LMFDB - Embedded newform 4864.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.59774$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +O(q^{10})$	$q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +1.74823 q^{11} +0.748228 q^{13} +4.15049 q^{15} +3.00000 q^{17} -1.00000 q^{19} +11.2897 q^{21} -1.94370 q^{23} -2.44724 q^{25} +1.94370 q^{27} -7.28966 q^{29} +0.654037 q^{31} +4.54143 q^{33} +6.94370 q^{35} +3.84951 q^{37} +1.94370 q^{39} +4.54143 q^{41} -10.4402 q^{43} +5.98868 q^{45} +7.59774 q^{47} +11.8874 q^{49} +7.79321 q^{51} +6.59774 q^{53} +2.79321 q^{55} -2.59774 q^{57} -6.74823 q^{59} +9.09419 q^{61} +16.2897 q^{63} +1.19547 q^{65} +7.40226 q^{67} -5.04921 q^{69} -1.69901 q^{71} -16.6848 q^{73} -6.35729 q^{75} +7.59774 q^{77} +0.654037 q^{79} -6.19547 q^{81} -12.5793 q^{83} +4.79321 q^{85} -18.9366 q^{87} -14.5414 q^{89} +3.25177 q^{91} +1.69901 q^{93} -1.59774 q^{95} -10.5793 q^{97} +6.55276 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9	$ 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9} + 2 q^{11} - q^{13} + 16 q^{15} + 9 q^{17} - 3 q^{19} + 7 q^{21} + 11 q^{23} + 3 q^{25} - 11 q^{27} + 5 q^{29} + 12 q^{31} - 10 q^{33} + 4 q^{35} + 8 q^{37} - 11 q^{39} - 10 q^{41} - 8 q^{43} - 16 q^{45} + 16 q^{47} + 2 q^{49} + 3 q^{51} + 13 q^{53} - 12 q^{55} - q^{57} - 17 q^{59} + 14 q^{61} + 22 q^{63} - 10 q^{65} + 29 q^{67} - 19 q^{69} + 2 q^{71} - 17 q^{73} - 43 q^{75} + 16 q^{77} + 12 q^{79} - 5 q^{81} + 16 q^{83} - 6 q^{85} - 27 q^{87} - 20 q^{89} + 13 q^{91} - 2 q^{93} + 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9 + 2 * q^11 - q^13 + 16 * q^15 + 9 * q^17 - 3 * q^19 + 7 * q^21 + 11 * q^23 + 3 * q^25 - 11 * q^27 + 5 * q^29 + 12 * q^31 - 10 * q^33 + 4 * q^35 + 8 * q^37 - 11 * q^39 - 10 * q^41 - 8 * q^43 - 16 * q^45 + 16 * q^47 + 2 * q^49 + 3 * q^51 + 13 * q^53 - 12 * q^55 - q^57 - 17 * q^59 + 14 * q^61 + 22 * q^63 - 10 * q^65 + 29 * q^67 - 19 * q^69 + 2 * q^71 - 17 * q^73 - 43 * q^75 + 16 * q^77 + 12 * q^79 - 5 * q^81 + 16 * q^83 - 6 * q^85 - 27 * q^87 - 20 * q^89 + 13 * q^91 - 2 * q^93 + 2 * q^95 + 22 * q^97 + 30 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.59774	1.49980	0.749902	−	0.661549i	$-0.230101\pi$
$3$	2.59774	1.49980	0.749902	+	0.661549i	$0.230101\pi$
$4$	0	0
$5$	1.59774	0.714529	0.357264	−	0.934003i	$-0.383709\pi$
$5$	1.59774	0.714529	0.357264	+	0.934003i	$0.383709\pi$
$6$	0	0
$7$	4.34596	1.64262	0.821310	−	0.570482i	$-0.193244\pi$
$7$	4.34596	1.64262	0.821310	+	0.570482i	$0.193244\pi$
$8$	0	0
$9$	3.74823	1.24941
$10$	0	0
$11$	1.74823	0.527111	0.263555	−	0.964644i	$-0.415105\pi$
$11$	1.74823	0.527111	0.263555	+	0.964644i	$0.415105\pi$
$12$	0	0
$13$	0.748228	0.207521	0.103761	−	0.994602i	$-0.466912\pi$
$13$	0.748228	0.207521	0.103761	+	0.994602i	$0.466912\pi$
$14$	0	0
$15$	4.15049	1.07165
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	11.2897	2.46361
$22$	0	0
$23$	−1.94370	−0.405289	−0.202645	−	0.979252i	$-0.564954\pi$
$23$	−1.94370	−0.405289	−0.202645	+	0.979252i	$0.564954\pi$
$24$	0	0
$25$	−2.44724	−0.489448
$26$	0	0
$27$	1.94370	0.374065
$28$	0	0
$29$	−7.28966	−1.35366	−0.676828	−	0.736141i	$-0.736646\pi$
$29$	−7.28966	−1.35366	−0.676828	+	0.736141i	$0.736646\pi$
$30$	0	0
$31$	0.654037	0.117468	0.0587342	−	0.998274i	$-0.481294\pi$
$31$	0.654037	0.117468	0.0587342	+	0.998274i	$0.481294\pi$
$32$	0	0
$33$	4.54143	0.790562
$34$	0	0
$35$	6.94370	1.17370
$36$	0	0
$37$	3.84951	0.632855	0.316428	−	0.948617i	$-0.397517\pi$
$37$	3.84951	0.632855	0.316428	+	0.948617i	$0.397517\pi$
$38$	0	0
$39$	1.94370	0.311241
$40$	0	0
$41$	4.54143	0.709253	0.354626	−	0.935008i	$-0.384608\pi$
$41$	4.54143	0.709253	0.354626	+	0.935008i	$0.384608\pi$
$42$	0	0
$43$	−10.4402	−1.59211	−0.796054	−	0.605225i	$-0.793083\pi$
$43$	−10.4402	−1.59211	−0.796054	+	0.605225i	$0.793083\pi$
$44$	0	0
$45$	5.98868	0.892739
$46$	0	0
$47$	7.59774	1.10824	0.554122	−	0.832436i	$-0.313054\pi$
$47$	7.59774	1.10824	0.554122	+	0.832436i	$0.313054\pi$
$48$	0	0
$49$	11.8874	1.69820
$50$	0	0
$51$	7.79321	1.09127
$52$	0	0
$53$	6.59774	0.906269	0.453134	−	0.891442i	$-0.350306\pi$
$53$	6.59774	0.906269	0.453134	+	0.891442i	$0.350306\pi$
$54$	0	0
$55$	2.79321	0.376636
$56$	0	0
$57$	−2.59774	−0.344078
$58$	0	0
$59$	−6.74823	−0.878544	−0.439272	−	0.898354i	$-0.644764\pi$
$59$	−6.74823	−0.878544	−0.439272	+	0.898354i	$0.644764\pi$
$60$	0	0
$61$	9.09419	1.16439	0.582196	−	0.813049i	$-0.302194\pi$
$61$	9.09419	1.16439	0.582196	+	0.813049i	$0.302194\pi$
$62$	0	0
$63$	16.2897	2.05230
$64$	0	0
$65$	1.19547	0.148280
$66$	0	0
$67$	7.40226	0.904331	0.452165	−	0.891934i	$-0.350652\pi$
$67$	7.40226	0.904331	0.452165	+	0.891934i	$0.350652\pi$
$68$	0	0
$69$	−5.04921	−0.607854
