LMFDB - Embedded newform 2888.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] 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("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-6,0,-2,0,-2,0,2,0,12,0,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.10564000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 24x^4 - 28x^3 - 21x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.26814$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26814 q^{3} +2.03239 q^{5} +4.86584 q^{7} +2.14447 q^{9} +0.629774 q^{11} -0.799892 q^{13} -4.60975 q^{15} -4.64783 q^{17} -11.0364 q^{21} -8.96303 q^{23} -0.869391 q^{25} +1.94047 q^{27} -6.31239 q^{29} -9.70568 q^{31} -1.42842 q^{33} +9.88928 q^{35} -8.43373 q^{37} +1.81427 q^{39} -5.66039 q^{41} -3.80014 q^{43} +4.35840 q^{45} +4.01565 q^{47} +16.6764 q^{49} +10.5419 q^{51} +1.25618 q^{53} +1.27995 q^{55} +10.2732 q^{59} -13.5642 q^{61} +10.4346 q^{63} -1.62569 q^{65} -1.26315 q^{67} +20.3294 q^{69} +8.22911 q^{71} -6.77791 q^{73} +1.97190 q^{75} +3.06438 q^{77} -6.88668 q^{79} -10.8347 q^{81} -0.202623 q^{83} -9.44620 q^{85} +14.3174 q^{87} +7.54055 q^{89} -3.89214 q^{91} +22.0139 q^{93} -11.0607 q^{97} +1.35053 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * 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
$3$	−2.26814	−1.30951	−0.654756	−	0.755840i	$-0.727229\pi$
$3$	−2.26814	−1.30951	−0.654756	+	0.755840i	$0.727229\pi$
$4$	0	0
$5$	2.03239	0.908912	0.454456	−	0.890769i	$-0.349834\pi$
$5$	2.03239	0.908912	0.454456	+	0.890769i	$0.349834\pi$
$6$	0	0
$7$	4.86584	1.83911	0.919556	−	0.392958i	$-0.128548\pi$
$7$	4.86584	1.83911	0.919556	+	0.392958i	$0.128548\pi$
$8$	0	0
$9$	2.14447	0.714823
$10$	0	0
$11$	0.629774	0.189884	0.0949420	−	0.995483i	$-0.469733\pi$
$11$	0.629774	0.189884	0.0949420	+	0.995483i	$0.469733\pi$
$12$	0	0
$13$	−0.799892	−0.221850	−0.110925	−	0.993829i	$-0.535381\pi$
$13$	−0.799892	−0.221850	−0.110925	+	0.993829i	$0.535381\pi$
$14$	0	0
$15$	−4.60975	−1.19023
$16$	0	0
$17$	−4.64783	−1.12726	−0.563632	−	0.826026i	$-0.690596\pi$
$17$	−4.64783	−1.12726	−0.563632	+	0.826026i	$0.690596\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−11.0364	−2.40834
$22$	0	0
$23$	−8.96303	−1.86892	−0.934460	−	0.356068i	$-0.884117\pi$
$23$	−8.96303	−1.86892	−0.934460	+	0.356068i	$0.884117\pi$
$24$	0	0
$25$	−0.869391	−0.173878
$26$	0	0
$27$	1.94047	0.373443
$28$	0	0
$29$	−6.31239	−1.17218	−0.586091	−	0.810246i	$-0.699334\pi$
$29$	−6.31239	−1.17218	−0.586091	+	0.810246i	$0.699334\pi$
$30$	0	0
$31$	−9.70568	−1.74319	−0.871596	−	0.490225i	$-0.836915\pi$
$31$	−9.70568	−1.74319	−0.871596	+	0.490225i	$0.836915\pi$
$32$	0	0
$33$	−1.42842	−0.248655
$34$	0	0
$35$	9.88928	1.67159
$36$	0	0
$37$	−8.43373	−1.38650	−0.693249	−	0.720699i	$-0.743821\pi$
$37$	−8.43373	−1.38650	−0.693249	+	0.720699i	$0.743821\pi$
$38$	0	0
$39$	1.81427	0.290516
$40$	0	0
$41$	−5.66039	−0.884004	−0.442002	−	0.897014i	$-0.645732\pi$
$41$	−5.66039	−0.884004	−0.442002	+	0.897014i	$0.645732\pi$
$42$	0	0
$43$	−3.80014	−0.579517	−0.289758	−	0.957100i	$-0.593575\pi$
$43$	−3.80014	−0.579517	−0.289758	+	0.957100i	$0.593575\pi$
$44$	0	0
$45$	4.35840	0.649711
$46$	0	0
$47$	4.01565	0.585743	0.292872	−	0.956152i	$-0.405389\pi$
$47$	4.01565	0.585743	0.292872	+	0.956152i	$0.405389\pi$
$48$	0	0
$49$	16.6764	2.38234
$50$	0	0
$51$	10.5419	1.47617
$52$	0	0
$53$	1.25618	0.172549	0.0862745	−	0.996271i	$-0.472504\pi$
$53$	1.25618	0.172549	0.0862745	+	0.996271i	$0.472504\pi$
$54$	0	0
$55$	1.27995	0.172588
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.2732	1.33746	0.668729	−	0.743506i	$-0.266838\pi$
$59$	10.2732	1.33746	0.668729	+	0.743506i	$0.266838\pi$
