LMFDB - Embedded newform 2520.2.k.a.1889.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1889,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2520.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.k (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:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.7
Character	$\chi$	$=$	2520.1889
Dual form			2520.2.k.a.1889.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.873162 - 2.05854i) q^{5} +(2.52372 + 0.794245i) q^{7} +O(q^{10})$	$q+(-0.873162 - 2.05854i) q^{5} +(2.52372 + 0.794245i) q^{7} +0.935722i q^{11} -6.43078 q^{13} +0.439826i q^{17} +3.43888i q^{19} +8.69904 q^{23} +(-3.47518 + 3.59488i) q^{25} +3.38512i q^{29} +6.60280i q^{31} +(-0.568633 - 5.88869i) q^{35} +9.91188i q^{37} +3.89198 q^{41} -5.85036i q^{43} -3.41686i q^{47} +(5.73835 + 4.00891i) q^{49} +7.31847 q^{53} +(1.92622 - 0.817037i) q^{55} +0.437881 q^{59} -12.7718i q^{61} +(5.61512 + 13.2380i) q^{65} +11.9199i q^{67} +7.68748i q^{71} +8.54529 q^{73} +(-0.743193 + 2.36150i) q^{77} -11.7710 q^{79} -11.2493i q^{83} +(0.905400 - 0.384040i) q^{85} -7.78403 q^{89} +(-16.2295 - 5.10762i) q^{91} +(7.07908 - 3.00270i) q^{95} +13.4398 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) $ 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$1$	$-1$

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
$5$	−0.873162	−	2.05854i	−0.390490	−	0.920607i
$5$	−0.873162	−	2.05854i	−0.390490	−	0.920607i
$6$		0			0
$7$	2.52372	+	0.794245i	0.953877	+	0.300196i
$7$	2.52372	+	0.794245i	0.953877	+	0.300196i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.935722i			0.282131i	0.990000	+	0.141065i	$0.0450528\pi$
$11$			0.935722i			0.282131i	−0.990000	+	0.141065i	$0.954947\pi$
$12$		0			0
$13$	−6.43078			−1.78358			−0.891789	−	0.452451i	$-0.850550\pi$
$13$	−6.43078			−1.78358			−0.891789	+	0.452451i	$0.850550\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.439826i			0.106674i	0.998577	+	0.0533368i	$0.0169857\pi$
$17$			0.439826i			0.106674i	−0.998577	+	0.0533368i	$0.983014\pi$
$18$		0			0
$19$			3.43888i			0.788934i	0.918910	+	0.394467i	$0.129071\pi$
$19$			3.43888i			0.788934i	−0.918910	+	0.394467i	$0.870929\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.69904			1.81388			0.906938	−	0.421264i	$-0.138413\pi$
$23$	8.69904			1.81388			0.906938	+	0.421264i	$0.138413\pi$
$24$		0			0
$25$	−3.47518	+	3.59488i	−0.695035	+	0.718976i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.38512i			0.628600i	0.949324	+	0.314300i	$0.101770\pi$
$29$			3.38512i			0.628600i	−0.949324	+	0.314300i	$0.898230\pi$
$30$		0			0
$31$			6.60280i			1.18590i	0.805240	+	0.592949i	$0.202036\pi$
$31$			6.60280i			1.18590i	−0.805240	+	0.592949i	$0.797964\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.568633	−	5.88869i	−0.0961165	−	0.995370i
$36$		0			0
$37$			9.91188i			1.62950i	0.579811	+	0.814751i	$0.303127\pi$
$37$			9.91188i			1.62950i	−0.579811	+	0.814751i	$0.696873\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.89198			0.607825			0.303913	−	0.952700i	$-0.401707\pi$
$41$	3.89198			0.607825			0.303913	+	0.952700i	$0.401707\pi$
$42$		0			0
$43$		−	5.85036i		−	0.892172i	−0.894990	−	0.446086i	$-0.852818\pi$
$43$		−	5.85036i		−	0.892172i	0.894990	−	0.446086i	$-0.147182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.41686i		−	0.498400i	−0.968452	−	0.249200i	$-0.919832\pi$
$47$		−	3.41686i		−	0.498400i	0.968452	−	0.249200i	$-0.0801677\pi$
$48$		0			0
$49$	5.73835	+	4.00891i	0.819764	+	0.572701i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.31847			1.00527			0.502634	−	0.864499i	$-0.332364\pi$
$53$	7.31847			1.00527			0.502634	+	0.864499i	$0.332364\pi$
$54$		0			0
$55$	1.92622	−	0.817037i	0.259732	−	0.110169i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.437881			0.0570072			0.0285036	−	0.999594i	$-0.490926\pi$
$59$	0.437881			0.0570072			0.0285036	+	0.999594i	$0.490926\pi$
$60$		0			0
$61$		−	12.7718i		−	1.63526i	−0.575747	−	0.817628i	$-0.695289\pi$
$61$		−	12.7718i		−	1.63526i	0.575747	−	0.817628i	$-0.304711\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.61512	+	13.2380i	0.696470	+	1.64198i
$66$		0			0
$67$			11.9199i			1.45625i	0.685446	+	0.728123i	$0.259607\pi$
$67$			11.9199i			1.45625i	−0.685446	+	0.728123i	$0.740393\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.68748i			0.912336i	0.889894	+	0.456168i	$0.150778\pi$
$71$			7.68748i			0.912336i	−0.889894	+	0.456168i	$0.849222\pi$
$72$		0			0
$73$	8.54529			1.00015			0.500075	−	0.865982i	$-0.333306\pi$
$73$	8.54529			1.00015			0.500075	+	0.865982i	$0.333306\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.743193	+	2.36150i	−0.0846947	+	0.269118i
$78$		0			0
$79$	−11.7710			−1.32435			−0.662173	−	0.749351i	$-0.730366\pi$
$79$	−11.7710			−1.32435			−0.662173	+	0.749351i	$0.730366\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.2493i		−	1.23477i	−0.786662	−	0.617384i	$-0.788193\pi$
$83$		−	11.2493i		−	1.23477i	0.786662	−	0.617384i	$-0.211807\pi$
$84$		0			0
$85$	0.905400	−	0.384040i	0.0982044	−	0.0416549i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.78403			−0.825106			−0.412553	−	0.910934i	$-0.635363\pi$
