LMFDB - Embedded newform 7488.2.a.cy.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(1,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$59.7919810335$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	7488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.622216 q^{5} +4.42864 q^{7} +5.80642 q^{11} -1.00000 q^{13} -2.00000 q^{17} +4.42864 q^{19} +8.85728 q^{23} -4.61285 q^{25} +2.00000 q^{29} -7.18421 q^{31} +2.75557 q^{35} -0.755569 q^{37} -3.37778 q^{41} +7.61285 q^{43} -1.80642 q^{47} +12.6128 q^{49} +4.75557 q^{53} +3.61285 q^{55} +11.0509 q^{59} +8.10171 q^{61} -0.622216 q^{65} -8.04149 q^{67} +7.05086 q^{71} +7.24443 q^{73} +25.7146 q^{77} -12.0000 q^{79} -3.05086 q^{83} -1.24443 q^{85} +1.86665 q^{89} -4.42864 q^{91} +2.75557 q^{95} -0.755569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} + 4 q^{11} - 3 q^{13} - 6 q^{17} + 13 q^{25} + 6 q^{29} - 8 q^{31} + 8 q^{35} - 2 q^{37} - 10 q^{41} - 4 q^{43} + 8 q^{47} + 11 q^{49} + 14 q^{53} - 16 q^{55} + 20 q^{59} - 2 q^{61} - 2 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 + 4 * q^11 - 3 * q^13 - 6 * q^17 + 13 * q^25 + 6 * q^29 - 8 * q^31 + 8 * q^35 - 2 * q^37 - 10 * q^41 - 4 * q^43 + 8 * q^47 + 11 * q^49 + 14 * q^53 - 16 * q^55 + 20 * q^59 - 2 * q^61 - 2 * q^65 + 16 * q^67 + 8 * q^71 + 22 * q^73 + 24 * q^77 - 36 * q^79 + 4 * q^83 - 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.622216	0.278263	0.139132	−	0.990274i	$-0.455569\pi$
$5$	0.622216	0.278263	0.139132	+	0.990274i	$0.455569\pi$
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.80642	1.75070	0.875351	−	0.483487i	$-0.160630\pi$
$11$	5.80642	1.75070	0.875351	+	0.483487i	$0.160630\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.42864	1.01600	0.508000	−	0.861357i	$-0.330385\pi$
$19$	4.42864	1.01600	0.508000	+	0.861357i	$0.330385\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.85728	1.84687	0.923435	−	0.383754i	$-0.125369\pi$
$23$	8.85728	1.84687	0.923435	+	0.383754i	$0.125369\pi$
$24$	0	0
$25$	−4.61285	−0.922570
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−7.18421	−1.29032	−0.645161	−	0.764047i	$-0.723210\pi$
$31$	−7.18421	−1.29032	−0.645161	+	0.764047i	$0.723210\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.75557	0.465776
$36$	0	0
$37$	−0.755569	−0.124215	−0.0621074	−	0.998069i	$-0.519782\pi$
$37$	−0.755569	−0.124215	−0.0621074	+	0.998069i	$0.519782\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.37778	−0.527521	−0.263761	−	0.964588i	$-0.584963\pi$
$41$	−3.37778	−0.527521	−0.263761	+	0.964588i	$0.584963\pi$
$42$	0	0
$43$	7.61285	1.16095	0.580474	−	0.814279i	$-0.302867\pi$
$43$	7.61285	1.16095	0.580474	+	0.814279i	$0.302867\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.80642	−0.263494	−0.131747	−	0.991283i	$-0.542059\pi$
$47$	−1.80642	−0.263494	−0.131747	+	0.991283i	$0.542059\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.75557	0.653228	0.326614	−	0.945158i	$-0.394092\pi$
$53$	4.75557	0.653228	0.326614	+	0.945158i	$0.394092\pi$
$54$	0	0
$55$	3.61285	0.487156
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.0509	1.43870	0.719349	−	0.694648i	$-0.244440\pi$
$59$	11.0509	1.43870	0.719349	+	0.694648i	$0.244440\pi$
$60$	0	0
$61$	8.10171	1.03732	0.518659	−	0.854981i	$-0.326431\pi$
$61$	8.10171	1.03732	0.518659	+	0.854981i	$0.326431\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.622216	−0.0771764
$66$	0	0
$67$	−8.04149	−0.982424	−0.491212	−	0.871040i	$-0.663446\pi$
$67$	−8.04149	−0.982424	−0.491212	+	0.871040i	$0.663446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.05086	0.836783	0.418391	−	0.908267i	$-0.362594\pi$
$71$	7.05086	0.836783	0.418391	+	0.908267i	$0.362594\pi$
$72$	0	0
$73$	7.24443	0.847897	0.423948	−	0.905686i	$-0.360644\pi$
$73$	7.24443	0.847897	0.423948	+	0.905686i	$0.360644\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25.7146	2.93045
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.05086	−0.334875	−0.167437	−	0.985883i	$-0.553549\pi$
$83$	−3.05086	−0.334875	−0.167437	+	0.985883i	$0.553549\pi$
$84$	0	0
$85$	−1.24443	−0.134978
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.86665	0.197864	0.0989321	−	0.995094i	$-0.468457\pi$
$89$	1.86665	0.197864	0.0989321	+	0.995094i	$0.468457\pi$
$90$	0	0
$91$	−4.42864	−0.464248
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.75557	0.282715
$96$	0	0
