LMFDB - Embedded newform 6336.2.a.cp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6336,2,Mod(1,6336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$50.5932147207$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{5} +0.828427 q^{7} +1.00000 q^{11} -4.82843 q^{13} -2.00000 q^{17} +2.00000 q^{19} -2.82843 q^{23} +3.00000 q^{25} -3.65685 q^{29} -9.65685 q^{31} +2.34315 q^{35} -7.65685 q^{37} -0.343146 q^{41} +0.343146 q^{43} -12.4853 q^{47} -6.31371 q^{49} +6.82843 q^{53} +2.82843 q^{55} -1.65685 q^{59} -3.17157 q^{61} -13.6569 q^{65} +11.3137 q^{67} +4.48528 q^{71} -13.3137 q^{73} +0.828427 q^{77} -4.82843 q^{79} +9.65685 q^{83} -5.65685 q^{85} -16.0000 q^{89} -4.00000 q^{91} +5.65685 q^{95} +5.31371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{19} + 6 q^{25} + 4 q^{29} - 8 q^{31} + 16 q^{35} - 4 q^{37} - 12 q^{41} + 12 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} + 8 q^{59} - 12 q^{61} - 16 q^{65}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 + 2 * q^11 - 4 * q^13 - 4 * q^17 + 4 * q^19 + 6 * q^25 + 4 * q^29 - 8 * q^31 + 16 * q^35 - 4 * q^37 - 12 * q^41 + 12 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 + 8 * q^59 - 12 * q^61 - 16 * q^65 - 8 * q^71 - 4 * q^73 - 4 * q^77 - 4 * q^79 + 8 * q^83 - 32 * q^89 - 8 * q^91 - 12 * q^97

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.82843	−1.33916	−0.669582	−	0.742738i	$-0.733527\pi$
$13$	−4.82843	−1.33916	−0.669582	+	0.742738i	$0.733527\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.34315	0.396064
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.343146	−0.0535904	−0.0267952	−	0.999641i	$-0.508530\pi$
$41$	−0.343146	−0.0535904	−0.0267952	+	0.999641i	$0.508530\pi$
$42$	0	0
$43$	0.343146	0.0523292	0.0261646	−	0.999658i	$-0.491671\pi$
$43$	0.343146	0.0523292	0.0261646	+	0.999658i	$0.491671\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.4853	−1.82117	−0.910583	−	0.413327i	$-0.864367\pi$
$47$	−12.4853	−1.82117	−0.910583	+	0.413327i	$0.864367\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.6569	−1.69392
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.828427	0.0944080
$78$	0	0
$79$	−4.82843	−0.543240	−0.271620	−	0.962405i	$-0.587559\pi$
$79$	−4.82843	−0.543240	−0.271620	+	0.962405i	$0.587559\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.65685	1.05998	0.529989	−	0.848005i	$-0.322196\pi$
$83$	9.65685	1.05998	0.529989	+	0.848005i	$0.322196\pi$
$84$	0	0
$85$	−5.65685	−0.613572
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	5.31371	0.539525	0.269763	−	0.962927i	$-0.413055\pi$
$97$	5.31371	0.539525	0.269763	+	0.962927i	$0.413055\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	11.1716	1.07004	0.535021	−	0.844839i	$-0.320304\pi$
$109$	11.1716	1.07004	0.535021	+	0.844839i	$0.320304\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.31371	−0.688016	−0.344008	−	0.938967i	$-0.611785\pi$
$113$	−7.31371	−0.688016	−0.344008	+	0.938967i	$0.611785\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.65685	−0.151884
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	11.1716	0.991317	0.495658	−	0.868518i	$-0.334927\pi$
$127$	11.1716	0.991317	0.495658	+	0.868518i	$0.334927\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.9706	1.13324	0.566622	−	0.823978i	$-0.308250\pi$
$131$	12.9706	1.13324	0.566622	+	0.823978i	$0.308250\pi$
$132$	0	0
$133$	1.65685	0.143667
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.34315	−0.541932	−0.270966	−	0.962589i	$-0.587343\pi$
$137$	−6.34315	−0.541932	−0.270966	+	0.962589i	$0.587343\pi$
$138$	0	0
$139$	−9.31371	−0.789978	−0.394989	−	0.918686i	$-0.629252\pi$
$139$	−9.31371	−0.789978	−0.394989	+	0.918686i	$0.629252\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.82843	−0.403773
$144$	0	0
$145$	−10.3431	−0.858952
