LMFDB - Embedded newform 2208.2.b.c.689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2208,2,Mod(689,2208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2208, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2208.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2208 = 2^{5} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2208.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.6309687663$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 552)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.1
Character	$\chi$	$=$	2208.689
Dual form			2208.2.b.c.689.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71832 - 0.217636i) q^{3} -1.34112i q^{5} -2.80797i q^{7} +(2.90527 + 0.747937i) q^{9} +O(q^{10})$	\(q+(-1.71832 - 0.217636i) q^{3} -1.34112i q^{5} -2.80797i q^{7} +(2.90527 + 0.747937i) q^{9} -6.17378i q^{11} -4.84265i q^{13} +(-0.291875 + 2.30447i) q^{15} -4.64639 q^{17} -5.42593 q^{19} +(-0.611113 + 4.82499i) q^{21} +(-3.02469 - 3.72173i) q^{23} +3.20141 q^{25} +(-4.82941 - 1.91749i) q^{27} +1.97237 q^{29} +2.08414 q^{31} +(-1.34363 + 10.6085i) q^{33} -3.76581 q^{35} +6.74349 q^{37} +(-1.05393 + 8.32124i) q^{39} +6.56520i q^{41} +8.76132 q^{43} +(1.00307 - 3.89631i) q^{45} +1.29516i q^{47} -0.884669 q^{49} +(7.98400 + 1.01122i) q^{51} -8.64336i q^{53} -8.27976 q^{55} +(9.32350 + 1.18088i) q^{57} +5.23804 q^{59} -8.20972 q^{61} +(2.10018 - 8.15790i) q^{63} -6.49456 q^{65} -1.61351 q^{67} +(4.38742 + 7.05341i) q^{69} +1.95795i q^{71} +8.57628 q^{73} +(-5.50105 - 0.696740i) q^{75} -17.3358 q^{77} +8.21280i q^{79} +(7.88118 + 4.34592i) q^{81} +10.6556i q^{83} +6.23135i q^{85} +(-3.38917 - 0.429258i) q^{87} -0.917940 q^{89} -13.5980 q^{91} +(-3.58123 - 0.453583i) q^{93} +7.27680i q^{95} -1.91006i q^{97} +(4.61760 - 17.9365i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 4 q^{9}+O(q^{10}) $ 80 * q - 4 * q^9	$ 80 q - 4 q^{9} - 144 q^{25} + 24 q^{31} - 68 q^{39} - 160 q^{49} - 32 q^{55} - 8 q^{73} + 12 q^{81} - 92 q^{87}+O(q^{100}) $ 80 * q - 4 * q^9 - 144 * q^25 + 24 * q^31 - 68 * q^39 - 160 * q^49 - 32 * q^55 - 8 * q^73 + 12 * q^81 - 92 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2208\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$737$	$1381$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.71832	−	0.217636i	−0.992074	−	0.125652i
$3$	−1.71832	−	0.217636i	−0.992074	−	0.125652i
$4$		0			0
$5$		−	1.34112i		−	0.599766i	−0.953976	−	0.299883i	$-0.903052\pi$
$5$		−	1.34112i		−	0.599766i	0.953976	−	0.299883i	$-0.0969476\pi$
$6$		0			0
$7$		−	2.80797i		−	1.06131i	−0.847588	−	0.530656i	$-0.821946\pi$
$7$		−	2.80797i		−	1.06131i	0.847588	−	0.530656i	$-0.178054\pi$
$8$		0			0
$9$	2.90527	+	0.747937i	0.968423	+	0.249312i
$10$		0			0
$11$		−	6.17378i		−	1.86146i	−0.365702	−	0.930732i	$-0.619171\pi$
$11$		−	6.17378i		−	1.86146i	0.365702	−	0.930732i	$-0.380829\pi$
$12$		0			0
$13$		−	4.84265i		−	1.34311i	−0.740955	−	0.671555i	$-0.765627\pi$
$13$		−	4.84265i		−	1.34311i	0.740955	−	0.671555i	$-0.234373\pi$
$14$		0			0
$15$	−0.291875	+	2.30447i	−0.0753618	+	0.595012i
$16$		0			0
$17$	−4.64639			−1.12692			−0.563458	−	0.826145i	$-0.690529\pi$
$17$	−4.64639			−1.12692			−0.563458	+	0.826145i	$0.690529\pi$
$18$		0			0
$19$	−5.42593			−1.24479			−0.622396	−	0.782702i	$-0.713841\pi$
$19$	−5.42593			−1.24479			−0.622396	+	0.782702i	$0.713841\pi$
$20$		0			0
$21$	−0.611113	+	4.82499i	−0.133356	+	1.05290i
$22$		0			0
$23$	−3.02469	−	3.72173i	−0.630692	−	0.776033i
$23$	−3.02469	−	3.72173i	−0.630692	−	0.776033i
$24$		0			0
$25$	3.20141			0.640281
$26$		0			0
$27$	−4.82941	−	1.91749i	−0.929421	−	0.369021i
$28$		0			0
$29$	1.97237			0.366260			0.183130	−	0.983089i	$-0.441377\pi$
$29$	1.97237			0.366260			0.183130	+	0.983089i	$0.441377\pi$
$30$		0			0
$31$	2.08414			0.374323			0.187161	−	0.982329i	$-0.440071\pi$
$31$	2.08414			0.374323			0.187161	+	0.982329i	$0.440071\pi$
$32$		0			0
$33$	−1.34363	+	10.6085i	−0.233897	+	1.84671i
$34$		0			0
$35$	−3.76581			−0.636538
$36$		0			0
$37$	6.74349			1.10862			0.554311	−	0.832309i	$-0.312982\pi$
$37$	6.74349			1.10862			0.554311	+	0.832309i	$0.312982\pi$
$38$		0			0
$39$	−1.05393	+	8.32124i	−0.168764	+	1.33246i
$40$		0			0
$41$			6.56520i			1.02531i	0.858594	+	0.512656i	$0.171338\pi$
$41$			6.56520i			1.02531i	−0.858594	+	0.512656i	$0.828662\pi$
$42$		0			0
$43$	8.76132			1.33609			0.668045	−	0.744121i	$-0.267132\pi$
$43$	8.76132			1.33609			0.668045	+	0.744121i	$0.267132\pi$
$44$		0			0
$45$	1.00307	−	3.89631i	0.149529	−	0.580827i
$46$		0			0
$47$			1.29516i			0.188919i	0.995529	+	0.0944595i	$0.0301123\pi$
$47$			1.29516i			0.188919i	−0.995529	+	0.0944595i	$0.969888\pi$
$48$		0			0
$49$	−0.884669			−0.126381
$50$		0			0
$51$	7.98400	+	1.01122i	1.11798	+	0.141599i
$52$		0			0
$53$		−	8.64336i		−	1.18726i	−0.804739	−	0.593628i	$-0.797695\pi$
$53$		−	8.64336i		−	1.18726i	0.804739	−	0.593628i	$-0.202305\pi$
$54$		0			0
$55$	−8.27976			−1.11644
$56$		0			0
$57$	9.32350	+	1.18088i	1.23493	+	0.156411i
$58$		0			0
$59$	5.23804			0.681934			0.340967	−	0.940075i	$-0.389246\pi$
$59$	5.23804			0.681934			0.340967	+	0.940075i	$0.389246\pi$
$60$		0			0
$61$	−8.20972			−1.05115			−0.525573	−	0.850748i	$-0.676149\pi$
$61$	−8.20972			−1.05115			−0.525573	+	0.850748i	$0.676149\pi$
$62$		0			0
$63$	2.10018	−	8.15790i	0.264598	−	1.02780i
$64$		0			0
$65$	−6.49456			−0.805551
$66$		0			0
$67$	−1.61351			−0.197121			−0.0985606	−	0.995131i	$-0.531424\pi$
$67$	−1.61351			−0.197121			−0.0985606	+	0.995131i	$0.531424\pi$
$68$		0			0
$69$	4.38742	+	7.05341i	0.528183	+	0.849131i
$70$		0			0
$71$			1.95795i			0.232365i	0.993228	+	0.116183i	$0.0370658\pi$
$71$			1.95795i			0.232365i	−0.993228	+	0.116183i	$0.962934\pi$
$72$		0			0
$73$	8.57628			1.00378			0.501889	−	0.864932i	$-0.332639\pi$
