LMFDB - Embedded newform 4140.2.n.b.2069.8

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			2069.8
Character	$\chi$	$=$	4140.2069
Dual form			4140.2.n.b.2069.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.83744 + 1.27428i) q^{5} +0.920125 q^{7} +O(q^{10})$	$q+(-1.83744 + 1.27428i) q^{5} +0.920125 q^{7} -4.88423 q^{11} -0.822839i q^{13} +5.93583i q^{17} +7.43704i q^{19} +(2.12845 - 4.29764i) q^{23} +(1.75240 - 4.68285i) q^{25} +2.63328i q^{29} +5.24089 q^{31} +(-1.69068 + 1.17250i) q^{35} +7.23616 q^{37} -1.93295i q^{41} -1.42434 q^{43} -13.2960 q^{47} -6.15337 q^{49} -0.0569429i q^{53} +(8.97450 - 6.22390i) q^{55} -2.50951i q^{59} -5.95419i q^{61} +(1.04853 + 1.51192i) q^{65} -5.00085 q^{67} -8.61372i q^{71} +3.83292i q^{73} -4.49410 q^{77} -5.42450i q^{79} +1.52731i q^{83} +(-7.56394 - 10.9068i) q^{85} -4.74561 q^{89} -0.757115i q^{91} +(-9.47690 - 13.6651i) q^{95} -12.6112 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.83744	+	1.27428i	−0.821730	+	0.569877i
$5$	−1.83744	+	1.27428i	−0.821730	+	0.569877i
$6$		0			0
$7$	0.920125			0.347775			0.173887	−	0.984766i	$-0.444367\pi$
$7$	0.920125			0.347775			0.173887	+	0.984766i	$0.444367\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.88423			−1.47265			−0.736326	−	0.676627i	$-0.763441\pi$
$11$	−4.88423			−1.47265			−0.736326	+	0.676627i	$0.763441\pi$
$12$		0			0
$13$		−	0.822839i		−	0.228214i	−0.993468	−	0.114107i	$-0.963599\pi$
$13$		−	0.822839i		−	0.228214i	0.993468	−	0.114107i	$-0.0364007\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.93583i			1.43965i	0.694156	+	0.719825i	$0.255778\pi$
$17$			5.93583i			1.43965i	−0.694156	+	0.719825i	$0.744222\pi$
$18$		0			0
$19$			7.43704i			1.70617i	0.521769	+	0.853087i	$0.325272\pi$
$19$			7.43704i			1.70617i	−0.521769	+	0.853087i	$0.674728\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.12845	−	4.29764i	0.443812	−	0.896120i
$23$	2.12845	−	4.29764i	0.443812	−	0.896120i
$24$		0			0
$25$	1.75240	−	4.68285i	0.350480	−	0.936570i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.63328i			0.488987i	0.969651	+	0.244494i	$0.0786217\pi$
$29$			2.63328i			0.488987i	−0.969651	+	0.244494i	$0.921378\pi$
$30$		0			0
$31$	5.24089			0.941292			0.470646	−	0.882322i	$-0.344021\pi$
$31$	5.24089			0.941292			0.470646	+	0.882322i	$0.344021\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.69068	+	1.17250i	−0.285777	+	0.198189i
$36$		0			0
$37$	7.23616			1.18962			0.594808	−	0.803868i	$-0.297228\pi$
$37$	7.23616			1.18962			0.594808	+	0.803868i	$0.297228\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.93295i		−	0.301877i	−0.988543	−	0.150938i	$-0.951771\pi$
$41$		−	1.93295i		−	0.301877i	0.988543	−	0.150938i	$-0.0482295\pi$
$42$		0			0
$43$	−1.42434			−0.217210			−0.108605	−	0.994085i	$-0.534638\pi$
$43$	−1.42434			−0.217210			−0.108605	+	0.994085i	$0.534638\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.2960			−1.93942			−0.969712	−	0.244249i	$-0.921458\pi$
$47$	−13.2960			−1.93942			−0.969712	+	0.244249i	$0.921458\pi$
$48$		0			0
$49$	−6.15337			−0.879053
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.0569429i		−	0.00782171i	−0.999992	−	0.00391086i	$-0.998755\pi$
$53$		−	0.0569429i		−	0.00782171i	0.999992	−	0.00391086i	$-0.00124487\pi$
$54$		0			0
$55$	8.97450	−	6.22390i	1.21012	−	0.839231i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.50951i		−	0.326710i	−0.986567	−	0.163355i	$-0.947768\pi$
$59$		−	2.50951i		−	0.326710i	0.986567	−	0.163355i	$-0.0522316\pi$
$60$		0			0
$61$		−	5.95419i		−	0.762356i	−0.924502	−	0.381178i	$-0.875519\pi$
$61$		−	5.95419i		−	0.762356i	0.924502	−	0.381178i	$-0.124481\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.04853	+	1.51192i	0.130054	+	0.187531i
$66$		0			0
$67$	−5.00085			−0.610951			−0.305475	−	0.952200i	$-0.598815\pi$
$67$	−5.00085			−0.610951			−0.305475	+	0.952200i	$0.598815\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.61372i		−	1.02226i	−0.859503	−	0.511130i	$-0.829227\pi$
$71$		−	8.61372i		−	1.02226i	0.859503	−	0.511130i	$-0.170773\pi$
$72$		0			0
$73$			3.83292i			0.448609i	0.974519	+	0.224305i	$0.0720111\pi$
$73$			3.83292i			0.448609i	−0.974519	+	0.224305i	$0.927989\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.49410			−0.512151
$78$		0			0
$79$		−	5.42450i		−	0.610304i	−0.952304	−	0.305152i	$-0.901293\pi$
$79$		−	5.42450i		−	0.610304i	0.952304	−	0.305152i	$-0.0987073\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.52731i			0.167644i	0.996481	+	0.0838220i	$0.0267127\pi$
$83$			1.52731i			0.167644i	−0.996481	+	0.0838220i	$0.973287\pi$
$84$		0			0
$85$	−7.56394	−	10.9068i	−0.820424	−	1.18300i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.74561			−0.503034			−0.251517	−	0.967853i	$-0.580929\pi$
$89$	−4.74561			−0.503034			−0.251517	+	0.967853i	$0.580929\pi$
$90$		0			0
$91$		−	0.757115i		−	0.0793672i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.47690	−	13.6651i	−0.972309	−	1.40201i
$96$		0			0
