LMFDB - Embedded newform 4140.2.s.b.737.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))

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

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

chi := DirichletCharacter("4140.737");

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.s (of order $4$, degree $2$, 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:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.20
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.83311 - 1.28050i) q^{5} +(-0.0692454 - 0.0692454i) q^{7} +O(q^{10})$	$q+(1.83311 - 1.28050i) q^{5} +(-0.0692454 - 0.0692454i) q^{7} +0.618138i q^{11} +(-2.47186 + 2.47186i) q^{13} +(2.38450 - 2.38450i) q^{17} -7.07775i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.72061 - 4.69462i) q^{25} -7.15848 q^{29} -3.54241 q^{31} +(-0.215604 - 0.0382657i) q^{35} +(-3.55794 - 3.55794i) q^{37} +6.73237i q^{41} +(-5.49894 + 5.49894i) q^{43} +(3.58115 - 3.58115i) q^{47} -6.99041i q^{49} +(-8.25513 - 8.25513i) q^{53} +(0.791529 + 1.13312i) q^{55} +1.16195 q^{59} +12.4990 q^{61} +(-1.36597 + 7.69642i) q^{65} +(-3.47497 - 3.47497i) q^{67} -0.149301i q^{71} +(6.67216 - 6.67216i) q^{73} +(0.0428033 - 0.0428033i) q^{77} +11.0119i q^{79} +(-8.06983 - 8.06983i) q^{83} +(1.31769 - 7.42441i) q^{85} -5.27752 q^{89} +0.342330 q^{91} +(-9.06309 - 12.9743i) q^{95} +(4.39996 + 4.39996i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$5$	1.83311	−	1.28050i	0.819794	−	0.572659i
$5$	1.83311	−	1.28050i	0.819794	−	0.572659i
$6$		0			0
$7$	−0.0692454	−	0.0692454i	−0.0261723	−	0.0261723i	0.693900	−	0.720072i	$-0.255891\pi$
$7$	−0.0692454	−	0.0692454i	−0.0261723	−	0.0261723i	−0.720072	+	0.693900i	$0.755891\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.618138i			0.186376i	0.995649	+	0.0931879i	$0.0297057\pi$
$11$			0.618138i			0.186376i	−0.995649	+	0.0931879i	$0.970294\pi$
$12$		0			0
$13$	−2.47186	+	2.47186i	−0.685569	+	0.685569i	−0.961249	−	0.275680i	$-0.911097\pi$
$13$	−2.47186	+	2.47186i	−0.685569	+	0.685569i	0.275680	+	0.961249i	$0.411097\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.38450	−	2.38450i	0.578325	−	0.578325i	−0.356116	−	0.934442i	$-0.615899\pi$
$17$	2.38450	−	2.38450i	0.578325	−	0.578325i	0.934442	+	0.356116i	$0.115899\pi$
$18$		0			0
$19$		−	7.07775i		−	1.62375i	−0.583834	−	0.811873i	$-0.698448\pi$
$19$		−	7.07775i		−	1.62375i	0.583834	−	0.811873i	$-0.301552\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$24$		0			0
$25$	1.72061	−	4.69462i	0.344123	−	0.938925i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.15848			−1.32930			−0.664648	−	0.747156i	$-0.731419\pi$
$29$	−7.15848			−1.32930			−0.664648	+	0.747156i	$0.731419\pi$
$30$		0			0
$31$	−3.54241			−0.636235			−0.318118	−	0.948051i	$-0.603051\pi$
$31$	−3.54241			−0.636235			−0.318118	+	0.948051i	$0.603051\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.215604	−	0.0382657i	−0.0364437	−	0.00646808i
$36$		0			0
$37$	−3.55794	−	3.55794i	−0.584921	−	0.584921i	0.351331	−	0.936251i	$-0.385729\pi$
$37$	−3.55794	−	3.55794i	−0.584921	−	0.584921i	−0.936251	+	0.351331i	$0.885729\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.73237i			1.05142i	0.850664	+	0.525710i	$0.176200\pi$
$41$			6.73237i			1.05142i	−0.850664	+	0.525710i	$0.823800\pi$
$42$		0			0
$43$	−5.49894	+	5.49894i	−0.838580	+	0.838580i	−0.988672	−	0.150092i	$-0.952043\pi$
$43$	−5.49894	+	5.49894i	−0.838580	+	0.838580i	0.150092	+	0.988672i	$0.452043\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.58115	−	3.58115i	0.522364	−	0.522364i	−0.395921	−	0.918285i	$-0.629574\pi$
$47$	3.58115	−	3.58115i	0.522364	−	0.522364i	0.918285	+	0.395921i	$0.129574\pi$
$48$		0			0
$49$		−	6.99041i		−	0.998630i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.25513	−	8.25513i	−1.13393	−	1.13393i	−0.989518	−	0.144411i	$-0.953871\pi$
$53$	−8.25513	−	8.25513i	−1.13393	−	1.13393i	−0.144411	−	0.989518i	$-0.546129\pi$
$54$		0			0
$55$	0.791529	+	1.13312i	0.106730	+	0.152790i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.16195			0.151273			0.0756363	−	0.997135i	$-0.475901\pi$
$59$	1.16195			0.151273			0.0756363	+	0.997135i	$0.475901\pi$
$60$		0			0
$61$	12.4990			1.60033			0.800164	−	0.599782i	$-0.204746\pi$
$61$	12.4990			1.60033			0.800164	+	0.599782i	$0.204746\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.36597	+	7.69642i	−0.169428	+	0.954623i
$66$		0			0
$67$	−3.47497	−	3.47497i	−0.424535	−	0.424535i	0.462227	−	0.886762i	$-0.347051\pi$
$67$	−3.47497	−	3.47497i	−0.424535	−	0.424535i	−0.886762	+	0.462227i	$0.847051\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.149301i		−	0.0177187i	−0.999961	−	0.00885936i	$-0.997180\pi$
$71$		−	0.149301i		−	0.0177187i	0.999961	−	0.00885936i	$-0.00282006\pi$
$72$		0			0
$73$	6.67216	−	6.67216i	0.780918	−	0.780918i	−0.199068	−	0.979986i	$-0.563791\pi$
$73$	6.67216	−	6.67216i	0.780918	−	0.780918i	0.979986	+	0.199068i	$0.0637915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0428033	−	0.0428033i	0.00487788	−	0.00487788i
$78$		0			0
$79$			11.0119i			1.23894i	0.785021	+	0.619469i	$0.212652\pi$
$79$			11.0119i			1.23894i	−0.785021	+	0.619469i	$0.787348\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.06983	−	8.06983i	−0.885779	−	0.885779i	0.108335	−	0.994114i	$-0.465448\pi$
$83$	−8.06983	−	8.06983i	−0.885779	−	0.885779i	−0.994114	+	0.108335i	$0.965448\pi$
$84$		0			0
$85$	1.31769	−	7.42441i	0.142924	−	0.805291i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.27752			−0.559416			−0.279708	−	0.960085i	$-0.590238\pi$