$60$	0	0
$61$	−13.5642	−1.73672	−0.868360	−	0.495935i	$-0.834825\pi$
$61$	−13.5642	−1.73672	−0.868360	+	0.495935i	$0.834825\pi$
$62$	0	0
$63$	10.4346	1.31464
$64$	0	0
$65$	−1.62569	−0.201642
$66$	0	0
$67$	−1.26315	−0.154318	−0.0771590	−	0.997019i	$-0.524585\pi$
$67$	−1.26315	−0.154318	−0.0771590	+	0.997019i	$0.524585\pi$
$68$	0	0
$69$	20.3294	2.44737
$70$	0	0
$71$	8.22911	0.976616	0.488308	−	0.872671i	$-0.337614\pi$
$71$	8.22911	0.976616	0.488308	+	0.872671i	$0.337614\pi$
$72$	0	0
$73$	−6.77791	−0.793295	−0.396647	−	0.917971i	$-0.629826\pi$
$73$	−6.77791	−0.793295	−0.396647	+	0.917971i	$0.629826\pi$
$74$	0	0
$75$	1.97190	0.227696
$76$	0	0
$77$	3.06438	0.349218
$78$	0	0
$79$	−6.88668	−0.774812	−0.387406	−	0.921909i	$-0.626629\pi$
$79$	−6.88668	−0.774812	−0.387406	+	0.921909i	$0.626629\pi$
$80$	0	0
$81$	−10.8347	−1.20385
$82$	0	0
$83$	−0.202623	−0.0222407	−0.0111204	−	0.999938i	$-0.503540\pi$
$83$	−0.202623	−0.0222407	−0.0111204	+	0.999938i	$0.503540\pi$
$84$	0	0
$85$	−9.44620	−1.02458
$86$	0	0
$87$	14.3174	1.53499
$88$	0	0
$89$	7.54055	0.799296	0.399648	−	0.916669i	$-0.369132\pi$
$89$	7.54055	0.799296	0.399648	+	0.916669i	$0.369132\pi$
$90$	0	0
$91$	−3.89214	−0.408008
$92$	0	0
$93$	22.0139	2.28273
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.0607	−1.12304	−0.561520	−	0.827463i	$-0.689783\pi$
$97$	−11.0607	−1.12304	−0.561520	+	0.827463i	$0.689783\pi$
$98$	0	0
$99$	1.35053	0.135733
$100$	0	0
$101$	−1.65818	−0.164995	−0.0824975	−	0.996591i	$-0.526290\pi$
$101$	−1.65818	−0.164995	−0.0824975	+	0.996591i	$0.526290\pi$
$102$	0	0
$103$	−1.85589	−0.182867	−0.0914333	−	0.995811i	$-0.529145\pi$
$103$	−1.85589	−0.182867	−0.0914333	+	0.995811i	$0.529145\pi$
$104$	0	0
$105$	−22.4303	−2.18897
$106$	0	0
$107$	14.8662	1.43717	0.718583	−	0.695441i	$-0.244791\pi$
$107$	14.8662	1.43717	0.718583	+	0.695441i	$0.244791\pi$
$108$	0	0
$109$	−0.309512	−0.0296459	−0.0148229	−	0.999890i	$-0.504718\pi$
$109$	−0.309512	−0.0296459	−0.0148229	+	0.999890i	$0.504718\pi$
$110$	0	0
$111$	19.1289	1.81564
$112$	0	0
$113$	4.95674	0.466291	0.233146	−	0.972442i	$-0.425098\pi$
$113$	4.95674	0.466291	0.233146	+	0.972442i	$0.425098\pi$
$114$	0	0
$115$	−18.2164	−1.69868
$116$	0	0
$117$	−1.71534	−0.158584
$118$	0	0
$119$	−22.6156	−2.07316
$120$	0	0
$121$	−10.6034	−0.963944
$122$	0	0
$123$	12.8386	1.15761
$124$	0	0
$125$	−11.9289	−1.06695
$126$	0	0
$127$	1.51273	0.134233	0.0671165	−	0.997745i	$-0.478620\pi$
$127$	1.51273	0.134233	0.0671165	+	0.997745i	$0.478620\pi$
$128$	0	0
$129$	8.61927	0.758884
$130$	0	0
$131$	0.887697	0.0775584	0.0387792	−	0.999248i	$-0.487653\pi$
$131$	0.887697	0.0775584	0.0387792	+	0.999248i	$0.487653\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.94379	0.339427
$136$	0	0
$137$	0.826203	0.0705873	0.0352937	−	0.999377i	$-0.488763\pi$
$137$	0.826203	0.0705873	0.0352937	+	0.999377i	$0.488763\pi$
$138$	0	0
$139$	19.4899	1.65311	0.826555	−	0.562856i	$-0.190297\pi$
$139$	19.4899	1.65311	0.826555	+	0.562856i	$0.190297\pi$
$140$	0	0
$141$	−9.10807	−0.767038
$142$	0	0
$143$	−0.503751	−0.0421258
$144$	0	0
$145$	−12.8292	−1.06541
$146$	0	0
$147$	−37.8243	−3.11970
$148$	0	0
$149$	9.48397	0.776957	0.388479	−	0.921458i	$-0.373001\pi$
$149$	9.48397	0.776957	0.388479	+	0.921458i	$0.373001\pi$
$150$	0	0
$151$	4.57575	0.372369	0.186185	−	0.982515i	$-0.440388\pi$
$151$	4.57575	0.372369	0.186185	+	0.982515i	$0.440388\pi$
$152$	0	0
$153$	−9.96712	−0.805794
$154$	0	0
$155$	−19.7257	−1.58441
$156$	0	0
$157$	4.20692	0.335749	0.167874	−	0.985808i	$-0.446310\pi$
$157$	4.20692	0.335749	0.167874	+	0.985808i	$0.446310\pi$
$158$	0	0
$159$	−2.84919	−0.225955
$160$	0	0
$161$	−43.6126	−3.43716
