LMFDB - Embedded newform 8280.2.p.a.1241.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.20
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.20278i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.20278i q^{7} +4.86348 q^{11} +3.00195 q^{13} -5.00429 q^{17} -0.784550i q^{19} +(4.48574 + 1.69651i) q^{23} +1.00000 q^{25} -2.73005i q^{29} +2.49369 q^{31} +1.20278i q^{35} +10.7764i q^{37} +11.9800i q^{41} -3.55733i q^{43} -2.32725i q^{47} +5.55332 q^{49} -4.40654 q^{53} -4.86348 q^{55} +3.40648i q^{59} +1.35028i q^{61} -3.00195 q^{65} +5.73004i q^{67} +2.08991i q^{71} +6.31279 q^{73} -5.84969i q^{77} -10.9085i q^{79} +9.22926 q^{83} +5.00429 q^{85} -1.52015 q^{89} -3.61069i q^{91} +0.784550i q^{95} -13.8588i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	1.20278i		−	0.454608i	−0.973824	−	0.227304i	$-0.927009\pi$
$7$		−	1.20278i		−	0.454608i	0.973824	−	0.227304i	$-0.0729911\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.86348			1.46639			0.733197	−	0.680016i	$-0.238027\pi$
$11$	4.86348			1.46639			0.733197	+	0.680016i	$0.238027\pi$
$12$		0			0
$13$	3.00195			0.832592			0.416296	−	0.909229i	$-0.363328\pi$
$13$	3.00195			0.832592			0.416296	+	0.909229i	$0.363328\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.00429			−1.21372			−0.606859	−	0.794810i	$-0.707571\pi$
$17$	−5.00429			−1.21372			−0.606859	+	0.794810i	$0.707571\pi$
$18$		0			0
$19$		−	0.784550i		−	0.179988i	−0.995942	−	0.0899940i	$-0.971315\pi$
$19$		−	0.784550i		−	0.179988i	0.995942	−	0.0899940i	$-0.0286848\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.48574	+	1.69651i	0.935341	+	0.353748i
$23$	4.48574	+	1.69651i	0.935341	+	0.353748i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.73005i		−	0.506958i	−0.967341	−	0.253479i	$-0.918425\pi$
$29$		−	2.73005i		−	0.506958i	0.967341	−	0.253479i	$-0.0815749\pi$
$30$		0			0
$31$	2.49369			0.447880			0.223940	−	0.974603i	$-0.428108\pi$
$31$	2.49369			0.447880			0.223940	+	0.974603i	$0.428108\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			1.20278i			0.203307i
$36$		0			0
$37$			10.7764i			1.77163i	0.464039	+	0.885815i	$0.346400\pi$
$37$			10.7764i			1.77163i	−0.464039	+	0.885815i	$0.653600\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			11.9800i			1.87096i	0.353384	+	0.935478i	$0.385031\pi$
$41$			11.9800i			1.87096i	−0.353384	+	0.935478i	$0.614969\pi$
$42$		0			0
$43$		−	3.55733i		−	0.542487i	−0.962511	−	0.271244i	$-0.912565\pi$
$43$		−	3.55733i		−	0.542487i	0.962511	−	0.271244i	$-0.0874349\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.32725i		−	0.339464i	−0.985490	−	0.169732i	$-0.945710\pi$
$47$		−	2.32725i		−	0.339464i	0.985490	−	0.169732i	$-0.0542903\pi$
$48$		0			0
$49$	5.55332			0.793332
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.40654			−0.605284			−0.302642	−	0.953104i	$-0.597869\pi$
$53$	−4.40654			−0.605284			−0.302642	+	0.953104i	$0.597869\pi$
$54$		0			0
$55$	−4.86348			−0.655792
$56$		0			0
$57$		0			0
$58$		0			0
$59$			3.40648i			0.443485i	0.975105	+	0.221743i	$0.0711745\pi$
$59$			3.40648i			0.443485i	−0.975105	+	0.221743i	$0.928826\pi$
$60$		0			0
$61$			1.35028i			0.172886i	0.996257	+	0.0864429i	$0.0275500\pi$
$61$			1.35028i			0.172886i	−0.996257	+	0.0864429i	$0.972450\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00195			−0.372346
$66$		0			0
$67$			5.73004i			0.700035i	0.936743	+	0.350018i	$0.113824\pi$
$67$			5.73004i			0.700035i	−0.936743	+	0.350018i	$0.886176\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.08991i			0.248027i	0.992281	+	0.124014i	$0.0395767\pi$
$71$			2.08991i			0.248027i	−0.992281	+	0.124014i	$0.960423\pi$
$72$		0			0
$73$	6.31279			0.738856			0.369428	−	0.929259i	$-0.379554\pi$
$73$	6.31279			0.738856			0.369428	+	0.929259i	$0.379554\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	5.84969i		−	0.666634i
$78$		0			0
$79$		−	10.9085i		−	1.22731i	−0.789576	−	0.613653i	$-0.789699\pi$
$79$		−	10.9085i		−	1.22731i	0.789576	−	0.613653i	$-0.210301\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.22926			1.01304			0.506522	−	0.862227i	$-0.330931\pi$
$83$	9.22926			1.01304			0.506522	+	0.862227i	$0.330931\pi$
$84$		0			0
$85$	5.00429			0.542791
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.52015			−0.161135			−0.0805676	−	0.996749i	$-0.525673\pi$
$89$	−1.52015			−0.161135			−0.0805676	+	0.996749i	$0.525673\pi$
$90$		0			0
$91$		−	3.61069i		−	0.378503i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			0.784550i			0.0804931i
$96$		0			0
$97$		−	13.8588i		−	1.40715i	−0.710621	−	0.703575i	$-0.751586\pi$
$97$		−	13.8588i		−	1.40715i	0.710621	−	0.703575i	$-0.248414\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.19858i		−	0.218767i	−0.994000	−	0.109383i	$-0.965112\pi$