$89$	−5.27752			−0.559416			−0.279708	+	0.960085i	$0.590238\pi$
$90$		0			0
$91$	0.342330			0.0358859
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.06309	−	12.9743i	−0.929853	−	1.33114i
$96$		0			0
$97$	4.39996	+	4.39996i	0.446749	+	0.446749i	0.894272	−	0.447524i	$-0.147694\pi$
$97$	4.39996	+	4.39996i	0.446749	+	0.446749i	−0.447524	+	0.894272i	$0.647694\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.4306i		−	1.33640i	−0.743983	−	0.668198i	$-0.767066\pi$
$101$		−	13.4306i		−	1.33640i	0.743983	−	0.668198i	$-0.232934\pi$
$102$		0			0
$103$	10.0290	−	10.0290i	0.988182	−	0.988182i	−0.0117492	−	0.999931i	$-0.503740\pi$
$103$	10.0290	−	10.0290i	0.988182	−	0.988182i	0.999931	+	0.0117492i	$0.00373996\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.71046	+	2.71046i	−0.262030	+	0.262030i	−0.825878	−	0.563848i	$-0.809320\pi$
$107$	−2.71046	+	2.71046i	−0.262030	+	0.262030i	0.563848	+	0.825878i	$0.309320\pi$
$108$		0			0
$109$		−	16.5642i		−	1.58657i	−0.608853	−	0.793283i	$-0.708370\pi$
$109$		−	16.5642i		−	1.58657i	0.608853	−	0.793283i	$-0.291630\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.21393	+	3.21393i	0.302341	+	0.302341i	0.841929	−	0.539588i	$-0.181420\pi$
$113$	3.21393	+	3.21393i	0.302341	+	0.302341i	−0.539588	+	0.841929i	$0.681420\pi$
$114$		0			0
$115$	−2.20166	−	0.390754i	−0.205306	−	0.0364380i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.330231			−0.0302722
$120$		0			0
$121$	10.6179			0.965264
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.85741	−	10.8090i	−0.255574	−	0.966789i
$126$		0			0
$127$	−0.701549	−	0.701549i	−0.0622524	−	0.0622524i	0.675295	−	0.737548i	$-0.264016\pi$
$127$	−0.701549	−	0.701549i	−0.0622524	−	0.0622524i	−0.737548	+	0.675295i	$0.764016\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			9.87376i			0.862674i	0.902191	+	0.431337i	$0.141958\pi$
$131$			9.87376i			0.862674i	−0.902191	+	0.431337i	$0.858042\pi$
$132$		0			0
$133$	−0.490102	+	0.490102i	−0.0424972	+	0.0424972i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.64421	+	6.64421i	−0.567653	+	0.567653i	−0.931470	−	0.363817i	$-0.881473\pi$
$137$	−6.64421	+	6.64421i	−0.567653	+	0.567653i	0.363817	+	0.931470i	$0.381473\pi$
$138$		0			0
$139$		−	6.59037i		−	0.558988i	−0.960147	−	0.279494i	$-0.909833\pi$
$139$		−	6.59037i		−	0.558988i	0.960147	−	0.279494i	$-0.0901667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.52795	−	1.52795i	−0.127774	−	0.127774i
$144$		0			0
$145$	−13.1223	+	9.16647i	−1.08975	+	0.761234i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.8867			1.05572			0.527858	−	0.849332i	$-0.322995\pi$
$149$	12.8867			1.05572			0.527858	+	0.849332i	$0.322995\pi$
$150$		0			0
$151$	−16.1015			−1.31032			−0.655159	−	0.755491i	$-0.727399\pi$
$151$	−16.1015			−1.31032			−0.655159	+	0.755491i	$0.727399\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.49364	+	4.53607i	−0.521582	+	0.364346i
$156$		0			0
$157$	5.89147	+	5.89147i	0.470191	+	0.470191i	0.901976	−	0.431786i	$-0.142116\pi$
$157$	5.89147	+	5.89147i	0.470191	+	0.470191i	−0.431786	+	0.901976i	$0.642116\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.0979278i			0.00771780i
$162$		0			0
$163$	−5.54054	+	5.54054i	−0.433968	+	0.433968i	−0.889976	−	0.456008i	$-0.849279\pi$
$163$	−5.54054	+	5.54054i	−0.433968	+	0.433968i	0.456008	+	0.889976i	$0.349279\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.3546	+	17.3546i	−1.34294	+	1.34294i	−0.449823	+	0.893117i	$0.648513\pi$
$167$	−17.3546	+	17.3546i	−1.34294	+	1.34294i	−0.893117	+	0.449823i	$0.851487\pi$
$168$		0			0
$169$			0.779857i			0.0599890i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.52006	−	3.52006i	−0.267625	−	0.267625i	0.560517	−	0.828143i	$-0.310602\pi$
$173$	−3.52006	−	3.52006i	−0.267625	−	0.267625i	−0.828143	+	0.560517i	$0.810602\pi$
$174$		0			0
$175$	−0.444226	+	0.205937i	−0.0335803	+	0.0155673i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.9451			−1.19180			−0.595898	−	0.803060i	$-0.703204\pi$
$179$	−15.9451			−1.19180			−0.595898	+	0.803060i	$0.703204\pi$
$180$		0			0
$181$	16.1816			1.20277			0.601384	−	0.798960i	$-0.294616\pi$
$181$	16.1816			1.20277			0.601384	+	0.798960i	$0.294616\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−11.0781	−	1.96615i	−0.814475	−	0.144554i
$186$		0			0
$187$	1.47395	+	1.47395i	0.107786	+	0.107786i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	20.8037i		−	1.50531i	−0.658417	−	0.752653i	$-0.728774\pi$
$191$		−	20.8037i		−	1.50531i	0.658417	−	0.752653i	$-0.271226\pi$
$192$		0			0
$193$	−13.7756	+	13.7756i	−0.991590	+	0.991590i	−0.999965	−	0.00837513i	$-0.997334\pi$
$193$	−13.7756	+	13.7756i	−0.991590	+	0.991590i	0.00837513	+	0.999965i	$0.497334\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.6560	+	12.6560i	−0.901701	+	0.901701i	−0.995583	−	0.0938821i	$-0.970072\pi$
$197$	−12.6560	+	12.6560i	−0.901701	+	0.901701i	0.0938821	+	0.995583i	$0.470072\pi$
$198$		0			0
$199$		−	15.9129i		−	1.12803i	−0.825763	−	0.564017i	$-0.809255\pi$
$199$		−	15.9129i		−	1.12803i	0.825763	−	0.564017i	$-0.190745\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.495692	+	0.495692i	0.0347908	+	0.0347908i
$204$		0			0
$205$	8.62083	+	12.3412i	0.602105	+	0.861947i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.37503			0.302627
$210$		0			0
$211$	8.06953			0.555529			0.277765	−	0.960649i	$-0.410406\pi$
$211$	8.06953			0.555529			0.277765	+	0.960649i	$0.410406\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.03876	+	17.1216i	−0.207242	+	1.16768i
