LMFDB - Embedded newform 6336.2.f.m.3169.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6336,2,Mod(3169,6336)]

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, 1, 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("6336.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

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:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.12
Root			$-0.342020 - 0.939693i$ of defining polynomial
Character	$\chi$	$=$	6336.3169
Dual form			6336.2.f.m.3169.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.93923i q^{5} +1.36808 q^{7} +O(q^{10})$	$q+3.93923i q^{5} +1.36808 q^{7} -1.00000i q^{11} +3.93923i q^{13} -5.06418 q^{17} +1.06418i q^{19} +1.20307 q^{23} -10.5175 q^{25} -6.03525i q^{29} +8.60640 q^{31} +5.38919i q^{35} +0.325008 q^{41} +12.5817i q^{43} -8.13127 q^{47} -5.12836 q^{49} -8.13127i q^{53} +3.93923 q^{55} +5.38919i q^{59} +11.8177i q^{61} -15.5175 q^{65} +7.51754i q^{67} -13.6036 q^{71} -2.73917 q^{73} -1.36808i q^{77} -5.56012 q^{79} -3.51754i q^{83} -19.9490i q^{85} +13.6459 q^{89} +5.38919i q^{91} -4.19204 q^{95} -10.9067 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 24 q^{17} - 36 q^{25} + 24 q^{41} + 12 q^{49} - 96 q^{65} + 24 q^{73} - 24 q^{97}+O(q^{100}) $ 12 * q - 24 * q^17 - 36 * q^25 + 24 * q^41 + 12 * q^49 - 96 * q^65 + 24 * q^73 - 24 * q^97

Display 100 coefficients 10 coefficients

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
$5$	3.93923i	1.76168i	0.473416	+	0.880839i	$0.343021\pi$
$5$	3.93923i	1.76168i	−0.473416	+	0.880839i	$0.656979\pi$
$6$	0	0
$7$	1.36808	0.517086	0.258543	−	0.966000i	$-0.416758\pi$
$7$	1.36808	0.517086	0.258543	+	0.966000i	$0.416758\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	3.93923i	1.09255i	0.837607	+	0.546273i	$0.183954\pi$
$13$	3.93923i	1.09255i	−0.837607	+	0.546273i	$0.816046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.06418	−1.22824	−0.614122	−	0.789211i	$-0.710490\pi$
$17$	−5.06418	−1.22824	−0.614122	+	0.789211i	$0.710490\pi$
$18$	0	0
$19$	1.06418i	0.244139i	0.992522	+	0.122070i	$0.0389531\pi$
$19$	1.06418i	0.244139i	−0.992522	+	0.122070i	$0.961047\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.20307	0.250857	0.125429	−	0.992103i	$-0.459969\pi$
$23$	1.20307	0.250857	0.125429	+	0.992103i	$0.459969\pi$
$24$	0	0
$25$	−10.5175	−2.10351
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.03525i	− 1.12072i	−0.828250	−	0.560359i	$-0.810663\pi$
$29$	− 6.03525i	− 1.12072i	0.828250	−	0.560359i	$-0.189337\pi$
$30$	0	0
$31$	8.60640	1.54576	0.772878	−	0.634555i	$-0.218817\pi$
$31$	8.60640	1.54576	0.772878	+	0.634555i	$0.218817\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.38919i	0.910939i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.325008	0.0507577	0.0253788	−	0.999678i	$-0.491921\pi$
$41$	0.325008	0.0507577	0.0253788	+	0.999678i	$0.491921\pi$
$42$	0	0
$43$	12.5817i	1.91869i	0.282229	+	0.959347i	$0.408926\pi$
$43$	12.5817i	1.91869i	−0.282229	+	0.959347i	$0.591074\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.13127	−1.18607	−0.593034	−	0.805177i	$-0.702070\pi$
$47$	−8.13127	−1.18607	−0.593034	+	0.805177i	$0.702070\pi$
$48$	0	0
$49$	−5.12836	−0.732622
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.13127i	− 1.11692i	−0.829533	−	0.558458i	$-0.811393\pi$
$53$	− 8.13127i	− 1.11692i	0.829533	−	0.558458i	$-0.188607\pi$
$54$	0	0
$55$	3.93923	0.531166
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.38919i	0.701612i	0.936448	+	0.350806i	$0.114092\pi$
$59$	5.38919i	0.701612i	−0.936448	+	0.350806i	$0.885908\pi$
$60$	0	0
$61$	11.8177i	1.51310i	0.653936	+	0.756550i	$0.273117\pi$
$61$	11.8177i	1.51310i	−0.653936	+	0.756550i	$0.726883\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.5175	−1.92471
$66$	0	0
$67$	7.51754i	0.918414i	0.888329	+	0.459207i	$0.151866\pi$
$67$	7.51754i	0.918414i	−0.888329	+	0.459207i	$0.848134\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.6036	−1.61445	−0.807225	−	0.590244i	$-0.799032\pi$
$71$	−13.6036	−1.61445	−0.807225	+	0.590244i	$0.799032\pi$
$72$	0	0
$73$	−2.73917	−0.320596	−0.160298	−	0.987069i	$-0.551245\pi$
$73$	−2.73917	−0.320596	−0.160298	+	0.987069i	$0.551245\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.36808i	− 0.155907i
$78$	0	0
$79$	−5.56012	−0.625563	−0.312781	−	0.949825i	$-0.601261\pi$
$79$	−5.56012	−0.625563	−0.312781	+	0.949825i	$0.601261\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.51754i	− 0.386100i	−0.981189	−	0.193050i	$-0.938162\pi$
