LMFDB - Embedded newform 4140.3.d.b.2161.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.12
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +6.51414i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +6.51414i q^{7} +16.5307i q^{11} -12.7134 q^{13} +15.5027i q^{17} -21.3188i q^{19} +(21.7442 - 7.49602i) q^{23} -5.00000 q^{25} +21.6211 q^{29} +61.2086 q^{31} +14.5660 q^{35} +29.9640i q^{37} +57.4591 q^{41} -61.2273i q^{43} -72.3919 q^{47} +6.56604 q^{49} -98.7521i q^{53} +36.9637 q^{55} -40.4289 q^{59} +48.2394i q^{61} +28.4279i q^{65} +70.9318i q^{67} +62.9249 q^{71} -13.0823 q^{73} -107.683 q^{77} +105.370i q^{79} -16.3211i q^{83} +34.6651 q^{85} +101.880i q^{89} -82.8166i q^{91} -47.6703 q^{95} +148.212i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) $ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			6.51414i			0.930591i	0.885155	+	0.465295i	$0.154052\pi$
$7$			6.51414i			0.930591i	−0.885155	+	0.465295i	$0.845948\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			16.5307i			1.50279i	0.659853	+	0.751394i	$0.270618\pi$
$11$			16.5307i			1.50279i	−0.659853	+	0.751394i	$0.729382\pi$
$12$		0			0
$13$	−12.7134			−0.977951			−0.488975	−	0.872298i	$-0.662629\pi$
$13$	−12.7134			−0.977951			−0.488975	+	0.872298i	$0.662629\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			15.5027i			0.911925i	0.889999	+	0.455962i	$0.150705\pi$
$17$			15.5027i			0.911925i	−0.889999	+	0.455962i	$0.849295\pi$
$18$		0			0
$19$		−	21.3188i		−	1.12204i	−0.827801	−	0.561021i	$-0.810409\pi$
$19$		−	21.3188i		−	1.12204i	0.827801	−	0.561021i	$-0.189591\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.7442	−	7.49602i	0.945399	−	0.325914i
$23$	21.7442	−	7.49602i	0.945399	−	0.325914i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	21.6211			0.745556			0.372778	−	0.927921i	$-0.378405\pi$
$29$	21.6211			0.745556			0.372778	+	0.927921i	$0.378405\pi$
$30$		0			0
$31$	61.2086			1.97447			0.987236	−	0.159264i	$-0.0509121\pi$
$31$	61.2086			1.97447			0.987236	+	0.159264i	$0.0509121\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	14.5660			0.416173
$36$		0			0
$37$			29.9640i			0.809838i	0.914352	+	0.404919i	$0.132700\pi$
$37$			29.9640i			0.809838i	−0.914352	+	0.404919i	$0.867300\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	57.4591			1.40144			0.700721	−	0.713436i	$-0.252862\pi$
$41$	57.4591			1.40144			0.700721	+	0.713436i	$0.252862\pi$
$42$		0			0
$43$		−	61.2273i		−	1.42389i	−0.702235	−	0.711945i	$-0.747814\pi$
$43$		−	61.2273i		−	1.42389i	0.702235	−	0.711945i	$-0.252186\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−72.3919			−1.54025			−0.770126	−	0.637892i	$-0.779807\pi$
$47$	−72.3919			−1.54025			−0.770126	+	0.637892i	$0.779807\pi$
$48$		0			0
$49$	6.56604			0.134001
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	98.7521i		−	1.86325i	−0.363426	−	0.931623i	$-0.618393\pi$
$53$		−	98.7521i		−	1.86325i	0.363426	−	0.931623i	$-0.381607\pi$
$54$		0			0
$55$	36.9637			0.672068
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−40.4289			−0.685236			−0.342618	−	0.939475i	$-0.611314\pi$
$59$	−40.4289			−0.685236			−0.342618	+	0.939475i	$0.611314\pi$
$60$		0			0
$61$			48.2394i			0.790810i	0.918507	+	0.395405i	$0.129396\pi$
$61$			48.2394i			0.790810i	−0.918507	+	0.395405i	$0.870604\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			28.4279i			0.437353i
$66$		0			0
$67$			70.9318i			1.05868i	0.848409	+	0.529342i	$0.177561\pi$
$67$			70.9318i			1.05868i	−0.848409	+	0.529342i	$0.822439\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	62.9249			0.886266			0.443133	−	0.896456i	$-0.353867\pi$
$71$	62.9249			0.886266			0.443133	+	0.896456i	$0.353867\pi$
$72$		0			0
$73$	−13.0823			−0.179209			−0.0896047	−	0.995977i	$-0.528560\pi$
$73$	−13.0823			−0.179209			−0.0896047	+	0.995977i	$0.528560\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−107.683			−1.39848
$78$		0			0
$79$			105.370i			1.33380i	0.745147	+	0.666900i	$0.232379\pi$
$79$			105.370i			1.33380i	−0.745147	+	0.666900i	$0.767621\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	16.3211i		−	0.196640i	−0.995155	−	0.0983200i	$-0.968653\pi$
$83$		−	16.3211i		−	0.196640i	0.995155	−	0.0983200i	$-0.0313469\pi$
$84$		0			0
$85$	34.6651			0.407825
$86$		0			0
$87$		0			0
$88$		0			0
$89$			101.880i			1.14472i	0.820002	+	0.572361i	$0.193972\pi$
$89$			101.880i			1.14472i	−0.820002	+	0.572361i	$0.806028\pi$
$90$		0			0
$91$		−	82.8166i		−	0.910072i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−47.6703			−0.501793
$96$		0			0
$97$			148.212i			1.52796i	0.645239	+	0.763981i	$0.276758\pi$
$97$			148.212i			1.52796i	−0.645239	+	0.763981i	$0.723242\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−125.726			−1.24481			−0.622404	−	0.782696i	$-0.713844\pi$
