83} -0.725363 q^{86} -1.00000i q^{88} -12.6891 q^{89} -12.8289 q^{91} -7.43259i q^{92} -8.70722 q^{94} -0.414446i q^{97} +15.1579i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{11} + 2 q^{14} + 6 q^{16} - 6 q^{19} + 4 q^{26} - 8 q^{29} + 20 q^{31} - 10 q^{34} - 22 q^{41} + 6 q^{44} + 6 q^{46} - 40 q^{49} - 2 q^{56} - 26 q^{59} + 28 q^{61} - 6 q^{64} - 14 q^{71}+ \cdots - 26 q^{94}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^11 + 2 * q^14 + 6 * q^16 - 6 * q^19 + 4 * q^26 - 8 * q^29 + 20 * q^31 - 10 * q^34 - 22 * q^41 + 6 * q^44 + 6 * q^46 - 40 * q^49 - 2 * q^56 - 26 * q^59 + 28 * q^61 - 6 * q^64 - 14 * q^71 + 2 * q^74 + 6 * q^76 - 58 * q^79 + 8 * q^86 - 36 * q^89 + 28 * q^91 - 26 * 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
$7$	4.70722i	1.77916i	0.456776	+	0.889582i	$0.349004\pi$
$7$	4.70722i	1.77916i	−0.456776	+	0.889582i	$0.650996\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.72536i	0.755880i	0.925830	+	0.377940i	$0.123367\pi$
$13$	2.72536i	0.755880i	−0.925830	+	0.377940i	$0.876633\pi$
$14$	4.70722	1.25806
$15$	0	0
$16$	1.00000	0.250000
$17$	2.70722i	0.656598i	0.944574	+	0.328299i	$0.106475\pi$
$17$	2.70722i	0.656598i	−0.944574	+	0.328299i	$0.893525\pi$
$18$	0	0
$19$	5.43259	1.24632	0.623160	−	0.782094i	$-0.285848\pi$
$19$	5.43259	1.24632	0.623160	+	0.782094i	$0.285848\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	0.213201i
$23$	7.43259i	1.54980i	0.632083	+	0.774901i	$0.282200\pi$
$23$	7.43259i	1.54980i	−0.632083	+	0.774901i	$0.717800\pi$
$24$	0	0
$25$	0	0
$26$	2.72536	0.534488
$27$	0	0
$28$	− 4.70722i	− 0.889582i
$29$	−5.70722	−1.05980	−0.529902	−	0.848059i	$-0.677771\pi$
$29$	−5.70722	−1.05980	−0.529902	+	0.848059i	$0.677771\pi$
$30$	0	0
$31$	7.70722	1.38426	0.692129	−	0.721774i	$-0.256673\pi$
$31$	7.70722	1.38426	0.692129	+	0.721774i	$0.256673\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	2.70722	0.464285
$35$	0	0
$36$	0	0
$37$	4.70722i	0.773863i	0.922109	+	0.386931i	$0.126465\pi$
$37$	4.70722i	0.773863i	−0.922109	+	0.386931i	$0.873535\pi$
$38$	− 5.43259i	− 0.881282i
$39$	0	0
$40$	0	0
$41$	−12.1579	−1.89875	−0.949376	−	0.314141i	$-0.898284\pi$
$41$	−12.1579	−1.89875	−0.949376	+	0.314141i	$0.898284\pi$
$42$	0	0
$43$	− 0.725363i	− 0.110617i	−0.998469	−	0.0553084i	$-0.982386\pi$
$43$	− 0.725363i	− 0.110617i	0.998469	−	0.0553084i	$-0.0176142\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	7.43259	1.09588
$47$	− 8.70722i	− 1.27008i	−0.772480	−	0.635040i	$-0.780984\pi$
$47$	− 8.70722i	− 1.27008i	0.772480	−	0.635040i	$-0.219016\pi$
$48$	0	0
$49$	−15.1579	−2.16542
$50$	0	0
$51$	0	0
$52$	− 2.72536i	− 0.377940i
$53$	− 3.45073i	− 0.473994i	−0.971510	−	0.236997i	$-0.923837\pi$
$53$	− 3.45073i	− 0.473994i	0.971510	−	0.236997i	$-0.0761631\pi$
$54$	0	0
$55$	0	0
$56$	−4.70722	−0.629029
$57$	0	0
$58$	5.70722i	0.749395i
$59$	−8.70722	−1.13358	−0.566792	−	0.823861i	$-0.691816\pi$
$59$	−8.70722	−1.13358	−0.566792	+	0.823861i	$0.691816\pi$
$60$	0	0
$61$	13.4144	1.71754	0.858772	−	0.512358i	$-0.171228\pi$
$61$	13.4144	1.71754	0.858772	+	0.512358i	$0.171228\pi$
$62$	− 7.70722i	− 0.978818i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 2.70722i	− 0.328299i
$69$	0	0
$70$	0	0
$71$	12.8470	1.52466	0.762331	−	0.647187i	$-0.224055\pi$
$71$	12.8470	1.52466	0.762331	+	0.647187i	$0.224055\pi$
$72$	0	0
$73$	− 10.8652i	− 1.27167i	−0.771824	−	0.635836i	$-0.780655\pi$
$73$	− 10.8652i	− 1.27167i	0.771824	−	0.635836i	$-0.219345\pi$
$74$	4.70722	0.547204
$75$	0	0
$76$	−5.43259	−0.623160
$77$	− 4.70722i	− 0.536438i
$78$	0	0
$79$	−5.29278	−0.595484	−0.297742	−	0.954646i	$-0.596234\pi$
$79$	−5.29278	−0.595484	−0.297742	+	0.954646i	$0.596234\pi$
$80$	0	0
$81$	0	0
$82$	12.1579i	1.34262i
$83$	− 2.25650i	− 0.247683i	−0.992302	−	0.123841i	$-0.960479\pi$
$83$	− 2.25650i	− 0.247683i	0.992302	−	0.123841i	$-0.0395214\pi$
$84$	0	0
$85$	0	0
$86$	−0.725363	−0.0782179
$87$	0	0
$88$	− 1.00000i	− 0.106600i
$89$	−12.6891	−1.34504	−0.672520	−	0.740079i	$-0.734788\pi$
$89$	−12.6891	−1.34504	−0.672520	+	0.740079i	$0.734788\pi$
$90$	0	0
$91$	−12.8289	−1.34483
$92$	− 7.43259i	− 0.774901i
$93$	0	0
$94$	−8.70722	−0.898081
$95$	0	0
$96$	0	0
$97$	− 0.414446i	− 0.0420806i	−0.999779	−	0.0210403i	$-0.993302\pi$
$97$	− 0.414446i	− 0.0420806i	0.999779	−	0.0210403i	$-0.00669784\pi$
$98$	15.1579i	1.53118i
$99$	0	0
$100$	0	0
$101$	0.450726	0.0448489	0.0224245	−	0.999749i	$-0.492861\pi$
$101$	0.450726	0.0448489	0.0224245	+	0.999749i	$0.492861\pi$
$102$	0	0
$103$	9.70722i	0.956481i	0.878229	+	0.478241i	$0.158725\pi$
$103$	9.70722i	0.956481i	−0.878229	+	0.478241i	$0.841275\pi$
