LMFDB - Embedded newform 2240.2.b.g.1121.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.10
Root			$0.500000 + 0.631151i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.g.1121.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.11677i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.48073 q^{9} +O(q^{10})$	$q+2.11677i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.48073 q^{9} -2.75846i q^{11} +3.19661i q^{13} +2.11677 q^{15} +0.352975 q^{17} -8.13580i q^{19} -2.11677i q^{21} -7.61119 q^{23} -1.00000 q^{25} +3.21595i q^{27} +2.84395i q^{29} -8.60555 q^{31} +5.83904 q^{33} +1.00000i q^{35} +8.38064i q^{37} -6.76650 q^{39} -6.89457 q^{41} -10.8504i q^{43} +1.48073i q^{45} -4.41384 q^{47} +1.00000 q^{49} +0.747168i q^{51} -10.4143i q^{53} -2.75846 q^{55} +17.2216 q^{57} -13.3743i q^{59} -4.55620i q^{61} +1.48073 q^{63} +3.19661 q^{65} +4.63807i q^{67} -16.1112i q^{69} +1.00299 q^{71} -12.0909 q^{73} -2.11677i q^{75} +2.75846i q^{77} +4.60775 q^{79} -11.2496 q^{81} +12.4945i q^{83} -0.352975i q^{85} -6.01999 q^{87} -5.61119 q^{89} -3.19661i q^{91} -18.2160i q^{93} -8.13580 q^{95} +10.5752 q^{97} +4.08455i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} + 16 q^{17} - 8 q^{23} - 12 q^{25} + 8 q^{31} + 72 q^{33} + 32 q^{39} + 32 q^{47} + 12 q^{49} - 8 q^{55} + 8 q^{57} + 20 q^{63} + 8 q^{65} - 48 q^{71} + 8 q^{73} + 16 q^{79} + 92 q^{81} + 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 + 16 * q^17 - 8 * q^23 - 12 * q^25 + 8 * q^31 + 72 * q^33 + 32 * q^39 + 32 * q^47 + 12 * q^49 - 8 * q^55 + 8 * q^57 + 20 * q^63 + 8 * q^65 - 48 * q^71 + 8 * q^73 + 16 * q^79 + 92 * q^81 + 32 * q^87 + 16 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.11677i	1.22212i	0.791584	+	0.611060i	$0.209257\pi$
$3$	2.11677i	1.22212i	−0.791584	+	0.611060i	$0.790743\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.48073	−0.493577
$10$	0	0
$11$	− 2.75846i	− 0.831708i	−0.909431	−	0.415854i	$-0.863483\pi$
$11$	− 2.75846i	− 0.831708i	0.909431	−	0.415854i	$-0.136517\pi$
$12$	0	0
$13$	3.19661i	0.886580i	0.896378	+	0.443290i	$0.146189\pi$
$13$	3.19661i	0.886580i	−0.896378	+	0.443290i	$0.853811\pi$
$14$	0	0
$15$	2.11677	0.546549
$16$	0	0
$17$	0.352975	0.0856090	0.0428045	−	0.999083i	$-0.486371\pi$
$17$	0.352975	0.0856090	0.0428045	+	0.999083i	$0.486371\pi$
$18$	0	0
$19$	− 8.13580i	− 1.86648i	−0.359254	−	0.933240i	$-0.616969\pi$
$19$	− 8.13580i	− 1.86648i	0.359254	−	0.933240i	$-0.383031\pi$
$20$	0	0
$21$	− 2.11677i	− 0.461918i
$22$	0	0
$23$	−7.61119	−1.58704	−0.793522	−	0.608542i	$-0.791755\pi$
$23$	−7.61119	−1.58704	−0.793522	+	0.608542i	$0.791755\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	3.21595i	0.618909i
$28$	0	0
$29$	2.84395i	0.528108i	0.964508	+	0.264054i	$0.0850597\pi$
$29$	2.84395i	0.528108i	−0.964508	+	0.264054i	$0.914940\pi$
$30$	0	0
$31$	−8.60555	−1.54560	−0.772801	−	0.634649i	$-0.781145\pi$
$31$	−8.60555	−1.54560	−0.772801	+	0.634649i	$0.781145\pi$
$32$	0	0
$33$	5.83904	1.01645
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	8.38064i	1.37777i	0.724871	+	0.688884i	$0.241899\pi$
$37$	8.38064i	1.37777i	−0.724871	+	0.688884i	$0.758101\pi$
$38$	0	0
$39$	−6.76650	−1.08351
$40$	0	0
$41$	−6.89457	−1.07675	−0.538376	−	0.842705i	$-0.680962\pi$
$41$	−6.89457	−1.07675	−0.538376	+	0.842705i	$0.680962\pi$
$42$	0	0
$43$	− 10.8504i	− 1.65467i	−0.561710	−	0.827334i	$-0.689856\pi$
$43$	− 10.8504i	− 1.65467i	0.561710	−	0.827334i	$-0.310144\pi$
$44$	0	0
$45$	1.48073i	0.220735i
$46$	0	0
$47$	−4.41384	−0.643825	−0.321912	−	0.946769i	$-0.604326\pi$
$47$	−4.41384	−0.643825	−0.321912	+	0.946769i	$0.604326\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.747168i	0.104624i
$52$	0	0
$53$	− 10.4143i	− 1.43051i	−0.698863	−	0.715255i	$-0.746311\pi$
$53$	− 10.4143i	− 1.43051i	0.698863	−	0.715255i	$-0.253689\pi$
$54$	0	0
$55$	−2.75846	−0.371951
$56$	0	0
$57$	17.2216	2.28106
$58$	0	0
$59$	− 13.3743i	− 1.74118i	−0.492010	−	0.870590i	$-0.663738\pi$
$59$	− 13.3743i	− 1.74118i	0.492010	−	0.870590i	$-0.336262\pi$
$60$	0	0
$61$	− 4.55620i	− 0.583362i	−0.956516	−	0.291681i	$-0.905785\pi$
$61$	− 4.55620i	− 0.583362i	0.956516	−	0.291681i	$-0.0942146\pi$
$62$	0	0
$63$	1.48073	0.186555
$64$	0	0
$65$	3.19661	0.396491
$66$	0	0
$67$	4.63807i	0.566630i	0.959027	+	0.283315i	$0.0914342\pi$
$67$	4.63807i	0.566630i	−0.959027	+	0.283315i	$0.908566\pi$
$68$	0	0
$69$	− 16.1112i	− 1.93956i
$70$	0	0
$71$	1.00299	0.119034	0.0595168	−	0.998227i	$-0.481044\pi$
$71$	1.00299	0.119034	0.0595168	+	0.998227i	$0.481044\pi$
$72$	0	0
$73$	−12.0909	−1.41513	−0.707565	−	0.706649i	$-0.750206\pi$
$73$	−12.0909	−1.41513	−0.707565	+	0.706649i	$0.750206\pi$
$74$	0	0
$75$	− 2.11677i	− 0.244424i
$76$	0	0
$77$	2.75846i	0.314356i
$78$	0	0
