LMFDB - Embedded newform 4140.2.s.b.737.5

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			737.5
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79146 + 1.33815i) q^{5} +(-2.07691 - 2.07691i) q^{7} +O(q^{10})$	$q+(-1.79146 + 1.33815i) q^{5} +(-2.07691 - 2.07691i) q^{7} +3.22627i q^{11} +(4.92009 - 4.92009i) q^{13} +(-3.44072 + 3.44072i) q^{17} +1.09153i q^{19} +(0.707107 + 0.707107i) q^{23} +(1.41868 - 4.79451i) q^{25} -6.42174 q^{29} -5.08128 q^{31} +(6.49993 + 0.941481i) q^{35} +(3.18217 + 3.18217i) q^{37} +3.93089i q^{41} +(-0.659981 + 0.659981i) q^{43} +(9.43041 - 9.43041i) q^{47} +1.62709i q^{49} +(-5.54110 - 5.54110i) q^{53} +(-4.31725 - 5.77975i) q^{55} +2.11321 q^{59} -1.31239 q^{61} +(-2.23032 + 15.3980i) q^{65} +(7.76215 + 7.76215i) q^{67} -4.31910i q^{71} +(-7.21024 + 7.21024i) q^{73} +(6.70067 - 6.70067i) q^{77} -10.2687i q^{79} +(0.143504 + 0.143504i) q^{83} +(1.55971 - 10.7681i) q^{85} +14.9640 q^{89} -20.4371 q^{91} +(-1.46064 - 1.95544i) q^{95} +(5.05998 + 5.05998i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.79146	+	1.33815i	−0.801167	+	0.598441i
$5$	−1.79146	+	1.33815i	−0.801167	+	0.598441i
$6$		0			0
$7$	−2.07691	−	2.07691i	−0.784997	−	0.784997i	0.195672	−	0.980669i	$-0.437311\pi$
$7$	−2.07691	−	2.07691i	−0.784997	−	0.784997i	−0.980669	+	0.195672i	$0.937311\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.22627i			0.972758i	0.873748	+	0.486379i	$0.161683\pi$
$11$			3.22627i			0.972758i	−0.873748	+	0.486379i	$0.838317\pi$
$12$		0			0
$13$	4.92009	−	4.92009i	1.36459	−	1.36459i	0.496620	−	0.867968i	$-0.334574\pi$
$13$	4.92009	−	4.92009i	1.36459	−	1.36459i	0.867968	−	0.496620i	$-0.165426\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.44072	+	3.44072i	−0.834498	+	0.834498i	−0.988128	−	0.153631i	$-0.950903\pi$
$17$	−3.44072	+	3.44072i	−0.834498	+	0.834498i	0.153631	+	0.988128i	$0.450903\pi$
$18$		0			0
$19$			1.09153i			0.250415i	0.992131	+	0.125207i	$0.0399596\pi$
$19$			1.09153i			0.250415i	−0.992131	+	0.125207i	$0.960040\pi$
$20$		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
$25$	1.41868	−	4.79451i	0.283737	−	0.958902i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.42174			−1.19249			−0.596244	−	0.802803i	$-0.703341\pi$
$29$	−6.42174			−1.19249			−0.596244	+	0.802803i	$0.703341\pi$
$30$		0			0
$31$	−5.08128			−0.912626			−0.456313	−	0.889819i	$-0.650830\pi$
$31$	−5.08128			−0.912626			−0.456313	+	0.889819i	$0.650830\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.49993	+	0.941481i	1.09869	+	0.159139i
$36$		0			0
$37$	3.18217	+	3.18217i	0.523145	+	0.523145i	0.918520	−	0.395375i	$-0.129385\pi$
$37$	3.18217	+	3.18217i	0.523145	+	0.523145i	−0.395375	+	0.918520i	$0.629385\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3.93089i			0.613901i	0.951725	+	0.306951i	$0.0993087\pi$
$41$			3.93089i			0.613901i	−0.951725	+	0.306951i	$0.900691\pi$
$42$		0			0
$43$	−0.659981	+	0.659981i	−0.100646	+	0.100646i	−0.755637	−	0.654991i	$-0.772673\pi$
$43$	−0.659981	+	0.659981i	−0.100646	+	0.100646i	0.654991	+	0.755637i	$0.272673\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.43041	−	9.43041i	1.37557	−	1.37557i	0.523608	−	0.851960i	$-0.324586\pi$
$47$	9.43041	−	9.43041i	1.37557	−	1.37557i	0.851960	−	0.523608i	$-0.175414\pi$
$48$		0			0
$49$			1.62709i			0.232441i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.54110	−	5.54110i	−0.761128	−	0.761128i	0.215398	−	0.976526i	$-0.430895\pi$
$53$	−5.54110	−	5.54110i	−0.761128	−	0.761128i	−0.976526	+	0.215398i	$0.930895\pi$
$54$		0			0
$55$	−4.31725	−	5.77975i	−0.582138	−	0.779342i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.11321			0.275117			0.137559	−	0.990494i	$-0.456074\pi$
$59$	2.11321			0.275117			0.137559	+	0.990494i	$0.456074\pi$
$60$		0			0
$61$	−1.31239			−0.168035			−0.0840174	−	0.996464i	$-0.526775\pi$
$61$	−1.31239			−0.168035			−0.0840174	+	0.996464i	$0.526775\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.23032	+	15.3980i	−0.276637	+	1.90989i
$66$		0			0
$67$	7.76215	+	7.76215i	0.948298	+	0.948298i	0.998728	−	0.0504298i	$-0.0160591\pi$
$67$	7.76215	+	7.76215i	0.948298	+	0.948298i	−0.0504298	+	0.998728i	$0.516059\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.31910i		−	0.512583i	−0.966600	−	0.256291i	$-0.917499\pi$
$71$		−	4.31910i		−	0.512583i	0.966600	−	0.256291i	$-0.0825007\pi$
$72$		0			0
$73$	−7.21024	+	7.21024i	−0.843895	+	0.843895i	−0.989363	−	0.145468i	$-0.953531\pi$
$73$	−7.21024	+	7.21024i	−0.843895	+	0.843895i	0.145468	+	0.989363i	$0.453531\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.70067	−	6.70067i	0.763612	−	0.763612i
$78$		0			0
$79$		−	10.2687i		−	1.15532i	−0.816277	−	0.577661i	$-0.803965\pi$
$79$		−	10.2687i		−	1.15532i	0.816277	−	0.577661i	$-0.196035\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.143504	+	0.143504i	0.0157516	+	0.0157516i	0.714939	−	0.699187i	$-0.246455\pi$
$83$	0.143504	+	0.143504i	0.0157516	+	0.0157516i	−0.699187	+	0.714939i	$0.746455\pi$
$84$		0			0
$85$	1.55971	−	10.7681i	0.169174	−	1.16797i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.9640			1.58618			0.793090	−	0.609105i	$-0.208471\pi$
$89$	14.9640			1.58618			0.793090	+	0.609105i	$0.208471\pi$
$90$		0			0
$91$	−20.4371			−2.14240
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.46064	−	1.95544i	−0.149859	−	0.200624i
$96$		0			0
$97$	5.05998	+	5.05998i	0.513763	+	0.513763i	0.915677	−	0.401914i	$-0.131655\pi$
