LMFDB - Embedded newform 6336.2.d.e.3455.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3455,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.3455"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1768034304.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 9x^{6} - 2x^{5} + 34x^{4} - 18x^{3} + 51x^{2} + 18x + 9 $ x^8 - 2x^7 + 9x^6 - 2x^5 + 34x^4 - 18x^3 + 51x^2 + 18*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1584)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3455.5
Root			$0.697356 - 1.20786i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.e.3455.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.558208i q^{5} -0.558208i q^{7} +1.00000 q^{11} +1.30678 q^{13} +5.94784i q^{17} -6.72638i q^{19} -4.99518 q^{23} +4.68840 q^{25} -3.46410i q^{29} -2.48374i q^{31} +0.311596 q^{35} -9.62209 q^{37} -2.48374i q^{41} -2.87532i q^{43} +6.92887 q^{47} +6.68840 q^{49} -7.48641i q^{53} +0.558208i q^{55} +3.32013 q^{59} -2.69322 q^{61} +0.729454i q^{65} -5.02118i q^{67} -4.99518 q^{71} +1.32013 q^{73} -0.558208i q^{77} +11.3375i q^{79} +11.6884 q^{83} -3.32013 q^{85} +1.56915i q^{89} -0.729454i q^{91} +3.75472 q^{95} +3.68840 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.558208i	0.249638i	0.992180	+	0.124819i	$0.0398350\pi$
$5$	0.558208i	0.249638i	−0.992180	+	0.124819i	$0.960165\pi$
$6$	0	0
$7$	− 0.558208i	− 0.210983i	−0.994420	−	0.105491i	$-0.966358\pi$
$7$	− 0.558208i	− 0.210983i	0.994420	−	0.105491i	$-0.0336415\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.30678	0.362435	0.181218	−	0.983443i	$-0.441996\pi$
$13$	1.30678	0.362435	0.181218	+	0.983443i	$0.441996\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.94784i	1.44256i	0.692642	+	0.721282i	$0.256447\pi$
$17$	5.94784i	1.44256i	−0.692642	+	0.721282i	$0.743553\pi$
$18$	0	0
$19$	− 6.72638i	− 1.54314i	−0.636146	−	0.771569i	$-0.719472\pi$
$19$	− 6.72638i	− 1.54314i	0.636146	−	0.771569i	$-0.280528\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.99518	−1.04157	−0.520784	−	0.853689i	$-0.674360\pi$
$23$	−4.99518	−1.04157	−0.520784	+	0.853689i	$0.674360\pi$
$24$	0	0
$25$	4.68840	0.937681
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	0	0
$31$	− 2.48374i	− 0.446093i	−0.974808	−	0.223046i	$-0.928400\pi$
$31$	− 2.48374i	− 0.446093i	0.974808	−	0.223046i	$-0.0716001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.311596	0.0526693
$36$	0	0
$37$	−9.62209	−1.58186	−0.790931	−	0.611905i	$-0.790403\pi$
$37$	−9.62209	−1.58186	−0.790931	+	0.611905i	$0.790403\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.48374i	− 0.387895i	−0.981012	−	0.193947i	$-0.937871\pi$
$41$	− 2.48374i	− 0.387895i	0.981012	−	0.193947i	$-0.0621291\pi$
$42$	0	0
$43$	− 2.87532i	− 0.438482i	−0.975671	−	0.219241i	$-0.929642\pi$
$43$	− 2.87532i	− 0.438482i	0.975671	−	0.219241i	$-0.0703580\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92887	1.01068	0.505340	−	0.862920i	$-0.331367\pi$
$47$	6.92887	1.01068	0.505340	+	0.862920i	$0.331367\pi$
$48$	0	0
$49$	6.68840	0.955486
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.48641i	− 1.02834i	−0.857689	−	0.514169i	$-0.828100\pi$
$53$	− 7.48641i	− 1.02834i	0.857689	−	0.514169i	$-0.171900\pi$
$54$	0	0
$55$	0.558208i	0.0752687i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.32013	0.432244	0.216122	−	0.976366i	$-0.430659\pi$
$59$	3.32013	0.432244	0.216122	+	0.976366i	$0.430659\pi$
$60$	0	0
$61$	−2.69322	−0.344832	−0.172416	−	0.985024i	$-0.555157\pi$
$61$	−2.69322	−0.344832	−0.172416	+	0.985024i	$0.555157\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.729454i	0.0904776i
$66$	0	0
$67$	− 5.02118i	− 0.613435i	−0.951801	−	0.306717i	$-0.900769\pi$
$67$	− 5.02118i	− 0.613435i	0.951801	−	0.306717i	$-0.0992306\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.99518	−0.592819	−0.296410	−	0.955061i	$-0.595789\pi$
$71$	−4.99518	−0.592819	−0.296410	+	0.955061i	$0.595789\pi$
$72$	0	0
$73$	1.32013	0.154510	0.0772548	−	0.997011i	$-0.475385\pi$
$73$	1.32013	0.154510	0.0772548	+	0.997011i	$0.475385\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.558208i	− 0.0636137i
$78$	0	0
$79$	11.3375i	1.27557i	0.770216	+	0.637783i	$0.220148\pi$
$79$	11.3375i	1.27557i	−0.770216	+	0.637783i	$0.779852\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.6884	1.28297	0.641485	−	0.767136i	$-0.278319\pi$
$83$	11.6884	1.28297	0.641485	+	0.767136i	$0.278319\pi$
$84$	0	0
$85$	−3.32013	−0.360119
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.56915i	0.166329i	0.996536	+	0.0831646i	$0.0265027\pi$
