LMFDB - Embedded newform 1568.2.e.c.783.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,2,Mod(783,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.783"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.1
Root			$-1.60021 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1568.783
Dual form			1568.2.e.c.783.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61313i q^{3} -3.20041 q^{5} -3.82843 q^{9} +2.00000 q^{11} -5.07517 q^{13} +8.36308i q^{15} -0.317025i q^{17} +4.77791i q^{19} -1.43488i q^{23} +5.24264 q^{25} +2.16478i q^{27} +9.37769i q^{29} -2.65131 q^{31} -5.22625i q^{33} -2.44949i q^{37} +13.2621i q^{39} +4.46088i q^{41} +8.82843 q^{43} +12.2525 q^{45} +5.30262 q^{47} -0.828427 q^{51} -3.46410i q^{53} -6.40083 q^{55} +12.4853 q^{57} -2.61313i q^{59} +3.97696 q^{61} +16.2426 q^{65} -12.0000 q^{67} -3.74952 q^{69} +1.39942i q^{73} -13.6997i q^{75} +11.8272i q^{79} -5.82843 q^{81} +8.47343i q^{83} +1.01461i q^{85} +24.5051 q^{87} +9.87285i q^{89} +6.92820i q^{93} -15.2913i q^{95} -3.82683i q^{97} -7.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 16 q^{11} + 8 q^{25} + 48 q^{43} + 16 q^{51} + 32 q^{57} + 96 q^{65} - 96 q^{67} - 24 q^{81} - 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 + 8 * q^25 + 48 * q^43 + 16 * q^51 + 32 * q^57 + 96 * q^65 - 96 * q^67 - 24 * q^81 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	− 2.61313i	− 1.50869i	−0.656479	−	0.754344i	$-0.727955\pi$
$3$	− 2.61313i	− 1.50869i	0.656479	−	0.754344i	$-0.272045\pi$
$4$	0	0
$5$	−3.20041	−1.43127	−0.715634	−	0.698475i	$-0.753862\pi$
$5$	−3.20041	−1.43127	−0.715634	+	0.698475i	$0.753862\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.82843	−1.27614
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−5.07517	−1.40760	−0.703800	−	0.710399i	$-0.748515\pi$
$13$	−5.07517	−1.40760	−0.703800	+	0.710399i	$0.748515\pi$
$14$	0	0
$15$	8.36308i	2.15934i
$16$	0	0
$17$	− 0.317025i	− 0.0768899i	−0.999261	−	0.0384450i	$-0.987760\pi$
$17$	− 0.317025i	− 0.0768899i	0.999261	−	0.0384450i	$-0.0122404\pi$
$18$	0	0
$19$	4.77791i	1.09613i	0.836436	+	0.548064i	$0.184635\pi$
$19$	4.77791i	1.09613i	−0.836436	+	0.548064i	$0.815365\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.43488i	− 0.299193i	−0.988747	−	0.149596i	$-0.952203\pi$
$23$	− 1.43488i	− 0.299193i	0.988747	−	0.149596i	$-0.0477974\pi$
$24$	0	0
$25$	5.24264	1.04853
$26$	0	0
$27$	2.16478i	0.416613i
$28$	0	0
$29$	9.37769i	1.74139i	0.491820	+	0.870697i	$0.336332\pi$
$29$	9.37769i	1.74139i	−0.491820	+	0.870697i	$0.663668\pi$
$30$	0	0
$31$	−2.65131	−0.476189	−0.238095	−	0.971242i	$-0.576523\pi$
$31$	−2.65131	−0.476189	−0.238095	+	0.971242i	$0.576523\pi$
$32$	0	0
$33$	− 5.22625i	− 0.909774i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.44949i	− 0.402694i	−0.979520	−	0.201347i	$-0.935468\pi$
$37$	− 2.44949i	− 0.402694i	0.979520	−	0.201347i	$-0.0645318\pi$
$38$	0	0
$39$	13.2621i	2.12363i
$40$	0	0
$41$	4.46088i	0.696673i	0.937370	+	0.348337i	$0.113253\pi$
$41$	4.46088i	0.696673i	−0.937370	+	0.348337i	$0.886747\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	0	0
$45$	12.2525	1.82650
$46$	0	0
$47$	5.30262	0.773466	0.386733	−	0.922192i	$-0.373603\pi$
$47$	5.30262	0.773466	0.386733	+	0.922192i	$0.373603\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	− 3.46410i	− 0.475831i	−0.971286	−	0.237915i	$-0.923536\pi$
$53$	− 3.46410i	− 0.475831i	0.971286	−	0.237915i	$-0.0764641\pi$
$54$	0	0
$55$	−6.40083	−0.863087
$56$	0	0
$57$	12.4853	1.65372
$58$	0	0
$59$	− 2.61313i	− 0.340200i	−0.985427	−	0.170100i	$-0.945591\pi$
$59$	− 2.61313i	− 0.340200i	0.985427	−	0.170100i	$-0.0544091\pi$
$60$	0	0
$61$	3.97696	0.509198	0.254599	−	0.967047i	$-0.418057\pi$
$61$	3.97696	0.509198	0.254599	+	0.967047i	$0.418057\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.2426	2.01465
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−3.74952	−0.451389
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	1.39942i	0.163789i	0.996641	+	0.0818947i	$0.0260971\pi$
$73$	1.39942i	0.163789i	−0.996641	+	0.0818947i	$0.973903\pi$
$74$	0	0
$75$	− 13.6997i	− 1.58190i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.8272i	1.33066i	0.746548	+	0.665331i	$0.231710\pi$
$79$	11.8272i	1.33066i	−0.746548	+	0.665331i	$0.768290\pi$
$80$	0	0
$81$	−5.82843	−0.647603
$82$	0	0
$83$	8.47343i	0.930080i	0.885290	+	0.465040i	$0.153960\pi$
$83$	8.47343i	0.930080i	−0.885290	+	0.465040i	$0.846040\pi$
$84$	0	0
$85$	1.01461i	0.110050i
$86$	0	0
$87$	24.5051	2.62722
$88$	0	0
$89$	9.87285i	1.04652i	0.852173	+	0.523260i	$0.175284\pi$
$89$	9.87285i	1.04652i	−0.852173	+	0.523260i	$0.824716\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	− 15.2913i	− 1.56885i