$97$	5.05998	+	5.05998i	0.513763	+	0.513763i	−0.401914	+	0.915677i	$0.631655\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			4.34542i			0.432386i	0.976351	+	0.216193i	$0.0693640\pi$
$101$			4.34542i			0.432386i	−0.976351	+	0.216193i	$0.930636\pi$
$102$		0			0
$103$	13.5424	−	13.5424i	1.33437	−	1.33437i	0.432955	−	0.901416i	$-0.357471\pi$
$103$	13.5424	−	13.5424i	1.33437	−	1.33437i	0.901416	−	0.432955i	$-0.142529\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.26237	+	4.26237i	−0.412059	+	0.412059i	−0.882455	−	0.470396i	$-0.844111\pi$
$107$	−4.26237	+	4.26237i	−0.412059	+	0.412059i	0.470396	+	0.882455i	$0.344111\pi$
$108$		0			0
$109$		−	5.29238i		−	0.506918i	−0.967346	−	0.253459i	$-0.918432\pi$
$109$		−	5.29238i		−	0.506918i	0.967346	−	0.253459i	$-0.0815682\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.0012	+	10.0012i	0.940830	+	0.940830i	0.998345	−	0.0575143i	$-0.0183175\pi$
$113$	10.0012	+	10.0012i	0.940830	+	0.940830i	−0.0575143	+	0.998345i	$0.518317\pi$
$114$		0			0
$115$	−2.21297	−	0.320538i	−0.206361	−	0.0298903i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.2921			1.31016
$120$		0			0
$121$	0.591160			0.0537418
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.87428	+	10.4876i	0.346526	+	0.938040i
$126$		0			0
$127$	−0.950873	−	0.950873i	−0.0843763	−	0.0843763i	0.663659	−	0.748035i	$-0.269003\pi$
$127$	−0.950873	−	0.950873i	−0.0843763	−	0.0843763i	−0.748035	+	0.663659i	$0.769003\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	9.03565i		−	0.789449i	−0.918800	−	0.394724i	$-0.870840\pi$
$131$		−	9.03565i		−	0.789449i	0.918800	−	0.394724i	$-0.129160\pi$
$132$		0			0
$133$	2.26701	−	2.26701i	0.196575	−	0.196575i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.5209	−	14.5209i	1.24060	−	1.24060i	0.280848	−	0.959752i	$-0.409384\pi$
$137$	14.5209	−	14.5209i	1.24060	−	1.24060i	0.959752	−	0.280848i	$-0.0906156\pi$
$138$		0			0
$139$			3.41041i			0.289267i	0.989485	+	0.144633i	$0.0462003\pi$
$139$			3.41041i			0.289267i	−0.989485	+	0.144633i	$0.953800\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.8736	+	15.8736i	1.32741	+	1.32741i
$144$		0			0
$145$	11.5043	−	8.59328i	0.955382	−	0.713633i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.97015			0.161401			0.0807003	−	0.996738i	$-0.474284\pi$
$149$	1.97015			0.161401			0.0807003	+	0.996738i	$0.474284\pi$
$150$		0			0
$151$	17.6629			1.43739			0.718693	−	0.695327i	$-0.244741\pi$
$151$	17.6629			1.43739			0.718693	+	0.695327i	$0.244741\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.10294	−	6.79954i	0.731165	−	0.546152i
$156$		0			0
$157$	10.2551	+	10.2551i	0.818444	+	0.818444i	0.985882	−	0.167439i	$-0.0535497\pi$
$157$	10.2551	+	10.2551i	0.818444	+	0.818444i	−0.167439	+	0.985882i	$0.553550\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.93719i		−	0.231483i
$162$		0			0
$163$	−2.74811	+	2.74811i	−0.215249	+	0.215249i	−0.806493	−	0.591244i	$-0.798637\pi$
$163$	−2.74811	+	2.74811i	−0.215249	+	0.215249i	0.591244	+	0.806493i	$0.298637\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.98842	−	4.98842i	0.386016	−	0.386016i	−0.487248	−	0.873264i	$-0.661999\pi$
$167$	4.98842	−	4.98842i	0.386016	−	0.386016i	0.873264	+	0.487248i	$0.161999\pi$
$168$		0			0
$169$		−	35.4146i		−	2.72420i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.2447	+	14.2447i	1.08300	+	1.08300i	0.996228	+	0.0867761i	$0.0276565\pi$
$173$	14.2447	+	14.2447i	1.08300	+	1.08300i	0.0867761	+	0.996228i	$0.472344\pi$
$174$		0			0
$175$	−12.9042	+	7.01128i	−0.975468	+	0.530003i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.5103			−0.785573			−0.392787	−	0.919630i	$-0.628489\pi$
$179$	−10.5103			−0.785573			−0.392787	+	0.919630i	$0.628489\pi$
$180$		0			0
$181$	16.0040			1.18957			0.594785	−	0.803885i	$-0.297237\pi$
$181$	16.0040			1.18957			0.594785	+	0.803885i	$0.297237\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.95897	−	1.44251i	−0.732198	−	0.106055i
$186$		0			0
$187$	−11.1007	−	11.1007i	−0.811764	−	0.811764i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.10514i			0.152323i	0.997096	+	0.0761613i	$0.0242664\pi$
$191$			2.10514i			0.152323i	−0.997096	+	0.0761613i	$0.975734\pi$
$192$		0			0
$193$	13.3295	−	13.3295i	0.959482	−	0.959482i	−0.0397288	−	0.999210i	$-0.512649\pi$
$193$	13.3295	−	13.3295i	0.959482	−	0.959482i	0.999210	+	0.0397288i	$0.0126494\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.5873	−	14.5873i	1.03930	−	1.03930i	0.0401091	−	0.999195i	$-0.487229\pi$
$197$	14.5873	−	14.5873i	1.03930	−	1.03930i	0.999195	−	0.0401091i	$-0.0127705\pi$
$198$		0			0
$199$		−	10.9827i		−	0.778545i	−0.921123	−	0.389273i	$-0.872726\pi$
$199$		−	10.9827i		−	0.778545i	0.921123	−	0.389273i	$-0.127274\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.3374	+	13.3374i	0.936099	+	0.936099i
$204$		0			0
$205$	−5.26014	−	7.04204i	−0.367384	−	0.491838i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.52159			−0.243593
$210$		0			0
$211$	21.0673			1.45033			0.725166	−	0.688574i	$-0.241763\pi$
$211$	21.0673			1.45033			0.725166	+	0.688574i	$0.241763\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.299175	−	2.06549i	0.0204036	−	0.140865i
$216$		0			0
$217$	10.5534	+	10.5534i	0.716408	+	0.716408i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			33.8573i			2.27749i
$222$		0			0
$223$	−15.0582	+	15.0582i	−1.00837	+	1.00837i	−0.00840967	+	0.999965i	$0.502677\pi$
