LMFDB - Embedded newform 8280.2.p.a.1241.17

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			1241.17
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -1.46186i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -1.46186i q^{7} -0.486025 q^{11} +5.34662 q^{13} -2.25784 q^{17} -6.43611i q^{19} +(1.32323 - 4.60967i) q^{23} +1.00000 q^{25} +3.84093i q^{29} -6.74149 q^{31} +1.46186i q^{35} +1.26105i q^{37} +4.64596i q^{41} -7.19611i q^{43} -5.27068i q^{47} +4.86296 q^{49} -5.75415 q^{53} +0.486025 q^{55} +12.4553i q^{59} -9.62475i q^{61} -5.34662 q^{65} -6.89880i q^{67} -8.76708i q^{71} +1.29306 q^{73} +0.710500i q^{77} +6.71117i q^{79} -5.08641 q^{83} +2.25784 q^{85} +4.45700 q^{89} -7.81601i q^{91} +6.43611i q^{95} +0.599189i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	1.46186i		−	0.552531i	−0.961081	−	0.276266i	$-0.910903\pi$
$7$		−	1.46186i		−	0.552531i	0.961081	−	0.276266i	$-0.0890970\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.486025			−0.146542			−0.0732710	−	0.997312i	$-0.523344\pi$
$11$	−0.486025			−0.146542			−0.0732710	+	0.997312i	$0.523344\pi$
$12$		0			0
$13$	5.34662			1.48289			0.741443	−	0.671016i	$-0.234142\pi$
$13$	5.34662			1.48289			0.741443	+	0.671016i	$0.234142\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.25784			−0.547606			−0.273803	−	0.961786i	$-0.588282\pi$
$17$	−2.25784			−0.547606			−0.273803	+	0.961786i	$0.588282\pi$
$18$		0			0
$19$		−	6.43611i		−	1.47655i	−0.674502	−	0.738273i	$-0.735642\pi$
$19$		−	6.43611i		−	1.47655i	0.674502	−	0.738273i	$-0.264358\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.32323	−	4.60967i	0.275912	−	0.961183i
$23$	1.32323	−	4.60967i	0.275912	−	0.961183i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.84093i			0.713243i	0.934249	+	0.356621i	$0.116071\pi$
$29$			3.84093i			0.713243i	−0.934249	+	0.356621i	$0.883929\pi$
$30$		0			0
$31$	−6.74149			−1.21081			−0.605403	−	0.795919i	$-0.706988\pi$
$31$	−6.74149			−1.21081			−0.605403	+	0.795919i	$0.706988\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			1.46186i			0.247100i
$36$		0			0
$37$			1.26105i			0.207315i	0.994613	+	0.103657i	$0.0330545\pi$
$37$			1.26105i			0.207315i	−0.994613	+	0.103657i	$0.966945\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.64596i			0.725577i	0.931871	+	0.362789i	$0.118175\pi$
$41$			4.64596i			0.725577i	−0.931871	+	0.362789i	$0.881825\pi$
$42$		0			0
$43$		−	7.19611i		−	1.09740i	−0.836021	−	0.548698i	$-0.815124\pi$
$43$		−	7.19611i		−	1.09740i	0.836021	−	0.548698i	$-0.184876\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.27068i		−	0.768807i	−0.923165	−	0.384404i	$-0.874407\pi$
$47$		−	5.27068i		−	0.768807i	0.923165	−	0.384404i	$-0.125593\pi$
$48$		0			0
$49$	4.86296			0.694709
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.75415			−0.790394			−0.395197	−	0.918596i	$-0.629324\pi$
$53$	−5.75415			−0.790394			−0.395197	+	0.918596i	$0.629324\pi$
$54$		0			0
$55$	0.486025			0.0655356
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.4553i			1.62154i	0.585363	+	0.810772i	$0.300952\pi$
$59$			12.4553i			1.62154i	−0.585363	+	0.810772i	$0.699048\pi$
$60$		0			0
$61$		−	9.62475i		−	1.23232i	−0.787620	−	0.616161i	$-0.788687\pi$
$61$		−	9.62475i		−	1.23232i	0.787620	−	0.616161i	$-0.211313\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.34662			−0.663166
$66$		0			0
$67$		−	6.89880i		−	0.842823i	−0.906870	−	0.421412i	$-0.861535\pi$
$67$		−	6.89880i		−	0.842823i	0.906870	−	0.421412i	$-0.138465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.76708i		−	1.04046i	−0.854026	−	0.520230i	$-0.825846\pi$
$71$		−	8.76708i		−	1.04046i	0.854026	−	0.520230i	$-0.174154\pi$
$72$		0			0
$73$	1.29306			0.151341			0.0756706	−	0.997133i	$-0.475890\pi$
$73$	1.29306			0.151341			0.0756706	+	0.997133i	$0.475890\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.710500i			0.0809690i
$78$		0			0
$79$			6.71117i			0.755065i	0.925996	+	0.377533i	$0.123227\pi$
$79$			6.71117i			0.755065i	−0.925996	+	0.377533i	$0.876773\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.08641			−0.558306			−0.279153	−	0.960247i	$-0.590054\pi$
$83$	−5.08641			−0.558306			−0.279153	+	0.960247i	$0.590054\pi$
$84$		0			0
$85$	2.25784			0.244897
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.45700			0.472441			0.236221	−	0.971699i	$-0.424091\pi$
$89$	4.45700			0.472441			0.236221	+	0.971699i	$0.424091\pi$
$90$		0			0
$91$		−	7.81601i		−	0.819341i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			6.43611i			0.660331i
$96$		0			0
$97$			0.599189i			0.0608384i	0.999537	+	0.0304192i	$0.00968423\pi$
$97$			0.599189i			0.0608384i	−0.999537	+	0.0304192i	$0.990316\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			4.63694i			0.461393i	0.973026	+	0.230696i	$0.0741004\pi$