$101$	−125.726			−1.24481			−0.622404	+	0.782696i	$0.713844\pi$
$102$		0			0
$103$			123.892i			1.20283i	0.798936	+	0.601416i	$0.205396\pi$
$103$			123.892i			1.20283i	−0.798936	+	0.601416i	$0.794604\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	65.4784i		−	0.611948i	−0.952040	−	0.305974i	$-0.901018\pi$
$107$		−	65.4784i		−	0.611948i	0.952040	−	0.305974i	$-0.0989821\pi$
$108$		0			0
$109$			104.447i			0.958231i	0.877752	+	0.479116i	$0.159043\pi$
$109$			104.447i			0.958231i	−0.877752	+	0.479116i	$0.840957\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			61.6217i			0.545325i	0.962110	+	0.272663i	$0.0879043\pi$
$113$			61.6217i			0.545325i	−0.962110	+	0.272663i	$0.912096\pi$
$114$		0			0
$115$	−16.7616	−	48.6215i	−0.145753	−	0.422795i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−100.987			−0.848629
$120$		0			0
$121$	−152.263			−1.25837
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	31.7368			0.249896			0.124948	−	0.992163i	$-0.460124\pi$
$127$	31.7368			0.249896			0.124948	+	0.992163i	$0.460124\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−62.8008			−0.479395			−0.239698	−	0.970848i	$-0.577048\pi$
$131$	−62.8008			−0.479395			−0.239698	+	0.970848i	$0.577048\pi$
$132$		0			0
$133$	138.874			1.04416
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	131.647i		−	0.960924i	−0.877016	−	0.480462i	$-0.840469\pi$
$137$		−	131.647i		−	0.960924i	0.877016	−	0.480462i	$-0.159531\pi$
$138$		0			0
$139$	−174.347			−1.25429			−0.627147	−	0.778901i	$-0.715778\pi$
$139$	−174.347			−1.25429			−0.627147	+	0.778901i	$0.715778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	210.160i		−	1.46965i
$144$		0			0
$145$		−	48.3463i		−	0.333423i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			115.373i			0.774313i	0.922014	+	0.387157i	$0.126543\pi$
$149$			115.373i			0.774313i	−0.922014	+	0.387157i	$0.873457\pi$
$150$		0			0
$151$	−213.709			−1.41529			−0.707645	−	0.706568i	$-0.750242\pi$
$151$	−213.709			−1.41529			−0.707645	+	0.706568i	$0.750242\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	136.867i		−	0.883011i
$156$		0			0
$157$			30.4657i			0.194049i	0.995282	+	0.0970246i	$0.0309326\pi$
$157$			30.4657i			0.194049i	−0.995282	+	0.0970246i	$0.969067\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	48.8301	+	141.645i	0.303292	+	0.879780i
$162$		0			0
$163$	−279.817			−1.71667			−0.858336	−	0.513089i	$-0.828501\pi$
$163$	−279.817			−1.71667			−0.858336	+	0.513089i	$0.828501\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−130.550			−0.781734			−0.390867	−	0.920447i	$-0.627825\pi$
$167$	−130.550			−0.781734			−0.390867	+	0.920447i	$0.627825\pi$
$168$		0			0
$169$	−7.37046			−0.0436122
$170$		0			0
$171$		0			0
$172$		0			0
$173$	92.6253			0.535406			0.267703	−	0.963501i	$-0.413735\pi$
$173$	92.6253			0.535406			0.267703	+	0.963501i	$0.413735\pi$
$174$		0			0
$175$		−	32.5707i		−	0.186118i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	191.091			1.06755			0.533773	−	0.845628i	$-0.320774\pi$
$179$	191.091			1.06755			0.533773	+	0.845628i	$0.320774\pi$
$180$		0			0
$181$		−	157.032i		−	0.867579i	−0.901014	−	0.433789i	$-0.857176\pi$
$181$		−	157.032i		−	0.867579i	0.901014	−	0.433789i	$-0.142824\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	67.0016			0.362171
$186$		0			0
$187$	−256.270			−1.37043
$188$		0			0
$189$		0			0
$190$		0			0
$191$			181.035i			0.947825i	0.880572	+	0.473912i	$0.157159\pi$
$191$			181.035i			0.947825i	−0.880572	+	0.473912i	$0.842841\pi$
$192$		0			0
$193$	134.038			0.694497			0.347248	−	0.937773i	$-0.387116\pi$
$193$	134.038			0.694497			0.347248	+	0.937773i	$0.387116\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	148.351			0.753052			0.376526	−	0.926406i	$-0.377119\pi$
$197$	148.351			0.753052			0.376526	+	0.926406i	$0.377119\pi$
$198$		0			0
$199$			305.068i			1.53301i	0.642241	+	0.766503i	$0.278005\pi$
$199$			305.068i			1.53301i	−0.642241	+	0.766503i	$0.721995\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			140.843i			0.693808i
$204$		0			0
$205$		−	128.482i		−	0.626744i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	352.414			1.68619
$210$		0			0
$211$	363.104			1.72087			0.860436	−	0.509558i	$-0.170191\pi$
$211$	363.104			1.72087			0.860436	+	0.509558i	$0.170191\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−136.908			−0.636783
$216$		0			0
$217$			398.721i			1.83743i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	197.092i		−	0.891818i
$222$		0			0
$223$	−103.775			−0.465360			−0.232680	−	0.972553i	$-0.574750\pi$
$223$	−103.775			−0.465360			−0.232680	+	0.972553i	$0.574750\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			256.904i			1.13174i	0.824496	+	0.565868i	$0.191459\pi$
