LMFDB - Embedded newform 6336.2.k.d.287.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(287,6336)] 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, 1, 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("6336.287"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.11843234091237376.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{11} + 8x^{10} - 4x^{9} - 12x^{7} + 56x^{6} - 36x^{5} - 108x^{3} + 648x^{2} - 972x + 729 $ x^12 - 4x^11 + 8x^10 - 4x^9 - 12x^7 + 56x^6 - 36x^5 - 108x^3 + 648x^2 - 972*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.8
Root			$1.43058 + 0.976448i$ of defining polynomial
Character	$\chi$	$=$	6336.287
Dual form			6336.2.k.d.287.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09174 q^{5} +0.908256i q^{7} -1.00000i q^{11} +3.39984i q^{13} -1.54396i q^{17} -2.69868 q^{19} -4.94380 q^{23} -3.80809 q^{25} +2.80809i q^{31} +0.991583i q^{35} -8.34363i q^{37} -8.34363i q^{41} -2.69868 q^{43} +1.85588 q^{47} +6.17507 q^{49} +1.09174 q^{53} -1.09174i q^{55} -11.7997i q^{59} -6.48776i q^{61} +3.71175i q^{65} -8.48528 q^{67} +1.85588 q^{71} -2.00000 q^{73} +0.908256 q^{77} -12.7079i q^{79} -8.62461i q^{83} -1.68561i q^{85} +5.64495i q^{89} -3.08792 q^{91} -2.94627 q^{95} +6.80809 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{5} + 36 q^{25} - 28 q^{49} + 8 q^{53} - 24 q^{73} + 16 q^{77}+O(q^{100}) $ 12 * q + 8 * q^5 + 36 * q^25 - 28 * q^49 + 8 * q^53 - 24 * q^73 + 16 * q^77

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
$5$	1.09174	0.488243	0.244121	−	0.969745i	$-0.421500\pi$
$5$	1.09174	0.488243	0.244121	+	0.969745i	$0.421500\pi$
$6$	0	0
$7$	0.908256i	0.343288i	0.985159	+	0.171644i	$0.0549079\pi$
$7$	0.908256i	0.343288i	−0.985159	+	0.171644i	$0.945092\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	3.39984i	0.942945i	0.881881	+	0.471472i	$0.156277\pi$
$13$	3.39984i	0.942945i	−0.881881	+	0.471472i	$0.843723\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.54396i	− 0.374465i	−0.982316	−	0.187233i	$-0.940048\pi$
$17$	− 1.54396i	− 0.374465i	0.982316	−	0.187233i	$-0.0599518\pi$
$18$	0	0
$19$	−2.69868	−0.619120	−0.309560	−	0.950880i	$-0.600182\pi$
$19$	−2.69868	−0.619120	−0.309560	+	0.950880i	$0.600182\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.94380	−1.03085	−0.515426	−	0.856934i	$-0.672366\pi$
$23$	−4.94380	−1.03085	−0.515426	+	0.856934i	$0.672366\pi$
$24$	0	0
$25$	−3.80809	−0.761619
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.80809i	0.504349i	0.967682	+	0.252174i	$0.0811456\pi$
$31$	2.80809i	0.504349i	−0.967682	+	0.252174i	$0.918854\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.991583i	0.167608i
$36$	0	0
$37$	− 8.34363i	− 1.37168i	−0.727750	−	0.685842i	$-0.759434\pi$
$37$	− 8.34363i	− 1.37168i	0.727750	−	0.685842i	$-0.240566\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.34363i	− 1.30306i	−0.758624	−	0.651528i	$-0.774128\pi$
$41$	− 8.34363i	− 1.30306i	0.758624	−	0.651528i	$-0.225872\pi$
$42$	0	0
$43$	−2.69868	−0.411545	−0.205772	−	0.978600i	$-0.565971\pi$
$43$	−2.69868	−0.411545	−0.205772	+	0.978600i	$0.565971\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.85588	0.270707	0.135354	−	0.990797i	$-0.456783\pi$
$47$	1.85588	0.270707	0.135354	+	0.990797i	$0.456783\pi$
$48$	0	0
$49$	6.17507	0.882153
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.09174	0.149963	0.0749813	−	0.997185i	$-0.476110\pi$
$53$	1.09174	0.149963	0.0749813	+	0.997185i	$0.476110\pi$
$54$	0	0
$55$	− 1.09174i	− 0.147211i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.7997i	− 1.53619i	−0.640338	−	0.768094i	$-0.721206\pi$
$59$	− 11.7997i	− 1.53619i	0.640338	−	0.768094i	$-0.278794\pi$
$60$	0	0
$61$	− 6.48776i	− 0.830672i	−0.909668	−	0.415336i	$-0.863664\pi$
$61$	− 6.48776i	− 0.830672i	0.909668	−	0.415336i	$-0.136336\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.71175i	0.460386i
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.85588	0.220252	0.110126	−	0.993918i	$-0.464875\pi$
$71$	1.85588	0.220252	0.110126	+	0.993918i	$0.464875\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.908256	0.103505
$78$	0	0
$79$	− 12.7079i	− 1.42975i	−0.699250	−	0.714877i	$-0.746483\pi$
$79$	− 12.7079i	− 1.42975i	0.699250	−	0.714877i	$-0.253517\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.62461i	− 0.946673i	−0.880882	−	0.473337i	$-0.843049\pi$
$83$	− 8.62461i	− 0.946673i	0.880882	−	0.473337i	$-0.156951\pi$
$84$	0	0