$97$	−12.6112			−1.28047			−0.640237	−	0.768177i	$-0.721164\pi$
$97$	−12.6112			−1.28047			−0.640237	+	0.768177i	$0.721164\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	8.41390i		−	0.837215i	−0.908167	−	0.418607i	$-0.862518\pi$
$101$		−	8.41390i		−	0.837215i	0.908167	−	0.418607i	$-0.137482\pi$
$102$		0			0
$103$	−12.8106			−1.26227			−0.631135	−	0.775673i	$-0.717410\pi$
$103$	−12.8106			−1.26227			−0.631135	+	0.775673i	$0.717410\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.1902i			1.46850i	0.678882	+	0.734248i	$0.262465\pi$
$107$			15.1902i			1.46850i	−0.678882	+	0.734248i	$0.737535\pi$
$108$		0			0
$109$		−	6.15023i		−	0.589085i	−0.955638	−	0.294543i	$-0.904833\pi$
$109$		−	6.15023i		−	0.589085i	0.955638	−	0.294543i	$-0.0951673\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.52402i		−	0.613728i	−0.951753	−	0.306864i	$-0.900720\pi$
$113$		−	6.52402i		−	0.613728i	0.951753	−	0.306864i	$-0.0992796\pi$
$114$		0			0
$115$	1.56551	+	10.6089i	0.145984	+	0.989287i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.46170i			0.500674i
$120$		0			0
$121$	12.8557			1.16870
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.74735	+	10.8375i	0.245731	+	0.969338i
$126$		0			0
$127$		−	8.61488i		−	0.764447i	−0.924070	−	0.382223i	$-0.875158\pi$
$127$		−	8.61488i		−	0.764447i	0.924070	−	0.382223i	$-0.124842\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.83952i			0.772312i	0.922433	+	0.386156i	$0.126197\pi$
$131$			8.83952i			0.772312i	−0.922433	+	0.386156i	$0.873803\pi$
$132$		0			0
$133$			6.84300i			0.593364i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	13.3407i		−	1.13978i	−0.821722	−	0.569888i	$-0.806987\pi$
$137$		−	13.3407i		−	1.13978i	0.821722	−	0.569888i	$-0.193013\pi$
$138$		0			0
$139$	7.78027			0.659914			0.329957	−	0.943996i	$-0.392966\pi$
$139$	7.78027			0.659914			0.329957	+	0.943996i	$0.392966\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4.01894i			0.336080i
$144$		0			0
$145$	−3.35554	−	4.83850i	−0.278663	−	0.401815i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.20934			−0.0990733			−0.0495367	−	0.998772i	$-0.515774\pi$
$149$	−1.20934			−0.0990733			−0.0495367	+	0.998772i	$0.515774\pi$
$150$		0			0
$151$	−17.3471			−1.41169			−0.705843	−	0.708368i	$-0.749432\pi$
$151$	−17.3471			−1.41169			−0.705843	+	0.708368i	$0.749432\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.62984	+	6.67838i	−0.773487	+	0.536421i
$156$		0			0
$157$	4.34577			0.346830			0.173415	−	0.984849i	$-0.444520\pi$
$157$	4.34577			0.346830			0.173415	+	0.984849i	$0.444520\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.95844	−	3.95437i	0.154347	−	0.311648i
$162$		0			0
$163$		−	8.00130i		−	0.626710i	−0.949636	−	0.313355i	$-0.898547\pi$
$163$		−	8.00130i		−	0.626710i	0.949636	−	0.313355i	$-0.101453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−17.8411			−1.38058			−0.690291	−	0.723532i	$-0.742518\pi$
$167$	−17.8411			−1.38058			−0.690291	+	0.723532i	$0.742518\pi$
$168$		0			0
$169$	12.3229			0.947918
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.3862			−1.16979			−0.584894	−	0.811110i	$-0.698864\pi$
$173$	−15.3862			−1.16979			−0.584894	+	0.811110i	$0.698864\pi$
$174$		0			0
$175$	1.61243	−	4.30881i	0.121888	−	0.325715i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.9209i		−	1.26473i	−0.774671	−	0.632365i	$-0.782084\pi$
$179$		−	16.9209i		−	1.26473i	0.774671	−	0.632365i	$-0.217916\pi$
$180$		0			0
$181$			6.71130i			0.498847i	0.968394	+	0.249424i	$0.0802412\pi$
$181$			6.71130i			0.498847i	−0.968394	+	0.249424i	$0.919759\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.2960	+	9.22092i	−0.977543	+	0.677936i
$186$		0			0
$187$		−	28.9920i		−	2.12010i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.73039			0.559351			0.279676	−	0.960095i	$-0.409773\pi$
$191$	7.73039			0.559351			0.279676	+	0.960095i	$0.409773\pi$
$192$		0			0
$193$		−	1.77188i		−	0.127542i	−0.997965	−	0.0637712i	$-0.979687\pi$
$193$		−	1.77188i		−	0.127542i	0.997965	−	0.0637712i	$-0.0203128\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.86375			0.702763			0.351382	−	0.936232i	$-0.385712\pi$
$197$	9.86375			0.702763			0.351382	+	0.936232i	$0.385712\pi$
$198$		0			0
$199$		−	3.71423i		−	0.263295i	−0.991297	−	0.131647i	$-0.957973\pi$
$199$		−	3.71423i		−	0.263295i	0.991297	−	0.131647i	$-0.0420266\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2.42294i			0.170057i
$204$		0			0
$205$	2.46313	+	3.55169i	0.172033	+	0.248061i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	36.3242i		−	2.51260i
$210$		0			0
$211$	7.93116			0.546004			0.273002	−	0.962014i	$-0.411984\pi$
$211$	7.93116			0.546004			0.273002	+	0.962014i	$0.411984\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.61714	−	1.81501i	0.178488	−	0.123783i
$216$		0			0
$217$	4.82227			0.327357
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.88423			0.328549
$222$		0			0
$223$		−	12.7833i		−	0.856030i	−0.903772	−	0.428015i	$-0.859213\pi$