$104$	−2.72536	−0.267244
$105$	0	0
$106$	−3.45073	−0.335164
$107$	14.5724i	1.40877i	0.709820	+	0.704383i	$0.248776\pi$
$107$	14.5724i	1.40877i	−0.709820	+	0.704383i	$0.751224\pi$
$108$	0	0
$109$	−14.6891	−1.40696	−0.703479	−	0.710716i	$-0.748371\pi$
$109$	−14.6891	−1.40696	−0.703479	+	0.710716i	$0.748371\pi$
$110$	0	0
$111$	0	0
$112$	4.70722i	0.444791i
$113$	5.41445i	0.509348i	0.967027	+	0.254674i	$0.0819682\pi$
$113$	5.41445i	0.509348i	−0.967027	+	0.254674i	$0.918032\pi$
$114$	0	0
$115$	0	0
$116$	5.70722	0.529902
$117$	0	0
$118$	8.70722i	0.801565i
$119$	−12.7435	−1.16820
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 13.4144i	− 1.21449i
$123$	0	0
$124$	−7.70722	−0.692129
$125$	0	0
$126$	0	0
$127$	− 12.1217i	− 1.07562i	−0.843065	−	0.537812i	$-0.819251\pi$
$127$	− 12.1217i	− 1.07562i	0.843065	−	0.537812i	$-0.180749\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	8.57240	0.748974	0.374487	−	0.927232i	$-0.377819\pi$
$131$	8.57240	0.748974	0.374487	+	0.927232i	$0.377819\pi$
$132$	0	0
$133$	25.5724i	2.21741i
$134$	2.00000	0.172774
$135$	0	0
$136$	−2.70722	−0.232142
$137$	0.689083i	0.0588723i	0.999567	+	0.0294362i	$0.00937118\pi$
$137$	0.689083i	0.0588723i	−0.999567	+	0.0294362i	$0.990629\pi$
$138$	0	0
$139$	−12.7254	−1.07935	−0.539676	−	0.841873i	$-0.681453\pi$
$139$	−12.7254	−1.07935	−0.539676	+	0.841873i	$0.681453\pi$
$140$	0	0
$141$	0	0
$142$	− 12.8470i	− 1.07810i
$143$	− 2.72536i	− 0.227906i
$144$	0	0
$145$	0	0
$146$	−10.8652	−0.899208
$147$	0	0
$148$	− 4.70722i	− 0.386931i
$149$	22.9868	1.88316	0.941578	−	0.336796i	$-0.109343\pi$
$149$	22.9868	1.88316	0.941578	+	0.336796i	$0.109343\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	5.43259i	0.440641i
$153$	0	0
$154$	−4.70722	−0.379319
$155$	0	0
$156$	0	0
$157$	2.58555i	0.206350i	0.994663	+	0.103175i	$0.0329001\pi$
$157$	2.58555i	0.206350i	−0.994663	+	0.103175i	$0.967100\pi$
$158$	5.29278i	0.421071i
$159$	0	0
$160$	0	0
$161$	−34.9868	−2.75735
$162$	0	0
$163$	7.45073i	0.583586i	0.956482	+	0.291793i	$0.0942518\pi$
$163$	7.45073i	0.583586i	−0.956482	+	0.291793i	$0.905748\pi$
$164$	12.1579	0.949376
$165$	0	0
$166$	−2.25650	−0.175138
$167$	− 14.8652i	− 1.15030i	−0.818047	−	0.575151i	$-0.804943\pi$
$167$	− 14.8652i	− 1.15030i	0.818047	−	0.575151i	$-0.195057\pi$
$168$	0	0
$169$	5.57240	0.428646
$170$	0	0
$171$	0	0
$172$	0.725363i	0.0553084i
$173$	− 12.4507i	− 0.946611i	−0.880898	−	0.473306i	$-0.843061\pi$
$173$	− 12.4507i	− 0.946611i	0.880898	−	0.473306i	$-0.156939\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	12.6891i	0.951087i
$179$	−20.1217	−1.50396	−0.751982	−	0.659184i	$-0.770902\pi$
$179$	−20.1217	−1.50396	−0.751982	+	0.659184i	$0.770902\pi$
$180$	0	0
$181$	−12.1217	−0.900997	−0.450498	−	0.892777i	$-0.648754\pi$
$181$	−12.1217	−0.900997	−0.450498	+	0.892777i	$0.648754\pi$
$182$	12.8289i	0.950941i
$183$	0	0
$184$	−7.43259	−0.547938
$185$	0	0
$186$	0	0
$187$	− 2.70722i	− 0.197972i
$188$	8.70722i	0.635040i
$189$	0	0
$190$	0	0
$191$	−19.4326	−1.40609	−0.703046	−	0.711144i	$-0.748177\pi$
$191$	−19.4326	−1.40609	−0.703046	+	0.711144i	$0.748177\pi$
$192$	0	0
$193$	4.58555i	0.330075i	0.986287	+	0.165038i	$0.0527746\pi$
$193$	4.58555i	0.330075i	−0.986287	+	0.165038i	$0.947225\pi$
$194$	−0.414446	−0.0297555
$195$	0	0
$196$	15.1579	1.08271
$197$	16.3782i	1.16690i	0.812150	+	0.583448i	$0.198297\pi$
$197$	16.3782i	1.16690i	−0.812150	+	0.583448i	$0.801703\pi$
$198$	0	0
$199$	−25.1217	−1.78083	−0.890414	−	0.455152i	$-0.849585\pi$
$199$	−25.1217	−1.78083	−0.890414	+	0.455152i	$0.849585\pi$
$200$	0	0
$201$	0	0
$202$	− 0.450726i	− 0.0317130i
$203$	− 26.8652i	− 1.88557i
$204$	0	0
$205$	0	0
$206$	9.70722	0.676334
$207$	0	0
$208$	2.72536i	0.188970i
$209$	−5.43259	−0.375780
$210$	0	0
$211$	−13.4507	−0.925986	−0.462993	−	0.886362i	$-0.653225\pi$
$211$	−13.4507	−0.925986	−0.462993	+	0.886362i	$0.653225\pi$
$212$	3.45073i	0.236997i
$213$	0	0
$214$	14.5724	0.996148
$215$	0	0
$216$	0	0
$217$	36.2796i	2.46282i
$218$	14.6891i	0.994870i
$219$	0	0
$220$	0	0
$221$	−7.37817	−0.496309
$222$	0	0
$223$	21.4144i	1.43402i	0.697064	+	0.717009i	$0.254489\pi$
$223$	21.4144i	1.43402i	−0.697064	+	0.717009i	$0.745511\pi$
$224$	4.70722	0.314515
$225$	0	0
$226$	5.41445	0.360164
$227$	− 25.0854i	− 1.66498i	−0.554043	−	0.832488i	$-0.686916\pi$
$227$	− 25.0854i	− 1.66498i	0.554043	−	0.832488i	$-0.313084\pi$
$228$	0	0
$229$	−12.6709	−0.837319	−0.418660	−	0.908143i	$-0.637500\pi$
$229$	−12.6709	−0.837319	−0.418660	+	0.908143i	$0.637500\pi$
$230$	0	0
$231$	0	0
$232$	− 5.70722i	− 0.374698i
$233$	0.743503i	0.0487085i	0.999703	+	0.0243543i	$0.00775297\pi$