$85$	− 1.68561i	− 0.182830i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.64495i	0.598364i	0.954196	+	0.299182i	$0.0967137\pi$
$89$	5.64495i	0.598364i	−0.954196	+	0.299182i	$0.903286\pi$
$90$	0	0
$91$	−3.08792	−0.323702
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.94627	−0.302281
$96$	0	0
$97$	6.80809	0.691257	0.345629	−	0.938371i	$-0.387666\pi$
$97$	6.80809	0.691257	0.345629	+	0.938371i	$0.387666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.6246	1.45520	0.727601	−	0.686000i	$-0.240635\pi$
$101$	14.6246	1.45520	0.727601	+	0.686000i	$0.240635\pi$
$102$	0	0
$103$	− 8.99158i	− 0.885967i	−0.896530	−	0.442983i	$-0.853920\pi$
$103$	− 8.99158i	− 0.885967i	0.896530	−	0.442983i	$-0.146080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.79968i	0.560676i	0.959901	+	0.280338i	$0.0904465\pi$
$107$	5.79968i	0.560676i	−0.959901	+	0.280338i	$0.909553\pi$
$108$	0	0
$109$	− 5.08545i	− 0.487097i	−0.969889	−	0.243549i	$-0.921688\pi$
$109$	− 5.08545i	− 0.487097i	0.969889	−	0.243549i	$-0.0783116\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.24264i	− 0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$113$	− 4.24264i	− 0.399114i	0.979886	−	0.199557i	$-0.0639503\pi$
$114$	0	0
$115$	−5.39736	−0.503307
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.40231	0.128550
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.61619	−0.860098
$126$	0	0
$127$	5.27523i	0.468101i	0.972224	+	0.234051i	$0.0751982\pi$
$127$	5.27523i	0.468101i	−0.972224	+	0.234051i	$0.924802\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.80809i	0.420085i	0.977692	+	0.210043i	$0.0673603\pi$
$131$	4.80809i	0.420085i	−0.977692	+	0.210043i	$0.932640\pi$
$132$	0	0
$133$	− 2.45109i	− 0.212537i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.7279i	1.08742i	0.839273	+	0.543710i	$0.182981\pi$
$137$	12.7279i	1.08742i	−0.839273	+	0.543710i	$0.817019\pi$
$138$	0	0
$139$	−8.87452	−0.752727	−0.376363	−	0.926472i	$-0.622826\pi$
$139$	−8.87452	−0.752727	−0.376363	+	0.926472i	$0.622826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.39984	0.284309
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.19191	0.589184	0.294592	−	0.955623i	$-0.404816\pi$
$149$	7.19191	0.589184	0.294592	+	0.955623i	$0.404816\pi$
$150$	0	0
$151$	12.7079i	1.03416i	0.855938	+	0.517078i	$0.172980\pi$
$151$	12.7079i	1.03416i	−0.855938	+	0.517078i	$0.827020\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.06572i	0.246245i
$156$	0	0
$157$	− 6.79967i	− 0.542673i	−0.962485	−	0.271336i	$-0.912534\pi$
$157$	− 6.79967i	− 0.542673i	0.962485	−	0.271336i	$-0.0874656\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.49023i	− 0.353880i
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.85340	−0.298185	−0.149093	−	0.988823i	$-0.547635\pi$
$167$	−3.85340	−0.298185	−0.149093	+	0.988823i	$0.547635\pi$
$168$	0	0
$169$	1.44112	0.110855
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.6162	1.64345	0.821724	−	0.569885i	$-0.193012\pi$
$173$	21.6162	1.64345	0.821724	+	0.569885i	$0.193012\pi$
$174$	0	0
$175$	− 3.45872i	− 0.261455i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 21.6162i	− 1.61567i	−0.589409	−	0.807835i	$-0.700639\pi$
$179$	− 21.6162i	− 1.61567i	0.589409	−	0.807835i	$-0.299361\pi$
$180$	0	0
$181$	− 18.5145i	− 1.37617i	−0.725628	−	0.688087i	$-0.758451\pi$
$181$	− 18.5145i	− 1.37617i	0.725628	−	0.688087i	$-0.241549\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 9.10911i	− 0.669715i
$186$	0	0
$187$	−1.54396	−0.112906
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.16532	−0.301392	−0.150696	−	0.988580i	$-0.548151\pi$
$191$	−4.16532	−0.301392	−0.150696	+	0.988580i	$0.548151\pi$
$192$	0	0
$193$	−5.43270	−0.391054	−0.195527	−	0.980698i	$-0.562642\pi$
$193$	−5.43270	−0.391054	−0.195527	+	0.980698i	$0.562642\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.3586	0.809264	0.404632	−	0.914480i	$-0.367400\pi$
$197$	11.3586	0.809264	0.404632	+	0.914480i	$0.367400\pi$
$198$	0	0
$199$	4.55047i	0.322574i	0.986908	+	0.161287i	$0.0515645\pi$
$199$	4.55047i	0.322574i	−0.986908	+	0.161287i	$0.948436\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 9.10911i	− 0.636208i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.69868i	0.186672i
$210$	0	0
$211$	−15.6742	−1.07906	−0.539528	−	0.841968i	$-0.681397\pi$
$211$	−15.6742	−1.07906	−0.539528	+	0.841968i	$0.681397\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.94627	−0.200934
$216$	0	0
$217$	−2.55047	−0.173137
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.24921	0.353100