$223$		−	12.7833i		−	0.856030i	0.903772	−	0.428015i	$-0.140787\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			17.6427i			1.17099i	0.810678	+	0.585493i	$0.199099\pi$
$227$			17.6427i			1.17099i	−0.810678	+	0.585493i	$0.800901\pi$
$228$		0			0
$229$		−	1.71027i		−	0.113018i	−0.998402	−	0.0565089i	$-0.982003\pi$
$229$		−	1.71027i		−	0.113018i	0.998402	−	0.0565089i	$-0.0179969\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.7197			−1.29188			−0.645940	−	0.763388i	$-0.723534\pi$
$233$	−19.7197			−1.29188			−0.645940	+	0.763388i	$0.723534\pi$
$234$		0			0
$235$	24.4307	−	16.9429i	1.59368	−	1.10523i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.3338i		−	1.25060i	−0.780385	−	0.625300i	$-0.784977\pi$
$239$		−	19.3338i		−	1.25060i	0.780385	−	0.625300i	$-0.215023\pi$
$240$		0			0
$241$		−	18.0586i		−	1.16326i	−0.813454	−	0.581629i	$-0.802416\pi$
$241$		−	18.0586i		−	1.16326i	0.813454	−	0.581629i	$-0.197584\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.3065	−	7.84114i	0.722344	−	0.500952i
$246$		0			0
$247$	6.11948			0.389373
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.1890			−1.14808			−0.574038	−	0.818828i	$-0.694624\pi$
$251$	−18.1890			−1.14808			−0.574038	+	0.818828i	$0.694624\pi$
$252$		0			0
$253$	−10.3958	+	20.9907i	−0.653581	+	1.31967i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.14415			0.196127			0.0980634	−	0.995180i	$-0.468735\pi$
$257$	3.14415			0.196127			0.0980634	+	0.995180i	$0.468735\pi$
$258$		0			0
$259$	6.65817			0.413718
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.06549i		−	0.0657011i	−0.999460	−	0.0328505i	$-0.989541\pi$
$263$		−	1.06549i		−	0.0657011i	0.999460	−	0.0328505i	$-0.0104585\pi$
$264$		0			0
$265$	0.0725615	+	0.104629i	0.00445742	+	0.00642733i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			18.9912i			1.15792i	0.815358	+	0.578958i	$0.196540\pi$
$269$			18.9912i			1.15792i	−0.815358	+	0.578958i	$0.803460\pi$
$270$		0			0
$271$	−2.05626			−0.124909			−0.0624544	−	0.998048i	$-0.519893\pi$
$271$	−2.05626			−0.124909			−0.0624544	+	0.998048i	$0.519893\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.55912	+	22.8721i	−0.516134	+	1.37924i
$276$		0			0
$277$		−	24.8236i		−	1.49150i	−0.666223	−	0.745752i	$-0.732090\pi$
$277$		−	24.8236i		−	1.49150i	0.666223	−	0.745752i	$-0.267910\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.1737			1.26312			0.631560	−	0.775327i	$-0.282415\pi$
$281$	21.1737			1.26312			0.631560	+	0.775327i	$0.282415\pi$
$282$		0			0
$283$	−23.1162			−1.37412			−0.687058	−	0.726603i	$-0.741098\pi$
$283$	−23.1162			−1.37412			−0.687058	+	0.726603i	$0.741098\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.77856i		−	0.104985i
$288$		0			0
$289$	−18.2341			−1.07259
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	11.4483i		−	0.668816i	−0.942428	−	0.334408i	$-0.891464\pi$
$293$		−	11.4483i		−	0.668816i	0.942428	−	0.334408i	$-0.108536\pi$
$294$		0			0
$295$	3.19782	+	4.61107i	0.186184	+	0.268467i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.53626	−	1.75137i	−0.204507	−	0.101284i
$300$		0			0
$301$	−1.31057			−0.0755400
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.58733	+	10.9405i	0.434449	+	0.626451i
$306$		0			0
$307$			3.90494i			0.222866i	0.993772	+	0.111433i	$0.0355441\pi$
$307$			3.90494i			0.222866i	−0.993772	+	0.111433i	$0.964456\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			8.78989i			0.498429i	0.968448	+	0.249214i	$0.0801724\pi$
$311$			8.78989i			0.498429i	−0.968448	+	0.249214i	$0.919828\pi$
$312$		0			0
$313$	−22.9154			−1.29526			−0.647628	−	0.761957i	$-0.724239\pi$
$313$	−22.9154			−1.29526			−0.647628	+	0.761957i	$0.724239\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	29.7362			1.67015			0.835076	−	0.550135i	$-0.185424\pi$
$317$	29.7362			1.67015			0.835076	+	0.550135i	$0.185424\pi$
$318$		0			0
$319$		−	12.8615i		−	0.720108i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−44.1450			−2.45629
$324$		0			0
$325$	−3.85323	−	1.44194i	−0.213739	−	0.0799845i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.2340			−0.674483
$330$		0			0
$331$	27.7485			1.52519			0.762597	−	0.646874i	$-0.223924\pi$
$331$	27.7485			1.52519			0.762597	+	0.646874i	$0.223924\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.18877	−	6.37250i	0.502036	−	0.348167i
$336$		0			0
$337$	−13.6017			−0.740930			−0.370465	−	0.928846i	$-0.620802\pi$
$337$	−13.6017			−0.740930			−0.370465	+	0.928846i	$0.620802\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−25.5977			−1.38619
$342$		0			0
$343$	−12.1027			−0.653487
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.2059			−0.601564			−0.300782	−	0.953693i	$-0.597248\pi$
$347$	−11.2059			−0.601564			−0.300782	+	0.953693i	$0.597248\pi$
$348$		0			0
$349$	5.91113			0.316416			0.158208	−	0.987406i	$-0.449428\pi$
$349$	5.91113			0.316416			0.158208	+	0.987406i	$0.449428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.7784			−1.05270			−0.526350	−	0.850268i	$-0.676440\pi$