$97$	−0.755569	−0.0767164	−0.0383582	−	0.999264i	$-0.512213\pi$
$97$	−0.755569	−0.0767164	−0.0383582	+	0.999264i	$0.512213\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.7146	−1.56366	−0.781828	−	0.623494i	$-0.785713\pi$
$101$	−15.7146	−1.56366	−0.781828	+	0.623494i	$0.785713\pi$
$102$	0	0
$103$	−18.1017	−1.78361	−0.891807	−	0.452416i	$-0.850562\pi$
$103$	−18.1017	−1.78361	−0.891807	+	0.452416i	$0.850562\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3461	1.48357	0.741784	−	0.670639i	$-0.233980\pi$
$107$	15.3461	1.48357	0.741784	+	0.670639i	$0.233980\pi$
$108$	0	0
$109$	−15.7146	−1.50518	−0.752591	−	0.658488i	$-0.771196\pi$
$109$	−15.7146	−1.50518	−0.752591	+	0.658488i	$0.771196\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.4701	−1.36124	−0.680618	−	0.732639i	$-0.738288\pi$
$113$	−14.4701	−1.36124	−0.680618	+	0.732639i	$0.738288\pi$
$114$	0	0
$115$	5.51114	0.513916
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.85728	−0.811945
$120$	0	0
$121$	22.7146	2.06496
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.98126	−0.534981
$126$	0	0
$127$	−20.8573	−1.85078	−0.925392	−	0.379011i	$-0.876264\pi$
$127$	−20.8573	−1.85078	−0.925392	+	0.379011i	$0.876264\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.3684	−0.905893	−0.452946	−	0.891538i	$-0.649627\pi$
$131$	−10.3684	−0.905893	−0.452946	+	0.891538i	$0.649627\pi$
$132$	0	0
$133$	19.6128	1.70065
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.8479	1.69572	0.847861	−	0.530219i	$-0.177890\pi$
$137$	19.8479	1.69572	0.847861	+	0.530219i	$0.177890\pi$
$138$	0	0
$139$	−6.75557	−0.573000	−0.286500	−	0.958080i	$-0.592492\pi$
$139$	−6.75557	−0.573000	−0.286500	+	0.958080i	$0.592492\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.80642	−0.485558
$144$	0	0
$145$	1.24443	0.103344
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.13335	−0.174771	−0.0873855	−	0.996175i	$-0.527851\pi$
$149$	−2.13335	−0.174771	−0.0873855	+	0.996175i	$0.527851\pi$
$150$	0	0
$151$	1.67307	0.136153	0.0680763	−	0.997680i	$-0.478314\pi$
$151$	1.67307	0.136153	0.0680763	+	0.997680i	$0.478314\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.47013	−0.359049
$156$	0	0
$157$	−15.1240	−1.20703	−0.603513	−	0.797353i	$-0.706233\pi$
$157$	−15.1240	−1.20703	−0.603513	+	0.797353i	$0.706233\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	39.2257	3.09142
$162$	0	0
$163$	15.7748	1.23558	0.617788	−	0.786345i	$-0.288029\pi$
$163$	15.7748	1.23558	0.617788	+	0.786345i	$0.288029\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.31756	−0.566250	−0.283125	−	0.959083i	$-0.591371\pi$
$167$	−7.31756	−0.566250	−0.283125	+	0.959083i	$0.591371\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.51114	0.571061	0.285531	−	0.958370i	$-0.407830\pi$
$173$	7.51114	0.571061	0.285531	+	0.958370i	$0.407830\pi$
$174$	0	0
$175$	−20.4286	−1.54426
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.7556	1.10288	0.551441	−	0.834214i	$-0.314078\pi$
$179$	14.7556	1.10288	0.551441	+	0.834214i	$0.314078\pi$
$180$	0	0
$181$	−4.10171	−0.304878	−0.152439	−	0.988313i	$-0.548713\pi$
$181$	−4.10171	−0.304878	−0.152439	+	0.988313i	$0.548713\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.470127	−0.0345644
$186$	0	0
$187$	−11.6128	−0.849216
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.266706	0.0192982	0.00964909	−	0.999953i	$-0.496929\pi$
$191$	0.266706	0.0192982	0.00964909	+	0.999953i	$0.496929\pi$
$192$	0	0
$193$	−26.2034	−1.88616	−0.943082	−	0.332561i	$-0.892087\pi$
$193$	−26.2034	−1.88616	−0.943082	+	0.332561i	$0.892087\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.8479	1.12912	0.564558	−	0.825393i	$-0.309046\pi$
$197$	15.8479	1.12912	0.564558	+	0.825393i	$0.309046\pi$
$198$	0	0
$199$	10.3684	0.734998	0.367499	−	0.930024i	$-0.380214\pi$
$199$	10.3684	0.734998	0.367499	+	0.930024i	$0.380214\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.85728	0.621659
$204$	0	0
$205$	−2.10171	−0.146790
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.7146	1.77871
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.73683	0.323049
$216$	0	0
$217$	−31.8163	−2.15983
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	−4.69535	−0.314424	−0.157212	−	0.987565i	$-0.550251\pi$
$223$	−4.69535	−0.314424	−0.157212	+	0.987565i	$0.550251\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.80642	−0.385386	−0.192693	−	0.981259i	$-0.561722\pi$