$222$	0	0
$223$	15.9832i	1.07031i	0.844753	+	0.535156i	$0.179747\pi$
$223$	15.9832i	1.07031i	−0.844753	+	0.535156i	$0.820253\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 22.1667i	− 1.47125i	−0.677388	−	0.735626i	$-0.736888\pi$
$227$	− 22.1667i	− 1.47125i	0.677388	−	0.735626i	$-0.263112\pi$
$228$	0	0
$229$	− 23.7702i	− 1.57078i	−0.619000	−	0.785391i	$-0.712462\pi$
$229$	− 23.7702i	− 1.57078i	0.619000	−	0.785391i	$-0.287538\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5195i	0.951202i	0.879661	+	0.475601i	$0.157769\pi$
$233$	14.5195i	0.951202i	−0.879661	+	0.475601i	$0.842231\pi$
$234$	0	0
$235$	2.02614	0.132171
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.0292	0.648738	0.324369	−	0.945931i	$-0.394848\pi$
$239$	10.0292	0.648738	0.324369	+	0.945931i	$0.394848\pi$
$240$	0	0
$241$	−15.9832	−1.02957	−0.514783	−	0.857320i	$-0.672128\pi$
$241$	−15.9832	−1.02957	−0.514783	+	0.857320i	$0.672128\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.74160	0.430705
$246$	0	0
$247$	− 9.17507i	− 0.583796i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 11.7997i	− 0.744789i	−0.928075	−	0.372395i	$-0.878537\pi$
$251$	− 11.7997i	− 0.744789i	0.928075	−	0.372395i	$-0.121463\pi$
$252$	0	0
$253$	4.94380i	0.310814i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.15472i	− 0.0720295i	−0.999351	−	0.0360148i	$-0.988534\pi$
$257$	− 1.15472i	− 0.0720295i	0.999351	−	0.0360148i	$-0.0114663\pi$
$258$	0	0
$259$	7.57815	0.470883
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.22957	0.199144	0.0995719	−	0.995030i	$-0.468253\pi$
$263$	3.22957	0.199144	0.0995719	+	0.995030i	$0.468253\pi$
$264$	0	0
$265$	1.19191	0.0732182
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.27523	0.199695	0.0998473	−	0.995003i	$-0.468165\pi$
$269$	3.27523	0.199695	0.0998473	+	0.995003i	$0.468165\pi$
$270$	0	0
$271$	12.9083i	0.784121i	0.919940	+	0.392060i	$0.128238\pi$
$271$	12.9083i	0.784121i	−0.919940	+	0.392060i	$0.871762\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.80809i	0.229637i
$276$	0	0
$277$	− 14.9730i	− 0.899643i	−0.893119	−	0.449821i	$-0.851488\pi$
$277$	− 14.9730i	− 0.899643i	0.893119	−	0.449821i	$-0.148512\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 0.765482i	− 0.0456648i	−0.999739	−	0.0228324i	$-0.992732\pi$
$281$	− 0.765482i	− 0.0456648i	0.999739	−	0.0228324i	$-0.00726842\pi$
$282$	0	0
$283$	−16.2980	−0.968817	−0.484408	−	0.874842i	$-0.660965\pi$
$283$	−16.2980	−0.968817	−0.484408	+	0.874842i	$0.660965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.57815	0.447324
$288$	0	0
$289$	14.6162	0.859776
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.16665	−0.243419	−0.121709	−	0.992566i	$-0.538838\pi$
$293$	−4.16665	−0.243419	−0.121709	+	0.992566i	$0.538838\pi$
$294$	0	0
$295$	− 12.8822i	− 0.750033i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16.8081i	− 0.972037i
$300$	0	0
$301$	− 2.45109i	− 0.141279i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7.08297i	− 0.405570i
$306$	0	0
$307$	−14.8957	−0.850143	−0.425072	−	0.905160i	$-0.639751\pi$
$307$	−14.8957	−0.850143	−0.425072	+	0.905160i	$0.639751\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.170266	−0.00965492	−0.00482746	−	0.999988i	$-0.501537\pi$
$311$	−0.170266	−0.00965492	−0.00482746	+	0.999988i	$0.501537\pi$
$312$	0	0
$313$	−15.6330	−0.883631	−0.441815	−	0.897106i	$-0.645665\pi$
$313$	−15.6330	−0.883631	−0.441815	+	0.897106i	$0.645665\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.34095	−0.356143	−0.178072	−	0.984018i	$-0.556986\pi$
$317$	−6.34095	−0.356143	−0.178072	+	0.984018i	$0.556986\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.16665i	0.231839i
$324$	0	0
$325$	− 12.9469i	− 0.718164i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.68561i	0.0929307i
$330$	0	0
$331$	−34.4363	−1.89279	−0.946395	−	0.323011i	$-0.895305\pi$
$331$	−34.4363	−1.89279	−0.946395	+	0.323011i	$0.895305\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.26376	−0.506133
$336$	0	0
$337$	−1.26604	−0.0689658	−0.0344829	−	0.999405i	$-0.510978\pi$
$337$	−1.26604	−0.0689658	−0.0344829	+	0.999405i	$0.510978\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.80809	0.152067
$342$	0	0
$343$	11.9663i	0.646121i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.9747i	0.803887i	0.915665	+	0.401943i	$0.131665\pi$
$347$	14.9747i	0.803887i	−0.915665	+	0.401943i	$0.868335\pi$
$348$	0	0
$349$	2.62136i	0.140318i	0.997536	+	0.0701590i	$0.0223507\pi$