$101$		−	2.19858i		−	0.218767i	0.994000	−	0.109383i	$-0.0348876\pi$
$102$		0			0
$103$		−	2.87680i		−	0.283460i	−0.989905	−	0.141730i	$-0.954734\pi$
$103$		−	2.87680i		−	0.283460i	0.989905	−	0.141730i	$-0.0452665\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.7482			−1.13574			−0.567872	−	0.823117i	$-0.692233\pi$
$107$	−11.7482			−1.13574			−0.567872	+	0.823117i	$0.692233\pi$
$108$		0			0
$109$			8.75927i			0.838986i	0.907758	+	0.419493i	$0.137792\pi$
$109$			8.75927i			0.838986i	−0.907758	+	0.419493i	$0.862208\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.16297			−0.109403			−0.0547015	−	0.998503i	$-0.517421\pi$
$113$	−1.16297			−0.109403			−0.0547015	+	0.998503i	$0.517421\pi$
$114$		0			0
$115$	−4.48574	−	1.69651i	−0.418297	−	0.158201i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.01905i			0.551765i
$120$		0			0
$121$	12.6534			1.15031
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−20.0359			−1.77790			−0.888948	−	0.458008i	$-0.848563\pi$
$127$	−20.0359			−1.77790			−0.888948	+	0.458008i	$0.848563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.10772i			0.184152i	0.995752	+	0.0920761i	$0.0293503\pi$
$131$			2.10772i			0.184152i	−0.995752	+	0.0920761i	$0.970650\pi$
$132$		0			0
$133$	−0.943640			−0.0818240
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.17189			0.100121			0.0500607	−	0.998746i	$-0.484059\pi$
$137$	1.17189			0.100121			0.0500607	+	0.998746i	$0.484059\pi$
$138$		0			0
$139$	8.66372			0.734847			0.367423	−	0.930054i	$-0.380240\pi$
$139$	8.66372			0.734847			0.367423	+	0.930054i	$0.380240\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.5999			1.22091
$144$		0			0
$145$			2.73005i			0.226719i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00264			−0.0821392			−0.0410696	−	0.999156i	$-0.513077\pi$
$149$	−1.00264			−0.0821392			−0.0410696	+	0.999156i	$0.513077\pi$
$150$		0			0
$151$	22.4899			1.83020			0.915101	−	0.403224i	$-0.132110\pi$
$151$	22.4899			1.83020			0.915101	+	0.403224i	$0.132110\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.49369			−0.200298
$156$		0			0
$157$		−	9.25115i		−	0.738322i	−0.929365	−	0.369161i	$-0.879645\pi$
$157$		−	9.25115i		−	0.738322i	0.929365	−	0.369161i	$-0.120355\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.04053	−	5.39535i	0.160816	−	0.425213i
$162$		0			0
$163$	19.4727			1.52522			0.762609	−	0.646859i	$-0.223918\pi$
$163$	19.4727			1.52522			0.762609	+	0.646859i	$0.223918\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.1255i			0.938302i	0.883118	+	0.469151i	$0.155440\pi$
$167$			12.1255i			0.938302i	−0.883118	+	0.469151i	$0.844560\pi$
$168$		0			0
$169$	−3.98828			−0.306791
$170$		0			0
$171$		0			0
$172$		0			0
$173$			23.6624i			1.79902i	0.436902	+	0.899509i	$0.356076\pi$
$173$			23.6624i			1.79902i	−0.436902	+	0.899509i	$0.643924\pi$
$174$		0			0
$175$		−	1.20278i		−	0.0909216i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.7900i			1.25494i	0.778640	+	0.627470i	$0.215910\pi$
$179$			16.7900i			1.25494i	−0.778640	+	0.627470i	$0.784090\pi$
$180$		0			0
$181$		−	9.09972i		−	0.676377i	−0.941078	−	0.338188i	$-0.890186\pi$
$181$		−	9.09972i		−	0.676377i	0.941078	−	0.338188i	$-0.109814\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	10.7764i		−	0.792297i
$186$		0			0
$187$	−24.3382			−1.77979
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.51735			0.109792			0.0548960	−	0.998492i	$-0.482517\pi$
$191$	1.51735			0.109792			0.0548960	+	0.998492i	$0.482517\pi$
$192$		0			0
$193$	−9.98572			−0.718788			−0.359394	−	0.933186i	$-0.617017\pi$
$193$	−9.98572			−0.718788			−0.359394	+	0.933186i	$0.617017\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.76240i		−	0.481801i	−0.970550	−	0.240901i	$-0.922557\pi$
$197$		−	6.76240i		−	0.481801i	0.970550	−	0.240901i	$-0.0774428\pi$
$198$		0			0
$199$			10.6065i			0.751873i	0.926645	+	0.375937i	$0.122679\pi$
$199$			10.6065i			0.751873i	−0.926645	+	0.375937i	$0.877321\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.28365			−0.230467
$204$		0			0
$205$		−	11.9800i		−	0.836717i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.81564i		−	0.263933i
$210$		0			0
$211$	9.55030			0.657470			0.328735	−	0.944422i	$-0.393378\pi$
$211$	9.55030			0.657470			0.328735	+	0.944422i	$0.393378\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.55733i			0.242608i
$216$		0			0
$217$		−	2.99936i		−	0.203610i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.0226			−1.01053
$222$		0			0
$223$	9.94870			0.666214			0.333107	−	0.942889i	$-0.391903\pi$
$223$	9.94870			0.666214			0.333107	+	0.942889i	$0.391903\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.66730			0.177035			0.0885174	−	0.996075i	$-0.471787\pi$
