LMFDB - Embedded newform 990.2.d.a.791.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [990,2,Mod(791,990)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(990, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("990.791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	990.d (of order $2$, degree $1$, 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:	$7.90518980011$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			791.8
Root			$2.18398i$ of defining polynomial
Character	$\chi$	$=$	990.791
Dual form			990.2.d.a.791.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{5} +3.08861i q^{7} -1.00000 q^{8} -1.00000i q^{10} +(-0.130093 - 3.31407i) q^{11} +3.78217i q^{13} -3.08861i q^{14} +1.00000 q^{16} +0.828427 q^{17} +2.50283i q^{19} +1.00000i q^{20} +(0.130093 + 3.31407i) q^{22} +2.69356i q^{23} -1.00000 q^{25} -3.78217i q^{26} +3.08861i q^{28} -4.36796 q^{29} -2.36796 q^{31} -1.00000 q^{32} -0.828427 q^{34} -3.08861 q^{35} +2.72065 q^{37} -2.50283i q^{38} -1.00000i q^{40} -10.2850 q^{41} -0.325600i q^{43} +(-0.130093 - 3.31407i) q^{44} -2.69356i q^{46} -5.67440i q^{47} -2.53953 q^{49} +1.00000 q^{50} +3.78217i q^{52} +12.5452i q^{53} +(3.31407 - 0.130093i) q^{55} -3.08861i q^{56} +4.36796 q^{58} +0.720655i q^{59} +3.98245i q^{61} +2.36796 q^{62} +1.00000 q^{64} -3.78217 q^{65} -9.52603 q^{67} +0.828427 q^{68} +3.08861 q^{70} +2.96330i q^{71} +0.367959i q^{73} -2.72065 q^{74} +2.50283i q^{76} +(10.2359 - 0.401807i) q^{77} +9.19639i q^{79} +1.00000i q^{80} +10.2850 q^{82} +2.11732 q^{83} +0.828427i q^{85} +0.325600i q^{86} +(0.130093 + 3.31407i) q^{88} +15.1558i q^{89} -11.6817 q^{91} +2.69356i q^{92} +5.67440i q^{94} -2.50283 q^{95} -8.23713 q^{97} +2.53953 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 16 q^{17} - 8 q^{25} - 8 q^{29} + 8 q^{31} - 8 q^{32} + 16 q^{34} + 24 q^{37} - 8 q^{41} - 16 q^{49} + 8 q^{50} + 12 q^{55} + 8 q^{58} - 8 q^{62} + 8 q^{64}+ \cdots + 16 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^16 - 16 * q^17 - 8 * q^25 - 8 * q^29 + 8 * q^31 - 8 * q^32 + 16 * q^34 + 24 * q^37 - 8 * q^41 - 16 * q^49 + 8 * q^50 + 12 * q^55 + 8 * q^58 - 8 * q^62 + 8 * q^64 + 8 * q^65 - 16 * q^68 - 24 * q^74 + 12 * q^77 + 8 * q^82 - 24 * q^83 + 24 * q^91 + 16 * q^95 - 8 * q^97 + 16 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/990\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$551$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			3.08861i			1.16739i	0.811974	+	0.583693i	$0.198393\pi$
$7$			3.08861i			1.16739i	−0.811974	+	0.583693i	$0.801607\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		−	1.00000i		−	0.316228i
$11$	−0.130093	−	3.31407i	−0.0392245	−	0.999230i
$11$	−0.130093	−	3.31407i	−0.0392245	−	0.999230i
$12$		0			0
$13$			3.78217i			1.04899i	0.851415	+	0.524493i	$0.175745\pi$
$13$			3.78217i			1.04899i	−0.851415	+	0.524493i	$0.824255\pi$
$14$		−	3.08861i		−	0.825467i
$15$		0			0
$16$	1.00000			0.250000
$17$	0.828427			0.200923			0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427			0.200923			0.100462	+	0.994941i	$0.467968\pi$
$18$		0			0
$19$			2.50283i			0.574188i	0.957902	+	0.287094i	$0.0926892\pi$
$19$			2.50283i			0.574188i	−0.957902	+	0.287094i	$0.907311\pi$
$20$			1.00000i			0.223607i
$21$		0			0
$22$	0.130093	+	3.31407i	0.0277359	+	0.706563i
$23$			2.69356i			0.561646i	0.959760	+	0.280823i	$0.0906074\pi$
$23$			2.69356i			0.561646i	−0.959760	+	0.280823i	$0.909393\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		−	3.78217i		−	0.741745i
$27$		0			0
$28$			3.08861i			0.583693i
$29$	−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	$0.568209\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	−0.828427			−0.142074
$35$	−3.08861			−0.522071
$36$		0			0
$37$	2.72065			0.447273			0.223636	−	0.974673i	$-0.428207\pi$
$37$	2.72065			0.447273			0.223636	+	0.974673i	$0.428207\pi$
$38$		−	2.50283i		−	0.406012i
$39$		0			0
$40$		−	1.00000i		−	0.158114i
$41$	−10.2850			−1.60625			−0.803123	−	0.595813i	$-0.796830\pi$
$41$	−10.2850			−1.60625			−0.803123	+	0.595813i	$0.796830\pi$
$42$		0			0
$43$		−	0.325600i		−	0.0496536i	−0.999692	−	0.0248268i	$-0.992097\pi$
$43$		−	0.325600i		−	0.0496536i	0.999692	−	0.0248268i	$-0.00790343\pi$
$44$	−0.130093	−	3.31407i	−0.0196123	−	0.499615i
$45$		0			0
$46$		−	2.69356i		−	0.397144i
$47$		−	5.67440i		−	0.827696i	−0.910346	−	0.413848i	$-0.864184\pi$
$47$		−	5.67440i		−	0.827696i	0.910346	−	0.413848i	$-0.135816\pi$
$48$		0			0
$49$	−2.53953			−0.362790
$50$	1.00000			0.141421
$51$		0			0
$52$			3.78217i			0.524493i
$53$			12.5452i			1.72321i	0.507576	+	0.861607i	$0.330542\pi$
$53$			12.5452i			1.72321i	−0.507576	+	0.861607i	$0.669458\pi$
$54$		0			0
$55$	3.31407	−	0.130093i	0.446869	−	0.0175417i
$56$		−	3.08861i		−	0.412733i
$57$		0			0
$58$	4.36796			0.573541
$59$			0.720655i			0.0938212i	0.998899	+	0.0469106i	$0.0149376\pi$
$59$			0.720655i			0.0938212i	−0.998899	+	0.0469106i	$0.985062\pi$
$60$		0			0
$61$			3.98245i			0.509901i	0.966954	+	0.254951i	$0.0820592\pi$
$61$			3.98245i			0.509901i	−0.966954	+	0.254951i	$0.917941\pi$
$62$	2.36796			0.300731
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.78217			−0.469121
$66$		0			0
$67$	−9.52603			−1.16379			−0.581895	−	0.813264i	$-0.697688\pi$
$67$	−9.52603			−1.16379			−0.581895	+	0.813264i	$0.697688\pi$
$68$	0.828427			0.100462
$69$		0			0
$70$	3.08861			0.369160
$71$			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
$73$			0.367959i			0.0430663i	0.999768	+	0.0215332i	$0.00685475\pi$