$73$	8.57628			1.00378			0.501889	+	0.864932i	$0.332639\pi$
$74$		0			0
$75$	−5.50105	−	0.696740i	−0.635207	−	0.0804526i
$76$		0			0
$77$	−17.3358			−1.97559
$78$		0			0
$79$			8.21280i			0.924012i	0.886877	+	0.462006i	$0.152870\pi$
$79$			8.21280i			0.924012i	−0.886877	+	0.462006i	$0.847130\pi$
$80$		0			0
$81$	7.88118	+	4.34592i	0.875687	+	0.482880i
$82$		0			0
$83$			10.6556i			1.16960i	0.811176	+	0.584802i	$0.198828\pi$
$83$			10.6556i			1.16960i	−0.811176	+	0.584802i	$0.801172\pi$
$84$		0			0
$85$			6.23135i			0.675885i
$86$		0			0
$87$	−3.38917	−	0.429258i	−0.363357	−	0.0460213i
$88$		0			0
$89$	−0.917940			−0.0973014			−0.0486507	−	0.998816i	$-0.515492\pi$
$89$	−0.917940			−0.0973014			−0.0486507	+	0.998816i	$0.515492\pi$
$90$		0			0
$91$	−13.5980			−1.42546
$92$		0			0
$93$	−3.58123	−	0.453583i	−0.371356	−	0.0470344i
$94$		0			0
$95$			7.27680i			0.746584i
$96$		0			0
$97$		−	1.91006i		−	0.193937i	−0.995287	−	0.0969685i	$-0.969085\pi$
$97$		−	1.91006i		−	0.193937i	0.995287	−	0.0969685i	$-0.0309146\pi$
$98$		0			0
$99$	4.61760	−	17.9365i	0.464086	−	1.80268i
$100$		0			0
$101$	−16.1544			−1.60742			−0.803712	−	0.595019i	$-0.797145\pi$
$101$	−16.1544			−1.60742			−0.803712	+	0.595019i	$0.797145\pi$
$102$		0			0
$103$		−	9.72456i		−	0.958189i	−0.877763	−	0.479095i	$-0.840965\pi$
$103$		−	9.72456i		−	0.958189i	0.877763	−	0.479095i	$-0.159035\pi$
$104$		0			0
$105$	6.47088	+	0.819574i	0.631493	+	0.0799823i
$106$		0			0
$107$			2.49280i			0.240988i	0.992714	+	0.120494i	$0.0384479\pi$
$107$			2.49280i			0.240988i	−0.992714	+	0.120494i	$0.961552\pi$
$108$		0			0
$109$	−15.8055			−1.51389			−0.756945	−	0.653479i	$-0.773309\pi$
$109$	−15.8055			−1.51389			−0.756945	+	0.653479i	$0.773309\pi$
$110$		0			0
$111$	−11.5875	−	1.46762i	−1.09984	−	0.139301i
$112$		0			0
$113$	−3.53474			−0.332520			−0.166260	−	0.986082i	$-0.553169\pi$
$113$	−3.53474			−0.332520			−0.166260	+	0.986082i	$0.553169\pi$
$114$		0			0
$115$	−4.99127	+	4.05646i	−0.465438	+	0.378267i
$116$		0			0
$117$	3.62200	−	14.0692i	0.334854	−	1.30070i
$118$		0			0
$119$			13.0469i			1.19601i
$120$		0			0
$121$	−27.1155			−2.46505
$122$		0			0
$123$	1.42882	−	11.2811i	0.128833	−	1.01719i
$124$		0			0
$125$		−	10.9990i		−	0.983784i
$126$		0			0
$127$	4.22479			0.374890			0.187445	−	0.982275i	$-0.439979\pi$
$127$	4.22479			0.374890			0.187445	+	0.982275i	$0.439979\pi$
$128$		0			0
$129$	−15.0548	−	1.90678i	−1.32550	−	0.167882i
$130$		0			0
$131$	−5.08510			−0.444287			−0.222143	−	0.975014i	$-0.571305\pi$
$131$	−5.08510			−0.444287			−0.222143	+	0.975014i	$0.571305\pi$
$132$		0			0
$133$			15.2358i			1.32111i
$134$		0			0
$135$	−2.57157	+	6.47681i	−0.221326	+	0.557435i
$136$		0			0
$137$	20.5616			1.75670			0.878350	−	0.478018i	$-0.158645\pi$
$137$	20.5616			1.75670			0.878350	+	0.478018i	$0.158645\pi$
$138$		0			0
$139$			13.6067i			1.15410i	0.816707	+	0.577052i	$0.195797\pi$
$139$			13.6067i			1.15410i	−0.816707	+	0.577052i	$0.804203\pi$
$140$		0			0
$141$	0.281874	−	2.22551i	0.0237380	−	0.187422i
$142$		0			0
$143$	−29.8974			−2.50015
$144$		0			0
$145$		−	2.64518i		−	0.219670i
$146$		0			0
$147$	1.52015	+	0.192535i	0.125380	+	0.0158801i
$148$		0			0
$149$		−	14.0535i		−	1.15130i	−0.817695	−	0.575652i	$-0.804748\pi$
$149$		−	14.0535i		−	1.15130i	0.817695	−	0.575652i	$-0.195252\pi$
$150$		0			0
$151$	5.49673			0.447318			0.223659	−	0.974667i	$-0.428200\pi$
$151$	5.49673			0.447318			0.223659	+	0.974667i	$0.428200\pi$
$152$		0			0
$153$	−13.4990	−	3.47521i	−1.09133	−	0.280954i
$154$		0			0
$155$		−	2.79507i		−	0.224506i
$156$		0			0
$157$	15.7631			1.25803			0.629016	−	0.777392i	$-0.283458\pi$
$157$	15.7631			1.25803			0.629016	+	0.777392i	$0.283458\pi$
$158$		0			0
$159$	−1.88110	+	14.8521i	−0.149181	+	1.17785i
$160$		0			0
$161$	−10.4505	+	8.49323i	−0.823613	+	0.669360i
$162$		0			0
$163$			13.2932i			1.04120i	0.853800	+	0.520601i	$0.174292\pi$
$163$			13.2932i			1.04120i	−0.853800	+	0.520601i	$0.825708\pi$
$164$		0			0
$165$	14.2273	+	1.80197i	1.10759	+	0.140283i
$166$		0			0
$167$			10.5750i			0.818318i	0.912463	+	0.409159i	$0.134178\pi$
$167$			10.5750i			0.818318i	−0.912463	+	0.409159i	$0.865822\pi$
$168$		0			0
$169$	−10.4512			−0.803942
$170$		0			0
$171$	−15.7638	−	4.05825i	−1.20549	−	0.310342i
$172$		0			0
$173$	15.5116			1.17933			0.589664	−	0.807649i	$-0.299260\pi$
$173$	15.5116			1.17933			0.589664	+	0.807649i	$0.299260\pi$
$174$		0			0
$175$		−	8.98944i		−	0.679538i
$176$		0			0
$177$	−9.00064	−	1.13998i	−0.676530	−	0.0856864i
$178$		0			0
$179$	9.95208			0.743853			0.371927	−	0.928262i	$-0.378697\pi$
$179$	9.95208			0.743853			0.371927	+	0.928262i	$0.378697\pi$
$180$		0			0
$181$	−0.0303315			−0.00225452			−0.00112726	−	0.999999i	$-0.500359\pi$
$181$	−0.0303315			−0.00225452			−0.00112726	+	0.999999i	$0.500359\pi$
$182$		0			0
$183$	14.1069	+	1.78673i	1.04282	+	0.132079i
$184$		0			0
$185$		−	9.04381i		−	0.664914i
$186$		0			0
$187$			28.6858i			2.09771i
$188$		0			0
$189$	−5.38424	+	13.5608i	−0.391646	+	0.986405i
$190$		0			0
$191$	24.9085			1.80232			0.901158	−	0.433490i	$-0.142718\pi$
$191$	24.9085			1.80232			0.901158	+	0.433490i	$0.142718\pi$
$192$		0			0
$193$	−4.83745			−0.348207			−0.174104	−	0.984727i	$-0.555703\pi$
$193$	−4.83745			−0.348207			−0.174104	+	0.984727i	$0.555703\pi$
$194$		0			0
$195$	11.1597	+	1.41345i	0.799166	+	0.101219i
$196$		0			0
$197$	−11.5674			−0.824146			−0.412073	−	0.911151i	$-0.635195\pi$
$197$	−11.5674			−0.824146			−0.412073	+	0.911151i	$0.635195\pi$
$198$		0			0
$199$		−	4.84445i		−	0.343414i	−0.985148	−	0.171707i	$-0.945072\pi$
$199$		−	4.84445i		−	0.343414i	0.985148	−	0.171707i	$-0.0549283\pi$
$200$		0			0
$201$	2.77253	+	0.351157i	0.195559	+	0.0247687i
$202$		0			0
$203$		−	5.53835i		−	0.388716i
$204$		0			0
$205$	8.80470			0.614947
$206$		0			0