$96$	0	0
$97$	− 3.82683i	− 0.388556i	−0.980946	−	0.194278i	$-0.937764\pi$
$97$	− 3.82683i	− 0.388556i	0.980946	−	0.194278i	$-0.0622364\pi$
$98$	0	0
$99$	−7.65685	−0.769543
$100$	0	0
$101$	7.72648	0.768813	0.384407	−	0.923164i	$-0.374406\pi$
$101$	7.72648	0.768813	0.384407	+	0.923164i	$0.374406\pi$
$102$	0	0
$103$	15.4530	1.52263	0.761313	−	0.648385i	$-0.224555\pi$
$103$	15.4530	1.52263	0.761313	+	0.648385i	$0.224555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	0.420266i	0.0402542i	0.999797	+	0.0201271i	$0.00640708\pi$
$109$	0.420266i	0.0402542i	−0.999797	+	0.0201271i	$0.993593\pi$
$110$	0	0
$111$	−6.40083	−0.607539
$112$	0	0
$113$	−16.4853	−1.55080	−0.775402	−	0.631467i	$-0.782453\pi$
$113$	−16.4853	−1.55080	−0.775402	+	0.631467i	$0.782453\pi$
$114$	0	0
$115$	4.59220i	0.428225i
$116$	0	0
$117$	19.4299	1.79630
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	11.6569	1.05106
$124$	0	0
$125$	−0.776550	−0.0694568
$126$	0	0
$127$	5.49333i	0.487454i	0.969844	+	0.243727i	$0.0783701\pi$
$127$	5.49333i	0.487454i	−0.969844	+	0.243727i	$0.921630\pi$
$128$	0	0
$129$	− 23.0698i	− 2.03118i
$130$	0	0
$131$	− 3.50981i	− 0.306653i	−0.988176	−	0.153327i	$-0.951001\pi$
$131$	− 3.50981i	− 0.306653i	0.988176	−	0.153327i	$-0.0489987\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 6.92820i	− 0.596285i
$136$	0	0
$137$	0.242641	0.0207302	0.0103651	−	0.999946i	$-0.496701\pi$
$137$	0.242641	0.0207302	0.0103651	+	0.999946i	$0.496701\pi$
$138$	0	0
$139$	5.04054i	0.427533i	0.976885	+	0.213767i	$0.0685732\pi$
$139$	5.04054i	0.427533i	−0.976885	+	0.213767i	$0.931427\pi$
$140$	0	0
$141$	− 13.8564i	− 1.16692i
$142$	0	0
$143$	−10.1503	−0.848814
$144$	0	0
$145$	− 30.0125i	− 2.49240i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.1903i	1.65405i	0.562165	+	0.827025i	$0.309969\pi$
$149$	20.1903i	1.65405i	−0.562165	+	0.827025i	$0.690031\pi$
$150$	0	0
$151$	− 15.2913i	− 1.24439i	−0.782863	−	0.622194i	$-0.786242\pi$
$151$	− 15.2913i	− 1.24439i	0.782863	−	0.622194i	$-0.213758\pi$
$152$	0	0
$153$	1.21371i	0.0981225i
$154$	0	0
$155$	8.48528	0.681554
$156$	0	0
$157$	−23.9560	−1.91190	−0.955948	−	0.293536i	$-0.905168\pi$
$157$	−23.9560	−1.91190	−0.955948	+	0.293536i	$0.905168\pi$
$158$	0	0
$159$	−9.05213	−0.717881
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.14214	−0.637741	−0.318871	−	0.947798i	$-0.603304\pi$
$163$	−8.14214	−0.637741	−0.318871	+	0.947798i	$0.603304\pi$
$164$	0	0
$165$	16.7262i	1.30213i
$166$	0	0
$167$	14.3548	1.11080	0.555402	−	0.831582i	$-0.312564\pi$
$167$	14.3548	1.11080	0.555402	+	0.831582i	$0.312564\pi$
$168$	0	0
$169$	12.7574	0.981335
$170$	0	0
$171$	− 18.2919i	− 1.39882i
$172$	0	0
$173$	6.94993	0.528393	0.264197	−	0.964469i	$-0.414893\pi$
$173$	6.94993	0.528393	0.264197	+	0.964469i	$0.414893\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.82843	−0.513256
$178$	0	0
$179$	20.4853	1.53114	0.765571	−	0.643352i	$-0.222457\pi$
$179$	20.4853	1.53114	0.765571	+	0.643352i	$0.222457\pi$
$180$	0	0
$181$	−2.10220	−0.156256	−0.0781278	−	0.996943i	$-0.524894\pi$
$181$	−2.10220	−0.156256	−0.0781278	+	0.996943i	$0.524894\pi$
$182$	0	0
$183$	− 10.3923i	− 0.768221i
$184$	0	0
$185$	7.83938i	0.576363i
$186$	0	0
$187$	− 0.634051i	− 0.0463664i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.8272i	0.855785i	0.903830	+	0.427892i	$0.140744\pi$
$191$	11.8272i	0.855785i	−0.903830	+	0.427892i	$0.859256\pi$
$192$	0	0
$193$	−14.1421	−1.01797	−0.508987	−	0.860774i	$-0.669980\pi$
$193$	−14.1421	−1.01797	−0.508987	+	0.860774i	$0.669980\pi$
$194$	0	0
$195$	− 42.4441i	− 3.03948i
$196$	0	0
$197$	26.5241i	1.88977i	0.327409	+	0.944883i	$0.393824\pi$
$197$	26.5241i	1.88977i	−0.327409	+	0.944883i	$0.606176\pi$
$198$	0	0
$199$	−2.65131	−0.187946	−0.0939731	−	0.995575i	$-0.529957\pi$
$199$	−2.65131	−0.187946	−0.0939731	+	0.995575i	$0.529957\pi$
$200$	0	0
$201$	31.3575i	2.21179i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 14.2767i	− 0.997126i
$206$	0	0
$207$	5.49333i	0.381813i
$208$	0	0
$209$	9.55582i	0.660990i
$210$	0	0
$211$	−20.4853	−1.41026	−0.705132	−	0.709076i	$-0.749112\pi$
$211$	−20.4853	−1.41026	−0.705132	+	0.709076i	$0.749112\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−28.2546	−1.92695
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.65685	0.247107
$220$	0	0
$221$	1.60896i	0.108230i
$222$	0	0
$223$	−24.5051	−1.64098	−0.820491	−	0.571659i	$-0.806300\pi$
$223$	−24.5051	−1.64098	−0.820491	+	0.571659i	$0.806300\pi$
$224$	0	0
$225$	−20.0711	−1.33807
$226$	0	0
$227$	23.2555i	1.54352i	0.635913	+	0.771761i	$0.280624\pi$
