LMFDB - Embedded newform 1170.2.e.a.469.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(469,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (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:	$9.34249703649$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.a.469.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +(-1.00000 + 2.00000i) q^{10} -2.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +4.00000 q^{19} +(2.00000 + 1.00000i) q^{20} +2.00000i q^{22} +(3.00000 + 4.00000i) q^{25} +1.00000 q^{26} -2.00000i q^{28} +4.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +(2.00000 - 4.00000i) q^{35} +6.00000i q^{37} -4.00000i q^{38} +(1.00000 - 2.00000i) q^{40} +6.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +8.00000i q^{47} +3.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -1.00000i q^{52} +2.00000i q^{53} +(4.00000 + 2.00000i) q^{55} -2.00000 q^{56} -4.00000i q^{58} +10.0000 q^{59} -14.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +(1.00000 - 2.00000i) q^{65} -16.0000i q^{67} +2.00000i q^{68} +(-4.00000 - 2.00000i) q^{70} +4.00000 q^{71} -8.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +8.00000 q^{79} +(-2.00000 - 1.00000i) q^{80} -6.00000i q^{82} +12.0000i q^{83} +(-2.00000 + 4.00000i) q^{85} -4.00000 q^{86} -2.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} +8.00000 q^{94} +(-8.00000 - 4.00000i) q^{95} +12.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{19} + 4 q^{20} + 6 q^{25} + 2 q^{26} + 8 q^{29} + 16 q^{31} - 4 q^{34} + 4 q^{35} + 2 q^{40} + 12 q^{41} + 4 q^{44} + 6 q^{49} + 8 q^{50} + 8 q^{55} - 4 q^{56} + 20 q^{59} - 28 q^{61} - 2 q^{64} + 2 q^{65} - 8 q^{70} + 8 q^{71} + 12 q^{74} - 8 q^{76} + 16 q^{79} - 4 q^{80} - 4 q^{85} - 8 q^{86} + 12 q^{89} - 4 q^{91} + 16 q^{94} - 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^10 - 4 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^20 + 6 * q^25 + 2 * q^26 + 8 * q^29 + 16 * q^31 - 4 * q^34 + 4 * q^35 + 2 * q^40 + 12 * q^41 + 4 * q^44 + 6 * q^49 + 8 * q^50 + 8 * q^55 - 4 * q^56 + 20 * q^59 - 28 * q^61 - 2 * q^64 + 2 * q^65 - 8 * q^70 + 8 * q^71 + 12 * q^74 - 8 * q^76 + 16 * q^79 - 4 * q^80 - 4 * q^85 - 8 * q^86 + 12 * q^89 - 4 * q^91 + 16 * q^94 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−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	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			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
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$	2.00000			0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$	2.00000	+	1.00000i	0.447214	+	0.223607i
$21$		0			0
$22$			2.00000i			0.426401i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	1.00000			0.196116
$27$		0			0
$28$		−	2.00000i		−	0.377964i
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−2.00000			−0.342997
$35$	2.00000	−	4.00000i	0.338062	−	0.676123i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		−	4.00000i		−	0.648886i
$39$		0			0
$40$	1.00000	−	2.00000i	0.158114	−	0.316228i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$		−	1.00000i		−	0.138675i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$	4.00000	+	2.00000i	0.539360	+	0.269680i
$56$	−2.00000			−0.267261
$57$		0			0
$58$		−	4.00000i		−	0.525226i
$59$	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
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$		−	8.00000i		−	1.01600i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.00000	−	2.00000i	0.124035	−	0.248069i
$66$		0			0
$67$		−	16.0000i		−	1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$		−	16.0000i		−	1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$			2.00000i			0.242536i
$69$		0			0
$70$	−4.00000	−	2.00000i	−0.478091	−	0.239046i
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$	−4.00000			−0.458831
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−	1.00000i	−0.223607	−	0.111803i
$81$		0			0
$82$		−	6.00000i		−	0.662589i
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	−2.00000	+	4.00000i	−0.216930	+	0.433861i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	2.00000i		−	0.213201i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$		0			0
$94$	8.00000			0.825137
$95$	−8.00000	−	4.00000i	−0.820783	−	0.410391i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	−3.00000	−	4.00000i	−0.300000	−	0.400000i