$89$	−7.78403			−0.825106			−0.412553	+	0.910934i	$0.635363\pi$
$90$		0			0
$91$	−16.2295	−	5.10762i	−1.70132	−	0.535424i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.07908	−	3.00270i	0.726299	−	0.308071i
$96$		0			0
$97$	13.4398			1.36460			0.682300	−	0.731072i	$-0.260980\pi$
$97$	13.4398			1.36460			0.682300	+	0.731072i	$0.260980\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.74457			−0.969621			−0.484811	−	0.874619i	$-0.661112\pi$
$101$	−9.74457			−0.969621			−0.484811	+	0.874619i	$0.661112\pi$
$102$		0			0
$103$	7.91809			0.780192			0.390096	−	0.920774i	$-0.372442\pi$
$103$	7.91809			0.780192			0.390096	+	0.920774i	$0.372442\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.4863			1.49712			0.748560	−	0.663068i	$-0.230746\pi$
$107$	15.4863			1.49712			0.748560	+	0.663068i	$0.230746\pi$
$108$		0			0
$109$	−7.24470			−0.693916			−0.346958	−	0.937881i	$-0.612785\pi$
$109$	−7.24470			−0.693916			−0.346958	+	0.937881i	$0.612785\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.2989			0.968838			0.484419	−	0.874836i	$-0.339031\pi$
$113$	10.2989			0.968838			0.484419	+	0.874836i	$0.339031\pi$
$114$		0			0
$115$	−7.59568	−	17.9073i	−0.708300	−	1.66987i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.349330	+	1.11000i	−0.0320230	+	0.101753i
$120$		0			0
$121$	10.1244			0.920402
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.4346	+	4.01488i	0.933298	+	0.359102i
$126$		0			0
$127$		−	0.299817i		−	0.0266044i	−0.999912	−	0.0133022i	$-0.995766\pi$
$127$		−	0.299817i		−	0.0266044i	0.999912	−	0.0133022i	$-0.00423435\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.36236			0.293771			0.146885	−	0.989154i	$-0.453075\pi$
$131$	3.36236			0.293771			0.146885	+	0.989154i	$0.453075\pi$
$132$		0			0
$133$	−2.73132	+	8.67879i	−0.236835	+	0.752547i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.36190			−0.372662			−0.186331	−	0.982487i	$-0.559660\pi$
$137$	−4.36190			−0.372662			−0.186331	+	0.982487i	$0.559660\pi$
$138$		0			0
$139$			15.4619i			1.31146i	0.754996	+	0.655730i	$0.227639\pi$
$139$			15.4619i			1.31146i	−0.754996	+	0.655730i	$0.772361\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.01743i		−	0.503203i
$144$		0			0
$145$	6.96840	−	2.95576i	0.578694	−	0.245462i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.2144i			1.08257i	0.840841	+	0.541283i	$0.182061\pi$
$149$			13.2144i			1.08257i	−0.840841	+	0.541283i	$0.817939\pi$
$150$		0			0
$151$	−5.46397			−0.444651			−0.222326	−	0.974972i	$-0.571365\pi$
$151$	−5.46397			−0.444651			−0.222326	+	0.974972i	$0.571365\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	13.5921	−	5.76532i	1.09175	−	0.463081i
$156$		0			0
$157$	−0.608353			−0.0485518			−0.0242759	−	0.999705i	$-0.507728\pi$
$157$	−0.608353			−0.0485518			−0.0242759	+	0.999705i	$0.507728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	21.9540	+	6.90917i	1.73022	+	0.544519i
$162$		0			0
$163$			12.1088i			0.948434i	0.880408	+	0.474217i	$0.157269\pi$
$163$			12.1088i			0.948434i	−0.880408	+	0.474217i	$0.842731\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.4948i			1.35379i	0.736082	+	0.676893i	$0.236674\pi$
$167$			17.4948i			1.35379i	−0.736082	+	0.676893i	$0.763326\pi$
$168$		0			0
$169$	28.3550			2.18115
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.03861i		−	0.0789644i	−0.999220	−	0.0394822i	$-0.987429\pi$
$173$		−	1.03861i		−	0.0789644i	0.999220	−	0.0394822i	$-0.0125708\pi$
$174$		0			0
$175$	−11.6256	+	6.31233i	−0.878812	+	0.477168i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.28428i			0.0959913i	0.998848	+	0.0479956i	$0.0152834\pi$
$179$			1.28428i			0.0959913i	−0.998848	+	0.0479956i	$0.984717\pi$
$180$		0			0
$181$			12.0173i			0.893241i	0.894723	+	0.446621i	$0.147373\pi$
$181$			12.0173i			0.893241i	−0.894723	+	0.446621i	$0.852627\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.4040	−	8.65468i	1.50013	−	0.636304i
$186$		0			0
$187$	−0.411555			−0.0300959
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.8750i		−	1.79989i	−0.436004	−	0.899945i	$-0.643607\pi$
$191$		−	24.8750i		−	1.79989i	0.436004	−	0.899945i	$-0.356393\pi$
$192$		0			0
$193$			17.9267i			1.29039i	0.764019	+	0.645194i	$0.223224\pi$
$193$			17.9267i			1.29039i	−0.764019	+	0.645194i	$0.776776\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.9727			1.35175			0.675875	−	0.737017i	$-0.263766\pi$
$197$	18.9727			1.35175			0.675875	+	0.737017i	$0.263766\pi$
$198$		0			0
$199$			7.97787i			0.565536i	0.959188	+	0.282768i	$0.0912527\pi$
$199$			7.97787i			0.565536i	−0.959188	+	0.282768i	$0.908747\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.68861	+	8.54309i	−0.188704	+	0.599608i
$204$		0			0
$205$	−3.39833	−	8.01180i	−0.237350	−	0.559568i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.21784			−0.222583
$210$		0			0
$211$	−10.8000			−0.743503			−0.371752	−	0.928332i	$-0.621243\pi$
$211$	−10.8000			−0.743503			−0.371752	+	0.928332i	$0.621243\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.0432	+	5.10832i	−0.821340	+	0.348384i
$216$		0			0
$217$	−5.24424	+	16.6636i	−0.356002	+	1.13120i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$	0.215277			0.0144160			0.00720799	−	0.999974i	$-0.497706\pi$