$353$	−19.7784			−1.05270			−0.526350	+	0.850268i	$0.676440\pi$
$354$		0			0
$355$	10.9763	+	15.8272i	0.582563	+	0.840022i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.4230			−1.60566			−0.802831	−	0.596206i	$-0.796674\pi$
$359$	−30.4230			−1.60566			−0.802831	+	0.596206i	$0.796674\pi$
$360$		0			0
$361$	−36.3095			−1.91103
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.88423	−	7.04278i	−0.255652	−	0.368636i
$366$		0			0
$367$	−12.6858			−0.662193			−0.331096	−	0.943597i	$-0.607419\pi$
$367$	−12.6858			−0.662193			−0.331096	+	0.943597i	$0.607419\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	0.0523946i		−	0.00272019i
$372$		0			0
$373$	−0.216486			−0.0112092			−0.00560462	−	0.999984i	$-0.501784\pi$
$373$	−0.216486			−0.0112092			−0.00560462	+	0.999984i	$0.501784\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.16676			0.111594
$378$		0			0
$379$			17.0078i			0.873633i	0.899551	+	0.436816i	$0.143894\pi$
$379$			17.0078i			0.873633i	−0.899551	+	0.436816i	$0.856106\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	13.6599i		−	0.697989i	−0.937125	−	0.348995i	$-0.886523\pi$
$383$		−	13.6599i		−	0.697989i	0.937125	−	0.348995i	$-0.113477\pi$
$384$		0			0
$385$	8.25766	−	5.72677i	0.420849	−	0.291863i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.9346			1.16283			0.581414	−	0.813608i	$-0.302500\pi$
$389$	22.9346			1.16283			0.581414	+	0.813608i	$0.302500\pi$
$390$		0			0
$391$	25.5101	+	12.6341i	1.29010	+	0.638935i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.91236	+	9.96721i	0.347798	+	0.501505i
$396$		0			0
$397$			36.2966i			1.82168i	0.412765	+	0.910838i	$0.364563\pi$
$397$			36.2966i			1.82168i	−0.412765	+	0.910838i	$0.635437\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.8778			−1.39215			−0.696075	−	0.717969i	$-0.745072\pi$
$401$	−27.8778			−1.39215			−0.696075	+	0.717969i	$0.745072\pi$
$402$		0			0
$403$		−	4.31241i		−	0.214816i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−35.3431			−1.75189
$408$		0			0
$409$	20.4187			1.00964			0.504820	−	0.863225i	$-0.331559\pi$
$409$	20.4187			1.00964			0.504820	+	0.863225i	$0.331559\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	2.30906i		−	0.113621i
$414$		0			0
$415$	−1.94623	−	2.80635i	−0.0955365	−	0.137758i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.30292			−0.356771			−0.178385	−	0.983961i	$-0.557087\pi$
$419$	−7.30292			−0.356771			−0.178385	+	0.983961i	$0.557087\pi$
$420$		0			0
$421$			19.1180i			0.931754i	0.884850	+	0.465877i	$0.154261\pi$
$421$			19.1180i			0.931754i	−0.884850	+	0.465877i	$0.845739\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	27.7966	+	10.4019i	1.34833	+	0.504568i
$426$		0			0
$427$		−	5.47860i		−	0.265128i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	28.3849			1.36725			0.683626	−	0.729832i	$-0.260402\pi$
$431$	28.3849			1.36725			0.683626	+	0.729832i	$0.260402\pi$
$432$		0			0
$433$	−23.1182			−1.11099			−0.555494	−	0.831520i	$-0.687471\pi$
$433$	−23.1182			−1.11099			−0.555494	+	0.831520i	$0.687471\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	31.9617	+	15.8294i	1.52894	+	0.757221i
$438$		0			0
$439$	16.0741			0.767175			0.383587	−	0.923505i	$-0.374688\pi$
$439$	16.0741			0.767175			0.383587	+	0.923505i	$0.374688\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.19817			−0.199461			−0.0997305	−	0.995014i	$-0.531798\pi$
$443$	−4.19817			−0.199461			−0.0997305	+	0.995014i	$0.531798\pi$
$444$		0			0
$445$	8.71979	−	6.04726i	0.413358	−	0.286667i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			21.4294i			1.01131i	0.862735	+	0.505657i	$0.168750\pi$
$449$			21.4294i			1.01131i	−0.862735	+	0.505657i	$0.831250\pi$
$450$		0			0
$451$			9.44099i			0.444559i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.964779	+	1.39116i	0.0452296	+	0.0652184i
$456$		0			0
$457$	37.3072			1.74516			0.872578	−	0.488474i	$-0.162446\pi$
$457$	37.3072			1.74516			0.872578	+	0.488474i	$0.162446\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	0.562985i		−	0.0262208i	−0.999914	−	0.0131104i	$-0.995827\pi$
$461$		−	0.562985i		−	0.0262208i	0.999914	−	0.0131104i	$-0.00417329\pi$
$462$		0			0
$463$		−	7.23264i		−	0.336129i	−0.985776	−	0.168065i	$-0.946248\pi$
$463$		−	7.23264i		−	0.336129i	0.985776	−	0.168065i	$-0.0537518\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	15.0064i		−	0.694412i	−0.937789	−	0.347206i	$-0.887130\pi$
$467$		−	15.0064i		−	0.694412i	0.937789	−	0.347206i	$-0.112870\pi$
$468$		0			0
$469$	−4.60140			−0.212473
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.95680			0.319874
$474$		0			0
$475$	34.8265	+	13.0327i	1.59795	+	0.597979i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.06441			−0.140017			−0.0700083	−	0.997546i	$-0.522303\pi$
$479$	−3.06441			−0.140017			−0.0700083	+	0.997546i	$0.522303\pi$
$480$		0			0
$481$		−	5.95419i		−	0.271488i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23.1724	−	16.0703i	1.05220	−	0.729713i
$486$		0			0