$227$	23.2555i	1.54352i	−0.635913	+	0.771761i	$0.719376\pi$
$228$	0	0
$229$	−17.5552	−1.16008	−0.580039	−	0.814589i	$-0.696963\pi$
$229$	−17.5552	−1.16008	−0.580039	+	0.814589i	$0.696963\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.2426	−1.32614	−0.663070	−	0.748558i	$-0.730747\pi$
$233$	−20.2426	−1.32614	−0.663070	+	0.748558i	$0.730747\pi$
$234$	0	0
$235$	−16.9706	−1.10704
$236$	0	0
$237$	30.9059	2.00756
$238$	0	0
$239$	− 13.2621i	− 0.857851i	−0.903340	−	0.428926i	$-0.858892\pi$
$239$	− 13.2621i	− 0.857851i	0.903340	−	0.428926i	$-0.141108\pi$
$240$	0	0
$241$	22.1187i	1.42479i	0.701778	+	0.712396i	$0.252390\pi$
$241$	22.1187i	1.42479i	−0.701778	+	0.712396i	$0.747610\pi$
$242$	0	0
$243$	21.7248i	1.39364i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 24.2487i	− 1.54291i
$248$	0	0
$249$	22.1421	1.40320
$250$	0	0
$251$	19.1886i	1.21117i	0.795780	+	0.605586i	$0.207061\pi$
$251$	19.1886i	1.21117i	−0.795780	+	0.605586i	$0.792939\pi$
$252$	0	0
$253$	− 2.86976i	− 0.180420i
$254$	0	0
$255$	2.65131	0.166031
$256$	0	0
$257$	8.15640i	0.508782i	0.967101	+	0.254391i	$0.0818751\pi$
$257$	8.15640i	0.508782i	−0.967101	+	0.254391i	$0.918125\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 35.9018i	− 2.22227i
$262$	0	0
$263$	8.95743i	0.552339i	0.961109	+	0.276169i	$0.0890651\pi$
$263$	8.95743i	0.552339i	−0.961109	+	0.276169i	$0.910935\pi$
$264$	0	0
$265$	11.0866i	0.681042i
$266$	0	0
$267$	25.7990	1.57887
$268$	0	0
$269$	−2.42386	−0.147785	−0.0738927	−	0.997266i	$-0.523542\pi$
$269$	−2.42386	−0.147785	−0.0738927	+	0.997266i	$0.523542\pi$
$270$	0	0
$271$	−13.2565	−0.805278	−0.402639	−	0.915359i	$-0.631907\pi$
$271$	−13.2565	−0.805278	−0.402639	+	0.915359i	$0.631907\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.4853	0.632286
$276$	0	0
$277$	− 16.7262i	− 1.00498i	−0.864584	−	0.502489i	$-0.832418\pi$
$277$	− 16.7262i	− 1.00498i	0.864584	−	0.502489i	$-0.167582\pi$
$278$	0	0
$279$	10.1503	0.607685
$280$	0	0
$281$	8.72792	0.520664	0.260332	−	0.965519i	$-0.416168\pi$
$281$	8.72792	0.520664	0.260332	+	0.965519i	$0.416168\pi$
$282$	0	0
$283$	− 0.185709i	− 0.0110393i	−0.999985	−	0.00551963i	$-0.998243\pi$
$283$	− 0.185709i	− 0.0110393i	0.999985	−	0.00551963i	$-0.00175696\pi$
$284$	0	0
$285$	−39.9581	−2.36691
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.8995	0.994088
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	5.07517	0.296495	0.148247	−	0.988950i	$-0.452637\pi$
$293$	5.07517	0.296495	0.148247	+	0.988950i	$0.452637\pi$
$294$	0	0
$295$	8.36308i	0.486917i
$296$	0	0
$297$	4.32957i	0.251227i
$298$	0	0
$299$	7.28225i	0.421143i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 20.1903i	− 1.15990i
$304$	0	0
$305$	−12.7279	−0.728799
$306$	0	0
$307$	6.04601i	0.345064i	0.985004	+	0.172532i	$0.0551948\pi$
$307$	6.04601i	0.345064i	−0.985004	+	0.172532i	$0.944805\pi$
$308$	0	0
$309$	− 40.3805i	− 2.29717i
$310$	0	0
$311$	−27.1564	−1.53990	−0.769949	−	0.638105i	$-0.779718\pi$
$311$	−27.1564	−1.53990	−0.769949	+	0.638105i	$0.779718\pi$
$312$	0	0
$313$	10.5838i	0.598233i	0.954217	+	0.299116i	$0.0966919\pi$
$313$	10.5838i	0.598233i	−0.954217	+	0.299116i	$0.903308\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.05845i	− 0.227945i	−0.993484	−	0.113973i	$-0.963642\pi$
$317$	− 4.05845i	− 0.227945i	0.993484	−	0.113973i	$-0.0363576\pi$
$318$	0	0
$319$	18.7554i	1.05010i
$320$	0	0
$321$	− 22.1731i	− 1.23758i
$322$	0	0
$323$	1.51472	0.0842812
$324$	0	0
$325$	−26.6073	−1.47591
$326$	0	0
$327$	1.09821	0.0607310
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.8284	−1.14483	−0.572417	−	0.819963i	$-0.693994\pi$
$331$	−20.8284	−1.14483	−0.572417	+	0.819963i	$0.693994\pi$
$332$	0	0
$333$	9.37769i	0.513894i
$334$	0	0
$335$	38.4050	2.09829
$336$	0	0
$337$	4.24264	0.231111	0.115556	−	0.993301i	$-0.463135\pi$
$337$	4.24264	0.231111	0.115556	+	0.993301i	$0.463135\pi$
$338$	0	0
$339$	43.0781i	2.33968i
$340$	0	0
$341$	−5.30262	−0.287153
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−6.97056	−0.374199	−0.187100	−	0.982341i	$-0.559909\pi$
$347$	−6.97056	−0.374199	−0.187100	+	0.982341i	$0.559909\pi$
$348$	0	0
$349$	−22.4029	−1.19920	−0.599600	−	0.800300i	$-0.704673\pi$
$349$	−22.4029	−1.19920	−0.599600	+	0.800300i	$0.704673\pi$
$350$	0	0
$351$	− 10.9867i	− 0.586424i
$352$	0	0
$353$	13.1200i	0.698308i	0.937065	+	0.349154i	$0.113531\pi$
$353$	13.1200i	0.698308i	−0.937065	+	0.349154i	$0.886469\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.3205i	0.914141i	0.889430	+	0.457071i	$0.151101\pi$
$359$	17.3205i	0.914141i	−0.889430	+	0.457071i	$0.848899\pi$
$360$	0	0
$361$	−3.82843	−0.201496
$362$	0	0
$363$	18.2919i	0.960075i
$364$	0	0