$223$	0.215277			0.0144160			0.00720799	+	0.999974i	$0.497706\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	17.0683i		−	1.13286i	−0.824110	−	0.566430i	$-0.808324\pi$
$227$		−	17.0683i		−	1.13286i	0.824110	−	0.566430i	$-0.191676\pi$
$228$		0			0
$229$			8.11009i			0.535930i	0.963429	+	0.267965i	$0.0863511\pi$
$229$			8.11009i			0.535930i	−0.963429	+	0.267965i	$0.913649\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.2671			−1.39325			−0.696626	−	0.717435i	$-0.745316\pi$
$233$	−21.2671			−1.39325			−0.696626	+	0.717435i	$0.745316\pi$
$234$		0			0
$235$	−7.03374	+	2.98347i	−0.458831	+	0.194620i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			11.9711i			0.774345i	0.922007	+	0.387172i	$0.126548\pi$
$239$			11.9711i			0.774345i	−0.922007	+	0.387172i	$0.873452\pi$
$240$		0			0
$241$			10.2698i			0.661533i	0.943713	+	0.330767i	$0.107307\pi$
$241$			10.2698i			0.661533i	−0.943713	+	0.330767i	$0.892693\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.24199	−	15.3130i	0.207123	−	0.978315i
$246$		0			0
$247$		−	22.1147i		−	1.40713i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.97654			0.503475			0.251738	−	0.967796i	$-0.418998\pi$
$251$	7.97654			0.503475			0.251738	+	0.967796i	$0.418998\pi$
$252$		0			0
$253$			8.13989i			0.511750i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.5949i		−	1.53419i	−0.641536	−	0.767093i	$-0.721703\pi$
$257$		−	24.5949i		−	1.53419i	0.641536	−	0.767093i	$-0.278297\pi$
$258$		0			0
$259$	−7.87246	+	25.0148i	−0.489171	+	1.55435i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.62250			0.223373			0.111687	−	0.993743i	$-0.464375\pi$
$263$	3.62250			0.223373			0.111687	+	0.993743i	$0.464375\pi$
$264$		0			0
$265$	−6.39021	−	15.0654i	−0.392547	−	0.925458i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.4305			−1.06276			−0.531378	−	0.847135i	$-0.678326\pi$
$269$	−17.4305			−1.06276			−0.531378	+	0.847135i	$0.678326\pi$
$270$		0			0
$271$		−	28.0986i		−	1.70687i	−0.521202	−	0.853433i	$-0.674516\pi$
$271$		−	28.0986i		−	1.70687i	0.521202	−	0.853433i	$-0.325484\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.36381	−	3.25180i	−0.202845	−	0.196091i
$276$		0			0
$277$		−	19.3858i		−	1.16478i	−0.812911	−	0.582389i	$-0.802118\pi$
$277$		−	19.3858i		−	1.16478i	0.812911	−	0.582389i	$-0.197882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.7178i			1.41489i	0.706770	+	0.707444i	$0.250152\pi$
$281$			23.7178i			1.41489i	−0.706770	+	0.707444i	$0.749848\pi$
$282$		0			0
$283$	−3.51150			−0.208737			−0.104369	−	0.994539i	$-0.533282\pi$
$283$	−3.51150			−0.208737			−0.104369	+	0.994539i	$0.533282\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.82228	+	3.09119i	0.579791	+	0.182467i
$288$		0			0
$289$	16.8066			0.988621
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.4094i			1.42602i	0.701156	+	0.713008i	$0.252667\pi$
$293$			24.4094i			1.42602i	−0.701156	+	0.713008i	$0.747333\pi$
$294$		0			0
$295$	−0.382341	−	0.901395i	−0.0222607	−	0.0524813i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−55.9417			−3.23519
$300$		0			0
$301$	4.64662	−	14.7647i	0.267827	−	0.851023i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−26.2912	+	11.1518i	−1.50543	+	0.638551i
$306$		0			0
$307$	−23.4207			−1.33669			−0.668344	−	0.743852i	$-0.732997\pi$
$307$	−23.4207			−1.33669			−0.668344	+	0.743852i	$0.732997\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.6767			−1.17247			−0.586233	−	0.810142i	$-0.699390\pi$
$311$	−20.6767			−1.17247			−0.586233	+	0.810142i	$0.699390\pi$
$312$		0			0
$313$	−12.3102			−0.695816			−0.347908	−	0.937529i	$-0.613108\pi$
$313$	−12.3102			−0.695816			−0.347908	+	0.937529i	$0.613108\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.8669			−0.947339			−0.473670	−	0.880703i	$-0.657071\pi$
$317$	−16.8669			−0.947339			−0.473670	+	0.880703i	$0.657071\pi$
$318$		0			0
$319$	−3.16753			−0.177348
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.51251			−0.0841584
$324$		0			0
$325$	22.3481	−	23.1179i	1.23965	−	1.28235i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.71382	−	8.62320i	0.149618	−	0.475413i
$330$		0			0
$331$	13.3777			0.735305			0.367653	−	0.929963i	$-0.380162\pi$
$331$	13.3777			0.735305			0.367653	+	0.929963i	$0.380162\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.5376	−	10.4080i	1.34063	−	0.568650i
$336$		0			0
$337$		−	0.185308i		−	0.0100944i	−0.999987	−	0.00504719i	$-0.998393\pi$
$337$		−	0.185308i		−	0.0100944i	0.999987	−	0.00504719i	$-0.00160658\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.17839			−0.334578
$342$		0			0
$343$	11.2979	+	14.6750i	0.610032	+	0.792377i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.4526			−0.561124			−0.280562	−	0.959836i	$-0.590521\pi$
$347$	−10.4526			−0.561124			−0.280562	+	0.959836i	$0.590521\pi$
$348$		0			0
$349$			21.1970i			1.13465i	0.823494	+	0.567325i	$0.192021\pi$
$349$			21.1970i			1.13465i	−0.823494	+	0.567325i	$0.807979\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			23.7680i			1.26504i	0.774543	+	0.632521i	$0.217980\pi$
$353$			23.7680i			1.26504i	−0.774543	+	0.632521i	$0.782020\pi$