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.6569	−1.28266	−0.641330	−	0.767265i	$-0.721617\pi$
$149$	−15.6569	−1.28266	−0.641330	+	0.767265i	$0.721617\pi$
$150$	0	0
$151$	14.4853	1.17880	0.589398	−	0.807843i	$-0.299365\pi$
$151$	14.4853	1.17880	0.589398	+	0.807843i	$0.299365\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−27.3137	−2.19389
$156$	0	0
$157$	7.65685	0.611083	0.305542	−	0.952179i	$-0.401162\pi$
$157$	7.65685	0.611083	0.305542	+	0.952179i	$0.401162\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.34315	−0.184666
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.6569	−1.49448	−0.747241	−	0.664553i	$-0.768622\pi$
$173$	−19.6569	−1.49448	−0.747241	+	0.664553i	$0.768622\pi$
$174$	0	0
$175$	2.48528	0.187870
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.6569	1.61871	0.809355	−	0.587320i	$-0.199817\pi$
$179$	21.6569	1.61871	0.809355	+	0.587320i	$0.199817\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−21.6569	−1.59224
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.82843	0.494088	0.247044	−	0.969004i	$-0.420541\pi$
$191$	6.82843	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	0	0
$199$	25.6569	1.81877	0.909383	−	0.415960i	$-0.136554\pi$
$199$	25.6569	1.81877	0.909383	+	0.415960i	$0.136554\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.02944	−0.212625
$204$	0	0
$205$	−0.970563	−0.0677870
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	15.6569	1.07786	0.538931	−	0.842350i	$-0.318828\pi$
$211$	15.6569	1.07786	0.538931	+	0.842350i	$0.318828\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.970563	0.0661918
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.65685	0.649590
$222$	0	0
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.6274	−1.76732	−0.883662	−	0.468126i	$-0.844929\pi$
$227$	−26.6274	−1.76732	−0.883662	+	0.468126i	$0.844929\pi$
$228$	0	0
$229$	18.9706	1.25361	0.626805	−	0.779176i	$-0.284362\pi$
$229$	18.9706	1.25361	0.626805	+	0.779176i	$0.284362\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.9706	1.76690	0.883450	−	0.468525i	$-0.155214\pi$
$233$	26.9706	1.76690	0.883450	+	0.468525i	$0.155214\pi$
$234$	0	0
$235$	−35.3137	−2.30361
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	3.65685	0.235559	0.117779	−	0.993040i	$-0.462422\pi$
$241$	3.65685	0.235559	0.117779	+	0.993040i	$0.462422\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−17.8579	−1.14090
$246$	0	0
$247$	−9.65685	−0.614451
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.3431	−0.652854	−0.326427	−	0.945222i	$-0.605845\pi$
$251$	−10.3431	−0.652854	−0.326427	+	0.945222i	$0.605845\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.31371	0.206703	0.103352	−	0.994645i	$-0.467043\pi$
$257$	3.31371	0.206703	0.103352	+	0.994645i	$0.467043\pi$
$258$	0	0
$259$	−6.34315	−0.394144
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.6569	1.82872	0.914360	−	0.404902i	$-0.132694\pi$
$263$	29.6569	1.82872	0.914360	+	0.404902i	$0.132694\pi$
$264$	0	0
$265$	19.3137	1.18643
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.48528	−0.273472	−0.136736	−	0.990607i	$-0.543661\pi$
$269$	−4.48528	−0.273472	−0.136736	+	0.990607i	$0.543661\pi$
$270$	0	0
$271$	−25.7990	−1.56718	−0.783589	−	0.621280i	$-0.786613\pi$
$271$	−25.7990	−1.56718	−0.783589	+	0.621280i	$0.786613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	31.4558	1.89000	0.944999	−	0.327072i	$-0.106062\pi$
$277$	31.4558	1.89000	0.944999	+	0.327072i	$0.106062\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.31371	−0.0783693	−0.0391846	−	0.999232i	$-0.512476\pi$
$281$	−1.31371	−0.0783693	−0.0391846	+	0.999232i	$0.512476\pi$
$282$	0	0
$283$	30.9706	1.84101	0.920504	−	0.390732i	$-0.127778\pi$
$283$	30.9706	1.84101	0.920504	+	0.390732i	$0.127778\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.284271	−0.0167800