$101$			4.63694i			0.461393i	−0.973026	+	0.230696i	$0.925900\pi$
$102$		0			0
$103$			11.8491i			1.16752i	0.811926	+	0.583761i	$0.198419\pi$
$103$			11.8491i			1.16752i	−0.811926	+	0.583761i	$0.801581\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.8131			1.14202			0.571010	−	0.820943i	$-0.306552\pi$
$107$	11.8131			1.14202			0.571010	+	0.820943i	$0.306552\pi$
$108$		0			0
$109$			8.42010i			0.806500i	0.915090	+	0.403250i	$0.132119\pi$
$109$			8.42010i			0.806500i	−0.915090	+	0.403250i	$0.867881\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.4034			−1.63717			−0.818585	−	0.574385i	$-0.805241\pi$
$113$	−17.4034			−1.63717			−0.818585	+	0.574385i	$0.805241\pi$
$114$		0			0
$115$	−1.32323	+	4.60967i	−0.123391	+	0.429854i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			3.30065i			0.302570i
$120$		0			0
$121$	−10.7638			−0.978525
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−0.360384			−0.0319789			−0.0159895	−	0.999872i	$-0.505090\pi$
$127$	−0.360384			−0.0319789			−0.0159895	+	0.999872i	$0.505090\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	6.18094i		−	0.540031i	−0.962856	−	0.270016i	$-0.912971\pi$
$131$		−	6.18094i		−	0.540031i	0.962856	−	0.270016i	$-0.0870289\pi$
$132$		0			0
$133$	−9.40870			−0.815838
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.40208			0.461531			0.230765	−	0.973009i	$-0.425877\pi$
$137$	5.40208			0.461531			0.230765	+	0.973009i	$0.425877\pi$
$138$		0			0
$139$	−15.4390			−1.30952			−0.654761	−	0.755836i	$-0.727231\pi$
$139$	−15.4390			−1.30952			−0.654761	+	0.755836i	$0.727231\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.59859			−0.217305
$144$		0			0
$145$		−	3.84093i		−	0.318972i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.09348			0.253428			0.126714	−	0.991939i	$-0.459557\pi$
$149$	3.09348			0.253428			0.126714	+	0.991939i	$0.459557\pi$
$150$		0			0
$151$	15.6406			1.27281			0.636406	−	0.771354i	$-0.280420\pi$
$151$	15.6406			1.27281			0.636406	+	0.771354i	$0.280420\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.74149			0.541489
$156$		0			0
$157$		−	14.2006i		−	1.13333i	−0.823948	−	0.566666i	$-0.808233\pi$
$157$		−	14.2006i		−	1.13333i	0.823948	−	0.566666i	$-0.191767\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.73870	−	1.93437i	−0.531084	−	0.152450i
$162$		0			0
$163$	−4.01314			−0.314333			−0.157167	−	0.987572i	$-0.550236\pi$
$163$	−4.01314			−0.314333			−0.157167	+	0.987572i	$0.550236\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.28445i			0.254159i	0.991893	+	0.127079i	$0.0405603\pi$
$167$			3.28445i			0.254159i	−0.991893	+	0.127079i	$0.959440\pi$
$168$		0			0
$169$	15.5863			1.19895
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	13.1227i		−	0.997699i	−0.866689	−	0.498850i	$-0.833756\pi$
$173$		−	13.1227i		−	0.997699i	0.866689	−	0.498850i	$-0.166244\pi$
$174$		0			0
$175$		−	1.46186i		−	0.110506i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.03400i		−	0.152029i	−0.997107	−	0.0760143i	$-0.975781\pi$
$179$		−	2.03400i		−	0.152029i	0.997107	−	0.0760143i	$-0.0242195\pi$
$180$		0			0
$181$			9.52408i			0.707919i	0.935261	+	0.353960i	$0.115165\pi$
$181$			9.52408i			0.707919i	−0.935261	+	0.353960i	$0.884835\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1.26105i		−	0.0927139i
$186$		0			0
$187$	1.09737			0.0802473
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.61633			0.189311			0.0946554	−	0.995510i	$-0.469825\pi$
$191$	2.61633			0.189311			0.0946554	+	0.995510i	$0.469825\pi$
$192$		0			0
$193$	−6.90467			−0.497009			−0.248505	−	0.968631i	$-0.579939\pi$
$193$	−6.90467			−0.497009			−0.248505	+	0.968631i	$0.579939\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.22801i			0.586222i	0.956078	+	0.293111i	$0.0946905\pi$
$197$			8.22801i			0.586222i	−0.956078	+	0.293111i	$0.905310\pi$
$198$		0			0
$199$		−	22.8525i		−	1.61997i	−0.586449	−	0.809986i	$-0.699475\pi$
$199$		−	22.8525i		−	1.61997i	0.586449	−	0.809986i	$-0.300525\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.61491			0.394089
$204$		0			0
$205$		−	4.64596i		−	0.324488i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.12811i			0.216376i
$210$		0			0
$211$	9.66998			0.665709			0.332854	−	0.942978i	$-0.391988\pi$
$211$	9.66998			0.665709			0.332854	+	0.942978i	$0.391988\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.19611i			0.490770i
$216$		0			0
$217$			9.85512i			0.669009i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0718			−0.812038
$222$		0			0
$223$	−16.3517			−1.09499			−0.547497	−	0.836808i	$-0.684419\pi$
$223$	−16.3517			−1.09499			−0.547497	+	0.836808i	$0.684419\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.2644			−1.14588			−0.572940	−	0.819597i	$-0.694197\pi$