$487$			10.1885i			0.461687i	0.972991	+	0.230843i	$0.0741485\pi$
$487$			10.1885i			0.461687i	−0.972991	+	0.230843i	$0.925851\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	37.2461i		−	1.68089i	−0.541895	−	0.840446i	$-0.682293\pi$
$491$		−	37.2461i		−	1.68089i	0.541895	−	0.840446i	$-0.317707\pi$
$492$		0			0
$493$	−15.6307			−0.703971
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.92570i		−	0.355516i
$498$		0			0
$499$	−11.1591			−0.499551			−0.249776	−	0.968304i	$-0.580357\pi$
$499$	−11.1591			−0.499551			−0.249776	+	0.968304i	$0.580357\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	17.5992i		−	0.784709i	−0.919814	−	0.392354i	$-0.871661\pi$
$503$		−	17.5992i		−	0.784709i	0.919814	−	0.392354i	$-0.128339\pi$
$504$		0			0
$505$	10.7217	+	15.4601i	0.477110	+	0.687964i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			20.9909i			0.930405i	0.885204	+	0.465202i	$0.154018\pi$
$509$			20.9909i			0.930405i	−0.885204	+	0.465202i	$0.845982\pi$
$510$		0			0
$511$			3.52677i			0.156015i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	23.5388	−	16.3244i	1.03724	−	0.719339i
$516$		0			0
$517$	64.9409			2.85610
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.9917			−0.744422			−0.372211	−	0.928148i	$-0.621400\pi$
$521$	−16.9917			−0.744422			−0.372211	+	0.928148i	$0.621400\pi$
$522$		0			0
$523$	−6.71018			−0.293416			−0.146708	−	0.989180i	$-0.546868\pi$
$523$	−6.71018			−0.293416			−0.146708	+	0.989180i	$0.546868\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			31.1090i			1.35513i
$528$		0			0
$529$	−13.9394	−	18.2946i	−0.606061	−	0.795418i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.59051			−0.0688926
$534$		0			0
$535$	−19.3567	−	27.9112i	−0.836862	−	1.20671i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	30.0545			1.29454
$540$		0			0
$541$	6.39426			0.274911			0.137455	−	0.990508i	$-0.456108\pi$
$541$	6.39426			0.274911			0.137455	+	0.990508i	$0.456108\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.83714	+	11.3007i	0.335706	+	0.484069i
$546$		0			0
$547$		−	36.0332i		−	1.54067i	−0.637641	−	0.770334i	$-0.720090\pi$
$547$		−	36.0332i		−	1.54067i	0.637641	−	0.770334i	$-0.279910\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−19.5838			−0.834297
$552$		0			0
$553$		−	4.99122i		−	0.212248i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.31632i			0.140517i	0.997529	+	0.0702584i	$0.0223824\pi$
$557$			3.31632i			0.140517i	−0.997529	+	0.0702584i	$0.977618\pi$
$558$		0			0
$559$			1.17200i			0.0495704i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.19388i		−	0.0924611i	−0.998931	−	0.0462306i	$-0.985279\pi$
$563$		−	2.19388i		−	0.0924611i	0.998931	−	0.0462306i	$-0.0147209\pi$
$564$		0			0
$565$	8.31345	+	11.9875i	0.349750	+	0.504318i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.5616			1.28121			0.640604	−	0.767871i	$-0.278684\pi$
$569$	30.5616			1.28121			0.640604	+	0.767871i	$0.278684\pi$
$570$		0			0
$571$		−	25.5356i		−	1.06863i	−0.845285	−	0.534316i	$-0.820569\pi$
$571$		−	25.5356i		−	1.06863i	0.845285	−	0.534316i	$-0.179431\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.3953	−	17.4984i	−0.683732	−	0.729733i
$576$		0			0
$577$			27.0029i			1.12415i	0.827087	+	0.562073i	$0.189996\pi$
$577$			27.0029i			1.12415i	−0.827087	+	0.562073i	$0.810004\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.40532i			0.0583023i
$582$		0			0
$583$			0.278122i			0.0115187i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.2333			0.917667			0.458833	−	0.888522i	$-0.348267\pi$
$587$	22.2333			0.917667			0.458833	+	0.888522i	$0.348267\pi$
$588$		0			0
$589$			38.9767i			1.60601i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.3861			0.919286			0.459643	−	0.888104i	$-0.347977\pi$
$593$	22.3861			0.919286			0.459643	+	0.888104i	$0.347977\pi$
$594$		0			0
$595$	−6.95977	−	10.0356i	−0.285323	−	0.411418i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			36.2194i			1.47989i	0.672670	+	0.739943i	$0.265147\pi$
$599$			36.2194i			1.47989i	−0.672670	+	0.739943i	$0.734853\pi$
$600$		0			0
$601$	−8.07460			−0.329370			−0.164685	−	0.986346i	$-0.552661\pi$
$601$	−8.07460			−0.329370			−0.164685	+	0.986346i	$0.552661\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.6217	+	16.3818i	−0.960357	+	0.666016i
$606$		0			0
$607$			36.5858i			1.48497i	0.669862	+	0.742486i	$0.266353\pi$
$607$			36.5858i			1.48497i	−0.669862	+	0.742486i	$0.733647\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			10.9405i			0.442605i
$612$		0			0
$613$	14.4367			0.583092			0.291546	−	0.956557i	$-0.405830\pi$
$613$	14.4367			0.583092			0.291546	+	0.956557i	$0.405830\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			33.7752i			1.35974i	0.733334	+	0.679869i	$0.237963\pi$
$617$			33.7752i			1.35974i	−0.733334	+	0.679869i	$0.762037\pi$
$618$		0			0
$619$		−	9.04108i		−	0.363392i	−0.983355	−	0.181696i	$-0.941841\pi$
$619$		−	9.04108i		−	0.363392i	0.983355	−	0.181696i	$-0.0581587\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.36655			−0.174942