$223$	−15.0582	+	15.0582i	−1.00837	+	1.00837i	−0.999965	+	0.00840967i	$0.997323\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.52586	+	5.52586i	−0.366764	+	0.366764i	−0.866296	−	0.499531i	$-0.833506\pi$
$227$	−5.52586	+	5.52586i	−0.366764	+	0.366764i	0.499531	+	0.866296i	$0.333506\pi$
$228$		0			0
$229$			21.2128i			1.40178i	0.713269	+	0.700891i	$0.247214\pi$
$229$			21.2128i			1.40178i	−0.713269	+	0.700891i	$0.752786\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.93354	+	6.93354i	0.454231	+	0.454231i	0.896756	−	0.442525i	$-0.145917\pi$
$233$	6.93354	+	6.93354i	0.454231	+	0.454231i	−0.442525	+	0.896756i	$0.645917\pi$
$234$		0			0
$235$	−4.27489	+	29.5136i	−0.278863	+	1.92525i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.2837			−1.05331			−0.526653	−	0.850081i	$-0.676553\pi$
$239$	−16.2837			−1.05331			−0.526653	+	0.850081i	$0.676553\pi$
$240$		0			0
$241$	−3.40648			−0.219430			−0.109715	−	0.993963i	$-0.534994\pi$
$241$	−3.40648			−0.219430			−0.109715	+	0.993963i	$0.534994\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.17729	−	2.91487i	−0.139102	−	0.186224i
$246$		0			0
$247$	5.37044	+	5.37044i	0.341713	+	0.341713i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	28.7535i		−	1.81490i	−0.420158	−	0.907451i	$-0.638025\pi$
$251$		−	28.7535i		−	1.81490i	0.420158	−	0.907451i	$-0.361975\pi$
$252$		0			0
$253$	−2.28132	+	2.28132i	−0.143425	+	0.143425i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.06985	−	2.06985i	0.129114	−	0.129114i	−0.639597	−	0.768711i	$-0.720899\pi$
$257$	2.06985	−	2.06985i	0.129114	−	0.129114i	0.768711	+	0.639597i	$0.220899\pi$
$258$		0			0
$259$		−	13.2181i		−	0.821335i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.35079	+	3.35079i	0.206619	+	0.206619i	0.802829	−	0.596210i	$-0.203327\pi$
$263$	3.35079	+	3.35079i	0.206619	+	0.206619i	−0.596210	+	0.802829i	$0.703327\pi$
$264$		0			0
$265$	17.3415	+	2.51183i	1.06528	+	0.154300i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.03191			0.428743			0.214372	−	0.976752i	$-0.431230\pi$
$269$	7.03191			0.428743			0.214372	+	0.976752i	$0.431230\pi$
$270$		0			0
$271$	−10.7733			−0.654432			−0.327216	−	0.944950i	$-0.606110\pi$
$271$	−10.7733			−0.654432			−0.327216	+	0.944950i	$0.606110\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.4684	+	4.57706i	0.932780	+	0.276007i
$276$		0			0
$277$	4.25480	+	4.25480i	0.255646	+	0.255646i	0.823281	−	0.567634i	$-0.192141\pi$
$277$	4.25480	+	4.25480i	0.255646	+	0.255646i	−0.567634	+	0.823281i	$0.692141\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.7221i			1.29583i	0.761711	+	0.647917i	$0.224360\pi$
$281$			21.7221i			1.29583i	−0.761711	+	0.647917i	$0.775640\pi$
$282$		0			0
$283$	7.94900	−	7.94900i	0.472519	−	0.472519i	−0.430210	−	0.902729i	$-0.641560\pi$
$283$	7.94900	−	7.94900i	0.472519	−	0.472519i	0.902729	+	0.430210i	$0.141560\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.16409	−	8.16409i	0.481911	−	0.481911i
$288$		0			0
$289$		−	6.67714i		−	0.392773i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.0812	−	13.0812i	−0.764213	−	0.764213i	0.212868	−	0.977081i	$-0.431720\pi$
$293$	−13.0812	−	13.0812i	−0.764213	−	0.764213i	−0.977081	+	0.212868i	$0.931720\pi$
$294$		0			0
$295$	−3.78575	+	2.82781i	−0.220415	+	0.164641i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.95806			0.402395
$300$		0			0
$301$	2.74144			0.158014
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.35111	−	1.75619i	0.134624	−	0.100559i
$306$		0			0
$307$	−15.9966	−	15.9966i	−0.912973	−	0.912973i	0.0835323	−	0.996505i	$-0.473380\pi$
$307$	−15.9966	−	15.9966i	−0.912973	−	0.912973i	−0.996505	+	0.0835323i	$0.973380\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.9516i		−	0.847825i	−0.905703	−	0.423913i	$-0.860656\pi$
$311$		−	14.9516i		−	0.847825i	0.905703	−	0.423913i	$-0.139344\pi$
$312$		0			0
$313$	−12.7743	+	12.7743i	−0.722047	+	0.722047i	−0.969022	−	0.246975i	$-0.920563\pi$
$313$	−12.7743	+	12.7743i	−0.722047	+	0.722047i	0.246975	+	0.969022i	$0.420563\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.8104	−	15.8104i	0.888000	−	0.888000i	−0.106331	−	0.994331i	$-0.533910\pi$
$317$	15.8104	−	15.8104i	0.888000	−	0.888000i	0.994331	+	0.106331i	$0.0339103\pi$
$318$		0			0
$319$		−	20.7183i		−	1.16000i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.75566	−	3.75566i	−0.208971	−	0.208971i
$324$		0			0
$325$	−16.6094	−	30.5695i	−0.921323	−	1.69569i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−39.1722			−2.15963
$330$		0			0
$331$	18.7571			1.03098			0.515491	−	0.856895i	$-0.327609\pi$
$331$	18.7571			1.03098			0.515491	+	0.856895i	$0.327609\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−24.2926	−	3.51865i	−1.32725	−	0.192245i
$336$		0			0
$337$	−8.63677	−	8.63677i	−0.470475	−	0.470475i	0.431593	−	0.902068i	$-0.357952\pi$
$337$	−8.63677	−	8.63677i	−0.470475	−	0.470475i	−0.902068	+	0.431593i	$0.857952\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	16.3936i		−	0.887764i
$342$		0			0
$343$	−11.1590	+	11.1590i	−0.602532	+	0.602532i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.6113	+	17.6113i	−0.945424	+	0.945424i	−0.998586	−	0.0531621i	$-0.983070\pi$
$347$	−17.6113	+	17.6113i	−0.945424	+	0.945424i	0.0531621	+	0.998586i	$0.483070\pi$
$348$		0			0
$349$			28.3085i			1.51532i	0.652650	+	0.757659i	$0.273657\pi$
$349$			28.3085i			1.51532i	−0.652650	+	0.757659i	$0.726343\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.72532	+	8.72532i	0.464402	+	0.464402i	0.900095	−	0.435693i	$-0.143497\pi$
