LMFDB - Embedded newform 4140.2.n.b.2069.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(2069,4140)]

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, 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("4140.2069");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.n (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2069.15
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.b.2069.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.55901 + 1.60296i) q^{5} -1.21571 q^{7} +O(q^{10})$	$q+(-1.55901 + 1.60296i) q^{5} -1.21571 q^{7} +2.73752 q^{11} +6.99737i q^{13} +0.391222i q^{17} -2.94625i q^{19} +(3.74030 + 3.00169i) q^{23} +(-0.138992 - 4.99807i) q^{25} -3.30404i q^{29} +7.96077 q^{31} +(1.89530 - 1.94874i) q^{35} -1.84588 q^{37} +9.35402i q^{41} +4.89960 q^{43} +2.87774 q^{47} -5.52205 q^{49} +2.95606i q^{53} +(-4.26782 + 4.38815i) q^{55} +8.16713i q^{59} -12.9163i q^{61} +(-11.2165 - 10.9089i) q^{65} -10.5479 q^{67} +14.5479i q^{71} -1.70779i q^{73} -3.32804 q^{77} -11.9911i q^{79} +12.9935i q^{83} +(-0.627114 - 0.609917i) q^{85} -15.1484 q^{89} -8.50678i q^{91} +(4.72274 + 4.59323i) q^{95} +13.6909 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) $ 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * 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
$5$	−1.55901	+	1.60296i	−0.697209	+	0.716868i
$5$	−1.55901	+	1.60296i	−0.697209	+	0.716868i
$6$		0			0
$7$	−1.21571			−0.459496			−0.229748	−	0.973250i	$-0.573790\pi$
$7$	−1.21571			−0.459496			−0.229748	+	0.973250i	$0.573790\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.73752			0.825394			0.412697	−	0.910868i	$-0.364587\pi$
$11$	2.73752			0.825394			0.412697	+	0.910868i	$0.364587\pi$
$12$		0			0
$13$			6.99737i			1.94072i	0.241663	+	0.970360i	$0.422307\pi$
$13$			6.99737i			1.94072i	−0.241663	+	0.970360i	$0.577693\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.391222i			0.0948852i	0.998874	+	0.0474426i	$0.0151071\pi$
$17$			0.391222i			0.0948852i	−0.998874	+	0.0474426i	$0.984893\pi$
$18$		0			0
$19$		−	2.94625i		−	0.675917i	−0.941161	−	0.337958i	$-0.890264\pi$
$19$		−	2.94625i		−	0.675917i	0.941161	−	0.337958i	$-0.109736\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.74030	+	3.00169i	0.779906	+	0.625896i
$23$	3.74030	+	3.00169i	0.779906	+	0.625896i
$24$		0			0
$25$	−0.138992	−	4.99807i	−0.0277984	−	0.999614i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.30404i		−	0.613545i	−0.951783	−	0.306772i	$-0.900751\pi$
$29$		−	3.30404i		−	0.613545i	0.951783	−	0.306772i	$-0.0992490\pi$
$30$		0			0
$31$	7.96077			1.42980			0.714898	−	0.699229i	$-0.246473\pi$
$31$	7.96077			1.42980			0.714898	+	0.699229i	$0.246473\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.89530	−	1.94874i	0.320365	−	0.329398i
$36$		0			0
$37$	−1.84588			−0.303461			−0.151731	−	0.988422i	$-0.548485\pi$
$37$	−1.84588			−0.303461			−0.151731	+	0.988422i	$0.548485\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.35402i			1.46085i	0.682991	+	0.730426i	$0.260679\pi$
$41$			9.35402i			1.46085i	−0.682991	+	0.730426i	$0.739321\pi$
$42$		0			0
$43$	4.89960			0.747182			0.373591	−	0.927593i	$-0.378126\pi$
$43$	4.89960			0.747182			0.373591	+	0.927593i	$0.378126\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.87774			0.419762			0.209881	−	0.977727i	$-0.432692\pi$
$47$	2.87774			0.419762			0.209881	+	0.977727i	$0.432692\pi$
$48$		0			0
$49$	−5.52205			−0.788864
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.95606i			0.406046i	0.979174	+	0.203023i	$0.0650766\pi$
$53$			2.95606i			0.406046i	−0.979174	+	0.203023i	$0.934923\pi$
$54$		0			0
$55$	−4.26782	+	4.38815i	−0.575472	+	0.591698i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.16713i			1.06327i	0.846974	+	0.531635i	$0.178422\pi$
$59$			8.16713i			1.06327i	−0.846974	+	0.531635i	$0.821578\pi$
$60$		0			0
$61$		−	12.9163i		−	1.65377i	−0.562375	−	0.826883i	$-0.690112\pi$
$61$		−	12.9163i		−	1.65377i	0.562375	−	0.826883i	$-0.309888\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−11.2165	−	10.9089i	−1.39124	−	1.35309i
$66$		0			0
$67$	−10.5479			−1.28863			−0.644315	−	0.764760i	$-0.722857\pi$
$67$	−10.5479			−1.28863			−0.644315	+	0.764760i	$0.722857\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.5479i			1.72652i	0.504762	+	0.863258i	$0.331580\pi$
$71$			14.5479i			1.72652i	−0.504762	+	0.863258i	$0.668420\pi$
$72$		0			0
$73$		−	1.70779i		−	0.199881i	−0.994993	−	0.0999406i	$-0.968135\pi$
$73$		−	1.70779i		−	0.199881i	0.994993	−	0.0999406i	$-0.0318653\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.32804			−0.379265
$78$		0			0
$79$		−	11.9911i		−	1.34911i	−0.738226	−	0.674554i	$-0.764336\pi$
$79$		−	11.9911i		−	1.34911i	0.738226	−	0.674554i	$-0.235664\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.9935i			1.42622i	0.701054	+	0.713108i	$0.252713\pi$
$83$			12.9935i			1.42622i	−0.701054	+	0.713108i	$0.747287\pi$
$84$		0			0
$85$	−0.627114	−	0.609917i	−0.0680201	−	0.0661548i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.1484			−1.60573			−0.802865	−	0.596161i	$-0.796692\pi$
