LMFDB - Embedded newform 1248.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(1,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.96533017226$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	1248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.622216 q^{5} -4.42864 q^{7} +1.00000 q^{9} -5.80642 q^{11} +1.00000 q^{13} -0.622216 q^{15} +2.00000 q^{17} +4.42864 q^{19} +4.42864 q^{21} +8.85728 q^{23} -4.61285 q^{25} -1.00000 q^{27} +2.00000 q^{29} +7.18421 q^{31} +5.80642 q^{33} -2.75557 q^{35} +0.755569 q^{37} -1.00000 q^{39} +3.37778 q^{41} +7.61285 q^{43} +0.622216 q^{45} -1.80642 q^{47} +12.6128 q^{49} -2.00000 q^{51} +4.75557 q^{53} -3.61285 q^{55} -4.42864 q^{57} -11.0509 q^{59} -8.10171 q^{61} -4.42864 q^{63} +0.622216 q^{65} -8.04149 q^{67} -8.85728 q^{69} +7.05086 q^{71} +7.24443 q^{73} +4.61285 q^{75} +25.7146 q^{77} +12.0000 q^{79} +1.00000 q^{81} +3.05086 q^{83} +1.24443 q^{85} -2.00000 q^{87} -1.86665 q^{89} -4.42864 q^{91} -7.18421 q^{93} +2.75557 q^{95} -0.755569 q^{97} -5.80642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - 4 q^{11} + 3 q^{13} - 2 q^{15} + 6 q^{17} + 13 q^{25} - 3 q^{27} + 6 q^{29} + 8 q^{31} + 4 q^{33} - 8 q^{35} + 2 q^{37} - 3 q^{39} + 10 q^{41} - 4 q^{43} + 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 - 4 * q^11 + 3 * q^13 - 2 * q^15 + 6 * q^17 + 13 * q^25 - 3 * q^27 + 6 * q^29 + 8 * q^31 + 4 * q^33 - 8 * q^35 + 2 * q^37 - 3 * q^39 + 10 * q^41 - 4 * q^43 + 2 * q^45 + 8 * q^47 + 11 * q^49 - 6 * q^51 + 14 * q^53 + 16 * q^55 - 20 * q^59 + 2 * q^61 + 2 * q^65 + 16 * q^67 + 8 * q^71 + 22 * q^73 - 13 * q^75 + 24 * q^77 + 36 * q^79 + 3 * q^81 - 4 * q^83 + 4 * q^85 - 6 * q^87 - 6 * q^89 - 8 * q^93 + 8 * q^95 - 2 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	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	$-0.815654\pi$
$7$	−4.42864	−1.67387	−0.836934	+	0.547304i	$0.815654\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.80642	−1.75070	−0.875351	−	0.483487i	$-0.839370\pi$
$11$	−5.80642	−1.75070	−0.875351	+	0.483487i	$0.839370\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.622216	−0.160655
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	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
$21$	4.42864	0.966408
$22$	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
$27$	−1.00000	−0.192450
$28$	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	$-0.276790\pi$
$31$	7.18421	1.29032	0.645161	+	0.764047i	$0.276790\pi$
$32$	0	0
$33$	5.80642	1.01077
$34$	0	0
$35$	−2.75557	−0.465776
$36$	0	0
$37$	0.755569	0.124215	0.0621074	−	0.998069i	$-0.480218\pi$
$37$	0.755569	0.124215	0.0621074	+	0.998069i	$0.480218\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.37778	0.527521	0.263761	−	0.964588i	$-0.415037\pi$
$41$	3.37778	0.527521	0.263761	+	0.964588i	$0.415037\pi$
$42$	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
$45$	0.622216	0.0927544
$46$	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
$51$	−2.00000	−0.280056
$52$	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
$57$	−4.42864	−0.586588
$58$	0	0
$59$	−11.0509	−1.43870	−0.719349	−	0.694648i	$-0.755560\pi$
$59$	−11.0509	−1.43870	−0.719349	+	0.694648i	$0.755560\pi$
$60$	0	0
$61$	−8.10171	−1.03732	−0.518659	−	0.854981i	$-0.673569\pi$
$61$	−8.10171	−1.03732	−0.518659	+	0.854981i	$0.673569\pi$
$62$	0	0
$63$	−4.42864	−0.557956
$64$	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