$353$	8.72532	+	8.72532i	0.464402	+	0.464402i	−0.435693	+	0.900095i	$0.643497\pi$
$354$		0			0
$355$	5.77962	+	7.73751i	0.306751	+	0.410664i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.2104			0.961107			0.480553	−	0.876965i	$-0.340436\pi$
$359$	18.2104			0.961107			0.480553	+	0.876965i	$0.340436\pi$
$360$		0			0
$361$	17.8086			0.937292
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.26847	−	22.5653i	0.171079	−	1.18112i
$366$		0			0
$367$	−0.0211932	−	0.0211932i	−0.00110627	−	0.00110627i	0.706553	−	0.707660i	$-0.250249\pi$
$367$	−0.0211932	−	0.0211932i	−0.00110627	−	0.00110627i	−0.707660	+	0.706553i	$0.750249\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			23.0167i			1.19497i
$372$		0			0
$373$	−7.42722	+	7.42722i	−0.384567	+	0.384567i	−0.872744	−	0.488178i	$-0.837662\pi$
$373$	−7.42722	+	7.42722i	−0.384567	+	0.384567i	0.488178	+	0.872744i	$0.337662\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−31.5956	+	31.5956i	−1.62725	+	1.62725i
$378$		0			0
$379$		−	15.8634i		−	0.814849i	−0.913239	−	0.407425i	$-0.866427\pi$
$379$		−	15.8634i		−	0.814849i	0.913239	−	0.407425i	$-0.133573\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.680180	+	0.680180i	0.0347556	+	0.0347556i	0.724271	−	0.689515i	$-0.242176\pi$
$383$	0.680180	+	0.680180i	0.0347556	+	0.0347556i	−0.689515	+	0.724271i	$0.742176\pi$
$384$		0			0
$385$	−3.03747	+	20.9705i	−0.154804	+	1.06876i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.9723			0.759127			0.379564	−	0.925166i	$-0.376074\pi$
$389$	14.9723			0.759127			0.379564	+	0.925166i	$0.376074\pi$
$390$		0			0
$391$	−4.86592			−0.246080
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.7412	+	18.3961i	0.691392	+	0.925606i
$396$		0			0
$397$	−23.5879	−	23.5879i	−1.18384	−	1.18384i	−0.978741	−	0.205100i	$-0.934248\pi$
$397$	−23.5879	−	23.5879i	−1.18384	−	1.18384i	−0.205100	−	0.978741i	$-0.565752\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	9.70223i		−	0.484506i	−0.970213	−	0.242253i	$-0.922114\pi$
$401$		−	9.70223i		−	0.484506i	0.970213	−	0.242253i	$-0.0778864\pi$
$402$		0			0
$403$	−25.0004	+	25.0004i	−1.24536	+	1.24536i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.2665	+	10.2665i	−0.508894	+	0.508894i
$408$		0			0
$409$		−	30.5760i		−	1.51189i	−0.654637	−	0.755944i	$-0.727178\pi$
$409$		−	30.5760i		−	1.51189i	0.654637	−	0.755944i	$-0.272822\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.38895	−	4.38895i	−0.215966	−	0.215966i
$414$		0			0
$415$	−0.449111	−	0.0650515i	−0.0220460	−	0.00319325i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.8248			−0.577681			−0.288840	−	0.957377i	$-0.593270\pi$
$419$	−11.8248			−0.577681			−0.288840	+	0.957377i	$0.593270\pi$
$420$		0			0
$421$	29.1695			1.42163			0.710816	−	0.703378i	$-0.248326\pi$
$421$	29.1695			1.42163			0.710816	+	0.703378i	$0.248326\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.6153	+	21.3779i	0.563424	+	1.03698i
$426$		0			0
$427$	2.72572	+	2.72572i	0.131907	+	0.131907i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.0979i			0.871745i	0.900009	+	0.435872i	$0.143560\pi$
$431$			18.0979i			0.871745i	−0.900009	+	0.435872i	$0.856440\pi$
$432$		0			0
$433$	−13.0949	+	13.0949i	−0.629302	+	0.629302i	−0.947892	−	0.318591i	$-0.896790\pi$
$433$	−13.0949	+	13.0949i	−0.629302	+	0.629302i	0.318591	+	0.947892i	$0.396790\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.771831	+	0.771831i	−0.0369217	+	0.0369217i
$438$		0			0
$439$		−	40.0592i		−	1.91192i	−0.293494	−	0.955961i	$-0.594818\pi$
$439$		−	40.0592i		−	1.91192i	0.293494	−	0.955961i	$-0.405182\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.50371	+	6.50371i	0.309000	+	0.309000i	0.844522	−	0.535521i	$-0.179885\pi$
$443$	6.50371	+	6.50371i	0.309000	+	0.309000i	−0.535521	+	0.844522i	$0.679885\pi$
$444$		0			0
$445$	−26.8074	+	20.0241i	−1.27079	+	0.949235i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.85394			−0.465036			−0.232518	−	0.972592i	$-0.574696\pi$
$449$	−9.85394			−0.465036			−0.232518	+	0.972592i	$0.574696\pi$
$450$		0			0
$451$	−12.6821			−0.597178
$452$		0			0
$453$		0			0
$454$		0			0
$455$	36.6124	−	27.3481i	1.71642	−	1.28210i
$456$		0			0
$457$	22.3000	+	22.3000i	1.04315	+	1.04315i	0.999026	+	0.0441258i	$0.0140502\pi$
$457$	22.3000	+	22.3000i	1.04315	+	1.04315i	0.0441258	+	0.999026i	$0.485950\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			16.7661i			0.780876i	0.920629	+	0.390438i	$0.127676\pi$
$461$			16.7661i			0.780876i	−0.920629	+	0.390438i	$0.872324\pi$
$462$		0			0
$463$	4.78426	−	4.78426i	0.222343	−	0.222343i	−0.587141	−	0.809485i	$-0.699747\pi$
$463$	4.78426	−	4.78426i	0.222343	−	0.222343i	0.809485	+	0.587141i	$0.199747\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.47320	−	1.47320i	0.0681714	−	0.0681714i	−0.672199	−	0.740370i	$-0.734650\pi$
$467$	1.47320	−	1.47320i	0.0681714	−	0.0681714i	0.740370	+	0.672199i	$0.234650\pi$
$468$		0			0
$469$		−	32.2425i		−	1.48882i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.12928	−	2.12928i	−0.0979044	−	0.0979044i
$474$		0			0
$475$	5.23337	+	1.54854i	0.240123	+	0.0710519i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.87686			−0.451285			−0.225643	−	0.974210i	$-0.572448\pi$
$479$	−9.87686			−0.451285			−0.225643	+	0.974210i	$0.572448\pi$
$480$		0			0
$481$	31.3131			1.42775
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.8358	−	2.29373i	−0.719067	−	0.104153i