$624$		0			0
$625$	−18.8582	−	16.4124i	−0.754328	−	0.656498i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			42.9526i			1.71263i
$630$		0			0
$631$		−	6.78615i		−	0.270152i	−0.990835	−	0.135076i	$-0.956872\pi$
$631$		−	6.78615i		−	0.270152i	0.990835	−	0.135076i	$-0.0431279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10.9778	+	15.8294i	0.435641	+	0.628169i
$636$		0			0
$637$			5.06323i			0.200613i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	38.0147			1.50149			0.750746	−	0.660591i	$-0.229694\pi$
$641$	38.0147			1.50149			0.750746	+	0.660591i	$0.229694\pi$
$642$		0			0
$643$	−36.0051			−1.41990			−0.709951	−	0.704251i	$-0.751283\pi$
$643$	−36.0051			−1.41990			−0.709951	+	0.704251i	$0.751283\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.5215			−0.610214			−0.305107	−	0.952318i	$-0.598692\pi$
$647$	−15.5215			−0.610214			−0.305107	+	0.952318i	$0.598692\pi$
$648$		0			0
$649$			12.2570i			0.481130i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	42.6898			1.67058			0.835291	−	0.549808i	$-0.185299\pi$
$653$	42.6898			1.67058			0.835291	+	0.549808i	$0.185299\pi$
$654$		0			0
$655$	−11.2641	−	16.2421i	−0.440123	−	0.634632i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	11.9124			0.464040			0.232020	−	0.972711i	$-0.425466\pi$
$659$	11.9124			0.464040			0.232020	+	0.972711i	$0.425466\pi$
$660$		0			0
$661$			14.8078i			0.575957i	0.957637	+	0.287979i	$0.0929832\pi$
$661$			14.8078i			0.575957i	−0.957637	+	0.287979i	$0.907017\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.71993	−	12.5736i	−0.338144	−	0.487585i
$666$		0			0
$667$	11.3169	+	5.60480i	0.438191	+	0.217019i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			29.0816i			1.12268i
$672$		0			0
$673$			33.3123i			1.28409i	0.766665	+	0.642047i	$0.221915\pi$
$673$			33.3123i			1.28409i	−0.766665	+	0.642047i	$0.778085\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	35.1145i		−	1.34956i	−0.738019	−	0.674780i	$-0.764238\pi$
$677$		−	35.1145i		−	1.34956i	0.738019	−	0.674780i	$-0.235762\pi$
$678$		0			0
$679$	−11.6039			−0.445316
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.1646			−1.34554			−0.672768	−	0.739854i	$-0.734895\pi$
$683$	−35.1646			−1.34554			−0.672768	+	0.739854i	$0.734895\pi$
$684$		0			0
$685$	16.9999	+	24.5128i	0.649532	+	0.936588i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.0468549			−0.00178503
$690$		0			0
$691$	−19.8239			−0.754137			−0.377068	−	0.926185i	$-0.623068\pi$
$691$	−19.8239			−0.754137			−0.377068	+	0.926185i	$0.623068\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.2958	+	9.91428i	−0.542271	+	0.376070i
$696$		0			0
$697$	11.4737			0.434597
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.89314			0.147042			0.0735210	−	0.997294i	$-0.476576\pi$
$701$	3.89314			0.147042			0.0735210	+	0.997294i	$0.476576\pi$
$702$		0			0
$703$			53.8155i			2.02969i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.74184i		−	0.291162i
$708$		0			0
$709$			47.6606i			1.78993i	0.446136	+	0.894965i	$0.352800\pi$
$709$			47.6606i			1.78993i	−0.446136	+	0.894965i	$0.647200\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.1550	−	22.5235i	0.417757	−	0.843510i
$714$		0			0
$715$	−5.12127	−	7.38457i	−0.191525	−	0.276167i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	33.8066i		−	1.26077i	−0.776282	−	0.630386i	$-0.782897\pi$
$719$		−	33.8066i		−	1.26077i	0.776282	−	0.630386i	$-0.217103\pi$
$720$		0			0
$721$	−11.7874			−0.438985
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.3312	+	4.61455i	0.457971	+	0.171380i
$726$		0			0
$727$	−44.2650			−1.64170			−0.820849	−	0.571145i	$-0.806500\pi$
$727$	−44.2650			−1.64170			−0.820849	+	0.571145i	$0.806500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.45464i		−	0.312706i
$732$		0			0
$733$	−52.1528			−1.92631			−0.963154	−	0.268952i	$-0.913323\pi$
$733$	−52.1528			−1.92631			−0.963154	+	0.268952i	$0.913323\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.4253			0.899717
$738$		0			0
$739$	1.52348			0.0560420			0.0280210	−	0.999607i	$-0.491079\pi$
$739$	1.52348			0.0560420			0.0280210	+	0.999607i	$0.491079\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20.2734i		−	0.743758i	−0.928281	−	0.371879i	$-0.878714\pi$
$743$		−	20.2734i		−	0.743758i	0.928281	−	0.371879i	$-0.121286\pi$
$744$		0			0
$745$	2.22210	−	1.54105i	0.0814115	−	0.0564596i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			13.9769i			0.510705i
$750$		0			0
$751$		−	43.2955i		−	1.57987i	−0.613188	−	0.789937i	$-0.710113\pi$
$751$		−	43.2955i		−	1.57987i	0.613188	−	0.789937i	$-0.289887\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	31.8743	−	22.1051i	1.16002	−	0.804488i
$756$		0			0
$757$	34.4783			1.25314			0.626568	−	0.779367i	$-0.284459\pi$
$757$	34.4783			1.25314			0.626568	+	0.779367i	$0.284459\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.8219i			0.827291i	0.910438	+	0.413646i	$0.135745\pi$