$83$	− 3.51754i	− 0.386100i	0.981189	−	0.193050i	$-0.0618380\pi$
$84$	0	0
$85$	− 19.9490i	− 2.16377i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.6459	1.44646	0.723231	−	0.690606i	$-0.242656\pi$
$89$	13.6459	1.44646	0.723231	+	0.690606i	$0.242656\pi$
$90$	0	0
$91$	5.38919i	0.564940i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.19204	−0.430094
$96$	0	0
$97$	−10.9067	−1.10741	−0.553705	−	0.832713i	$-0.686787\pi$
$97$	−10.9067	−1.10741	−0.553705	+	0.832713i	$0.686787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.62911i	− 0.361110i	−0.983565	−	0.180555i	$-0.942211\pi$
$101$	− 3.62911i	− 0.361110i	0.983565	−	0.180555i	$-0.0577894\pi$
$102$	0	0
$103$	−11.3426	−1.11762	−0.558808	−	0.829297i	$-0.688741\pi$
$103$	−11.3426	−1.11762	−0.558808	+	0.829297i	$0.688741\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.90673i	− 0.861046i	−0.902580	−	0.430523i	$-0.858329\pi$
$107$	− 8.90673i	− 0.861046i	0.902580	−	0.430523i	$-0.141671\pi$
$108$	0	0
$109$	16.0097i	1.53345i	0.641973	+	0.766727i	$0.278116\pi$
$109$	16.0097i	1.53345i	−0.641973	+	0.766727i	$0.721884\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.51754	0.330902	0.165451	−	0.986218i	$-0.447092\pi$
$113$	3.51754	0.330902	0.165451	+	0.986218i	$0.447092\pi$
$114$	0	0
$115$	4.73917i	0.441930i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 21.7349i	− 1.94403i
$126$	0	0
$127$	1.36808	0.121398	0.0606988	−	0.998156i	$-0.480667\pi$
$127$	1.36808	0.121398	0.0606988	+	0.998156i	$0.480667\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.6459i	− 1.19225i	−0.802893	−	0.596124i	$-0.796707\pi$
$131$	− 13.6459i	− 1.19225i	0.802893	−	0.596124i	$-0.203293\pi$
$132$	0	0
$133$	1.45588i	0.126241i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.2959	−1.22138	−0.610691	−	0.791869i	$-0.709108\pi$
$137$	−14.2959	−1.22138	−0.610691	+	0.791869i	$0.709108\pi$
$138$	0	0
$139$	− 13.2317i	− 1.12230i	−0.827714	−	0.561151i	$-0.810359\pi$
$139$	− 13.2317i	− 1.12230i	0.827714	−	0.561151i	$-0.189641\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.93923	0.329415
$144$	0	0
$145$	23.7743	1.97434
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.72167i	0.796430i	0.917292	+	0.398215i	$0.130370\pi$
$149$	9.72167i	0.796430i	−0.917292	+	0.398215i	$0.869630\pi$
$150$	0	0
$151$	21.3170	1.73476	0.867378	−	0.497649i	$-0.165803\pi$
$151$	21.3170	1.73476	0.867378	+	0.497649i	$0.165803\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	33.9026i	2.72312i
$156$	0	0
$157$	0.950259i	0.0758389i	0.999281	+	0.0379195i	$0.0120730\pi$
$157$	0.950259i	0.0758389i	−0.999281	+	0.0379195i	$0.987927\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.64590	0.129715
$162$	0	0
$163$	− 22.3851i	− 1.75333i	−0.481098	−	0.876667i	$-0.659762\pi$
$163$	− 22.3851i	− 1.75333i	0.481098	−	0.876667i	$-0.340238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.78590	−0.138197	−0.0690986	−	0.997610i	$-0.522012\pi$
$167$	−1.78590	−0.138197	−0.0690986	+	0.997610i	$0.522012\pi$
$168$	0	0
$169$	−2.51754	−0.193657
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.82115i	− 0.594631i	−0.954779	−	0.297316i	$-0.903909\pi$
$173$	− 7.82115i	− 0.594631i	0.954779	−	0.297316i	$-0.0960914\pi$
$174$	0	0
$175$	−14.3888	−1.08769
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.2567i	1.51406i	0.653381	+	0.757029i	$0.273350\pi$
$179$	20.2567i	1.51406i	−0.653381	+	0.757029i	$0.726650\pi$
$180$	0	0
$181$	− 3.35640i	− 0.249479i	−0.992190	−	0.124740i	$-0.960190\pi$
$181$	− 3.35640i	− 0.249479i	0.992190	−	0.124740i	$-0.0398095\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.06418i	0.370329i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.62565	0.551773	0.275886	−	0.961190i	$-0.411029\pi$
$191$	7.62565	0.551773	0.275886	+	0.961190i	$0.411029\pi$
$192$	0	0
$193$	5.51754	0.397161	0.198581	−	0.980085i	$-0.436367\pi$
$193$	5.51754	0.397161	0.198581	+	0.980085i	$0.436367\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.4855i	− 1.24579i	−0.782305	−	0.622896i	$-0.785956\pi$
$197$	− 17.4855i	− 1.24579i	0.782305	−	0.622896i	$-0.214044\pi$
$198$	0	0
$199$	−7.15052	−0.506887	−0.253444	−	0.967350i	$-0.581563\pi$
$199$	−7.15052	−0.506887	−0.253444	+	0.967350i	$0.581563\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 8.25671i	− 0.579508i
$204$	0	0
$205$	1.28028i	0.0894186i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.06418	0.0736107
$210$	0	0
$211$	9.71419i	0.668753i	0.942440	+	0.334376i	$0.108526\pi$
