LMFDB - Embedded newform 4950.2.f.c.4949.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(4949,4950)]

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([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4950.4949");

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.f (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:	$8$
Coefficient field:	8.0.4328587264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 109x^{4} + 260x^{2} + 196 $ x^8 + 18x^6 + 109x^4 + 260*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 990)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4949.1
Root			$-1.18398i$ of defining polynomial
Character	$\chi$	$=$	4950.4949
Dual form			4950.2.f.c.4949.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.08861 q^{7} +1.00000i q^{8} +(0.130093 - 3.31407i) q^{11} +3.78217 q^{13} +3.08861i q^{14} +1.00000 q^{16} +0.828427i q^{17} +2.50283i q^{19} +(-3.31407 - 0.130093i) q^{22} -2.69356 q^{23} -3.78217i q^{26} +3.08861 q^{28} -4.36796 q^{29} -2.36796 q^{31} -1.00000i q^{32} +0.828427 q^{34} -2.72065i q^{37} +2.50283 q^{38} +10.2850 q^{41} -0.325600 q^{43} +(-0.130093 + 3.31407i) q^{44} +2.69356i q^{46} -5.67440 q^{47} +2.53953 q^{49} -3.78217 q^{52} -12.5452 q^{53} -3.08861i q^{56} +4.36796i q^{58} -0.720655i q^{59} -3.98245i q^{61} +2.36796i q^{62} -1.00000 q^{64} +9.52603i q^{67} -0.828427i q^{68} +2.96330i q^{71} +0.367959 q^{73} -2.72065 q^{74} -2.50283i q^{76} +(-0.401807 + 10.2359i) q^{77} +9.19639i q^{79} -10.2850i q^{82} -2.11732i q^{83} +0.325600i q^{86} +(3.31407 + 0.130093i) q^{88} -15.1558i q^{89} -11.6817 q^{91} +2.69356 q^{92} +5.67440i q^{94} +8.23713i q^{97} -2.53953i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{13} + 8 q^{16} - 12 q^{22} - 8 q^{23} - 8 q^{29} + 8 q^{31} - 16 q^{34} - 16 q^{38} + 8 q^{41} - 16 q^{43} - 32 q^{47} + 16 q^{49} + 8 q^{52} - 24 q^{53} - 8 q^{64} - 24 q^{73} - 24 q^{74}+ \cdots + 8 q^{92}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^13 + 8 * q^16 - 12 * q^22 - 8 * q^23 - 8 * q^29 + 8 * q^31 - 16 * q^34 - 16 * q^38 + 8 * q^41 - 16 * q^43 - 32 * q^47 + 16 * q^49 + 8 * q^52 - 24 * q^53 - 8 * q^64 - 24 * q^73 - 24 * q^74 - 28 * q^77 + 12 * q^88 + 24 * q^91 + 8 * q^92

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
$7$	−3.08861			−1.16739			−0.583693	−	0.811974i	$-0.698393\pi$
$7$	−3.08861			−1.16739			−0.583693	+	0.811974i	$0.698393\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$		0			0
$11$	0.130093	−	3.31407i	0.0392245	−	0.999230i
$11$	0.130093	−	3.31407i	0.0392245	−	0.999230i
$12$		0			0
$13$	3.78217			1.04899			0.524493	−	0.851415i	$-0.324255\pi$
$13$	3.78217			1.04899			0.524493	+	0.851415i	$0.324255\pi$
$14$			3.08861i			0.825467i
$15$		0			0
$16$	1.00000			0.250000
$17$			0.828427i			0.200923i	0.994941	+	0.100462i	$0.0320319\pi$
$17$			0.828427i			0.200923i	−0.994941	+	0.100462i	$0.967968\pi$
$18$		0			0
$19$			2.50283i			0.574188i	0.957902	+	0.287094i	$0.0926892\pi$
$19$			2.50283i			0.574188i	−0.957902	+	0.287094i	$0.907311\pi$
$20$		0			0
$21$		0			0
$22$	−3.31407	−	0.130093i	−0.706563	−	0.0277359i
$23$	−2.69356			−0.561646			−0.280823	−	0.959760i	$-0.590607\pi$
$23$	−2.69356			−0.561646			−0.280823	+	0.959760i	$0.590607\pi$
$24$		0			0
$25$		0			0
$26$		−	3.78217i		−	0.741745i
$27$		0			0
$28$	3.08861			0.583693
$29$	−4.36796			−0.811110			−0.405555	−	0.914071i	$-0.632922\pi$
$29$	−4.36796			−0.811110			−0.405555	+	0.914071i	$0.632922\pi$
$30$		0			0
$31$	−2.36796			−0.425298			−0.212649	−	0.977129i	$-0.568209\pi$
$31$	−2.36796			−0.425298			−0.212649	+	0.977129i	$0.568209\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	0.828427			0.142074
$35$		0			0
$36$		0			0
$37$		−	2.72065i		−	0.447273i	−0.974673	−	0.223636i	$-0.928207\pi$
$37$		−	2.72065i		−	0.447273i	0.974673	−	0.223636i	$-0.0717928\pi$
$38$	2.50283			0.406012
$39$		0			0
$40$		0			0
$41$	10.2850			1.60625			0.803123	−	0.595813i	$-0.203170\pi$
$41$	10.2850			1.60625			0.803123	+	0.595813i	$0.203170\pi$
$42$		0			0
$43$	−0.325600			−0.0496536			−0.0248268	−	0.999692i	$-0.507903\pi$
$43$	−0.325600			−0.0496536			−0.0248268	+	0.999692i	$0.507903\pi$
$44$	−0.130093	+	3.31407i	−0.0196123	+	0.499615i
$45$		0			0
$46$			2.69356i			0.397144i
$47$	−5.67440			−0.827696			−0.413848	−	0.910346i	$-0.635816\pi$
$47$	−5.67440			−0.827696			−0.413848	+	0.910346i	$0.635816\pi$
$48$		0			0
$49$	2.53953			0.362790
$50$		0			0
$51$		0			0
$52$	−3.78217			−0.524493
$53$	−12.5452			−1.72321			−0.861607	−	0.507576i	$-0.830542\pi$
$53$	−12.5452			−1.72321			−0.861607	+	0.507576i	$0.830542\pi$
$54$		0			0
$55$		0			0
$56$		−	3.08861i		−	0.412733i
$57$		0			0
$58$			4.36796i			0.573541i
$59$		−	0.720655i		−	0.0938212i	−0.998899	−	0.0469106i	$-0.985062\pi$
$59$		−	0.720655i		−	0.0938212i	0.998899	−	0.0469106i	$-0.0149376\pi$
$60$		0			0
$61$		−	3.98245i		−	0.509901i	−0.966954	−	0.254951i	$-0.917941\pi$
$61$		−	3.98245i		−	0.509901i	0.966954	−	0.254951i	$-0.0820592\pi$
$62$			2.36796i			0.300731i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$			9.52603i			1.16379i	0.813264	+	0.581895i	$0.197688\pi$
$67$			9.52603i			1.16379i	−0.813264	+	0.581895i	$0.802312\pi$
$68$		−	0.828427i		−	0.100462i
$69$		0			0
$70$		0			0
$71$			2.96330i			0.351678i	0.984419	+	0.175839i	$0.0562639\pi$
$71$			2.96330i			0.351678i	−0.984419	+	0.175839i	$0.943736\pi$
$72$		0			0
$73$	0.367959			0.0430663			0.0215332	−	0.999768i	$-0.493145\pi$
$73$	0.367959			0.0430663			0.0215332	+	0.999768i	$0.493145\pi$
$74$	−2.72065			−0.316270