$227$	2.66730			0.177035			0.0885174	+	0.996075i	$0.471787\pi$
$228$		0			0
$229$		−	15.6647i		−	1.03515i	−0.855637	−	0.517576i	$-0.826835\pi$
$229$		−	15.6647i		−	1.03515i	0.855637	−	0.517576i	$-0.173165\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	13.1941i		−	0.864372i	−0.901784	−	0.432186i	$-0.857742\pi$
$233$		−	13.1941i		−	0.864372i	0.901784	−	0.432186i	$-0.142258\pi$
$234$		0			0
$235$			2.32725i			0.151813i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			15.2725i			0.987896i	0.869492	+	0.493948i	$0.164447\pi$
$239$			15.2725i			0.987896i	−0.869492	+	0.493948i	$0.835553\pi$
$240$		0			0
$241$		−	13.2443i		−	0.853137i	−0.904455	−	0.426569i	$-0.859722\pi$
$241$		−	13.2443i		−	0.853137i	0.904455	−	0.426569i	$-0.140278\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.55332			−0.354789
$246$		0			0
$247$		−	2.35518i		−	0.149857i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.64797			−0.545855			−0.272928	−	0.962035i	$-0.587992\pi$
$251$	−8.64797			−0.545855			−0.272928	+	0.962035i	$0.587992\pi$
$252$		0			0
$253$	21.8163	+	8.25096i	1.37158	+	0.518734i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			23.1626i			1.44484i	0.691452	+	0.722422i	$0.256971\pi$
$257$			23.1626i			1.44484i	−0.691452	+	0.722422i	$0.743029\pi$
$258$		0			0
$259$	12.9616			0.805397
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.1903			−1.24499			−0.622494	−	0.782624i	$-0.713881\pi$
$263$	−20.1903			−1.24499			−0.622494	+	0.782624i	$0.713881\pi$
$264$		0			0
$265$	4.40654			0.270691
$266$		0			0
$267$		0			0
$268$		0			0
$269$			13.6410i			0.831704i	0.909432	+	0.415852i	$0.136517\pi$
$269$			13.6410i			0.831704i	−0.909432	+	0.415852i	$0.863483\pi$
$270$		0			0
$271$	16.8608			1.02422			0.512110	−	0.858920i	$-0.328864\pi$
$271$	16.8608			1.02422			0.512110	+	0.858920i	$0.328864\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.86348			0.293279
$276$		0			0
$277$	3.54073			0.212742			0.106371	−	0.994327i	$-0.466077\pi$
$277$	3.54073			0.212742			0.106371	+	0.994327i	$0.466077\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.9011			0.948578			0.474289	−	0.880369i	$-0.342705\pi$
$281$	15.9011			0.948578			0.474289	+	0.880369i	$0.342705\pi$
$282$		0			0
$283$			20.7179i			1.23155i	0.787922	+	0.615775i	$0.211157\pi$
$283$			20.7179i			1.23155i	−0.787922	+	0.615775i	$0.788843\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.4093			0.850551
$288$		0			0
$289$	8.04288			0.473110
$290$		0			0
$291$		0			0
$292$		0			0
$293$	27.5012			1.60664			0.803318	−	0.595550i	$-0.203066\pi$
$293$	27.5012			1.60664			0.803318	+	0.595550i	$0.203066\pi$
$294$		0			0
$295$		−	3.40648i		−	0.198333i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	13.4660	+	5.09285i	0.778757	+	0.294527i
$300$		0			0
$301$	−4.27868			−0.246619
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	1.35028i		−	0.0773169i
$306$		0			0
$307$	15.1061			0.862150			0.431075	−	0.902316i	$-0.358134\pi$
$307$	15.1061			0.862150			0.431075	+	0.902316i	$0.358134\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.1362i		−	0.744885i	−0.928055	−	0.372442i	$-0.878520\pi$
$311$		−	13.1362i		−	0.744885i	0.928055	−	0.372442i	$-0.121480\pi$
$312$		0			0
$313$		−	14.1254i		−	0.798413i	−0.916861	−	0.399206i	$-0.869286\pi$
$313$		−	14.1254i		−	0.798413i	0.916861	−	0.399206i	$-0.130714\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.58183i		−	0.145010i	−0.997368	−	0.0725051i	$-0.976901\pi$
$317$		−	2.58183i		−	0.145010i	0.997368	−	0.0725051i	$-0.0230993\pi$
$318$		0			0
$319$		−	13.2776i		−	0.743401i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.92611i			0.218455i
$324$		0			0
$325$	3.00195			0.166518
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.79917			−0.154323
$330$		0			0
$331$	−15.9217			−0.875137			−0.437569	−	0.899185i	$-0.644160\pi$
$331$	−15.9217			−0.875137			−0.437569	+	0.899185i	$0.644160\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	5.73004i		−	0.313065i
$336$		0			0
$337$		−	6.33159i		−	0.344904i	−0.985018	−	0.172452i	$-0.944831\pi$
$337$		−	6.33159i		−	0.344904i	0.985018	−	0.172452i	$-0.0551689\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.1280			0.656769
$342$		0			0
$343$		−	15.0989i		−	0.815263i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5.30495i		−	0.284785i	−0.989810	−	0.142392i	$-0.954521\pi$
$347$		−	5.30495i		−	0.284785i	0.989810	−	0.142392i	$-0.0454795\pi$
$348$		0			0
$349$	33.1525			1.77461			0.887305	−	0.461183i	$-0.152575\pi$
$349$	33.1525			1.77461			0.887305	+	0.461183i	$0.152575\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			4.89279i			0.260417i	0.991487	+	0.130209i	$0.0415647\pi$