$233$	0.743503i	0.0487085i	−0.999703	+	0.0243543i	$0.992247\pi$
$234$	0	0
$235$	0	0
$236$	8.70722	0.566792
$237$	0	0
$238$	12.7435i	0.826039i
$239$	1.96372	0.127022	0.0635112	−	0.997981i	$-0.479770\pi$
$239$	1.96372	0.127022	0.0635112	+	0.997981i	$0.479770\pi$
$240$	0	0
$241$	5.45073	0.351112	0.175556	−	0.984469i	$-0.443828\pi$
$241$	5.45073	0.351112	0.175556	+	0.984469i	$0.443828\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	0	0
$244$	−13.4144	−0.858772
$245$	0	0
$246$	0	0
$247$	14.8058i	0.942069i
$248$	7.70722i	0.489409i
$249$	0	0
$250$	0	0
$251$	1.96372	0.123949	0.0619745	−	0.998078i	$-0.480260\pi$
$251$	1.96372	0.123949	0.0619745	+	0.998078i	$0.480260\pi$
$252$	0	0
$253$	− 7.43259i	− 0.467283i
$254$	−12.1217	−0.760581
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.7254i	1.04330i	0.853160	+	0.521650i	$0.174683\pi$
$257$	16.7254i	1.04330i	−0.853160	+	0.521650i	$0.825317\pi$
$258$	0	0
$259$	−22.1579	−1.37683
$260$	0	0
$261$	0	0
$262$	− 8.57240i	− 0.529604i
$263$	− 19.3782i	− 1.19491i	−0.801903	−	0.597454i	$-0.796179\pi$
$263$	− 19.3782i	− 1.19491i	0.801903	−	0.597454i	$-0.203821\pi$
$264$	0	0
$265$	0	0
$266$	25.5724	1.56794
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	13.9637	0.851383	0.425692	−	0.904868i	$-0.360031\pi$
$269$	13.9637	0.851383	0.425692	+	0.904868i	$0.360031\pi$
$270$	0	0
$271$	23.0231	1.39856	0.699278	−	0.714850i	$-0.253505\pi$
$271$	23.0231	1.39856	0.699278	+	0.714850i	$0.253505\pi$
$272$	2.70722i	0.164150i
$273$	0	0
$274$	0.689083	0.0416290
$275$	0	0
$276$	0	0
$277$	− 9.41445i	− 0.565659i	−0.959170	−	0.282830i	$-0.908727\pi$
$277$	− 9.41445i	− 0.565659i	0.959170	−	0.282830i	$-0.0912731\pi$
$278$	12.7254i	0.763217i
$279$	0	0
$280$	0	0
$281$	20.1217	1.20036	0.600179	−	0.799866i	$-0.295096\pi$
$281$	20.1217	1.20036	0.600179	+	0.799866i	$0.295096\pi$
$282$	0	0
$283$	7.29278i	0.433511i	0.976226	+	0.216755i	$0.0695474\pi$
$283$	7.29278i	0.433511i	−0.976226	+	0.216755i	$0.930453\pi$
$284$	−12.8470	−0.762331
$285$	0	0
$286$	−2.72536	−0.161154
$287$	− 57.2302i	− 3.37819i
$288$	0	0
$289$	9.67094	0.568879
$290$	0	0
$291$	0	0
$292$	10.8652i	0.635836i
$293$	10.9868i	0.641858i	0.947103	+	0.320929i	$0.103995\pi$
$293$	10.9868i	0.641858i	−0.947103	+	0.320929i	$0.896005\pi$
$294$	0	0
$295$	0	0
$296$	−4.70722	−0.273602
$297$	0	0
$298$	− 22.9868i	− 1.33159i
$299$	−20.2565	−1.17146
$300$	0	0
$301$	3.41445	0.196805
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	5.43259	0.311580
$305$	0	0
$306$	0	0
$307$	16.2796i	0.929127i	0.885540	+	0.464563i	$0.153789\pi$
$307$	16.2796i	0.929127i	−0.885540	+	0.464563i	$0.846211\pi$
$308$	4.70722i	0.268219i
$309$	0	0
$310$	0	0
$311$	−6.68908	−0.379303	−0.189652	−	0.981851i	$-0.560736\pi$
$311$	−6.68908	−0.379303	−0.189652	+	0.981851i	$0.560736\pi$
$312$	0	0
$313$	2.67094i	0.150971i	0.997147	+	0.0754853i	$0.0240506\pi$
$313$	2.67094i	0.150971i	−0.997147	+	0.0754853i	$0.975949\pi$
$314$	2.58555	0.145911
$315$	0	0
$316$	5.29278	0.297742
$317$	27.4507i	1.54179i	0.636964	+	0.770893i	$0.280190\pi$
$317$	27.4507i	1.54179i	−0.636964	+	0.770893i	$0.719810\pi$
$318$	0	0
$319$	5.70722	0.319543
$320$	0	0
$321$	0	0
$322$	34.9868i	1.94974i
$323$	14.7072i	0.818332i
$324$	0	0
$325$	0	0
$326$	7.45073	0.412658
$327$	0	0
$328$	− 12.1579i	− 0.671310i
$329$	40.9868	2.25968
$330$	0	0
$331$	0.512994	0.0281967	0.0140983	−	0.999901i	$-0.495512\pi$
$331$	0.512994	0.0281967	0.0140983	+	0.999901i	$0.495512\pi$
$332$	2.25650i	0.123841i
$333$	0	0
$334$	−14.8652	−0.813386
$335$	0	0
$336$	0	0
$337$	4.54927i	0.247815i	0.992294	+	0.123907i	$0.0395426\pi$
$337$	4.54927i	0.247815i	−0.992294	+	0.123907i	$0.960457\pi$
$338$	− 5.57240i	− 0.303098i
$339$	0	0
$340$	0	0
$341$	−7.70722	−0.417370
$342$	0	0
$343$	− 38.4013i	− 2.07347i
$344$	0.725363	0.0391090
$345$	0	0
$346$	−12.4507	−0.669355
$347$	28.2433i	1.51618i	0.652149	+	0.758091i	$0.273868\pi$
$347$	28.2433i	1.51618i	−0.652149	+	0.758091i	$0.726132\pi$
$348$	0	0
$349$	1.23836	0.0662877	0.0331439	−	0.999451i	$-0.489448\pi$
$349$	1.23836	0.0662877	0.0331439	+	0.999451i	$0.489448\pi$
$350$	0	0
$351$	0	0
$352$	1.00000i	0.0533002i
$353$	− 1.86019i	− 0.0990080i	−0.998774	−	0.0495040i	$-0.984236\pi$
$353$	− 1.86019i	− 0.0990080i	0.998774	−	0.0495040i	$-0.0157640\pi$
$354$	0	0
$355$	0	0
$356$	12.6891	0.672520
$357$	0	0
$358$	20.1217i	1.06346i
$359$	21.4507	1.13213	0.566063	−	0.824362i	$-0.308466\pi$
$359$	21.4507	1.13213	0.566063	+	0.824362i	$0.308466\pi$
$360$	0	0
$361$	10.5130	0.553315
$362$	12.1217i	0.637101i
$363$	0	0
$364$	12.8289	0.672417
$365$	0	0
$366$	0	0