$73$			0.367959i			0.0430663i	−0.999768	+	0.0215332i	$0.993145\pi$
$74$	−2.72065			−0.316270
$75$		0			0
$76$			2.50283i			0.287094i
$77$	10.2359	−	0.401807i	1.16649	−	0.0457902i
$78$		0			0
$79$			9.19639i			1.03467i	0.855782	+	0.517337i	$0.173077\pi$
$79$			9.19639i			1.03467i	−0.855782	+	0.517337i	$0.826923\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$	10.2850			1.13579
$83$	2.11732			0.232406			0.116203	−	0.993225i	$-0.462928\pi$
$83$	2.11732			0.232406			0.116203	+	0.993225i	$0.462928\pi$
$84$		0			0
$85$			0.828427i			0.0898555i
$86$			0.325600i			0.0351104i
$87$		0			0
$88$	0.130093	+	3.31407i	0.0138680	+	0.353281i
$89$			15.1558i			1.60651i	0.595635	+	0.803255i	$0.296900\pi$
$89$			15.1558i			1.60651i	−0.595635	+	0.803255i	$0.703100\pi$
$90$		0			0
$91$	−11.6817			−1.22457
$92$			2.69356i			0.280823i
$93$		0			0
$94$			5.67440i			0.585270i
$95$	−2.50283			−0.256785
$96$		0			0
$97$	−8.23713			−0.836354			−0.418177	−	0.908366i	$-0.637331\pi$
$97$	−8.23713			−0.836354			−0.418177	+	0.908366i	$0.637331\pi$
$98$	2.53953			0.256531
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−8.91314			−0.886891			−0.443445	−	0.896301i	$-0.646244\pi$
$101$	−8.91314			−0.886891			−0.443445	+	0.896301i	$0.646244\pi$
$102$		0			0
$103$	10.4023			1.02497			0.512486	−	0.858696i	$-0.328725\pi$
$103$	10.4023			1.02497			0.512486	+	0.858696i	$0.328725\pi$
$104$		−	3.78217i		−	0.370873i
$105$		0			0
$106$		−	12.5452i		−	1.21850i
$107$	6.69760			0.647481			0.323741	−	0.946146i	$-0.395059\pi$
$107$	6.69760			0.647481			0.323741	+	0.946146i	$0.395059\pi$
$108$		0			0
$109$		−	11.3313i		−	1.08534i	−0.839947	−	0.542669i	$-0.817414\pi$
$109$		−	11.3313i		−	1.08534i	0.839947	−	0.542669i	$-0.182586\pi$
$110$	−3.31407	+	0.130093i	−0.315984	+	0.0124039i
$111$		0			0
$112$			3.08861i			0.291847i
$113$			5.04801i			0.474877i	0.971403	+	0.237439i	$0.0763078\pi$
$113$			5.04801i			0.474877i	−0.971403	+	0.237439i	$0.923692\pi$
$114$		0			0
$115$	−2.69356			−0.251176
$116$	−4.36796			−0.405555
$117$		0			0
$118$		−	0.720655i		−	0.0663416i
$119$			2.55869i			0.234555i
$120$		0			0
$121$	−10.9662	+	0.862276i	−0.996923	+	0.0783887i
$122$		−	3.98245i		−	0.360554i
$123$		0			0
$124$	−2.36796			−0.212649
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			12.2602i			1.08792i	0.839113	+	0.543958i	$0.183075\pi$
$127$			12.2602i			1.08792i	−0.839113	+	0.543958i	$0.816925\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	3.78217			0.331718
$131$	−12.7703			−1.11574			−0.557872	−	0.829927i	$-0.688382\pi$
$131$	−12.7703			−1.11574			−0.557872	+	0.829927i	$0.688382\pi$
$132$		0			0
$133$	−7.73026			−0.670299
$134$	9.52603			0.822923
$135$		0			0
$136$	−0.828427			−0.0710370
$137$			7.35607i			0.628471i	0.949345	+	0.314236i	$0.101748\pi$
$137$			7.35607i			0.628471i	−0.949345	+	0.314236i	$0.898252\pi$
$138$		0			0
$139$		−	2.13487i		−	0.181077i	−0.995893	−	0.0905386i	$-0.971141\pi$
$139$		−	2.13487i		−	0.181077i	0.995893	−	0.0905386i	$-0.0288588\pi$
$140$	−3.08861			−0.261035
$141$		0			0
$142$		−	2.96330i		−	0.248674i
$143$	12.5344	−	0.492034i	1.04818	−	0.0411460i
$144$		0			0
$145$		−	4.36796i		−	0.362739i
$146$		−	0.367959i		−	0.0304525i
$147$		0			0
$148$	2.72065			0.223636
$149$	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	$-0.760495\pi$
$151$		−	16.7958i		−	1.36682i	0.730033	−	0.683412i	$-0.239505\pi$
$152$		−	2.50283i		−	0.203006i
$153$		0			0
$154$	−10.2359	+	0.401807i	−0.824831	+	0.0323785i
$155$		−	2.36796i		−	0.190199i
$156$		0			0
$157$	−22.1191			−1.76529			−0.882647	−	0.470036i	$-0.844241\pi$
$157$	−22.1191			−1.76529			−0.882647	+	0.470036i	$0.844241\pi$
$158$		−	9.19639i		−	0.731625i
$159$		0			0
$160$		−	1.00000i		−	0.0790569i
$161$	−8.31936			−0.655658
$162$		0			0
$163$	25.3059			1.98211			0.991056	−	0.133446i	$-0.0426042\pi$
$163$	25.3059			1.98211			0.991056	+	0.133446i	$0.0426042\pi$
$164$	−10.2850			−0.803123
$165$		0			0
$166$	−2.11732			−0.164336
$167$	14.1095			1.09183			0.545914	−	0.837841i	$-0.316182\pi$
$167$	14.1095			1.09183			0.545914	+	0.837841i	$0.316182\pi$
$168$		0			0
$169$	−1.30483			−0.100371
$170$		−	0.828427i		−	0.0635375i
$171$		0			0
$172$		−	0.325600i		−	0.0248268i
$173$	−16.2020			−1.23182			−0.615909	−	0.787817i	$-0.711211\pi$
$173$	−16.2020			−1.23182			−0.615909	+	0.787817i	$0.711211\pi$
$174$		0			0
$175$		−	3.08861i		−	0.233477i
$176$	−0.130093	−	3.31407i	−0.00980613	−	0.249808i
$177$		0			0
$178$		−	15.1558i		−	1.13597i
$179$		−	7.02871i		−	0.525350i	−0.964884	−	0.262675i	$-0.915395\pi$
$179$		−	7.02871i		−	0.525350i	0.964884	−	0.262675i	$-0.0846048\pi$
$180$		0			0
$181$	8.64341			0.642459			0.321230	−	0.947001i	$-0.395904\pi$
$181$	8.64341			0.642459			0.321230	+	0.947001i	$0.395904\pi$
$182$	11.6817			0.865903
$183$		0			0
$184$		−	2.69356i		−	0.198572i
$185$			2.72065i			0.200026i
$186$		0			0
$187$	−0.107773	−	2.74547i	−0.00788111	−	0.200768i
$188$		−	5.67440i		−	0.413848i
$189$		0			0
$190$	2.50283			0.181574
$191$			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
$193$			5.67036i			0.408161i	0.978954	+	0.204081i	$0.0654205\pi$
$193$			5.67036i			0.408161i	−0.978954	+	0.204081i	$0.934579\pi$
$194$	8.23713			0.591391
$195$		0			0
$196$	−2.53953			−0.181395
$197$	20.6299			1.46982			0.734910	−	0.678165i	$-0.237224\pi$
$197$	20.6299			1.46982			0.734910	+	0.678165i	$0.237224\pi$
$198$		0			0
$199$	15.4335			1.09405			0.547026	−	0.837115i	$-0.315760\pi$