$70$	0	0
$71$	−1.69901	−0.201636	−0.100818	−	0.994905i	$-0.532146\pi$
$71$	−1.69901	−0.201636	−0.100818	+	0.994905i	$0.532146\pi$
$72$	0	0
$73$	−16.6848	−1.95281	−0.976406	−	0.215941i	$-0.930718\pi$
$73$	−16.6848	−1.95281	−0.976406	+	0.215941i	$0.930718\pi$
$74$	0	0
$75$	−6.35729	−0.734076
$76$	0	0
$77$	7.59774	0.865842
$78$	0	0
$79$	0.654037	0.0735849	0.0367924	−	0.999323i	$-0.488286\pi$
$79$	0.654037	0.0735849	0.0367924	+	0.999323i	$0.488286\pi$
$80$	0	0
$81$	−6.19547	−0.688386
$82$	0	0
$83$	−12.5793	−1.38076	−0.690380	−	0.723447i	$-0.742557\pi$
$83$	−12.5793	−1.38076	−0.690380	+	0.723447i	$0.742557\pi$
$84$	0	0
$85$	4.79321	0.519896
$86$	0	0
$87$	−18.9366	−2.03022
$88$	0	0
$89$	−14.5414	−1.54139	−0.770694	−	0.637205i	$-0.780091\pi$
$89$	−14.5414	−1.54139	−0.770694	+	0.637205i	$0.780091\pi$
$90$	0	0
$91$	3.25177	0.340878
$92$	0	0
$93$	1.69901	0.176180
$94$	0	0
$95$	−1.59774	−0.163924
$96$	0	0
$97$	−10.5793	−1.07417	−0.537084	−	0.843529i	$-0.680474\pi$
$97$	−10.5793	−1.07417	−0.537084	+	0.843529i	$0.680474\pi$
$98$	0	0
$99$	6.55276	0.658577
$100$	0	0
$101$	−4.80453	−0.478069	−0.239034	−	0.971011i	$-0.576831\pi$
$101$	−4.80453	−0.478069	−0.239034	+	0.971011i	$0.576831\pi$
$102$	0	0
$103$	−12.5793	−1.23948	−0.619739	−	0.784808i	$-0.712761\pi$
$103$	−12.5793	−1.23948	−0.619739	+	0.784808i	$0.712761\pi$
$104$	0	0
$105$	18.0379	1.76032
$106$	0	0
$107$	5.44015	0.525920	0.262960	−	0.964807i	$-0.415301\pi$
$107$	5.44015	0.525920	0.262960	+	0.964807i	$0.415301\pi$
$108$	0	0
$109$	14.7861	1.41625	0.708127	−	0.706085i	$-0.249541\pi$
$109$	14.7861	1.41625	0.708127	+	0.706085i	$0.249541\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	14.0379	1.32057	0.660287	−	0.751014i	$-0.270435\pi$
$113$	14.0379	1.32057	0.660287	+	0.751014i	$0.270435\pi$
$114$	0	0
$115$	−3.10552	−0.289591
$116$	0	0
$117$	2.80453	0.259279
$118$	0	0
$119$	13.0379	1.19518
$120$	0	0
$121$	−7.94370	−0.722154
$122$	0	0
$123$	11.7974	1.06374
$124$	0	0
$125$	−11.8987	−1.06425
$126$	0	0
$127$	7.54852	0.669823	0.334911	−	0.942250i	$-0.391294\pi$
$127$	7.54852	0.669823	0.334911	+	0.942250i	$0.391294\pi$
$128$	0	0
$129$	−27.1208	−2.38785
$130$	0	0
$131$	0.0492139	0.00429984	0.00214992	−	0.999998i	$-0.499316\pi$
$131$	0.0492139	0.00429984	0.00214992	+	0.999998i	$0.499316\pi$
$132$	0	0
$133$	−4.34596	−0.376843
$134$	0	0
$135$	3.10552	0.267280
$136$	0	0
$137$	17.3909	1.48581	0.742904	−	0.669398i	$-0.233448\pi$
$137$	17.3909	1.48581	0.742904	+	0.669398i	$0.233448\pi$
$138$	0	0
$139$	−8.25177	−0.699906	−0.349953	−	0.936767i	$-0.613802\pi$
$139$	−8.25177	−0.699906	−0.349953	+	0.936767i	$0.613802\pi$
$140$	0	0
$141$	19.7369	1.66215
$142$	0	0
$143$	1.30807	0.109387
$144$	0	0
$145$	−11.6469	−0.967226
$146$	0	0
$147$	30.8803	2.54696
$148$	0	0
$149$	−6.28966	−0.515269	−0.257635	−	0.966242i	$-0.582943\pi$
$149$	−6.28966	−0.515269	−0.257635	+	0.966242i	$0.582943\pi$
$150$	0	0
$151$	22.1884	1.80566	0.902832	−	0.429992i	$-0.141484\pi$
$151$	22.1884	1.80566	0.902832	+	0.429992i	$0.141484\pi$
$152$	0	0
$153$	11.2447	0.909079
$154$	0	0
$155$	1.04498	0.0839346
$156$	0	0
$157$	−8.99291	−0.717713	−0.358856	−	0.933393i	$-0.616833\pi$
$157$	−8.99291	−0.717713	−0.358856	+	0.933393i	$0.616833\pi$
$158$	0	0
$159$	17.1392	1.35922
$160$	0	0
$161$	−8.44724	−0.665736
$162$	0	0
$163$	−16.9929	−1.33099	−0.665494	−	0.746403i	$-0.731779\pi$
$163$	−16.9929	−1.33099	−0.665494	+	0.746403i	$0.731779\pi$
$164$	0	0
$165$	7.25601	0.564879
$166$	0	0
$167$	−16.3389	−1.26434	−0.632170	−	0.774830i	$-0.717836\pi$
$167$	−16.3389	−1.26434	−0.632170	+	0.774830i	$0.717836\pi$
$168$	0	0
$169$	−12.4402	−0.956935
$170$	0	0
$171$	−3.74823	−0.286634
$172$	0	0
$173$	−23.3839	−1.77784	−0.888921	−	0.458061i	$-0.848544\pi$
$173$	−23.3839	−1.77784	−0.888921	+	0.458061i	$0.848544\pi$
$174$	0	0
$175$	−10.6356	−0.803978
$176$	0	0
$177$	−17.5301	−1.31764
$178$	0	0
$179$	−4.54143	−0.339443	−0.169721	−	0.985492i	$-0.554287\pi$
$179$	−4.54143	−0.339443	−0.169721	+	0.985492i	$0.554287\pi$
$180$	0	0
$181$	−12.0900	−0.898639	−0.449320	−	0.893371i	$-0.648334\pi$
$181$	−12.0900	−0.898639	−0.449320	+	0.893371i	$0.648334\pi$
$182$	0	0
$183$	23.6243	1.74636
$184$	0	0
$185$	6.15049	0.452193
$186$	0	0
$187$	5.24468	0.383529
$188$	0	0
$189$	8.44724	0.614446
$190$	0	0
$191$	12.5485	0.907979	0.453990	−	0.891007i	$-0.350000\pi$
$191$	12.5485	0.907979	0.453990	+	0.891007i	$0.350000\pi$
$192$	0	0
$193$	10.7298	0.772349	0.386175	−	0.922426i	$-0.373796\pi$
$193$	10.7298	0.772349	0.386175	+	0.922426i	$0.373796\pi$
$194$	0	0
$195$	3.10552	0.222391
$196$	0	0
$197$	9.38385	0.668572	0.334286	−	0.942472i	$-0.391505\pi$
$197$	9.38385	0.668572	0.334286	+	0.942472i	$0.391505\pi$
$198$	0	0
