LMFDB - Embedded newform 90.2.c.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,2,Mod(19,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("90.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	90.c (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:	$0.718653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	90.19
Dual form			90.2.c.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} -2.00000 q^{11} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +(-2.00000 + 1.00000i) q^{20} +2.00000i q^{22} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{26} +2.00000i q^{28} -8.00000 q^{31} -1.00000i q^{32} +2.00000 q^{34} +(-2.00000 - 4.00000i) q^{35} -2.00000i q^{37} +(1.00000 + 2.00000i) q^{40} -2.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -6.00000i q^{52} -6.00000i q^{53} +(-4.00000 + 2.00000i) q^{55} +2.00000 q^{56} +10.0000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +(6.00000 + 12.0000i) q^{65} +8.00000i q^{67} -2.00000i q^{68} +(-4.00000 + 2.00000i) q^{70} -12.0000 q^{71} -4.00000i q^{73} -2.00000 q^{74} +4.00000i q^{77} +(2.00000 - 1.00000i) q^{80} +2.00000i q^{82} +4.00000i q^{83} +(2.00000 + 4.00000i) q^{85} -4.00000 q^{86} -2.00000i q^{88} -10.0000 q^{89} +12.0000 q^{91} -4.00000i q^{92} -8.00000 q^{94} +8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^5	$ 2 q - 2 q^{4} + 4 q^{5} - 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{20} + 6 q^{25} + 12 q^{26} - 16 q^{31} + 4 q^{34} - 4 q^{35} + 2 q^{40} - 4 q^{41} + 4 q^{44} + 8 q^{46} + 6 q^{49} - 8 q^{50} - 8 q^{55} + 4 q^{56} + 20 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{65} - 8 q^{70} - 24 q^{71} - 4 q^{74} + 4 q^{80} + 4 q^{85} - 8 q^{86} - 20 q^{89} + 24 q^{91} - 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^20 + 6 * q^25 + 12 * q^26 - 16 * q^31 + 4 * q^34 - 4 * q^35 + 2 * q^40 - 4 * q^41 + 4 * q^44 + 8 * q^46 + 6 * q^49 - 8 * q^50 - 8 * q^55 + 4 * q^56 + 20 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^65 - 8 * q^70 - 24 * q^71 - 4 * q^74 + 4 * q^80 + 4 * q^85 - 8 * q^86 - 20 * q^89 + 24 * q^91 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000			−0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−2.00000	+	1.00000i	−0.447214	+	0.223607i
$21$		0			0
$22$			2.00000i			0.426401i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	6.00000			1.17670
$27$		0			0
$28$			2.00000i			0.377964i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	2.00000			0.342997
$35$	−2.00000	−	4.00000i	−0.338062	−	0.676123i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	4.00000			0.589768
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$		−	6.00000i		−	0.832050i
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$	2.00000			0.267261
$57$		0			0
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$			8.00000i			1.01600i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	6.00000	+	12.0000i	0.744208	+	1.48842i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		−	2.00000i		−	0.242536i
$69$		0			0
$70$	−4.00000	+	2.00000i	−0.478091	+	0.239046i
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−2.00000			−0.232495
$75$		0			0
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	2.00000	−	1.00000i	0.223607	−	0.111803i
$81$		0			0
$82$			2.00000i			0.220863i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	2.00000	+	4.00000i	0.216930	+	0.433861i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	2.00000i		−	0.213201i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		−	4.00000i		−	0.417029i
$93$		0			0
$94$	−8.00000			−0.825137
$95$		0			0
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$		−	14.0000i		−	1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$		−	14.0000i		−	1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−6.00000			−0.582772
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	2.00000	+	4.00000i	0.190693	+	0.381385i
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	4.00000	+	8.00000i	0.373002	+	0.746004i
$116$		0			0
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$	4.00000			0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$	8.00000			0.718421
