LMFDB - Embedded newform 1848.2.v.e.881.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1848,2,Mod(881,1848)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1848.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.9
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.e.881.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.12980 - 1.31284i) q^{3} -1.68437 q^{5} +(-2.40047 + 1.11255i) q^{7} +(-0.447111 + 2.96649i) q^{9} +1.00000i q^{11} -4.48628i q^{13} +(1.90300 + 2.21131i) q^{15} -4.63878 q^{17} +0.439499i q^{19} +(4.17264 + 1.89448i) q^{21} +3.90671i q^{23} -2.16290 q^{25} +(4.39969 - 2.76456i) q^{27} +0.885307i q^{29} +1.28536i q^{31} +(1.31284 - 1.12980i) q^{33} +(4.04327 - 1.87394i) q^{35} -0.462685 q^{37} +(-5.88977 + 5.06859i) q^{39} -2.71558 q^{41} +6.77465 q^{43} +(0.753100 - 4.99667i) q^{45} +11.4640 q^{47} +(4.52447 - 5.34127i) q^{49} +(5.24089 + 6.08999i) q^{51} +8.17336i q^{53} -1.68437i q^{55} +(0.576993 - 0.496546i) q^{57} +5.54560 q^{59} -10.1658i q^{61} +(-2.22709 - 7.61840i) q^{63} +7.55654i q^{65} +4.56513 q^{67} +(5.12890 - 4.41380i) q^{69} +2.24037i q^{71} +0.389877i q^{73} +(2.44364 + 2.83955i) q^{75} +(-1.11255 - 2.40047i) q^{77} +12.2758 q^{79} +(-8.60018 - 2.65270i) q^{81} +2.78586 q^{83} +7.81342 q^{85} +(1.16227 - 1.00022i) q^{87} +1.78140 q^{89} +(4.99120 + 10.7692i) q^{91} +(1.68748 - 1.45220i) q^{93} -0.740279i q^{95} -15.4243i q^{97} +(-2.96649 - 0.447111i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} + 4 q^{9} - 16 q^{15} + 6 q^{21} + 68 q^{25} - 68 q^{37} - 32 q^{39} + 120 q^{43} - 64 q^{49} + 56 q^{51} - 44 q^{57} + 42 q^{63} + 4 q^{67} - 56 q^{79} - 28 q^{81} - 44 q^{91} - 28 q^{93}+O(q^{100}) $ 32 * q + 4 * q^7 + 4 * q^9 - 16 * q^15 + 6 * q^21 + 68 * q^25 - 68 * q^37 - 32 * q^39 + 120 * q^43 - 64 * q^49 + 56 * q^51 - 44 * q^57 + 42 * q^63 + 4 * q^67 - 56 * q^79 - 28 * q^81 - 44 * q^91 - 28 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

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
$3$	−1.12980	−	1.31284i	−0.652289	−	0.757970i
$3$	−1.12980	−	1.31284i	−0.652289	−	0.757970i
$4$		0			0
$5$	−1.68437			−0.753273			−0.376636	−	0.926361i	$-0.622919\pi$
$5$	−1.68437			−0.753273			−0.376636	+	0.926361i	$0.622919\pi$
$6$		0			0
$7$	−2.40047	+	1.11255i	−0.907291	+	0.420504i
$7$	−2.40047	+	1.11255i	−0.907291	+	0.420504i
$8$		0			0
$9$	−0.447111	+	2.96649i	−0.149037	+	0.988832i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$		−	4.48628i		−	1.24427i	−0.782910	−	0.622134i	$-0.786266\pi$
$13$		−	4.48628i		−	1.24427i	0.782910	−	0.622134i	$-0.213734\pi$
$14$		0			0
$15$	1.90300	+	2.21131i	0.491352	+	0.570958i
$16$		0			0
$17$	−4.63878			−1.12507			−0.562535	−	0.826774i	$-0.690174\pi$
$17$	−4.63878			−1.12507			−0.562535	+	0.826774i	$0.690174\pi$
$18$		0			0
$19$			0.439499i			0.100828i	0.998728	+	0.0504140i	$0.0160541\pi$
$19$			0.439499i			0.100828i	−0.998728	+	0.0504140i	$0.983946\pi$
$20$		0			0
$21$	4.17264	+	1.89448i	0.910545	+	0.413409i
$22$		0			0
$23$			3.90671i			0.814605i	0.913293	+	0.407303i	$0.133531\pi$
$23$			3.90671i			0.814605i	−0.913293	+	0.407303i	$0.866469\pi$
$24$		0			0
$25$	−2.16290			−0.432580
$26$		0			0
$27$	4.39969	−	2.76456i	0.846720	−	0.532039i
$28$		0			0
$29$			0.885307i			0.164397i	0.996616	+	0.0821987i	$0.0261942\pi$
$29$			0.885307i			0.164397i	−0.996616	+	0.0821987i	$0.973806\pi$
$30$		0			0
$31$			1.28536i			0.230858i	0.993316	+	0.115429i	$0.0368243\pi$
$31$			1.28536i			0.230858i	−0.993316	+	0.115429i	$0.963176\pi$
$32$		0			0
$33$	1.31284	−	1.12980i	0.228537	−	0.196673i
$34$		0			0
$35$	4.04327	−	1.87394i	0.683437	−	0.316754i
$36$		0			0
$37$	−0.462685			−0.0760649			−0.0380324	−	0.999277i	$-0.512109\pi$
$37$	−0.462685			−0.0760649			−0.0380324	+	0.999277i	$0.512109\pi$
$38$		0			0
$39$	−5.88977	+	5.06859i	−0.943118	+	0.811623i
$40$		0			0
$41$	−2.71558			−0.424103			−0.212051	−	0.977259i	$-0.568014\pi$
$41$	−2.71558			−0.424103			−0.212051	+	0.977259i	$0.568014\pi$
$42$		0			0
$43$	6.77465			1.03312			0.516562	−	0.856250i	$-0.327211\pi$
$43$	6.77465			1.03312			0.516562	+	0.856250i	$0.327211\pi$
$44$		0			0
$45$	0.753100	−	4.99667i	0.112265	−	0.744860i
$46$		0			0
$47$	11.4640			1.67219			0.836096	−	0.548584i	$-0.184833\pi$
$47$	11.4640			1.67219			0.836096	+	0.548584i	$0.184833\pi$
$48$		0			0
$49$	4.52447	−	5.34127i	0.646353	−	0.763038i
$50$		0			0
$51$	5.24089	+	6.08999i	0.733871	+	0.852769i
$52$		0			0
$53$			8.17336i			1.12270i	0.827580	+	0.561348i	$0.189717\pi$
$53$			8.17336i			1.12270i	−0.827580	+	0.561348i	$0.810283\pi$
$54$		0			0
$55$		−	1.68437i		−	0.227120i
$56$		0			0
$57$	0.576993	−	0.496546i	0.0764246	−	0.0657691i
$58$		0			0
$59$	5.54560			0.721975			0.360988	−	0.932571i	$-0.382440\pi$
$59$	5.54560			0.721975			0.360988	+	0.932571i	$0.382440\pi$
$60$		0			0
$61$		−	10.1658i		−	1.30160i	−0.759250	−	0.650800i	$-0.774434\pi$
$61$		−	10.1658i		−	1.30160i	0.759250	−	0.650800i	$-0.225566\pi$
$62$		0			0
$63$	−2.22709	−	7.61840i	−0.280587	−	0.959828i
$64$		0			0
$65$			7.55654i			0.937274i
$66$		0			0
$67$	4.56513			0.557719			0.278860	−	0.960332i	$-0.410044\pi$
$67$	4.56513			0.557719			0.278860	+	0.960332i	$0.410044\pi$
$68$		0			0
$69$	5.12890	−	4.41380i	0.617446	−	0.531359i
$70$		0			0
$71$			2.24037i			0.265883i	0.991124	+	0.132941i	$0.0424422\pi$
$71$			2.24037i			0.265883i	−0.991124	+	0.132941i	$0.957558\pi$
$72$		0			0
$73$			0.389877i			0.0456317i	0.999740	+	0.0228158i	$0.00726314\pi$
$73$			0.389877i			0.0456317i	−0.999740	+	0.0228158i	$0.992737\pi$
$74$		0			0
$75$	2.44364	+	2.83955i	0.282168	+	0.327883i
$76$		0			0
$77$	−1.11255	−	2.40047i	−0.126787	−	0.273558i
$78$		0			0
$79$	12.2758			1.38114			0.690568	−	0.723268i	$-0.257361\pi$
