LMFDB - Embedded newform 6048.2.j.c.5615.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (of order $2$, degree $1$, not 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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.12
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.436221 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-0.436221 q^{5} -1.00000i q^{7} -2.73863i q^{11} -1.64657i q^{13} -5.22231i q^{17} +5.99963 q^{19} -7.68296 q^{23} -4.80971 q^{25} +4.94791 q^{29} -9.77950i q^{31} +0.436221i q^{35} +3.81676i q^{37} +9.74668i q^{41} +8.84195 q^{43} +4.54668 q^{47} -1.00000 q^{49} -9.94845 q^{53} +1.19465i q^{55} -13.2892i q^{59} +6.26145i q^{61} +0.718269i q^{65} -9.42171 q^{67} -9.76751 q^{71} -9.14079 q^{73} -2.73863 q^{77} +5.19984i q^{79} +0.227071i q^{83} +2.27808i q^{85} +2.46567i q^{89} -1.64657 q^{91} -2.61717 q^{95} -3.37287 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) $ 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.436221	−0.195084	−0.0975421	−	0.995231i	$-0.531098\pi$
$5$	−0.436221	−0.195084	−0.0975421	+	0.995231i	$0.531098\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.73863i	− 0.825727i	−0.910793	−	0.412863i	$-0.864529\pi$
$11$	− 2.73863i	− 0.825727i	0.910793	−	0.412863i	$-0.135471\pi$
$12$	0	0
$13$	− 1.64657i	− 0.456676i	−0.973582	−	0.228338i	$-0.926671\pi$
$13$	− 1.64657i	− 0.456676i	0.973582	−	0.228338i	$-0.0733291\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.22231i	− 1.26660i	−0.773908	−	0.633298i	$-0.781701\pi$
$17$	− 5.22231i	− 1.26660i	0.773908	−	0.633298i	$-0.218299\pi$
$18$	0	0
$19$	5.99963	1.37641	0.688204	−	0.725517i	$-0.258399\pi$
$19$	5.99963	1.37641	0.688204	+	0.725517i	$0.258399\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.68296	−1.60201	−0.801004	−	0.598659i	$-0.795700\pi$
$23$	−7.68296	−1.60201	−0.801004	+	0.598659i	$0.795700\pi$
$24$	0	0
$25$	−4.80971	−0.961942
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.94791	0.918804	0.459402	−	0.888228i	$-0.348064\pi$
$29$	4.94791	0.918804	0.459402	+	0.888228i	$0.348064\pi$
$30$	0	0
$31$	− 9.77950i	− 1.75645i	−0.478248	−	0.878225i	$-0.658728\pi$
$31$	− 9.77950i	− 1.75645i	0.478248	−	0.878225i	$-0.341272\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.436221i	0.0737349i
$36$	0	0
$37$	3.81676i	0.627472i	0.949510	+	0.313736i	$0.101581\pi$
$37$	3.81676i	0.627472i	−0.949510	+	0.313736i	$0.898419\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.74668i	1.52218i	0.648649	+	0.761088i	$0.275334\pi$
$41$	9.74668i	1.52218i	−0.648649	+	0.761088i	$0.724666\pi$
$42$	0	0
$43$	8.84195	1.34838	0.674192	−	0.738556i	$-0.264492\pi$
$43$	8.84195	1.34838	0.674192	+	0.738556i	$0.264492\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.54668	0.663202	0.331601	−	0.943420i	$-0.392411\pi$
$47$	4.54668	0.663202	0.331601	+	0.943420i	$0.392411\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.94845	−1.36652	−0.683262	−	0.730173i	$-0.739439\pi$
$53$	−9.94845	−1.36652	−0.683262	+	0.730173i	$0.739439\pi$
$54$	0	0
$55$	1.19465i	0.161086i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.2892i	− 1.73011i	−0.501681	−	0.865053i	$-0.667285\pi$
$59$	− 13.2892i	− 1.73011i	0.501681	−	0.865053i	$-0.332715\pi$
$60$	0	0
$61$	6.26145i	0.801696i	0.916145	+	0.400848i	$0.131284\pi$
$61$	6.26145i	0.801696i	−0.916145	+	0.400848i	$0.868716\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.718269i	0.0890903i
$66$	0	0
$67$	−9.42171	−1.15104	−0.575522	−	0.817786i	$-0.695201\pi$
$67$	−9.42171	−1.15104	−0.575522	+	0.817786i	$0.695201\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.76751	−1.15919	−0.579595	−	0.814905i	$-0.696789\pi$
$71$	−9.76751	−1.15919	−0.579595	+	0.814905i	$0.696789\pi$
$72$	0	0
$73$	−9.14079	−1.06985	−0.534924	−	0.844900i	$-0.679660\pi$
$73$	−9.14079	−1.06985	−0.534924	+	0.844900i	$0.679660\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.73863	−0.312095
$78$	0	0
$79$	5.19984i	0.585028i	0.956261	+	0.292514i	$0.0944917\pi$
$79$	5.19984i	0.585028i	−0.956261	+	0.292514i	$0.905508\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.227071i	0.0249243i	0.999922	+	0.0124622i	$0.00396693\pi$
$83$	0.227071i	0.0249243i	−0.999922	+	0.0124622i	$0.996033\pi$
$84$	0	0
$85$	2.27808i	0.247093i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.46567i	0.261360i	0.991425	+	0.130680i	$0.0417161\pi$
$89$	2.46567i	0.261360i	−0.991425	+	0.130680i	$0.958284\pi$
$90$	0	0
$91$	−1.64657	−0.172607
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.61717	−0.268516
