LMFDB - Embedded newform 6048.2.j.c.5615.19

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			5615.19
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.206991 q^{5} -1.00000i q^{7} +O(q^{10})$	$q+0.206991 q^{5} -1.00000i q^{7} +6.42190i q^{11} +3.52336i q^{13} -3.90616i q^{17} +5.48691 q^{19} +0.880512 q^{23} -4.95715 q^{25} -3.20814 q^{29} -0.0631261i q^{31} -0.206991i q^{35} +9.91837i q^{37} +5.94221i q^{41} +1.13960 q^{43} -6.81266 q^{47} -1.00000 q^{49} -12.6324 q^{53} +1.32928i q^{55} -4.23338i q^{59} -12.3006i q^{61} +0.729305i q^{65} +5.74666 q^{67} +10.1795 q^{71} +8.27618 q^{73} +6.42190 q^{77} +4.86426i q^{79} +7.51539i q^{83} -0.808542i q^{85} -4.44682i q^{89} +3.52336 q^{91} +1.13574 q^{95} -17.3929 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.206991	0.0925694	0.0462847	−	0.998928i	$-0.485262\pi$
$5$	0.206991	0.0925694	0.0462847	+	0.998928i	$0.485262\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.42190i	1.93628i	0.250418	+	0.968138i	$0.419432\pi$
$11$	6.42190i	1.93628i	−0.250418	+	0.968138i	$0.580568\pi$
$12$	0	0
$13$	3.52336i	0.977204i	0.872507	+	0.488602i	$0.162493\pi$
$13$	3.52336i	0.977204i	−0.872507	+	0.488602i	$0.837507\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.90616i	− 0.947383i	−0.880691	−	0.473692i	$-0.842921\pi$
$17$	− 3.90616i	− 0.947383i	0.880691	−	0.473692i	$-0.157079\pi$
$18$	0	0
$19$	5.48691	1.25878	0.629392	−	0.777088i	$-0.283304\pi$
$19$	5.48691	1.25878	0.629392	+	0.777088i	$0.283304\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.880512	0.183599	0.0917997	−	0.995777i	$-0.470738\pi$
$23$	0.880512	0.183599	0.0917997	+	0.995777i	$0.470738\pi$
$24$	0	0
$25$	−4.95715	−0.991431
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.20814	−0.595737	−0.297868	−	0.954607i	$-0.596276\pi$
$29$	−3.20814	−0.595737	−0.297868	+	0.954607i	$0.596276\pi$
$30$	0	0
$31$	− 0.0631261i	− 0.0113378i	−0.999984	−	0.00566890i	$-0.998196\pi$
$31$	− 0.0631261i	− 0.0113378i	0.999984	−	0.00566890i	$-0.00180448\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.206991i	− 0.0349879i
$36$	0	0
$37$	9.91837i	1.63057i	0.579060	+	0.815285i	$0.303420\pi$
$37$	9.91837i	1.63057i	−0.579060	+	0.815285i	$0.696580\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.94221i	0.928017i	0.885831	+	0.464009i	$0.153589\pi$
$41$	5.94221i	0.928017i	−0.885831	+	0.464009i	$0.846411\pi$
$42$	0	0
$43$	1.13960	0.173788	0.0868938	−	0.996218i	$-0.472306\pi$
$43$	1.13960	0.173788	0.0868938	+	0.996218i	$0.472306\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.81266	−0.993729	−0.496864	−	0.867828i	$-0.665515\pi$
$47$	−6.81266	−0.993729	−0.496864	+	0.867828i	$0.665515\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.6324	−1.73520	−0.867598	−	0.497265i	$-0.834338\pi$
$53$	−12.6324	−1.73520	−0.867598	+	0.497265i	$0.834338\pi$
$54$	0	0
$55$	1.32928i	0.179240i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.23338i	− 0.551139i	−0.961281	−	0.275570i	$-0.911134\pi$
$59$	− 4.23338i	− 0.551139i	0.961281	−	0.275570i	$-0.0888664\pi$
$60$	0	0
$61$	− 12.3006i	− 1.57493i	−0.616359	−	0.787465i	$-0.711393\pi$
$61$	− 12.3006i	− 1.57493i	0.616359	−	0.787465i	$-0.288607\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.729305i	0.0904591i
$66$	0	0
$67$	5.74666	0.702067	0.351033	−	0.936363i	$-0.385830\pi$
$67$	5.74666	0.702067	0.351033	+	0.936363i	$0.385830\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.1795	1.20809	0.604043	−	0.796952i	$-0.293556\pi$
$71$	10.1795	1.20809	0.604043	+	0.796952i	$0.293556\pi$
$72$	0	0
$73$	8.27618	0.968653	0.484327	−	0.874887i	$-0.339065\pi$
$73$	8.27618	0.968653	0.484327	+	0.874887i	$0.339065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.42190	0.731843
$78$	0	0
$79$	4.86426i	0.547272i	0.961833	+	0.273636i	$0.0882264\pi$
$79$	4.86426i	0.547272i	−0.961833	+	0.273636i	$0.911774\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.51539i	0.824921i	0.910976	+	0.412460i	$0.135331\pi$
$83$	7.51539i	0.824921i	−0.910976	+	0.412460i	$0.864669\pi$
$84$	0	0
$85$	− 0.808542i	− 0.0876987i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.44682i	− 0.471362i	−0.971830	−	0.235681i	$-0.924268\pi$
$89$	− 4.44682i	− 0.471362i	0.971830	−	0.235681i	$-0.0757320\pi$
$90$	0	0
$91$	3.52336	0.369348
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.13574	0.116525
$96$	0	0
$97$	−17.3929	−1.76599	−0.882993	−	0.469387i	$-0.844475\pi$