$354$		0			0
$355$	15.8250	−	6.71242i	0.839903	−	0.356258i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	9.21037i		−	0.486105i	−0.970013	−	0.243052i	$-0.921851\pi$
$359$		−	9.21037i		−	0.486105i	0.970013	−	0.243052i	$-0.0781487\pi$
$360$		0			0
$361$	7.17407			0.377583
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.46142	−	17.5908i	−0.390549	−	0.920746i
$366$		0			0
$367$	−8.12205			−0.423968			−0.211984	−	0.977273i	$-0.567992\pi$
$367$	−8.12205			−0.423968			−0.211984	+	0.977273i	$0.567992\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	18.4698	+	5.81266i	0.958903	+	0.301778i
$372$		0			0
$373$		−	17.7417i		−	0.918629i	−0.888274	−	0.459315i	$-0.848095\pi$
$373$		−	17.7417i		−	0.918629i	0.888274	−	0.459315i	$-0.151905\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	21.7689i		−	1.12116i
$378$		0			0
$379$	−12.7421			−0.654517			−0.327258	−	0.944935i	$-0.606125\pi$
$379$	−12.7421			−0.654517			−0.327258	+	0.944935i	$0.606125\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	30.4519i		−	1.55602i	−0.628251	−	0.778011i	$-0.716229\pi$
$383$		−	30.4519i		−	1.55602i	0.628251	−	0.778011i	$-0.283771\pi$
$384$		0			0
$385$	5.51018	−	0.532083i	0.280825	−	0.0271175i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			5.79727i			0.293933i	0.989141	+	0.146967i	$0.0469510\pi$
$389$			5.79727i			0.293933i	−0.989141	+	0.146967i	$0.953049\pi$
$390$		0			0
$391$			3.82607i			0.193493i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.2780	+	24.2312i	0.517144	+	1.21920i
$396$		0			0
$397$	−3.74526			−0.187969			−0.0939845	−	0.995574i	$-0.529960\pi$
$397$	−3.74526			−0.187969			−0.0939845	+	0.995574i	$0.529960\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.3290i			0.515807i	0.966171	+	0.257904i	$0.0830317\pi$
$401$			10.3290i			0.515807i	−0.966171	+	0.257904i	$0.916968\pi$
$402$		0			0
$403$		−	42.4612i		−	2.11514i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.27477			−0.459733
$408$		0			0
$409$			1.64604i			0.0813913i	0.999172	+	0.0406957i	$0.0129574\pi$
$409$			1.64604i			0.0813913i	−0.999172	+	0.0406957i	$0.987043\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.10509	+	0.347785i	0.0543779	+	0.0171134i
$414$		0			0
$415$	−23.1571	+	9.82243i	−1.13674	+	0.482164i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.4352			0.851763			0.425882	−	0.904779i	$-0.359964\pi$
$419$	17.4352			0.851763			0.425882	+	0.904779i	$0.359964\pi$
$420$		0			0
$421$	−2.89595			−0.141140			−0.0705701	−	0.997507i	$-0.522482\pi$
$421$	−2.89595			−0.141140			−0.0705701	+	0.997507i	$0.522482\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.58112	−	1.52847i	−0.0766957	−	0.0741418i
$426$		0			0
$427$	10.1439	−	32.2324i	0.490898	−	1.55983i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	32.9082i		−	1.58513i	−0.609786	−	0.792566i	$-0.708744\pi$
$431$		−	32.9082i		−	1.58513i	0.609786	−	0.792566i	$-0.291256\pi$
$432$		0			0
$433$	−9.15690			−0.440053			−0.220026	−	0.975494i	$-0.570614\pi$
$433$	−9.15690			−0.440053			−0.220026	+	0.975494i	$0.570614\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			29.9150i			1.43103i
$438$		0			0
$439$			11.5050i			0.549105i	0.961572	+	0.274552i	$0.0885297\pi$
$439$			11.5050i			0.549105i	−0.961572	+	0.274552i	$0.911470\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−24.5008			−1.16407			−0.582035	−	0.813164i	$-0.697743\pi$
$443$	−24.5008			−1.16407			−0.582035	+	0.813164i	$0.697743\pi$
$444$		0			0
$445$	6.79672	+	16.0237i	0.322196	+	0.759598i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.63242i		−	0.407389i	−0.979034	−	0.203695i	$-0.934705\pi$
$449$		−	8.63242i		−	0.407389i	0.979034	−	0.203695i	$-0.0652950\pi$
$450$		0			0
$451$			3.64181i			0.171486i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.65676	+	37.8689i	0.171431	+	1.77532i
$456$		0			0
$457$		−	15.4408i		−	0.722291i	−0.932509	−	0.361146i	$-0.882386\pi$
$457$		−	15.4408i		−	0.722291i	0.932509	−	0.361146i	$-0.117614\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	28.3288			1.31940			0.659701	−	0.751528i	$-0.270683\pi$
$461$	28.3288			1.31940			0.659701	+	0.751528i	$0.270683\pi$
$462$		0			0
$463$		−	30.5982i		−	1.42202i	−0.703182	−	0.711010i	$-0.748238\pi$
$463$		−	30.5982i		−	1.42202i	0.703182	−	0.711010i	$-0.251762\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			17.8389i			0.825485i	0.910848	+	0.412743i	$0.135429\pi$
$467$			17.8389i			0.825485i	−0.910848	+	0.412743i	$0.864571\pi$
$468$		0			0
$469$	−9.46732	+	30.0825i	−0.437160	+	1.38908i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.47432			0.251709
$474$		0			0
$475$	−12.3624	−	11.9507i	−0.567225	−	0.548337i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.63091			0.0745182			0.0372591	−	0.999306i	$-0.488137\pi$
$479$	1.63091			0.0745182			0.0372591	+	0.999306i	$0.488137\pi$
$480$		0			0
$481$		−	63.7411i		−	2.90635i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.7351	−	27.6663i	−0.532863	−	1.25626i
$486$		0			0
$487$			16.1569i			0.732139i	0.930587	+	0.366070i	$0.119297\pi$
$487$			16.1569i			0.732139i	−0.930587	+	0.366070i	$0.880703\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.4958i		−	1.33113i	−0.746341	−	0.665564i	$-0.768191\pi$