$211$	9.71419i	0.668753i	−0.942440	+	0.334376i	$0.891474\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−49.5623	−3.38012
$216$	0	0
$217$	11.7743	0.799288
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 19.9490i	− 1.34191i
$222$	0	0
$223$	11.3426	0.759554	0.379777	−	0.925078i	$-0.376001\pi$
$223$	11.3426	0.759554	0.379777	+	0.925078i	$0.376001\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.5134i	1.89250i	0.323433	+	0.946251i	$0.395163\pi$
$227$	28.5134i	1.89250i	−0.323433	+	0.946251i	$0.604837\pi$
$228$	0	0
$229$	− 21.2292i	− 1.40287i	−0.712734	−	0.701434i	$-0.752543\pi$
$229$	− 21.2292i	− 1.40287i	0.712734	−	0.701434i	$-0.247457\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.325008	−0.0212920	−0.0106460	−	0.999943i	$-0.503389\pi$
$233$	−0.325008	−0.0212920	−0.0106460	+	0.999943i	$0.503389\pi$
$234$	0	0
$235$	− 32.0310i	− 2.08947i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0705	−0.780776	−0.390388	−	0.920650i	$-0.627659\pi$
$239$	−12.0705	−0.780776	−0.390388	+	0.920650i	$0.627659\pi$
$240$	0	0
$241$	−2.73917	−0.176445	−0.0882227	−	0.996101i	$-0.528119\pi$
$241$	−2.73917	−0.176445	−0.0882227	+	0.996101i	$0.528119\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 20.2018i	− 1.29064i
$246$	0	0
$247$	−4.19204	−0.266733
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.3892i	1.09760i	0.835955	+	0.548798i	$0.184914\pi$
$251$	17.3892i	1.09760i	−0.835955	+	0.548798i	$0.815086\pi$
$252$	0	0
$253$	− 1.20307i	− 0.0756364i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.3892	−1.08471	−0.542354	−	0.840150i	$-0.682467\pi$
$257$	−17.3892	−1.08471	−0.542354	+	0.840150i	$0.682467\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.19204	−0.258492	−0.129246	−	0.991613i	$-0.541256\pi$
$263$	−4.19204	−0.258492	−0.129246	+	0.991613i	$0.541256\pi$
$264$	0	0
$265$	32.0310	1.96765
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.0318i	0.611649i	0.952088	+	0.305825i	$0.0989321\pi$
$269$	10.0318i	0.611649i	−0.952088	+	0.305825i	$0.901068\pi$
$270$	0	0
$271$	−4.72448	−0.286992	−0.143496	−	0.989651i	$-0.545834\pi$
$271$	−4.72448	−0.286992	−0.143496	+	0.989651i	$0.545834\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.5175i	0.634232i
$276$	0	0
$277$	0.252811i	0.0151899i	0.999971	+	0.00759497i	$0.00241758\pi$
$277$	0.252811i	0.0151899i	−0.999971	+	0.00759497i	$0.997582\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.2276	−1.32599	−0.662994	−	0.748625i	$-0.730715\pi$
$281$	−22.2276	−1.32599	−0.662994	+	0.748625i	$0.730715\pi$
$282$	0	0
$283$	− 9.06418i	− 0.538809i	−0.963027	−	0.269405i	$-0.913173\pi$
$283$	− 9.06418i	− 0.538809i	0.963027	−	0.269405i	$-0.0868269\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.444637	0.0262461
$288$	0	0
$289$	8.64590	0.508582
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 19.3860i	− 1.13254i	−0.824218	−	0.566272i	$-0.808385\pi$
$293$	− 19.3860i	− 1.13254i	0.824218	−	0.566272i	$-0.191615\pi$
$294$	0	0
$295$	−21.2292	−1.23601
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.73917i	0.274073i
$300$	0	0
$301$	17.2128i	0.992130i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−46.5526	−2.66560
$306$	0	0
$307$	− 15.3601i	− 0.876647i	−0.898817	−	0.438323i	$-0.855572\pi$
$307$	− 15.3601i	− 0.876647i	0.898817	−	0.438323i	$-0.144428\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.4877	0.651406	0.325703	−	0.945472i	$-0.394399\pi$
$311$	11.4877	0.651406	0.325703	+	0.945472i	$0.394399\pi$
$312$	0	0
$313$	14.1676	0.800798	0.400399	−	0.916341i	$-0.368872\pi$
$313$	14.1676	0.800798	0.400399	+	0.916341i	$0.368872\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.75151i	0.491534i	0.969329	+	0.245767i	$0.0790398\pi$
$317$	8.75151i	0.491534i	−0.969329	+	0.245767i	$0.920960\pi$
$318$	0	0
$319$	−6.03525	−0.337909
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.38919i	− 0.299862i
$324$	0	0
$325$	− 41.4310i	− 2.29818i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.1242	−0.613299
$330$	0	0
$331$	4.90673i	0.269698i	0.990866	+	0.134849i	$0.0430549\pi$
$331$	4.90673i	0.269698i	−0.990866	+	0.134849i	$0.956945\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.6133	−1.61795
$336$	0	0
$337$	−26.9959	−1.47056	−0.735280	−	0.677764i	$-0.762949\pi$
$337$	−26.9959	−1.47056	−0.735280	+	0.677764i	$0.762949\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8.60640i	− 0.466063i
$342$	0	0
$343$	−16.5926	−0.895914
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.8093i	− 1.65393i	−0.562252	−	0.826966i	$-0.690065\pi$