$97$	−17.3929	−1.76599	−0.882993	+	0.469387i	$0.844475\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.0273	−1.89329	−0.946643	−	0.322284i	$-0.895550\pi$
$101$	−19.0273	−1.89329	−0.946643	+	0.322284i	$0.895550\pi$
$102$	0	0
$103$	− 10.7903i	− 1.06320i	−0.846995	−	0.531601i	$-0.821591\pi$
$103$	− 10.7903i	− 1.06320i	0.846995	−	0.531601i	$-0.178409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.97087i	0.480553i	0.970705	+	0.240276i	$0.0772380\pi$
$107$	4.97087i	0.480553i	−0.970705	+	0.240276i	$0.922762\pi$
$108$	0	0
$109$	2.84810i	0.272798i	0.990654	+	0.136399i	$0.0435530\pi$
$109$	2.84810i	0.272798i	−0.990654	+	0.136399i	$0.956447\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.88561i	0.271455i	0.990746	+	0.135728i	$0.0433372\pi$
$113$	2.88561i	0.271455i	−0.990746	+	0.135728i	$0.956663\pi$
$114$	0	0
$115$	0.182258	0.0169957
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.90616	−0.358077
$120$	0	0
$121$	−30.2408	−2.74916
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.06105	−0.184345
$126$	0	0
$127$	7.23082i	0.641631i	0.947142	+	0.320816i	$0.103957\pi$
$127$	7.23082i	0.641631i	−0.947142	+	0.320816i	$0.896043\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.8026i	1.11857i	0.828975	+	0.559286i	$0.188924\pi$
$131$	12.8026i	1.11857i	−0.828975	+	0.559286i	$0.811076\pi$
$132$	0	0
$133$	− 5.48691i	− 0.475776i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.363873i	− 0.0310878i	−0.999879	−	0.0155439i	$-0.995052\pi$
$137$	− 0.363873i	− 0.0310878i	0.999879	−	0.0155439i	$-0.00494798\pi$
$138$	0	0
$139$	9.32338	0.790799	0.395399	−	0.918509i	$-0.370606\pi$
$139$	9.32338	0.790799	0.395399	+	0.918509i	$0.370606\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−22.6266	−1.89214
$144$	0	0
$145$	−0.664057	−0.0551470
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.82519	−0.395295	−0.197648	−	0.980273i	$-0.563330\pi$
$149$	−4.82519	−0.395295	−0.197648	+	0.980273i	$0.563330\pi$
$150$	0	0
$151$	− 12.7969i	− 1.04140i	−0.853740	−	0.520700i	$-0.825671\pi$
$151$	− 12.7969i	− 1.04140i	0.853740	−	0.520700i	$-0.174329\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.0130666i	− 0.00104953i
$156$	0	0
$157$	− 15.5455i	− 1.24067i	−0.784337	−	0.620335i	$-0.786997\pi$
$157$	− 15.5455i	− 1.24067i	0.784337	−	0.620335i	$-0.213003\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.880512i	− 0.0693940i
$162$	0	0
$163$	−18.4630	−1.44613	−0.723067	−	0.690778i	$-0.757268\pi$
$163$	−18.4630	−1.44613	−0.723067	+	0.690778i	$0.757268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.5320	−0.969754	−0.484877	−	0.874582i	$-0.661136\pi$
$167$	−12.5320	−0.969754	−0.484877	+	0.874582i	$0.661136\pi$
$168$	0	0
$169$	0.585951	0.0450731
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.49171	0.417527	0.208763	−	0.977966i	$-0.433056\pi$
$173$	5.49171	0.417527	0.208763	+	0.977966i	$0.433056\pi$
$174$	0	0
$175$	4.95715i	0.374726i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.5830i	1.08998i	0.838441	+	0.544992i	$0.183467\pi$
$179$	14.5830i	1.08998i	−0.838441	+	0.544992i	$0.816533\pi$
$180$	0	0
$181$	− 17.2753i	− 1.28407i	−0.766677	−	0.642033i	$-0.778091\pi$
$181$	− 17.2753i	− 1.28407i	0.766677	−	0.642033i	$-0.221909\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.05302i	0.150941i
$186$	0	0
$187$	25.0850	1.83440
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.29828	0.528085	0.264042	−	0.964511i	$-0.414944\pi$
$191$	7.29828	0.528085	0.264042	+	0.964511i	$0.414944\pi$
$192$	0	0
$193$	12.1106	0.871742	0.435871	−	0.900009i	$-0.356440\pi$
$193$	12.1106	0.871742	0.435871	+	0.900009i	$0.356440\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.3925	−1.23917	−0.619584	−	0.784931i	$-0.712699\pi$
$197$	−17.3925	−1.23917	−0.619584	+	0.784931i	$0.712699\pi$
$198$	0	0
$199$	− 3.07583i	− 0.218040i	−0.994040	−	0.109020i	$-0.965229\pi$
$199$	− 3.07583i	− 0.218040i	0.994040	−	0.109020i	$-0.0347712\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.20814i	0.225167i
$204$	0	0
$205$	1.22999i	0.0859060i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	35.2364i	2.43735i
$210$	0	0
$211$	−9.57154	−0.658932	−0.329466	−	0.944167i	$-0.606869\pi$
$211$	−9.57154	−0.658932	−0.329466	+	0.944167i	$0.606869\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.235888	0.0160874
$216$	0	0
$217$	−0.0631261	−0.00428528
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.7628	0.925786
$222$	0	0
$223$	26.5694i	1.77922i	0.456722	+	0.889610i	$0.349024\pi$