$199$	15.4335			1.09405			0.547026	+	0.837115i	$0.315760\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	8.91314			0.627127
$203$		−	13.4909i		−	0.946878i
$204$		0			0
$205$		−	10.2850i		−	0.718335i
$206$	−10.4023			−0.724764
$207$		0			0
$208$			3.78217i			0.262246i
$209$	8.29455	−	0.325600i	0.573746	−	0.0225222i
$210$		0			0
$211$		−	24.8821i		−	1.71295i	−0.516185	−	0.856477i	$-0.672648\pi$
$211$		−	24.8821i		−	1.71295i	0.516185	−	0.856477i	$-0.327352\pi$
$212$			12.5452i			0.861607i
$213$		0			0
$214$	−6.69760			−0.457838
$215$	0.325600			0.0222058
$216$		0			0
$217$		−	7.31371i		−	0.496487i
$218$			11.3313i			0.767449i
$219$		0			0
$220$	3.31407	−	0.130093i	0.223435	−	0.00877087i
$221$			3.13325i			0.210765i
$222$		0			0
$223$	9.05601			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$223$	9.05601			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$224$		−	3.08861i		−	0.206367i
$225$		0			0
$226$		−	5.04801i		−	0.335789i
$227$	−17.1365			−1.13739			−0.568694	−	0.822549i	$-0.692551\pi$
$227$	−17.1365			−1.13739			−0.568694	+	0.822549i	$0.692551\pi$
$228$		0			0
$229$	25.6682			1.69620			0.848100	−	0.529836i	$-0.177747\pi$
$229$	25.6682			1.69620			0.848100	+	0.529836i	$0.177747\pi$
$230$	2.69356			0.177608
$231$		0			0
$232$	4.36796			0.286771
$233$	16.1772			1.05981			0.529903	−	0.848058i	$-0.322228\pi$
$233$	16.1772			1.05981			0.529903	+	0.848058i	$0.322228\pi$
$234$		0			0
$235$	5.67440			0.370157
$236$			0.720655i			0.0469106i
$237$		0			0
$238$		−	2.55869i		−	0.165855i
$239$	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	$-0.763929\pi$
$241$		−	20.9730i		−	1.35099i	0.737363	−	0.675496i	$-0.236071\pi$
$242$	10.9662	−	0.862276i	0.704931	−	0.0554292i
$243$		0			0
$244$			3.98245i			0.254951i
$245$		−	2.53953i		−	0.162245i
$246$		0			0
$247$	−9.46612			−0.602315
$248$	2.36796			0.150366
$249$		0			0
$250$			1.00000i			0.0632456i
$251$		−	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
$253$	8.92665	−	0.350413i	0.561214	−	0.0220303i
$254$		−	12.2602i		−	0.769272i
$255$		0			0
$256$	1.00000			0.0625000
$257$			8.77828i			0.547574i	0.961790	+	0.273787i	$0.0882763\pi$
$257$			8.77828i			0.547574i	−0.961790	+	0.273787i	$0.911724\pi$
$258$		0			0
$259$			8.40305i			0.522140i
$260$	−3.78217			−0.234560
$261$		0			0
$262$	12.7703			0.788951
$263$	−6.98084			−0.430457			−0.215229	−	0.976564i	$-0.569050\pi$
$263$	−6.98084			−0.430457			−0.215229	+	0.976564i	$0.569050\pi$
$264$		0			0
$265$	−12.5452			−0.770645
$266$	7.73026			0.473973
$267$		0			0
$268$	−9.52603			−0.581895
$269$		−	1.74936i		−	0.106661i	−0.998577	−	0.0533303i	$-0.983016\pi$
$269$		−	1.74936i		−	0.106661i	0.998577	−	0.0533303i	$-0.0169836\pi$
$270$		0			0
$271$		−	20.6760i		−	1.25598i	−0.778222	−	0.627989i	$-0.783878\pi$
$271$		−	20.6760i		−	1.25598i	0.778222	−	0.627989i	$-0.216122\pi$
$272$	0.828427			0.0502308
$273$		0			0
$274$		−	7.35607i		−	0.444396i
$275$	0.130093	+	3.31407i	0.00784491	+	0.199846i
$276$		0			0
$277$			28.7878i			1.72969i	0.502037	+	0.864846i	$0.332584\pi$
$277$			28.7878i			1.72969i	−0.502037	+	0.864846i	$0.667416\pi$
$278$			2.13487i			0.128041i
$279$		0			0
$280$	3.08861			0.184580
$281$	5.67991			0.338835			0.169418	−	0.985544i	$-0.445811\pi$
$281$	5.67991			0.338835			0.169418	+	0.985544i	$0.445811\pi$
$282$		0			0
$283$		−	32.8956i		−	1.95544i	−0.209912	−	0.977720i	$-0.567318\pi$
$283$		−	32.8956i		−	1.95544i	0.209912	−	0.977720i	$-0.432682\pi$
$284$			2.96330i			0.175839i
$285$		0			0
$286$	−12.5344	+	0.492034i	−0.741174	+	0.0290946i
$287$		−	31.7664i		−	1.87511i
$288$		0			0
$289$	−16.3137			−0.959630
$290$			4.36796i			0.256495i
$291$		0			0
$292$			0.367959i			0.0215332i
$293$	−4.88585			−0.285434			−0.142717	−	0.989764i	$-0.545584\pi$
$293$	−4.88585			−0.285434			−0.142717	+	0.989764i	$0.545584\pi$
$294$		0			0
$295$	−0.720655			−0.0419581
$296$	−2.72065			−0.158135
$297$		0			0
$298$	−22.0496			−1.27730
$299$	−10.1875			−0.589159
$300$		0			0
$301$	1.00565			0.0579649
$302$			16.7958i			0.966491i
$303$		0			0
$304$			2.50283i			0.143547i
$305$	−3.98245			−0.228035
$306$		0			0
$307$			10.3720i			0.591961i	0.955194	+	0.295981i	$0.0956464\pi$
$307$			10.3720i			0.591961i	−0.955194	+	0.295981i	$0.904354\pi$
$308$	10.2359	−	0.401807i	0.583244	−	0.0228951i
$309$		0			0
$310$			2.36796i			0.134491i
$311$		−	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
$313$	−32.5023			−1.83714			−0.918569	−	0.395260i	$-0.870655\pi$
$313$	−32.5023			−1.83714			−0.918569	+	0.395260i	$0.870655\pi$
$314$	22.1191			1.24825
$315$		0			0
$316$			9.19639i			0.517337i
$317$		−	15.7416i		−	0.884135i	−0.896982	−	0.442067i	$-0.854245\pi$
$317$		−	15.7416i		−	0.884135i	0.896982	−	0.442067i	$-0.145755\pi$
$318$		0			0
$319$	0.568241	+	14.4757i	0.0318154	+	0.810485i
$320$			1.00000i			0.0559017i
$321$		0			0
$322$	8.31936			0.463620
$323$			2.07341i			0.115368i
$324$		0			0
$325$		−	3.78217i		−	0.209797i
$326$	−25.3059			−1.40157
$327$		0			0
$328$	10.2850			0.567894
$329$	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	$0.223206\pi$
$332$	2.11732			0.116203
$333$		0			0
$334$	−14.1095			−0.772039
$335$		−	9.52603i		−	0.520462i
$336$		0			0
$337$		−	4.67279i		−	0.254543i	−0.991868	−	0.127271i	$-0.959378\pi$
$337$		−	4.67279i		−	0.254543i	0.991868	−	0.127271i	$-0.0406219\pi$
$338$	1.30483			0.0709732
$339$		0			0
$340$			0.828427i			0.0449278i
$341$	0.308055	+	7.84759i	0.0166821	+	0.424971i
$342$		0			0
$343$			13.7767i			0.743870i