$227$			256.904i			1.13174i	−0.824496	+	0.565868i	$0.808541\pi$
$228$		0			0
$229$		−	147.159i		−	0.642614i	−0.946975	−	0.321307i	$-0.895878\pi$
$229$		−	147.159i		−	0.642614i	0.946975	−	0.321307i	$-0.104122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	257.832			1.10658			0.553288	−	0.832990i	$-0.313373\pi$
$233$	257.832			1.10658			0.553288	+	0.832990i	$0.313373\pi$
$234$		0			0
$235$			161.873i			0.688822i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−234.246			−0.980107			−0.490054	−	0.871692i	$-0.663023\pi$
$239$	−234.246			−0.980107			−0.490054	+	0.871692i	$0.663023\pi$
$240$		0			0
$241$			164.967i			0.684509i	0.939607	+	0.342255i	$0.111191\pi$
$241$			164.967i			0.684509i	−0.939607	+	0.342255i	$0.888809\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	14.6821i		−	0.0599270i
$246$		0			0
$247$			271.034i			1.09730i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			74.5087i			0.296848i	0.988924	+	0.148424i	$0.0474200\pi$
$251$			74.5087i			0.296848i	−0.988924	+	0.148424i	$0.952580\pi$
$252$		0			0
$253$	123.914	+	359.446i	0.489780	+	1.42074i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	111.698			0.434623			0.217312	−	0.976102i	$-0.430271\pi$
$257$	111.698			0.434623			0.217312	+	0.976102i	$0.430271\pi$
$258$		0			0
$259$	−195.190			−0.753628
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	318.510i		−	1.21106i	−0.795821	−	0.605532i	$-0.792960\pi$
$263$		−	318.510i		−	1.21106i	0.795821	−	0.605532i	$-0.207040\pi$
$264$		0			0
$265$	−220.816			−0.833269
$266$		0			0
$267$		0			0
$268$		0			0
$269$	259.585			0.964998			0.482499	−	0.875896i	$-0.339729\pi$
$269$	259.585			0.964998			0.482499	+	0.875896i	$0.339729\pi$
$270$		0			0
$271$	30.6023			0.112924			0.0564618	−	0.998405i	$-0.482018\pi$
$271$	30.6023			0.112924			0.0564618	+	0.998405i	$0.482018\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	82.6534i		−	0.300558i
$276$		0			0
$277$	−448.590			−1.61946			−0.809729	−	0.586804i	$-0.800386\pi$
$277$	−448.590			−1.61946			−0.809729	+	0.586804i	$0.800386\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	236.179i		−	0.840494i	−0.907410	−	0.420247i	$-0.861943\pi$
$281$		−	236.179i		−	0.840494i	0.907410	−	0.420247i	$-0.138057\pi$
$282$		0			0
$283$		−	81.4361i		−	0.287760i	−0.989595	−	0.143880i	$-0.954042\pi$
$283$		−	81.4361i		−	0.287760i	0.989595	−	0.143880i	$-0.0459579\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			374.296i			1.30417i
$288$		0			0
$289$	48.6656			0.168393
$290$		0			0
$291$		0			0
$292$		0			0
$293$			59.1544i			0.201892i	0.994892	+	0.100946i	$0.0321869\pi$
$293$			59.1544i			0.201892i	−0.994892	+	0.100946i	$0.967813\pi$
$294$		0			0
$295$			90.4018i			0.306447i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−276.442	+	95.2996i	−0.924554	+	0.318728i
$300$		0			0
$301$	398.843			1.32506
$302$		0			0
$303$		0			0
$304$		0			0
$305$	107.867			0.353661
$306$		0			0
$307$	130.636			0.425523			0.212762	−	0.977104i	$-0.431754\pi$
$307$	130.636			0.425523			0.212762	+	0.977104i	$0.431754\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−142.419			−0.457938			−0.228969	−	0.973434i	$-0.573535\pi$
$311$	−142.419			−0.457938			−0.228969	+	0.973434i	$0.573535\pi$
$312$		0			0
$313$			71.5400i			0.228562i	0.993448	+	0.114281i	$0.0364565\pi$
$313$			71.5400i			0.228562i	−0.993448	+	0.114281i	$0.963544\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−127.889			−0.403435			−0.201717	−	0.979444i	$-0.564652\pi$
$317$	−127.889			−0.403435			−0.201717	+	0.979444i	$0.564652\pi$
$318$		0			0
$319$			357.412i			1.12041i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	330.500			1.02322
$324$		0			0
$325$	63.5668			0.195590
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	471.570i		−	1.43334i
$330$		0			0
$331$	−392.565			−1.18600			−0.592999	−	0.805203i	$-0.702056\pi$
$331$	−392.565			−1.18600			−0.592999	+	0.805203i	$0.702056\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	158.608			0.473458
$336$		0			0
$337$		−	158.242i		−	0.469561i	−0.972048	−	0.234780i	$-0.924563\pi$
$337$		−	158.242i		−	0.469561i	0.972048	−	0.234780i	$-0.0754371\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1011.82i			2.96721i
$342$		0			0
$343$			361.965i			1.05529i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	111.993			0.322747			0.161373	−	0.986893i	$-0.448408\pi$
$347$	111.993			0.322747			0.161373	+	0.986893i	$0.448408\pi$
$348$		0			0
$349$	14.5348			0.0416470			0.0208235	−	0.999783i	$-0.493371\pi$
$349$	14.5348			0.0416470			0.0208235	+	0.999783i	$0.493371\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−613.599			−1.73824			−0.869120	−	0.494601i	$-0.835314\pi$