$69$	−8.85728	−1.06629
$70$	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
$75$	4.61285	0.532646
$76$	0	0
$77$	25.7146	2.93045
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.05086	0.334875	0.167437	−	0.985883i	$-0.446451\pi$
$83$	3.05086	0.334875	0.167437	+	0.985883i	$0.446451\pi$
$84$	0	0
$85$	1.24443	0.134978
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−1.86665	−0.197864	−0.0989321	−	0.995094i	$-0.531543\pi$
$89$	−1.86665	−0.197864	−0.0989321	+	0.995094i	$0.531543\pi$
$90$	0	0
$91$	−4.42864	−0.464248
$92$	0	0
$93$	−7.18421	−0.744968
$94$	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
$99$	−5.80642	−0.583568
$100$	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	$-0.149438\pi$
$103$	18.1017	1.78361	0.891807	+	0.452416i	$0.149438\pi$
$104$	0	0
$105$	2.75557	0.268916
$106$	0	0
$107$	−15.3461	−1.48357	−0.741784	−	0.670639i	$-0.766020\pi$
$107$	−15.3461	−1.48357	−0.741784	+	0.670639i	$0.766020\pi$
$108$	0	0
$109$	15.7146	1.50518	0.752591	−	0.658488i	$-0.228804\pi$
$109$	15.7146	1.50518	0.752591	+	0.658488i	$0.228804\pi$
$110$	0	0
$111$	−0.755569	−0.0717154
$112$	0	0
$113$	14.4701	1.36124	0.680618	−	0.732639i	$-0.261712\pi$
$113$	14.4701	1.36124	0.680618	+	0.732639i	$0.261712\pi$
$114$	0	0
$115$	5.51114	0.513916
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−8.85728	−0.811945
$120$	0	0
$121$	22.7146	2.06496
$122$	0	0
$123$	−3.37778	−0.304565
$124$	0	0
$125$	−5.98126	−0.534981
$126$	0	0
$127$	20.8573	1.85078	0.925392	−	0.379011i	$-0.123736\pi$
$127$	20.8573	1.85078	0.925392	+	0.379011i	$0.123736\pi$
$128$	0	0
$129$	−7.61285	−0.670274
$130$	0	0
$131$	10.3684	0.905893	0.452946	−	0.891538i	$-0.350373\pi$
$131$	10.3684	0.905893	0.452946	+	0.891538i	$0.350373\pi$
$132$	0	0
$133$	−19.6128	−1.70065
$134$	0	0
$135$	−0.622216	−0.0535518
$136$	0	0
$137$	−19.8479	−1.69572	−0.847861	−	0.530219i	$-0.822110\pi$
$137$	−19.8479	−1.69572	−0.847861	+	0.530219i	$0.822110\pi$
$138$	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
$141$	1.80642	0.152128
$142$	0	0
$143$	−5.80642	−0.485558
$144$	0	0
$145$	1.24443	0.103344
$146$	0	0
$147$	−12.6128	−1.04029
$148$	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	$-0.521686\pi$
$151$	−1.67307	−0.136153	−0.0680763	+	0.997680i	$0.521686\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	4.47013	0.359049
$156$	0	0
$157$	15.1240	1.20703	0.603513	−	0.797353i	$-0.293767\pi$
$157$	15.1240	1.20703	0.603513	+	0.797353i	$0.293767\pi$
$158$	0	0
$159$	−4.75557	−0.377141
$160$	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
$165$	3.61285	0.281260
$166$	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
$171$	4.42864	0.338667
$172$	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
$177$	11.0509	0.830633
$178$	0	0
$179$	−14.7556	−1.10288	−0.551441	−	0.834214i	$-0.685922\pi$
$179$	−14.7556	−1.10288	−0.551441	+	0.834214i	$0.685922\pi$
$180$	0	0
$181$	4.10171	0.304878	0.152439	−	0.988313i	$-0.451287\pi$
$181$	4.10171	0.304878	0.152439	+	0.988313i	$0.451287\pi$
$182$	0	0
$183$	8.10171	0.598896
$184$	0	0
$185$	0.470127	0.0345644
$186$	0	0
$187$	−11.6128	−0.849216
$188$	0	0
$189$	4.42864	0.322136
$190$	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
$195$	−0.622216	−0.0445578
$196$	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	$-0.619786\pi$
$199$	−10.3684	−0.734998	−0.367499	+	0.930024i	$0.619786\pi$
$200$	0	0