$227$	−5.80642	−0.385386	−0.192693	+	0.981259i	$0.561722\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.24443	−0.474598	−0.237299	−	0.971437i	$-0.576262\pi$
$233$	−7.24443	−0.474598	−0.237299	+	0.971437i	$0.576262\pi$
$234$	0	0
$235$	−1.12399	−0.0733207
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.6637	−0.689778	−0.344889	−	0.938644i	$-0.612083\pi$
$239$	−10.6637	−0.689778	−0.344889	+	0.938644i	$0.612083\pi$
$240$	0	0
$241$	−5.73329	−0.369314	−0.184657	−	0.982803i	$-0.559117\pi$
$241$	−5.73329	−0.369314	−0.184657	+	0.982803i	$0.559117\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.84791	0.501385
$246$	0	0
$247$	−4.42864	−0.281788
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.6128	1.49043	0.745215	−	0.666824i	$-0.232347\pi$
$251$	23.6128	1.49043	0.745215	+	0.666824i	$0.232347\pi$
$252$	0	0
$253$	51.4291	3.23332
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2444	0.701408	0.350704	−	0.936486i	$-0.385942\pi$
$257$	11.2444	0.701408	0.350704	+	0.936486i	$0.385942\pi$
$258$	0	0
$259$	−3.34614	−0.207919
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.38715	−0.270523	−0.135262	−	0.990810i	$-0.543188\pi$
$263$	−4.38715	−0.270523	−0.135262	+	0.990810i	$0.543188\pi$
$264$	0	0
$265$	2.95899	0.181769
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.51114	0.457962	0.228981	−	0.973431i	$-0.426461\pi$
$269$	7.51114	0.457962	0.228981	+	0.973431i	$0.426461\pi$
$270$	0	0
$271$	−6.06022	−0.368132	−0.184066	−	0.982914i	$-0.558926\pi$
$271$	−6.06022	−0.368132	−0.184066	+	0.982914i	$0.558926\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−26.7841	−1.61514
$276$	0	0
$277$	−0.488863	−0.0293729	−0.0146865	−	0.999892i	$-0.504675\pi$
$277$	−0.488863	−0.0293729	−0.0146865	+	0.999892i	$0.504675\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.0923	−0.781024	−0.390512	−	0.920598i	$-0.627702\pi$
$281$	−13.0923	−0.781024	−0.390512	+	0.920598i	$0.627702\pi$
$282$	0	0
$283$	−23.3461	−1.38778	−0.693892	−	0.720079i	$-0.744106\pi$
$283$	−23.3461	−1.38778	−0.693892	+	0.720079i	$0.744106\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.9590	−0.883001
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.0923	0.764863	0.382431	−	0.923984i	$-0.375087\pi$
$293$	13.0923	0.764863	0.382431	+	0.923984i	$0.375087\pi$
$294$	0	0
$295$	6.87601	0.400337
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.85728	−0.512230
$300$	0	0
$301$	33.7146	1.94327
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.04101	0.288647
$306$	0	0
$307$	32.0415	1.82870	0.914352	−	0.404920i	$-0.132701\pi$
$307$	32.0415	1.82870	0.914352	+	0.404920i	$0.132701\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0830	−1.36562	−0.682810	−	0.730596i	$-0.739242\pi$
$311$	−24.0830	−1.36562	−0.682810	+	0.730596i	$0.739242\pi$
$312$	0	0
$313$	27.7146	1.56652	0.783260	−	0.621695i	$-0.213556\pi$
$313$	27.7146	1.56652	0.783260	+	0.621695i	$0.213556\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.0923	−0.960002	−0.480001	−	0.877268i	$-0.659364\pi$
$317$	−17.0923	−0.960002	−0.480001	+	0.877268i	$0.659364\pi$
$318$	0	0
$319$	11.6128	0.650195
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.85728	−0.492832
$324$	0	0
$325$	4.61285	0.255875
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	7.18421	0.394880	0.197440	−	0.980315i	$-0.436737\pi$
$331$	7.18421	0.394880	0.197440	+	0.980315i	$0.436737\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.00354	−0.273373
$336$	0	0
$337$	0.285442	0.0155490	0.00777451	−	0.999970i	$-0.497525\pi$
$337$	0.285442	0.0155490	0.00777451	+	0.999970i	$0.497525\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−41.7146	−2.25897
$342$	0	0
$343$	24.8573	1.34217
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.2034	−1.29931	−0.649654	−	0.760230i	$-0.725086\pi$
$347$	−24.2034	−1.29931	−0.649654	+	0.760230i	$0.725086\pi$
$348$	0	0
$349$	12.7556	0.682790	0.341395	−	0.939920i	$-0.389101\pi$
$349$	12.7556	0.682790	0.341395	+	0.939920i	$0.389101\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.0923	−1.12263	−0.561316	−	0.827602i	$-0.689705\pi$
$353$	−21.0923	−1.12263	−0.561316	+	0.827602i	$0.689705\pi$
$354$	0	0
$355$	4.38715	0.232846
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.1338	0.798733	0.399366	−	0.916791i	$-0.369230\pi$
$359$	15.1338	0.798733	0.399366	+	0.916791i	$0.369230\pi$