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	12.0000	−	6.00000i	1.05247	−	0.526235i
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$	8.00000			0.691095
$135$		0			0
$136$	−2.00000			−0.171499
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	2.00000	+	4.00000i	0.169031	+	0.338062i
$141$		0			0
$142$			12.0000i			1.00702i
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$		0			0
$146$	−4.00000			−0.331042
$147$		0			0
$148$			2.00000i			0.164399i
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$	4.00000			0.322329
$155$	−16.0000	+	8.00000i	−1.28515	+	0.642575i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$	8.00000			0.630488
$162$		0			0
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	2.00000			0.156174
$165$		0			0
$166$	4.00000			0.310460
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$	4.00000	−	2.00000i	0.306786	−	0.153393i
$171$		0			0
$172$			4.00000i			0.304997i
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$	−2.00000			−0.150756
$177$		0			0
$178$			10.0000i			0.749532i
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		−	12.0000i		−	0.889499i
$183$		0			0
$184$	−4.00000			−0.294884
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$		−	4.00000i		−	0.292509i
$188$			8.00000i			0.583460i
$189$		0			0
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	8.00000			0.574367
$195$		0			0
$196$	−3.00000			−0.214286
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$		0			0
$202$		−	8.00000i		−	0.562878i
$203$		0			0
$204$		0			0
$205$	−4.00000	+	2.00000i	−0.279372	+	0.139686i
$206$	−14.0000			−0.975426
$207$		0			0
$208$			6.00000i			0.416025i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	12.0000			0.820303
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$			16.0000i			1.08615i
$218$			10.0000i			0.677285i
$219$		0			0
$220$	4.00000	−	2.00000i	0.269680	−	0.134840i
$221$	−12.0000			−0.807207
$222$		0			0
$223$			26.0000i			1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$			26.0000i			1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	−2.00000			−0.133631
$225$		0			0
$226$	−6.00000			−0.399114
$227$		−	28.0000i		−	1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$		−	28.0000i		−	1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	8.00000	−	4.00000i	0.527504	−	0.263752i
$231$		0			0
$232$		0			0
$233$			14.0000i			0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$			14.0000i			0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$		0			0
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$	−10.0000			−0.650945
$237$		0			0
$238$		−	4.00000i		−	0.259281i
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$			7.00000i			0.449977i
$243$		0			0
$244$	−2.00000			−0.128037
$245$	6.00000	−	3.00000i	0.383326	−	0.191663i
$246$		0			0
$247$		0			0
$248$		−	8.00000i		−	0.508001i
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$	−2.00000			−0.125491
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$	−6.00000	−	12.0000i	−0.372104	−	0.744208i
$261$		0			0
$262$		−	18.0000i		−	1.11204i
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$		0			0
$268$		−	8.00000i		−	0.488678i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$			2.00000i			0.121268i
$273$		0			0
$274$	−18.0000			−1.08742
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		−	20.0000i		−	1.19952i
$279$		0			0
$280$	4.00000	−	2.00000i	0.239046	−	0.119523i
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	12.0000			0.712069
$285$		0			0
$286$	−12.0000			−0.709575
$287$			4.00000i			0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$			4.00000i			0.234082i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	20.0000	−	10.0000i	1.16445	−	0.582223i
$296$	2.00000			0.116248
$297$		0			0
$298$			20.0000i			1.15857i
$299$	−24.0000			−1.38796
$300$		0			0
$301$	−8.00000			−0.461112
$302$			8.00000i			0.460348i
$303$		0			0
$304$		0			0
$305$	4.00000	−	2.00000i	0.229039	−	0.114520i
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		−	4.00000i		−	0.227921i
$309$		0			0
$310$	8.00000	+	16.0000i	0.454369	+	0.908739i