$201$	8.04149	0.567203
$202$	0	0
$203$	−8.85728	−0.621659
$204$	0	0
$205$	2.10171	0.146790
$206$	0	0
$207$	8.85728	0.615623
$208$	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
$213$	−7.05086	−0.483117
$214$	0	0
$215$	4.73683	0.323049
$216$	0	0
$217$	−31.8163	−2.15983
$218$	0	0
$219$	−7.24443	−0.489533
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	4.69535	0.314424	0.157212	−	0.987565i	$-0.449749\pi$
$223$	4.69535	0.314424	0.157212	+	0.987565i	$0.449749\pi$
$224$	0	0
$225$	−4.61285	−0.307523
$226$	0	0
$227$	5.80642	0.385386	0.192693	−	0.981259i	$-0.438278\pi$
$227$	5.80642	0.385386	0.192693	+	0.981259i	$0.438278\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−25.7146	−1.69189
$232$	0	0
$233$	7.24443	0.474598	0.237299	−	0.971437i	$-0.423738\pi$
$233$	7.24443	0.474598	0.237299	+	0.971437i	$0.423738\pi$
$234$	0	0
$235$	−1.12399	−0.0733207
$236$	0	0
$237$	−12.0000	−0.779484
$238$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.84791	0.501385
$246$	0	0
$247$	4.42864	0.281788
$248$	0	0
$249$	−3.05086	−0.193340
$250$	0	0
$251$	−23.6128	−1.49043	−0.745215	−	0.666824i	$-0.767653\pi$
$251$	−23.6128	−1.49043	−0.745215	+	0.666824i	$0.767653\pi$
$252$	0	0
$253$	−51.4291	−3.23332
$254$	0	0
$255$	−1.24443	−0.0779293
$256$	0	0
$257$	−11.2444	−0.701408	−0.350704	−	0.936486i	$-0.614058\pi$
$257$	−11.2444	−0.701408	−0.350704	+	0.936486i	$0.614058\pi$
$258$	0	0
$259$	−3.34614	−0.207919
$260$	0	0
$261$	2.00000	0.123797
$262$	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
$267$	1.86665	0.114237
$268$	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	$-0.441074\pi$
$271$	6.06022	0.368132	0.184066	+	0.982914i	$0.441074\pi$
$272$	0	0
$273$	4.42864	0.268033
$274$	0	0
$275$	26.7841	1.61514
$276$	0	0
$277$	0.488863	0.0293729	0.0146865	−	0.999892i	$-0.495325\pi$
$277$	0.488863	0.0293729	0.0146865	+	0.999892i	$0.495325\pi$
$278$	0	0
$279$	7.18421	0.430107
$280$	0	0
$281$	13.0923	0.781024	0.390512	−	0.920598i	$-0.372298\pi$
$281$	13.0923	0.781024	0.390512	+	0.920598i	$0.372298\pi$
$282$	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
$285$	−2.75557	−0.163226
$286$	0	0
$287$	−14.9590	−0.883001
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0.755569	0.0442922
$292$	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
$297$	5.80642	0.336923
$298$	0	0
$299$	8.85728	0.512230
$300$	0	0
$301$	−33.7146	−1.94327
$302$	0	0
$303$	15.7146	0.902778
$304$	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
$309$	−18.1017	−1.02977
$310$	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
$315$	−2.75557	−0.155259
$316$	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
$321$	15.3461	0.856538
$322$	0	0
$323$	8.85728	0.492832
$324$	0	0
$325$	−4.61285	−0.255875
$326$	0	0
$327$	−15.7146	−0.869017
$328$	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
$333$	0.755569	0.0414049
$334$	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
$339$	−14.4701	−0.785909
$340$	0	0
$341$	−41.7146	−2.25897
$342$	0	0
$343$	−24.8573	−1.34217
$344$	0	0
$345$	−5.51114	−0.296710
$346$	0	0
$347$	24.2034	1.29931	0.649654	−	0.760230i	$-0.274914\pi$
$347$	24.2034	1.29931	0.649654	+	0.760230i	$0.274914\pi$
$348$	0	0
$349$	−12.7556	−0.682790	−0.341395	−	0.939920i	$-0.610899\pi$
$349$	−12.7556	−0.682790	−0.341395	+	0.939920i	$0.610899\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	21.0923	1.12263	0.561316	−	0.827602i	$-0.310295\pi$
