LMFDB - Embedded newform 8280.2.p.a.1241.39

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			1241.39
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.84853i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.84853i q^{7} -2.95839 q^{11} -0.451593 q^{13} +4.38605 q^{17} -7.84690i q^{19} +(3.10541 + 3.65464i) q^{23} +1.00000 q^{25} +3.01776i q^{29} -6.09850 q^{31} -2.84853i q^{35} -0.619475i q^{37} -5.35531i q^{41} +3.75365i q^{43} +5.23505i q^{47} -1.11412 q^{49} +14.0102 q^{53} +2.95839 q^{55} -11.8371i q^{59} +9.81751i q^{61} +0.451593 q^{65} +12.3590i q^{67} -13.6631i q^{71} +1.07832 q^{73} -8.42705i q^{77} +8.71942i q^{79} +3.20932 q^{83} -4.38605 q^{85} -3.64156 q^{89} -1.28638i q^{91} +7.84690i q^{95} +8.46148i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			2.84853i			1.07664i	0.842740	+	0.538322i	$0.180941\pi$
$7$			2.84853i			1.07664i	−0.842740	+	0.538322i	$0.819059\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.95839			−0.891987			−0.445994	−	0.895036i	$-0.647150\pi$
$11$	−2.95839			−0.891987			−0.445994	+	0.895036i	$0.647150\pi$
$12$		0			0
$13$	−0.451593			−0.125249			−0.0626246	−	0.998037i	$-0.519947\pi$
$13$	−0.451593			−0.125249			−0.0626246	+	0.998037i	$0.519947\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.38605			1.06377			0.531887	−	0.846815i	$-0.321483\pi$
$17$	4.38605			1.06377			0.531887	+	0.846815i	$0.321483\pi$
$18$		0			0
$19$		−	7.84690i		−	1.80020i	−0.435681	−	0.900101i	$-0.643492\pi$
$19$		−	7.84690i		−	1.80020i	0.435681	−	0.900101i	$-0.356508\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.10541	+	3.65464i	0.647523	+	0.762046i
$23$	3.10541	+	3.65464i	0.647523	+	0.762046i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.01776i			0.560383i	0.959944	+	0.280192i	$0.0903980\pi$
$29$			3.01776i			0.560383i	−0.959944	+	0.280192i	$0.909602\pi$
$30$		0			0
$31$	−6.09850			−1.09532			−0.547662	−	0.836700i	$-0.684482\pi$
$31$	−6.09850			−1.09532			−0.547662	+	0.836700i	$0.684482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.84853i		−	0.481489i
$36$		0			0
$37$		−	0.619475i		−	0.101841i	−0.998703	−	0.0509205i	$-0.983784\pi$
$37$		−	0.619475i		−	0.101841i	0.998703	−	0.0509205i	$-0.0162155\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.35531i		−	0.836359i	−0.908364	−	0.418180i	$-0.862668\pi$
$41$		−	5.35531i		−	0.836359i	0.908364	−	0.418180i	$-0.137332\pi$
$42$		0			0
$43$			3.75365i			0.572426i	0.958166	+	0.286213i	$0.0923965\pi$
$43$			3.75365i			0.572426i	−0.958166	+	0.286213i	$0.907604\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.23505i			0.763610i	0.924243	+	0.381805i	$0.124697\pi$
$47$			5.23505i			0.763610i	−0.924243	+	0.381805i	$0.875303\pi$
$48$		0			0
$49$	−1.11412			−0.159160
$50$		0			0
$51$		0			0
$52$		0			0
$53$	14.0102			1.92445			0.962227	−	0.272248i	$-0.0877671\pi$
$53$	14.0102			1.92445			0.962227	+	0.272248i	$0.0877671\pi$
$54$		0			0
$55$	2.95839			0.398909
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.8371i		−	1.54106i	−0.637404	−	0.770530i	$-0.719992\pi$
$59$		−	11.8371i		−	1.54106i	0.637404	−	0.770530i	$-0.280008\pi$
$60$		0			0
$61$			9.81751i			1.25700i	0.777808	+	0.628502i	$0.216332\pi$
$61$			9.81751i			1.25700i	−0.777808	+	0.628502i	$0.783668\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.451593			0.0560132
$66$		0			0
$67$			12.3590i			1.50990i	0.655785	+	0.754948i	$0.272338\pi$
$67$			12.3590i			1.50990i	−0.655785	+	0.754948i	$0.727662\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.6631i		−	1.62151i	−0.585385	−	0.810756i	$-0.699056\pi$
$71$		−	13.6631i		−	1.62151i	0.585385	−	0.810756i	$-0.300944\pi$
$72$		0			0
$73$	1.07832			0.126207			0.0631037	−	0.998007i	$-0.479900\pi$
$73$	1.07832			0.126207			0.0631037	+	0.998007i	$0.479900\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	8.42705i		−	0.960352i
$78$		0			0
$79$			8.71942i			0.981011i	0.871438	+	0.490506i	$0.163188\pi$
$79$			8.71942i			0.981011i	−0.871438	+	0.490506i	$0.836812\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.20932			0.352269			0.176134	−	0.984366i	$-0.443641\pi$
$83$	3.20932			0.352269			0.176134	+	0.984366i	$0.443641\pi$
$84$		0			0
$85$	−4.38605			−0.475734
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.64156			−0.386004			−0.193002	−	0.981198i	$-0.561822\pi$
$89$	−3.64156			−0.386004			−0.193002	+	0.981198i	$0.561822\pi$
$90$		0			0
$91$		−	1.28638i		−	0.134849i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			7.84690i			0.805075i
$96$		0			0
$97$			8.46148i			0.859133i	0.903035	+	0.429567i	$0.141334\pi$
$97$			8.46148i			0.859133i	−0.903035	+	0.429567i	$0.858666\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			10.2407i			1.01899i	0.860474	+	0.509495i	$0.170168\pi$