$227$	−17.2644			−1.14588			−0.572940	+	0.819597i	$0.694197\pi$
$228$		0			0
$229$		−	25.1696i		−	1.66326i	−0.555334	−	0.831628i	$-0.687409\pi$
$229$		−	25.1696i		−	1.66326i	0.555334	−	0.831628i	$-0.312591\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	6.17521i		−	0.404551i	−0.979329	−	0.202276i	$-0.935166\pi$
$233$		−	6.17521i		−	0.404551i	0.979329	−	0.202276i	$-0.0648337\pi$
$234$		0			0
$235$			5.27068i			0.343821i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	0.436912i		−	0.0282615i	−0.999900	−	0.0141307i	$-0.995502\pi$
$239$		−	0.436912i		−	0.0282615i	0.999900	−	0.0141307i	$-0.00449811\pi$
$240$		0			0
$241$		−	18.2979i		−	1.17867i	−0.807889	−	0.589335i	$-0.799390\pi$
$241$		−	18.2979i		−	1.17867i	0.807889	−	0.589335i	$-0.200610\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.86296			−0.310683
$246$		0			0
$247$		−	34.4114i		−	2.18955i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.6945			−0.864391			−0.432196	−	0.901780i	$-0.642261\pi$
$251$	−13.6945			−0.864391			−0.432196	+	0.901780i	$0.642261\pi$
$252$		0			0
$253$	−0.643120	+	2.24041i	−0.0404326	+	0.140854i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.4046i			1.33518i	0.744528	+	0.667591i	$0.232674\pi$
$257$			21.4046i			1.33518i	−0.744528	+	0.667591i	$0.767326\pi$
$258$		0			0
$259$	1.84347			0.114548
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.4004			−0.702977			−0.351489	−	0.936192i	$-0.614324\pi$
$263$	−11.4004			−0.702977			−0.351489	+	0.936192i	$0.614324\pi$
$264$		0			0
$265$	5.75415			0.353475
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	12.7191i		−	0.775499i	−0.921765	−	0.387749i	$-0.873253\pi$
$269$		−	12.7191i		−	0.775499i	0.921765	−	0.387749i	$-0.126747\pi$
$270$		0			0
$271$	−13.4722			−0.818378			−0.409189	−	0.912450i	$-0.634188\pi$
$271$	−13.4722			−0.818378			−0.409189	+	0.912450i	$0.634188\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.486025			−0.0293084
$276$		0			0
$277$	2.09459			0.125852			0.0629259	−	0.998018i	$-0.479957\pi$
$277$	2.09459			0.125852			0.0629259	+	0.998018i	$0.479957\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.7912			−1.35961			−0.679805	−	0.733393i	$-0.737936\pi$
$281$	−22.7912			−1.35961			−0.679805	+	0.733393i	$0.737936\pi$
$282$		0			0
$283$		−	24.5435i		−	1.45896i	−0.684003	−	0.729479i	$-0.739763\pi$
$283$		−	24.5435i		−	1.45896i	0.684003	−	0.729479i	$-0.260237\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.79175			0.400904
$288$		0			0
$289$	−11.9022			−0.700127
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.23272			−0.247278			−0.123639	−	0.992327i	$-0.539457\pi$
$293$	−4.23272			−0.247278			−0.123639	+	0.992327i	$0.539457\pi$
$294$		0			0
$295$		−	12.4553i		−	0.725176i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.07478	−	24.6462i	0.409145	−	1.42532i
$300$		0			0
$301$	−10.5197			−0.606346
$302$		0			0
$303$		0			0
$304$		0			0
$305$			9.62475i			0.551111i
$306$		0			0
$307$	−6.09476			−0.347846			−0.173923	−	0.984759i	$-0.555644\pi$
$307$	−6.09476			−0.347846			−0.173923	+	0.984759i	$0.555644\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2.42782i			0.137669i	0.997628	+	0.0688344i	$0.0219280\pi$
$311$			2.42782i			0.137669i	−0.997628	+	0.0688344i	$0.978072\pi$
$312$		0			0
$313$			6.94954i			0.392811i	0.980523	+	0.196405i	$0.0629269\pi$
$313$			6.94954i			0.392811i	−0.980523	+	0.196405i	$0.937073\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.66049i			0.317925i	0.987285	+	0.158962i	$0.0508148\pi$
$317$			5.66049i			0.317925i	−0.987285	+	0.158962i	$0.949185\pi$
$318$		0			0
$319$		−	1.86679i		−	0.104520i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			14.5317i			0.808566i
$324$		0			0
$325$	5.34662			0.296577
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.70500			−0.424790
$330$		0			0
$331$	8.26461			0.454264			0.227132	−	0.973864i	$-0.427065\pi$
$331$	8.26461			0.454264			0.227132	+	0.973864i	$0.427065\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			6.89880i			0.376922i
$336$		0			0
$337$			4.40421i			0.239912i	0.992779	+	0.119956i	$0.0382754\pi$
$337$			4.40421i			0.239912i	−0.992779	+	0.119956i	$0.961725\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.27653			0.177434
$342$		0			0
$343$		−	17.3420i		−	0.936380i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			17.9070i			0.961299i	0.876913	+	0.480650i	$0.159599\pi$
$347$			17.9070i			0.961299i	−0.876913	+	0.480650i	$0.840401\pi$
$348$		0			0
$349$	−26.4606			−1.41640			−0.708202	−	0.706010i	$-0.750494\pi$
$349$	−26.4606			−1.41640			−0.708202	+	0.706010i	$0.750494\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	16.3188i		−	0.868564i	−0.900777	−	0.434282i	$-0.857002\pi$