$353$	21.0923	1.12263	0.561316	+	0.827602i	$0.310295\pi$
$354$	0	0
$355$	4.38715	0.232846
$356$	0	0
$357$	8.85728	0.468777
$358$	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
$363$	−22.7146	−1.19221
$364$	0	0
$365$	4.50760	0.235938
$366$	0	0
$367$	18.9590	0.989651	0.494826	−	0.868992i	$-0.335232\pi$
$367$	18.9590	0.989651	0.494826	+	0.868992i	$0.335232\pi$
$368$	0	0
$369$	3.37778	0.175840
$370$	0	0
$371$	−21.0607	−1.09342
$372$	0	0
$373$	2.97773	0.154181	0.0770904	−	0.997024i	$-0.475437\pi$
$373$	2.97773	0.154181	0.0770904	+	0.997024i	$0.475437\pi$
$374$	0	0
$375$	5.98126	0.308871
$376$	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
$381$	−20.8573	−1.06855
$382$	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
$387$	7.61285	0.386983
$388$	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
$393$	−10.3684	−0.523017
$394$	0	0
$395$	7.46659	0.375685
$396$	0	0
$397$	3.24443	0.162833	0.0814167	−	0.996680i	$-0.474056\pi$
$397$	3.24443	0.162833	0.0814167	+	0.996680i	$0.474056\pi$
$398$	0	0
$399$	19.6128	0.981870
$400$	0	0
$401$	−30.6035	−1.52826	−0.764132	−	0.645059i	$-0.776833\pi$
$401$	−30.6035	−1.52826	−0.764132	+	0.645059i	$0.776833\pi$
$402$	0	0
$403$	7.18421	0.357871
$404$	0	0
$405$	0.622216	0.0309181
$406$	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
$411$	19.8479	0.979025
$412$	0	0
$413$	48.9403	2.40819
$414$	0	0
$415$	1.89829	0.0931834
$416$	0	0
$417$	6.75557	0.330822
$418$	0	0
$419$	23.8796	1.16659	0.583296	−	0.812259i	$-0.301763\pi$
$419$	23.8796	1.16659	0.583296	+	0.812259i	$0.301763\pi$
$420$	0	0
$421$	−8.28544	−0.403808	−0.201904	−	0.979405i	$-0.564713\pi$
$421$	−8.28544	−0.403808	−0.201904	+	0.979405i	$0.564713\pi$
$422$	0	0
$423$	−1.80642	−0.0878313
$424$	0	0
$425$	−9.22570	−0.447512
$426$	0	0
$427$	35.8796	1.73633
$428$	0	0
$429$	5.80642	0.280337
$430$	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
$435$	−1.24443	−0.0596659
$436$	0	0
$437$	39.2257	1.87642
$438$	0	0
$439$	−4.59057	−0.219096	−0.109548	−	0.993982i	$-0.534940\pi$
$439$	−4.59057	−0.219096	−0.109548	+	0.993982i	$0.534940\pi$
$440$	0	0
$441$	12.6128	0.600612
$442$	0	0
$443$	10.3684	0.492618	0.246309	−	0.969191i	$-0.420782\pi$
$443$	10.3684	0.492618	0.246309	+	0.969191i	$0.420782\pi$
$444$	0	0
$445$	−1.16146	−0.0550583
$446$	0	0
$447$	2.13335	0.100904
$448$	0	0
$449$	10.3368	0.487823	0.243911	−	0.969798i	$-0.421569\pi$
$449$	10.3368	0.487823	0.243911	+	0.969798i	$0.421569\pi$
$450$	0	0
$451$	−19.6128	−0.923533
$452$	0	0
$453$	1.67307	0.0786077
$454$	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
$459$	−2.00000	−0.0933520
$460$	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	$-0.855507\pi$
$463$	−38.6766	−1.79745	−0.898727	+	0.438508i	$0.855507\pi$
$464$	0	0
$465$	−4.47013	−0.207297
$466$	0	0
$467$	−10.1017	−0.467451	−0.233726	−	0.972303i	$-0.575092\pi$
$467$	−10.1017	−0.467451	−0.233726	+	0.972303i	$0.575092\pi$
$468$	0	0
$469$	35.6128	1.64445
$470$	0	0
$471$	−15.1240	−0.696876
$472$	0	0
$473$	−44.2034	−2.03248
$474$	0	0
$475$	−20.4286	−0.937330
$476$	0	0
$477$	4.75557	0.217743
$478$	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
$483$	39.2257	1.78483
$484$	0	0
$485$	−0.470127	−0.0213474
$486$	0	0
$487$	−25.7560	−1.16712	−0.583559	−	0.812071i	$-0.698340\pi$
$487$	−25.7560	−1.16712	−0.583559	+	0.812071i	$0.698340\pi$
$488$	0	0
$489$	−15.7748	−0.713360
$490$	0	0