$491$		−	29.4958i		−	1.33113i	0.746341	−	0.665564i	$-0.231809\pi$
$492$		0			0
$493$	−1.48886			−0.0670550
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.10574	+	19.4011i	−0.273880	+	0.870257i
$498$		0			0
$499$	41.1053			1.84013			0.920064	−	0.391768i	$-0.128137\pi$
$499$	41.1053			1.84013			0.920064	+	0.391768i	$0.128137\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	0.713025i		−	0.0317922i	−0.999874	−	0.0158961i	$-0.994940\pi$
$503$		−	0.713025i		−	0.0317922i	0.999874	−	0.0158961i	$-0.00506010\pi$
$504$		0			0
$505$	8.50859	+	20.0596i	0.378627	+	0.892640i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−35.8985			−1.59117			−0.795585	−	0.605841i	$-0.792837\pi$
$509$	−35.8985			−1.59117			−0.795585	+	0.605841i	$0.792837\pi$
$510$		0			0
$511$	21.5659	+	6.78706i	0.954021	+	0.300242i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.91377	−	16.2997i	−0.304657	−	0.718250i
$516$		0			0
$517$	3.19723			0.140614
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.02640			−0.0887783			−0.0443892	−	0.999014i	$-0.514134\pi$
$521$	−2.02640			−0.0887783			−0.0443892	+	0.999014i	$0.514134\pi$
$522$		0			0
$523$	39.1259			1.71086			0.855428	−	0.517922i	$-0.173294\pi$
$523$	39.1259			1.71086			0.855428	+	0.517922i	$0.173294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.90408			−0.126504
$528$		0			0
$529$	52.6734			2.29015
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−25.0285			−1.08410
$534$		0			0
$535$	−13.5221	−	31.8792i	−0.584610	−	1.37826i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.75123	+	5.36950i	−0.161577	+	0.231281i
$540$		0			0
$541$	−41.6320			−1.78990			−0.894950	−	0.446167i	$-0.852789\pi$
$541$	−41.6320			−1.78990			−0.894950	+	0.446167i	$0.852789\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.32580	+	14.9135i	0.270967	+	0.638824i
$546$		0			0
$547$		−	12.0851i		−	0.516721i	−0.966049	−	0.258360i	$-0.916818\pi$
$547$		−	12.0851i		−	0.516721i	0.966049	−	0.258360i	$-0.0831822\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.6410			−0.495924
$552$		0			0
$553$	−29.7069	−	9.34910i	−1.26326	−	0.397564i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.52975			0.107189			0.0535945	−	0.998563i	$-0.482932\pi$
$557$	2.52975			0.107189			0.0535945	+	0.998563i	$0.482932\pi$
$558$		0			0
$559$			37.6224i			1.59126i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	38.8051i		−	1.63544i	−0.575615	−	0.817721i	$-0.695237\pi$
$563$		−	38.8051i		−	1.63544i	0.575615	−	0.817721i	$-0.304763\pi$
$564$		0			0
$565$	−8.99260	−	21.2007i	−0.378321	−	0.891919i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.1694i			0.594012i	0.954876	+	0.297006i	$0.0959881\pi$
$569$			14.1694i			0.594012i	−0.954876	+	0.297006i	$0.904012\pi$
$570$		0			0
$571$	−12.3951			−0.518718			−0.259359	−	0.965781i	$-0.583511\pi$
$571$	−12.3951			−0.518718			−0.259359	+	0.965781i	$0.583511\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.2307	+	31.2720i	−1.26071	+	1.30413i
$576$		0			0
$577$	23.2459			0.967740			0.483870	−	0.875140i	$-0.339231\pi$
$577$	23.2459			0.967740			0.483870	+	0.875140i	$0.339231\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.93467	−	28.3900i	0.370673	−	1.17782i
$582$		0			0
$583$			6.84805i			0.283617i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			38.6784i			1.59643i	0.602374	+	0.798214i	$0.294222\pi$
$587$			38.6784i			1.59643i	−0.602374	+	0.798214i	$0.705778\pi$
$588$		0			0
$589$	−22.7063			−0.935596
$590$		0			0
$591$		0			0
$592$		0			0
$593$			14.7303i			0.604902i	0.953165	+	0.302451i	$0.0978049\pi$
$593$			14.7303i			0.604902i	−0.953165	+	0.302451i	$0.902195\pi$
$594$		0			0
$595$	2.59000	−	0.250100i	0.106180	−	0.0102531i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.83171i		−	0.0748417i	−0.999300	−	0.0374208i	$-0.988086\pi$
$599$		−	1.83171i		−	0.0748417i	0.999300	−	0.0374208i	$-0.0119142\pi$
$600$		0			0
$601$			2.42573i			0.0989475i	0.998775	+	0.0494737i	$0.0157544\pi$
$601$			2.42573i			0.0989475i	−0.998775	+	0.0494737i	$0.984246\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.84026	−	20.8415i	−0.359408	−	0.847329i
$606$		0			0
$607$	2.20006			0.0892979			0.0446489	−	0.999003i	$-0.485783\pi$
$607$	2.20006			0.0892979			0.0446489	+	0.999003i	$0.485783\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			21.9731i			0.888936i
$612$		0			0
$613$		−	41.8447i		−	1.69009i	−0.534693	−	0.845046i	$-0.679573\pi$
$613$		−	41.8447i		−	1.69009i	0.534693	−	0.845046i	$-0.320427\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−42.5694			−1.71378			−0.856889	−	0.515501i	$-0.827606\pi$
$617$	−42.5694			−1.71378			−0.856889	+	0.515501i	$0.827606\pi$
$618$		0			0
$619$		−	2.67764i		−	0.107623i	−0.998551	−	0.0538117i	$-0.982863\pi$
$619$		−	2.67764i		−	0.107623i	0.998551	−	0.0538117i	$-0.0171371\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.6447	−	6.18243i	−0.787050	−	0.247694i
$624$		0			0
$625$	−0.846307	−	24.9857i	−0.0338523	−	0.999427i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.35950			−0.173825
$630$		0			0
$631$	9.91314			0.394636			0.197318	−	0.980340i	$-0.436777\pi$