$223$	26.5694i	1.77922i	−0.456722	+	0.889610i	$0.650976\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.7345i	1.37620i	0.725616	+	0.688100i	$0.241555\pi$
$227$	20.7345i	1.37620i	−0.725616	+	0.688100i	$0.758445\pi$
$228$	0	0
$229$	19.2625i	1.27290i	0.771316	+	0.636452i	$0.219599\pi$
$229$	19.2625i	1.27290i	−0.771316	+	0.636452i	$0.780401\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.0038i	1.63805i	0.573756	+	0.819027i	$0.305486\pi$
$233$	25.0038i	1.63805i	−0.573756	+	0.819027i	$0.694514\pi$
$234$	0	0
$235$	−1.41016	−0.0919889
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.0439	−0.973107	−0.486553	−	0.873651i	$-0.661746\pi$
$239$	−15.0439	−0.973107	−0.486553	+	0.873651i	$0.661746\pi$
$240$	0	0
$241$	−3.23274	−0.208239	−0.104120	−	0.994565i	$-0.533203\pi$
$241$	−3.23274	−0.208239	−0.104120	+	0.994565i	$0.533203\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.206991	−0.0132242
$246$	0	0
$247$	19.3324i	1.23009i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.4258i	0.658071i	0.944318	+	0.329036i	$0.106724\pi$
$251$	10.4258i	0.658071i	−0.944318	+	0.329036i	$0.893276\pi$
$252$	0	0
$253$	5.65456i	0.355499i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.12452i	0.506794i	0.967362	+	0.253397i	$0.0815479\pi$
$257$	8.12452i	0.506794i	−0.967362	+	0.253397i	$0.918452\pi$
$258$	0	0
$259$	9.91837	0.616297
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.7639	−0.910382	−0.455191	−	0.890394i	$-0.650429\pi$
$263$	−14.7639	−0.910382	−0.455191	+	0.890394i	$0.650429\pi$
$264$	0	0
$265$	−2.61480	−0.160626
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.8961	1.63989	0.819943	−	0.572445i	$-0.194005\pi$
$269$	26.8961	1.63989	0.819943	+	0.572445i	$0.194005\pi$
$270$	0	0
$271$	− 17.5914i	− 1.06860i	−0.845295	−	0.534301i	$-0.820575\pi$
$271$	− 17.5914i	− 1.06860i	0.845295	−	0.534301i	$-0.179425\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 31.8343i	− 1.91968i
$276$	0	0
$277$	24.8383i	1.49239i	0.665729	+	0.746194i	$0.268121\pi$
$277$	24.8383i	1.49239i	−0.665729	+	0.746194i	$0.731879\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.13370i	0.246596i	0.992370	+	0.123298i	$0.0393471\pi$
$281$	4.13370i	0.246596i	−0.992370	+	0.123298i	$0.960653\pi$
$282$	0	0
$283$	23.9297	1.42247	0.711236	−	0.702954i	$-0.248136\pi$
$283$	23.9297	1.42247	0.711236	+	0.702954i	$0.248136\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.94221	0.350758
$288$	0	0
$289$	1.74190	0.102465
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.0225	−0.702360	−0.351180	−	0.936308i	$-0.614219\pi$
$293$	−12.0225	−0.702360	−0.351180	+	0.936308i	$0.614219\pi$
$294$	0	0
$295$	− 0.876273i	− 0.0510186i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.10236i	0.179414i
$300$	0	0
$301$	− 1.13960i	− 0.0656855i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.54612i	− 0.145790i
$306$	0	0
$307$	5.82060	0.332199	0.166100	−	0.986109i	$-0.446883\pi$
$307$	5.82060	0.332199	0.166100	+	0.986109i	$0.446883\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.7884	−0.725161	−0.362580	−	0.931952i	$-0.618104\pi$
$311$	−12.7884	−0.725161	−0.362580	+	0.931952i	$0.618104\pi$
$312$	0	0
$313$	1.74881	0.0988483	0.0494242	−	0.998778i	$-0.484261\pi$
$313$	1.74881	0.0988483	0.0494242	+	0.998778i	$0.484261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.5938	−1.71832	−0.859159	−	0.511709i	$-0.829013\pi$
$317$	−30.5938	−1.71832	−0.859159	+	0.511709i	$0.829013\pi$
$318$	0	0
$319$	− 20.6024i	− 1.15351i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 21.4328i	− 1.19255i
$324$	0	0
$325$	− 17.4658i	− 0.968830i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.81266i	0.375594i
$330$	0	0
$331$	20.7269	1.13925	0.569627	−	0.821904i	$-0.307088\pi$
$331$	20.7269	1.13925	0.569627	+	0.821904i	$0.307088\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.18951	0.0649899
$336$	0	0
$337$	19.9716	1.08792	0.543961	−	0.839110i	$-0.316924\pi$
$337$	19.9716	1.08792	0.543961	+	0.839110i	$0.316924\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.405390	0.0219531
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.6443i	− 0.571414i	−0.958317	−	0.285707i	$-0.907772\pi$
$347$	− 10.6443i	− 0.571414i	0.958317	−	0.285707i	$-0.0922284\pi$
$348$	0	0
$349$	21.1575i	1.13254i	0.824221	+	0.566268i	$0.191613\pi$
$349$	21.1575i	1.13254i	−0.824221	+	0.566268i	$0.808387\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.85498i	0.418079i	0.977907	+	0.209039i	$0.0670337\pi$
$353$	7.85498i	0.418079i	−0.977907	+	0.209039i	$0.932966\pi$
$354$	0	0