$101$	−4.00000			−0.398015			−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000			−0.398015			−0.199007	+	0.979998i	$0.563772\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$	2.00000			0.194257
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$	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	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$	−4.00000			−0.371391
$117$		0			0
$118$		−	10.0000i		−	0.920575i
$119$	4.00000			0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$			14.0000i			1.26750i
$123$		0			0
$124$	−8.00000			−0.718421
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−2.00000	−	1.00000i	−0.175412	−	0.0877058i
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$	−16.0000			−1.38219
$135$		0			0
$136$	2.00000			0.171499
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$	−2.00000	+	4.00000i	−0.169031	+	0.338062i
$141$		0			0
$142$		−	4.00000i		−	0.335673i
$143$		−	2.00000i		−	0.167248i
$144$		0			0
$145$	−8.00000	−	4.00000i	−0.664364	−	0.332182i
$146$	−8.00000			−0.662085
$147$		0			0
$148$		−	6.00000i		−	0.493197i
$149$	12.0000			0.983078			0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000			0.983078			0.491539	+	0.870855i	$0.336434\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$			4.00000i			0.324443i
$153$		0			0
$154$	−4.00000			−0.322329
$155$	−16.0000	−	8.00000i	−1.28515	−	0.642575i
$156$		0			0
$157$			10.0000i			0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$			10.0000i			0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	−1.00000	+	2.00000i	−0.0790569	+	0.158114i
$161$		0			0
$162$		0			0
$163$			8.00000i			0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$			8.00000i			0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	12.0000			0.931381
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	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
$178$		−	6.00000i		−	0.449719i
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$			2.00000i			0.148250i
$183$		0			0
$184$		0			0
$185$	6.00000	−	12.0000i	0.441129	−	0.882258i
$186$		0			0
$187$			4.00000i			0.292509i
$188$		−	8.00000i		−	0.583460i
$189$		0			0
$190$	−4.00000	+	8.00000i	−0.290191	+	0.580381i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	12.0000			0.861550
$195$		0			0
$196$	−3.00000			−0.214286
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$			4.00000i			0.281439i
$203$			8.00000i			0.561490i
$204$		0			0
$205$	−12.0000	−	6.00000i	−0.838116	−	0.419058i
$206$	−6.00000			−0.418040
$207$		0			0
$208$			1.00000i			0.0693375i
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		−	2.00000i		−	0.137361i
$213$		0			0
$214$	4.00000			0.273434
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		−	18.0000i		−	1.21911i
$219$		0			0
$220$	−4.00000	−	2.00000i	−0.269680	−	0.134840i
$221$	2.00000			0.134535
$222$		0			0
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	2.00000			0.133631
$225$		0			0
$226$	6.00000			0.399114
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$			4.00000i			0.262613i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$	−10.0000			−0.650945
$237$		0			0
$238$		−	4.00000i		−	0.259281i
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$			7.00000i			0.449977i
$243$		0			0
$244$	14.0000			0.896258
$245$	−6.00000	−	3.00000i	−0.383326	−	0.191663i
$246$		0			0
$247$			4.00000i			0.254514i
$248$			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	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		0			0
$254$	6.00000			0.376473
$255$		0			0
$256$	1.00000			0.0625000
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	−12.0000			−0.745644
$260$	−1.00000	+	2.00000i	−0.0620174	+	0.124035i
$261$		0			0
$262$		−	6.00000i		−	0.370681i
$263$		−	8.00000i		−	0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$		−	8.00000i		−	0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$		0			0
$265$	2.00000	−	4.00000i	0.122859	−	0.245718i
$266$	8.00000			0.490511
$267$		0			0
$268$			16.0000i			0.977356i
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		−	2.00000i		−	0.121268i
$273$		0			0
$274$	14.0000			0.845771
$275$	−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	$-0.0191369\pi$
$278$			4.00000i			0.239904i