$96$	0	0
$97$	−3.37287	−0.342463	−0.171232	−	0.985231i	$-0.554775\pi$
$97$	−3.37287	−0.342463	−0.171232	+	0.985231i	$0.554775\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.20021	0.119425	0.0597125	−	0.998216i	$-0.480982\pi$
$101$	1.20021	0.119425	0.0597125	+	0.998216i	$0.480982\pi$
$102$	0	0
$103$	10.7004i	1.05434i	0.849761	+	0.527169i	$0.176746\pi$
$103$	10.7004i	1.05434i	−0.849761	+	0.527169i	$0.823254\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.49655i	− 0.144676i	−0.997380	−	0.0723382i	$-0.976954\pi$
$107$	− 1.49655i	− 0.144676i	0.997380	−	0.0723382i	$-0.0230461\pi$
$108$	0	0
$109$	18.4352i	1.76578i	0.469583	+	0.882888i	$0.344404\pi$
$109$	18.4352i	1.76578i	−0.469583	+	0.882888i	$0.655596\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.25692i	− 0.494529i	−0.968948	−	0.247265i	$-0.920468\pi$
$113$	− 5.25692i	− 0.494529i	0.968948	−	0.247265i	$-0.0795317\pi$
$114$	0	0
$115$	3.35147	0.312526
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.22231	−0.478729
$120$	0	0
$121$	3.49993	0.318175
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.27921	0.382744
$126$	0	0
$127$	− 19.7677i	− 1.75410i	−0.480396	−	0.877051i	$-0.659507\pi$
$127$	− 19.7677i	− 1.75410i	0.480396	−	0.877051i	$-0.340493\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 9.25252i	− 0.808397i	−0.914671	−	0.404198i	$-0.867551\pi$
$131$	− 9.25252i	− 0.808397i	0.914671	−	0.404198i	$-0.132449\pi$
$132$	0	0
$133$	− 5.99963i	− 0.520234i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.53255i	0.216370i	0.994131	+	0.108185i	$0.0345039\pi$
$137$	2.53255i	0.216370i	−0.994131	+	0.108185i	$0.965496\pi$
$138$	0	0
$139$	19.7831	1.67798	0.838991	−	0.544146i	$-0.183146\pi$
$139$	19.7831	1.67798	0.838991	+	0.544146i	$0.183146\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.50934	−0.377090
$144$	0	0
$145$	−2.15839	−0.179244
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.16461	−0.668871	−0.334435	−	0.942419i	$-0.608546\pi$
$149$	−8.16461	−0.668871	−0.334435	+	0.942419i	$0.608546\pi$
$150$	0	0
$151$	− 14.8741i	− 1.21044i	−0.796058	−	0.605220i	$-0.793085\pi$
$151$	− 14.8741i	− 1.21044i	0.796058	−	0.605220i	$-0.206915\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.26603i	0.342656i
$156$	0	0
$157$	4.92207i	0.392824i	0.980521	+	0.196412i	$0.0629290\pi$
$157$	4.92207i	0.392824i	−0.980521	+	0.196412i	$0.937071\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.68296i	0.605502i
$162$	0	0
$163$	5.87250	0.459970	0.229985	−	0.973194i	$-0.426132\pi$
$163$	5.87250	0.459970	0.229985	+	0.973194i	$0.426132\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.1181	−1.71155	−0.855777	−	0.517345i	$-0.826920\pi$
$167$	−22.1181	−1.71155	−0.855777	+	0.517345i	$0.826920\pi$
$168$	0	0
$169$	10.2888	0.791447
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.1920	−1.83928	−0.919642	−	0.392758i	$-0.871521\pi$
$173$	−24.1920	−1.83928	−0.919642	+	0.392758i	$0.871521\pi$
$174$	0	0
$175$	4.80971i	0.363580i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.41287i	0.404577i	0.979326	+	0.202288i	$0.0648378\pi$
$179$	5.41287i	0.404577i	−0.979326	+	0.202288i	$0.935162\pi$
$180$	0	0
$181$	− 6.19672i	− 0.460598i	−0.973120	−	0.230299i	$-0.926030\pi$
$181$	− 6.19672i	− 0.460598i	0.973120	−	0.230299i	$-0.0739705\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.66495i	− 0.122410i
$186$	0	0
$187$	−14.3020	−1.04586
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.6191	−0.985444	−0.492722	−	0.870187i	$-0.663998\pi$
$191$	−13.6191	−0.985444	−0.492722	+	0.870187i	$0.663998\pi$
$192$	0	0
$193$	5.89278	0.424172	0.212086	−	0.977251i	$-0.431974\pi$
$193$	5.89278	0.424172	0.212086	+	0.977251i	$0.431974\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.0323	−1.35600	−0.677998	−	0.735064i	$-0.737152\pi$
$197$	−19.0323	−1.35600	−0.677998	+	0.735064i	$0.737152\pi$
$198$	0	0
$199$	5.91398i	0.419231i	0.977784	+	0.209616i	$0.0672212\pi$
$199$	5.91398i	0.419231i	−0.977784	+	0.209616i	$0.932779\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.94791i	− 0.347275i
$204$	0	0
$205$	− 4.25171i	− 0.296952i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 16.4307i	− 1.13654i
$210$	0	0
$211$	−12.9334	−0.890369	−0.445185	−	0.895439i	$-0.646862\pi$
$211$	−12.9334	−0.890369	−0.445185	+	0.895439i	$0.646862\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.85705	−0.263049
$216$	0	0
$217$	−9.77950	−0.663876
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.59890	−0.578424
$222$	0	0