$101$			10.2407i			1.01899i	−0.860474	+	0.509495i	$0.829832\pi$
$102$		0			0
$103$		−	8.25352i		−	0.813244i	−0.913597	−	0.406622i	$-0.866707\pi$
$103$		−	8.25352i		−	0.813244i	0.913597	−	0.406622i	$-0.133293\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.4938			1.88453			0.942266	−	0.334865i	$-0.108691\pi$
$107$	19.4938			1.88453			0.942266	+	0.334865i	$0.108691\pi$
$108$		0			0
$109$		−	1.36398i		−	0.130646i	−0.997864	−	0.0653228i	$-0.979192\pi$
$109$		−	1.36398i		−	0.130646i	0.997864	−	0.0653228i	$-0.0208077\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.411889			−0.0387472			−0.0193736	−	0.999812i	$-0.506167\pi$
$113$	−0.411889			−0.0387472			−0.0193736	+	0.999812i	$0.506167\pi$
$114$		0			0
$115$	−3.10541	−	3.65464i	−0.289581	−	0.340797i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			12.4938i			1.14531i
$120$		0			0
$121$	−2.24795			−0.204359
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−17.8566			−1.58452			−0.792259	−	0.610185i	$-0.791095\pi$
$127$	−17.8566			−1.58452			−0.792259	+	0.610185i	$0.791095\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	19.5945i		−	1.71198i	−0.516996	−	0.855988i	$-0.672950\pi$
$131$		−	19.5945i		−	1.71198i	0.516996	−	0.855988i	$-0.327050\pi$
$132$		0			0
$133$	22.3521			1.93818
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.542882			−0.0463815			−0.0231908	−	0.999731i	$-0.507383\pi$
$137$	−0.542882			−0.0463815			−0.0231908	+	0.999731i	$0.507383\pi$
$138$		0			0
$139$	8.44914			0.716647			0.358323	−	0.933598i	$-0.383349\pi$
$139$	8.44914			0.716647			0.358323	+	0.933598i	$0.383349\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.33599			0.111721
$144$		0			0
$145$		−	3.01776i		−	0.250611i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.10938			0.664346			0.332173	−	0.943218i	$-0.392218\pi$
$149$	8.10938			0.664346			0.332173	+	0.943218i	$0.392218\pi$
$150$		0			0
$151$	1.74008			0.141606			0.0708030	−	0.997490i	$-0.477444\pi$
$151$	1.74008			0.141606			0.0708030	+	0.997490i	$0.477444\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.09850			0.489844
$156$		0			0
$157$		−	1.69896i		−	0.135592i	−0.997699	−	0.0677958i	$-0.978403\pi$
$157$		−	1.69896i		−	0.135592i	0.997699	−	0.0677958i	$-0.0215966\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.4104	+	8.84586i	−0.820451	+	0.697151i
$162$		0			0
$163$	−21.8082			−1.70815			−0.854077	−	0.520147i	$-0.825877\pi$
$163$	−21.8082			−1.70815			−0.854077	+	0.520147i	$0.825877\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.2292i			1.33324i	0.745399	+	0.666619i	$0.232259\pi$
$167$			17.2292i			1.33324i	−0.745399	+	0.666619i	$0.767741\pi$
$168$		0			0
$169$	−12.7961			−0.984313
$170$		0			0
$171$		0			0
$172$		0			0
$173$			12.4800i			0.948838i	0.880299	+	0.474419i	$0.157342\pi$
$173$			12.4800i			0.948838i	−0.880299	+	0.474419i	$0.842658\pi$
$174$		0			0
$175$			2.84853i			0.215329i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.6646i		−	0.871850i	−0.899983	−	0.435925i	$-0.856421\pi$
$179$		−	11.6646i		−	0.871850i	0.899983	−	0.435925i	$-0.143579\pi$
$180$		0			0
$181$			23.7617i			1.76619i	0.469194	+	0.883095i	$0.344544\pi$
$181$			23.7617i			1.76619i	−0.469194	+	0.883095i	$0.655456\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.619475i			0.0455447i
$186$		0			0
$187$	−12.9756			−0.948873
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−18.6185			−1.34719			−0.673594	−	0.739102i	$-0.735250\pi$
$191$	−18.6185			−1.34719			−0.673594	+	0.739102i	$0.735250\pi$
$192$		0			0
$193$	−6.08797			−0.438221			−0.219111	−	0.975700i	$-0.570316\pi$
$193$	−6.08797			−0.438221			−0.219111	+	0.975700i	$0.570316\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.5354i			1.03560i	0.855501	+	0.517801i	$0.173249\pi$
$197$			14.5354i			1.03560i	−0.855501	+	0.517801i	$0.826751\pi$
$198$		0			0
$199$			23.0638i			1.63495i	0.575965	+	0.817475i	$0.304627\pi$
$199$			23.0638i			1.63495i	−0.575965	+	0.817475i	$0.695373\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.59617			−0.603333
$204$		0			0
$205$			5.35531i			0.374031i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			23.2142i			1.60576i
$210$		0			0
$211$	6.20881			0.427432			0.213716	−	0.976896i	$-0.431443\pi$
$211$	6.20881			0.427432			0.213716	+	0.976896i	$0.431443\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.75365i		−	0.255997i
$216$		0			0
$217$		−	17.3718i		−	1.17927i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.98071			−0.133237
$222$		0			0
$223$	4.90111			0.328202			0.164101	−	0.986444i	$-0.447528\pi$
$223$	4.90111			0.328202			0.164101	+	0.986444i	$0.447528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	24.1700			1.60422			0.802109	−	0.597178i	$-0.203712\pi$