$79$	12.2758			1.38114			0.690568	+	0.723268i	$0.257361\pi$
$80$		0			0
$81$	−8.60018	−	2.65270i	−0.955576	−	0.294745i
$82$		0			0
$83$	2.78586			0.305788			0.152894	−	0.988243i	$-0.451141\pi$
$83$	2.78586			0.305788			0.152894	+	0.988243i	$0.451141\pi$
$84$		0			0
$85$	7.81342			0.847484
$86$		0			0
$87$	1.16227	−	1.00022i	0.124608	−	0.107235i
$88$		0			0
$89$	1.78140			0.188828			0.0944142	−	0.995533i	$-0.469902\pi$
$89$	1.78140			0.188828			0.0944142	+	0.995533i	$0.469902\pi$
$90$		0			0
$91$	4.99120	+	10.7692i	0.523220	+	1.12891i
$92$		0			0
$93$	1.68748	−	1.45220i	0.174983	−	0.150586i
$94$		0			0
$95$		−	0.740279i		−	0.0759510i
$96$		0			0
$97$		−	15.4243i		−	1.56610i	−0.621957	−	0.783052i	$-0.713662\pi$
$97$		−	15.4243i		−	1.56610i	0.621957	−	0.783052i	$-0.286338\pi$
$98$		0			0
$99$	−2.96649	−	0.447111i	−0.298144	−	0.0449363i
$100$		0			0
$101$	−5.71466			−0.568630			−0.284315	−	0.958731i	$-0.591766\pi$
$101$	−5.71466			−0.568630			−0.284315	+	0.958731i	$0.591766\pi$
$102$		0			0
$103$		−	4.93058i		−	0.485825i	−0.970048	−	0.242912i	$-0.921897\pi$
$103$		−	4.93058i		−	0.485825i	0.970048	−	0.242912i	$-0.0781028\pi$
$104$		0			0
$105$	−7.02827	−	3.19100i	−0.685889	−	0.311410i
$106$		0			0
$107$		−	20.3066i		−	1.96312i	−0.191161	−	0.981559i	$-0.561225\pi$
$107$		−	20.3066i		−	1.96312i	0.191161	−	0.981559i	$-0.438775\pi$
$108$		0			0
$109$	14.9381			1.43081			0.715405	−	0.698710i	$-0.246242\pi$
$109$	14.9381			1.43081			0.715405	+	0.698710i	$0.246242\pi$
$110$		0			0
$111$	0.522740	+	0.607432i	0.0496163	+	0.0576549i
$112$		0			0
$113$		−	10.7484i		−	1.01113i	−0.862789	−	0.505563i	$-0.831285\pi$
$113$		−	10.7484i		−	1.01113i	0.862789	−	0.505563i	$-0.168715\pi$
$114$		0			0
$115$		−	6.58034i		−	0.613620i
$116$		0			0
$117$	13.3085	+	2.00586i	1.23037	+	0.185442i
$118$		0			0
$119$	11.1352	−	5.16087i	1.02077	−	0.473096i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	3.06806	+	3.56513i	0.276638	+	0.321457i
$124$		0			0
$125$	12.0650			1.07912
$126$		0			0
$127$	−16.7931			−1.49014			−0.745072	−	0.666984i	$-0.767585\pi$
$127$	−16.7931			−1.49014			−0.745072	+	0.666984i	$0.767585\pi$
$128$		0			0
$129$	−7.65399	−	8.89405i	−0.673896	−	0.783078i
$130$		0			0
$131$	−13.8878			−1.21338			−0.606692	−	0.794937i	$-0.707504\pi$
$131$	−13.8878			−1.21338			−0.606692	+	0.794937i	$0.707504\pi$
$132$		0			0
$133$	−0.488964	−	1.05500i	−0.0423986	−	0.0914804i
$134$		0			0
$135$	−7.41069	+	4.65653i	−0.637811	+	0.400770i
$136$		0			0
$137$		−	7.60846i		−	0.650035i	−0.945708	−	0.325017i	$-0.894630\pi$
$137$		−	7.60846i		−	0.650035i	0.945708	−	0.325017i	$-0.105370\pi$
$138$		0			0
$139$			16.4781i			1.39765i	0.715292	+	0.698825i	$0.246294\pi$
$139$			16.4781i			1.39765i	−0.715292	+	0.698825i	$0.753706\pi$
$140$		0			0
$141$	−12.9520	−	15.0504i	−1.09075	−	1.26747i
$142$		0			0
$143$	4.48628			0.375161
$144$		0			0
$145$		−	1.49118i		−	0.123836i
$146$		0			0
$147$	−12.1240	+	0.0946350i	−0.999970	+	0.00780537i
$148$		0			0
$149$			13.9231i			1.14062i	0.821428	+	0.570312i	$0.193178\pi$
$149$			13.9231i			1.14062i	−0.821428	+	0.570312i	$0.806822\pi$
$150$		0			0
$151$	−11.4496			−0.931759			−0.465879	−	0.884848i	$-0.654262\pi$
$151$	−11.4496			−0.931759			−0.465879	+	0.884848i	$0.654262\pi$
$152$		0			0
$153$	2.07405	−	13.7609i	0.167677	−	1.11250i
$154$		0			0
$155$		−	2.16502i		−	0.173899i
$156$		0			0
$157$			17.9900i			1.43576i	0.696167	+	0.717879i	$0.254887\pi$
$157$			17.9900i			1.43576i	−0.696167	+	0.717879i	$0.745113\pi$
$158$		0			0
$159$	10.7303	−	9.23425i	0.850971	−	0.732323i
$160$		0			0
$161$	−4.34640	−	9.37793i	−0.342545	−	0.739084i
$162$		0			0
$163$	14.1943			1.11178			0.555892	−	0.831255i	$-0.312377\pi$
$163$	14.1943			1.11178			0.555892	+	0.831255i	$0.312377\pi$
$164$		0			0
$165$	−2.21131	+	1.90300i	−0.172150	+	0.148148i
$166$		0			0
$167$	16.6473			1.28820			0.644102	−	0.764940i	$-0.277231\pi$
$167$	16.6473			1.28820			0.644102	+	0.764940i	$0.277231\pi$
$168$		0			0
$169$	−7.12666			−0.548205
$170$		0			0
$171$	−1.30377	−	0.196505i	−0.0997019	−	0.0150271i
$172$		0			0
$173$	−3.05227			−0.232060			−0.116030	−	0.993246i	$-0.537017\pi$
$173$	−3.05227			−0.232060			−0.116030	+	0.993246i	$0.537017\pi$
$174$		0			0
$175$	5.19197	−	2.40633i	0.392476	−	0.181902i
$176$		0			0
$177$	−6.26541	−	7.28050i	−0.470937	−	0.547236i
$178$		0			0
$179$			12.3581i			0.923691i	0.886960	+	0.461845i	$0.152813\pi$
$179$			12.3581i			0.923691i	−0.886960	+	0.461845i	$0.847187\pi$
$180$		0			0
$181$		−	0.319529i		−	0.0237504i	−0.999929	−	0.0118752i	$-0.996220\pi$
$181$		−	0.319529i		−	0.0237504i	0.999929	−	0.0118752i	$-0.00378008\pi$
$182$		0			0
$183$	−13.3461	+	11.4853i	−0.986573	+	0.849019i
$184$		0			0
$185$	0.779332			0.0572976
$186$		0			0
$187$		−	4.63878i		−	0.339221i
$188$		0			0
$189$	−7.48559	+	11.5311i	−0.544497	+	0.838763i
$190$		0			0
$191$			21.1850i			1.53290i	0.642306	+	0.766448i	$0.277978\pi$
$191$			21.1850i			1.53290i	−0.642306	+	0.766448i	$0.722022\pi$
$192$		0			0
$193$	26.8293			1.93122			0.965608	−	0.260001i	$-0.0837229\pi$
$193$	26.8293			1.93122			0.965608	+	0.260001i	$0.0837229\pi$
$194$		0			0
$195$	9.92055	−	8.53737i	0.710425	−	0.611374i
$196$		0			0
$197$		−	4.76470i		−	0.339471i	−0.985490	−	0.169735i	$-0.945709\pi$
$197$		−	4.76470i		−	0.339471i	0.985490	−	0.169735i	$-0.0542913\pi$
$198$		0			0
$199$			21.5564i			1.52809i	0.645161	+	0.764047i	$0.276790\pi$
$199$			21.5564i			1.52809i	−0.645161	+	0.764047i	$0.723210\pi$
$200$		0			0
$201$	−5.15768	−	5.99329i	−0.363794	−	0.422734i
$202$		0			0
$203$	−0.984946	−	2.12515i	−0.0691297	−	0.149156i
$204$		0			0
$205$	4.57404			0.319465
$206$		0			0
$207$	−11.5892	−	1.74673i	−0.805508	−	0.121406i
$208$		0			0
$209$	−0.439499			−0.0304008