$353$			4.89279i			0.260417i	−0.991487	+	0.130209i	$0.958435\pi$
$354$		0			0
$355$		−	2.08991i		−	0.110921i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.99007			0.474478			0.237239	−	0.971451i	$-0.423758\pi$
$359$	8.99007			0.474478			0.237239	+	0.971451i	$0.423758\pi$
$360$		0			0
$361$	18.3845			0.967604
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.31279			−0.330426
$366$		0			0
$367$		−	7.44106i		−	0.388420i	−0.980960	−	0.194210i	$-0.937786\pi$
$367$		−	7.44106i		−	0.388420i	0.980960	−	0.194210i	$-0.0622143\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.30009i			0.275167i
$372$		0			0
$373$			1.86325i			0.0964753i	0.998836	+	0.0482377i	$0.0153605\pi$
$373$			1.86325i			0.0964753i	−0.998836	+	0.0482377i	$0.984640\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.19549i		−	0.422089i
$378$		0			0
$379$			11.1078i			0.570567i	0.958443	+	0.285284i	$0.0920878\pi$
$379$			11.1078i			0.570567i	−0.958443	+	0.285284i	$0.907912\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.2482			0.625851			0.312926	−	0.949778i	$-0.398691\pi$
$383$	12.2482			0.625851			0.312926	+	0.949778i	$0.398691\pi$
$384$		0			0
$385$			5.84969i			0.298128i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.5105			0.938521			0.469261	−	0.883060i	$-0.344520\pi$
$389$	18.5105			0.938521			0.469261	+	0.883060i	$0.344520\pi$
$390$		0			0
$391$	−22.4479	−	8.48984i	−1.13524	−	0.429350i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			10.9085i			0.548868i
$396$		0			0
$397$	18.3370			0.920307			0.460153	−	0.887839i	$-0.347794\pi$
$397$	18.3370			0.920307			0.460153	+	0.887839i	$0.347794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	35.7511			1.78532			0.892662	−	0.450726i	$-0.148835\pi$
$401$	35.7511			1.78532			0.892662	+	0.450726i	$0.148835\pi$
$402$		0			0
$403$	7.48594			0.372901
$404$		0			0
$405$		0			0
$406$		0			0
$407$			52.4108i			2.59791i
$408$		0			0
$409$	26.0658			1.28887			0.644435	−	0.764659i	$-0.277093\pi$
$409$	26.0658			1.28887			0.644435	+	0.764659i	$0.277093\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.09724			0.201612
$414$		0			0
$415$	−9.22926			−0.453047
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.9037			1.21663			0.608314	−	0.793697i	$-0.291846\pi$
$419$	24.9037			1.21663			0.608314	+	0.793697i	$0.291846\pi$
$420$		0			0
$421$		−	19.7292i		−	0.961543i	−0.876846	−	0.480771i	$-0.840357\pi$
$421$		−	19.7292i		−	0.961543i	0.876846	−	0.480771i	$-0.159643\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.00429			−0.242744
$426$		0			0
$427$	1.62409			0.0785952
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.2738			−0.543040			−0.271520	−	0.962433i	$-0.587526\pi$
$431$	−11.2738			−0.543040			−0.271520	+	0.962433i	$0.587526\pi$
$432$		0			0
$433$			24.0943i			1.15790i	0.815363	+	0.578950i	$0.196537\pi$
$433$			24.0943i			1.15790i	−0.815363	+	0.578950i	$0.803463\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.33100	−	3.51928i	0.0636703	−	0.168350i
$438$		0			0
$439$	−4.47604			−0.213630			−0.106815	−	0.994279i	$-0.534065\pi$
$439$	−4.47604			−0.213630			−0.106815	+	0.994279i	$0.534065\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			7.69535i			0.365617i	0.983149	+	0.182808i	$0.0585188\pi$
$443$			7.69535i			0.365617i	−0.983149	+	0.182808i	$0.941481\pi$
$444$		0			0
$445$	1.52015			0.0720618
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.71418i		−	0.0808971i	−0.999182	−	0.0404486i	$-0.987121\pi$
$449$		−	1.71418i		−	0.0808971i	0.999182	−	0.0404486i	$-0.0128787\pi$
$450$		0			0
$451$			58.2643i			2.74356i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.61069i			0.169272i
$456$		0			0
$457$			4.34550i			0.203274i	0.994822	+	0.101637i	$0.0324080\pi$
$457$			4.34550i			0.203274i	−0.994822	+	0.101637i	$0.967592\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	27.5689i		−	1.28401i	−0.766699	−	0.642007i	$-0.778102\pi$
$461$		−	27.5689i		−	1.28401i	0.766699	−	0.642007i	$-0.221898\pi$
$462$		0			0
$463$	−7.52391			−0.349666			−0.174833	−	0.984598i	$-0.555939\pi$
$463$	−7.52391			−0.349666			−0.174833	+	0.984598i	$0.555939\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18.1486			−0.839817			−0.419908	−	0.907566i	$-0.637938\pi$
$467$	−18.1486			−0.839817			−0.419908	+	0.907566i	$0.637938\pi$
$468$		0			0
$469$	6.89197			0.318242
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	17.3010i		−	0.795500i
$474$		0			0
$475$		−	0.784550i		−	0.0359976i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.4305			1.93870			0.969349	−	0.245689i	$-0.0790141\pi$
$479$	42.4305			1.93870			0.969349	+	0.245689i	$0.0790141\pi$
$480$		0			0
$481$			32.3502i			1.47504i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			13.8588i			0.629296i