$162$	0	0
$163$	−2.14191	−0.167767	−0.0838837	−	0.996476i	$-0.526732\pi$
$163$	−2.14191	−0.167767	−0.0838837	+	0.996476i	$0.526732\pi$
$164$	0	0
$165$	−2.90310	−0.226006
$166$	0	0
$167$	−7.92896	−0.613561	−0.306781	−	0.951780i	$-0.599252\pi$
$167$	−7.92896	−0.613561	−0.306781	+	0.951780i	$0.599252\pi$
$168$	0	0
$169$	−12.3602	−0.950783
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.562006	−0.0427285	−0.0213643	−	0.999772i	$-0.506801\pi$
$173$	−0.562006	−0.0427285	−0.0213643	+	0.999772i	$0.506801\pi$
$174$	0	0
$175$	−4.23031	−0.319782
$176$	0	0
$177$	−23.3011	−1.75142
$178$	0	0
$179$	−2.71190	−0.202697	−0.101348	−	0.994851i	$-0.532316\pi$
$179$	−2.71190	−0.202697	−0.101348	+	0.994851i	$0.532316\pi$
$180$	0	0
$181$	21.0287	1.56305	0.781527	−	0.623872i	$-0.214441\pi$
$181$	21.0287	1.56305	0.781527	+	0.623872i	$0.214441\pi$
$182$	0	0
$183$	30.7656	2.27426
$184$	0	0
$185$	−17.1406	−1.26020
$186$	0	0
$187$	−2.92708	−0.214049
$188$	0	0
$189$	9.44199	0.686804
$190$	0	0
$191$	6.84240	0.495099	0.247550	−	0.968875i	$-0.420375\pi$
$191$	6.84240	0.495099	0.247550	+	0.968875i	$0.420375\pi$
$192$	0	0
$193$	−15.9865	−1.15073	−0.575365	−	0.817897i	$-0.695140\pi$
$193$	−15.9865	−1.15073	−0.575365	+	0.817897i	$0.695140\pi$
$194$	0	0
$195$	3.68730	0.264053
$196$	0	0
$197$	19.6602	1.40073	0.700366	−	0.713784i	$-0.253020\pi$
$197$	19.6602	1.40073	0.700366	+	0.713784i	$0.253020\pi$
$198$	0	0
$199$	14.2790	1.01221	0.506107	−	0.862471i	$-0.331084\pi$
$199$	14.2790	1.01221	0.506107	+	0.862471i	$0.331084\pi$
$200$	0	0
$201$	2.86500	0.202081
$202$	0	0
$203$	−30.7150	−2.15577
$204$	0	0
$205$	−11.5041	−0.803482
$206$	0	0
$207$	−19.2209	−1.33595
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.70444	−0.117339	−0.0586693	−	0.998277i	$-0.518686\pi$
$211$	−1.70444	−0.117339	−0.0586693	+	0.998277i	$0.518686\pi$
$212$	0	0
$213$	−18.6648	−1.27889
$214$	0	0
$215$	−7.72338	−0.526730
$216$	0	0
$217$	−47.2262	−3.20593
$218$	0	0
$219$	15.3733	1.03883
$220$	0	0
$221$	3.71776	0.250084
$222$	0	0
$223$	−13.8833	−0.929693	−0.464847	−	0.885391i	$-0.653891\pi$
$223$	−13.8833	−0.929693	−0.464847	+	0.885391i	$0.653891\pi$
$224$	0	0
$225$	−1.86438	−0.124292
$226$	0	0
$227$	11.1983	0.743258	0.371629	−	0.928381i	$-0.378799\pi$
$227$	11.1983	0.743258	0.371629	+	0.928381i	$0.378799\pi$
$228$	0	0
$229$	−22.9335	−1.51549	−0.757744	−	0.652552i	$-0.773698\pi$
$229$	−22.9335	−1.51549	−0.757744	+	0.652552i	$0.773698\pi$
$230$	0	0
$231$	−6.95044	−0.457306
$232$	0	0
$233$	9.37923	0.614454	0.307227	−	0.951636i	$-0.400599\pi$
$233$	9.37923	0.614454	0.307227	+	0.951636i	$0.400599\pi$
$234$	0	0
$235$	8.16137	0.532389
$236$	0	0
$237$	15.6200	1.01463
$238$	0	0
$239$	−15.4897	−1.00195	−0.500973	−	0.865463i	$-0.667024\pi$
$239$	−15.4897	−1.00195	−0.500973	+	0.865463i	$0.667024\pi$
$240$	0	0
$241$	21.1993	1.36557	0.682785	−	0.730620i	$-0.260769\pi$
$241$	21.1993	1.36557	0.682785	+	0.730620i	$0.260769\pi$
$242$	0	0
$243$	18.7531	1.20302
$244$	0	0
$245$	33.8929	2.16534
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.459577	0.0291245
$250$	0	0
$251$	14.0898	0.889340	0.444670	−	0.895694i	$-0.353321\pi$
$251$	14.0898	0.889340	0.444670	+	0.895694i	$0.353321\pi$
$252$	0	0
$253$	−5.64468	−0.354878
$254$	0	0
$255$	21.4253	1.34171
$256$	0	0
$257$	−10.8367	−0.675974	−0.337987	−	0.941151i	$-0.609746\pi$
$257$	−10.8367	−0.675974	−0.337987	+	0.941151i	$0.609746\pi$
$258$	0	0
$259$	−41.0372	−2.54992
$260$	0	0
$261$	−13.5367	−0.837902
$262$	0	0
$263$	−10.4251	−0.642839	−0.321419	−	0.946937i	$-0.604160\pi$
$263$	−10.4251	−0.642839	−0.321419	+	0.946937i	$0.604160\pi$
$264$	0	0
$265$	2.55304	0.156832
$266$	0	0
$267$	−17.1030	−1.04669
$268$	0	0