$207$	−6.00393	−	13.0749i	−0.417302	−	0.908768i
$208$		0			0
$209$			33.4985i			2.31714i
$210$		0			0
$211$		−	25.5581i		−	1.75949i	−0.475443	−	0.879747i	$-0.657712\pi$
$211$		−	25.5581i		−	1.75949i	0.475443	−	0.879747i	$-0.342288\pi$
$212$		0			0
$213$	0.426119	−	3.36438i	0.0291972	−	0.230524i
$214$		0			0
$215$		−	11.7500i		−	0.801340i
$216$		0			0
$217$		−	5.85219i		−	0.397273i
$218$		0			0
$219$	−14.7368	−	1.86650i	−0.995822	−	0.126127i
$220$		0			0
$221$			22.5008i			1.51357i
$222$		0			0
$223$	−22.9173			−1.53466			−0.767328	−	0.641255i	$-0.778414\pi$
$223$	−22.9173			−1.53466			−0.767328	+	0.641255i	$0.778414\pi$
$224$		0			0
$225$	9.30095	+	2.39445i	0.620063	+	0.159630i
$226$		0			0
$227$		−	2.32217i		−	0.154128i	−0.997026	−	0.0770640i	$-0.975445\pi$
$227$		−	2.32217i		−	0.154128i	0.997026	−	0.0770640i	$-0.0245546\pi$
$228$		0			0
$229$	−11.9304			−0.788385			−0.394192	−	0.919028i	$-0.628976\pi$
$229$	−11.9304			−0.788385			−0.394192	+	0.919028i	$0.628976\pi$
$230$		0			0
$231$	29.7884	+	3.77288i	1.95993	+	0.248237i
$232$		0			0
$233$		−	27.4486i		−	1.79822i	−0.437726	−	0.899109i	$-0.644216\pi$
$233$		−	27.4486i		−	1.79822i	0.437726	−	0.899109i	$-0.355784\pi$
$234$		0			0
$235$	1.73696			0.113307
$236$		0			0
$237$	1.78740	−	14.1122i	0.116104	−	0.916688i
$238$		0			0
$239$		−	7.36946i		−	0.476691i	−0.971180	−	0.238345i	$-0.923395\pi$
$239$		−	7.36946i		−	0.476691i	0.971180	−	0.238345i	$-0.0766050\pi$
$240$		0			0
$241$		−	24.7816i		−	1.59632i	−0.602443	−	0.798162i	$-0.705806\pi$
$241$		−	24.7816i		−	1.59632i	0.602443	−	0.798162i	$-0.294194\pi$
$242$		0			0
$243$	−12.5966	−	9.18292i	−0.808072	−	0.589084i
$244$		0			0
$245$			1.18644i			0.0757991i
$246$		0			0
$247$			26.2759i			1.67189i
$248$		0			0
$249$	2.31904	−	18.3098i	0.146963	−	1.16033i
$250$		0			0
$251$			17.1403i			1.08189i	0.841059	+	0.540944i	$0.181933\pi$
$251$			17.1403i			1.08189i	−0.841059	+	0.540944i	$0.818067\pi$
$252$		0			0
$253$	−22.9771	+	18.6738i	−1.44456	+	1.17401i
$254$		0			0
$255$	1.35616	−	10.7075i	0.0849263	−	0.670528i
$256$		0			0
$257$		−	10.2467i		−	0.639171i	−0.947557	−	0.319586i	$-0.896456\pi$
$257$		−	10.2467i		−	0.639171i	0.947557	−	0.319586i	$-0.103544\pi$
$258$		0			0
$259$		−	18.9355i		−	1.17659i
$260$		0			0
$261$	5.73027	+	1.47521i	0.354695	+	0.0913131i
$262$		0			0
$263$	−0.385612			−0.0237779			−0.0118889	−	0.999929i	$-0.503784\pi$
$263$	−0.385612			−0.0237779			−0.0118889	+	0.999929i	$0.503784\pi$
$264$		0			0
$265$	−11.5917			−0.712076
$266$		0			0
$267$	1.57732	+	0.199776i	0.0965302	+	0.0122261i
$268$		0			0
$269$	7.40157			0.451282			0.225641	−	0.974211i	$-0.427552\pi$
$269$	7.40157			0.451282			0.225641	+	0.974211i	$0.427552\pi$
$270$		0			0
$271$	−5.54348			−0.336742			−0.168371	−	0.985724i	$-0.553851\pi$
$271$	−5.54348			−0.336742			−0.168371	+	0.985724i	$0.553851\pi$
$272$		0			0
$273$	23.3657	+	2.95941i	1.41416	+	0.179112i
$274$		0			0
$275$		−	19.7648i		−	1.19186i
$276$		0			0
$277$			9.38881i			0.564119i	0.959397	+	0.282059i	$0.0910175\pi$
$277$			9.38881i			0.564119i	−0.959397	+	0.282059i	$0.908982\pi$
$278$		0			0
$279$	6.05499	+	1.55881i	0.362503	+	0.0933232i
$280$		0			0
$281$	−7.05346			−0.420774			−0.210387	−	0.977618i	$-0.567472\pi$
$281$	−7.05346			−0.420774			−0.210387	+	0.977618i	$0.567472\pi$
$282$		0			0
$283$	0.239678			0.0142474			0.00712370	−	0.999975i	$-0.497732\pi$
$283$	0.239678			0.0142474			0.00712370	+	0.999975i	$0.497732\pi$
$284$		0			0
$285$	1.58369	−	12.5039i	0.0938098	−	0.740667i
$286$		0			0
$287$	18.4349			1.08818
$288$		0			0
$289$	4.58894			0.269938
$290$		0			0
$291$	−0.415697	+	3.28210i	−0.0243686	+	0.192400i
$292$		0			0
$293$			14.2081i			0.830048i	0.909810	+	0.415024i	$0.136227\pi$
$293$			14.2081i			0.830048i	−0.909810	+	0.415024i	$0.863773\pi$
$294$		0			0
$295$		−	7.02482i		−	0.409001i
$296$		0			0
$297$	−11.8381	+	29.8157i	−0.686919	+	1.73008i
$298$		0			0
$299$	−18.0230	+	14.6475i	−1.04230	+	0.847088i
$300$		0			0
$301$		−	24.6015i		−	1.41801i
$302$		0			0
$303$	27.7585	+	3.51578i	1.59468	+	0.201976i
$304$		0			0
$305$			11.0102i			0.630442i
$306$		0			0
$307$		−	22.5656i		−	1.28789i	−0.765073	−	0.643943i	$-0.777297\pi$
$307$		−	22.5656i		−	1.28789i	0.765073	−	0.643943i	$-0.222703\pi$
$308$		0			0
$309$	−2.11641	+	16.7099i	−0.120398	+	0.950595i
$310$		0			0
$311$		−	14.2260i		−	0.806682i	−0.915050	−	0.403341i	$-0.867849\pi$
$311$		−	14.2260i		−	0.806682i	0.915050	−	0.403341i	$-0.132151\pi$
$312$		0			0
$313$		−	9.28451i		−	0.524792i	−0.964960	−	0.262396i	$-0.915487\pi$
$313$		−	9.28451i		−	0.524792i	0.964960	−	0.262396i	$-0.0845126\pi$
$314$		0			0
$315$	−10.9407	−	2.81659i	−0.616438	−	0.158697i
$316$		0			0
$317$	−4.11851			−0.231319			−0.115659	−	0.993289i	$-0.536898\pi$
$317$	−4.11851			−0.231319			−0.115659	+	0.993289i	$0.536898\pi$
$318$		0			0
$319$		−	12.1770i		−	0.681780i
$320$		0			0
$321$	0.542523	−	4.28344i	0.0302807	−	0.239078i
$322$		0			0
$323$	25.2110			1.40278
$324$		0			0
$325$		−	15.5033i		−	0.859968i
$326$		0			0
$327$	27.1589	+	3.43983i	1.50189	+	0.190223i
$328$		0			0
$329$	3.63677			0.200502
$330$		0			0
$331$			23.4632i			1.28965i	0.764329	+	0.644827i	$0.223071\pi$
$331$			23.4632i			1.28965i	−0.764329	+	0.644827i	$0.776929\pi$
$332$		0			0
$333$	19.5917	+	5.04370i	1.07362	+	0.276393i
$334$		0			0
$335$			2.16390i			0.118227i
$336$		0			0
$337$			7.32876i			0.399223i	0.979875	+	0.199611i	$0.0639680\pi$
$337$			7.32876i			0.399223i	−0.979875	+	0.199611i	$0.936032\pi$
$338$		0			0
$339$	6.07382	+	0.769285i	0.329885	+	0.0417818i
$340$		0			0
$341$		−	12.8670i		−	0.696788i
$342$		0			0
$343$		−	17.1716i		−	0.927181i
$344$		0			0
$345$	9.45944	−	5.88404i	0.509279	−	0.316786i
$346$		0			0
$347$	19.7499			1.06023			0.530115	−	0.847926i	$-0.322149\pi$
$347$	19.7499			1.06023			0.530115	+	0.847926i	$0.322149\pi$