$631$	9.91314			0.394636			0.197318	+	0.980340i	$0.436777\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.617185	+	0.261788i	−0.0244922	+	0.0103888i
$636$		0			0
$637$	−36.9021	−	25.7804i	−1.46211	−	1.02146i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			8.66124i			0.342099i	0.985262	+	0.171049i	$0.0547157\pi$
$641$			8.66124i			0.342099i	−0.985262	+	0.171049i	$0.945284\pi$
$642$		0			0
$643$	20.3934			0.804238			0.402119	−	0.915588i	$-0.368274\pi$
$643$	20.3934			0.804238			0.402119	+	0.915588i	$0.368274\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	31.2584i		−	1.22889i	−0.788958	−	0.614447i	$-0.789379\pi$
$647$		−	31.2584i		−	1.22889i	0.788958	−	0.614447i	$-0.210621\pi$
$648$		0			0
$649$			0.409735i			0.0160835i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	33.1374			1.29677			0.648383	−	0.761314i	$-0.275446\pi$
$653$	33.1374			1.29677			0.648383	+	0.761314i	$0.275446\pi$
$654$		0			0
$655$	−2.93589	−	6.92156i	−0.114715	−	0.270448i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	29.3642i		−	1.14387i	−0.820300	−	0.571934i	$-0.806194\pi$
$659$		−	29.3642i		−	1.14387i	0.820300	−	0.571934i	$-0.193806\pi$
$660$		0			0
$661$			17.6230i			0.685455i	0.939435	+	0.342728i	$0.111351\pi$
$661$			17.6230i			0.685455i	−0.939435	+	0.342728i	$0.888649\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.2505	−	1.95546i	0.785282	−	0.0758296i
$666$		0			0
$667$			29.4473i			1.14020i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.9508			0.461356
$672$		0			0
$673$		−	15.3669i		−	0.592350i	−0.955134	−	0.296175i	$-0.904289\pi$
$673$		−	15.3669i		−	0.592350i	0.955134	−	0.296175i	$-0.0957112\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	31.3938i		−	1.20656i	−0.797529	−	0.603280i	$-0.793860\pi$
$677$		−	31.3938i		−	1.20656i	0.797529	−	0.603280i	$-0.206140\pi$
$678$		0			0
$679$	33.9182	+	10.6745i	1.30166	+	0.409648i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.89761			0.378721			0.189361	−	0.981908i	$-0.439358\pi$
$683$	9.89761			0.378721			0.189361	+	0.981908i	$0.439358\pi$
$684$		0			0
$685$	3.80864	+	8.97914i	0.145521	+	0.343075i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−47.0635			−1.79298
$690$		0			0
$691$		−	1.69577i		−	0.0645103i	−0.999480	−	0.0322551i	$-0.989731\pi$
$691$		−	1.69577i		−	0.0645103i	0.999480	−	0.0322551i	$-0.0102689\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	31.8289	−	13.5007i	1.20734	−	0.512112i
$696$		0			0
$697$			1.71179i			0.0648388i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	40.6337i		−	1.53471i	−0.641220	−	0.767357i	$-0.721571\pi$
$701$		−	40.6337i		−	1.53471i	0.641220	−	0.767357i	$-0.278429\pi$
$702$		0			0
$703$	−34.0858			−1.28557
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.5926	−	7.73958i	−0.924900	−	0.291077i
$708$		0			0
$709$	−19.4616			−0.730895			−0.365447	−	0.930832i	$-0.619084\pi$
$709$	−19.4616			−0.730895			−0.365447	+	0.930832i	$0.619084\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			57.4380i			2.15107i
$714$		0			0
$715$	−12.3871	+	5.25419i	−0.463252	+	0.196496i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.1417			−1.27327			−0.636636	−	0.771165i	$-0.719675\pi$
$719$	−34.1417			−1.27327			−0.636636	+	0.771165i	$0.719675\pi$
$720$		0			0
$721$	19.9830	+	6.28890i	0.744208	+	0.234211i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.1691	−	11.7639i	−0.451948	−	0.436899i
$726$		0			0
$727$	−31.1532			−1.15541			−0.577704	−	0.816246i	$-0.696051\pi$
$727$	−31.1532			−1.15541			−0.577704	+	0.816246i	$0.696051\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.57314			0.0951711
$732$		0			0
$733$	−9.48005			−0.350154			−0.175077	−	0.984555i	$-0.556017\pi$
$733$	−9.48005			−0.350154			−0.175077	+	0.984555i	$0.556017\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.1537			−0.410852
$738$		0			0
$739$	23.3230			0.857951			0.428975	−	0.903316i	$-0.358875\pi$
$739$	23.3230			0.857951			0.428975	+	0.903316i	$0.358875\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−44.3215			−1.62600			−0.812998	−	0.582266i	$-0.802166\pi$
$743$	−44.3215			−1.62600			−0.812998	+	0.582266i	$0.802166\pi$
$744$		0			0
$745$	27.2024	−	11.5383i	0.996618	−	0.422731i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	39.0832	+	12.2999i	1.42807	+	0.449430i
$750$		0			0
$751$	−12.2679			−0.447663			−0.223831	−	0.974628i	$-0.571856\pi$
$751$	−12.2679			−0.447663			−0.223831	+	0.974628i	$0.571856\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.77093	+	11.2478i	0.173632	+	0.409349i
$756$		0			0
$757$			7.04824i			0.256173i	0.991763	+	0.128086i	$0.0408835\pi$
$757$			7.04824i			0.256173i	−0.991763	+	0.128086i	$0.959117\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−47.0344			−1.70499			−0.852497	−	0.522731i	$-0.824913\pi$
$761$	−47.0344			−1.70499			−0.852497	+	0.522731i	$0.824913\pi$
$762$		0			0
$763$	−18.2836	−	5.75407i	−0.661911	−	0.208311i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.81592			−0.101677
$768$		0			0
$769$			50.0737i			1.80570i	0.429952	+	0.902852i	$0.358531\pi$
$769$			50.0737i			1.80570i	−0.429952	+	0.902852i	$0.641469\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			12.5132i			0.450067i	0.974351	+	0.225034i	$0.0722492\pi$