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−4.68629	−0.272846
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.6569	0.789796
$300$	0	0
$301$	0.284271	0.0163851
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.97056	−0.513653
$306$	0	0
$307$	13.3137	0.759853	0.379927	−	0.925017i	$-0.375949\pi$
$307$	13.3137	0.759853	0.379927	+	0.925017i	$0.375949\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.8284	−1.52130	−0.760650	−	0.649162i	$-0.775120\pi$
$311$	−26.8284	−1.52130	−0.760650	+	0.649162i	$0.775120\pi$
$312$	0	0
$313$	25.3137	1.43082	0.715408	−	0.698707i	$-0.246241\pi$
$313$	25.3137	1.43082	0.715408	+	0.698707i	$0.246241\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.8284	−0.608185	−0.304093	−	0.952643i	$-0.598353\pi$
$317$	−10.8284	−0.608185	−0.304093	+	0.952643i	$0.598353\pi$
$318$	0	0
$319$	−3.65685	−0.204745
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−14.4853	−0.803499
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.3431	−0.570236
$330$	0	0
$331$	−18.6274	−1.02386	−0.511928	−	0.859029i	$-0.671068\pi$
$331$	−18.6274	−1.02386	−0.511928	+	0.859029i	$0.671068\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	32.0000	1.74835
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.65685	−0.522948
$342$	0	0
$343$	−11.0294	−0.595534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.9706	0.696296	0.348148	−	0.937440i	$-0.386811\pi$
$347$	12.9706	0.696296	0.348148	+	0.937440i	$0.386811\pi$
$348$	0	0
$349$	13.7990	0.738643	0.369321	−	0.929302i	$-0.379590\pi$
$349$	13.7990	0.738643	0.369321	+	0.929302i	$0.379590\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.00000	−0.425797	−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000	−0.425797	−0.212899	+	0.977074i	$0.568290\pi$
$354$	0	0
$355$	12.6863	0.673318
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−37.6569	−1.97105
$366$	0	0
$367$	−3.31371	−0.172974	−0.0864871	−	0.996253i	$-0.527564\pi$
$367$	−3.31371	−0.172974	−0.0864871	+	0.996253i	$0.527564\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	−20.1421	−1.04292	−0.521460	−	0.853276i	$-0.674612\pi$
$373$	−20.1421	−1.04292	−0.521460	+	0.853276i	$0.674612\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.6569	0.909374
$378$	0	0
$379$	−0.686292	−0.0352524	−0.0176262	−	0.999845i	$-0.505611\pi$
$379$	−0.686292	−0.0352524	−0.0176262	+	0.999845i	$0.505611\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.4853	1.65992	0.829960	−	0.557823i	$-0.188363\pi$
$383$	32.4853	1.65992	0.829960	+	0.557823i	$0.188363\pi$
$384$	0	0
$385$	2.34315	0.119418
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−21.1716	−1.07344	−0.536721	−	0.843760i	$-0.680337\pi$
$389$	−21.1716	−1.07344	−0.536721	+	0.843760i	$0.680337\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.6569	−0.687151
$396$	0	0
$397$	−6.68629	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$397$	−6.68629	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.6569	−1.28124	−0.640621	−	0.767857i	$-0.721323\pi$
$401$	−25.6569	−1.28124	−0.640621	+	0.767857i	$0.721323\pi$
$402$	0	0
$403$	46.6274	2.32268
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.65685	−0.379536
$408$	0	0
$409$	−12.3431	−0.610329	−0.305165	−	0.952300i	$-0.598711\pi$
$409$	−12.3431	−0.610329	−0.305165	+	0.952300i	$0.598711\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.37258	−0.0675404
$414$	0	0
$415$	27.3137	1.34078
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.2843	−1.77260	−0.886301	−	0.463109i	$-0.846734\pi$
$419$	−36.2843	−1.77260	−0.886301	+	0.463109i	$0.846734\pi$
$420$	0	0
$421$	−15.6569	−0.763068	−0.381534	−	0.924355i	$-0.624604\pi$
$421$	−15.6569	−0.763068	−0.381534	+	0.924355i	$0.624604\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	−2.62742	−0.127150