$349$	2.62136i	0.140318i	−0.997536	+	0.0701590i	$0.977649\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 24.6417i	− 1.31154i	−0.754959	−	0.655772i	$-0.772343\pi$
$353$	− 24.6417i	− 1.31154i	0.754959	−	0.655772i	$-0.227657\pi$
$354$	0	0
$355$	2.02614	0.107536
$356$	0	0
$357$	0	0
$358$	0	0
$359$	37.3696	1.97229	0.986145	−	0.165883i	$-0.0530474\pi$
$359$	37.3696	1.97229	0.986145	+	0.165883i	$0.0530474\pi$
$360$	0	0
$361$	−11.7171	−0.616691
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.18349	−0.114289
$366$	0	0
$367$	− 0.0168341i	0 0.000878733i	−1.00000		0.000439366i	$-0.999860\pi$
$367$	− 0.0168341i	0 0.000878733i	1.00000		0.000439366i	$-0.000139855\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.991583i	0.0514804i
$372$	0	0
$373$	− 27.9485i	− 1.44712i	−0.690261	−	0.723560i	$-0.742504\pi$
$373$	− 27.9485i	− 1.44712i	0.690261	−	0.723560i	$-0.257496\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.30944	−0.118628	−0.0593140	−	0.998239i	$-0.518891\pi$
$379$	−2.30944	−0.118628	−0.0593140	+	0.998239i	$0.518891\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.16532	−0.212838	−0.106419	−	0.994321i	$-0.533938\pi$
$383$	−4.16532	−0.212838	−0.106419	+	0.994321i	$0.533938\pi$
$384$	0	0
$385$	0.991583	0.0505357
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.7248	1.05079	0.525394	−	0.850859i	$-0.323918\pi$
$389$	20.7248	1.05079	0.525394	+	0.850859i	$0.323918\pi$
$390$	0	0
$391$	7.63302i	0.386019i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 13.8738i	− 0.698067i
$396$	0	0
$397$	21.6024i	1.08420i	0.840315	+	0.542098i	$0.182370\pi$
$397$	21.6024i	1.08420i	−0.840315	+	0.542098i	$0.817630\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.9250i	1.24469i	0.782742	+	0.622346i	$0.213820\pi$
$401$	24.9250i	1.24469i	−0.782742	+	0.622346i	$0.786180\pi$
$402$	0	0
$403$	−9.54706	−0.475573
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.34363	−0.413578
$408$	0	0
$409$	5.43270	0.268630	0.134315	−	0.990939i	$-0.457117\pi$
$409$	5.43270	0.268630	0.134315	+	0.990939i	$0.457117\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.7171	0.527355
$414$	0	0
$415$	− 9.41586i	− 0.462207i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6.55047i	− 0.320011i	−0.987116	−	0.160006i	$-0.948849\pi$
$419$	− 6.55047i	− 0.320011i	0.987116	−	0.160006i	$-0.0511512\pi$
$420$	0	0
$421$	− 5.53901i	− 0.269955i	−0.990849	−	0.134977i	$-0.956904\pi$
$421$	− 5.53901i	− 0.269955i	0.990849	−	0.134977i	$-0.0430962\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.87954i	0.285200i
$426$	0	0
$427$	5.89254	0.285160
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.7116	−1.47932	−0.739662	−	0.672979i	$-0.765015\pi$
$431$	−30.7116	−1.47932	−0.739662	+	0.672979i	$0.765015\pi$
$432$	0	0
$433$	−9.98317	−0.479760	−0.239880	−	0.970803i	$-0.577108\pi$
$433$	−9.98317	−0.479760	−0.239880	+	0.970803i	$0.577108\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	13.3417	0.638221
$438$	0	0
$439$	− 5.07491i	− 0.242212i	−0.992640	−	0.121106i	$-0.961356\pi$
$439$	− 5.07491i	− 0.242212i	0.992640	−	0.121106i	$-0.0386441\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 2.38381i	− 0.113258i	−0.998395	−	0.0566292i	$-0.981965\pi$
$443$	− 2.38381i	− 0.113258i	0.998395	−	0.0566292i	$-0.0180353\pi$
$444$	0	0
$445$	6.16284i	0.292147i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.4446i	0.587298i	0.955913	+	0.293649i	$0.0948697\pi$
$449$	12.4446i	0.587298i	−0.955913	+	0.293649i	$0.905130\pi$
$450$	0	0
$451$	−8.34363	−0.392886
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.37122	−0.158045
$456$	0	0
$457$	27.0489	1.26529	0.632647	−	0.774440i	$-0.281969\pi$
$457$	27.0489	1.26529	0.632647	+	0.774440i	$0.281969\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.200323	0.00932997	0.00466499	−	0.999989i	$-0.498515\pi$
$461$	0.200323	0.00932997	0.00466499	+	0.999989i	$0.498515\pi$
$462$	0	0
$463$	− 16.9916i	− 0.789666i	−0.918753	−	0.394833i	$-0.870802\pi$
$463$	− 16.9916i	− 0.789666i	0.918753	−	0.394833i	$-0.129198\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.26604i	− 0.151134i	−0.997141	−	0.0755672i	$-0.975923\pi$
$467$	− 3.26604i	− 0.151134i	0.997141	−	0.0755672i	$-0.0240768\pi$
$468$	0	0
$469$	− 7.70680i	− 0.355867i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.69868i	0.124085i
$474$	0	0
$475$	10.2768	0.471533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.8289	0.768933	0.384466	−	0.923139i	$-0.374385\pi$
$479$	16.8289	0.768933	0.384466	+	0.923139i	$0.374385\pi$