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$		−	4.00000i		−	0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$		−	4.00000i		−	0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	−22.0000			−1.24153
$315$		0			0
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$		0			0
$320$	−2.00000	+	1.00000i	−0.111803	+	0.0559017i
$321$		0			0
$322$		−	8.00000i		−	0.445823i
$323$		0			0
$324$		0			0
$325$	24.0000	+	18.0000i	1.33128	+	0.998460i
$326$	16.0000			0.886158
$327$		0			0
$328$		−	2.00000i		−	0.110432i
$329$	−16.0000			−0.882109
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		−	4.00000i		−	0.219529i
$333$		0			0
$334$	12.0000			0.656611
$335$	8.00000	+	16.0000i	0.437087	+	0.874173i
$336$		0			0
$337$			28.0000i			1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$			28.0000i			1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$			23.0000i			1.25104i
$339$		0			0
$340$	−2.00000	−	4.00000i	−0.108465	−	0.216930i
$341$	16.0000			0.866449
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$	4.00000			0.215666
$345$		0			0
$346$	14.0000			0.752645
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$	−6.00000	+	8.00000i	−0.320713	+	0.427618i
$351$		0			0
$352$			2.00000i			0.106600i
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$		0			0
$355$	−24.0000	+	12.0000i	−1.27379	+	0.636894i
$356$	10.0000			0.529999
$357$		0			0
$358$		−	10.0000i		−	0.528516i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		−	2.00000i		−	0.105118i
$363$		0			0
$364$	−12.0000			−0.628971
$365$	−4.00000	−	8.00000i	−0.209370	−	0.418739i
$366$		0			0
$367$		−	2.00000i		−	0.104399i	−0.998637	−	0.0521996i	$-0.983377\pi$
$367$		−	2.00000i		−	0.104399i	0.998637	−	0.0521996i	$-0.0166232\pi$
$368$			4.00000i			0.208514i
$369$		0			0
$370$	−4.00000	+	2.00000i	−0.207950	+	0.103975i
$371$	−12.0000			−0.623009
$372$		0			0
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	−4.00000			−0.206835
$375$		0			0
$376$	8.00000			0.412568
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$			12.0000i			0.613973i
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$	4.00000	+	8.00000i	0.203859	+	0.407718i
$386$	−4.00000			−0.203595
$387$		0			0
$388$		−	8.00000i		−	0.406138i
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$			3.00000i			0.151523i
$393$		0			0
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		0			0
$403$		−	48.0000i		−	2.39105i
$404$	−8.00000			−0.398015
$405$		0			0
$406$		0			0
$407$			4.00000i			0.198273i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	+	4.00000i	0.0987730	+	0.197546i
$411$		0			0
$412$			14.0000i			0.689730i
$413$		−	20.0000i		−	0.984136i
$414$		0			0
$415$	4.00000	+	8.00000i	0.196352	+	0.392705i
$416$	6.00000			0.294174
$417$		0			0
$418$		0			0
$419$	−10.0000			−0.488532			−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000			−0.488532			−0.244266	+	0.969708i	$0.578547\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$	6.00000			0.291386
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		−	12.0000i		−	0.580042i
$429$		0			0
$430$	−8.00000	+	4.00000i	−0.385794	+	0.192897i
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	16.0000			0.768025
$435$		0			0
$436$	10.0000			0.478913
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−2.00000	−	4.00000i	−0.0953463	−	0.190693i
$441$		0			0
$442$			12.0000i			0.570782i
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$		0			0
$445$	−20.0000	+	10.0000i	−0.948091	+	0.474045i
$446$	26.0000			1.23114
$447$		0			0
$448$			2.00000i			0.0944911i
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$			6.00000i			0.282216i
$453$		0			0
$454$	−28.0000			−1.31411
$455$	24.0000	−	12.0000i	1.12514	−	0.562569i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		−	10.0000i		−	0.467269i
$459$		0			0
$460$	−4.00000	−	8.00000i	−0.186501	−	0.373002i
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$		0			0
$466$	14.0000			0.648537
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	−16.0000	+	8.00000i	−0.738025	+	0.369012i
$471$		0			0
$472$			10.0000i			0.460287i
$473$			8.00000i			0.367840i
$474$		0			0
$475$		0			0
$476$	−4.00000			−0.183340
$477$		0			0
$478$		−	20.0000i		−	0.914779i