$227$	24.1700			1.60422			0.802109	+	0.597178i	$0.203712\pi$
$228$		0			0
$229$		−	6.21941i		−	0.410990i	−0.978658	−	0.205495i	$-0.934120\pi$
$229$		−	6.21941i		−	0.410990i	0.978658	−	0.205495i	$-0.0658805\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			25.7214i			1.68506i	0.538648	+	0.842531i	$0.318935\pi$
$233$			25.7214i			1.68506i	−0.538648	+	0.842531i	$0.681065\pi$
$234$		0			0
$235$		−	5.23505i		−	0.341497i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.9786i			1.74510i	0.488527	+	0.872549i	$0.337534\pi$
$239$			26.9786i			1.74510i	−0.488527	+	0.872549i	$0.662466\pi$
$240$		0			0
$241$		−	3.41127i		−	0.219739i	−0.993946	−	0.109870i	$-0.964957\pi$
$241$		−	3.41127i		−	0.219739i	0.993946	−	0.109870i	$-0.0350433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.11412			0.0711787
$246$		0			0
$247$			3.54360i			0.225474i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.41740			−0.215704			−0.107852	−	0.994167i	$-0.534397\pi$
$251$	−3.41740			−0.215704			−0.107852	+	0.994167i	$0.534397\pi$
$252$		0			0
$253$	−9.18701	−	10.8118i	−0.577582	−	0.679735i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			22.7443i			1.41875i	0.704832	+	0.709375i	$0.251023\pi$
$257$			22.7443i			1.41875i	−0.704832	+	0.709375i	$0.748977\pi$
$258$		0			0
$259$	1.76459			0.109646
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−27.9026			−1.72055			−0.860274	−	0.509833i	$-0.829707\pi$
$263$	−27.9026			−1.72055			−0.860274	+	0.509833i	$0.829707\pi$
$264$		0			0
$265$	−14.0102			−0.860642
$266$		0			0
$267$		0			0
$268$		0			0
$269$			18.4394i			1.12427i	0.827046	+	0.562134i	$0.190019\pi$
$269$			18.4394i			1.12427i	−0.827046	+	0.562134i	$0.809981\pi$
$270$		0			0
$271$	−11.9202			−0.724100			−0.362050	−	0.932159i	$-0.617923\pi$
$271$	−11.9202			−0.724100			−0.362050	+	0.932159i	$0.617923\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.95839			−0.178397
$276$		0			0
$277$	−23.4021			−1.40609			−0.703046	−	0.711144i	$-0.748177\pi$
$277$	−23.4021			−1.40609			−0.703046	+	0.711144i	$0.748177\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.66269			−0.0991880			−0.0495940	−	0.998769i	$-0.515793\pi$
$281$	−1.66269			−0.0991880			−0.0495940	+	0.998769i	$0.515793\pi$
$282$		0			0
$283$			30.1800i			1.79401i	0.442018	+	0.897006i	$0.354263\pi$
$283$			30.1800i			1.79401i	−0.442018	+	0.897006i	$0.645737\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.2548			0.900461
$288$		0			0
$289$	2.23746			0.131615
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.6235			0.971158			0.485579	−	0.874193i	$-0.338609\pi$
$293$	16.6235			0.971158			0.485579	+	0.874193i	$0.338609\pi$
$294$		0			0
$295$			11.8371i			0.689183i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.40238	−	1.65041i	−0.0811018	−	0.0954457i
$300$		0			0
$301$	−10.6924			−0.616298
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	9.81751i		−	0.562149i
$306$		0			0
$307$	−10.2948			−0.587556			−0.293778	−	0.955874i	$-0.594913\pi$
$307$	−10.2948			−0.587556			−0.293778	+	0.955874i	$0.594913\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.06273i		−	0.287081i	−0.989644	−	0.143540i	$-0.954151\pi$
$311$		−	5.06273i		−	0.287081i	0.989644	−	0.143540i	$-0.0458487\pi$
$312$		0			0
$313$			14.8335i			0.838438i	0.907885	+	0.419219i	$0.137696\pi$
$313$			14.8335i			0.838438i	−0.907885	+	0.419219i	$0.862304\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.9847i		−	1.01012i	−0.863084	−	0.505060i	$-0.831470\pi$
$317$		−	17.9847i		−	1.01012i	0.863084	−	0.505060i	$-0.168530\pi$
$318$		0			0
$319$		−	8.92769i		−	0.499855i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	34.4169i		−	1.91501i
$324$		0			0
$325$	−0.451593			−0.0250499
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.9122			−0.822136
$330$		0			0
$331$	−18.8433			−1.03572			−0.517859	−	0.855466i	$-0.673271\pi$
$331$	−18.8433			−1.03572			−0.517859	+	0.855466i	$0.673271\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	12.3590i		−	0.675246i
$336$		0			0
$337$		−	20.4634i		−	1.11471i	−0.830273	−	0.557356i	$-0.811816\pi$
$337$		−	20.4634i		−	1.11471i	0.830273	−	0.557356i	$-0.188184\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.0417			0.977014
$342$		0			0
$343$			16.7661i			0.905284i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.6403i		−	1.10803i	−0.832507	−	0.554014i	$-0.813095\pi$
$347$		−	20.6403i		−	1.10803i	0.832507	−	0.554014i	$-0.186905\pi$
$348$		0			0
$349$	36.6859			1.96375			0.981877	−	0.189521i	$-0.0606936\pi$
$349$	36.6859			1.96375			0.981877	+	0.189521i	$0.0606936\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.82424i		−	0.150319i	−0.997172	−	0.0751595i	$-0.976053\pi$