$344$			0.325600i			0.0175552i
$345$		0			0
$346$	16.2020			0.871027
$347$	33.3849			1.79220			0.896098	−	0.443856i	$-0.146390\pi$
$347$	33.3849			1.79220			0.896098	+	0.443856i	$0.146390\pi$
$348$		0			0
$349$			5.48938i			0.293840i	0.989148	+	0.146920i	$0.0469360\pi$
$349$			5.48938i			0.293840i	−0.989148	+	0.146920i	$0.953064\pi$
$350$			3.08861i			0.165093i
$351$		0			0
$352$	0.130093	+	3.31407i	0.00693398	+	0.176641i
$353$			12.4078i			0.660402i	0.943911	+	0.330201i	$0.107117\pi$
$353$			12.4078i			0.660402i	−0.943911	+	0.330201i	$0.892883\pi$
$354$		0			0
$355$	−2.96330			−0.157275
$356$			15.1558i			0.803255i
$357$		0			0
$358$			7.02871i			0.371479i
$359$	−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
$362$	−8.64341			−0.454287
$363$		0			0
$364$	−11.6817			−0.612286
$365$	−0.367959			−0.0192598
$366$		0			0
$367$	10.7590			0.561614			0.280807	−	0.959764i	$-0.409398\pi$
$367$	10.7590			0.561614			0.280807	+	0.959764i	$0.409398\pi$
$368$			2.69356i			0.140411i
$369$		0			0
$370$		−	2.72065i		−	0.141440i
$371$	−38.7472			−2.01166
$372$		0			0
$373$			10.3600i			0.536419i	0.963361	+	0.268209i	$0.0864319\pi$
$373$			10.3600i			0.536419i	−0.963361	+	0.268209i	$0.913568\pi$
$374$	0.107773	+	2.74547i	0.00557279	+	0.141965i
$375$		0			0
$376$			5.67440i			0.292635i
$377$		−	16.5204i		−	0.850842i
$378$		0			0
$379$	20.8206			1.06948			0.534742	−	0.845015i	$-0.320409\pi$
$379$	20.8206			1.06948			0.534742	+	0.845015i	$0.320409\pi$
$380$	−2.50283			−0.128392
$381$		0			0
$382$		−	15.6145i		−	0.798907i
$383$			15.6010i			0.797173i	0.917131	+	0.398587i	$0.130499\pi$
$383$			15.6010i			0.797173i	−0.917131	+	0.398587i	$0.869501\pi$
$384$		0			0
$385$	0.401807	+	10.2359i	0.0204780	+	0.521669i
$386$		−	5.67036i		−	0.288614i
$387$		0			0
$388$	−8.23713			−0.418177
$389$			24.3928i			1.23676i	0.785878	+	0.618381i	$0.212211\pi$
$389$			24.3928i			1.23676i	−0.785878	+	0.618381i	$0.787789\pi$
$390$		0			0
$391$			2.23142i			0.112848i
$392$	2.53953			0.128266
$393$		0			0
$394$	−20.6299			−1.03932
$395$	−9.19639			−0.462720
$396$		0			0
$397$	24.5931			1.23429			0.617145	−	0.786849i	$-0.288289\pi$
$397$	24.5931			1.23429			0.617145	+	0.786849i	$0.288289\pi$
$398$	−15.4335			−0.773612
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			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
$403$		−	8.95603i		−	0.446132i
$404$	−8.91314			−0.443445
$405$		0			0
$406$			13.4909i			0.669544i
$407$	−0.353938	−	9.01645i	−0.0175441	−	0.446929i
$408$		0			0
$409$			22.9322i			1.13393i	0.823743	+	0.566963i	$0.191882\pi$
$409$			22.9322i			1.13393i	−0.823743	+	0.566963i	$0.808118\pi$
$410$			10.2850i			0.507940i
$411$		0			0
$412$	10.4023			0.512486
$413$	−2.22582			−0.109526
$414$		0			0
$415$			2.11732i			0.103935i
$416$		−	3.78217i		−	0.185436i
$417$		0			0
$418$	−8.29455	+	0.325600i	−0.405700	+	0.0159256i
$419$		−	10.4622i		−	0.511113i	−0.966794	−	0.255557i	$-0.917741\pi$
$419$		−	10.4622i		−	0.511113i	0.966794	−	0.255557i	$-0.0822587\pi$
$420$		0			0
$421$	4.04640			0.197209			0.0986047	−	0.995127i	$-0.468562\pi$
$421$	4.04640			0.197209			0.0986047	+	0.995127i	$0.468562\pi$
$422$			24.8821i			1.21124i
$423$		0			0
$424$		−	12.5452i		−	0.609248i
$425$	−0.828427			−0.0401846
$426$		0			0
$427$	−12.3003			−0.595251
$428$	6.69760			0.323741
$429$		0			0
$430$	−0.325600			−0.0157018
$431$	25.9266			1.24884			0.624420	−	0.781089i	$-0.285335\pi$
$431$	25.9266			1.24884			0.624420	+	0.781089i	$0.285335\pi$
$432$		0			0
$433$	−2.10953			−0.101378			−0.0506888	−	0.998714i	$-0.516142\pi$
$433$	−2.10953			−0.101378			−0.0506888	+	0.998714i	$0.516142\pi$
$434$			7.31371i			0.351069i
$435$		0			0
$436$		−	11.3313i		−	0.542669i
$437$	−6.74151			−0.322490
$438$		0			0
$439$			18.5374i			0.884741i	0.896832	+	0.442371i	$0.145862\pi$
$439$			18.5374i			0.884741i	−0.896832	+	0.442371i	$0.854138\pi$
$440$	−3.31407	+	0.130093i	−0.157992	+	0.00620194i
$441$		0			0
$442$		−	3.13325i		−	0.149034i
$443$		−	12.3545i		−	0.586978i	−0.955962	−	0.293489i	$-0.905184\pi$
$443$		−	12.3545i		−	0.586978i	0.955962	−	0.293489i	$-0.0948164\pi$
$444$		0			0
$445$	−15.1558			−0.718453
$446$	−9.05601			−0.428814
$447$		0			0
$448$			3.08861i			0.145923i
$449$			13.8038i			0.651440i	0.945466	+	0.325720i	$0.105607\pi$
$449$			13.8038i			0.651440i	−0.945466	+	0.325720i	$0.894393\pi$
$450$		0			0
$451$	1.33801	+	34.0852i	0.0630043	+	1.60501i
$452$			5.04801i			0.237439i
$453$		0			0
$454$	17.1365			0.804255
$455$		−	11.6817i		−	0.547645i
$456$		0			0
$457$			19.1964i			0.897969i	0.893540	+	0.448985i	$0.148214\pi$
$457$			19.1964i			0.897969i	−0.893540	+	0.448985i	$0.851786\pi$
$458$	−25.6682			−1.19939
$459$		0			0
$460$	−2.69356			−0.125588
$461$	−36.3218			−1.69168			−0.845839	−	0.533439i	$-0.820899\pi$
$461$	−36.3218			−1.69168			−0.845839	+	0.533439i	$0.820899\pi$
$462$		0			0
$463$	−13.6473			−0.634244			−0.317122	−	0.948385i	$-0.602716\pi$
$463$	−13.6473			−0.634244			−0.317122	+	0.948385i	$0.602716\pi$
$464$	−4.36796			−0.202777
$465$		0			0
$466$	−16.1772			−0.749396
$467$			5.40622i			0.250170i	0.992146	+	0.125085i	$0.0399203\pi$
$467$			5.40622i			0.250170i	−0.992146	+	0.125085i	$0.960080\pi$
$468$		0			0
$469$		−	29.4222i		−	1.35859i
$470$	−5.67440			−0.261741
$471$		0			0
$472$		−	0.720655i		−	0.0331708i
$473$	−1.07906	+	0.0423583i	−0.0496154	+	0.00194764i
$474$		0			0
$475$		−	2.50283i		−	0.114838i
$476$			2.55869i			0.117277i
$477$		0			0
$478$	−15.6842			−0.717376