$355$	2.10707	0.111832
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0102	1.16165	0.580826	−	0.814028i	$-0.302730\pi$
$359$	22.0102	1.16165	0.580826	+	0.814028i	$0.302730\pi$
$360$	0	0
$361$	11.1062	0.584537
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.71310	0.0896676
$366$	0	0
$367$	5.61196i	0.292942i	0.989215	+	0.146471i	$0.0467915\pi$
$367$	5.61196i	0.292942i	−0.989215	+	0.146471i	$0.953209\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.6324i	0.655843i
$372$	0	0
$373$	21.5334i	1.11496i	0.830192	+	0.557478i	$0.188231\pi$
$373$	21.5334i	1.11496i	−0.830192	+	0.557478i	$0.811769\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11.3034i	− 0.582156i
$378$	0	0
$379$	−36.2973	−1.86447	−0.932233	−	0.361858i	$-0.882143\pi$
$379$	−36.2973	−1.86447	−0.932233	+	0.361858i	$0.882143\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.4362	0.635462	0.317731	−	0.948181i	$-0.397079\pi$
$383$	12.4362	0.635462	0.317731	+	0.948181i	$0.397079\pi$
$384$	0	0
$385$	1.32928	0.0677463
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.47648	−0.277669	−0.138834	−	0.990316i	$-0.544336\pi$
$389$	−5.47648	−0.277669	−0.138834	+	0.990316i	$0.544336\pi$
$390$	0	0
$391$	− 3.43942i	− 0.173939i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.00686i	0.0506606i
$396$	0	0
$397$	9.58947i	0.481282i	0.970614	+	0.240641i	$0.0773576\pi$
$397$	9.58947i	0.481282i	−0.970614	+	0.240641i	$0.922642\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.76702i	0.0882408i	0.999026	+	0.0441204i	$0.0140485\pi$
$401$	1.76702i	0.0882408i	−0.999026	+	0.0441204i	$0.985951\pi$
$402$	0	0
$403$	0.222416	0.0110793
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−63.6948	−3.15723
$408$	0	0
$409$	−5.87490	−0.290495	−0.145248	−	0.989395i	$-0.546398\pi$
$409$	−5.87490	−0.290495	−0.145248	+	0.989395i	$0.546398\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.23338	−0.208311
$414$	0	0
$415$	1.55562i	0.0763624i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 18.9459i	− 0.925567i	−0.886471	−	0.462783i	$-0.846851\pi$
$419$	− 18.9459i	− 0.925567i	0.886471	−	0.462783i	$-0.153149\pi$
$420$	0	0
$421$	− 25.2598i	− 1.23109i	−0.788103	−	0.615544i	$-0.788936\pi$
$421$	− 25.2598i	− 1.23109i	0.788103	−	0.615544i	$-0.211064\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.3634i	0.939265i
$426$	0	0
$427$	−12.3006	−0.595268
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.5774	−1.81004	−0.905020	−	0.425369i	$-0.860144\pi$
$431$	−37.5774	−1.81004	−0.905020	+	0.425369i	$0.860144\pi$
$432$	0	0
$433$	21.1040	1.01419	0.507097	−	0.861889i	$-0.330719\pi$
$433$	21.1040	1.01419	0.507097	+	0.861889i	$0.330719\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.83129	0.231112
$438$	0	0
$439$	− 2.88405i	− 0.137648i	−0.997629	−	0.0688240i	$-0.978075\pi$
$439$	− 2.88405i	− 0.137648i	0.997629	−	0.0688240i	$-0.0219247\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.5978i	1.64379i	0.569640	+	0.821895i	$0.307083\pi$
$443$	34.5978i	1.64379i	−0.569640	+	0.821895i	$0.692917\pi$
$444$	0	0
$445$	− 0.920453i	− 0.0436337i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14.8661i	− 0.701575i	−0.936455	−	0.350787i	$-0.885914\pi$
$449$	− 14.8661i	− 0.701575i	0.936455	−	0.350787i	$-0.114086\pi$
$450$	0	0
$451$	−38.1603	−1.79690
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.729305	0.0341903
$456$	0	0
$457$	11.0037	0.514730	0.257365	−	0.966314i	$-0.417146\pi$
$457$	11.0037	0.514730	0.257365	+	0.966314i	$0.417146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.9687	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$461$	−32.9687	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$462$	0	0
$463$	12.3348i	0.573245i	0.958044	+	0.286622i	$0.0925325\pi$
$463$	12.3348i	0.573245i	−0.958044	+	0.286622i	$0.907468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.5264i	1.55142i	0.631091	+	0.775709i	$0.282607\pi$
$467$	33.5264i	1.55142i	−0.631091	+	0.775709i	$0.717393\pi$
$468$	0	0
$469$	− 5.74666i	− 0.265356i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.31840i	0.336501i
$474$	0	0
$475$	−27.1995	−1.24800
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.65791	−0.441281	−0.220641	−	0.975355i	$-0.570815\pi$
$479$	−9.65791	−0.441281	−0.220641	+	0.975355i	$0.570815\pi$
$480$	0	0
$481$	−34.9460	−1.59340
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.60019	−0.163476
$486$	0	0
$487$	− 25.1008i	− 1.13742i	−0.822537	−	0.568712i	$-0.807442\pi$