$347$	− 30.8093i	− 1.65393i	0.562252	−	0.826966i	$-0.309935\pi$
$348$	0	0
$349$	− 3.31899i	− 0.177662i	−0.996047	−	0.0888308i	$-0.971687\pi$
$349$	− 3.31899i	− 0.177662i	0.996047	−	0.0888308i	$-0.0283131\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	34.3269	1.82704	0.913518	−	0.406799i	$-0.133355\pi$
$353$	34.3269	1.82704	0.913518	+	0.406799i	$0.133355\pi$
$354$	0	0
$355$	− 53.5877i	− 2.84414i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.8495	1.15317	0.576586	−	0.817037i	$-0.304385\pi$
$359$	21.8495	1.15317	0.576586	+	0.817037i	$0.304385\pi$
$360$	0	0
$361$	17.8675	0.940396
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 10.7902i	− 0.564786i
$366$	0	0
$367$	−16.8149	−0.877730	−0.438865	−	0.898553i	$-0.644619\pi$
$367$	−16.8149	−0.877730	−0.438865	+	0.898553i	$0.644619\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11.1242i	− 0.577542i
$372$	0	0
$373$	− 26.2943i	− 1.36147i	−0.732530	−	0.680735i	$-0.761661\pi$
$373$	− 26.2943i	− 1.36147i	0.732530	−	0.680735i	$-0.238339\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.7743	1.22444
$378$	0	0
$379$	15.3500i	0.788476i	0.919008	+	0.394238i	$0.128991\pi$
$379$	15.3500i	0.788476i	−0.919008	+	0.394238i	$0.871009\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.67539	−0.341097	−0.170548	−	0.985349i	$-0.554554\pi$
$383$	−6.67539	−0.341097	−0.170548	+	0.985349i	$0.554554\pi$
$384$	0	0
$385$	5.38919	0.274658
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.53309i	− 0.0777308i	−0.999244	−	0.0388654i	$-0.987626\pi$
$389$	− 1.53309i	− 0.0777308i	0.999244	−	0.0388654i	$-0.0123744\pi$
$390$	0	0
$391$	−6.09256	−0.308114
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 21.9026i	− 1.10204i
$396$	0	0
$397$	− 32.0195i	− 1.60701i	−0.595297	−	0.803506i	$-0.702966\pi$
$397$	− 32.0195i	− 1.60701i	0.595297	−	0.803506i	$-0.297034\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.6067	−0.979112	−0.489556	−	0.871972i	$-0.662841\pi$
$401$	−19.6067	−0.979112	−0.489556	+	0.871972i	$0.662841\pi$
$402$	0	0
$403$	33.9026i	1.68881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7.55674	−0.373657	−0.186828	−	0.982393i	$-0.559821\pi$
$409$	−7.55674	−0.373657	−0.186828	+	0.982393i	$0.559821\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.37284i	0.362794i
$414$	0	0
$415$	13.8564	0.680184
$416$	0	0
$417$	0	0
$418$	0	0
$419$	31.4593i	1.53689i	0.639916	+	0.768445i	$0.278969\pi$
$419$	31.4593i	1.53689i	−0.639916	+	0.768445i	$0.721031\pi$
$420$	0	0
$421$	6.42258i	0.313017i	0.987677	+	0.156509i	$0.0500239\pi$
$421$	6.42258i	0.313017i	−0.987677	+	0.156509i	$0.949976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	53.2627	2.58362
$426$	0	0
$427$	16.1676i	0.782403i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.4918	−1.80592	−0.902958	−	0.429729i	$-0.858609\pi$
$431$	−37.4918	−1.80592	−0.902958	+	0.429729i	$0.858609\pi$
$432$	0	0
$433$	2.08915	0.100398	0.0501992	−	0.998739i	$-0.484014\pi$
$433$	2.08915	0.100398	0.0501992	+	0.998739i	$0.484014\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.28028i	0.0612441i
$438$	0	0
$439$	35.6181	1.69996	0.849979	−	0.526816i	$-0.176614\pi$
$439$	35.6181	1.69996	0.849979	+	0.526816i	$0.176614\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.0351i	0.904384i	0.891921	+	0.452192i	$0.149358\pi$
$443$	19.0351i	0.904384i	−0.891921	+	0.452192i	$0.850642\pi$
$444$	0	0
$445$	53.7543i	2.54820i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.6459	−1.77662	−0.888310	−	0.459245i	$-0.848120\pi$
$449$	−37.6459	−1.77662	−0.888310	+	0.459245i	$0.848120\pi$
$450$	0	0
$451$	− 0.325008i	− 0.0153040i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−21.2292	−0.995242
$456$	0	0
$457$	−37.2918	−1.74444	−0.872218	−	0.489117i	$-0.837319\pi$
$457$	−37.2918	−1.74444	−0.872218	+	0.489117i	$0.837319\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.31553i	0.340718i	0.985382	+	0.170359i	$0.0544928\pi$
$461$	7.31553i	0.340718i	−0.985382	+	0.170359i	$0.945507\pi$
$462$	0	0
$463$	21.6272	1.00510	0.502550	−	0.864548i	$-0.332395\pi$
$463$	21.6272	1.00510	0.502550	+	0.864548i	$0.332395\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.96492i	0.229749i	0.993380	+	0.114875i	$0.0366466\pi$
$467$	4.96492i	0.229749i	−0.993380	+	0.114875i	$0.963353\pi$
$468$	0	0
$469$	10.2846i	0.474899i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.5817	0.578508
$474$	0	0
$475$	− 11.1925i	− 0.513549i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.5471	1.21297	0.606485	−	0.795095i	$-0.292579\pi$