$89$	−15.1484			−1.60573			−0.802865	+	0.596161i	$0.796692\pi$
$90$		0			0
$91$		−	8.50678i		−	0.891753i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.72274	+	4.59323i	0.484543	+	0.471255i
$96$		0			0
$97$	13.6909			1.39010			0.695049	−	0.718963i	$-0.255383\pi$
$97$	13.6909			1.39010			0.695049	+	0.718963i	$0.255383\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.21591i		−	0.718009i	−0.933336	−	0.359005i	$-0.883116\pi$
$101$		−	7.21591i		−	0.718009i	0.933336	−	0.359005i	$-0.116884\pi$
$102$		0			0
$103$	−18.9655			−1.86873			−0.934365	−	0.356316i	$-0.884033\pi$
$103$	−18.9655			−1.86873			−0.934365	+	0.356316i	$0.884033\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.17749i		−	0.403853i	−0.979401	−	0.201927i	$-0.935280\pi$
$107$		−	4.17749i		−	0.403853i	0.979401	−	0.201927i	$-0.0647203\pi$
$108$		0			0
$109$			12.3782i			1.18562i	0.805343	+	0.592809i	$0.201981\pi$
$109$			12.3782i			1.18562i	−0.805343	+	0.592809i	$0.798019\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.88175i		−	0.365165i	−0.983191	−	0.182582i	$-0.941554\pi$
$113$		−	3.88175i		−	0.365165i	0.983191	−	0.182582i	$-0.0584456\pi$
$114$		0			0
$115$	−10.6428	+	1.31590i	−0.992443	+	0.122709i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.475613i		−	0.0435993i
$120$		0			0
$121$	−3.50598			−0.318725
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.22842	+	7.56923i	0.735972	+	0.677012i
$126$		0			0
$127$			7.06851i			0.627229i	0.949550	+	0.313615i	$0.101540\pi$
$127$			7.06851i			0.627229i	−0.949550	+	0.313615i	$0.898460\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			1.98669i			0.173578i	0.996227	+	0.0867889i	$0.0276606\pi$
$131$			1.98669i			0.173578i	−0.996227	+	0.0867889i	$0.972339\pi$
$132$		0			0
$133$			3.58179i			0.310581i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			13.4934i			1.15282i	0.817160	+	0.576411i	$0.195547\pi$
$137$			13.4934i			1.15282i	−0.817160	+	0.576411i	$0.804453\pi$
$138$		0			0
$139$	2.96667			0.251630			0.125815	−	0.992054i	$-0.459845\pi$
$139$	2.96667			0.251630			0.125815	+	0.992054i	$0.459845\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			19.1554i			1.60186i
$144$		0			0
$145$	5.29626	+	5.15102i	0.439830	+	0.427769i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.85554			0.479704			0.239852	−	0.970809i	$-0.422901\pi$
$149$	5.85554			0.479704			0.239852	+	0.970809i	$0.422901\pi$
$150$		0			0
$151$	1.14041			0.0928049			0.0464024	−	0.998923i	$-0.485224\pi$
$151$	1.14041			0.0928049			0.0464024	+	0.998923i	$0.485224\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.4109	+	12.7608i	−0.996867	+	1.02497i
$156$		0			0
$157$	−16.9077			−1.34938			−0.674692	−	0.738099i	$-0.735724\pi$
$157$	−16.9077			−1.34938			−0.674692	+	0.738099i	$0.735724\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.54712	−	3.64919i	−0.358364	−	0.287597i
$162$		0			0
$163$			9.28818i			0.727506i	0.931495	+	0.363753i	$0.118505\pi$
$163$			9.28818i			0.727506i	−0.931495	+	0.363753i	$0.881495\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16.5641			1.28177			0.640885	−	0.767637i	$-0.278567\pi$
$167$	16.5641			1.28177			0.640885	+	0.767637i	$0.278567\pi$
$168$		0			0
$169$	−35.9631			−2.76640
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.5167			1.40780			0.703900	−	0.710299i	$-0.251440\pi$
$173$	18.5167			1.40780			0.703900	+	0.710299i	$0.251440\pi$
$174$		0			0
$175$	0.168974	+	6.07621i	0.0127732	+	0.459318i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	8.89279i		−	0.664678i	−0.943160	−	0.332339i	$-0.892162\pi$
$179$		−	8.89279i		−	0.664678i	0.943160	−	0.332339i	$-0.107838\pi$
$180$		0			0
$181$			21.4231i			1.59237i	0.605056	+	0.796183i	$0.293151\pi$
$181$			21.4231i			1.59237i	−0.605056	+	0.796183i	$0.706849\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.87774	−	2.95889i	0.211576	−	0.217542i
$186$		0			0
$187$			1.07098i			0.0783176i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−23.8622			−1.72661			−0.863303	−	0.504686i	$-0.831608\pi$
$191$	−23.8622			−1.72661			−0.863303	+	0.504686i	$0.831608\pi$
$192$		0			0
$193$			10.6503i			0.766626i	0.923619	+	0.383313i	$0.125217\pi$
$193$			10.6503i			0.766626i	−0.923619	+	0.383313i	$0.874783\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.9273			1.27727			0.638633	−	0.769512i	$-0.279500\pi$
$197$	17.9273			1.27727			0.638633	+	0.769512i	$0.279500\pi$
$198$		0			0
$199$		−	20.2721i		−	1.43705i	−0.695500	−	0.718526i	$-0.744817\pi$
$199$		−	20.2721i		−	1.43705i	0.695500	−	0.718526i	$-0.255183\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			4.01676i			0.281921i
$204$		0			0
$205$	−14.9942	−	14.5830i	−1.04724	−	1.01852i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	8.06543i		−	0.557898i
$210$		0			0
$211$	−19.9786			−1.37539			−0.687693	−	0.726002i	$-0.741376\pi$
$211$	−19.9786			−1.37539			−0.687693	+	0.726002i	$0.741376\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.63851	+	7.85389i	−0.520942	+	0.535631i
$216$		0			0