$199$	11.2404	0.796814	0.398407	−	0.917209i	$-0.369563\pi$
$199$	11.2404	0.796814	0.398407	+	0.917209i	$0.369563\pi$
$200$	0	0
$201$	19.2291	1.35632
$202$	0	0
$203$	−31.6806	−2.22354
$204$	0	0
$205$	7.25601	0.506782
$206$	0	0
$207$	−7.28543	−0.506372
$208$	0	0
$209$	−1.74823	−0.120927
$210$	0	0
$211$	−5.44015	−0.374516	−0.187258	−	0.982311i	$-0.559960\pi$
$211$	−5.44015	−0.374516	−0.187258	+	0.982311i	$0.559960\pi$
$212$	0	0
$213$	−4.41359	−0.302414
$214$	0	0
$215$	−16.6806	−1.13761
$216$	0	0
$217$	2.84242	0.192956
$218$	0	0
$219$	−43.3428	−2.92883
$220$	0	0
$221$	2.24468	0.150994
$222$	0	0
$223$	25.4217	1.70237	0.851183	−	0.524869i	$-0.175886\pi$
$223$	25.4217	1.70237	0.851183	+	0.524869i	$0.175886\pi$
$224$	0	0
$225$	−9.17282	−0.611521
$226$	0	0
$227$	3.74114	0.248308	0.124154	−	0.992263i	$-0.460378\pi$
$227$	3.74114	0.248308	0.124154	+	0.992263i	$0.460378\pi$
$228$	0	0
$229$	−5.29675	−0.350019	−0.175010	−	0.984567i	$-0.555996\pi$
$229$	−5.29675	−0.350019	−0.175010	+	0.984567i	$0.555996\pi$
$230$	0	0
$231$	19.7369	1.29859
$232$	0	0
$233$	9.93661	0.650969	0.325485	−	0.945547i	$-0.394473\pi$
$233$	9.93661	0.650969	0.325485	+	0.945547i	$0.394473\pi$
$234$	0	0
$235$	12.1392	0.791872
$236$	0	0
$237$	1.69901	0.110363
$238$	0	0
$239$	27.8424	1.80098	0.900488	−	0.434880i	$-0.143209\pi$
$239$	27.8424	1.80098	0.900488	+	0.434880i	$0.143209\pi$
$240$	0	0
$241$	5.84951	0.376800	0.188400	−	0.982092i	$-0.439670\pi$
$241$	5.84951	0.376800	0.188400	+	0.982092i	$0.439670\pi$
$242$	0	0
$243$	−21.9253	−1.40651
$244$	0	0
$245$	18.9929	1.21341
$246$	0	0
$247$	−0.748228	−0.0476086
$248$	0	0
$249$	−32.6778	−2.07087
$250$	0	0
$251$	15.2447	0.962236	0.481118	−	0.876656i	$-0.340231\pi$
$251$	15.2447	0.962236	0.481118	+	0.876656i	$0.340231\pi$
$252$	0	0
$253$	−3.39803	−0.213632
$254$	0	0
$255$	12.4515	0.779742
$256$	0	0
$257$	7.58641	0.473227	0.236614	−	0.971604i	$-0.423962\pi$
$257$	7.58641	0.473227	0.236614	+	0.971604i	$0.423962\pi$
$258$	0	0
$259$	16.7298	1.03954
$260$	0	0
$261$	−27.3233	−1.69127
$262$	0	0
$263$	−2.79321	−0.172236	−0.0861182	−	0.996285i	$-0.527446\pi$
$263$	−2.79321	−0.172236	−0.0861182	+	0.996285i	$0.527446\pi$
$264$	0	0
$265$	10.5414	0.647555
$266$	0	0
$267$	−37.7748	−2.31178
$268$	0	0
$269$	25.0308	1.52615	0.763077	−	0.646307i	$-0.223687\pi$
$269$	25.0308	1.52615	0.763077	+	0.646307i	$0.223687\pi$
$270$	0	0
$271$	−27.5301	−1.67234	−0.836168	−	0.548474i	$-0.815209\pi$
$271$	−27.5301	−1.67234	−0.836168	+	0.548474i	$0.815209\pi$
$272$	0	0
$273$	8.44724	0.511250
$274$	0	0
$275$	−4.27834	−0.257993
$276$	0	0
$277$	−10.4023	−0.625012	−0.312506	−	0.949916i	$-0.601168\pi$
$277$	−10.4023	−0.625012	−0.312506	+	0.949916i	$0.601168\pi$
$278$	0	0
$279$	2.45148	0.146766
$280$	0	0
$281$	−13.8874	−0.828453	−0.414226	−	0.910174i	$-0.635948\pi$
$281$	−13.8874	−0.828453	−0.414226	+	0.910174i	$0.635948\pi$
$282$	0	0
$283$	27.3346	1.62488	0.812438	−	0.583048i	$-0.198140\pi$
$283$	27.3346	1.62488	0.812438	+	0.583048i	$0.198140\pi$
$284$	0	0
$285$	−4.15049	−0.245854
$286$	0	0
$287$	19.7369	1.16503
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−27.4823	−1.61104
$292$	0	0
$293$	−10.1321	−0.591923	−0.295961	−	0.955200i	$-0.595640\pi$
$293$	−10.1321	−0.591923	−0.295961	+	0.955200i	$0.595640\pi$
$294$	0	0
$295$	−10.7819	−0.627745
$296$	0	0
$297$	3.39803	0.197174
$298$	0	0
$299$	−1.45433	−0.0841061
$300$	0	0
$301$	−45.3725	−2.61523
$302$	0	0
$303$	−12.4809	−0.717009
$304$	0	0
$305$	14.5301	0.831992
$306$	0	0
$307$	17.1208	0.977133	0.488567	−	0.872527i	$-0.337520\pi$
$307$	17.1208	0.977133	0.488567	+	0.872527i	$0.337520\pi$
$308$	0	0
$309$	−32.6778	−1.85897
$310$	0	0
$311$	31.2404	1.77148	0.885742	−	0.464179i	$-0.153650\pi$
$311$	31.2404	1.77148	0.885742	+	0.464179i	$0.153650\pi$
$312$	0	0
$313$	−25.2291	−1.42603	−0.713017	−	0.701147i	$-0.752672\pi$
$313$	−25.2291	−1.42603	−0.713017	+	0.701147i	$0.752672\pi$
$314$	0	0
$315$	26.0266	1.46643
$316$	0	0
$317$	−11.6427	−0.653920	−0.326960	−	0.945038i	$-0.606024\pi$
$317$	−11.6427	−0.653920	−0.326960	+	0.945038i	$0.606024\pi$
$318$	0	0
$319$	−12.7440	−0.713527
$320$	0	0
$321$	14.1321	0.788776
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−1.83110	−0.101571
$326$	0	0
$327$	38.4104	2.12410
$328$	0	0
$329$	33.0195	1.82042
$330$	0	0
$331$	12.5977	0.692434	0.346217	−	0.938154i	$-0.387466\pi$
$331$	12.5977	0.692434	0.346217	+	0.938154i	$0.387466\pi$
$332$	0	0
$333$	14.4288	0.790695
$334$	0	0
$335$	11.8269	0.646170
$336$	0	0
$337$	−5.49646	−0.299411	−0.149706	−	0.988731i	$-0.547833\pi$
$337$	−5.49646	−0.299411	−0.149706	+	0.988731i	$0.547833\pi$
$338$	0	0
$339$	36.4667	1.98060
$340$	0	0
$341$	1.14341	0.0619189
$342$	0	0
$343$	21.2404	1.14688