$210$		0			0
$211$	27.7602			1.91109			0.955544	−	0.294848i	$-0.0952689\pi$
$211$	27.7602			1.91109			0.955544	+	0.294848i	$0.0952689\pi$
$212$		0			0
$213$	2.94125	−	2.53116i	0.201531	−	0.173432i
$214$		0			0
$215$	−11.4110			−0.778225
$216$		0			0
$217$	−1.43003	−	3.08547i	−0.0970766	−	0.209455i
$218$		0			0
$219$	0.511847	−	0.440483i	0.0345874	−	0.0297651i
$220$		0			0
$221$			20.8109i			1.39989i
$222$		0			0
$223$			2.67499i			0.179131i	0.995981	+	0.0895654i	$0.0285478\pi$
$223$			2.67499i			0.179131i	−0.995981	+	0.0895654i	$0.971452\pi$
$224$		0			0
$225$	0.967056	−	6.41624i	0.0644704	−	0.427749i
$226$		0			0
$227$	7.82767			0.519541			0.259770	−	0.965670i	$-0.416353\pi$
$227$	7.82767			0.519541			0.259770	+	0.965670i	$0.416353\pi$
$228$		0			0
$229$			16.0327i			1.05947i	0.848164	+	0.529734i	$0.177708\pi$
$229$			16.0327i			1.05947i	−0.848164	+	0.529734i	$0.822292\pi$
$230$		0			0
$231$	−1.89448	+	4.17264i	−0.124648	+	0.274540i
$232$		0			0
$233$		−	0.975484i		−	0.0639061i	−0.999489	−	0.0319530i	$-0.989827\pi$
$233$		−	0.975484i		−	0.0639061i	0.999489	−	0.0319530i	$-0.0101727\pi$
$234$		0			0
$235$	−19.3095			−1.25962
$236$		0			0
$237$	−13.8692	−	16.1162i	−0.900900	−	1.04686i
$238$		0			0
$239$		−	1.78695i		−	0.115588i	−0.998329	−	0.0577942i	$-0.981593\pi$
$239$		−	1.78695i		−	0.115588i	0.998329	−	0.0577942i	$-0.0184067\pi$
$240$		0			0
$241$			11.3850i			0.733374i	0.930344	+	0.366687i	$0.119508\pi$
$241$			11.3850i			0.733374i	−0.930344	+	0.366687i	$0.880492\pi$
$242$		0			0
$243$	6.23389	+	14.2877i	0.399904	+	0.916557i
$244$		0			0
$245$	−7.62088	+	8.99667i	−0.486880	+	0.574776i
$246$		0			0
$247$	1.97171			0.125457
$248$		0			0
$249$	−3.14746	−	3.65740i	−0.199462	−	0.231778i
$250$		0			0
$251$	7.42838			0.468875			0.234438	−	0.972131i	$-0.424675\pi$
$251$	7.42838			0.468875			0.234438	+	0.972131i	$0.424675\pi$
$252$		0			0
$253$	−3.90671			−0.245613
$254$		0			0
$255$	−8.82759	−	10.2578i	−0.552805	−	0.642368i
$256$		0			0
$257$	−1.40337			−0.0875397			−0.0437699	−	0.999042i	$-0.513937\pi$
$257$	−1.40337			−0.0875397			−0.0437699	+	0.999042i	$0.513937\pi$
$258$		0			0
$259$	1.11066	−	0.514759i	0.0690130	−	0.0319856i
$260$		0			0
$261$	−2.62626	−	0.395830i	−0.162561	−	0.0245013i
$262$		0			0
$263$		−	0.486075i		−	0.0299726i	−0.999888	−	0.0149863i	$-0.995230\pi$
$263$		−	0.486075i		−	0.0299726i	0.999888	−	0.0149863i	$-0.00477047\pi$
$264$		0			0
$265$		−	13.7669i		−	0.845697i
$266$		0			0
$267$	−2.01263	−	2.33870i	−0.123171	−	0.143126i
$268$		0			0
$269$	−20.5487			−1.25288			−0.626439	−	0.779471i	$-0.715488\pi$
$269$	−20.5487			−1.25288			−0.626439	+	0.779471i	$0.715488\pi$
$270$		0			0
$271$			9.54878i			0.580047i	0.957019	+	0.290024i	$0.0936632\pi$
$271$			9.54878i			0.580047i	−0.957019	+	0.290024i	$0.906337\pi$
$272$		0			0
$273$	8.49915	−	18.7196i	0.514392	−	1.13296i
$274$		0			0
$275$		−	2.16290i		−	0.130428i
$276$		0			0
$277$	23.8813			1.43489			0.717444	−	0.696616i	$-0.245312\pi$
$277$	23.8813			1.43489			0.717444	+	0.696616i	$0.245312\pi$
$278$		0			0
$279$	−3.81302	−	0.574699i	−0.228280	−	0.0344063i
$280$		0			0
$281$		−	20.1241i		−	1.20051i	−0.799810	−	0.600253i	$-0.795067\pi$
$281$		−	20.1241i		−	1.20051i	0.799810	−	0.600253i	$-0.204933\pi$
$282$		0			0
$283$			6.50788i			0.386853i	0.981115	+	0.193427i	$0.0619601\pi$
$283$			6.50788i			0.386853i	−0.981115	+	0.193427i	$0.938040\pi$
$284$		0			0
$285$	−0.971869	+	0.836366i	−0.0575686	+	0.0495420i
$286$		0			0
$287$	6.51866	−	3.02122i	0.384785	−	0.178337i
$288$		0			0
$289$	4.51830			0.265782
$290$		0			0
$291$	−20.2497	+	17.4264i	−1.18706	+	1.02155i
$292$		0			0
$293$	13.9406			0.814417			0.407209	−	0.913335i	$-0.366502\pi$
$293$	13.9406			0.814417			0.407209	+	0.913335i	$0.366502\pi$
$294$		0			0
$295$	−9.34083			−0.543844
$296$		0			0
$297$	2.76456	+	4.39969i	0.160416	+	0.255296i
$298$		0			0
$299$	17.5266			1.01359
$300$		0			0
$301$	−16.2623	+	7.53713i	−0.937345	+	0.434433i
$302$		0			0
$303$	6.45641	+	7.50245i	0.370911	+	0.431004i
$304$		0			0
$305$			17.1230i			0.980459i
$306$		0			0
$307$			0.0330843i			0.00188822i	1.00000		0.000944111i	$0.000300520\pi$
$307$			0.0330843i			0.00188822i	−1.00000		0.000944111i	$0.999699\pi$
$308$		0			0
$309$	−6.47308	+	5.57057i	−0.368241	+	0.316898i
$310$		0			0
$311$	−3.78417			−0.214581			−0.107290	−	0.994228i	$-0.534217\pi$
$311$	−3.78417			−0.214581			−0.107290	+	0.994228i	$0.534217\pi$
$312$		0			0
$313$		−	10.4717i		−	0.591898i	−0.955204	−	0.295949i	$-0.904364\pi$
$313$		−	10.4717i		−	0.591898i	0.955204	−	0.295949i	$-0.0956358\pi$
$314$		0			0
$315$	3.75125	+	12.8322i	0.211359	+	0.723013i
$316$		0			0
$317$		−	3.28784i		−	0.184663i	−0.995728	−	0.0923316i	$-0.970568\pi$
$317$		−	3.28784i		−	0.184663i	0.995728	−	0.0923316i	$-0.0294320\pi$
$318$		0			0
$319$	−0.885307			−0.0495677
$320$		0			0
$321$	−26.6594	+	22.9424i	−1.48798	+	1.28052i
$322$		0			0
$323$		−	2.03874i		−	0.113439i
$324$		0			0
$325$			9.70337i			0.538246i
$326$		0			0
$327$	−16.8770	−	19.6114i	−0.933303	−	1.08451i
$328$		0			0
$329$	−27.5189	+	12.7542i	−1.51716	+	0.703163i
$330$		0			0
$331$	−24.4468			−1.34372			−0.671860	−	0.740678i	$-0.734504\pi$
$331$	−24.4468			−1.34372			−0.671860	+	0.740678i	$0.734504\pi$
$332$		0			0
$333$	0.206871	−	1.37255i	0.0113365	−	0.0752154i
$334$		0			0
$335$	−7.68936			−0.420115
$336$		0			0
$337$	26.5672			1.44721			0.723605	−	0.690214i	$-0.242484\pi$
$337$	26.5672			1.44721			0.723605	+	0.690214i	$0.242484\pi$
$338$		0			0
$339$	−14.1110	+	12.1436i	−0.766404	+	0.659547i
$340$		0			0
$341$	−1.28536			−0.0696063
$342$		0			0
$343$	−4.91843	+	17.8552i	−0.265570	+	0.964092i
$344$		0			0
$345$	−8.63895	+	7.43446i	−0.465106	+	0.400258i
$346$		0			0