$486$		0			0
$487$	−27.2808	−	27.2808i	−1.23621	−	1.23621i	−0.961537	−	0.274675i	$-0.911430\pi$
$487$	−27.2808	−	27.2808i	−1.23621	−	1.23621i	−0.274675	−	0.961537i	$-0.588570\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	16.2576i		−	0.733697i	−0.930281	−	0.366849i	$-0.880437\pi$
$491$		−	16.2576i		−	0.733697i	0.930281	−	0.366849i	$-0.119563\pi$
$492$		0			0
$493$	22.0954	−	22.0954i	0.995128	−	0.995128i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.97037	+	8.97037i	−0.402376	+	0.402376i
$498$		0			0
$499$		−	22.5006i		−	1.00727i	−0.863918	−	0.503633i	$-0.831997\pi$
$499$		−	22.5006i		−	1.00727i	0.863918	−	0.503633i	$-0.168003\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.90262	−	3.90262i	−0.174009	−	0.174009i	0.614729	−	0.788738i	$-0.289265\pi$
$503$	−3.90262	−	3.90262i	−0.174009	−	0.174009i	−0.788738	+	0.614729i	$0.789265\pi$
$504$		0			0
$505$	−5.81485	−	7.78467i	−0.258757	−	0.346413i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.58821			−0.0703961			−0.0351980	−	0.999380i	$-0.511206\pi$
$509$	−1.58821			−0.0703961			−0.0351980	+	0.999380i	$0.511206\pi$
$510$		0			0
$511$	29.9500			1.32491
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.13888	+	42.3825i	−0.270512	+	1.86760i
$516$		0			0
$517$	30.4251	+	30.4251i	1.33809	+	1.33809i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	13.3814i		−	0.586249i	−0.956074	−	0.293125i	$-0.905305\pi$
$521$		−	13.3814i		−	0.586249i	0.956074	−	0.293125i	$-0.0946951\pi$
$522$		0			0
$523$	17.7895	−	17.7895i	0.777881	−	0.777881i	−0.201589	−	0.979470i	$-0.564611\pi$
$523$	17.7895	−	17.7895i	0.777881	−	0.777881i	0.979470	+	0.201589i	$0.0646106\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.4833	−	17.4833i	0.761584	−	0.761584i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.3403	+	19.3403i	0.837723	+	0.837723i
$534$		0			0
$535$	1.93217	−	13.3396i	0.0835350	−	0.576721i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.24942			−0.226109
$540$		0			0
$541$	−7.05304			−0.303234			−0.151617	−	0.988439i	$-0.548448\pi$
$541$	−7.05304			−0.303234			−0.151617	+	0.988439i	$0.548448\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.08202	+	9.48110i	0.303360	+	0.406126i
$546$		0			0
$547$	28.0260	+	28.0260i	1.19830	+	1.19830i	0.974674	+	0.223629i	$0.0717903\pi$
$547$	28.0260	+	28.0260i	1.19830	+	1.19830i	0.223629	+	0.974674i	$0.428210\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7.00955i		−	0.298617i
$552$		0			0
$553$	−21.3272	+	21.3272i	−0.906925	+	0.906925i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.6010	+	21.6010i	−0.915263	+	0.915263i	−0.996680	−	0.0814174i	$-0.974055\pi$
$557$	−21.6010	+	21.6010i	−0.915263	+	0.915263i	0.0814174	+	0.996680i	$0.474055\pi$
$558$		0			0
$559$			6.49433i			0.274681i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.43394	−	8.43394i	−0.355448	−	0.355448i	0.506684	−	0.862132i	$-0.330871\pi$
$563$	−8.43394	−	8.43394i	−0.355448	−	0.355448i	−0.862132	+	0.506684i	$0.830871\pi$
$564$		0			0
$565$	−31.2998	−	4.53362i	−1.31679	−	0.190731i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.5316			0.525352			0.262676	−	0.964884i	$-0.415395\pi$
$569$	12.5316			0.525352			0.262676	+	0.964884i	$0.415395\pi$
$570$		0			0
$571$	−29.1831			−1.22127			−0.610637	−	0.791911i	$-0.709086\pi$
$571$	−29.1831			−1.22127			−0.610637	+	0.791911i	$0.709086\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.39339	−	2.38707i	0.183217	−	0.0995477i
$576$		0			0
$577$	17.5195	+	17.5195i	0.729346	+	0.729346i	0.970489	−	0.241143i	$-0.0775224\pi$
$577$	17.5195	+	17.5195i	0.729346	+	0.729346i	−0.241143	+	0.970489i	$0.577522\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.596087i		−	0.0247299i
$582$		0			0
$583$	17.8771	−	17.8771i	0.740394	−	0.740394i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.4525	−	27.4525i	1.13309	−	1.13309i	0.143426	−	0.989661i	$-0.454188\pi$
$587$	27.4525	−	27.4525i	1.13309	−	1.13309i	0.989661	−	0.143426i	$-0.0458118\pi$
$588$		0			0
$589$		−	5.54639i		−	0.228535i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.8610	−	24.8610i	−1.02092	−	1.02092i	−0.999776	−	0.0211428i	$-0.993270\pi$
$593$	−24.8610	−	24.8610i	−1.02092	−	1.02092i	−0.0211428	−	0.999776i	$-0.506730\pi$
$594$		0			0
$595$	−25.6038	+	19.1251i	−1.04965	+	0.784051i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	40.7591			1.66537			0.832685	−	0.553747i	$-0.186802\pi$
$599$	40.7591			1.66537			0.832685	+	0.553747i	$0.186802\pi$
$600$		0			0
$601$	−0.0947750			−0.00386595			−0.00193298	−	0.999998i	$-0.500615\pi$
$601$	−0.0947750			−0.00386595			−0.00193298	+	0.999998i	$0.500615\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.05904	+	0.791063i	−0.0430562	+	0.0321613i
$606$		0			0
$607$	−11.8813	−	11.8813i	−0.482246	−	0.482246i	0.423602	−	0.905848i	$-0.360766\pi$
$607$	−11.8813	−	11.8813i	−0.482246	−	0.482246i	−0.905848	+	0.423602i	$0.860766\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	92.7970i		−	3.75416i
$612$		0			0
$613$	−8.27740	+	8.27740i	−0.334321	+	0.334321i	−0.854225	−	0.519904i	$-0.825968\pi$
$613$	−8.27740	+	8.27740i	−0.334321	+	0.334321i	0.519904	+	0.854225i	$0.325968\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.98713	−	2.98713i	0.120257	−	0.120257i	−0.644417	−	0.764674i	$-0.722900\pi$
$617$	2.98713	−	2.98713i	0.120257	−	0.120257i	0.764674	+	0.644417i	$0.222900\pi$
$618$		0			0