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		−	22.0000i		−	1.00207i
$483$		0			0
$484$	7.00000			0.318182
$485$	8.00000	+	16.0000i	0.363261	+	0.726523i
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$			2.00000i			0.0905357i
$489$		0			0
$490$	−3.00000	−	6.00000i	−0.135526	−	0.271052i
$491$	18.0000			0.812329			0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000			0.812329			0.406164	+	0.913800i	$0.366866\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−8.00000			−0.359211
$497$			24.0000i			1.07655i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−2.00000	+	11.0000i	−0.0894427	+	0.491935i
$501$		0			0
$502$		−	18.0000i		−	0.803379i
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	16.0000	−	8.00000i	0.711991	−	0.355995i
$506$	−8.00000			−0.355643
$507$		0			0
$508$			2.00000i			0.0887357i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−18.0000			−0.793946
$515$	−14.0000	−	28.0000i	−0.616914	−	1.23383i
$516$		0			0
$517$			16.0000i			0.703679i
$518$			4.00000i			0.175750i
$519$		0			0
$520$	−12.0000	+	6.00000i	−0.526235	+	0.263117i
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−18.0000			−0.786334
$525$		0			0
$526$	4.00000			0.174408
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$	−12.0000	+	6.00000i	−0.521247	+	0.260623i
$531$		0			0
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	12.0000	+	24.0000i	0.518805	+	1.03761i
$536$	−8.00000			−0.345547
$537$		0			0
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	−38.0000			−1.63375			−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000			−1.63375			−0.816874	+	0.576816i	$0.804295\pi$
$542$			8.00000i			0.343629i
$543$		0			0
$544$	2.00000			0.0857493
$545$	−20.0000	+	10.0000i	−0.856706	+	0.428353i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	8.00000	+	6.00000i	0.341121	+	0.255841i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−2.00000			−0.0849719
$555$		0			0
$556$	−20.0000			−0.848189
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$	−2.00000	−	4.00000i	−0.0845154	−	0.169031i
$561$		0			0
$562$		−	18.0000i		−	0.759284i
$563$			44.0000i			1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$			44.0000i			1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$		0			0
$565$	−6.00000	−	12.0000i	−0.252422	−	0.504844i
$566$	16.0000			0.672530
$567$		0			0
$568$		−	12.0000i		−	0.503509i
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$	−8.00000			−0.334790			−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000			−0.334790			−0.167395	+	0.985890i	$0.553535\pi$
$572$			12.0000i			0.501745i
$573$		0			0
$574$	4.00000			0.166957
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$			12.0000i			0.496989i
$584$	4.00000			0.165521
$585$		0			0
$586$	−6.00000			−0.247858
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$		0			0
$590$	−10.0000	−	20.0000i	−0.411693	−	0.823387i
$591$		0			0
$592$		−	2.00000i		−	0.0821995i
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$	8.00000	−	4.00000i	0.327968	−	0.163984i
$596$	20.0000			0.819232
$597$		0			0
$598$			24.0000i			0.981433i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$			8.00000i			0.326056i
$603$		0			0
$604$	8.00000			0.325515
$605$	−14.0000	+	7.00000i	−0.569181	+	0.284590i
$606$		0			0
$607$		−	22.0000i		−	0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$		−	22.0000i		−	0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$		0			0
$609$		0			0
$610$	−2.00000	−	4.00000i	−0.0809776	−	0.161955i
$611$	48.0000			1.94187
$612$		0			0
$613$			26.0000i			1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$			26.0000i			1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$	−4.00000			−0.161165
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$	16.0000	−	8.00000i	0.642575	−	0.321288i
$621$		0			0
$622$			12.0000i			0.481156i
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−4.00000			−0.159872
$627$		0			0
$628$			22.0000i			0.877896i
$629$	4.00000			0.159490
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		0			0
$634$	2.00000			0.0794301
$635$	−2.00000	−	4.00000i	−0.0793676	−	0.158735i
$636$		0			0