$217$	−9.67800			−0.656985
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.73752			−0.184146
$222$		0			0
$223$			14.6489i			0.980963i	0.871451	+	0.490482i	$0.163179\pi$
$223$			14.6489i			0.980963i	−0.871451	+	0.490482i	$0.836821\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			17.4187i			1.15612i	0.815995	+	0.578059i	$0.196190\pi$
$227$			17.4187i			1.15612i	−0.815995	+	0.578059i	$0.803810\pi$
$228$		0			0
$229$			8.28098i			0.547222i	0.961840	+	0.273611i	$0.0882182\pi$
$229$			8.28098i			0.547222i	−0.961840	+	0.273611i	$0.911782\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−27.7224			−1.81616			−0.908078	−	0.418800i	$-0.862451\pi$
$233$	−27.7224			−1.81616			−0.908078	+	0.418800i	$0.862451\pi$
$234$		0			0
$235$	−4.48643	+	4.61292i	−0.292662	+	0.300914i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			25.5089i			1.65004i	0.565107	+	0.825018i	$0.308835\pi$
$239$			25.5089i			1.65004i	−0.565107	+	0.825018i	$0.691165\pi$
$240$		0			0
$241$		−	13.4544i		−	0.866675i	−0.901232	−	0.433338i	$-0.857336\pi$
$241$		−	13.4544i		−	0.866675i	0.901232	−	0.433338i	$-0.142664\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.60891	−	8.85164i	0.550003	−	0.565511i
$246$		0			0
$247$	20.6160			1.31177
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.7914			−0.807385			−0.403693	−	0.914895i	$-0.632274\pi$
$251$	−12.7914			−0.807385			−0.403693	+	0.914895i	$0.632274\pi$
$252$		0			0
$253$	10.2391	+	8.21720i	0.643730	+	0.516611i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.361953			0.0225780			0.0112890	−	0.999936i	$-0.496407\pi$
$257$	0.361953			0.0225780			0.0112890	+	0.999936i	$0.496407\pi$
$258$		0			0
$259$	2.24406			0.139439
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.5816i		−	1.02247i	−0.859442	−	0.511233i	$-0.829189\pi$
$263$		−	16.5816i		−	1.02247i	0.859442	−	0.511233i	$-0.170811\pi$
$264$		0			0
$265$	−4.73846	−	4.60852i	−0.291081	−	0.283099i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.28389i		−	0.383136i	−0.981479	−	0.191568i	$-0.938643\pi$
$269$		−	6.28389i		−	0.383136i	0.981479	−	0.191568i	$-0.0613572\pi$
$270$		0			0
$271$	−5.63929			−0.342562			−0.171281	−	0.985222i	$-0.554791\pi$
$271$	−5.63929			−0.342562			−0.171281	+	0.985222i	$0.554791\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.380493	−	13.6823i	−0.0229446	−	0.825075i
$276$		0			0
$277$		−	6.50942i		−	0.391113i	−0.980692	−	0.195556i	$-0.937349\pi$
$277$		−	6.50942i		−	0.391113i	0.980692	−	0.195556i	$-0.0626513\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−26.2192			−1.56411			−0.782053	−	0.623212i	$-0.785828\pi$
$281$	−26.2192			−1.56411			−0.782053	+	0.623212i	$0.785828\pi$
$282$		0			0
$283$	17.6516			1.04928			0.524638	−	0.851325i	$-0.324201\pi$
$283$	17.6516			1.04928			0.524638	+	0.851325i	$0.324201\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	11.3718i		−	0.671256i
$288$		0			0
$289$	16.8469			0.990997
$290$		0			0
$291$		0			0
$292$		0			0
$293$			18.9531i			1.10725i	0.832765	+	0.553626i	$0.186756\pi$
$293$			18.9531i			1.10725i	−0.832765	+	0.553626i	$0.813244\pi$
$294$		0			0
$295$	−13.0916	−	12.7326i	−0.762224	−	0.741322i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−21.0040	+	26.1722i	−1.21469	+	1.51358i
$300$		0			0
$301$	−5.95650			−0.343327
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.7044	+	20.1366i	1.18553	+	1.15302i
$306$		0			0
$307$		−	25.7698i		−	1.47076i	−0.677655	−	0.735380i	$-0.737004\pi$
$307$		−	25.7698i		−	1.47076i	0.677655	−	0.735380i	$-0.262996\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.09108i		−	0.345394i	−0.984975	−	0.172697i	$-0.944752\pi$
$311$		−	6.09108i		−	0.345394i	0.984975	−	0.172697i	$-0.0552481\pi$
$312$		0			0
$313$	1.62159			0.0916577			0.0458288	−	0.998949i	$-0.485407\pi$
$313$	1.62159			0.0916577			0.0458288	+	0.998949i	$0.485407\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.39354			−0.302931			−0.151466	−	0.988463i	$-0.548399\pi$
$317$	−5.39354			−0.302931			−0.151466	+	0.988463i	$0.548399\pi$
$318$		0			0
$319$		−	9.04488i		−	0.506416i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.15264			0.0641345
$324$		0			0
$325$	34.9733	−	0.972577i	1.93997	−	0.0539489i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.49851			−0.192879
$330$		0			0
$331$	−18.6809			−1.02679			−0.513397	−	0.858151i	$-0.671613\pi$
$331$	−18.6809			−1.02679			−0.513397	+	0.858151i	$0.671613\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.4442	−	16.9079i	0.898445	−	0.923777i
$336$		0			0
$337$	−20.4134			−1.11199			−0.555995	−	0.831186i	$-0.687663\pi$
$337$	−20.4134			−1.11199			−0.555995	+	0.831186i	$0.687663\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	21.7928			1.18015
$342$		0			0
$343$	15.2232			0.821975
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.7612			−0.685059			−0.342530	−	0.939507i	$-0.611284\pi$
$347$	−12.7612			−0.685059			−0.342530	+	0.939507i	$0.611284\pi$
$348$		0			0
$349$	19.1162			1.02327			0.511633	−	0.859204i	$-0.329041\pi$