$216$		0			0
$217$	0.245296	+	0.245296i	0.0166518	+	0.0166518i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			11.7883i			0.792964i
$222$		0			0
$223$	14.3187	−	14.3187i	0.958850	−	0.958850i	−0.0403364	−	0.999186i	$-0.512843\pi$
$223$	14.3187	−	14.3187i	0.958850	−	0.958850i	0.999186	+	0.0403364i	$0.0128429\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.27832	+	6.27832i	−0.416706	+	0.416706i	−0.884067	−	0.467360i	$-0.845205\pi$
$227$	−6.27832	+	6.27832i	−0.416706	+	0.416706i	0.467360	+	0.884067i	$0.345205\pi$
$228$		0			0
$229$		−	18.0423i		−	1.19227i	−0.802884	−	0.596135i	$-0.796702\pi$
$229$		−	18.0423i		−	1.19227i	0.802884	−	0.596135i	$-0.203298\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.942382	+	0.942382i	0.0617375	+	0.0617375i	0.737301	−	0.675564i	$-0.236100\pi$
$233$	0.942382	+	0.942382i	0.0617375	+	0.0617375i	−0.675564	+	0.737301i	$0.736100\pi$
$234$		0			0
$235$	1.97898	−	11.1503i	0.129094	−	0.727367i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.6146			−1.52750			−0.763750	−	0.645512i	$-0.776644\pi$
$239$	−23.6146			−1.52750			−0.763750	+	0.645512i	$0.776644\pi$
$240$		0			0
$241$	24.1604			1.55631			0.778153	−	0.628075i	$-0.216157\pi$
$241$	24.1604			1.55631			0.778153	+	0.628075i	$0.216157\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.95125	−	12.8142i	−0.571875	−	0.818670i
$246$		0			0
$247$	17.4952	+	17.4952i	1.11319	+	1.11319i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5.41151i			0.341572i	0.985308	+	0.170786i	$0.0546306\pi$
$251$			5.41151i			0.341572i	−0.985308	+	0.170786i	$0.945369\pi$
$252$		0			0
$253$	0.437090	−	0.437090i	0.0274796	−	0.0274796i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.5038	−	14.5038i	0.904721	−	0.904721i	−0.0911193	−	0.995840i	$-0.529044\pi$
$257$	14.5038	−	14.5038i	0.904721	−	0.904721i	0.995840	+	0.0911193i	$0.0290445\pi$
$258$		0			0
$259$			0.492742i			0.0306175i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.7543	−	13.7543i	−0.848129	−	0.848129i	0.141770	−	0.989900i	$-0.454721\pi$
$263$	−13.7543	−	13.7543i	−0.848129	−	0.848129i	−0.989900	+	0.141770i	$0.954721\pi$
$264$		0			0
$265$	−25.7033	−	4.56186i	−1.57894	−	0.280233i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.09381			0.432517			0.216259	−	0.976336i	$-0.430615\pi$
$269$	7.09381			0.432517			0.216259	+	0.976336i	$0.430615\pi$
$270$		0			0
$271$	−15.8122			−0.960525			−0.480262	−	0.877125i	$-0.659459\pi$
$271$	−15.8122			−0.960525			−0.480262	+	0.877125i	$0.659459\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.90193	+	1.06358i	0.174993	+	0.0641361i
$276$		0			0
$277$	3.69026	+	3.69026i	0.221726	+	0.221726i	0.809225	−	0.587499i	$-0.199887\pi$
$277$	3.69026	+	3.69026i	0.221726	+	0.221726i	−0.587499	+	0.809225i	$0.699887\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.2143i		−	1.08657i	−0.839547	−	0.543287i	$-0.817180\pi$
$281$		−	18.2143i		−	1.08657i	0.839547	−	0.543287i	$-0.182820\pi$
$282$		0			0
$283$	1.59265	−	1.59265i	0.0946735	−	0.0946735i	−0.658184	−	0.752857i	$-0.728675\pi$
$283$	1.59265	−	1.59265i	0.0946735	−	0.0946735i	0.752857	+	0.658184i	$0.228675\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.466186	−	0.466186i	0.0275181	−	0.0275181i
$288$		0			0
$289$			5.62835i			0.331080i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.8323	−	15.8323i	−0.924933	−	0.924933i	0.0724400	−	0.997373i	$-0.476921\pi$
$293$	−15.8323	−	15.8323i	−0.924933	−	0.924933i	−0.997373	+	0.0724400i	$0.976921\pi$
$294$		0			0
$295$	2.12998	−	1.48788i	0.124012	−	0.0866276i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.49573			0.202163
$300$		0			0
$301$	0.761553			0.0438952
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.9120	−	16.0050i	1.31194	−	0.916442i
$306$		0			0
$307$	−15.9654	−	15.9654i	−0.911194	−	0.911194i	0.0851721	−	0.996366i	$-0.472856\pi$
$307$	−15.9654	−	15.9654i	−0.911194	−	0.911194i	−0.996366	+	0.0851721i	$0.972856\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.61089i		−	0.318165i	−0.987265	−	0.159082i	$-0.949146\pi$
$311$		−	5.61089i		−	0.318165i	0.987265	−	0.159082i	$-0.0508535\pi$
$312$		0			0
$313$	16.6813	−	16.6813i	0.942881	−	0.942881i	−0.0555734	−	0.998455i	$-0.517699\pi$
$313$	16.6813	−	16.6813i	0.942881	−	0.942881i	0.998455	+	0.0555734i	$0.0176987\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.265337	+	0.265337i	−0.0149028	+	0.0149028i	−0.714519	−	0.699616i	$-0.753354\pi$
$317$	−0.265337	+	0.265337i	−0.0149028	+	0.0149028i	0.699616	+	0.714519i	$0.253354\pi$
$318$		0			0
$319$		−	4.42493i		−	0.247749i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.8769	−	16.8769i	−0.939054	−	0.939054i
$324$		0			0
$325$	7.35132	+	15.8575i	0.407778	+	0.879618i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.495956			−0.0273430
$330$		0			0
$331$	−5.94338			−0.326678			−0.163339	−	0.986570i	$-0.552226\pi$
$331$	−5.94338			−0.326678			−0.163339	+	0.986570i	$0.552226\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.8197	−	1.92030i	−0.591145	−	0.104917i
$336$		0			0
$337$	9.72424	+	9.72424i	0.529713	+	0.529713i	0.920487	−	0.390774i	$-0.127792\pi$
$337$	9.72424	+	9.72424i	0.529713	+	0.529713i	−0.390774	+	0.920487i	$0.627792\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.18970i		−	0.118579i
$342$		0			0
$343$	−0.968772	+	0.968772i	−0.0523088	+	0.0523088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.5048	+	21.5048i	−1.15444	+	1.15444i	−0.168787	+	0.985653i	$0.553985\pi$