$353$		−	16.3188i		−	0.868564i	0.900777	−	0.434282i	$-0.142998\pi$
$354$		0			0
$355$			8.76708i			0.465308i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.1236			−1.37875			−0.689374	−	0.724405i	$-0.742114\pi$
$359$	−26.1236			−1.37875			−0.689374	+	0.724405i	$0.742114\pi$
$360$		0			0
$361$	−22.4236			−1.18019
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.29306			−0.0676818
$366$		0			0
$367$		−	5.51561i		−	0.287913i	−0.989584	−	0.143956i	$-0.954018\pi$
$367$		−	5.51561i		−	0.287913i	0.989584	−	0.143956i	$-0.0459825\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			8.41177i			0.436717i
$372$		0			0
$373$		−	17.7114i		−	0.917060i	−0.888679	−	0.458530i	$-0.848376\pi$
$373$		−	17.7114i		−	0.917060i	0.888679	−	0.458530i	$-0.151624\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			20.5360i			1.05766i
$378$		0			0
$379$		−	0.634288i		−	0.0325812i	−0.999867	−	0.0162906i	$-0.994814\pi$
$379$		−	0.634288i		−	0.0325812i	0.999867	−	0.0162906i	$-0.00518569\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.88446			0.249584			0.124792	−	0.992183i	$-0.460174\pi$
$383$	4.88446			0.249584			0.124792	+	0.992183i	$0.460174\pi$
$384$		0			0
$385$		−	0.710500i		−	0.0362105i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.7527			−1.25501			−0.627506	−	0.778612i	$-0.715924\pi$
$389$	−24.7527			−1.25501			−0.627506	+	0.778612i	$0.715924\pi$
$390$		0			0
$391$	−2.98763	+	10.4079i	−0.151091	+	0.526350i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	6.71117i		−	0.337676i
$396$		0			0
$397$	18.7037			0.938709			0.469355	−	0.883010i	$-0.344487\pi$
$397$	18.7037			0.938709			0.469355	+	0.883010i	$0.344487\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.2731			−1.31202			−0.656009	−	0.754753i	$-0.727757\pi$
$401$	−26.2731			−1.31202			−0.656009	+	0.754753i	$0.727757\pi$
$402$		0			0
$403$	−36.0442			−1.79549
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.612899i		−	0.0303803i
$408$		0			0
$409$	34.3901			1.70048			0.850241	−	0.526393i	$-0.176456\pi$
$409$	34.3901			1.70048			0.850241	+	0.526393i	$0.176456\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	18.2079			0.895953
$414$		0			0
$415$	5.08641			0.249682
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.87218			−0.286875			−0.143437	−	0.989659i	$-0.545816\pi$
$419$	−5.87218			−0.286875			−0.143437	+	0.989659i	$0.545816\pi$
$420$		0			0
$421$			20.1851i			0.983762i	0.870663	+	0.491881i	$0.163690\pi$
$421$			20.1851i			0.983762i	−0.870663	+	0.491881i	$0.836310\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.25784			−0.109521
$426$		0			0
$427$	−14.0700			−0.680897
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.08689			−0.389532			−0.194766	−	0.980850i	$-0.562395\pi$
$431$	−8.08689			−0.389532			−0.194766	+	0.980850i	$0.562395\pi$
$432$		0			0
$433$			14.0055i			0.673063i	0.941672	+	0.336532i	$0.109254\pi$
$433$			14.0055i			0.673063i	−0.941672	+	0.336532i	$0.890746\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−29.6684	−	8.51643i	−1.41923	−	0.407396i
$438$		0			0
$439$	−31.2021			−1.48919			−0.744596	−	0.667515i	$-0.767358\pi$
$439$	−31.2021			−1.48919			−0.744596	+	0.667515i	$0.767358\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	0.214910i		−	0.0102107i	−0.999987	−	0.00510533i	$-0.998375\pi$
$443$		−	0.214910i		−	0.0102107i	0.999987	−	0.00510533i	$-0.00162509\pi$
$444$		0			0
$445$	−4.45700			−0.211282
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	37.6204i		−	1.77542i	−0.460406	−	0.887709i	$-0.652296\pi$
$449$		−	37.6204i		−	1.77542i	0.460406	−	0.887709i	$-0.347704\pi$
$450$		0			0
$451$		−	2.25805i		−	0.106327i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			7.81601i			0.366420i
$456$		0			0
$457$		−	38.4178i		−	1.79711i	−0.438861	−	0.898555i	$-0.644618\pi$
$457$		−	38.4178i		−	1.79711i	0.438861	−	0.898555i	$-0.355382\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			33.7178i			1.57039i	0.619246	+	0.785197i	$0.287438\pi$
$461$			33.7178i			1.57039i	−0.619246	+	0.785197i	$0.712562\pi$
$462$		0			0
$463$	8.45400			0.392891			0.196445	−	0.980515i	$-0.437060\pi$
$463$	8.45400			0.392891			0.196445	+	0.980515i	$0.437060\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.5903			0.999082			0.499541	−	0.866290i	$-0.333502\pi$
$467$	21.5903			0.999082			0.499541	+	0.866290i	$0.333502\pi$
$468$		0			0
$469$	−10.0851			−0.465686
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3.49749i			0.160815i
$474$		0			0
$475$		−	6.43611i		−	0.295309i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.53531			−0.161533			−0.0807663	−	0.996733i	$-0.525737\pi$
$479$	−3.53531			−0.161533			−0.0807663	+	0.996733i	$0.525737\pi$