$344$	0	0
$345$	−8.06731	−0.434329
$346$	0	0
$347$	10.3417	0.555173	0.277586	−	0.960701i	$-0.410465\pi$
$347$	10.3417	0.555173	0.277586	+	0.960701i	$0.410465\pi$
$348$	0	0
$349$	−30.6806	−1.64230	−0.821148	−	0.570716i	$-0.806666\pi$
$349$	−30.6806	−1.64230	−0.821148	+	0.570716i	$0.806666\pi$
$350$	0	0
$351$	1.45433	0.0776264
$352$	0	0
$353$	23.3276	1.24160	0.620800	−	0.783969i	$-0.286808\pi$
$353$	23.3276	1.24160	0.620800	+	0.783969i	$0.286808\pi$
$354$	0	0
$355$	−2.71457	−0.144075
$356$	0	0
$357$	33.8690	1.79254
$358$	0	0
$359$	12.1576	0.641653	0.320826	−	0.947138i	$-0.396039\pi$
$359$	12.1576	0.641653	0.320826	+	0.947138i	$0.396039\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.6356	−1.08309
$364$	0	0
$365$	−26.6580	−1.39534
$366$	0	0
$367$	9.08287	0.474122	0.237061	−	0.971495i	$-0.423816\pi$
$367$	9.08287	0.474122	0.237061	+	0.971495i	$0.423816\pi$
$368$	0	0
$369$	17.0223	0.886147
$370$	0	0
$371$	28.6735	1.48865
$372$	0	0
$373$	−18.1321	−0.938844	−0.469422	−	0.882974i	$-0.655538\pi$
$373$	−18.1321	−0.938844	−0.469422	+	0.882974i	$0.655538\pi$
$374$	0	0
$375$	−30.9097	−1.59617
$376$	0	0
$377$	−5.45433	−0.280912
$378$	0	0
$379$	23.6427	1.21444	0.607222	−	0.794532i	$-0.292284\pi$
$379$	23.6427	1.21444	0.607222	+	0.794532i	$0.292284\pi$
$380$	0	0
$381$	19.6091	1.00460
$382$	0	0
$383$	−21.9253	−1.12033	−0.560165	−	0.828381i	$-0.689262\pi$
$383$	−21.9253	−1.12033	−0.560165	+	0.828381i	$0.689262\pi$
$384$	0	0
$385$	12.1392	0.618669
$386$	0	0
$387$	−39.1321	−1.98920
$388$	0	0
$389$	−29.2599	−1.48354	−0.741769	−	0.670656i	$-0.766013\pi$
$389$	−29.2599	−1.48354	−0.741769	+	0.670656i	$0.766013\pi$
$390$	0	0
$391$	−5.83110	−0.294891
$392$	0	0
$393$	0.127845	0.00644892
$394$	0	0
$395$	1.04498	0.0525785
$396$	0	0
$397$	−6.10128	−0.306214	−0.153107	−	0.988210i	$-0.548928\pi$
$397$	−6.10128	−0.306214	−0.153107	+	0.988210i	$0.548928\pi$
$398$	0	0
$399$	−11.2897	−0.565190
$400$	0	0
$401$	16.0379	0.800894	0.400447	−	0.916320i	$-0.368855\pi$
$401$	16.0379	0.800894	0.400447	+	0.916320i	$0.368855\pi$
$402$	0	0
$403$	0.489369	0.0243772
$404$	0	0
$405$	−9.89872	−0.491871
$406$	0	0
$407$	6.72982	0.333585
$408$	0	0
$409$	−25.5343	−1.26259	−0.631296	−	0.775542i	$-0.717477\pi$
$409$	−25.5343	−1.26259	−0.631296	+	0.775542i	$0.717477\pi$
$410$	0	0
$411$	45.1771	2.22842
$412$	0	0
$413$	−29.3276	−1.44311
$414$	0	0
$415$	−20.0984	−0.986593
$416$	0	0
$417$	−21.4359	−1.04972
$418$	0	0
$419$	−35.1586	−1.71761	−0.858806	−	0.512301i	$-0.828793\pi$
$419$	−35.1586	−1.71761	−0.858806	+	0.512301i	$0.828793\pi$
$420$	0	0
$421$	2.89872	0.141275	0.0706375	−	0.997502i	$-0.477497\pi$
$421$	2.89872	0.141275	0.0706375	+	0.997502i	$0.477497\pi$
$422$	0	0
$423$	28.4780	1.38465
$424$	0	0
$425$	−7.34173	−0.356126
$426$	0	0
$427$	39.5230	1.91265
$428$	0	0
$429$	3.39803	0.164058
$430$	0	0
$431$	25.9774	1.25129	0.625643	−	0.780110i	$-0.284837\pi$
$431$	25.9774	1.25129	0.625643	+	0.780110i	$0.284837\pi$
$432$	0	0
$433$	30.9324	1.48652	0.743258	−	0.669005i	$-0.233280\pi$
$433$	30.9324	1.48652	0.743258	+	0.669005i	$0.233280\pi$
$434$	0	0
$435$	−30.2557	−1.45065
$436$	0	0
$437$	1.94370	0.0929797
$438$	0	0
$439$	10.1278	0.483376	0.241688	−	0.970354i	$-0.422299\pi$
$439$	10.1278	0.483376	0.241688	+	0.970354i	$0.422299\pi$
$440$	0	0
$441$	44.5567	2.12175
$442$	0	0
$443$	−14.3276	−0.680723	−0.340361	−	0.940295i	$-0.610549\pi$
$443$	−14.3276	−0.680723	−0.340361	+	0.940295i	$0.610549\pi$
$444$	0	0
$445$	−23.2334	−1.10137
$446$	0	0
$447$	−16.3389	−0.772802
$448$	0	0
$449$	6.61615	0.312235	0.156118	−	0.987738i	$-0.450102\pi$
$449$	6.61615	0.312235	0.156118	+	0.987738i	$0.450102\pi$
$450$	0	0
$451$	7.93946	0.373855
$452$	0	0
$453$	57.6395	2.70814
$454$	0	0
$455$	5.19547	0.243567
$456$	0	0
$457$	38.5864	1.80500	0.902498	−	0.430694i	$-0.141731\pi$
$457$	38.5864	1.80500	0.902498	+	0.430694i	$0.141731\pi$
$458$	0	0
$459$	5.83110	0.272172
$460$	0	0
$461$	−33.1841	−1.54554	−0.772770	−	0.634686i	$-0.781129\pi$
$461$	−33.1841	−1.54554	−0.772770	+	0.634686i	$0.781129\pi$
$462$	0	0
$463$	13.2826	0.617294	0.308647	−	0.951177i	$-0.400124\pi$
$463$	13.2826	0.617294	0.308647	+	0.951177i	$0.400124\pi$
$464$	0	0
$465$	2.71457	0.125885
$466$	0	0
$467$	20.8311	0.963948	0.481974	−	0.876185i	$-0.339920\pi$
$467$	20.8311	0.963948	0.481974	+	0.876185i	$0.339920\pi$
$468$	0	0
$469$	32.1700	1.48547
$470$	0	0
$471$	−23.3612	−1.07643
$472$	0	0
$473$	−18.2518	−0.839217
$474$	0	0
$475$	2.44724	0.112287
$476$	0	0
$477$	24.7298	1.13230
$478$	0	0
$479$	25.6848	1.17357	0.586785	−	0.809743i	$-0.300393\pi$
$479$	25.6848	1.17357	0.586785	+	0.809743i	$0.300393\pi$
$480$	0	0
$481$	2.88031	0.131331
$482$	0	0
$483$	−21.9437	−0.998473
$484$	0	0
$485$	−16.9030	−0.767524