$348$		0			0
$349$			33.5713i			1.79703i	0.438945	+	0.898514i	$0.355352\pi$
$349$			33.5713i			1.79703i	−0.438945	+	0.898514i	$0.644648\pi$
$350$		0			0
$351$	−9.28572	+	23.3872i	−0.495635	+	1.24831i
$352$		0			0
$353$			6.67543i			0.355297i	0.984094	+	0.177649i	$0.0568490\pi$
$353$			6.67543i			0.355297i	−0.984094	+	0.177649i	$0.943151\pi$
$354$		0			0
$355$	2.62583			0.139365
$356$		0			0
$357$	2.83947	−	22.4188i	0.150281	−	1.18653i
$358$		0			0
$359$	−11.4626			−0.604972			−0.302486	−	0.953154i	$-0.597816\pi$
$359$	−11.4626			−0.604972			−0.302486	+	0.953154i	$0.597816\pi$
$360$		0			0
$361$	10.4407			0.549509
$362$		0			0
$363$	46.5932	+	5.90131i	2.44551	+	0.309738i
$364$		0			0
$365$		−	11.5018i		−	0.602031i
$366$		0			0
$367$			5.41598i			0.282712i	0.989959	+	0.141356i	$0.0451462\pi$
$367$			5.41598i			0.282712i	−0.989959	+	0.141356i	$0.954854\pi$
$368$		0			0
$369$	−4.91036	+	19.0737i	−0.255623	+	0.992936i
$370$		0			0
$371$	−24.2702			−1.26005
$372$		0			0
$373$	13.1527			0.681018			0.340509	−	0.940241i	$-0.389401\pi$
$373$	13.1527			0.681018			0.340509	+	0.940241i	$0.389401\pi$
$374$		0			0
$375$	−2.39378	+	18.8999i	−0.123614	+	0.975987i
$376$		0			0
$377$		−	9.55150i		−	0.491927i
$378$		0			0
$379$	−4.98683			−0.256156			−0.128078	−	0.991764i	$-0.540881\pi$
$379$	−4.98683			−0.256156			−0.128078	+	0.991764i	$0.540881\pi$
$380$		0			0
$381$	−7.25956	−	0.919465i	−0.371918	−	0.0471056i
$382$		0			0
$383$	−6.42719			−0.328414			−0.164207	−	0.986426i	$-0.552506\pi$
$383$	−6.42719			−0.328414			−0.164207	+	0.986426i	$0.552506\pi$
$384$		0			0
$385$			23.2493i			1.18489i
$386$		0			0
$387$	25.4540	+	6.55292i	1.29390	+	0.333103i
$388$		0			0
$389$			14.3879i			0.729498i	0.931106	+	0.364749i	$0.118845\pi$
$389$			14.3879i			0.729498i	−0.931106	+	0.364749i	$0.881155\pi$
$390$		0			0
$391$	14.0539	+	17.2926i	0.710736	+	0.874524i
$392$		0			0
$393$	8.73784	+	1.10670i	0.440766	+	0.0558255i
$394$		0			0
$395$	11.0143			0.554190
$396$		0			0
$397$		−	5.15731i		−	0.258838i	−0.991590	−	0.129419i	$-0.958689\pi$
$397$		−	5.15731i		−	0.258838i	0.991590	−	0.129419i	$-0.0413112\pi$
$398$		0			0
$399$	3.31586	−	26.1801i	0.166000	−	1.31064i
$400$		0			0
$401$	17.0512			0.851496			0.425748	−	0.904842i	$-0.360011\pi$
$401$	17.0512			0.851496			0.425748	+	0.904842i	$0.360011\pi$
$402$		0			0
$403$		−	10.0928i		−	0.502756i
$404$		0			0
$405$	5.82838	−	10.5696i	0.289615	−	0.525207i
$406$		0			0
$407$		−	41.6328i		−	2.06366i
$408$		0			0
$409$	−0.175936			−0.00869950			−0.00434975	−	0.999991i	$-0.501385\pi$
$409$	−0.175936			−0.00869950			−0.00434975	+	0.999991i	$0.501385\pi$
$410$		0			0
$411$	−35.3316	−	4.47495i	−1.74278	−	0.220733i
$412$		0			0
$413$		−	14.7082i		−	0.723744i
$414$		0			0
$415$	14.2904			0.701488
$416$		0			0
$417$	2.96130	−	23.3807i	0.145015	−	1.14496i
$418$		0			0
$419$			15.1092i			0.738133i	0.929403	+	0.369066i	$0.120323\pi$
$419$			15.1092i			0.738133i	−0.929403	+	0.369066i	$0.879677\pi$
$420$		0			0
$421$	−14.4108			−0.702337			−0.351169	−	0.936312i	$-0.614216\pi$
$421$	−14.4108			−0.702337			−0.351169	+	0.936312i	$0.614216\pi$
$422$		0			0
$423$	−0.968700	+	3.76280i	−0.0470998	+	0.182953i
$424$		0			0
$425$	−14.8750			−0.721543
$426$		0			0
$427$			23.0526i			1.11559i
$428$		0			0
$429$	51.3735	+	6.50675i	2.48033	+	0.314149i
$430$		0			0
$431$	14.2732			0.687514			0.343757	−	0.939059i	$-0.388300\pi$
$431$	14.2732			0.687514			0.343757	+	0.939059i	$0.388300\pi$
$432$		0			0
$433$			14.5780i			0.700573i	0.936643	+	0.350287i	$0.113916\pi$
$433$			14.5780i			0.700573i	−0.936643	+	0.350287i	$0.886084\pi$
$434$		0			0
$435$	−0.575685	+	4.54527i	−0.0276020	+	0.217929i
$436$		0			0
$437$	16.4118	+	20.1938i	0.785081	+	0.966001i
$438$		0			0
$439$	−26.6570			−1.27227			−0.636134	−	0.771578i	$-0.719468\pi$
$439$	−26.6570			−1.27227			−0.636134	+	0.771578i	$0.719468\pi$
$440$		0			0
$441$	−2.57020	−	0.661676i	−0.122391	−	0.0315084i
$442$		0			0
$443$	18.0639			0.858244			0.429122	−	0.903247i	$-0.358823\pi$
$443$	18.0639			0.858244			0.429122	+	0.903247i	$0.358823\pi$
$444$		0			0
$445$			1.23106i			0.0583580i
$446$		0			0
$447$	−3.05853	+	24.1484i	−0.144664	+	1.14218i
$448$		0			0
$449$		−	3.51992i		−	0.166115i	−0.996545	−	0.0830577i	$-0.973531\pi$
$449$		−	3.51992i		−	0.166115i	0.996545	−	0.0830577i	$-0.0264686\pi$
$450$		0			0
$451$	40.5321			1.90858
$452$		0			0
$453$	−9.44517	−	1.19629i	−0.443773	−	0.0562064i
$454$		0			0
$455$			18.2365i			0.854940i
$456$		0			0
$457$		−	8.18565i		−	0.382908i	−0.981502	−	0.191454i	$-0.938680\pi$
$457$		−	8.18565i		−	0.382908i	0.981502	−	0.191454i	$-0.0613204\pi$
$458$		0			0
$459$	22.4393	+	8.90940i	1.04738	+	0.415855i
$460$		0			0
$461$	4.67228			0.217610			0.108805	−	0.994063i	$-0.465298\pi$
$461$	4.67228			0.217610			0.108805	+	0.994063i	$0.465298\pi$
$462$		0			0
$463$	−34.0823			−1.58394			−0.791969	−	0.610562i	$-0.790944\pi$
$463$	−34.0823			−1.58394			−0.791969	+	0.610562i	$0.790944\pi$
$464$		0			0
$465$	−0.608308	+	4.80284i	−0.0282096	+	0.222726i
$466$		0			0
$467$			5.14166i			0.237928i	0.992899	+	0.118964i	$0.0379572\pi$
$467$			5.14166i			0.237928i	−0.992899	+	0.118964i	$0.962043\pi$
$468$		0			0
$469$			4.53067i			0.209207i
$470$		0			0
$471$	−27.0861	−	3.43061i	−1.24806	−	0.158074i
$472$		0			0
$473$		−	54.0905i		−	2.48708i
$474$		0			0
$475$	−17.3706			−0.797018
$476$		0			0
$477$	6.46469	−	25.1113i	0.295998	−	1.14977i
$478$		0			0
$479$	34.5989			1.58086			0.790432	−	0.612550i	$-0.209856\pi$
$479$	34.5989			1.58086			0.790432	+	0.612550i	$0.209856\pi$
$480$		0			0
$481$		−	32.6563i		−	1.48900i
$482$		0			0
$483$	19.8057	−	12.3197i	0.901192	−	0.560567i
$484$		0			0
$485$	−2.56161			−0.116317
$486$		0			0
$487$	15.8570			0.718549			0.359274	−	0.933232i	$-0.383024\pi$