$486$		0			0
$487$	29.2399			1.32499			0.662493	−	0.749068i	$-0.269498\pi$
$487$	29.2399			1.32499			0.662493	+	0.749068i	$0.269498\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			25.1344i			1.13430i	0.823615	+	0.567149i	$0.191954\pi$
$491$			25.1344i			1.13430i	−0.823615	+	0.567149i	$0.808046\pi$
$492$		0			0
$493$			13.6620i			0.615304i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.51371			0.112755
$498$		0			0
$499$	−1.87499			−0.0839359			−0.0419680	−	0.999119i	$-0.513363\pi$
$499$	−1.87499			−0.0839359			−0.0419680	+	0.999119i	$0.513363\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.00738			−0.312444			−0.156222	−	0.987722i	$-0.549932\pi$
$503$	−7.00738			−0.312444			−0.156222	+	0.987722i	$0.549932\pi$
$504$		0			0
$505$			2.19858i			0.0978355i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	21.8299i		−	0.967594i	−0.875180	−	0.483797i	$-0.839257\pi$
$509$		−	21.8299i		−	0.967594i	0.875180	−	0.483797i	$-0.160743\pi$
$510$		0			0
$511$		−	7.59289i		−	0.335890i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			2.87680i			0.126767i
$516$		0			0
$517$		−	11.3185i		−	0.497789i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00895			0.0442029			0.0221014	−	0.999756i	$-0.492964\pi$
$521$	1.00895			0.0442029			0.0221014	+	0.999756i	$0.492964\pi$
$522$		0			0
$523$		−	0.536079i		−	0.0234411i	−0.999931	−	0.0117206i	$-0.996269\pi$
$523$		−	0.536079i		−	0.0234411i	0.999931	−	0.0117206i	$-0.00373085\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.4791			−0.543600
$528$		0			0
$529$	17.2437	+	15.2202i	0.749725	+	0.661749i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			35.9633i			1.55774i
$534$		0			0
$535$	11.7482			0.507920
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.0085			1.16334
$540$		0			0
$541$	18.9635			0.815304			0.407652	−	0.913137i	$-0.366348\pi$
$541$	18.9635			0.815304			0.407652	+	0.913137i	$0.366348\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.75927i		−	0.375206i
$546$		0			0
$547$	37.0833			1.58557			0.792783	−	0.609503i	$-0.208631\pi$
$547$	37.0833			1.58557			0.792783	+	0.609503i	$0.208631\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.14186			−0.0912464
$552$		0			0
$553$	−13.1206			−0.557943
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.7384			−1.42954			−0.714772	−	0.699358i	$-0.753469\pi$
$557$	−33.7384			−1.42954			−0.714772	+	0.699358i	$0.753469\pi$
$558$		0			0
$559$		−	10.6789i		−	0.451670i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.69643			−0.155786			−0.0778929	−	0.996962i	$-0.524819\pi$
$563$	−3.69643			−0.155786			−0.0778929	+	0.996962i	$0.524819\pi$
$564$		0			0
$565$	1.16297			0.0489265
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.497897			0.0208729			0.0104365	−	0.999946i	$-0.496678\pi$
$569$	0.497897			0.0208729			0.0104365	+	0.999946i	$0.496678\pi$
$570$		0			0
$571$		−	26.0981i		−	1.09217i	−0.837729	−	0.546087i	$-0.816117\pi$
$571$		−	26.0981i		−	1.09217i	0.837729	−	0.546087i	$-0.183883\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.48574	+	1.69651i	0.187068	+	0.0707495i
$576$		0			0
$577$	13.7422			0.572094			0.286047	−	0.958216i	$-0.407659\pi$
$577$	13.7422			0.572094			0.286047	+	0.958216i	$0.407659\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	11.1008i		−	0.460537i
$582$		0			0
$583$	−21.4311			−0.887585
$584$		0			0
$585$		0			0
$586$		0			0
$587$			29.2518i			1.20735i	0.797229	+	0.603676i	$0.206298\pi$
$587$			29.2518i			1.20735i	−0.797229	+	0.603676i	$0.793702\pi$
$588$		0			0
$589$		−	1.95642i		−	0.0806131i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.92385i		−	0.161133i	−0.996749	−	0.0805666i	$-0.974327\pi$
$593$		−	3.92385i		−	0.161133i	0.996749	−	0.0805666i	$-0.0256730\pi$
$594$		0			0
$595$		−	6.01905i		−	0.246757i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	3.46473i		−	0.141565i	−0.997492	−	0.0707825i	$-0.977450\pi$
$599$		−	3.46473i		−	0.141565i	0.997492	−	0.0707825i	$-0.0225496\pi$
$600$		0			0
$601$	−30.9047			−1.26063			−0.630315	−	0.776340i	$-0.717074\pi$
$601$	−30.9047			−1.26063			−0.630315	+	0.776340i	$0.717074\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.6534			−0.514436
$606$		0			0
$607$	−24.4383			−0.991919			−0.495959	−	0.868346i	$-0.665184\pi$
$607$	−24.4383			−0.991919			−0.495959	+	0.868346i	$0.665184\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.98629i		−	0.282635i
$612$		0			0
$613$		−	36.4942i		−	1.47399i	−0.675900	−	0.736993i	$-0.736245\pi$
$613$		−	36.4942i		−	1.47399i	0.675900	−	0.736993i	$-0.263755\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.1740			−1.13424			−0.567121	−	0.823635i	$-0.691943\pi$