$223$	0.758557i	0.0507967i	0.999677	+	0.0253984i	$0.00808542\pi$
$223$	0.758557i	0.0507967i	−0.999677	+	0.0253984i	$0.991915\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.60986i	− 0.239595i	−0.992798	−	0.119797i	$-0.961776\pi$
$227$	− 3.60986i	− 0.239595i	0.992798	−	0.119797i	$-0.0382245\pi$
$228$	0	0
$229$	− 3.47625i	− 0.229717i	−0.993382	−	0.114859i	$-0.963358\pi$
$229$	− 3.47625i	− 0.229717i	0.993382	−	0.114859i	$-0.0366415\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 22.9786i	− 1.50538i	−0.658376	−	0.752689i	$-0.728756\pi$
$233$	− 22.9786i	− 1.50538i	0.658376	−	0.752689i	$-0.271244\pi$
$234$	0	0
$235$	−1.98336	−0.129380
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.4478	−0.675813	−0.337906	−	0.941180i	$-0.609719\pi$
$239$	−10.4478	−0.675813	−0.337906	+	0.941180i	$0.609719\pi$
$240$	0	0
$241$	−10.0939	−0.650208	−0.325104	−	0.945678i	$-0.605399\pi$
$241$	−10.0939	−0.650208	−0.325104	+	0.945678i	$0.605399\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.436221	0.0278692
$246$	0	0
$247$	− 9.87880i	− 0.628573i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.0568i	1.13974i	0.821736	+	0.569868i	$0.193006\pi$
$251$	18.0568i	1.13974i	−0.821736	+	0.569868i	$0.806994\pi$
$252$	0	0
$253$	21.0408i	1.32282i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.77643i	− 0.235567i	−0.993039	−	0.117784i	$-0.962421\pi$
$257$	− 3.77643i	− 0.235567i	0.993039	−	0.117784i	$-0.0375789\pi$
$258$	0	0
$259$	3.81676	0.237162
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.1460	1.18060	0.590298	−	0.807185i	$-0.299010\pi$
$263$	19.1460	1.18060	0.590298	+	0.807185i	$0.299010\pi$
$264$	0	0
$265$	4.33973	0.266587
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.9031	−0.725747	−0.362873	−	0.931838i	$-0.618204\pi$
$269$	−11.9031	−0.725747	−0.362873	+	0.931838i	$0.618204\pi$
$270$	0	0
$271$	− 26.4765i	− 1.60833i	−0.594403	−	0.804167i	$-0.702612\pi$
$271$	− 26.4765i	− 1.60833i	0.594403	−	0.804167i	$-0.297388\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.1720i	0.794302i
$276$	0	0
$277$	− 9.49587i	− 0.570552i	−0.958445	−	0.285276i	$-0.907915\pi$
$277$	− 9.49587i	− 0.570552i	0.958445	−	0.285276i	$-0.0920852\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 16.7754i	− 1.00073i	−0.865813	−	0.500367i	$-0.833198\pi$
$281$	− 16.7754i	− 1.00073i	0.865813	−	0.500367i	$-0.166802\pi$
$282$	0	0
$283$	−13.6841	−0.813437	−0.406719	−	0.913554i	$-0.633327\pi$
$283$	−13.6841	−0.813437	−0.406719	+	0.913554i	$0.633327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.74668	0.575328
$288$	0	0
$289$	−10.2725	−0.604268
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.46088	−0.552710	−0.276355	−	0.961056i	$-0.589127\pi$
$293$	−9.46088	−0.552710	−0.276355	+	0.961056i	$0.589127\pi$
$294$	0	0
$295$	5.79703i	0.337516i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.6505i	0.731599i
$300$	0	0
$301$	− 8.84195i	− 0.509641i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.73138i	− 0.156398i
$306$	0	0
$307$	12.4055	0.708019	0.354009	−	0.935242i	$-0.384818\pi$
$307$	12.4055	0.708019	0.354009	+	0.935242i	$0.384818\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.70862	0.0968870	0.0484435	−	0.998826i	$-0.484574\pi$
$311$	1.70862	0.0968870	0.0484435	+	0.998826i	$0.484574\pi$
$312$	0	0
$313$	11.9889	0.677652	0.338826	−	0.940849i	$-0.389970\pi$
$313$	11.9889	0.677652	0.338826	+	0.940849i	$0.389970\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6470	1.38431	0.692155	−	0.721748i	$-0.256661\pi$
$317$	24.6470	1.38431	0.692155	+	0.721748i	$0.256661\pi$
$318$	0	0
$319$	− 13.5505i	− 0.758681i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 31.3319i	− 1.74336i
$324$	0	0
$325$	7.91952i	0.439296i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.54668i	− 0.250667i
$330$	0	0
$331$	5.66221	0.311223	0.155612	−	0.987818i	$-0.450265\pi$
$331$	5.66221	0.311223	0.155612	+	0.987818i	$0.450265\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.10995	0.224551
$336$	0	0
$337$	−32.2480	−1.75666	−0.878331	−	0.478053i	$-0.841343\pi$
$337$	−32.2480	−1.75666	−0.878331	+	0.478053i	$0.841343\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−26.7824	−1.45035
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.3357i	− 1.09168i	−0.837889	−	0.545840i	$-0.816211\pi$
$347$	− 20.3357i	− 1.09168i	0.837889	−	0.545840i	$-0.183789\pi$
$348$	0	0
$349$	− 8.27539i	− 0.442971i	−0.975164	−	0.221486i	$-0.928909\pi$