$487$	− 25.1008i	− 1.13742i	0.822537	−	0.568712i	$-0.192558\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 10.5524i	− 0.476224i	−0.971238	−	0.238112i	$-0.923471\pi$
$491$	− 10.5524i	− 0.476224i	0.971238	−	0.238112i	$-0.0765285\pi$
$492$	0	0
$493$	12.5315i	0.564391i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.1795i	− 0.456613i
$498$	0	0
$499$	11.6619	0.522059	0.261029	−	0.965331i	$-0.415938\pi$
$499$	11.6619	0.522059	0.261029	+	0.965331i	$0.415938\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.0202	1.29394	0.646972	−	0.762513i	$-0.276035\pi$
$503$	29.0202	1.29394	0.646972	+	0.762513i	$0.276035\pi$
$504$	0	0
$505$	−3.93849	−0.175260
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.0894	0.979096	0.489548	−	0.871976i	$-0.337162\pi$
$509$	22.0894	0.979096	0.489548	+	0.871976i	$0.337162\pi$
$510$	0	0
$511$	− 8.27618i	− 0.366116i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.23350i	− 0.0984199i
$516$	0	0
$517$	− 43.7502i	− 1.92413i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 14.5460i	− 0.637273i	−0.947877	−	0.318636i	$-0.896775\pi$
$521$	− 14.5460i	− 0.637273i	0.947877	−	0.318636i	$-0.103225\pi$
$522$	0	0
$523$	−31.6350	−1.38330	−0.691651	−	0.722232i	$-0.743116\pi$
$523$	−31.6350	−1.38330	−0.691651	+	0.722232i	$0.743116\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.246581	−0.0107412
$528$	0	0
$529$	−22.2247	−0.966291
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20.9365	−0.906862
$534$	0	0
$535$	1.02893i	0.0444844i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.42190i	− 0.276611i
$540$	0	0
$541$	− 22.7160i	− 0.976639i	−0.872665	−	0.488319i	$-0.837610\pi$
$541$	− 22.7160i	− 0.976639i	0.872665	−	0.488319i	$-0.162390\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.589532i	0.0252528i
$546$	0	0
$547$	22.2601	0.951774	0.475887	−	0.879506i	$-0.342127\pi$
$547$	22.2601	0.951774	0.475887	+	0.879506i	$0.342127\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.6028	−0.749904
$552$	0	0
$553$	4.86426	0.206849
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.99670	−0.126974	−0.0634872	−	0.997983i	$-0.520222\pi$
$557$	−2.99670	−0.126974	−0.0634872	+	0.997983i	$0.520222\pi$
$558$	0	0
$559$	4.01522i	0.169826i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.4913i	0.568589i	0.958737	+	0.284295i	$0.0917594\pi$
$563$	13.4913i	0.568589i	−0.958737	+	0.284295i	$0.908241\pi$
$564$	0	0
$565$	0.597296i	0.0251284i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.7445i	1.70810i	0.520191	+	0.854050i	$0.325861\pi$
$569$	40.7445i	1.70810i	−0.520191	+	0.854050i	$0.674139\pi$
$570$	0	0
$571$	−18.0110	−0.753739	−0.376869	−	0.926266i	$-0.622999\pi$
$571$	−18.0110	−0.753739	−0.376869	+	0.926266i	$0.622999\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.36483	−0.182026
$576$	0	0
$577$	29.3495	1.22184	0.610919	−	0.791693i	$-0.290800\pi$
$577$	29.3495	1.22184	0.610919	+	0.791693i	$0.290800\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.51539	0.311791
$582$	0	0
$583$	− 81.1242i	− 3.35982i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.1732i	0.461168i	0.973052	+	0.230584i	$0.0740637\pi$
$587$	11.1732i	0.461168i	−0.973052	+	0.230584i	$0.925936\pi$
$588$	0	0
$589$	− 0.346368i	− 0.0142718i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	29.9792i	1.23110i	0.788099	+	0.615549i	$0.211066\pi$
$593$	29.9792i	1.23110i	−0.788099	+	0.615549i	$0.788934\pi$
$594$	0	0
$595$	−0.808542	−0.0331470
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.38388	−0.260838	−0.130419	−	0.991459i	$-0.541632\pi$
$599$	−6.38388	−0.260838	−0.130419	+	0.991459i	$0.541632\pi$
$600$	0	0
$601$	−23.5445	−0.960402	−0.480201	−	0.877159i	$-0.659436\pi$
$601$	−23.5445	−0.960402	−0.480201	+	0.877159i	$0.659436\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.25958	−0.254488
$606$	0	0
$607$	33.9599i	1.37839i	0.724577	+	0.689194i	$0.242035\pi$
$607$	33.9599i	1.37839i	−0.724577	+	0.689194i	$0.757965\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 24.0034i	− 0.971076i
$612$	0	0
$613$	20.6042i	0.832197i	0.909320	+	0.416098i	$0.136603\pi$
$613$	20.6042i	0.832197i	−0.909320	+	0.416098i	$0.863397\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 37.4489i	− 1.50764i	−0.657083	−	0.753818i	$-0.728210\pi$
$617$	− 37.4489i	− 1.50764i	0.657083	−	0.753818i	$-0.271790\pi$
$618$	0	0
$619$	−14.2766	−0.573825	−0.286913	−	0.957957i	$-0.592629\pi$
$619$	−14.2766	−0.573825	−0.286913	+	0.957957i	$0.592629\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.44682	−0.178158