$761$			22.8219i			0.827291i	−0.910438	+	0.413646i	$0.864255\pi$
$762$		0			0
$763$		−	5.65898i		−	0.204869i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.06492			−0.0745599
$768$		0			0
$769$		−	21.6752i		−	0.781628i	−0.920470	−	0.390814i	$-0.872194\pi$
$769$		−	21.6752i		−	0.781628i	0.920470	−	0.390814i	$-0.127806\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	30.6086i		−	1.10091i	−0.834864	−	0.550457i	$-0.814454\pi$
$773$		−	30.6086i		−	1.10091i	0.834864	−	0.550457i	$-0.185546\pi$
$774$		0			0
$775$	9.18413	−	24.5423i	0.329904	−	0.881586i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.3754			0.515054
$780$		0			0
$781$			42.0714i			1.50543i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.98511	+	5.53775i	−0.285001	+	0.197651i
$786$		0			0
$787$	−19.3287			−0.688994			−0.344497	−	0.938787i	$-0.611950\pi$
$787$	−19.3287			−0.688994			−0.344497	+	0.938787i	$0.611950\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.00291i		−	0.213439i
$792$		0			0
$793$	−4.89934			−0.173981
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.01839i			0.142339i	0.997464	+	0.0711693i	$0.0226731\pi$
$797$			4.01839i			0.142339i	−0.997464	+	0.0711693i	$0.977327\pi$
$798$		0			0
$799$		−	78.9229i		−	2.79209i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	18.7209i		−	0.660645i
$804$		0			0
$805$	1.44046	+	9.76153i	0.0507697	+	0.344049i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	53.8434i		−	1.89303i	−0.322653	−	0.946517i	$-0.604575\pi$
$809$		−	53.8434i		−	1.89303i	0.322653	−	0.946517i	$-0.395425\pi$
$810$		0			0
$811$	−47.5340			−1.66914			−0.834572	−	0.550899i	$-0.814285\pi$
$811$	−47.5340			−1.66914			−0.834572	+	0.550899i	$0.814285\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.1959	+	14.7019i	0.357148	+	0.514986i
$816$		0			0
$817$		−	10.5929i		−	0.370597i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			37.7574i			1.31774i	0.752256	+	0.658871i	$0.228966\pi$
$821$			37.7574i			1.31774i	−0.752256	+	0.658871i	$0.771034\pi$
$822$		0			0
$823$			32.4174i			1.13000i	0.825091	+	0.565000i	$0.191124\pi$
$823$			32.4174i			1.13000i	−0.825091	+	0.565000i	$0.808876\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	49.5864i		−	1.72429i	−0.506663	−	0.862144i	$-0.669121\pi$
$827$		−	49.5864i		−	1.72429i	0.506663	−	0.862144i	$-0.330879\pi$
$828$		0			0
$829$	−45.0656			−1.56519			−0.782597	−	0.622529i	$-0.786105\pi$
$829$	−45.0656			−1.56519			−0.782597	+	0.622529i	$0.786105\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	36.5254i		−	1.26553i
$834$		0			0
$835$	32.7819	−	22.7346i	1.13447	−	0.786762i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.75090			0.233067			0.116534	−	0.993187i	$-0.462822\pi$
$839$	6.75090			0.233067			0.116534	+	0.993187i	$0.462822\pi$
$840$		0			0
$841$	22.0658			0.760891
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−22.6427	+	15.7029i	−0.778933	+	0.540197i
$846$		0			0
$847$	11.8289			0.406445
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.4018	−	31.0984i	0.527967	−	1.06604i
$852$		0			0
$853$			51.1378i			1.75092i	0.483287	+	0.875462i	$0.339443\pi$
$853$			51.1378i			1.75092i	−0.483287	+	0.875462i	$0.660557\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	34.1285			1.16581			0.582903	−	0.812542i	$-0.301917\pi$
$857$	34.1285			1.16581			0.582903	+	0.812542i	$0.301917\pi$
$858$		0			0
$859$	−33.1274			−1.13029			−0.565146	−	0.824991i	$-0.691180\pi$
$859$	−33.1274			−1.13029			−0.565146	+	0.824991i	$0.691180\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	47.5520			1.61869			0.809345	−	0.587333i	$-0.199822\pi$
$863$	47.5520			1.61869			0.809345	+	0.587333i	$0.199822\pi$
$864$		0			0
$865$	28.2712	−	19.6063i	0.961250	−	0.666636i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			26.4945i			0.898765i
$870$		0			0
$871$			4.11489i			0.139428i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.52791	+	9.97188i	0.0854589	+	0.337111i
$876$		0			0
$877$		−	13.6140i		−	0.459712i	−0.973225	−	0.229856i	$-0.926175\pi$
$877$		−	13.6140i		−	0.459712i	0.973225	−	0.229856i	$-0.0738255\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	51.2726			1.72742			0.863709	−	0.503991i	$-0.168135\pi$
$881$	51.2726			1.72742			0.863709	+	0.503991i	$0.168135\pi$
$882$		0			0
$883$			50.6073i			1.70307i	0.524298	+	0.851535i	$0.324328\pi$
$883$			50.6073i			1.70307i	−0.524298	+	0.851535i	$0.675672\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.7197			0.662122			0.331061	−	0.943609i	$-0.392593\pi$
$887$	19.7197			0.662122			0.331061	+	0.943609i	$0.392593\pi$
$888$		0			0
$889$		−	7.92677i		−	0.265855i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	98.8830i		−	3.30899i
$894$		0			0
$895$	21.5621	+	31.0913i	0.720741	+	1.03927i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			13.8007i			0.460280i
$900$		0			0
$901$	0.338003			0.0112605
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.55211	−	12.3316i	−0.284282	−	0.409918i
$906$		0			0