$279$		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	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−4.00000			−0.237356
$285$		0			0
$286$	−2.00000			−0.118262
$287$			12.0000i			0.708338i
$288$		0			0
$289$	13.0000			0.764706
$290$	−4.00000	+	8.00000i	−0.234888	+	0.469776i
$291$		0			0
$292$			8.00000i			0.468165i
$293$			26.0000i			1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	−20.0000	−	10.0000i	−1.16445	−	0.582223i
$296$	−6.00000			−0.348743
$297$		0			0
$298$		−	12.0000i		−	0.695141i
$299$		0			0
$300$		0			0
$301$	8.00000			0.461112
$302$			16.0000i			0.920697i
$303$		0			0
$304$	4.00000			0.229416
$305$	28.0000	+	14.0000i	1.60328	+	0.801638i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$			4.00000i			0.227921i
$309$		0			0
$310$	−8.00000	+	16.0000i	−0.454369	+	0.908739i
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$			20.0000i			1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$			20.0000i			1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$	10.0000			0.564333
$315$		0			0
$316$	−8.00000			−0.450035
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$	2.00000	+	1.00000i	0.111803	+	0.0559017i
$321$		0			0
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$	8.00000			0.443079
$327$		0			0
$328$			6.00000i			0.331295i
$329$	−16.0000			−0.882109
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		−	12.0000i		−	0.658586i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−16.0000	+	32.0000i	−0.874173	+	1.74835i
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$			1.00000i			0.0543928i
$339$		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	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$	6.00000	+	8.00000i	0.320713	+	0.427618i
$351$		0			0
$352$			2.00000i			0.106600i
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$	−8.00000	−	4.00000i	−0.424596	−	0.212298i
$356$	−6.00000			−0.317999
$357$		0			0
$358$			18.0000i			0.951330i
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$			6.00000i			0.315353i
$363$		0			0
$364$	2.00000			0.104828
$365$	−8.00000	+	16.0000i	−0.418739	+	0.837478i
$366$		0			0
$367$		−	10.0000i		−	0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$		−	10.0000i		−	0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$		0			0
$369$		0			0
$370$	−12.0000	−	6.00000i	−0.623850	−	0.311925i
$371$	−4.00000			−0.207670
$372$		0			0
$373$			22.0000i			1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$			22.0000i			1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	4.00000			0.206835
$375$		0			0
$376$	−8.00000			−0.412568
$377$			4.00000i			0.206010i
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$	8.00000	+	4.00000i	0.410391	+	0.205196i
$381$		0			0
$382$		−	12.0000i		−	0.613973i
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$	−4.00000	+	8.00000i	−0.203859	+	0.407718i
$386$		0			0
$387$		0			0
$388$		−	12.0000i		−	0.609208i
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$		0			0
$391$		0			0
$392$			3.00000i			0.151523i
$393$		0			0
$394$	−10.0000			−0.503793
$395$	−16.0000	−	8.00000i	−0.805047	−	0.402524i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$		−	24.0000i		−	1.20301i
$399$		0			0
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$			8.00000i			0.398508i
$404$	4.00000			0.199007
$405$		0			0
$406$	8.00000			0.397033
$407$		−	12.0000i		−	0.594818i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$	−6.00000	+	12.0000i	−0.296319	+	0.592638i
$411$		0			0
$412$			6.00000i			0.295599i
$413$			20.0000i			0.984136i
$414$		0			0
$415$	12.0000	−	24.0000i	0.589057	−	1.17811i
$416$	1.00000			0.0490290
$417$		0			0
$418$			8.00000i			0.391293i
$419$	−14.0000			−0.683945			−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000			−0.683945			−0.341972	+	0.939710i	$0.611095\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$			4.00000i			0.194717i
$423$		0			0
$424$	−2.00000			−0.0971286
$425$	8.00000	−	6.00000i	0.388057	−	0.291043i
$426$		0			0
$427$		−	28.0000i		−	1.35501i
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$	8.00000	+	4.00000i	0.385794	+	0.192897i
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$		−	12.0000i		−	0.576683i	−0.957528	−	0.288342i	$-0.906896\pi$
$433$		−	12.0000i		−	0.576683i	0.957528	−	0.288342i	$-0.0931039\pi$