$487$	15.8570			0.718549			0.359274	+	0.933232i	$0.383024\pi$
$488$		0			0
$489$	2.89307	−	22.8420i	0.130829	−	1.03295i
$490$		0			0
$491$	−22.7212			−1.02540			−0.512698	−	0.858569i	$-0.671354\pi$
$491$	−22.7212			−1.02540			−0.512698	+	0.858569i	$0.671354\pi$
$492$		0			0
$493$	−9.16440			−0.412744
$494$		0			0
$495$	−24.0549	−	6.19273i	−1.08119	−	0.278343i
$496$		0			0
$497$	5.49784			0.246612
$498$		0			0
$499$			3.47554i			0.155587i	0.996970	+	0.0777933i	$0.0247874\pi$
$499$			3.47554i			0.155587i	−0.996970	+	0.0777933i	$0.975213\pi$
$500$		0			0
$501$	2.30150	−	18.1713i	0.102823	−	0.811832i
$502$		0			0
$503$	16.9913			0.757606			0.378803	−	0.925477i	$-0.376336\pi$
$503$	16.9913			0.757606			0.378803	+	0.925477i	$0.376336\pi$
$504$		0			0
$505$			21.6649i			0.964077i
$506$		0			0
$507$	17.9586	+	2.27456i	0.797570	+	0.101017i
$508$		0			0
$509$	23.7927			1.05459			0.527295	−	0.849682i	$-0.323206\pi$
$509$	23.7927			1.05459			0.527295	+	0.849682i	$0.323206\pi$
$510$		0			0
$511$		−	24.0819i		−	1.06532i
$512$		0			0
$513$	26.2040	+	10.4041i	1.15694	+	0.459354i
$514$		0			0
$515$	−13.0418			−0.574689
$516$		0			0
$517$	7.99605			0.351666
$518$		0			0
$519$	−26.6540	−	3.37589i	−1.16998	−	0.148185i
$520$		0			0
$521$	40.4306			1.77130			0.885649	−	0.464355i	$-0.153714\pi$
$521$	40.4306			1.77130			0.885649	+	0.464355i	$0.153714\pi$
$522$		0			0
$523$	−37.7446			−1.65046			−0.825228	−	0.564799i	$-0.808954\pi$
$523$	−37.7446			−1.65046			−0.825228	+	0.564799i	$0.808954\pi$
$524$		0			0
$525$	−1.95642	+	15.4468i	−0.0853853	+	0.674152i
$526$		0			0
$527$	−9.68373			−0.421830
$528$		0			0
$529$	−4.70248	+	22.5141i	−0.204456	+	0.978876i
$530$		0			0
$531$	15.2179	+	3.91772i	0.660401	+	0.170015i
$532$		0			0
$533$	31.7930			1.37711
$534$		0			0
$535$	3.34314			0.144537
$536$		0			0
$537$	−17.1009	−	2.16593i	−0.737958	−	0.0934667i
$538$		0			0
$539$			5.46175i			0.235254i
$540$		0			0
$541$			1.12183i			0.0482313i	0.999709	+	0.0241157i	$0.00767700\pi$
$541$			1.12183i			0.0482313i	−0.999709	+	0.0241157i	$0.992323\pi$
$542$		0			0
$543$	0.0521193	+	0.00660121i	0.00223665	+	0.000283285i
$544$		0			0
$545$			21.1970i			0.907979i
$546$		0			0
$547$		−	24.3932i		−	1.04298i	−0.853258	−	0.521489i	$-0.825377\pi$
$547$		−	24.3932i		−	1.04298i	0.853258	−	0.521489i	$-0.174623\pi$
$548$		0			0
$549$	−23.8514	−	6.14035i	−1.01795	−	0.262064i
$550$		0			0
$551$	−10.7019			−0.455918
$552$		0			0
$553$	23.0612			0.980664
$554$		0			0
$555$	−1.96825	+	15.5402i	−0.0835478	+	0.659644i
$556$		0			0
$557$			7.71934i			0.327079i	0.986537	+	0.163540i	$0.0522911\pi$
$557$			7.71934i			0.327079i	−0.986537	+	0.163540i	$0.947709\pi$
$558$		0			0
$559$		−	42.4280i		−	1.79451i
$560$		0			0
$561$	6.24305	−	49.2914i	0.263582	−	2.08109i
$562$		0			0
$563$		−	2.37078i		−	0.0999163i	−0.998751	−	0.0499581i	$-0.984091\pi$
$563$		−	2.37078i		−	0.0999163i	0.998751	−	0.0499581i	$-0.0159088\pi$
$564$		0			0
$565$			4.74050i			0.199434i
$566$		0			0
$567$	12.2032	−	22.1301i	0.512485	−	0.929376i
$568$		0			0
$569$	8.45368			0.354397			0.177198	−	0.984175i	$-0.443297\pi$
$569$	8.45368			0.354397			0.177198	+	0.984175i	$0.443297\pi$
$570$		0			0
$571$	−11.0497			−0.462416			−0.231208	−	0.972904i	$-0.574268\pi$
$571$	−11.0497			−0.462416			−0.231208	+	0.972904i	$0.574268\pi$
$572$		0			0
$573$	−42.8009	−	5.42098i	−1.78803	−	0.226465i
$574$		0			0
$575$	−9.68327	−	11.9148i	−0.403820	−	0.496880i
$576$		0			0
$577$	24.9723			1.03961			0.519804	−	0.854285i	$-0.326005\pi$
$577$	24.9723			1.03961			0.519804	+	0.854285i	$0.326005\pi$
$578$		0			0
$579$	8.31231	+	1.05280i	0.345448	+	0.0437530i
$580$		0			0
$581$	29.9205			1.24131
$582$		0			0
$583$	−53.3622			−2.21003
$584$		0			0
$585$	−18.8684	−	4.85752i	−0.780114	−	0.200834i
$586$		0			0
$587$	20.3561			0.840187			0.420093	−	0.907481i	$-0.361997\pi$
$587$	20.3561			0.840187			0.420093	+	0.907481i	$0.361997\pi$
$588$		0			0
$589$	−11.3084			−0.465954
$590$		0			0
$591$	19.8766	+	2.51749i	0.817614	+	0.103556i
$592$		0			0
$593$		−	7.15453i		−	0.293801i	−0.989151	−	0.146901i	$-0.953070\pi$
$593$		−	7.15453i		−	0.293801i	0.989151	−	0.146901i	$-0.0469298\pi$
$594$		0			0
$595$	17.4974			0.717324
$596$		0			0
$597$	−1.05433	+	8.32433i	−0.0431507	+	0.340692i
$598$		0			0
$599$		−	29.8254i		−	1.21863i	−0.792927	−	0.609316i	$-0.791444\pi$
$599$		−	29.8254i		−	1.21863i	0.792927	−	0.609316i	$-0.208556\pi$
$600$		0			0
$601$	23.2672			0.949090			0.474545	−	0.880231i	$-0.342613\pi$
$601$	23.2672			0.949090			0.474545	+	0.880231i	$0.342613\pi$
$602$		0			0
$603$	−4.68767	−	1.20680i	−0.190897	−	0.0491448i
$604$		0			0
$605$			36.3651i			1.47845i
$606$		0			0
$607$	45.1275			1.83167			0.915833	−	0.401558i	$-0.131531\pi$
$607$	45.1275			1.83167			0.915833	+	0.401558i	$0.131531\pi$
$608$		0			0
$609$	−1.20534	+	9.51667i	−0.0488429	+	0.385635i
$610$		0			0
$611$	6.27202			0.253739
$612$		0			0
$613$	−20.5759			−0.831054			−0.415527	−	0.909581i	$-0.636403\pi$
$613$	−20.5759			−0.831054			−0.415527	+	0.909581i	$0.636403\pi$
$614$		0			0
$615$	−15.1293	−	1.91622i	−0.610073	−	0.0772693i
$616$		0			0
$617$	24.4777			0.985434			0.492717	−	0.870189i	$-0.336004\pi$
$617$	24.4777			0.985434			0.492717	+	0.870189i	$0.336004\pi$
$618$		0			0
$619$	−21.6956			−0.872020			−0.436010	−	0.899942i	$-0.643609\pi$
$619$	−21.6956			−0.872020			−0.436010	+	0.899942i	$0.643609\pi$
$620$		0			0
$621$	7.47113	+	23.7736i	0.299806	+	0.954000i
$622$		0			0
$623$			2.57754i			0.103267i
$624$		0			0
$625$	1.25603			0.0502412
$626$		0			0
$627$	7.29046	−	57.5612i	0.291153	−	2.29877i
$628$		0			0
$629$	−31.3329			−1.24932
$630$		0			0
$631$		−	37.8901i		−	1.50838i	−0.656656	−	0.754190i	$-0.728030\pi$
$631$		−	37.8901i		−	1.50838i	0.656656	−	0.754190i	$-0.271970\pi$
$632$		0			0
$633$	−5.56236	+	43.9171i	−0.221084	+	1.74555i