$367$	26.0231i	1.35840i	0.733955	+	0.679198i	$0.237672\pi$
$367$	26.0231i	1.35840i	−0.733955	+	0.679198i	$0.762328\pi$
$368$	7.43259i	0.387450i
$369$	0	0
$370$	0	0
$371$	16.2433	0.843312
$372$	0	0
$373$	33.4144i	1.73013i	0.501656	+	0.865067i	$0.332724\pi$
$373$	33.4144i	1.73013i	−0.501656	+	0.865067i	$0.667276\pi$
$374$	−2.70722	−0.139987
$375$	0	0
$376$	8.70722	0.449041
$377$	− 15.5543i	− 0.801085i
$378$	0	0
$379$	10.9015	0.559970	0.279985	−	0.960004i	$-0.409671\pi$
$379$	10.9015	0.559970	0.279985	+	0.960004i	$0.409671\pi$
$380$	0	0
$381$	0	0
$382$	19.4326i	0.994258i
$383$	− 26.9687i	− 1.37804i	−0.724744	−	0.689018i	$-0.758042\pi$
$383$	− 26.9687i	− 1.37804i	0.724744	−	0.689018i	$-0.241958\pi$
$384$	0	0
$385$	0	0
$386$	4.58555	0.233399
$387$	0	0
$388$	0.414446i	0.0210403i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−20.1217	−1.01760
$392$	− 15.1579i	− 0.765592i
$393$	0	0
$394$	16.3782	0.825120
$395$	0	0
$396$	0	0
$397$	1.41445i	0.0709890i	0.999370	+	0.0354945i	$0.0113006\pi$
$397$	1.41445i	0.0709890i	−0.999370	+	0.0354945i	$0.988699\pi$
$398$	25.1217i	1.25924i
$399$	0	0
$400$	0	0
$401$	−7.59054	−0.379053	−0.189527	−	0.981876i	$-0.560695\pi$
$401$	−7.59054	−0.379053	−0.189527	+	0.981876i	$0.560695\pi$
$402$	0	0
$403$	21.0050i	1.04633i
$404$	−0.450726	−0.0224245
$405$	0	0
$406$	−26.8652	−1.33330
$407$	− 4.70722i	− 0.233328i
$408$	0	0
$409$	17.6941	0.874915	0.437458	−	0.899239i	$-0.355879\pi$
$409$	17.6941	0.874915	0.437458	+	0.899239i	$0.355879\pi$
$410$	0	0
$411$	0	0
$412$	− 9.70722i	− 0.478241i
$413$	− 40.9868i	− 2.01683i
$414$	0	0
$415$	0	0
$416$	2.72536	0.133622
$417$	0	0
$418$	5.43259i	0.265716i
$419$	−26.1217	−1.27613	−0.638064	−	0.769984i	$-0.720264\pi$
$419$	−26.1217	−1.27613	−0.638064	+	0.769984i	$0.720264\pi$
$420$	0	0
$421$	28.1217	1.37057	0.685283	−	0.728277i	$-0.259678\pi$
$421$	28.1217	1.37057	0.685283	+	0.728277i	$0.259678\pi$
$422$	13.4507i	0.654771i
$423$	0	0
$424$	3.45073	0.167582
$425$	0	0
$426$	0	0
$427$	63.1448i	3.05579i
$428$	− 14.5724i	− 0.704383i
$429$	0	0
$430$	0	0
$431$	−6.31590	−0.304226	−0.152113	−	0.988363i	$-0.548608\pi$
$431$	−6.31590	−0.304226	−0.152113	+	0.988363i	$0.548608\pi$
$432$	0	0
$433$	− 4.25650i	− 0.204554i	−0.994756	−	0.102277i	$-0.967387\pi$
$433$	− 4.25650i	− 0.204554i	0.994756	−	0.102277i	$-0.0326128\pi$
$434$	36.2796	1.74148
$435$	0	0
$436$	14.6891	0.703479
$437$	40.3782i	1.93155i
$438$	0	0
$439$	−14.7435	−0.703669	−0.351835	−	0.936062i	$-0.614442\pi$
$439$	−14.7435	−0.703669	−0.351835	+	0.936062i	$0.614442\pi$
$440$	0	0
$441$	0	0
$442$	7.37817i	0.350944i
$443$	− 30.1579i	− 1.43285i	−0.697665	−	0.716424i	$-0.745778\pi$
$443$	− 30.1579i	− 1.43285i	0.697665	−	0.716424i	$-0.254222\pi$
$444$	0	0
$445$	0	0
$446$	21.4144	1.01400
$447$	0	0
$448$	− 4.70722i	− 0.222395i
$449$	3.55426	0.167736	0.0838678	−	0.996477i	$-0.473273\pi$
$449$	3.55426	0.167736	0.0838678	+	0.996477i	$0.473273\pi$
$450$	0	0
$451$	12.1579	0.572495
$452$	− 5.41445i	− 0.254674i
$453$	0	0
$454$	−25.0854	−1.17732
$455$	0	0
$456$	0	0
$457$	− 22.9015i	− 1.07128i	−0.844445	−	0.535642i	$-0.820070\pi$
$457$	− 22.9015i	− 1.07128i	0.844445	−	0.535642i	$-0.179930\pi$
$458$	12.6709i	0.592074i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 14.0231i	− 0.651710i	−0.945420	−	0.325855i	$-0.894348\pi$
$463$	− 14.0231i	− 0.651710i	0.945420	−	0.325855i	$-0.105652\pi$
$464$	−5.70722	−0.264951
$465$	0	0
$466$	0.743503	0.0344421
$467$	28.5592i	1.32156i	0.750578	+	0.660782i	$0.229775\pi$
$467$	28.5592i	1.32156i	−0.750578	+	0.660782i	$0.770225\pi$
$468$	0	0
$469$	−9.41445	−0.434719
$470$	0	0
$471$	0	0
$472$	− 8.70722i	− 0.400782i
$473$	0.725363i	0.0333522i
$474$	0	0
$475$	0	0
$476$	12.7435	0.584098
$477$	0	0
$478$	− 1.96372i	− 0.0898185i
$479$	−13.4870	−0.616237	−0.308119	−	0.951348i	$-0.599699\pi$
$479$	−13.4870	−0.616237	−0.308119	+	0.951348i	$0.599699\pi$
$480$	0	0
$481$	−12.8289	−0.584947
$482$	− 5.45073i	− 0.248274i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 40.0231i	− 1.81362i	−0.421539	−	0.906810i	$-0.638510\pi$
$487$	− 40.0231i	− 1.81362i	0.421539	−	0.906810i	$-0.361490\pi$
$488$	13.4144i	0.607243i
$489$	0	0
$490$	0	0
$491$	−14.2565	−0.643387	−0.321693	−	0.946844i	$-0.604252\pi$
$491$	−14.2565	−0.643387	−0.321693	+	0.946844i	$0.604252\pi$
$492$	0	0
$493$	− 15.4507i	− 0.695866i
$494$	14.8058	0.666143
$495$	0	0
$496$	7.70722	0.346065
$497$	60.4738i	2.71262i
$498$	0	0
$499$	−16.8652	−0.754989	−0.377494	−	0.926012i	$-0.623214\pi$
$499$	−16.8652	−0.754989	−0.377494	+	0.926012i	$0.623214\pi$
$500$	0	0
$501$	0	0
$502$	− 1.96372i	− 0.0876451i
$503$	20.8652i	0.930332i	0.885223	+	0.465166i	$0.154005\pi$