$75$		0			0
$76$		−	2.50283i		−	0.287094i
$77$	−0.401807	+	10.2359i	−0.0457902	+	1.16649i
$78$		0			0
$79$			9.19639i			1.03467i	0.855782	+	0.517337i	$0.173077\pi$
$79$			9.19639i			1.03467i	−0.855782	+	0.517337i	$0.826923\pi$
$80$		0			0
$81$		0			0
$82$		−	10.2850i		−	1.13579i
$83$		−	2.11732i		−	0.232406i	−0.993225	−	0.116203i	$-0.962928\pi$
$83$		−	2.11732i		−	0.232406i	0.993225	−	0.116203i	$-0.0370724\pi$
$84$		0			0
$85$		0			0
$86$			0.325600i			0.0351104i
$87$		0			0
$88$	3.31407	+	0.130093i	0.353281	+	0.0138680i
$89$		−	15.1558i		−	1.60651i	−0.595635	−	0.803255i	$-0.703100\pi$
$89$		−	15.1558i		−	1.60651i	0.595635	−	0.803255i	$-0.296900\pi$
$90$		0			0
$91$	−11.6817			−1.22457
$92$	2.69356			0.280823
$93$		0			0
$94$			5.67440i			0.585270i
$95$		0			0
$96$		0			0
$97$			8.23713i			0.836354i	0.908366	+	0.418177i	$0.137331\pi$
$97$			8.23713i			0.836354i	−0.908366	+	0.418177i	$0.862669\pi$
$98$		−	2.53953i		−	0.256531i
$99$		0			0
$100$		0			0
$101$	8.91314			0.886891			0.443445	−	0.896301i	$-0.353756\pi$
$101$	8.91314			0.886891			0.443445	+	0.896301i	$0.353756\pi$
$102$		0			0
$103$			10.4023i			1.02497i	0.858696	+	0.512486i	$0.171275\pi$
$103$			10.4023i			1.02497i	−0.858696	+	0.512486i	$0.828725\pi$
$104$			3.78217i			0.370873i
$105$		0			0
$106$			12.5452i			1.21850i
$107$			6.69760i			0.647481i	0.946146	+	0.323741i	$0.104941\pi$
$107$			6.69760i			0.647481i	−0.946146	+	0.323741i	$0.895059\pi$
$108$		0			0
$109$		−	11.3313i		−	1.08534i	−0.839947	−	0.542669i	$-0.817414\pi$
$109$		−	11.3313i		−	1.08534i	0.839947	−	0.542669i	$-0.182586\pi$
$110$		0			0
$111$		0			0
$112$	−3.08861			−0.291847
$113$	−5.04801			−0.474877			−0.237439	−	0.971403i	$-0.576308\pi$
$113$	−5.04801			−0.474877			−0.237439	+	0.971403i	$0.576308\pi$
$114$		0			0
$115$		0			0
$116$	4.36796			0.405555
$117$		0			0
$118$	−0.720655			−0.0663416
$119$		−	2.55869i		−	0.234555i
$120$		0			0
$121$	−10.9662	−	0.862276i	−0.996923	−	0.0783887i
$122$	−3.98245			−0.360554
$123$		0			0
$124$	2.36796			0.212649
$125$		0			0
$126$		0			0
$127$	−12.2602			−1.08792			−0.543958	−	0.839113i	$-0.683075\pi$
$127$	−12.2602			−1.08792			−0.543958	+	0.839113i	$0.683075\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	12.7703			1.11574			0.557872	−	0.829927i	$-0.311618\pi$
$131$	12.7703			1.11574			0.557872	+	0.829927i	$0.311618\pi$
$132$		0			0
$133$		−	7.73026i		−	0.670299i
$134$	9.52603			0.822923
$135$		0			0
$136$	−0.828427			−0.0710370
$137$	7.35607			0.628471			0.314236	−	0.949345i	$-0.398252\pi$
$137$	7.35607			0.628471			0.314236	+	0.949345i	$0.398252\pi$
$138$		0			0
$139$		−	2.13487i		−	0.181077i	−0.995893	−	0.0905386i	$-0.971141\pi$
$139$		−	2.13487i		−	0.181077i	0.995893	−	0.0905386i	$-0.0288588\pi$
$140$		0			0
$141$		0			0
$142$	2.96330			0.248674
$143$	0.492034	−	12.5344i	0.0411460	−	1.04818i
$144$		0			0
$145$		0			0
$146$		−	0.367959i		−	0.0304525i
$147$		0			0
$148$			2.72065i			0.223636i
$149$	22.0496			1.80638			0.903188	−	0.429245i	$-0.141220\pi$
$149$	22.0496			1.80638			0.903188	+	0.429245i	$0.141220\pi$
$150$		0			0
$151$			16.7958i			1.36682i	0.730033	+	0.683412i	$0.239505\pi$
$151$			16.7958i			1.36682i	−0.730033	+	0.683412i	$0.760495\pi$
$152$	−2.50283			−0.203006
$153$		0			0
$154$	10.2359	+	0.401807i	0.824831	+	0.0323785i
$155$		0			0
$156$		0			0
$157$			22.1191i			1.76529i	0.470036	+	0.882647i	$0.344241\pi$
$157$			22.1191i			1.76529i	−0.470036	+	0.882647i	$0.655759\pi$
$158$	9.19639			0.731625
$159$		0			0
$160$		0			0
$161$	8.31936			0.655658
$162$		0			0
$163$			25.3059i			1.98211i	0.133446	+	0.991056i	$0.457396\pi$
$163$			25.3059i			1.98211i	−0.133446	+	0.991056i	$0.542604\pi$
$164$	−10.2850			−0.803123
$165$		0			0
$166$	−2.11732			−0.164336
$167$			14.1095i			1.09183i	0.837841	+	0.545914i	$0.183818\pi$
$167$			14.1095i			1.09183i	−0.837841	+	0.545914i	$0.816182\pi$
$168$		0			0
$169$	1.30483			0.100371
$170$		0			0
$171$		0			0
$172$	0.325600			0.0248268
$173$			16.2020i			1.23182i	0.787817	+	0.615909i	$0.211211\pi$
$173$			16.2020i			1.23182i	−0.787817	+	0.615909i	$0.788789\pi$
$174$		0			0
$175$		0			0
$176$	0.130093	−	3.31407i	0.00980613	−	0.249808i
$177$		0			0
$178$	−15.1558			−1.13597
$179$			7.02871i			0.525350i	0.964884	+	0.262675i	$0.0846048\pi$
$179$			7.02871i			0.525350i	−0.964884	+	0.262675i	$0.915395\pi$
$180$		0			0
$181$	8.64341			0.642459			0.321230	−	0.947001i	$-0.395904\pi$
$181$	8.64341			0.642459			0.321230	+	0.947001i	$0.395904\pi$
$182$			11.6817i			0.865903i
$183$		0			0
$184$		−	2.69356i		−	0.198572i
$185$		0			0
$186$		0			0
$187$	2.74547	+	0.107773i	0.200768	+	0.00788111i
$188$	5.67440			0.413848
$189$		0			0
$190$		0			0
$191$			15.6145i			1.12983i	0.825151	+	0.564913i	$0.191090\pi$
$191$			15.6145i			1.12983i	−0.825151	+	0.564913i	$0.808910\pi$
$192$		0			0
$193$	5.67036			0.408161			0.204081	−	0.978954i	$-0.434579\pi$
$193$	5.67036			0.408161			0.204081	+	0.978954i	$0.434579\pi$
$194$	8.23713			0.591391
$195$		0			0
$196$	−2.53953			−0.181395
$197$			20.6299i			1.46982i	0.678165	+	0.734910i	$0.262776\pi$
$197$			20.6299i			1.46982i	−0.678165	+	0.734910i	$0.737224\pi$
$198$		0			0
$199$	−15.4335			−1.09405			−0.547026	−	0.837115i	$-0.684240\pi$
$199$	−15.4335			−1.09405			−0.547026	+	0.837115i	$0.684240\pi$
$200$		0			0
$201$		0			0
$202$		−	8.91314i		−	0.627127i
$203$	13.4909			0.946878
$204$		0			0
$205$		0			0
$206$	10.4023			0.724764
$207$		0			0
$208$	3.78217			0.262246
$209$	8.29455	+	0.325600i	0.573746	+	0.0225222i
$210$		0			0