$480$	0	0
$481$	28.3670	1.29342
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.43270	0.337501
$486$	0	0
$487$	− 7.41586i	− 0.336045i	−0.985783	−	0.168022i	$-0.946262\pi$
$487$	− 7.41586i	− 0.336045i	0.985783	−	0.168022i	$-0.0537381\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 8.18349i	− 0.369316i	−0.982803	−	0.184658i	$-0.940882\pi$
$491$	− 8.18349i	− 0.369316i	0.982803	−	0.184658i	$-0.0591177\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.68561i	0.0756099i
$498$	0	0
$499$	−24.3941	−1.09203	−0.546014	−	0.837776i	$-0.683856\pi$
$499$	−24.3941	−1.09203	−0.546014	+	0.837776i	$0.683856\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.94132	0.309498	0.154749	−	0.987954i	$-0.450543\pi$
$503$	6.94132	0.309498	0.154749	+	0.987954i	$0.450543\pi$
$504$	0	0
$505$	15.9663	0.710492
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.7248	−0.918609	−0.459305	−	0.888279i	$-0.651901\pi$
$509$	−20.7248	−0.918609	−0.459305	+	0.888279i	$0.651901\pi$
$510$	0	0
$511$	− 1.81651i	− 0.0803577i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 9.81651i	− 0.432567i
$516$	0	0
$517$	− 1.85588i	− 0.0816213i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.1302i	0.619057i	0.950890	+	0.309528i	$0.100171\pi$
$521$	14.1302i	0.619057i	−0.950890	+	0.309528i	$0.899829\pi$
$522$	0	0
$523$	−28.4951	−1.24600	−0.623001	−	0.782221i	$-0.714087\pi$
$523$	−28.4951	−1.24600	−0.623001	+	0.782221i	$0.714087\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.33558	0.188861
$528$	0	0
$529$	1.44112	0.0626572
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.3670	1.22871
$534$	0	0
$535$	6.33177i	0.273746i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.17507i	− 0.265979i
$540$	0	0
$541$	5.08545i	0.218640i	0.994007	+	0.109320i	$0.0348674\pi$
$541$	5.08545i	0.218640i	−0.994007	+	0.109320i	$0.965133\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.55201i	− 0.237822i
$546$	0	0
$547$	2.07485	0.0887141	0.0443571	−	0.999016i	$-0.485876\pi$
$547$	2.07485	0.0887141	0.0443571	+	0.999016i	$0.485876\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11.5420	0.490818
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.1667	−1.19346	−0.596730	−	0.802442i	$-0.703534\pi$
$557$	−28.1667	−1.19346	−0.596730	+	0.802442i	$0.703534\pi$
$558$	0	0
$559$	− 9.17507i	− 0.388064i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.26604i	0.137647i	0.997629	+	0.0688237i	$0.0219246\pi$
$563$	3.26604i	0.137647i	−0.997629	+	0.0688237i	$0.978075\pi$
$564$	0	0
$565$	− 4.63188i	− 0.194865i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 4.91518i	− 0.206055i	−0.994679	−	0.103028i	$-0.967147\pi$
$569$	− 4.91518i	− 0.206055i	0.994679	−	0.103028i	$-0.0328530\pi$
$570$	0	0
$571$	35.4494	1.48351	0.741755	−	0.670671i	$-0.233994\pi$
$571$	35.4494	1.48351	0.741755	+	0.670671i	$0.233994\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.8264	0.785117
$576$	0	0
$577$	−42.4074	−1.76545	−0.882723	−	0.469895i	$-0.844292\pi$
$577$	−42.4074	−1.76545	−0.882723	+	0.469895i	$0.844292\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.83335	0.324982
$582$	0	0
$583$	− 1.09174i	− 0.0452154i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.1667i	− 1.16256i	−0.813703	−	0.581281i	$-0.802552\pi$
$587$	− 28.1667i	− 1.16256i	0.813703	−	0.581281i	$-0.197448\pi$
$588$	0	0
$589$	− 7.57815i	− 0.312252i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 26.4332i	− 1.08548i	−0.839900	−	0.542741i	$-0.817387\pi$
$593$	− 26.4332i	− 1.08548i	0.839900	−	0.542741i	$-0.182613\pi$
$594$	0	0
$595$	1.53096	0.0627634
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.8019	−1.29939	−0.649696	−	0.760194i	$-0.725104\pi$
$599$	−31.8019	−1.29939	−0.649696	+	0.760194i	$0.725104\pi$
$600$	0	0
$601$	6.56730	0.267886	0.133943	−	0.990989i	$-0.457236\pi$
$601$	6.56730	0.267886	0.133943	+	0.990989i	$0.457236\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.09174	−0.0443857
$606$	0	0
$607$	− 13.4756i	− 0.546956i	−0.961878	−	0.273478i	$-0.911826\pi$
$607$	− 13.4756i	− 0.546956i	0.961878	−	0.273478i	$-0.0881741\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.30967i	0.255262i
$612$	0	0
$613$	− 40.1456i	− 1.62146i	−0.585417	−	0.810732i	$-0.699069\pi$
$613$	− 40.1456i	− 1.62146i	0.585417	−	0.810732i	$-0.300931\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.247589i	0.00996756i	0.999988	+	0.00498378i	$0.00158639\pi$
$617$	0.247589i	0.00996756i	−0.999988	+	0.00498378i	$0.998414\pi$
$618$	0	0