$503$	20.8652i	0.930332i	−0.885223	+	0.465166i	$0.845995\pi$
$504$	0	0
$505$	0	0
$506$	−7.43259	−0.330419
$507$	0	0
$508$	12.1217i	0.537812i
$509$	−25.6941	−1.13887	−0.569435	−	0.822037i	$-0.692838\pi$
$509$	−25.6941	−1.13887	−0.569435	+	0.822037i	$0.692838\pi$
$510$	0	0
$511$	51.1448	2.26251
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	16.7254	0.737724
$515$	0	0
$516$	0	0
$517$	8.70722i	0.382943i
$518$	22.1579i	0.973564i
$519$	0	0
$520$	0	0
$521$	19.5905	0.858277	0.429138	−	0.903239i	$-0.358817\pi$
$521$	19.5905	0.858277	0.429138	+	0.903239i	$0.358817\pi$
$522$	0	0
$523$	5.43259i	0.237550i	0.992921	+	0.118775i	$0.0378968\pi$
$523$	5.43259i	0.237550i	−0.992921	+	0.118775i	$0.962103\pi$
$524$	−8.57240	−0.374487
$525$	0	0
$526$	−19.3782	−0.844928
$527$	20.8652i	0.908901i
$528$	0	0
$529$	−32.2433	−1.40188
$530$	0	0
$531$	0	0
$532$	− 25.5724i	− 1.10870i
$533$	− 33.1348i	− 1.43523i
$534$	0	0
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	0	0
$538$	− 13.9637i	− 0.602019i
$539$	15.1579	0.652899
$540$	0	0
$541$	−33.5180	−1.44105	−0.720525	−	0.693429i	$-0.756099\pi$
$541$	−33.5180	−1.44105	−0.720525	+	0.693429i	$0.756099\pi$
$542$	− 23.0231i	− 0.988928i
$543$	0	0
$544$	2.70722	0.116071
$545$	0	0
$546$	0	0
$547$	− 36.5411i	− 1.56238i	−0.624291	−	0.781192i	$-0.714612\pi$
$547$	− 36.5411i	− 1.56238i	0.624291	−	0.781192i	$-0.285388\pi$
$548$	− 0.689083i	− 0.0294362i
$549$	0	0
$550$	0	0
$551$	−31.0050	−1.32086
$552$	0	0
$553$	− 24.9143i	− 1.05946i
$554$	−9.41445	−0.399981
$555$	0	0
$556$	12.7254	0.539676
$557$	− 46.8520i	− 1.98518i	−0.121497	−	0.992592i	$-0.538770\pi$
$557$	− 46.8520i	− 1.98518i	0.121497	−	0.992592i	$-0.461230\pi$
$558$	0	0
$559$	1.97688	0.0836130
$560$	0	0
$561$	0	0
$562$	− 20.1217i	− 0.848781i
$563$	28.2433i	1.19031i	0.803609	+	0.595157i	$0.202910\pi$
$563$	28.2433i	1.19031i	−0.803609	+	0.595157i	$0.797090\pi$
$564$	0	0
$565$	0	0
$566$	7.29278	0.306538
$567$	0	0
$568$	12.8470i	0.539050i
$569$	15.6087	0.654350	0.327175	−	0.944964i	$-0.393903\pi$
$569$	15.6087	0.654350	0.327175	+	0.944964i	$0.393903\pi$
$570$	0	0
$571$	9.00498	0.376847	0.188423	−	0.982088i	$-0.439662\pi$
$571$	9.00498	0.376847	0.188423	+	0.982088i	$0.439662\pi$
$572$	2.72536i	0.113953i
$573$	0	0
$574$	−57.2302	−2.38874
$575$	0	0
$576$	0	0
$577$	2.74350i	0.114214i	0.998368	+	0.0571068i	$0.0181875\pi$
$577$	2.74350i	0.114214i	−0.998368	+	0.0571068i	$0.981812\pi$
$578$	− 9.67094i	− 0.402258i
$579$	0	0
$580$	0	0
$581$	10.6218	0.440668
$582$	0	0
$583$	3.45073i	0.142914i
$584$	10.8652	0.449604
$585$	0	0
$586$	10.9868	0.453862
$587$	2.39132i	0.0987005i	0.998782	+	0.0493503i	$0.0157151\pi$
$587$	2.39132i	0.0987005i	−0.998782	+	0.0493503i	$0.984285\pi$
$588$	0	0
$589$	41.8702	1.72523
$590$	0	0
$591$	0	0
$592$	4.70722i	0.193466i
$593$	5.41445i	0.222345i	0.993801	+	0.111172i	$0.0354606\pi$
$593$	5.41445i	0.222345i	−0.993801	+	0.111172i	$0.964539\pi$
$594$	0	0
$595$	0	0
$596$	−22.9868	−0.941578
$597$	0	0
$598$	20.2565i	0.828350i
$599$	27.6087	1.12806	0.564030	−	0.825754i	$-0.309250\pi$
$599$	27.6087	1.12806	0.564030	+	0.825754i	$0.309250\pi$
$600$	0	0
$601$	2.58555	0.105467	0.0527335	−	0.998609i	$-0.483207\pi$
$601$	2.58555	0.105467	0.0527335	+	0.998609i	$0.483207\pi$
$602$	− 3.41445i	− 0.139162i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	− 19.3419i	− 0.785063i	−0.919739	−	0.392531i	$-0.871599\pi$
$607$	− 19.3419i	− 0.785063i	0.919739	−	0.392531i	$-0.128401\pi$
$608$	− 5.43259i	− 0.220320i
$609$	0	0
$610$	0	0
$611$	23.7303	0.960027
$612$	0	0
$613$	− 9.17111i	− 0.370418i	−0.982699	−	0.185209i	$-0.940704\pi$
$613$	− 9.17111i	− 0.370418i	0.982699	−	0.185209i	$-0.0592961\pi$
$614$	16.2796	0.656992
$615$	0	0
$616$	4.70722	0.189659
$617$	35.6268i	1.43428i	0.696928	+	0.717141i	$0.254550\pi$
$617$	35.6268i	1.43428i	−0.696928	+	0.717141i	$0.745450\pi$
$618$	0	0
$619$	48.5955	1.95322	0.976609	−	0.215021i	$-0.0689821\pi$
$619$	48.5955	1.95322	0.976609	+	0.215021i	$0.0689821\pi$
$620$	0	0
$621$	0	0
$622$	6.68908i	0.268208i
$623$	− 59.7303i	− 2.39305i
$624$	0	0
$625$	0	0
$626$	2.67094	0.106752
$627$	0	0
$628$	− 2.58555i	− 0.103175i
$629$	−12.7435	−0.508117
$630$	0	0
$631$	−39.1448	−1.55833	−0.779165	−	0.626819i	$-0.784356\pi$
$631$	−39.1448	−1.55833	−0.779165	+	0.626819i	$0.784356\pi$
$632$	− 5.29278i	− 0.210535i
$633$	0	0
$634$	27.4507	1.09021
$635$	0	0
$636$	0	0
$637$	− 41.3109i	− 1.63680i
$638$	− 5.70722i	− 0.225951i
$639$	0	0
$640$	0	0
$641$	27.5543	1.08833	0.544164	−	0.838979i	$-0.316847\pi$
$641$	27.5543	1.08833	0.544164	+	0.838979i	$0.316847\pi$
$642$	0	0