$479$	26.5471	1.21297	0.606485	+	0.795095i	$0.292579\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 42.9641i	− 1.95090i
$486$	0	0
$487$	−10.5069	−0.476114	−0.238057	−	0.971251i	$-0.576511\pi$
$487$	−10.5069	−0.476114	−0.238057	+	0.971251i	$0.576511\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.7202i	1.20586i	0.797792	+	0.602932i	$0.206001\pi$
$491$	26.7202i	1.20586i	−0.797792	+	0.602932i	$0.793999\pi$
$492$	0	0
$493$	30.5636i	1.37651i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.6108	−0.834809
$498$	0	0
$499$	33.8135i	1.51370i	0.653590	+	0.756849i	$0.273262\pi$
$499$	33.8135i	1.51370i	−0.653590	+	0.756849i	$0.726738\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.6133	−1.32039	−0.660197	−	0.751093i	$-0.729527\pi$
$503$	−29.6133	−1.32039	−0.660197	+	0.751093i	$0.729527\pi$
$504$	0	0
$505$	14.2959	0.636160
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.65895i	0.117856i	0.998262	+	0.0589280i	$0.0187682\pi$
$509$	2.65895i	0.117856i	−0.998262	+	0.0589280i	$0.981232\pi$
$510$	0	0
$511$	−3.74741	−0.165775
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 44.6810i	− 1.96888i
$516$	0	0
$517$	8.13127i	0.357613i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.6851	1.38815	0.694075	−	0.719903i	$-0.255814\pi$
$521$	31.6851	1.38815	0.694075	+	0.719903i	$0.255814\pi$
$522$	0	0
$523$	− 18.4534i	− 0.806909i	−0.915000	−	0.403455i	$-0.867809\pi$
$523$	− 18.4534i	− 0.806909i	0.915000	−	0.403455i	$-0.132191\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.5844	−1.89856
$528$	0	0
$529$	−21.5526	−0.937071
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.28028i	0.0554551i
$534$	0	0
$535$	35.0857	1.51689
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.12836i	0.220894i
$540$	0	0
$541$	14.1092i	0.606603i	0.952895	+	0.303301i	$0.0980889\pi$
$541$	14.1092i	0.606603i	−0.952895	+	0.303301i	$0.901911\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−63.0660	−2.70145
$546$	0	0
$547$	36.0411i	1.54100i	0.637437	+	0.770502i	$0.279995\pi$
$547$	36.0411i	1.54100i	−0.637437	+	0.770502i	$0.720005\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.42258	0.273611
$552$	0	0
$553$	−7.60670	−0.323470
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 43.5270i	− 1.84430i	−0.386832	−	0.922150i	$-0.626431\pi$
$557$	− 43.5270i	− 1.84430i	0.386832	−	0.922150i	$-0.373569\pi$
$558$	0	0
$559$	−49.5623	−2.09626
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 19.0351i	− 0.802233i	−0.916027	−	0.401117i	$-0.868622\pi$
$563$	− 19.0351i	− 0.802233i	0.916027	−	0.401117i	$-0.131378\pi$
$564$	0	0
$565$	13.8564i	0.582943i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.00599	−0.377551	−0.188775	−	0.982020i	$-0.560452\pi$
$569$	−9.00599	−0.377551	−0.188775	+	0.982020i	$0.560452\pi$
$570$	0	0
$571$	3.93170i	0.164537i	0.996610	+	0.0822683i	$0.0262165\pi$
$571$	3.93170i	0.164537i	−0.996610	+	0.0822683i	$0.973784\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.6533	−0.527681
$576$	0	0
$577$	−2.99588	−0.124720	−0.0623601	−	0.998054i	$-0.519863\pi$
$577$	−2.99588	−0.124720	−0.0623601	+	0.998054i	$0.519863\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.81228i	− 0.199647i
$582$	0	0
$583$	−8.13127	−0.336763
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.8675i	0.613648i	0.951766	+	0.306824i	$0.0992664\pi$
$587$	14.8675i	0.613648i	−0.951766	+	0.306824i	$0.900734\pi$
$588$	0	0
$589$	9.15874i	0.377379i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−46.8776	−1.92503	−0.962517	−	0.271222i	$-0.912572\pi$
$593$	−46.8776	−1.92503	−0.962517	+	0.271222i	$0.912572\pi$
$594$	0	0
$595$	− 27.2918i	− 1.11885i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.2472	0.418689	0.209345	−	0.977842i	$-0.432867\pi$
$599$	10.2472	0.418689	0.209345	+	0.977842i	$0.432867\pi$
$600$	0	0
$601$	−42.0310	−1.71448	−0.857239	−	0.514918i	$-0.827822\pi$
$601$	−42.0310	−1.71448	−0.857239	+	0.514918i	$0.827822\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.93923i	− 0.160153i
$606$	0	0
$607$	16.4650	0.668292	0.334146	−	0.942521i	$-0.391552\pi$
$607$	16.4650	0.668292	0.334146	+	0.942521i	$0.391552\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 32.0310i	− 1.29583i
$612$	0	0
$613$	11.3121i	0.456890i	0.973557	+	0.228445i	$0.0733641\pi$
$613$	11.3121i	0.456890i	−0.973557	+	0.228445i	$0.926636\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.2526	1.33870	0.669349	−	0.742948i	$-0.266573\pi$
$617$	33.2526	1.33870	0.669349	+	0.742948i	$0.266573\pi$