$486$	0	0
$487$	9.51063	0.430968	0.215484	−	0.976507i	$-0.430867\pi$
$487$	9.51063	0.430968	0.215484	+	0.976507i	$0.430867\pi$
$488$	0	0
$489$	−44.1431	−1.99622
$490$	0	0
$491$	−6.89448	−0.311144	−0.155572	−	0.987825i	$-0.549722\pi$
$491$	−6.89448	−0.311144	−0.155572	+	0.987825i	$0.549722\pi$
$492$	0	0
$493$	−21.8690	−0.984930
$494$	0	0
$495$	10.4696	0.470572
$496$	0	0
$497$	−7.38385	−0.331211
$498$	0	0
$499$	−29.1321	−1.30413	−0.652066	−	0.758163i	$-0.726097\pi$
$499$	−29.1321	−1.30413	−0.652066	+	0.758163i	$0.726097\pi$
$500$	0	0
$501$	−42.4441	−1.89626
$502$	0	0
$503$	13.1533	0.586479	0.293239	−	0.956039i	$-0.405267\pi$
$503$	13.1533	0.586479	0.293239	+	0.956039i	$0.405267\pi$
$504$	0	0
$505$	−7.67637	−0.341594
$506$	0	0
$507$	−32.3162	−1.43521
$508$	0	0
$509$	32.5651	1.44342	0.721712	−	0.692193i	$-0.243355\pi$
$509$	32.5651	1.44342	0.721712	+	0.692193i	$0.243355\pi$
$510$	0	0
$511$	−72.5117	−3.20773
$512$	0	0
$513$	−1.94370	−0.0858164
$514$	0	0
$515$	−20.0984	−0.885643
$516$	0	0
$517$	13.2826	0.584167
$518$	0	0
$519$	−60.7451	−2.66641
$520$	0	0
$521$	13.0450	0.571511	0.285755	−	0.958303i	$-0.407756\pi$
$521$	13.0450	0.571511	0.285755	+	0.958303i	$0.407756\pi$
$522$	0	0
$523$	−11.9437	−0.522261	−0.261131	−	0.965303i	$-0.584095\pi$
$523$	−11.9437	−0.522261	−0.261131	+	0.965303i	$0.584095\pi$
$524$	0	0
$525$	−27.6285	−1.20581
$526$	0	0
$527$	1.96211	0.0854709
$528$	0	0
$529$	−19.2220	−0.835741
$530$	0	0
$531$	−25.2939	−1.09766
$532$	0	0
$533$	3.39803	0.147185
$534$	0	0
$535$	8.69193	0.375785
$536$	0	0
$537$	−11.7974	−0.509097
$538$	0	0
$539$	20.7819	0.895139
$540$	0	0
$541$	1.48513	0.0638508	0.0319254	−	0.999490i	$-0.489836\pi$
$541$	1.48513	0.0638508	0.0319254	+	0.999490i	$0.489836\pi$
$542$	0	0
$543$	−31.4065	−1.34778
$544$	0	0
$545$	23.6243	1.01195
$546$	0	0
$547$	−6.01418	−0.257148	−0.128574	−	0.991700i	$-0.541040\pi$
$547$	−6.01418	−0.257148	−0.128574	+	0.991700i	$0.541040\pi$
$548$	0	0
$549$	34.0871	1.45480
$550$	0	0
$551$	7.28966	0.310550
$552$	0	0
$553$	2.84242	0.120872
$554$	0	0
$555$	15.9774	0.678201
$556$	0	0
$557$	−27.3584	−1.15921	−0.579605	−	0.814897i	$-0.696793\pi$
$557$	−27.3584	−1.15921	−0.579605	+	0.814897i	$0.696793\pi$
$558$	0	0
$559$	−7.81162	−0.330396
$560$	0	0
$561$	13.6243	0.575218
$562$	0	0
$563$	21.1434	0.891088	0.445544	−	0.895260i	$-0.353010\pi$
$563$	21.1434	0.891088	0.445544	+	0.895260i	$0.353010\pi$
$564$	0	0
$565$	22.4288	0.943588
$566$	0	0
$567$	−26.9253	−1.13076
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−9.18130	−0.384225	−0.192113	−	0.981373i	$-0.561534\pi$
$571$	−9.18130	−0.384225	−0.192113	+	0.981373i	$0.561534\pi$
$572$	0	0
$573$	32.5977	1.36179
$574$	0	0
$575$	4.75670	0.198368
$576$	0	0
$577$	−4.79744	−0.199720	−0.0998601	−	0.995001i	$-0.531840\pi$
$577$	−4.79744	−0.199720	−0.0998601	+	0.995001i	$0.531840\pi$
$578$	0	0
$579$	27.8732	1.15837
$580$	0	0
$581$	−54.6693	−2.26806
$582$	0	0
$583$	11.5343	0.477704
$584$	0	0
$585$	4.48090	0.185262
$586$	0	0
$587$	1.64980	0.0680945	0.0340473	−	0.999420i	$-0.489160\pi$
$587$	1.64980	0.0680945	0.0340473	+	0.999420i	$0.489160\pi$
$588$	0	0
$589$	−0.654037	−0.0269491
$590$	0	0
$591$	24.3768	1.00273
$592$	0	0
$593$	−19.8874	−0.816678	−0.408339	−	0.912830i	$-0.633892\pi$
$593$	−19.8874	−0.816678	−0.408339	+	0.912830i	$0.633892\pi$
$594$	0	0
$595$	20.8311	0.853992
$596$	0	0
$597$	29.1997	1.19506
$598$	0	0
$599$	−2.25462	−0.0921214	−0.0460607	−	0.998939i	$-0.514667\pi$
$599$	−2.25462	−0.0921214	−0.0460607	+	0.998939i	$0.514667\pi$
$600$	0	0
$601$	24.7677	1.01030	0.505148	−	0.863033i	$-0.331438\pi$
$601$	24.7677	1.01030	0.505148	+	0.863033i	$0.331438\pi$
$602$	0	0
$603$	27.7454	1.12988
$604$	0	0
$605$	−12.6919	−0.516000
$606$	0	0
$607$	−45.9774	−1.86616	−0.933082	−	0.359665i	$-0.882891\pi$
$607$	−45.9774	−1.86616	−0.933082	+	0.359665i	$0.882891\pi$
$608$	0	0
$609$	−82.2978	−3.33488
$610$	0	0
$611$	5.68484	0.229984
$612$	0	0
$613$	−0.326485	−0.0131866	−0.00659330	−	0.999978i	$-0.502099\pi$
$613$	−0.326485	−0.0131866	−0.00659330	+	0.999978i	$0.502099\pi$
$614$	0	0
$615$	18.8492	0.760073
$616$	0	0
$617$	−37.9140	−1.52636	−0.763179	−	0.646187i	$-0.776363\pi$
$617$	−37.9140	−1.52636	−0.763179	+	0.646187i	$0.776363\pi$
$618$	0	0
$619$	4.76771	0.191630	0.0958151	−	0.995399i	$-0.469454\pi$
$619$	4.76771	0.191630	0.0958151	+	0.995399i	$0.469454\pi$
$620$	0	0
$621$	−3.77796	−0.151604
$622$	0	0
$623$	−63.1965	−2.53192
$624$	0	0
$625$	−6.77479	−0.270992
$626$	0	0
$627$	−4.54143	−0.181367
$628$	0	0
$629$	11.5485	0.460470
$630$	0	0
$631$	10.9958	0.437734	0.218867	−	0.975755i	$-0.429764\pi$
$631$	10.9958	0.437734	0.218867	+	0.975755i	$0.429764\pi$
$632$	0	0
$633$	−14.1321	−0.561700