$619$	−30.5699	−1.22871	−0.614354	−	0.789030i	$-0.710583\pi$
$619$	−30.5699	−1.22871	−0.614354	+	0.789030i	$0.710583\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.12706	−0.205411
$624$	0	0
$625$	8.54205	0.341682
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.8822	−0.513648
$630$	0	0
$631$	3.54205i	0.141007i	0.997512	+	0.0705034i	$0.0224606\pi$
$631$	3.54205i	0.141007i	−0.997512	+	0.0705034i	$0.977539\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.75921i	0.228547i
$636$	0	0
$637$	20.9942i	0.831822i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 21.8370i	− 0.862511i	−0.902230	−	0.431256i	$-0.858071\pi$
$641$	− 21.8370i	− 0.862511i	0.902230	−	0.431256i	$-0.141929\pi$
$642$	0	0
$643$	16.9706	0.669254	0.334627	−	0.942351i	$-0.391390\pi$
$643$	16.9706	0.669254	0.334627	+	0.942351i	$0.391390\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.0251	−1.80943	−0.904717	−	0.426013i	$-0.859918\pi$
$647$	−46.0251	−1.80943	−0.904717	+	0.426013i	$0.859918\pi$
$648$	0	0
$649$	−11.7997	−0.463178
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.1086	−0.434712	−0.217356	−	0.976092i	$-0.569743\pi$
$653$	−11.1086	−0.434712	−0.217356	+	0.976092i	$0.569743\pi$
$654$	0	0
$655$	5.24921i	0.205104i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.23238i	0.281733i	0.990029	+	0.140867i	$0.0449889\pi$
$659$	7.23238i	0.281733i	−0.990029	+	0.140867i	$0.955011\pi$
$660$	0	0
$661$	− 11.4186i	− 0.444130i	−0.975032	−	0.222065i	$-0.928720\pi$
$661$	− 11.4186i	− 0.444130i	0.975032	−	0.222065i	$-0.0712798\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2.67597i	− 0.103770i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.48776	−0.250457
$672$	0	0
$673$	−41.9663	−1.61768	−0.808842	−	0.588027i	$-0.799905\pi$
$673$	−41.9663	−1.61768	−0.808842	+	0.588027i	$0.799905\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.3586	0.897742	0.448871	−	0.893596i	$-0.351826\pi$
$677$	23.3586	0.897742	0.448871	+	0.893596i	$0.351826\pi$
$678$	0	0
$679$	6.18349i	0.237301i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.1835i	− 0.542716i	−0.962478	−	0.271358i	$-0.912527\pi$
$683$	− 14.1835i	− 0.542716i	0.962478	−	0.271358i	$-0.0874728\pi$
$684$	0	0
$685$	13.8956i	0.530925i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.71175i	0.141406i
$690$	0	0
$691$	7.70680	0.293181	0.146590	−	0.989197i	$-0.453170\pi$
$691$	7.70680	0.293181	0.146590	+	0.989197i	$0.453170\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.68871	−0.367514
$696$	0	0
$697$	−12.8822	−0.487949
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25.7828	−0.973805	−0.486902	−	0.873456i	$-0.661873\pi$
$701$	−25.7828	−0.973805	−0.486902	+	0.873456i	$0.661873\pi$
$702$	0	0
$703$	22.5168i	0.849237i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.2829i	0.499554i
$708$	0	0
$709$	− 10.1709i	− 0.381976i	−0.981592	−	0.190988i	$-0.938831\pi$
$709$	− 10.1709i	− 0.381976i	0.981592	−	0.190988i	$-0.0611691\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13.8826i	− 0.519909i
$714$	0	0
$715$	3.71175	0.138812
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.8019	−1.18601	−0.593006	−	0.805198i	$-0.702059\pi$
$719$	−31.8019	−1.18601	−0.593006	+	0.805198i	$0.702059\pi$
$720$	0	0
$721$	8.16665	0.304142
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.0657i	0.632933i	0.948604	+	0.316466i	$0.102496\pi$
$727$	17.0657i	0.632933i	−0.948604	+	0.316466i	$0.897504\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.16665i	0.154109i
$732$	0	0
$733$	48.6309i	1.79622i	0.439769	+	0.898111i	$0.355060\pi$
$733$	48.6309i	1.79622i	−0.439769	+	0.898111i	$0.644940\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.48528i	0.312559i
$738$	0	0
$739$	−2.07485	−0.0763245	−0.0381623	−	0.999272i	$-0.512150\pi$
$739$	−2.07485	−0.0763245	−0.0381623	+	0.999272i	$0.512150\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.6333	1.71081	0.855406	−	0.517959i	$-0.173308\pi$
$743$	46.6333	1.71081	0.855406	+	0.517959i	$0.173308\pi$
$744$	0	0
$745$	7.85172	0.287665
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.26759	−0.192473
$750$	0	0
$751$	− 7.90903i	− 0.288605i	−0.989534	−	0.144302i	$-0.953906\pi$
$751$	− 7.90903i	− 0.288605i	0.989534	−	0.144302i	$-0.0460938\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	13.8738i	0.504920i
$756$	0	0
$757$	29.6498i	1.07764i	0.842421	+	0.538820i	$0.181130\pi$
$757$	29.6498i	1.07764i	−0.842421	+	0.538820i	$0.818870\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 4.34858i	− 0.157636i	−0.996889	−	0.0788180i	$-0.974885\pi$