$353$		−	2.82424i		−	0.150319i	0.997172	−	0.0751595i	$-0.0239466\pi$
$354$		0			0
$355$			13.6631i			0.725162i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.383141			−0.0202214			−0.0101107	−	0.999949i	$-0.503218\pi$
$359$	−0.383141			−0.0202214			−0.0101107	+	0.999949i	$0.503218\pi$
$360$		0			0
$361$	−42.5738			−2.24073
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.07832			−0.0564417
$366$		0			0
$367$			3.34271i			0.174488i	0.996187	+	0.0872439i	$0.0278059\pi$
$367$			3.34271i			0.174488i	−0.996187	+	0.0872439i	$0.972194\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			39.9086i			2.07195i
$372$		0			0
$373$		−	22.4610i		−	1.16299i	−0.813550	−	0.581494i	$-0.802468\pi$
$373$		−	22.4610i		−	1.16299i	0.813550	−	0.581494i	$-0.197532\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.36280i		−	0.0701876i
$378$		0			0
$379$			16.3221i			0.838411i	0.907891	+	0.419205i	$0.137691\pi$
$379$			16.3221i			0.838411i	−0.907891	+	0.419205i	$0.862309\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.2685			−0.575794			−0.287897	−	0.957661i	$-0.592956\pi$
$383$	−11.2685			−0.575794			−0.287897	+	0.957661i	$0.592956\pi$
$384$		0			0
$385$			8.42705i			0.429482i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.17467			0.363770			0.181885	−	0.983320i	$-0.441780\pi$
$389$	7.17467			0.363770			0.181885	+	0.983320i	$0.441780\pi$
$390$		0			0
$391$	13.6205	+	16.0295i	0.688818	+	0.810645i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	8.71942i		−	0.438722i
$396$		0			0
$397$	34.9127			1.75222			0.876109	−	0.482113i	$-0.160130\pi$
$397$	34.9127			1.75222			0.876109	+	0.482113i	$0.160130\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.6127			1.12922			0.564611	−	0.825357i	$-0.309026\pi$
$401$	22.6127			1.12922			0.564611	+	0.825357i	$0.309026\pi$
$402$		0			0
$403$	2.75404			0.137188
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.83265i			0.0908409i
$408$		0			0
$409$	−4.70826			−0.232808			−0.116404	−	0.993202i	$-0.537137\pi$
$409$	−4.70826			−0.232808			−0.116404	+	0.993202i	$0.537137\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	33.7183			1.65917
$414$		0			0
$415$	−3.20932			−0.157539
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.32551			0.260168			0.130084	−	0.991503i	$-0.458475\pi$
$419$	5.32551			0.260168			0.130084	+	0.991503i	$0.458475\pi$
$420$		0			0
$421$		−	11.4267i		−	0.556905i	−0.960450	−	0.278452i	$-0.910179\pi$
$421$		−	11.4267i		−	0.556905i	0.960450	−	0.278452i	$-0.0898214\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.38605			0.212755
$426$		0			0
$427$	−27.9655			−1.35334
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−19.4968			−0.939129			−0.469565	−	0.882898i	$-0.655589\pi$
$431$	−19.4968			−0.939129			−0.469565	+	0.882898i	$0.655589\pi$
$432$		0			0
$433$			22.9884i			1.10475i	0.833595	+	0.552375i	$0.186279\pi$
$433$			22.9884i			1.10475i	−0.833595	+	0.552375i	$0.813721\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	28.6776	−	24.3679i	1.37184	−	1.16567i
$438$		0			0
$439$	2.56767			0.122548			0.0612740	−	0.998121i	$-0.480484\pi$
$439$	2.56767			0.122548			0.0612740	+	0.998121i	$0.480484\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			41.1892i			1.95696i	0.206347	+	0.978479i	$0.433842\pi$
$443$			41.1892i			1.95696i	−0.206347	+	0.978479i	$0.566158\pi$
$444$		0			0
$445$	3.64156			0.172626
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	24.9921i		−	1.17945i	−0.807605	−	0.589724i	$-0.799237\pi$
$449$		−	24.9921i		−	1.17945i	0.807605	−	0.589724i	$-0.200763\pi$
$450$		0			0
$451$			15.8431i			0.746022i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1.28638i			0.0603062i
$456$		0			0
$457$			9.01138i			0.421535i	0.977536	+	0.210767i	$0.0675962\pi$
$457$			9.01138i			0.421535i	−0.977536	+	0.210767i	$0.932404\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.98035i			0.418257i	0.977888	+	0.209128i	$0.0670627\pi$
$461$			8.98035i			0.418257i	−0.977888	+	0.209128i	$0.932937\pi$
$462$		0			0
$463$	7.13347			0.331520			0.165760	−	0.986166i	$-0.446992\pi$
$463$	7.13347			0.331520			0.165760	+	0.986166i	$0.446992\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.9368			1.20021			0.600107	−	0.799920i	$-0.295125\pi$
$467$	25.9368			1.20021			0.600107	+	0.799920i	$0.295125\pi$
$468$		0			0
$469$	−35.2051			−1.62562
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.1047i		−	0.510596i
$474$		0			0
$475$		−	7.84690i		−	0.360040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.278689			−0.0127336			−0.00636681	−	0.999980i	$-0.502027\pi$
$479$	−0.278689			−0.0127336			−0.00636681	+	0.999980i	$0.502027\pi$
$480$		0			0