$347$	−21.5048	+	21.5048i	−1.15444	+	1.15444i	−0.985653	+	0.168787i	$0.946015\pi$
$348$		0			0
$349$		−	23.5042i		−	1.25815i	−0.777345	−	0.629075i	$-0.783434\pi$
$349$		−	23.5042i		−	1.25815i	0.777345	−	0.629075i	$-0.216566\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.75835	+	9.75835i	0.519385	+	0.519385i	0.917385	−	0.398000i	$-0.130296\pi$
$353$	9.75835	+	9.75835i	0.519385	+	0.519385i	−0.398000	+	0.917385i	$0.630296\pi$
$354$		0			0
$355$	−0.191180	−	0.273685i	−0.0101468	−	0.0145257i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	33.4998			1.76805			0.884025	−	0.467439i	$-0.154823\pi$
$359$	33.4998			1.76805			0.884025	+	0.467439i	$0.154823\pi$
$360$		0			0
$361$	−31.0945			−1.63655
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.68710	−	20.7746i	0.192992	−	1.08739i
$366$		0			0
$367$	−10.3916	−	10.3916i	−0.542437	−	0.542437i	0.381805	−	0.924243i	$-0.375302\pi$
$367$	−10.3916	−	10.3916i	−0.542437	−	0.542437i	−0.924243	+	0.381805i	$0.875302\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.14326i			0.0593551i
$372$		0			0
$373$	26.8169	−	26.8169i	1.38853	−	1.38853i	0.560112	−	0.828417i	$-0.310758\pi$
$373$	26.8169	−	26.8169i	1.38853	−	1.38853i	0.828417	−	0.560112i	$-0.189242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.6947	−	17.6947i	0.911325	−	0.911325i
$378$		0			0
$379$		−	5.90370i		−	0.303253i	−0.988438	−	0.151626i	$-0.951549\pi$
$379$		−	5.90370i		−	0.303253i	0.988438	−	0.151626i	$-0.0484511\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	22.8064	+	22.8064i	1.16535	+	1.16535i	0.983286	+	0.182067i	$0.0582787\pi$
$383$	22.8064	+	22.8064i	1.16535	+	1.16535i	0.182067	+	0.983286i	$0.441721\pi$
$384$		0			0
$385$	0.0236535	−	0.133273i	0.00120549	−	0.00679222i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−29.5229			−1.49687			−0.748435	−	0.663208i	$-0.769194\pi$
$389$	−29.5229			−1.49687			−0.748435	+	0.663208i	$0.769194\pi$
$390$		0			0
$391$	−3.37219			−0.170539
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.1008	+	20.1861i	0.709489	+	1.01567i
$396$		0			0
$397$	12.6398	+	12.6398i	0.634375	+	0.634375i	0.949162	−	0.314787i	$-0.101933\pi$
$397$	12.6398	+	12.6398i	0.634375	+	0.634375i	−0.314787	+	0.949162i	$0.601933\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			0.984088i			0.0491430i	0.999698	+	0.0245715i	$0.00782214\pi$
$401$			0.984088i			0.0491430i	−0.999698	+	0.0245715i	$0.992178\pi$
$402$		0			0
$403$	8.75632	−	8.75632i	0.436184	−	0.436184i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.19930	−	2.19930i	0.109015	−	0.109015i
$408$		0			0
$409$			3.35198i			0.165745i	0.996560	+	0.0828723i	$0.0264094\pi$
$409$			3.35198i			0.165745i	−0.996560	+	0.0828723i	$0.973591\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.0804595	−	0.0804595i	−0.00395915	−	0.00395915i
$414$		0			0
$415$	−25.1264	−	4.45946i	−1.23341	−	0.218906i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	26.8691			1.31264			0.656322	−	0.754481i	$-0.272111\pi$
$419$	26.8691			1.31264			0.656322	+	0.754481i	$0.272111\pi$
$420$		0			0
$421$	8.42259			0.410492			0.205246	−	0.978710i	$-0.434201\pi$
$421$	8.42259			0.410492			0.205246	+	0.978710i	$0.434201\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.09151	−	15.2971i	−0.343989	−	0.742019i
$426$		0			0
$427$	−0.865496	−	0.865496i	−0.0418843	−	0.0418843i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2.34030i			0.112728i	0.998410	+	0.0563642i	$0.0179508\pi$
$431$			2.34030i			0.112728i	−0.998410	+	0.0563642i	$0.982049\pi$
$432$		0			0
$433$	−1.70123	+	1.70123i	−0.0817560	+	0.0817560i	−0.746802	−	0.665046i	$-0.768412\pi$
$433$	−1.70123	+	1.70123i	−0.0817560	+	0.0817560i	0.665046	+	0.746802i	$0.268412\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.00472	+	5.00472i	−0.239408	+	0.239408i
$438$		0			0
$439$		−	18.6859i		−	0.891830i	−0.895075	−	0.445915i	$-0.852878\pi$
$439$		−	18.6859i		−	0.891830i	0.895075	−	0.445915i	$-0.147122\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.82411	+	4.82411i	0.229200	+	0.229200i	0.812359	−	0.583158i	$-0.198183\pi$
$443$	4.82411	+	4.82411i	0.229200	+	0.229200i	−0.583158	+	0.812359i	$0.698183\pi$
$444$		0			0
$445$	−9.67430	+	6.75790i	−0.458606	+	0.320355i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.922816			−0.0435504			−0.0217752	−	0.999763i	$-0.506932\pi$
$449$	−0.922816			−0.0435504			−0.0217752	+	0.999763i	$0.506932\pi$
$450$		0			0
$451$	−4.16153			−0.195959
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.627529	−	0.438355i	0.0294190	−	0.0205504i
$456$		0			0
$457$	−6.72359	−	6.72359i	−0.314516	−	0.314516i	0.532140	−	0.846656i	$-0.321388\pi$
$457$	−6.72359	−	6.72359i	−0.314516	−	0.314516i	−0.846656	+	0.532140i	$0.821388\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.4393i			1.46428i	0.681156	+	0.732138i	$0.261477\pi$
$461$			31.4393i			1.46428i	−0.681156	+	0.732138i	$0.738523\pi$
$462$		0			0
$463$	−21.0869	+	21.0869i	−0.979991	+	0.979991i	−0.999804	−	0.0198125i	$-0.993693\pi$
$463$	−21.0869	+	21.0869i	−0.979991	+	0.979991i	0.0198125	+	0.999804i	$0.493693\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.9350	+	24.9350i	−1.15386	+	1.15386i	−0.168082	+	0.985773i	$0.553757\pi$
$467$	−24.9350	+	24.9350i	−1.15386	+	1.15386i	−0.985773	+	0.168082i	$0.946243\pi$
$468$		0			0
$469$			0.481252i			0.0222221i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.39911	−	3.39911i	−0.156291	−	0.156291i
$474$		0			0