$634$	0	0
$635$	12.0605	0.478608
$636$	0	0
$637$	8.89448	0.352412
$638$	0	0
$639$	−6.36829	−0.251926
$640$	0	0
$641$	−25.1586	−0.993707	−0.496853	−	0.867834i	$-0.665511\pi$
$641$	−25.1586	−0.993707	−0.496853	+	0.867834i	$0.665511\pi$
$642$	0	0
$643$	36.9069	1.45547	0.727733	−	0.685861i	$-0.240574\pi$
$643$	36.9069	1.45547	0.727733	+	0.685861i	$0.240574\pi$
$644$	0	0
$645$	−43.3318	−1.70619
$646$	0	0
$647$	17.3531	0.682219	0.341109	−	0.940024i	$-0.389197\pi$
$647$	17.3531	0.682219	0.341109	+	0.940024i	$0.389197\pi$
$648$	0	0
$649$	−11.7974	−0.463090
$650$	0	0
$651$	7.38385	0.289396
$652$	0	0
$653$	14.4780	0.566570	0.283285	−	0.959036i	$-0.408576\pi$
$653$	14.4780	0.566570	0.283285	+	0.959036i	$0.408576\pi$
$654$	0	0
$655$	0.0786309	0.00307236
$656$	0	0
$657$	−62.5386	−2.43986
$658$	0	0
$659$	47.5301	1.85151	0.925755	−	0.378124i	$-0.123431\pi$
$659$	47.5301	1.85151	0.925755	+	0.378124i	$0.123431\pi$
$660$	0	0
$661$	−8.22204	−0.319800	−0.159900	−	0.987133i	$-0.551117\pi$
$661$	−8.22204	−0.319800	−0.159900	+	0.987133i	$0.551117\pi$
$662$	0	0
$663$	5.83110	0.226461
$664$	0	0
$665$	−6.94370	−0.269265
$666$	0	0
$667$	14.1689	0.548622
$668$	0	0
$669$	66.0390	2.55321
$670$	0	0
$671$	15.8987	0.613763
$672$	0	0
$673$	15.9016	0.612961	0.306480	−	0.951877i	$-0.400849\pi$
$673$	15.9016	0.612961	0.306480	+	0.951877i	$0.400849\pi$
$674$	0	0
$675$	−4.75670	−0.183086
$676$	0	0
$677$	21.9816	0.844821	0.422411	−	0.906405i	$-0.361184\pi$
$677$	21.9816	0.844821	0.422411	+	0.906405i	$0.361184\pi$
$678$	0	0
$679$	−45.9774	−1.76445
$680$	0	0
$681$	9.71849	0.372413
$682$	0	0
$683$	1.30807	0.0500520	0.0250260	−	0.999687i	$-0.492033\pi$
$683$	1.30807	0.0500520	0.0250260	+	0.999687i	$0.492033\pi$
$684$	0	0
$685$	27.7861	1.06165
$686$	0	0
$687$	−13.7596	−0.524960
$688$	0	0
$689$	4.93661	0.188070
$690$	0	0
$691$	−44.2291	−1.68256	−0.841278	−	0.540603i	$-0.818196\pi$
$691$	−44.2291	−1.68256	−0.841278	+	0.540603i	$0.818196\pi$
$692$	0	0
$693$	28.4780	1.08179
$694$	0	0
$695$	−13.1841	−0.500103
$696$	0	0
$697$	13.6243	0.516057
$698$	0	0
$699$	25.8127	0.976325
$700$	0	0
$701$	24.5793	0.928348	0.464174	−	0.885744i	$-0.346351\pi$
$701$	24.5793	0.928348	0.464174	+	0.885744i	$0.346351\pi$
$702$	0	0
$703$	−3.84951	−0.145187
$704$	0	0
$705$	31.5343	1.18765
$706$	0	0
$707$	−20.8803	−0.785285
$708$	0	0
$709$	9.39803	0.352950	0.176475	−	0.984305i	$-0.443530\pi$
$709$	9.39803	0.352950	0.176475	+	0.984305i	$0.443530\pi$
$710$	0	0
$711$	2.45148	0.0919376
$712$	0	0
$713$	−1.27125	−0.0476087
$714$	0	0
$715$	2.08995	0.0781599
$716$	0	0
$717$	72.3272	2.70111
$718$	0	0
$719$	24.3460	0.907951	0.453976	−	0.891014i	$-0.350005\pi$
$719$	24.3460	0.907951	0.453976	+	0.891014i	$0.350005\pi$
$720$	0	0
$721$	−54.6693	−2.03599
$722$	0	0
$723$	15.1955	0.565126
$724$	0	0
$725$	17.8396	0.662545
$726$	0	0
$727$	−29.1137	−1.07977	−0.539883	−	0.841740i	$-0.681532\pi$
$727$	−29.1137	−1.07977	−0.539883	+	0.841740i	$0.681532\pi$
$728$	0	0
$729$	−38.3697	−1.42110
$730$	0	0
$731$	−31.3205	−1.15843
$732$	0	0
$733$	4.91713	0.181618	0.0908092	−	0.995868i	$-0.471055\pi$
$733$	4.91713	0.181618	0.0908092	+	0.995868i	$0.471055\pi$
$734$	0	0
$735$	49.3386	1.81988
$736$	0	0
$737$	12.9408	0.476682
$738$	0	0
$739$	−51.7114	−1.90223	−0.951117	−	0.308830i	$-0.900063\pi$
$739$	−51.7114	−1.90223	−0.951117	+	0.308830i	$0.900063\pi$
$740$	0	0
$741$	−1.94370	−0.0714035
$742$	0	0
$743$	18.0379	0.661746	0.330873	−	0.943675i	$-0.392657\pi$
$743$	18.0379	0.661746	0.330873	+	0.943675i	$0.392657\pi$
$744$	0	0
$745$	−10.0492	−0.368175
$746$	0	0
$747$	−47.1502	−1.72513
$748$	0	0
$749$	23.6427	0.863886
$750$	0	0
$751$	−10.9171	−0.398372	−0.199186	−	0.979962i	$-0.563830\pi$
$751$	−10.9171	−0.398372	−0.199186	+	0.979962i	$0.563830\pi$
$752$	0	0
$753$	39.6017	1.44316
$754$	0	0
$755$	35.4512	1.29020
$756$	0	0
$757$	−33.4710	−1.21652	−0.608261	−	0.793737i	$-0.708133\pi$
$757$	−33.4710	−1.21652	−0.608261	+	0.793737i	$0.708133\pi$
$758$	0	0
$759$	−8.82718	−0.320406
$760$	0	0
$761$	−51.3399	−1.86107	−0.930536	−	0.366201i	$-0.880658\pi$
$761$	−51.3399	−1.86107	−0.930536	+	0.366201i	$0.880658\pi$
$762$	0	0
$763$	64.2599	2.32637
$764$	0	0
$765$	17.9660	0.649563
$766$	0	0
$767$	−5.04921	−0.182317
$768$	0	0
$769$	16.7974	0.605731	0.302866	−	0.953033i	$-0.402057\pi$
$769$	16.7974	0.605731	0.302866	+	0.953033i	$0.402057\pi$
$770$	0	0
$771$	19.7075	0.709748
$772$	0	0
$773$	−8.62145	−0.310092	−0.155046	−	0.987907i	$-0.549553\pi$
$773$	−8.62145	−0.310092	−0.155046	+	0.987907i	$0.549553\pi$
$774$	0	0
$775$	−1.60059	−0.0574948
$776$	0	0
$777$	43.4596	1.55911
$778$	0	0
$779$	−4.54143	−0.162714
$780$	0	0
$781$	−2.97026	−0.106284
$782$	0	0
$783$	−14.1689	−0.506355
$784$	0	0