$491$	−27.8163	−1.25533	−0.627665	−	0.778483i	$-0.715989\pi$
$491$	−27.8163	−1.25533	−0.627665	+	0.778483i	$0.715989\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−3.61285	−0.162385
$496$	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
$501$	7.31756	0.326925
$502$	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
$507$	−1.00000	−0.0444116
$508$	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
$513$	−4.42864	−0.195529
$514$	0	0
$515$	11.2632	0.496314
$516$	0	0
$517$	10.4889	0.461300
$518$	0	0
$519$	−7.51114	−0.329702
$520$	0	0
$521$	35.9813	1.57637	0.788184	−	0.615440i	$-0.211022\pi$
$521$	35.9813	1.57637	0.788184	+	0.615440i	$0.211022\pi$
$522$	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
$525$	−20.4286	−0.891579
$526$	0	0
$527$	14.3684	0.625898
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	−11.0509	−0.479566
$532$	0	0
$533$	3.37778	0.146308
$534$	0	0
$535$	−9.54861	−0.412822
$536$	0	0
$537$	14.7556	0.636750
$538$	0	0
$539$	−73.2355	−3.15448
$540$	0	0
$541$	19.5111	0.838849	0.419425	−	0.907790i	$-0.362232\pi$
$541$	19.5111	0.838849	0.419425	+	0.907790i	$0.362232\pi$
$542$	0	0
$543$	−4.10171	−0.176021
$544$	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
$549$	−8.10171	−0.345773
$550$	0	0
$551$	8.85728	0.377333
$552$	0	0
$553$	−53.1437	−2.25990
$554$	0	0
$555$	−0.470127	−0.0199558
$556$	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
$561$	11.6128	0.490295
$562$	0	0
$563$	27.4924	1.15867	0.579333	−	0.815091i	$-0.303313\pi$
$563$	27.4924	1.15867	0.579333	+	0.815091i	$0.303313\pi$
$564$	0	0
$565$	9.00354	0.378782
$566$	0	0
$567$	−4.42864	−0.185985
$568$	0	0
$569$	−13.4924	−0.565631	−0.282815	−	0.959174i	$-0.591268\pi$
$569$	−13.4924	−0.565631	−0.282815	+	0.959174i	$0.591268\pi$
$570$	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
$573$	−0.266706	−0.0111418
$574$	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
$579$	26.2034	1.08898
$580$	0	0
$581$	−13.5111	−0.560536
$582$	0	0
$583$	−27.6128	−1.14361
$584$	0	0
$585$	0.622216	0.0257255
$586$	0	0
$587$	17.3363	0.715546	0.357773	−	0.933809i	$-0.383536\pi$
$587$	17.3363	0.715546	0.357773	+	0.933809i	$0.383536\pi$
$588$	0	0
$589$	31.8163	1.31097
$590$	0	0
$591$	−15.8479	−0.651896
$592$	0	0
$593$	−18.8069	−0.772307	−0.386153	−	0.922435i	$-0.626196\pi$
$593$	−18.8069	−0.772307	−0.386153	+	0.922435i	$0.626196\pi$
$594$	0	0
$595$	−5.51114	−0.225935
$596$	0	0
$597$	10.3684	0.424351
$598$	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
$603$	−8.04149	−0.327475
$604$	0	0
$605$	14.1334	0.574603
$606$	0	0
$607$	−17.2444	−0.699930	−0.349965	−	0.936763i	$-0.613807\pi$
$607$	−17.2444	−0.699930	−0.349965	+	0.936763i	$0.613807\pi$
$608$	0	0
$609$	8.85728	0.358915
$610$	0	0
$611$	−1.80642	−0.0730801
$612$	0	0
$613$	23.9813	0.968594	0.484297	−	0.874904i	$-0.339075\pi$
$613$	23.9813	0.968594	0.484297	+	0.874904i	$0.339075\pi$
$614$	0	0
$615$	−2.10171	−0.0847491
$616$	0	0
$617$	6.40006	0.257657	0.128828	−	0.991667i	$-0.458878\pi$
$617$	6.40006	0.257657	0.128828	+	0.991667i	$0.458878\pi$
$618$	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
$621$	−8.85728	−0.355430
$622$	0	0
$623$	8.26671	0.331199
$624$	0	0
$625$	19.3426	0.773704
$626$	0	0
$627$	25.7146	1.02694
$628$	0	0
$629$	1.51114	0.0602530
$630$	0	0
$631$	−23.1842	−0.922949	−0.461474	−	0.887154i	$-0.652679\pi$
$631$	−23.1842	−0.922949	−0.461474	+	0.887154i	$0.652679\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	12.9777	0.515005