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.3431	−0.498212	−0.249106	−	0.968476i	$-0.580137\pi$
$431$	−10.3431	−0.498212	−0.249106	+	0.968476i	$0.580137\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.65685	−0.270604
$438$	0	0
$439$	−20.1421	−0.961332	−0.480666	−	0.876904i	$-0.659605\pi$
$439$	−20.1421	−0.961332	−0.480666	+	0.876904i	$0.659605\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.9706	−0.996342	−0.498171	−	0.867079i	$-0.665995\pi$
$443$	−20.9706	−0.996342	−0.498171	+	0.867079i	$0.665995\pi$
$444$	0	0
$445$	−45.2548	−2.14528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.3431	−0.865667	−0.432833	−	0.901474i	$-0.642486\pi$
$449$	−18.3431	−0.865667	−0.432833	+	0.901474i	$0.642486\pi$
$450$	0	0
$451$	−0.343146	−0.0161581
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.3137	−0.530395
$456$	0	0
$457$	27.6569	1.29373	0.646867	−	0.762603i	$-0.276079\pi$
$457$	27.6569	1.29373	0.646867	+	0.762603i	$0.276079\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.31371	−0.247484	−0.123742	−	0.992314i	$-0.539490\pi$
$461$	−5.31371	−0.247484	−0.123742	+	0.992314i	$0.539490\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.686292	0.0317578	0.0158789	−	0.999874i	$-0.494945\pi$
$467$	0.686292	0.0317578	0.0158789	+	0.999874i	$0.494945\pi$
$468$	0	0
$469$	9.37258	0.432786
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.343146	0.0157779
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.9706	1.14093	0.570467	−	0.821320i	$-0.306762\pi$
$479$	24.9706	1.14093	0.570467	+	0.821320i	$0.306762\pi$
$480$	0	0
$481$	36.9706	1.68571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.0294	0.682452
$486$	0	0
$487$	9.65685	0.437594	0.218797	−	0.975770i	$-0.429787\pi$
$487$	9.65685	0.437594	0.218797	+	0.975770i	$0.429787\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.6274	−1.20168	−0.600839	−	0.799370i	$-0.705167\pi$
$491$	−26.6274	−1.20168	−0.600839	+	0.799370i	$0.705167\pi$
$492$	0	0
$493$	7.31371	0.329393
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.71573	0.166673
$498$	0	0
$499$	−43.3137	−1.93899	−0.969494	−	0.245115i	$-0.921174\pi$
$499$	−43.3137	−1.93899	−0.969494	+	0.245115i	$0.921174\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.3137	−0.861156	−0.430578	−	0.902553i	$-0.641690\pi$
$503$	−19.3137	−0.861156	−0.430578	+	0.902553i	$0.641690\pi$
$504$	0	0
$505$	−39.5980	−1.76209
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−39.7990	−1.76406	−0.882030	−	0.471194i	$-0.843823\pi$
$509$	−39.7990	−1.76406	−0.882030	+	0.471194i	$0.843823\pi$
$510$	0	0
$511$	−11.0294	−0.487914
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.4853	−0.549102
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.9706	−1.26922	−0.634612	−	0.772831i	$-0.718840\pi$
$521$	−28.9706	−1.26922	−0.634612	+	0.772831i	$0.718840\pi$
$522$	0	0
$523$	−10.9706	−0.479709	−0.239855	−	0.970809i	$-0.577100\pi$
$523$	−10.9706	−0.479709	−0.239855	+	0.970809i	$0.577100\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.3137	0.841318
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.65685	0.0717663
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.31371	−0.271951
$540$	0	0
$541$	19.1716	0.824250	0.412125	−	0.911127i	$-0.364787\pi$
$541$	19.1716	0.824250	0.412125	+	0.911127i	$0.364787\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	31.5980	1.35351
$546$	0	0
$547$	2.68629	0.114858	0.0574288	−	0.998350i	$-0.481710\pi$
$547$	2.68629	0.114858	0.0574288	+	0.998350i	$0.481710\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.31371	−0.311574
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.3137	0.733605	0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137	0.733605	0.366803	+	0.930299i	$0.380452\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.3137	1.65688	0.828438	−	0.560081i	$-0.189230\pi$