$79$	4.60775	0.518412	0.259206	−	0.965822i	$-0.416539\pi$
$79$	4.60775	0.518412	0.259206	+	0.965822i	$0.416539\pi$
$80$	0	0
$81$	−11.2496	−1.24996
$82$	0	0
$83$	12.4945i	1.37145i	0.727862	+	0.685723i	$0.240514\pi$
$83$	12.4945i	1.37145i	−0.727862	+	0.685723i	$0.759486\pi$
$84$	0	0
$85$	− 0.352975i	− 0.0382855i
$86$	0	0
$87$	−6.01999	−0.645411
$88$	0	0
$89$	−5.61119	−0.594785	−0.297393	−	0.954755i	$-0.596117\pi$
$89$	−5.61119	−0.594785	−0.297393	+	0.954755i	$0.596117\pi$
$90$	0	0
$91$	− 3.19661i	− 0.335096i
$92$	0	0
$93$	− 18.2160i	− 1.88891i
$94$	0	0
$95$	−8.13580	−0.834715
$96$	0	0
$97$	10.5752	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$97$	10.5752	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$98$	0	0
$99$	4.08455i	0.410512i
$100$	0	0
$101$	− 1.05057i	− 0.104536i	−0.998633	−	0.0522680i	$-0.983355\pi$
$101$	− 1.05057i	− 0.104536i	0.998633	−	0.0522680i	$-0.0166450\pi$
$102$	0	0
$103$	4.41384	0.434909	0.217454	−	0.976071i	$-0.430225\pi$
$103$	4.41384	0.434909	0.217454	+	0.976071i	$0.430225\pi$
$104$	0	0
$105$	−2.11677	−0.206576
$106$	0	0
$107$	− 10.9502i	− 1.05859i	−0.848437	−	0.529296i	$-0.822456\pi$
$107$	− 10.9502i	− 1.05859i	0.848437	−	0.529296i	$-0.177544\pi$
$108$	0	0
$109$	− 8.01699i	− 0.767889i	−0.923356	−	0.383944i	$-0.874565\pi$
$109$	− 8.01699i	− 0.767889i	0.923356	−	0.383944i	$-0.125435\pi$
$110$	0	0
$111$	−17.7399	−1.68380
$112$	0	0
$113$	3.76571	0.354248	0.177124	−	0.984189i	$-0.443321\pi$
$113$	3.76571	0.354248	0.177124	+	0.984189i	$0.443321\pi$
$114$	0	0
$115$	7.61119i	0.709748i
$116$	0	0
$117$	− 4.73333i	− 0.437596i
$118$	0	0
$119$	−0.352975	−0.0323572
$120$	0	0
$121$	3.39088	0.308262
$122$	0	0
$123$	− 14.5943i	− 1.31592i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−5.19501	−0.460983	−0.230491	−	0.973074i	$-0.574033\pi$
$127$	−5.19501	−0.460983	−0.230491	+	0.973074i	$0.574033\pi$
$128$	0	0
$129$	22.9678	2.02220
$130$	0	0
$131$	9.97612i	0.871618i	0.900039	+	0.435809i	$0.143538\pi$
$131$	9.97612i	0.871618i	−0.900039	+	0.435809i	$0.856462\pi$
$132$	0	0
$133$	8.13580i	0.705463i
$134$	0	0
$135$	3.21595	0.276785
$136$	0	0
$137$	−4.55620	−0.389263	−0.194631	−	0.980876i	$-0.562351\pi$
$137$	−4.55620	−0.389263	−0.194631	+	0.980876i	$0.562351\pi$
$138$	0	0
$139$	13.1407i	1.11458i	0.830318	+	0.557290i	$0.188159\pi$
$139$	13.1407i	1.11458i	−0.830318	+	0.557290i	$0.811841\pi$
$140$	0	0
$141$	− 9.34310i	− 0.786831i
$142$	0	0
$143$	8.81773	0.737376
$144$	0	0
$145$	2.84395	0.236177
$146$	0	0
$147$	2.11677i	0.174589i
$148$	0	0
$149$	10.2829i	0.842407i	0.906966	+	0.421204i	$0.138392\pi$
$149$	10.2829i	0.842407i	−0.906966	+	0.421204i	$0.861608\pi$
$150$	0	0
$151$	20.8793	1.69914	0.849568	−	0.527478i	$-0.176862\pi$
$151$	20.8793	1.69914	0.849568	+	0.527478i	$0.176862\pi$
$152$	0	0
$153$	−0.522661	−0.0422547
$154$	0	0
$155$	8.60555i	0.691214i
$156$	0	0
$157$	− 1.91163i	− 0.152565i	−0.997086	−	0.0762825i	$-0.975695\pi$
$157$	− 1.91163i	− 0.152565i	0.997086	−	0.0762825i	$-0.0243051\pi$
$158$	0	0
$159$	22.0447	1.74826
$160$	0	0
$161$	7.61119	0.599846
$162$	0	0
$163$	− 11.5443i	− 0.904220i	−0.891962	−	0.452110i	$-0.850672\pi$
$163$	− 11.5443i	− 0.904220i	0.891962	−	0.452110i	$-0.149328\pi$
$164$	0	0
$165$	− 5.83904i	− 0.454569i
$166$	0	0
$167$	−19.2537	−1.48989	−0.744947	−	0.667123i	$-0.767525\pi$
$167$	−19.2537	−1.48989	−0.744947	+	0.667123i	$0.767525\pi$
$168$	0	0
$169$	2.78168	0.213975
$170$	0	0
$171$	12.0469i	0.921252i
$172$	0	0
$173$	− 9.74226i	− 0.740691i	−0.928894	−	0.370345i	$-0.879239\pi$
$173$	− 9.74226i	− 0.740691i	0.928894	−	0.370345i	$-0.120761\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	28.3103	2.12793
$178$	0	0
$179$	− 15.8677i	− 1.18601i	−0.805200	−	0.593004i	$-0.797942\pi$
$179$	− 15.8677i	− 1.18601i	0.805200	−	0.593004i	$-0.202058\pi$
$180$	0	0
$181$	− 24.9835i	− 1.85701i	−0.371318	−	0.928506i	$-0.621094\pi$
$181$	− 24.9835i	− 1.85701i	0.371318	−	0.928506i	$-0.378906\pi$
$182$	0	0
$183$	9.64446	0.712939
$184$	0	0
$185$	8.38064	0.616157
$186$	0	0
$187$	− 0.973669i	− 0.0712017i
$188$	0	0
$189$	− 3.21595i	− 0.233926i
$190$	0	0
$191$	−15.8579	−1.14743	−0.573717	−	0.819053i	$-0.694499\pi$
$191$	−15.8579	−1.14743	−0.573717	+	0.819053i	$0.694499\pi$
$192$	0	0
$193$	−1.32279	−0.0952166	−0.0476083	−	0.998866i	$-0.515160\pi$
$193$	−1.32279	−0.0952166	−0.0476083	+	0.998866i	$0.515160\pi$
$194$	0	0
$195$	6.76650i	0.484559i
$196$	0	0
$197$	− 7.63076i	− 0.543669i	−0.962344	−	0.271834i	$-0.912370\pi$
$197$	− 7.63076i	− 0.543669i	0.962344	−	0.271834i	$-0.0876303\pi$
$198$	0	0
$199$	−4.72643	−0.335048	−0.167524	−	0.985868i	$-0.553577\pi$
$199$	−4.72643	−0.335048	−0.167524	+	0.985868i	$0.553577\pi$
$200$	0	0
$201$	−9.81774	−0.692490
$202$	0	0
$203$	− 2.84395i	− 0.199606i
$204$	0	0