$773$			12.5132i			0.450067i	−0.974351	+	0.225034i	$0.927751\pi$
$774$		0			0
$775$	−23.7363	−	22.9459i	−0.852632	−	0.824241i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			13.3841i			0.479534i
$780$		0			0
$781$	−7.19335			−0.257398
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.531191	+	1.25232i	0.0189590	+	0.0446972i
$786$		0			0
$787$	−14.9877			−0.534254			−0.267127	−	0.963661i	$-0.586074\pi$
$787$	−14.9877			−0.534254			−0.267127	+	0.963661i	$0.586074\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	25.9915	+	8.17984i	0.924152	+	0.290842i
$792$		0			0
$793$			82.1324i			2.91661i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	16.6176i		−	0.588627i	−0.955709	−	0.294313i	$-0.904909\pi$
$797$		−	16.6176i		−	0.588627i	0.955709	−	0.294313i	$-0.0950910\pi$
$798$		0			0
$799$	1.50282			0.0531661
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7.99602i			0.282173i
$804$		0			0
$805$	−4.94656	−	51.2260i	−0.174343	−	1.80548i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			37.0727i			1.30341i	0.758475	+	0.651703i	$0.225945\pi$
$809$			37.0727i			1.30341i	−0.758475	+	0.651703i	$0.774055\pi$
$810$		0			0
$811$			2.13592i			0.0750022i	0.999297	+	0.0375011i	$0.0119398\pi$
$811$			2.13592i			0.0750022i	−0.999297	+	0.0375011i	$0.988060\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.9264	−	10.5729i	0.873135	−	0.370354i
$816$		0			0
$817$	20.1187			0.703865
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	13.4820i		−	0.470524i	−0.971932	−	0.235262i	$-0.924405\pi$
$821$		−	13.4820i		−	0.470524i	0.971932	−	0.235262i	$-0.0755948\pi$
$822$		0			0
$823$		−	12.8738i		−	0.448754i	−0.974502	−	0.224377i	$-0.927965\pi$
$823$		−	12.8738i		−	0.448754i	0.974502	−	0.224377i	$-0.0720347\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.485040			0.0168665			0.00843325	−	0.999964i	$-0.497316\pi$
$827$	0.485040			0.0168665			0.00843325	+	0.999964i	$0.497316\pi$
$828$		0			0
$829$			49.5130i			1.71966i	0.510582	+	0.859829i	$0.329430\pi$
$829$			49.5130i			1.71966i	−0.510582	+	0.859829i	$0.670570\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.76322	+	2.52388i	−0.0610921	+	0.0874471i
$834$		0			0
$835$	36.0137	−	15.2758i	1.24630	−	0.528640i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.8718			−0.858671			−0.429335	−	0.903145i	$-0.641252\pi$
$839$	−24.8718			−0.858671			−0.429335	+	0.903145i	$0.641252\pi$
$840$		0			0
$841$	17.5410			0.604862
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.7585	−	58.3699i	−0.851718	−	2.00798i
$846$		0			0
$847$	25.5512	+	8.04127i	0.877951	+	0.276301i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			86.2238i			2.95572i
$852$		0			0
$853$	−20.0889			−0.687832			−0.343916	−	0.939000i	$-0.611754\pi$
$853$	−20.0889			−0.687832			−0.343916	+	0.939000i	$0.611754\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.24183i			0.281535i	0.990043	+	0.140768i	$0.0449571\pi$
$857$			8.24183i			0.281535i	−0.990043	+	0.140768i	$0.955043\pi$
$858$		0			0
$859$		−	17.4964i		−	0.596969i	−0.954414	−	0.298484i	$-0.903519\pi$
$859$		−	17.4964i		−	0.596969i	0.954414	−	0.298484i	$-0.0964811\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−31.8624			−1.08461			−0.542305	−	0.840182i	$-0.682448\pi$
$863$	−31.8624			−1.08461			−0.542305	+	0.840182i	$0.682448\pi$
$864$		0			0
$865$	−2.13803	+	0.906879i	−0.0726952	+	0.0308348i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	11.0144i		−	0.373639i
$870$		0			0
$871$		−	76.6543i		−	2.59733i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	23.1452	+	18.4201i	0.782451	+	0.622712i
$876$		0			0
$877$		−	21.5352i		−	0.727193i	−0.931557	−	0.363596i	$-0.881549\pi$
$877$		−	21.5352i		−	0.727193i	0.931557	−	0.363596i	$-0.118451\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.7466			0.901115			0.450557	−	0.892747i	$-0.351225\pi$
$881$	26.7466			0.901115			0.450557	+	0.892747i	$0.351225\pi$
$882$		0			0
$883$			47.0512i			1.58340i	0.610910	+	0.791700i	$0.290804\pi$
$883$			47.0512i			1.58340i	−0.610910	+	0.791700i	$0.709196\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28.5612i			0.958990i	0.877545	+	0.479495i	$0.159180\pi$
$887$			28.5612i			0.958990i	−0.877545	+	0.479495i	$0.840820\pi$
$888$		0			0
$889$	0.238128	−	0.756654i	0.00798655	−	0.0253774i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.7502			0.393205
$894$		0			0
$895$	2.64373	−	1.12138i	0.0883703	−	0.0374836i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.3512			−0.745456
$900$		0			0
$901$			3.21885i			0.107236i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24.7382	−	10.4931i	0.822324	−	0.348802i
$906$		0			0
$907$		−	4.28727i		−	0.142356i	−0.997464	−	0.0711782i	$-0.977324\pi$
$907$		−	4.28727i		−	0.142356i	0.997464	−	0.0711782i	$-0.0226759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	35.8771i		−	1.18866i	−0.804221	−	0.594330i	$-0.797417\pi$
$911$		−	35.8771i		−	1.18866i	0.804221	−	0.594330i	$-0.202583\pi$
$912$		0			0
$913$	10.5262			0.348366
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.48567	+	2.67054i	0.280221	+	0.0881890i
$918$		0			0