$624$	0	0
$625$	24.3592	0.974366
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38.7427	1.54477
$630$	0	0
$631$	7.00448i	0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$631$	7.00448i	0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.49672i	0.0593954i
$636$	0	0
$637$	− 3.52336i	− 0.139601i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 9.23092i	− 0.364599i	−0.983243	−	0.182300i	$-0.941646\pi$
$641$	− 9.23092i	− 0.364599i	0.983243	−	0.182300i	$-0.0583541\pi$
$642$	0	0
$643$	−1.48603	−0.0586031	−0.0293016	−	0.999571i	$-0.509328\pi$
$643$	−1.48603	−0.0586031	−0.0293016	+	0.999571i	$0.509328\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.2654	0.639457	0.319729	−	0.947509i	$-0.396408\pi$
$647$	16.2654	0.639457	0.319729	+	0.947509i	$0.396408\pi$
$648$	0	0
$649$	27.1863	1.06716
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.92310	−0.270922	−0.135461	−	0.990783i	$-0.543252\pi$
$653$	−6.92310	−0.270922	−0.135461	+	0.990783i	$0.543252\pi$
$654$	0	0
$655$	2.65003i	0.103545i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.3057i	− 0.440407i	−0.975454	−	0.220203i	$-0.929328\pi$
$659$	− 11.3057i	− 0.440407i	0.975454	−	0.220203i	$-0.0706721\pi$
$660$	0	0
$661$	− 18.9877i	− 0.738536i	−0.929323	−	0.369268i	$-0.879608\pi$
$661$	− 18.9877i	− 0.738536i	0.929323	−	0.369268i	$-0.120392\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.13574i	− 0.0440422i
$666$	0	0
$667$	−2.82481	−0.109377
$668$	0	0
$669$	0	0
$670$	0	0
$671$	78.9932	3.04950
$672$	0	0
$673$	17.6164	0.679061	0.339531	−	0.940595i	$-0.389732\pi$
$673$	17.6164	0.679061	0.339531	+	0.940595i	$0.389732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.42579	0.285396	0.142698	−	0.989766i	$-0.454422\pi$
$677$	7.42579	0.285396	0.142698	+	0.989766i	$0.454422\pi$
$678$	0	0
$679$	17.3929i	0.667480i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 22.8832i	− 0.875600i	−0.899072	−	0.437800i	$-0.855758\pi$
$683$	− 22.8832i	− 0.875600i	0.899072	−	0.437800i	$-0.144242\pi$
$684$	0	0
$685$	− 0.0753186i	− 0.00287778i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 44.5086i	− 1.69564i
$690$	0	0
$691$	−33.6439	−1.27987	−0.639936	−	0.768428i	$-0.721039\pi$
$691$	−33.6439	−1.27987	−0.639936	+	0.768428i	$0.721039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.92986	0.0732038
$696$	0	0
$697$	23.2112	0.879188
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−35.4474	−1.33883	−0.669414	−	0.742890i	$-0.733455\pi$
$701$	−35.4474	−1.33883	−0.669414	+	0.742890i	$0.733455\pi$
$702$	0	0
$703$	54.4212i	2.05253i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.0273i	0.715595i
$708$	0	0
$709$	− 8.71118i	− 0.327155i	−0.986530	−	0.163578i	$-0.947697\pi$
$709$	− 8.71118i	− 0.327155i	0.986530	−	0.163578i	$-0.0523034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.0555833i	− 0.00208161i
$714$	0	0
$715$	−4.68352	−0.175154
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.821108	−0.0306222	−0.0153111	−	0.999883i	$-0.504874\pi$
$719$	−0.821108	−0.0306222	−0.0153111	+	0.999883i	$0.504874\pi$
$720$	0	0
$721$	−10.7903	−0.401853
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.9032	0.590632
$726$	0	0
$727$	28.0509i	1.04035i	0.854059	+	0.520176i	$0.174134\pi$
$727$	28.0509i	1.04035i	−0.854059	+	0.520176i	$0.825866\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.45147i	− 0.164643i
$732$	0	0
$733$	21.9677i	0.811395i	0.914007	+	0.405697i	$0.132971\pi$
$733$	21.9677i	0.811395i	−0.914007	+	0.405697i	$0.867029\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.9045i	1.35939i
$738$	0	0
$739$	18.6044	0.684374	0.342187	−	0.939632i	$-0.388832\pi$
$739$	18.6044	0.684374	0.342187	+	0.939632i	$0.388832\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.0849	1.06702	0.533512	−	0.845793i	$-0.320872\pi$
$743$	29.0849	1.06702	0.533512	+	0.845793i	$0.320872\pi$
$744$	0	0
$745$	−0.998774	−0.0365922
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.97087	0.181632
$750$	0	0
$751$	− 14.4693i	− 0.527991i	−0.964524	−	0.263995i	$-0.914960\pi$
$751$	− 14.4693i	− 0.527991i	0.964524	−	0.263995i	$-0.0850403\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.64886i	− 0.0964018i
$756$	0	0
$757$	13.6988i	0.497891i	0.968517	+	0.248946i	$0.0800840\pi$
$757$	13.6988i	0.497891i	−0.968517	+	0.248946i	$0.919916\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 46.5579i	− 1.68772i	−0.536562	−	0.843861i	$-0.680277\pi$