$479$	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
$482$			20.9730i			0.955296i
$483$		0			0
$484$	−10.9662	+	0.862276i	−0.498461	+	0.0391943i
$485$		−	8.23713i		−	0.374029i
$486$		0			0
$487$	−10.4271			−0.472499			−0.236249	−	0.971692i	$-0.575918\pi$
$487$	−10.4271			−0.472499			−0.236249	+	0.971692i	$0.575918\pi$
$488$		−	3.98245i		−	0.180277i
$489$		0			0
$490$			2.53953i			0.114724i
$491$	−0.434928			−0.0196280			−0.00981401	−	0.999952i	$-0.503124\pi$
$491$	−0.434928			−0.0196280			−0.00981401	+	0.999952i	$0.503124\pi$
$492$		0			0
$493$	−3.61854			−0.162971
$494$	9.46612			0.425901
$495$		0			0
$496$	−2.36796			−0.106324
$497$	−9.15247			−0.410545
$498$		0			0
$499$	11.3002			0.505867			0.252933	−	0.967484i	$-0.418605\pi$
$499$	11.3002			0.505867			0.252933	+	0.967484i	$0.418605\pi$
$500$		−	1.00000i		−	0.0447214i
$501$		0			0
$502$			20.8550i			0.930804i
$503$	7.07906			0.315640			0.157820	−	0.987468i	$-0.449553\pi$
$503$	7.07906			0.315640			0.157820	+	0.987468i	$0.449553\pi$
$504$		0			0
$505$		−	8.91314i		−	0.396630i
$506$	−8.92665	+	0.350413i	−0.396838	+	0.0155778i
$507$		0			0
$508$			12.2602i			0.543958i
$509$		−	5.51472i		−	0.244436i	−0.992503	−	0.122218i	$-0.960999\pi$
$509$		−	5.51472i		−	0.244436i	0.992503	−	0.122218i	$-0.0390006\pi$
$510$		0			0
$511$	−1.13648			−0.0502750
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	8.77828i		−	0.387193i
$515$			10.4023i			0.458381i
$516$		0			0
$517$	−18.8054	+	0.738200i	−0.827059	+	0.0324660i
$518$		−	8.40305i		−	0.369209i
$519$		0			0
$520$	3.78217			0.165859
$521$			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
$523$			35.8629i			1.56818i	0.620649	+	0.784088i	$0.286869\pi$
$523$			35.8629i			1.56818i	−0.620649	+	0.784088i	$0.713131\pi$
$524$	−12.7703			−0.557872
$525$		0			0
$526$	6.98084			0.304379
$527$	−1.96168			−0.0854522
$528$		0			0
$529$	15.7447			0.684554
$530$	12.5452			0.544928
$531$		0			0
$532$	−7.73026			−0.335149
$533$		−	38.8996i		−	1.68493i
$534$		0			0
$535$			6.69760i			0.289562i
$536$	9.52603			0.411462
$537$		0			0
$538$			1.74936i			0.0754204i
$539$	0.330375	+	8.41619i	0.0142303	+	0.362511i
$540$		0			0
$541$		−	13.1380i		−	0.564848i	−0.959290	−	0.282424i	$-0.908861\pi$
$541$		−	13.1380i		−	0.564848i	0.959290	−	0.282424i	$-0.0911386\pi$
$542$			20.6760i			0.888111i
$543$		0			0
$544$	−0.828427			−0.0355185
$545$	11.3313			0.485378
$546$		0			0
$547$		−	28.3401i		−	1.21174i	−0.795565	−	0.605868i	$-0.792826\pi$
$547$		−	28.3401i		−	1.21174i	0.795565	−	0.605868i	$-0.207174\pi$
$548$			7.35607i			0.314236i
$549$		0			0
$550$	−0.130093	−	3.31407i	−0.00554719	−	0.141313i
$551$		−	10.9322i		−	0.465729i
$552$		0			0
$553$	−28.4041			−1.20786
$554$		−	28.7878i		−	1.22308i
$555$		0			0
$556$		−	2.13487i		−	0.0905386i
$557$	−25.1996			−1.06774			−0.533871	−	0.845566i	$-0.679263\pi$
$557$	−25.1996			−1.06774			−0.533871	+	0.845566i	$0.679263\pi$
$558$		0			0
$559$	1.23148			0.0520859
$560$	−3.08861			−0.130518
$561$		0			0
$562$	−5.67991			−0.239593
$563$	25.7664			1.08592			0.542962	−	0.839757i	$-0.317303\pi$
$563$	25.7664			1.08592			0.542962	+	0.839757i	$0.317303\pi$
$564$		0			0
$565$	−5.04801			−0.212371
$566$			32.8956i			1.38271i
$567$		0			0
$568$		−	2.96330i		−	0.124337i
$569$	−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	$0.314943\pi$
$571$			39.9396i			1.67142i	−0.549172	+	0.835710i	$0.685057\pi$
$572$	12.5344	−	0.492034i	0.524089	−	0.0205730i
$573$		0			0
$574$			31.7664i			1.32590i
$575$		−	2.69356i		−	0.112329i
$576$		0			0
$577$	−20.8532			−0.868132			−0.434066	−	0.900881i	$-0.642921\pi$
$577$	−20.8532			−0.868132			−0.434066	+	0.900881i	$0.642921\pi$
$578$	16.3137			0.678561
$579$		0			0
$580$		−	4.36796i		−	0.181370i
$581$			6.53959i			0.271308i
$582$		0			0
$583$	41.5757	−	1.63204i	1.72189	−	0.0675922i
$584$		−	0.367959i		−	0.0152262i
$585$		0			0
$586$	4.88585			0.201832
$587$		−	14.3467i		−	0.592150i	−0.955165	−	0.296075i	$-0.904322\pi$
$587$		−	14.3467i		−	0.592150i	0.955165	−	0.296075i	$-0.0956778\pi$
$588$		0			0
$589$		−	5.92659i		−	0.244201i
$590$	0.720655			0.0296689
$591$		0			0
$592$	2.72065			0.111818
$593$	−5.73026			−0.235314			−0.117657	−	0.993054i	$-0.537538\pi$
$593$	−5.73026			−0.235314			−0.117657	+	0.993054i	$0.537538\pi$
$594$		0			0
$595$	−2.55869			−0.104896
$596$	22.0496			0.903188
$597$		0			0
$598$	10.1875			0.416598
$599$		−	5.52199i		−	0.225622i	−0.993616	−	0.112811i	$-0.964015\pi$
$599$		−	5.52199i		−	0.225622i	0.993616	−	0.112811i	$-0.0359855\pi$
$600$		0			0
$601$		−	13.4197i		−	0.547402i	−0.961815	−	0.273701i	$-0.911752\pi$
$601$		−	13.4197i		−	0.547402i	0.961815	−	0.273701i	$-0.0882479\pi$
$602$	−1.00565			−0.0409874
$603$		0			0
$604$		−	16.7958i		−	0.683412i
$605$	−0.862276	−	10.9662i	−0.0350565	−	0.445837i
$606$		0			0
$607$			33.9858i			1.37944i	0.724076	+	0.689720i	$0.242266\pi$
$607$			33.9858i			1.37944i	−0.724076	+	0.689720i	$0.757734\pi$
$608$		−	2.50283i		−	0.101503i
$609$		0			0
$610$	3.98245			0.161245
$611$	21.4616			0.868242
$612$		0			0
$613$			26.7304i			1.07963i	0.841783	+	0.539815i	$0.181506\pi$
$613$			26.7304i			1.07963i	−0.841783	+	0.539815i	$0.818494\pi$
$614$		−	10.3720i		−	0.418580i
$615$		0			0
$616$	−10.2359	+	0.401807i	−0.412416	+	0.0161893i
$617$			30.9906i			1.24763i	0.781570	+	0.623817i	$0.214419\pi$
$617$			30.9906i			1.24763i	−0.781570	+	0.623817i	$0.785581\pi$
$618$		0			0
$619$	12.1198			0.487136			0.243568	−	0.969884i	$-0.421682\pi$