$481$			0.279750i			0.0127555i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.46148i		−	0.384216i
$486$		0			0
$487$	31.6247			1.43305			0.716527	−	0.697559i	$-0.245731\pi$
$487$	31.6247			1.43305			0.716527	+	0.697559i	$0.245731\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			17.4654i			0.788202i	0.919067	+	0.394101i	$0.128944\pi$
$491$			17.4654i			0.788202i	−0.919067	+	0.394101i	$0.871056\pi$
$492$		0			0
$493$			13.2360i			0.596121i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	38.9197			1.74579
$498$		0			0
$499$	25.4931			1.14123			0.570613	−	0.821219i	$-0.306706\pi$
$499$	25.4931			1.14123			0.570613	+	0.821219i	$0.306706\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−34.1952			−1.52469			−0.762345	−	0.647170i	$-0.775952\pi$
$503$	−34.1952			−1.52469			−0.762345	+	0.647170i	$0.775952\pi$
$504$		0			0
$505$		−	10.2407i		−	0.455706i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			35.9594i			1.59387i	0.604064	+	0.796936i	$0.293547\pi$
$509$			35.9594i			1.59387i	−0.604064	+	0.796936i	$0.706453\pi$
$510$		0			0
$511$			3.07162i			0.135880i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			8.25352i			0.363694i
$516$		0			0
$517$		−	15.4873i		−	0.681131i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	25.6426			1.12342			0.561710	−	0.827334i	$-0.310144\pi$
$521$	25.6426			1.12342			0.561710	+	0.827334i	$0.310144\pi$
$522$		0			0
$523$		−	14.5928i		−	0.638098i	−0.947738	−	0.319049i	$-0.896637\pi$
$523$		−	14.5928i		−	0.638098i	0.947738	−	0.319049i	$-0.103363\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.7484			−1.16518
$528$		0			0
$529$	−3.71284	+	22.6983i	−0.161428	+	0.986884i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2.41842i			0.104753i
$534$		0			0
$535$	−19.4938			−0.842789
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.29601			0.141969
$540$		0			0
$541$	20.6370			0.887253			0.443627	−	0.896212i	$-0.353692\pi$
$541$	20.6370			0.887253			0.443627	+	0.896212i	$0.353692\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.36398i			0.0584264i
$546$		0			0
$547$	8.00106			0.342101			0.171050	−	0.985262i	$-0.445284\pi$
$547$	8.00106			0.342101			0.171050	+	0.985262i	$0.445284\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	23.6800			1.00880
$552$		0			0
$553$	−24.8375			−1.05620
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.8992			−1.01264			−0.506321	−	0.862345i	$-0.668995\pi$
$557$	−23.8992			−1.01264			−0.506321	+	0.862345i	$0.668995\pi$
$558$		0			0
$559$		−	1.69512i		−	0.0716959i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.6858			−0.618934			−0.309467	−	0.950910i	$-0.600151\pi$
$563$	−14.6858			−0.618934			−0.309467	+	0.950910i	$0.600151\pi$
$564$		0			0
$565$	0.411889			0.0173283
$566$		0			0
$567$		0			0
$568$		0			0
$569$	39.9436			1.67452			0.837261	−	0.546803i	$-0.184156\pi$
$569$	39.9436			1.67452			0.837261	+	0.546803i	$0.184156\pi$
$570$		0			0
$571$			25.0900i			1.04998i	0.851107	+	0.524992i	$0.175932\pi$
$571$			25.0900i			1.04998i	−0.851107	+	0.524992i	$0.824068\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.10541	+	3.65464i	0.129505	+	0.152409i
$576$		0			0
$577$	0.519605			0.0216315			0.0108157	−	0.999942i	$-0.496557\pi$
$577$	0.519605			0.0216315			0.0108157	+	0.999942i	$0.496557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			9.14185i			0.379268i
$582$		0			0
$583$	−41.4477			−1.71659
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	2.43276i		−	0.100411i	−0.998739	−	0.0502054i	$-0.984012\pi$
$587$		−	2.43276i		−	0.100411i	0.998739	−	0.0502054i	$-0.0159876\pi$
$588$		0			0
$589$			47.8543i			1.97180i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	3.69741i		−	0.151834i	−0.997114	−	0.0759172i	$-0.975812\pi$
$593$		−	3.69741i		−	0.151834i	0.997114	−	0.0759172i	$-0.0241885\pi$
$594$		0			0
$595$		−	12.4938i		−	0.512196i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.88862i		−	0.199744i	−0.995000	−	0.0998718i	$-0.968157\pi$
$599$		−	4.88862i		−	0.199744i	0.995000	−	0.0998718i	$-0.0318433\pi$
$600$		0			0
$601$	27.9886			1.14168			0.570838	−	0.821062i	$-0.306618\pi$
$601$	27.9886			1.14168			0.570838	+	0.821062i	$0.306618\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.24795			0.0913921
$606$		0			0
$607$	−22.5025			−0.913348			−0.456674	−	0.889634i	$-0.650959\pi$
$607$	−22.5025			−0.913348			−0.456674	+	0.889634i	$0.650959\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.36411i		−	0.0956417i
$612$		0			0
$613$			26.4302i			1.06751i	0.845641	+	0.533753i	$0.179219\pi$
$613$			26.4302i			1.06751i	−0.845641	+	0.533753i	$0.820781\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.74830			−0.110642			−0.0553211	−	0.998469i	$-0.517618\pi$
$617$	−2.74830			−0.110642			−0.0553211	+	0.998469i	$0.517618\pi$