$618$	0	0
$619$	36.9377i	1.48465i	0.670039	+	0.742326i	$0.266277\pi$
$619$	36.9377i	1.48465i	−0.670039	+	0.742326i	$0.733723\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.6687	0.747945
$624$	0	0
$625$	33.0310	1.32124
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	43.4766	1.73078	0.865389	−	0.501101i	$-0.167071\pi$
$631$	43.4766	1.73078	0.865389	+	0.501101i	$0.167071\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.38919i	0.213863i
$636$	0	0
$637$	− 20.2018i	− 0.800424i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.74329	−0.147851	−0.0739255	−	0.997264i	$-0.523553\pi$
$641$	−3.74329	−0.147851	−0.0739255	+	0.997264i	$0.523553\pi$
$642$	0	0
$643$	33.6459i	1.32686i	0.748236	+	0.663432i	$0.230901\pi$
$643$	33.6459i	1.32686i	−0.748236	+	0.663432i	$0.769099\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.70869	−0.0671756	−0.0335878	−	0.999436i	$-0.510693\pi$
$647$	−1.70869	−0.0671756	−0.0335878	+	0.999436i	$0.510693\pi$
$648$	0	0
$649$	5.38919	0.211544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3385i	1.38290i	0.722424	+	0.691450i	$0.243028\pi$
$653$	35.3385i	1.38290i	−0.722424	+	0.691450i	$0.756972\pi$
$654$	0	0
$655$	53.7543	2.10036
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.1284i	0.861998i	0.902352	+	0.430999i	$0.141839\pi$
$659$	22.1284i	0.861998i	−0.902352	+	0.430999i	$0.858161\pi$
$660$	0	0
$661$	11.2349i	0.436985i	0.975839	+	0.218493i	$0.0701140\pi$
$661$	11.2349i	0.436985i	−0.975839	+	0.218493i	$0.929886\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.73505	−0.222396
$666$	0	0
$667$	− 7.26083i	− 0.281140i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.8177	0.456217
$672$	0	0
$673$	−5.29179	−0.203984	−0.101992	−	0.994785i	$-0.532522\pi$
$673$	−5.29179	−0.203984	−0.101992	+	0.994785i	$0.532522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.2537i	0.509380i	0.967023	+	0.254690i	$0.0819734\pi$
$677$	13.2537i	0.509380i	−0.967023	+	0.254690i	$0.918027\pi$
$678$	0	0
$679$	−14.9213	−0.572626
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30.1593i	1.15401i	0.816739	+	0.577007i	$0.195780\pi$
$683$	30.1593i	1.15401i	−0.816739	+	0.577007i	$0.804220\pi$
$684$	0	0
$685$	− 56.3149i	− 2.15168i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	32.0310	1.22028
$690$	0	0
$691$	− 10.6108i	− 0.403654i	−0.979421	−	0.201827i	$-0.935312\pi$
$691$	− 10.6108i	− 0.403654i	0.979421	−	0.201827i	$-0.0646879\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	52.1229	1.97713
$696$	0	0
$697$	−1.64590	−0.0623428
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.4829i	1.30240i	0.758905	+	0.651201i	$0.225734\pi$
$701$	34.4829i	1.30240i	−0.758905	+	0.651201i	$0.774266\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.96492i	− 0.186725i
$708$	0	0
$709$	46.0303i	1.72870i	0.502887	+	0.864352i	$0.332271\pi$
$709$	46.0303i	1.72870i	−0.502887	+	0.864352i	$0.667729\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.3541	0.387764
$714$	0	0
$715$	15.5175i	0.580323i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.7887	0.961757	0.480878	−	0.876787i	$-0.340318\pi$
$719$	25.7887	0.961757	0.480878	+	0.876787i	$0.340318\pi$
$720$	0	0
$721$	−15.5175	−0.577903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	63.4760i	2.35744i
$726$	0	0
$727$	7.15052	0.265198	0.132599	−	0.991170i	$-0.457668\pi$
$727$	7.15052	0.265198	0.132599	+	0.991170i	$0.457668\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 63.7161i	− 2.35662i
$732$	0	0
$733$	− 26.8000i	− 0.989879i	−0.868927	−	0.494940i	$-0.835190\pi$
$733$	− 26.8000i	− 0.989879i	0.868927	−	0.494940i	$-0.164810\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.51754	0.276912
$738$	0	0
$739$	40.0993i	1.47508i	0.675306	+	0.737538i	$0.264012\pi$
$739$	40.0993i	1.47508i	−0.675306	+	0.737538i	$0.735988\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.8205	−1.31413	−0.657063	−	0.753835i	$-0.728202\pi$
$743$	−35.8205	−1.31413	−0.657063	+	0.753835i	$0.728202\pi$
$744$	0	0
$745$	−38.2959	−1.40305
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.1851i	− 0.445235i
$750$	0	0
$751$	12.1782	0.444389	0.222195	−	0.975002i	$-0.428678\pi$
$751$	12.1782	0.444389	0.222195	+	0.975002i	$0.428678\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	83.9728i	3.05608i
$756$	0	0
$757$	45.6605i	1.65956i	0.558092	+	0.829779i	$0.311534\pi$
$757$	45.6605i	1.65956i	−0.558092	+	0.829779i	$0.688466\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	19.5858	0.709986	0.354993	−	0.934869i	$-0.384483\pi$