$480$		0			0
$481$			6.74233i			0.307424i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.599189i		−	0.0272078i
$486$		0			0
$487$	24.3037			1.10131			0.550654	−	0.834734i	$-0.314379\pi$
$487$	24.3037			1.10131			0.550654	+	0.834734i	$0.314379\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	19.2922i		−	0.870643i	−0.900275	−	0.435321i	$-0.856635\pi$
$491$		−	19.2922i		−	0.870643i	0.900275	−	0.435321i	$-0.143365\pi$
$492$		0			0
$493$		−	8.67220i		−	0.390576i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.8162			−0.574887
$498$		0			0
$499$	−8.18868			−0.366576			−0.183288	−	0.983059i	$-0.558674\pi$
$499$	−8.18868			−0.366576			−0.183288	+	0.983059i	$0.558674\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	43.2428			1.92810			0.964050	−	0.265721i	$-0.0856101\pi$
$503$	43.2428			1.92810			0.964050	+	0.265721i	$0.0856101\pi$
$504$		0			0
$505$		−	4.63694i		−	0.206341i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	0.693652i		−	0.0307456i	−0.999882	−	0.0153728i	$-0.995106\pi$
$509$		−	0.693652i		−	0.0307456i	0.999882	−	0.0153728i	$-0.00489351\pi$
$510$		0			0
$511$		−	1.89027i		−	0.0836208i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	11.8491i		−	0.522132i
$516$		0			0
$517$			2.56168i			0.112663i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.9342			0.917144			0.458572	−	0.888657i	$-0.348361\pi$
$521$	20.9342			0.917144			0.458572	+	0.888657i	$0.348361\pi$
$522$		0			0
$523$		−	33.7306i		−	1.47494i	−0.675381	−	0.737469i	$-0.736021\pi$
$523$		−	33.7306i		−	1.47494i	0.675381	−	0.737469i	$-0.263979\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.2212			0.663046
$528$		0			0
$529$	−19.4981	−	12.1993i	−0.847746	−	0.530403i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			24.8402i			1.07595i
$534$		0			0
$535$	−11.8131			−0.510727
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.36352			−0.101804
$540$		0			0
$541$	−7.62262			−0.327722			−0.163861	−	0.986483i	$-0.552395\pi$
$541$	−7.62262			−0.327722			−0.163861	+	0.986483i	$0.552395\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.42010i		−	0.360678i
$546$		0			0
$547$	4.90098			0.209551			0.104775	−	0.994496i	$-0.466588\pi$
$547$	4.90098			0.209551			0.104775	+	0.994496i	$0.466588\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	24.7207			1.05314
$552$		0			0
$553$	9.81079			0.417197
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.0973			−1.10578			−0.552890	−	0.833254i	$-0.686475\pi$
$557$	−26.0973			−1.10578			−0.552890	+	0.833254i	$0.686475\pi$
$558$		0			0
$559$		−	38.4748i		−	1.62731i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.8699			−1.21672			−0.608361	−	0.793660i	$-0.708173\pi$
$563$	−28.8699			−1.21672			−0.608361	+	0.793660i	$0.708173\pi$
$564$		0			0
$565$	17.4034			0.732165
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−25.5843			−1.07255			−0.536275	−	0.844043i	$-0.680169\pi$
$569$	−25.5843			−1.07255			−0.536275	+	0.844043i	$0.680169\pi$
$570$		0			0
$571$		−	21.6870i		−	0.907571i	−0.891111	−	0.453785i	$-0.850073\pi$
$571$		−	21.6870i		−	0.907571i	0.891111	−	0.453785i	$-0.149927\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.32323	−	4.60967i	0.0551823	−	0.192237i
$576$		0			0
$577$	−19.3665			−0.806237			−0.403119	−	0.915148i	$-0.632074\pi$
$577$	−19.3665			−0.806237			−0.403119	+	0.915148i	$0.632074\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			7.43562i			0.308482i
$582$		0			0
$583$	2.79666			0.115826
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	15.1591i		−	0.625684i	−0.949805	−	0.312842i	$-0.898719\pi$
$587$		−	15.1591i		−	0.625684i	0.949805	−	0.312842i	$-0.101281\pi$
$588$		0			0
$589$			43.3890i			1.78781i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19.7749i		−	0.812059i	−0.913860	−	0.406030i	$-0.866913\pi$
$593$		−	19.7749i		−	0.812059i	0.913860	−	0.406030i	$-0.133087\pi$
$594$		0			0
$595$		−	3.30065i		−	0.135313i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	32.0332i		−	1.30884i	−0.756130	−	0.654421i	$-0.772912\pi$
$599$		−	32.0332i		−	1.30884i	0.756130	−	0.654421i	$-0.227088\pi$
$600$		0			0
$601$	22.5416			0.919491			0.459746	−	0.888051i	$-0.347941\pi$
$601$	22.5416			0.919491			0.459746	+	0.888051i	$0.347941\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	10.7638			0.437610
$606$		0			0
$607$	39.5657			1.60592			0.802961	−	0.596031i	$-0.203257\pi$
$607$	39.5657			1.60592			0.802961	+	0.596031i	$0.203257\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	28.1803i		−	1.14005i
$612$		0			0
$613$		−	1.33551i		−	0.0539409i	−0.999636	−	0.0269705i	$-0.991414\pi$
$613$		−	1.33551i		−	0.0539409i	0.999636	−	0.0269705i	$-0.00858601\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.3779			0.860641			0.430320	−	0.902676i	$-0.358401\pi$