$89$	1.56915i	0.166329i	−0.996536	+	0.0831646i	$0.973497\pi$
$90$	0	0
$91$	− 0.729454i	− 0.0764676i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.75472	0.385226
$96$	0	0
$97$	3.68840	0.374501	0.187250	−	0.982312i	$-0.440042\pi$
$97$	3.68840	0.374501	0.187250	+	0.982312i	$0.440042\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.85261i	− 0.980371i	−0.871618	−	0.490185i	$-0.836929\pi$
$101$	− 9.85261i	− 0.980371i	0.871618	−	0.490185i	$-0.163071\pi$
$102$	0	0
$103$	− 5.56088i	− 0.547930i	−0.961740	−	0.273965i	$-0.911665\pi$
$103$	− 5.56088i	− 0.547930i	0.961740	−	0.273965i	$-0.0883352\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.93369	0.766979	0.383489	−	0.923545i	$-0.374722\pi$
$107$	7.93369	0.766979	0.383489	+	0.923545i	$0.374722\pi$
$108$	0	0
$109$	−10.6173	−1.01695	−0.508475	−	0.861077i	$-0.669791\pi$
$109$	−10.6173	−1.01695	−0.508475	+	0.861077i	$0.669791\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.21012i	0.866415i	0.901294	+	0.433208i	$0.142618\pi$
$113$	9.21012i	0.866415i	−0.901294	+	0.433208i	$0.857382\pi$
$114$	0	0
$115$	− 2.78835i	− 0.260015i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.32013	0.304356
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.40814i	0.483719i
$126$	0	0
$127$	− 7.48641i	− 0.664312i	−0.943225	−	0.332156i	$-0.892224\pi$
$127$	− 7.48641i	− 0.664312i	0.943225	−	0.332156i	$-0.107776\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.62209	−0.665945	−0.332973	−	0.942936i	$-0.608052\pi$
$131$	−7.62209	−0.665945	−0.332973	+	0.942936i	$0.608052\pi$
$132$	0	0
$133$	−3.75472	−0.325575
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.53663i	0.558462i	0.960224	+	0.279231i	$0.0900795\pi$
$137$	6.53663i	0.558462i	−0.960224	+	0.279231i	$0.909921\pi$
$138$	0	0
$139$	− 16.3892i	− 1.39012i	−0.718953	−	0.695058i	$-0.755379\pi$
$139$	− 16.3892i	− 1.39012i	0.718953	−	0.695058i	$-0.244621\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.30678	0.109278
$144$	0	0
$145$	1.93369	0.160584
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.1301i	1.64912i	0.565776	+	0.824559i	$0.308577\pi$
$149$	20.1301i	1.64912i	−0.565776	+	0.824559i	$0.691423\pi$
$150$	0	0
$151$	− 10.1599i	− 0.826801i	−0.910549	−	0.413401i	$-0.864341\pi$
$151$	− 10.1599i	− 0.826801i	0.910549	−	0.413401i	$-0.135659\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.38644	0.111362
$156$	0	0
$157$	3.25382	0.259683	0.129841	−	0.991535i	$-0.458553\pi$
$157$	3.25382	0.259683	0.129841	+	0.991535i	$0.458553\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.78835i	0.219753i
$162$	0	0
$163$	− 17.6463i	− 1.38217i	−0.722775	−	0.691083i	$-0.757134\pi$
$163$	− 17.6463i	− 1.38217i	0.722775	−	0.691083i	$-0.242866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0085	1.62569	0.812845	−	0.582481i	$-0.197918\pi$
$167$	21.0085	1.62569	0.812845	+	0.582481i	$0.197918\pi$
$168$	0	0
$169$	−11.2923	−0.868641
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.43158i	− 0.641041i	−0.947242	−	0.320521i	$-0.896142\pi$
$173$	− 8.43158i	− 0.641041i	0.947242	−	0.320521i	$-0.103858\pi$
$174$	0	0
$175$	− 2.61710i	− 0.197834i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.6702	1.39548	0.697739	−	0.716352i	$-0.254189\pi$
$179$	18.6702	1.39548	0.697739	+	0.716352i	$0.254189\pi$
$180$	0	0
$181$	19.6125	1.45778	0.728891	−	0.684629i	$-0.240036\pi$
$181$	19.6125	1.45778	0.728891	+	0.684629i	$0.240036\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.37113i	− 0.394893i
$186$	0	0
$187$	5.94784i	0.434949i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6258	−1.56479	−0.782394	−	0.622783i	$-0.786002\pi$
$191$	−21.6258	−1.56479	−0.782394	+	0.622783i	$0.786002\pi$
$192$	0	0
$193$	10.0567	0.723896	0.361948	−	0.932198i	$-0.382112\pi$
$193$	10.0567	0.723896	0.361948	+	0.932198i	$0.382112\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0854i	− 0.861052i	−0.902578	−	0.430526i	$-0.858328\pi$
$197$	− 12.0854i	− 0.861052i	0.902578	−	0.430526i	$-0.141672\pi$
$198$	0	0
$199$	− 26.5186i	− 1.87985i	−0.341380	−	0.939925i	$-0.610894\pi$
$199$	− 26.5186i	− 1.87985i	0.341380	−	0.939925i	$-0.389106\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.93369	−0.135718
$204$	0	0
$205$	1.38644	0.0968333
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.72638i	− 0.465273i
$210$	0	0
$211$	− 6.45427i	− 0.444331i	−0.975009	−	0.222165i	$-0.928688\pi$
$211$	− 6.45427i	− 0.444331i	0.975009	−	0.222165i	$-0.0713125\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.60502	0.109462
