LMFDB - Embedded newform 1568.2.e.c.783.5

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)

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")

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);

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			783.5
Root			$-0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1568.783
Dual form			1568.2.e.c.783.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239i q^{3} -1.32565 q^{5} +1.82843 q^{9} +2.00000 q^{11} -5.85172 q^{13} -1.43488i q^{15} -4.46088i q^{17} +4.14386i q^{19} +8.36308i q^{23} -3.24264 q^{25} +5.22625i q^{27} +4.47871i q^{29} +6.40083 q^{31} +2.16478i q^{33} +2.44949i q^{37} -6.33386i q^{39} -0.317025i q^{41} +3.17157 q^{43} -2.42386 q^{45} -12.8017 q^{47} +4.82843 q^{51} -3.46410i q^{53} -2.65131 q^{55} -4.48528 q^{57} +1.08239i q^{59} -9.60124 q^{61} +7.75736 q^{65} -12.0000 q^{67} -9.05213 q^{69} +7.07401i q^{73} -3.50981i q^{75} +2.02922i q^{79} -0.171573 q^{81} +5.67459i q^{83} +5.91359i q^{85} -4.84772 q^{87} +12.7486i q^{89} +6.92820i q^{93} -5.49333i q^{95} +9.23880i q^{97} +3.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.08239i	0.624919i	0.949931	+	0.312460i	$0.101153\pi$
$3$	1.08239i	0.624919i	−0.949931	+	0.312460i	$0.898847\pi$
$4$	0	0
$5$	−1.32565	−0.592851	−0.296425	−	0.955056i	$-0.595795\pi$
$5$	−1.32565	−0.592851	−0.296425	+	0.955056i	$0.595795\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.82843	0.609476
$10$	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
$13$	−5.85172	−1.62298	−0.811488	−	0.584369i	$-0.801342\pi$
$13$	−5.85172	−1.62298	−0.811488	+	0.584369i	$0.801342\pi$
$14$	0	0
$15$	− 1.43488i	− 0.370484i
$16$	0	0
$17$	− 4.46088i	− 1.08192i	−0.841047	−	0.540962i	$-0.818060\pi$
$17$	− 4.46088i	− 1.08192i	0.841047	−	0.540962i	$-0.181940\pi$
$18$	0	0
$19$	4.14386i	0.950667i	0.879806	+	0.475333i	$0.157673\pi$
$19$	4.14386i	0.950667i	−0.879806	+	0.475333i	$0.842327\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.36308i	1.74382i	0.489664	+	0.871911i	$0.337120\pi$
$23$	8.36308i	1.74382i	−0.489664	+	0.871911i	$0.662880\pi$
$24$	0	0
$25$	−3.24264	−0.648528
$26$	0	0
$27$	5.22625i	1.00579i
$28$	0	0
$29$	4.47871i	0.831676i	0.909439	+	0.415838i	$0.136512\pi$
$29$	4.47871i	0.831676i	−0.909439	+	0.415838i	$0.863488\pi$
$30$	0	0
$31$	6.40083	1.14962	0.574811	−	0.818286i	$-0.305076\pi$
$31$	6.40083	1.14962	0.574811	+	0.818286i	$0.305076\pi$
$32$	0	0
$33$	2.16478i	0.376841i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.44949i	0.402694i	0.979520	+	0.201347i	$0.0645318\pi$
$37$	2.44949i	0.402694i	−0.979520	+	0.201347i	$0.935468\pi$
$38$	0	0
$39$	− 6.33386i	− 1.01423i
$40$	0	0
$41$	− 0.317025i	− 0.0495110i	−0.999694	−	0.0247555i	$-0.992119\pi$
$41$	− 0.317025i	− 0.0495110i	0.999694	−	0.0247555i	$-0.00788073\pi$
$42$	0	0
$43$	3.17157	0.483660	0.241830	−	0.970319i	$-0.422252\pi$
$43$	3.17157	0.483660	0.241830	+	0.970319i	$0.422252\pi$
$44$	0	0
$45$	−2.42386	−0.361328
$46$	0	0
$47$	−12.8017	−1.86731	−0.933656	−	0.358170i	$-0.883401\pi$
$47$	−12.8017	−1.86731	−0.933656	+	0.358170i	$0.883401\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.82843	0.676115
$52$	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
$55$	−2.65131	−0.357502
$56$	0	0
$57$	−4.48528	−0.594090
$58$	0	0
$59$	1.08239i	0.140915i	0.997515	+	0.0704577i	$0.0224460\pi$
$59$	1.08239i	0.140915i	−0.997515	+	0.0704577i	$0.977554\pi$
$60$	0	0
$61$	−9.60124	−1.22931	−0.614656	−	0.788795i	$-0.710705\pi$
$61$	−9.60124	−1.22931	−0.614656	+	0.788795i	$0.710705\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.75736	0.962182
$66$	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
$69$	−9.05213	−1.08975
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	7.07401i	0.827950i	0.910288	+	0.413975i	$0.135860\pi$
$73$	7.07401i	0.827950i	−0.910288	+	0.413975i	$0.864140\pi$
$74$	0	0
$75$	− 3.50981i	− 0.405278i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.02922i	0.228306i	0.993463	+	0.114153i	$0.0364153\pi$
$79$	2.02922i	0.228306i	−0.993463	+	0.114153i	$0.963585\pi$
$80$	0	0
$81$	−0.171573	−0.0190637
$82$	0	0
$83$	5.67459i	0.622868i	0.950268	+	0.311434i	$0.100809\pi$
$83$	5.67459i	0.622868i	−0.950268	+	0.311434i	$0.899191\pi$
$84$	0	0
$85$	5.91359i	0.641419i
$86$	0	0
$87$	−4.84772	−0.519731
$88$	0	0
$89$	12.7486i	1.35135i	0.737200	+	0.675675i	$0.236148\pi$
$89$	12.7486i	1.35135i	−0.737200	+	0.675675i	$0.763852\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	− 5.49333i	− 0.563603i
$96$	0	0
$97$	9.23880i	0.938058i	0.883183	+	0.469029i	$0.155396\pi$