$617$	21.3779			0.860641			0.430320	+	0.902676i	$0.358401\pi$
$618$		0			0
$619$			8.06463i			0.324145i	0.986779	+	0.162072i	$0.0518178\pi$
$619$			8.06463i			0.324145i	−0.986779	+	0.162072i	$0.948182\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	6.51551i		−	0.261039i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	2.84724i		−	0.113527i
$630$		0			0
$631$		−	16.8120i		−	0.669277i	−0.942347	−	0.334639i	$-0.891386\pi$
$631$		−	16.8120i		−	0.669277i	0.942347	−	0.334639i	$-0.108614\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.360384			0.0143014
$636$		0			0
$637$	26.0004			1.03017
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.87011			0.350348			0.175174	−	0.984537i	$-0.443951\pi$
$641$	8.87011			0.350348			0.175174	+	0.984537i	$0.443951\pi$
$642$		0			0
$643$			6.66969i			0.263027i	0.991314	+	0.131514i	$0.0419837\pi$
$643$			6.66969i			0.263027i	−0.991314	+	0.131514i	$0.958016\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.64490i			0.182610i	0.995823	+	0.0913049i	$0.0291038\pi$
$647$			4.64490i			0.182610i	−0.995823	+	0.0913049i	$0.970896\pi$
$648$		0			0
$649$		−	6.05359i		−	0.237624i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			20.0101i			0.783056i	0.920166	+	0.391528i	$0.128053\pi$
$653$			20.0101i			0.783056i	−0.920166	+	0.391528i	$0.871947\pi$
$654$		0			0
$655$			6.18094i			0.241509i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.65790			0.376218			0.188109	−	0.982148i	$-0.439764\pi$
$659$	9.65790			0.376218			0.188109	+	0.982148i	$0.439764\pi$
$660$		0			0
$661$			27.4152i			1.06633i	0.846012	+	0.533163i	$0.178997\pi$
$661$			27.4152i			1.06633i	−0.846012	+	0.533163i	$0.821003\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.40870			0.364854
$666$		0			0
$667$	17.7054	+	5.08242i	0.685557	+	0.196792i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			4.67787i			0.180587i
$672$		0			0
$673$	12.1641			0.468894			0.234447	−	0.972129i	$-0.424672\pi$
$673$	12.1641			0.468894			0.234447	+	0.972129i	$0.424672\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−37.0926			−1.42559			−0.712793	−	0.701375i	$-0.752570\pi$
$677$	−37.0926			−1.42559			−0.712793	+	0.701375i	$0.752570\pi$
$678$		0			0
$679$	0.875931			0.0336151
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7.22665i		−	0.276520i	−0.990396	−	0.138260i	$-0.955849\pi$
$683$		−	7.22665i		−	0.276520i	0.990396	−	0.138260i	$-0.0441510\pi$
$684$		0			0
$685$	−5.40208			−0.206403
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.7653			−1.17206
$690$		0			0
$691$	36.1066			1.37356			0.686781	−	0.726865i	$-0.259023\pi$
$691$	36.1066			1.37356			0.686781	+	0.726865i	$0.259023\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.4390			0.585636
$696$		0			0
$697$		−	10.4898i		−	0.397331i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−52.3663			−1.97785			−0.988924	−	0.148422i	$-0.952581\pi$
$701$	−52.3663			−1.97785			−0.988924	+	0.148422i	$0.952581\pi$
$702$		0			0
$703$	8.11623			0.306110
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.77856			0.254934
$708$		0			0
$709$			20.5649i			0.772331i	0.922430	+	0.386165i	$0.126201\pi$
$709$			20.5649i			0.772331i	−0.922430	+	0.386165i	$0.873799\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.92051	+	31.0760i	−0.334076	+	1.16381i
$714$		0			0
$715$	2.59859			0.0971817
$716$		0			0
$717$		0			0
$718$		0			0
$719$			7.21873i			0.269213i	0.990899	+	0.134607i	$0.0429770\pi$
$719$			7.21873i			0.269213i	−0.990899	+	0.134607i	$0.957023\pi$
$720$		0			0
$721$	17.3217			0.645092
$722$		0			0
$723$		0			0
$724$		0			0
$725$			3.84093i			0.142649i
$726$		0			0
$727$		−	18.0275i		−	0.668601i	−0.942466	−	0.334301i	$-0.891500\pi$
$727$		−	18.0275i		−	0.668601i	0.942466	−	0.334301i	$-0.108500\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			16.2476i			0.600941i
$732$		0			0
$733$			27.3842i			1.01146i	0.862693	+	0.505729i	$0.168776\pi$
$733$			27.3842i			1.01146i	−0.862693	+	0.505729i	$0.831224\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.35299i			0.123509i
$738$		0			0
$739$	−17.3639			−0.638743			−0.319372	−	0.947630i	$-0.603472\pi$
$739$	−17.3639			−0.638743			−0.319372	+	0.947630i	$0.603472\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	42.0374			1.54220			0.771102	−	0.636712i	$-0.219706\pi$
$743$	42.0374			1.54220			0.771102	+	0.636712i	$0.219706\pi$
$744$		0			0
$745$	−3.09348			−0.113336
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	17.2692i		−	0.631002i
$750$		0			0
$751$		−	7.86549i		−	0.287016i	−0.989649	−	0.143508i	$-0.954162\pi$
$751$		−	7.86549i		−	0.287016i	0.989649	−	0.143508i	$-0.0458382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.6406			−0.569219