$475$	−33.2273	−	12.1781i	−1.52458	−	0.558768i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.11536			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$479$	−5.11536			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$480$		0			0
$481$	17.5894			0.802008
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.6998	+	2.43146i	0.622076	+	0.110407i
$486$		0			0
$487$	20.0676	+	20.0676i	0.909349	+	0.909349i	0.996220	−	0.0868708i	$-0.0276867\pi$
$487$	20.0676	+	20.0676i	0.909349	+	0.909349i	−0.0868708	+	0.996220i	$0.527687\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.17230i		−	0.323681i	−0.986817	−	0.161841i	$-0.948257\pi$
$491$		−	7.17230i		−	0.323681i	0.986817	−	0.161841i	$-0.0517431\pi$
$492$		0			0
$493$	−17.0694	+	17.0694i	−0.768766	+	0.768766i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0103384	+	0.0103384i	−0.000463740	+	0.000463740i
$498$		0			0
$499$			7.28501i			0.326122i	0.986616	+	0.163061i	$0.0521367\pi$
$499$			7.28501i			0.326122i	−0.986616	+	0.163061i	$0.947863\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.194485	+	0.194485i	0.00867165	+	0.00867165i	0.711429	−	0.702758i	$-0.248048\pi$
$503$	0.194485	+	0.194485i	0.00867165	+	0.00867165i	−0.702758	+	0.711429i	$0.748048\pi$
$504$		0			0
$505$	−17.1980	−	24.6199i	−0.765300	−	1.09557i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.9097			1.14843			0.574214	−	0.818706i	$-0.305308\pi$
$509$	25.9097			1.14843			0.574214	+	0.818706i	$0.305308\pi$
$510$		0			0
$511$	−0.924034			−0.0408768
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.54209	−	31.2263i	0.244214	−	1.37600i
$516$		0			0
$517$	2.21365	+	2.21365i	0.0973560	+	0.0973560i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			20.9650i			0.918493i	0.888309	+	0.459247i	$0.151881\pi$
$521$			20.9650i			0.918493i	−0.888309	+	0.459247i	$0.848119\pi$
$522$		0			0
$523$	−31.8074	+	31.8074i	−1.39084	+	1.39084i	−0.567394	+	0.823446i	$0.692048\pi$
$523$	−31.8074	+	31.8074i	−1.39084	+	1.39084i	−0.823446	+	0.567394i	$0.807952\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.44686	+	8.44686i	−0.367951	+	0.367951i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−16.6414	−	16.6414i	−0.720821	−	0.720821i
$534$		0			0
$535$	−1.49782	+	8.43933i	−0.0647566	+	0.364864i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.32104			0.186120
$540$		0			0
$541$	14.4485			0.621190			0.310595	−	0.950542i	$-0.399472\pi$
$541$	14.4485			0.621190			0.310595	+	0.950542i	$0.399472\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−21.2106	−	30.3641i	−0.908562	−	1.30066i
$546$		0			0
$547$	10.5864	+	10.5864i	0.452641	+	0.452641i	0.896230	−	0.443590i	$-0.146295\pi$
$547$	10.5864	+	10.5864i	0.452641	+	0.452641i	−0.443590	+	0.896230i	$0.646295\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			50.6659i			2.15844i
$552$		0			0
$553$	0.762525	−	0.762525i	0.0324259	−	0.0324259i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.61536	+	3.61536i	−0.153188	+	0.153188i	−0.779540	−	0.626352i	$-0.784547\pi$
$557$	−3.61536	+	3.61536i	−0.153188	+	0.153188i	0.626352	+	0.779540i	$0.284547\pi$
$558$		0			0
$559$		−	27.1852i		−	1.14981i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.92357	−	1.92357i	−0.0810687	−	0.0810687i	0.665410	−	0.746478i	$-0.268257\pi$
$563$	−1.92357	−	1.92357i	−0.0810687	−	0.0810687i	−0.746478	+	0.665410i	$0.768257\pi$
$564$		0			0
$565$	10.0070	+	1.77605i	0.420996	+	0.0747189i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.7087			0.616622			0.308311	−	0.951286i	$-0.400236\pi$
$569$	14.7087			0.616622			0.308311	+	0.951286i	$0.400236\pi$
$570$		0			0
$571$	40.7014			1.70330			0.851651	−	0.524110i	$-0.175602\pi$
$571$	40.7014			1.70330			0.851651	+	0.524110i	$0.175602\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.53626	+	2.10294i	−0.189175	+	0.0876987i
$576$		0			0
$577$	−29.0869	−	29.0869i	−1.21090	−	1.21090i	−0.970731	−	0.240171i	$-0.922797\pi$
$577$	−29.0869	−	29.0869i	−1.21090	−	1.21090i	−0.240171	−	0.970731i	$-0.577203\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.11760i			0.0463658i
$582$		0			0
$583$	5.10281	−	5.10281i	0.211337	−	0.211337i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	30.0160	−	30.0160i	1.23889	−	1.23889i	0.278437	−	0.960455i	$-0.410184\pi$
$587$	30.0160	−	30.0160i	1.23889	−	1.23889i	0.960455	−	0.278437i	$-0.0898163\pi$
$588$		0			0
$589$			25.0723i			1.03308i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.8167	+	14.8167i	0.608450	+	0.608450i	0.942541	−	0.334091i	$-0.108429\pi$
$593$	14.8167	+	14.8167i	0.608450	+	0.608450i	−0.334091	+	0.942541i	$0.608429\pi$
$594$		0			0
$595$	−0.605351	+	0.422862i	−0.0248170	+	0.0173357i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	39.0690			1.59632			0.798159	−	0.602447i	$-0.205808\pi$
$599$	39.0690			1.59632			0.798159	+	0.602447i	$0.205808\pi$
$600$		0			0
$601$	25.6025			1.04435			0.522174	−	0.852839i	$-0.325121\pi$
$601$	25.6025			1.04435			0.522174	+	0.852839i	$0.325121\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.4638	−	13.5963i	0.791317	−	0.552767i
$606$		0			0
$607$	5.09318	+	5.09318i	0.206726	+	0.206726i	0.802874	−	0.596149i	$-0.203303\pi$
$607$	5.09318	+	5.09318i	0.206726	+	0.206726i	−0.596149	+	0.802874i	$0.703303\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			17.7042i			0.716234i
$612$		0			0
$613$	−23.9850	+	23.9850i	−0.968745	+	0.968745i	−0.999526	−	0.0307808i	$-0.990201\pi$
$613$	−23.9850	+	23.9850i	−0.968745	+	0.968745i	0.0307808	+	0.999526i	$0.490201\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.9147	−	17.9147i	0.721220	−	0.721220i	−0.247634	−	0.968854i	$-0.579653\pi$