$216$	0	0
$217$	−1.38644	−0.0941178
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.77251i	0.522836i
$222$	0	0
$223$	− 6.19875i	− 0.415099i	−0.978225	−	0.207549i	$-0.933451\pi$
$223$	− 6.19875i	− 0.415099i	0.978225	−	0.207549i	$-0.0665488\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.29343	0.0858478	0.0429239	−	0.999078i	$-0.486333\pi$
$227$	1.29343	0.0858478	0.0429239	+	0.999078i	$0.486333\pi$
$228$	0	0
$229$	3.38644	0.223782	0.111891	−	0.993720i	$-0.464309\pi$
$229$	3.38644	0.223782	0.111891	+	0.993720i	$0.464309\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.44446i	0.291167i	0.989346	+	0.145583i	$0.0465059\pi$
$233$	4.44446i	0.291167i	−0.989346	+	0.145583i	$0.953494\pi$
$234$	0	0
$235$	3.86775i	0.252304i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.3683	0.929406	0.464703	−	0.885467i	$-0.346161\pi$
$239$	14.3683	0.929406	0.464703	+	0.885467i	$0.346161\pi$
$240$	0	0
$241$	−18.6306	−1.20010	−0.600052	−	0.799961i	$-0.704853\pi$
$241$	−18.6306	−1.20010	−0.600052	+	0.799961i	$0.704853\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.73352i	0.238526i
$246$	0	0
$247$	− 8.78989i	− 0.559287i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.3009	−1.47074	−0.735369	−	0.677667i	$-0.762991\pi$
$251$	−23.3009	−1.47074	−0.735369	+	0.677667i	$0.762991\pi$
$252$	0	0
$253$	−4.99518	−0.314044
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 26.5843i	− 1.65828i	−0.559037	−	0.829142i	$-0.688829\pi$
$257$	− 26.5843i	− 1.65828i	0.559037	−	0.829142i	$-0.311171\pi$
$258$	0	0
$259$	5.37113i	0.333745i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.9989	1.17152	0.585761	−	0.810484i	$-0.300796\pi$
$263$	18.9989	1.17152	0.585761	+	0.810484i	$0.300796\pi$
$264$	0	0
$265$	4.17897	0.256712
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 0.998871i	− 0.0609022i	−0.999536	−	0.0304511i	$-0.990306\pi$
$269$	− 0.998871i	− 0.0609022i	0.999536	−	0.0304511i	$-0.00969439\pi$
$270$	0	0
$271$	3.96861i	0.241076i	0.992709	+	0.120538i	$0.0384619\pi$
$271$	3.96861i	0.241076i	−0.992709	+	0.120538i	$0.961538\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.68840	0.282721
$276$	0	0
$277$	14.6932	0.882830	0.441415	−	0.897303i	$-0.354477\pi$
$277$	14.6932	0.882830	0.441415	+	0.897303i	$0.354477\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.2003i	− 0.727808i	−0.931436	−	0.363904i	$-0.881444\pi$
$281$	− 12.2003i	− 0.727808i	0.931436	−	0.363904i	$-0.118556\pi$
$282$	0	0
$283$	7.84279i	0.466206i	0.972452	+	0.233103i	$0.0748879\pi$
$283$	7.84279i	0.466206i	−0.972452	+	0.233103i	$0.925112\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.38644	−0.0818391
$288$	0	0
$289$	−18.3768	−1.08099
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.2434i	0.832106i	0.909340	+	0.416053i	$0.136587\pi$
$293$	14.2434i	0.832106i	−0.909340	+	0.416053i	$0.863413\pi$
$294$	0	0
$295$	1.85332i	0.107905i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.52760	−0.377501
$300$	0	0
$301$	−1.60502	−0.0925120
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.50338i	− 0.0860831i
$306$	0	0
$307$	− 19.4664i	− 1.11100i	−0.831515	−	0.555502i	$-0.812526\pi$
$307$	− 19.4664i	− 1.11100i	0.831515	−	0.555502i	$-0.187474\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.0048	1.07766	0.538832	−	0.842413i	$-0.318866\pi$
$311$	19.0048	1.07766	0.538832	+	0.842413i	$0.318866\pi$
$312$	0	0
$313$	16.6039	0.938509	0.469255	−	0.883063i	$-0.344523\pi$
$313$	16.6039	0.938509	0.469255	+	0.883063i	$0.344523\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.96861i	0.222899i	0.993770	+	0.111450i	$0.0355494\pi$
$317$	3.96861i	0.222899i	−0.993770	+	0.111450i	$0.964451\pi$
$318$	0	0
$319$	− 3.46410i	− 0.193952i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	40.0074	2.22607
$324$	0	0
$325$	6.12671	0.339849
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.86775i	− 0.213236i
$330$	0	0
$331$	1.28497i	0.0706285i	0.999376	+	0.0353142i	$0.0112432\pi$
$331$	1.28497i	0.0706285i	−0.999376	+	0.0353142i	$0.988757\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.80286	0.153137
$336$	0	0
$337$	−33.9807	−1.85105	−0.925524	−	0.378688i	$-0.876375\pi$
$337$	−33.9807	−1.85105	−0.925524	+	0.378688i	$0.876375\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.48374i	− 0.134502i
$342$	0	0
$343$	− 7.64097i	− 0.412574i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.3094	1.41236	0.706181	−	0.708031i	$-0.250416\pi$