$619$	12.1198			0.487136			0.243568	+	0.969884i	$0.421682\pi$
$620$		−	2.36796i		−	0.0950995i
$621$		0			0
$622$			33.4408i			1.34085i
$623$	−46.8104			−1.87542
$624$		0			0
$625$	1.00000			0.0400000
$626$	32.5023			1.29905
$627$		0			0
$628$	−22.1191			−0.882647
$629$	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	$0.396332\pi$
$632$		−	9.19639i		−	0.365813i
$633$		0			0
$634$			15.7416i			0.625178i
$635$	−12.2602			−0.486531
$636$		0			0
$637$		−	9.60495i		−	0.380562i
$638$	−0.568241	−	14.4757i	−0.0224969	−	0.573100i
$639$		0			0
$640$		−	1.00000i		−	0.0395285i
$641$			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
$643$	20.1038			0.792817			0.396409	−	0.918074i	$-0.370256\pi$
$643$	20.1038			0.792817			0.396409	+	0.918074i	$0.370256\pi$
$644$	−8.31936			−0.327829
$645$		0			0
$646$		−	2.07341i		−	0.0815772i
$647$		−	19.7208i		−	0.775305i	−0.921806	−	0.387652i	$-0.873286\pi$
$647$		−	19.7208i		−	0.775305i	0.921806	−	0.387652i	$-0.126714\pi$
$648$		0			0
$649$	2.38830	−	0.0937521i	0.0937490	−	0.00368009i
$650$			3.78217i			0.148349i
$651$		0			0
$652$	25.3059			0.991056
$653$		−	42.0993i		−	1.64747i	−0.566974	−	0.823736i	$-0.691886\pi$
$653$		−	42.0993i		−	1.64747i	0.566974	−	0.823736i	$-0.308114\pi$
$654$		0			0
$655$		−	12.7703i		−	0.498976i
$656$	−10.2850			−0.401562
$657$		0			0
$658$	−17.5260			−0.683236
$659$	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	$0.153194\pi$
$662$	−27.8015			−1.08053
$663$		0			0
$664$	−2.11732			−0.0821681
$665$		−	7.73026i		−	0.299767i
$666$		0			0
$667$		−	11.7654i		−	0.455556i
$668$	14.1095			0.545914
$669$		0			0
$670$			9.52603i			0.368022i
$671$	13.1981	−	0.518090i	0.509509	−	0.0200006i
$672$		0			0
$673$		−	24.5374i		−	0.945847i	−0.881104	−	0.472923i	$-0.843199\pi$
$673$		−	24.5374i		−	0.945847i	0.881104	−	0.472923i	$-0.156801\pi$
$674$			4.67279i			0.179989i
$675$		0			0
$676$	−1.30483			−0.0501857
$677$	−17.3162			−0.665515			−0.332758	−	0.943012i	$-0.607979\pi$
$677$	−17.3162			−0.665515			−0.332758	+	0.943012i	$0.607979\pi$
$678$		0			0
$679$		−	25.4413i		−	0.976348i
$680$		−	0.828427i		−	0.0317687i
$681$		0			0
$682$	−0.308055	−	7.84759i	−0.0117960	−	0.300500i
$683$			28.8397i			1.10352i	0.834002	+	0.551761i	$0.186044\pi$
$683$			28.8397i			1.10352i	−0.834002	+	0.551761i	$0.813956\pi$
$684$		0			0
$685$	−7.35607			−0.281061
$686$		−	13.7767i		−	0.525995i
$687$		0			0
$688$		−	0.325600i		−	0.0124134i
$689$	−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	$0.652983\pi$
$692$	−16.2020			−0.615909
$693$		0			0
$694$	−33.3849			−1.26727
$695$	2.13487			0.0809802
$696$		0			0
$697$	−8.52037			−0.322732
$698$		−	5.48938i		−	0.207776i
$699$		0			0
$700$		−	3.08861i		−	0.116739i
$701$	−1.85538			−0.0700767			−0.0350384	−	0.999386i	$-0.511155\pi$
$701$	−1.85538			−0.0700767			−0.0350384	+	0.999386i	$0.511155\pi$
$702$		0			0
$703$			6.80933i			0.256819i
$704$	−0.130093	−	3.31407i	−0.00490307	−	0.124904i
$705$		0			0
$706$		−	12.4078i		−	0.466975i
$707$		−	27.5293i		−	1.03534i
$708$		0			0
$709$	18.1198			0.680504			0.340252	−	0.940334i	$-0.389488\pi$
$709$	18.1198			0.680504			0.340252	+	0.940334i	$0.389488\pi$
$710$	2.96330			0.111210
$711$		0			0
$712$		−	15.1558i		−	0.567987i
$713$		−	6.37824i		−	0.238867i
$714$		0			0
$715$	0.492034	+	12.5344i	0.0184010	+	0.468760i
$716$		−	7.02871i		−	0.262675i
$717$		0			0
$718$	26.9322			1.00510
$719$			15.1788i			0.566075i	0.959109	+	0.283038i	$0.0913421\pi$
$719$			15.1788i			0.566075i	−0.959109	+	0.283038i	$0.908658\pi$
$720$		0			0
$721$			32.1287i			1.19654i
$722$	−12.7359			−0.473980
$723$		0			0
$724$	8.64341			0.321230
$725$	4.36796			0.162222
$726$		0			0
$727$	9.92732			0.368184			0.184092	−	0.982909i	$-0.441066\pi$
$727$	9.92732			0.368184			0.184092	+	0.982909i	$0.441066\pi$
$728$	11.6817			0.432951
$729$		0			0
$730$	0.367959			0.0136188
$731$		−	0.269736i		−	0.00997655i
$732$		0			0
$733$		−	17.6355i		−	0.651381i	−0.945476	−	0.325691i	$-0.894403\pi$
$733$		−	17.6355i		−	0.651381i	0.945476	−	0.325691i	$-0.105597\pi$
$734$	−10.7590			−0.397121
$735$		0			0
$736$		−	2.69356i		−	0.0992859i
$737$	1.23927	+	31.5699i	0.0456491	+	1.16289i
$738$		0			0
$739$		−	16.7808i		−	0.617290i	−0.951177	−	0.308645i	$-0.900124\pi$
$739$		−	16.7808i		−	0.617290i	0.951177	−	0.308645i	$-0.0998755\pi$
$740$			2.72065i			0.100013i
$741$		0			0
$742$	38.7472			1.42246
$743$	28.1343			1.03215			0.516074	−	0.856544i	$-0.327393\pi$
$743$	28.1343			1.03215			0.516074	+	0.856544i	$0.327393\pi$
$744$		0			0
$745$			22.0496i			0.807836i
$746$		−	10.3600i		−	0.379305i
$747$		0			0
$748$	−0.107773	−	2.74547i	−0.00394056	−	0.100384i
$749$			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	$0.726664\pi$
$752$		−	5.67440i		−	0.206924i
$753$		0			0
$754$			16.5204i			0.601636i
$755$	16.7958			0.611263
$756$		0			0
$757$	5.39915			0.196236			0.0981178	−	0.995175i	$-0.468718\pi$
$757$	5.39915			0.196236			0.0981178	+	0.995175i	$0.468718\pi$
$758$	−20.8206			−0.756239
$759$		0			0
$760$	2.50283			0.0907871
$761$	46.3888			1.68159			0.840797	−	0.541351i	$-0.182087\pi$
$761$	46.3888			1.68159			0.840797	+	0.541351i	$0.182087\pi$
$762$		0			0
$763$	34.9979			1.26701
$764$			15.6145i			0.564913i
$765$		0			0
$766$		−	15.6010i		−	0.563687i
$767$	−2.72564			−0.0984171
$768$		0			0
$769$			0.838705i			0.0302445i	0.999886	+	0.0151222i	$0.00481374\pi$
$769$			0.838705i			0.0302445i	−0.999886	+	0.0151222i	$0.995186\pi$
$770$	−0.401807	−	10.2359i	−0.0144801	−	0.368876i
$771$		0			0