$785$	−14.3683	−0.512826
$786$	0	0
$787$	−25.0124	−0.891595	−0.445798	−	0.895134i	$-0.647080\pi$
$787$	−25.0124	−0.891595	−0.445798	+	0.895134i	$0.647080\pi$
$788$	0	0
$789$	−7.25601	−0.258321
$790$	0	0
$791$	61.0082	2.16920
$792$	0	0
$793$	6.80453	0.241636
$794$	0	0
$795$	27.3839	0.971205
$796$	0	0
$797$	48.0574	1.70228	0.851140	−	0.524939i	$-0.175912\pi$
$797$	48.0574	1.70228	0.851140	+	0.524939i	$0.175912\pi$
$798$	0	0
$799$	22.7932	0.806366
$800$	0	0
$801$	−54.5046	−1.92583
$802$	0	0
$803$	−29.1689	−1.02935
$804$	0	0
$805$	−13.4965	−0.475688
$806$	0	0
$807$	65.0234	2.28893
$808$	0	0
$809$	6.20965	0.218320	0.109160	−	0.994024i	$-0.465184\pi$
$809$	6.20965	0.218320	0.109160	+	0.994024i	$0.465184\pi$
$810$	0	0
$811$	−26.6130	−0.934508	−0.467254	−	0.884123i	$-0.654757\pi$
$811$	−26.6130	−0.934508	−0.467254	+	0.884123i	$0.654757\pi$
$812$	0	0
$813$	−71.5159	−2.50817
$814$	0	0
$815$	−27.1502	−0.951029
$816$	0	0
$817$	10.4402	0.365255
$818$	0	0
$819$	12.1884	0.425897
$820$	0	0
$821$	−25.4993	−0.889932	−0.444966	−	0.895547i	$-0.646784\pi$
$821$	−25.4993	−0.889932	−0.444966	+	0.895547i	$0.646784\pi$
$822$	0	0
$823$	34.0166	1.18575	0.592873	−	0.805296i	$-0.297994\pi$
$823$	34.0166	1.18575	0.592873	+	0.805296i	$0.297994\pi$
$824$	0	0
$825$	−11.1140	−0.386939
$826$	0	0
$827$	−25.6048	−0.890367	−0.445183	−	0.895439i	$-0.646862\pi$
$827$	−25.6048	−0.890367	−0.445183	+	0.895439i	$0.646862\pi$
$828$	0	0
$829$	−19.0350	−0.661114	−0.330557	−	0.943786i	$-0.607237\pi$
$829$	−19.0350	−0.661114	−0.330557	+	0.943786i	$0.607237\pi$
$830$	0	0
$831$	−27.0223	−0.937394
$832$	0	0
$833$	35.6622	1.23562
$834$	0	0
$835$	−26.1052	−0.903408
$836$	0	0
$837$	1.27125	0.0439408
$838$	0	0
$839$	−15.6848	−0.541501	−0.270750	−	0.962650i	$-0.587272\pi$
$839$	−15.6848	−0.541501	−0.270750	+	0.962650i	$0.587272\pi$
$840$	0	0
$841$	24.1392	0.832385
$842$	0	0
$843$	−36.0758	−1.24252
$844$	0	0
$845$	−19.8761	−0.683758
$846$	0	0
$847$	−34.5230	−1.18623
$848$	0	0
$849$	71.0082	2.43699
$850$	0	0
$851$	−7.48228	−0.256489
$852$	0	0
$853$	−3.57224	−0.122311	−0.0611555	−	0.998128i	$-0.519479\pi$
$853$	−3.57224	−0.122311	−0.0611555	+	0.998128i	$0.519479\pi$
$854$	0	0
$855$	−5.98868	−0.204808
$856$	0	0
$857$	56.7309	1.93789	0.968945	−	0.247276i	$-0.0795355\pi$
$857$	56.7309	1.93789	0.968945	+	0.247276i	$0.0795355\pi$
$858$	0	0
$859$	−18.6059	−0.634825	−0.317412	−	0.948288i	$-0.602814\pi$
$859$	−18.6059	−0.634825	−0.317412	+	0.948288i	$0.602814\pi$
$860$	0	0
$861$	51.2713	1.74732
$862$	0	0
$863$	−17.9395	−0.610666	−0.305333	−	0.952246i	$-0.598768\pi$
$863$	−17.9395	−0.610666	−0.305333	+	0.952246i	$0.598768\pi$
$864$	0	0
$865$	−37.3612	−1.27032
$866$	0	0
$867$	−20.7819	−0.705790
$868$	0	0
$869$	1.14341	0.0387874
$870$	0	0
$871$	5.53858	0.187668
$872$	0	0
$873$	−39.6537	−1.34208
$874$	0	0
$875$	−51.7114	−1.74816
$876$	0	0
$877$	57.6201	1.94569	0.972846	−	0.231455i	$-0.0743485\pi$
$877$	57.6201	1.94569	0.972846	+	0.231455i	$0.0743485\pi$
$878$	0	0
$879$	−26.3205	−0.887767
$880$	0	0
$881$	15.3431	0.516923	0.258461	−	0.966022i	$-0.416785\pi$
$881$	15.3431	0.516923	0.258461	+	0.966022i	$0.416785\pi$
$882$	0	0
$883$	−17.8608	−0.601065	−0.300532	−	0.953772i	$-0.597164\pi$
$883$	−17.8608	−0.601065	−0.300532	+	0.953772i	$0.597164\pi$
$884$	0	0
$885$	−28.0085	−0.941495
$886$	0	0
$887$	−49.1445	−1.65011	−0.825055	−	0.565053i	$-0.808856\pi$
$887$	−49.1445	−1.65011	−0.825055	+	0.565053i	$0.808856\pi$
$888$	0	0
$889$	32.8056	1.10026
$890$	0	0
$891$	−10.8311	−0.362855
$892$	0	0
$893$	−7.59774	−0.254249
$894$	0	0
$895$	−7.25601	−0.242542
$896$	0	0
$897$	−3.77796	−0.126143
$898$	0	0
$899$	−4.76771	−0.159012
$900$	0	0
$901$	19.7932	0.659407
$902$	0	0
$903$	−117.866	−3.92233
$904$	0	0
$905$	−19.3165	−0.642104
$906$	0	0
$907$	9.19971	0.305471	0.152736	−	0.988267i	$-0.451192\pi$
$907$	9.19971	0.305471	0.152736	+	0.988267i	$0.451192\pi$
$908$	0	0
$909$	−18.0085	−0.597303
$910$	0	0
$911$	25.2939	0.838024	0.419012	−	0.907981i	$-0.362376\pi$
$911$	25.2939	0.838024	0.419012	+	0.907981i	$0.362376\pi$
$912$	0	0
$913$	−21.9915	−0.727813
$914$	0	0
$915$	37.7454	1.24782
$916$	0	0
$917$	0.213882	0.00706301
$918$	0	0
$919$	44.8155	1.47833	0.739164	−	0.673525i	$-0.235221\pi$
$919$	44.8155	1.47833	0.739164	+	0.673525i	$0.235221\pi$
$920$	0	0
$921$	44.4752	1.46551
$922$	0	0
$923$	−1.27125	−0.0418437
$924$	0	0
$925$	−9.42068	−0.309750
$926$	0	0
$927$	−47.1502	−1.54861
$928$	0	0
$929$	−5.65119	−0.185409	−0.0927047	−	0.995694i	$-0.529551\pi$
$929$	−5.65119	−0.185409	−0.0927047	+	0.995694i	$0.529551\pi$
$930$	0	0
$931$	−11.8874	−0.389594
$932$	0	0
$933$	81.1544	2.65688
$934$	0	0
$935$	8.37962	0.274043
$936$	0	0
$937$	−8.58641	−0.280506	−0.140253	−	0.990116i	$-0.544792\pi$