$636$	0	0
$637$	12.6128	0.499739
$638$	0	0
$639$	7.05086	0.278928
$640$	0	0
$641$	−3.51114	−0.138682	−0.0693408	−	0.997593i	$-0.522090\pi$
$641$	−3.51114	−0.138682	−0.0693408	+	0.997593i	$0.522090\pi$
$642$	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
$645$	−4.73683	−0.186513
$646$	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
$651$	31.8163	1.24698
$652$	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
$657$	7.24443	0.282632
$658$	0	0
$659$	1.51114	0.0588656	0.0294328	−	0.999567i	$-0.490630\pi$
$659$	1.51114	0.0588656	0.0294328	+	0.999567i	$0.490630\pi$
$660$	0	0
$661$	−24.1847	−0.940675	−0.470338	−	0.882487i	$-0.655868\pi$
$661$	−24.1847	−0.940675	−0.470338	+	0.882487i	$0.655868\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−12.2034	−0.473228
$666$	0	0
$667$	17.7146	0.685910
$668$	0	0
$669$	−4.69535	−0.181533
$670$	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
$675$	4.61285	0.177549
$676$	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
$681$	−5.80642	−0.222503
$682$	0	0
$683$	30.1561	1.15389	0.576946	−	0.816783i	$-0.304244\pi$
$683$	30.1561	1.15389	0.576946	+	0.816783i	$0.304244\pi$
$684$	0	0
$685$	−12.3497	−0.471857
$686$	0	0
$687$	−6.00000	−0.228914
$688$	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
$693$	25.7146	0.976815
$694$	0	0
$695$	−4.20342	−0.159445
$696$	0	0
$697$	6.75557	0.255885
$698$	0	0
$699$	−7.24443	−0.274010
$700$	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
$705$	1.12399	0.0423317
$706$	0	0
$707$	69.5941	2.61736
$708$	0	0
$709$	−1.22570	−0.0460320	−0.0230160	−	0.999735i	$-0.507327\pi$
$709$	−1.22570	−0.0460320	−0.0230160	+	0.999735i	$0.507327\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	63.6325	2.38306
$714$	0	0
$715$	−3.61285	−0.135113
$716$	0	0
$717$	10.6637	0.398243
$718$	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
$723$	5.73329	0.213223
$724$	0	0
$725$	−9.22570	−0.342634
$726$	0	0
$727$	49.0607	1.81956	0.909780	−	0.415090i	$-0.136250\pi$
$727$	49.0607	1.81956	0.909780	+	0.415090i	$0.136250\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.2257	0.563143
$732$	0	0
$733$	−53.4291	−1.97345	−0.986725	−	0.162402i	$-0.948076\pi$
$733$	−53.4291	−1.97345	−0.986725	+	0.162402i	$0.948076\pi$
$734$	0	0
$735$	−7.84791	−0.289475
$736$	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
$741$	−4.42864	−0.162690
$742$	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
$747$	3.05086	0.111625
$748$	0	0
$749$	67.9625	2.48330
$750$	0	0
$751$	21.8983	0.799080	0.399540	−	0.916716i	$-0.369170\pi$
$751$	21.8983	0.799080	0.399540	+	0.916716i	$0.369170\pi$
$752$	0	0
$753$	23.6128	0.860500
$754$	0	0
$755$	−1.04101	−0.0378863
$756$	0	0
$757$	−36.8385	−1.33892	−0.669460	−	0.742848i	$-0.733474\pi$
$757$	−36.8385	−1.33892	−0.669460	+	0.742848i	$0.733474\pi$
$758$	0	0
$759$	51.4291	1.86676
$760$	0	0
$761$	38.8069	1.40675	0.703375	−	0.710819i	$-0.251676\pi$
$761$	38.8069	1.40675	0.703375	+	0.710819i	$0.251676\pi$
$762$	0	0
$763$	−69.5941	−2.51948
$764$	0	0
$765$	1.24443	0.0449925
$766$	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
$771$	11.2444	0.404958
$772$	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
$777$	3.34614	0.120042
$778$	0	0
$779$	14.9590	0.535961
$780$	0	0
$781$	−40.9403	−1.46496
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	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