$211$			24.8821i			1.71295i	0.516185	+	0.856477i	$0.327352\pi$
$211$			24.8821i			1.71295i	−0.516185	+	0.856477i	$0.672648\pi$
$212$	12.5452			0.861607
$213$		0			0
$214$	6.69760			0.457838
$215$		0			0
$216$		0			0
$217$	7.31371			0.496487
$218$	−11.3313			−0.767449
$219$		0			0
$220$		0			0
$221$			3.13325i			0.210765i
$222$		0			0
$223$			9.05601i			0.606435i	0.952921	+	0.303218i	$0.0980609\pi$
$223$			9.05601i			0.606435i	−0.952921	+	0.303218i	$0.901939\pi$
$224$			3.08861i			0.206367i
$225$		0			0
$226$			5.04801i			0.335789i
$227$		−	17.1365i		−	1.13739i	−0.822549	−	0.568694i	$-0.807449\pi$
$227$		−	17.1365i		−	1.13739i	0.822549	−	0.568694i	$-0.192551\pi$
$228$		0			0
$229$	−25.6682			−1.69620			−0.848100	−	0.529836i	$-0.822253\pi$
$229$	−25.6682			−1.69620			−0.848100	+	0.529836i	$0.822253\pi$
$230$		0			0
$231$		0			0
$232$		−	4.36796i		−	0.286771i
$233$		−	16.1772i		−	1.05981i	−0.848058	−	0.529903i	$-0.822228\pi$
$233$		−	16.1772i		−	1.05981i	0.848058	−	0.529903i	$-0.177772\pi$
$234$		0			0
$235$		0			0
$236$			0.720655i			0.0469106i
$237$		0			0
$238$	−2.55869			−0.165855
$239$	15.6842			1.01452			0.507262	−	0.861792i	$-0.330658\pi$
$239$	15.6842			1.01452			0.507262	+	0.861792i	$0.330658\pi$
$240$		0			0
$241$			20.9730i			1.35099i	0.737363	+	0.675496i	$0.236071\pi$
$241$			20.9730i			1.35099i	−0.737363	+	0.675496i	$0.763929\pi$
$242$	−0.862276	+	10.9662i	−0.0554292	+	0.704931i
$243$		0			0
$244$			3.98245i			0.254951i
$245$		0			0
$246$		0			0
$247$			9.46612i			0.602315i
$248$		−	2.36796i		−	0.150366i
$249$		0			0
$250$		0			0
$251$		−	20.8550i		−	1.31636i	−0.752862	−	0.658178i	$-0.771327\pi$
$251$		−	20.8550i		−	1.31636i	0.752862	−	0.658178i	$-0.228673\pi$
$252$		0			0
$253$	−0.350413	+	8.92665i	−0.0220303	+	0.561214i
$254$			12.2602i			0.769272i
$255$		0			0
$256$	1.00000			0.0625000
$257$	8.77828			0.547574			0.273787	−	0.961790i	$-0.411724\pi$
$257$	8.77828			0.547574			0.273787	+	0.961790i	$0.411724\pi$
$258$		0			0
$259$			8.40305i			0.522140i
$260$		0			0
$261$		0			0
$262$		−	12.7703i		−	0.788951i
$263$			6.98084i			0.430457i	0.976564	+	0.215229i	$0.0690496\pi$
$263$			6.98084i			0.430457i	−0.976564	+	0.215229i	$0.930950\pi$
$264$		0			0
$265$		0			0
$266$	−7.73026			−0.473973
$267$		0			0
$268$		−	9.52603i		−	0.581895i
$269$			1.74936i			0.106661i	0.998577	+	0.0533303i	$0.0169836\pi$
$269$			1.74936i			0.106661i	−0.998577	+	0.0533303i	$0.983016\pi$
$270$		0			0
$271$			20.6760i			1.25598i	0.778222	+	0.627989i	$0.216122\pi$
$271$			20.6760i			1.25598i	−0.778222	+	0.627989i	$0.783878\pi$
$272$			0.828427i			0.0502308i
$273$		0			0
$274$		−	7.35607i		−	0.444396i
$275$		0			0
$276$		0			0
$277$	−28.7878			−1.72969			−0.864846	−	0.502037i	$-0.832584\pi$
$277$	−28.7878			−1.72969			−0.864846	+	0.502037i	$0.832584\pi$
$278$	−2.13487			−0.128041
$279$		0			0
$280$		0			0
$281$	−5.67991			−0.338835			−0.169418	−	0.985544i	$-0.554189\pi$
$281$	−5.67991			−0.338835			−0.169418	+	0.985544i	$0.554189\pi$
$282$		0			0
$283$	−32.8956			−1.95544			−0.977720	−	0.209912i	$-0.932682\pi$
$283$	−32.8956			−1.95544			−0.977720	+	0.209912i	$0.932682\pi$
$284$		−	2.96330i		−	0.175839i
$285$		0			0
$286$	−12.5344	−	0.492034i	−0.741174	−	0.0290946i
$287$	−31.7664			−1.87511
$288$		0			0
$289$	16.3137			0.959630
$290$		0			0
$291$		0			0
$292$	−0.367959			−0.0215332
$293$			4.88585i			0.285434i	0.989764	+	0.142717i	$0.0455839\pi$
$293$			4.88585i			0.285434i	−0.989764	+	0.142717i	$0.954416\pi$
$294$		0			0
$295$		0			0
$296$	2.72065			0.158135
$297$		0			0
$298$		−	22.0496i		−	1.27730i
$299$	−10.1875			−0.589159
$300$		0			0
$301$	1.00565			0.0579649
$302$	16.7958			0.966491
$303$		0			0
$304$			2.50283i			0.143547i
$305$		0			0
$306$		0			0
$307$	−10.3720			−0.591961			−0.295981	−	0.955194i	$-0.595646\pi$
$307$	−10.3720			−0.591961			−0.295981	+	0.955194i	$0.595646\pi$
$308$	0.401807	−	10.2359i	0.0228951	−	0.583244i
$309$		0			0
$310$		0			0
$311$		−	33.4408i		−	1.89625i	−0.317893	−	0.948126i	$-0.602975\pi$
$311$		−	33.4408i		−	1.89625i	0.317893	−	0.948126i	$-0.397025\pi$
$312$		0			0
$313$		−	32.5023i		−	1.83714i	−0.395260	−	0.918569i	$-0.629345\pi$
$313$		−	32.5023i		−	1.83714i	0.395260	−	0.918569i	$-0.370655\pi$
$314$	22.1191			1.24825
$315$		0			0
$316$		−	9.19639i		−	0.517337i
$317$	−15.7416			−0.884135			−0.442067	−	0.896982i	$-0.645755\pi$
$317$	−15.7416			−0.884135			−0.442067	+	0.896982i	$0.645755\pi$
$318$		0			0
$319$	−0.568241	+	14.4757i	−0.0318154	+	0.810485i
$320$		0			0
$321$		0			0
$322$		−	8.31936i		−	0.463620i
$323$	−2.07341			−0.115368
$324$		0			0
$325$		0			0
$326$	25.3059			1.40157
$327$		0			0
$328$			10.2850i			0.567894i
$329$	17.5260			0.966241
$330$		0			0
$331$	27.8015			1.52811			0.764054	−	0.645153i	$-0.223206\pi$
$331$	27.8015			1.52811			0.764054	+	0.645153i	$0.223206\pi$
$332$			2.11732i			0.116203i
$333$		0			0
$334$	14.1095			0.772039
$335$		0			0
$336$		0			0
$337$	4.67279			0.254543			0.127271	−	0.991868i	$-0.459378\pi$
$337$	4.67279			0.254543			0.127271	+	0.991868i	$0.459378\pi$
$338$		−	1.30483i		−	0.0709732i
$339$		0			0
$340$		0			0
$341$	−0.308055	+	7.84759i	−0.0166821	+	0.424971i
$342$		0			0
$343$	13.7767			0.743870
$344$		−	0.325600i		−	0.0175552i
$345$		0			0
$346$	16.2020			0.871027
$347$			33.3849i			1.79220i	0.443856	+	0.896098i	$0.353610\pi$
$347$			33.3849i			1.79220i	−0.443856	+	0.896098i	$0.646390\pi$
$348$		0			0
$349$			5.48938i			0.293840i	0.989148	+	0.146920i	$0.0469360\pi$
$349$			5.48938i			0.293840i	−0.989148	+	0.146920i	$0.953064\pi$