$353$	−613.599			−1.73824			−0.869120	+	0.494601i	$0.835314\pi$
$354$		0			0
$355$		−	140.704i		−	0.396350i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	37.2630i		−	0.103797i	−0.998652	−	0.0518983i	$-0.983473\pi$
$359$		−	37.2630i		−	0.103797i	0.998652	−	0.0518983i	$-0.0165272\pi$
$360$		0			0
$361$	−93.4917			−0.258980
$362$		0			0
$363$		0			0
$364$		0			0
$365$			29.2529i			0.0801449i
$366$		0			0
$367$		−	68.0517i		−	0.185427i	−0.995693	−	0.0927135i	$-0.970446\pi$
$367$		−	68.0517i		−	0.185427i	0.995693	−	0.0927135i	$-0.0295541\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	643.284			1.73392
$372$		0			0
$373$			378.573i			1.01494i	0.861669	+	0.507470i	$0.169419\pi$
$373$			378.573i			1.01494i	−0.861669	+	0.507470i	$0.830581\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−274.877			−0.729117
$378$		0			0
$379$			618.326i			1.63147i	0.578428	+	0.815733i	$0.303666\pi$
$379$			618.326i			1.63147i	−0.578428	+	0.815733i	$0.696334\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	127.236i		−	0.332209i	−0.986108	−	0.166105i	$-0.946881\pi$
$383$		−	127.236i		−	0.332209i	0.986108	−	0.166105i	$-0.0531189\pi$
$384$		0			0
$385$			240.787i			0.625420i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	152.707i		−	0.392564i	−0.980547	−	0.196282i	$-0.937113\pi$
$389$		−	152.707i		−	0.392564i	0.980547	−	0.196282i	$-0.0628868\pi$
$390$		0			0
$391$	116.209	+	337.094i	0.297209	+	0.862133i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	235.615			0.596493
$396$		0			0
$397$	212.688			0.535737			0.267869	−	0.963455i	$-0.413681\pi$
$397$	212.688			0.535737			0.267869	+	0.963455i	$0.413681\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			435.957i			1.08717i	0.839353	+	0.543587i	$0.182934\pi$
$401$			435.957i			1.08717i	−0.839353	+	0.543587i	$0.817066\pi$
$402$		0			0
$403$	−778.167			−1.93094
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−495.325			−1.21702
$408$		0			0
$409$	452.953			1.10747			0.553733	−	0.832694i	$-0.313203\pi$
$409$	452.953			1.10747			0.553733	+	0.832694i	$0.313203\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	263.360i		−	0.637674i
$414$		0			0
$415$	−36.4951			−0.0879401
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	453.068i		−	1.08131i	−0.841245	−	0.540654i	$-0.818177\pi$
$419$		−	453.068i		−	1.08131i	0.841245	−	0.540654i	$-0.181823\pi$
$420$		0			0
$421$			244.793i			0.581457i	0.956806	+	0.290729i	$0.0938977\pi$
$421$			244.793i			0.581457i	−0.956806	+	0.290729i	$0.906102\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	77.5136i		−	0.182385i
$426$		0			0
$427$	−314.238			−0.735920
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	90.0099i		−	0.208840i	−0.994533	−	0.104420i	$-0.966701\pi$
$431$		−	90.0099i		−	0.208840i	0.994533	−	0.104420i	$-0.0332986\pi$
$432$		0			0
$433$			570.075i			1.31657i	0.752769	+	0.658285i	$0.228718\pi$
$433$			570.075i			1.31657i	−0.752769	+	0.658285i	$0.771282\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−159.806	−	463.560i	−0.365689	−	1.06078i
$438$		0			0
$439$	265.282			0.604287			0.302144	−	0.953262i	$-0.402298\pi$
$439$	265.282			0.604287			0.302144	+	0.953262i	$0.402298\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−704.098			−1.58939			−0.794694	−	0.607011i	$-0.792368\pi$
$443$	−704.098			−1.58939			−0.794694	+	0.607011i	$0.792368\pi$
$444$		0			0
$445$	227.811			0.511935
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−563.750			−1.25557			−0.627784	−	0.778387i	$-0.716038\pi$
$449$	−563.750			−1.25557			−0.627784	+	0.778387i	$0.716038\pi$
$450$		0			0
$451$			949.838i			2.10607i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−185.183			−0.406997
$456$		0			0
$457$			566.245i			1.23905i	0.784978	+	0.619524i	$0.212674\pi$
$457$			566.245i			1.23905i	−0.784978	+	0.619524i	$0.787326\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−292.661			−0.634840			−0.317420	−	0.948285i	$-0.602817\pi$
$461$	−292.661			−0.634840			−0.317420	+	0.948285i	$0.602817\pi$
$462$		0			0
$463$	177.935			0.384309			0.192154	−	0.981365i	$-0.438453\pi$
$463$	177.935			0.384309			0.192154	+	0.981365i	$0.438453\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	563.656i		−	1.20697i	−0.797373	−	0.603486i	$-0.793778\pi$
$467$		−	563.656i		−	1.20697i	0.797373	−	0.603486i	$-0.206222\pi$
$468$		0			0
$469$	−462.059			−0.985201
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1012.13			2.13981
$474$		0			0
$475$			106.594i			0.224409i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	42.7175i		−	0.0891806i	−0.999005	−	0.0445903i	$-0.985802\pi$
$479$		−	42.7175i		−	0.0891806i	0.999005	−	0.0445903i	$-0.0141982\pi$
$480$		0			0
$481$		−	380.943i		−	0.791982i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	331.413			0.683325