$789$	4.38715	0.156187
$790$	0	0
$791$	−64.0830	−2.27853
$792$	0	0
$793$	−8.10171	−0.287700
$794$	0	0
$795$	−2.95899	−0.104945
$796$	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
$801$	−1.86665	−0.0659547
$802$	0	0
$803$	−42.0642	−1.48441
$804$	0	0
$805$	−24.4068	−0.860228
$806$	0	0
$807$	−7.51114	−0.264405
$808$	0	0
$809$	−36.9590	−1.29941	−0.649704	−	0.760187i	$-0.725107\pi$
$809$	−36.9590	−1.29941	−0.649704	+	0.760187i	$0.725107\pi$
$810$	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
$813$	−6.06022	−0.212541
$814$	0	0
$815$	9.81532	0.343816
$816$	0	0
$817$	33.7146	1.17952
$818$	0	0
$819$	−4.42864	−0.154749
$820$	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	$-0.250140\pi$
$823$	40.5531	1.41359	0.706796	+	0.707417i	$0.250140\pi$
$824$	0	0
$825$	−26.7841	−0.932504
$826$	0	0
$827$	−36.4987	−1.26918	−0.634592	−	0.772847i	$-0.718832\pi$
$827$	−36.4987	−1.26918	−0.634592	+	0.772847i	$0.718832\pi$
$828$	0	0
$829$	40.8385	1.41838	0.709191	−	0.705017i	$-0.249061\pi$
$829$	40.8385	1.41838	0.709191	+	0.705017i	$0.249061\pi$
$830$	0	0
$831$	−0.488863	−0.0169585
$832$	0	0
$833$	25.2257	0.874019
$834$	0	0
$835$	−4.55310	−0.157567
$836$	0	0
$837$	−7.18421	−0.248323
$838$	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
$843$	−13.0923	−0.450924
$844$	0	0
$845$	0.622216	0.0214049
$846$	0	0
$847$	−100.595	−3.45647
$848$	0	0
$849$	23.3461	0.801238
$850$	0	0
$851$	6.69228	0.229409
$852$	0	0
$853$	29.4924	1.00980	0.504900	−	0.863178i	$-0.331529\pi$
$853$	29.4924	1.00980	0.504900	+	0.863178i	$0.331529\pi$
$854$	0	0
$855$	2.75557	0.0942385
$856$	0	0
$857$	8.01874	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$857$	8.01874	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$858$	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
$861$	14.9590	0.509801
$862$	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
$867$	13.0000	0.441503
$868$	0	0
$869$	−69.6771	−2.36363
$870$	0	0
$871$	−8.04149	−0.272475
$872$	0	0
$873$	−0.755569	−0.0255721
$874$	0	0
$875$	26.4889	0.895487
$876$	0	0
$877$	28.1847	0.951729	0.475865	−	0.879519i	$-0.342135\pi$
$877$	28.1847	0.951729	0.475865	+	0.879519i	$0.342135\pi$
$878$	0	0
$879$	−13.0923	−0.441594
$880$	0	0
$881$	−9.52987	−0.321070	−0.160535	−	0.987030i	$-0.551322\pi$
$881$	−9.52987	−0.321070	−0.160535	+	0.987030i	$0.551322\pi$
$882$	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
$885$	6.87601	0.231135
$886$	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
$891$	−5.80642	−0.194523
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−9.18115	−0.306892
$896$	0	0
$897$	−8.85728	−0.295736
$898$	0	0
$899$	14.3684	0.479214
$900$	0	0
$901$	9.51114	0.316862
$902$	0	0
$903$	33.7146	1.12195
$904$	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
$909$	−15.7146	−0.521219
$910$	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
$915$	5.04101	0.166651
$916$	0	0
$917$	−45.9180	−1.51635
$918$	0	0
$919$	−12.5334	−0.413439	−0.206720	−	0.978400i	$-0.566279\pi$
$919$	−12.5334	−0.413439	−0.206720	+	0.978400i	$0.566279\pi$
$920$	0	0
$921$	−32.0415	−1.05580
$922$	0	0
$923$	7.05086	0.232082
$924$	0	0
$925$	−3.48532	−0.114597
$926$	0	0
$927$	18.1017	0.594538
$928$	0	0
$929$	−45.0291	−1.47736	−0.738678	−	0.674059i	$-0.764549\pi$
$929$	−45.0291	−1.47736	−0.738678	+	0.674059i	$0.764549\pi$
$930$	0	0
$931$	55.8578	1.83066
$932$	0	0
$933$	24.0830	0.788441