$97$	9.23880i	0.938058i	−0.883183	+	0.469029i	$0.844604\pi$
$98$	0	0
$99$	3.65685	0.367528
$100$	0	0
$101$	−0.549104	−0.0546379	−0.0273189	−	0.999627i	$-0.508697\pi$
$101$	−0.549104	−0.0546379	−0.0273189	+	0.999627i	$0.508697\pi$
$102$	0	0
$103$	−1.09821	−0.108210	−0.0541048	−	0.998535i	$-0.517231\pi$
$103$	−1.09821	−0.108210	−0.0541048	+	0.998535i	$0.517231\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	− 14.2767i	− 1.36746i	−0.729736	−	0.683729i	$-0.760357\pi$
$109$	− 14.2767i	− 1.36746i	0.729736	−	0.683729i	$-0.239643\pi$
$110$	0	0
$111$	−2.65131	−0.251651
$112$	0	0
$113$	0.485281	0.0456514	0.0228257	−	0.999739i	$-0.492734\pi$
$113$	0.485281	0.0456514	0.0228257	+	0.999739i	$0.492734\pi$
$114$	0	0
$115$	− 11.0866i	− 1.03383i
$116$	0	0
$117$	−10.6994	−0.989164
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0.343146	0.0309404
$124$	0	0
$125$	10.9269	0.977331
$126$	0	0
$127$	15.2913i	1.35688i	0.734655	+	0.678441i	$0.237344\pi$
$127$	15.2913i	1.35688i	−0.734655	+	0.678441i	$0.762656\pi$
$128$	0	0
$129$	3.43289i	0.302249i
$130$	0	0
$131$	13.6997i	1.19695i	0.801143	+	0.598473i	$0.204226\pi$
$131$	13.6997i	1.19695i	−0.801143	+	0.598473i	$0.795774\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 6.92820i	− 0.596285i
$136$	0	0
$137$	−8.24264	−0.704216	−0.352108	−	0.935959i	$-0.614535\pi$
$137$	−8.24264	−0.704216	−0.352108	+	0.935959i	$0.614535\pi$
$138$	0	0
$139$	− 17.3952i	− 1.47544i	−0.675106	−	0.737721i	$-0.735902\pi$
$139$	− 17.3952i	− 1.47544i	0.675106	−	0.737721i	$-0.264098\pi$
$140$	0	0
$141$	− 13.8564i	− 1.16692i
$142$	0	0
$143$	−11.7034	−0.978691
$144$	0	0
$145$	− 5.93723i	− 0.493060i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.594346i	0.0486907i	0.999704	+	0.0243454i	$0.00775013\pi$
$149$	0.594346i	0.0486907i	−0.999704	+	0.0243454i	$0.992250\pi$
$150$	0	0
$151$	− 5.49333i	− 0.447040i	−0.974699	−	0.223520i	$-0.928245\pi$
$151$	− 5.49333i	− 0.447040i	0.974699	−	0.223520i	$-0.0717549\pi$
$152$	0	0
$153$	− 8.15640i	− 0.659406i
$154$	0	0
$155$	−8.48528	−0.681554
$156$	0	0
$157$	12.5742	1.00353	0.501765	−	0.865004i	$-0.332684\pi$
$157$	12.5742	1.00353	0.501765	+	0.865004i	$0.332684\pi$
$158$	0	0
$159$	3.74952	0.297356
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.1421	1.57765	0.788827	−	0.614615i	$-0.210689\pi$
$163$	20.1421	1.57765	0.788827	+	0.614615i	$0.210689\pi$
$164$	0	0
$165$	− 2.86976i	− 0.223410i
$166$	0	0
$167$	−16.5512	−1.28077	−0.640384	−	0.768055i	$-0.721225\pi$
$167$	−16.5512	−1.28077	−0.640384	+	0.768055i	$0.721225\pi$
$168$	0	0
$169$	21.2426	1.63405
$170$	0	0
$171$	7.57675i	0.579408i
$172$	0	0
$173$	10.3778	0.789009	0.394504	−	0.918894i	$-0.370916\pi$
$173$	10.3778	0.789009	0.394504	+	0.918894i	$0.370916\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.17157	−0.0880608
$178$	0	0
$179$	3.51472	0.262702	0.131351	−	0.991336i	$-0.458068\pi$
$179$	3.51472	0.262702	0.131351	+	0.991336i	$0.458068\pi$
$180$	0	0
$181$	14.1273	1.05007	0.525037	−	0.851079i	$-0.324051\pi$
$181$	14.1273	1.05007	0.525037	+	0.851079i	$0.324051\pi$
$182$	0	0
$183$	− 10.3923i	− 0.768221i
$184$	0	0
$185$	− 3.24718i	− 0.238737i
$186$	0	0
$187$	− 8.92177i	− 0.652424i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.02922i	0.146829i	0.997301	+	0.0734147i	$0.0233897\pi$
$191$	2.02922i	0.146829i	−0.997301	+	0.0734147i	$0.976610\pi$
$192$	0	0
$193$	14.1421	1.01797	0.508987	−	0.860774i	$-0.330020\pi$
$193$	14.1421	1.01797	0.508987	+	0.860774i	$0.330020\pi$
$194$	0	0
$195$	8.39651i	0.601286i
$196$	0	0
$197$	− 12.6677i	− 0.902537i	−0.892388	−	0.451269i	$-0.850972\pi$
$197$	− 12.6677i	− 0.902537i	0.892388	−	0.451269i	$-0.149028\pi$
$198$	0	0
$199$	6.40083	0.453742	0.226871	−	0.973925i	$-0.427150\pi$
$199$	6.40083	0.453742	0.226871	+	0.973925i	$0.427150\pi$
$200$	0	0
$201$	− 12.9887i	− 0.916153i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.420266i	0.0293527i
$206$	0	0
$207$	15.2913i	1.06282i
$208$	0	0
$209$	8.28772i	0.573274i
$210$	0	0
$211$	−3.51472	−0.241963	−0.120982	−	0.992655i	$-0.538604\pi$
$211$	−3.51472	−0.241963	−0.120982	+	0.992655i	$0.538604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.20441	−0.286738
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−7.65685	−0.517402
$220$	0	0
$221$	26.1039i	1.75594i
$222$	0	0
$223$	4.84772	0.324628	0.162314	−	0.986739i	$-0.448104\pi$
$223$	4.84772	0.324628	0.162314	+	0.986739i	$0.448104\pi$
$224$	0	0
$225$	−5.92893	−0.395262
$226$	0	0
$227$	11.7975i	0.783029i	0.920172	+	0.391515i	$0.128049\pi$
$227$	11.7975i	0.783029i	−0.920172	+	0.391515i	$0.871951\pi$
$228$	0	0
$229$	15.2255	1.00613	0.503065	−	0.864249i	$-0.332206\pi$