$563$	39.3137	1.65688	0.828438	+	0.560081i	$0.189230\pi$
$564$	0	0
$565$	−20.6863	−0.870279
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.31371	0.390451	0.195225	−	0.980758i	$-0.437456\pi$
$569$	9.31371	0.390451	0.195225	+	0.980758i	$0.437456\pi$
$570$	0	0
$571$	28.6274	1.19802	0.599010	−	0.800742i	$-0.295561\pi$
$571$	28.6274	1.19802	0.599010	+	0.800742i	$0.295561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.48528	−0.353861
$576$	0	0
$577$	−28.6274	−1.19177	−0.595887	−	0.803068i	$-0.703200\pi$
$577$	−28.6274	−1.19177	−0.595887	+	0.803068i	$0.703200\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	6.82843	0.282805
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.65685	0.0683857	0.0341928	−	0.999415i	$-0.489114\pi$
$587$	1.65685	0.0683857	0.0341928	+	0.999415i	$0.489114\pi$
$588$	0	0
$589$	−19.3137	−0.795807
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−4.68629	−0.192119
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.51472	−0.307043	−0.153522	−	0.988145i	$-0.549061\pi$
$599$	−7.51472	−0.307043	−0.153522	+	0.988145i	$0.549061\pi$
$600$	0	0
$601$	−17.3137	−0.706241	−0.353120	−	0.935578i	$-0.614879\pi$
$601$	−17.3137	−0.706241	−0.353120	+	0.935578i	$0.614879\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.82843	0.114992
$606$	0	0
$607$	−36.8284	−1.49482	−0.747410	−	0.664363i	$-0.768703\pi$
$607$	−36.8284	−1.49482	−0.747410	+	0.664363i	$0.768703\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	60.2843	2.43884
$612$	0	0
$613$	4.82843	0.195018	0.0975092	−	0.995235i	$-0.468912\pi$
$613$	4.82843	0.195018	0.0975092	+	0.995235i	$0.468912\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.3137	−1.26064	−0.630321	−	0.776334i	$-0.717077\pi$
$617$	−31.3137	−1.26064	−0.630321	+	0.776334i	$0.717077\pi$
$618$	0	0
$619$	−0.686292	−0.0275844	−0.0137922	−	0.999905i	$-0.504390\pi$
$619$	−0.686292	−0.0275844	−0.0137922	+	0.999905i	$0.504390\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.2548	−0.531044
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.3137	0.610598
$630$	0	0
$631$	11.0294	0.439075	0.219537	−	0.975604i	$-0.429545\pi$
$631$	11.0294	0.439075	0.219537	+	0.975604i	$0.429545\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	31.5980	1.25393
$636$	0	0
$637$	30.4853	1.20787
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.2843	1.11716	0.558581	−	0.829450i	$-0.311346\pi$
$641$	28.2843	1.11716	0.558581	+	0.829450i	$0.311346\pi$
$642$	0	0
$643$	−34.6274	−1.36557	−0.682786	−	0.730618i	$-0.739232\pi$
$643$	−34.6274	−1.36557	−0.682786	+	0.730618i	$0.739232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.7990	0.463866	0.231933	−	0.972732i	$-0.425495\pi$
$647$	11.7990	0.463866	0.231933	+	0.972732i	$0.425495\pi$
$648$	0	0
$649$	−1.65685	−0.0650372
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.1127	1.06100	0.530501	−	0.847684i	$-0.322004\pi$
$653$	27.1127	1.06100	0.530501	+	0.847684i	$0.322004\pi$
$654$	0	0
$655$	36.6863	1.43345
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.686292	−0.0267341	−0.0133671	−	0.999911i	$-0.504255\pi$
$659$	−0.686292	−0.0267341	−0.0133671	+	0.999911i	$0.504255\pi$
$660$	0	0
$661$	−11.6569	−0.453399	−0.226700	−	0.973965i	$-0.572794\pi$
$661$	−11.6569	−0.453399	−0.226700	+	0.973965i	$0.572794\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.68629	0.181727
$666$	0	0
$667$	10.3431	0.400488
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.17157	−0.122437
$672$	0	0
$673$	−8.34315	−0.321605	−0.160802	−	0.986987i	$-0.551408\pi$
$673$	−8.34315	−0.321605	−0.160802	+	0.986987i	$0.551408\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.9411	−0.920132	−0.460066	−	0.887885i	$-0.652174\pi$
$677$	−23.9411	−0.920132	−0.460066	+	0.887885i	$0.652174\pi$