$617$	−28.1740			−1.13424			−0.567121	+	0.823635i	$0.691943\pi$
$618$		0			0
$619$			24.3339i			0.978061i	0.872267	+	0.489030i	$0.162649\pi$
$619$			24.3339i			0.978061i	−0.872267	+	0.489030i	$0.837351\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.82840i			0.0732533i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	53.9282i		−	2.15026i
$630$		0			0
$631$			6.30204i			0.250880i	0.992101	+	0.125440i	$0.0400343\pi$
$631$			6.30204i			0.250880i	−0.992101	+	0.125440i	$0.959966\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20.0359			0.795099
$636$		0			0
$637$	16.6708			0.660521
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−35.4671			−1.40087			−0.700434	−	0.713718i	$-0.747010\pi$
$641$	−35.4671			−1.40087			−0.700434	+	0.713718i	$0.747010\pi$
$642$		0			0
$643$		−	4.60976i		−	0.181791i	−0.995860	−	0.0908957i	$-0.971027\pi$
$643$		−	4.60976i		−	0.181791i	0.995860	−	0.0908957i	$-0.0289730\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.13783i			0.162675i	0.996687	+	0.0813375i	$0.0259192\pi$
$647$			4.13783i			0.162675i	−0.996687	+	0.0813375i	$0.974081\pi$
$648$		0			0
$649$			16.5673i			0.650324i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	44.3574i		−	1.73584i	−0.496704	−	0.867920i	$-0.665457\pi$
$653$		−	44.3574i		−	1.73584i	0.496704	−	0.867920i	$-0.334543\pi$
$654$		0			0
$655$		−	2.10772i		−	0.0823553i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.9865			−0.700655			−0.350327	−	0.936627i	$-0.613930\pi$
$659$	−17.9865			−0.700655			−0.350327	+	0.936627i	$0.613930\pi$
$660$		0			0
$661$		−	30.9280i		−	1.20296i	−0.798887	−	0.601481i	$-0.794578\pi$
$661$		−	30.9280i		−	1.20296i	0.798887	−	0.601481i	$-0.205422\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.943640			0.0365928
$666$		0			0
$667$	4.63158	−	12.2463i	0.179335	−	0.474179i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			6.56707i			0.253519i
$672$		0			0
$673$	10.9869			0.423512			0.211756	−	0.977323i	$-0.432082\pi$
$673$	10.9869			0.423512			0.211756	+	0.977323i	$0.432082\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.8262			−0.646684			−0.323342	−	0.946282i	$-0.604806\pi$
$677$	−16.8262			−0.646684			−0.323342	+	0.946282i	$0.604806\pi$
$678$		0			0
$679$	−16.6691			−0.639701
$680$		0			0
$681$		0			0
$682$		0			0
$683$			29.0452i			1.11138i	0.831388	+	0.555692i	$0.187547\pi$
$683$			29.0452i			1.11138i	−0.831388	+	0.555692i	$0.812453\pi$
$684$		0			0
$685$	−1.17189			−0.0447756
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.2282			−0.503955
$690$		0			0
$691$	−4.23850			−0.161240			−0.0806201	−	0.996745i	$-0.525690\pi$
$691$	−4.23850			−0.161240			−0.0806201	+	0.996745i	$0.525690\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.66372			−0.328634
$696$		0			0
$697$		−	59.9512i		−	2.27081i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.7688			−1.19989			−0.599946	−	0.800040i	$-0.704811\pi$
$701$	−31.7688			−1.19989			−0.599946	+	0.800040i	$0.704811\pi$
$702$		0			0
$703$	8.45462			0.318872
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.64441			−0.0994531
$708$		0			0
$709$		−	1.78116i		−	0.0668928i	−0.999441	−	0.0334464i	$-0.989352\pi$
$709$		−	1.78116i		−	0.0668928i	0.999441	−	0.0334464i	$-0.0106483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.1860	+	4.23058i	0.418921	+	0.158437i
$714$		0			0
$715$	−14.5999			−0.546007
$716$		0			0
$717$		0			0
$718$		0			0
$719$			5.09166i			0.189887i	0.995483	+	0.0949434i	$0.0302670\pi$
$719$			5.09166i			0.189887i	−0.995483	+	0.0949434i	$0.969733\pi$
$720$		0			0
$721$	−3.46016			−0.128863
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	2.73005i		−	0.101392i
$726$		0			0
$727$		−	1.48269i		−	0.0549898i	−0.999622	−	0.0274949i	$-0.991247\pi$
$727$		−	1.48269i		−	0.0549898i	0.999622	−	0.0274949i	$-0.00875301\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			17.8019i			0.658426i
$732$		0			0
$733$		−	23.3604i		−	0.862837i	−0.902152	−	0.431418i	$-0.858013\pi$
$733$		−	23.3604i		−	0.862837i	0.902152	−	0.431418i	$-0.141987\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			27.8679i			1.02653i
$738$		0			0
$739$	−40.4263			−1.48711			−0.743553	−	0.668676i	$-0.766861\pi$
$739$	−40.4263			−1.48711			−0.743553	+	0.668676i	$0.766861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−42.5956			−1.56268			−0.781340	−	0.624105i	$-0.785464\pi$
$743$	−42.5956			−1.56268			−0.781340	+	0.624105i	$0.785464\pi$
$744$		0			0
$745$	1.00264			0.0367338
$746$		0			0
$747$		0			0
$748$		0			0
$749$			14.1305i			0.516318i
$750$		0			0
$751$		−	1.82117i		−	0.0664554i	−0.999448	−	0.0332277i	$-0.989421\pi$
$751$		−	1.82117i		−	0.0664554i	0.999448	−	0.0332277i	$-0.0105786\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−22.4899			−0.818492