$637$			18.0000i			0.713186i
$638$		0			0
$639$		0			0
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		−	24.0000i		−	0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$		−	24.0000i		−	0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$	−8.00000			−0.315244
$645$		0			0
$646$		0			0
$647$		−	48.0000i		−	1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$		−	48.0000i		−	1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$	18.0000	−	24.0000i	0.706018	−	0.941357i
$651$		0			0
$652$		−	16.0000i		−	0.626608i
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$	36.0000	−	18.0000i	1.40664	−	0.703318i
$656$	−2.00000			−0.0780869
$657$		0			0
$658$			16.0000i			0.623745i
$659$	−50.0000			−1.94772			−0.973862	−	0.227142i	$-0.927062\pi$
$659$	−50.0000			−1.94772			−0.973862	+	0.227142i	$0.927062\pi$
$660$		0			0
$661$	2.00000			0.0777910			0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000			0.0777910			0.0388955	+	0.999243i	$0.487616\pi$
$662$			8.00000i			0.310929i
$663$		0			0
$664$	−4.00000			−0.155230
$665$		0			0
$666$		0			0
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	16.0000	−	8.00000i	0.618134	−	0.309067i
$671$	−4.00000			−0.154418
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$	28.0000			1.07852
$675$		0			0
$676$	23.0000			0.884615
$677$			2.00000i			0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$			2.00000i			0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$	−4.00000	+	2.00000i	−0.153393	+	0.0766965i
$681$		0			0
$682$		−	16.0000i		−	0.612672i
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	−18.0000	−	36.0000i	−0.687745	−	1.37549i
$686$	−20.0000			−0.763604
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	36.0000			1.37149
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		−	14.0000i		−	0.532200i
$693$		0			0
$694$	12.0000			0.455514
$695$	40.0000	−	20.0000i	1.51729	−	0.758643i
$696$		0			0
$697$		−	4.00000i		−	0.151511i
$698$			10.0000i			0.378506i
$699$		0			0
$700$	8.00000	+	6.00000i	0.302372	+	0.226779i
$701$	−32.0000			−1.20862			−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000			−1.20862			−0.604312	+	0.796748i	$0.706552\pi$
$702$		0			0
$703$		0			0
$704$	2.00000			0.0753778
$705$		0			0
$706$	14.0000			0.526897
$707$		−	16.0000i		−	0.601742i
$708$		0			0
$709$	−30.0000			−1.12667			−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000			−1.12667			−0.563337	+	0.826227i	$0.690483\pi$
$710$	12.0000	+	24.0000i	0.450352	+	0.900704i
$711$		0			0
$712$		−	10.0000i		−	0.374766i
$713$		−	32.0000i		−	1.19841i
$714$		0			0
$715$	−12.0000	−	24.0000i	−0.448775	−	0.897549i
$716$	−10.0000			−0.373718
$717$		0			0
$718$		0			0
$719$	−40.0000			−1.49175			−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000			−1.49175			−0.745874	+	0.666087i	$0.767968\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$			19.0000i			0.707107i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$		0			0
$726$		0			0
$727$			18.0000i			0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$			18.0000i			0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$			12.0000i			0.444750i
$729$		0			0
$730$	−8.00000	+	4.00000i	−0.296093	+	0.148047i
$731$	8.00000			0.295891
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−2.00000			−0.0738213
$735$		0			0
$736$	4.00000			0.147442
$737$		−	16.0000i		−	0.589368i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$	2.00000	+	4.00000i	0.0735215	+	0.147043i
$741$		0			0
$742$			12.0000i			0.440534i
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	−40.0000	+	20.0000i	−1.46549	+	0.732743i
$746$	6.00000			0.219676
$747$		0			0
$748$			4.00000i			0.146254i
$749$	24.0000			0.876941
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		−	8.00000i		−	0.291730i
$753$		0			0
$754$		0			0
$755$	−16.0000	+	8.00000i	−0.582300	+	0.291150i
$756$		0			0
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$			20.0000i			0.726433i
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$			20.0000i			0.724049i
$764$	12.0000			0.434145
$765$		0			0
$766$	−16.0000			−0.578103
$767$			60.0000i			2.16647i
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$	8.00000	−	4.00000i	0.288300	−	0.144150i