$617$	17.9147	−	17.9147i	0.721220	−	0.721220i	0.968854	+	0.247634i	$0.0796531\pi$
$618$		0			0
$619$		−	0.141070i		−	0.00567006i	−0.999996	−	0.00283503i	$-0.999098\pi$
$619$		−	0.141070i		−	0.00567006i	0.999996	−	0.00283503i	$-0.000902420\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.365445	+	0.365445i	0.0146412	+	0.0146412i
$624$		0			0
$625$	−19.0790	−	16.1553i	−0.763159	−	0.646211i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.9678			−0.676549
$630$		0			0
$631$	20.2324			0.805438			0.402719	−	0.915324i	$-0.368065\pi$
$631$	20.2324			0.805438			0.402719	+	0.915324i	$0.368065\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.18436	−	0.387683i	−0.0866836	−	0.0153847i
$636$		0			0
$637$	17.2793	+	17.2793i	0.684630	+	0.684630i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13.0114i		−	0.513919i	−0.966422	−	0.256959i	$-0.917279\pi$
$641$		−	13.0114i		−	0.513919i	0.966422	−	0.256959i	$-0.0827207\pi$
$642$		0			0
$643$	−11.8616	+	11.8616i	−0.467774	+	0.467774i	−0.901193	−	0.433419i	$-0.857307\pi$
$643$	−11.8616	+	11.8616i	−0.467774	+	0.467774i	0.433419	+	0.901193i	$0.357307\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.16036	+	5.16036i	−0.202875	+	0.202875i	−0.801230	−	0.598356i	$-0.795821\pi$
$647$	−5.16036	+	5.16036i	−0.202875	+	0.202875i	0.598356	+	0.801230i	$0.295821\pi$
$648$		0			0
$649$			0.718243i			0.0281935i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.01069	+	1.01069i	0.0395512	+	0.0395512i	0.726606	−	0.687055i	$-0.241097\pi$
$653$	1.01069	+	1.01069i	0.0395512	+	0.0395512i	−0.687055	+	0.726606i	$0.741097\pi$
$654$		0			0
$655$	12.6434	+	18.0997i	0.494018	+	0.707215i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.5807			−0.451120			−0.225560	−	0.974229i	$-0.572421\pi$
$659$	−11.5807			−0.451120			−0.225560	+	0.974229i	$0.572421\pi$
$660$		0			0
$661$	15.7168			0.611313			0.305656	−	0.952142i	$-0.401124\pi$
$661$	15.7168			0.611313			0.305656	+	0.952142i	$0.401124\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.270835	+	1.52599i	−0.0105025	+	0.0591753i
$666$		0			0
$667$	5.06181	+	5.06181i	0.195994	+	0.195994i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			7.72608i			0.298262i
$672$		0			0
$673$	−23.0122	+	23.0122i	−0.887053	+	0.887053i	−0.994239	−	0.107186i	$-0.965816\pi$
$673$	−23.0122	+	23.0122i	−0.887053	+	0.887053i	0.107186	+	0.994239i	$0.465816\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.51608	−	6.51608i	0.250433	−	0.250433i	−0.570715	−	0.821148i	$-0.693334\pi$
$677$	6.51608	−	6.51608i	0.250433	−	0.250433i	0.821148	+	0.570715i	$0.193334\pi$
$678$		0			0
$679$		−	0.609355i		−	0.0233849i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.72705	+	8.72705i	0.333931	+	0.333931i	0.854077	−	0.520146i	$-0.174123\pi$
$683$	8.72705	+	8.72705i	0.333931	+	0.333931i	−0.520146	+	0.854077i	$0.674123\pi$
$684$		0			0
$685$	−3.67165	+	20.6875i	−0.140287	+	0.790430i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	40.8110			1.55477
$690$		0			0
$691$	18.8047			0.715364			0.357682	−	0.933843i	$-0.383567\pi$
$691$	18.8047			0.715364			0.357682	+	0.933843i	$0.383567\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.43900	−	12.0809i	−0.320110	−	0.458255i
$696$		0			0
$697$	16.0533	+	16.0533i	0.608062	+	0.608062i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.1394i		−	1.25166i	−0.779960	−	0.625830i	$-0.784761\pi$
$701$		−	33.1394i		−	1.25166i	0.779960	−	0.625830i	$-0.215239\pi$
$702$		0			0
$703$	−25.1822	+	25.1822i	−0.949763	+	0.949763i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.930009	+	0.930009i	−0.0349766	+	0.0349766i
$708$		0			0
$709$			16.8686i			0.633513i	0.948507	+	0.316757i	$0.102594\pi$
$709$			16.8686i			0.633513i	−0.948507	+	0.316757i	$0.897406\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.50486	+	2.50486i	0.0938078	+	0.0938078i
$714$		0			0
$715$	−4.75745	−	0.844358i	−0.177919	−	0.0315772i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−38.8408			−1.44852			−0.724258	−	0.689529i	$-0.757818\pi$
$719$	−38.8408			−1.44852			−0.724258	+	0.689529i	$0.757818\pi$
$720$		0			0
$721$	−1.38892			−0.0517260
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.3170	+	33.6064i	−0.457441	+	1.24811i
$726$		0			0
$727$	−15.9312	−	15.9312i	−0.590857	−	0.590857i	0.347006	−	0.937863i	$-0.387198\pi$
$727$	−15.9312	−	15.9312i	−0.590857	−	0.590857i	−0.937863	+	0.347006i	$0.887198\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			26.2244i			0.969945i
$732$		0			0
$733$	10.9450	−	10.9450i	0.404264	−	0.404264i	−0.475469	−	0.879732i	$-0.657722\pi$
$733$	10.9450	−	10.9450i	0.404264	−	0.404264i	0.879732	+	0.475469i	$0.157722\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.14801	−	2.14801i	0.0791231	−	0.0791231i
$738$		0			0
$739$		−	48.6714i		−	1.79040i	−0.445660	−	0.895202i	$-0.647031\pi$
$739$		−	48.6714i		−	1.79040i	0.445660	−	0.895202i	$-0.352969\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	35.4624	+	35.4624i	1.30099	+	1.30099i	0.927726	+	0.373262i	$0.121761\pi$
$743$	35.4624	+	35.4624i	1.30099	+	1.30099i	0.373262	+	0.927726i	$0.378239\pi$
$744$		0			0
$745$	23.6227	−	16.5014i	0.865470	−	0.604566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.375374			0.0137159
$750$		0			0
$751$	−33.1271			−1.20882			−0.604412	−	0.796672i	$-0.706592\pi$
$751$	−33.1271			−1.20882			−0.604412	+	0.796672i	$0.706592\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.5158	+	20.6180i	−1.07419	+	0.750366i