$205$	6.89457i	0.481538i
$206$	0	0
$207$	11.2701	0.783329
$208$	0	0
$209$	−22.4423	−1.55237
$210$	0	0
$211$	− 5.17365i	− 0.356169i	−0.984015	−	0.178084i	$-0.943010\pi$
$211$	− 5.17365i	− 0.356169i	0.984015	−	0.178084i	$-0.0569900\pi$
$212$	0	0
$213$	2.12311i	0.145473i
$214$	0	0
$215$	−10.8504	−0.739990
$216$	0	0
$217$	8.60555	0.584183
$218$	0	0
$219$	− 25.5936i	− 1.72946i
$220$	0	0
$221$	1.12832i	0.0758993i
$222$	0	0
$223$	−2.91739	−0.195363	−0.0976816	−	0.995218i	$-0.531143\pi$
$223$	−2.91739	−0.195363	−0.0976816	+	0.995218i	$0.531143\pi$
$224$	0	0
$225$	1.48073	0.0987155
$226$	0	0
$227$	5.79337i	0.384520i	0.981344	+	0.192260i	$0.0615817\pi$
$227$	5.79337i	0.384520i	−0.981344	+	0.192260i	$0.938418\pi$
$228$	0	0
$229$	− 1.22092i	− 0.0806806i	−0.999186	−	0.0403403i	$-0.987156\pi$
$229$	− 1.22092i	− 0.0806806i	0.999186	−	0.0403403i	$-0.0128442\pi$
$230$	0	0
$231$	−5.83904	−0.384181
$232$	0	0
$233$	9.24435	0.605618	0.302809	−	0.953051i	$-0.402076\pi$
$233$	9.24435	0.605618	0.302809	+	0.953051i	$0.402076\pi$
$234$	0	0
$235$	4.41384i	0.287927i
$236$	0	0
$237$	9.75356i	0.633562i
$238$	0	0
$239$	−5.80744	−0.375652	−0.187826	−	0.982202i	$-0.560144\pi$
$239$	−5.80744	−0.375652	−0.187826	+	0.982202i	$0.560144\pi$
$240$	0	0
$241$	−15.0398	−0.968796	−0.484398	−	0.874848i	$-0.660961\pi$
$241$	−15.0398	−0.968796	−0.484398	+	0.874848i	$0.660961\pi$
$242$	0	0
$243$	− 14.1651i	− 0.908691i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	26.0070	1.65478
$248$	0	0
$249$	−26.4480	−1.67607
$250$	0	0
$251$	27.3811i	1.72828i	0.503249	+	0.864141i	$0.332138\pi$
$251$	27.3811i	1.72828i	−0.503249	+	0.864141i	$0.667862\pi$
$252$	0	0
$253$	20.9952i	1.31996i
$254$	0	0
$255$	0.747168	0.0467895
$256$	0	0
$257$	19.5971	1.22244	0.611218	−	0.791463i	$-0.290680\pi$
$257$	19.5971	1.22244	0.611218	+	0.791463i	$0.290680\pi$
$258$	0	0
$259$	− 8.38064i	− 0.520748i
$260$	0	0
$261$	− 4.21113i	− 0.260662i
$262$	0	0
$263$	−20.9722	−1.29320	−0.646602	−	0.762828i	$-0.723810\pi$
$263$	−20.9722	−1.29320	−0.646602	+	0.762828i	$0.723810\pi$
$264$	0	0
$265$	−10.4143	−0.639744
$266$	0	0
$267$	− 11.8776i	− 0.726899i
$268$	0	0
$269$	− 10.9729i	− 0.669027i	−0.942391	−	0.334514i	$-0.891428\pi$
$269$	− 10.9729i	− 0.669027i	0.942391	−	0.334514i	$-0.108572\pi$
$270$	0	0
$271$	5.97878	0.363185	0.181592	−	0.983374i	$-0.441875\pi$
$271$	5.97878	0.363185	0.181592	+	0.983374i	$0.441875\pi$
$272$	0	0
$273$	6.76650	0.409527
$274$	0	0
$275$	2.75846i	0.166342i
$276$	0	0
$277$	24.9094i	1.49666i	0.663327	+	0.748329i	$0.269144\pi$
$277$	24.9094i	1.49666i	−0.663327	+	0.748329i	$0.730856\pi$
$278$	0	0
$279$	12.7425	0.762874
$280$	0	0
$281$	−3.87004	−0.230867	−0.115434	−	0.993315i	$-0.536826\pi$
$281$	−3.87004	−0.230867	−0.115434	+	0.993315i	$0.536826\pi$
$282$	0	0
$283$	28.6332i	1.70207i	0.525111	+	0.851034i	$0.324024\pi$
$283$	28.6332i	1.70207i	−0.525111	+	0.851034i	$0.675976\pi$
$284$	0	0
$285$	− 17.2216i	− 1.02012i
$286$	0	0
$287$	6.89457	0.406974
$288$	0	0
$289$	−16.8754	−0.992671
$290$	0	0
$291$	22.3854i	1.31225i
$292$	0	0
$293$	5.67968i	0.331811i	0.986142	+	0.165905i	$0.0530546\pi$
$293$	5.67968i	0.331811i	−0.986142	+	0.165905i	$0.946945\pi$
$294$	0	0
$295$	−13.3743	−0.778679
$296$	0	0
$297$	8.87107	0.514752
$298$	0	0
$299$	− 24.3300i	− 1.40704i
$300$	0	0
$301$	10.8504i	0.625406i
$302$	0	0
$303$	2.22383	0.127755
$304$	0	0
$305$	−4.55620	−0.260888
$306$	0	0
$307$	− 2.05614i	− 0.117350i	−0.998277	−	0.0586750i	$-0.981312\pi$
$307$	− 2.05614i	− 0.117350i	0.998277	−	0.0586750i	$-0.0186876\pi$
$308$	0	0
$309$	9.34310i	0.531511i
$310$	0	0
$311$	−12.8643	−0.729465	−0.364733	−	0.931112i	$-0.618840\pi$
$311$	−12.8643	−0.729465	−0.364733	+	0.931112i	$0.618840\pi$
$312$	0	0
$313$	−14.5372	−0.821690	−0.410845	−	0.911705i	$-0.634766\pi$
$313$	−14.5372	−0.821690	−0.410845	+	0.911705i	$0.634766\pi$
$314$	0	0
$315$	− 1.48073i	− 0.0834298i
$316$	0	0
$317$	− 1.09285i	− 0.0613803i	−0.999529	−	0.0306902i	$-0.990229\pi$
$317$	− 1.09285i	− 0.0613803i	0.999529	−	0.0306902i	$-0.00977052\pi$
$318$	0	0
$319$	7.84492	0.439231
$320$	0	0
$321$	23.1790	1.29373
$322$	0	0
$323$	− 2.87173i	− 0.159788i
$324$	0	0
$325$	− 3.19661i	− 0.177316i
$326$	0	0
$327$	16.9702	0.938452
$328$	0	0
$329$	4.41384	0.243343
$330$	0	0
$331$	− 4.29270i	− 0.235948i	−0.993017	−	0.117974i	$-0.962360\pi$
$331$	− 4.29270i	− 0.235948i	0.993017	−	0.117974i	$-0.0376400\pi$
$332$	0	0
$333$	− 12.4095i	− 0.680036i
$334$	0	0
$335$	4.63807	0.253405
$336$	0	0
$337$	22.5730	1.22963	0.614816	−	0.788671i	$-0.289230\pi$
$337$	22.5730	1.22963	0.614816	+	0.788671i	$0.289230\pi$
$338$	0	0
$339$	7.97115i	0.432934i
$340$	0	0
$341$	23.7381i	1.28549i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−16.1112	−0.867397
$346$	0	0