$347$	26.3094	1.41236	0.706181	+	0.708031i	$0.250416\pi$
$348$	0	0
$349$	8.68359	0.464822	0.232411	−	0.972618i	$-0.425339\pi$
$349$	8.68359	0.464822	0.232411	+	0.972618i	$0.425339\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 5.35906i	− 0.285234i	−0.989778	−	0.142617i	$-0.954448\pi$
$353$	− 5.35906i	− 0.285234i	0.989778	−	0.142617i	$-0.0455517\pi$
$354$	0	0
$355$	− 2.78835i	− 0.147990i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.3672	1.76105	0.880526	−	0.473998i	$-0.157190\pi$
$359$	33.3672	1.76105	0.880526	+	0.473998i	$0.157190\pi$
$360$	0	0
$361$	−26.2442	−1.38127
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.736907i	0.0385715i
$366$	0	0
$367$	− 27.3629i	− 1.42833i	−0.699977	−	0.714165i	$-0.746807\pi$
$367$	− 27.3629i	− 1.42833i	0.699977	−	0.714165i	$-0.253193\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.17897	−0.216961
$372$	0	0
$373$	19.2971	0.999168	0.499584	−	0.866265i	$-0.333486\pi$
$373$	19.2971	0.999168	0.499584	+	0.866265i	$0.333486\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.52682i	− 0.233143i
$378$	0	0
$379$	8.48528i	0.435860i	0.975964	+	0.217930i	$0.0699304\pi$
$379$	8.48528i	0.435860i	−0.975964	+	0.217930i	$0.930070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.0219	0.665387	0.332694	−	0.943035i	$-0.392043\pi$
$383$	13.0219	0.665387	0.332694	+	0.943035i	$0.392043\pi$
$384$	0	0
$385$	0.311596	0.0158804
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.45778i	0.124614i	0.998057	+	0.0623072i	$0.0198458\pi$
$389$	2.45778i	0.124614i	−0.998057	+	0.0623072i	$0.980154\pi$
$390$	0	0
$391$	− 29.7106i	− 1.50253i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.32866	−0.318430
$396$	0	0
$397$	10.8759	0.545846	0.272923	−	0.962036i	$-0.412010\pi$
$397$	10.8759	0.545846	0.272923	+	0.962036i	$0.412010\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 34.0143i	− 1.69860i	−0.527914	−	0.849298i	$-0.677026\pi$
$401$	− 34.0143i	− 1.69860i	0.527914	−	0.849298i	$-0.322974\pi$
$402$	0	0
$403$	− 3.24570i	− 0.161680i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.62209	−0.476949
$408$	0	0
$409$	−20.5643	−1.01684	−0.508420	−	0.861109i	$-0.669770\pi$
$409$	−20.5643	−1.01684	−0.508420	+	0.861109i	$0.669770\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.85332i	− 0.0911960i
$414$	0	0
$415$	6.52456i	0.320278i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.64026	0.324398	0.162199	−	0.986758i	$-0.448141\pi$
$419$	6.64026	0.324398	0.162199	+	0.986758i	$0.448141\pi$
$420$	0	0
$421$	−14.8492	−0.723706	−0.361853	−	0.932235i	$-0.617856\pi$
$421$	−14.8492	−0.723706	−0.361853	+	0.932235i	$0.617856\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	27.8859i	1.35266i
$426$	0	0
$427$	1.50338i	0.0727535i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.6392	1.23499	0.617497	−	0.786573i	$-0.288147\pi$
$431$	25.6392	1.23499	0.617497	+	0.786573i	$0.288147\pi$
$432$	0	0
$433$	39.9807	1.92135	0.960676	−	0.277673i	$-0.0895631\pi$
$433$	39.9807	1.92135	0.960676	+	0.277673i	$0.0895631\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.5995i	1.60728i
$438$	0	0
$439$	− 12.3927i	− 0.591473i	−0.955270	−	0.295736i	$-0.904435\pi$
$439$	− 12.3927i	− 0.591473i	0.955270	−	0.295736i	$-0.0955650\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.5943	−1.45358	−0.726789	−	0.686860i	$-0.758988\pi$
$443$	−30.5943	−1.45358	−0.726789	+	0.686860i	$0.758988\pi$
$444$	0	0
$445$	−0.875910	−0.0415221
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.0786i	1.70266i	0.524634	+	0.851328i	$0.324202\pi$
$449$	36.0786i	1.70266i	−0.524634	+	0.851328i	$0.675798\pi$
$450$	0	0
$451$	− 2.48374i	− 0.116955i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.407187	0.0190892
$456$	0	0
$457$	7.22712	0.338070	0.169035	−	0.985610i	$-0.445935\pi$
$457$	7.22712	0.338070	0.169035	+	0.985610i	$0.445935\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.11797i	0.0986439i	0.998783	+	0.0493220i	$0.0157060\pi$
$461$	2.11797i	0.0986439i	−0.998783	+	0.0493220i	$0.984294\pi$
$462$	0	0
$463$	11.6818i	0.542899i	0.962453	+	0.271449i	$0.0875030\pi$
$463$	11.6818i	0.542899i	−0.962453	+	0.271449i	$0.912497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.0567	0.928112	0.464056	−	0.885806i	$-0.346394\pi$
$467$	20.0567	0.928112	0.464056	+	0.885806i	$0.346394\pi$
$468$	0	0
$469$	−2.80286	−0.129424
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.87532i	− 0.132207i
$474$	0	0
$475$	− 31.5360i	− 1.44697i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−42.3757	−1.93620	−0.968098	−	0.250573i	$-0.919381\pi$