$934$	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
$939$	−27.7146	−0.904430
$940$	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
$945$	2.75557	0.0896387
$946$	0	0
$947$	−43.9911	−1.42952	−0.714759	−	0.699370i	$-0.753464\pi$
$947$	−43.9911	−1.42952	−0.714759	+	0.699370i	$0.753464\pi$
$948$	0	0
$949$	7.24443	0.235164
$950$	0	0
$951$	17.0923	0.554257
$952$	0	0
$953$	−18.4701	−0.598306	−0.299153	−	0.954205i	$-0.596704\pi$
$953$	−18.4701	−0.598306	−0.299153	+	0.954205i	$0.596704\pi$
$954$	0	0
$955$	0.165949	0.00536998
$956$	0	0
$957$	11.6128	0.375390
$958$	0	0
$959$	87.8992	2.83841
$960$	0	0
$961$	20.6128	0.664931
$962$	0	0
$963$	−15.3461	−0.494522
$964$	0	0
$965$	−16.3042	−0.524850
$966$	0	0
$967$	−50.2895	−1.61720	−0.808600	−	0.588359i	$-0.799774\pi$
$967$	−50.2895	−1.61720	−0.808600	+	0.588359i	$0.799774\pi$
$968$	0	0
$969$	−8.85728	−0.284537
$970$	0	0
$971$	35.2257	1.13045	0.565223	−	0.824938i	$-0.308790\pi$
$971$	35.2257	1.13045	0.565223	+	0.824938i	$0.308790\pi$
$972$	0	0
$973$	29.9180	0.959126
$974$	0	0
$975$	4.61285	0.147729
$976$	0	0
$977$	−58.8069	−1.88140	−0.940700	−	0.339240i	$-0.889830\pi$
$977$	−58.8069	−1.88140	−0.940700	+	0.339240i	$0.889830\pi$
$978$	0	0
$979$	10.8385	0.346401
$980$	0	0
$981$	15.7146	0.501727
$982$	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
$987$	−8.00000	−0.254643
$988$	0	0
$989$	67.4291	2.14412
$990$	0	0
$991$	35.4924	1.12745	0.563727	−	0.825961i	$-0.309367\pi$
$991$	35.4924	1.12745	0.563727	+	0.825961i	$0.309367\pi$
$992$	0	0
$993$	−7.18421	−0.227984
$994$	0	0
$995$	−6.45139	−0.204523
$996$	0	0
$997$	49.4291	1.56544	0.782718	−	0.622377i	$-0.213833\pi$
$997$	49.4291	1.56544	0.782718	+	0.622377i	$0.213833\pi$
$998$	0	0
$999$	−0.755569	−0.0239051

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.622216 q^{5} -4.42864 q^{7} +1.00000 q^{9} -5.80642 q^{11} +1.00000 q^{13} -0.622216 q^{15} +2.00000 q^{17} +4.42864 q^{19} +4.42864 q^{21} +8.85728 q^{23} -4.61285 q^{25} -1.00000 q^{27} +2.00000 q^{29} +7.18421 q^{31} +5.80642 q^{33} -2.75557 q^{35} +0.755569 q^{37} -1.00000 q^{39} +3.37778 q^{41} +7.61285 q^{43} +0.622216 q^{45} -1.80642 q^{47} +12.6128 q^{49} -2.00000 q^{51} +4.75557 q^{53} -3.61285 q^{55} -4.42864 q^{57} -11.0509 q^{59} -8.10171 q^{61} -4.42864 q^{63} +0.622216 q^{65} -8.04149 q^{67} -8.85728 q^{69} +7.05086 q^{71} +7.24443 q^{73} +4.61285 q^{75} +25.7146 q^{77} +12.0000 q^{79} +1.00000 q^{81} +3.05086 q^{83} +1.24443 q^{85} -2.00000 q^{87} -1.86665 q^{89} -4.42864 q^{91} -7.18421 q^{93} +2.75557 q^{95} -0.755569 q^{97} -5.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - 4 q^{11} + 3 q^{13} - 2 q^{15} + 6 q^{17} + 13 q^{25} - 3 q^{27} + 6 q^{29} + 8 q^{31} + 4 q^{33} - 8 q^{35} + 2 q^{37} - 3 q^{39} + 10 q^{41} - 4 q^{43} + 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 - 4 * q^11 + 3 * q^13 - 2 * q^15 + 6 * q^17 + 13 * q^25 - 3 * q^27 + 6 * q^29 + 8 * q^31 + 4 * q^33 - 8 * q^35 + 2 * q^37 - 3 * q^39 + 10 * q^41 - 4 * q^43 + 2 * q^45 + 8 * q^47 + 11 * q^49 - 6 * q^51 + 14 * q^53 + 16 * q^55 - 20 * q^59 + 2 * q^61 + 2 * q^65 + 16 * q^67 + 8 * q^71 + 22 * q^73 - 13 * q^75 + 24 * q^77 + 36 * q^79 + 3 * q^81 - 4 * q^83 + 4 * q^85 - 6 * q^87 - 6 * q^89 - 8 * q^93 + 8 * q^95 - 2 * q^97 - 4 * q^99

Introduction

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Database

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