$772$			5.67036i			0.204081i
$773$		−	4.12304i		−	0.148295i	−0.997247	−	0.0741476i	$-0.976376\pi$
$773$		−	4.12304i		−	0.148295i	0.997247	−	0.0741476i	$-0.0236236\pi$
$774$		0			0
$775$	2.36796			0.0850596
$776$	8.23713			0.295696
$777$		0			0
$778$		−	24.3928i		−	0.874523i
$779$		−	25.7416i		−	0.922287i
$780$		0			0
$781$	9.82057	−	0.385504i	0.351408	−	0.0137944i
$782$		−	2.23142i		−	0.0797953i
$783$		0			0
$784$	−2.53953			−0.0906976
$785$		−	22.1191i		−	0.789464i
$786$		0			0
$787$		−	35.7130i		−	1.27303i	−0.771264	−	0.636516i	$-0.780375\pi$
$787$		−	35.7130i		−	1.27303i	0.771264	−	0.636516i	$-0.219625\pi$
$788$	20.6299			0.734910
$789$		0			0
$790$	9.19639			0.327193
$791$	−15.5914			−0.554365
$792$		0			0
$793$	−15.0623			−0.534879
$794$	−24.5931			−0.872775
$795$		0			0
$796$	15.4335			0.547026
$797$		−	3.89398i		−	0.137932i	−0.997619	−	0.0689660i	$-0.978030\pi$
$797$		−	3.89398i		−	0.137932i	0.997619	−	0.0689660i	$-0.0219700\pi$
$798$		0			0
$799$		−	4.70083i		−	0.166303i
$800$	1.00000			0.0353553
$801$		0			0
$802$		−	19.9459i		−	0.704314i
$803$	1.21944	−	0.0478689i	0.0430332	−	0.00168926i
$804$		0			0
$805$		−	8.31936i		−	0.293219i
$806$			8.95603i			0.315463i
$807$		0			0
$808$	8.91314			0.313563
$809$	−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	$0.372044\pi$
$811$			52.4159i			1.84057i	−0.391247	+	0.920286i	$0.627956\pi$
$812$		−	13.4909i		−	0.473439i
$813$		0			0
$814$	0.353938	+	9.01645i	0.0124055	+	0.316026i
$815$			25.3059i			0.886428i
$816$		0			0
$817$	0.814921			0.0285105
$818$		−	22.9322i		−	0.801807i
$819$		0			0
$820$		−	10.2850i		−	0.359168i
$821$	40.0418			1.39747			0.698735	−	0.715381i	$-0.253747\pi$
$821$	40.0418			1.39747			0.698735	+	0.715381i	$0.253747\pi$
$822$		0			0
$823$	−40.4158			−1.40881			−0.704404	−	0.709800i	$-0.748785\pi$
$823$	−40.4158			−1.40881			−0.704404	+	0.709800i	$0.748785\pi$
$824$	−10.4023			−0.362382
$825$		0			0
$826$	2.22582			0.0774463
$827$	11.8749			0.412930			0.206465	−	0.978454i	$-0.433804\pi$
$827$	11.8749			0.412930			0.206465	+	0.978454i	$0.433804\pi$
$828$		0			0
$829$	14.6132			0.507536			0.253768	−	0.967265i	$-0.418330\pi$
$829$	14.6132			0.507536			0.253768	+	0.967265i	$0.418330\pi$
$830$		−	2.11732i		−	0.0734933i
$831$		0			0
$832$			3.78217i			0.131123i
$833$	−2.10382			−0.0728929
$834$		0			0
$835$			14.1095i			0.488280i
$836$	8.29455	−	0.325600i	0.286873	−	0.0112611i
$837$		0			0
$838$			10.4622i			0.361412i
$839$			13.7917i			0.476143i	0.971248	+	0.238072i	$0.0765153\pi$
$839$			13.7917i			0.476143i	−0.971248	+	0.238072i	$0.923485\pi$
$840$		0			0
$841$	−9.92094			−0.342101
$842$	−4.04640			−0.139448
$843$		0			0
$844$		−	24.8821i		−	0.856477i
$845$		−	1.30483i		−	0.0448874i
$846$		0			0
$847$	−2.66324	−	33.8702i	−0.0915099	−	1.16379i
$848$			12.5452i			0.430803i
$849$		0			0
$850$	0.828427			0.0284148
$851$			7.32824i			0.251209i
$852$		0			0
$853$			3.96719i			0.135834i	0.997691	+	0.0679170i	$0.0216353\pi$
$853$			3.96719i			0.135834i	−0.997691	+	0.0679170i	$0.978365\pi$
$854$	12.3003			0.420906
$855$		0			0
$856$	−6.69760			−0.228919
$857$	33.5994			1.14773			0.573867	−	0.818948i	$-0.305442\pi$
$857$	33.5994			1.14773			0.573867	+	0.818948i	$0.305442\pi$
$858$		0			0
$859$	−29.4799			−1.00584			−0.502921	−	0.864332i	$-0.667741\pi$
$859$	−29.4799			−1.00584			−0.502921	+	0.864332i	$0.667741\pi$
$860$	0.325600			0.0111029
$861$		0			0
$862$	−25.9266			−0.883063
$863$		−	15.8491i		−	0.539511i	−0.962929	−	0.269756i	$-0.913057\pi$
$863$		−	15.8491i		−	0.539511i	0.962929	−	0.269756i	$-0.0869429\pi$
$864$		0			0
$865$		−	16.2020i		−	0.550886i
$866$	2.10953			0.0716847
$867$		0			0
$868$		−	7.31371i		−	0.248243i
$869$	30.4775	−	1.19639i	1.03388	−	0.0405846i
$870$		0			0
$871$		−	36.0291i		−	1.22080i
$872$			11.3313i			0.383725i
$873$		0			0
$874$	6.74151			0.228035
$875$	3.08861			0.104414
$876$		0			0
$877$		−	41.9243i		−	1.41568i	−0.706371	−	0.707842i	$-0.749669\pi$
$877$		−	41.9243i		−	1.41568i	0.706371	−	0.707842i	$-0.250331\pi$
$878$		−	18.5374i		−	0.625607i
$879$		0			0
$880$	3.31407	−	0.130093i	0.111717	−	0.00438544i
$881$			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
$883$	−12.4357			−0.418493			−0.209247	−	0.977863i	$-0.567101\pi$
$883$	−12.4357			−0.418493			−0.209247	+	0.977863i	$0.567101\pi$
$884$			3.13325i			0.105383i
$885$		0			0
$886$			12.3545i			0.415056i
$887$	−51.5193			−1.72985			−0.864924	−	0.501903i	$-0.832633\pi$
$887$	−51.5193			−1.72985			−0.864924	+	0.501903i	$0.832633\pi$
$888$		0			0
$889$	−37.8670			−1.27002
$890$	15.1558			0.508023
$891$		0			0
$892$	9.05601			0.303218
$893$	14.2020			0.475253
$894$		0			0
$895$	7.02871			0.234944
$896$		−	3.08861i		−	0.103183i
$897$		0			0
$898$		−	13.8038i		−	0.460637i
$899$	10.3431			0.344963
$900$		0			0
$901$			10.3928i			0.346233i
$902$	−1.33801	−	34.0852i	−0.0445508	−	1.13491i
$903$		0			0
$904$		−	5.04801i		−	0.167894i
$905$			8.64341i			0.287317i
$906$		0			0
$907$	−24.6242			−0.817633			−0.408816	−	0.912617i	$-0.634058\pi$
$907$	−24.6242			−0.817633			−0.408816	+	0.912617i	$0.634058\pi$
$908$	−17.1365			−0.568694
$909$		0			0
$910$			11.6817i			0.387243i
$911$		−	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
$913$	−0.275449	−	7.01696i	−0.00911603	−	0.232228i
$914$		−	19.1964i		−	0.634960i
$915$		0			0
$916$	25.6682			0.848100
$917$		−	39.4425i		−	1.30250i
$918$		0			0
$919$		−	36.0634i		−	1.18962i	−0.803865	−	0.594811i	$-0.797227\pi$
$919$		−	36.0634i		−	1.18962i	0.803865	−	0.594811i	$-0.202773\pi$