$434$	16.0000			0.768025
$435$		0			0
$436$	−18.0000			−0.862044
$437$		0			0
$438$		0			0
$439$	−32.0000			−1.52728			−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000			−1.52728			−0.763638	+	0.645644i	$0.776589\pi$
$440$	−2.00000	+	4.00000i	−0.0953463	+	0.190693i
$441$		0			0
$442$		−	2.00000i		−	0.0951303i
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$	−12.0000	−	6.00000i	−0.568855	−	0.284427i
$446$	14.0000			0.662919
$447$		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
$451$	−12.0000			−0.565058
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$	−12.0000			−0.563188
$455$	4.00000	+	2.00000i	0.187523	+	0.0937614i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		−	22.0000i		−	1.02799i
$459$		0			0
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	4.00000			0.185695
$465$		0			0
$466$	−14.0000			−0.648537
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	32.0000			1.47762
$470$	−16.0000	−	8.00000i	−0.738025	−	0.369012i
$471$		0			0
$472$			10.0000i			0.460287i
$473$			8.00000i			0.367840i
$474$		0			0
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$	−4.00000			−0.183340
$477$		0			0
$478$			12.0000i			0.548867i
$479$	36.0000			1.64488			0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000			1.64488			0.822441	+	0.568850i	$0.192612\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$			18.0000i			0.819878i
$483$		0			0
$484$	7.00000			0.318182
$485$	12.0000	−	24.0000i	0.544892	−	1.08978i
$486$		0			0
$487$		−	26.0000i		−	1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$		−	26.0000i		−	1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$		−	14.0000i		−	0.633750i
$489$		0			0
$490$	−3.00000	+	6.00000i	−0.135526	+	0.271052i
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$	4.00000			0.179969
$495$		0			0
$496$	8.00000			0.359211
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−36.0000			−1.61158			−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000			−1.61158			−0.805791	+	0.592200i	$0.798259\pi$
$500$	2.00000	+	11.0000i	0.0894427	+	0.491935i
$501$		0			0
$502$			18.0000i			0.803379i
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$	8.00000	+	4.00000i	0.355995	+	0.177998i
$506$		0			0
$507$		0			0
$508$		−	6.00000i		−	0.266207i
$509$	−40.0000			−1.77297			−0.886484	−	0.462758i	$-0.846860\pi$
$509$	−40.0000			−1.77297			−0.886484	+	0.462758i	$0.846860\pi$
$510$		0			0
$511$	16.0000			0.707798
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	18.0000			0.793946
$515$	−6.00000	+	12.0000i	−0.264392	+	0.528783i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$			12.0000i			0.527250i
$519$		0			0
$520$	2.00000	+	1.00000i	0.0877058	+	0.0438529i
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$		−	16.0000i		−	0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$	−8.00000			−0.348817
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	23.0000			1.00000
$530$	−4.00000	−	2.00000i	−0.173749	−	0.0868744i
$531$		0			0
$532$		−	8.00000i		−	0.346844i
$533$			6.00000i			0.259889i
$534$		0			0
$535$	4.00000	−	8.00000i	0.172935	−	0.345870i
$536$	16.0000			0.691095
$537$		0			0
$538$			20.0000i			0.862261i
$539$	−6.00000			−0.258438
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		−	16.0000i		−	0.687259i
$543$		0			0
$544$	−2.00000			−0.0857493
$545$	−36.0000	−	18.0000i	−1.54207	−	0.771035i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		−	14.0000i		−	0.598050i
$549$		0			0
$550$	−8.00000	+	6.00000i	−0.341121	+	0.255841i
$551$	16.0000			0.681623
$552$		0			0
$553$			16.0000i			0.680389i
$554$	−2.00000			−0.0849719
$555$		0			0
$556$	4.00000			0.169638
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$	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	$-0.622217\pi$
$563$		−	44.0000i		−	1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$		0			0
$565$	6.00000	−	12.0000i	0.252422	−	0.504844i
$566$	−16.0000			−0.672530
$567$		0			0
$568$			4.00000i			0.167836i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$			2.00000i			0.0836242i
$573$		0			0
$574$	12.0000			0.500870
$575$		0			0
$576$		0			0
$577$			28.0000i			1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$577$			28.0000i			1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$	8.00000	+	4.00000i	0.332182	+	0.166091i