$907$	−38.7358			−1.28620			−0.643100	−	0.765782i	$-0.722352\pi$
$907$	−38.7358			−1.28620			−0.643100	+	0.765782i	$0.722352\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.1748			0.569025			0.284512	−	0.958672i	$-0.408168\pi$
$911$	17.1748			0.569025			0.284512	+	0.958672i	$0.408168\pi$
$912$		0			0
$913$		−	7.45973i		−	0.246881i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8.13346i			0.268591i
$918$		0			0
$919$			32.4693i			1.07106i	0.844516	+	0.535531i	$0.179888\pi$
$919$			32.4693i			1.07106i	−0.844516	+	0.535531i	$0.820112\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.08771			−0.233295
$924$		0			0
$925$	12.6806	−	33.8858i	0.416936	−	1.11416i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			37.4936i			1.23013i	0.788478	+	0.615063i	$0.210870\pi$
$929$			37.4936i			1.23013i	−0.788478	+	0.615063i	$0.789130\pi$
$930$		0			0
$931$		−	45.7628i		−	1.49982i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	36.9440	+	53.2711i	1.20820	+	1.74215i
$936$		0			0
$937$	55.5965			1.81626			0.908130	−	0.418689i	$-0.137510\pi$
$937$	55.5965			1.81626			0.908130	+	0.418689i	$0.137510\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.73387			−0.121721			−0.0608604	−	0.998146i	$-0.519384\pi$
$941$	−3.73387			−0.121721			−0.0608604	+	0.998146i	$0.519384\pi$
$942$		0			0
$943$	−8.30714	−	4.11419i	−0.270518	−	0.133977i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.3862			−0.499983			−0.249992	−	0.968248i	$-0.580428\pi$
$947$	−15.3862			−0.499983			−0.249992	+	0.968248i	$0.580428\pi$
$948$		0			0
$949$	3.15388			0.102379
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.94928i			0.127930i	0.997952	+	0.0639649i	$0.0203746\pi$
$953$			3.94928i			0.127930i	−0.997952	+	0.0639649i	$0.979625\pi$
$954$		0			0
$955$	−14.2042	+	9.85071i	−0.459636	+	0.318762i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	12.2751i		−	0.396385i
$960$		0			0
$961$	−3.53307			−0.113970
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.25787	+	3.25572i	0.0726836	+	0.104805i
$966$		0			0
$967$		−	11.7622i		−	0.378247i	−0.981953	−	0.189124i	$-0.939435\pi$
$967$		−	11.7622i		−	0.378247i	0.981953	−	0.189124i	$-0.0605647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	45.2259			1.45137			0.725684	−	0.688028i	$-0.241524\pi$
$971$	45.2259			1.45137			0.725684	+	0.688028i	$0.241524\pi$
$972$		0			0
$973$	7.15882			0.229501
$974$		0			0
$975$		0			0
$976$		0			0
$977$			38.0320i			1.21675i	0.793650	+	0.608375i	$0.208178\pi$
$977$			38.0320i			1.21675i	−0.793650	+	0.608375i	$0.791822\pi$
$978$		0			0
$979$	23.1787			0.740793
$980$		0			0
$981$		0			0
$982$		0			0
$983$			23.3545i			0.744892i	0.928054	+	0.372446i	$0.121481\pi$
$983$			23.3545i			0.744892i	−0.928054	+	0.372446i	$0.878519\pi$
$984$		0			0
$985$	−18.1241	+	12.5692i	−0.577481	+	0.400489i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.03163	+	6.12130i	−0.0964004	+	0.194646i
$990$		0			0
$991$	−16.2121			−0.514993			−0.257497	−	0.966279i	$-0.582898\pi$
$991$	−16.2121			−0.514993			−0.257497	+	0.966279i	$0.582898\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.73298	+	6.82469i	0.150046	+	0.216357i
$996$		0			0
$997$		−	33.8538i		−	1.07216i	−0.844167	−	0.536081i	$-0.819904\pi$
$997$		−	33.8538i		−	1.07216i	0.844167	−	0.536081i	$-0.180096\pi$
$998$		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.2069.8	yes	32
3.2	odd	2	inner	4140.2.n.b.2069.26	yes	32
5.4	even	2	inner	4140.2.n.b.2069.5	✓	32
15.14	odd	2	inner	4140.2.n.b.2069.27	yes	32
23.22	odd	2	inner	4140.2.n.b.2069.25	yes	32
69.68	even	2	inner	4140.2.n.b.2069.7	yes	32
115.114	odd	2	inner	4140.2.n.b.2069.28	yes	32
345.344	even	2	inner	4140.2.n.b.2069.6	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.n.b.2069.5	✓	32	5.4	even	2	inner
4140.2.n.b.2069.6	yes	32	345.344	even	2	inner
4140.2.n.b.2069.7	yes	32	69.68	even	2	inner
4140.2.n.b.2069.8	yes	32	1.1	even	1	trivial
4140.2.n.b.2069.25	yes	32	23.22	odd	2	inner
4140.2.n.b.2069.26	yes	32	3.2	odd	2	inner
4140.2.n.b.2069.27	yes	32	15.14	odd	2	inner
4140.2.n.b.2069.28	yes	32	115.114	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)

\(f(q)\)	\(=\)	\(q+(-1.83744 + 1.27428i) q^{5} +0.920125 q^{7} +O(q^{10})\)	\(q+(-1.83744 + 1.27428i) q^{5} +0.920125 q^{7} -4.88423 q^{11} -0.822839i q^{13} +5.93583i q^{17} +7.43704i q^{19} +(2.12845 - 4.29764i) q^{23} +(1.75240 - 4.68285i) q^{25} +2.63328i q^{29} +5.24089 q^{31} +(-1.69068 + 1.17250i) q^{35} +7.23616 q^{37} -1.93295i q^{41} -1.42434 q^{43} -13.2960 q^{47} -6.15337 q^{49} -0.0569429i q^{53} +(8.97450 - 6.22390i) q^{55} -2.50951i q^{59} -5.95419i q^{61} +(1.04853 + 1.51192i) q^{65} -5.00085 q^{67} -8.61372i q^{71} +3.83292i q^{73} -4.49410 q^{77} -5.42450i q^{79} +1.52731i q^{83} +(-7.56394 - 10.9068i) q^{85} -4.74561 q^{89} -0.757115i q^{91} +(-9.47690 - 13.6651i) q^{95} -12.6112 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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

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