$756$		0			0
$757$	6.86936	+	6.86936i	0.249671	+	0.249671i	0.820836	−	0.571164i	$-0.193508\pi$
$757$	6.86936	+	6.86936i	0.249671	+	0.249671i	−0.571164	+	0.820836i	$0.693508\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			9.72840i			0.352654i	0.984332	+	0.176327i	$0.0564216\pi$
$761$			9.72840i			0.352654i	−0.984332	+	0.176327i	$0.943578\pi$
$762$		0			0
$763$	−1.14700	+	1.14700i	−0.0415241	+	0.0415241i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.87216	+	2.87216i	−0.103708	+	0.103708i
$768$		0			0
$769$		−	8.21353i		−	0.296188i	−0.988973	−	0.148094i	$-0.952686\pi$
$769$		−	8.21353i		−	0.296188i	0.988973	−	0.148094i	$-0.0473137\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29.6176	−	29.6176i	−1.06527	−	1.06527i	−0.997716	−	0.0675543i	$-0.978480\pi$
$773$	−29.6176	−	29.6176i	−1.06527	−	1.06527i	−0.0675543	−	0.997716i	$-0.521520\pi$
$774$		0			0
$775$	−6.09512	+	16.6303i	−0.218943	+	0.597377i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	47.6500			1.70724
$780$		0			0
$781$	0.0922884			0.00330234
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.3438	+	3.25568i	0.654718	+	0.116200i
$786$		0			0
$787$	31.9784	+	31.9784i	1.13991	+	1.13991i	0.988467	+	0.151440i	$0.0483909\pi$
$787$	31.9784	+	31.9784i	1.13991	+	1.13991i	0.151440	+	0.988467i	$0.451609\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.445100i		−	0.0158259i
$792$		0			0
$793$	−30.8956	+	30.8956i	−1.09714	+	1.09714i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.8212	+	26.8212i	−0.950056	+	0.950056i	−0.998811	−	0.0487552i	$-0.984475\pi$
$797$	−26.8212	+	26.8212i	−0.950056	+	0.950056i	0.0487552	+	0.998811i	$0.484475\pi$
$798$		0			0
$799$		−	17.0785i		−	0.604193i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.12432	+	4.12432i	0.145544	+	0.145544i
$804$		0			0
$805$	0.125397	+	0.179513i	0.00441967	+	0.00632700i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	43.7948			1.53974			0.769872	−	0.638198i	$-0.220320\pi$
$809$	43.7948			1.53974			0.769872	+	0.638198i	$0.220320\pi$
$810$		0			0
$811$	6.26886			0.220129			0.110065	−	0.993924i	$-0.464894\pi$
$811$	6.26886			0.220129			0.110065	+	0.993924i	$0.464894\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.06175	+	17.2511i	−0.107248	+	0.604280i
$816$		0			0
$817$	38.9201	+	38.9201i	1.36164	+	1.36164i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.43593i		−	0.119915i	−0.998201	−	0.0599573i	$-0.980904\pi$
$821$		−	3.43593i		−	0.119915i	0.998201	−	0.0599573i	$-0.0190965\pi$
$822$		0			0
$823$	−1.60532	+	1.60532i	−0.0559579	+	0.0559579i	−0.734532	−	0.678574i	$-0.762598\pi$
$823$	−1.60532	+	1.60532i	−0.0559579	+	0.0559579i	0.678574	+	0.734532i	$0.262598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.8186	−	39.8186i	1.38463	−	1.38463i	0.548439	−	0.836190i	$-0.315222\pi$
$827$	39.8186	−	39.8186i	1.38463	−	1.38463i	0.836190	−	0.548439i	$-0.184778\pi$
$828$		0			0
$829$			25.9670i			0.901872i	0.892556	+	0.450936i	$0.148910\pi$
$829$			25.9670i			0.901872i	−0.892556	+	0.450936i	$0.851090\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.6686	−	16.6686i	−0.577533	−	0.577533i
$834$		0			0
$835$	−9.59032	+	54.0357i	−0.331887	+	1.86998i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.2013			0.593856			0.296928	−	0.954900i	$-0.404038\pi$
$839$	17.2013			0.593856			0.296928	+	0.954900i	$0.404038\pi$
$840$		0			0
$841$	22.2439			0.767030
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.998611	+	1.42957i	0.0343532	+	0.0491786i
$846$		0			0
$847$	−0.735242	−	0.735242i	−0.0252632	−	0.0252632i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			5.03168i			0.172484i
$852$		0			0
$853$	−23.9819	+	23.9819i	−0.821125	+	0.821125i	−0.986269	−	0.165144i	$-0.947191\pi$
$853$	−23.9819	+	23.9819i	−0.821125	+	0.821125i	0.165144	+	0.986269i	$0.447191\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.328399	+	0.328399i	−0.0112179	+	0.0112179i	−0.712693	−	0.701476i	$-0.752525\pi$
$857$	−0.328399	+	0.328399i	−0.0112179	+	0.0112179i	0.701476	+	0.712693i	$0.252525\pi$
$858$		0			0
$859$		−	44.1890i		−	1.50771i	−0.657041	−	0.753855i	$-0.728192\pi$
$859$		−	44.1890i		−	1.50771i	0.657041	−	0.753855i	$-0.271808\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.3122	+	28.3122i	0.963760	+	0.963760i	0.999366	−	0.0356057i	$-0.0113361\pi$
$863$	28.3122	+	28.3122i	0.963760	+	0.963760i	−0.0356057	+	0.999366i	$0.511336\pi$
$864$		0			0
$865$	−10.9601	−	1.94522i	−0.372656	−	0.0661394i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.80689			−0.230908
$870$		0			0
$871$	17.1793			0.582097
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.550614	+	0.946339i	−0.0186142	+	0.0319921i
$876$		0			0
$877$	−17.3914	−	17.3914i	−0.587265	−	0.587265i	0.349625	−	0.936890i	$-0.386309\pi$
$877$	−17.3914	−	17.3914i	−0.587265	−	0.587265i	−0.936890	+	0.349625i	$0.886309\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.21128i		−	0.242954i	−0.992594	−	0.121477i	$-0.961237\pi$
$881$		−	7.21128i		−	0.242954i	0.992594	−	0.121477i	$-0.0387631\pi$
$882$		0			0
$883$	26.0775	−	26.0775i	0.877579	−	0.877579i	−0.115705	−	0.993284i	$-0.536913\pi$
$883$	26.0775	−	26.0775i	0.877579	−	0.877579i	0.993284	+	0.115705i	$0.0369127\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−30.0318	+	30.0318i	−1.00837	+	1.00837i	−0.00840587	+	0.999965i	$0.502676\pi$
$887$	−30.0318	+	30.0318i	−1.00837	+	1.00837i	−0.999965	+	0.00840587i	$0.997324\pi$
$888$		0			0
$889$			0.0971582i			0.00325858i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.3465	−	25.3465i	−0.848187	−	0.848187i