$347$		−	20.8583i		−	1.11973i	−0.828583	−	0.559866i	$-0.810853\pi$
$347$		−	20.8583i		−	1.11973i	0.828583	−	0.559866i	$-0.189147\pi$
$348$		0			0
$349$		−	2.40855i		−	0.128927i	−0.997920	−	0.0644634i	$-0.979466\pi$
$349$		−	2.40855i		−	0.128927i	0.997920	−	0.0644634i	$-0.0205336\pi$
$350$		0			0
$351$	−12.4026	−	19.7382i	−0.661999	−	1.05355i
$352$		0			0
$353$	−9.15710			−0.487383			−0.243692	−	0.969853i	$-0.578358\pi$
$353$	−9.15710			−0.487383			−0.243692	+	0.969853i	$0.578358\pi$
$354$		0			0
$355$		−	3.77360i		−	0.200282i
$356$		0			0
$357$	−19.3560	−	8.78807i	−1.02443	−	0.465114i
$358$		0			0
$359$			24.0224i			1.26785i	0.773393	+	0.633927i	$0.218558\pi$
$359$			24.0224i			1.26785i	−0.773393	+	0.633927i	$0.781442\pi$
$360$		0			0
$361$	18.8068			0.989834
$362$		0			0
$363$	1.12980	+	1.31284i	0.0592990	+	0.0689064i
$364$		0			0
$365$		−	0.656697i		−	0.0343731i
$366$		0			0
$367$			0.130678i			0.00682135i	0.999994	+	0.00341067i	$0.00108565\pi$
$367$			0.130678i			0.00682135i	−0.999994	+	0.00341067i	$0.998914\pi$
$368$		0			0
$369$	1.21417	−	8.05576i	0.0632070	−	0.419366i
$370$		0			0
$371$	−9.09325	−	19.6199i	−0.472098	−	1.01861i
$372$		0			0
$373$	6.03624			0.312544			0.156272	−	0.987714i	$-0.450052\pi$
$373$	6.03624			0.312544			0.156272	+	0.987714i	$0.450052\pi$
$374$		0			0
$375$	−13.6310	−	15.8394i	−0.703901	−	0.817943i
$376$		0			0
$377$	3.97173			0.204555
$378$		0			0
$379$	3.85601			0.198070			0.0990349	−	0.995084i	$-0.468424\pi$
$379$	3.85601			0.198070			0.0990349	+	0.995084i	$0.468424\pi$
$380$		0			0
$381$	18.9728	+	22.0467i	0.972006	+	1.12948i
$382$		0			0
$383$	−27.3546			−1.39776			−0.698879	−	0.715240i	$-0.746317\pi$
$383$	−27.3546			−1.39776			−0.698879	+	0.715240i	$0.746317\pi$
$384$		0			0
$385$	1.87394	+	4.04327i	0.0955049	+	0.206064i
$386$		0			0
$387$	−3.02902	+	20.0970i	−0.153974	+	1.02159i
$388$		0			0
$389$		−	6.62052i		−	0.335674i	−0.985815	−	0.167837i	$-0.946322\pi$
$389$		−	6.62052i		−	0.335674i	0.985815	−	0.167837i	$-0.0536782\pi$
$390$		0			0
$391$		−	18.1224i		−	0.916488i
$392$		0			0
$393$	15.6904	+	18.2325i	0.791478	+	0.919709i
$394$		0			0
$395$	−20.6770			−1.04037
$396$		0			0
$397$			27.3381i			1.37206i	0.727574	+	0.686029i	$0.240648\pi$
$397$			27.3381i			1.37206i	−0.727574	+	0.686029i	$0.759352\pi$
$398$		0			0
$399$	−0.832622	+	1.83387i	−0.0416832	+	0.0918085i
$400$		0			0
$401$		−	10.0945i		−	0.504096i	−0.967715	−	0.252048i	$-0.918896\pi$
$401$		−	10.0945i		−	0.504096i	0.967715	−	0.252048i	$-0.0811041\pi$
$402$		0			0
$403$	5.76649			0.287249
$404$		0			0
$405$	14.4859	+	4.46813i	0.719809	+	0.222023i
$406$		0			0
$407$		−	0.462685i		−	0.0229344i
$408$		0			0
$409$			27.6161i			1.36553i	0.730639	+	0.682764i	$0.239222\pi$
$409$			27.6161i			1.36553i	−0.730639	+	0.682764i	$0.760778\pi$
$410$		0			0
$411$	−9.98871	+	8.59603i	−0.492707	+	0.424011i
$412$		0			0
$413$	−13.3120	+	6.16974i	−0.655042	+	0.303593i
$414$		0			0
$415$	−4.69242			−0.230342
$416$		0			0
$417$	21.6331	−	18.6169i	1.05938	−	0.911673i
$418$		0			0
$419$	39.3531			1.92252			0.961262	−	0.275637i	$-0.0888888\pi$
$419$	39.3531			1.92252			0.961262	+	0.275637i	$0.0888888\pi$
$420$		0			0
$421$	−11.4509			−0.558080			−0.279040	−	0.960279i	$-0.590016\pi$
$421$	−11.4509			−0.558080			−0.279040	+	0.960279i	$0.590016\pi$
$422$		0			0
$423$	−5.12566	+	34.0078i	−0.249218	+	1.65352i
$424$		0			0
$425$	10.0332			0.486683
$426$		0			0
$427$	11.3100	+	24.4027i	0.547327	+	1.18093i
$428$		0			0
$429$	−5.06859	−	5.88977i	−0.244714	−	0.284361i
$430$		0			0
$431$		−	32.9444i		−	1.58688i	−0.608651	−	0.793438i	$-0.708289\pi$
$431$		−	32.9444i		−	1.58688i	0.608651	−	0.793438i	$-0.291711\pi$
$432$		0			0
$433$		−	25.8943i		−	1.24440i	−0.782859	−	0.622200i	$-0.786239\pi$
$433$		−	25.8943i		−	1.24440i	0.782859	−	0.622200i	$-0.213761\pi$
$434$		0			0
$435$	−1.95769	+	1.68474i	−0.0938640	+	0.0807769i
$436$		0			0
$437$	−1.71700			−0.0821351
$438$		0			0
$439$			0.583243i			0.0278367i	0.999903	+	0.0139183i	$0.00443049\pi$
$439$			0.583243i			0.0278367i	−0.999903	+	0.0139183i	$0.995570\pi$
$440$		0			0
$441$	13.8219	+	15.8100i	0.658186	+	0.752856i
$442$		0			0
$443$			22.2184i			1.05563i	0.849359	+	0.527815i	$0.176989\pi$
$443$			22.2184i			1.05563i	−0.849359	+	0.527815i	$0.823011\pi$
$444$		0			0
$445$	−3.00054			−0.142239
$446$		0			0
$447$	18.2788	−	15.7303i	0.864559	−	0.744018i
$448$		0			0
$449$		−	0.319547i		−	0.0150803i	−0.999972	−	0.00754017i	$-0.997600\pi$
$449$		−	0.319547i		−	0.0150803i	0.999972	−	0.00754017i	$-0.00240013\pi$
$450$		0			0
$451$		−	2.71558i		−	0.127872i
$452$		0			0
$453$	12.9358	+	15.0316i	0.607776	+	0.706245i
$454$		0			0
$455$	−8.40702	−	18.1392i	−0.394127	−	0.850380i
$456$		0			0
$457$	−34.6727			−1.62192			−0.810960	−	0.585101i	$-0.801055\pi$
$457$	−34.6727			−1.62192			−0.810960	+	0.585101i	$0.801055\pi$
$458$		0			0
$459$	−20.4092	+	12.8242i	−0.952619	+	0.598581i
$460$		0			0
$461$	7.89425			0.367672			0.183836	−	0.982957i	$-0.441148\pi$
$461$	7.89425			0.367672			0.183836	+	0.982957i	$0.441148\pi$
$462$		0			0
$463$	−5.76395			−0.267873			−0.133937	−	0.990990i	$-0.542762\pi$
$463$	−5.76395			−0.267873			−0.133937	+	0.990990i	$0.542762\pi$
$464$		0			0
$465$	−2.84234	+	2.44604i	−0.131810	+	0.113432i
$466$		0			0
$467$	−37.5418			−1.73723			−0.868613	−	0.495490i	$-0.834988\pi$
$467$	−37.5418			−1.73723			−0.868613	+	0.495490i	$0.834988\pi$
$468$		0			0
$469$	−10.9584	+	5.07892i	−0.506014	+	0.234523i
$470$		0			0
$471$	23.6180	−	20.3251i	1.08826	−	0.936530i
$472$		0			0
$473$			6.77465i			0.311499i
$474$		0			0
$475$		−	0.950593i		−	0.0436162i
$476$		0			0
$477$	−24.2462	−	3.65440i	−1.11016	−	0.167323i
$478$		0			0
$479$	−18.5673			−0.848362			−0.424181	−	0.905577i	$-0.639438\pi$