$360$	0	0
$361$	0.612848	0.0322551
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.50760	0.235938
$366$	0	0
$367$	−18.9590	−0.989651	−0.494826	−	0.868992i	$-0.664768\pi$
$367$	−18.9590	−0.989651	−0.494826	+	0.868992i	$0.664768\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	21.0607	1.09342
$372$	0	0
$373$	−2.97773	−0.154181	−0.0770904	−	0.997024i	$-0.524563\pi$
$373$	−2.97773	−0.154181	−0.0770904	+	0.997024i	$0.524563\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	14.1432	0.726487	0.363244	−	0.931694i	$-0.381669\pi$
$379$	14.1432	0.726487	0.363244	+	0.931694i	$0.381669\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.8064	1.31865	0.659323	−	0.751860i	$-0.270843\pi$
$383$	25.8064	1.31865	0.659323	+	0.751860i	$0.270843\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.9590	−0.657047	−0.328523	−	0.944496i	$-0.606551\pi$
$389$	−12.9590	−0.657047	−0.328523	+	0.944496i	$0.606551\pi$
$390$	0	0
$391$	−17.7146	−0.895864
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.46659	−0.375685
$396$	0	0
$397$	−3.24443	−0.162833	−0.0814167	−	0.996680i	$-0.525944\pi$
$397$	−3.24443	−0.162833	−0.0814167	+	0.996680i	$0.525944\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.6035	1.52826	0.764132	−	0.645059i	$-0.223167\pi$
$401$	30.6035	1.52826	0.764132	+	0.645059i	$0.223167\pi$
$402$	0	0
$403$	7.18421	0.357871
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.38715	−0.217463
$408$	0	0
$409$	−8.75557	−0.432935	−0.216468	−	0.976290i	$-0.569454\pi$
$409$	−8.75557	−0.432935	−0.216468	+	0.976290i	$0.569454\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	48.9403	2.40819
$414$	0	0
$415$	−1.89829	−0.0931834
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.8796	−1.16659	−0.583296	−	0.812259i	$-0.698237\pi$
$419$	−23.8796	−1.16659	−0.583296	+	0.812259i	$0.698237\pi$
$420$	0	0
$421$	8.28544	0.403808	0.201904	−	0.979405i	$-0.435287\pi$
$421$	8.28544	0.403808	0.201904	+	0.979405i	$0.435287\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.22570	0.447512
$426$	0	0
$427$	35.8796	1.73633
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.0098	0.674830	0.337415	−	0.941356i	$-0.390447\pi$
$431$	14.0098	0.674830	0.337415	+	0.941356i	$0.390447\pi$
$432$	0	0
$433$	27.1240	1.30350	0.651748	−	0.758436i	$-0.274036\pi$
$433$	27.1240	1.30350	0.651748	+	0.758436i	$0.274036\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	39.2257	1.87642
$438$	0	0
$439$	4.59057	0.219096	0.109548	−	0.993982i	$-0.465060\pi$
$439$	4.59057	0.219096	0.109548	+	0.993982i	$0.465060\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.3684	−0.492618	−0.246309	−	0.969191i	$-0.579218\pi$
$443$	−10.3684	−0.492618	−0.246309	+	0.969191i	$0.579218\pi$
$444$	0	0
$445$	1.16146	0.0550583
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.3368	−0.487823	−0.243911	−	0.969798i	$-0.578431\pi$
$449$	−10.3368	−0.487823	−0.243911	+	0.969798i	$0.578431\pi$
$450$	0	0
$451$	−19.6128	−0.923533
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.75557	−0.129183
$456$	0	0
$457$	5.52987	0.258677	0.129338	−	0.991601i	$-0.458715\pi$
$457$	5.52987	0.258677	0.129338	+	0.991601i	$0.458715\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.8666	−0.832133	−0.416066	−	0.909334i	$-0.636592\pi$
$461$	−17.8666	−0.832133	−0.416066	+	0.909334i	$0.636592\pi$
$462$	0	0
$463$	38.6766	1.79745	0.898727	−	0.438508i	$-0.144493\pi$
$463$	38.6766	1.79745	0.898727	+	0.438508i	$0.144493\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.1017	0.467451	0.233726	−	0.972303i	$-0.424908\pi$
$467$	10.1017	0.467451	0.233726	+	0.972303i	$0.424908\pi$
$468$	0	0
$469$	−35.6128	−1.64445
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.2034	2.03248
$474$	0	0
$475$	−20.4286	−0.937330
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.8671	1.41035	0.705177	−	0.709031i	$-0.250867\pi$
$479$	30.8671	1.41035	0.705177	+	0.709031i	$0.250867\pi$
$480$	0	0
$481$	0.755569	0.0344510
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.470127	−0.0213474
$486$	0	0
$487$	25.7560	1.16712	0.583559	−	0.812071i	$-0.301660\pi$
$487$	25.7560	1.16712	0.583559	+	0.812071i	$0.301660\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.8163	1.25533	0.627665	−	0.778483i	$-0.284011\pi$
$491$	27.8163	1.25533	0.627665	+	0.778483i	$0.284011\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.2257	1.40066