$937$	−8.58641	−0.280506	−0.140253	+	0.990116i	$0.544792\pi$
$938$	0	0
$939$	−65.5386	−2.13877
$940$	0	0
$941$	34.8155	1.13495	0.567477	−	0.823389i	$-0.307920\pi$
$941$	34.8155	1.13495	0.567477	+	0.823389i	$0.307920\pi$
$942$	0	0
$943$	−8.82718	−0.287452
$944$	0	0
$945$	13.4965	0.439040
$946$	0	0
$947$	−18.6551	−0.606209	−0.303105	−	0.952957i	$-0.598023\pi$
$947$	−18.6551	−0.606209	−0.303105	+	0.952957i	$0.598023\pi$
$948$	0	0
$949$	−12.4841	−0.405250
$950$	0	0
$951$	−30.2447	−0.980751
$952$	0	0
$953$	37.7227	1.22196	0.610980	−	0.791646i	$-0.290776\pi$
$953$	37.7227	1.22196	0.610980	+	0.791646i	$0.290776\pi$
$954$	0	0
$955$	20.0492	0.648777
$956$	0	0
$957$	−33.1055	−1.07015
$958$	0	0
$959$	75.5804	2.44062
$960$	0	0
$961$	−30.5722	−0.986201
$962$	0	0
$963$	20.3909	0.657089
$964$	0	0
$965$	17.1434	0.551866
$966$	0	0
$967$	−26.3683	−0.847947	−0.423974	−	0.905675i	$-0.639365\pi$
$967$	−26.3683	−0.847947	−0.423974	+	0.905675i	$0.639365\pi$
$968$	0	0
$969$	−7.79321	−0.250354
$970$	0	0
$971$	0.978737	0.0314092	0.0157046	−	0.999877i	$-0.495001\pi$
$971$	0.978737	0.0314092	0.0157046	+	0.999877i	$0.495001\pi$
$972$	0	0
$973$	−35.8619	−1.14968
$974$	0	0
$975$	−4.75670	−0.152336
$976$	0	0
$977$	−0.254623	−0.00814611	−0.00407306	−	0.999992i	$-0.501296\pi$
$977$	−0.254623	−0.00814611	−0.00407306	+	0.999992i	$0.501296\pi$
$978$	0	0
$979$	−25.4217	−0.812482
$980$	0	0
$981$	55.4217	1.76948
$982$	0	0
$983$	−49.3754	−1.57483	−0.787415	−	0.616423i	$-0.788581\pi$
$983$	−49.3754	−1.57483	−0.787415	+	0.616423i	$0.788581\pi$
$984$	0	0
$985$	14.9929	0.477714
$986$	0	0
$987$	85.7759	2.73028
$988$	0	0
$989$	20.2925	0.645264
$990$	0	0
$991$	37.6101	1.19473	0.597363	−	0.801971i	$-0.296215\pi$
$991$	37.6101	1.19473	0.597363	+	0.801971i	$0.296215\pi$
$992$	0	0
$993$	32.7256	1.03851
$994$	0	0
$995$	17.9593	0.569347
$996$	0	0
$997$	19.3867	0.613983	0.306992	−	0.951712i	$-0.400678\pi$
$997$	19.3867	0.613983	0.306992	+	0.951712i	$0.400678\pi$
$998$	0	0
$999$	7.48228	0.236729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.be.1.3		3
4.3	odd	2		4864.2.a.bc.1.1		3
8.3	odd	2		4864.2.a.bf.1.3		3
8.5	even	2		4864.2.a.bd.1.1		3
16.3	odd	4		2432.2.c.g.1217.2	yes	6
16.5	even	4		2432.2.c.f.1217.2	✓	6
16.11	odd	4		2432.2.c.g.1217.5	yes	6
16.13	even	4		2432.2.c.f.1217.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.2	✓	6	16.5	even	4
2432.2.c.f.1217.5	yes	6	16.13	even	4
2432.2.c.g.1217.2	yes	6	16.3	odd	4
2432.2.c.g.1217.5	yes	6	16.11	odd	4
4864.2.a.bc.1.1		3	4.3	odd	2
4864.2.a.bd.1.1		3	8.5	even	2
4864.2.a.be.1.3		3	1.1	even	1	trivial
4864.2.a.bf.1.3		3	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +O(q^{10})\)	\(q+2.59774 q^{3} +1.59774 q^{5} +4.34596 q^{7} +3.74823 q^{9} +1.74823 q^{11} +0.748228 q^{13} +4.15049 q^{15} +3.00000 q^{17} -1.00000 q^{19} +11.2897 q^{21} -1.94370 q^{23} -2.44724 q^{25} +1.94370 q^{27} -7.28966 q^{29} +0.654037 q^{31} +4.54143 q^{33} +6.94370 q^{35} +3.84951 q^{37} +1.94370 q^{39} +4.54143 q^{41} -10.4402 q^{43} +5.98868 q^{45} +7.59774 q^{47} +11.8874 q^{49} +7.79321 q^{51} +6.59774 q^{53} +2.79321 q^{55} -2.59774 q^{57} -6.74823 q^{59} +9.09419 q^{61} +16.2897 q^{63} +1.19547 q^{65} +7.40226 q^{67} -5.04921 q^{69} -1.69901 q^{71} -16.6848 q^{73} -6.35729 q^{75} +7.59774 q^{77} +0.654037 q^{79} -6.19547 q^{81} -12.5793 q^{83} +4.79321 q^{85} -18.9366 q^{87} -14.5414 q^{89} +3.25177 q^{91} +1.69901 q^{93} -1.59774 q^{95} -10.5793 q^{97} +6.55276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9	\( 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9} + 2 q^{11} - q^{13} + 16 q^{15} + 9 q^{17} - 3 q^{19} + 7 q^{21} + 11 q^{23} + 3 q^{25} - 11 q^{27} + 5 q^{29} + 12 q^{31} - 10 q^{33} + 4 q^{35} + 8 q^{37} - 11 q^{39} - 10 q^{41} - 8 q^{43} - 16 q^{45} + 16 q^{47} + 2 q^{49} + 3 q^{51} + 13 q^{53} - 12 q^{55} - q^{57} - 17 q^{59} + 14 q^{61} + 22 q^{63} - 10 q^{65} + 29 q^{67} - 19 q^{69} + 2 q^{71} - 17 q^{73} - 43 q^{75} + 16 q^{77} + 12 q^{79} - 5 q^{81} + 16 q^{83} - 6 q^{85} - 27 q^{87} - 20 q^{89} + 13 q^{91} - 2 q^{93} + 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9 + 2 * q^11 - q^13 + 16 * q^15 + 9 * q^17 - 3 * q^19 + 7 * q^21 + 11 * q^23 + 3 * q^25 - 11 * q^27 + 5 * q^29 + 12 * q^31 - 10 * q^33 + 4 * q^35 + 8 * q^37 - 11 * q^39 - 10 * q^41 - 8 * q^43 - 16 * q^45 + 16 * q^47 + 2 * q^49 + 3 * q^51 + 13 * q^53 - 12 * q^55 - q^57 - 17 * q^59 + 14 * q^61 + 22 * q^63 - 10 * q^65 + 29 * q^67 - 19 * q^69 + 2 * q^71 - 17 * q^73 - 43 * q^75 + 16 * q^77 + 12 * q^79 - 5 * q^81 + 16 * q^83 - 6 * q^85 - 27 * q^87 - 20 * q^89 + 13 * q^91 - 2 * q^93 + 2 * q^95 + 22 * q^97 + 30 * q^99

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