$581$	−24.0000			−0.995688
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$	8.00000			0.331042
$585$		0			0
$586$	26.0000			1.07405
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$	−10.0000	+	20.0000i	−0.411693	+	0.823387i
$591$		0			0
$592$			6.00000i			0.246598i
$593$			34.0000i			1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$			34.0000i			1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$		0			0
$595$	−8.00000	−	4.00000i	−0.327968	−	0.163984i
$596$	−12.0000			−0.491539
$597$		0			0
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		−	8.00000i		−	0.326056i
$603$		0			0
$604$	16.0000			0.651031
$605$	14.0000	+	7.00000i	0.569181	+	0.284590i
$606$		0			0
$607$			18.0000i			0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$			18.0000i			0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$		−	4.00000i		−	0.162221i
$609$		0			0
$610$	14.0000	−	28.0000i	0.566843	−	1.13369i
$611$	−8.00000			−0.323645
$612$		0			0
$613$			2.00000i			0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$			2.00000i			0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−28.0000			−1.12999
$615$		0			0
$616$	4.00000			0.161165
$617$		−	46.0000i		−	1.85189i	−0.377658	−	0.925945i	$-0.623271\pi$
$617$		−	46.0000i		−	1.85189i	0.377658	−	0.925945i	$-0.376729\pi$
$618$		0			0
$619$	24.0000			0.964641			0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000			0.964641			0.482321	+	0.875995i	$0.339794\pi$
$620$	16.0000	+	8.00000i	0.642575	+	0.321288i
$621$		0			0
$622$			20.0000i			0.801927i
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	20.0000			0.799361
$627$		0			0
$628$		−	10.0000i		−	0.399043i
$629$	12.0000			0.478471
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$	18.0000			0.714871
$635$	6.00000	−	12.0000i	0.238103	−	0.476205i
$636$		0			0
$637$			3.00000i			0.118864i
$638$			8.00000i			0.316723i
$639$		0			0
$640$	1.00000	−	2.00000i	0.0395285	−	0.0790569i
$641$	38.0000			1.50091			0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000			1.50091			0.750455	+	0.660922i	$0.229834\pi$
$642$		0			0
$643$		−	32.0000i		−	1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$		−	32.0000i		−	1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$		0			0
$645$		0			0
$646$	−8.00000			−0.314756
$647$		−	36.0000i		−	1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$		−	36.0000i		−	1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$	3.00000	+	4.00000i	0.117670	+	0.156893i
$651$		0			0
$652$		−	8.00000i		−	0.313304i
$653$		−	34.0000i		−	1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$		−	34.0000i		−	1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$		0			0
$655$	−12.0000	−	6.00000i	−0.468879	−	0.234439i
$656$	6.00000			0.234261
$657$		0			0
$658$			16.0000i			0.623745i
$659$	10.0000			0.389545			0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000			0.389545			0.194772	+	0.980848i	$0.437603\pi$
$660$		0			0
$661$	30.0000			1.16686			0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000			1.16686			0.583432	+	0.812162i	$0.301709\pi$
$662$			12.0000i			0.466393i
$663$		0			0
$664$	−12.0000			−0.465690
$665$	8.00000	−	16.0000i	0.310227	−	0.620453i
$666$		0			0
$667$		0			0
$668$			12.0000i			0.464294i
$669$		0			0
$670$	32.0000	+	16.0000i	1.23627	+	0.618134i
$671$	28.0000			1.08093
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$	20.0000			0.770371
$675$		0			0
$676$	1.00000			0.0384615
$677$			10.0000i			0.384331i	0.981363	+	0.192166i	$0.0615511\pi$
$677$			10.0000i			0.384331i	−0.981363	+	0.192166i	$0.938449\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$	−4.00000	−	2.00000i	−0.153393	−	0.0766965i
$681$		0			0
$682$			16.0000i			0.612672i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	14.0000	−	28.0000i	0.534913	−	1.06983i
$686$	20.0000			0.763604
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	−2.00000			−0.0761939
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$		−	14.0000i		−	0.532200i
$693$		0			0
$694$	−12.0000			−0.455514
$695$	8.00000	+	4.00000i	0.303457	+	0.151729i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$			14.0000i			0.529908i
$699$		0			0
$700$	8.00000	−	6.00000i	0.302372	−	0.226779i