$347$	− 20.0687i	− 1.07735i	−0.842515	−	0.538673i	$-0.818926\pi$
$347$	− 20.0687i	− 1.07735i	0.842515	−	0.538673i	$-0.181074\pi$
$348$	0	0
$349$	35.1098i	1.87939i	0.342019	+	0.939693i	$0.388889\pi$
$349$	35.1098i	1.87939i	−0.342019	+	0.939693i	$0.611111\pi$
$350$	0	0
$351$	−10.2801	−0.548713
$352$	0	0
$353$	14.4023	0.766558	0.383279	−	0.923633i	$-0.374795\pi$
$353$	14.4023	0.766558	0.383279	+	0.923633i	$0.374795\pi$
$354$	0	0
$355$	− 1.00299i	− 0.0532334i
$356$	0	0
$357$	− 0.747168i	− 0.0395443i
$358$	0	0
$359$	33.1017	1.74704	0.873520	−	0.486788i	$-0.161832\pi$
$359$	33.1017	1.74704	0.873520	+	0.486788i	$0.161832\pi$
$360$	0	0
$361$	−47.1912	−2.48375
$362$	0	0
$363$	7.17773i	0.376733i
$364$	0	0
$365$	12.0909i	0.632865i
$366$	0	0
$367$	−9.41775	−0.491603	−0.245801	−	0.969320i	$-0.579051\pi$
$367$	−9.41775	−0.491603	−0.245801	+	0.969320i	$0.579051\pi$
$368$	0	0
$369$	10.2090	0.531460
$370$	0	0
$371$	10.4143i	0.540682i
$372$	0	0
$373$	17.0885i	0.884811i	0.896815	+	0.442405i	$0.145875\pi$
$373$	17.0885i	0.884811i	−0.896815	+	0.442405i	$0.854125\pi$
$374$	0	0
$375$	−2.11677	−0.109310
$376$	0	0
$377$	−9.09099	−0.468210
$378$	0	0
$379$	17.2743i	0.887321i	0.896195	+	0.443660i	$0.146320\pi$
$379$	17.2743i	0.887321i	−0.896195	+	0.443660i	$0.853680\pi$
$380$	0	0
$381$	− 10.9967i	− 0.563376i
$382$	0	0
$383$	16.8496	0.860977	0.430488	−	0.902596i	$-0.358341\pi$
$383$	16.8496	0.860977	0.430488	+	0.902596i	$0.358341\pi$
$384$	0	0
$385$	2.75846	0.140584
$386$	0	0
$387$	16.0665i	0.816707i
$388$	0	0
$389$	− 34.9832i	− 1.77372i	−0.462037	−	0.886860i	$-0.652881\pi$
$389$	− 34.9832i	− 1.77372i	0.462037	−	0.886860i	$-0.347119\pi$
$390$	0	0
$391$	−2.68656	−0.135865
$392$	0	0
$393$	−21.1172	−1.06522
$394$	0	0
$395$	− 4.60775i	− 0.231841i
$396$	0	0
$397$	9.54676i	0.479138i	0.970879	+	0.239569i	$0.0770062\pi$
$397$	9.54676i	0.479138i	−0.970879	+	0.239569i	$0.922994\pi$
$398$	0	0
$399$	−17.2216	−0.862161
$400$	0	0
$401$	−28.1150	−1.40400	−0.701998	−	0.712179i	$-0.747708\pi$
$401$	−28.1150	−1.40400	−0.701998	+	0.712179i	$0.747708\pi$
$402$	0	0
$403$	− 27.5086i	− 1.37030i
$404$	0	0
$405$	11.2496i	0.558999i
$406$	0	0
$407$	23.1177	1.14590
$408$	0	0
$409$	−2.86022	−0.141429	−0.0707143	−	0.997497i	$-0.522528\pi$
$409$	−2.86022	−0.141429	−0.0707143	+	0.997497i	$0.522528\pi$
$410$	0	0
$411$	− 9.64446i	− 0.475726i
$412$	0	0
$413$	13.3743i	0.658104i
$414$	0	0
$415$	12.4945	0.613329
$416$	0	0
$417$	−27.8159	−1.36215
$418$	0	0
$419$	12.4149i	0.606507i	0.952910	+	0.303254i	$0.0980729\pi$
$419$	12.4149i	0.606507i	−0.952910	+	0.303254i	$0.901927\pi$
$420$	0	0
$421$	11.5455i	0.562693i	0.959606	+	0.281347i	$0.0907810\pi$
$421$	11.5455i	0.562693i	−0.959606	+	0.281347i	$0.909219\pi$
$422$	0	0
$423$	6.53572	0.317777
$424$	0	0
$425$	−0.352975	−0.0171218
$426$	0	0
$427$	4.55620i	0.220490i
$428$	0	0
$429$	18.6652i	0.901162i
$430$	0	0
$431$	−12.1960	−0.587460	−0.293730	−	0.955888i	$-0.594897\pi$
$431$	−12.1960	−0.587460	−0.293730	+	0.955888i	$0.594897\pi$
$432$	0	0
$433$	2.08637	0.100264	0.0501322	−	0.998743i	$-0.484036\pi$
$433$	2.08637	0.100264	0.0501322	+	0.998743i	$0.484036\pi$
$434$	0	0
$435$	6.01999i	0.288637i
$436$	0	0
$437$	61.9231i	2.96219i
$438$	0	0
$439$	−28.3495	−1.35305	−0.676524	−	0.736420i	$-0.736515\pi$
$439$	−28.3495	−1.35305	−0.676524	+	0.736420i	$0.736515\pi$
$440$	0	0
$441$	−1.48073	−0.0705111
$442$	0	0
$443$	− 15.5156i	− 0.737167i	−0.929595	−	0.368583i	$-0.879843\pi$
$443$	− 15.5156i	− 0.737167i	0.929595	−	0.368583i	$-0.120157\pi$
$444$	0	0
$445$	5.61119i	0.265996i
$446$	0	0
$447$	−21.7666	−1.02952
$448$	0	0
$449$	27.7544	1.30981	0.654906	−	0.755711i	$-0.272708\pi$
$449$	27.7544	1.30981	0.654906	+	0.755711i	$0.272708\pi$
$450$	0	0
$451$	19.0184i	0.895543i
$452$	0	0
$453$	44.1968i	2.07655i
$454$	0	0
$455$	−3.19661	−0.149859
$456$	0	0
$457$	−29.9596	−1.40145	−0.700726	−	0.713431i	$-0.747140\pi$
$457$	−29.9596	−1.40145	−0.700726	+	0.713431i	$0.747140\pi$
$458$	0	0
$459$	1.13515i	0.0529842i
$460$	0	0
$461$	− 2.85433i	− 0.132939i	−0.997788	−	0.0664697i	$-0.978826\pi$
$461$	− 2.85433i	− 0.132939i	0.997788	−	0.0664697i	$-0.0211736\pi$
$462$	0	0
$463$	−3.54862	−0.164918	−0.0824591	−	0.996594i	$-0.526277\pi$
$463$	−3.54862	−0.164918	−0.0824591	+	0.996594i	$0.526277\pi$
$464$	0	0
$465$	−18.2160	−0.844747
$466$	0	0
$467$	− 42.4238i	− 1.96314i	−0.191100	−	0.981571i	$-0.561205\pi$
$467$	− 42.4238i	− 1.96314i	0.191100	−	0.981571i	$-0.438795\pi$
$468$	0	0
$469$	− 4.63807i	− 0.214166i
$470$	0	0
$471$	4.04650	0.186453
$472$	0	0
$473$	−29.9304	−1.37620
$474$	0	0
$475$	8.13580i	0.373296i
$476$	0	0
$477$	15.4207i	0.706068i
$478$	0	0
$479$	−28.5040	−1.30238	−0.651192	−	0.758913i	$-0.725731\pi$
$479$	−28.5040	−1.30238	−0.651192	+	0.758913i	$0.725731\pi$
$480$	0	0