$619$			25.7419i			1.03466i	0.855787	+	0.517328i	$0.173073\pi$
$619$			25.7419i			1.03466i	−0.855787	+	0.517328i	$0.826927\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−31.0788	−	31.0788i	−1.24515	−	1.24515i
$624$		0			0
$625$	−20.9747	−	13.6038i	−0.838987	−	0.544152i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−21.8979			−0.873127
$630$		0			0
$631$	−33.3196			−1.32643			−0.663215	−	0.748429i	$-0.730809\pi$
$631$	−33.3196			−1.32643			−0.663215	+	0.748429i	$0.730809\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.97587	+	0.431039i	0.118094	+	0.0171053i
$636$		0			0
$637$	8.00541	+	8.00541i	0.317186	+	0.317186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17.5390i		−	0.692749i	−0.938096	−	0.346375i	$-0.887413\pi$
$641$		−	17.5390i		−	0.692749i	0.938096	−	0.346375i	$-0.112587\pi$
$642$		0			0
$643$	8.97642	−	8.97642i	0.353996	−	0.353996i	−0.507598	−	0.861594i	$-0.669467\pi$
$643$	8.97642	−	8.97642i	0.353996	−	0.353996i	0.861594	+	0.507598i	$0.169467\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.46482	+	8.46482i	−0.332786	+	0.332786i	−0.853644	−	0.520857i	$-0.825612\pi$
$647$	−8.46482	+	8.46482i	−0.332786	+	0.332786i	0.520857	+	0.853644i	$0.325612\pi$
$648$		0			0
$649$			6.81781i			0.267622i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.27605	−	5.27605i	−0.206468	−	0.206468i	0.596296	−	0.802764i	$-0.296638\pi$
$653$	−5.27605	−	5.27605i	−0.206468	−	0.206468i	−0.802764	+	0.596296i	$0.796638\pi$
$654$		0			0
$655$	12.0911	+	16.1870i	0.472438	+	0.632480i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.6579			−1.50590			−0.752949	−	0.658078i	$-0.771369\pi$
$659$	−38.6579			−1.50590			−0.752949	+	0.658078i	$0.771369\pi$
$660$		0			0
$661$	18.4444			0.717406			0.358703	−	0.933452i	$-0.383219\pi$
$661$	18.4444			0.717406			0.358703	+	0.933452i	$0.383219\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.02766	+	7.09489i	−0.0398509	+	0.275128i
$666$		0			0
$667$	−4.54086	−	4.54086i	−0.175823	−	0.175823i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.23414i		−	0.163457i
$672$		0			0
$673$	−18.6806	+	18.6806i	−0.720085	+	0.720085i	−0.968622	−	0.248537i	$-0.920050\pi$
$673$	−18.6806	+	18.6806i	−0.720085	+	0.720085i	0.248537	+	0.968622i	$0.420050\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.2105	−	20.2105i	0.776752	−	0.776752i	−0.202525	−	0.979277i	$-0.564915\pi$
$677$	20.2105	−	20.2105i	0.776752	−	0.776752i	0.979277	+	0.202525i	$0.0649146\pi$
$678$		0			0
$679$		−	21.0182i		−	0.806605i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.59310	−	1.59310i	−0.0609583	−	0.0609583i	0.675970	−	0.736929i	$-0.263725\pi$
$683$	−1.59310	−	1.59310i	−0.0609583	−	0.0609583i	−0.736929	+	0.675970i	$0.763725\pi$
$684$		0			0
$685$	−6.58243	+	45.4447i	−0.251502	+	1.73635i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−54.5254			−2.07725
$690$		0			0
$691$	33.4395			1.27210			0.636049	−	0.771649i	$-0.280568\pi$
$691$	33.4395			1.27210			0.636049	+	0.771649i	$0.280568\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.56365	−	6.10962i	−0.173109	−	0.231751i
$696$		0			0
$697$	−13.5251	−	13.5251i	−0.512299	−	0.512299i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3.10675i			0.117340i	0.998277	+	0.0586702i	$0.0186860\pi$
$701$			3.10675i			0.117340i	−0.998277	+	0.0586702i	$0.981314\pi$
$702$		0			0
$703$	−3.47344	+	3.47344i	−0.131003	+	0.131003i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.02504	−	9.02504i	0.339422	−	0.339422i
$708$		0			0
$709$			51.6094i			1.93823i	0.246606	+	0.969116i	$0.420685\pi$
$709$			51.6094i			1.93823i	−0.246606	+	0.969116i	$0.579315\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.59301	−	3.59301i	−0.134559	−	0.134559i
$714$		0			0
$715$	−49.6782	−	7.19563i	−1.85786	−	0.269101i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.4648			−0.800501			−0.400251	−	0.916406i	$-0.631077\pi$
$719$	−21.4648			−0.800501			−0.400251	+	0.916406i	$0.631077\pi$
$720$		0			0
$721$	−56.2525			−2.09495
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.11043	+	30.7891i	−0.338353	+	1.14348i
$726$		0			0
$727$	2.11843	+	2.11843i	0.0785683	+	0.0785683i	0.745299	−	0.666731i	$-0.232307\pi$
$727$	2.11843	+	2.11843i	0.0785683	+	0.0785683i	−0.666731	+	0.745299i	$0.732307\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4.54162i		−	0.167978i
$732$		0			0
$733$	23.2780	−	23.2780i	0.859793	−	0.859793i	−0.131520	−	0.991313i	$-0.541986\pi$
$733$	23.2780	−	23.2780i	0.859793	−	0.859793i	0.991313	+	0.131520i	$0.0419858\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.0428	+	25.0428i	−0.922464	+	0.922464i
$738$		0			0
$739$		−	31.1982i		−	1.14765i	−0.818979	−	0.573823i	$-0.805460\pi$
$739$		−	31.1982i		−	1.14765i	0.818979	−	0.573823i	$-0.194540\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	31.0202	+	31.0202i	1.13802	+	1.13802i	0.988805	+	0.149215i	$0.0476747\pi$
$743$	31.0202	+	31.0202i	1.13802	+	1.13802i	0.149215	+	0.988805i	$0.452325\pi$
$744$		0			0
$745$	−3.52945	+	2.63636i	−0.129309	+	0.0965888i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.7051			0.646930
$750$		0			0
$751$	−33.4332			−1.22000			−0.609998	−	0.792403i	$-0.708830\pi$
$751$	−33.4332			−1.22000			−0.609998	+	0.792403i	$0.708830\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.6425	+	23.6357i	−1.15159	+	0.860191i
$756$		0			0
$757$	17.2431	+	17.2431i	0.626711	+	0.626711i	0.947239	−	0.320528i	$-0.103860\pi$