$365$	− 4.47871i	− 0.234427i
$366$	0	0
$367$	−17.0061	−0.887709	−0.443855	−	0.896099i	$-0.646389\pi$
$367$	−17.0061	−0.887709	−0.443855	+	0.896099i	$0.646389\pi$
$368$	0	0
$369$	− 17.0782i	− 0.889054i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.20361i	0.476545i	0.971198	+	0.238273i	$0.0765811\pi$
$373$	9.20361i	0.476545i	−0.971198	+	0.238273i	$0.923419\pi$
$374$	0	0
$375$	2.02922i	0.104789i
$376$	0	0
$377$	− 47.5934i	− 2.45118i
$378$	0	0
$379$	−17.3137	−0.889345	−0.444673	−	0.895693i	$-0.646680\pi$
$379$	−17.3137	−0.889345	−0.444673	+	0.895693i	$0.646680\pi$
$380$	0	0
$381$	14.3548	0.735416
$382$	0	0
$383$	5.30262	0.270951	0.135476	−	0.990781i	$-0.456744\pi$
$383$	5.30262	0.270951	0.135476	+	0.990781i	$0.456744\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−33.7990	−1.71810
$388$	0	0
$389$	3.63818i	0.184463i	0.995738	+	0.0922316i	$0.0294000\pi$
$389$	3.63818i	0.184463i	−0.995738	+	0.0922316i	$0.970600\pi$
$390$	0	0
$391$	−0.454893	−0.0230049
$392$	0	0
$393$	−9.17157	−0.462645
$394$	0	0
$395$	− 37.8519i	− 1.90453i
$396$	0	0
$397$	24.2776	1.21846	0.609230	−	0.792994i	$-0.291479\pi$
$397$	24.2776	1.21846	0.609230	+	0.792994i	$0.291479\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.75736	0.387384	0.193692	−	0.981062i	$-0.437954\pi$
$401$	7.75736	0.387384	0.193692	+	0.981062i	$0.437954\pi$
$402$	0	0
$403$	13.4558	0.670283
$404$	0	0
$405$	18.6534	0.926893
$406$	0	0
$407$	− 4.89898i	− 0.242833i
$408$	0	0
$409$	3.82683i	0.189225i	0.995514	+	0.0946124i	$0.0301612\pi$
$409$	3.82683i	0.189225i	−0.995514	+	0.0946124i	$0.969839\pi$
$410$	0	0
$411$	− 0.634051i	− 0.0312754i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 27.1185i	− 1.33119i
$416$	0	0
$417$	13.1716	0.645015
$418$	0	0
$419$	− 39.8309i	− 1.94587i	−0.231082	−	0.972934i	$-0.574226\pi$
$419$	− 39.8309i	− 1.94587i	0.231082	−	0.972934i	$-0.425774\pi$
$420$	0	0
$421$	− 26.5241i	− 1.29271i	−0.763038	−	0.646353i	$-0.776293\pi$
$421$	− 26.5241i	− 1.29271i	0.763038	−	0.646353i	$-0.223707\pi$
$422$	0	0
$423$	−20.3007	−0.987053
$424$	0	0
$425$	− 1.66205i	− 0.0806213i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	26.5241i	1.28060i
$430$	0	0
$431$	− 7.52255i	− 0.362348i	−0.983451	−	0.181174i	$-0.942010\pi$
$431$	− 7.52255i	− 0.362348i	0.983451	−	0.181174i	$-0.0579898\pi$
$432$	0	0
$433$	5.17186i	0.248544i	0.992248	+	0.124272i	$0.0396595\pi$
$433$	5.17186i	0.248544i	−0.992248	+	0.124272i	$0.960341\pi$
$434$	0	0
$435$	−78.4264	−3.76026
$436$	0	0
$437$	6.85572	0.327953
$438$	0	0
$439$	35.7536	1.70643	0.853214	−	0.521561i	$-0.174650\pi$
$439$	35.7536	1.70643	0.853214	+	0.521561i	$0.174650\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	28.2843	1.34383	0.671913	−	0.740630i	$-0.265473\pi$
$443$	28.2843	1.34383	0.671913	+	0.740630i	$0.265473\pi$
$444$	0	0
$445$	− 31.5972i	− 1.49785i
$446$	0	0
$447$	52.7597	2.49545
$448$	0	0
$449$	−5.65685	−0.266963	−0.133482	−	0.991051i	$-0.542616\pi$
$449$	−5.65685	−0.266963	−0.133482	+	0.991051i	$0.542616\pi$
$450$	0	0
$451$	8.92177i	0.420110i
$452$	0	0
$453$	−39.9581	−1.87739
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.1716	−0.990364	−0.495182	−	0.868789i	$-0.664899\pi$
$457$	−21.1716	−0.990364	−0.495182	+	0.868789i	$0.664899\pi$
$458$	0	0
$459$	0.686292	0.0320333
$460$	0	0
$461$	−26.6073	−1.23923	−0.619613	−	0.784908i	$-0.712710\pi$
$461$	−26.6073	−1.23923	−0.619613	+	0.784908i	$0.712710\pi$
$462$	0	0
$463$	− 18.7554i	− 0.871637i	−0.900035	−	0.435818i	$-0.856459\pi$
$463$	− 18.7554i	− 0.871637i	0.900035	−	0.435818i	$-0.143541\pi$
$464$	0	0
$465$	− 22.1731i	− 1.02825i
$466$	0	0
$467$	− 0.0769232i	− 0.00355958i	−0.999998	−	0.00177979i	$-0.999433\pi$
$467$	− 0.0769232i	− 0.00355958i	0.999998	−	0.00177979i	$-0.000566526\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	62.6000i	2.88446i
$472$	0	0
$473$	17.6569	0.811863
$474$	0	0
$475$	25.0489i	1.14932i
$476$	0	0
$477$	13.2621i	0.607228i
$478$	0	0
$479$	11.7034	0.534744	0.267372	−	0.963593i	$-0.413845\pi$
$479$	11.7034	0.534744	0.267372	+	0.963593i	$0.413845\pi$
$480$	0	0
$481$	12.4316i	0.566831i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.2474i	0.556128i
$486$	0	0
$487$	− 34.0467i	− 1.54280i	−0.636349	−	0.771401i	$-0.719556\pi$
$487$	− 34.0467i	− 1.54280i	0.636349	−	0.771401i	$-0.280444\pi$
$488$	0	0
$489$	21.2764i	0.962153i
$490$	0	0
$491$	31.7990	1.43507	0.717534	−	0.696523i	$-0.245271\pi$
$491$	31.7990	1.43507	0.717534	+	0.696523i	$0.245271\pi$
$492$	0	0
$493$	2.97297	0.133896
$494$	0	0
$495$	24.5051	1.10142
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.48528	0.379853	0.189927	−	0.981798i	$-0.439175\pi$