$350$		0			0
$351$		0			0
$352$	−3.31407	−	0.130093i	−0.176641	−	0.00693398i
$353$	−12.4078			−0.660402			−0.330201	−	0.943911i	$-0.607117\pi$
$353$	−12.4078			−0.660402			−0.330201	+	0.943911i	$0.607117\pi$
$354$		0			0
$355$		0			0
$356$			15.1558i			0.803255i
$357$		0			0
$358$	7.02871			0.371479
$359$	−26.9322			−1.42143			−0.710715	−	0.703480i	$-0.751628\pi$
$359$	−26.9322			−1.42143			−0.710715	+	0.703480i	$0.751628\pi$
$360$		0			0
$361$	12.7359			0.670308
$362$		−	8.64341i		−	0.454287i
$363$		0			0
$364$	11.6817			0.612286
$365$		0			0
$366$		0			0
$367$		−	10.7590i		−	0.561614i	−0.959764	−	0.280807i	$-0.909398\pi$
$367$		−	10.7590i		−	0.561614i	0.959764	−	0.280807i	$-0.0906021\pi$
$368$	−2.69356			−0.140411
$369$		0			0
$370$		0			0
$371$	38.7472			2.01166
$372$		0			0
$373$	10.3600			0.536419			0.268209	−	0.963361i	$-0.413568\pi$
$373$	10.3600			0.536419			0.268209	+	0.963361i	$0.413568\pi$
$374$	0.107773	−	2.74547i	0.00557279	−	0.141965i
$375$		0			0
$376$		−	5.67440i		−	0.292635i
$377$	−16.5204			−0.850842
$378$		0			0
$379$	−20.8206			−1.06948			−0.534742	−	0.845015i	$-0.679591\pi$
$379$	−20.8206			−1.06948			−0.534742	+	0.845015i	$0.679591\pi$
$380$		0			0
$381$		0			0
$382$	15.6145			0.798907
$383$	−15.6010			−0.797173			−0.398587	−	0.917131i	$-0.630499\pi$
$383$	−15.6010			−0.797173			−0.398587	+	0.917131i	$0.630499\pi$
$384$		0			0
$385$		0			0
$386$		−	5.67036i		−	0.288614i
$387$		0			0
$388$		−	8.23713i		−	0.418177i
$389$		−	24.3928i		−	1.23676i	−0.785878	−	0.618381i	$-0.787789\pi$
$389$		−	24.3928i		−	1.23676i	0.785878	−	0.618381i	$-0.212211\pi$
$390$		0			0
$391$		−	2.23142i		−	0.112848i
$392$			2.53953i			0.128266i
$393$		0			0
$394$	20.6299			1.03932
$395$		0			0
$396$		0			0
$397$		−	24.5931i		−	1.23429i	−0.786849	−	0.617145i	$-0.788289\pi$
$397$		−	24.5931i		−	1.23429i	0.786849	−	0.617145i	$-0.211711\pi$
$398$			15.4335i			0.773612i
$399$		0			0
$400$		0			0
$401$			19.9459i			0.996050i	0.867163	+	0.498025i	$0.165941\pi$
$401$			19.9459i			0.996050i	−0.867163	+	0.498025i	$0.834059\pi$
$402$		0			0
$403$	−8.95603			−0.446132
$404$	−8.91314			−0.443445
$405$		0			0
$406$		−	13.4909i		−	0.669544i
$407$	−9.01645	−	0.353938i	−0.446929	−	0.0175441i
$408$		0			0
$409$			22.9322i			1.13393i	0.823743	+	0.566963i	$0.191882\pi$
$409$			22.9322i			1.13393i	−0.823743	+	0.566963i	$0.808118\pi$
$410$		0			0
$411$		0			0
$412$		−	10.4023i		−	0.512486i
$413$			2.22582i			0.109526i
$414$		0			0
$415$		0			0
$416$		−	3.78217i		−	0.185436i
$417$		0			0
$418$	0.325600	−	8.29455i	0.0159256	−	0.405700i
$419$			10.4622i			0.511113i	0.966794	+	0.255557i	$0.0822587\pi$
$419$			10.4622i			0.511113i	−0.966794	+	0.255557i	$0.917741\pi$
$420$		0			0
$421$	4.04640			0.197209			0.0986047	−	0.995127i	$-0.468562\pi$
$421$	4.04640			0.197209			0.0986047	+	0.995127i	$0.468562\pi$
$422$	24.8821			1.21124
$423$		0			0
$424$		−	12.5452i		−	0.609248i
$425$		0			0
$426$		0			0
$427$			12.3003i			0.595251i
$428$		−	6.69760i		−	0.323741i
$429$		0			0
$430$		0			0
$431$	−25.9266			−1.24884			−0.624420	−	0.781089i	$-0.714665\pi$
$431$	−25.9266			−1.24884			−0.624420	+	0.781089i	$0.714665\pi$
$432$		0			0
$433$		−	2.10953i		−	0.101378i	−0.998714	−	0.0506888i	$-0.983858\pi$
$433$		−	2.10953i		−	0.101378i	0.998714	−	0.0506888i	$-0.0161417\pi$
$434$		−	7.31371i		−	0.351069i
$435$		0			0
$436$			11.3313i			0.542669i
$437$		−	6.74151i		−	0.322490i
$438$		0			0
$439$			18.5374i			0.884741i	0.896832	+	0.442371i	$0.145862\pi$
$439$			18.5374i			0.884741i	−0.896832	+	0.442371i	$0.854138\pi$
$440$		0			0
$441$		0			0
$442$	3.13325			0.149034
$443$	12.3545			0.586978			0.293489	−	0.955962i	$-0.405184\pi$
$443$	12.3545			0.586978			0.293489	+	0.955962i	$0.405184\pi$
$444$		0			0
$445$		0			0
$446$	9.05601			0.428814
$447$		0			0
$448$	3.08861			0.145923
$449$		−	13.8038i		−	0.651440i	−0.945466	−	0.325720i	$-0.894393\pi$
$449$		−	13.8038i		−	0.651440i	0.945466	−	0.325720i	$-0.105607\pi$
$450$		0			0
$451$	1.33801	−	34.0852i	0.0630043	−	1.60501i
$452$	5.04801			0.237439
$453$		0			0
$454$	−17.1365			−0.804255
$455$		0			0
$456$		0			0
$457$	−19.1964			−0.897969			−0.448985	−	0.893540i	$-0.648214\pi$
$457$	−19.1964			−0.897969			−0.448985	+	0.893540i	$0.648214\pi$
$458$			25.6682i			1.19939i
$459$		0			0
$460$		0			0
$461$	36.3218			1.69168			0.845839	−	0.533439i	$-0.179101\pi$
$461$	36.3218			1.69168			0.845839	+	0.533439i	$0.179101\pi$
$462$		0			0
$463$		−	13.6473i		−	0.634244i	−0.948385	−	0.317122i	$-0.897284\pi$
$463$		−	13.6473i		−	0.634244i	0.948385	−	0.317122i	$-0.102716\pi$
$464$	−4.36796			−0.202777
$465$		0			0
$466$	−16.1772			−0.749396
$467$	5.40622			0.250170			0.125085	−	0.992146i	$-0.460080\pi$
$467$	5.40622			0.250170			0.125085	+	0.992146i	$0.460080\pi$
$468$		0			0
$469$		−	29.4222i		−	1.35859i
$470$		0			0
$471$		0			0
$472$	0.720655			0.0331708
$473$	−0.0423583	+	1.07906i	−0.00194764	+	0.0496154i
$474$		0			0
$475$		0			0
$476$			2.55869i			0.117277i
$477$		0			0
$478$		−	15.6842i		−	0.717376i
$479$	4.67030			0.213391			0.106696	−	0.994292i	$-0.465973\pi$
$479$	4.67030			0.213391			0.106696	+	0.994292i	$0.465973\pi$
$480$		0			0
$481$		−	10.2900i		−	0.469183i
$482$	20.9730			0.955296
$483$		0			0
$484$	10.9662	+	0.862276i	0.498461	+	0.0391943i
$485$		0			0
$486$		0			0
$487$			10.4271i			0.472499i	0.971692	+	0.236249i	$0.0759182\pi$
$487$			10.4271i			0.472499i	−0.971692	+	0.236249i	$0.924082\pi$
$488$	3.98245			0.180277
$489$		0			0
$490$		0			0