$757$	17.2431	+	17.2431i	0.626711	+	0.626711i	−0.320528	+	0.947239i	$0.603860\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	9.73407i		−	0.352860i	−0.984313	−	0.176430i	$-0.943545\pi$
$761$		−	9.73407i		−	0.352860i	0.984313	−	0.176430i	$-0.0564549\pi$
$762$		0			0
$763$	−10.9918	+	10.9918i	−0.397929	+	0.397929i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.3972	−	10.3972i	0.375421	−	0.375421i
$768$		0			0
$769$			15.6710i			0.565111i	0.959251	+	0.282555i	$0.0911821\pi$
$769$			15.6710i			0.565111i	−0.959251	+	0.282555i	$0.908818\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.8704	−	16.8704i	−0.606785	−	0.606785i	0.335320	−	0.942104i	$-0.391156\pi$
$773$	−16.8704	−	16.8704i	−0.606785	−	0.606785i	−0.942104	+	0.335320i	$0.891156\pi$
$774$		0			0
$775$	−7.20874	+	24.3623i	−0.258946	+	0.875119i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.29069			−0.153730
$780$		0			0
$781$	13.9346			0.498619
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−32.0945	−	4.64872i	−1.14550	−	0.165920i
$786$		0			0
$787$	30.3273	+	30.3273i	1.08105	+	1.08105i	0.996411	+	0.0846416i	$0.0269745\pi$
$787$	30.3273	+	30.3273i	1.08105	+	1.08105i	0.0846416	+	0.996411i	$0.473025\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	41.5430i		−	1.47710i
$792$		0			0
$793$	−6.45710	+	6.45710i	−0.229298	+	0.229298i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.5433	−	11.5433i	0.408884	−	0.408884i	−0.472465	−	0.881349i	$-0.656636\pi$
$797$	11.5433	−	11.5433i	0.408884	−	0.408884i	0.881349	+	0.472465i	$0.156636\pi$
$798$		0			0
$799$			64.8949i			2.29582i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.2622	−	23.2622i	−0.820905	−	0.820905i
$804$		0			0
$805$	3.93041	+	5.26187i	0.138529	+	0.185457i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.6886			−1.07895			−0.539477	−	0.842000i	$-0.681378\pi$
$809$	−30.6886			−1.07895			−0.539477	+	0.842000i	$0.681378\pi$
$810$		0			0
$811$	39.2923			1.37974			0.689870	−	0.723933i	$-0.257668\pi$
$811$	39.2923			1.37974			0.689870	+	0.723933i	$0.257668\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.24574	−	8.60054i	0.0436365	−	0.301264i
$816$		0			0
$817$	−0.720391	−	0.720391i	−0.0252033	−	0.0252033i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			32.4437i			1.13229i	0.824305	+	0.566147i	$0.191566\pi$
$821$			32.4437i			1.13229i	−0.824305	+	0.566147i	$0.808434\pi$
$822$		0			0
$823$	−12.8530	+	12.8530i	−0.448027	+	0.448027i	−0.894698	−	0.446671i	$-0.852609\pi$
$823$	−12.8530	+	12.8530i	−0.448027	+	0.448027i	0.446671	+	0.894698i	$0.352609\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.6467	−	25.6467i	0.891825	−	0.891825i	−0.102870	−	0.994695i	$-0.532803\pi$
$827$	25.6467	−	25.6467i	0.891825	−	0.891825i	0.994695	+	0.102870i	$0.0328026\pi$
$828$		0			0
$829$		−	4.25217i		−	0.147684i	−0.997270	−	0.0738420i	$-0.976474\pi$
$829$		−	4.25217i		−	0.147684i	0.997270	−	0.0738420i	$-0.0235260\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.59835	−	5.59835i	−0.193971	−	0.193971i
$834$		0			0
$835$	−2.26130	+	15.6119i	−0.0782554	+	0.540270i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0668			−0.416593			−0.208296	−	0.978066i	$-0.566792\pi$
$839$	−12.0668			−0.416593			−0.208296	+	0.978066i	$0.566792\pi$
$840$		0			0
$841$	12.2388			0.422027
$842$		0			0
$843$		0			0
$844$		0			0
$845$	47.3902	+	63.4440i	1.63027	+	2.18254i
$846$		0			0
$847$	−1.22778	−	1.22778i	−0.0421872	−	0.0421872i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			4.50026i			0.154267i
$852$		0			0
$853$	−16.5521	+	16.5521i	−0.566734	+	0.566734i	−0.931212	−	0.364478i	$-0.881247\pi$
$853$	−16.5521	+	16.5521i	−0.566734	+	0.566734i	0.364478	+	0.931212i	$0.381247\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.4281	+	37.4281i	−1.27852	+	1.27852i	−0.337024	+	0.941496i	$0.609420\pi$
$857$	−37.4281	+	37.4281i	−1.27852	+	1.27852i	−0.941496	+	0.337024i	$0.890580\pi$
$858$		0			0
$859$			8.01433i			0.273445i	0.990609	+	0.136723i	$0.0436569\pi$
$859$			8.01433i			0.273445i	−0.990609	+	0.136723i	$0.956343\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.333122	−	0.333122i	−0.0113396	−	0.0113396i	0.701414	−	0.712754i	$-0.252552\pi$
$863$	−0.333122	−	0.333122i	−0.0113396	−	0.0113396i	−0.712754	+	0.701414i	$0.752552\pi$
$864$		0			0
$865$	−44.5805	−	6.45725i	−1.51578	−	0.219553i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	33.1297			1.12385
$870$		0			0
$871$	76.3810			2.58807
$872$		0			0
$873$		0			0
$874$		0			0
$875$	13.7353	−	29.8283i	0.464337	−	1.00838i
$876$		0			0
$877$	26.0535	+	26.0535i	0.879765	+	0.879765i	0.993510	−	0.113745i	$-0.0362846\pi$
$877$	26.0535	+	26.0535i	0.879765	+	0.879765i	−0.113745	+	0.993510i	$0.536285\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			18.3712i			0.618943i	0.950909	+	0.309471i	$0.100152\pi$
$881$			18.3712i			0.618943i	−0.950909	+	0.309471i	$0.899848\pi$
$882$		0			0
$883$	−5.69116	+	5.69116i	−0.191523	+	0.191523i	−0.796354	−	0.604831i	$-0.793241\pi$
$883$	−5.69116	+	5.69116i	−0.191523	+	0.191523i	0.604831	+	0.796354i	$0.293241\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.36609	−	9.36609i	0.314483	−	0.314483i	−0.532161	−	0.846643i	$-0.678620\pi$
$887$	9.36609	−	9.36609i	0.314483	−	0.314483i	0.846643	+	0.532161i	$0.178620\pi$
$888$		0			0
$889$			3.94975i			0.132470i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.2936	+	10.2936i	0.344463	+	0.344463i
$894$		0			0