$499$	8.48528	0.379853	0.189927	+	0.981798i	$0.439175\pi$
$500$	0	0
$501$	− 37.5108i	− 1.67586i
$502$	0	0
$503$	−24.5051	−1.09263	−0.546314	−	0.837580i	$-0.683969\pi$
$503$	−24.5051	−1.09263	−0.546314	+	0.837580i	$0.683969\pi$
$504$	0	0
$505$	−24.7279	−1.10038
$506$	0	0
$507$	− 33.3366i	− 1.48053i
$508$	0	0
$509$	14.1273	0.626182	0.313091	−	0.949723i	$-0.398636\pi$
$509$	14.1273	0.626182	0.313091	+	0.949723i	$0.398636\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−10.3431	−0.456661
$514$	0	0
$515$	−49.4558	−2.17928
$516$	0	0
$517$	10.6052	0.466418
$518$	0	0
$519$	− 18.1610i	− 0.797181i
$520$	0	0
$521$	− 18.6089i	− 0.815271i	−0.913145	−	0.407636i	$-0.866353\pi$
$521$	− 18.6089i	− 0.815271i	0.913145	−	0.407636i	$-0.133647\pi$
$522$	0	0
$523$	− 32.8113i	− 1.43474i	−0.696693	−	0.717369i	$-0.745346\pi$
$523$	− 32.8113i	− 1.43474i	0.696693	−	0.717369i	$-0.254654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.840532i	0.0366141i
$528$	0	0
$529$	20.9411	0.910484
$530$	0	0
$531$	10.0042i	0.434144i
$532$	0	0
$533$	− 22.6398i	− 0.980637i
$534$	0	0
$535$	−27.1564	−1.17407
$536$	0	0
$537$	− 53.5306i	− 2.31002i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 32.8580i	− 1.41267i	−0.707875	−	0.706337i	$-0.750346\pi$
$541$	− 32.8580i	− 1.41267i	0.707875	−	0.706337i	$-0.249654\pi$
$542$	0	0
$543$	5.49333i	0.235741i
$544$	0	0
$545$	− 1.34502i	− 0.0576145i
$546$	0	0
$547$	15.1716	0.648690	0.324345	−	0.945939i	$-0.394856\pi$
$547$	15.1716	0.648690	0.324345	+	0.945939i	$0.394856\pi$
$548$	0	0
$549$	−15.2255	−0.649809
$550$	0	0
$551$	−44.8058	−1.90879
$552$	0	0
$553$	0	0
$554$	0	0
$555$	20.4853	0.869552
$556$	0	0
$557$	− 16.7262i	− 0.708710i	−0.935111	−	0.354355i	$-0.884700\pi$
$557$	− 16.7262i	− 0.708710i	0.935111	−	0.354355i	$-0.115300\pi$
$558$	0	0
$559$	−44.8058	−1.89508
$560$	0	0
$561$	−1.65685	−0.0699524
$562$	0	0
$563$	− 16.7611i	− 0.706398i	−0.935548	−	0.353199i	$-0.885094\pi$
$563$	− 16.7611i	− 0.706398i	0.935548	−	0.353199i	$-0.114906\pi$
$564$	0	0
$565$	52.7597	2.21962
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.1005	0.591124	0.295562	−	0.955324i	$-0.404493\pi$
$569$	14.1005	0.591124	0.295562	+	0.955324i	$0.404493\pi$
$570$	0	0
$571$	23.6569	0.990009	0.495004	−	0.868891i	$-0.335166\pi$
$571$	23.6569	0.990009	0.495004	+	0.868891i	$0.335166\pi$
$572$	0	0
$573$	30.9059	1.29111
$574$	0	0
$575$	− 7.52255i	− 0.313712i
$576$	0	0
$577$	2.11039i	0.0878567i	0.999035	+	0.0439284i	$0.0139873\pi$
$577$	2.11039i	0.0878567i	−0.999035	+	0.0439284i	$0.986013\pi$
$578$	0	0
$579$	36.9552i	1.53580i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 6.92820i	− 0.286937i
$584$	0	0
$585$	−62.1838	−2.57098
$586$	0	0
$587$	− 43.2638i	− 1.78569i	−0.450365	−	0.892845i	$-0.648706\pi$
$587$	− 43.2638i	− 1.78569i	0.450365	−	0.892845i	$-0.351294\pi$
$588$	0	0
$589$	− 12.6677i	− 0.521964i
$590$	0	0
$591$	69.3109	2.85107
$592$	0	0
$593$	12.3003i	0.505111i	0.967582	+	0.252556i	$0.0812711\pi$
$593$	12.3003i	0.505111i	−0.967582	+	0.252556i	$0.918729\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.92820i	0.283552i
$598$	0	0
$599$	35.4815i	1.44974i	0.688887	+	0.724868i	$0.258099\pi$
$599$	35.4815i	1.44974i	−0.688887	+	0.724868i	$0.741901\pi$
$600$	0	0
$601$	35.1843i	1.43520i	0.696456	+	0.717600i	$0.254759\pi$
$601$	35.1843i	1.43520i	−0.696456	+	0.717600i	$0.745241\pi$
$602$	0	0
$603$	45.9411	1.87087
$604$	0	0
$605$	22.4029	0.910807
$606$	0	0
$607$	25.6033	1.03921	0.519603	−	0.854408i	$-0.326080\pi$
$607$	25.6033	1.03921	0.519603	+	0.854408i	$0.326080\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.9117	−1.08873
$612$	0	0
$613$	26.9444i	1.08827i	0.838997	+	0.544137i	$0.183143\pi$
$613$	26.9444i	1.08827i	−0.838997	+	0.544137i	$0.816857\pi$
$614$	0	0
$615$	−37.3067	−1.50435
$616$	0	0
$617$	−9.21320	−0.370910	−0.185455	−	0.982653i	$-0.559376\pi$
$617$	−9.21320	−0.370910	−0.185455	+	0.982653i	$0.559376\pi$
$618$	0	0
$619$	29.1158i	1.17026i	0.810938	+	0.585131i	$0.198957\pi$
$619$	29.1158i	1.17026i	−0.810938	+	0.585131i	$0.801043\pi$
$620$	0	0
$621$	3.10620	0.124648
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.7279	−0.949117
$626$	0	0
$627$	24.9706	0.997228
$628$	0	0
$629$	−0.776550	−0.0309631
$630$	0	0
$631$	29.7420i	1.18401i	0.805934	+	0.592006i	$0.201664\pi$
$631$	29.7420i	1.18401i	−0.805934	+	0.592006i	$0.798336\pi$
$632$	0	0
$633$	53.5306i	2.12765i
$634$	0	0
$635$	− 17.5809i	− 0.697677i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.9289	−0.668653	−0.334326	−	0.942457i	$-0.608509\pi$
$641$	−16.9289	−0.668653	−0.334326	+	0.942457i	$0.608509\pi$
$642$	0	0
$643$	− 36.3981i	− 1.43540i	−0.696353	−	0.717700i	$-0.745195\pi$