$761$	19.5858	0.709986	0.354993	+	0.934869i	$0.384483\pi$
$762$	0	0
$763$	21.9026i	0.792928i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.2292	−0.766544
$768$	0	0
$769$	−32.5526	−1.17388	−0.586939	−	0.809631i	$-0.699667\pi$
$769$	−32.5526	−1.17388	−0.586939	+	0.809631i	$0.699667\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5.21951i	− 0.187733i	−0.995585	−	0.0938664i	$-0.970077\pi$
$773$	− 5.21951i	− 0.187733i	0.995585	−	0.0938664i	$-0.0299226\pi$
$774$	0	0
$775$	−90.5182	−3.25151
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.345866i	0.0123919i
$780$	0	0
$781$	13.6036i	0.486775i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.74329	−0.133604
$786$	0	0
$787$	3.93170i	0.140150i	0.997542	+	0.0700750i	$0.0223239\pi$
$787$	3.93170i	0.140150i	−0.997542	+	0.0700750i	$0.977676\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.81228	0.171105
$792$	0	0
$793$	−46.5526	−1.65313
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.6962i	− 0.697674i	−0.937183	−	0.348837i	$-0.886577\pi$
$797$	− 19.6962i	− 0.697674i	0.937183	−	0.348837i	$-0.113423\pi$
$798$	0	0
$799$	41.1782	1.45678
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.73917i	0.0966632i
$804$	0	0
$805$	6.48357i	0.228516i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14.3952	−0.506107	−0.253054	−	0.967452i	$-0.581435\pi$
$809$	−14.3952	−0.506107	−0.253054	+	0.967452i	$0.581435\pi$
$810$	0	0
$811$	− 15.3601i	− 0.539366i	−0.962949	−	0.269683i	$-0.913081\pi$
$811$	− 15.3601i	− 0.539366i	0.962949	−	0.269683i	$-0.0869189\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	88.1799	3.08881
$816$	0	0
$817$	−13.3892	−0.468428
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.41501i	0.188985i	0.995526	+	0.0944927i	$0.0301229\pi$
$821$	5.41501i	0.188985i	−0.995526	+	0.0944927i	$0.969877\pi$
$822$	0	0
$823$	34.4723	1.20163	0.600815	−	0.799388i	$-0.294843\pi$
$823$	34.4723	1.20163	0.600815	+	0.799388i	$0.294843\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.6459i	1.30908i	0.756029	+	0.654538i	$0.227137\pi$
$827$	37.6459i	1.30908i	−0.756029	+	0.654538i	$0.772863\pi$
$828$	0	0
$829$	− 44.8646i	− 1.55821i	−0.626892	−	0.779107i	$-0.715673\pi$
$829$	− 44.8646i	− 1.55821i	0.626892	−	0.779107i	$-0.284327\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.9709	0.899838
$834$	0	0
$835$	− 7.03508i	− 0.243459i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.8272	1.54761	0.773804	−	0.633425i	$-0.218352\pi$
$839$	44.8272	1.54761	0.773804	+	0.633425i	$0.218352\pi$
$840$	0	0
$841$	−7.42427	−0.256009
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.91718i	− 0.341161i
$846$	0	0
$847$	−1.36808	−0.0470078
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 6.89080i	− 0.235936i	−0.993017	−	0.117968i	$-0.962362\pi$
$853$	− 6.89080i	− 0.235936i	0.993017	−	0.117968i	$-0.0376381\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.1533	1.13250	0.566248	−	0.824235i	$-0.308394\pi$
$857$	33.1533	1.13250	0.566248	+	0.824235i	$0.308394\pi$
$858$	0	0
$859$	− 34.6418i	− 1.18196i	−0.806685	−	0.590981i	$-0.798741\pi$
$859$	− 34.6418i	− 1.18196i	0.806685	−	0.590981i	$-0.201259\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.5318	0.698911	0.349455	−	0.936953i	$-0.386367\pi$
$863$	20.5318	0.698911	0.349455	+	0.936953i	$0.386367\pi$
$864$	0	0
$865$	30.8093	1.04755
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.56012i	0.188614i
$870$	0	0
$871$	−29.6133	−1.00341
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 29.7351i	− 1.00523i
$876$	0	0
$877$	− 34.7182i	− 1.17235i	−0.810184	−	0.586176i	$-0.800633\pi$
$877$	− 34.7182i	− 1.17235i	0.810184	−	0.586176i	$-0.199367\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.83656	−0.230330	−0.115165	−	0.993346i	$-0.536740\pi$
$881$	−6.83656	−0.230330	−0.115165	+	0.993346i	$0.536740\pi$
$882$	0	0
$883$	6.55262i	0.220513i	0.993903	+	0.110257i	$0.0351673\pi$
$883$	6.55262i	0.220513i	−0.993903	+	0.110257i	$0.964833\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.5800	1.16108	0.580542	−	0.814230i	$-0.302841\pi$
$887$	34.5800	1.16108	0.580542	+	0.814230i	$0.302841\pi$
$888$	0	0
$889$	1.87164	0.0627729
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 8.65312i	− 0.289566i
$894$	0	0
$895$	−79.7959	−2.66728
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 51.9418i	− 1.73236i
$900$	0	0
$901$	41.1782i	1.37185i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.2216	0.439502
$906$	0	0
$907$	30.8675i	1.02494i	0.858705	+	0.512470i	$0.171269\pi$