$634$		0			0
$635$		−	5.66594i		−	0.224846i
$636$		0			0
$637$			4.28414i			0.169744i
$638$		0			0
$639$	−1.46442	+	5.68836i	−0.0579316	+	0.225028i
$640$		0			0
$641$	−20.7530			−0.819696			−0.409848	−	0.912154i	$-0.634418\pi$
$641$	−20.7530			−0.819696			−0.409848	+	0.912154i	$0.634418\pi$
$642$		0			0
$643$	1.87244			0.0738417			0.0369209	−	0.999318i	$-0.488245\pi$
$643$	1.87244			0.0738417			0.0369209	+	0.999318i	$0.488245\pi$
$644$		0			0
$645$	−2.55721	+	20.1902i	−0.100690	+	0.794989i
$646$		0			0
$647$			16.8036i			0.660617i	0.943873	+	0.330308i	$0.107153\pi$
$647$			16.8036i			0.660617i	−0.943873	+	0.330308i	$0.892847\pi$
$648$		0			0
$649$		−	32.3385i		−	1.26940i
$650$		0			0
$651$	−1.27365	+	10.0560i	−0.0499181	+	0.394124i
$652$		0			0
$653$	−19.3214			−0.756106			−0.378053	−	0.925784i	$-0.623406\pi$
$653$	−19.3214			−0.756106			−0.378053	+	0.925784i	$0.623406\pi$
$654$		0			0
$655$			6.81971i			0.266468i
$656$		0			0
$657$	24.9164	+	6.41452i	0.972082	+	0.250254i
$658$		0			0
$659$		−	23.3970i		−	0.911419i	−0.890129	−	0.455709i	$-0.849386\pi$
$659$		−	23.3970i		−	0.911419i	0.890129	−	0.455709i	$-0.150614\pi$
$660$		0			0
$661$	23.4727			0.912984			0.456492	−	0.889728i	$-0.349106\pi$
$661$	23.4727			0.912984			0.456492	+	0.889728i	$0.349106\pi$
$662$		0			0
$663$	4.89698	−	38.6637i	0.190183	−	1.50157i
$664$		0			0
$665$	20.4330			0.792358
$666$		0			0
$667$	−5.96581	−	7.34062i	−0.230997	−	0.284230i
$668$		0			0
$669$	39.3793	+	4.98762i	1.52249	+	0.192833i
$670$		0			0
$671$			50.6850i			1.95667i
$672$		0			0
$673$	−26.3870			−1.01714			−0.508572	−	0.861019i	$-0.669827\pi$
$673$	−26.3870			−1.01714			−0.508572	+	0.861019i	$0.669827\pi$
$674$		0			0
$675$	−15.4609	−	6.13866i	−0.595091	−	0.236277i
$676$		0			0
$677$			19.2132i			0.738422i	0.929346	+	0.369211i	$0.120372\pi$
$677$			19.2132i			0.738422i	−0.929346	+	0.369211i	$0.879628\pi$
$678$		0			0
$679$	−5.36337			−0.205827
$680$		0			0
$681$	−0.505388	+	3.99024i	−0.0193665	+	0.152906i
$682$		0			0
$683$	16.2143			0.620422			0.310211	−	0.950668i	$-0.399600\pi$
$683$	16.2143			0.620422			0.310211	+	0.950668i	$0.399600\pi$
$684$		0			0
$685$		−	27.5756i		−	1.05361i
$686$		0			0
$687$	20.5003	+	2.59649i	0.782136	+	0.0990621i
$688$		0			0
$689$	−41.8567			−1.59461
$690$		0			0
$691$			10.2750i			0.390880i	0.980716	+	0.195440i	$0.0626134\pi$
$691$			10.2750i			0.390880i	−0.980716	+	0.195440i	$0.937387\pi$
$692$		0			0
$693$	−50.3650	−	12.9660i	−1.91321	−	0.492539i
$694$		0			0
$695$	18.2482			0.692192
$696$		0			0
$697$		−	30.5045i		−	1.15544i
$698$		0			0
$699$	−5.97379	+	47.1656i	−0.225950	+	1.78397i
$700$		0			0
$701$			37.9687i			1.43406i	0.697044	+	0.717029i	$0.254498\pi$
$701$			37.9687i			1.43406i	−0.697044	+	0.717029i	$0.745502\pi$
$702$		0			0
$703$	−36.5897			−1.38001
$704$		0			0
$705$	−2.98467	−	0.378025i	−0.112409	−	0.0142373i
$706$		0			0
$707$			45.3610i			1.70598i
$708$		0			0
$709$	−31.3658			−1.17797			−0.588983	−	0.808146i	$-0.700471\pi$
$709$	−31.3658			−1.17797			−0.588983	+	0.808146i	$0.700471\pi$
$710$		0			0
$711$	−6.14265	+	23.8604i	−0.230367	+	0.894834i
$712$		0			0
$713$	−6.30388	−	7.75660i	−0.236082	−	0.290487i
$714$		0			0
$715$			40.0959i			1.49950i
$716$		0			0
$717$	−1.60386	+	12.6631i	−0.0598972	+	0.472913i
$718$		0			0
$719$			13.6827i			0.510279i	0.966904	+	0.255140i	$0.0821214\pi$
$719$			13.6827i			0.510279i	−0.966904	+	0.255140i	$0.917879\pi$
$720$		0			0
$721$	−27.3062			−1.01694
$722$		0			0
$723$	−5.39336	+	42.5828i	−0.200581	+	1.58367i
$724$		0			0
$725$	6.31436			0.234509
$726$		0			0
$727$		−	7.42941i		−	0.275542i	−0.990464	−	0.137771i	$-0.956006\pi$
$727$		−	7.42941i		−	0.275542i	0.990464	−	0.137771i	$-0.0439937\pi$
$728$		0			0
$729$	19.6465	+	18.5207i	0.727647	+	0.685951i
$730$		0			0
$731$	−40.7085			−1.50566
$732$		0			0
$733$	22.0117			0.813021			0.406511	−	0.913646i	$-0.366745\pi$
$733$	22.0117			0.813021			0.406511	+	0.913646i	$0.366745\pi$
$734$		0			0
$735$	0.258213	−	2.03869i	0.00952431	−	0.0751984i
$736$		0			0
$737$			9.96143i			0.366934i
$738$		0			0
$739$		−	8.66574i		−	0.318774i	−0.987216	−	0.159387i	$-0.949048\pi$
$739$		−	8.66574i		−	0.318774i	0.987216	−	0.159387i	$-0.0509518\pi$
$740$		0			0
$741$	5.71856	−	45.1504i	0.210077	−	1.65864i
$742$		0			0
$743$	−45.6054			−1.67310			−0.836550	−	0.547891i	$-0.815431\pi$
$743$	−45.6054			−1.67310			−0.836550	+	0.547891i	$0.815431\pi$
$744$		0			0
$745$	−18.8473			−0.690513
$746$		0			0
$747$	−7.96971	+	30.9574i	−0.291597	+	1.13267i
$748$		0			0
$749$	6.99970			0.255764
$750$		0			0
$751$			50.6579i			1.84853i	0.381748	+	0.924266i	$0.375322\pi$
$751$			50.6579i			1.84853i	−0.381748	+	0.924266i	$0.624678\pi$
$752$		0			0
$753$	3.73035	−	29.4526i	0.135941	−	1.07331i
$754$		0			0
$755$		−	7.37176i		−	0.268286i
$756$		0			0
$757$	2.49892			0.0908248			0.0454124	−	0.998968i	$-0.485540\pi$
$757$	2.49892			0.0908248			0.0454124	+	0.998968i	$0.485540\pi$
$758$		0			0
$759$	43.5462	−	27.0869i	1.58063	−	0.983194i
$760$		0			0
$761$			35.7987i			1.29770i	0.760916	+	0.648851i	$0.224750\pi$
$761$			35.7987i			1.29770i	−0.760916	+	0.648851i	$0.775250\pi$
$762$		0			0
$763$			44.3812i			1.60671i
$764$		0			0
$765$	−4.66066	+	18.1038i	−0.168506	+	0.654543i
$766$		0			0
$767$		−	25.3660i		−	0.915912i
$768$		0			0
$769$			20.1663i			0.727216i	0.931552	+	0.363608i	$0.118455\pi$
$769$			20.1663i			0.727216i	−0.931552	+	0.363608i	$0.881545\pi$
$770$		0			0
$771$	−2.23005	+	17.6071i	−0.0803131	+	0.634105i
$772$		0			0
$773$		−	26.0350i		−	0.936415i	−0.883619	−	0.468207i	$-0.844900\pi$
$773$		−	26.0350i		−	0.936415i	0.883619	−	0.468207i	$-0.155100\pi$
$774$		0			0
$775$	6.67218			0.239672
$776$		0			0
$777$	−4.12104	+	32.5373i	−0.147841	+	1.16727i
$778$		0			0
$779$		−	35.6223i		−	1.27630i