$761$	− 4.34858i	− 0.157636i	0.996889	−	0.0788180i	$-0.0251146\pi$
$762$	0	0
$763$	4.61888	0.167215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.1170	1.44854
$768$	0	0
$769$	18.1667	0.655106	0.327553	−	0.944833i	$-0.393776\pi$
$769$	18.1667	0.655106	0.327553	+	0.944833i	$0.393776\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.52444	0.306603	0.153301	−	0.988179i	$-0.451009\pi$
$773$	8.52444	0.306603	0.153301	+	0.988179i	$0.451009\pi$
$774$	0	0
$775$	− 10.6935i	− 0.384121i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.5168i	0.806748i
$780$	0	0
$781$	− 1.85588i	− 0.0664085i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 7.42350i	− 0.264956i
$786$	0	0
$787$	40.2229	1.43379	0.716896	−	0.697180i	$-0.245562\pi$
$787$	40.2229	1.43379	0.716896	+	0.697180i	$0.245562\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.85340	0.137011
$792$	0	0
$793$	22.0573	0.783278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.1086	−0.393486	−0.196743	−	0.980455i	$-0.563036\pi$
$797$	−11.1086	−0.393486	−0.196743	+	0.980455i	$0.563036\pi$
$798$	0	0
$799$	− 2.86540i	− 0.101370i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	− 4.90218i	− 0.172779i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 47.3988i	− 1.66645i	−0.552931	−	0.833227i	$-0.686491\pi$
$809$	− 47.3988i	− 1.66645i	0.552931	−	0.833227i	$-0.313509\pi$
$810$	0	0
$811$	37.4755	1.31594	0.657972	−	0.753043i	$-0.271415\pi$
$811$	37.4755	1.31594	0.657972	+	0.753043i	$0.271415\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.26376	−0.324495
$816$	0	0
$817$	7.28288	0.254796
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.4900	1.44801	0.724006	−	0.689794i	$-0.242299\pi$
$821$	41.4900	1.44801	0.724006	+	0.689794i	$0.242299\pi$
$822$	0	0
$823$	3.98317i	0.138844i	0.997587	+	0.0694222i	$0.0221156\pi$
$823$	3.98317i	0.138844i	−0.997587	+	0.0694222i	$0.977884\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.3586i	1.43818i	0.694918	+	0.719089i	$0.255441\pi$
$827$	41.3586i	1.43818i	−0.694918	+	0.719089i	$0.744559\pi$
$828$	0	0
$829$	3.37122i	0.117087i	0.998285	+	0.0585436i	$0.0186457\pi$
$829$	3.37122i	0.117087i	−0.998285	+	0.0585436i	$0.981354\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 9.53406i	− 0.330336i
$834$	0	0
$835$	−4.20693	−0.145587
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.0268	0.415210	0.207605	−	0.978213i	$-0.433433\pi$
$839$	12.0268	0.415210	0.207605	+	0.978213i	$0.433433\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.57333	0.0541242
$846$	0	0
$847$	− 0.908256i	− 0.0312080i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41.2492i	1.41400i
$852$	0	0
$853$	− 31.6603i	− 1.08403i	−0.840369	−	0.542014i	$-0.817662\pi$
$853$	− 31.6603i	− 1.08403i	0.840369	−	0.542014i	$-0.182338\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.4283i	1.03941i	0.854346	+	0.519705i	$0.173958\pi$
$857$	30.4283i	1.03941i	−0.854346	+	0.519705i	$0.826042\pi$
$858$	0	0
$859$	12.8209	0.437442	0.218721	−	0.975787i	$-0.429811\pi$
$859$	12.8209	0.437442	0.218721	+	0.975787i	$0.429811\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.1114	−1.16117	−0.580583	−	0.814201i	$-0.697175\pi$
$863$	−34.1114	−1.16117	−0.580583	+	0.814201i	$0.697175\pi$
$864$	0	0
$865$	23.5994	0.802402
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.7079	−0.431087
$870$	0	0
$871$	− 28.8486i	− 0.977496i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 8.73396i	− 0.295262i
$876$	0	0
$877$	30.2580i	1.02174i	0.859658	+	0.510870i	$0.170677\pi$
$877$	30.2580i	1.02174i	−0.859658	+	0.510870i	$0.829323\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.0440i	0.877444i	0.898623	+	0.438722i	$0.144569\pi$
$881$	26.0440i	0.877444i	−0.898623	+	0.438722i	$0.855431\pi$
$882$	0	0
$883$	2.30944	0.0777189	0.0388595	−	0.999245i	$-0.487628\pi$
$883$	2.30944	0.0777189	0.0388595	+	0.999245i	$0.487628\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.08297	0.237823	0.118911	−	0.992905i	$-0.462060\pi$
$887$	7.08297	0.237823	0.118911	+	0.992905i	$0.462060\pi$
$888$	0	0
$889$	−4.79126	−0.160694
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.00842	−0.167600
$894$	0	0
$895$	− 23.5994i	− 0.788839i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 1.68561i	− 0.0561558i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 20.2131i	− 0.671907i
$906$	0	0
$907$	−7.42350	−0.246493	−0.123247	−	0.992376i	$-0.539331\pi$