$229$	15.2255	1.00613	0.503065	+	0.864249i	$0.332206\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.7574	−0.770250	−0.385125	−	0.922864i	$-0.625842\pi$
$233$	−11.7574	−0.770250	−0.385125	+	0.922864i	$0.625842\pi$
$234$	0	0
$235$	16.9706	1.10704
$236$	0	0
$237$	−2.19642	−0.142673
$238$	0	0
$239$	6.33386i	0.409703i	0.978793	+	0.204852i	$0.0656712\pi$
$239$	6.33386i	0.409703i	−0.978793	+	0.204852i	$0.934329\pi$
$240$	0	0
$241$	− 16.8155i	− 1.08318i	−0.840641	−	0.541592i	$-0.817822\pi$
$241$	− 16.8155i	− 1.08318i	0.840641	−	0.541592i	$-0.182178\pi$
$242$	0	0
$243$	15.4930i	0.993879i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 24.2487i	− 1.54291i
$248$	0	0
$249$	−6.14214	−0.389242
$250$	0	0
$251$	− 20.1940i	− 1.27464i	−0.770601	−	0.637318i	$-0.780044\pi$
$251$	− 20.1940i	− 1.27464i	0.770601	−	0.637318i	$-0.219956\pi$
$252$	0	0
$253$	16.7262i	1.05156i
$254$	0	0
$255$	−6.40083	−0.400835
$256$	0	0
$257$	1.21371i	0.0757090i	0.999283	+	0.0378545i	$0.0120523\pi$
$257$	1.21371i	0.0757090i	−0.999283	+	0.0378545i	$0.987948\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.18900i	0.506886i
$262$	0	0
$263$	18.7554i	1.15651i	0.815857	+	0.578253i	$0.196265\pi$
$263$	18.7554i	1.15651i	−0.815857	+	0.578253i	$0.803735\pi$
$264$	0	0
$265$	4.59220i	0.282097i
$266$	0	0
$267$	−13.7990	−0.844484
$268$	0	0
$269$	−12.2525	−0.747051	−0.373525	−	0.927620i	$-0.621851\pi$
$269$	−12.2525	−0.747051	−0.373525	+	0.927620i	$0.621851\pi$
$270$	0	0
$271$	32.0041	1.94411	0.972056	−	0.234749i	$-0.0754267\pi$
$271$	32.0041	1.94411	0.972056	+	0.234749i	$0.0754267\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.48528	−0.391077
$276$	0	0
$277$	2.86976i	0.172427i	0.996277	+	0.0862135i	$0.0274767\pi$
$277$	2.86976i	0.172427i	−0.996277	+	0.0862135i	$0.972523\pi$
$278$	0	0
$279$	11.7034	0.700667
$280$	0	0
$281$	−16.7279	−0.997904	−0.498952	−	0.866630i	$-0.666282\pi$
$281$	−16.7279	−0.997904	−0.498952	+	0.866630i	$0.666282\pi$
$282$	0	0
$283$	− 15.2304i	− 0.905354i	−0.891675	−	0.452677i	$-0.850469\pi$
$283$	− 15.2304i	− 0.905354i	0.891675	−	0.452677i	$-0.149531\pi$
$284$	0	0
$285$	5.94593	0.352207
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.89949	−0.170559
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	5.85172	0.341861	0.170931	−	0.985283i	$-0.445323\pi$
$293$	5.85172	0.341861	0.170931	+	0.985283i	$0.445323\pi$
$294$	0	0
$295$	− 1.43488i	− 0.0835418i
$296$	0	0
$297$	10.4525i	0.606516i
$298$	0	0
$299$	− 48.9384i	− 2.83018i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 0.594346i	− 0.0341443i
$304$	0	0
$305$	12.7279	0.728799
$306$	0	0
$307$	21.9874i	1.25489i	0.778662	+	0.627444i	$0.215899\pi$
$307$	21.9874i	1.25489i	−0.778662	+	0.627444i	$0.784101\pi$
$308$	0	0
$309$	− 1.18869i	− 0.0676223i
$310$	0	0
$311$	11.2485	0.637847	0.318923	−	0.947781i	$-0.396679\pi$
$311$	11.2485	0.637847	0.318923	+	0.947781i	$0.396679\pi$
$312$	0	0
$313$	− 15.0991i	− 0.853451i	−0.904381	−	0.426726i	$-0.859667\pi$
$313$	− 15.0991i	− 0.853451i	0.904381	−	0.426726i	$-0.140333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.6544i	− 1.32856i	−0.747483	−	0.664281i	$-0.768738\pi$
$317$	− 23.6544i	− 1.32856i	0.747483	−	0.664281i	$-0.231262\pi$
$318$	0	0
$319$	8.95743i	0.501520i
$320$	0	0
$321$	− 9.18440i	− 0.512623i
$322$	0	0
$323$	18.4853	1.02855
$324$	0	0
$325$	18.9750	1.05255
$326$	0	0
$327$	15.4530	0.854551
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.1716	−0.833905	−0.416953	−	0.908928i	$-0.636902\pi$
$331$	−15.1716	−0.833905	−0.416953	+	0.908928i	$0.636902\pi$
$332$	0	0
$333$	4.47871i	0.245432i
$334$	0	0
$335$	15.9079	0.869139
$336$	0	0
$337$	−4.24264	−0.231111	−0.115556	−	0.993301i	$-0.536865\pi$
$337$	−4.24264	−0.231111	−0.115556	+	0.993301i	$0.536865\pi$
$338$	0	0
$339$	0.525265i	0.0285285i
$340$	0	0
$341$	12.8017	0.693248
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	26.9706	1.44786	0.723928	−	0.689876i	$-0.242335\pi$
$347$	26.9706	1.44786	0.723928	+	0.689876i	$0.242335\pi$
$348$	0	0
$349$	−9.27958	−0.496725	−0.248362	−	0.968667i	$-0.579892\pi$
$349$	−9.27958	−0.496725	−0.248362	+	0.968667i	$0.579892\pi$
$350$	0	0
$351$	− 30.5826i	− 1.63238i
$352$	0	0
$353$	20.5880i	1.09579i	0.836548	+	0.547894i	$0.184570\pi$
$353$	20.5880i	1.09579i	−0.836548	+	0.547894i	$0.815430\pi$
$354$	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
$361$	1.82843	0.0962330
$362$	0	0
$363$	− 7.57675i	− 0.397676i
$364$	0	0
$365$	− 9.37769i	− 0.490851i
$366$	0	0
$367$	22.9520	1.19808	0.599042	−	0.800718i	$-0.295548\pi$
$367$	22.9520	1.19808	0.599042	+	0.800718i	$0.295548\pi$