$756$		0			0
$757$			13.2726i			0.482400i	0.970475	+	0.241200i	$0.0775410\pi$
$757$			13.2726i			0.482400i	−0.970475	+	0.241200i	$0.922459\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	45.7260i		−	1.65756i	−0.559571	−	0.828782i	$-0.689034\pi$
$761$		−	45.7260i		−	1.65756i	0.559571	−	0.828782i	$-0.310966\pi$
$762$		0			0
$763$	12.3090			0.445616
$764$		0			0
$765$		0			0
$766$		0			0
$767$			66.5938i			2.40456i
$768$		0			0
$769$			14.7461i			0.531756i	0.964007	+	0.265878i	$0.0856619\pi$
$769$			14.7461i			0.531756i	−0.964007	+	0.265878i	$0.914338\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−33.3346			−1.19896			−0.599481	−	0.800389i	$-0.704626\pi$
$773$	−33.3346			−1.19896			−0.599481	+	0.800389i	$0.704626\pi$
$774$		0			0
$775$	−6.74149			−0.242161
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.9019			1.07135
$780$		0			0
$781$			4.26102i			0.152471i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			14.2006i			0.506841i
$786$		0			0
$787$			0.245113i			0.00873733i	0.999990	+	0.00436866i	$0.00139059\pi$
$787$			0.245113i			0.00873733i	−0.999990	+	0.00436866i	$0.998609\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			25.4413i			0.904588i
$792$		0			0
$793$		−	51.4599i		−	1.82739i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.3212			0.613549			0.306775	−	0.951782i	$-0.400750\pi$
$797$	17.3212			0.613549			0.306775	+	0.951782i	$0.400750\pi$
$798$		0			0
$799$			11.9003i			0.421004i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.628459			−0.0221778
$804$		0			0
$805$	6.73870	+	1.93437i	0.237508	+	0.0681776i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	19.8231i		−	0.696943i	−0.937319	−	0.348471i	$-0.886701\pi$
$809$		−	19.8231i		−	0.696943i	0.937319	−	0.348471i	$-0.113299\pi$
$810$		0			0
$811$	−30.9612			−1.08720			−0.543598	−	0.839346i	$-0.682938\pi$
$811$	−30.9612			−1.08720			−0.543598	+	0.839346i	$0.682938\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.01314			0.140574
$816$		0			0
$817$	−46.3150			−1.62036
$818$		0			0
$819$		0			0
$820$		0			0
$821$			45.3124i			1.58141i	0.612195	+	0.790707i	$0.290287\pi$
$821$			45.3124i			1.58141i	−0.612195	+	0.790707i	$0.709713\pi$
$822$		0			0
$823$	18.7597			0.653922			0.326961	−	0.945038i	$-0.393975\pi$
$823$	18.7597			0.653922			0.326961	+	0.945038i	$0.393975\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	42.0146			1.46099			0.730495	−	0.682918i	$-0.239289\pi$
$827$	42.0146			1.46099			0.730495	+	0.682918i	$0.239289\pi$
$828$		0			0
$829$	−42.2342			−1.46685			−0.733427	−	0.679768i	$-0.762081\pi$
$829$	−42.2342			−1.46685			−0.733427	+	0.679768i	$0.762081\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.9798			−0.380427
$834$		0			0
$835$		−	3.28445i		−	0.113663i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.1568			0.454225			0.227112	−	0.973869i	$-0.427072\pi$
$839$	13.1568			0.454225			0.227112	+	0.973869i	$0.427072\pi$
$840$		0			0
$841$	14.2473			0.491285
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.5863			−0.536186
$846$		0			0
$847$			15.7351i			0.540666i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.81301	+	1.66865i	0.199267	+	0.0572005i
$852$		0			0
$853$	−43.0005			−1.47231			−0.736155	−	0.676813i	$-0.763361\pi$
$853$	−43.0005			−1.47231			−0.736155	+	0.676813i	$0.763361\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			55.3285i			1.88999i	0.327090	+	0.944993i	$0.393932\pi$
$857$			55.3285i			1.88999i	−0.327090	+	0.944993i	$0.606068\pi$
$858$		0			0
$859$	−13.7190			−0.468085			−0.234043	−	0.972226i	$-0.575195\pi$
$859$	−13.7190			−0.468085			−0.234043	+	0.972226i	$0.575195\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30.3285i		−	1.03239i	−0.856470	−	0.516197i	$-0.827347\pi$
$863$		−	30.3285i		−	1.03239i	0.856470	−	0.516197i	$-0.172653\pi$
$864$		0			0
$865$			13.1227i			0.446185i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	3.26179i		−	0.110649i
$870$		0			0
$871$		−	36.8853i		−	1.24981i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			1.46186i			0.0494199i
$876$		0			0
$877$	7.68113			0.259373			0.129687	−	0.991555i	$-0.458603\pi$
$877$	7.68113			0.259373			0.129687	+	0.991555i	$0.458603\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.0052			1.11197			0.555987	−	0.831191i	$-0.312341\pi$
$881$	33.0052			1.11197			0.555987	+	0.831191i	$0.312341\pi$
$882$		0			0
$883$	24.0592			0.809655			0.404828	−	0.914393i	$-0.367332\pi$
$883$	24.0592			0.809655			0.404828	+	0.914393i	$0.367332\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			15.7323i			0.528238i	0.964490	+	0.264119i	$0.0850812\pi$
$887$			15.7323i			0.528238i	−0.964490	+	0.264119i	$0.914919\pi$