$349$	− 8.27539i	− 0.442971i	0.975164	−	0.221486i	$-0.0710906\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.23298i	0.438197i	0.975703	+	0.219099i	$0.0703117\pi$
$353$	8.23298i	0.438197i	−0.975703	+	0.219099i	$0.929688\pi$
$354$	0	0
$355$	4.26080	0.226140
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.5802	−0.558403	−0.279202	−	0.960232i	$-0.590070\pi$
$359$	−10.5802	−0.558403	−0.279202	+	0.960232i	$0.590070\pi$
$360$	0	0
$361$	16.9955	0.894502
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.98741	0.208710
$366$	0	0
$367$	23.7595i	1.24023i	0.784509	+	0.620117i	$0.212915\pi$
$367$	23.7595i	1.24023i	−0.784509	+	0.620117i	$0.787085\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.94845i	0.516498i
$372$	0	0
$373$	12.1442i	0.628802i	0.949290	+	0.314401i	$0.101804\pi$
$373$	12.1442i	0.628802i	−0.949290	+	0.314401i	$0.898196\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.14708i	− 0.419596i
$378$	0	0
$379$	32.1386	1.65085	0.825426	−	0.564511i	$-0.190935\pi$
$379$	32.1386	1.65085	0.825426	+	0.564511i	$0.190935\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.0530	−1.53564	−0.767819	−	0.640667i	$-0.778658\pi$
$383$	−30.0530	−1.53564	−0.767819	+	0.640667i	$0.778658\pi$
$384$	0	0
$385$	1.19465	0.0608849
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.8723	0.652652	0.326326	−	0.945257i	$-0.394189\pi$
$389$	12.8723	0.652652	0.326326	+	0.945257i	$0.394189\pi$
$390$	0	0
$391$	40.1228i	2.02910i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.26828i	− 0.114130i
$396$	0	0
$397$	− 20.8323i	− 1.04554i	−0.852473	−	0.522771i	$-0.824898\pi$
$397$	− 20.8323i	− 1.04554i	0.852473	−	0.522771i	$-0.175102\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8270i	0.790361i	0.918604	+	0.395180i	$0.129318\pi$
$401$	15.8270i	0.790361i	−0.918604	+	0.395180i	$0.870682\pi$
$402$	0	0
$403$	−16.1026	−0.802129
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4527	0.518120
$408$	0	0
$409$	−3.95136	−0.195382	−0.0976910	−	0.995217i	$-0.531146\pi$
$409$	−3.95136	−0.195382	−0.0976910	+	0.995217i	$0.531146\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.2892	−0.653919
$414$	0	0
$415$	− 0.0990534i	− 0.00486234i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.56977i	0.174395i	0.996191	+	0.0871974i	$0.0277911\pi$
$419$	3.56977i	0.174395i	−0.996191	+	0.0871974i	$0.972209\pi$
$420$	0	0
$421$	16.3767i	0.798152i	0.916918	+	0.399076i	$0.130669\pi$
$421$	16.3767i	0.798152i	−0.916918	+	0.399076i	$0.869331\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.1178i	1.21839i
$426$	0	0
$427$	6.26145	0.303013
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.7503	−1.38485	−0.692426	−	0.721489i	$-0.743458\pi$
$431$	−28.7503	−1.38485	−0.692426	+	0.721489i	$0.743458\pi$
$432$	0	0
$433$	−23.6276	−1.13547	−0.567735	−	0.823212i	$-0.692180\pi$
$433$	−23.6276	−1.13547	−0.567735	+	0.823212i	$0.692180\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−46.0949	−2.20502
$438$	0	0
$439$	15.4891i	0.739255i	0.929180	+	0.369628i	$0.120515\pi$
$439$	15.4891i	0.739255i	−0.929180	+	0.369628i	$0.879485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.7640i	1.03404i	0.855974	+	0.517019i	$0.172958\pi$
$443$	21.7640i	1.03404i	−0.855974	+	0.517019i	$0.827042\pi$
$444$	0	0
$445$	− 1.07558i	− 0.0509872i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 28.9358i	− 1.36557i	−0.730621	−	0.682783i	$-0.760769\pi$
$449$	− 28.9358i	− 1.36557i	0.730621	−	0.682783i	$-0.239231\pi$
$450$	0	0
$451$	26.6925	1.25690
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.718269	0.0336730
$456$	0	0
$457$	26.2466	1.22776	0.613882	−	0.789397i	$-0.289607\pi$
$457$	26.2466	1.22776	0.613882	+	0.789397i	$0.289607\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.47471	−0.348132	−0.174066	−	0.984734i	$-0.555691\pi$
$461$	−7.47471	−0.348132	−0.174066	+	0.984734i	$0.555691\pi$
$462$	0	0
$463$	8.09410i	0.376165i	0.982153	+	0.188082i	$0.0602272\pi$
$463$	8.09410i	0.376165i	−0.982153	+	0.188082i	$0.939773\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.0617i	0.743246i	0.928384	+	0.371623i	$0.121199\pi$
$467$	16.0617i	0.743246i	−0.928384	+	0.371623i	$0.878801\pi$
$468$	0	0
$469$	9.42171i	0.435054i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 24.2148i	− 1.11340i
$474$	0	0
$475$	−28.8565	−1.32403
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.6989	1.44836	0.724181	−	0.689610i	$-0.242218\pi$
$479$	31.6989	1.44836	0.724181	+	0.689610i	$0.242218\pi$
$480$	0	0
$481$	6.28456	0.286551
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.47132	0.0668092