$701$	−12.0000			−0.453234			−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000			−0.453234			−0.226617	+	0.973984i	$0.572767\pi$
$702$		0			0
$703$			24.0000i			0.905177i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−34.0000			−1.27961
$707$		−	8.00000i		−	0.300871i
$708$		0			0
$709$	−34.0000			−1.27690			−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000			−1.27690			−0.638448	+	0.769665i	$0.720423\pi$
$710$	−4.00000	+	8.00000i	−0.150117	+	0.300235i
$711$		0			0
$712$			6.00000i			0.224860i
$713$		0			0
$714$		0			0
$715$	−2.00000	+	4.00000i	−0.0747958	+	0.149592i
$716$	18.0000			0.672692
$717$		0			0
$718$		−	32.0000i		−	1.19423i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$			3.00000i			0.111648i
$723$		0			0
$724$	6.00000			0.222988
$725$	12.0000	+	16.0000i	0.445669	+	0.594225i
$726$		0			0
$727$			34.0000i			1.26099i	0.776193	+	0.630495i	$0.217148\pi$
$727$			34.0000i			1.26099i	−0.776193	+	0.630495i	$0.782852\pi$
$728$		−	2.00000i		−	0.0741249i
$729$		0			0
$730$	16.0000	+	8.00000i	0.592187	+	0.296093i
$731$	−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	$-0.0832450\pi$
$734$	−10.0000			−0.369107
$735$		0			0
$736$		0			0
$737$			32.0000i			1.17874i
$738$		0			0
$739$	−28.0000			−1.03000			−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000			−1.03000			−0.514998	+	0.857191i	$0.672207\pi$
$740$	−6.00000	+	12.0000i	−0.220564	+	0.441129i
$741$		0			0
$742$			4.00000i			0.146845i
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−24.0000	−	12.0000i	−0.879292	−	0.439646i
$746$	22.0000			0.805477
$747$		0			0
$748$		−	4.00000i		−	0.146254i
$749$	−8.00000			−0.292314
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$			8.00000i			0.291730i
$753$		0			0
$754$	4.00000			0.145671
$755$	32.0000	+	16.0000i	1.16460	+	0.582300i
$756$		0			0
$757$		−	26.0000i		−	0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$		−	26.0000i		−	0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$			8.00000i			0.290573i
$759$		0			0
$760$	4.00000	−	8.00000i	0.145095	−	0.290191i
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$			36.0000i			1.30329i
$764$	−12.0000			−0.434145
$765$		0			0
$766$	24.0000			0.867155
$767$			10.0000i			0.361079i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$	8.00000	+	4.00000i	0.288300	+	0.144150i
$771$		0			0
$772$		0			0
$773$			22.0000i			0.791285i	0.918405	+	0.395643i	$0.129478\pi$
$773$			22.0000i			0.791285i	−0.918405	+	0.395643i	$0.870522\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$	−12.0000			−0.430775
$777$		0			0
$778$		−	16.0000i		−	0.573628i
$779$	24.0000			0.859889
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$		0			0
$784$	3.00000			0.107143
$785$	10.0000	−	20.0000i	0.356915	−	0.713831i
$786$		0			0
$787$		−	8.00000i		−	0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$		−	8.00000i		−	0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$			10.0000i			0.356235i
$789$		0			0
$790$	−8.00000	+	16.0000i	−0.284627	+	0.569254i
$791$	−12.0000			−0.426671
$792$		0			0
$793$		−	14.0000i		−	0.497155i
$794$	−18.0000			−0.638796
$795$		0			0
$796$	−24.0000			−0.850657
$797$		−	14.0000i		−	0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$		−	14.0000i		−	0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$		0			0
$799$	16.0000			0.566039
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$		−	18.0000i		−	0.635602i
$803$			16.0000i			0.564628i
$804$		0			0
$805$		0			0
$806$	8.00000			0.281788
$807$		0			0
$808$		−	4.00000i		−	0.140720i
$809$	−22.0000			−0.773479			−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000			−0.773479			−0.386739	+	0.922189i	$0.626399\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		−	8.00000i		−	0.280745i
$813$		0			0
$814$	−12.0000			−0.420600
$815$	8.00000	−	16.0000i	0.280228	−	0.560456i
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$		−	26.0000i		−	0.909069i
$819$		0			0
$820$	12.0000	+	6.00000i	0.419058	+	0.209529i
$821$	−24.0000			−0.837606			−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000			−0.837606			−0.418803	+	0.908077i	$0.637550\pi$
$822$		0			0
$823$		−	34.0000i		−	1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$		−	34.0000i		−	1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	6.00000			0.209020
$825$		0			0
$826$	20.0000			0.695889