$479$	−18.5673			−0.848362			−0.424181	+	0.905577i	$0.639438\pi$
$480$		0			0
$481$			2.07573i			0.0946451i
$482$		0			0
$483$	−7.40118	+	16.3013i	−0.336765	+	0.741735i
$484$		0			0
$485$			25.9803i			1.17970i
$486$		0			0
$487$	−0.718171			−0.0325434			−0.0162717	−	0.999868i	$-0.505180\pi$
$487$	−0.718171			−0.0325434			−0.0162717	+	0.999868i	$0.505180\pi$
$488$		0			0
$489$	−16.0367	−	18.6349i	−0.725204	−	0.842698i
$490$		0			0
$491$		−	19.0980i		−	0.861881i	−0.902380	−	0.430940i	$-0.858182\pi$
$491$		−	19.0980i		−	0.861881i	0.902380	−	0.430940i	$-0.141818\pi$
$492$		0			0
$493$		−	4.10675i		−	0.184959i
$494$		0			0
$495$	4.99667	+	0.753100i	0.224584	+	0.0338493i
$496$		0			0
$497$	−2.49252	−	5.37793i	−0.111805	−	0.241233i
$498$		0			0
$499$	5.97171			0.267331			0.133665	−	0.991027i	$-0.457325\pi$
$499$	5.97171			0.267331			0.133665	+	0.991027i	$0.457325\pi$
$500$		0			0
$501$	−18.8081	−	21.8552i	−0.840282	−	0.976420i
$502$		0			0
$503$	3.82442			0.170523			0.0852613	−	0.996359i	$-0.472828\pi$
$503$	3.82442			0.170523			0.0852613	+	0.996359i	$0.472828\pi$
$504$		0			0
$505$	9.62559			0.428333
$506$		0			0
$507$	8.05169	+	9.35619i	0.357588	+	0.415523i
$508$		0			0
$509$	24.1349			1.06976			0.534881	−	0.844927i	$-0.320356\pi$
$509$	24.1349			1.06976			0.534881	+	0.844927i	$0.320356\pi$
$510$		0			0
$511$	−0.433757	−	0.935887i	−0.0191883	−	0.0414012i
$512$		0			0
$513$	1.21502	+	1.93366i	0.0536444	+	0.0853731i
$514$		0			0
$515$			8.30492i			0.365959i
$516$		0			0
$517$			11.4640i			0.504185i
$518$		0			0
$519$	3.44845	+	4.00715i	0.151370	+	0.175894i
$520$		0			0
$521$	−27.7449			−1.21552			−0.607762	−	0.794119i	$-0.707933\pi$
$521$	−27.7449			−1.21552			−0.607762	+	0.794119i	$0.707933\pi$
$522$		0			0
$523$		−	32.2482i		−	1.41012i	−0.709149	−	0.705059i	$-0.750921\pi$
$523$		−	32.2482i		−	1.41012i	0.709149	−	0.705059i	$-0.249079\pi$
$524$		0			0
$525$	−9.02501	−	4.09757i	−0.393884	−	0.178833i
$526$		0			0
$527$		−	5.96251i		−	0.259731i
$528$		0			0
$529$	7.73761			0.336418
$530$		0			0
$531$	−2.47950	+	16.4510i	−0.107601	+	0.713912i
$532$		0			0
$533$			12.1829i			0.527698i
$534$		0			0
$535$			34.2039i			1.47876i
$536$		0			0
$537$	16.2243	−	13.9622i	0.700130	−	0.602514i
$538$		0			0
$539$	5.34127	+	4.52447i	0.230065	+	0.194883i
$540$		0			0
$541$	−32.8203			−1.41105			−0.705527	−	0.708683i	$-0.749290\pi$
$541$	−32.8203			−1.41105			−0.705527	+	0.708683i	$0.749290\pi$
$542$		0			0
$543$	−0.419491	+	0.361003i	−0.0180021	+	0.0154921i
$544$		0			0
$545$	−25.1613			−1.07779
$546$		0			0
$547$	24.3011			1.03904			0.519520	−	0.854458i	$-0.326111\pi$
$547$	24.3011			1.03904			0.519520	+	0.854458i	$0.326111\pi$
$548$		0			0
$549$	30.1568	+	4.54524i	1.28706	+	0.193986i
$550$		0			0
$551$	−0.389092			−0.0165759
$552$		0			0
$553$	−29.4676	+	13.6574i	−1.25309	+	0.580773i
$554$		0			0
$555$	−0.880488	−	1.02314i	−0.0373746	−	0.0434299i
$556$		0			0
$557$		−	27.4493i		−	1.16307i	−0.813523	−	0.581533i	$-0.802453\pi$
$557$		−	27.4493i		−	1.16307i	0.813523	−	0.581533i	$-0.197547\pi$
$558$		0			0
$559$		−	30.3930i		−	1.28548i
$560$		0			0
$561$	−6.08999	+	5.24089i	−0.257120	+	0.221271i
$562$		0			0
$563$	5.83605			0.245960			0.122980	−	0.992409i	$-0.460755\pi$
$563$	5.83605			0.245960			0.122980	+	0.992409i	$0.460755\pi$
$564$		0			0
$565$			18.1043i			0.761654i
$566$		0			0
$567$	23.5957	−	3.20039i	0.990927	−	0.134404i
$568$		0			0
$569$		−	5.62414i		−	0.235776i	−0.993027	−	0.117888i	$-0.962388\pi$
$569$		−	5.62414i		−	0.235776i	0.993027	−	0.117888i	$-0.0376124\pi$
$570$		0			0
$571$	26.3368			1.10216			0.551081	−	0.834452i	$-0.314216\pi$
$571$	26.3368			1.10216			0.551081	+	0.834452i	$0.314216\pi$
$572$		0			0
$573$	27.8126	−	23.9348i	1.16189	−	0.999892i
$574$		0			0
$575$		−	8.44983i		−	0.352382i
$576$		0			0
$577$		−	9.69194i		−	0.403481i	−0.979439	−	0.201740i	$-0.935340\pi$
$577$		−	9.69194i		−	0.403481i	0.979439	−	0.201740i	$-0.0646597\pi$
$578$		0			0
$579$	−30.3117	−	35.2227i	−1.25971	−	1.46380i
$580$		0			0
$581$	−6.68737	+	3.09940i	−0.277439	+	0.128585i
$582$		0			0
$583$	−8.17336			−0.338506
$584$		0			0
$585$	−22.4164	−	3.37861i	−0.926806	−	0.139688i
$586$		0			0
$587$	29.4357			1.21494			0.607470	−	0.794342i	$-0.292184\pi$
$587$	29.4357			1.21494			0.607470	+	0.794342i	$0.292184\pi$
$588$		0			0
$589$	−0.564916			−0.0232769
$590$		0			0
$591$	−6.25530	+	5.38315i	−0.257308	+	0.221433i
$592$		0			0
$593$	−21.5449			−0.884743			−0.442372	−	0.896832i	$-0.645863\pi$
$593$	−21.5449			−0.884743			−0.442372	+	0.896832i	$0.645863\pi$
$594$		0			0
$595$	−18.7559	+	8.69281i	−0.768915	+	0.356370i
$596$		0			0
$597$	28.3002	−	24.3544i	1.15825	−	0.996759i
$598$		0			0
$599$		−	8.91553i		−	0.364279i	−0.983273	−	0.182139i	$-0.941698\pi$
$599$		−	8.91553i		−	0.364279i	0.983273	−	0.182139i	$-0.0583022\pi$
$600$		0			0
$601$			10.6715i			0.435300i	0.976027	+	0.217650i	$0.0698391\pi$
$601$			10.6715i			0.435300i	−0.976027	+	0.217650i	$0.930161\pi$
$602$		0			0
$603$	−2.04112	+	13.5424i	−0.0831208	+	0.551490i
$604$		0			0
$605$	1.68437			0.0684793
$606$		0			0
$607$		−	38.2957i		−	1.55438i	−0.629268	−	0.777188i	$-0.716645\pi$
$607$		−	38.2957i		−	1.55438i	0.629268	−	0.777188i	$-0.283355\pi$
$608$		0			0
$609$	−1.67719	+	3.69407i	−0.0679634	+	0.149691i
$610$		0			0
$611$		−	51.4305i		−	2.08066i
$612$		0			0
$613$	−14.2896			−0.577153			−0.288577	−	0.957457i	$-0.593182\pi$
$613$	−14.2896			−0.577153			−0.288577	+	0.957457i	$0.593182\pi$
$614$		0			0
$615$	−5.16775	−	6.00500i	−0.208384	−	0.242145i
$616$		0			0
$617$			41.9821i			1.69014i	0.534658	+	0.845069i	$0.320440\pi$
$617$			41.9821i			1.69014i	−0.534658	+	0.845069i	$0.679560\pi$
$618$		0			0
$619$		−	29.0832i		−	1.16895i	−0.811411	−	0.584476i	$-0.801300\pi$