$618$		0			0
$619$		−	0.463520i		−	0.0186304i	−0.999957	−	0.00931522i	$-0.997035\pi$
$619$		−	0.463520i		−	0.0186304i	0.999957	−	0.00931522i	$-0.00296517\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	10.3731i		−	0.415589i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	2.71705i		−	0.108336i
$630$		0			0
$631$			0.514097i			0.0204659i	0.999948	+	0.0102329i	$0.00325731\pi$
$631$			0.514097i			0.0204659i	−0.999948	+	0.0102329i	$0.996743\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.8566			0.708618
$636$		0			0
$637$	0.503130			0.0199347
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.15293			−0.124533			−0.0622665	−	0.998060i	$-0.519833\pi$
$641$	−3.15293			−0.124533			−0.0622665	+	0.998060i	$0.519833\pi$
$642$		0			0
$643$			14.1082i			0.556374i	0.960527	+	0.278187i	$0.0897335\pi$
$643$			14.1082i			0.556374i	−0.960527	+	0.278187i	$0.910266\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.62870i			0.181973i	0.995852	+	0.0909865i	$0.0290020\pi$
$647$			4.62870i			0.181973i	−0.995852	+	0.0909865i	$0.970998\pi$
$648$		0			0
$649$			35.0187i			1.37461i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			34.2118i			1.33881i	0.742897	+	0.669406i	$0.233451\pi$
$653$			34.2118i			1.33881i	−0.742897	+	0.669406i	$0.766549\pi$
$654$		0			0
$655$			19.5945i			0.765619i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−45.9387			−1.78952			−0.894760	−	0.446548i	$-0.852653\pi$
$659$	−45.9387			−1.78952			−0.894760	+	0.446548i	$0.852653\pi$
$660$		0			0
$661$		−	31.0803i		−	1.20888i	−0.796650	−	0.604441i	$-0.793397\pi$
$661$		−	31.0803i		−	1.20888i	0.796650	−	0.604441i	$-0.206603\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−22.3521			−0.866778
$666$		0			0
$667$	−11.0288	+	9.37138i	−0.427038	+	0.362861i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	29.0440i		−	1.12123i
$672$		0			0
$673$	38.3898			1.47982			0.739910	−	0.672706i	$-0.234868\pi$
$673$	38.3898			1.47982			0.739910	+	0.672706i	$0.234868\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	37.4158			1.43801			0.719003	−	0.695007i	$-0.244599\pi$
$677$	37.4158			1.43801			0.719003	+	0.695007i	$0.244599\pi$
$678$		0			0
$679$	−24.1028			−0.924980
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7.21233i		−	0.275972i	−0.990434	−	0.137986i	$-0.955937\pi$
$683$		−	7.21233i		−	0.275972i	0.990434	−	0.137986i	$-0.0440629\pi$
$684$		0			0
$685$	0.542882			0.0207424
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.32692			−0.241036
$690$		0			0
$691$	−35.5977			−1.35420			−0.677101	−	0.735890i	$-0.736764\pi$
$691$	−35.5977			−1.35420			−0.677101	+	0.735890i	$0.736764\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.44914			−0.320494
$696$		0			0
$697$		−	23.4887i		−	0.889698i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.5389			0.586894			0.293447	−	0.955975i	$-0.405198\pi$
$701$	15.5389			0.586894			0.293447	+	0.955975i	$0.405198\pi$
$702$		0			0
$703$	−4.86096			−0.183335
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−29.1710			−1.09709
$708$		0			0
$709$		−	40.3904i		−	1.51689i	−0.651735	−	0.758447i	$-0.725959\pi$
$709$		−	40.3904i		−	1.51689i	0.651735	−	0.758447i	$-0.274041\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.9384	−	22.2879i	−0.709247	−	0.834687i
$714$		0			0
$715$	−1.33599			−0.0499630
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	2.35988i		−	0.0880086i	−0.999031	−	0.0440043i	$-0.985988\pi$
$719$		−	2.35988i		−	0.0880086i	0.999031	−	0.0440043i	$-0.0140115\pi$
$720$		0			0
$721$	23.5104			0.875573
$722$		0			0
$723$		0			0
$724$		0			0
$725$			3.01776i			0.112077i
$726$		0			0
$727$		−	11.8636i		−	0.439996i	−0.975500	−	0.219998i	$-0.929395\pi$
$727$		−	11.8636i		−	0.439996i	0.975500	−	0.219998i	$-0.0706051\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			16.4637i			0.608932i
$732$		0			0
$733$		−	7.17262i		−	0.264927i	−0.991188	−	0.132463i	$-0.957711\pi$
$733$		−	7.17262i		−	0.264927i	0.991188	−	0.132463i	$-0.0422887\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	36.5628i		−	1.34681i
$738$		0			0
$739$	49.6293			1.82564			0.912822	−	0.408357i	$-0.133898\pi$
$739$	49.6293			1.82564			0.912822	+	0.408357i	$0.133898\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	30.8004			1.12996			0.564979	−	0.825105i	$-0.308884\pi$
$743$	30.8004			1.12996			0.564979	+	0.825105i	$0.308884\pi$
$744$		0			0
$745$	−8.10938			−0.297105
$746$		0			0
$747$		0			0
$748$		0			0
$749$			55.5285i			2.02897i
$750$		0			0
$751$			19.3155i			0.704833i	0.935843	+	0.352416i	$0.114640\pi$
$751$			19.3155i			0.704833i	−0.935843	+	0.352416i	$0.885360\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.74008			−0.0633281