$895$	18.8287	−	14.0643i	0.629375	−	0.470119i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.6307			1.08829
$900$		0			0
$901$	38.1308			1.27032
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−28.6706	+	21.4159i	−0.953044	+	0.711887i
$906$		0			0
$907$	15.8287	+	15.8287i	0.525584	+	0.525584i	0.919252	−	0.393669i	$-0.128794\pi$
$907$	15.8287	+	15.8287i	0.525584	+	0.525584i	−0.393669	+	0.919252i	$0.628794\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	27.6363i		−	0.915633i	−0.889047	−	0.457816i	$-0.848632\pi$
$911$		−	27.6363i		−	0.915633i	0.889047	−	0.457816i	$-0.151368\pi$
$912$		0			0
$913$	−0.462982	+	0.462982i	−0.0153225	+	0.0153225i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.7662	+	18.7662i	−0.619715	+	0.619715i
$918$		0			0
$919$		−	33.9429i		−	1.11967i	−0.828603	−	0.559836i	$-0.810864\pi$
$919$		−	33.9429i		−	1.11967i	0.828603	−	0.559836i	$-0.189136\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−21.2504	−	21.2504i	−0.699465	−	0.699465i
$924$		0			0
$925$	19.7714	−	10.7424i	0.650080	−	0.353209i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−35.7087			−1.17156			−0.585781	−	0.810469i	$-0.699212\pi$
$929$	−35.7087			−1.17156			−0.585781	+	0.810469i	$0.699212\pi$
$930$		0			0
$931$	−1.77602			−0.0582067
$932$		0			0
$933$		0			0
$934$		0			0
$935$	34.7410	+	5.03205i	1.13615	+	0.164566i
$936$		0			0
$937$	27.1663	+	27.1663i	0.887485	+	0.887485i	0.994281	−	0.106796i	$-0.0340592\pi$
$937$	27.1663	+	27.1663i	0.887485	+	0.887485i	−0.106796	+	0.994281i	$0.534059\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	17.2931i		−	0.563738i	−0.959453	−	0.281869i	$-0.909046\pi$
$941$		−	17.2931i		−	0.563738i	0.959453	−	0.281869i	$-0.0909544\pi$
$942$		0			0
$943$	−2.77956	+	2.77956i	−0.0905148	+	0.0905148i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.6872	−	19.6872i	0.639747	−	0.639747i	−0.310746	−	0.950493i	$-0.600579\pi$
$947$	19.6872	−	19.6872i	0.639747	−	0.639747i	0.950493	+	0.310746i	$0.100579\pi$
$948$		0			0
$949$			70.9501i			2.30314i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.38568	+	7.38568i	0.239246	+	0.239246i	0.816538	−	0.577292i	$-0.195891\pi$
$953$	7.38568	+	7.38568i	0.239246	+	0.239246i	−0.577292	+	0.816538i	$0.695891\pi$
$954$		0			0
$955$	−2.81700	−	3.77128i	−0.0911561	−	0.122036i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−60.3169			−1.94773
$960$		0			0
$961$	−5.18055			−0.167115
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.04240	+	41.7164i	−0.194512	+	1.34290i
$966$		0			0
$967$	36.2008	+	36.2008i	1.16414	+	1.16414i	0.983560	+	0.180580i	$0.0577973\pi$
$967$	36.2008	+	36.2008i	1.16414	+	1.16414i	0.180580	+	0.983560i	$0.442203\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	7.32294i		−	0.235004i	−0.993073	−	0.117502i	$-0.962511\pi$
$971$		−	7.32294i		−	0.235004i	0.993073	−	0.117502i	$-0.0374887\pi$
$972$		0			0
$973$	7.08310	−	7.08310i	0.227074	−	0.227074i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.4203	+	17.4203i	−0.557325	+	0.557325i	−0.928545	−	0.371220i	$-0.878939\pi$
$977$	−17.4203	+	17.4203i	−0.557325	+	0.557325i	0.371220	+	0.928545i	$0.378939\pi$
$978$		0			0
$979$			48.2779i			1.54297i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−39.3561	−	39.3561i	−1.25526	−	1.25526i	−0.953328	−	0.301936i	$-0.902367\pi$
$983$	−39.3561	−	39.3561i	−1.25526	−	1.25526i	−0.301936	−	0.953328i	$-0.597633\pi$
$984$		0			0
$985$	−6.61257	+	45.6528i	−0.210694	+	1.45462i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.933354			−0.0296789
$990$		0			0
$991$	−34.3377			−1.09077			−0.545386	−	0.838185i	$-0.683617\pi$
$991$	−34.3377			−1.09077			−0.545386	+	0.838185i	$0.683617\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.6966	+	19.6752i	0.465913	+	0.623745i
$996$		0			0
$997$	−14.1018	−	14.1018i	−0.446609	−	0.446609i	0.447616	−	0.894226i	$-0.352273\pi$
$997$	−14.1018	−	14.1018i	−0.446609	−	0.446609i	−0.894226	+	0.447616i	$0.852273\pi$
$998$		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.737.5	✓	44
3.2	odd	2	inner	4140.2.s.b.737.18	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.18	yes	44
15.8	even	4	inner	4140.2.s.b.2393.5	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.5	✓	44	1.1	even	1	trivial
4140.2.s.b.737.18	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.5	yes	44	15.8	even	4	inner
4140.2.s.b.2393.18	yes	44	5.3	odd	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)

\(f(q)\)	\(=\)	\(q+(-1.79146 + 1.33815i) q^{5} +(-2.07691 - 2.07691i) q^{7} +O(q^{10})\)	\(q+(-1.79146 + 1.33815i) q^{5} +(-2.07691 - 2.07691i) q^{7} +3.22627i q^{11} +(4.92009 - 4.92009i) q^{13} +(-3.44072 + 3.44072i) q^{17} +1.09153i q^{19} +(0.707107 + 0.707107i) q^{23} +(1.41868 - 4.79451i) q^{25} -6.42174 q^{29} -5.08128 q^{31} +(6.49993 + 0.941481i) q^{35} +(3.18217 + 3.18217i) q^{37} +3.93089i q^{41} +(-0.659981 + 0.659981i) q^{43} +(9.43041 - 9.43041i) q^{47} +1.62709i q^{49} +(-5.54110 - 5.54110i) q^{53} +(-4.31725 - 5.77975i) q^{55} +2.11321 q^{59} -1.31239 q^{61} +(-2.23032 + 15.3980i) q^{65} +(7.76215 + 7.76215i) q^{67} -4.31910i q^{71} +(-7.21024 + 7.21024i) q^{73} +(6.70067 - 6.70067i) q^{77} -10.2687i q^{79} +(0.143504 + 0.143504i) q^{83} +(1.55971 - 10.7681i) q^{85} +14.9640 q^{89} -20.4371 q^{91} +(-1.46064 - 1.95544i) q^{95} +(5.05998 + 5.05998i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 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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