$894$		0			0
$895$	−29.2292	+	20.4178i	−0.977026	+	0.682493i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25.3583			0.845746
$900$		0			0
$901$	−39.3686			−1.31156
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29.6627	−	20.7206i	0.986021	−	0.688776i
$906$		0			0
$907$	38.8877	+	38.8877i	1.29125	+	1.29125i	0.934017	+	0.357229i	$0.116278\pi$
$907$	38.8877	+	38.8877i	1.29125	+	1.29125i	0.357229	+	0.934017i	$0.383722\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			40.7956i			1.35162i	0.737077	+	0.675809i	$0.236205\pi$
$911$			40.7956i			1.35162i	−0.737077	+	0.675809i	$0.763795\pi$
$912$		0			0
$913$	4.98827	−	4.98827i	0.165088	−	0.165088i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.683713	−	0.683713i	0.0225782	−	0.0225782i
$918$		0			0
$919$		−	36.9356i		−	1.21839i	−0.793019	−	0.609197i	$-0.791492\pi$
$919$		−	36.9356i		−	1.21839i	0.793019	−	0.609197i	$-0.208508\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.369050	+	0.369050i	0.0121474	+	0.0121474i
$924$		0			0
$925$	−22.8250	+	10.5813i	−0.750481	+	0.347912i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	52.2712			1.71496			0.857481	−	0.514516i	$-0.172028\pi$
$929$	52.2712			1.71496			0.857481	+	0.514516i	$0.172028\pi$
$930$		0			0
$931$	−49.4763			−1.62152
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.58931	+	0.814517i	0.150087	+	0.0266376i
$936$		0			0
$937$	17.5678	+	17.5678i	0.573916	+	0.573916i	0.933220	−	0.359305i	$-0.116986\pi$
$937$	17.5678	+	17.5678i	0.573916	+	0.573916i	−0.359305	+	0.933220i	$0.616986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			38.7808i			1.26422i	0.774879	+	0.632109i	$0.217811\pi$
$941$			38.7808i			1.26422i	−0.774879	+	0.632109i	$0.782189\pi$
$942$		0			0
$943$	4.76050	−	4.76050i	0.155023	−	0.155023i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.7743	+	16.7743i	−0.545093	+	0.545093i	−0.925017	−	0.379925i	$-0.875950\pi$
$947$	−16.7743	+	16.7743i	−0.545093	+	0.545093i	0.379925	+	0.925017i	$0.375950\pi$
$948$		0			0
$949$			32.9852i			1.07075i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.35307	+	9.35307i	0.302976	+	0.302976i	0.842177	−	0.539201i	$-0.181274\pi$
$953$	9.35307	+	9.35307i	0.302976	+	0.302976i	−0.539201	+	0.842177i	$0.681274\pi$
$954$		0			0
$955$	−26.6393	−	38.1356i	−0.862027	−	1.23404i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.920162			0.0297136
$960$		0			0
$961$	−18.4513			−0.595204
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.61253	+	42.8920i	−0.245056	+	1.38074i
$966$		0			0
$967$	−9.45808	−	9.45808i	−0.304151	−	0.304151i	0.538484	−	0.842636i	$-0.318997\pi$
$967$	−9.45808	−	9.45808i	−0.304151	−	0.304151i	−0.842636	+	0.538484i	$0.818997\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.34790i		−	0.0432562i	−0.999766	−	0.0216281i	$-0.993115\pi$
$971$		−	1.34790i		−	0.0432562i	0.999766	−	0.0216281i	$-0.00688498\pi$
$972$		0			0
$973$	−0.456353	+	0.456353i	−0.0146300	+	0.0146300i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.0136991	+	0.0136991i	−0.000438275	+	0.000438275i	−0.707326	−	0.706888i	$-0.750099\pi$
$977$	−0.0136991	+	0.0136991i	−0.000438275	+	0.000438275i	0.706888	+	0.707326i	$0.250099\pi$
$978$		0			0
$979$		−	3.26224i		−	0.104262i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.8906	+	19.8906i	0.634413	+	0.634413i	0.949172	−	0.314759i	$-0.101924\pi$
$983$	19.8906	+	19.8906i	0.634413	+	0.634413i	−0.314759	+	0.949172i	$0.601924\pi$
$984$		0			0
$985$	−6.99381	+	39.4059i	−0.222841	+	1.25558i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.77668			0.247284
$990$		0			0
$991$	−34.1343			−1.08431			−0.542155	−	0.840278i	$-0.682392\pi$
$991$	−34.1343			−1.08431			−0.542155	+	0.840278i	$0.682392\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.3765	−	29.1701i	−0.645980	−	0.924756i
$996$		0			0
$997$	13.1742	+	13.1742i	0.417232	+	0.417232i	0.884248	−	0.467017i	$-0.154671\pi$
$997$	13.1742	+	13.1742i	0.417232	+	0.417232i	−0.467017	+	0.884248i	$0.654671\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.20	yes	44
3.2	odd	2	inner	4140.2.s.b.737.3	✓	44
5.3	odd	4	inner	4140.2.s.b.2393.3	yes	44
15.8	even	4	inner	4140.2.s.b.2393.20	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.3	✓	44	3.2	odd	2	inner
4140.2.s.b.737.20	yes	44	1.1	even	1	trivial
4140.2.s.b.2393.3	yes	44	5.3	odd	4	inner
4140.2.s.b.2393.20	yes	44	15.8	even	4	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.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.83311 - 1.28050i) q^{5} +(-0.0692454 - 0.0692454i) q^{7} +O(q^{10})\)	\(q+(1.83311 - 1.28050i) q^{5} +(-0.0692454 - 0.0692454i) q^{7} +0.618138i q^{11} +(-2.47186 + 2.47186i) q^{13} +(2.38450 - 2.38450i) q^{17} -7.07775i q^{19} +(-0.707107 - 0.707107i) q^{23} +(1.72061 - 4.69462i) q^{25} -7.15848 q^{29} -3.54241 q^{31} +(-0.215604 - 0.0382657i) q^{35} +(-3.55794 - 3.55794i) q^{37} +6.73237i q^{41} +(-5.49894 + 5.49894i) q^{43} +(3.58115 - 3.58115i) q^{47} -6.99041i q^{49} +(-8.25513 - 8.25513i) q^{53} +(0.791529 + 1.13312i) q^{55} +1.16195 q^{59} +12.4990 q^{61} +(-1.36597 + 7.69642i) q^{65} +(-3.47497 - 3.47497i) q^{67} -0.149301i q^{71} +(6.67216 - 6.67216i) q^{73} +(0.0428033 - 0.0428033i) q^{77} +11.0119i q^{79} +(-8.06983 - 8.06983i) q^{83} +(1.31769 - 7.42441i) q^{85} -5.27752 q^{89} +0.342330 q^{91} +(-9.06309 - 12.9743i) q^{95} +(4.39996 + 4.39996i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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