$678$	0	0
$679$	4.40202	0.168934
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.9706	−1.56770	−0.783848	−	0.620953i	$-0.786746\pi$
$683$	−40.9706	−1.56770	−0.783848	+	0.620953i	$0.786746\pi$
$684$	0	0
$685$	−17.9411	−0.685495
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.9706	−1.25608
$690$	0	0
$691$	45.9411	1.74768	0.873841	−	0.486211i	$-0.161621\pi$
$691$	45.9411	1.74768	0.873841	+	0.486211i	$0.161621\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−26.3431	−0.999252
$696$	0	0
$697$	0.686292	0.0259951
$698$	0	0
$699$	0	0
$700$	0	0
$701$	44.6274	1.68555	0.842777	−	0.538263i	$-0.180919\pi$
$701$	44.6274	1.68555	0.842777	+	0.538263i	$0.180919\pi$
$702$	0	0
$703$	−15.3137	−0.577567
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.5980	−0.436187
$708$	0	0
$709$	39.6569	1.48934	0.744672	−	0.667430i	$-0.232606\pi$
$709$	39.6569	1.48934	0.744672	+	0.667430i	$0.232606\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.3137	1.02291
$714$	0	0
$715$	−13.6569	−0.510737
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.51472	−0.131077	−0.0655384	−	0.997850i	$-0.520876\pi$
$719$	−3.51472	−0.131077	−0.0655384	+	0.997850i	$0.520876\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.9706	−0.407436
$726$	0	0
$727$	20.9706	0.777755	0.388878	−	0.921289i	$-0.372863\pi$
$727$	20.9706	0.777755	0.388878	+	0.921289i	$0.372863\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.686292	−0.0253834
$732$	0	0
$733$	25.7990	0.952907	0.476454	−	0.879200i	$-0.341922\pi$
$733$	25.7990	0.952907	0.476454	+	0.879200i	$0.341922\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.3137	0.416746
$738$	0	0
$739$	−34.9706	−1.28641	−0.643206	−	0.765693i	$-0.722396\pi$
$739$	−34.9706	−1.28641	−0.643206	+	0.765693i	$0.722396\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.97056	0.329098	0.164549	−	0.986369i	$-0.447383\pi$
$743$	8.97056	0.329098	0.164549	+	0.986369i	$0.447383\pi$
$744$	0	0
$745$	−44.2843	−1.62245
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.31371	0.121080
$750$	0	0
$751$	43.5980	1.59091	0.795456	−	0.606011i	$-0.207231\pi$
$751$	43.5980	1.59091	0.795456	+	0.606011i	$0.207231\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	40.9706	1.49107
$756$	0	0
$757$	0.343146	0.0124718	0.00623592	−	0.999981i	$-0.498015\pi$
$757$	0.343146	0.0124718	0.00623592	+	0.999981i	$0.498015\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.6274	−1.18274	−0.591371	−	0.806399i	$-0.701413\pi$
$761$	−32.6274	−1.18274	−0.591371	+	0.806399i	$0.701413\pi$
$762$	0	0
$763$	9.25483	0.335047
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−27.6569	−0.997332	−0.498666	−	0.866794i	$-0.666177\pi$
$769$	−27.6569	−0.997332	−0.498666	+	0.866794i	$0.666177\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.5147	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$773$	19.5147	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$774$	0	0
$775$	−28.9706	−1.04065
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.686292	−0.0245889
$780$	0	0
$781$	4.48528	0.160496
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.6569	0.772966
$786$	0	0
$787$	−31.6569	−1.12844	−0.564222	−	0.825623i	$-0.690824\pi$
$787$	−31.6569	−1.12844	−0.564222	+	0.825623i	$0.690824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.05887	−0.215429
$792$	0	0
$793$	15.3137	0.543806
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.1421	1.06769	0.533845	−	0.845583i	$-0.320747\pi$
$797$	30.1421	1.06769	0.533845	+	0.845583i	$0.320747\pi$
$798$	0	0
$799$	24.9706	0.883395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.3137	−0.469831
$804$	0	0
$805$	−6.62742	−0.233586
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41.5980	−1.46251	−0.731254	−	0.682106i	$-0.761064\pi$