$919$	4.09899			0.135213			0.0676066	−	0.997712i	$-0.478464\pi$
$919$	4.09899			0.135213			0.0676066	+	0.997712i	$0.478464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	49.4365i		−	1.62722i
$924$		0			0
$925$	−35.6320	−	34.4455i	−1.17157	−	1.13256i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.3045			−0.534933			−0.267467	−	0.963567i	$-0.586187\pi$
$929$	−16.3045			−0.534933			−0.267467	+	0.963567i	$0.586187\pi$
$930$		0			0
$931$	−13.7862	+	19.7335i	−0.451824	+	0.646740i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.359354	+	0.847203i	0.0117521	+	0.0277065i
$936$		0			0
$937$	35.5393			1.16102			0.580509	−	0.814254i	$-0.302854\pi$
$937$	35.5393			1.16102			0.580509	+	0.814254i	$0.302854\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.3369			−0.434770			−0.217385	−	0.976086i	$-0.569753\pi$
$941$	−13.3369			−0.434770			−0.217385	+	0.976086i	$0.569753\pi$
$942$		0			0
$943$	33.8565			1.10252
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.3088			0.562460			0.281230	−	0.959640i	$-0.409258\pi$
$947$	17.3088			0.562460			0.281230	+	0.959640i	$0.409258\pi$
$948$		0			0
$949$	−54.9529			−1.78385
$950$		0			0
$951$		0			0
$952$		0			0
$953$	35.8082			1.15994			0.579971	−	0.814637i	$-0.303064\pi$
$953$	35.8082			1.15994			0.579971	+	0.814637i	$0.303064\pi$
$954$		0			0
$955$	−51.2061	+	21.7199i	−1.65699	+	0.702839i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.0082	−	3.46442i	−0.355474	−	0.111872i
$960$		0			0
$961$	−12.5970			−0.406354
$962$		0			0
$963$		0			0
$964$		0			0
$965$	36.9027	−	15.6529i	1.18794	−	0.503884i
$966$		0			0
$967$			52.3351i			1.68298i	0.540269	+	0.841492i	$0.318323\pi$
$967$			52.3351i			1.68298i	−0.540269	+	0.841492i	$0.681677\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−33.6316			−1.07929			−0.539644	−	0.841893i	$-0.681441\pi$
$971$	−33.6316			−1.07929			−0.539644	+	0.841893i	$0.681441\pi$
$972$		0			0
$973$	−12.2805	+	39.0215i	−0.393696	+	1.25097i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.77412			0.280709			0.140355	−	0.990101i	$-0.455176\pi$
$977$	8.77412			0.280709			0.140355	+	0.990101i	$0.455176\pi$
$978$		0			0
$979$		−	7.28369i		−	0.232788i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	5.65880i		−	0.180488i	−0.995920	−	0.0902439i	$-0.971235\pi$
$983$		−	5.65880i		−	0.180488i	0.995920	−	0.0902439i	$-0.0287647\pi$
$984$		0			0
$985$	−16.5662	−	39.0561i	−0.527845	−	1.24443i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	50.8926i		−	1.61829i
$990$		0			0
$991$	12.4795			0.396426			0.198213	−	0.980159i	$-0.436486\pi$
$991$	12.4795			0.396426			0.198213	+	0.980159i	$0.436486\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.4228	−	6.96598i	0.520637	−	0.220836i
$996$		0			0
$997$	19.5614			0.619515			0.309758	−	0.950816i	$-0.399752\pi$
$997$	19.5614			0.619515			0.309758	+	0.950816i	$0.399752\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.k.a.1889.7	✓	24
3.2	odd	2		2520.2.k.b.1889.18	yes	24
4.3	odd	2		5040.2.k.i.1889.7		24
5.4	even	2		2520.2.k.b.1889.8	yes	24
7.6	odd	2	inner	2520.2.k.a.1889.18	yes	24
12.11	even	2		5040.2.k.h.1889.18		24
15.14	odd	2	inner	2520.2.k.a.1889.17	yes	24
20.19	odd	2		5040.2.k.h.1889.8		24
21.20	even	2		2520.2.k.b.1889.7	yes	24
28.27	even	2		5040.2.k.i.1889.18		24
35.34	odd	2		2520.2.k.b.1889.17	yes	24
60.59	even	2		5040.2.k.i.1889.17		24
84.83	odd	2		5040.2.k.h.1889.7		24
105.104	even	2	inner	2520.2.k.a.1889.8	yes	24
140.139	even	2		5040.2.k.h.1889.17		24
420.419	odd	2		5040.2.k.i.1889.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.k.a.1889.7	✓	24	1.1	even	1	trivial
2520.2.k.a.1889.8	yes	24	105.104	even	2	inner
2520.2.k.a.1889.17	yes	24	15.14	odd	2	inner
2520.2.k.a.1889.18	yes	24	7.6	odd	2	inner
2520.2.k.b.1889.7	yes	24	21.20	even	2
2520.2.k.b.1889.8	yes	24	5.4	even	2
2520.2.k.b.1889.17	yes	24	35.34	odd	2
2520.2.k.b.1889.18	yes	24	3.2	odd	2
5040.2.k.h.1889.7		24	84.83	odd	2
5040.2.k.h.1889.8		24	20.19	odd	2
5040.2.k.h.1889.17		24	140.139	even	2
5040.2.k.h.1889.18		24	12.11	even	2
5040.2.k.i.1889.7		24	4.3	odd	2
5040.2.k.i.1889.8		24	420.419	odd	2
5040.2.k.i.1889.17		24	60.59	even	2
5040.2.k.i.1889.18		24	28.27	even	2

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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.873162 - 2.05854i) q^{5} +(2.52372 + 0.794245i) q^{7} +O(q^{10})\)	\(q+(-0.873162 - 2.05854i) q^{5} +(2.52372 + 0.794245i) q^{7} +0.935722i q^{11} -6.43078 q^{13} +0.439826i q^{17} +3.43888i q^{19} +8.69904 q^{23} +(-3.47518 + 3.59488i) q^{25} +3.38512i q^{29} +6.60280i q^{31} +(-0.568633 - 5.88869i) q^{35} +9.91188i q^{37} +3.89198 q^{41} -5.85036i q^{43} -3.41686i q^{47} +(5.73835 + 4.00891i) q^{49} +7.31847 q^{53} +(1.92622 - 0.817037i) q^{55} +0.437881 q^{59} -12.7718i q^{61} +(5.61512 + 13.2380i) q^{65} +11.9199i q^{67} +7.68748i q^{71} +8.54529 q^{73} +(-0.743193 + 2.36150i) q^{77} -11.7710 q^{79} -11.2493i q^{83} +(0.905400 - 0.384040i) q^{85} -7.78403 q^{89} +(-16.2295 - 5.10762i) q^{91} +(7.07908 - 3.00270i) q^{95} +13.4398 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) \) 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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