$349$	19.1162			1.02327			0.511633	+	0.859204i	$0.329041\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.8401			−1.00276			−0.501380	−	0.865227i	$-0.667174\pi$
$353$	−18.8401			−1.00276			−0.501380	+	0.865227i	$0.667174\pi$
$354$		0			0
$355$	−23.3198	−	22.6803i	−1.23768	−	1.20374i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.2899			−0.859748			−0.429874	−	0.902889i	$-0.641442\pi$
$359$	−16.2899			−0.859748			−0.429874	+	0.902889i	$0.641442\pi$
$360$		0			0
$361$	10.3196			0.543136
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.73752	+	2.66245i	0.143288	+	0.139359i
$366$		0			0
$367$	−26.8454			−1.40132			−0.700658	−	0.713497i	$-0.747110\pi$
$367$	−26.8454			−1.40132			−0.700658	+	0.713497i	$0.747110\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	3.59371i		−	0.186576i
$372$		0			0
$373$	30.1882			1.56309			0.781544	−	0.623850i	$-0.214433\pi$
$373$	30.1882			1.56309			0.781544	+	0.623850i	$0.214433\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	23.1196			1.19072
$378$		0			0
$379$		−	31.8864i		−	1.63790i	−0.573868	−	0.818948i	$-0.694558\pi$
$379$		−	31.8864i		−	1.63790i	0.573868	−	0.818948i	$-0.305442\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			20.8594i			1.06586i	0.846158	+	0.532932i	$0.178910\pi$
$383$			20.8594i			1.06586i	−0.846158	+	0.532932i	$0.821090\pi$
$384$		0			0
$385$	5.18843	−	5.33473i	0.264427	−	0.271883i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.9398			1.41660			0.708302	−	0.705909i	$-0.249461\pi$
$389$	27.9398			1.41660			0.708302	+	0.705909i	$0.249461\pi$
$390$		0			0
$391$	−1.17433	+	1.46329i	−0.0593883	+	0.0740015i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	19.2214	+	18.6943i	0.967132	+	0.940611i
$396$		0			0
$397$		−	0.0173642i		−	0.000871486i	−1.00000		0.000435743i	$-0.999861\pi$
$397$		−	0.0173642i		−	0.000871486i	1.00000		0.000435743i	$-0.000138701\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.6179			0.779921			0.389961	−	0.920832i	$-0.372489\pi$
$401$	15.6179			0.779921			0.389961	+	0.920832i	$0.372489\pi$
$402$		0			0
$403$			55.7044i			2.77484i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.05314			−0.250475
$408$		0			0
$409$	−15.5255			−0.767685			−0.383842	−	0.923399i	$-0.625399\pi$
$409$	−15.5255			−0.767685			−0.383842	+	0.923399i	$0.625399\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	9.92888i		−	0.488568i
$414$		0			0
$415$	−20.8281	−	20.2569i	−1.02241	−	0.994372i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.4486			0.705860			0.352930	−	0.935650i	$-0.385185\pi$
$419$	14.4486			0.705860			0.352930	+	0.935650i	$0.385185\pi$
$420$		0			0
$421$			15.3048i			0.745910i	0.927849	+	0.372955i	$0.121655\pi$
$421$			15.3048i			0.745910i	−0.927849	+	0.372955i	$0.878345\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.95535	−	0.0543766i	0.0948485	−	0.00263765i
$426$		0			0
$427$			15.7025i			0.759898i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.09723			−0.101020			−0.0505101	−	0.998724i	$-0.516085\pi$
$431$	−2.09723			−0.101020			−0.0505101	+	0.998724i	$0.516085\pi$
$432$		0			0
$433$	−14.8785			−0.715014			−0.357507	−	0.933911i	$-0.616373\pi$
$433$	−14.8785			−0.715014			−0.357507	+	0.933911i	$0.616373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.84375	−	11.0199i	0.423054	−	0.527152i
$438$		0			0
$439$	21.0823			1.00620			0.503101	−	0.864228i	$-0.332192\pi$
$439$	21.0823			1.00620			0.503101	+	0.864228i	$0.332192\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.3629			−0.777424			−0.388712	−	0.921359i	$-0.627080\pi$
$443$	−16.3629			−0.777424			−0.388712	+	0.921359i	$0.627080\pi$
$444$		0			0
$445$	23.6165	−	24.2824i	1.11953	−	1.15110i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	15.7204i		−	0.741892i	−0.928655	−	0.370946i	$-0.879034\pi$
$449$		−	15.7204i		−	0.741892i	0.928655	−	0.370946i	$-0.120966\pi$
$450$		0			0
$451$			25.6068i			1.20578i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.6361	+	13.2621i	0.639269	+	0.621738i
$456$		0			0
$457$	−1.13286			−0.0529929			−0.0264964	−	0.999649i	$-0.508435\pi$
$457$	−1.13286			−0.0529929			−0.0264964	+	0.999649i	$0.508435\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.8726i		−	0.552964i	−0.961019	−	0.276482i	$-0.910831\pi$
$461$		−	11.8726i		−	0.552964i	0.961019	−	0.276482i	$-0.0891686\pi$
$462$		0			0
$463$		−	25.1156i		−	1.16722i	−0.812033	−	0.583611i	$-0.801639\pi$
$463$		−	25.1156i		−	1.16722i	0.812033	−	0.583611i	$-0.198361\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			30.6360i			1.41766i	0.705377	+	0.708832i	$0.250778\pi$
$467$			30.6360i			1.41766i	−0.705377	+	0.708832i	$0.749222\pi$
$468$		0			0
$469$	12.8232			0.592120
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.4128			0.616720
$474$		0			0
$475$	−14.7256	+	0.409505i	−0.675656	+	0.0187894i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.0763			0.734547			0.367273	−	0.930113i	$-0.380291\pi$
$479$	16.0763			0.734547			0.367273	+	0.930113i	$0.380291\pi$
$480$		0			0
$481$		−	12.9163i		−	0.588933i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−21.3442	+	21.9460i	−0.969189	+	0.996516i