$486$	0	0
$487$	− 30.5363i	− 1.38373i	−0.722025	−	0.691867i	$-0.756789\pi$
$487$	− 30.5363i	− 1.38373i	0.722025	−	0.691867i	$-0.243211\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 41.2201i	− 1.86024i	−0.367259	−	0.930119i	$-0.619704\pi$
$491$	− 41.2201i	− 1.86024i	0.367259	−	0.930119i	$-0.380296\pi$
$492$	0	0
$493$	− 25.8395i	− 1.16375i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.76751i	0.438133i
$498$	0	0
$499$	−4.03676	−0.180710	−0.0903552	−	0.995910i	$-0.528800\pi$
$499$	−4.03676	−0.180710	−0.0903552	+	0.995910i	$0.528800\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−40.2393	−1.79418	−0.897092	−	0.441845i	$-0.854324\pi$
$503$	−40.2393	−1.79418	−0.897092	+	0.441845i	$0.854324\pi$
$504$	0	0
$505$	−0.523556	−0.0232979
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.68050	−0.429081	−0.214540	−	0.976715i	$-0.568825\pi$
$509$	−9.68050	−0.429081	−0.214540	+	0.976715i	$0.568825\pi$
$510$	0	0
$511$	9.14079i	0.404365i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.66773i	− 0.205685i
$516$	0	0
$517$	− 12.4517i	− 0.547624i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3.92629i	− 0.172014i	−0.996295	−	0.0860069i	$-0.972589\pi$
$521$	− 3.92629i	− 0.172014i	0.996295	−	0.0860069i	$-0.0274107\pi$
$522$	0	0
$523$	−20.8709	−0.912623	−0.456311	−	0.889820i	$-0.650830\pi$
$523$	−20.8709	−0.912623	−0.456311	+	0.889820i	$0.650830\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−51.0716	−2.22471
$528$	0	0
$529$	36.0279	1.56643
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.0486	0.695141
$534$	0	0
$535$	0.652825i	0.0282241i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.73863i	0.117961i
$540$	0	0
$541$	− 3.18975i	− 0.137138i	−0.997646	−	0.0685691i	$-0.978157\pi$
$541$	− 3.18975i	− 0.137138i	0.997646	−	0.0685691i	$-0.0218434\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.04185i	− 0.344475i
$546$	0	0
$547$	−27.9137	−1.19350	−0.596751	−	0.802426i	$-0.703542\pi$
$547$	−27.9137	−1.19350	−0.596751	+	0.802426i	$0.703542\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.6856	1.26465
$552$	0	0
$553$	5.19984	0.221120
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.0025	−0.593304	−0.296652	−	0.954986i	$-0.595870\pi$
$557$	−14.0025	−0.593304	−0.296652	+	0.954986i	$0.595870\pi$
$558$	0	0
$559$	− 14.5589i	− 0.615775i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.8839i	0.585137i	0.956245	+	0.292569i	$0.0945100\pi$
$563$	13.8839i	0.585137i	−0.956245	+	0.292569i	$0.905490\pi$
$564$	0	0
$565$	2.29318i	0.0964749i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 14.7512i	− 0.618401i	−0.950997	−	0.309200i	$-0.899939\pi$
$569$	− 14.7512i	− 0.618401i	0.950997	−	0.309200i	$-0.100061\pi$
$570$	0	0
$571$	−4.28135	−0.179169	−0.0895845	−	0.995979i	$-0.528554\pi$
$571$	−4.28135	−0.179169	−0.0895845	+	0.995979i	$0.528554\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	36.9528	1.54104
$576$	0	0
$577$	11.1951	0.466059	0.233030	−	0.972470i	$-0.425136\pi$
$577$	11.1951	0.466059	0.233030	+	0.972470i	$0.425136\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.227071	0.00942051
$582$	0	0
$583$	27.2451i	1.12838i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.45377i	0.0600035i	0.999550	+	0.0300018i	$0.00955129\pi$
$587$	1.45377i	0.0600035i	−0.999550	+	0.0300018i	$0.990449\pi$
$588$	0	0
$589$	− 58.6733i	− 2.41759i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.1827i	0.541347i	0.962671	+	0.270674i	$0.0872464\pi$
$593$	13.1827i	0.541347i	−0.962671	+	0.270674i	$0.912754\pi$
$594$	0	0
$595$	2.27808	0.0933924
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.4852	0.591848	0.295924	−	0.955211i	$-0.404372\pi$
$599$	14.4852	0.591848	0.295924	+	0.955211i	$0.404372\pi$
$600$	0	0
$601$	−27.7364	−1.13139	−0.565695	−	0.824614i	$-0.691392\pi$
$601$	−27.7364	−1.13139	−0.565695	+	0.824614i	$0.691392\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.52674	−0.0620709
$606$	0	0
$607$	− 18.5137i	− 0.751447i	−0.926732	−	0.375723i	$-0.877394\pi$
$607$	− 18.5137i	− 0.751447i	0.926732	−	0.375723i	$-0.122606\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.48643i	− 0.302868i
$612$	0	0
$613$	25.3069i	1.02213i	0.859541	+	0.511067i	$0.170750\pi$
$613$	25.3069i	1.02213i	−0.859541	+	0.511067i	$0.829250\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 47.8249i	− 1.92536i	−0.270647	−	0.962679i	$-0.587238\pi$
$617$	− 47.8249i	− 1.92536i	0.270647	−	0.962679i	$-0.412762\pi$
$618$	0	0
$619$	−3.22411	−0.129588	−0.0647939	−	0.997899i	$-0.520639\pi$