$619$		−	29.0832i		−	1.16895i	0.811411	−	0.584476i	$-0.198700\pi$
$620$		0			0
$621$	10.8003	+	17.1883i	0.433402	+	0.689743i
$622$		0			0
$623$	−4.27620	+	1.98190i	−0.171322	+	0.0794031i
$624$		0			0
$625$	−9.50735			−0.380294
$626$		0			0
$627$	0.496546	+	0.576993i	0.0198301	+	0.0230429i
$628$		0			0
$629$	2.14629			0.0855783
$630$		0			0
$631$	42.0921			1.67566			0.837831	−	0.545930i	$-0.183823\pi$
$631$	42.0921			1.67566			0.837831	+	0.545930i	$0.183823\pi$
$632$		0			0
$633$	−31.3634	−	36.4447i	−1.24658	−	1.44855i
$634$		0			0
$635$	28.2857			1.12249
$636$		0			0
$637$	−23.9624	−	20.2980i	−0.949425	−	0.804237i
$638$		0			0
$639$	−6.64604	−	1.00169i	−0.262913	−	0.0396263i
$640$		0			0
$641$			3.70860i			0.146481i	0.997314	+	0.0732404i	$0.0233340\pi$
$641$			3.70860i			0.146481i	−0.997314	+	0.0732404i	$0.976666\pi$
$642$		0			0
$643$		−	26.2231i		−	1.03414i	−0.855943	−	0.517069i	$-0.827023\pi$
$643$		−	26.2231i		−	1.03414i	0.855943	−	0.517069i	$-0.172977\pi$
$644$		0			0
$645$	12.8921	+	14.9809i	0.507628	+	0.589871i
$646$		0			0
$647$	−39.9863			−1.57202			−0.786012	−	0.618212i	$-0.787858\pi$
$647$	−39.9863			−1.57202			−0.786012	+	0.618212i	$0.787858\pi$
$648$		0			0
$649$			5.54560i			0.217684i
$650$		0			0
$651$	−2.43509	+	5.36336i	−0.0954387	+	0.210207i
$652$		0			0
$653$		−	17.3435i		−	0.678704i	−0.940659	−	0.339352i	$-0.889792\pi$
$653$		−	17.3435i		−	0.678704i	0.940659	−	0.339352i	$-0.110208\pi$
$654$		0			0
$655$	23.3922			0.914009
$656$		0			0
$657$	−1.15657	−	0.174318i	−0.0451220	−	0.00680080i
$658$		0			0
$659$		−	0.108834i		−	0.00423958i	−0.999998	−	0.00211979i	$-0.999325\pi$
$659$		−	0.108834i		−	0.00423958i	0.999998	−	0.00211979i	$-0.000674751\pi$
$660$		0			0
$661$			31.1145i			1.21021i	0.796144	+	0.605107i	$0.206870\pi$
$661$			31.1145i			1.21021i	−0.796144	+	0.605107i	$0.793130\pi$
$662$		0			0
$663$	27.3214	−	23.5121i	1.06107	−	0.913133i
$664$		0			0
$665$	0.823596	+	1.77701i	0.0319377	+	0.0689097i
$666$		0			0
$667$	−3.45864			−0.133919
$668$		0			0
$669$	3.51185	−	3.02220i	0.135776	−	0.116845i
$670$		0			0
$671$	10.1658			0.392447
$672$		0			0
$673$	−15.4056			−0.593841			−0.296920	−	0.954902i	$-0.595960\pi$
$673$	−15.4056			−0.593841			−0.296920	+	0.954902i	$0.595960\pi$
$674$		0			0
$675$	−9.51608	+	5.97946i	−0.366274	+	0.230150i
$676$		0			0
$677$	39.9871			1.53683			0.768414	−	0.639953i	$-0.221046\pi$
$677$	39.9871			1.53683			0.768414	+	0.639953i	$0.221046\pi$
$678$		0			0
$679$	17.1603	+	37.0256i	0.658552	+	1.42091i
$680$		0			0
$681$	−8.84369	−	10.2765i	−0.338891	−	0.393796i
$682$		0			0
$683$		−	20.3682i		−	0.779369i	−0.920948	−	0.389684i	$-0.872584\pi$
$683$		−	20.3682i		−	0.779369i	0.920948	−	0.389684i	$-0.127416\pi$
$684$		0			0
$685$			12.8155i			0.489653i
$686$		0			0
$687$	21.0484	−	18.1137i	0.803045	−	0.691080i
$688$		0			0
$689$	36.6679			1.39694
$690$		0			0
$691$			32.2967i			1.22863i	0.789062	+	0.614313i	$0.210567\pi$
$691$			32.2967i			1.22863i	−0.789062	+	0.614313i	$0.789433\pi$
$692$		0			0
$693$	7.61840	−	2.22709i	0.289399	−	0.0846003i
$694$		0			0
$695$		−	27.7551i		−	1.05281i
$696$		0			0
$697$	12.5970			0.477145
$698$		0			0
$699$	−1.28066	+	1.10210i	−0.0484389	+	0.0416853i
$700$		0			0
$701$			7.51685i			0.283908i	0.989873	+	0.141954i	$0.0453385\pi$
$701$			7.51685i			0.283908i	−0.989873	+	0.141954i	$0.954662\pi$
$702$		0			0
$703$		−	0.203349i		−	0.00766947i
$704$		0			0
$705$	21.8159	+	25.3504i	0.821634	+	0.954751i
$706$		0			0
$707$	13.7178	−	6.35783i	0.515913	−	0.239111i
$708$		0			0
$709$	29.2608			1.09891			0.549456	−	0.835523i	$-0.314835\pi$
$709$	29.2608			1.09891			0.549456	+	0.835523i	$0.314835\pi$
$710$		0			0
$711$	−5.48864	+	36.4161i	−0.205840	+	1.36571i
$712$		0			0
$713$	−5.02154			−0.188058
$714$		0			0
$715$	−7.55654			−0.282599
$716$		0			0
$717$	−2.34599	+	2.01890i	−0.0876126	+	0.0753971i
$718$		0			0
$719$	−18.0916			−0.674702			−0.337351	−	0.941379i	$-0.609531\pi$
$719$	−18.0916			−0.674702			−0.337351	+	0.941379i	$0.609531\pi$
$720$		0			0
$721$	5.48551	+	11.8357i	0.204291	+	0.440785i
$722$		0			0
$723$	14.9467	−	12.8628i	0.555875	−	0.478372i
$724$		0			0
$725$		−	1.91483i		−	0.0711150i
$726$		0			0
$727$			27.7259i			1.02830i	0.857701	+	0.514149i	$0.171892\pi$
$727$			27.7259i			1.02830i	−0.857701	+	0.514149i	$0.828108\pi$
$728$		0			0
$729$	11.7145	−	24.3263i	0.433869	−	0.900976i
$730$		0			0
$731$	−31.4261			−1.16234
$732$		0			0
$733$			16.5678i			0.611946i	0.952040	+	0.305973i	$0.0989817\pi$
$733$			16.5678i			0.611946i	−0.952040	+	0.305973i	$0.901018\pi$
$734$		0			0
$735$	20.4213	−	0.159400i	0.753250	−	0.00587957i
$736$		0			0
$737$			4.56513i			0.168159i
$738$		0			0
$739$	14.7944			0.544220			0.272110	−	0.962266i	$-0.412278\pi$
$739$	14.7944			0.544220			0.272110	+	0.962266i	$0.412278\pi$
$740$		0			0
$741$	−2.22764	−	2.58855i	−0.0818344	−	0.0950928i
$742$		0			0
$743$			47.5654i			1.74500i	0.488610	+	0.872502i	$0.337504\pi$
$743$			47.5654i			1.74500i	−0.488610	+	0.872502i	$0.662496\pi$
$744$		0			0
$745$		−	23.4516i		−	0.859202i
$746$		0			0
$747$	−1.24559	+	8.26424i	−0.0455737	+	0.302373i
$748$		0			0
$749$	22.5921	+	48.7454i	0.825498	+	1.78112i
$750$		0			0
$751$	36.0295			1.31474			0.657368	−	0.753570i	$-0.271670\pi$
$751$	36.0295			1.31474			0.657368	+	0.753570i	$0.271670\pi$
$752$		0			0
$753$	−8.39257	−	9.75229i	−0.305842	−	0.355393i
$754$		0			0
$755$	19.2854			0.701868
$756$		0			0
$757$	−32.0938			−1.16647			−0.583234	−	0.812304i	$-0.698213\pi$
$757$	−32.0938			−1.16647			−0.583234	+	0.812304i	$0.698213\pi$
$758$		0			0
$759$	4.41380	+	5.12890i	0.160211	+	0.186167i
$760$		0			0
$761$	−17.8725			−0.647879			−0.323940	−	0.946078i	$-0.605007\pi$