$368$	0	0
$369$	− 0.579658i	− 0.0301758i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 29.9882i	− 1.55273i	−0.630283	−	0.776366i	$-0.717061\pi$
$373$	− 29.9882i	− 1.55273i	0.630283	−	0.776366i	$-0.282939\pi$
$374$	0	0
$375$	11.8272i	0.610753i
$376$	0	0
$377$	− 26.2082i	− 1.34979i
$378$	0	0
$379$	5.31371	0.272947	0.136473	−	0.990644i	$-0.456423\pi$
$379$	5.31371	0.272947	0.136473	+	0.990644i	$0.456423\pi$
$380$	0	0
$381$	−16.5512	−0.847942
$382$	0	0
$383$	−12.8017	−0.654134	−0.327067	−	0.945001i	$-0.606060\pi$
$383$	−12.8017	−0.654134	−0.327067	+	0.945001i	$0.606060\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.79899	0.294779
$388$	0	0
$389$	37.9310i	1.92318i	0.274490	+	0.961590i	$0.411491\pi$
$389$	37.9310i	1.92318i	−0.274490	+	0.961590i	$0.588509\pi$
$390$	0	0
$391$	37.3067	1.88668
$392$	0	0
$393$	−14.8284	−0.747995
$394$	0	0
$395$	− 2.69005i	− 0.135351i
$396$	0	0
$397$	13.8056	0.692886	0.346443	−	0.938071i	$-0.387389\pi$
$397$	13.8056	0.692886	0.346443	+	0.938071i	$0.387389\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.2426	0.811119	0.405559	−	0.914069i	$-0.367077\pi$
$401$	16.2426	0.811119	0.405559	+	0.914069i	$0.367077\pi$
$402$	0	0
$403$	−37.4558	−1.86581
$404$	0	0
$405$	0.227446	0.0113019
$406$	0	0
$407$	4.89898i	0.242833i
$408$	0	0
$409$	− 9.23880i	− 0.456829i	−0.973564	−	0.228415i	$-0.926646\pi$
$409$	− 9.23880i	− 0.456829i	0.973564	−	0.228415i	$-0.0733541\pi$
$410$	0	0
$411$	− 8.92177i	− 0.440078i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 7.52255i	− 0.369267i
$416$	0	0
$417$	18.8284	0.922032
$418$	0	0
$419$	7.31411i	0.357318i	0.983911	+	0.178659i	$0.0571759\pi$
$419$	7.31411i	0.357318i	−0.983911	+	0.178659i	$0.942824\pi$
$420$	0	0
$421$	12.6677i	0.617387i	0.951162	+	0.308693i	$0.0998917\pi$
$421$	12.6677i	0.617387i	−0.951162	+	0.308693i	$0.900108\pi$
$422$	0	0
$423$	−23.4069	−1.13808
$424$	0	0
$425$	14.4650i	0.701658i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 12.6677i	− 0.611603i
$430$	0	0
$431$	− 27.1185i	− 1.30625i	−0.757250	−	0.653125i	$-0.773457\pi$
$431$	− 27.1185i	− 1.30625i	0.757250	−	0.653125i	$-0.226543\pi$
$432$	0	0
$433$	− 28.1647i	− 1.35351i	−0.736208	−	0.676755i	$-0.763386\pi$
$433$	− 28.1647i	− 1.35351i	0.736208	−	0.676755i	$-0.236614\pi$
$434$	0	0
$435$	6.42641	0.308123
$436$	0	0
$437$	−34.6554	−1.65779
$438$	0	0
$439$	22.3087	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$439$	22.3087	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.2843	−1.34383	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843	−1.34383	−0.671913	+	0.740630i	$0.734527\pi$
$444$	0	0
$445$	− 16.9002i	− 0.801148i
$446$	0	0
$447$	−0.643315	−0.0304278
$448$	0	0
$449$	5.65685	0.266963	0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685	0.266963	0.133482	+	0.991051i	$0.457384\pi$
$450$	0	0
$451$	− 0.634051i	− 0.0298563i
$452$	0	0
$453$	5.94593	0.279364
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.8284	−1.25498	−0.627490	−	0.778624i	$-0.715918\pi$
$457$	−26.8284	−1.25498	−0.627490	+	0.778624i	$0.715918\pi$
$458$	0	0
$459$	23.3137	1.08819
$460$	0	0
$461$	18.9750	0.883755	0.441878	−	0.897075i	$-0.354313\pi$
$461$	18.9750	0.883755	0.441878	+	0.897075i	$0.354313\pi$
$462$	0	0
$463$	− 8.95743i	− 0.416287i	−0.978098	−	0.208143i	$-0.933258\pi$
$463$	− 8.95743i	− 0.416287i	0.978098	−	0.208143i	$-0.0667421\pi$
$464$	0	0
$465$	− 9.18440i	− 0.425916i
$466$	0	0
$467$	36.7695i	1.70149i	0.525580	+	0.850744i	$0.323848\pi$
$467$	36.7695i	1.70149i	−0.525580	+	0.850744i	$0.676152\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	13.6102i	0.627126i
$472$	0	0
$473$	6.34315	0.291658
$474$	0	0
$475$	− 13.4370i	− 0.616534i
$476$	0	0
$477$	− 6.33386i	− 0.290007i
$478$	0	0
$479$	−10.1503	−0.463781	−0.231890	−	0.972742i	$-0.574491\pi$
$479$	−10.1503	−0.463781	−0.231890	+	0.972742i	$0.574491\pi$
$480$	0	0
$481$	− 14.3337i	− 0.653562i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 12.2474i	− 0.556128i
$486$	0	0
$487$	− 14.4508i	− 0.654826i	−0.944881	−	0.327413i	$-0.893823\pi$
$487$	− 14.4508i	− 0.654826i	0.944881	−	0.327413i	$-0.106177\pi$
$488$	0	0
$489$	21.8017i	0.985907i
$490$	0	0
$491$	−7.79899	−0.351963	−0.175982	−	0.984393i	$-0.556310\pi$
$491$	−7.79899	−0.351963	−0.175982	+	0.984393i	$0.556310\pi$
$492$	0	0
$493$	19.9790	0.899810
$494$	0	0
$495$	−4.84772	−0.217889
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.48528	−0.379853	−0.189927	−	0.981798i	$-0.560825\pi$
$499$	−8.48528	−0.379853	−0.189927	+	0.981798i	$0.560825\pi$
$500$	0	0
$501$	− 17.9149i	− 0.800377i
$502$	0	0
$503$	4.84772	0.216149	0.108075	−	0.994143i	$-0.465531\pi$