$619$	−3.22411	−0.129588	−0.0647939	+	0.997899i	$0.520639\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.46567	0.0987848
$624$	0	0
$625$	22.1819	0.887275
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.9323	0.794754
$630$	0	0
$631$	22.6025i	0.899791i	0.893081	+	0.449896i	$0.148539\pi$
$631$	22.6025i	0.899791i	−0.893081	+	0.449896i	$0.851461\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.62311i	0.342198i
$636$	0	0
$637$	1.64657i	0.0652394i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.4383i	0.609775i	0.952388	+	0.304888i	$0.0986189\pi$
$641$	15.4383i	0.609775i	−0.952388	+	0.304888i	$0.901381\pi$
$642$	0	0
$643$	−1.41036	−0.0556191	−0.0278095	−	0.999613i	$-0.508853\pi$
$643$	−1.41036	−0.0556191	−0.0278095	+	0.999613i	$0.508853\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.811574	0.0319063	0.0159531	−	0.999873i	$-0.494922\pi$
$647$	0.811574	0.0319063	0.0159531	+	0.999873i	$0.494922\pi$
$648$	0	0
$649$	−36.3941	−1.42860
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.82168	0.384352	0.192176	−	0.981360i	$-0.438446\pi$
$653$	9.82168	0.384352	0.192176	+	0.981360i	$0.438446\pi$
$654$	0	0
$655$	4.03615i	0.157705i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.1456i	− 0.434170i	−0.976153	−	0.217085i	$-0.930345\pi$
$659$	− 11.1456i	− 0.434170i	0.976153	−	0.217085i	$-0.0696548\pi$
$660$	0	0
$661$	15.3989i	0.598947i	0.954105	+	0.299474i	$0.0968111\pi$
$661$	15.3989i	0.598947i	−0.954105	+	0.299474i	$0.903189\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.61717i	0.101489i
$666$	0	0
$667$	−38.0146	−1.47193
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.1478	0.661982
$672$	0	0
$673$	20.4112	0.786795	0.393398	−	0.919368i	$-0.371300\pi$
$673$	20.4112	0.786795	0.393398	+	0.919368i	$0.371300\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	50.5124	1.94135	0.970674	−	0.240401i	$-0.0772789\pi$
$677$	50.5124	1.94135	0.970674	+	0.240401i	$0.0772789\pi$
$678$	0	0
$679$	3.37287i	0.129439i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33.3625i	− 1.27658i	−0.769796	−	0.638291i	$-0.779642\pi$
$683$	− 33.3625i	− 1.27658i	0.769796	−	0.638291i	$-0.220358\pi$
$684$	0	0
$685$	− 1.10475i	− 0.0422104i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.3808i	0.624059i
$690$	0	0
$691$	−24.3714	−0.927133	−0.463567	−	0.886062i	$-0.653431\pi$
$691$	−24.3714	−0.927133	−0.463567	+	0.886062i	$0.653431\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.62982	−0.327348
$696$	0	0
$697$	50.9002	1.92798
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.1271	1.28896	0.644481	−	0.764620i	$-0.277073\pi$
$701$	34.1271	1.28896	0.644481	+	0.764620i	$0.277073\pi$
$702$	0	0
$703$	22.8991i	0.863658i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.20021i	− 0.0451384i
$708$	0	0
$709$	− 22.8145i	− 0.856818i	−0.903585	−	0.428409i	$-0.859074\pi$
$709$	− 22.8145i	− 0.856818i	0.903585	−	0.428409i	$-0.140926\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	75.1355i	2.81385i
$714$	0	0
$715$	1.96707	0.0735642
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.18175	−0.0813654	−0.0406827	−	0.999172i	$-0.512953\pi$
$719$	−2.18175	−0.0813654	−0.0406827	+	0.999172i	$0.512953\pi$
$720$	0	0
$721$	10.7004	0.398502
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−23.7980	−0.883837
$726$	0	0
$727$	15.3017i	0.567507i	0.958897	+	0.283754i	$0.0915798\pi$
$727$	15.3017i	0.567507i	−0.958897	+	0.283754i	$0.908420\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 46.1754i	− 1.70786i
$732$	0	0
$733$	15.8711i	0.586213i	0.956080	+	0.293106i	$0.0946890\pi$
$733$	15.8711i	0.586213i	−0.956080	+	0.293106i	$0.905311\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.8025i	0.950449i
$738$	0	0
$739$	−31.1319	−1.14521	−0.572603	−	0.819833i	$-0.694066\pi$
$739$	−31.1319	−1.14521	−0.572603	+	0.819833i	$0.694066\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.6629	−0.537930	−0.268965	−	0.963150i	$-0.586682\pi$
$743$	−14.6629	−0.537930	−0.268965	+	0.963150i	$0.586682\pi$
$744$	0	0
$745$	3.56158	0.130486
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.49655	−0.0546826
$750$	0	0
$751$	− 8.46100i	− 0.308746i	−0.988013	−	0.154373i	$-0.950664\pi$
$751$	− 8.46100i	− 0.308746i	0.988013	−	0.154373i	$-0.0493358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.48842i	0.236138i
$756$	0	0
$757$	− 39.6473i	− 1.44101i	−0.693451	−	0.720504i	$-0.743911\pi$
$757$	− 39.6473i	− 1.44101i	0.693451	−	0.720504i	$-0.256089\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.7301i	1.80271i	0.433076	+	0.901357i	$0.357428\pi$