$756$		0			0
$757$		−	44.4725i		−	1.61638i	−0.588921	−	0.808190i	$-0.700447\pi$
$757$		−	44.4725i		−	1.61638i	0.588921	−	0.808190i	$-0.299553\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			35.6433i			1.29207i	0.763308	+	0.646035i	$0.223574\pi$
$761$			35.6433i			1.29207i	−0.763308	+	0.646035i	$0.776426\pi$
$762$		0			0
$763$	3.88534			0.140659
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.34555i			0.193017i
$768$		0			0
$769$			24.9184i			0.898580i	0.893386	+	0.449290i	$0.148323\pi$
$769$			24.9184i			0.898580i	−0.893386	+	0.449290i	$0.851677\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−38.5019			−1.38482			−0.692408	−	0.721506i	$-0.743450\pi$
$773$	−38.5019			−1.38482			−0.692408	+	0.721506i	$0.743450\pi$
$774$		0			0
$775$	−6.09850			−0.219065
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−42.0226			−1.50562
$780$		0			0
$781$			40.4207i			1.44637i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.69896i			0.0606384i
$786$		0			0
$787$			27.3157i			0.973701i	0.873485	+	0.486851i	$0.161854\pi$
$787$			27.3157i			0.973701i	−0.873485	+	0.486851i	$0.838146\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.17328i		−	0.0417169i
$792$		0			0
$793$		−	4.43352i		−	0.157439i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.8131			−0.808080			−0.404040	−	0.914741i	$-0.632394\pi$
$797$	−22.8131			−0.808080			−0.404040	+	0.914741i	$0.632394\pi$
$798$		0			0
$799$			22.9612i			0.812309i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.19008			−0.112575
$804$		0			0
$805$	10.4104	−	8.84586i	0.366917	−	0.311775i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			49.7903i			1.75053i	0.483640	+	0.875267i	$0.339314\pi$
$809$			49.7903i			1.75053i	−0.483640	+	0.875267i	$0.660686\pi$
$810$		0			0
$811$	−30.4119			−1.06791			−0.533953	−	0.845514i	$-0.679294\pi$
$811$	−30.4119			−1.06791			−0.533953	+	0.845514i	$0.679294\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	21.8082			0.763909
$816$		0			0
$817$	29.4545			1.03048
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	47.9871i		−	1.67476i	−0.546620	−	0.837381i	$-0.684086\pi$
$821$		−	47.9871i		−	1.67476i	0.546620	−	0.837381i	$-0.315914\pi$
$822$		0			0
$823$	−1.99002			−0.0693678			−0.0346839	−	0.999398i	$-0.511042\pi$
$823$	−1.99002			−0.0693678			−0.0346839	+	0.999398i	$0.511042\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−14.1678			−0.492662			−0.246331	−	0.969186i	$-0.579225\pi$
$827$	−14.1678			−0.492662			−0.246331	+	0.969186i	$0.579225\pi$
$828$		0			0
$829$	−21.3812			−0.742599			−0.371300	−	0.928513i	$-0.621088\pi$
$829$	−21.3812			−0.742599			−0.371300	+	0.928513i	$0.621088\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.88660			−0.169311
$834$		0			0
$835$		−	17.2292i		−	0.596242i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	39.4300			1.36127			0.680637	−	0.732621i	$-0.261703\pi$
$839$	39.4300			1.36127			0.680637	+	0.732621i	$0.261703\pi$
$840$		0			0
$841$	19.8931			0.685970
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.7961			0.440198
$846$		0			0
$847$		−	6.40335i		−	0.220022i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.26396	−	1.92372i	0.0776076	−	0.0659444i
$852$		0			0
$853$	20.2082			0.691915			0.345957	−	0.938250i	$-0.387554\pi$
$853$	20.2082			0.691915			0.345957	+	0.938250i	$0.387554\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	13.0925i		−	0.447230i	−0.974678	−	0.223615i	$-0.928214\pi$
$857$		−	13.0925i		−	0.447230i	0.974678	−	0.223615i	$-0.0717859\pi$
$858$		0			0
$859$	−47.0652			−1.60584			−0.802922	−	0.596084i	$-0.796723\pi$
$859$	−47.0652			−1.60584			−0.802922	+	0.596084i	$0.796723\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			28.6173i			0.974143i	0.873362	+	0.487071i	$0.161935\pi$
$863$			28.6173i			0.974143i	−0.873362	+	0.487071i	$0.838065\pi$
$864$		0			0
$865$		−	12.4800i		−	0.424333i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	25.7954i		−	0.875049i
$870$		0			0
$871$		−	5.58125i		−	0.189113i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.84853i		−	0.0962979i
$876$		0			0
$877$	16.7826			0.566709			0.283354	−	0.959015i	$-0.408553\pi$
$877$	16.7826			0.566709			0.283354	+	0.959015i	$0.408553\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22.7646			0.766960			0.383480	−	0.923549i	$-0.374726\pi$
$881$	22.7646			0.766960			0.383480	+	0.923549i	$0.374726\pi$
$882$		0			0
$883$	−9.13488			−0.307413			−0.153707	−	0.988117i	$-0.549121\pi$
$883$	−9.13488			−0.307413			−0.153707	+	0.988117i	$0.549121\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	23.4650i		−	0.787877i	−0.919137	−	0.393939i	$-0.871112\pi$
$887$		−	23.4650i		−	0.787877i	0.919137	−	0.393939i	$-0.128888\pi$