$491$	0.434928			0.0196280			0.00981401	−	0.999952i	$-0.496876\pi$
$491$	0.434928			0.0196280			0.00981401	+	0.999952i	$0.496876\pi$
$492$		0			0
$493$		−	3.61854i		−	0.162971i
$494$	9.46612			0.425901
$495$		0			0
$496$	−2.36796			−0.106324
$497$		−	9.15247i		−	0.410545i
$498$		0			0
$499$	−11.3002			−0.505867			−0.252933	−	0.967484i	$-0.581395\pi$
$499$	−11.3002			−0.505867			−0.252933	+	0.967484i	$0.581395\pi$
$500$		0			0
$501$		0			0
$502$	−20.8550			−0.930804
$503$		−	7.07906i		−	0.315640i	−0.987468	−	0.157820i	$-0.949553\pi$
$503$		−	7.07906i		−	0.315640i	0.987468	−	0.157820i	$-0.0504465\pi$
$504$		0			0
$505$		0			0
$506$	8.92665	+	0.350413i	0.396838	+	0.0155778i
$507$		0			0
$508$	12.2602			0.543958
$509$			5.51472i			0.244436i	0.992503	+	0.122218i	$0.0390006\pi$
$509$			5.51472i			0.244436i	−0.992503	+	0.122218i	$0.960999\pi$
$510$		0			0
$511$	−1.13648			−0.0502750
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	8.77828i		−	0.387193i
$515$		0			0
$516$		0			0
$517$	−0.738200	+	18.8054i	−0.0324660	+	0.827059i
$518$	8.40305			0.369209
$519$		0			0
$520$		0			0
$521$			15.4255i			0.675804i	0.941181	+	0.337902i	$0.109717\pi$
$521$			15.4255i			0.675804i	−0.941181	+	0.337902i	$0.890283\pi$
$522$		0			0
$523$	35.8629			1.56818			0.784088	−	0.620649i	$-0.213131\pi$
$523$	35.8629			1.56818			0.784088	+	0.620649i	$0.213131\pi$
$524$	−12.7703			−0.557872
$525$		0			0
$526$	6.98084			0.304379
$527$		−	1.96168i		−	0.0854522i
$528$		0			0
$529$	−15.7447			−0.684554
$530$		0			0
$531$		0			0
$532$			7.73026i			0.335149i
$533$	38.8996			1.68493
$534$		0			0
$535$		0			0
$536$	−9.52603			−0.411462
$537$		0			0
$538$	1.74936			0.0754204
$539$	0.330375	−	8.41619i	0.0142303	−	0.362511i
$540$		0			0
$541$			13.1380i			0.564848i	0.959290	+	0.282424i	$0.0911386\pi$
$541$			13.1380i			0.564848i	−0.959290	+	0.282424i	$0.908861\pi$
$542$	20.6760			0.888111
$543$		0			0
$544$	0.828427			0.0355185
$545$		0			0
$546$		0			0
$547$	28.3401			1.21174			0.605868	−	0.795565i	$-0.292826\pi$
$547$	28.3401			1.21174			0.605868	+	0.795565i	$0.292826\pi$
$548$	−7.35607			−0.314236
$549$		0			0
$550$		0			0
$551$		−	10.9322i		−	0.465729i
$552$		0			0
$553$		−	28.4041i		−	1.20786i
$554$			28.7878i			1.22308i
$555$		0			0
$556$			2.13487i			0.0905386i
$557$		−	25.1996i		−	1.06774i	−0.845566	−	0.533871i	$-0.820737\pi$
$557$		−	25.1996i		−	1.06774i	0.845566	−	0.533871i	$-0.179263\pi$
$558$		0			0
$559$	−1.23148			−0.0520859
$560$		0			0
$561$		0			0
$562$			5.67991i			0.239593i
$563$		−	25.7664i		−	1.08592i	−0.839757	−	0.542962i	$-0.817303\pi$
$563$		−	25.7664i		−	1.08592i	0.839757	−	0.542962i	$-0.182697\pi$
$564$		0			0
$565$		0			0
$566$			32.8956i			1.38271i
$567$		0			0
$568$	−2.96330			−0.124337
$569$	−31.3212			−1.31305			−0.656526	−	0.754303i	$-0.727975\pi$
$569$	−31.3212			−1.31305			−0.656526	+	0.754303i	$0.727975\pi$
$570$		0			0
$571$		−	39.9396i		−	1.67142i	−0.549172	−	0.835710i	$-0.685057\pi$
$571$		−	39.9396i		−	1.67142i	0.549172	−	0.835710i	$-0.314943\pi$
$572$	−0.492034	+	12.5344i	−0.0205730	+	0.524089i
$573$		0			0
$574$			31.7664i			1.32590i
$575$		0			0
$576$		0			0
$577$			20.8532i			0.868132i	0.900881	+	0.434066i	$0.142921\pi$
$577$			20.8532i			0.868132i	−0.900881	+	0.434066i	$0.857079\pi$
$578$		−	16.3137i		−	0.678561i
$579$		0			0
$580$		0			0
$581$			6.53959i			0.271308i
$582$		0			0
$583$	−1.63204	+	41.5757i	−0.0675922	+	1.72189i
$584$			0.367959i			0.0152262i
$585$		0			0
$586$	4.88585			0.201832
$587$	−14.3467			−0.592150			−0.296075	−	0.955165i	$-0.595678\pi$
$587$	−14.3467			−0.592150			−0.296075	+	0.955165i	$0.595678\pi$
$588$		0			0
$589$		−	5.92659i		−	0.244201i
$590$		0			0
$591$		0			0
$592$		−	2.72065i		−	0.111818i
$593$			5.73026i			0.235314i	0.993054	+	0.117657i	$0.0375383\pi$
$593$			5.73026i			0.235314i	−0.993054	+	0.117657i	$0.962462\pi$
$594$		0			0
$595$		0			0
$596$	−22.0496			−0.903188
$597$		0			0
$598$			10.1875i			0.416598i
$599$			5.52199i			0.225622i	0.993616	+	0.112811i	$0.0359855\pi$
$599$			5.52199i			0.225622i	−0.993616	+	0.112811i	$0.964015\pi$
$600$		0			0
$601$			13.4197i			0.547402i	0.961815	+	0.273701i	$0.0882479\pi$
$601$			13.4197i			0.547402i	−0.961815	+	0.273701i	$0.911752\pi$
$602$		−	1.00565i		−	0.0409874i
$603$		0			0
$604$		−	16.7958i		−	0.683412i
$605$		0			0
$606$		0			0
$607$	−33.9858			−1.37944			−0.689720	−	0.724076i	$-0.742266\pi$
$607$	−33.9858			−1.37944			−0.689720	+	0.724076i	$0.742266\pi$
$608$	2.50283			0.101503
$609$		0			0
$610$		0			0
$611$	−21.4616			−0.868242
$612$		0			0
$613$	26.7304			1.07963			0.539815	−	0.841783i	$-0.318494\pi$
$613$	26.7304			1.07963			0.539815	+	0.841783i	$0.318494\pi$
$614$			10.3720i			0.418580i
$615$		0			0
$616$	−10.2359	−	0.401807i	−0.412416	−	0.0161893i
$617$	30.9906			1.24763			0.623817	−	0.781570i	$-0.285581\pi$
$617$	30.9906			1.24763			0.623817	+	0.781570i	$0.285581\pi$
$618$		0			0
$619$	−12.1198			−0.487136			−0.243568	−	0.969884i	$-0.578318\pi$
$619$	−12.1198			−0.487136			−0.243568	+	0.969884i	$0.578318\pi$
$620$		0			0
$621$		0			0
$622$	−33.4408			−1.34085
$623$			46.8104i			1.87542i
$624$		0			0
$625$		0			0
$626$	−32.5023			−1.29905
$627$		0			0
$628$		−	22.1191i		−	0.882647i
$629$	2.25386			0.0898674
$630$		0			0
$631$	16.0744			0.639913			0.319957	−	0.947432i	$-0.396332\pi$
$631$	16.0744			0.639913			0.319957	+	0.947432i	$0.396332\pi$
$632$	−9.19639			−0.365813
$633$		0			0
$634$			15.7416i			0.625178i
$635$		0			0
$636$		0			0
$637$	9.60495			0.380562
$638$	14.4757	+	0.568241i	0.573100	+	0.0224969i