$771$		0			0
$772$			4.00000i			0.143963i
$773$			54.0000i			1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$			54.0000i			1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$	−8.00000			−0.287183
$777$		0			0
$778$		−	20.0000i		−	0.717035i
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$			8.00000i			0.286079i
$783$		0			0
$784$	3.00000			0.107143
$785$	−22.0000	−	44.0000i	−0.785214	−	1.57043i
$786$		0			0
$787$		−	32.0000i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		−	22.0000i		−	0.783718i
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$			12.0000i			0.426132i
$794$	−2.00000			−0.0709773
$795$		0			0
$796$		0			0
$797$			2.00000i			0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$			2.00000i			0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$		0			0
$799$	16.0000			0.566039
$800$	−4.00000	−	3.00000i	−0.141421	−	0.106066i
$801$		0			0
$802$			22.0000i			0.776847i
$803$			8.00000i			0.282314i
$804$		0			0
$805$	16.0000	−	8.00000i	0.563926	−	0.281963i
$806$	−48.0000			−1.69073
$807$		0			0
$808$			8.00000i			0.281439i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	52.0000			1.82597			0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000			1.82597			0.912983	+	0.407997i	$0.133772\pi$
$812$		0			0
$813$		0			0
$814$	4.00000			0.140200
$815$	16.0000	+	32.0000i	0.560456	+	1.12091i
$816$		0			0
$817$		0			0
$818$		−	10.0000i		−	0.349642i
$819$		0			0
$820$	4.00000	−	2.00000i	0.139686	−	0.0698430i
$821$	8.00000			0.279202			0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000			0.279202			0.139601	+	0.990208i	$0.455418\pi$
$822$		0			0
$823$			6.00000i			0.209147i	0.994517	+	0.104573i	$0.0333477\pi$
$823$			6.00000i			0.209147i	−0.994517	+	0.104573i	$0.966652\pi$
$824$	14.0000			0.487713
$825$		0			0
$826$	−20.0000			−0.695889
$827$		−	28.0000i		−	0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$		−	28.0000i		−	0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$	8.00000	−	4.00000i	0.277684	−	0.138842i
$831$		0			0
$832$		−	6.00000i		−	0.208013i
$833$			6.00000i			0.207888i
$834$		0			0
$835$	12.0000	+	24.0000i	0.415277	+	0.830554i
$836$		0			0
$837$		0			0
$838$			10.0000i			0.345444i
$839$	40.0000			1.38095			0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000			1.38095			0.690477	+	0.723355i	$0.257401\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		−	22.0000i		−	0.758170i
$843$		0			0
$844$	−12.0000			−0.413057
$845$	−46.0000	+	23.0000i	−1.58245	+	0.791224i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		−	6.00000i		−	0.206041i
$849$		0			0
$850$	6.00000	−	8.00000i	0.205798	−	0.274398i
$851$	8.00000			0.274236
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−4.00000			−0.136877
$855$		0			0
$856$	−12.0000			−0.410152
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$	4.00000	+	8.00000i	0.136399	+	0.272798i
$861$		0			0
$862$			32.0000i			1.08992i
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$	14.0000	+	28.0000i	0.476014	+	0.952029i
$866$	−4.00000			−0.135926
$867$		0			0
$868$		−	16.0000i		−	0.543075i
$869$		0			0
$870$		0			0
$871$	−48.0000			−1.62642
$872$		−	10.0000i		−	0.338643i
$873$		0			0
$874$		0			0
$875$	−22.0000	−	4.00000i	−0.743736	−	0.135225i
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$		0			0
$879$		0			0
$880$	−4.00000	+	2.00000i	−0.134840	+	0.0674200i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	12.0000			0.403604
$885$		0			0
$886$	−36.0000			−1.20944
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	−4.00000			−0.134156
$890$	10.0000	+	20.0000i	0.335201	+	0.670402i
$891$		0			0
$892$		−	26.0000i		−	0.870544i
$893$		0			0
$894$		0			0
$895$	20.0000	−	10.0000i	0.668526	−	0.334263i
$896$	2.00000			0.0668153
$897$		0			0
$898$		−	30.0000i		−	1.00111i
$899$		0			0
$900$		0			0
$901$	12.0000			0.399778
$902$		−	4.00000i		−	0.133185i
$903$		0			0
$904$	6.00000			0.199557
$905$	4.00000	−	2.00000i	0.132964	−	0.0664822i
$906$		0			0
$907$		−	12.0000i		−	0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$		−	12.0000i		−	0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$			28.0000i			0.929213i