$486$		0			0
$487$			2.99877i			0.135887i	0.997689	+	0.0679436i	$0.0216438\pi$
$487$			2.99877i			0.135887i	−0.997689	+	0.0679436i	$0.978356\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.4568i			0.652425i	0.945296	+	0.326213i	$0.105773\pi$
$491$			14.4568i			0.652425i	−0.945296	+	0.326213i	$0.894227\pi$
$492$		0			0
$493$	1.29261			0.0582163
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	17.6860i		−	0.793327i
$498$		0			0
$499$	2.94634			0.131896			0.0659480	−	0.997823i	$-0.478993\pi$
$499$	2.94634			0.131896			0.0659480	+	0.997823i	$0.478993\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			1.06093i			0.0473047i	0.999720	+	0.0236523i	$0.00752948\pi$
$503$			1.06093i			0.0473047i	−0.999720	+	0.0236523i	$0.992471\pi$
$504$		0			0
$505$	11.5668	+	11.2496i	0.514718	+	0.500603i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.5277i			0.599606i	0.954001	+	0.299803i	$0.0969209\pi$
$509$			13.5277i			0.599606i	−0.954001	+	0.299803i	$0.903079\pi$
$510$		0			0
$511$			2.07618i			0.0918446i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	29.5674	−	30.4011i	1.30290	−	1.33963i
$516$		0			0
$517$	7.87789			0.346469
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.8579			1.87764			0.938819	−	0.344410i	$-0.111921\pi$
$521$	42.8579			1.87764			0.938819	+	0.344410i	$0.111921\pi$
$522$		0			0
$523$	31.9651			1.39774			0.698868	−	0.715251i	$-0.253687\pi$
$523$	31.9651			1.39774			0.698868	+	0.715251i	$0.253687\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			3.11443i			0.135666i
$528$		0			0
$529$	4.97966	+	22.4545i	0.216507	+	0.976281i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−65.4535			−2.83511
$534$		0			0
$535$	6.69637	+	6.51274i	0.289509	+	0.281570i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.1167			−0.651123
$540$		0			0
$541$	8.48281			0.364705			0.182352	−	0.983233i	$-0.441629\pi$
$541$	8.48281			0.364705			0.182352	+	0.983233i	$0.441629\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−19.8419	−	19.2977i	−0.849932	−	0.826624i
$546$		0			0
$547$			18.2068i			0.778465i	0.921139	+	0.389233i	$0.127260\pi$
$547$			18.2068i			0.778465i	−0.921139	+	0.389233i	$0.872740\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.73454			−0.414705
$552$		0			0
$553$			14.5778i			0.619909i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.758851i		−	0.0321536i	−0.999871	−	0.0160768i	$-0.994882\pi$
$557$		−	0.758851i		−	0.0321536i	0.999871	−	0.0160768i	$-0.00511762\pi$
$558$		0			0
$559$			34.2843i			1.45007i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			14.3844i			0.606230i	0.952954	+	0.303115i	$0.0980266\pi$
$563$			14.3844i			0.606230i	−0.952954	+	0.303115i	$0.901973\pi$
$564$		0			0
$565$	6.22231	+	6.05168i	0.261775	+	0.254596i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.59604			−0.0669095			−0.0334548	−	0.999440i	$-0.510651\pi$
$569$	−1.59604			−0.0669095			−0.0334548	+	0.999440i	$0.510651\pi$
$570$		0			0
$571$			0.106176i			0.00444333i	0.999998	+	0.00222167i	$0.000707179\pi$
$571$			0.106176i			0.00444333i	−0.999998	+	0.00222167i	$0.999293\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.4828	−	19.1115i	0.603974	−	0.797004i
$576$		0			0
$577$		−	34.0755i		−	1.41858i	−0.704916	−	0.709291i	$-0.749015\pi$
$577$		−	34.0755i		−	1.41858i	0.704916	−	0.709291i	$-0.250985\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	15.7963i		−	0.655341i
$582$		0			0
$583$			8.09227i			0.335148i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.7927			0.858208			0.429104	−	0.903255i	$-0.358829\pi$
$587$	20.7927			0.858208			0.429104	+	0.903255i	$0.358829\pi$
$588$		0			0
$589$		−	23.4544i		−	0.966423i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−30.2505			−1.24224			−0.621120	−	0.783716i	$-0.713322\pi$
$593$	−30.2505			−1.24224			−0.621120	+	0.783716i	$0.713322\pi$
$594$		0			0
$595$	0.762390	+	0.741484i	0.0312550	+	0.0303979i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			0.878003i			0.0358742i	0.999839	+	0.0179371i	$0.00570986\pi$
$599$			0.878003i			0.0358742i	−0.999839	+	0.0179371i	$0.994290\pi$
$600$		0			0
$601$	−22.5102			−0.918212			−0.459106	−	0.888382i	$-0.651830\pi$
$601$	−22.5102			−0.918212			−0.459106	+	0.888382i	$0.651830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.46584	−	5.61996i	0.222218	−	0.228484i
$606$		0			0
$607$			19.5755i			0.794544i	0.917701	+	0.397272i	$0.130043\pi$
$607$			19.5755i			0.794544i	−0.917701	+	0.397272i	$0.869957\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			20.1366i			0.814641i
$612$		0			0
$613$	19.4086			0.783904			0.391952	−	0.919986i	$-0.371800\pi$
$613$	19.4086			0.783904			0.391952	+	0.919986i	$0.371800\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	39.9822i		−	1.60962i	−0.593530	−	0.804812i	$-0.702266\pi$
$617$		−	39.9822i		−	1.60962i	0.593530	−	0.804812i	$-0.297734\pi$
$618$		0			0
$619$			29.8654i			1.20039i	0.799854	+	0.600195i	$0.204910\pi$