$907$	30.8675i	1.02494i	−0.858705	+	0.512470i	$0.828731\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.1395	−1.29675	−0.648375	−	0.761321i	$-0.724551\pi$
$911$	−39.1395	−1.29675	−0.648375	+	0.761321i	$0.724551\pi$
$912$	0	0
$913$	−3.51754	−0.116414
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 18.6687i	− 0.616494i
$918$	0	0
$919$	33.8932	1.11803	0.559016	−	0.829157i	$-0.311179\pi$
$919$	33.8932	1.11803	0.559016	+	0.829157i	$0.311179\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 53.5877i	− 1.76386i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.3310	−1.48726	−0.743631	−	0.668590i	$-0.766898\pi$
$929$	−45.3310	−1.48726	−0.743631	+	0.668590i	$0.766898\pi$
$930$	0	0
$931$	− 5.45748i	− 0.178862i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.9490	−0.652401
$936$	0	0
$937$	39.8135	1.30065	0.650324	−	0.759657i	$-0.274633\pi$
$937$	39.8135	1.30065	0.650324	+	0.759657i	$0.274633\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29.5560i	0.963499i	0.876309	+	0.481749i	$0.159998\pi$
$941$	29.5560i	0.963499i	−0.876309	+	0.481749i	$0.840002\pi$
$942$	0	0
$943$	0.391007	0.0127329
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.7701i	1.19487i	0.801918	+	0.597434i	$0.203813\pi$
$947$	36.7701i	1.19487i	−0.801918	+	0.597434i	$0.796187\pi$
$948$	0	0
$949$	− 10.7902i	− 0.350265i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	19.9317	0.645651	0.322826	−	0.946458i	$-0.395367\pi$
$953$	19.9317	0.645651	0.322826	+	0.946458i	$0.395367\pi$
$954$	0	0
$955$	30.0392i	0.972046i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.5580	−0.631559
$960$	0	0
$961$	43.0702	1.38936
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21.7349i	0.699670i
$966$	0	0
$967$	8.91652	0.286736	0.143368	−	0.989669i	$-0.454207\pi$
$967$	8.91652	0.286736	0.143368	+	0.989669i	$0.454207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.0351i	0.610865i	0.952214	+	0.305432i	$0.0988010\pi$
$971$	19.0351i	0.610865i	−0.952214	+	0.305432i	$0.901199\pi$
$972$	0	0
$973$	− 18.1021i	− 0.580326i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.4201	1.19718	0.598588	−	0.801057i	$-0.295729\pi$
$977$	37.4201	1.19718	0.598588	+	0.801057i	$0.295729\pi$
$978$	0	0
$979$	− 13.6459i	− 0.436125i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−61.1646	−1.95085	−0.975424	−	0.220338i	$-0.929284\pi$
$983$	−61.1646	−1.95085	−0.975424	+	0.220338i	$0.929284\pi$
$984$	0	0
$985$	68.8795	2.19468
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.1367i	0.481319i
$990$	0	0
$991$	19.5510	0.621059	0.310530	−	0.950564i	$-0.399494\pi$
$991$	19.5510	0.621059	0.310530	+	0.950564i	$0.399494\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 28.1676i	− 0.892972i
$996$	0	0
$997$	− 3.82462i	− 0.121127i	−0.998164	−	0.0605634i	$-0.980710\pi$
$997$	− 3.82462i	− 0.121127i	0.998164	−	0.0605634i	$-0.0192897\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.f.m.3169.12	yes	12
3.2	odd	2		6336.2.f.o.3169.2	yes	12
4.3	odd	2	inner	6336.2.f.m.3169.11	yes	12
8.3	odd	2	inner	6336.2.f.m.3169.1	✓	12
8.5	even	2	inner	6336.2.f.m.3169.2	yes	12
12.11	even	2		6336.2.f.o.3169.1	yes	12
24.5	odd	2		6336.2.f.o.3169.12	yes	12
24.11	even	2		6336.2.f.o.3169.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6336.2.f.m.3169.1	✓	12	8.3	odd	2	inner
6336.2.f.m.3169.2	yes	12	8.5	even	2	inner
6336.2.f.m.3169.11	yes	12	4.3	odd	2	inner
6336.2.f.m.3169.12	yes	12	1.1	even	1	trivial
6336.2.f.o.3169.1	yes	12	12.11	even	2
6336.2.f.o.3169.2	yes	12	3.2	odd	2
6336.2.f.o.3169.11	yes	12	24.11	even	2
6336.2.f.o.3169.12	yes	12	24.5	odd	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.93923i q^{5} +1.36808 q^{7} +O(q^{10})\)	\(q+3.93923i q^{5} +1.36808 q^{7} -1.00000i q^{11} +3.93923i q^{13} -5.06418 q^{17} +1.06418i q^{19} +1.20307 q^{23} -10.5175 q^{25} -6.03525i q^{29} +8.60640 q^{31} +5.38919i q^{35} +0.325008 q^{41} +12.5817i q^{43} -8.13127 q^{47} -5.12836 q^{49} -8.13127i q^{53} +3.93923 q^{55} +5.38919i q^{59} +11.8177i q^{61} -15.5175 q^{65} +7.51754i q^{67} -13.6036 q^{71} -2.73917 q^{73} -1.36808i q^{77} -5.56012 q^{79} -3.51754i q^{83} -19.9490i q^{85} +13.6459 q^{89} +5.38919i q^{91} -4.19204 q^{95} -10.9067 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 24 q^{17} - 36 q^{25} + 24 q^{41} + 12 q^{49} - 96 q^{65} + 24 q^{73} - 24 q^{97}+O(q^{100}) \) 12 * q - 24 * q^17 - 36 * q^25 + 24 * q^41 + 12 * q^49 - 96 * q^65 + 24 * q^73 - 24 * q^97

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