$498$	0	0
$499$	−0.815792	−0.0365199	−0.0182599	−	0.999833i	$-0.505813\pi$
$499$	−0.815792	−0.0365199	−0.0182599	+	0.999833i	$0.505813\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.8385	−1.19667	−0.598336	−	0.801245i	$-0.704171\pi$
$503$	−26.8385	−1.19667	−0.598336	+	0.801245i	$0.704171\pi$
$504$	0	0
$505$	−9.77784	−0.435108
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.60348	−0.292694	−0.146347	−	0.989233i	$-0.546752\pi$
$509$	−6.60348	−0.292694	−0.146347	+	0.989233i	$0.546752\pi$
$510$	0	0
$511$	32.0830	1.41927
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.2632	−0.496314
$516$	0	0
$517$	−10.4889	−0.461300
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.9813	−1.57637	−0.788184	−	0.615440i	$-0.788978\pi$
$521$	−35.9813	−1.57637	−0.788184	+	0.615440i	$0.788978\pi$
$522$	0	0
$523$	34.9590	1.52865	0.764325	−	0.644831i	$-0.223072\pi$
$523$	34.9590	1.52865	0.764325	+	0.644831i	$0.223072\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.3684	0.625898
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.37778	0.146308
$534$	0	0
$535$	9.54861	0.412822
$536$	0	0
$537$	0	0
$538$	0	0
$539$	73.2355	3.15448
$540$	0	0
$541$	−19.5111	−0.838849	−0.419425	−	0.907790i	$-0.637768\pi$
$541$	−19.5111	−0.838849	−0.419425	+	0.907790i	$0.637768\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.77784	−0.418837
$546$	0	0
$547$	4.26671	0.182431	0.0912156	−	0.995831i	$-0.470925\pi$
$547$	4.26671	0.182431	0.0912156	+	0.995831i	$0.470925\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.85728	0.377333
$552$	0	0
$553$	−53.1437	−2.25990
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.86665	0.248578	0.124289	−	0.992246i	$-0.460335\pi$
$557$	5.86665	0.248578	0.124289	+	0.992246i	$0.460335\pi$
$558$	0	0
$559$	−7.61285	−0.321989
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.4924	−1.15867	−0.579333	−	0.815091i	$-0.696687\pi$
$563$	−27.4924	−1.15867	−0.579333	+	0.815091i	$0.696687\pi$
$564$	0	0
$565$	−9.00354	−0.378782
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.4924	0.565631	0.282815	−	0.959174i	$-0.408732\pi$
$569$	13.4924	0.565631	0.282815	+	0.959174i	$0.408732\pi$
$570$	0	0
$571$	−23.8796	−0.999328	−0.499664	−	0.866219i	$-0.666543\pi$
$571$	−23.8796	−0.999328	−0.499664	+	0.866219i	$0.666543\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−40.8573	−1.70387
$576$	0	0
$577$	32.4514	1.35097	0.675485	−	0.737374i	$-0.263935\pi$
$577$	32.4514	1.35097	0.675485	+	0.737374i	$0.263935\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−13.5111	−0.560536
$582$	0	0
$583$	27.6128	1.14361
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.3363	−0.715546	−0.357773	−	0.933809i	$-0.616464\pi$
$587$	−17.3363	−0.715546	−0.357773	+	0.933809i	$0.616464\pi$
$588$	0	0
$589$	−31.8163	−1.31097
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.8069	0.772307	0.386153	−	0.922435i	$-0.373804\pi$
$593$	18.8069	0.772307	0.386153	+	0.922435i	$0.373804\pi$
$594$	0	0
$595$	−5.51114	−0.225935
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.8163	−1.29998	−0.649989	−	0.759944i	$-0.725226\pi$
$599$	−31.8163	−1.29998	−0.649989	+	0.759944i	$0.725226\pi$
$600$	0	0
$601$	−4.28544	−0.174807	−0.0874034	−	0.996173i	$-0.527857\pi$
$601$	−4.28544	−0.174807	−0.0874034	+	0.996173i	$0.527857\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.1334	0.574603
$606$	0	0
$607$	17.2444	0.699930	0.349965	−	0.936763i	$-0.386193\pi$
$607$	17.2444	0.699930	0.349965	+	0.936763i	$0.386193\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.80642	0.0730801
$612$	0	0
$613$	−23.9813	−0.968594	−0.484297	−	0.874904i	$-0.660925\pi$
$613$	−23.9813	−0.968594	−0.484297	+	0.874904i	$0.660925\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.40006	−0.257657	−0.128828	−	0.991667i	$-0.541122\pi$
$617$	−6.40006	−0.257657	−0.128828	+	0.991667i	$0.541122\pi$
$618$	0	0
$619$	8.04149	0.323215	0.161607	−	0.986855i	$-0.448332\pi$
$619$	8.04149	0.323215	0.161607	+	0.986855i	$0.448332\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.26671	0.331199
$624$	0	0
$625$	19.3426	0.773704
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.51114	0.0602530
$630$	0	0
$631$	23.1842	0.922949	0.461474	−	0.887154i	$-0.347321\pi$
$631$	23.1842	0.922949	0.461474	+	0.887154i	$0.347321\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.9777	−0.515005