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$	−24.0000	−	12.0000i	−0.833052	−	0.416526i
$831$		0			0
$832$		−	1.00000i		−	0.0346688i
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	−12.0000	+	24.0000i	−0.415277	+	0.830554i
$836$	8.00000			0.276686
$837$		0			0
$838$			14.0000i			0.483622i
$839$	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
$841$	−13.0000			−0.448276
$842$		−	34.0000i		−	1.17172i
$843$		0			0
$844$	4.00000			0.137686
$845$	2.00000	+	1.00000i	0.0688021	+	0.0344010i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$			2.00000i			0.0686803i
$849$		0			0
$850$	−6.00000	−	8.00000i	−0.205798	−	0.274398i
$851$		0			0
$852$		0			0
$853$			18.0000i			0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$			18.0000i			0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	−28.0000			−0.958140
$855$		0			0
$856$	−4.00000			−0.136717
$857$		−	34.0000i		−	1.16142i	−0.814111	−	0.580709i	$-0.802775\pi$
$857$		−	34.0000i		−	1.16142i	0.814111	−	0.580709i	$-0.197225\pi$
$858$		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
$862$			16.0000i			0.544962i
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$	14.0000	−	28.0000i	0.476014	−	0.952029i
$866$	−12.0000			−0.407777
$867$		0			0
$868$		−	16.0000i		−	0.543075i
$869$	−16.0000			−0.542763
$870$		0			0
$871$	16.0000			0.542139
$872$			18.0000i			0.609557i
$873$		0			0
$874$		0			0
$875$	22.0000	−	4.00000i	0.743736	−	0.135225i
$876$		0			0
$877$			42.0000i			1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$			42.0000i			1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$			32.0000i			1.07995i
$879$		0			0
$880$	4.00000	+	2.00000i	0.134840	+	0.0674200i
$881$	10.0000			0.336909			0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		0			0
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	−2.00000			−0.0672673
$885$		0			0
$886$	4.00000			0.134383
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$	−6.00000	+	12.0000i	−0.201120	+	0.402241i
$891$		0			0
$892$		−	14.0000i		−	0.468755i
$893$			32.0000i			1.07084i
$894$		0			0
$895$	36.0000	+	18.0000i	1.20335	+	0.601674i
$896$	−2.00000			−0.0668153
$897$		0			0
$898$		−	30.0000i		−	1.00111i
$899$	32.0000			1.06726
$900$		0			0
$901$	4.00000			0.133259
$902$			12.0000i			0.399556i
$903$		0			0
$904$	−6.00000			−0.199557
$905$	12.0000	+	6.00000i	0.398893	+	0.199447i
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$	2.00000	−	4.00000i	0.0662994	−	0.132599i
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		−	24.0000i		−	0.794284i
$914$	−28.0000			−0.926158
$915$		0			0
$916$	−22.0000			−0.726900
$917$			12.0000i			0.396275i
$918$		0			0
$919$	48.0000			1.58337			0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000			1.58337			0.791687	+	0.610927i	$0.209203\pi$
$920$		0			0
$921$		0			0
$922$			20.0000i			0.658665i
$923$			4.00000i			0.131662i
$924$		0			0
$925$	−24.0000	+	18.0000i	−0.789115	+	0.591836i
$926$	−6.00000			−0.197172
$927$		0			0
$928$		−	4.00000i		−	0.131306i
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$		0			0
$931$	12.0000			0.393284
$932$			14.0000i			0.458585i
$933$		0			0
$934$	−28.0000			−0.916188
$935$	4.00000	−	8.00000i	0.130814	−	0.261628i
$936$		0			0
$937$			56.0000i			1.82944i	0.404088	+	0.914720i	$0.367589\pi$
$937$			56.0000i			1.82944i	−0.404088	+	0.914720i	$0.632411\pi$
$938$		−	32.0000i		−	1.04484i
$939$		0			0
$940$	−8.00000	+	16.0000i	−0.260931	+	0.521862i
$941$	44.0000			1.43436			0.717180	−	0.696888i	$-0.245433\pi$
$941$	44.0000			1.43436			0.717180	+	0.696888i	$0.245433\pi$
$942$		0			0
$943$		0			0
$944$	10.0000			0.325472
$945$		0			0
$946$	8.00000			0.260102
$947$			36.0000i			1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$			36.0000i			1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$		0			0
$949$	8.00000			0.259691
$950$	16.0000	−	12.0000i	0.519109	−	0.389331i
$951$		0			0
$952$			4.00000i			0.129641i
$953$			6.00000i			0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$			6.00000i			0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$		0			0
$955$	−24.0000	−	12.0000i	−0.776622	−	0.388311i
$956$	12.0000			0.388108
$957$		0			0
$958$		−	36.0000i		−	1.16311i
$959$	−28.0000			−0.904167
$960$		0			0
$961$	33.0000			1.06452
$962$			6.00000i			0.193448i