$761$	− 46.5579i	− 1.68772i	0.536562	−	0.843861i	$-0.319723\pi$
$762$	0	0
$763$	2.84810	0.103108
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.9157	0.538575
$768$	0	0
$769$	50.6427	1.82622	0.913111	−	0.407711i	$-0.133673\pi$
$769$	50.6427	1.82622	0.913111	+	0.407711i	$0.133673\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.66928	0.239877	0.119939	−	0.992781i	$-0.461730\pi$
$773$	6.66928	0.239877	0.119939	+	0.992781i	$0.461730\pi$
$774$	0	0
$775$	0.312926i	0.0112406i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.6044i	1.16817i
$780$	0	0
$781$	65.3718i	2.33919i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3.21779i	− 0.114848i
$786$	0	0
$787$	0.759732	0.0270815	0.0135408	−	0.999908i	$-0.495690\pi$
$787$	0.759732	0.0270815	0.0135408	+	0.999908i	$0.495690\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.88561	0.102600
$792$	0	0
$793$	43.3394	1.53903
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.1758	0.927194	0.463597	−	0.886046i	$-0.346559\pi$
$797$	26.1758	0.927194	0.463597	+	0.886046i	$0.346559\pi$
$798$	0	0
$799$	26.6114i	0.941442i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	53.1488i	1.87558i
$804$	0	0
$805$	− 0.182258i	− 0.00642376i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 14.6876i	− 0.516388i	−0.966093	−	0.258194i	$-0.916873\pi$
$809$	− 14.6876i	− 0.516388i	0.966093	−	0.258194i	$-0.0831273\pi$
$810$	0	0
$811$	4.13912	0.145344	0.0726721	−	0.997356i	$-0.476847\pi$
$811$	4.13912	0.145344	0.0726721	+	0.997356i	$0.476847\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.82168	−0.133868
$816$	0	0
$817$	6.25289	0.218761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.7760	1.07409	0.537045	−	0.843554i	$-0.319540\pi$
$821$	30.7760	1.07409	0.537045	+	0.843554i	$0.319540\pi$
$822$	0	0
$823$	− 28.3750i	− 0.989090i	−0.869152	−	0.494545i	$-0.835335\pi$
$823$	− 28.3750i	− 0.989090i	0.869152	−	0.494545i	$-0.164665\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 26.2522i	− 0.912880i	−0.889754	−	0.456440i	$-0.849124\pi$
$827$	− 26.2522i	− 0.912880i	0.889754	−	0.456440i	$-0.150876\pi$
$828$	0	0
$829$	8.40833i	0.292033i	0.989282	+	0.146017i	$0.0466453\pi$
$829$	8.40833i	0.292033i	−0.989282	+	0.146017i	$0.953355\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.90616i	0.135340i
$834$	0	0
$835$	−2.59401	−0.0897695
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.3099	0.355936	0.177968	−	0.984036i	$-0.443048\pi$
$839$	10.3099	0.355936	0.177968	+	0.984036i	$0.443048\pi$
$840$	0	0
$841$	−18.7078	−0.645098
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.121287	0.00417239
$846$	0	0
$847$	30.2408i	1.03909i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.73324i	0.299372i
$852$	0	0
$853$	− 42.6992i	− 1.46199i	−0.682382	−	0.730996i	$-0.739056\pi$
$853$	− 42.6992i	− 1.46199i	0.682382	−	0.730996i	$-0.260944\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.06990i	− 0.173185i	−0.996244	−	0.0865923i	$-0.972402\pi$
$857$	− 5.06990i	− 0.173185i	0.996244	−	0.0865923i	$-0.0275977\pi$
$858$	0	0
$859$	−10.9220	−0.372654	−0.186327	−	0.982488i	$-0.559658\pi$
$859$	−10.9220	−0.372654	−0.186327	+	0.982488i	$0.559658\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1030	1.29704	0.648520	−	0.761197i	$-0.275388\pi$
$863$	38.1030	1.29704	0.648520	+	0.761197i	$0.275388\pi$
$864$	0	0
$865$	1.13674	0.0386502
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−31.2378	−1.05967
$870$	0	0
$871$	20.2476i	0.686062i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.06105i	0.0696760i
$876$	0	0
$877$	− 26.1674i	− 0.883610i	−0.897111	−	0.441805i	$-0.854338\pi$
$877$	− 26.1674i	− 0.883610i	0.897111	−	0.441805i	$-0.145662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 29.2903i	− 0.986817i	−0.869798	−	0.493408i	$-0.835751\pi$
$881$	− 29.2903i	− 0.986817i	0.869798	−	0.493408i	$-0.164249\pi$
$882$	0	0
$883$	49.0887	1.65197	0.825983	−	0.563696i	$-0.190621\pi$
$883$	49.0887	1.65197	0.825983	+	0.563696i	$0.190621\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.0617	0.673606	0.336803	−	0.941575i	$-0.390654\pi$
$887$	20.0617	0.673606	0.336803	+	0.941575i	$0.390654\pi$
$888$	0	0
$889$	7.23082	0.242514
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−37.3805	−1.25089
$894$	0	0
$895$	3.01855i	0.100899i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.202518i	0.00675434i
$900$	0	0
$901$	49.3443i	1.64390i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.57585i	− 0.118865i
$906$	0	0