$269$	−0.423672	−0.0258318	−0.0129159	−	0.999917i	$-0.504111\pi$
$269$	−0.423672	−0.0258318	−0.0129159	+	0.999917i	$0.504111\pi$
$270$	0	0
$271$	−19.2009	−1.16637	−0.583187	−	0.812338i	$-0.698195\pi$
$271$	−19.2009	−1.16637	−0.583187	+	0.812338i	$0.698195\pi$
$272$	0	0
$273$	8.82793	0.534291
$274$	0	0
$275$	−0.547520	−0.0330167
$276$	0	0
$277$	−12.2083	−0.733523	−0.366761	−	0.930315i	$-0.619533\pi$
$277$	−12.2083	−0.733523	−0.366761	+	0.930315i	$0.619533\pi$
$278$	0	0
$279$	−20.8135	−1.24607
$280$	0	0
$281$	−15.4949	−0.924346	−0.462173	−	0.886790i	$-0.652930\pi$
$281$	−15.4949	−0.924346	−0.462173	+	0.886790i	$0.652930\pi$
$282$	0	0
$283$	17.8134	1.05890	0.529448	−	0.848342i	$-0.322399\pi$
$283$	17.8134	1.05890	0.529448	+	0.848342i	$0.322399\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.5425	−1.62578
$288$	0	0
$289$	4.60229	0.270723
$290$	0	0
$291$	25.0871	1.47063
$292$	0	0
$293$	20.9653	1.22481	0.612404	−	0.790545i	$-0.290202\pi$
$293$	20.9653	1.22481	0.612404	+	0.790545i	$0.290202\pi$
$294$	0	0
$295$	20.8792	1.21563
$296$	0	0
$297$	1.22206	0.0709109
$298$	0	0
$299$	7.16945	0.414620
$300$	0	0
$301$	−18.4909	−1.06580
$302$	0	0
$303$	3.76098	0.216063
$304$	0	0
$305$	−27.5678	−1.57853
$306$	0	0
$307$	0.237600	0.0135605	0.00678027	−	0.999977i	$-0.497842\pi$
$307$	0.237600	0.0135605	0.00678027	+	0.999977i	$0.497842\pi$
$308$	0	0
$309$	4.20943	0.239466
$310$	0	0
$311$	−26.9139	−1.52614	−0.763072	−	0.646313i	$-0.776310\pi$
$311$	−26.9139	−1.52614	−0.763072	+	0.646313i	$0.776310\pi$
$312$	0	0
$313$	1.73918	0.0983043	0.0491522	−	0.998791i	$-0.484348\pi$
$313$	1.73918	0.0983043	0.0491522	+	0.998791i	$0.484348\pi$
$314$	0	0
$315$	21.2072	1.19489
$316$	0	0
$317$	16.9887	0.954179	0.477089	−	0.878855i	$-0.341692\pi$
$317$	16.9887	0.954179	0.477089	+	0.878855i	$0.341692\pi$
$318$	0	0
$319$	−3.97538	−0.222578
$320$	0	0
$321$	−33.7186	−1.88199
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.695419	0.0385749
$326$	0	0
$327$	0.702017	0.0388216
$328$	0	0
$329$	19.5395	1.07725
$330$	0	0
$331$	−11.9459	−0.656609	−0.328304	−	0.944572i	$-0.606477\pi$
$331$	−11.9459	−0.656609	−0.328304	+	0.944572i	$0.606477\pi$
$332$	0	0
$333$	−18.0859	−0.991100
$334$	0	0
$335$	−2.56721	−0.140261
$336$	0	0
$337$	24.2710	1.32212	0.661062	−	0.750331i	$-0.270106\pi$
$337$	24.2710	1.32212	0.661062	+	0.750331i	$0.270106\pi$
$338$	0	0
$339$	−11.2426	−0.610614
$340$	0	0
$341$	−6.11239	−0.331004
$342$	0	0
$343$	47.0836	2.54227
$344$	0	0
$345$	41.3173	2.22445
$346$	0	0
$347$	0.443192	0.0237918	0.0118959	−	0.999929i	$-0.496213\pi$
$347$	0.443192	0.0237918	0.0118959	+	0.999929i	$0.496213\pi$
$348$	0	0
$349$	−3.16489	−0.169413	−0.0847064	−	0.996406i	$-0.526995\pi$
$349$	−3.16489	−0.169413	−0.0847064	+	0.996406i	$0.526995\pi$
$350$	0	0
$351$	−1.55216	−0.0828484
$352$	0	0
$353$	−2.13050	−0.113395	−0.0566974	−	0.998391i	$-0.518057\pi$
$353$	−2.13050	−0.113395	−0.0566974	+	0.998391i	$0.518057\pi$
$354$	0	0
$355$	16.7248	0.887658
$356$	0	0
$357$	51.2953	2.71484
$358$	0	0
$359$	7.55493	0.398734	0.199367	−	0.979925i	$-0.436111\pi$
$359$	7.55493	0.398734	0.199367	+	0.979925i	$0.436111\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	24.0500	1.26230
$364$	0	0
$365$	−13.7754	−0.721035
$366$	0	0
$367$	11.9830	0.625507	0.312754	−	0.949834i	$-0.398749\pi$
$367$	11.9830	0.625507	0.312754	+	0.949834i	$0.398749\pi$
$368$	0	0
$369$	−12.1385	−0.631906
$370$	0	0
$371$	6.11235	0.317337
$372$	0	0
$373$	13.4062	0.694145	0.347073	−	0.937838i	$-0.387176\pi$
$373$	13.4062	0.694145	0.347073	+	0.937838i	$0.387176\pi$
$374$	0	0
$375$	27.0564	1.39719
$376$	0	0
$377$	5.04923	0.260049
$378$	0	0