$636$	0	0
$637$	−12.6128	−0.499739
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.51114	0.138682	0.0693408	−	0.997593i	$-0.477910\pi$
$641$	3.51114	0.138682	0.0693408	+	0.997593i	$0.477910\pi$
$642$	0	0
$643$	−49.7560	−1.96219	−0.981093	−	0.193535i	$-0.938005\pi$
$643$	−49.7560	−1.96219	−0.981093	+	0.193535i	$0.938005\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.9403	0.980503	0.490251	−	0.871581i	$-0.336905\pi$
$647$	24.9403	0.980503	0.490251	+	0.871581i	$0.336905\pi$
$648$	0	0
$649$	64.1659	2.51873
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.7146	0.458426	0.229213	−	0.973376i	$-0.426385\pi$
$653$	11.7146	0.458426	0.229213	+	0.973376i	$0.426385\pi$
$654$	0	0
$655$	−6.45139	−0.252077
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.51114	−0.0588656	−0.0294328	−	0.999567i	$-0.509370\pi$
$659$	−1.51114	−0.0588656	−0.0294328	+	0.999567i	$0.509370\pi$
$660$	0	0
$661$	24.1847	0.940675	0.470338	−	0.882487i	$-0.344132\pi$
$661$	24.1847	0.940675	0.470338	+	0.882487i	$0.344132\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.2034	0.473228
$666$	0	0
$667$	17.7146	0.685910
$668$	0	0
$669$	0	0
$670$	0	0
$671$	47.0420	1.81603
$672$	0	0
$673$	−8.83854	−0.340701	−0.170350	−	0.985384i	$-0.554490\pi$
$673$	−8.83854	−0.340701	−0.170350	+	0.985384i	$0.554490\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.2444	1.50829	0.754143	−	0.656710i	$-0.228053\pi$
$677$	39.2444	1.50829	0.754143	+	0.656710i	$0.228053\pi$
$678$	0	0
$679$	−3.34614	−0.128413
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.1561	−1.15389	−0.576946	−	0.816783i	$-0.695756\pi$
$683$	−30.1561	−1.15389	−0.576946	+	0.816783i	$0.695756\pi$
$684$	0	0
$685$	12.3497	0.471857
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.75557	−0.181173
$690$	0	0
$691$	−21.5526	−0.819900	−0.409950	−	0.912108i	$-0.634454\pi$
$691$	−21.5526	−0.819900	−0.409950	+	0.912108i	$0.634454\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.20342	−0.159445
$696$	0	0
$697$	6.75557	0.255885
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.755569	−0.0285374	−0.0142687	−	0.999898i	$-0.504542\pi$
$701$	−0.755569	−0.0285374	−0.0142687	+	0.999898i	$0.504542\pi$
$702$	0	0
$703$	−3.34614	−0.126202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−69.5941	−2.61736
$708$	0	0
$709$	1.22570	0.0460320	0.0230160	−	0.999735i	$-0.492673\pi$
$709$	1.22570	0.0460320	0.0230160	+	0.999735i	$0.492673\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−63.6325	−2.38306
$714$	0	0
$715$	−3.61285	−0.135113
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.3461	−0.423140	−0.211570	−	0.977363i	$-0.567858\pi$
$719$	−11.3461	−0.423140	−0.211570	+	0.977363i	$0.567858\pi$
$720$	0	0
$721$	−80.1659	−2.98554
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.22570	−0.342634
$726$	0	0
$727$	−49.0607	−1.81956	−0.909780	−	0.415090i	$-0.863750\pi$
$727$	−49.0607	−1.81956	−0.909780	+	0.415090i	$0.863750\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.2257	−0.563143
$732$	0	0
$733$	53.4291	1.97345	0.986725	−	0.162402i	$-0.0519241\pi$
$733$	53.4291	1.97345	0.986725	+	0.162402i	$0.0519241\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−46.6923	−1.71993
$738$	0	0
$739$	21.0192	0.773204	0.386602	−	0.922247i	$-0.373649\pi$
$739$	21.0192	0.773204	0.386602	+	0.922247i	$0.373649\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.56199	0.167363	0.0836816	−	0.996493i	$-0.473332\pi$
$743$	4.56199	0.167363	0.0836816	+	0.996493i	$0.473332\pi$
$744$	0	0
$745$	−1.32741	−0.0486324
$746$	0	0
$747$	0	0
$748$	0	0
$749$	67.9625	2.48330
$750$	0	0
$751$	−21.8983	−0.799080	−0.399540	−	0.916716i	$-0.630830\pi$
$751$	−21.8983	−0.799080	−0.399540	+	0.916716i	$0.630830\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.04101	0.0378863
$756$	0	0
$757$	36.8385	1.33892	0.669460	−	0.742848i	$-0.266526\pi$
$757$	36.8385	1.33892	0.669460	+	0.742848i	$0.266526\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.8069	−1.40675	−0.703375	−	0.710819i	$-0.748324\pi$
$761$	−38.8069	−1.40675	−0.703375	+	0.710819i	$0.748324\pi$
$762$	0	0
$763$	−69.5941	−2.51948
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.0509	−0.399023
$768$	0	0
$769$	24.6923	0.890426	0.445213	−	0.895425i	$-0.353128\pi$
$769$	24.6923	0.890426	0.445213	+	0.895425i	$0.353128\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.58120	0.272677	0.136338	−	0.990662i	$-0.456467\pi$