$643$	17.6941i	0.697786i	0.937163	+	0.348893i	$0.113442\pi$
$643$	17.6941i	0.697786i	−0.937163	+	0.348893i	$0.886558\pi$
$644$	34.9868	1.37867
$645$	0	0
$646$	14.7072	0.578648
$647$	− 3.60868i	− 0.141872i	−0.997481	−	0.0709358i	$-0.977401\pi$
$647$	− 3.60868i	− 0.141872i	0.997481	−	0.0709358i	$-0.0225986\pi$
$648$	0	0
$649$	8.70722	0.341788
$650$	0	0
$651$	0	0
$652$	− 7.45073i	− 0.291793i
$653$	20.2796i	0.793603i	0.917904	+	0.396801i	$0.129880\pi$
$653$	20.2796i	0.793603i	−0.917904	+	0.396801i	$0.870120\pi$
$654$	0	0
$655$	0	0
$656$	−12.1579	−0.474688
$657$	0	0
$658$	− 40.9868i	− 1.59783i
$659$	−13.9868	−0.544850	−0.272425	−	0.962177i	$-0.587826\pi$
$659$	−13.9868	−0.544850	−0.272425	+	0.962177i	$0.587826\pi$
$660$	0	0
$661$	−15.2565	−0.593409	−0.296704	−	0.954969i	$-0.595888\pi$
$661$	−15.2565	−0.593409	−0.296704	+	0.954969i	$0.595888\pi$
$662$	− 0.512994i	− 0.0199381i
$663$	0	0
$664$	2.25650	0.0875691
$665$	0	0
$666$	0	0
$667$	− 42.4194i	− 1.64249i
$668$	14.8652i	0.575151i
$669$	0	0
$670$	0	0
$671$	−13.4144	−0.517859
$672$	0	0
$673$	21.7303i	0.837643i	0.908069	+	0.418822i	$0.137557\pi$
$673$	21.7303i	0.837643i	−0.908069	+	0.418822i	$0.862443\pi$
$674$	4.54927	0.175231
$675$	0	0
$676$	−5.57240	−0.214323
$677$	10.8058i	0.415299i	0.978203	+	0.207650i	$0.0665814\pi$
$677$	10.8058i	0.415299i	−0.978203	+	0.207650i	$0.933419\pi$
$678$	0	0
$679$	1.95089	0.0748683
$680$	0	0
$681$	0	0
$682$	7.70722i	0.295125i
$683$	− 7.05940i	− 0.270121i	−0.990837	−	0.135060i	$-0.956877\pi$
$683$	− 7.05940i	− 0.270121i	0.990837	−	0.135060i	$-0.0431228\pi$
$684$	0	0
$685$	0	0
$686$	−38.4013	−1.46617
$687$	0	0
$688$	− 0.725363i	− 0.0276542i
$689$	9.40448	0.358282
$690$	0	0
$691$	0.305935	0.0116383	0.00581916	−	0.999983i	$-0.498148\pi$
$691$	0.305935	0.0116383	0.00581916	+	0.999983i	$0.498148\pi$
$692$	12.4507i	0.473306i
$693$	0	0
$694$	28.2433	1.07210
$695$	0	0
$696$	0	0
$697$	− 32.9143i	− 1.24672i
$698$	− 1.23836i	− 0.0468725i
$699$	0	0
$700$	0	0
$701$	24.7666	0.935423	0.467711	−	0.883881i	$-0.345079\pi$
$701$	24.7666	0.935423	0.467711	+	0.883881i	$0.345079\pi$
$702$	0	0
$703$	25.5724i	0.964481i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−1.86019	−0.0700092
$707$	2.12167i	0.0797936i
$708$	0	0
$709$	−48.7172	−1.82961	−0.914806	−	0.403893i	$-0.867657\pi$
$709$	−48.7172	−1.82961	−0.914806	+	0.403893i	$0.867657\pi$
$710$	0	0
$711$	0	0
$712$	− 12.6891i	− 0.475543i
$713$	57.2846i	2.14533i
$714$	0	0
$715$	0	0
$716$	20.1217	0.751982
$717$	0	0
$718$	− 21.4507i	− 0.800534i
$719$	−14.5493	−0.542596	−0.271298	−	0.962495i	$-0.587453\pi$
$719$	−14.5493	−0.542596	−0.271298	+	0.962495i	$0.587453\pi$
$720$	0	0
$721$	−45.6941	−1.70174
$722$	− 10.5130i	− 0.391253i
$723$	0	0
$724$	12.1217	0.450498
$725$	0	0
$726$	0	0
$727$	− 16.8783i	− 0.625983i	−0.949756	−	0.312991i	$-0.898669\pi$
$727$	− 16.8783i	− 0.625983i	0.949756	−	0.312991i	$-0.101331\pi$
$728$	− 12.8289i	− 0.475470i
$729$	0	0
$730$	0	0
$731$	1.96372	0.0726308
$732$	0	0
$733$	− 25.6268i	− 0.946548i	−0.880915	−	0.473274i	$-0.843072\pi$
$733$	− 25.6268i	− 0.946548i	0.880915	−	0.473274i	$-0.156928\pi$
$734$	26.0231	0.960531
$735$	0	0
$736$	7.43259	0.273969
$737$	− 2.00000i	− 0.0736709i
$738$	0	0
$739$	11.2202	0.412742	0.206371	−	0.978474i	$-0.433835\pi$
$739$	11.2202	0.412742	0.206371	+	0.978474i	$0.433835\pi$
$740$	0	0
$741$	0	0
$742$	− 16.2433i	− 0.596312i
$743$	− 13.4870i	− 0.494790i	−0.968915	−	0.247395i	$-0.920425\pi$
$743$	− 13.4870i	− 0.494790i	0.968915	−	0.247395i	$-0.0795746\pi$
$744$	0	0
$745$	0	0
$746$	33.4144	1.22339
$747$	0	0
$748$	2.70722i	0.0989859i
$749$	−68.5955	−2.50643
$750$	0	0
$751$	−22.6087	−0.825002	−0.412501	−	0.910957i	$-0.635345\pi$
$751$	−22.6087	−0.825002	−0.412501	+	0.910957i	$0.635345\pi$
$752$	− 8.70722i	− 0.317520i
$753$	0	0
$754$	−15.5543	−0.566453
$755$	0	0
$756$	0	0
$757$	27.3782i	0.995076i	0.867442	+	0.497538i	$0.165763\pi$
$757$	27.3782i	0.995076i	−0.867442	+	0.497538i	$0.834237\pi$
$758$	− 10.9015i	− 0.395959i
$759$	0	0
$760$	0	0
$761$	10.5130	0.381096	0.190548	−	0.981678i	$-0.438974\pi$
$761$	10.5130	0.381096	0.190548	+	0.981678i	$0.438974\pi$
$762$	0	0
$763$	− 69.1448i	− 2.50321i
$764$	19.4326	0.703046
$765$	0	0
$766$	−26.9687	−0.974419
$767$	− 23.7303i	− 0.856853i
$768$	0	0
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	0	0
$772$	− 4.58555i	− 0.165038i
$773$	− 11.4144i	− 0.410549i	−0.978704	−	0.205275i	$-0.934191\pi$
$773$	− 11.4144i	− 0.410549i	0.978704	−	0.205275i	$-0.0658087\pi$
$774$	0	0
$775$	0	0
$776$	0.414446	0.0148778
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−66.0491	−2.36645
$780$	0	0
$781$	−12.8470	−0.459703