$780$		0			0
$781$	12.0879			0.432540
$782$		0			0
$783$	−9.52539	−	3.78200i	−0.340410	−	0.135158i
$784$		0			0
$785$		−	21.1402i		−	0.754525i
$786$		0			0
$787$	8.07737			0.287927			0.143964	−	0.989583i	$-0.454015\pi$
$787$	8.07737			0.287927			0.143964	+	0.989583i	$0.454015\pi$
$788$		0			0
$789$	0.662607	+	0.0839230i	0.0235894	+	0.00298774i
$790$		0			0
$791$			9.92542i			0.352907i
$792$		0			0
$793$			39.7568i			1.41180i
$794$		0			0
$795$	19.9184	+	2.52278i	0.706432	+	0.0894737i
$796$		0			0
$797$		−	37.2035i		−	1.31782i	−0.752223	−	0.658908i	$-0.771019\pi$
$797$		−	37.2035i		−	1.31782i	0.752223	−	0.658908i	$-0.228981\pi$
$798$		0			0
$799$		−	6.01783i		−	0.212896i
$800$		0			0
$801$	−2.66686	−	0.686561i	−0.0942289	−	0.0242584i
$802$		0			0
$803$		−	52.9480i		−	1.86850i
$804$		0			0
$805$	11.3904	+	14.0153i	0.401459	+	0.493975i
$806$		0			0
$807$	−12.7183	−	1.61085i	−0.447705	−	0.0567045i
$808$		0			0
$809$		−	1.81783i		−	0.0639116i	−0.999489	−	0.0319558i	$-0.989826\pi$
$809$		−	1.81783i		−	0.0639116i	0.999489	−	0.0319558i	$-0.0101736\pi$
$810$		0			0
$811$		−	25.6084i		−	0.899233i	−0.893222	−	0.449617i	$-0.851561\pi$
$811$		−	25.6084i		−	0.899233i	0.893222	−	0.449617i	$-0.148439\pi$
$812$		0			0
$813$	9.52549	+	1.20646i	0.334074	+	0.0423124i
$814$		0			0
$815$	17.8277			0.624477
$816$		0			0
$817$	−47.5383			−1.66315
$818$		0			0
$819$	−39.5058	−	10.1704i	−1.38045	−	0.355384i
$820$		0			0
$821$	−5.10198			−0.178060			−0.0890301	−	0.996029i	$-0.528377\pi$
$821$	−5.10198			−0.178060			−0.0890301	+	0.996029i	$0.528377\pi$
$822$		0			0
$823$	10.6288			0.370498			0.185249	−	0.982692i	$-0.440691\pi$
$823$	10.6288			0.370498			0.185249	+	0.982692i	$0.440691\pi$
$824$		0			0
$825$	−4.30152	+	33.9623i	−0.149760	+	1.18241i
$826$		0			0
$827$			15.8531i			0.551267i	0.961263	+	0.275634i	$0.0888877\pi$
$827$			15.8531i			0.551267i	−0.961263	+	0.275634i	$0.911112\pi$
$828$		0			0
$829$			19.4000i			0.673791i	0.941542	+	0.336896i	$0.109377\pi$
$829$			19.4000i			0.673791i	−0.941542	+	0.336896i	$0.890623\pi$
$830$		0			0
$831$	2.04334	−	16.1330i	0.0708827	−	0.559648i
$832$		0			0
$833$	4.11052			0.142421
$834$		0			0
$835$	14.1823			0.490799
$836$		0			0
$837$	−10.0652	−	3.99631i	−0.347903	−	0.138133i
$838$		0			0
$839$	−18.3937			−0.635023			−0.317511	−	0.948254i	$-0.602847\pi$
$839$	−18.3937			−0.635023			−0.317511	+	0.948254i	$0.602847\pi$
$840$		0			0
$841$	−25.1098			−0.865854
$842$		0			0
$843$	12.1201	+	1.53508i	0.417439	+	0.0528711i
$844$		0			0
$845$			14.0163i			0.482177i
$846$		0			0
$847$			76.1394i			2.61618i
$848$		0			0
$849$	−0.411845	−	0.0521625i	−0.0141345	−	0.00179021i
$850$		0			0
$851$	−20.3970	−	25.0974i	−0.699199	−	0.860328i
$852$		0			0
$853$		−	33.2062i		−	1.13696i	−0.822697	−	0.568479i	$-0.807532\pi$
$853$		−	33.2062i		−	1.13696i	0.822697	−	0.568479i	$-0.192468\pi$
$854$		0			0
$855$	−5.44259	+	21.1411i	−0.186133	+	0.723009i
$856$		0			0
$857$		−	27.1763i		−	0.928324i	−0.885750	−	0.464162i	$-0.846355\pi$
$857$		−	27.1763i		−	0.928324i	0.885750	−	0.464162i	$-0.153645\pi$
$858$		0			0
$859$			1.63700i			0.0558536i	0.999610	+	0.0279268i	$0.00889053\pi$
$859$			1.63700i			0.0558536i	−0.999610	+	0.0279268i	$0.991109\pi$
$860$		0			0
$861$	−31.6770	−	4.01208i	−1.07955	−	0.136731i
$862$		0			0
$863$		−	26.5108i		−	0.902440i	−0.892413	−	0.451220i	$-0.850989\pi$
$863$		−	26.5108i		−	0.902440i	0.892413	−	0.451220i	$-0.149011\pi$
$864$		0			0
$865$		−	20.8029i		−	0.707321i
$866$		0			0
$867$	−7.88529	−	0.998718i	−0.267798	−	0.0339182i
$868$		0			0
$869$	50.7040			1.72001
$870$		0			0
$871$			7.81365i			0.264755i
$872$		0			0
$873$	1.42860	−	5.54923i	0.0483509	−	0.187813i
$874$		0			0
$875$	−30.8849			−1.04410
$876$		0			0
$877$		−	34.5725i		−	1.16743i	−0.811958	−	0.583716i	$-0.801598\pi$
$877$		−	34.5725i		−	1.16743i	0.811958	−	0.583716i	$-0.198402\pi$
$878$		0			0
$879$	3.09220	−	24.4142i	0.104297	−	0.823469i
$880$		0			0
$881$	−18.6056			−0.626839			−0.313420	−	0.949615i	$-0.601475\pi$
$881$	−18.6056			−0.626839			−0.313420	+	0.949615i	$0.601475\pi$
$882$		0			0
$883$		−	13.2184i		−	0.444834i	−0.974952	−	0.222417i	$-0.928605\pi$
$883$		−	13.2184i		−	0.444834i	0.974952	−	0.222417i	$-0.0713946\pi$
$884$		0			0
$885$	−1.52885	+	12.0709i	−0.0513918	+	0.405759i
$886$		0			0
$887$			4.96093i			0.166572i	0.996526	+	0.0832859i	$0.0265415\pi$
$887$			4.96093i			0.166572i	−0.996526	+	0.0832859i	$0.973459\pi$
$888$		0			0
$889$		−	11.8631i		−	0.397874i
$890$		0			0
$891$	26.8307	−	48.6567i	0.898863	−	1.63006i
$892$		0			0
$893$		−	7.02746i		−	0.235165i
$894$		0			0
$895$		−	13.3469i		−	0.446138i
$896$		0			0
$897$	34.1572	−	21.2467i	1.14048	−	0.709407i
$898$		0			0
$899$	4.11070			0.137099
$900$		0			0
$901$			40.1604i			1.33794i
$902$		0			0
$903$	−5.35416	+	42.2733i	−0.178175	+	1.40677i
$904$		0			0
$905$			0.0406781i			0.00135218i
$906$		0			0
$907$	3.89499			0.129331			0.0646655	−	0.997907i	$-0.479402\pi$
$907$	3.89499			0.129331			0.0646655	+	0.997907i	$0.479402\pi$
$908$		0			0
$909$	−46.9329	−	12.0825i	−1.55667	−	0.400750i
$910$		0			0
$911$	−36.4367			−1.20720			−0.603600	−	0.797287i	$-0.706268\pi$
$911$	−36.4367			−1.20720			−0.603600	+	0.797287i	$0.706268\pi$
$912$		0			0
$913$	65.7853			2.17717
$914$		0			0
$915$	2.39621	−	18.9191i	0.0792163	−	0.625445i
$916$		0			0
$917$			14.2788i			0.471527i
$918$		0			0
$919$			2.53453i			0.0836063i	0.999126	+	0.0418032i	$0.0133102\pi$
$919$			2.53453i			0.0836063i	−0.999126	+	0.0418032i	$0.986690\pi$
$920$		0			0
$921$	−4.91108	+	38.7750i	−0.161825	+	1.27768i
$922$		0			0
$923$	9.48164			0.312092
$924$		0			0
$925$	21.5886			0.709830
$926$		0			0
$927$	7.27336	−	28.2525i	0.238888	−	0.927933i
$928$		0			0
$929$		−	55.4181i		−	1.81821i	−0.416569	−	0.909104i	$-0.636768\pi$