$481$	−26.7896	−1.22150
$482$	0	0
$483$	16.1112i	0.733084i
$484$	0	0
$485$	− 10.5752i	− 0.480196i
$486$	0	0
$487$	13.9416	0.631753	0.315877	−	0.948800i	$-0.397701\pi$
$487$	13.9416	0.631753	0.315877	+	0.948800i	$0.397701\pi$
$488$	0	0
$489$	24.4367	1.10506
$490$	0	0
$491$	26.6255i	1.20159i	0.799402	+	0.600797i	$0.205150\pi$
$491$	26.6255i	1.20159i	−0.799402	+	0.600797i	$0.794850\pi$
$492$	0	0
$493$	1.00384i	0.0452108i
$494$	0	0
$495$	4.08455	0.183587
$496$	0	0
$497$	−1.00299	−0.0449905
$498$	0	0
$499$	− 0.169740i	− 0.00759861i	−0.999993	−	0.00379930i	$-0.998791\pi$
$499$	− 0.169740i	− 0.00759861i	0.999993	−	0.00379930i	$-0.00120936\pi$
$500$	0	0
$501$	− 40.7557i	− 1.82083i
$502$	0	0
$503$	−23.7632	−1.05955	−0.529775	−	0.848138i	$-0.677724\pi$
$503$	−23.7632	−1.05955	−0.529775	+	0.848138i	$0.677724\pi$
$504$	0	0
$505$	−1.05057	−0.0467499
$506$	0	0
$507$	5.88818i	0.261503i
$508$	0	0
$509$	8.86968i	0.393141i	0.980490	+	0.196571i	$0.0629805\pi$
$509$	8.86968i	0.393141i	−0.980490	+	0.196571i	$0.937019\pi$
$510$	0	0
$511$	12.0909	0.534869
$512$	0	0
$513$	26.1643	1.15518
$514$	0	0
$515$	− 4.41384i	− 0.194497i
$516$	0	0
$517$	12.1754i	0.535474i
$518$	0	0
$519$	20.6222	0.905213
$520$	0	0
$521$	−12.4381	−0.544925	−0.272463	−	0.962166i	$-0.587838\pi$
$521$	−12.4381	−0.544925	−0.272463	+	0.962166i	$0.587838\pi$
$522$	0	0
$523$	28.8943i	1.26346i	0.775189	+	0.631730i	$0.217655\pi$
$523$	28.8943i	1.26346i	−0.775189	+	0.631730i	$0.782345\pi$
$524$	0	0
$525$	2.11677i	0.0923836i
$526$	0	0
$527$	−3.03754	−0.132317
$528$	0	0
$529$	34.9303	1.51871
$530$	0	0
$531$	19.8037i	0.859407i
$532$	0	0
$533$	− 22.0393i	− 0.954627i
$534$	0	0
$535$	−10.9502	−0.473417
$536$	0	0
$537$	33.5883	1.44944
$538$	0	0
$539$	− 2.75846i	− 0.118815i
$540$	0	0
$541$	− 10.4825i	− 0.450678i	−0.974280	−	0.225339i	$-0.927651\pi$
$541$	− 10.4825i	− 0.450678i	0.974280	−	0.225339i	$-0.0723489\pi$
$542$	0	0
$543$	52.8845	2.26949
$544$	0	0
$545$	−8.01699	−0.343410
$546$	0	0
$547$	3.63268i	0.155322i	0.996980	+	0.0776612i	$0.0247452\pi$
$547$	3.63268i	0.155322i	−0.996980	+	0.0776612i	$0.975255\pi$
$548$	0	0
$549$	6.74652i	0.287934i
$550$	0	0
$551$	23.1378	0.985703
$552$	0	0
$553$	−4.60775	−0.195941
$554$	0	0
$555$	17.7399i	0.753018i
$556$	0	0
$557$	− 10.5852i	− 0.448511i	−0.974530	−	0.224256i	$-0.928005\pi$
$557$	− 10.5852i	− 0.448511i	0.974530	−	0.224256i	$-0.0719950\pi$
$558$	0	0
$559$	34.6845	1.46700
$560$	0	0
$561$	2.06104	0.0870170
$562$	0	0
$563$	− 1.40438i	− 0.0591876i	−0.999562	−	0.0295938i	$-0.990579\pi$
$563$	− 1.40438i	− 0.0591876i	0.999562	−	0.0295938i	$-0.00942138\pi$
$564$	0	0
$565$	− 3.76571i	− 0.158425i
$566$	0	0
$567$	11.2496	0.472440
$568$	0	0
$569$	18.0761	0.757789	0.378895	−	0.925440i	$-0.376304\pi$
$569$	18.0761	0.757789	0.378895	+	0.925440i	$0.376304\pi$
$570$	0	0
$571$	2.78360i	0.116490i	0.998302	+	0.0582450i	$0.0185505\pi$
$571$	2.78360i	0.116490i	−0.998302	+	0.0582450i	$0.981450\pi$
$572$	0	0
$573$	− 33.5675i	− 1.40230i
$574$	0	0
$575$	7.61119	0.317409
$576$	0	0
$577$	30.0337	1.25032	0.625160	−	0.780496i	$-0.285034\pi$
$577$	30.0337	1.25032	0.625160	+	0.780496i	$0.285034\pi$
$578$	0	0
$579$	− 2.80005i	− 0.116366i
$580$	0	0
$581$	− 12.4945i	− 0.518358i
$582$	0	0
$583$	−28.7274	−1.18977
$584$	0	0
$585$	−4.73333	−0.195699
$586$	0	0
$587$	4.30742i	0.177786i	0.996041	+	0.0888932i	$0.0283330\pi$
$587$	4.30742i	0.177786i	−0.996041	+	0.0888932i	$0.971667\pi$
$588$	0	0
$589$	70.0130i	2.88483i
$590$	0	0
$591$	16.1526	0.664429
$592$	0	0
$593$	−20.8581	−0.856540	−0.428270	−	0.903651i	$-0.640877\pi$
$593$	−20.8581	−0.856540	−0.428270	+	0.903651i	$0.640877\pi$
$594$	0	0
$595$	0.352975i	0.0144706i
$596$	0	0
$597$	− 10.0048i	− 0.409469i
$598$	0	0
$599$	34.1829	1.39667	0.698337	−	0.715769i	$-0.253924\pi$
$599$	34.1829	1.39667	0.698337	+	0.715769i	$0.253924\pi$
$600$	0	0
$601$	−38.0319	−1.55135	−0.775676	−	0.631131i	$-0.782591\pi$
$601$	−38.0319	−1.55135	−0.775676	+	0.631131i	$0.782591\pi$
$602$	0	0
$603$	− 6.86773i	− 0.279676i
$604$	0	0
$605$	− 3.39088i	− 0.137859i
$606$	0	0
$607$	24.4185	0.991117	0.495559	−	0.868574i	$-0.334963\pi$
$607$	24.4185	0.991117	0.495559	+	0.868574i	$0.334963\pi$
$608$	0	0
$609$	6.01999	0.243942
$610$	0	0
$611$	− 14.1093i	− 0.570802i
$612$	0	0
$613$	20.7849i	0.839494i	0.907641	+	0.419747i	$0.137881\pi$
$613$	20.7849i	0.839494i	−0.907641	+	0.419747i	$0.862119\pi$
$614$	0	0
$615$	−14.5943	−0.588497
$616$	0	0
$617$	−28.6356	−1.15283	−0.576413	−	0.817159i	$-0.695548\pi$
$617$	−28.6356	−1.15283	−0.576413	+	0.817159i	$0.695548\pi$
$618$	0	0
$619$	44.6416i	1.79430i	0.441730	+	0.897148i	$0.354365\pi$
$619$	44.6416i	1.79430i	−0.441730	+	0.897148i	$0.645635\pi$
$620$	0	0
$621$	− 24.4772i	− 0.982236i
$622$	0	0
$623$	5.61119	0.224808