$619$			29.8654i			1.20039i	−0.799854	+	0.600195i	$0.795090\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.4161			0.737826
$624$		0			0
$625$	−24.9614	+	1.38938i	−0.998454	+	0.0555753i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.722149i		−	0.0287940i
$630$		0			0
$631$		−	7.18510i		−	0.286034i	−0.989720	−	0.143017i	$-0.954320\pi$
$631$		−	7.18510i		−	0.286034i	0.989720	−	0.143017i	$-0.0456804\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.3306	−	11.0199i	−0.449640	−	0.437310i
$636$		0			0
$637$		−	38.6398i		−	1.53096i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.3137			0.407365			0.203683	−	0.979037i	$-0.434709\pi$
$641$	10.3137			0.407365			0.203683	+	0.979037i	$0.434709\pi$
$642$		0			0
$643$	−24.4473			−0.964107			−0.482054	−	0.876142i	$-0.660109\pi$
$643$	−24.4473			−0.964107			−0.482054	+	0.876142i	$0.660109\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.3595			−0.446590			−0.223295	−	0.974751i	$-0.571681\pi$
$647$	−11.3595			−0.446590			−0.223295	+	0.974751i	$0.571681\pi$
$648$		0			0
$649$			22.3577i			0.877616i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.7299			−1.43735			−0.718676	−	0.695345i	$-0.755252\pi$
$653$	−36.7299			−1.43735			−0.718676	+	0.695345i	$0.755252\pi$
$654$		0			0
$655$	−3.18459	−	3.09726i	−0.124432	−	0.121020i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.22349			−0.281387			−0.140694	−	0.990053i	$-0.544933\pi$
$659$	−7.22349			−0.281387			−0.140694	+	0.990053i	$0.544933\pi$
$660$		0			0
$661$		−	35.1450i		−	1.36698i	−0.729959	−	0.683491i	$-0.760461\pi$
$661$		−	35.1450i		−	1.36698i	0.729959	−	0.683491i	$-0.239539\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.74149	−	5.58404i	−0.222645	−	0.216540i
$666$		0			0
$667$	9.91772	−	12.3581i	0.384016	−	0.478507i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	35.3587i		−	1.36501i
$672$		0			0
$673$		−	3.90352i		−	0.150470i	−0.997166	−	0.0752349i	$-0.976029\pi$
$673$		−	3.90352i		−	0.150470i	0.997166	−	0.0752349i	$-0.0239707\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	43.7055i		−	1.67974i	−0.542787	−	0.839870i	$-0.682631\pi$
$677$		−	43.7055i		−	1.67974i	0.542787	−	0.839870i	$-0.317369\pi$
$678$		0			0
$679$	−16.6441			−0.638744
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.323411			−0.0123750			−0.00618750	−	0.999981i	$-0.501970\pi$
$683$	−0.323411			−0.0123750			−0.00618750	+	0.999981i	$0.501970\pi$
$684$		0			0
$685$	−21.6295	−	21.0364i	−0.826420	−	0.803758i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−20.6846			−0.788021
$690$		0			0
$691$	38.1535			1.45143			0.725715	−	0.687996i	$-0.241509\pi$
$691$	38.1535			1.45143			0.725715	+	0.687996i	$0.241509\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.62506	+	4.75547i	−0.175439	+	0.180385i
$696$		0			0
$697$	−3.65950			−0.138613
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.16633			0.308438			0.154219	−	0.988037i	$-0.450714\pi$
$701$	8.16633			0.308438			0.154219	+	0.988037i	$0.450714\pi$
$702$		0			0
$703$			5.43844i			0.205115i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			8.77246i			0.329922i
$708$		0			0
$709$		−	6.35439i		−	0.238644i	−0.992856	−	0.119322i	$-0.961928\pi$
$709$		−	6.35439i		−	0.238644i	0.992856	−	0.119322i	$-0.0380721\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29.7757	+	23.8958i	1.11511	+	0.894905i
$714$		0			0
$715$	−30.7055	−	29.8635i	−1.14832	−	1.11683i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	35.2797i		−	1.31571i	−0.753144	−	0.657856i	$-0.771464\pi$
$719$		−	35.2797i		−	1.31571i	0.753144	−	0.657856i	$-0.228536\pi$
$720$		0			0
$721$	23.0566			0.858674
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.5138	+	0.459235i	−0.613308	+	0.0170556i
$726$		0			0
$727$	−2.32193			−0.0861154			−0.0430577	−	0.999073i	$-0.513710\pi$
$727$	−2.32193			−0.0861154			−0.0430577	+	0.999073i	$0.513710\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1.91683i			0.0708965i
$732$		0			0
$733$	48.5901			1.79472			0.897358	−	0.441304i	$-0.145484\pi$
$733$	48.5901			1.79472			0.897358	+	0.441304i	$0.145484\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−28.8751			−1.06363
$738$		0			0
$739$	−26.8179			−0.986512			−0.493256	−	0.869884i	$-0.664193\pi$
$739$	−26.8179			−0.986512			−0.493256	+	0.869884i	$0.664193\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.33417i		−	0.0489458i	−0.999700	−	0.0244729i	$-0.992209\pi$
$743$		−	1.33417i		−	0.0489458i	0.999700	−	0.0244729i	$-0.00779074\pi$
$744$		0			0
$745$	−9.12882	+	9.38622i	−0.334454	+	0.343884i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			5.07863i			0.185569i
$750$		0			0
$751$			16.4951i			0.601916i	0.953637	+	0.300958i	$0.0973065\pi$
$751$			16.4951i			0.601916i	−0.953637	+	0.300958i	$0.902694\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.77790	+	1.82803i	−0.0647044	+	0.0665288i
$756$		0			0
$757$	9.93942			0.361254			0.180627	−	0.983552i	$-0.442187\pi$
$757$	9.93942			0.361254			0.180627	+	0.983552i	$0.442187\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			15.4267i			0.559219i	0.960114	+	0.279609i	$0.0902049\pi$