$761$	49.7301i	1.80271i	−0.433076	+	0.901357i	$0.642572\pi$
$762$	0	0
$763$	18.4352	0.667401
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.8816	−0.790098
$768$	0	0
$769$	5.62877	0.202979	0.101489	−	0.994837i	$-0.467639\pi$
$769$	5.62877	0.202979	0.101489	+	0.994837i	$0.467639\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.3999	0.913570	0.456785	−	0.889577i	$-0.349001\pi$
$773$	25.3999	0.913570	0.456785	+	0.889577i	$0.349001\pi$
$774$	0	0
$775$	47.0366i	1.68960i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	58.4764i	2.09514i
$780$	0	0
$781$	26.7495i	0.957174i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.14711i	− 0.0766338i
$786$	0	0
$787$	31.5311	1.12396	0.561982	−	0.827150i	$-0.310039\pi$
$787$	31.5311	1.12396	0.561982	+	0.827150i	$0.310039\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.25692	−0.186915
$792$	0	0
$793$	10.3099	0.366115
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.3724	−1.32380	−0.661899	−	0.749593i	$-0.730249\pi$
$797$	−37.3724	−1.32380	−0.661899	+	0.749593i	$0.730249\pi$
$798$	0	0
$799$	− 23.7442i	− 0.840010i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.0332i	0.883402i
$804$	0	0
$805$	− 3.35147i	− 0.118124i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.37230i	− 0.153722i	−0.997042	−	0.0768610i	$-0.975510\pi$
$809$	− 4.37230i	− 0.153722i	0.997042	−	0.0768610i	$-0.0244898\pi$
$810$	0	0
$811$	29.1067	1.02208	0.511038	−	0.859558i	$-0.329261\pi$
$811$	29.1067	1.02208	0.511038	+	0.859558i	$0.329261\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.56171	−0.0897328
$816$	0	0
$817$	53.0484	1.85593
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.1785	1.57674	0.788370	−	0.615201i	$-0.210925\pi$
$821$	45.1785	1.57674	0.788370	+	0.615201i	$0.210925\pi$
$822$	0	0
$823$	30.6242i	1.06749i	0.845644	+	0.533747i	$0.179217\pi$
$823$	30.6242i	1.06749i	−0.845644	+	0.533747i	$0.820783\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0993i	0.420734i	0.977623	+	0.210367i	$0.0674658\pi$
$827$	12.0993i	0.420734i	−0.977623	+	0.210367i	$0.932534\pi$
$828$	0	0
$829$	− 15.2105i	− 0.528283i	−0.964484	−	0.264142i	$-0.914911\pi$
$829$	− 15.2105i	− 0.528283i	0.964484	−	0.264142i	$-0.0850886\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.22231i	0.180942i
$834$	0	0
$835$	9.64841	0.333897
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.2155	1.45744	0.728721	−	0.684811i	$-0.240115\pi$
$839$	42.2155	1.45744	0.728721	+	0.684811i	$0.240115\pi$
$840$	0	0
$841$	−4.51816	−0.155799
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.48820	−0.154399
$846$	0	0
$847$	− 3.49993i	− 0.120259i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 29.3240i	− 1.00521i
$852$	0	0
$853$	− 49.2536i	− 1.68641i	−0.537592	−	0.843205i	$-0.680666\pi$
$853$	− 49.2536i	− 1.68641i	0.537592	−	0.843205i	$-0.319334\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 25.9243i	− 0.885557i	−0.896631	−	0.442779i	$-0.853993\pi$
$857$	− 25.9243i	− 0.885557i	0.896631	−	0.442779i	$-0.146007\pi$
$858$	0	0
$859$	−9.25940	−0.315927	−0.157963	−	0.987445i	$-0.550493\pi$
$859$	−9.25940	−0.315927	−0.157963	+	0.987445i	$0.550493\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.21443	0.211542	0.105771	−	0.994391i	$-0.466269\pi$
$863$	6.21443	0.211542	0.105771	+	0.994391i	$0.466269\pi$
$864$	0	0
$865$	10.5531	0.358815
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.2404	0.483073
$870$	0	0
$871$	15.5135i	0.525655i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.27921i	− 0.144664i
$876$	0	0
$877$	15.1038i	0.510020i	0.966938	+	0.255010i	$0.0820787\pi$
$877$	15.1038i	0.510020i	−0.966938	+	0.255010i	$0.917921\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.59677i	0.0874874i	0.999043	+	0.0437437i	$0.0139285\pi$
$881$	2.59677i	0.0874874i	−0.999043	+	0.0437437i	$0.986072\pi$
$882$	0	0
$883$	−11.0661	−0.372404	−0.186202	−	0.982511i	$-0.559618\pi$
$883$	−11.0661	−0.372404	−0.186202	+	0.982511i	$0.559618\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.38595	0.247996	0.123998	−	0.992282i	$-0.460428\pi$
$887$	7.38595	0.247996	0.123998	+	0.992282i	$0.460428\pi$
$888$	0	0
$889$	−19.7677	−0.662989
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.2784	0.912837
$894$	0	0
$895$	− 2.36121i	− 0.0789265i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 48.3881i	− 1.61383i
$900$	0	0
$901$	51.9539i	1.73084i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.70314i	0.0898555i