$639$		0			0
$640$		0			0
$641$			31.4560i			1.24244i	0.783637	+	0.621220i	$0.213362\pi$
$641$			31.4560i			1.24244i	−0.783637	+	0.621220i	$0.786638\pi$
$642$		0			0
$643$			20.1038i			0.792817i	0.918074	+	0.396409i	$0.129744\pi$
$643$			20.1038i			0.792817i	−0.918074	+	0.396409i	$0.870256\pi$
$644$	−8.31936			−0.327829
$645$		0			0
$646$			2.07341i			0.0815772i
$647$	−19.7208			−0.775305			−0.387652	−	0.921806i	$-0.626714\pi$
$647$	−19.7208			−0.775305			−0.387652	+	0.921806i	$0.626714\pi$
$648$		0			0
$649$	−2.38830	−	0.0937521i	−0.0937490	−	0.00368009i
$650$		0			0
$651$		0			0
$652$		−	25.3059i		−	0.991056i
$653$	42.0993			1.64747			0.823736	−	0.566974i	$-0.191886\pi$
$653$	42.0993			1.64747			0.823736	+	0.566974i	$0.191886\pi$
$654$		0			0
$655$		0			0
$656$	10.2850			0.401562
$657$		0			0
$658$		−	17.5260i		−	0.683236i
$659$	18.5141			0.721205			0.360602	−	0.932720i	$-0.382571\pi$
$659$	18.5141			0.721205			0.360602	+	0.932720i	$0.382571\pi$
$660$		0			0
$661$	45.5789			1.77281			0.886406	−	0.462908i	$-0.153194\pi$
$661$	45.5789			1.77281			0.886406	+	0.462908i	$0.153194\pi$
$662$		−	27.8015i		−	1.08053i
$663$		0			0
$664$	2.11732			0.0821681
$665$		0			0
$666$		0			0
$667$	11.7654			0.455556
$668$		−	14.1095i		−	0.545914i
$669$		0			0
$670$		0			0
$671$	−13.1981	−	0.518090i	−0.509509	−	0.0200006i
$672$		0			0
$673$	−24.5374			−0.945847			−0.472923	−	0.881104i	$-0.656801\pi$
$673$	−24.5374			−0.945847			−0.472923	+	0.881104i	$0.656801\pi$
$674$		−	4.67279i		−	0.179989i
$675$		0			0
$676$	−1.30483			−0.0501857
$677$		−	17.3162i		−	0.665515i	−0.943012	−	0.332758i	$-0.892021\pi$
$677$		−	17.3162i		−	0.665515i	0.943012	−	0.332758i	$-0.107979\pi$
$678$		0			0
$679$		−	25.4413i		−	0.976348i
$680$		0			0
$681$		0			0
$682$	7.84759	+	0.308055i	0.300500	+	0.0117960i
$683$	−28.8397			−1.10352			−0.551761	−	0.834002i	$-0.686044\pi$
$683$	−28.8397			−1.10352			−0.551761	+	0.834002i	$0.686044\pi$
$684$		0			0
$685$		0			0
$686$		−	13.7767i		−	0.525995i
$687$		0			0
$688$	−0.325600			−0.0124134
$689$	−47.4481			−1.80763
$690$		0			0
$691$	−24.3059			−0.924638			−0.462319	−	0.886714i	$-0.652983\pi$
$691$	−24.3059			−0.924638			−0.462319	+	0.886714i	$0.652983\pi$
$692$		−	16.2020i		−	0.615909i
$693$		0			0
$694$	33.3849			1.26727
$695$		0			0
$696$		0			0
$697$			8.52037i			0.322732i
$698$	5.48938			0.207776
$699$		0			0
$700$		0			0
$701$	1.85538			0.0700767			0.0350384	−	0.999386i	$-0.488845\pi$
$701$	1.85538			0.0700767			0.0350384	+	0.999386i	$0.488845\pi$
$702$		0			0
$703$	6.80933			0.256819
$704$	−0.130093	+	3.31407i	−0.00490307	+	0.124904i
$705$		0			0
$706$			12.4078i			0.466975i
$707$	−27.5293			−1.03534
$708$		0			0
$709$	−18.1198			−0.680504			−0.340252	−	0.940334i	$-0.610512\pi$
$709$	−18.1198			−0.680504			−0.340252	+	0.940334i	$0.610512\pi$
$710$		0			0
$711$		0			0
$712$	15.1558			0.567987
$713$	6.37824			0.238867
$714$		0			0
$715$		0			0
$716$		−	7.02871i		−	0.262675i
$717$		0			0
$718$			26.9322i			1.00510i
$719$		−	15.1788i		−	0.566075i	−0.959109	−	0.283038i	$-0.908658\pi$
$719$		−	15.1788i		−	0.566075i	0.959109	−	0.283038i	$-0.0913421\pi$
$720$		0			0
$721$		−	32.1287i		−	1.19654i
$722$		−	12.7359i		−	0.473980i
$723$		0			0
$724$	−8.64341			−0.321230
$725$		0			0
$726$		0			0
$727$		−	9.92732i		−	0.368184i	−0.982909	−	0.184092i	$-0.941066\pi$
$727$		−	9.92732i		−	0.368184i	0.982909	−	0.184092i	$-0.0589344\pi$
$728$		−	11.6817i		−	0.432951i
$729$		0			0
$730$		0			0
$731$		−	0.269736i		−	0.00997655i
$732$		0			0
$733$	−17.6355			−0.651381			−0.325691	−	0.945476i	$-0.605597\pi$
$733$	−17.6355			−0.651381			−0.325691	+	0.945476i	$0.605597\pi$
$734$	−10.7590			−0.397121
$735$		0			0
$736$			2.69356i			0.0992859i
$737$	31.5699	+	1.23927i	1.16289	+	0.0456491i
$738$		0			0
$739$		−	16.7808i		−	0.617290i	−0.951177	−	0.308645i	$-0.900124\pi$
$739$		−	16.7808i		−	0.617290i	0.951177	−	0.308645i	$-0.0998755\pi$
$740$		0			0
$741$		0			0
$742$		−	38.7472i		−	1.42246i
$743$		−	28.1343i		−	1.03215i	−0.856544	−	0.516074i	$-0.827393\pi$
$743$		−	28.1343i		−	1.03215i	0.856544	−	0.516074i	$-0.172607\pi$
$744$		0			0
$745$		0			0
$746$		−	10.3600i		−	0.379305i
$747$		0			0
$748$	−2.74547	−	0.107773i	−0.100384	−	0.00394056i
$749$		−	20.6863i		−	0.755861i
$750$		0			0
$751$	−35.8128			−1.30683			−0.653413	−	0.757001i	$-0.726664\pi$
$751$	−35.8128			−1.30683			−0.653413	+	0.757001i	$0.726664\pi$
$752$	−5.67440			−0.206924
$753$		0			0
$754$			16.5204i			0.601636i
$755$		0			0
$756$		0			0
$757$		−	5.39915i		−	0.196236i	−0.995175	−	0.0981178i	$-0.968718\pi$
$757$		−	5.39915i		−	0.196236i	0.995175	−	0.0981178i	$-0.0312822\pi$
$758$			20.8206i			0.756239i
$759$		0			0
$760$		0			0
$761$	−46.3888			−1.68159			−0.840797	−	0.541351i	$-0.817913\pi$
$761$	−46.3888			−1.68159			−0.840797	+	0.541351i	$0.817913\pi$
$762$		0			0
$763$			34.9979i			1.26701i
$764$		−	15.6145i		−	0.564913i
$765$		0			0
$766$			15.6010i			0.563687i
$767$		−	2.72564i		−	0.0984171i
$768$		0			0
$769$			0.838705i			0.0302445i	0.999886	+	0.0151222i	$0.00481374\pi$
$769$			0.838705i			0.0302445i	−0.999886	+	0.0151222i	$0.995186\pi$
$770$		0			0
$771$		0			0
$772$	−5.67036			−0.204081
$773$	4.12304			0.148295			0.0741476	−	0.997247i	$-0.476376\pi$
$773$	4.12304			0.148295			0.0741476	+	0.997247i	$0.476376\pi$
$774$		0			0
$775$		0			0
$776$	−8.23713			−0.295696
$777$		0			0
$778$	−24.3928			−0.874523
$779$			25.7416i			0.922287i
$780$		0			0
$781$	9.82057	+	0.385504i	0.351408	+	0.0137944i
$782$	−2.23142			−0.0797953