$379$	−30.7731	−1.58071	−0.790355	−	0.612649i	$-0.790104\pi$
$379$	−30.7731	−1.58071	−0.790355	+	0.612649i	$0.790104\pi$
$380$	0	0
$381$	−3.43108	−0.175780
$382$	0	0
$383$	−16.1959	−0.827570	−0.413785	−	0.910375i	$-0.635793\pi$
$383$	−16.1959	−0.827570	−0.413785	+	0.910375i	$0.635793\pi$
$384$	0	0
$385$	6.22801	0.317409
$386$	0	0
$387$	−8.14929	−0.414252
$388$	0	0
$389$	17.9684	0.911032	0.455516	−	0.890228i	$-0.349455\pi$
$389$	17.9684	0.911032	0.455516	+	0.890228i	$0.349455\pi$
$390$	0	0
$391$	41.6586	2.10677
$392$	0	0
$393$	−2.01342	−0.101564
$394$	0	0
$395$	−13.9964	−0.704237
$396$	0	0
$397$	−12.1580	−0.610194	−0.305097	−	0.952321i	$-0.598689\pi$
$397$	−12.1580	−0.610194	−0.305097	+	0.952321i	$0.598689\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.0990	1.40320	0.701598	−	0.712573i	$-0.252470\pi$
$401$	28.0990	1.40320	0.701598	+	0.712573i	$0.252470\pi$
$402$	0	0
$403$	7.76350	0.386727
$404$	0	0
$405$	−22.0203	−1.09420
$406$	0	0
$407$	−5.31135	−0.263274
$408$	0	0
$409$	7.46974	0.369355	0.184677	−	0.982799i	$-0.440876\pi$
$409$	7.46974	0.369355	0.184677	+	0.982799i	$0.440876\pi$
$410$	0	0
$411$	−1.87395	−0.0924350
$412$	0	0
$413$	49.9878	2.45974
$414$	0	0
$415$	−0.411808	−0.0202149
$416$	0	0
$417$	−44.2058	−2.16477
$418$	0	0
$419$	28.3307	1.38405	0.692024	−	0.721875i	$-0.256719\pi$
$419$	28.3307	1.38405	0.692024	+	0.721875i	$0.256719\pi$
$420$	0	0
$421$	34.0098	1.65754	0.828769	−	0.559592i	$-0.189042\pi$
$421$	34.0098	1.65754	0.828769	+	0.559592i	$0.189042\pi$
$422$	0	0
$423$	8.61144	0.418703
$424$	0	0
$425$	4.04078	0.196006
$426$	0	0
$427$	−66.0012	−3.19402
$428$	0	0
$429$	1.14258	0.0551643
$430$	0	0
$431$	−25.4279	−1.22482	−0.612409	−	0.790541i	$-0.709799\pi$
$431$	−25.4279	−1.22482	−0.612409	+	0.790541i	$0.709799\pi$
$432$	0	0
$433$	−13.4144	−0.644655	−0.322328	−	0.946628i	$-0.604465\pi$
$433$	−13.4144	−0.644655	−0.322328	+	0.946628i	$0.604465\pi$
$434$	0	0
$435$	29.0985	1.39517
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.51697	0.406493	0.203247	−	0.979128i	$-0.434851\pi$
$439$	8.51697	0.406493	0.203247	+	0.979128i	$0.434851\pi$
$440$	0	0
$441$	35.7619	1.70295
$442$	0	0
$443$	−14.6107	−0.694177	−0.347088	−	0.937832i	$-0.612830\pi$
$443$	−14.6107	−0.694177	−0.347088	+	0.937832i	$0.612830\pi$
$444$	0	0
$445$	15.3253	0.726490
$446$	0	0
$447$	−21.5110	−1.01744
$448$	0	0
$449$	−4.09508	−0.193259	−0.0966293	−	0.995320i	$-0.530806\pi$
$449$	−4.09508	−0.193259	−0.0966293	+	0.995320i	$0.530806\pi$
$450$	0	0
$451$	−3.56477	−0.167858
$452$	0	0
$453$	−10.3784	−0.487622
$454$	0	0
$455$	−7.91035	−0.370843
$456$	0	0
$457$	28.5352	1.33482	0.667410	−	0.744691i	$-0.267403\pi$
$457$	28.5352	1.33482	0.667410	+	0.744691i	$0.267403\pi$
$458$	0	0
$459$	−9.01895	−0.420969
$460$	0	0
$461$	20.6494	0.961740	0.480870	−	0.876792i	$-0.340321\pi$
$461$	20.6494	0.961740	0.480870	+	0.876792i	$0.340321\pi$
$462$	0	0
$463$	−15.1413	−0.703675	−0.351838	−	0.936061i	$-0.614443\pi$
$463$	−15.1413	−0.703675	−0.351838	+	0.936061i	$0.614443\pi$
$464$	0	0
$465$	44.7408	2.07480
$466$	0	0
$467$	−33.9214	−1.56969	−0.784847	−	0.619689i	$-0.787259\pi$
$467$	−33.9214	−1.56969	−0.784847	+	0.619689i	$0.787259\pi$
$468$	0	0
$469$	−6.14626	−0.283808
$470$	0	0
$471$	−9.54190	−0.439667
$472$	0	0
$473$	−2.39323	−0.110041
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.69383	0.123342
$478$	0	0
$479$	−15.7155	−0.718061	−0.359030	−	0.933326i	$-0.616893\pi$
$479$	−15.7155	−0.718061	−0.359030	+	0.933326i	$0.616893\pi$
$480$	0	0
$481$	6.74608	0.307595
$482$	0	0
$483$	98.9196	4.50100
$484$	0	0
$485$	−22.4796	−1.02074
$486$	0	0
$487$	−21.5863	−0.978169	−0.489084	−	0.872236i	$-0.662669\pi$
$487$	−21.5863	−0.978169	−0.489084	+	0.872236i	$0.662669\pi$