$486$		0			0
$487$	160.343			0.329246			0.164623	−	0.986357i	$-0.447359\pi$
$487$	160.343			0.329246			0.164623	+	0.986357i	$0.447359\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	945.862			1.92640			0.963199	−	0.268788i	$-0.0866232\pi$
$491$	945.862			1.92640			0.963199	+	0.268788i	$0.0866232\pi$
$492$		0			0
$493$			335.186i			0.679891i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			409.901i			0.824751i
$498$		0			0
$499$	−362.924			−0.727303			−0.363652	−	0.931535i	$-0.618470\pi$
$499$	−362.924			−0.727303			−0.363652	+	0.931535i	$0.618470\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	180.887i		−	0.359616i	−0.983702	−	0.179808i	$-0.942452\pi$
$503$		−	180.887i		−	0.359616i	0.983702	−	0.179808i	$-0.0575476\pi$
$504$		0			0
$505$			281.131i			0.556695i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−979.030			−1.92344			−0.961719	−	0.274036i	$-0.911641\pi$
$509$	−979.030			−1.92344			−0.961719	+	0.274036i	$0.911641\pi$
$510$		0			0
$511$		−	85.2198i		−	0.166771i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	277.030			0.537923
$516$		0			0
$517$		−	1196.69i		−	2.31467i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			299.224i			0.574326i	0.957882	+	0.287163i	$0.0927121\pi$
$521$			299.224i			0.574326i	−0.957882	+	0.287163i	$0.907288\pi$
$522$		0			0
$523$		−	401.828i		−	0.768313i	−0.923268	−	0.384156i	$-0.874492\pi$
$523$		−	401.828i		−	0.768313i	0.923268	−	0.384156i	$-0.125508\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			948.901i			1.80057i
$528$		0			0
$529$	416.619	−	325.990i	0.787560	−	0.616238i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−730.498			−1.37054
$534$		0			0
$535$	−146.414			−0.273671
$536$		0			0
$537$		0			0
$538$		0			0
$539$			108.541i			0.201375i
$540$		0			0
$541$	−814.502			−1.50555			−0.752775	−	0.658278i	$-0.771285\pi$
$541$	−814.502			−1.50555			−0.752775	+	0.658278i	$0.771285\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	233.551			0.428534
$546$		0			0
$547$	391.919			0.716489			0.358244	−	0.933628i	$-0.383375\pi$
$547$	391.919			0.716489			0.358244	+	0.933628i	$0.383375\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	460.937i		−	0.836546i
$552$		0			0
$553$	−686.396			−1.24122
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	715.529i		−	1.28461i	−0.766448	−	0.642306i	$-0.777978\pi$
$557$		−	715.529i		−	1.28461i	0.766448	−	0.642306i	$-0.222022\pi$
$558$		0			0
$559$			778.405i			1.39249i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			662.966i			1.17756i	0.808294	+	0.588780i	$0.200391\pi$
$563$			662.966i			1.17756i	−0.808294	+	0.588780i	$0.799609\pi$
$564$		0			0
$565$	137.790			0.243877
$566$		0			0
$567$		0			0
$568$		0			0
$569$			34.8235i			0.0612012i	0.999532	+	0.0306006i	$0.00974200\pi$
$569$			34.8235i			0.0612012i	−0.999532	+	0.0306006i	$0.990258\pi$
$570$		0			0
$571$			871.118i			1.52560i	0.646634	+	0.762801i	$0.276176\pi$
$571$			871.118i			1.52560i	−0.646634	+	0.762801i	$0.723824\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−108.721	+	37.4801i	−0.189080	+	0.0651828i
$576$		0			0
$577$	904.475			1.56755			0.783774	−	0.621046i	$-0.213292\pi$
$577$	904.475			1.56755			0.783774	+	0.621046i	$0.213292\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	106.318			0.182991
$582$		0			0
$583$	1632.44			2.80007
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−666.375			−1.13522			−0.567611	−	0.823297i	$-0.692132\pi$
$587$	−666.375			−1.13522			−0.567611	+	0.823297i	$0.692132\pi$
$588$		0			0
$589$		−	1304.90i		−	2.21544i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−529.331			−0.892633			−0.446316	−	0.894875i	$-0.647264\pi$
$593$	−529.331			−0.892633			−0.446316	+	0.894875i	$0.647264\pi$
$594$		0			0
$595$			225.813i			0.379518i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−398.632			−0.665495			−0.332748	−	0.943016i	$-0.607976\pi$
$599$	−398.632			−0.665495			−0.332748	+	0.943016i	$0.607976\pi$
$600$		0			0
$601$	−334.299			−0.556238			−0.278119	−	0.960547i	$-0.589711\pi$
$601$	−334.299			−0.556238			−0.278119	+	0.960547i	$0.589711\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			340.471i			0.562762i
$606$		0			0
$607$	−550.446			−0.906831			−0.453415	−	0.891299i	$-0.649795\pi$
$607$	−550.446			−0.906831			−0.453415	+	0.891299i	$0.649795\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	920.344			1.50629
$612$		0			0
$613$		−	802.435i		−	1.30903i	−0.756050	−	0.654514i	$-0.772873\pi$
$613$		−	802.435i		−	1.30903i	0.756050	−	0.654514i	$-0.227127\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	630.122i		−	1.02127i	−0.859798	−	0.510634i	$-0.829411\pi$
$617$		−	630.122i		−	1.02127i	0.859798	−	0.510634i	$-0.170589\pi$