$783$		0			0
$784$	2.53953			0.0906976
$785$		0			0
$786$		0			0
$787$	35.7130			1.27303			0.636516	−	0.771264i	$-0.280375\pi$
$787$	35.7130			1.27303			0.636516	+	0.771264i	$0.280375\pi$
$788$		−	20.6299i		−	0.734910i
$789$		0			0
$790$		0			0
$791$	15.5914			0.554365
$792$		0			0
$793$		−	15.0623i		−	0.534879i
$794$	−24.5931			−0.872775
$795$		0			0
$796$	15.4335			0.547026
$797$	−3.89398			−0.137932			−0.0689660	−	0.997619i	$-0.521970\pi$
$797$	−3.89398			−0.137932			−0.0689660	+	0.997619i	$0.521970\pi$
$798$		0			0
$799$		−	4.70083i		−	0.166303i
$800$		0			0
$801$		0			0
$802$	19.9459			0.704314
$803$	0.0478689	−	1.21944i	0.00168926	−	0.0430332i
$804$		0			0
$805$		0			0
$806$			8.95603i			0.315463i
$807$		0			0
$808$			8.91314i			0.313563i
$809$	−0.767052			−0.0269681			−0.0134841	−	0.999909i	$-0.504292\pi$
$809$	−0.767052			−0.0269681			−0.0134841	+	0.999909i	$0.504292\pi$
$810$		0			0
$811$		−	52.4159i		−	1.84057i	−0.391247	−	0.920286i	$-0.627956\pi$
$811$		−	52.4159i		−	1.84057i	0.391247	−	0.920286i	$-0.372044\pi$
$812$	−13.4909			−0.473439
$813$		0			0
$814$	−0.353938	+	9.01645i	−0.0124055	+	0.316026i
$815$		0			0
$816$		0			0
$817$		−	0.814921i		−	0.0285105i
$818$	22.9322			0.801807
$819$		0			0
$820$		0			0
$821$	−40.0418			−1.39747			−0.698735	−	0.715381i	$-0.746253\pi$
$821$	−40.0418			−1.39747			−0.698735	+	0.715381i	$0.746253\pi$
$822$		0			0
$823$		−	40.4158i		−	1.40881i	−0.709800	−	0.704404i	$-0.751215\pi$
$823$		−	40.4158i		−	1.40881i	0.709800	−	0.704404i	$-0.248785\pi$
$824$	−10.4023			−0.362382
$825$		0			0
$826$	2.22582			0.0774463
$827$			11.8749i			0.412930i	0.978454	+	0.206465i	$0.0661960\pi$
$827$			11.8749i			0.412930i	−0.978454	+	0.206465i	$0.933804\pi$
$828$		0			0
$829$	−14.6132			−0.507536			−0.253768	−	0.967265i	$-0.581670\pi$
$829$	−14.6132			−0.507536			−0.253768	+	0.967265i	$0.581670\pi$
$830$		0			0
$831$		0			0
$832$	−3.78217			−0.131123
$833$			2.10382i			0.0728929i
$834$		0			0
$835$		0			0
$836$	−8.29455	−	0.325600i	−0.286873	−	0.0112611i
$837$		0			0
$838$	10.4622			0.361412
$839$		−	13.7917i		−	0.476143i	−0.971248	−	0.238072i	$-0.923485\pi$
$839$		−	13.7917i		−	0.476143i	0.971248	−	0.238072i	$-0.0765153\pi$
$840$		0			0
$841$	−9.92094			−0.342101
$842$		−	4.04640i		−	0.139448i
$843$		0			0
$844$		−	24.8821i		−	0.856477i
$845$		0			0
$846$		0			0
$847$	33.8702	+	2.66324i	1.16379	+	0.0915099i
$848$	−12.5452			−0.430803
$849$		0			0
$850$		0			0
$851$			7.32824i			0.251209i
$852$		0			0
$853$	3.96719			0.135834			0.0679170	−	0.997691i	$-0.478365\pi$
$853$	3.96719			0.135834			0.0679170	+	0.997691i	$0.478365\pi$
$854$	12.3003			0.420906
$855$		0			0
$856$	−6.69760			−0.228919
$857$			33.5994i			1.14773i	0.818948	+	0.573867i	$0.194558\pi$
$857$			33.5994i			1.14773i	−0.818948	+	0.573867i	$0.805442\pi$
$858$		0			0
$859$	29.4799			1.00584			0.502921	−	0.864332i	$-0.332259\pi$
$859$	29.4799			1.00584			0.502921	+	0.864332i	$0.332259\pi$
$860$		0			0
$861$		0			0
$862$			25.9266i			0.883063i
$863$	15.8491			0.539511			0.269756	−	0.962929i	$-0.413057\pi$
$863$	15.8491			0.539511			0.269756	+	0.962929i	$0.413057\pi$
$864$		0			0
$865$		0			0
$866$	−2.10953			−0.0716847
$867$		0			0
$868$	−7.31371			−0.248243
$869$	30.4775	+	1.19639i	1.03388	+	0.0405846i
$870$		0			0
$871$			36.0291i			1.22080i
$872$	11.3313			0.383725
$873$		0			0
$874$	−6.74151			−0.228035
$875$		0			0
$876$		0			0
$877$	41.9243			1.41568			0.707842	−	0.706371i	$-0.249669\pi$
$877$	41.9243			1.41568			0.707842	+	0.706371i	$0.249669\pi$
$878$	18.5374			0.625607
$879$		0			0
$880$		0			0
$881$			14.5206i			0.489211i	0.969623	+	0.244605i	$0.0786584\pi$
$881$			14.5206i			0.489211i	−0.969623	+	0.244605i	$0.921342\pi$
$882$		0			0
$883$		−	12.4357i		−	0.418493i	−0.977863	−	0.209247i	$-0.932899\pi$
$883$		−	12.4357i		−	0.418493i	0.977863	−	0.209247i	$-0.0671011\pi$
$884$		−	3.13325i		−	0.105383i
$885$		0			0
$886$		−	12.3545i		−	0.415056i
$887$		−	51.5193i		−	1.72985i	−0.501903	−	0.864924i	$-0.667367\pi$
$887$		−	51.5193i		−	1.72985i	0.501903	−	0.864924i	$-0.332633\pi$
$888$		0			0
$889$	37.8670			1.27002
$890$		0			0
$891$		0			0
$892$		−	9.05601i		−	0.303218i
$893$		−	14.2020i		−	0.475253i
$894$		0			0
$895$		0			0
$896$		−	3.08861i		−	0.103183i
$897$		0			0
$898$	−13.8038			−0.460637
$899$	10.3431			0.344963
$900$		0			0
$901$		−	10.3928i		−	0.346233i
$902$	−34.0852	−	1.33801i	−1.13491	−	0.0445508i
$903$		0			0
$904$		−	5.04801i		−	0.167894i
$905$		0			0
$906$		0			0
$907$			24.6242i			0.817633i	0.912617	+	0.408816i	$0.134058\pi$
$907$			24.6242i			0.817633i	−0.912617	+	0.408816i	$0.865942\pi$
$908$			17.1365i			0.568694i
$909$		0			0
$910$		0			0
$911$		−	31.4756i		−	1.04283i	−0.853302	−	0.521416i	$-0.825404\pi$
$911$		−	31.4756i		−	1.04283i	0.853302	−	0.521416i	$-0.174596\pi$
$912$		0			0
$913$	−7.01696	−	0.275449i	−0.232228	−	0.00911603i
$914$			19.1964i			0.634960i
$915$		0			0
$916$	25.6682			0.848100
$917$	−39.4425			−1.30250
$918$		0			0
$919$		−	36.0634i		−	1.18962i	−0.803865	−	0.594811i	$-0.797227\pi$
$919$		−	36.0634i		−	1.18962i	0.803865	−	0.594811i	$-0.202773\pi$
$920$		0			0
$921$		0			0
$922$		−	36.3218i		−	1.19620i
$923$			11.2077i			0.368906i
$924$		0			0
$925$		0			0
$926$	−13.6473			−0.448478
$927$		0			0
$928$			4.36796i			0.143385i
$929$			8.01902i			0.263095i	0.991310	+	0.131548i	$0.0419946\pi$
$929$			8.01902i			0.263095i	−0.991310	+	0.131548i	$0.958005\pi$
$930$		0			0
$931$			6.35601i			0.208310i
$932$			16.1772i			0.529903i
$933$		0			0
$934$		−	5.40622i		−	0.176897i