$907$	−17.0142	−0.564948	−0.282474	−	0.959275i	$-0.591155\pi$
$907$	−17.0142	−0.564948	−0.282474	+	0.959275i	$0.591155\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.54462	0.283096	0.141548	−	0.989931i	$-0.454792\pi$
$911$	8.54462	0.283096	0.141548	+	0.989931i	$0.454792\pi$
$912$	0	0
$913$	−48.2631	−1.59727
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.8026	0.422780
$918$	0	0
$919$	− 6.49841i	− 0.214363i	−0.994239	−	0.107181i	$-0.965817\pi$
$919$	− 6.49841i	− 0.214363i	0.994239	−	0.107181i	$-0.0341826\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.8660i	1.18055i
$924$	0	0
$925$	− 49.1669i	− 1.61660i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.9216i	0.555181i	0.960699	+	0.277591i	$0.0895358\pi$
$929$	16.9216i	0.555181i	−0.960699	+	0.277591i	$0.910464\pi$
$930$	0	0
$931$	−5.48691	−0.179826
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.19237	0.169809
$936$	0	0
$937$	14.4249	0.471240	0.235620	−	0.971845i	$-0.424288\pi$
$937$	14.4249	0.471240	0.235620	+	0.971845i	$0.424288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−58.5364	−1.90823	−0.954116	−	0.299436i	$-0.903201\pi$
$941$	−58.5364	−1.90823	−0.954116	+	0.299436i	$0.903201\pi$
$942$	0	0
$943$	5.23218i	0.170383i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.3006i	− 0.497201i	−0.968606	−	0.248601i	$-0.920029\pi$
$947$	− 15.3006i	− 0.497201i	0.968606	−	0.248601i	$-0.0799707\pi$
$948$	0	0
$949$	29.1599i	0.946571i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 14.9133i	− 0.483089i	−0.970390	−	0.241544i	$-0.922346\pi$
$953$	− 14.9133i	− 0.483089i	0.970390	−	0.241544i	$-0.0776539\pi$
$954$	0	0
$955$	1.51068	0.0488845
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.363873	−0.0117501
$960$	0	0
$961$	30.9960	0.999871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.50680	0.0806966
$966$	0	0
$967$	− 27.1896i	− 0.874358i	−0.899375	−	0.437179i	$-0.855978\pi$
$967$	− 27.1896i	− 0.874358i	0.899375	−	0.437179i	$-0.144022\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 35.0892i	− 1.12607i	−0.826434	−	0.563034i	$-0.809634\pi$
$971$	− 35.0892i	− 1.12607i	0.826434	−	0.563034i	$-0.190366\pi$
$972$	0	0
$973$	− 9.32338i	− 0.298894i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.0438i	− 1.02517i	−0.858636	−	0.512586i	$-0.828688\pi$
$977$	− 32.0438i	− 1.02517i	0.858636	−	0.512586i	$-0.171312\pi$
$978$	0	0
$979$	28.5570	0.912687
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43.5719	−1.38973	−0.694864	−	0.719142i	$-0.744535\pi$
$983$	−43.5719	−1.38973	−0.694864	+	0.719142i	$0.744535\pi$
$984$	0	0
$985$	−3.60011	−0.114709
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.00343	0.0319073
$990$	0	0
$991$	− 15.2929i	− 0.485794i	−0.970052	−	0.242897i	$-0.921902\pi$
$991$	− 15.2929i	− 0.485794i	0.970052	−	0.242897i	$-0.0780978\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 0.636670i	− 0.0201838i
$996$	0	0
$997$	− 29.8050i	− 0.943933i	−0.881616	−	0.471967i	$-0.843544\pi$
$997$	− 29.8050i	− 0.943933i	0.881616	−	0.471967i	$-0.156456\pi$
$998$	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.5615.19		32
3.2	odd	2	inner	6048.2.j.c.5615.14		32
4.3	odd	2		1512.2.j.c.323.10	yes	32
8.3	odd	2	inner	6048.2.j.c.5615.13		32
8.5	even	2		1512.2.j.c.323.24	yes	32
12.11	even	2		1512.2.j.c.323.23	yes	32
24.5	odd	2		1512.2.j.c.323.9	✓	32
24.11	even	2	inner	6048.2.j.c.5615.20		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.9	✓	32	24.5	odd	2
1512.2.j.c.323.10	yes	32	4.3	odd	2
1512.2.j.c.323.23	yes	32	12.11	even	2
1512.2.j.c.323.24	yes	32	8.5	even	2
6048.2.j.c.5615.13		32	8.3	odd	2	inner
6048.2.j.c.5615.14		32	3.2	odd	2	inner
6048.2.j.c.5615.19		32	1.1	even	1	trivial
6048.2.j.c.5615.20		32	24.11	even	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)

\(f(q)\)	\(=\)	\(q+0.206991 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+0.206991 q^{5} -1.00000i q^{7} +6.42190i q^{11} +3.52336i q^{13} -3.90616i q^{17} +5.48691 q^{19} +0.880512 q^{23} -4.95715 q^{25} -3.20814 q^{29} -0.0631261i q^{31} -0.206991i q^{35} +9.91837i q^{37} +5.94221i q^{41} +1.13960 q^{43} -6.81266 q^{47} -1.00000 q^{49} -12.6324 q^{53} +1.32928i q^{55} -4.23338i q^{59} -12.3006i q^{61} +0.729305i q^{65} +5.74666 q^{67} +10.1795 q^{71} +8.27618 q^{73} +6.42190 q^{77} +4.86426i q^{79} +7.51539i q^{83} -0.808542i q^{85} -4.44682i q^{89} +3.52336 q^{91} +1.13574 q^{95} -17.3929 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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