$920$	2.69356			0.0888040
$921$		0			0
$922$	36.3218			1.19620
$923$	−11.2077			−0.368906
$924$		0			0
$925$	−2.72065			−0.0894546
$926$	13.6473			0.448478
$927$		0			0
$928$	4.36796			0.143385
$929$		−	8.01902i		−	0.263095i	−0.991310	−	0.131548i	$-0.958005\pi$
$929$		−	8.01902i		−	0.263095i	0.991310	−	0.131548i	$-0.0419946\pi$
$930$		0			0
$931$		−	6.35601i		−	0.208310i
$932$	16.1772			0.529903
$933$		0			0
$934$		−	5.40622i		−	0.176897i
$935$	2.74547	−	0.107773i	0.0897864	−	0.00352454i
$936$		0			0
$937$			3.31122i			0.108173i	0.998536	+	0.0540865i	$0.0172247\pi$
$937$			3.31122i			0.108173i	−0.998536	+	0.0540865i	$0.982775\pi$
$938$			29.4222i			0.960669i
$939$		0			0
$940$	5.67440			0.185079
$941$	52.5352			1.71260			0.856299	−	0.516480i	$-0.172758\pi$
$941$	52.5352			1.71260			0.856299	+	0.516480i	$0.172758\pi$
$942$		0			0
$943$		−	27.7033i		−	0.902142i
$944$			0.720655i			0.0234553i
$945$		0			0
$946$	1.07906	−	0.0423583i	0.0350834	−	0.00137719i
$947$			22.7472i			0.739185i	0.929194	+	0.369593i	$0.120503\pi$
$947$			22.7472i			0.739185i	−0.929194	+	0.369593i	$0.879497\pi$
$948$		0			0
$949$	−1.39168			−0.0451759
$950$			2.50283i			0.0812024i
$951$		0			0
$952$		−	2.55869i		−	0.0829277i
$953$	16.6976			0.540888			0.270444	−	0.962736i	$-0.412829\pi$
$953$	16.6976			0.540888			0.270444	+	0.962736i	$0.412829\pi$
$954$		0			0
$955$	−15.6145			−0.505273
$956$	15.6842			0.507262
$957$		0			0
$958$	−4.67030			−0.150891
$959$	−22.7200			−0.733669
$960$		0			0
$961$	−25.3928			−0.819122
$962$		−	10.2900i		−	0.331762i
$963$		0			0
$964$		−	20.9730i		−	0.675496i
$965$	−5.67036			−0.182535
$966$		0			0
$967$		−	31.4115i		−	1.01013i	−0.863083	−	0.505063i	$-0.831469\pi$
$967$		−	31.4115i		−	1.01013i	0.863083	−	0.505063i	$-0.168531\pi$
$968$	10.9662	−	0.862276i	0.352465	−	0.0277146i
$969$		0			0
$970$			8.23713i			0.264478i
$971$		−	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
$973$	6.59378			0.211387
$974$	10.4271			0.334107
$975$		0			0
$976$			3.98245i			0.127475i
$977$			49.0515i			1.56930i	0.619941	+	0.784649i	$0.287157\pi$
$977$			49.0515i			1.56930i	−0.619941	+	0.784649i	$0.712843\pi$
$978$		0			0
$979$	50.2274	−	1.97166i	1.60527	−	0.0630146i
$980$		−	2.53953i		−	0.0811224i
$981$		0			0
$982$	0.434928			0.0138791
$983$			42.1487i			1.34433i	0.740400	+	0.672167i	$0.234636\pi$
$983$			42.1487i			1.34433i	−0.740400	+	0.672167i	$0.765364\pi$
$984$		0			0
$985$			20.6299i			0.657323i
$986$	3.61854			0.115238
$987$		0			0
$988$	−9.46612			−0.301157
$989$	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	$0.621279\pi$
$992$	2.36796			0.0751828
$993$		0			0
$994$	9.15247			0.290299
$995$			15.4335i			0.489275i
$996$		0			0
$997$		−	21.6436i		−	0.685458i	−0.939434	−	0.342729i	$-0.888649\pi$
$997$		−	21.6436i		−	0.685458i	0.939434	−	0.342729i	$-0.111351\pi$
$998$	−11.3002			−0.357702
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.d.a.791.1	✓	8	33.32	even	2	inner
990.2.d.a.791.8	yes	8	1.1	even	1	trivial
990.2.d.b.791.4	yes	8	3.2	odd	2
990.2.d.b.791.5	yes	8	11.10	odd	2
4950.2.d.h.4751.2		8	15.14	odd	2
4950.2.d.h.4751.7		8	55.54	odd	2
4950.2.d.m.4751.2		8	5.4	even	2
4950.2.d.m.4751.7		8	165.164	even	2
4950.2.f.c.4949.1		8	55.43	even	4
4950.2.f.c.4949.5		8	15.2	even	4
4950.2.f.d.4949.1		8	5.2	odd	4
4950.2.f.d.4949.5		8	165.98	odd	4
4950.2.f.e.4949.4		8	165.32	odd	4
4950.2.f.e.4949.8		8	5.3	odd	4
4950.2.f.f.4949.4		8	15.8	even	4
4950.2.f.f.4949.8		8	55.32	even	4
7920.2.f.a.3761.4		8	132.131	odd	2
7920.2.f.a.3761.5		8	4.3	odd	2
7920.2.f.b.3761.1		8	12.11	even	2
7920.2.f.b.3761.8		8	44.43	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	990.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000i q^{5} +3.08861i q^{7} -1.00000 q^{8} -1.00000i q^{10} +(-0.130093 - 3.31407i) q^{11} +3.78217i q^{13} -3.08861i q^{14} +1.00000 q^{16} +0.828427 q^{17} +2.50283i q^{19} +1.00000i q^{20} +(0.130093 + 3.31407i) q^{22} +2.69356i q^{23} -1.00000 q^{25} -3.78217i q^{26} +3.08861i q^{28} -4.36796 q^{29} -2.36796 q^{31} -1.00000 q^{32} -0.828427 q^{34} -3.08861 q^{35} +2.72065 q^{37} -2.50283i q^{38} -1.00000i q^{40} -10.2850 q^{41} -0.325600i q^{43} +(-0.130093 - 3.31407i) q^{44} -2.69356i q^{46} -5.67440i q^{47} -2.53953 q^{49} +1.00000 q^{50} +3.78217i q^{52} +12.5452i q^{53} +(3.31407 - 0.130093i) q^{55} -3.08861i q^{56} +4.36796 q^{58} +0.720655i q^{59} +3.98245i q^{61} +2.36796 q^{62} +1.00000 q^{64} -3.78217 q^{65} -9.52603 q^{67} +0.828427 q^{68} +3.08861 q^{70} +2.96330i q^{71} +0.367959i q^{73} -2.72065 q^{74} +2.50283i q^{76} +(10.2359 - 0.401807i) q^{77} +9.19639i q^{79} +1.00000i q^{80} +10.2850 q^{82} +2.11732 q^{83} +0.828427i q^{85} +0.325600i q^{86} +(0.130093 + 3.31407i) q^{88} +15.1558i q^{89} -11.6817 q^{91} +2.69356i q^{92} +5.67440i q^{94} -2.50283 q^{95} -8.23713 q^{97} +2.53953 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 16 q^{17} - 8 q^{25} - 8 q^{29} + 8 q^{31} - 8 q^{32} + 16 q^{34} + 24 q^{37} - 8 q^{41} - 16 q^{49} + 8 q^{50} + 12 q^{55} + 8 q^{58} - 8 q^{62} + 8 q^{64}+ \cdots + 16 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 8 * q^16 - 16 * q^17 - 8 * q^25 - 8 * q^29 + 8 * q^31 - 8 * q^32 + 16 * q^34 + 24 * q^37 - 8 * q^41 - 16 * q^49 + 8 * q^50 + 12 * q^55 + 8 * q^58 - 8 * q^62 + 8 * q^64 + 8 * q^65 - 16 * q^68 - 24 * q^74 + 12 * q^77 + 8 * q^82 - 24 * q^83 + 24 * q^91 + 16 * q^95 - 8 * q^97 + 16 * q^98

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