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 47.5053i	− 1.89718i
$628$	0	0
$629$	2.95816i	0.117949i
$630$	0	0
$631$	5.18140	0.206268	0.103134	−	0.994667i	$-0.467113\pi$
$631$	5.18140	0.206268	0.103134	+	0.994667i	$0.467113\pi$
$632$	0	0
$633$	10.9515	0.435281
$634$	0	0
$635$	5.19501i	0.206158i
$636$	0	0
$637$	3.19661i	0.126654i
$638$	0	0
$639$	−1.48517	−0.0587523
$640$	0	0
$641$	9.88682	0.390506	0.195253	−	0.980753i	$-0.437447\pi$
$641$	9.88682	0.390506	0.195253	+	0.980753i	$0.437447\pi$
$642$	0	0
$643$	4.47663i	0.176541i	0.996097	+	0.0882706i	$0.0281340\pi$
$643$	4.47663i	0.176541i	−0.996097	+	0.0882706i	$0.971866\pi$
$644$	0	0
$645$	− 22.9678i	− 0.904357i
$646$	0	0
$647$	33.6534	1.32305	0.661525	−	0.749923i	$-0.269909\pi$
$647$	33.6534	1.32305	0.661525	+	0.749923i	$0.269909\pi$
$648$	0	0
$649$	−36.8924	−1.44815
$650$	0	0
$651$	18.2160i	0.713941i
$652$	0	0
$653$	− 41.1791i	− 1.61146i	−0.592281	−	0.805731i	$-0.701772\pi$
$653$	− 41.1791i	− 1.61146i	0.592281	−	0.805731i	$-0.298228\pi$
$654$	0	0
$655$	9.97612	0.389799
$656$	0	0
$657$	17.9033	0.698476
$658$	0	0
$659$	− 21.2602i	− 0.828179i	−0.910236	−	0.414090i	$-0.864100\pi$
$659$	− 21.2602i	− 0.828179i	0.910236	−	0.414090i	$-0.135900\pi$
$660$	0	0
$661$	44.2330i	1.72047i	0.509902	+	0.860233i	$0.329682\pi$
$661$	44.2330i	1.72047i	−0.509902	+	0.860233i	$0.670318\pi$
$662$	0	0
$663$	−2.38841	−0.0927580
$664$	0	0
$665$	8.13580	0.315493
$666$	0	0
$667$	− 21.6458i	− 0.838130i
$668$	0	0
$669$	− 6.17546i	− 0.238757i
$670$	0	0
$671$	−12.5681	−0.485187
$672$	0	0
$673$	26.6801	1.02844	0.514222	−	0.857657i	$-0.328081\pi$
$673$	26.6801	1.02844	0.514222	+	0.857657i	$0.328081\pi$
$674$	0	0
$675$	− 3.21595i	− 0.123782i
$676$	0	0
$677$	− 43.8203i	− 1.68415i	−0.539360	−	0.842075i	$-0.681334\pi$
$677$	− 43.8203i	− 1.68415i	0.539360	−	0.842075i	$-0.318666\pi$
$678$	0	0
$679$	−10.5752	−0.405840
$680$	0	0
$681$	−12.2633	−0.469929
$682$	0	0
$683$	31.7419i	1.21457i	0.794485	+	0.607284i	$0.207741\pi$
$683$	31.7419i	1.21457i	−0.794485	+	0.607284i	$0.792259\pi$
$684$	0	0
$685$	4.55620i	0.174084i
$686$	0	0
$687$	2.58441	0.0986014
$688$	0	0
$689$	33.2904	1.26826
$690$	0	0
$691$	8.98976i	0.341986i	0.985272	+	0.170993i	$0.0546976\pi$
$691$	8.98976i	0.341986i	−0.985272	+	0.170993i	$0.945302\pi$
$692$	0	0
$693$	− 4.08455i	− 0.155159i
$694$	0	0
$695$	13.1407	0.498455
$696$	0	0
$697$	−2.43361	−0.0921796
$698$	0	0
$699$	19.5682i	0.740138i
$700$	0	0
$701$	− 34.7670i	− 1.31313i	−0.754269	−	0.656565i	$-0.772009\pi$
$701$	− 34.7670i	− 1.31313i	0.754269	−	0.656565i	$-0.227991\pi$
$702$	0	0
$703$	68.1832	2.57158
$704$	0	0
$705$	−9.34310	−0.351882
$706$	0	0
$707$	1.05057i	0.0395109i
$708$	0	0
$709$	49.5111i	1.85943i	0.368283	+	0.929714i	$0.379946\pi$
$709$	49.5111i	1.85943i	−0.368283	+	0.929714i	$0.620054\pi$
$710$	0	0
$711$	−6.82284	−0.255876
$712$	0	0
$713$	65.4985	2.45294
$714$	0	0
$715$	− 8.81773i	− 0.329765i
$716$	0	0
$717$	− 12.2930i	− 0.459092i
$718$	0	0
$719$	−16.8285	−0.627598	−0.313799	−	0.949489i	$-0.601602\pi$
$719$	−16.8285	−0.627598	−0.313799	+	0.949489i	$0.601602\pi$
$720$	0	0
$721$	−4.41384	−0.164380
$722$	0	0
$723$	− 31.8358i	− 1.18398i
$724$	0	0
$725$	− 2.84395i	− 0.105622i
$726$	0	0
$727$	33.0753	1.22670	0.613348	−	0.789813i	$-0.289822\pi$
$727$	33.0753	1.22670	0.613348	+	0.789813i	$0.289822\pi$
$728$	0	0
$729$	−3.76461	−0.139430
$730$	0	0
$731$	− 3.82992i	− 0.141655i
$732$	0	0
$733$	14.7585i	0.545117i	0.962139	+	0.272558i	$0.0878698\pi$
$733$	14.7585i	0.545117i	−0.962139	+	0.272558i	$0.912130\pi$
$734$	0	0
$735$	2.11677	0.0780784
$736$	0	0
$737$	12.7939	0.471271
$738$	0	0
$739$	− 29.4607i	− 1.08373i	−0.840466	−	0.541865i	$-0.817719\pi$
$739$	− 29.4607i	− 1.08373i	0.840466	−	0.541865i	$-0.182281\pi$
$740$	0	0
$741$	55.0509i	2.02235i
$742$	0	0
$743$	−48.7721	−1.78927	−0.894637	−	0.446795i	$-0.852566\pi$
$743$	−48.7721	−1.78927	−0.894637	+	0.446795i	$0.852566\pi$
$744$	0	0
$745$	10.2829	0.376736
$746$	0	0
$747$	− 18.5010i	− 0.676915i
$748$	0	0
$749$	10.9502i	0.400110i
$750$	0	0
$751$	−19.2115	−0.701037	−0.350518	−	0.936556i	$-0.613995\pi$
$751$	−19.2115	−0.701037	−0.350518	+	0.936556i	$0.613995\pi$
$752$	0	0
$753$	−57.9597	−2.11217
$754$	0	0
$755$	− 20.8793i	− 0.759877i
$756$	0	0
$757$	− 21.2029i	− 0.770634i	−0.922784	−	0.385317i	$-0.874092\pi$
$757$	− 21.2029i	− 0.770634i	0.922784	−	0.385317i	$-0.125908\pi$
$758$	0	0
$759$	−44.4421	−1.61315
$760$	0	0
$761$	−39.2674	−1.42344	−0.711721	−	0.702462i	$-0.752084\pi$
$761$	−39.2674	−1.42344	−0.711721	+	0.702462i	$0.752084\pi$
$762$	0	0
$763$	8.01699i	0.290235i
$764$	0	0
$765$	0.522661i	0.0188969i
$766$	0	0
$767$	42.7523	1.54370
$768$	0	0
$769$	49.2543	1.77615	0.888077	−	0.459694i	$-0.152041\pi$
$769$	49.2543	1.77615	0.888077	+	0.459694i	$0.152041\pi$