$888$		0			0
$889$			0.526832i			0.0176694i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.9227			−1.13518
$894$		0			0
$895$			2.03400i			0.0679893i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	25.8936i		−	0.863599i
$900$		0			0
$901$	12.9920			0.432825
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	9.52408i		−	0.316591i
$906$		0			0
$907$		−	14.6457i		−	0.486302i	−0.969988	−	0.243151i	$-0.921819\pi$
$907$		−	14.6457i		−	0.486302i	0.969988	−	0.243151i	$-0.0781811\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.4342			−1.07459			−0.537297	−	0.843393i	$-0.680554\pi$
$911$	−32.4342			−1.07459			−0.537297	+	0.843393i	$0.680554\pi$
$912$		0			0
$913$	2.47212			0.0818152
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.03567			−0.298384
$918$		0			0
$919$		−	29.9938i		−	0.989405i	−0.869062	−	0.494703i	$-0.835277\pi$
$919$		−	29.9938i		−	0.989405i	0.869062	−	0.494703i	$-0.164723\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	46.8742i		−	1.54288i
$924$		0			0
$925$			1.26105i			0.0414629i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	20.8411i		−	0.683774i	−0.939741	−	0.341887i	$-0.888934\pi$
$929$		−	20.8411i		−	0.683774i	0.939741	−	0.341887i	$-0.111066\pi$
$930$		0			0
$931$		−	31.2986i		−	1.02577i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.09737			−0.0358877
$936$		0			0
$937$			5.24077i			0.171208i	0.996329	+	0.0856042i	$0.0272821\pi$
$937$			5.24077i			0.171208i	−0.996329	+	0.0856042i	$0.972718\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.10059			−0.101076			−0.0505381	−	0.998722i	$-0.516094\pi$
$941$	−3.10059			−0.101076			−0.0505381	+	0.998722i	$0.516094\pi$
$942$		0			0
$943$	21.4164	+	6.14765i	0.697412	+	0.200195i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.0407i			0.391271i	0.980677	+	0.195636i	$0.0626769\pi$
$947$			12.0407i			0.391271i	−0.980677	+	0.195636i	$0.937323\pi$
$948$		0			0
$949$	6.91350			0.224422
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.72757			−0.217927			−0.108964	−	0.994046i	$-0.534753\pi$
$953$	−6.72757			−0.217927			−0.108964	+	0.994046i	$0.534753\pi$
$954$		0			0
$955$	−2.61633			−0.0846624
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	7.89708i		−	0.255010i
$960$		0			0
$961$	14.4477			0.466054
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.90467			0.222269
$966$		0			0
$967$	52.5223			1.68900			0.844502	−	0.535553i	$-0.179897\pi$
$967$	52.5223			1.68900			0.844502	+	0.535553i	$0.179897\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.0184			1.22007			0.610034	−	0.792375i	$-0.291156\pi$
$971$	38.0184			1.22007			0.610034	+	0.792375i	$0.291156\pi$
$972$		0			0
$973$			22.5697i			0.723552i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.3636			0.331560			0.165780	−	0.986163i	$-0.446986\pi$
$977$	10.3636			0.331560			0.165780	+	0.986163i	$0.446986\pi$
$978$		0			0
$979$	−2.16621			−0.0692324
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−56.1704			−1.79156			−0.895778	−	0.444501i	$-0.853381\pi$
$983$	−56.1704			−1.79156			−0.895778	+	0.444501i	$0.853381\pi$
$984$		0			0
$985$		−	8.22801i		−	0.262166i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−33.1717	−	9.52207i	−1.05480	−	0.302784i
$990$		0			0
$991$	−36.7900			−1.16867			−0.584336	−	0.811512i	$-0.698645\pi$
$991$	−36.7900			−1.16867			−0.584336	+	0.811512i	$0.698645\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			22.8525i			0.724474i
$996$		0			0
$997$	−11.8674			−0.375844			−0.187922	−	0.982184i	$-0.560175\pi$
$997$	−11.8674			−0.375844			−0.187922	+	0.982184i	$0.560175\pi$
$998$		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.1241.17	✓	48
3.2	odd	2		8280.2.p.b.1241.17	yes	48
23.22	odd	2		8280.2.p.b.1241.32	yes	48
69.68	even	2	inner	8280.2.p.a.1241.32	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.17	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.32	yes	48	69.68	even	2	inner
8280.2.p.b.1241.17	yes	48	3.2	odd	2
8280.2.p.b.1241.32	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -1.46186i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -1.46186i q^{7} -0.486025 q^{11} +5.34662 q^{13} -2.25784 q^{17} -6.43611i q^{19} +(1.32323 - 4.60967i) q^{23} +1.00000 q^{25} +3.84093i q^{29} -6.74149 q^{31} +1.46186i q^{35} +1.26105i q^{37} +4.64596i q^{41} -7.19611i q^{43} -5.27068i q^{47} +4.86296 q^{49} -5.75415 q^{53} +0.486025 q^{55} +12.4553i q^{59} -9.62475i q^{61} -5.34662 q^{65} -6.89880i q^{67} -8.76708i q^{71} +1.29306 q^{73} +0.710500i q^{77} +6.71117i q^{79} -5.08641 q^{83} +2.25784 q^{85} +4.45700 q^{89} -7.81601i q^{91} +6.43611i q^{95} +0.599189i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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