$643$	− 36.3981i	− 1.43540i	0.696353	−	0.717700i	$-0.254805\pi$
$644$	0	0
$645$	73.8329i	2.90717i
$646$	0	0
$647$	11.7034	0.460110	0.230055	−	0.973178i	$-0.426109\pi$
$647$	11.7034	0.460110	0.230055	+	0.973178i	$0.426109\pi$
$648$	0	0
$649$	− 5.22625i	− 0.205148i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.0613i	1.37205i	0.727576	+	0.686027i	$0.240647\pi$
$653$	35.0613i	1.37205i	−0.727576	+	0.686027i	$0.759353\pi$
$654$	0	0
$655$	11.2328i	0.438903i
$656$	0	0
$657$	− 5.35757i	− 0.209019i
$658$	0	0
$659$	−25.5147	−0.993912	−0.496956	−	0.867776i	$-0.665549\pi$
$659$	−25.5147	−0.993912	−0.496956	+	0.867776i	$0.665549\pi$
$660$	0	0
$661$	20.8498	0.810963	0.405481	−	0.914103i	$-0.367104\pi$
$661$	20.8498	0.810963	0.405481	+	0.914103i	$0.367104\pi$
$662$	0	0
$663$	4.20441	0.163286
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.4558	0.521012
$668$	0	0
$669$	64.0349i	2.47573i
$670$	0	0
$671$	7.95393	0.307058
$672$	0	0
$673$	30.3848	1.17125	0.585624	−	0.810583i	$-0.300850\pi$
$673$	30.3848	1.17125	0.585624	+	0.810583i	$0.300850\pi$
$674$	0	0
$675$	11.3492i	0.436830i
$676$	0	0
$677$	1.00400	0.0385867	0.0192934	−	0.999814i	$-0.493858\pi$
$677$	1.00400	0.0385867	0.0192934	+	0.999814i	$0.493858\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	60.7696	2.32869
$682$	0	0
$683$	14.1421	0.541134	0.270567	−	0.962701i	$-0.412789\pi$
$683$	14.1421	0.541134	0.270567	+	0.962701i	$0.412789\pi$
$684$	0	0
$685$	−0.776550	−0.0296705
$686$	0	0
$687$	45.8739i	1.75020i
$688$	0	0
$689$	17.5809i	0.669779i
$690$	0	0
$691$	19.9314i	0.758226i	0.925350	+	0.379113i	$0.123771\pi$
$691$	19.9314i	0.758226i	−0.925350	+	0.379113i	$0.876229\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 16.1318i	− 0.611915i
$696$	0	0
$697$	1.41421	0.0535672
$698$	0	0
$699$	52.8966i	2.00073i
$700$	0	0
$701$	6.15978i	0.232652i	0.993211	+	0.116326i	$0.0371117\pi$
$701$	6.15978i	0.232652i	−0.993211	+	0.116326i	$0.962888\pi$
$702$	0	0
$703$	11.7034	0.441404
$704$	0	0
$705$	44.3462i	1.67018i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 36.7423i	− 1.37989i	−0.723863	−	0.689944i	$-0.757635\pi$
$709$	− 36.7423i	− 1.37989i	0.723863	−	0.689944i	$-0.242365\pi$
$710$	0	0
$711$	− 45.2795i	− 1.69811i
$712$	0	0
$713$	3.80430i	0.142472i
$714$	0	0
$715$	32.4853	1.21488
$716$	0	0
$717$	−34.6554	−1.29423
$718$	0	0
$719$	−29.3528	−1.09468	−0.547338	−	0.836912i	$-0.684359\pi$
$719$	−29.3528	−1.09468	−0.547338	+	0.836912i	$0.684359\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	57.7990	2.14957
$724$	0	0
$725$	49.1639i	1.82590i
$726$	0	0
$727$	43.0643	1.59716	0.798582	−	0.601886i	$-0.205584\pi$
$727$	43.0643	1.59716	0.798582	+	0.601886i	$0.205584\pi$
$728$	0	0
$729$	39.2843	1.45497
$730$	0	0
$731$	− 2.79884i	− 0.103519i
$732$	0	0
$733$	−7.40482	−0.273503	−0.136752	−	0.990605i	$-0.543666\pi$
$733$	−7.40482	−0.273503	−0.136752	+	0.990605i	$0.543666\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−15.1716	−0.558095	−0.279048	−	0.960277i	$-0.590019\pi$
$739$	−15.1716	−0.558095	−0.279048	+	0.960277i	$0.590019\pi$
$740$	0	0
$741$	−63.3649	−2.32777
$742$	0	0
$743$	13.2621i	0.486538i	0.969959	+	0.243269i	$0.0782197\pi$
$743$	13.2621i	0.486538i	−0.969959	+	0.243269i	$0.921780\pi$
$744$	0	0
$745$	− 64.6172i	− 2.36739i
$746$	0	0
$747$	− 32.4399i	− 1.18691i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 35.7277i	− 1.30372i	−0.758338	−	0.651862i	$-0.773988\pi$
$751$	− 35.7277i	− 1.30372i	0.758338	−	0.651862i	$-0.226012\pi$
$752$	0	0
$753$	50.1421	1.82728
$754$	0	0
$755$	48.9384i	1.78105i
$756$	0	0
$757$	35.9018i	1.30487i	0.757843	+	0.652437i	$0.226253\pi$
$757$	35.9018i	1.30487i	−0.757843	+	0.652437i	$0.773747\pi$
$758$	0	0
$759$	−7.49903	−0.272198
$760$	0	0
$761$	− 21.8561i	− 0.792282i	−0.918190	−	0.396141i	$-0.870349\pi$
$761$	− 21.8561i	− 0.792282i	0.918190	−	0.396141i	$-0.129651\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 3.88437i	− 0.140440i
$766$	0	0
$767$	13.2621i	0.478865i
$768$	0	0
$769$	46.1940i	1.66580i	0.553425	+	0.832899i	$0.313320\pi$
$769$	46.1940i	1.66580i	−0.553425	+	0.832899i	$0.686680\pi$
$770$	0	0
$771$	21.3137	0.767594
$772$	0	0
$773$	31.0001	1.11500	0.557499	−	0.830178i	$-0.311761\pi$
$773$	31.0001	1.11500	0.557499	+	0.830178i	$0.311761\pi$
$774$	0	0
$775$	−13.8999	−0.499298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−21.3137	−0.763643
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−20.3007	−0.725487
$784$	0	0
$785$	76.6690	2.73644
$786$	0	0
$787$	38.8255i	1.38398i	0.721908	+	0.691989i	$0.243265\pi$
$787$	38.8255i	1.38398i	−0.721908	+	0.691989i	$0.756735\pi$
$788$	0	0
$789$	23.4069	0.833307