$761$	−17.8725			−0.647879			−0.323940	+	0.946078i	$0.605007\pi$
$762$		0			0
$763$	−35.8584	+	16.6194i	−1.29816	+	0.601661i
$764$		0			0
$765$	−3.49346	+	23.1785i	−0.126306	+	0.838019i
$766$		0			0
$767$		−	24.8791i		−	0.898331i
$768$		0			0
$769$		−	40.9792i		−	1.47775i	−0.673843	−	0.738875i	$-0.735358\pi$
$769$		−	40.9792i		−	1.47775i	0.673843	−	0.738875i	$-0.264642\pi$
$770$		0			0
$771$	1.58552	+	1.84240i	0.0571012	+	0.0663525i
$772$		0			0
$773$	5.96160			0.214424			0.107212	−	0.994236i	$-0.465808\pi$
$773$	5.96160			0.214424			0.107212	+	0.994236i	$0.465808\pi$
$774$		0			0
$775$		−	2.78011i		−	0.0998645i
$776$		0			0
$777$	−1.93062	−	0.876546i	−0.0692605	−	0.0314459i
$778$		0			0
$779$		−	1.19350i		−	0.0427614i
$780$		0			0
$781$	−2.24037			−0.0801666
$782$		0			0
$783$	2.44748	+	3.89507i	0.0874658	+	0.139199i
$784$		0			0
$785$		−	30.3018i		−	1.08152i
$786$		0			0
$787$		−	29.9974i		−	1.06929i	−0.845077	−	0.534645i	$-0.820445\pi$
$787$		−	29.9974i		−	1.06929i	0.845077	−	0.534645i	$-0.179555\pi$
$788$		0			0
$789$	−0.638139	+	0.549166i	−0.0227184	+	0.0195508i
$790$		0			0
$791$	11.9581	+	25.8012i	0.425182	+	0.917386i
$792$		0			0
$793$	−45.6066			−1.61954
$794$		0			0
$795$	−18.0738	+	15.5539i	−0.641013	+	0.551639i
$796$		0			0
$797$	44.6607			1.58196			0.790982	−	0.611839i	$-0.209570\pi$
$797$	44.6607			1.58196			0.790982	+	0.611839i	$0.209570\pi$
$798$		0			0
$799$	−53.1788			−1.88133
$800$		0			0
$801$	−0.796485	+	5.28453i	−0.0281424	+	0.186720i
$802$		0			0
$803$	−0.389877			−0.0137585
$804$		0			0
$805$	7.32095	+	15.7959i	0.258029	+	0.556732i
$806$		0			0
$807$	23.2159	+	26.9772i	0.817239	+	0.949643i
$808$		0			0
$809$		−	14.0773i		−	0.494933i	−0.968896	−	0.247467i	$-0.920402\pi$
$809$		−	14.0773i		−	0.494933i	0.968896	−	0.247467i	$-0.0795980\pi$
$810$		0			0
$811$		−	44.2560i		−	1.55404i	−0.629477	−	0.777019i	$-0.716731\pi$
$811$		−	44.2560i		−	1.55404i	0.629477	−	0.777019i	$-0.283269\pi$
$812$		0			0
$813$	12.5360	−	10.7882i	0.439658	−	0.378359i
$814$		0			0
$815$	−23.9084			−0.837476
$816$		0			0
$817$			2.97745i			0.104168i
$818$		0			0
$819$	−34.1782	+	9.99136i	−1.19428	+	0.349126i
$820$		0			0
$821$		−	1.90237i		−	0.0663932i	−0.999449	−	0.0331966i	$-0.989431\pi$
$821$		−	1.90237i		−	0.0663932i	0.999449	−	0.0331966i	$-0.0105687\pi$
$822$		0			0
$823$	32.1076			1.11920			0.559600	−	0.828763i	$-0.310955\pi$
$823$	32.1076			1.11920			0.559600	+	0.828763i	$0.310955\pi$
$824$		0			0
$825$	−2.83955	+	2.44364i	−0.0988604	+	0.0850767i
$826$		0			0
$827$			12.9825i			0.451446i	0.974192	+	0.225723i	$0.0724744\pi$
$827$			12.9825i			0.451446i	−0.974192	+	0.225723i	$0.927526\pi$
$828$		0			0
$829$		−	24.6568i		−	0.856365i	−0.903692	−	0.428183i	$-0.859154\pi$
$829$		−	24.6568i		−	0.856365i	0.903692	−	0.428183i	$-0.140846\pi$
$830$		0			0
$831$	−26.9811	−	31.3524i	−0.935963	−	1.08760i
$832$		0			0
$833$	−20.9880	+	24.7770i	−0.727193	+	0.858471i
$834$		0			0
$835$	−28.0401			−0.970369
$836$		0			0
$837$	3.55345	+	5.65519i	0.122825	+	0.195472i
$838$		0			0
$839$	15.6311			0.539647			0.269823	−	0.962910i	$-0.413035\pi$
$839$	15.6311			0.539647			0.269823	+	0.962910i	$0.413035\pi$
$840$		0			0
$841$	28.2162			0.972974
$842$		0			0
$843$	−26.4198	+	22.7362i	−0.909947	+	0.783077i
$844$		0			0
$845$	12.0039			0.412948
$846$		0			0
$847$	2.40047	−	1.11255i	0.0824810	−	0.0382276i
$848$		0			0
$849$	8.54382	−	7.35259i	0.293223	−	0.252340i
$850$		0			0
$851$		−	1.80757i		−	0.0619629i
$852$		0			0
$853$		−	6.39059i		−	0.218810i	−0.993997	−	0.109405i	$-0.965106\pi$
$853$		−	6.39059i		−	0.218810i	0.993997	−	0.109405i	$-0.0348945\pi$
$854$		0			0
$855$	2.19603	+	0.330987i	0.0751028	+	0.0113195i
$856$		0			0
$857$	5.92927			0.202540			0.101270	−	0.994859i	$-0.467709\pi$
$857$	5.92927			0.202540			0.101270	+	0.994859i	$0.467709\pi$
$858$		0			0
$859$			30.3899i			1.03689i	0.855111	+	0.518445i	$0.173489\pi$
$859$			30.3899i			1.03689i	−0.855111	+	0.518445i	$0.826511\pi$
$860$		0			0
$861$	−11.3312	−	5.14461i	−0.386165	−	0.175328i
$862$		0			0
$863$		−	19.6054i		−	0.667375i	−0.942684	−	0.333688i	$-0.891707\pi$
$863$		−	19.6054i		−	0.667375i	0.942684	−	0.333688i	$-0.108293\pi$
$864$		0			0
$865$	5.14115			0.174804
$866$		0			0
$867$	−5.10477	−	5.93182i	−0.173367	−	0.201455i
$868$		0			0
$869$			12.2758i			0.416428i
$870$		0			0
$871$		−	20.4804i		−	0.693953i
$872$		0			0
$873$	45.7562	+	6.89638i	1.54861	+	0.233407i
$874$		0			0
$875$	−28.9615	+	13.4229i	−0.979079	+	0.453775i
$876$		0			0
$877$	−0.709577			−0.0239607			−0.0119804	−	0.999928i	$-0.503814\pi$
$877$	−0.709577			−0.0239607			−0.0119804	+	0.999928i	$0.503814\pi$
$878$		0			0
$879$	−15.7500	−	18.3018i	−0.531236	−	0.617304i
$880$		0			0
$881$	36.3383			1.22427			0.612134	−	0.790754i	$-0.290311\pi$
$881$	36.3383			1.22427			0.612134	+	0.790754i	$0.290311\pi$
$882$		0			0
$883$	9.56741			0.321969			0.160985	−	0.986957i	$-0.448533\pi$
$883$	9.56741			0.321969			0.160985	+	0.986957i	$0.448533\pi$
$884$		0			0
$885$	10.5533	+	12.2630i	0.354744	+	0.412218i
$886$		0			0
$887$	35.1585			1.18051			0.590253	−	0.807218i	$-0.299028\pi$
$887$	35.1585			1.18051			0.590253	+	0.807218i	$0.299028\pi$
$888$		0			0
$889$	40.3112	−	18.6831i	1.35199	−	0.626611i
$890$		0			0
$891$	2.65270	−	8.60018i	0.0888689	−	0.288117i
$892$		0			0
$893$			5.03840i			0.168604i
$894$		0			0
$895$		−	20.8157i		−	0.695791i
$896$		0			0
$897$	−19.8015	−	23.0096i	−0.661153	−	0.768269i
$898$		0			0
$899$	−1.13794			−0.0379524
$900$		0			0
$901$		−	37.9144i		−	1.26311i
$902$		0			0
$903$	28.2682	+	12.8344i	0.940707	+	0.427103i
$904$		0			0
$905$			0.538205i			0.0178905i
$906$		0			0
$907$	32.4473			1.07739			0.538697	−	0.842499i	$-0.318917\pi$
$907$	32.4473			1.07739			0.538697	+	0.842499i	$0.318917\pi$