$503$	4.84772	0.216149	0.108075	+	0.994143i	$0.465531\pi$
$504$	0	0
$505$	0.727922	0.0323921
$506$	0	0
$507$	22.9929i	1.02115i
$508$	0	0
$509$	2.10220	0.0931786	0.0465893	−	0.998914i	$-0.485165\pi$
$509$	2.10220	0.0931786	0.0465893	+	0.998914i	$0.485165\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−21.6569	−0.956173
$514$	0	0
$515$	1.45584	0.0641522
$516$	0	0
$517$	−25.6033	−1.12603
$518$	0	0
$519$	11.2328i	0.493067i
$520$	0	0
$521$	3.11586i	0.136508i	0.997668	+	0.0682542i	$0.0217429\pi$
$521$	3.11586i	0.136508i	−0.997668	+	0.0682542i	$0.978257\pi$
$522$	0	0
$523$	− 20.0852i	− 0.878267i	−0.898422	−	0.439133i	$-0.855286\pi$
$523$	− 20.0852i	− 0.878267i	0.898422	−	0.439133i	$-0.144714\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.5533i	− 1.24380i
$528$	0	0
$529$	−46.9411	−2.04092
$530$	0	0
$531$	1.97908i	0.0858846i
$532$	0	0
$533$	1.85514i	0.0803552i
$534$	0	0
$535$	11.2485	0.486317
$536$	0	0
$537$	3.80430i	0.164168i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	25.9298i	1.11481i	0.830241	+	0.557404i	$0.188203\pi$
$541$	25.9298i	1.11481i	−0.830241	+	0.557404i	$0.811797\pi$
$542$	0	0
$543$	15.2913i	0.656212i
$544$	0	0
$545$	18.9259i	0.810698i
$546$	0	0
$547$	20.8284	0.890559	0.445280	−	0.895392i	$-0.353104\pi$
$547$	20.8284	0.890559	0.445280	+	0.895392i	$0.353104\pi$
$548$	0	0
$549$	−17.5552	−0.749236
$550$	0	0
$551$	−18.5592	−0.790647
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.51472	0.149191
$556$	0	0
$557$	2.86976i	0.121595i	0.998150	+	0.0607977i	$0.0193645\pi$
$557$	2.86976i	0.121595i	−0.998150	+	0.0607977i	$0.980636\pi$
$558$	0	0
$559$	−18.5592	−0.784969
$560$	0	0
$561$	9.65685	0.407713
$562$	0	0
$563$	3.88123i	0.163574i	0.996650	+	0.0817871i	$0.0260628\pi$
$563$	3.88123i	0.163574i	−0.996650	+	0.0817871i	$0.973937\pi$
$564$	0	0
$565$	−0.643315	−0.0270645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.8995	1.42114	0.710570	−	0.703626i	$-0.248437\pi$
$569$	33.8995	1.42114	0.710570	+	0.703626i	$0.248437\pi$
$570$	0	0
$571$	12.3431	0.516545	0.258272	−	0.966072i	$-0.416847\pi$
$571$	12.3431	0.516545	0.258272	+	0.966072i	$0.416847\pi$
$572$	0	0
$573$	−2.19642	−0.0917566
$574$	0	0
$575$	− 27.1185i	− 1.13092i
$576$	0	0
$577$	− 20.7737i	− 0.864820i	−0.901677	−	0.432410i	$-0.857663\pi$
$577$	− 20.7737i	− 0.864820i	0.901677	−	0.432410i	$-0.142337\pi$
$578$	0	0
$579$	15.3073i	0.636151i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 6.92820i	− 0.286937i
$584$	0	0
$585$	14.1838	0.586427
$586$	0	0
$587$	− 15.7557i	− 0.650306i	−0.945661	−	0.325153i	$-0.894584\pi$
$587$	− 15.7557i	− 0.650306i	0.945661	−	0.325153i	$-0.105416\pi$
$588$	0	0
$589$	26.5241i	1.09291i
$590$	0	0
$591$	13.7114	0.564013
$592$	0	0
$593$	− 3.56420i	− 0.146364i	−0.997319	−	0.0731821i	$-0.976685\pi$
$593$	− 3.56420i	− 0.146364i	0.997319	−	0.0731821i	$-0.0233154\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.92820i	0.283552i
$598$	0	0
$599$	6.08767i	0.248736i	0.992236	+	0.124368i	$0.0396903\pi$
$599$	6.08767i	0.248736i	−0.992236	+	0.124368i	$0.960310\pi$
$600$	0	0
$601$	− 22.2275i	− 0.906679i	−0.891338	−	0.453339i	$-0.850233\pi$
$601$	− 22.2275i	− 0.906679i	0.891338	−	0.453339i	$-0.149767\pi$
$602$	0	0
$603$	−21.9411	−0.893512
$604$	0	0
$605$	9.27958	0.377269
$606$	0	0
$607$	10.6052	0.430453	0.215227	−	0.976564i	$-0.430951\pi$
$607$	10.6052	0.430453	0.215227	+	0.976564i	$0.430951\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	74.9117	3.03060
$612$	0	0
$613$	− 26.9444i	− 1.08827i	−0.838997	−	0.544137i	$-0.816857\pi$
$613$	− 26.9444i	− 1.08827i	0.838997	−	0.544137i	$-0.183143\pi$
$614$	0	0
$615$	−0.454893	−0.0183430
$616$	0	0
$617$	33.2132	1.33711	0.668557	−	0.743661i	$-0.266912\pi$
$617$	33.2132	1.33711	0.668557	+	0.743661i	$0.266912\pi$
$618$	0	0
$619$	18.5545i	0.745769i	0.927878	+	0.372884i	$0.121631\pi$
$619$	18.5545i	0.745769i	−0.927878	+	0.372884i	$0.878369\pi$
$620$	0	0
$621$	−43.7076	−1.75392
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.72792	0.0691169
$626$	0	0
$627$	−8.97056	−0.358250
$628$	0	0
$629$	10.9269	0.435684
$630$	0	0
$631$	39.5400i	1.57406i	0.616913	+	0.787031i	$0.288383\pi$
$631$	39.5400i	1.57406i	−0.616913	+	0.787031i	$0.711617\pi$
$632$	0	0
$633$	− 3.80430i	− 0.151208i
$634$	0	0
$635$	− 20.2710i	− 0.804428i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31.0711	−1.22723	−0.613617	−	0.789604i	$-0.710286\pi$
$641$	−31.0711	−1.22723	−0.613617	+	0.789604i	$0.710286\pi$
$642$	0	0
$643$	30.3839i	1.19822i	0.800665	+	0.599112i	$0.204480\pi$
$643$	30.3839i	1.19822i	−0.800665	+	0.599112i	$0.795520\pi$