$935$		0			0
$936$		0			0
$937$	−3.31122			−0.108173			−0.0540865	−	0.998536i	$-0.517225\pi$
$937$	−3.31122			−0.108173			−0.0540865	+	0.998536i	$0.517225\pi$
$938$	−29.4222			−0.960669
$939$		0			0
$940$		0			0
$941$	−52.5352			−1.71260			−0.856299	−	0.516480i	$-0.827242\pi$
$941$	−52.5352			−1.71260			−0.856299	+	0.516480i	$0.827242\pi$
$942$		0			0
$943$	−27.7033			−0.902142
$944$		−	0.720655i		−	0.0234553i
$945$		0			0
$946$	1.07906	+	0.0423583i	0.0350834	+	0.00137719i
$947$	22.7472			0.739185			0.369593	−	0.929194i	$-0.379497\pi$
$947$	22.7472			0.739185			0.369593	+	0.929194i	$0.379497\pi$
$948$		0			0
$949$	1.39168			0.0451759
$950$		0			0
$951$		0			0
$952$	2.55869			0.0829277
$953$		−	16.6976i		−	0.540888i	−0.962736	−	0.270444i	$-0.912829\pi$
$953$		−	16.6976i		−	0.540888i	0.962736	−	0.270444i	$-0.0871705\pi$
$954$		0			0
$955$		0			0
$956$	−15.6842			−0.507262
$957$		0			0
$958$		−	4.67030i		−	0.150891i
$959$	−22.7200			−0.733669
$960$		0			0
$961$	−25.3928			−0.819122
$962$	−10.2900			−0.331762
$963$		0			0
$964$		−	20.9730i		−	0.675496i
$965$		0			0
$966$		0			0
$967$	31.4115			1.01013			0.505063	−	0.863083i	$-0.331469\pi$
$967$	31.4115			1.01013			0.505063	+	0.863083i	$0.331469\pi$
$968$	0.862276	−	10.9662i	0.0277146	−	0.352465i
$969$		0			0
$970$		0			0
$971$		−	12.3662i		−	0.396850i	−0.980116	−	0.198425i	$-0.936417\pi$
$971$		−	12.3662i		−	0.396850i	0.980116	−	0.198425i	$-0.0635827\pi$
$972$		0			0
$973$			6.59378i			0.211387i
$974$	10.4271			0.334107
$975$		0			0
$976$		−	3.98245i		−	0.127475i
$977$	49.0515			1.56930			0.784649	−	0.619941i	$-0.212843\pi$
$977$	49.0515			1.56930			0.784649	+	0.619941i	$0.212843\pi$
$978$		0			0
$979$	−50.2274	−	1.97166i	−1.60527	−	0.0630146i
$980$		0			0
$981$		0			0
$982$		−	0.434928i		−	0.0138791i
$983$	−42.1487			−1.34433			−0.672167	−	0.740400i	$-0.734636\pi$
$983$	−42.1487			−1.34433			−0.672167	+	0.740400i	$0.734636\pi$
$984$		0			0
$985$		0			0
$986$	−3.61854			−0.115238
$987$		0			0
$988$		−	9.46612i		−	0.301157i
$989$	0.877024			0.0278877
$990$		0			0
$991$	−23.4122			−0.743714			−0.371857	−	0.928290i	$-0.621279\pi$
$991$	−23.4122			−0.743714			−0.371857	+	0.928290i	$0.621279\pi$
$992$			2.36796i			0.0751828i
$993$		0			0
$994$	−9.15247			−0.290299
$995$		0			0
$996$		0			0
$997$	21.6436			0.685458			0.342729	−	0.939434i	$-0.388649\pi$
$997$	21.6436			0.685458			0.342729	+	0.939434i	$0.388649\pi$
$998$			11.3002i			0.357702i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.f.c.4949.1		8
3.2	odd	2		4950.2.f.d.4949.5		8
5.2	odd	4		990.2.d.b.791.5	yes	8
5.3	odd	4		4950.2.d.h.4751.7		8
5.4	even	2		4950.2.f.f.4949.8		8
11.10	odd	2		4950.2.f.e.4949.8		8
15.2	even	4		990.2.d.a.791.1	✓	8
15.8	even	4		4950.2.d.m.4751.7		8
15.14	odd	2		4950.2.f.e.4949.4		8
20.7	even	4		7920.2.f.b.3761.8		8
33.32	even	2		4950.2.f.f.4949.4		8
55.32	even	4		990.2.d.a.791.8	yes	8
55.43	even	4		4950.2.d.m.4751.2		8
55.54	odd	2		4950.2.f.d.4949.1		8
60.47	odd	4		7920.2.f.a.3761.4		8
165.32	odd	4		990.2.d.b.791.4	yes	8
165.98	odd	4		4950.2.d.h.4751.2		8
165.164	even	2	inner	4950.2.f.c.4949.5		8
220.87	odd	4		7920.2.f.a.3761.5		8
660.527	even	4		7920.2.f.b.3761.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.d.a.791.1	✓	8	15.2	even	4
990.2.d.a.791.8	yes	8	55.32	even	4
990.2.d.b.791.4	yes	8	165.32	odd	4
990.2.d.b.791.5	yes	8	5.2	odd	4
4950.2.d.h.4751.2		8	165.98	odd	4
4950.2.d.h.4751.7		8	5.3	odd	4
4950.2.d.m.4751.2		8	55.43	even	4
4950.2.d.m.4751.7		8	15.8	even	4
4950.2.f.c.4949.1		8	1.1	even	1	trivial
4950.2.f.c.4949.5		8	165.164	even	2	inner
4950.2.f.d.4949.1		8	55.54	odd	2
4950.2.f.d.4949.5		8	3.2	odd	2
4950.2.f.e.4949.4		8	15.14	odd	2
4950.2.f.e.4949.8		8	11.10	odd	2
4950.2.f.f.4949.4		8	33.32	even	2
4950.2.f.f.4949.8		8	5.4	even	2
7920.2.f.a.3761.4		8	60.47	odd	4
7920.2.f.a.3761.5		8	220.87	odd	4
7920.2.f.b.3761.1		8	660.527	even	4
7920.2.f.b.3761.8		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.08861 q^{7} +1.00000i q^{8} +(0.130093 - 3.31407i) q^{11} +3.78217 q^{13} +3.08861i q^{14} +1.00000 q^{16} +0.828427i q^{17} +2.50283i q^{19} +(-3.31407 - 0.130093i) q^{22} -2.69356 q^{23} -3.78217i q^{26} +3.08861 q^{28} -4.36796 q^{29} -2.36796 q^{31} -1.00000i q^{32} +0.828427 q^{34} -2.72065i q^{37} +2.50283 q^{38} +10.2850 q^{41} -0.325600 q^{43} +(-0.130093 + 3.31407i) q^{44} +2.69356i q^{46} -5.67440 q^{47} +2.53953 q^{49} -3.78217 q^{52} -12.5452 q^{53} -3.08861i q^{56} +4.36796i q^{58} -0.720655i q^{59} -3.98245i q^{61} +2.36796i q^{62} -1.00000 q^{64} +9.52603i q^{67} -0.828427i q^{68} +2.96330i q^{71} +0.367959 q^{73} -2.72065 q^{74} -2.50283i q^{76} +(-0.401807 + 10.2359i) q^{77} +9.19639i q^{79} -10.2850i q^{82} -2.11732i q^{83} +0.325600i q^{86} +(3.31407 + 0.130093i) q^{88} -15.1558i q^{89} -11.6817 q^{91} +2.69356 q^{92} +5.67440i q^{94} +8.23713i q^{97} -2.53953i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{13} + 8 q^{16} - 12 q^{22} - 8 q^{23} - 8 q^{29} + 8 q^{31} - 16 q^{34} - 16 q^{38} + 8 q^{41} - 16 q^{43} - 32 q^{47} + 16 q^{49} + 8 q^{52} - 24 q^{53} - 8 q^{64} - 24 q^{73} - 24 q^{74}+ \cdots + 8 q^{92}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^13 + 8 * q^16 - 12 * q^22 - 8 * q^23 - 8 * q^29 + 8 * q^31 - 16 * q^34 - 16 * q^38 + 8 * q^41 - 16 * q^43 - 32 * q^47 + 16 * q^49 + 8 * q^52 - 24 * q^53 - 8 * q^64 - 24 * q^73 - 24 * q^74 - 28 * q^77 + 12 * q^88 + 24 * q^91 + 8 * q^92

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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