$770$	0	0
$771$	41.4827i	1.49396i
$772$	0	0
$773$	44.9758i	1.61767i	0.588038	+	0.808833i	$0.299900\pi$
$773$	44.9758i	1.61767i	−0.588038	+	0.808833i	$0.700100\pi$
$774$	0	0
$775$	8.60555	0.309120
$776$	0	0
$777$	17.7399	0.636416
$778$	0	0
$779$	56.0928i	2.00973i
$780$	0	0
$781$	− 2.76672i	− 0.0990012i
$782$	0	0
$783$	−9.14598	−0.326851
$784$	0	0
$785$	−1.91163	−0.0682292
$786$	0	0
$787$	− 35.4295i	− 1.26292i	−0.775407	−	0.631462i	$-0.782455\pi$
$787$	− 35.4295i	− 1.26292i	0.775407	−	0.631462i	$-0.217545\pi$
$788$	0	0
$789$	− 44.3935i	− 1.58045i
$790$	0	0
$791$	−3.76571	−0.133893
$792$	0	0
$793$	14.5644	0.517198
$794$	0	0
$795$	− 22.0447i	− 0.781844i
$796$	0	0
$797$	26.6378i	0.943560i	0.881716	+	0.471780i	$0.156388\pi$
$797$	26.6378i	0.943560i	−0.881716	+	0.471780i	$0.843612\pi$
$798$	0	0
$799$	−1.55798	−0.0551172
$800$	0	0
$801$	8.30868	0.293573
$802$	0	0
$803$	33.3522i	1.17697i
$804$	0	0
$805$	− 7.61119i	− 0.268259i
$806$	0	0
$807$	23.2271	0.817632
$808$	0	0
$809$	26.3909	0.927854	0.463927	−	0.885873i	$-0.346440\pi$
$809$	26.3909	0.927854	0.463927	+	0.885873i	$0.346440\pi$
$810$	0	0
$811$	35.7496i	1.25534i	0.778481	+	0.627668i	$0.215991\pi$
$811$	35.7496i	1.25534i	−0.778481	+	0.627668i	$0.784009\pi$
$812$	0	0
$813$	12.6557i	0.443855i
$814$	0	0
$815$	−11.5443	−0.404379
$816$	0	0
$817$	−88.2766	−3.08841
$818$	0	0
$819$	4.73333i	0.165396i
$820$	0	0
$821$	− 40.0310i	− 1.39709i	−0.715566	−	0.698545i	$-0.753831\pi$
$821$	− 40.0310i	− 1.39709i	0.715566	−	0.698545i	$-0.246169\pi$
$822$	0	0
$823$	44.3049	1.54437	0.772185	−	0.635397i	$-0.219164\pi$
$823$	44.3049	1.54437	0.772185	+	0.635397i	$0.219164\pi$
$824$	0	0
$825$	−5.83904	−0.203289
$826$	0	0
$827$	21.4886i	0.747231i	0.927584	+	0.373615i	$0.121882\pi$
$827$	21.4886i	0.747231i	−0.927584	+	0.373615i	$0.878118\pi$
$828$	0	0
$829$	− 33.3650i	− 1.15881i	−0.815038	−	0.579407i	$-0.803284\pi$
$829$	− 33.3650i	− 1.15881i	0.815038	−	0.579407i	$-0.196716\pi$
$830$	0	0
$831$	−52.7275	−1.82910
$832$	0	0
$833$	0.352975	0.0122299
$834$	0	0
$835$	19.2537i	0.666301i
$836$	0	0
$837$	− 27.6750i	− 0.956587i
$838$	0	0
$839$	47.7176	1.64739	0.823697	−	0.567030i	$-0.191908\pi$
$839$	47.7176	1.64739	0.823697	+	0.567030i	$0.191908\pi$
$840$	0	0
$841$	20.9120	0.721102
$842$	0	0
$843$	− 8.19201i	− 0.282148i
$844$	0	0
$845$	− 2.78168i	− 0.0956926i
$846$	0	0
$847$	−3.39088	−0.116512
$848$	0	0
$849$	−60.6100	−2.08013
$850$	0	0
$851$	− 63.7867i	− 2.18658i
$852$	0	0
$853$	− 19.5053i	− 0.667848i	−0.942600	−	0.333924i	$-0.891627\pi$
$853$	− 19.5053i	− 0.667848i	0.942600	−	0.333924i	$-0.108373\pi$
$854$	0	0
$855$	12.0469	0.411997
$856$	0	0
$857$	−11.5286	−0.393810	−0.196905	−	0.980423i	$-0.563089\pi$
$857$	−11.5286	−0.393810	−0.196905	+	0.980423i	$0.563089\pi$
$858$	0	0
$859$	− 6.58708i	− 0.224748i	−0.993666	−	0.112374i	$-0.964154\pi$
$859$	− 6.58708i	− 0.224748i	0.993666	−	0.112374i	$-0.0358455\pi$
$860$	0	0
$861$	14.5943i	0.497371i
$862$	0	0
$863$	−43.5380	−1.48205	−0.741026	−	0.671476i	$-0.765661\pi$
$863$	−43.5380	−1.48205	−0.741026	+	0.671476i	$0.765661\pi$
$864$	0	0
$865$	−9.74226	−0.331247
$866$	0	0
$867$	− 35.7214i	− 1.21316i
$868$	0	0
$869$	− 12.7103i	− 0.431167i
$870$	0	0
$871$	−14.8261	−0.502363
$872$	0	0
$873$	−15.6591	−0.529980
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	− 53.4928i	− 1.80632i	−0.429300	−	0.903162i	$-0.641240\pi$
$877$	− 53.4928i	− 1.80632i	0.429300	−	0.903162i	$-0.358760\pi$
$878$	0	0
$879$	−12.0226	−0.405513
$880$	0	0
$881$	29.4775	0.993122	0.496561	−	0.868002i	$-0.334596\pi$
$881$	29.4775	0.993122	0.496561	+	0.868002i	$0.334596\pi$
$882$	0	0
$883$	52.4996i	1.76675i	0.468663	+	0.883377i	$0.344736\pi$
$883$	52.4996i	1.76675i	−0.468663	+	0.883377i	$0.655264\pi$
$884$	0	0
$885$	− 28.3103i	− 0.951639i
$886$	0	0
$887$	−48.0708	−1.61406	−0.807030	−	0.590511i	$-0.798926\pi$
$887$	−48.0708	−1.61406	−0.807030	+	0.590511i	$0.798926\pi$
$888$	0	0
$889$	5.19501	0.174235
$890$	0	0
$891$	31.0317i	1.03960i
$892$	0	0
$893$	35.9101i	1.20169i
$894$	0	0
$895$	−15.8677	−0.530399
$896$	0	0
$897$	51.5012	1.71957
$898$	0	0
$899$	− 24.4737i	− 0.816244i
$900$	0	0
$901$	− 3.67598i	− 0.122465i
$902$	0	0
$903$	−22.9678	−0.764321
$904$	0	0
$905$	−24.9835	−0.830481
$906$	0	0
$907$	− 36.9764i	− 1.22778i	−0.789390	−	0.613891i	$-0.789603\pi$
$907$	− 36.9764i	− 1.22778i	0.789390	−	0.613891i	$-0.210397\pi$
$908$	0	0
$909$	1.55562i	0.0515966i
$910$	0	0
$911$	−37.5655	−1.24460	−0.622300	−	0.782779i	$-0.713801\pi$
$911$	−37.5655	−1.24460	−0.622300	+	0.782779i	$0.713801\pi$
$912$	0	0
$913$	34.4655	1.14064
$914$	0	0
$915$	− 9.64446i	− 0.318836i
$916$	0	0
$917$	− 9.97612i	− 0.329441i
$918$	0	0
$919$	33.8991	1.11823	0.559114	−	0.829091i	$-0.311141\pi$