$756$		0			0
$757$		−	19.4350i		−	0.706378i	−0.935552	−	0.353189i	$-0.885097\pi$
$757$		−	19.4350i		−	0.706378i	0.935552	−	0.353189i	$-0.114903\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			47.1772i			1.71017i	0.518487	+	0.855085i	$0.326495\pi$
$761$			47.1772i			1.71017i	−0.518487	+	0.855085i	$0.673505\pi$
$762$		0			0
$763$	10.5355			0.381410
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10.2261i			0.369242i
$768$		0			0
$769$			33.7622i			1.21749i	0.793364	+	0.608747i	$0.208328\pi$
$769$			33.7622i			1.21749i	−0.793364	+	0.608747i	$0.791672\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.3866			1.38067			0.690335	−	0.723490i	$-0.257463\pi$
$773$	38.3866			1.38067			0.690335	+	0.723490i	$0.257463\pi$
$774$		0			0
$775$	2.49369			0.0895760
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.39888			0.336750
$780$		0			0
$781$			10.1643i			0.363706i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.25115i			0.330188i
$786$		0			0
$787$		−	22.2660i		−	0.793698i	−0.917884	−	0.396849i	$-0.870104\pi$
$787$		−	22.2660i		−	0.793698i	0.917884	−	0.396849i	$-0.129896\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1.39880i			0.0497355i
$792$		0			0
$793$			4.05348i			0.143943i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.53418			0.125187			0.0625936	−	0.998039i	$-0.480063\pi$
$797$	3.53418			0.125187			0.0625936	+	0.998039i	$0.480063\pi$
$798$		0			0
$799$			11.6462i			0.412014i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	30.7021			1.08345
$804$		0			0
$805$	−2.04053	+	5.39535i	−0.0719193	+	0.190161i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			42.9574i			1.51030i	0.655551	+	0.755151i	$0.272437\pi$
$809$			42.9574i			1.51030i	−0.655551	+	0.755151i	$0.727563\pi$
$810$		0			0
$811$	−37.1607			−1.30489			−0.652444	−	0.757837i	$-0.726256\pi$
$811$	−37.1607			−1.30489			−0.652444	+	0.757837i	$0.726256\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.4727			−0.682099
$816$		0			0
$817$	−2.79090			−0.0976412
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	31.4497i		−	1.09760i	−0.835954	−	0.548800i	$-0.815085\pi$
$821$		−	31.4497i		−	1.09760i	0.835954	−	0.548800i	$-0.184915\pi$
$822$		0			0
$823$	25.1442			0.876473			0.438236	−	0.898860i	$-0.355603\pi$
$823$	25.1442			0.876473			0.438236	+	0.898860i	$0.355603\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−54.8423			−1.90705			−0.953527	−	0.301308i	$-0.902577\pi$
$827$	−54.8423			−1.90705			−0.953527	+	0.301308i	$0.902577\pi$
$828$		0			0
$829$	−34.9251			−1.21300			−0.606499	−	0.795084i	$-0.707427\pi$
$829$	−34.9251			−1.21300			−0.606499	+	0.795084i	$0.707427\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−27.7904			−0.962881
$834$		0			0
$835$		−	12.1255i		−	0.419621i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.482660			−0.0166633			−0.00833164	−	0.999965i	$-0.502652\pi$
$839$	−0.482660			−0.0166633			−0.00833164	+	0.999965i	$0.502652\pi$
$840$		0			0
$841$	21.5468			0.742993
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.98828			0.137201
$846$		0			0
$847$		−	15.2193i		−	0.522941i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.2823	+	48.3401i	−0.626710	+	1.65708i
$852$		0			0
$853$	−3.56862			−0.122187			−0.0610936	−	0.998132i	$-0.519459\pi$
$853$	−3.56862			−0.122187			−0.0610936	+	0.998132i	$0.519459\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			46.1569i			1.57669i	0.615234	+	0.788345i	$0.289062\pi$
$857$			46.1569i			1.57669i	−0.615234	+	0.788345i	$0.710938\pi$
$858$		0			0
$859$	−2.09000			−0.0713098			−0.0356549	−	0.999364i	$-0.511352\pi$
$859$	−2.09000			−0.0713098			−0.0356549	+	0.999364i	$0.511352\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	17.4318i		−	0.593385i	−0.954973	−	0.296693i	$-0.904116\pi$
$863$		−	17.4318i		−	0.593385i	0.954973	−	0.296693i	$-0.0958836\pi$
$864$		0			0
$865$		−	23.6624i		−	0.804545i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	53.0535i		−	1.79972i
$870$		0			0
$871$			17.2013i			0.582844i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1.20278i			0.0406614i
$876$		0			0
$877$	42.4178			1.43235			0.716174	−	0.697922i	$-0.245892\pi$
$877$	42.4178			1.43235			0.716174	+	0.697922i	$0.245892\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.64849			−0.325066			−0.162533	−	0.986703i	$-0.551966\pi$
$881$	−9.64849			−0.325066			−0.162533	+	0.986703i	$0.551966\pi$
$882$		0			0
$883$	−4.96411			−0.167056			−0.0835278	−	0.996505i	$-0.526619\pi$
$883$	−4.96411			−0.167056			−0.0835278	+	0.996505i	$0.526619\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			50.2713i			1.68794i	0.536387	+	0.843972i	$0.319789\pi$
$887$			50.2713i			1.68794i	−0.536387	+	0.843972i	$0.680211\pi$