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.1838	−0.716747
$794$	0	0
$795$	28.9706	1.02748
$796$	0	0
$797$	13.4840	0.477627	0.238814	−	0.971065i	$-0.423241\pi$
$797$	13.4840	0.477627	0.238814	+	0.971065i	$0.423241\pi$
$798$	0	0
$799$	− 1.68106i	− 0.0594718i
$800$	0	0
$801$	− 37.7975i	− 1.33551i
$802$	0	0
$803$	2.79884i	0.0987687i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.33386i	0.222962i
$808$	0	0
$809$	−15.4558	−0.543399	−0.271699	−	0.962382i	$-0.587586\pi$
$809$	−15.4558	−0.543399	−0.271699	+	0.962382i	$0.587586\pi$
$810$	0	0
$811$	28.4818i	1.00013i	0.865988	+	0.500065i	$0.166690\pi$
$811$	28.4818i	1.00013i	−0.865988	+	0.500065i	$0.833310\pi$
$812$	0	0
$813$	34.6410i	1.21491i
$814$	0	0
$815$	26.0582	0.912779
$816$	0	0
$817$	42.1814i	1.47574i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.9149i	0.625233i	0.949879	+	0.312616i	$0.101205\pi$
$821$	17.9149i	0.625233i	−0.949879	+	0.312616i	$0.898795\pi$
$822$	0	0
$823$	− 20.7846i	− 0.724506i	−0.932080	−	0.362253i	$-0.882008\pi$
$823$	− 20.7846i	− 0.724506i	0.932080	−	0.362253i	$-0.117992\pi$
$824$	0	0
$825$	− 27.3994i	− 0.953923i
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−16.7786	−0.582745	−0.291373	−	0.956610i	$-0.594112\pi$
$829$	−16.7786	−0.582745	−0.291373	+	0.956610i	$0.594112\pi$
$830$	0	0
$831$	−43.7076	−1.51620
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−45.9411	−1.58986
$836$	0	0
$837$	− 5.73951i	− 0.198387i
$838$	0	0
$839$	−30.4510	−1.05129	−0.525643	−	0.850705i	$-0.676175\pi$
$839$	−30.4510	−1.05129	−0.525643	+	0.850705i	$0.676175\pi$
$840$	0	0
$841$	−58.9411	−2.03245
$842$	0	0
$843$	− 22.8072i	− 0.785520i
$844$	0	0
$845$	−40.8288	−1.40455
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.485281	−0.0166548
$850$	0	0
$851$	−3.51472	−0.120483
$852$	0	0
$853$	−32.5532	−1.11460	−0.557301	−	0.830311i	$-0.688163\pi$
$853$	−32.5532	−1.11460	−0.557301	+	0.830311i	$0.688163\pi$
$854$	0	0
$855$	58.5416i	2.00208i
$856$	0	0
$857$	− 34.4734i	− 1.17759i	−0.808283	−	0.588794i	$-0.799603\pi$
$857$	− 34.4734i	− 1.17759i	0.808283	−	0.588794i	$-0.200397\pi$
$858$	0	0
$859$	− 44.5319i	− 1.51941i	−0.650268	−	0.759705i	$-0.725343\pi$
$859$	− 44.5319i	− 1.51941i	0.650268	−	0.759705i	$-0.274657\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.05845i	0.138151i	0.997611	+	0.0690756i	$0.0220050\pi$
$863$	4.05845i	0.138151i	−0.997611	+	0.0690756i	$0.977995\pi$
$864$	0	0
$865$	−22.2426	−0.756272
$866$	0	0
$867$	− 44.1605i	− 1.49977i
$868$	0	0
$869$	23.6544i	0.802419i
$870$	0	0
$871$	60.9021	2.06359
$872$	0	0
$873$	14.6508i	0.495853i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.1915i	− 1.08703i	−0.839399	−	0.543515i	$-0.817093\pi$
$877$	− 32.1915i	− 1.08703i	0.839399	−	0.543515i	$-0.182907\pi$
$878$	0	0
$879$	− 13.2621i	− 0.447318i
$880$	0	0
$881$	− 34.8448i	− 1.17395i	−0.809605	−	0.586975i	$-0.800319\pi$
$881$	− 34.8448i	− 1.17395i	0.809605	−	0.586975i	$-0.199681\pi$
$882$	0	0
$883$	−38.4264	−1.29315	−0.646576	−	0.762850i	$-0.723800\pi$
$883$	−38.4264	−1.29315	−0.646576	+	0.762850i	$0.723800\pi$
$884$	0	0
$885$	21.8538	0.734607
$886$	0	0
$887$	39.9581	1.34166	0.670830	−	0.741611i	$-0.265938\pi$
$887$	39.9581	1.34166	0.670830	+	0.741611i	$0.265938\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.6569	−0.390519
$892$	0	0
$893$	25.3354i	0.847818i
$894$	0	0
$895$	−65.5614	−2.19147
$896$	0	0
$897$	19.0294	0.635374
$898$	0	0
$899$	− 24.8632i	− 0.829233i
$900$	0	0
$901$	−1.09821	−0.0365866
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.72792	0.223644
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	−29.5803	−0.981115
$910$	0	0
$911$	− 5.49333i	− 0.182002i	−0.995851	−	0.0910010i	$-0.970993\pi$
$911$	− 5.49333i	− 0.182002i	0.995851	−	0.0910010i	$-0.0290066\pi$
$912$	0	0
$913$	16.9469i	0.560859i
$914$	0	0
$915$	33.2597i	1.09953i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 17.9149i	− 0.590957i	−0.955349	−	0.295478i	$-0.904521\pi$
$919$	− 17.9149i	− 0.590957i	0.955349	−	0.295478i	$-0.0954790\pi$
$920$	0	0
$921$	15.7990	0.520594
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 12.8418i	− 0.422236i
$926$	0	0
$927$	−59.1605	−1.94309
$928$	0	0
$929$	1.58513i	0.0520063i	0.999662	+	0.0260032i	$0.00827800\pi$
$929$	1.58513i	0.0520063i	−0.999662	+	0.0260032i	$0.991722\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	70.9631i	2.32323i
$934$	0	0
$935$	2.02922i	0.0663627i
$936$	0	0
$937$	− 16.9694i	− 0.554366i	−0.960817	−	0.277183i	$-0.910599\pi$
$937$	− 16.9694i	− 0.554366i	0.960817	−	0.277183i	$-0.0894008\pi$
$938$	0	0
$939$	27.6569	0.902547
$940$	0	0
$941$	−2.42386	−0.0790157	−0.0395078	−	0.999219i	$-0.512579\pi$