$908$		0			0
$909$	2.55509	−	16.9525i	0.0847468	−	0.562279i
$910$		0			0
$911$			39.2924i			1.30181i	0.759157	+	0.650907i	$0.225611\pi$
$911$			39.2924i			1.30181i	−0.759157	+	0.650907i	$0.774389\pi$
$912$		0			0
$913$			2.78586i			0.0921986i
$914$		0			0
$915$	22.4798	−	19.3455i	0.743159	−	0.639543i
$916$		0			0
$917$	33.3372	−	15.4509i	1.10089	−	0.510232i
$918$		0			0
$919$	−10.9754			−0.362046			−0.181023	−	0.983479i	$-0.557941\pi$
$919$	−10.9754			−0.362046			−0.181023	+	0.983479i	$0.557941\pi$
$920$		0			0
$921$	0.0434345	−	0.0373786i	0.00143122	−	0.00123167i
$922$		0			0
$923$	10.0509			0.330830
$924$		0			0
$925$	1.00074			0.0329042
$926$		0			0
$927$	14.6266	+	2.20452i	0.480399	+	0.0724059i
$928$		0			0
$929$	−3.33274			−0.109344			−0.0546718	−	0.998504i	$-0.517411\pi$
$929$	−3.33274			−0.109344			−0.0546718	+	0.998504i	$0.517411\pi$
$930$		0			0
$931$	2.34748	+	1.98850i	0.0769356	+	0.0651705i
$932$		0			0
$933$	4.27535	+	4.96802i	0.139969	+	0.162646i
$934$		0			0
$935$			7.81342i			0.255526i
$936$		0			0
$937$		−	22.7480i		−	0.743145i	−0.928404	−	0.371572i	$-0.878819\pi$
$937$		−	22.7480i		−	0.743145i	0.928404	−	0.371572i	$-0.121181\pi$
$938$		0			0
$939$	−13.7477	+	11.8310i	−0.448641	+	0.386089i
$940$		0			0
$941$	−24.0624			−0.784412			−0.392206	−	0.919877i	$-0.628288\pi$
$941$	−24.0624			−0.784412			−0.392206	+	0.919877i	$0.628288\pi$
$942$		0			0
$943$		−	10.6090i		−	0.345476i
$944$		0			0
$945$	12.6085	−	19.4226i	0.410155	−	0.631817i
$946$		0			0
$947$		−	18.7369i		−	0.608867i	−0.952534	−	0.304433i	$-0.901533\pi$
$947$		−	18.7369i		−	0.608867i	0.952534	−	0.304433i	$-0.0984671\pi$
$948$		0			0
$949$	1.74910			0.0567781
$950$		0			0
$951$	−4.31641	+	3.71459i	−0.139969	+	0.120454i
$952$		0			0
$953$			18.3763i			0.595267i	0.954680	+	0.297633i	$0.0961973\pi$
$953$			18.3763i			0.595267i	−0.954680	+	0.297633i	$0.903803\pi$
$954$		0			0
$955$		−	35.6834i		−	1.15469i
$956$		0			0
$957$	1.00022	+	1.16227i	0.0323325	+	0.0375708i
$958$		0			0
$959$	8.46478	+	18.2639i	0.273342	+	0.589770i
$960$		0			0
$961$	29.3478			0.946705
$962$		0			0
$963$	60.2396	+	9.07932i	1.94119	+	0.292577i
$964$		0			0
$965$	−45.1905			−1.45473
$966$		0			0
$967$	2.42754			0.0780645			0.0390323	−	0.999238i	$-0.487572\pi$
$967$	2.42754			0.0780645			0.0390323	+	0.999238i	$0.487572\pi$
$968$		0			0
$969$	−2.67655	+	2.30337i	−0.0859830	+	0.0739948i
$970$		0			0
$971$	−10.9178			−0.350368			−0.175184	−	0.984536i	$-0.556052\pi$
$971$	−10.9178			−0.350368			−0.175184	+	0.984536i	$0.556052\pi$
$972$		0			0
$973$	−18.3326	−	39.5550i	−0.587717	−	1.26808i
$974$		0			0
$975$	12.7390	−	10.9629i	0.407974	−	0.351092i
$976$		0			0
$977$			55.2872i			1.76879i	0.466736	+	0.884397i	$0.345430\pi$
$977$			55.2872i			1.76879i	−0.466736	+	0.884397i	$0.654570\pi$
$978$		0			0
$979$			1.78140i			0.0569339i
$980$		0			0
$981$	−6.67899	+	44.3138i	−0.213244	+	1.41483i
$982$		0			0
$983$	21.3288			0.680283			0.340142	−	0.940374i	$-0.389525\pi$
$983$	21.3288			0.680283			0.340142	+	0.940374i	$0.389525\pi$
$984$		0			0
$985$			8.02551i			0.255714i
$986$		0			0
$987$	47.8350	+	21.7182i	1.52261	+	0.691299i
$988$		0			0
$989$			26.4666i			0.841589i
$990$		0			0
$991$	−27.1534			−0.862555			−0.431278	−	0.902219i	$-0.641937\pi$
$991$	−27.1534			−0.862555			−0.431278	+	0.902219i	$0.641937\pi$
$992$		0			0
$993$	27.6200	+	32.0948i	0.876494	+	1.01850i
$994$		0			0
$995$		−	36.3089i		−	1.15107i
$996$		0			0
$997$			29.4497i			0.932681i	0.884605	+	0.466341i	$0.154428\pi$
$997$			29.4497i			0.932681i	−0.884605	+	0.466341i	$0.845572\pi$
$998$		0			0
$999$	−2.03567	+	1.27912i	−0.0644056	+	0.0404695i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.e.881.9	✓	32
3.2	odd	2	inner	1848.2.v.e.881.23	yes	32
7.6	odd	2	inner	1848.2.v.e.881.24	yes	32
21.20	even	2	inner	1848.2.v.e.881.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.e.881.9	✓	32	1.1	even	1	trivial
1848.2.v.e.881.10	yes	32	21.20	even	2	inner
1848.2.v.e.881.23	yes	32	3.2	odd	2	inner
1848.2.v.e.881.24	yes	32	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.12980 - 1.31284i) q^{3} -1.68437 q^{5} +(-2.40047 + 1.11255i) q^{7} +(-0.447111 + 2.96649i) q^{9} +1.00000i q^{11} -4.48628i q^{13} +(1.90300 + 2.21131i) q^{15} -4.63878 q^{17} +0.439499i q^{19} +(4.17264 + 1.89448i) q^{21} +3.90671i q^{23} -2.16290 q^{25} +(4.39969 - 2.76456i) q^{27} +0.885307i q^{29} +1.28536i q^{31} +(1.31284 - 1.12980i) q^{33} +(4.04327 - 1.87394i) q^{35} -0.462685 q^{37} +(-5.88977 + 5.06859i) q^{39} -2.71558 q^{41} +6.77465 q^{43} +(0.753100 - 4.99667i) q^{45} +11.4640 q^{47} +(4.52447 - 5.34127i) q^{49} +(5.24089 + 6.08999i) q^{51} +8.17336i q^{53} -1.68437i q^{55} +(0.576993 - 0.496546i) q^{57} +5.54560 q^{59} -10.1658i q^{61} +(-2.22709 - 7.61840i) q^{63} +7.55654i q^{65} +4.56513 q^{67} +(5.12890 - 4.41380i) q^{69} +2.24037i q^{71} +0.389877i q^{73} +(2.44364 + 2.83955i) q^{75} +(-1.11255 - 2.40047i) q^{77} +12.2758 q^{79} +(-8.60018 - 2.65270i) q^{81} +2.78586 q^{83} +7.81342 q^{85} +(1.16227 - 1.00022i) q^{87} +1.78140 q^{89} +(4.99120 + 10.7692i) q^{91} +(1.68748 - 1.45220i) q^{93} -0.740279i q^{95} -15.4243i q^{97} +(-2.96649 - 0.447111i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} + 4 q^{9} - 16 q^{15} + 6 q^{21} + 68 q^{25} - 68 q^{37} - 32 q^{39} + 120 q^{43} - 64 q^{49} + 56 q^{51} - 44 q^{57} + 42 q^{63} + 4 q^{67} - 56 q^{79} - 28 q^{81} - 44 q^{91} - 28 q^{93}+O(q^{100}) \) 32 * q + 4 * q^7 + 4 * q^9 - 16 * q^15 + 6 * q^21 + 68 * q^25 - 68 * q^37 - 32 * q^39 + 120 * q^43 - 64 * q^49 + 56 * q^51 - 44 * q^57 + 42 * q^63 + 4 * q^67 - 56 * q^79 - 28 * q^81 - 44 * q^91 - 28 * q^93

Introduction

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