$809$	−41.5980	−1.46251	−0.731254	+	0.682106i	$0.761064\pi$
$810$	0	0
$811$	12.3431	0.433426	0.216713	−	0.976235i	$-0.430466\pi$
$811$	12.3431	0.433426	0.216713	+	0.976235i	$0.430466\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.3137	−0.396302
$816$	0	0
$817$	0.686292	0.0240103
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.6274	1.69711	0.848554	−	0.529108i	$-0.177473\pi$
$821$	48.6274	1.69711	0.848554	+	0.529108i	$0.177473\pi$
$822$	0	0
$823$	−35.3137	−1.23096	−0.615479	−	0.788153i	$-0.711038\pi$
$823$	−35.3137	−1.23096	−0.615479	+	0.788153i	$0.711038\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.9706	−0.451031	−0.225515	−	0.974240i	$-0.572407\pi$
$827$	−12.9706	−0.451031	−0.225515	+	0.974240i	$0.572407\pi$
$828$	0	0
$829$	−20.6274	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$829$	−20.6274	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.6274	0.437514
$834$	0	0
$835$	−45.2548	−1.56611
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.17157	−0.178543	−0.0892713	−	0.996007i	$-0.528454\pi$
$839$	−5.17157	−0.178543	−0.0892713	+	0.996007i	$0.528454\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	29.1716	1.00353
$846$	0	0
$847$	0.828427	0.0284651
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.6569	0.742387
$852$	0	0
$853$	−15.8579	−0.542963	−0.271481	−	0.962444i	$-0.587514\pi$
$853$	−15.8579	−0.542963	−0.271481	+	0.962444i	$0.587514\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.9706	0.511385	0.255692	−	0.966758i	$-0.417697\pi$
$857$	14.9706	0.511385	0.255692	+	0.966758i	$0.417697\pi$
$858$	0	0
$859$	13.9411	0.475665	0.237833	−	0.971306i	$-0.423563\pi$
$859$	13.9411	0.475665	0.237833	+	0.971306i	$0.423563\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.7696	0.979327	0.489663	−	0.871912i	$-0.337120\pi$
$863$	28.7696	0.979327	0.489663	+	0.871912i	$0.337120\pi$
$864$	0	0
$865$	−55.5980	−1.89039
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.82843	−0.163793
$870$	0	0
$871$	−54.6274	−1.85098
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.68629	−0.158426
$876$	0	0
$877$	−23.4558	−0.792048	−0.396024	−	0.918240i	$-0.629610\pi$
$877$	−23.4558	−0.792048	−0.396024	+	0.918240i	$0.629610\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.6274	0.358047	0.179023	−	0.983845i	$-0.442706\pi$
$881$	10.6274	0.358047	0.179023	+	0.983845i	$0.442706\pi$
$882$	0	0
$883$	28.6863	0.965371	0.482685	−	0.875794i	$-0.339662\pi$
$883$	28.6863	0.965371	0.482685	+	0.875794i	$0.339662\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.6569	−1.53301	−0.766504	−	0.642240i	$-0.778005\pi$
$887$	−45.6569	−1.53301	−0.766504	+	0.642240i	$0.778005\pi$
$888$	0	0
$889$	9.25483	0.310397
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24.9706	−0.835608
$894$	0	0
$895$	61.2548	2.04752
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35.3137	1.17778
$900$	0	0
$901$	−13.6569	−0.454976
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.9706	0.564121
$906$	0	0
$907$	−47.3137	−1.57103	−0.785513	−	0.618845i	$-0.787601\pi$
$907$	−47.3137	−1.57103	−0.785513	+	0.618845i	$0.787601\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11.5147	−0.381500	−0.190750	−	0.981639i	$-0.561092\pi$
$911$	−11.5147	−0.381500	−0.190750	+	0.981639i	$0.561092\pi$
$912$	0	0
$913$	9.65685	0.319595
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.7452	0.354837
$918$	0	0
$919$	51.4558	1.69737	0.848686	−	0.528897i	$-0.177394\pi$
$919$	51.4558	1.69737	0.848686	+	0.528897i	$0.177394\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.6569	−0.712844
$924$	0	0
$925$	−22.9706	−0.755267
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17.6569	−0.579303	−0.289651	−	0.957132i	$-0.593539\pi$
$929$	−17.6569	−0.579303	−0.289651	+	0.957132i	$0.593539\pi$
$930$	0	0
$931$	−12.6274	−0.413847