$773$	7.58120	0.272677	0.136338	+	0.990662i	$0.456467\pi$
$774$	0	0
$775$	33.1397	1.19041
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.9590	−0.535961
$780$	0	0
$781$	40.9403	1.46496
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.41038	−0.335871
$786$	0	0
$787$	13.0192	0.464085	0.232042	−	0.972706i	$-0.425459\pi$
$787$	13.0192	0.464085	0.232042	+	0.972706i	$0.425459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−64.0830	−2.27853
$792$	0	0
$793$	−8.10171	−0.287700
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.2444	0.823360	0.411680	−	0.911328i	$-0.364942\pi$
$797$	23.2444	0.823360	0.411680	+	0.911328i	$0.364942\pi$
$798$	0	0
$799$	3.61285	0.127813
$800$	0	0
$801$	0	0
$802$	0	0
$803$	42.0642	1.48441
$804$	0	0
$805$	24.4068	0.860228
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.9590	1.29941	0.649704	−	0.760187i	$-0.274893\pi$
$809$	36.9590	1.29941	0.649704	+	0.760187i	$0.274893\pi$
$810$	0	0
$811$	−8.54909	−0.300199	−0.150099	−	0.988671i	$-0.547959\pi$
$811$	−8.54909	−0.300199	−0.150099	+	0.988671i	$0.547959\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.81532	0.343816
$816$	0	0
$817$	33.7146	1.17952
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.07007	0.0722459	0.0361229	−	0.999347i	$-0.488499\pi$
$821$	2.07007	0.0722459	0.0361229	+	0.999347i	$0.488499\pi$
$822$	0	0
$823$	−40.5531	−1.41359	−0.706796	−	0.707417i	$-0.749860\pi$
$823$	−40.5531	−1.41359	−0.706796	+	0.707417i	$0.749860\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.4987	1.26918	0.634592	−	0.772847i	$-0.281168\pi$
$827$	36.4987	1.26918	0.634592	+	0.772847i	$0.281168\pi$
$828$	0	0
$829$	−40.8385	−1.41838	−0.709191	−	0.705017i	$-0.750939\pi$
$829$	−40.8385	−1.41838	−0.709191	+	0.705017i	$0.750939\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−25.2257	−0.874019
$834$	0	0
$835$	−4.55310	−0.157567
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.72345	0.0595001	0.0297500	−	0.999557i	$-0.490529\pi$
$839$	1.72345	0.0595001	0.0297500	+	0.999557i	$0.490529\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.622216	0.0214049
$846$	0	0
$847$	100.595	3.45647
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.69228	−0.229409
$852$	0	0
$853$	−29.4924	−1.00980	−0.504900	−	0.863178i	$-0.668471\pi$
$853$	−29.4924	−1.00980	−0.504900	+	0.863178i	$0.668471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.01874	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$857$	−8.01874	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$858$	0	0
$859$	−55.4291	−1.89122	−0.945609	−	0.325307i	$-0.894532\pi$
$859$	−55.4291	−1.89122	−0.945609	+	0.325307i	$0.894532\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.6227	−0.872207	−0.436103	−	0.899897i	$-0.643642\pi$
$863$	−25.6227	−0.872207	−0.436103	+	0.899897i	$0.643642\pi$
$864$	0	0
$865$	4.67355	0.158905
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−69.6771	−2.36363
$870$	0	0
$871$	8.04149	0.272475
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−26.4889	−0.895487
$876$	0	0
$877$	−28.1847	−0.951729	−0.475865	−	0.879519i	$-0.657865\pi$
$877$	−28.1847	−0.951729	−0.475865	+	0.879519i	$0.657865\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.52987	0.321070	0.160535	−	0.987030i	$-0.448678\pi$
$881$	9.52987	0.321070	0.160535	+	0.987030i	$0.448678\pi$
$882$	0	0
$883$	−17.8350	−0.600196	−0.300098	−	0.953908i	$-0.597019\pi$
$883$	−17.8350	−0.600196	−0.300098	+	0.953908i	$0.597019\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.73329	−0.259659	−0.129829	−	0.991536i	$-0.541443\pi$
$887$	−7.73329	−0.259659	−0.129829	+	0.991536i	$0.541443\pi$
$888$	0	0
$889$	−92.3694	−3.09797
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	9.18115	0.306892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.3684	−0.479214
$900$	0	0
$901$	−9.51114	−0.316862
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.55215	−0.0848363
$906$	0	0
$907$	36.0830	1.19812	0.599058	−	0.800706i	$-0.295542\pi$
$907$	36.0830	1.19812	0.599058	+	0.800706i	$0.295542\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	58.9215	1.95216	0.976078	−	0.217418i	$-0.0697636\pi$
$911$	58.9215	1.95216	0.976078	+	0.217418i	$0.0697636\pi$
$912$	0	0
$913$	−17.7146	−0.586266
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−45.9180	−1.51635
$918$	0	0