$941$	−2.42386	−0.0790157	−0.0395078	+	0.999219i	$0.512579\pi$
$942$	0	0
$943$	6.40083	0.208440
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−51.4558	−1.67209	−0.836045	−	0.548661i	$-0.815138\pi$
$947$	−51.4558	−1.67209	−0.836045	+	0.548661i	$0.815138\pi$
$948$	0	0
$949$	− 7.10228i	− 0.230550i
$950$	0	0
$951$	−10.6052	−0.343898
$952$	0	0
$953$	−41.3137	−1.33828	−0.669141	−	0.743135i	$-0.733338\pi$
$953$	−41.3137	−1.33828	−0.669141	+	0.743135i	$0.733338\pi$
$954$	0	0
$955$	− 37.8519i	− 1.22486i
$956$	0	0
$957$	49.0102	1.58427
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.9706	−0.773244
$962$	0	0
$963$	−32.4853	−1.04682
$964$	0	0
$965$	45.2607	1.45699
$966$	0	0
$967$	21.0308i	0.676305i	0.941091	+	0.338152i	$0.109802\pi$
$967$	21.0308i	0.676305i	−0.941091	+	0.338152i	$0.890198\pi$
$968$	0	0
$969$	− 3.95815i	− 0.127154i
$970$	0	0
$971$	24.7862i	0.795428i	0.917509	+	0.397714i	$0.130196\pi$
$971$	24.7862i	0.795428i	−0.917509	+	0.397714i	$0.869804\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	69.5282i	2.22669i
$976$	0	0
$977$	37.2132	1.19056	0.595278	−	0.803520i	$-0.297042\pi$
$977$	37.2132	1.19056	0.595278	+	0.803520i	$0.297042\pi$
$978$	0	0
$979$	19.7457i	0.631075i
$980$	0	0
$981$	− 1.60896i	− 0.0513701i
$982$	0	0
$983$	26.0582	0.831127	0.415564	−	0.909564i	$-0.363584\pi$
$983$	26.0582	0.831127	0.415564	+	0.909564i	$0.363584\pi$
$984$	0	0
$985$	− 84.8881i	− 2.70476i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 12.6677i	− 0.402810i
$990$	0	0
$991$	32.2636i	1.02489i	0.858721	+	0.512444i	$0.171260\pi$
$991$	32.2636i	1.02489i	−0.858721	+	0.512444i	$0.828740\pi$
$992$	0	0
$993$	54.4273i	1.72720i
$994$	0	0
$995$	8.48528	0.269002
$996$	0	0
$997$	−5.39683	−0.170919	−0.0854596	−	0.996342i	$-0.527236\pi$
$997$	−5.39683	−0.170919	−0.0854596	+	0.996342i	$0.527236\pi$
$998$	0	0
$999$	5.30262	0.167767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.e.c.783.1		8
4.3	odd	2		392.2.e.c.195.6	yes	8
7.2	even	3		1568.2.q.d.815.4		8
7.3	odd	6		1568.2.q.c.1391.4		8
7.4	even	3		1568.2.q.c.1391.1		8
7.5	odd	6		1568.2.q.d.815.1		8
7.6	odd	2	inner	1568.2.e.c.783.8		8
8.3	odd	2	inner	1568.2.e.c.783.2		8
8.5	even	2		392.2.e.c.195.8	yes	8
28.3	even	6		392.2.m.c.19.3		8
28.11	odd	6		392.2.m.c.19.4		8
28.19	even	6		392.2.m.e.227.2		8
28.23	odd	6		392.2.m.e.227.1		8
28.27	even	2		392.2.e.c.195.5	✓	8
56.3	even	6		1568.2.q.d.1391.4		8
56.5	odd	6		392.2.m.c.227.4		8
56.11	odd	6		1568.2.q.d.1391.1		8
56.13	odd	2		392.2.e.c.195.7	yes	8
56.19	even	6		1568.2.q.c.815.1		8
56.27	even	2	inner	1568.2.e.c.783.7		8
56.37	even	6		392.2.m.c.227.3		8
56.45	odd	6		392.2.m.e.19.1		8
56.51	odd	6		1568.2.q.c.815.4		8
56.53	even	6		392.2.m.e.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.e.c.195.5	✓	8	28.27	even	2
392.2.e.c.195.6	yes	8	4.3	odd	2
392.2.e.c.195.7	yes	8	56.13	odd	2
392.2.e.c.195.8	yes	8	8.5	even	2
392.2.m.c.19.3		8	28.3	even	6
392.2.m.c.19.4		8	28.11	odd	6
392.2.m.c.227.3		8	56.37	even	6
392.2.m.c.227.4		8	56.5	odd	6
392.2.m.e.19.1		8	56.45	odd	6
392.2.m.e.19.2		8	56.53	even	6
392.2.m.e.227.1		8	28.23	odd	6
392.2.m.e.227.2		8	28.19	even	6
1568.2.e.c.783.1		8	1.1	even	1	trivial
1568.2.e.c.783.2		8	8.3	odd	2	inner
1568.2.e.c.783.7		8	56.27	even	2	inner
1568.2.e.c.783.8		8	7.6	odd	2	inner
1568.2.q.c.815.1		8	56.19	even	6
1568.2.q.c.815.4		8	56.51	odd	6
1568.2.q.c.1391.1		8	7.4	even	3
1568.2.q.c.1391.4		8	7.3	odd	6
1568.2.q.d.815.1		8	7.5	odd	6
1568.2.q.d.815.4		8	7.2	even	3
1568.2.q.d.1391.1		8	56.11	odd	6
1568.2.q.d.1391.4		8	56.3	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61313i q^{3} -3.20041 q^{5} -3.82843 q^{9} +2.00000 q^{11} -5.07517 q^{13} +8.36308i q^{15} -0.317025i q^{17} +4.77791i q^{19} -1.43488i q^{23} +5.24264 q^{25} +2.16478i q^{27} +9.37769i q^{29} -2.65131 q^{31} -5.22625i q^{33} -2.44949i q^{37} +13.2621i q^{39} +4.46088i q^{41} +8.82843 q^{43} +12.2525 q^{45} +5.30262 q^{47} -0.828427 q^{51} -3.46410i q^{53} -6.40083 q^{55} +12.4853 q^{57} -2.61313i q^{59} +3.97696 q^{61} +16.2426 q^{65} -12.0000 q^{67} -3.74952 q^{69} +1.39942i q^{73} -13.6997i q^{75} +11.8272i q^{79} -5.82843 q^{81} +8.47343i q^{83} +1.01461i q^{85} +24.5051 q^{87} +9.87285i q^{89} +6.92820i q^{93} -15.2913i q^{95} -3.82683i q^{97} -7.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 16 q^{11} + 8 q^{25} + 48 q^{43} + 16 q^{51} + 32 q^{57} + 96 q^{65} - 96 q^{67} - 24 q^{81} - 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 + 8 * q^25 + 48 * q^43 + 16 * q^51 + 32 * q^57 + 96 * q^65 - 96 * q^67 - 24 * q^81 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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