$479$	−42.3757	−1.93620	−0.968098	+	0.250573i	$0.919381\pi$
$480$	0	0
$481$	−12.5739	−0.573323
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.05890i	0.0934896i
$486$	0	0
$487$	4.20101i	0.190366i	0.995460	+	0.0951829i	$0.0303436\pi$
$487$	4.20101i	0.190366i	−0.995460	+	0.0951829i	$0.969656\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.9508	−1.17114	−0.585571	−	0.810621i	$-0.699130\pi$
$491$	−25.9508	−1.17114	−0.585571	+	0.810621i	$0.699130\pi$
$492$	0	0
$493$	20.6039	0.927954
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.78835i	0.125075i
$498$	0	0
$499$	30.7196i	1.37520i	0.726091	+	0.687598i	$0.241335\pi$
$499$	30.7196i	1.37520i	−0.726091	+	0.687598i	$0.758665\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.9722	−1.02428	−0.512140	−	0.858902i	$-0.671147\pi$
$503$	−22.9722	−1.02428	−0.512140	+	0.858902i	$0.671147\pi$
$504$	0	0
$505$	5.49980	0.244738
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.4592i	0.995488i	0.867324	+	0.497744i	$0.165838\pi$
$509$	22.4592i	0.995488i	−0.867324	+	0.497744i	$0.834162\pi$
$510$	0	0
$511$	− 0.736907i	− 0.0325988i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.10413	0.136784
$516$	0	0
$517$	6.92887	0.304731
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.4786i	− 0.809560i	−0.914414	−	0.404780i	$-0.867348\pi$
$521$	− 18.4786i	− 0.809560i	0.914414	−	0.404780i	$-0.132652\pi$
$522$	0	0
$523$	20.5828i	0.900022i	0.893023	+	0.450011i	$0.148580\pi$
$523$	20.5828i	0.900022i	−0.893023	+	0.450011i	$0.851420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.7729	0.643517
$528$	0	0
$529$	1.95186	0.0848633
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.24570i	− 0.140587i
$534$	0	0
$535$	4.42864i	0.191467i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.68840	0.288090
$540$	0	0
$541$	22.7499	0.978095	0.489047	−	0.872257i	$-0.337345\pi$
$541$	22.7499	0.978095	0.489047	+	0.872257i	$0.337345\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.92664i	− 0.253870i
$546$	0	0
$547$	− 25.4891i	− 1.08984i	−0.838489	−	0.544918i	$-0.816561\pi$
$547$	− 25.4891i	− 1.08984i	0.838489	−	0.544918i	$-0.183439\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.3009	−0.992650
$552$	0	0
$553$	6.32866	0.269122
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.5654i	− 1.37984i	−0.723885	−	0.689921i	$-0.757645\pi$
$557$	− 32.5654i	− 1.37984i	0.723885	−	0.689921i	$-0.242355\pi$
$558$	0	0
$559$	− 3.75740i	− 0.158921i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.8652	−1.76441	−0.882203	−	0.470868i	$-0.843941\pi$
$563$	−41.8652	−1.76441	−0.882203	+	0.470868i	$0.843941\pi$
$564$	0	0
$565$	−5.14116	−0.216290
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.3841i	1.14800i	0.818855	+	0.574000i	$0.194609\pi$
$569$	27.3841i	1.14800i	−0.818855	+	0.574000i	$0.805391\pi$
$570$	0	0
$571$	− 6.78753i	− 0.284049i	−0.989863	−	0.142025i	$-0.954639\pi$
$571$	− 6.78753i	− 0.284049i	0.989863	−	0.142025i	$-0.0453613\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−23.4194	−0.976658
$576$	0	0
$577$	−6.93259	−0.288607	−0.144304	−	0.989533i	$-0.546094\pi$
$577$	−6.93259	−0.288607	−0.144304	+	0.989533i	$0.546094\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.52456i	− 0.270684i
$582$	0	0
$583$	− 7.48641i	− 0.310055i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.78995	0.362800	0.181400	−	0.983409i	$-0.441937\pi$
$587$	8.78995	0.362800	0.181400	+	0.983409i	$0.441937\pi$
$588$	0	0
$589$	−16.7066	−0.688382
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.8239i	0.978329i	0.872192	+	0.489164i	$0.162698\pi$
$593$	23.8239i	0.978329i	−0.872192	+	0.489164i	$0.837302\pi$
$594$	0	0
$595$	1.85332i	0.0759788i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.1560	1.15042	0.575211	−	0.818005i	$-0.304920\pi$
$599$	28.1560	1.15042	0.575211	+	0.818005i	$0.304920\pi$
$600$	0	0
$601$	0.537612	0.0219297	0.0109648	−	0.999940i	$-0.496510\pi$
$601$	0.537612	0.0219297	0.0109648	+	0.999940i	$0.496510\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.558208i	0.0226944i
$606$	0	0
$607$	27.5953i	1.12006i	0.828473	+	0.560029i	$0.189210\pi$
$607$	27.5953i	1.12006i	−0.828473	+	0.560029i	$0.810790\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.05450	0.366306
$612$	0	0
$613$	−27.4565	−1.10896	−0.554478	−	0.832198i	$-0.687082\pi$
$613$	−27.4565	−1.10896	−0.554478	+	0.832198i	$0.687082\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 44.8360i	− 1.80503i	−0.430659	−	0.902515i	$-0.641719\pi$