$618$		0			0
$619$			916.573i			1.48073i	0.672204	+	0.740366i	$0.265348\pi$
$619$			916.573i			1.48073i	−0.672204	+	0.740366i	$0.734652\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−663.662			−1.06527
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−464.524			−0.738512
$630$		0			0
$631$		−	677.089i		−	1.07304i	−0.843887	−	0.536520i	$-0.819738\pi$
$631$		−	677.089i		−	1.07304i	0.843887	−	0.536520i	$-0.180262\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	70.9657i		−	0.111757i
$636$		0			0
$637$	−83.4764			−0.131046
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	322.508i		−	0.503132i	−0.967840	−	0.251566i	$-0.919054\pi$
$641$		−	322.508i		−	0.503132i	0.967840	−	0.251566i	$-0.0809456\pi$
$642$		0			0
$643$			393.766i			0.612388i	0.951969	+	0.306194i	$0.0990557\pi$
$643$			393.766i			0.612388i	−0.951969	+	0.306194i	$0.900944\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1203.70			1.86044			0.930218	−	0.367007i	$-0.119617\pi$
$647$	1203.70			1.86044			0.930218	+	0.367007i	$0.119617\pi$
$648$		0			0
$649$		−	668.318i		−	1.02977i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	393.335			0.602351			0.301176	−	0.953569i	$-0.402621\pi$
$653$	393.335			0.602351			0.301176	+	0.953569i	$0.402621\pi$
$654$		0			0
$655$			140.427i			0.214392i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			714.155i			1.08369i	0.840477	+	0.541847i	$0.182275\pi$
$659$			714.155i			1.08369i	−0.840477	+	0.541847i	$0.817725\pi$
$660$		0			0
$661$			464.968i			0.703431i	0.936107	+	0.351716i	$0.114402\pi$
$661$			464.968i			0.703431i	−0.936107	+	0.351716i	$0.885598\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	310.531i		−	0.466964i
$666$		0			0
$667$	470.134	−	162.072i	0.704849	−	0.242987i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−797.430			−1.18842
$672$		0			0
$673$	−1212.60			−1.80178			−0.900892	−	0.434043i	$-0.857087\pi$
$673$	−1212.60			−1.80178			−0.900892	+	0.434043i	$0.857087\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			310.574i			0.458751i	0.973338	+	0.229375i	$0.0736683\pi$
$677$			310.574i			0.458751i	−0.973338	+	0.229375i	$0.926332\pi$
$678$		0			0
$679$	−965.475			−1.42191
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.11327			−0.0104147			−0.00520737	−	0.999986i	$-0.501658\pi$
$683$	−7.11327			−0.0104147			−0.00520737	+	0.999986i	$0.501658\pi$
$684$		0			0
$685$	−294.371			−0.429738
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1255.47i			1.82216i
$690$		0			0
$691$	−800.467			−1.15842			−0.579209	−	0.815179i	$-0.696639\pi$
$691$	−800.467			−1.15842			−0.579209	+	0.815179i	$0.696639\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			389.852i			0.560938i
$696$		0			0
$697$			890.773i			1.27801i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	798.207i		−	1.13867i	−0.822106	−	0.569334i	$-0.807201\pi$
$701$		−	798.207i		−	1.13867i	0.822106	−	0.569334i	$-0.192799\pi$
$702$		0			0
$703$	638.797			0.908673
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	818.994i		−	1.15841i
$708$		0			0
$709$		−	500.723i		−	0.706239i	−0.935578	−	0.353120i	$-0.885121\pi$
$709$		−	500.723i		−	0.706239i	0.935578	−	0.353120i	$-0.114879\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1330.93	−	458.821i	1.86666	−	0.643508i
$714$		0			0
$715$	−469.933			−0.657249
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1121.08			1.55923			0.779613	−	0.626262i	$-0.215416\pi$
$719$	1121.08			1.55923			0.779613	+	0.626262i	$0.215416\pi$
$720$		0			0
$721$	−807.047			−1.11934
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−108.106			−0.149111
$726$		0			0
$727$			955.980i			1.31497i	0.753469	+	0.657483i	$0.228379\pi$
$727$			955.980i			1.31497i	−0.753469	+	0.657483i	$0.771621\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	949.190			1.29848
$732$		0			0
$733$		−	484.751i		−	0.661325i	−0.943749	−	0.330662i	$-0.892728\pi$
$733$		−	484.751i		−	0.661325i	0.943749	−	0.330662i	$-0.107272\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1172.55			−1.59098
$738$		0			0
$739$	−164.280			−0.222301			−0.111150	−	0.993804i	$-0.535453\pi$
$739$	−164.280			−0.222301			−0.111150	+	0.993804i	$0.535453\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	56.6353i		−	0.0762252i	−0.999273	−	0.0381126i	$-0.987865\pi$
$743$		−	56.6353i		−	0.0762252i	0.999273	−	0.0381126i	$-0.0121345\pi$
$744$		0			0
$745$	257.981			0.346283
$746$		0			0
$747$		0			0
$748$		0			0
$749$	426.535			0.569473
$750$		0			0
$751$			1362.45i			1.81417i	0.420943	+	0.907087i	$0.361699\pi$
$751$			1362.45i			1.81417i	−0.420943	+	0.907087i	$0.638301\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			477.868i			0.632937i
$756$		0			0