$888$		0			0
$889$			24.0987i			0.808245i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.82584			−0.0610995
$894$		0			0
$895$		−	16.7900i		−	0.561227i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.80791i		−	0.227057i
$900$		0			0
$901$	22.0516			0.734644
$902$		0			0
$903$		0			0
$904$		0			0
$905$			9.09972i			0.302485i
$906$		0			0
$907$		−	9.65806i		−	0.320691i	−0.987061	−	0.160345i	$-0.948739\pi$
$907$		−	9.65806i		−	0.320691i	0.987061	−	0.160345i	$-0.0512608\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.9025			0.858187			0.429093	−	0.903260i	$-0.358833\pi$
$911$	25.9025			0.858187			0.429093	+	0.903260i	$0.358833\pi$
$912$		0			0
$913$	44.8863			1.48552
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.53512			0.0837170
$918$		0			0
$919$			17.8669i			0.589374i	0.955594	+	0.294687i	$0.0952155\pi$
$919$			17.8669i			0.589374i	−0.955594	+	0.294687i	$0.904785\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.27382i			0.206505i
$924$		0			0
$925$			10.7764i			0.354326i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	51.7767i		−	1.69874i	−0.527798	−	0.849370i	$-0.676982\pi$
$929$		−	51.7767i		−	1.69874i	0.527798	−	0.849370i	$-0.323018\pi$
$930$		0			0
$931$		−	4.35686i		−	0.142790i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.3382			0.795946
$936$		0			0
$937$		−	14.4272i		−	0.471316i	−0.971836	−	0.235658i	$-0.924275\pi$
$937$		−	14.4272i		−	0.471316i	0.971836	−	0.235658i	$-0.0757245\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	31.4096			1.02392			0.511962	−	0.859008i	$-0.328919\pi$
$941$	31.4096			1.02392			0.511962	+	0.859008i	$0.328919\pi$
$942$		0			0
$943$	−20.3242	+	53.7390i	−0.661846	+	1.74998i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	32.7967i		−	1.06575i	−0.846194	−	0.532874i	$-0.821112\pi$
$947$		−	32.7967i		−	1.06575i	0.846194	−	0.532874i	$-0.178888\pi$
$948$		0			0
$949$	18.9507			0.615165
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−23.4104			−0.758338			−0.379169	−	0.925327i	$-0.623790\pi$
$953$	−23.4104			−0.758338			−0.379169	+	0.925327i	$0.623790\pi$
$954$		0			0
$955$	−1.51735			−0.0491004
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.40953i		−	0.0455159i
$960$		0			0
$961$	−24.7815			−0.799403
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.98572			0.321452
$966$		0			0
$967$	−58.2697			−1.87383			−0.936914	−	0.349560i	$-0.886331\pi$
$967$	−58.2697			−1.87383			−0.936914	+	0.349560i	$0.886331\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.1704			1.06449			0.532244	−	0.846591i	$-0.321349\pi$
$971$	33.1704			1.06449			0.532244	+	0.846591i	$0.321349\pi$
$972$		0			0
$973$		−	10.4205i		−	0.334067i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.86664			−0.0917121			−0.0458560	−	0.998948i	$-0.514602\pi$
$977$	−2.86664			−0.0917121			−0.0458560	+	0.998948i	$0.514602\pi$
$978$		0			0
$979$	−7.39320			−0.236288
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.9851			0.796902			0.398451	−	0.917190i	$-0.369548\pi$
$983$	24.9851			0.796902			0.398451	+	0.917190i	$0.369548\pi$
$984$		0			0
$985$			6.76240i			0.215468i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.03505	−	15.9572i	0.191903	−	0.507410i
$990$		0			0
$991$	4.97514			0.158041			0.0790203	−	0.996873i	$-0.474821\pi$
$991$	4.97514			0.158041			0.0790203	+	0.996873i	$0.474821\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	10.6065i		−	0.336248i
$996$		0			0
$997$	35.8925			1.13673			0.568364	−	0.822777i	$-0.307577\pi$
$997$	35.8925			1.13673			0.568364	+	0.822777i	$0.307577\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.20	✓	48
3.2	odd	2		8280.2.p.b.1241.20	yes	48
23.22	odd	2		8280.2.p.b.1241.29	yes	48
69.68	even	2	inner	8280.2.p.a.1241.29	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.20	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.29	yes	48	69.68	even	2	inner
8280.2.p.b.1241.20	yes	48	3.2	odd	2
8280.2.p.b.1241.29	yes	48	23.22	odd	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.20278i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.20278i q^{7} +4.86348 q^{11} +3.00195 q^{13} -5.00429 q^{17} -0.784550i q^{19} +(4.48574 + 1.69651i) q^{23} +1.00000 q^{25} -2.73005i q^{29} +2.49369 q^{31} +1.20278i q^{35} +10.7764i q^{37} +11.9800i q^{41} -3.55733i q^{43} -2.32725i q^{47} +5.55332 q^{49} -4.40654 q^{53} -4.86348 q^{55} +3.40648i q^{59} +1.35028i q^{61} -3.00195 q^{65} +5.73004i q^{67} +2.08991i q^{71} +6.31279 q^{73} -5.84969i q^{77} -10.9085i q^{79} +9.22926 q^{83} +5.00429 q^{85} -1.52015 q^{89} -3.61069i q^{91} +0.784550i q^{95} -13.8588i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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