$919$	12.5334	0.413439	0.206720	−	0.978400i	$-0.433721\pi$
$919$	12.5334	0.413439	0.206720	+	0.978400i	$0.433721\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.05086	−0.232082
$924$	0	0
$925$	3.48532	0.114597
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.0291	1.47736	0.738678	−	0.674059i	$-0.235451\pi$
$929$	45.0291	1.47736	0.738678	+	0.674059i	$0.235451\pi$
$930$	0	0
$931$	55.8578	1.83066
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.22570	−0.236306
$936$	0	0
$937$	0.876015	0.0286182	0.0143091	−	0.999898i	$-0.495445\pi$
$937$	0.876015	0.0286182	0.0143091	+	0.999898i	$0.495445\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.888922	0.0289780	0.0144890	−	0.999895i	$-0.495388\pi$
$941$	0.888922	0.0289780	0.0144890	+	0.999895i	$0.495388\pi$
$942$	0	0
$943$	−29.9180	−0.974263
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.9911	1.42952	0.714759	−	0.699370i	$-0.246536\pi$
$947$	43.9911	1.42952	0.714759	+	0.699370i	$0.246536\pi$
$948$	0	0
$949$	−7.24443	−0.235164
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.4701	0.598306	0.299153	−	0.954205i	$-0.403296\pi$
$953$	18.4701	0.598306	0.299153	+	0.954205i	$0.403296\pi$
$954$	0	0
$955$	0.165949	0.00536998
$956$	0	0
$957$	0	0
$958$	0	0
$959$	87.8992	2.83841
$960$	0	0
$961$	20.6128	0.664931
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.3042	−0.524850
$966$	0	0
$967$	50.2895	1.61720	0.808600	−	0.588359i	$-0.200226\pi$
$967$	50.2895	1.61720	0.808600	+	0.588359i	$0.200226\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.2257	−1.13045	−0.565223	−	0.824938i	$-0.691210\pi$
$971$	−35.2257	−1.13045	−0.565223	+	0.824938i	$0.691210\pi$
$972$	0	0
$973$	−29.9180	−0.959126
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.8069	1.88140	0.940700	−	0.339240i	$-0.110170\pi$
$977$	58.8069	1.88140	0.940700	+	0.339240i	$0.110170\pi$
$978$	0	0
$979$	10.8385	0.346401
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47.2168	−1.50598	−0.752991	−	0.658031i	$-0.771390\pi$
$983$	−47.2168	−1.50598	−0.752991	+	0.658031i	$0.771390\pi$
$984$	0	0
$985$	9.86082	0.314192
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.4291	2.14412
$990$	0	0
$991$	−35.4924	−1.12745	−0.563727	−	0.825961i	$-0.690633\pi$
$991$	−35.4924	−1.12745	−0.563727	+	0.825961i	$0.690633\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.45139	0.204523
$996$	0	0
$997$	−49.4291	−1.56544	−0.782718	−	0.622377i	$-0.786167\pi$
$997$	−49.4291	−1.56544	−0.782718	+	0.622377i	$0.786167\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.a.cy.1.2		3
3.2	odd	2		2496.2.a.bk.1.2		3
4.3	odd	2		7488.2.a.cx.1.2		3
8.3	odd	2		3744.2.a.ba.1.2		3
8.5	even	2		3744.2.a.z.1.2		3
12.11	even	2		2496.2.a.bl.1.2		3
24.5	odd	2		1248.2.a.p.1.2	yes	3
24.11	even	2		1248.2.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.o.1.2	✓	3	24.11	even	2
1248.2.a.p.1.2	yes	3	24.5	odd	2
2496.2.a.bk.1.2		3	3.2	odd	2
2496.2.a.bl.1.2		3	12.11	even	2
3744.2.a.z.1.2		3	8.5	even	2
3744.2.a.ba.1.2		3	8.3	odd	2
7488.2.a.cx.1.2		3	4.3	odd	2
7488.2.a.cy.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

\(f(q)\)	\(=\)	\(q+0.622216 q^{5} +4.42864 q^{7} +5.80642 q^{11} -1.00000 q^{13} -2.00000 q^{17} +4.42864 q^{19} +8.85728 q^{23} -4.61285 q^{25} +2.00000 q^{29} -7.18421 q^{31} +2.75557 q^{35} -0.755569 q^{37} -3.37778 q^{41} +7.61285 q^{43} -1.80642 q^{47} +12.6128 q^{49} +4.75557 q^{53} +3.61285 q^{55} +11.0509 q^{59} +8.10171 q^{61} -0.622216 q^{65} -8.04149 q^{67} +7.05086 q^{71} +7.24443 q^{73} +25.7146 q^{77} -12.0000 q^{79} -3.05086 q^{83} -1.24443 q^{85} +1.86665 q^{89} -4.42864 q^{91} +2.75557 q^{95} -0.755569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} + 4 q^{11} - 3 q^{13} - 6 q^{17} + 13 q^{25} + 6 q^{29} - 8 q^{31} + 8 q^{35} - 2 q^{37} - 10 q^{41} - 4 q^{43} + 8 q^{47} + 11 q^{49} + 14 q^{53} - 16 q^{55} + 20 q^{59} - 2 q^{61} - 2 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 + 4 * q^11 - 3 * q^13 - 6 * q^17 + 13 * q^25 + 6 * q^29 - 8 * q^31 + 8 * q^35 - 2 * q^37 - 10 * q^41 - 4 * q^43 + 8 * q^47 + 11 * q^49 + 14 * q^53 - 16 * q^55 + 20 * q^59 - 2 * q^61 - 2 * q^65 + 16 * q^67 + 8 * q^71 + 22 * q^73 + 24 * q^77 - 36 * q^79 + 4 * q^83 - 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

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Database

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