$907$	−7.42350	−0.246493	−0.123247	+	0.992376i	$0.539331\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.3167	0.772516	0.386258	−	0.922391i	$-0.373767\pi$
$911$	23.3167	0.772516	0.386258	+	0.922391i	$0.373767\pi$
$912$	0	0
$913$	−8.62461	−0.285433
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.36698	−0.144210
$918$	0	0
$919$	− 4.14063i	− 0.136587i	−0.997665	−	0.0682934i	$-0.978245\pi$
$919$	− 4.14063i	− 0.136587i	0.997665	−	0.0682934i	$-0.0217554\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.30967i	0.207685i
$924$	0	0
$925$	31.7733i	1.04470i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.9884i	1.34479i	0.740195	+	0.672393i	$0.234733\pi$
$929$	40.9884i	1.34479i	−0.740195	+	0.672393i	$0.765267\pi$
$930$	0	0
$931$	−16.6645	−0.546159
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.68561	−0.0551253
$936$	0	0
$937$	−27.9832	−0.914170	−0.457085	−	0.889423i	$-0.651107\pi$
$937$	−27.9832	−0.914170	−0.457085	+	0.889423i	$0.651107\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.5909	−1.38842	−0.694212	−	0.719770i	$-0.744247\pi$
$941$	−42.5909	−1.38842	−0.694212	+	0.719770i	$0.744247\pi$
$942$	0	0
$943$	41.2492i	1.34326i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 52.8486i	− 1.71735i	−0.512523	−	0.858674i	$-0.671289\pi$
$947$	− 52.8486i	− 1.71735i	0.512523	−	0.858674i	$-0.328711\pi$
$948$	0	0
$949$	− 6.79967i	− 0.220727i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.4478i	− 0.694762i	−0.937724	−	0.347381i	$-0.887071\pi$
$953$	− 21.4478i	− 0.694762i	0.937724	−	0.347381i	$-0.112929\pi$
$954$	0	0
$955$	−4.54746	−0.147152
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.5602	−0.373299
$960$	0	0
$961$	23.1146	0.745633
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.93112	−0.190929
$966$	0	0
$967$	22.8914i	0.736138i	0.929798	+	0.368069i	$0.119981\pi$
$967$	22.8914i	0.736138i	−0.929798	+	0.368069i	$0.880019\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	34.4984i	1.10711i	0.832814	+	0.553553i	$0.186729\pi$
$971$	34.4984i	1.10711i	−0.832814	+	0.553553i	$0.813271\pi$
$972$	0	0
$973$	− 8.06033i	− 0.258402i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.3011i	0.777462i	0.921351	+	0.388731i	$0.127086\pi$
$977$	24.3011i	0.777462i	−0.921351	+	0.388731i	$0.872914\pi$
$978$	0	0
$979$	5.64495	0.180413
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.2287	−0.645197	−0.322598	−	0.946536i	$-0.604556\pi$
$983$	−20.2287	−0.645197	−0.322598	+	0.946536i	$0.604556\pi$
$984$	0	0
$985$	12.4006	0.395117
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.3417	0.424242
$990$	0	0
$991$	7.17507i	0.227924i	0.993485	+	0.113962i	$0.0363542\pi$
$991$	7.17507i	0.227924i	−0.993485	+	0.113962i	$0.963646\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.96795i	0.157495i
$996$	0	0
$997$	− 1.71423i	− 0.0542901i	−0.999632	−	0.0271450i	$-0.991358\pi$
$997$	− 1.71423i	− 0.0542901i	0.999632	−	0.0271450i	$-0.00864160\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.k.d.287.8	yes	12
3.2	odd	2		6336.2.k.c.287.8	yes	12
4.3	odd	2	inner	6336.2.k.d.287.6	yes	12
8.3	odd	2		6336.2.k.c.287.5	✓	12
8.5	even	2		6336.2.k.c.287.7	yes	12
12.11	even	2		6336.2.k.c.287.6	yes	12
24.5	odd	2	inner	6336.2.k.d.287.7	yes	12
24.11	even	2	inner	6336.2.k.d.287.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6336.2.k.c.287.5	✓	12	8.3	odd	2
6336.2.k.c.287.6	yes	12	12.11	even	2
6336.2.k.c.287.7	yes	12	8.5	even	2
6336.2.k.c.287.8	yes	12	3.2	odd	2
6336.2.k.d.287.5	yes	12	24.11	even	2	inner
6336.2.k.d.287.6	yes	12	4.3	odd	2	inner
6336.2.k.d.287.7	yes	12	24.5	odd	2	inner
6336.2.k.d.287.8	yes	12	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.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.09174 q^{5} +0.908256i q^{7} -1.00000i q^{11} +3.39984i q^{13} -1.54396i q^{17} -2.69868 q^{19} -4.94380 q^{23} -3.80809 q^{25} +2.80809i q^{31} +0.991583i q^{35} -8.34363i q^{37} -8.34363i q^{41} -2.69868 q^{43} +1.85588 q^{47} +6.17507 q^{49} +1.09174 q^{53} -1.09174i q^{55} -11.7997i q^{59} -6.48776i q^{61} +3.71175i q^{65} -8.48528 q^{67} +1.85588 q^{71} -2.00000 q^{73} +0.908256 q^{77} -12.7079i q^{79} -8.62461i q^{83} -1.68561i q^{85} +5.64495i q^{89} -3.08792 q^{91} -2.94627 q^{95} +6.80809 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{5} + 36 q^{25} - 28 q^{49} + 8 q^{53} - 24 q^{73} + 16 q^{77}+O(q^{100}) \) 12 * q + 8 * q^5 + 36 * q^25 - 28 * q^49 + 8 * q^53 - 24 * q^73 + 16 * q^77

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more