$909$		0			0
$910$	−12.0000	−	24.0000i	−0.397796	−	0.795592i
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		−	8.00000i		−	0.264761i
$914$	−32.0000			−1.05847
$915$		0			0
$916$	−10.0000			−0.330409
$917$		−	36.0000i		−	1.18882i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−8.00000	+	4.00000i	−0.263752	+	0.131876i
$921$		0			0
$922$			12.0000i			0.395199i
$923$		−	72.0000i		−	2.36991i
$924$		0			0
$925$	−8.00000	−	6.00000i	−0.263038	−	0.197279i
$926$	6.00000			0.197172
$927$		0			0
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$		0			0
$932$		−	14.0000i		−	0.458585i
$933$		0			0
$934$	12.0000			0.392652
$935$	−4.00000	−	8.00000i	−0.130814	−	0.261628i
$936$		0			0
$937$			8.00000i			0.261349i	0.991425	+	0.130674i	$0.0417142\pi$
$937$			8.00000i			0.261349i	−0.991425	+	0.130674i	$0.958286\pi$
$938$		−	16.0000i		−	0.522419i
$939$		0			0
$940$	8.00000	+	16.0000i	0.260931	+	0.521862i
$941$	28.0000			0.912774			0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000			0.912774			0.456387	+	0.889781i	$0.349143\pi$
$942$		0			0
$943$		−	8.00000i		−	0.260516i
$944$	10.0000			0.325472
$945$		0			0
$946$	8.00000			0.260102
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$		0			0
$952$			4.00000i			0.129641i
$953$		−	46.0000i		−	1.49009i	−0.667016	−	0.745043i	$-0.732429\pi$
$953$		−	46.0000i		−	1.49009i	0.667016	−	0.745043i	$-0.267571\pi$
$954$		0			0
$955$	−24.0000	+	12.0000i	−0.776622	+	0.388311i
$956$	−20.0000			−0.646846
$957$		0			0
$958$		−	20.0000i		−	0.646171i
$959$	−36.0000			−1.16250
$960$		0			0
$961$	33.0000			1.06452
$962$		−	12.0000i		−	0.386896i
$963$		0			0
$964$	−22.0000			−0.708572
$965$	−4.00000	−	8.00000i	−0.128765	−	0.257529i
$966$		0			0
$967$			38.0000i			1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$			38.0000i			1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$		−	7.00000i		−	0.224989i
$969$		0			0
$970$	16.0000	−	8.00000i	0.513729	−	0.256865i
$971$	18.0000			0.577647			0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000			0.577647			0.288824	+	0.957382i	$0.406736\pi$
$972$		0			0
$973$		−	40.0000i		−	1.28234i
$974$	18.0000			0.576757
$975$		0			0
$976$	2.00000			0.0640184
$977$			42.0000i			1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		0			0
$979$	20.0000			0.639203
$980$	−6.00000	+	3.00000i	−0.191663	+	0.0958315i
$981$		0			0
$982$		−	18.0000i		−	0.574403i
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	22.0000	+	44.0000i	0.700978	+	1.40196i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.0000			0.508770
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$			8.00000i			0.254000i
$993$		0			0
$994$	24.0000			0.761234
$995$		0			0
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.2.c.a.19.1		2
3.2	odd	2		30.2.c.a.19.2	yes	2
4.3	odd	2		720.2.f.f.289.1		2
5.2	odd	4		450.2.a.f.1.1		1
5.3	odd	4		450.2.a.b.1.1		1
5.4	even	2	inner	90.2.c.a.19.2		2
8.3	odd	2		2880.2.f.c.1729.2		2
8.5	even	2		2880.2.f.e.1729.2		2
9.2	odd	6		810.2.i.e.109.2		4
9.4	even	3		810.2.i.b.379.2		4
9.5	odd	6		810.2.i.e.379.1		4
9.7	even	3		810.2.i.b.109.1		4
12.11	even	2		240.2.f.a.49.2		2
15.2	even	4		150.2.a.a.1.1		1
15.8	even	4		150.2.a.c.1.1		1
15.14	odd	2		30.2.c.a.19.1	✓	2
20.3	even	4		3600.2.a.bg.1.1		1
20.7	even	4		3600.2.a.o.1.1		1
20.19	odd	2		720.2.f.f.289.2		2
21.2	odd	6		1470.2.n.h.949.2		4
21.5	even	6		1470.2.n.a.949.2		4
21.11	odd	6		1470.2.n.h.79.1		4
21.17	even	6		1470.2.n.a.79.1		4
21.20	even	2		1470.2.g.g.589.2		2
24.5	odd	2		960.2.f.h.769.2		2
24.11	even	2		960.2.f.i.769.1		2
40.19	odd	2		2880.2.f.c.1729.1		2
40.29	even	2		2880.2.f.e.1729.1		2
45.4	even	6		810.2.i.b.379.1		4
45.14	odd	6		810.2.i.e.379.2		4
45.29	odd	6		810.2.i.e.109.1		4
45.34	even	6		810.2.i.b.109.2		4
48.5	odd	4		3840.2.d.g.2689.1		2
48.11	even	4		3840.2.d.x.2689.1		2
48.29	odd	4		3840.2.d.y.2689.2		2
48.35	even	4		3840.2.d.j.2689.2		2
60.23	odd	4		1200.2.a.g.1.1		1
60.47	odd	4		1200.2.a.m.1.1		1
60.59	even	2		240.2.f.a.49.1		2
105.44	odd	6		1470.2.n.h.949.1		4
105.59	even	6		1470.2.n.a.79.2		4
105.62	odd	4		7350.2.a.bg.1.1		1
105.74	odd	6		1470.2.n.h.79.2		4