$644$	0	0
$645$	− 4.55082i	− 0.179188i
$646$	0	0
$647$	−10.1503	−0.399051	−0.199526	−	0.979893i	$-0.563940\pi$
$647$	−10.1503	−0.399051	−0.199526	+	0.979893i	$0.563940\pi$
$648$	0	0
$649$	2.16478i	0.0849752i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.3643i	0.796918i	0.917186	+	0.398459i	$0.130455\pi$
$653$	20.3643i	0.796918i	−0.917186	+	0.398459i	$0.869545\pi$
$654$	0	0
$655$	− 18.1610i	− 0.709611i
$656$	0	0
$657$	12.9343i	0.504616i
$658$	0	0
$659$	−42.4853	−1.65499	−0.827496	−	0.561472i	$-0.810235\pi$
$659$	−42.4853	−1.65499	−0.827496	+	0.561472i	$0.810235\pi$
$660$	0	0
$661$	31.1334	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$661$	31.1334	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$662$	0	0
$663$	−28.2546	−1.09732
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−37.4558	−1.45030
$668$	0	0
$669$	5.24714i	0.202866i
$670$	0	0
$671$	−19.2025	−0.741303
$672$	0	0
$673$	−6.38478	−0.246115	−0.123058	−	0.992400i	$-0.539270\pi$
$673$	−6.38478	−0.246115	−0.123058	+	0.992400i	$0.539270\pi$
$674$	0	0
$675$	− 16.9469i	− 0.652285i
$676$	0	0
$677$	−29.5803	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$677$	−29.5803	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.7696	−0.489330
$682$	0	0
$683$	−14.1421	−0.541134	−0.270567	−	0.962701i	$-0.587211\pi$
$683$	−14.1421	−0.541134	−0.270567	+	0.962701i	$0.587211\pi$
$684$	0	0
$685$	10.9269	0.417495
$686$	0	0
$687$	16.4800i	0.628750i
$688$	0	0
$689$	20.2710i	0.772262i
$690$	0	0
$691$	40.7276i	1.54935i	0.632358	+	0.774676i	$0.282087\pi$
$691$	40.7276i	1.54935i	−0.632358	+	0.774676i	$0.717913\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.0600i	0.874716i
$696$	0	0
$697$	−1.41421	−0.0535672
$698$	0	0
$699$	− 12.7261i	− 0.481344i
$700$	0	0
$701$	− 47.7290i	− 1.80270i	−0.433092	−	0.901350i	$-0.642578\pi$
$701$	− 47.7290i	− 1.80270i	0.433092	−	0.901350i	$-0.357422\pi$
$702$	0	0
$703$	−10.1503	−0.382827
$704$	0	0
$705$	18.3688i	0.691809i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.7423i	1.37989i	0.723863	+	0.689944i	$0.242365\pi$
$709$	36.7423i	1.37989i	−0.723863	+	0.689944i	$0.757635\pi$
$710$	0	0
$711$	3.71029i	0.139147i
$712$	0	0
$713$	53.5306i	2.00474i
$714$	0	0
$715$	15.5147	0.580218
$716$	0	0
$717$	−6.85572	−0.256031
$718$	0	0
$719$	−19.6574	−0.733096	−0.366548	−	0.930399i	$-0.619460\pi$
$719$	−19.6574	−0.733096	−0.366548	+	0.930399i	$0.619460\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.2010	0.676903
$724$	0	0
$725$	− 14.5229i	− 0.539365i
$726$	0	0
$727$	−49.6535	−1.84155	−0.920773	−	0.390098i	$-0.872441\pi$
$727$	−49.6535	−1.84155	−0.920773	+	0.390098i	$0.872441\pi$
$728$	0	0
$729$	−17.2843	−0.640158
$730$	0	0
$731$	− 14.1480i	− 0.523283i
$732$	0	0
$733$	26.9290	0.994644	0.497322	−	0.867566i	$-0.334317\pi$
$733$	26.9290	0.994644	0.497322	+	0.867566i	$0.334317\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−20.8284	−0.766186	−0.383093	−	0.923710i	$-0.625141\pi$
$739$	−20.8284	−0.766186	−0.383093	+	0.923710i	$0.625141\pi$
$740$	0	0
$741$	26.2466	0.964194
$742$	0	0
$743$	− 6.33386i	− 0.232367i	−0.993228	−	0.116183i	$-0.962934\pi$
$743$	− 6.33386i	− 0.232367i	0.993228	−	0.116183i	$-0.0370660\pi$
$744$	0	0
$745$	− 0.787897i	− 0.0288663i
$746$	0	0
$747$	10.3756i	0.379623i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	42.6559i	1.55654i	0.627931	+	0.778269i	$0.283902\pi$
$751$	42.6559i	1.55654i	−0.627931	+	0.778269i	$0.716098\pi$
$752$	0	0
$753$	21.8579	0.796545
$754$	0	0
$755$	7.28225i	0.265028i
$756$	0	0
$757$	− 8.18900i	− 0.297634i	−0.988865	−	0.148817i	$-0.952453\pi$
$757$	− 8.18900i	− 0.297634i	0.988865	−	0.148817i	$-0.0475466\pi$
$758$	0	0
$759$	−18.1043	−0.657143
$760$	0	0
$761$	− 4.72352i	− 0.171227i	−0.996328	−	0.0856137i	$-0.972715\pi$
$761$	− 4.72352i	− 0.171227i	0.996328	−	0.0856137i	$-0.0272851\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.8126i	0.390929i
$766$	0	0
$767$	− 6.33386i	− 0.228702i
$768$	0	0
$769$	19.1342i	0.689996i	0.938603	+	0.344998i	$0.112120\pi$
$769$	19.1342i	0.689996i	−0.938603	+	0.344998i	$0.887880\pi$
$770$	0	0
$771$	−1.31371	−0.0473121
$772$	0	0
$773$	42.8368	1.54073	0.770366	−	0.637601i	$-0.220073\pi$
$773$	42.8368	1.54073	0.770366	+	0.637601i	$0.220073\pi$
$774$	0	0
$775$	−20.7556	−0.745562
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.31371	0.0470685
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−23.4069	−0.836494
$784$	0	0
$785$	−16.6690	−0.594944
$786$	0	0
$787$	− 46.6967i	− 1.66456i	−0.554357	−	0.832279i	$-0.687036\pi$
$787$	− 46.6967i	− 1.66456i	0.554357	−	0.832279i	$-0.312964\pi$
$788$	0	0