$617$	− 44.8360i	− 1.80503i	0.430659	−	0.902515i	$-0.358281\pi$
$618$	0	0
$619$	− 9.20550i	− 0.370000i	−0.982738	−	0.185000i	$-0.940771\pi$
$619$	− 9.20550i	− 0.370000i	0.982738	−	0.185000i	$-0.0592286\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.875910	0.0350926
$624$	0	0
$625$	20.4232	0.816926
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 57.2307i	− 2.28194i
$630$	0	0
$631$	− 44.3251i	− 1.76455i	−0.470732	−	0.882276i	$-0.656010\pi$
$631$	− 44.3251i	− 1.76455i	0.470732	−	0.882276i	$-0.343990\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.17897	0.165837
$636$	0	0
$637$	8.74027	0.346302
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.6561i	0.776370i	0.921581	+	0.388185i	$0.126898\pi$
$641$	19.6561i	0.776370i	−0.921581	+	0.388185i	$0.873102\pi$
$642$	0	0
$643$	− 27.6350i	− 1.08982i	−0.838496	−	0.544908i	$-0.816564\pi$
$643$	− 27.6350i	− 1.08982i	0.838496	−	0.544908i	$-0.183436\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.3153	1.27045	0.635223	−	0.772329i	$-0.280908\pi$
$647$	32.3153	1.27045	0.635223	+	0.772329i	$0.280908\pi$
$648$	0	0
$649$	3.32013	0.130326
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0306i	0.705591i	0.935700	+	0.352796i	$0.114769\pi$
$653$	18.0306i	0.705591i	−0.935700	+	0.352796i	$0.885231\pi$
$654$	0	0
$655$	− 4.25471i	− 0.166245i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.77288	−0.108016	−0.0540081	−	0.998540i	$-0.517200\pi$
$659$	−2.77288	−0.108016	−0.0540081	+	0.998540i	$0.517200\pi$
$660$	0	0
$661$	−25.0849	−0.975688	−0.487844	−	0.872931i	$-0.662216\pi$
$661$	−25.0849	−0.975688	−0.487844	+	0.872931i	$0.662216\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.09591i	− 0.0812760i
$666$	0	0
$667$	17.3038i	0.670007i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.69322	−0.103971
$672$	0	0
$673$	21.3276	0.822117	0.411059	−	0.911609i	$-0.365159\pi$
$673$	21.3276	0.822117	0.411059	+	0.911609i	$0.365159\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.0953i	1.04136i	0.853753	+	0.520678i	$0.174321\pi$
$677$	27.0953i	1.04136i	−0.853753	+	0.520678i	$0.825679\pi$
$678$	0	0
$679$	− 2.05890i	− 0.0790132i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	50.5450	1.93405	0.967026	−	0.254677i	$-0.0819691\pi$
$683$	50.5450	1.93405	0.967026	+	0.254677i	$0.0819691\pi$
$684$	0	0
$685$	−3.64879	−0.139413
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 9.78308i	− 0.372706i
$690$	0	0
$691$	− 29.4883i	− 1.12179i	−0.827888	−	0.560894i	$-0.810458\pi$
$691$	− 29.4883i	− 1.12179i	0.827888	−	0.560894i	$-0.189542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.14859	0.347026
$696$	0	0
$697$	14.7729	0.559563
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.8950i	0.562576i	0.959623	+	0.281288i	$0.0907616\pi$
$701$	14.8950i	0.562576i	−0.959623	+	0.281288i	$0.909238\pi$
$702$	0	0
$703$	64.7218i	2.44103i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.49980	−0.206841
$708$	0	0
$709$	−11.8577	−0.445327	−0.222663	−	0.974895i	$-0.571475\pi$
$709$	−11.8577	−0.445327	−0.222663	+	0.974895i	$0.571475\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.4067i	0.464636i
$714$	0	0
$715$	0.729454i	0.0272800i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−53.1352	−1.98161	−0.990805	−	0.135297i	$-0.956801\pi$
$719$	−53.1352	−1.98161	−0.990805	+	0.135297i	$0.956801\pi$
$720$	0	0
$721$	−3.10413	−0.115604
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 16.2411i	− 0.603180i
$726$	0	0
$727$	− 13.7416i	− 0.509646i	−0.966988	−	0.254823i	$-0.917983\pi$
$727$	− 13.7416i	− 0.509646i	0.966988	−	0.254823i	$-0.0820172\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.1019	0.632538
$732$	0	0
$733$	−33.3709	−1.23258	−0.616291	−	0.787518i	$-0.711365\pi$
$733$	−33.3709	−1.23258	−0.616291	+	0.787518i	$0.711365\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.02118i	− 0.184958i
$738$	0	0
$739$	− 49.2193i	− 1.81056i	−0.424813	−	0.905281i	$-0.639660\pi$
$739$	− 49.2193i	− 1.81056i	0.424813	−	0.905281i	$-0.360340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.8941	−0.436351	−0.218176	−	0.975910i	$-0.570011\pi$
$743$	−11.8941	−0.436351	−0.218176	+	0.975910i	$0.570011\pi$
$744$	0	0
$745$	−11.2367	−0.411683
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.42864i	− 0.161819i
$750$	0	0
$751$	2.98556i	0.108944i	0.998515	+	0.0544722i	$0.0173476\pi$
$751$	2.98556i	0.108944i	−0.998515	+	0.0544722i	$0.982652\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.67134	0.206401