$488$	0	0
$489$	4.85816	0.219694
$490$	0	0
$491$	28.4492	1.28389	0.641947	−	0.766749i	$-0.278127\pi$
$491$	28.4492	1.28389	0.641947	+	0.766749i	$0.278127\pi$
$492$	0	0
$493$	29.3389	1.32136
$494$	0	0
$495$	2.74481	0.123370
$496$	0	0
$497$	40.0415	1.79611
$498$	0	0
$499$	5.89187	0.263756	0.131878	−	0.991266i	$-0.457899\pi$
$499$	5.89187	0.263756	0.131878	+	0.991266i	$0.457899\pi$
$500$	0	0
$501$	17.9840	0.803466
$502$	0	0
$503$	21.8571	0.974560	0.487280	−	0.873246i	$-0.337989\pi$
$503$	21.8571	0.974560	0.487280	+	0.873246i	$0.337989\pi$
$504$	0	0
$505$	−3.37007	−0.149966
$506$	0	0
$507$	28.0346	1.24506
$508$	0	0
$509$	−7.15691	−0.317224	−0.158612	−	0.987341i	$-0.550702\pi$
$509$	−7.15691	−0.317224	−0.158612	+	0.987341i	$0.550702\pi$
$510$	0	0
$511$	−32.9802	−1.45896
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.77190	−0.166210
$516$	0	0
$517$	2.52895	0.111223
$518$	0	0
$519$	1.27471	0.0559535
$520$	0	0
$521$	8.87178	0.388680	0.194340	−	0.980934i	$-0.437744\pi$
$521$	8.87178	0.388680	0.194340	+	0.980934i	$0.437744\pi$
$522$	0	0
$523$	−19.2138	−0.840160	−0.420080	−	0.907487i	$-0.637998\pi$
$523$	−19.2138	−0.840160	−0.420080	+	0.907487i	$0.637998\pi$
$524$	0	0
$525$	9.59495	0.418758
$526$	0	0
$527$	45.1103	1.96504
$528$	0	0
$529$	57.3358	2.49286
$530$	0	0
$531$	22.0306	0.956046
$532$	0	0
$533$	4.52770	0.196116
$534$	0	0
$535$	30.2139	1.30626
$536$	0	0
$537$	6.15097	0.265434
$538$	0	0
$539$	10.5023	0.452368
$540$	0	0
$541$	−3.67916	−0.158179	−0.0790897	−	0.996868i	$-0.525201\pi$
$541$	−3.67916	−0.158179	−0.0790897	+	0.996868i	$0.525201\pi$
$542$	0	0
$543$	−47.6961	−2.04684
$544$	0	0
$545$	−0.629049	−0.0269455
$546$	0	0
$547$	−4.38138	−0.187334	−0.0936672	−	0.995604i	$-0.529859\pi$
$547$	−4.38138	−0.187334	−0.0936672	+	0.995604i	$0.529859\pi$
$548$	0	0
$549$	−29.0880	−1.24145
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−33.5095	−1.42497
$554$	0	0
$555$	38.8774	1.65025
$556$	0	0
$557$	−36.1073	−1.52992	−0.764959	−	0.644079i	$-0.777241\pi$
$557$	−36.1073	−1.52992	−0.764959	+	0.644079i	$0.777241\pi$
$558$	0	0
$559$	3.03971	0.128566
$560$	0	0
$561$	6.63903	0.280300
$562$	0	0
$563$	−30.5369	−1.28698	−0.643488	−	0.765456i	$-0.722513\pi$
$563$	−30.5369	−1.28698	−0.643488	+	0.765456i	$0.722513\pi$
$564$	0	0
$565$	10.0740	0.423818
$566$	0	0
$567$	−52.7197	−2.21402
$568$	0	0
$569$	18.7715	0.786942	0.393471	−	0.919337i	$-0.371274\pi$
$569$	18.7715	0.786942	0.393471	+	0.919337i	$0.371274\pi$
$570$	0	0
$571$	42.2926	1.76989	0.884945	−	0.465696i	$-0.154196\pi$
$571$	42.2926	1.76989	0.884945	+	0.465696i	$0.154196\pi$
$572$	0	0
$573$	−15.5195	−0.648338
$574$	0	0
$575$	7.79237	0.324964
$576$	0	0
$577$	−39.4541	−1.64249	−0.821247	−	0.570572i	$-0.806721\pi$
$577$	−39.4541	−1.64249	−0.821247	+	0.570572i	$0.806721\pi$
$578$	0	0
$579$	36.2596	1.50690
$580$	0	0
$581$	−0.985928	−0.0409032
$582$	0	0
$583$	0.791107	0.0327643
$584$	0	0
$585$	−3.48625	−0.144139
$586$	0	0
$587$	1.58742	0.0655196	0.0327598	−	0.999463i	$-0.489570\pi$
$587$	1.58742	0.0655196	0.0327598	+	0.999463i	$0.489570\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−44.5921	−1.83428
$592$	0	0
$593$	−25.7532	−1.05756	−0.528779	−	0.848759i	$-0.677350\pi$
$593$	−25.7532	−1.05756	−0.528779	+	0.848759i	$0.677350\pi$
$594$	0	0
$595$	−45.9636	−1.88433
$596$	0	0
$597$	−32.3869	−1.32551
$598$	0	0
$599$	44.0105	1.79822	0.899109	−	0.437724i	$-0.144215\pi$
$599$	44.0105	1.79822	0.899109	+	0.437724i	$0.144215\pi$
$600$	0	0
$601$	−5.99288	−0.244455	−0.122227	−	0.992502i	$-0.539004\pi$
$601$	−5.99288	−0.244455	−0.122227	+	0.992502i	$0.539004\pi$
$602$	0	0
$603$	−2.70878	−0.110310
$604$	0	0