$789$	−20.3007	−0.722723
$790$	0	0
$791$	0	0
$792$	0	0
$793$	56.1838	1.99514
$794$	0	0
$795$	−4.97056	−0.176288
$796$	0	0
$797$	−50.6575	−1.79438	−0.897190	−	0.441644i	$-0.854395\pi$
$797$	−50.6575	−1.79438	−0.897190	+	0.441644i	$0.854395\pi$
$798$	0	0
$799$	57.1067i	2.02029i
$800$	0	0
$801$	23.3099i	0.823615i
$802$	0	0
$803$	14.1480i	0.499273i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.2621i	− 0.466847i
$808$	0	0
$809$	35.4558	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$809$	35.4558	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$810$	0	0
$811$	9.63274i	0.338251i	0.985594	+	0.169126i	$0.0540944\pi$
$811$	9.63274i	0.338251i	−0.985594	+	0.169126i	$0.945906\pi$
$812$	0	0
$813$	34.6410i	1.21491i
$814$	0	0
$815$	−26.7015	−0.935313
$816$	0	0
$817$	13.1426i	0.459800i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.5108i	1.30913i	0.756004	+	0.654567i	$0.227149\pi$
$821$	37.5108i	1.30913i	−0.756004	+	0.654567i	$0.772851\pi$
$822$	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
$825$	− 7.01962i	− 0.244392i
$826$	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
$829$	4.29862	0.149297	0.0746486	−	0.997210i	$-0.476216\pi$
$829$	4.29862	0.149297	0.0746486	+	0.997210i	$0.476216\pi$
$830$	0	0
$831$	−3.10620	−0.107753
$832$	0	0
$833$	0	0
$834$	0	0
$835$	21.9411	0.759304
$836$	0	0
$837$	33.4523i	1.15628i
$838$	0	0
$839$	−35.1103	−1.21214	−0.606072	−	0.795410i	$-0.707255\pi$
$839$	−35.1103	−1.21214	−0.606072	+	0.795410i	$0.707255\pi$
$840$	0	0
$841$	8.94113	0.308315
$842$	0	0
$843$	− 18.1062i	− 0.623610i
$844$	0	0
$845$	−28.1604	−0.968747
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.4853	0.565773
$850$	0	0
$851$	−20.4853	−0.702226
$852$	0	0
$853$	−20.9830	−0.718445	−0.359223	−	0.933252i	$-0.616958\pi$
$853$	−20.9830	−0.718445	−0.359223	+	0.933252i	$0.616958\pi$
$854$	0	0
$855$	− 10.0441i	− 0.343503i
$856$	0	0
$857$	− 5.62020i	− 0.191982i	−0.995382	−	0.0959912i	$-0.969398\pi$
$857$	− 5.62020i	− 0.191982i	0.995382	−	0.0959912i	$-0.0306021\pi$
$858$	0	0
$859$	− 33.5992i	− 1.14639i	−0.819419	−	0.573195i	$-0.805704\pi$
$859$	− 33.5992i	− 1.14639i	0.819419	−	0.573195i	$-0.194296\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.6544i	0.805204i	0.915375	+	0.402602i	$0.131894\pi$
$863$	23.6544i	0.805204i	−0.915375	+	0.402602i	$0.868106\pi$
$864$	0	0
$865$	−13.7574	−0.467764
$866$	0	0
$867$	− 3.13839i	− 0.106585i
$868$	0	0
$869$	4.05845i	0.137673i
$870$	0	0
$871$	70.2207	2.37934
$872$	0	0
$873$	16.8925i	0.571723i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37.0905i	− 1.25246i	−0.779639	−	0.626229i	$-0.784598\pi$
$877$	− 37.0905i	− 1.25246i	0.779639	−	0.626229i	$-0.215402\pi$
$878$	0	0
$879$	6.33386i	0.213636i
$880$	0	0
$881$	− 36.0810i	− 1.21560i	−0.794090	−	0.607800i	$-0.792052\pi$
$881$	− 36.0810i	− 1.21560i	0.794090	−	0.607800i	$-0.207948\pi$
$882$	0	0
$883$	46.4264	1.56237	0.781186	−	0.624298i	$-0.214615\pi$
$883$	46.4264	1.56237	0.781186	+	0.624298i	$0.214615\pi$
$884$	0	0
$885$	1.55310	0.0522069
$886$	0	0
$887$	−5.94593	−0.199645	−0.0998224	−	0.995005i	$-0.531827\pi$
$887$	−5.94593	−0.199645	−0.0998224	+	0.995005i	$0.531827\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.343146	−0.0114958
$892$	0	0
$893$	− 53.0482i	− 1.77519i
$894$	0	0
$895$	−4.65930	−0.155743
$896$	0	0
$897$	52.9706	1.76864
$898$	0	0
$899$	28.6675i	0.956113i
$900$	0	0
$901$	−15.4530	−0.514813
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.7279	−0.622537
$906$	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
$909$	−1.00400	−0.0333005
$910$	0	0
$911$	− 15.2913i	− 0.506623i	−0.967385	−	0.253311i	$-0.918480\pi$
$911$	− 15.2913i	− 0.506623i	0.967385	−	0.253311i	$-0.0815197\pi$
$912$	0	0
$913$	11.3492i	0.375603i
$914$	0	0
$915$	13.7766i	0.455440i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 37.5108i	− 1.23737i	−0.785641	−	0.618683i	$-0.787666\pi$
$919$	− 37.5108i	− 1.23737i	0.785641	−	0.618683i	$-0.212334\pi$
$920$	0	0
$921$	−23.7990	−0.784203
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 7.94282i	− 0.261158i
$926$	0	0
$927$	−2.00799	−0.0659512
$928$	0	0
$929$	22.3044i	0.731784i	0.930657	+	0.365892i	$0.119236\pi$
$929$	22.3044i	0.731784i	−0.930657	+	0.365892i	$0.880764\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.1753i	0.398603i
$934$	0	0
$935$	11.8272i	0.386790i
$936$	0	0
$937$	51.4202i	1.67983i	0.542721	+	0.839913i	$0.317394\pi$
$937$	51.4202i	1.67983i	−0.542721	+	0.839913i	$0.682606\pi$
$938$	0	0
$939$	16.3431	0.533338
$940$	0	0
$941$	−12.2525	−0.399422	−0.199711	−	0.979855i	$-0.564000\pi$