$756$	0	0
$757$	−25.6028	−0.930551	−0.465275	−	0.885166i	$-0.654045\pi$
$757$	−25.6028	−0.930551	−0.465275	+	0.885166i	$0.654045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.52457i	0.0552657i	0.999618	+	0.0276329i	$0.00879693\pi$
$761$	1.52457i	0.0552657i	−0.999618	+	0.0276329i	$0.991203\pi$
$762$	0	0
$763$	5.92664i	0.214559i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.33868	0.156661
$768$	0	0
$769$	21.1116	0.761302	0.380651	−	0.924719i	$-0.375700\pi$
$769$	21.1116	0.761302	0.380651	+	0.924719i	$0.375700\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 24.7532i	− 0.890311i	−0.895453	−	0.445156i	$-0.853148\pi$
$773$	− 24.7532i	− 0.890311i	0.895453	−	0.445156i	$-0.146852\pi$
$774$	0	0
$775$	− 11.6448i	− 0.418293i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.7066	−0.598575
$780$	0	0
$781$	−4.99518	−0.178742
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.81631i	0.0648267i
$786$	0	0
$787$	6.68936i	0.238450i	0.992867	+	0.119225i	$0.0380410\pi$
$787$	6.68936i	0.238450i	−0.992867	+	0.119225i	$0.961959\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.14116	0.182799
$792$	0	0
$793$	−3.51944	−0.124979
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 50.0775i	− 1.77384i	−0.461926	−	0.886918i	$-0.652842\pi$
$797$	− 50.0775i	− 1.77384i	0.461926	−	0.886918i	$-0.347158\pi$
$798$	0	0
$799$	41.2118i	1.45797i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.32013	0.0465864
$804$	0	0
$805$	−1.55648	−0.0548586
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 11.2411i	− 0.395217i	−0.980281	−	0.197608i	$-0.936683\pi$
$809$	− 11.2411i	− 0.395217i	0.980281	−	0.197608i	$-0.0633175\pi$
$810$	0	0
$811$	50.8987i	1.78730i	0.448769	+	0.893648i	$0.351862\pi$
$811$	50.8987i	1.78730i	−0.448769	+	0.893648i	$0.648138\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.85031	0.345041
$816$	0	0
$817$	−19.3405	−0.676638
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.27279i	0.0793208i	0.999213	+	0.0396604i	$0.0126276\pi$
$821$	2.27279i	0.0793208i	−0.999213	+	0.0396604i	$0.987372\pi$
$822$	0	0
$823$	49.5730i	1.72801i	0.503486	+	0.864003i	$0.332051\pi$
$823$	49.5730i	1.72801i	−0.503486	+	0.864003i	$0.667949\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.1768	−1.46663	−0.733315	−	0.679889i	$-0.762028\pi$
$827$	−42.1768	−1.46663	−0.733315	+	0.679889i	$0.762028\pi$
$828$	0	0
$829$	9.11156	0.316458	0.158229	−	0.987402i	$-0.449422\pi$
$829$	9.11156	0.316458	0.158229	+	0.987402i	$0.449422\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	39.7816i	1.37835i
$834$	0	0
$835$	11.7271i	0.405834i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.3549	−0.771778	−0.385889	−	0.922545i	$-0.626105\pi$
$839$	−22.3549	−0.771778	−0.385889	+	0.922545i	$0.626105\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 6.30346i	− 0.216846i
$846$	0	0
$847$	− 0.558208i	− 0.0191802i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.0641	1.64762
$852$	0	0
$853$	−49.8882	−1.70814	−0.854069	−	0.520160i	$-0.825872\pi$
$853$	−49.8882	−1.70814	−0.854069	+	0.520160i	$0.825872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0650i	1.02700i	0.858089	+	0.513501i	$0.171652\pi$
$857$	30.0650i	1.02700i	−0.858089	+	0.513501i	$0.828348\pi$
$858$	0	0
$859$	54.2147i	1.84978i	0.380233	+	0.924891i	$0.375844\pi$
$859$	54.2147i	1.84978i	−0.380233	+	0.924891i	$0.624156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.0711	−0.581108	−0.290554	−	0.956859i	$-0.593840\pi$
$863$	−17.0711	−0.581108	−0.290554	+	0.956859i	$0.593840\pi$
$864$	0	0
$865$	4.70657	0.160028
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.3375i	0.384597i
$870$	0	0
$871$	− 6.56157i	− 0.222330i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.01887	0.102056
$876$	0	0
$877$	−11.9203	−0.402521	−0.201261	−	0.979538i	$-0.564504\pi$
$877$	−11.9203	−0.402521	−0.201261	+	0.979538i	$0.564504\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.0929i	1.01386i	0.861989	+	0.506928i	$0.169219\pi$
$881$	30.0929i	1.01386i	−0.861989	+	0.506928i	$0.830781\pi$
$882$	0	0
$883$	− 35.4686i	− 1.19361i	−0.802385	−	0.596807i	$-0.796436\pi$
$883$	− 35.4686i	− 1.19361i	0.802385	−	0.596807i	$-0.203564\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.7633	0.831469	0.415734	−	0.909486i	$-0.363525\pi$
$887$	24.7633	0.831469	0.415734	+	0.909486i	$0.363525\pi$
$888$	0	0
$889$	−4.17897	−0.140158
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 46.6062i	− 1.55962i
$894$	0	0