$919$	33.8991	1.11823	0.559114	+	0.829091i	$0.311141\pi$
$920$	0	0
$921$	4.35238	0.143416
$922$	0	0
$923$	3.20618i	0.105533i
$924$	0	0
$925$	− 8.38064i	− 0.275554i
$926$	0	0
$927$	−6.53572	−0.214661
$928$	0	0
$929$	15.4446	0.506722	0.253361	−	0.967372i	$-0.418464\pi$
$929$	15.4446	0.506722	0.253361	+	0.967372i	$0.418464\pi$
$930$	0	0
$931$	− 8.13580i	− 0.266640i
$932$	0	0
$933$	− 27.2307i	− 0.891494i
$934$	0	0
$935$	−0.973669	−0.0318424
$936$	0	0
$937$	−9.38232	−0.306507	−0.153254	−	0.988187i	$-0.548975\pi$
$937$	−9.38232	−0.306507	−0.153254	+	0.988187i	$0.548975\pi$
$938$	0	0
$939$	− 30.7719i	− 1.00420i
$940$	0	0
$941$	3.82635i	0.124735i	0.998053	+	0.0623677i	$0.0198651\pi$
$941$	3.82635i	0.124735i	−0.998053	+	0.0623677i	$0.980135\pi$
$942$	0	0
$943$	52.4759	1.70885
$944$	0	0
$945$	−3.21595	−0.104615
$946$	0	0
$947$	− 11.2607i	− 0.365923i	−0.983120	−	0.182961i	$-0.941432\pi$
$947$	− 11.2607i	− 0.365923i	0.983120	−	0.182961i	$-0.0585683\pi$
$948$	0	0
$949$	− 38.6498i	− 1.25463i
$950$	0	0
$951$	2.31331	0.0750141
$952$	0	0
$953$	26.3451	0.853403	0.426701	−	0.904393i	$-0.359676\pi$
$953$	26.3451	0.853403	0.426701	+	0.904393i	$0.359676\pi$
$954$	0	0
$955$	15.8579i	0.513148i
$956$	0	0
$957$	16.6059i	0.536794i
$958$	0	0
$959$	4.55620	0.147128
$960$	0	0
$961$	43.0554	1.38888
$962$	0	0
$963$	16.2143i	0.522498i
$964$	0	0
$965$	1.32279i	0.0425822i
$966$	0	0
$967$	21.5947	0.694439	0.347220	−	0.937784i	$-0.387126\pi$
$967$	21.5947	0.694439	0.347220	+	0.937784i	$0.387126\pi$
$968$	0	0
$969$	6.07881	0.195280
$970$	0	0
$971$	15.8259i	0.507877i	0.967220	+	0.253939i	$0.0817262\pi$
$971$	15.8259i	0.507877i	−0.967220	+	0.253939i	$0.918274\pi$
$972$	0	0
$973$	− 13.1407i	− 0.421272i
$974$	0	0
$975$	6.76650	0.216702
$976$	0	0
$977$	−4.34022	−0.138856	−0.0694280	−	0.997587i	$-0.522117\pi$
$977$	−4.34022	−0.138856	−0.0694280	+	0.997587i	$0.522117\pi$
$978$	0	0
$979$	15.4783i	0.494688i
$980$	0	0
$981$	11.8710i	0.379013i
$982$	0	0
$983$	−59.2922	−1.89113	−0.945563	−	0.325438i	$-0.894488\pi$
$983$	−59.2922	−1.89113	−0.945563	+	0.325438i	$0.894488\pi$
$984$	0	0
$985$	−7.63076	−0.243136
$986$	0	0
$987$	9.34310i	0.297394i
$988$	0	0
$989$	82.5844i	2.62603i
$990$	0	0
$991$	−1.13935	−0.0361927	−0.0180964	−	0.999836i	$-0.505761\pi$
$991$	−1.13935	−0.0361927	−0.0180964	+	0.999836i	$0.505761\pi$
$992$	0	0
$993$	9.08667	0.288357
$994$	0	0
$995$	4.72643i	0.149838i
$996$	0	0
$997$	5.67968i	0.179877i	0.995947	+	0.0899387i	$0.0286671\pi$
$997$	5.67968i	0.179877i	−0.995947	+	0.0899387i	$0.971333\pi$
$998$	0	0
$999$	−26.9517	−0.852714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.g.1121.10	yes	12
4.3	odd	2		2240.2.b.h.1121.3	yes	12
8.3	odd	2		2240.2.b.h.1121.10	yes	12
8.5	even	2	inner	2240.2.b.g.1121.3	✓	12
16.3	odd	4		8960.2.a.cb.1.5		6
16.5	even	4		8960.2.a.ch.1.5		6
16.11	odd	4		8960.2.a.ce.1.2		6
16.13	even	4		8960.2.a.cc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.3	✓	12	8.5	even	2	inner
2240.2.b.g.1121.10	yes	12	1.1	even	1	trivial
2240.2.b.h.1121.3	yes	12	4.3	odd	2
2240.2.b.h.1121.10	yes	12	8.3	odd	2
8960.2.a.cb.1.5		6	16.3	odd	4
8960.2.a.cc.1.2		6	16.13	even	4
8960.2.a.ce.1.2		6	16.11	odd	4
8960.2.a.ch.1.5		6	16.5	even	4

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.11677i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.48073 q^{9} +O(q^{10})\)	\(q+2.11677i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.48073 q^{9} -2.75846i q^{11} +3.19661i q^{13} +2.11677 q^{15} +0.352975 q^{17} -8.13580i q^{19} -2.11677i q^{21} -7.61119 q^{23} -1.00000 q^{25} +3.21595i q^{27} +2.84395i q^{29} -8.60555 q^{31} +5.83904 q^{33} +1.00000i q^{35} +8.38064i q^{37} -6.76650 q^{39} -6.89457 q^{41} -10.8504i q^{43} +1.48073i q^{45} -4.41384 q^{47} +1.00000 q^{49} +0.747168i q^{51} -10.4143i q^{53} -2.75846 q^{55} +17.2216 q^{57} -13.3743i q^{59} -4.55620i q^{61} +1.48073 q^{63} +3.19661 q^{65} +4.63807i q^{67} -16.1112i q^{69} +1.00299 q^{71} -12.0909 q^{73} -2.11677i q^{75} +2.75846i q^{77} +4.60775 q^{79} -11.2496 q^{81} +12.4945i q^{83} -0.352975i q^{85} -6.01999 q^{87} -5.61119 q^{89} -3.19661i q^{91} -18.2160i q^{93} -8.13580 q^{95} +10.5752 q^{97} +4.08455i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} + 16 q^{17} - 8 q^{23} - 12 q^{25} + 8 q^{31} + 72 q^{33} + 32 q^{39} + 32 q^{47} + 12 q^{49} - 8 q^{55} + 8 q^{57} + 20 q^{63} + 8 q^{65} - 48 q^{71} + 8 q^{73} + 16 q^{79} + 92 q^{81} + 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 + 16 * q^17 - 8 * q^23 - 12 * q^25 + 8 * q^31 + 72 * q^33 + 32 * q^39 + 32 * q^47 + 12 * q^49 - 8 * q^55 + 8 * q^57 + 20 * q^63 + 8 * q^65 - 48 * q^71 + 8 * q^73 + 16 * q^79 + 92 * q^81 + 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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