$888$		0			0
$889$		−	50.8651i		−	1.70596i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	41.0789			1.37465
$894$		0			0
$895$			11.6646i			0.389903i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	18.4038i		−	0.613801i
$900$		0			0
$901$	61.4496			2.04718
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	23.7617i		−	0.789864i
$906$		0			0
$907$		−	20.5124i		−	0.681102i	−0.940226	−	0.340551i	$-0.889386\pi$
$907$		−	20.5124i		−	0.681102i	0.940226	−	0.340551i	$-0.110614\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26.1273			−0.865635			−0.432818	−	0.901482i	$-0.642481\pi$
$911$	−26.1273			−0.865635			−0.432818	+	0.901482i	$0.642481\pi$
$912$		0			0
$913$	−9.49441			−0.314219
$914$		0			0
$915$		0			0
$916$		0			0
$917$	55.8154			1.84319
$918$		0			0
$919$		−	7.65624i		−	0.252556i	−0.991995	−	0.126278i	$-0.959697\pi$
$919$		−	7.65624i		−	0.252556i	0.991995	−	0.126278i	$-0.0403031\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.17016i			0.203093i
$924$		0			0
$925$		−	0.619475i		−	0.0203682i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	2.02895i		−	0.0665676i	−0.999446	−	0.0332838i	$-0.989403\pi$
$929$		−	2.02895i		−	0.0665676i	0.999446	−	0.0332838i	$-0.0105965\pi$
$930$		0			0
$931$			8.74241i			0.286521i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.9756			0.424349
$936$		0			0
$937$			11.6881i			0.381835i	0.981606	+	0.190918i	$0.0611463\pi$
$937$			11.6881i			0.381835i	−0.981606	+	0.190918i	$0.938854\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	26.1472			0.852376			0.426188	−	0.904635i	$-0.359856\pi$
$941$	26.1472			0.852376			0.426188	+	0.904635i	$0.359856\pi$
$942$		0			0
$943$	19.5718	−	16.6304i	0.637344	−	0.541562i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	41.4956i		−	1.34842i	−0.738538	−	0.674212i	$-0.764483\pi$
$947$		−	41.4956i		−	1.34842i	0.738538	−	0.674212i	$-0.235517\pi$
$948$		0			0
$949$	−0.486960			−0.0158074
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.80342			−0.317564			−0.158782	−	0.987314i	$-0.550757\pi$
$953$	−9.80342			−0.317564			−0.158782	+	0.987314i	$0.550757\pi$
$954$		0			0
$955$	18.6185			0.602481
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.54642i		−	0.0499363i
$960$		0			0
$961$	6.19174			0.199733
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.08797			0.195979
$966$		0			0
$967$	−40.4501			−1.30079			−0.650393	−	0.759597i	$-0.725396\pi$
$967$	−40.4501			−1.30079			−0.650393	+	0.759597i	$0.725396\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.391596			−0.0125669			−0.00628346	−	0.999980i	$-0.502000\pi$
$971$	−0.391596			−0.0125669			−0.00628346	+	0.999980i	$0.502000\pi$
$972$		0			0
$973$			24.0676i			0.771573i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.00774			0.0322403			0.0161202	−	0.999870i	$-0.494869\pi$
$977$	1.00774			0.0322403			0.0161202	+	0.999870i	$0.494869\pi$
$978$		0			0
$979$	10.7731			0.344311
$980$		0			0
$981$		0			0
$982$		0			0
$983$	52.5336			1.67556			0.837781	−	0.546007i	$-0.183853\pi$
$983$	52.5336			1.67556			0.837781	+	0.546007i	$0.183853\pi$
$984$		0			0
$985$		−	14.5354i		−	0.463135i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.7182	+	11.6566i	−0.436215	+	0.370659i
$990$		0			0
$991$	54.0797			1.71790			0.858949	−	0.512061i	$-0.171118\pi$
$991$	54.0797			1.71790			0.858949	+	0.512061i	$0.171118\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	23.0638i		−	0.731172i
$996$		0			0
$997$	−60.6485			−1.92076			−0.960379	−	0.278699i	$-0.910097\pi$
$997$	−60.6485			−1.92076			−0.960379	+	0.278699i	$0.910097\pi$
$998$		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.1241.39	yes	48
3.2	odd	2		8280.2.p.b.1241.39	yes	48
23.22	odd	2		8280.2.p.b.1241.10	yes	48
69.68	even	2	inner	8280.2.p.a.1241.10	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.10	✓	48	69.68	even	2	inner
8280.2.p.a.1241.39	yes	48	1.1	even	1	trivial
8280.2.p.b.1241.10	yes	48	23.22	odd	2
8280.2.p.b.1241.39	yes	48	3.2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.84853i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.84853i q^{7} -2.95839 q^{11} -0.451593 q^{13} +4.38605 q^{17} -7.84690i q^{19} +(3.10541 + 3.65464i) q^{23} +1.00000 q^{25} +3.01776i q^{29} -6.09850 q^{31} -2.84853i q^{35} -0.619475i q^{37} -5.35531i q^{41} +3.75365i q^{43} +5.23505i q^{47} -1.11412 q^{49} +14.0102 q^{53} +2.95839 q^{55} -11.8371i q^{59} +9.81751i q^{61} +0.451593 q^{65} +12.3590i q^{67} -13.6631i q^{71} +1.07832 q^{73} -8.42705i q^{77} +8.71942i q^{79} +3.20932 q^{83} -4.38605 q^{85} -3.64156 q^{89} -1.28638i q^{91} +7.84690i q^{95} +8.46148i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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