$757$		−	386.183i		−	0.510149i	−0.966921	−	0.255074i	$-0.917900\pi$
$757$		−	386.183i		−	0.510149i	0.966921	−	0.255074i	$-0.0821000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	814.442			1.07023			0.535113	−	0.844780i	$-0.320269\pi$
$761$	814.442			1.07023			0.535113	+	0.844780i	$0.320269\pi$
$762$		0			0
$763$	−680.383			−0.891721
$764$		0			0
$765$		0			0
$766$		0			0
$767$	513.988			0.670127
$768$		0			0
$769$		−	137.193i		−	0.178405i	−0.996014	−	0.0892023i	$-0.971568\pi$
$769$		−	137.193i		−	0.178405i	0.996014	−	0.0892023i	$-0.0284318\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			501.606i			0.648908i	0.945901	+	0.324454i	$0.105181\pi$
$773$			501.606i			0.648908i	−0.945901	+	0.324454i	$0.894819\pi$
$774$		0			0
$775$	−306.043			−0.394894
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1224.96i		−	1.57248i
$780$		0			0
$781$			1040.19i			1.33187i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	68.1235			0.0867815
$786$		0			0
$787$			38.9567i			0.0495002i	0.999694	+	0.0247501i	$0.00787901\pi$
$787$			38.9567i			0.0495002i	−0.999694	+	0.0247501i	$0.992121\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−401.412			−0.507475
$792$		0			0
$793$		−	613.285i		−	0.773373i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	489.799i		−	0.614554i	−0.951620	−	0.307277i	$-0.900582\pi$
$797$		−	489.799i		−	0.614554i	0.951620	−	0.307277i	$-0.0994177\pi$
$798$		0			0
$799$		−	1122.27i		−	1.40459i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	216.259i		−	0.269314i
$804$		0			0
$805$	316.727	−	109.187i	0.393450	−	0.135636i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−707.420			−0.874437			−0.437219	−	0.899355i	$-0.644036\pi$
$809$	−707.420			−0.874437			−0.437219	+	0.899355i	$0.644036\pi$
$810$		0			0
$811$	1111.67			1.37074			0.685369	−	0.728196i	$-0.259641\pi$
$811$	1111.67			1.37074			0.685369	+	0.728196i	$0.259641\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			625.691i			0.767719i
$816$		0			0
$817$	−1305.29			−1.59767
$818$		0			0
$819$		0			0
$820$		0			0
$821$	485.620			0.591498			0.295749	−	0.955266i	$-0.404431\pi$
$821$	485.620			0.591498			0.295749	+	0.955266i	$0.404431\pi$
$822$		0			0
$823$	−1142.96			−1.38878			−0.694388	−	0.719601i	$-0.744325\pi$
$823$	−1142.96			−1.38878			−0.694388	+	0.719601i	$0.744325\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1377.32i			1.66545i	0.553690	+	0.832723i	$0.313219\pi$
$827$			1377.32i			1.66545i	−0.553690	+	0.832723i	$0.686781\pi$
$828$		0			0
$829$	−302.774			−0.365228			−0.182614	−	0.983185i	$-0.558456\pi$
$829$	−302.774			−0.365228			−0.182614	+	0.983185i	$0.558456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			101.792i			0.122199i
$834$		0			0
$835$			291.918i			0.349602i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	503.628i		−	0.600272i	−0.953896	−	0.300136i	$-0.902968\pi$
$839$		−	503.628i		−	0.600272i	0.953896	−	0.300136i	$-0.0970320\pi$
$840$		0			0
$841$	−373.527			−0.444146
$842$		0			0
$843$		0			0
$844$		0			0
$845$			16.4808i			0.0195040i
$846$		0			0
$847$		−	991.863i		−	1.17103i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	224.611	+	651.543i	0.263937	+	0.765621i
$852$		0			0
$853$	117.568			0.137829			0.0689143	−	0.997623i	$-0.478046\pi$
$853$	117.568			0.137829			0.0689143	+	0.997623i	$0.478046\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	749.643			0.874730			0.437365	−	0.899284i	$-0.355912\pi$
$857$	749.643			0.874730			0.437365	+	0.899284i	$0.355912\pi$
$858$		0			0
$859$	1423.17			1.65677			0.828387	−	0.560157i	$-0.189259\pi$
$859$	1423.17			1.65677			0.828387	+	0.560157i	$0.189259\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−882.162			−1.02220			−0.511102	−	0.859520i	$-0.670762\pi$
$863$	−882.162			−1.02220			−0.511102	+	0.859520i	$0.670762\pi$
$864$		0			0
$865$		−	207.116i		−	0.239441i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1741.84			−2.00442
$870$		0			0
$871$		−	901.781i		−	1.03534i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−72.8302			−0.0832346
$876$		0			0
$877$	97.5525			0.111234			0.0556172	−	0.998452i	$-0.482287\pi$
$877$	97.5525			0.111234			0.0556172	+	0.998452i	$0.482287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			316.666i			0.359439i	0.983718	+	0.179719i	$0.0575190\pi$
$881$			316.666i			0.359439i	−0.983718	+	0.179719i	$0.942481\pi$
$882$		0			0
$883$	859.931			0.973874			0.486937	−	0.873437i	$-0.338114\pi$
$883$	859.931			0.973874			0.486937	+	0.873437i	$0.338114\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−715.067			−0.806164			−0.403082	−	0.915164i	$-0.632061\pi$
$887$	−715.067			−0.806164			−0.403082	+	0.915164i	$0.632061\pi$
$888$		0			0
$889$			206.738i			0.232551i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1543.31i			1.72823i