$782$	20.1217i	0.719549i
$783$	0	0
$784$	−15.1579	−0.541355
$785$	0	0
$786$	0	0
$787$	5.02312i	0.179055i	0.995984	+	0.0895275i	$0.0285357\pi$
$787$	5.02312i	0.179055i	−0.995984	+	0.0895275i	$0.971464\pi$
$788$	− 16.3782i	− 0.583448i
$789$	0	0
$790$	0	0
$791$	−25.4870	−0.906214
$792$	0	0
$793$	36.5592i	1.29826i
$794$	1.41445	0.0501968
$795$	0	0
$796$	25.1217	0.890414
$797$	− 26.8652i	− 0.951613i	−0.879550	−	0.475807i	$-0.842156\pi$
$797$	− 26.8652i	− 0.951613i	0.879550	−	0.475807i	$-0.157844\pi$
$798$	0	0
$799$	23.5724	0.833931
$800$	0	0
$801$	0	0
$802$	7.59054i	0.268031i
$803$	10.8652i	0.383424i
$804$	0	0
$805$	0	0
$806$	21.0050	0.739869
$807$	0	0
$808$	0.450726i	0.0158565i
$809$	9.87833	0.347304	0.173652	−	0.984807i	$-0.444443\pi$
$809$	9.87833	0.347304	0.173652	+	0.984807i	$0.444443\pi$
$810$	0	0
$811$	55.7795	1.95868	0.979341	−	0.202217i	$-0.0648146\pi$
$811$	55.7795	1.95868	0.979341	+	0.202217i	$0.0648146\pi$
$812$	26.8652i	0.942783i
$813$	0	0
$814$	−4.70722	−0.164988
$815$	0	0
$816$	0	0
$817$	− 3.94060i	− 0.137864i
$818$	− 17.6941i	− 0.618658i
$819$	0	0
$820$	0	0
$821$	22.8520	0.797541	0.398770	−	0.917051i	$-0.369437\pi$
$821$	22.8520	0.797541	0.398770	+	0.917051i	$0.369437\pi$
$822$	0	0
$823$	32.2433i	1.12393i	0.827160	+	0.561966i	$0.189955\pi$
$823$	32.2433i	1.12393i	−0.827160	+	0.561966i	$0.810045\pi$
$824$	−9.70722	−0.338167
$825$	0	0
$826$	−40.9868	−1.42611
$827$	1.08539i	0.0377427i	0.999822	+	0.0188713i	$0.00600729\pi$
$827$	1.08539i	0.0377427i	−0.999822	+	0.0188713i	$0.993993\pi$
$828$	0	0
$829$	−44.5230	−1.54635	−0.773173	−	0.634195i	$-0.781332\pi$
$829$	−44.5230	−1.54635	−0.773173	+	0.634195i	$0.781332\pi$
$830$	0	0
$831$	0	0
$832$	− 2.72536i	− 0.0944850i
$833$	− 41.0360i	− 1.42181i
$834$	0	0
$835$	0	0
$836$	5.43259	0.187890
$837$	0	0
$838$	26.1217i	0.902358i
$839$	31.1085	1.07399	0.536993	−	0.843587i	$-0.319560\pi$
$839$	31.1085	1.07399	0.536993	+	0.843587i	$0.319560\pi$
$840$	0	0
$841$	3.57240	0.123186
$842$	− 28.1217i	− 0.969137i
$843$	0	0
$844$	13.4507	0.462993
$845$	0	0
$846$	0	0
$847$	4.70722i	0.161742i
$848$	− 3.45073i	− 0.118498i
$849$	0	0
$850$	0	0
$851$	−34.9868	−1.19933
$852$	0	0
$853$	− 26.9015i	− 0.921088i	−0.887637	−	0.460544i	$-0.847654\pi$
$853$	− 26.9015i	− 0.921088i	0.887637	−	0.460544i	$-0.152346\pi$
$854$	63.1448	2.16077
$855$	0	0
$856$	−14.5724	−0.498074
$857$	1.32906i	0.0453997i	0.999742	+	0.0226999i	$0.00722621\pi$
$857$	1.32906i	0.0453997i	−0.999742	+	0.0226999i	$0.992774\pi$
$858$	0	0
$859$	−12.8289	−0.437716	−0.218858	−	0.975757i	$-0.570233\pi$
$859$	−12.8289	−0.437716	−0.218858	+	0.975757i	$0.570233\pi$
$860$	0	0
$861$	0	0
$862$	6.31590i	0.215120i
$863$	11.0413i	0.375849i	0.982183	+	0.187924i	$0.0601760\pi$
$863$	11.0413i	0.375849i	−0.982183	+	0.187924i	$0.939824\pi$
$864$	0	0
$865$	0	0
$866$	−4.25650	−0.144642
$867$	0	0
$868$	− 36.2796i	− 1.23141i
$869$	5.29278	0.179545
$870$	0	0
$871$	−5.45073	−0.184691
$872$	− 14.6891i	− 0.497435i
$873$	0	0
$874$	40.3782	1.36581
$875$	0	0
$876$	0	0
$877$	38.6891i	1.30644i	0.757169	+	0.653219i	$0.226582\pi$
$877$	38.6891i	1.30644i	−0.757169	+	0.653219i	$0.773418\pi$
$878$	14.7435i	0.497569i
$879$	0	0
$880$	0	0
$881$	43.2746	1.45796	0.728980	−	0.684535i	$-0.239995\pi$
$881$	43.2746	1.45796	0.728980	+	0.684535i	$0.239995\pi$
$882$	0	0
$883$	− 29.1811i	− 0.982021i	−0.871154	−	0.491011i	$-0.836628\pi$
$883$	− 29.1811i	− 0.982021i	0.871154	−	0.491011i	$-0.163372\pi$
$884$	7.37817	0.248155
$885$	0	0
$886$	−30.1579	−1.01318
$887$	− 19.1711i	− 0.643703i	−0.946790	−	0.321851i	$-0.895695\pi$
$887$	− 19.1711i	− 0.643703i	0.946790	−	0.321851i	$-0.104305\pi$
$888$	0	0
$889$	57.0594	1.91371
$890$	0	0
$891$	0	0
$892$	− 21.4144i	− 0.717009i
$893$	− 47.3027i	− 1.58293i
$894$	0	0
$895$	0	0
$896$	−4.70722	−0.157257
$897$	0	0
$898$	− 3.55426i	− 0.118607i
$899$	−43.9868	−1.46704
$900$	0	0
$901$	9.34189	0.311223
$902$	− 12.1579i	− 0.404815i
$903$	0	0
$904$	−5.41445	−0.180082
$905$	0	0
$906$	0	0
$907$	24.8289i	0.824430i	0.911087	+	0.412215i	$0.135245\pi$
$907$	24.8289i	0.824430i	−0.911087	+	0.412215i	$0.864755\pi$
$908$	25.0854i	0.832488i
$909$	0	0
$910$	0	0
$911$	3.60868	0.119561	0.0597804	−	0.998212i	$-0.480960\pi$
$911$	3.60868	0.119561	0.0597804	+	0.998212i	$0.480960\pi$
$912$	0	0
$913$	2.25650i	0.0746791i
$914$	−22.9015	−0.757513
$915$	0	0
$916$	12.6709	0.418660
$917$	40.3522i	1.33255i
$918$	0	0
$919$	24.5984	0.811426	0.405713	−	0.914001i	$-0.367023\pi$
$919$	24.5984	0.811426	0.405713	+	0.914001i	$0.367023\pi$
$920$	0	0
$921$	0	0
$922$	18.0000i	0.592798i
$923$	35.0128i	1.15246i
$924$	0	0
$925$	0	0