$761$			15.4267i			0.559219i	−0.960114	+	0.279609i	$0.909795\pi$
$762$		0			0
$763$		−	15.0483i		−	0.544787i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−57.1484			−2.06351
$768$		0			0
$769$			28.4021i			1.02421i	0.858924	+	0.512103i	$0.171133\pi$
$769$			28.4021i			1.02421i	−0.858924	+	0.512103i	$0.828867\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	44.4726i		−	1.59957i	−0.600287	−	0.799785i	$-0.704947\pi$
$773$		−	44.4726i		−	1.59957i	0.600287	−	0.799785i	$-0.295053\pi$
$774$		0			0
$775$	−1.10648	−	39.7885i	−0.0397460	−	1.42924i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.5593			0.987415
$780$		0			0
$781$			39.8252i			1.42506i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	26.3593	−	27.1025i	0.940803	−	0.967330i
$786$		0			0
$787$	−22.4314			−0.799594			−0.399797	−	0.916604i	$-0.630919\pi$
$787$	−22.4314			−0.799594			−0.399797	+	0.916604i	$0.630919\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.71909i			0.167792i
$792$		0			0
$793$	90.3802			3.20950
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.18044i		−	0.112657i	−0.998412	−	0.0563284i	$-0.982061\pi$
$797$		−	3.18044i		−	0.112657i	0.998412	−	0.0563284i	$-0.0179394\pi$
$798$		0			0
$799$			1.12584i			0.0398292i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.67510i		−	0.164981i
$804$		0			0
$805$	12.9385	−	1.59976i	0.456023	−	0.0563841i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	11.3367i		−	0.398578i	−0.979941	−	0.199289i	$-0.936137\pi$
$809$		−	11.3367i		−	0.398578i	0.979941	−	0.199289i	$-0.0638632\pi$
$810$		0			0
$811$	26.6106			0.934424			0.467212	−	0.884145i	$-0.345259\pi$
$811$	26.6106			0.934424			0.467212	+	0.884145i	$0.345259\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−14.8886	−	14.4803i	−0.521526	−	0.507224i
$816$		0			0
$817$		−	14.4355i		−	0.505033i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			7.15531i			0.249722i	0.992174	+	0.124861i	$0.0398485\pi$
$821$			7.15531i			0.249722i	−0.992174	+	0.124861i	$0.960152\pi$
$822$		0			0
$823$			27.1559i			0.946594i	0.880903	+	0.473297i	$0.156936\pi$
$823$			27.1559i			0.946594i	−0.880903	+	0.473297i	$0.843064\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	45.2505i		−	1.57351i	−0.617262	−	0.786757i	$-0.711758\pi$
$827$		−	45.2505i		−	1.57351i	0.617262	−	0.786757i	$-0.288242\pi$
$828$		0			0
$829$	36.1757			1.25643			0.628216	−	0.778039i	$-0.283785\pi$
$829$	36.1757			1.25643			0.628216	+	0.778039i	$0.283785\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.16034i		−	0.0748515i
$834$		0			0
$835$	−25.8236	+	26.5517i	−0.893662	+	0.918860i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.2859			1.08011			0.540055	−	0.841630i	$-0.318403\pi$
$839$	31.2859			1.08011			0.540055	+	0.841630i	$0.318403\pi$
$840$		0			0
$841$	18.0833			0.623563
$842$		0			0
$843$		0			0
$844$		0			0
$845$	56.0668	−	57.6477i	1.92876	−	1.98314i
$846$		0			0
$847$	4.26226			0.146453
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.90415	−	5.54078i	−0.236671	−	0.189935i
$852$		0			0
$853$			35.6476i			1.22055i	0.792190	+	0.610275i	$0.208941\pi$
$853$			35.6476i			1.22055i	−0.792190	+	0.610275i	$0.791059\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	31.9633			1.09185			0.545923	−	0.837835i	$-0.316179\pi$
$857$	31.9633			1.09185			0.545923	+	0.837835i	$0.316179\pi$
$858$		0			0
$859$	−9.82627			−0.335268			−0.167634	−	0.985849i	$-0.553613\pi$
$859$	−9.82627			−0.335268			−0.167634	+	0.985849i	$0.553613\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	50.8297			1.73026			0.865132	−	0.501544i	$-0.167234\pi$
$863$	50.8297			1.73026			0.865132	+	0.501544i	$0.167234\pi$
$864$		0			0
$865$	−28.8677	+	29.6817i	−0.981532	+	1.00921i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	32.8260i		−	1.11355i
$870$		0			0
$871$		−	73.8074i		−	2.50087i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.0034	−	9.20200i	−0.338176	−	0.311084i
$876$		0			0
$877$		−	31.2256i		−	1.05441i	−0.849737	−	0.527207i	$-0.823239\pi$
$877$		−	31.2256i		−	1.05441i	0.849737	−	0.527207i	$-0.176761\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5.15796			0.173776			0.0868880	−	0.996218i	$-0.472308\pi$
$881$	5.15796			0.173776			0.0868880	+	0.996218i	$0.472308\pi$
$882$		0			0
$883$			20.8664i			0.702212i	0.936336	+	0.351106i	$0.114194\pi$
$883$			20.8664i			0.702212i	−0.936336	+	0.351106i	$0.885806\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27.7224			0.930828			0.465414	−	0.885093i	$-0.345905\pi$
$887$	27.7224			0.930828			0.465414	+	0.885093i	$0.345905\pi$
$888$		0			0
$889$		−	8.59328i		−	0.288209i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.47857i		−	0.283724i
$894$		0			0
$895$	14.2548	+	13.8639i	0.476486	+	0.463420i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	26.3027i		−	0.877244i
$900$		0			0
$901$	−1.15647			−0.0385277
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−34.3405	−	33.3988i	−1.14152	−	1.11021i
$906$		0			0
$907$	0.816241			0.0271028			0.0135514	−	0.999908i	$-0.495686\pi$