105.83	odd	4		7350.2.a.cc.1.1		1
105.89	even	6		1470.2.n.a.949.1		4
105.104	even	2		1470.2.g.g.589.1		2
120.29	odd	2		960.2.f.h.769.1		2
120.53	even	4		4800.2.a.l.1.1		1
120.59	even	2		960.2.f.i.769.2		2
120.77	even	4		4800.2.a.cg.1.1		1
120.83	odd	4		4800.2.a.cj.1.1		1
120.107	odd	4		4800.2.a.m.1.1		1
240.29	odd	4		3840.2.d.g.2689.2		2
240.59	even	4		3840.2.d.j.2689.1		2
240.149	odd	4		3840.2.d.y.2689.1		2
240.179	even	4		3840.2.d.x.2689.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.c.a.19.1	✓	2	15.14	odd	2
30.2.c.a.19.2	yes	2	3.2	odd	2
90.2.c.a.19.1		2	1.1	even	1	trivial
90.2.c.a.19.2		2	5.4	even	2	inner
150.2.a.a.1.1		1	15.2	even	4
150.2.a.c.1.1		1	15.8	even	4
240.2.f.a.49.1		2	60.59	even	2
240.2.f.a.49.2		2	12.11	even	2
450.2.a.b.1.1		1	5.3	odd	4
450.2.a.f.1.1		1	5.2	odd	4
720.2.f.f.289.1		2	4.3	odd	2
720.2.f.f.289.2		2	20.19	odd	2
810.2.i.b.109.1		4	9.7	even	3
810.2.i.b.109.2		4	45.34	even	6
810.2.i.b.379.1		4	45.4	even	6
810.2.i.b.379.2		4	9.4	even	3
810.2.i.e.109.1		4	45.29	odd	6
810.2.i.e.109.2		4	9.2	odd	6
810.2.i.e.379.1		4	9.5	odd	6
810.2.i.e.379.2		4	45.14	odd	6
960.2.f.h.769.1		2	120.29	odd	2
960.2.f.h.769.2		2	24.5	odd	2
960.2.f.i.769.1		2	24.11	even	2
960.2.f.i.769.2		2	120.59	even	2
1200.2.a.g.1.1		1	60.23	odd	4
1200.2.a.m.1.1		1	60.47	odd	4
1470.2.g.g.589.1		2	105.104	even	2
1470.2.g.g.589.2		2	21.20	even	2
1470.2.n.a.79.1		4	21.17	even	6
1470.2.n.a.79.2		4	105.59	even	6
1470.2.n.a.949.1		4	105.89	even	6
1470.2.n.a.949.2		4	21.5	even	6
1470.2.n.h.79.1		4	21.11	odd	6
1470.2.n.h.79.2		4	105.74	odd	6
1470.2.n.h.949.1		4	105.44	odd	6
1470.2.n.h.949.2		4	21.2	odd	6
2880.2.f.c.1729.1		2	40.19	odd	2
2880.2.f.c.1729.2		2	8.3	odd	2
2880.2.f.e.1729.1		2	40.29	even	2
2880.2.f.e.1729.2		2	8.5	even	2
3600.2.a.o.1.1		1	20.7	even	4
3600.2.a.bg.1.1		1	20.3	even	4
3840.2.d.g.2689.1		2	48.5	odd	4
3840.2.d.g.2689.2		2	240.29	odd	4
3840.2.d.j.2689.1		2	240.59	even	4
3840.2.d.j.2689.2		2	48.35	even	4
3840.2.d.x.2689.1		2	48.11	even	4
3840.2.d.x.2689.2		2	240.179	even	4
3840.2.d.y.2689.1		2	240.149	odd	4
3840.2.d.y.2689.2		2	48.29	odd	4
4800.2.a.l.1.1		1	120.53	even	4
4800.2.a.m.1.1		1	120.107	odd	4
4800.2.a.cg.1.1		1	120.77	even	4
4800.2.a.cj.1.1		1	120.83	odd	4
7350.2.a.bg.1.1		1	105.62	odd	4
7350.2.a.cc.1.1		1	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} -2.00000 q^{11} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +(-2.00000 + 1.00000i) q^{20} +2.00000i q^{22} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{26} +2.00000i q^{28} -8.00000 q^{31} -1.00000i q^{32} +2.00000 q^{34} +(-2.00000 - 4.00000i) q^{35} -2.00000i q^{37} +(1.00000 + 2.00000i) q^{40} -2.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +4.00000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -6.00000i q^{52} -6.00000i q^{53} +(-4.00000 + 2.00000i) q^{55} +2.00000 q^{56} +10.0000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +(6.00000 + 12.0000i) q^{65} +8.00000i q^{67} -2.00000i q^{68} +(-4.00000 + 2.00000i) q^{70} -12.0000 q^{71} -4.00000i q^{73} -2.00000 q^{74} +4.00000i q^{77} +(2.00000 - 1.00000i) q^{80} +2.00000i q^{82} +4.00000i q^{83} +(2.00000 + 4.00000i) q^{85} -4.00000 q^{86} -2.00000i q^{88} -10.0000 q^{89} +12.0000 q^{91} -4.00000i q^{92} -8.00000 q^{94} +8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^5	\( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{20} + 6 q^{25} + 12 q^{26} - 16 q^{31} + 4 q^{34} - 4 q^{35} + 2 q^{40} - 4 q^{41} + 4 q^{44} + 8 q^{46} + 6 q^{49} - 8 q^{50} - 8 q^{55} + 4 q^{56} + 20 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{65} - 8 q^{70} - 24 q^{71} - 4 q^{74} + 4 q^{80} + 4 q^{85} - 8 q^{86} - 20 q^{89} + 24 q^{91} - 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^5 - 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^20 + 6 * q^25 + 12 * q^26 - 16 * q^31 + 4 * q^34 - 4 * q^35 + 2 * q^40 - 4 * q^41 + 4 * q^44 + 8 * q^46 + 6 * q^49 - 8 * q^50 - 8 * q^55 + 4 * q^56 + 20 * q^59 + 4 * q^61 - 2 * q^64 + 12 * q^65 - 8 * q^70 - 24 * q^71 - 4 * q^74 + 4 * q^80 + 4 * q^85 - 8 * q^86 - 20 * q^89 + 24 * q^91 - 16 * q^94

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