$895$	10.4219i	0.348365i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.60392	−0.286957
$900$	0	0
$901$	44.5280	1.48344
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.9478i	0.363918i
$906$	0	0
$907$	30.3696i	1.00841i	0.863585	+	0.504203i	$0.168214\pi$
$907$	30.3696i	1.00841i	−0.863585	+	0.504203i	$0.831786\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.987748	−0.0327256	−0.0163628	−	0.999866i	$-0.505209\pi$
$911$	−0.987748	−0.0327256	−0.0163628	+	0.999866i	$0.505209\pi$
$912$	0	0
$913$	11.6884	0.386830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.25471i	0.140503i
$918$	0	0
$919$	2.38739i	0.0787528i	0.999224	+	0.0393764i	$0.0125371\pi$
$919$	2.38739i	0.0787528i	−0.999224	+	0.0393764i	$0.987463\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.52760	−0.214859
$924$	0	0
$925$	−45.1123	−1.48328
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 28.7560i	− 0.943454i	−0.881745	−	0.471727i	$-0.843631\pi$
$929$	− 28.7560i	− 0.943454i	0.881745	−	0.471727i	$-0.156369\pi$
$930$	0	0
$931$	− 44.9887i	− 1.47445i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.32013	−0.108580
$936$	0	0
$937$	25.9241	0.846902	0.423451	−	0.905919i	$-0.360819\pi$
$937$	25.9241	0.846902	0.423451	+	0.905919i	$0.360819\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	54.0342i	1.76146i	0.473616	+	0.880732i	$0.342949\pi$
$941$	54.0342i	1.76146i	−0.473616	+	0.880732i	$0.657051\pi$
$942$	0	0
$943$	12.4067i	0.404019i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.60392	−0.279590	−0.139795	−	0.990180i	$-0.544644\pi$
$947$	−8.60392	−0.279590	−0.139795	+	0.990180i	$0.544644\pi$
$948$	0	0
$949$	1.72512	0.0559997
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.9797i	1.00353i	0.865004	+	0.501766i	$0.167316\pi$
$953$	30.9797i	1.00353i	−0.865004	+	0.501766i	$0.832684\pi$
$954$	0	0
$955$	− 12.0717i	− 0.390631i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.64879	0.117826
$960$	0	0
$961$	24.8310	0.801001
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.61371i	0.180712i
$966$	0	0
$967$	5.52569i	0.177694i	0.996045	+	0.0888470i	$0.0283182\pi$
$967$	5.52569i	0.177694i	−0.996045	+	0.0888470i	$0.971682\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.1423	−0.325481	−0.162740	−	0.986669i	$-0.552033\pi$
$971$	−10.1423	−0.325481	−0.162740	+	0.986669i	$0.552033\pi$
$972$	0	0
$973$	−9.14859	−0.293290
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.6032i	− 1.36300i	−0.731820	−	0.681498i	$-0.761329\pi$
$977$	− 42.6032i	− 1.36300i	0.731820	−	0.681498i	$-0.238671\pi$
$978$	0	0
$979$	1.56915i	0.0501502i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.8796	−1.62281	−0.811404	−	0.584486i	$-0.801296\pi$
$983$	−50.8796	−1.62281	−0.811404	+	0.584486i	$0.801296\pi$
$984$	0	0
$985$	6.74618	0.214951
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.3627i	0.456708i
$990$	0	0
$991$	29.1172i	0.924937i	0.886636	+	0.462468i	$0.153036\pi$
$991$	29.1172i	0.924937i	−0.886636	+	0.462468i	$0.846964\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.8029	0.469282
$996$	0	0
$997$	43.1153	1.36547	0.682737	−	0.730664i	$-0.260789\pi$
$997$	43.1153	1.36547	0.682737	+	0.730664i	$0.260789\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.d.e.3455.5		8
3.2	odd	2		6336.2.d.c.3455.4		8
4.3	odd	2		6336.2.d.c.3455.5		8
8.3	odd	2		1584.2.d.d.287.4	yes	8
8.5	even	2		1584.2.d.c.287.4	✓	8
12.11	even	2	inner	6336.2.d.e.3455.4		8
24.5	odd	2		1584.2.d.d.287.5	yes	8
24.11	even	2		1584.2.d.c.287.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1584.2.d.c.287.4	✓	8	8.5	even	2
1584.2.d.c.287.5	yes	8	24.11	even	2
1584.2.d.d.287.4	yes	8	8.3	odd	2
1584.2.d.d.287.5	yes	8	24.5	odd	2
6336.2.d.c.3455.4		8	3.2	odd	2
6336.2.d.c.3455.5		8	4.3	odd	2
6336.2.d.e.3455.4		8	12.11	even	2	inner
6336.2.d.e.3455.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.558208i q^{5} -0.558208i q^{7} +1.00000 q^{11} +1.30678 q^{13} +5.94784i q^{17} -6.72638i q^{19} -4.99518 q^{23} +4.68840 q^{25} -3.46410i q^{29} -2.48374i q^{31} +0.311596 q^{35} -9.62209 q^{37} -2.48374i q^{41} -2.87532i q^{43} +6.92887 q^{47} +6.68840 q^{49} -7.48641i q^{53} +0.558208i q^{55} +3.32013 q^{59} -2.69322 q^{61} +0.729454i q^{65} -5.02118i q^{67} -4.99518 q^{71} +1.32013 q^{73} -0.558208i q^{77} +11.3375i q^{79} +11.6884 q^{83} -3.32013 q^{85} +1.56915i q^{89} -0.729454i q^{91} +3.75472 q^{95} +3.68840 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * q^95 - 24 * q^97

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