$907$	0.816241			0.0271028			0.0135514	+	0.999908i	$0.495686\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14.2500			−0.472122			−0.236061	−	0.971738i	$-0.575857\pi$
$911$	−14.2500			−0.472122			−0.236061	+	0.971738i	$0.575857\pi$
$912$		0			0
$913$			35.5699i			1.17719i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.41524i		−	0.0797583i
$918$		0			0
$919$			35.8892i			1.18388i	0.805984	+	0.591938i	$0.201637\pi$
$919$			35.8892i			1.18388i	−0.805984	+	0.591938i	$0.798363\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−101.797			−3.35069
$924$		0			0
$925$	0.256563	+	9.22585i	0.00843573	+	0.303344i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	4.73060i		−	0.155206i	−0.996984	−	0.0776029i	$-0.975273\pi$
$929$		−	4.73060i		−	0.155206i	0.996984	−	0.0776029i	$-0.0247266\pi$
$930$		0			0
$931$			16.2693i			0.533206i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.71674	−	1.66966i	−0.0561434	−	0.0546038i
$936$		0			0
$937$	−13.0761			−0.427177			−0.213588	−	0.976924i	$-0.568515\pi$
$937$	−13.0761			−0.427177			−0.213588	+	0.976924i	$0.568515\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	37.7022			1.22906			0.614528	−	0.788895i	$-0.289346\pi$
$941$	37.7022			1.22906			0.614528	+	0.788895i	$0.289346\pi$
$942$		0			0
$943$	−28.0779	+	34.9868i	−0.914343	+	1.13933i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.5167			0.601713			0.300856	−	0.953669i	$-0.402728\pi$
$947$	18.5167			0.601713			0.300856	+	0.953669i	$0.402728\pi$
$948$		0			0
$949$	11.9500			0.387914
$950$		0			0
$951$		0			0
$952$		0			0
$953$			51.4367i			1.66620i	0.553124	+	0.833099i	$0.313436\pi$
$953$			51.4367i			1.66620i	−0.553124	+	0.833099i	$0.686564\pi$
$954$		0			0
$955$	37.2013	−	38.2502i	1.20381	−	1.23775i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	16.4041i		−	0.529717i
$960$		0			0
$961$	32.3739			1.04432
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.0721	−	16.6039i	−0.549569	−	0.534499i
$966$		0			0
$967$		−	13.0661i		−	0.420176i	−0.977682	−	0.210088i	$-0.932625\pi$
$967$		−	13.0661i		−	0.420176i	0.977682	−	0.210088i	$-0.0673750\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	34.4356			1.10509			0.552546	−	0.833482i	$-0.313656\pi$
$971$	34.4356			1.10509			0.552546	+	0.833482i	$0.313656\pi$
$972$		0			0
$973$	−3.60662			−0.115623
$974$		0			0
$975$		0			0
$976$		0			0
$977$			50.9556i			1.63022i	0.579310	+	0.815108i	$0.303322\pi$
$977$			50.9556i			1.63022i	−0.579310	+	0.815108i	$0.696678\pi$
$978$		0			0
$979$	−41.4691			−1.32536
$980$		0			0
$981$		0			0
$982$		0			0
$983$			41.2121i			1.31446i	0.753690	+	0.657230i	$0.228272\pi$
$983$			41.2121i			1.31446i	−0.753690	+	0.657230i	$0.771728\pi$
$984$		0			0
$985$	−27.9488	+	28.7368i	−0.890521	+	0.915630i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.3260	+	14.7071i	0.582732	+	0.467659i
$990$		0			0
$991$	−9.20850			−0.292518			−0.146259	−	0.989246i	$-0.546723\pi$
$991$	−9.20850			−0.292518			−0.146259	+	0.989246i	$0.546723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	32.4955	+	31.6044i	1.03018	+	1.00193i
$996$		0			0
$997$		−	22.3782i		−	0.708723i	−0.935108	−	0.354362i	$-0.884698\pi$
$997$		−	22.3782i		−	0.708723i	0.935108	−	0.354362i	$-0.115302\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.n.b.2069.15	yes	32
3.2	odd	2	inner	4140.2.n.b.2069.17	yes	32
5.4	even	2	inner	4140.2.n.b.2069.14	yes	32
15.14	odd	2	inner	4140.2.n.b.2069.20	yes	32
23.22	odd	2	inner	4140.2.n.b.2069.18	yes	32
69.68	even	2	inner	4140.2.n.b.2069.16	yes	32
115.114	odd	2	inner	4140.2.n.b.2069.19	yes	32
345.344	even	2	inner	4140.2.n.b.2069.13	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.b.2069.13	✓	32	345.344	even	2	inner
4140.2.n.b.2069.14	yes	32	5.4	even	2	inner
4140.2.n.b.2069.15	yes	32	1.1	even	1	trivial
4140.2.n.b.2069.16	yes	32	69.68	even	2	inner
4140.2.n.b.2069.17	yes	32	3.2	odd	2	inner
4140.2.n.b.2069.18	yes	32	23.22	odd	2	inner
4140.2.n.b.2069.19	yes	32	115.114	odd	2	inner
4140.2.n.b.2069.20	yes	32	15.14	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.55901 + 1.60296i) q^{5} -1.21571 q^{7} +O(q^{10})\)	\(q+(-1.55901 + 1.60296i) q^{5} -1.21571 q^{7} +2.73752 q^{11} +6.99737i q^{13} +0.391222i q^{17} -2.94625i q^{19} +(3.74030 + 3.00169i) q^{23} +(-0.138992 - 4.99807i) q^{25} -3.30404i q^{29} +7.96077 q^{31} +(1.89530 - 1.94874i) q^{35} -1.84588 q^{37} +9.35402i q^{41} +4.89960 q^{43} +2.87774 q^{47} -5.52205 q^{49} +2.95606i q^{53} +(-4.26782 + 4.38815i) q^{55} +8.16713i q^{59} -12.9163i q^{61} +(-11.2165 - 10.9089i) q^{65} -10.5479 q^{67} +14.5479i q^{71} -1.70779i q^{73} -3.32804 q^{77} -11.9911i q^{79} +12.9935i q^{83} +(-0.627114 - 0.609917i) q^{85} -15.1484 q^{89} -8.50678i q^{91} +(4.72274 + 4.59323i) q^{95} +13.6909 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 56 q^{25} + 16 q^{31} - 96 q^{49} - 16 q^{55} - 40 q^{85}+O(q^{100}) \) 32 * q + 56 * q^25 + 16 * q^31 - 96 * q^49 - 16 * q^55 - 40 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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