$941$	−12.2525	−0.399422	−0.199711	+	0.979855i	$0.564000\pi$
$942$	0	0
$943$	2.65131	0.0863385
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.544156	−0.0176827	−0.00884135	−	0.999961i	$-0.502814\pi$
$947$	−0.544156	−0.0176827	−0.00884135	+	0.999961i	$0.502814\pi$
$948$	0	0
$949$	− 41.3951i	− 1.34374i
$950$	0	0
$951$	25.6033	0.830244
$952$	0	0
$953$	−18.6863	−0.605308	−0.302654	−	0.953100i	$-0.597873\pi$
$953$	−18.6863	−0.605308	−0.302654	+	0.953100i	$0.597873\pi$
$954$	0	0
$955$	− 2.69005i	− 0.0870479i
$956$	0	0
$957$	−9.69545	−0.313409
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.97056	0.321631
$962$	0	0
$963$	−15.5147	−0.499955
$964$	0	0
$965$	−18.7476	−0.603506
$966$	0	0
$967$	− 27.9590i	− 0.899101i	−0.893255	−	0.449550i	$-0.851584\pi$
$967$	− 27.9590i	− 0.899101i	0.893255	−	0.449550i	$-0.148416\pi$
$968$	0	0
$969$	20.0083i	0.642760i
$970$	0	0
$971$	8.10201i	0.260006i	0.991514	+	0.130003i	$0.0414987\pi$
$971$	8.10201i	0.260006i	−0.991514	+	0.130003i	$0.958501\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	20.5384i	0.657756i
$976$	0	0
$977$	−5.21320	−0.166785	−0.0833926	−	0.996517i	$-0.526576\pi$
$977$	−5.21320	−0.166785	−0.0833926	+	0.996517i	$0.526576\pi$
$978$	0	0
$979$	25.4972i	0.814894i
$980$	0	0
$981$	− 26.1039i	− 0.833432i
$982$	0	0
$983$	−26.7015	−0.851646	−0.425823	−	0.904807i	$-0.640015\pi$
$983$	−26.7015	−0.851646	−0.425823	+	0.904807i	$0.640015\pi$
$984$	0	0
$985$	16.7930i	0.535070i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.5241i	0.843418i
$990$	0	0
$991$	− 46.1200i	− 1.46505i	−0.680739	−	0.732526i	$-0.738341\pi$
$991$	− 46.1200i	− 1.46505i	0.680739	−	0.732526i	$-0.261659\pi$
$992$	0	0
$993$	− 16.4216i	− 0.521123i
$994$	0	0
$995$	−8.48528	−0.269002
$996$	0	0
$997$	−32.2316	−1.02078	−0.510392	−	0.859942i	$-0.670500\pi$
$997$	−32.2316	−1.02078	−0.510392	+	0.859942i	$0.670500\pi$
$998$	0	0
$999$	−12.8017	−0.405026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.e.c.783.5		8
4.3	odd	2		392.2.e.c.195.3	yes	8
7.2	even	3		1568.2.q.d.815.2		8
7.3	odd	6		1568.2.q.c.1391.2		8
7.4	even	3		1568.2.q.c.1391.3		8
7.5	odd	6		1568.2.q.d.815.3		8
7.6	odd	2	inner	1568.2.e.c.783.4		8
8.3	odd	2	inner	1568.2.e.c.783.6		8
8.5	even	2		392.2.e.c.195.1	✓	8
28.3	even	6		392.2.m.c.19.2		8
28.11	odd	6		392.2.m.c.19.1		8
28.19	even	6		392.2.m.e.227.3		8
28.23	odd	6		392.2.m.e.227.4		8
28.27	even	2		392.2.e.c.195.4	yes	8
56.3	even	6		1568.2.q.d.1391.2		8
56.5	odd	6		392.2.m.c.227.1		8
56.11	odd	6		1568.2.q.d.1391.3		8
56.13	odd	2		392.2.e.c.195.2	yes	8
56.19	even	6		1568.2.q.c.815.3		8
56.27	even	2	inner	1568.2.e.c.783.3		8
56.37	even	6		392.2.m.c.227.2		8
56.45	odd	6		392.2.m.e.19.4		8
56.51	odd	6		1568.2.q.c.815.2		8
56.53	even	6		392.2.m.e.19.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.e.c.195.1	✓	8	8.5	even	2
392.2.e.c.195.2	yes	8	56.13	odd	2
392.2.e.c.195.3	yes	8	4.3	odd	2
392.2.e.c.195.4	yes	8	28.27	even	2
392.2.m.c.19.1		8	28.11	odd	6
392.2.m.c.19.2		8	28.3	even	6
392.2.m.c.227.1		8	56.5	odd	6
392.2.m.c.227.2		8	56.37	even	6
392.2.m.e.19.3		8	56.53	even	6
392.2.m.e.19.4		8	56.45	odd	6
392.2.m.e.227.3		8	28.19	even	6
392.2.m.e.227.4		8	28.23	odd	6
1568.2.e.c.783.3		8	56.27	even	2	inner
1568.2.e.c.783.4		8	7.6	odd	2	inner
1568.2.e.c.783.5		8	1.1	even	1	trivial
1568.2.e.c.783.6		8	8.3	odd	2	inner
1568.2.q.c.815.2		8	56.51	odd	6
1568.2.q.c.815.3		8	56.19	even	6
1568.2.q.c.1391.2		8	7.3	odd	6
1568.2.q.c.1391.3		8	7.4	even	3
1568.2.q.d.815.2		8	7.2	even	3
1568.2.q.d.815.3		8	7.5	odd	6
1568.2.q.d.1391.2		8	56.3	even	6
1568.2.q.d.1391.3		8	56.11	odd	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)

\(f(q)\)	\(=\)	\(q+1.08239i q^{3} -1.32565 q^{5} +1.82843 q^{9} +2.00000 q^{11} -5.85172 q^{13} -1.43488i q^{15} -4.46088i q^{17} +4.14386i q^{19} +8.36308i q^{23} -3.24264 q^{25} +5.22625i q^{27} +4.47871i q^{29} +6.40083 q^{31} +2.16478i q^{33} +2.44949i q^{37} -6.33386i q^{39} -0.317025i q^{41} +3.17157 q^{43} -2.42386 q^{45} -12.8017 q^{47} +4.82843 q^{51} -3.46410i q^{53} -2.65131 q^{55} -4.48528 q^{57} +1.08239i q^{59} -9.60124 q^{61} +7.75736 q^{65} -12.0000 q^{67} -9.05213 q^{69} +7.07401i q^{73} -3.50981i q^{75} +2.02922i q^{79} -0.171573 q^{81} +5.67459i q^{83} +5.91359i q^{85} -4.84772 q^{87} +12.7486i q^{89} +6.92820i q^{93} -5.49333i q^{95} +9.23880i q^{97} +3.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 16 q^{11} + 8 q^{25} + 48 q^{43} + 16 q^{51} + 32 q^{57} + 96 q^{65} - 96 q^{67} - 24 q^{81} - 16 q^{99}+O(q^{100}) \) 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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