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.65685	−0.184999
$936$	0	0
$937$	19.9411	0.651448	0.325724	−	0.945465i	$-0.394392\pi$
$937$	19.9411	0.651448	0.325724	+	0.945465i	$0.394392\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.6274	1.32442	0.662208	−	0.749320i	$-0.269620\pi$
$941$	40.6274	1.32442	0.662208	+	0.749320i	$0.269620\pi$
$942$	0	0
$943$	0.970563	0.0316059
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	0	0
$949$	64.2843	2.08676
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43.6569	1.41418	0.707092	−	0.707121i	$-0.250007\pi$
$953$	43.6569	1.41418	0.707092	+	0.707121i	$0.250007\pi$
$954$	0	0
$955$	19.3137	0.624977
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.25483	−0.169687
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.2843	0.395445
$966$	0	0
$967$	−17.5147	−0.563235	−0.281618	−	0.959527i	$-0.590871\pi$
$967$	−17.5147	−0.563235	−0.281618	+	0.959527i	$0.590871\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.65685	0.309903	0.154952	−	0.987922i	$-0.450478\pi$
$971$	9.65685	0.309903	0.154952	+	0.987922i	$0.450478\pi$
$972$	0	0
$973$	−7.71573	−0.247355
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.34315	0.202935	0.101468	−	0.994839i	$-0.467646\pi$
$977$	6.34315	0.202935	0.101468	+	0.994839i	$0.467646\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38.1421	−1.21655	−0.608273	−	0.793728i	$-0.708137\pi$
$983$	−38.1421	−1.21655	−0.608273	+	0.793728i	$0.708137\pi$
$984$	0	0
$985$	37.6569	1.19985
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.970563	−0.0308621
$990$	0	0
$991$	−16.2843	−0.517287	−0.258643	−	0.965973i	$-0.583275\pi$
$991$	−16.2843	−0.517287	−0.258643	+	0.965973i	$0.583275\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	72.5685	2.30058
$996$	0	0
$997$	7.17157	0.227126	0.113563	−	0.993531i	$-0.463774\pi$
$997$	7.17157	0.227126	0.113563	+	0.993531i	$0.463774\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.a.cp.1.2		2
3.2	odd	2		6336.2.a.co.1.1		2
4.3	odd	2		6336.2.a.cq.1.2		2
8.3	odd	2		792.2.a.j.1.1	yes	2
8.5	even	2		1584.2.a.u.1.1		2
12.11	even	2		6336.2.a.cr.1.1		2
24.5	odd	2		1584.2.a.v.1.2		2
24.11	even	2		792.2.a.i.1.2	✓	2
88.43	even	2		8712.2.a.bi.1.1		2
264.131	odd	2		8712.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.a.i.1.2	✓	2	24.11	even	2
792.2.a.j.1.1	yes	2	8.3	odd	2
1584.2.a.u.1.1		2	8.5	even	2
1584.2.a.v.1.2		2	24.5	odd	2
6336.2.a.co.1.1		2	3.2	odd	2
6336.2.a.cp.1.2		2	1.1	even	1	trivial
6336.2.a.cq.1.2		2	4.3	odd	2
6336.2.a.cr.1.1		2	12.11	even	2
8712.2.a.bh.1.2		2	264.131	odd	2
8712.2.a.bi.1.1		2	88.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} +0.828427 q^{7} +1.00000 q^{11} -4.82843 q^{13} -2.00000 q^{17} +2.00000 q^{19} -2.82843 q^{23} +3.00000 q^{25} -3.65685 q^{29} -9.65685 q^{31} +2.34315 q^{35} -7.65685 q^{37} -0.343146 q^{41} +0.343146 q^{43} -12.4853 q^{47} -6.31371 q^{49} +6.82843 q^{53} +2.82843 q^{55} -1.65685 q^{59} -3.17157 q^{61} -13.6569 q^{65} +11.3137 q^{67} +4.48528 q^{71} -13.3137 q^{73} +0.828427 q^{77} -4.82843 q^{79} +9.65685 q^{83} -5.65685 q^{85} -16.0000 q^{89} -4.00000 q^{91} +5.65685 q^{95} +5.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} + 2 q^{11} - 4 q^{13} - 4 q^{17} + 4 q^{19} + 6 q^{25} + 4 q^{29} - 8 q^{31} + 16 q^{35} - 4 q^{37} - 12 q^{41} + 12 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} + 8 q^{59} - 12 q^{61} - 16 q^{65}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 + 2 * q^11 - 4 * q^13 - 4 * q^17 + 4 * q^19 + 6 * q^25 + 4 * q^29 - 8 * q^31 + 16 * q^35 - 4 * q^37 - 12 * q^41 + 12 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 + 8 * q^59 - 12 * q^61 - 16 * q^65 - 8 * q^71 - 4 * q^73 - 4 * q^77 - 4 * q^79 + 8 * q^83 - 32 * q^89 - 8 * q^91 - 12 * q^97

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