$906$	0	0
$907$	−5.58114	−0.185319	−0.0926594	−	0.995698i	$-0.529537\pi$
$907$	−5.58114	−0.185319	−0.0926594	+	0.995698i	$0.529537\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.02450	0.232732	0.116366	−	0.993206i	$-0.462875\pi$
$911$	7.02450	0.232732	0.116366	+	0.993206i	$0.462875\pi$
$912$	0	0
$913$	0.621864	0.0205807
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.25252	−0.305545
$918$	0	0
$919$	− 25.7961i	− 0.850935i	−0.904974	−	0.425468i	$-0.860110\pi$
$919$	− 25.7961i	− 0.850935i	0.904974	−	0.425468i	$-0.139890\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.0829i	0.529374i
$924$	0	0
$925$	− 18.3575i	− 0.603591i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 10.1558i	− 0.333201i	−0.986024	−	0.166600i	$-0.946721\pi$
$929$	− 10.1558i	− 0.333201i	0.986024	−	0.166600i	$-0.0532790\pi$
$930$	0	0
$931$	−5.99963	−0.196630
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.23882	0.204031
$936$	0	0
$937$	19.1504	0.625616	0.312808	−	0.949816i	$-0.398730\pi$
$937$	19.1504	0.625616	0.312808	+	0.949816i	$0.398730\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.3260	0.467016	0.233508	−	0.972355i	$-0.424980\pi$
$941$	14.3260	0.467016	0.233508	+	0.972355i	$0.424980\pi$
$942$	0	0
$943$	− 74.8833i	− 2.43854i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.67662i	0.249456i	0.992191	+	0.124728i	$0.0398059\pi$
$947$	7.67662i	0.249456i	−0.992191	+	0.124728i	$0.960194\pi$
$948$	0	0
$949$	15.0509i	0.488574i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.6546i	1.70565i	0.522197	+	0.852825i	$0.325113\pi$
$953$	52.6546i	1.70565i	−0.522197	+	0.852825i	$0.674887\pi$
$954$	0	0
$955$	5.94095	0.192245
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.53255	0.0817803
$960$	0	0
$961$	−64.6386	−2.08512
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.57056	−0.0827492
$966$	0	0
$967$	50.1578i	1.61297i	0.591257	+	0.806483i	$0.298632\pi$
$967$	50.1578i	1.61297i	−0.591257	+	0.806483i	$0.701368\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 54.8144i	− 1.75908i	−0.475828	−	0.879538i	$-0.657852\pi$
$971$	− 54.8144i	− 1.75908i	0.475828	−	0.879538i	$-0.342148\pi$
$972$	0	0
$973$	− 19.7831i	− 0.634217i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 0.0603583i	− 0.00193103i	−1.00000		0.000965516i	$-0.999693\pi$
$977$	− 0.0603583i	− 0.00193103i	1.00000		0.000965516i	$-0.000307333\pi$
$978$	0	0
$979$	6.75254	0.215812
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.40354	0.236136	0.118068	−	0.993005i	$-0.462330\pi$
$983$	7.40354	0.236136	0.118068	+	0.993005i	$0.462330\pi$
$984$	0	0
$985$	8.30230	0.264533
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−67.9323	−2.16012
$990$	0	0
$991$	− 35.6861i	− 1.13361i	−0.823853	−	0.566803i	$-0.808180\pi$
$991$	− 35.6861i	− 1.13361i	0.823853	−	0.566803i	$-0.191820\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.57981i	− 0.0817853i
$996$	0	0
$997$	− 52.4345i	− 1.66062i	−0.557304	−	0.830308i	$-0.688164\pi$
$997$	− 52.4345i	− 1.66062i	0.557304	−	0.830308i	$-0.311836\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.c.5615.12		32
3.2	odd	2	inner	6048.2.j.c.5615.21		32
4.3	odd	2		1512.2.j.c.323.13	✓	32
8.3	odd	2	inner	6048.2.j.c.5615.22		32
8.5	even	2		1512.2.j.c.323.19	yes	32
12.11	even	2		1512.2.j.c.323.20	yes	32
24.5	odd	2		1512.2.j.c.323.14	yes	32
24.11	even	2	inner	6048.2.j.c.5615.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.13	✓	32	4.3	odd	2
1512.2.j.c.323.14	yes	32	24.5	odd	2
1512.2.j.c.323.19	yes	32	8.5	even	2
1512.2.j.c.323.20	yes	32	12.11	even	2
6048.2.j.c.5615.11		32	24.11	even	2	inner
6048.2.j.c.5615.12		32	1.1	even	1	trivial
6048.2.j.c.5615.21		32	3.2	odd	2	inner
6048.2.j.c.5615.22		32	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.436221 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-0.436221 q^{5} -1.00000i q^{7} -2.73863i q^{11} -1.64657i q^{13} -5.22231i q^{17} +5.99963 q^{19} -7.68296 q^{23} -4.80971 q^{25} +4.94791 q^{29} -9.77950i q^{31} +0.436221i q^{35} +3.81676i q^{37} +9.74668i q^{41} +8.84195 q^{43} +4.54668 q^{47} -1.00000 q^{49} -9.94845 q^{53} +1.19465i q^{55} -13.2892i q^{59} +6.26145i q^{61} +0.718269i q^{65} -9.42171 q^{67} -9.76751 q^{71} -9.14079 q^{73} -2.73863 q^{77} +5.19984i q^{79} +0.227071i q^{83} +2.27808i q^{85} +2.46567i q^{89} -1.64657 q^{91} -2.61717 q^{95} -3.37287 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) \) 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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