$926$	−14.0231	−0.460828
$927$	0	0
$928$	5.70722i	0.187349i
$929$	28.7254	0.942449	0.471224	−	0.882013i	$-0.343812\pi$
$929$	28.7254	0.942449	0.471224	+	0.882013i	$0.343812\pi$
$930$	0	0
$931$	−82.3469	−2.69881
$932$	− 0.743503i	− 0.0243543i
$933$	0	0
$934$	28.5592	0.934487
$935$	0	0
$936$	0	0
$937$	− 54.8652i	− 1.79237i	−0.443684	−	0.896184i	$-0.646329\pi$
$937$	− 54.8652i	− 1.79237i	0.443684	−	0.896184i	$-0.353671\pi$
$938$	9.41445i	0.307393i
$939$	0	0
$940$	0	0
$941$	−1.64496	−0.0536240	−0.0268120	−	0.999640i	$-0.508536\pi$
$941$	−1.64496	−0.0536240	−0.0268120	+	0.999640i	$0.508536\pi$
$942$	0	0
$943$	− 90.3650i	− 2.94269i
$944$	−8.70722	−0.283396
$945$	0	0
$946$	0.725363	0.0235836
$947$	12.6347i	0.410571i	0.978702	+	0.205286i	$0.0658124\pi$
$947$	12.6347i	0.410571i	−0.978702	+	0.205286i	$0.934188\pi$
$948$	0	0
$949$	29.6115	0.961231
$950$	0	0
$951$	0	0
$952$	− 12.7435i	− 0.413019i
$953$	− 26.1217i	− 0.846164i	−0.906091	−	0.423082i	$-0.860948\pi$
$953$	− 26.1217i	− 0.846164i	0.906091	−	0.423082i	$-0.139052\pi$
$954$	0	0
$955$	0	0
$956$	−1.96372	−0.0635112
$957$	0	0
$958$	13.4870i	0.435745i
$959$	−3.24367	−0.104743
$960$	0	0
$961$	28.4013	0.916171
$962$	12.8289i	0.413620i
$963$	0	0
$964$	−5.45073	−0.175556
$965$	0	0
$966$	0	0
$967$	− 55.7666i	− 1.79333i	−0.442706	−	0.896667i	$-0.645981\pi$
$967$	− 55.7666i	− 1.79333i	0.442706	−	0.896667i	$-0.354019\pi$
$968$	1.00000i	0.0321412i
$969$	0	0
$970$	0	0
$971$	−49.8520	−1.59983	−0.799914	−	0.600115i	$-0.795121\pi$
$971$	−49.8520	−1.59983	−0.799914	+	0.600115i	$0.795121\pi$
$972$	0	0
$973$	− 59.9011i	− 1.92034i
$974$	−40.0231	−1.28242
$975$	0	0
$976$	13.4144	0.429386
$977$	52.2433i	1.67141i	0.549177	+	0.835706i	$0.314941\pi$
$977$	52.2433i	1.67141i	−0.549177	+	0.835706i	$0.685059\pi$
$978$	0	0
$979$	12.6891	0.405545
$980$	0	0
$981$	0	0
$982$	14.2565i	0.454943i
$983$	3.18925i	0.101721i	0.998706	+	0.0508606i	$0.0161964\pi$
$983$	3.18925i	0.101721i	−0.998706	+	0.0508606i	$0.983804\pi$
$984$	0	0
$985$	0	0
$986$	−15.4507	−0.492051
$987$	0	0
$988$	− 14.8058i	− 0.471034i
$989$	5.39132	0.171434
$990$	0	0
$991$	7.14479	0.226962	0.113481	−	0.993540i	$-0.463800\pi$
$991$	7.14479	0.226962	0.113481	+	0.993540i	$0.463800\pi$
$992$	− 7.70722i	− 0.244705i
$993$	0	0
$994$	60.4738	1.91811
$995$	0	0
$996$	0	0
$997$	43.1448i	1.36641i	0.730227	+	0.683205i	$0.239414\pi$
$997$	43.1448i	1.36641i	−0.730227	+	0.683205i	$0.760586\pi$
$998$	16.8652i	0.533858i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.c.bd.199.3		6
3.2	odd	2		4950.2.c.be.199.6		6
5.2	odd	4		4950.2.a.ci.1.1	yes	3
5.3	odd	4		4950.2.a.ch.1.3	yes	3
5.4	even	2	inner	4950.2.c.bd.199.4		6
15.2	even	4		4950.2.a.cg.1.1	✓	3
15.8	even	4		4950.2.a.cj.1.3	yes	3
15.14	odd	2		4950.2.c.be.199.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4950.2.a.cg.1.1	✓	3	15.2	even	4
4950.2.a.ch.1.3	yes	3	5.3	odd	4
4950.2.a.ci.1.1	yes	3	5.2	odd	4
4950.2.a.cj.1.3	yes	3	15.8	even	4
4950.2.c.bd.199.3		6	1.1	even	1	trivial
4950.2.c.bd.199.4		6	5.4	even	2	inner
4950.2.c.be.199.1		6	15.14	odd	2
4950.2.c.be.199.6		6	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +4.70722i q^{7} +1.00000i q^{8} -1.00000 q^{11} +2.72536i q^{13} +4.70722 q^{14} +1.00000 q^{16} +2.70722i q^{17} +5.43259 q^{19} +1.00000i q^{22} +7.43259i q^{23} +2.72536 q^{26} -4.70722i q^{28} -5.70722 q^{29} +7.70722 q^{31} -1.00000i q^{32} +2.70722 q^{34} +4.70722i q^{37} -5.43259i q^{38} -12.1579 q^{41} -0.725363i q^{43} +1.00000 q^{44} +7.43259 q^{46} -8.70722i q^{47} -15.1579 q^{49} -2.72536i q^{52} -3.45073i q^{53} -4.70722 q^{56} +5.70722i q^{58} -8.70722 q^{59} +13.4144 q^{61} -7.70722i q^{62} -1.00000 q^{64} +2.00000i q^{67} -2.70722i q^{68} +12.8470 q^{71} -10.8652i q^{73} +4.70722 q^{74} -5.43259 q^{76} -4.70722i q^{77} -5.29278 q^{79} +12.1579i q^{82} -2.25650i q^{83} -0.725363 q^{86} -1.00000i q^{88} -12.6891 q^{89} -12.8289 q^{91} -7.43259i q^{92} -8.70722 q^{94} -0.414446i q^{97} +15.1579i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{11} + 2 q^{14} + 6 q^{16} - 6 q^{19} + 4 q^{26} - 8 q^{29} + 20 q^{31} - 10 q^{34} - 22 q^{41} + 6 q^{44} + 6 q^{46} - 40 q^{49} - 2 q^{56} - 26 q^{59} + 28 q^{61} - 6 q^{64} - 14 q^{71}+ \cdots - 26 q^{94}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^11 + 2 * q^14 + 6 * q^16 - 6 * q^19 + 4 * q^26 - 8 * q^29 + 20 * q^31 - 10 * q^34 - 22 * q^41 + 6 * q^44 + 6 * q^46 - 40 * q^49 - 2 * q^56 - 26 * q^59 + 28 * q^61 - 6 * q^64 - 14 * q^71 + 2 * q^74 + 6 * q^76 - 58 * q^79 + 8 * q^86 - 36 * q^89 + 28 * q^91 - 26 * q^94

Introduction

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