$605$	−21.5502	−0.876141
$606$	0	0
$607$	−11.2566	−0.456891	−0.228446	−	0.973557i	$-0.573364\pi$
$607$	−11.2566	−0.456891	−0.228446	+	0.973557i	$0.573364\pi$
$608$	0	0
$609$	69.6661	2.82301
$610$	0	0
$611$	−3.21209	−0.129947
$612$	0	0
$613$	−7.48695	−0.302395	−0.151198	−	0.988504i	$-0.548313\pi$
$613$	−7.48695	−0.302395	−0.151198	+	0.988504i	$0.548313\pi$
$614$	0	0
$615$	26.0930	1.05217
$616$	0	0
$617$	−19.1951	−0.772765	−0.386383	−	0.922339i	$-0.626276\pi$
$617$	−19.1951	−0.772765	−0.386383	+	0.922339i	$0.626276\pi$
$618$	0	0
$619$	−21.4504	−0.862165	−0.431082	−	0.902313i	$-0.641868\pi$
$619$	−21.4504	−0.862165	−0.431082	+	0.902313i	$0.641868\pi$
$620$	0	0
$621$	−17.3925	−0.697935
$622$	0	0
$623$	36.6911	1.47000
$624$	0	0
$625$	−19.8972	−0.795888
$626$	0	0
$627$	0	0
$628$	0	0
$629$	39.1985	1.56295
$630$	0	0
$631$	43.8325	1.74494	0.872472	−	0.488664i	$-0.162516\pi$
$631$	43.8325	1.74494	0.872472	+	0.488664i	$0.162516\pi$
$632$	0	0
$633$	3.86592	0.153656
$634$	0	0
$635$	3.07445	0.122006
$636$	0	0
$637$	−13.3393	−0.528522
$638$	0	0
$639$	17.6471	0.698107
$640$	0	0
$641$	−33.3519	−1.31732	−0.658661	−	0.752440i	$-0.728877\pi$
$641$	−33.3519	−1.31732	−0.658661	+	0.752440i	$0.728877\pi$
$642$	0	0
$643$	23.5928	0.930410	0.465205	−	0.885203i	$-0.345981\pi$
$643$	23.5928	0.930410	0.465205	+	0.885203i	$0.345981\pi$
$644$	0	0
$645$	17.5177	0.689759
$646$	0	0
$647$	4.78026	0.187931	0.0939657	−	0.995575i	$-0.470046\pi$
$647$	4.78026	0.187931	0.0939657	+	0.995575i	$0.470046\pi$
$648$	0	0
$649$	6.46980	0.253962
$650$	0	0
$651$	107.116	4.19820
$652$	0	0
$653$	46.1083	1.80436	0.902179	−	0.431363i	$-0.141967\pi$
$653$	46.1083	1.80436	0.902179	+	0.431363i	$0.141967\pi$
$654$	0	0
$655$	1.80415	0.0704938
$656$	0	0
$657$	−14.5350	−0.567065
$658$	0	0
$659$	34.6130	1.34833	0.674165	−	0.738581i	$-0.264504\pi$
$659$	34.6130	1.34833	0.674165	+	0.738581i	$0.264504\pi$
$660$	0	0
$661$	−26.8551	−1.04454	−0.522271	−	0.852780i	$-0.674915\pi$
$661$	−26.8551	−1.04454	−0.522271	+	0.852780i	$0.674915\pi$
$662$	0	0
$663$	−8.43241	−0.327488
$664$	0	0
$665$	0	0
$666$	0	0
$667$	56.5781	2.19071
$668$	0	0
$669$	31.4892	1.21744
$670$	0	0
$671$	−8.54239	−0.329775
$672$	0	0
$673$	6.83470	0.263458	0.131729	−	0.991286i	$-0.457947\pi$
$673$	6.83470	0.263458	0.131729	+	0.991286i	$0.457947\pi$
$674$	0	0
$675$	−1.68702	−0.0649336
$676$	0	0
$677$	−33.1909	−1.27563	−0.637815	−	0.770189i	$-0.720162\pi$
$677$	−33.1909	−1.27563	−0.637815	+	0.770189i	$0.720162\pi$
$678$	0	0
$679$	−53.8193	−2.06540
$680$	0	0
$681$	−25.3994	−0.973306
$682$	0	0
$683$	−19.7485	−0.755657	−0.377829	−	0.925876i	$-0.623329\pi$
$683$	−19.7485	−0.755657	−0.377829	+	0.925876i	$0.623329\pi$
$684$	0	0
$685$	1.67917	0.0641577
$686$	0	0
$687$	52.0164	1.98455
$688$	0	0
$689$	−1.00481	−0.0382800
$690$	0	0
$691$	−31.9365	−1.21492	−0.607460	−	0.794350i	$-0.707812\pi$
$691$	−31.9365	−1.21492	−0.607460	+	0.794350i	$0.707812\pi$
$692$	0	0
$693$	6.57146	0.249629
$694$	0	0
$695$	39.6110	1.50253
$696$	0	0
$697$	26.3085	0.996506
$698$	0	0
$699$	−21.2734	−0.804635
$700$	0	0
$701$	−26.9241	−1.01691	−0.508454	−	0.861089i	$-0.669783\pi$
$701$	−26.9241	−1.01691	−0.508454	+	0.861089i	$0.669783\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−18.5111	−0.697170
$706$	0	0
$707$	−8.06842	−0.303444
$708$	0	0
$709$	−20.3773	−0.765286	−0.382643	−	0.923896i	$-0.624986\pi$
$709$	−20.3773	−0.765286	−0.382643	+	0.923896i	$0.624986\pi$
$710$	0	0
$711$	−14.7683	−0.553854
$712$	0	0
$713$	86.9923	3.25789
$714$	0	0
$715$	−1.02382	−0.0382887
$716$	0	0
$717$	35.1328	1.31206
$718$	0	0
$719$	−12.3258	−0.459675	−0.229837	−	0.973229i	$-0.573819\pi$
$719$	−12.3258	−0.459675	−0.229837	+	0.973229i	$0.573819\pi$