$894$		0			0
$895$		−	427.292i		−	0.477421i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1323.40			1.47208
$900$		0			0
$901$	1530.93			1.69914
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−351.134			−0.387993
$906$		0			0
$907$			33.5263i			0.0369639i	0.999829	+	0.0184819i	$0.00588332\pi$
$907$			33.5263i			0.0369639i	−0.999829	+	0.0184819i	$0.994117\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1619.17i		−	1.77736i	−0.458530	−	0.888679i	$-0.651624\pi$
$911$		−	1619.17i		−	1.77736i	0.458530	−	0.888679i	$-0.348376\pi$
$912$		0			0
$913$	269.799			0.295508
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	409.093i		−	0.446121i
$918$		0			0
$919$			1653.87i			1.79964i	0.436265	+	0.899818i	$0.356301\pi$
$919$			1653.87i			1.79964i	−0.436265	+	0.899818i	$0.643699\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−799.987			−0.866725
$924$		0			0
$925$		−	149.820i		−	0.161968i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	269.149			0.289719			0.144859	−	0.989452i	$-0.453727\pi$
$929$	269.149			0.289719			0.144859	+	0.989452i	$0.453727\pi$
$930$		0			0
$931$		−	139.980i		−	0.150355i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			573.038i			0.612875i
$936$		0			0
$937$		−	864.044i		−	0.922139i	−0.887364	−	0.461069i	$-0.847466\pi$
$937$		−	864.044i		−	0.922139i	0.887364	−	0.461069i	$-0.152534\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	437.442i		−	0.464870i	−0.972612	−	0.232435i	$-0.925331\pi$
$941$		−	437.442i		−	0.464870i	0.972612	−	0.232435i	$-0.0746692\pi$
$942$		0			0
$943$	1249.40	−	430.714i	1.32492	−	0.456749i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1455.44			−1.53690			−0.768450	−	0.639910i	$-0.778971\pi$
$947$	−1455.44			−1.53690			−0.768450	+	0.639910i	$0.778971\pi$
$948$		0			0
$949$	166.320			0.175258
$950$		0			0
$951$		0			0
$952$		0			0
$953$			480.588i			0.504290i	0.967689	+	0.252145i	$0.0811360\pi$
$953$			480.588i			0.504290i	−0.967689	+	0.252145i	$0.918864\pi$
$954$		0			0
$955$	404.806			0.423880
$956$		0			0
$957$		0			0
$958$		0			0
$959$	857.564			0.894227
$960$		0			0
$961$	2785.50			2.89854
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	299.718i		−	0.310588i
$966$		0			0
$967$	764.269			0.790350			0.395175	−	0.918606i	$-0.370684\pi$
$967$	764.269			0.790350			0.395175	+	0.918606i	$0.370684\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			844.488i			0.869710i	0.900501	+	0.434855i	$0.143200\pi$
$971$			844.488i			0.869710i	−0.900501	+	0.434855i	$0.856800\pi$
$972$		0			0
$973$		−	1135.72i		−	1.16724i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1491.04i		−	1.52614i	−0.646316	−	0.763070i	$-0.723691\pi$
$977$		−	1491.04i		−	1.52614i	0.646316	−	0.763070i	$-0.276309\pi$
$978$		0			0
$979$	−1684.15			−1.72028
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1751.78i		−	1.78207i	−0.453934	−	0.891035i	$-0.649980\pi$
$983$		−	1751.78i		−	1.78207i	0.453934	−	0.891035i	$-0.350020\pi$
$984$		0			0
$985$		−	331.724i		−	0.336775i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−458.961	−	1331.34i	−0.464066	−	1.34615i
$990$		0			0
$991$	869.484			0.877380			0.438690	−	0.898638i	$-0.355443\pi$
$991$	869.484			0.877380			0.438690	+	0.898638i	$0.355443\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	682.153			0.685581
$996$		0			0
$997$	916.915			0.919674			0.459837	−	0.888003i	$-0.347908\pi$
$997$	916.915			0.919674			0.459837	+	0.888003i	$0.347908\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.b.2161.12	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.28	yes	32
23.22	odd	2	inner	4140.3.d.b.2161.21	yes	32
69.68	even	2	inner	4140.3.d.b.2161.5	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.5	✓	32	69.68	even	2	inner
4140.3.d.b.2161.12	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.21	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.28	yes	32	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +6.51414i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +6.51414i q^{7} +16.5307i q^{11} -12.7134 q^{13} +15.5027i q^{17} -21.3188i q^{19} +(21.7442 - 7.49602i) q^{23} -5.00000 q^{25} +21.6211 q^{29} +61.2086 q^{31} +14.5660 q^{35} +29.9640i q^{37} +57.4591 q^{41} -61.2273i q^{43} -72.3919 q^{47} +6.56604 q^{49} -98.7521i q^{53} +36.9637 q^{55} -40.4289 q^{59} +48.2394i q^{61} +28.4279i q^{65} +70.9318i q^{67} +62.9249 q^{71} -13.0823 q^{73} -107.683 q^{77} +105.370i q^{79} -16.3211i q^{83} +34.6651 q^{85} +101.880i q^{89} -82.8166i q^{91} -47.6703 q^{95} +148.212i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) \) 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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