$963$		0			0
$964$	18.0000			0.579741
$965$		0			0
$966$		0			0
$967$			58.0000i			1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$			58.0000i			1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$		−	7.00000i		−	0.224989i
$969$		0			0
$970$	−24.0000	−	12.0000i	−0.770594	−	0.385297i
$971$	−50.0000			−1.60458			−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000			−1.60458			−0.802288	+	0.596937i	$0.796384\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$	−26.0000			−0.833094
$975$		0			0
$976$	−14.0000			−0.448129
$977$			18.0000i			0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$			18.0000i			0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$	6.00000	+	3.00000i	0.191663	+	0.0958315i
$981$		0			0
$982$		−	22.0000i		−	0.702048i
$983$			56.0000i			1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$			56.0000i			1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$		0			0
$985$	−10.0000	+	20.0000i	−0.318626	+	0.637253i
$986$	−8.00000			−0.254772
$987$		0			0
$988$		−	4.00000i		−	0.127257i
$989$		0			0
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		−	8.00000i		−	0.254000i
$993$		0			0
$994$	8.00000			0.253745
$995$	−48.0000	−	24.0000i	−1.52170	−	0.760851i
$996$		0			0
$997$		−	14.0000i		−	0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$		−	14.0000i		−	0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$			36.0000i			1.13956i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.a.469.1		2
3.2	odd	2		390.2.e.c.79.2	yes	2
5.2	odd	4		5850.2.a.bj.1.1		1
5.3	odd	4		5850.2.a.t.1.1		1
5.4	even	2	inner	1170.2.e.a.469.2		2
12.11	even	2		3120.2.l.g.1249.2		2
15.2	even	4		1950.2.a.c.1.1		1
15.8	even	4		1950.2.a.z.1.1		1
15.14	odd	2		390.2.e.c.79.1	✓	2
60.59	even	2		3120.2.l.g.1249.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.c.79.1	✓	2	15.14	odd	2
390.2.e.c.79.2	yes	2	3.2	odd	2
1170.2.e.a.469.1		2	1.1	even	1	trivial
1170.2.e.a.469.2		2	5.4	even	2	inner
1950.2.a.c.1.1		1	15.2	even	4
1950.2.a.z.1.1		1	15.8	even	4
3120.2.l.g.1249.1		2	60.59	even	2
3120.2.l.g.1249.2		2	12.11	even	2
5850.2.a.t.1.1		1	5.3	odd	4
5850.2.a.bj.1.1		1	5.2	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +(-1.00000 + 2.00000i) q^{10} -2.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +4.00000 q^{19} +(2.00000 + 1.00000i) q^{20} +2.00000i q^{22} +(3.00000 + 4.00000i) q^{25} +1.00000 q^{26} -2.00000i q^{28} +4.00000 q^{29} +8.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} +(2.00000 - 4.00000i) q^{35} +6.00000i q^{37} -4.00000i q^{38} +(1.00000 - 2.00000i) q^{40} +6.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +8.00000i q^{47} +3.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -1.00000i q^{52} +2.00000i q^{53} +(4.00000 + 2.00000i) q^{55} -2.00000 q^{56} -4.00000i q^{58} +10.0000 q^{59} -14.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +(1.00000 - 2.00000i) q^{65} -16.0000i q^{67} +2.00000i q^{68} +(-4.00000 - 2.00000i) q^{70} +4.00000 q^{71} -8.00000i q^{73} +6.00000 q^{74} -4.00000 q^{76} -4.00000i q^{77} +8.00000 q^{79} +(-2.00000 - 1.00000i) q^{80} -6.00000i q^{82} +12.0000i q^{83} +(-2.00000 + 4.00000i) q^{85} -4.00000 q^{86} -2.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} +8.00000 q^{94} +(-8.00000 - 4.00000i) q^{95} +12.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^5	\( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{10} - 4 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{20} + 6 q^{25} + 2 q^{26} + 8 q^{29} + 16 q^{31} - 4 q^{34} + 4 q^{35} + 2 q^{40} + 12 q^{41} + 4 q^{44} + 6 q^{49} + 8 q^{50} + 8 q^{55} - 4 q^{56} + 20 q^{59} - 28 q^{61} - 2 q^{64} + 2 q^{65} - 8 q^{70} + 8 q^{71} + 12 q^{74} - 8 q^{76} + 16 q^{79} - 4 q^{80} - 4 q^{85} - 8 q^{86} + 12 q^{89} - 4 q^{91} + 16 q^{94} - 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^10 - 4 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 4 * q^20 + 6 * q^25 + 2 * q^26 + 8 * q^29 + 16 * q^31 - 4 * q^34 + 4 * q^35 + 2 * q^40 + 12 * q^41 + 4 * q^44 + 6 * q^49 + 8 * q^50 + 8 * q^55 - 4 * q^56 + 20 * q^59 - 28 * q^61 - 2 * q^64 + 2 * q^65 - 8 * q^70 + 8 * q^71 + 12 * q^74 - 8 * q^76 + 16 * q^79 - 4 * q^80 - 4 * q^85 - 8 * q^86 + 12 * q^89 - 4 * q^91 + 16 * q^94 - 16 * q^95

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