$929$		−	55.4181i		−	1.81821i	0.416569	−	0.909104i	$-0.363232\pi$
$930$		0			0
$931$	4.80015			0.157318
$932$		0			0
$933$	−3.09609	+	24.4449i	−0.101361	+	0.800289i
$934$		0			0
$935$	38.4710			1.25814
$936$		0			0
$937$			18.4826i			0.603800i	0.953340	+	0.301900i	$0.0976209\pi$
$937$			18.4826i			0.603800i	−0.953340	+	0.301900i	$0.902379\pi$
$938$		0			0
$939$	−2.02064	+	15.9538i	−0.0659411	+	0.520632i
$940$		0			0
$941$		−	46.6958i		−	1.52224i	−0.648610	−	0.761121i	$-0.724650\pi$
$941$		−	46.6958i		−	1.52224i	0.648610	−	0.761121i	$-0.275350\pi$
$942$		0			0
$943$	24.4339	−	19.8577i	0.795676	−	0.646656i
$944$		0			0
$945$	18.1867	+	7.22089i	0.591612	+	0.234896i
$946$		0			0
$947$	22.2910			0.724362			0.362181	−	0.932108i	$-0.382032\pi$
$947$	22.2910			0.724362			0.362181	+	0.932108i	$0.382032\pi$
$948$		0			0
$949$		−	41.5319i		−	1.34818i
$950$		0			0
$951$	7.07694	+	0.896335i	0.229485	+	0.0290657i
$952$		0			0
$953$	−29.3186			−0.949721			−0.474861	−	0.880061i	$-0.657501\pi$
$953$	−29.3186			−0.949721			−0.474861	+	0.880061i	$0.657501\pi$
$954$		0			0
$955$		−	33.4052i		−	1.08097i
$956$		0			0
$957$	−2.65014	+	20.9240i	−0.0856670	+	0.676376i
$958$		0			0
$959$		−	57.7364i		−	1.86441i
$960$		0			0
$961$	−26.6564			−0.859883
$962$		0			0
$963$	−1.86446	+	7.24226i	−0.0600814	+	0.233379i
$964$		0			0
$965$			6.48759i			0.208843i
$966$		0			0
$967$	44.5365			1.43220			0.716098	−	0.698000i	$-0.245926\pi$
$967$	44.5365			1.43220			0.716098	+	0.698000i	$0.245926\pi$
$968$		0			0
$969$	−43.3206	−	5.48681i	−1.39166	−	0.176262i
$970$		0			0
$971$		−	20.5639i		−	0.659926i	−0.943994	−	0.329963i	$-0.892964\pi$
$971$		−	20.5639i		−	0.659926i	0.943994	−	0.329963i	$-0.107036\pi$
$972$		0			0
$973$	38.2071			1.22486
$974$		0			0
$975$	−3.37407	+	26.6397i	−0.108057	+	0.853152i
$976$		0			0
$977$	−28.3080			−0.905655			−0.452827	−	0.891598i	$-0.649585\pi$
$977$	−28.3080			−0.905655			−0.452827	+	0.891598i	$0.649585\pi$
$978$		0			0
$979$			5.66715i			0.181123i
$980$		0			0
$981$	−45.9191	−	11.8215i	−1.46609	−	0.377431i
$982$		0			0
$983$	−44.0813			−1.40597			−0.702987	−	0.711203i	$-0.748151\pi$
$983$	−44.0813			−1.40597			−0.702987	+	0.711203i	$0.748151\pi$
$984$		0			0
$985$			15.5133i			0.494294i
$986$		0			0
$987$	−6.24915	−	0.791491i	−0.198913	−	0.0251935i
$988$		0			0
$989$	−26.5003	−	32.6072i	−0.842661	−	1.03685i
$990$		0			0
$991$	36.8884			1.17180			0.585900	−	0.810384i	$-0.300741\pi$
$991$	36.8884			1.17180			0.585900	+	0.810384i	$0.300741\pi$
$992$		0			0
$993$	5.10643	−	40.3174i	0.162048	−	1.27943i
$994$		0			0
$995$	−6.49698			−0.205968
$996$		0			0
$997$			12.4632i			0.394713i	0.980332	+	0.197357i	$0.0632357\pi$
$997$			12.4632i			0.394713i	−0.980332	+	0.197357i	$0.936764\pi$
$998$		0			0
$999$	−32.5671	−	12.9306i	−1.03038	−	0.409105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2208.2.b.c.689.1		80
3.2	odd	2	inner	2208.2.b.c.689.78		80
4.3	odd	2		552.2.b.c.413.19	yes	80
8.3	odd	2		552.2.b.c.413.64	yes	80
8.5	even	2	inner	2208.2.b.c.689.80		80
12.11	even	2		552.2.b.c.413.62	yes	80
23.22	odd	2	inner	2208.2.b.c.689.2		80
24.5	odd	2	inner	2208.2.b.c.689.3		80
24.11	even	2		552.2.b.c.413.17	✓	80
69.68	even	2	inner	2208.2.b.c.689.77		80
92.91	even	2		552.2.b.c.413.20	yes	80
184.45	odd	2	inner	2208.2.b.c.689.79		80
184.91	even	2		552.2.b.c.413.63	yes	80
276.275	odd	2		552.2.b.c.413.61	yes	80
552.275	odd	2		552.2.b.c.413.18	yes	80
552.413	even	2	inner	2208.2.b.c.689.4		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.b.c.413.17	✓	80	24.11	even	2
552.2.b.c.413.18	yes	80	552.275	odd	2
552.2.b.c.413.19	yes	80	4.3	odd	2
552.2.b.c.413.20	yes	80	92.91	even	2
552.2.b.c.413.61	yes	80	276.275	odd	2
552.2.b.c.413.62	yes	80	12.11	even	2
552.2.b.c.413.63	yes	80	184.91	even	2
552.2.b.c.413.64	yes	80	8.3	odd	2
2208.2.b.c.689.1		80	1.1	even	1	trivial
2208.2.b.c.689.2		80	23.22	odd	2	inner
2208.2.b.c.689.3		80	24.5	odd	2	inner
2208.2.b.c.689.4		80	552.413	even	2	inner
2208.2.b.c.689.77		80	69.68	even	2	inner
2208.2.b.c.689.78		80	3.2	odd	2	inner
2208.2.b.c.689.79		80	184.45	odd	2	inner
2208.2.b.c.689.80		80	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 2208 = 2^{5} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2208.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.71832 - 0.217636i) q^{3} -1.34112i q^{5} -2.80797i q^{7} +(2.90527 + 0.747937i) q^{9} +O(q^{10})\)	\(q+(-1.71832 - 0.217636i) q^{3} -1.34112i q^{5} -2.80797i q^{7} +(2.90527 + 0.747937i) q^{9} -6.17378i q^{11} -4.84265i q^{13} +(-0.291875 + 2.30447i) q^{15} -4.64639 q^{17} -5.42593 q^{19} +(-0.611113 + 4.82499i) q^{21} +(-3.02469 - 3.72173i) q^{23} +3.20141 q^{25} +(-4.82941 - 1.91749i) q^{27} +1.97237 q^{29} +2.08414 q^{31} +(-1.34363 + 10.6085i) q^{33} -3.76581 q^{35} +6.74349 q^{37} +(-1.05393 + 8.32124i) q^{39} +6.56520i q^{41} +8.76132 q^{43} +(1.00307 - 3.89631i) q^{45} +1.29516i q^{47} -0.884669 q^{49} +(7.98400 + 1.01122i) q^{51} -8.64336i q^{53} -8.27976 q^{55} +(9.32350 + 1.18088i) q^{57} +5.23804 q^{59} -8.20972 q^{61} +(2.10018 - 8.15790i) q^{63} -6.49456 q^{65} -1.61351 q^{67} +(4.38742 + 7.05341i) q^{69} +1.95795i q^{71} +8.57628 q^{73} +(-5.50105 - 0.696740i) q^{75} -17.3358 q^{77} +8.21280i q^{79} +(7.88118 + 4.34592i) q^{81} +10.6556i q^{83} +6.23135i q^{85} +(-3.38917 - 0.429258i) q^{87} -0.917940 q^{89} -13.5980 q^{91} +(-3.58123 - 0.453583i) q^{93} +7.27680i q^{95} -1.91006i q^{97} +(4.61760 - 17.9365i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{9}+O(q^{10}) \) 80 * q - 4 * q^9	\( 80 q - 4 q^{9} - 144 q^{25} + 24 q^{31} - 68 q^{39} - 160 q^{49} - 32 q^{55} - 8 q^{73} + 12 q^{81} - 92 q^{87}+O(q^{100}) \) 80 * q - 4 * q^9 - 144 * q^25 + 24 * q^31 - 68 * q^39 - 160 * q^49 - 32 * q^55 - 8 * q^73 + 12 * q^81 - 92 * q^87

Introduction

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