$720$	0	0
$721$	−9.03047	−0.336312
$722$	0	0
$723$	−48.0831	−1.78823
$724$	0	0
$725$	5.48793	0.203817
$726$	0	0
$727$	−25.2787	−0.937535	−0.468767	−	0.883322i	$-0.655302\pi$
$727$	−25.2787	−0.937535	−0.468767	+	0.883322i	$0.655302\pi$
$728$	0	0
$729$	−10.0308	−0.371512
$730$	0	0
$731$	17.6624	0.653268
$732$	0	0
$733$	−2.52985	−0.0934421	−0.0467210	−	0.998908i	$-0.514877\pi$
$733$	−2.52985	−0.0934421	−0.0467210	+	0.998908i	$0.514877\pi$
$734$	0	0
$735$	−76.8738	−2.83553
$736$	0	0
$737$	−0.795497	−0.0293025
$738$	0	0
$739$	−9.54456	−0.351102	−0.175551	−	0.984470i	$-0.556171\pi$
$739$	−9.54456	−0.351102	−0.175551	+	0.984470i	$0.556171\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.3100	0.598357	0.299178	−	0.954197i	$-0.403287\pi$
$743$	16.3100	0.598357	0.299178	+	0.954197i	$0.403287\pi$
$744$	0	0
$745$	19.2751	0.706186
$746$	0	0
$747$	−0.434518	−0.0158982
$748$	0	0
$749$	72.3363	2.64311
$750$	0	0
$751$	−10.2920	−0.375562	−0.187781	−	0.982211i	$-0.560129\pi$
$751$	−10.2920	−0.375562	−0.187781	+	0.982211i	$0.560129\pi$
$752$	0	0
$753$	−31.9577	−1.16460
$754$	0	0
$755$	9.29971	0.338451
$756$	0	0
$757$	−8.09240	−0.294123	−0.147062	−	0.989127i	$-0.546982\pi$
$757$	−8.09240	−0.294123	−0.147062	+	0.989127i	$0.546982\pi$
$758$	0	0
$759$	12.8029	0.464717
$760$	0	0
$761$	−22.3727	−0.811008	−0.405504	−	0.914093i	$-0.632904\pi$
$761$	−22.3727	−0.811008	−0.405504	+	0.914093i	$0.632904\pi$
$762$	0	0
$763$	−1.50603	−0.0545221
$764$	0	0
$765$	−20.2571	−0.732396
$766$	0	0
$767$	−8.21746	−0.296715
$768$	0	0
$769$	−21.6013	−0.778961	−0.389481	−	0.921035i	$-0.627346\pi$
$769$	−21.6013	−0.778961	−0.389481	+	0.921035i	$0.627346\pi$
$770$	0	0
$771$	24.5792	0.885197
$772$	0	0
$773$	−50.6542	−1.82190	−0.910952	−	0.412513i	$-0.864651\pi$
$773$	−50.6542	−1.82190	−0.910952	+	0.412513i	$0.864651\pi$
$774$	0	0
$775$	8.43803	0.303103
$776$	0	0
$777$	93.0781	3.33916
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.18248	0.185444
$782$	0	0
$783$	−12.2490	−0.437743
$784$	0	0
$785$	8.55010	0.305166
$786$	0	0
$787$	−46.3223	−1.65121	−0.825605	−	0.564248i	$-0.809166\pi$
$787$	−46.3223	−1.65121	−0.825605	+	0.564248i	$0.809166\pi$
$788$	0	0
$789$	23.6456	0.841805
$790$	0	0
$791$	24.1187	0.857562
$792$	0	0
$793$	10.8499	0.385291
$794$	0	0
$795$	−5.79066	−0.205373
$796$	0	0
$797$	−0.926931	−0.0328336	−0.0164168	−	0.999865i	$-0.505226\pi$
$797$	−0.926931	−0.0328336	−0.0164168	+	0.999865i	$0.505226\pi$
$798$	0	0
$799$	−18.6641	−0.660287
$800$	0	0
$801$	16.1705	0.571355
$802$	0	0
$803$	−4.26855	−0.150634
$804$	0	0
$805$	−88.6378	−3.12407
$806$	0	0
$807$	0.960949	0.0338270
$808$	0	0
$809$	−46.1715	−1.62330	−0.811651	−	0.584142i	$-0.801431\pi$
$809$	−46.1715	−1.62330	−0.811651	+	0.584142i	$0.801431\pi$
$810$	0	0
$811$	22.9121	0.804552	0.402276	−	0.915518i	$-0.368219\pi$
$811$	22.9121	0.804552	0.402276	+	0.915518i	$0.368219\pi$
$812$	0	0
$813$	43.5505	1.52738
$814$	0	0
$815$	−4.35320	−0.152486
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−8.34658	−0.291653
$820$	0	0
$821$	−10.2391	−0.357348	−0.178674	−	0.983908i	$-0.557181\pi$
$821$	−10.2391	−0.357348	−0.178674	+	0.983908i	$0.557181\pi$
$822$	0	0
$823$	−2.87285	−0.100141	−0.0500706	−	0.998746i	$-0.515945\pi$
$823$	−2.87285	−0.100141	−0.0500706	+	0.998746i	$0.515945\pi$
$824$	0	0
$825$	1.24185	0.0432358
$826$	0	0
$827$	−35.2840	−1.22695	−0.613473	−	0.789716i	$-0.710228\pi$
$827$	−35.2840	−1.22695	−0.613473	+	0.789716i	$0.710228\pi$
$828$	0	0
$829$	−35.9700	−1.24929	−0.624645	−	0.780909i	$-0.714756\pi$
$829$	−35.9700	−1.24929	−0.624645	+	0.780909i	$0.714756\pi$
$830$	0	0
$831$	27.6901	0.960557
$832$	0	0
$833$	−77.5088	−2.68552
$834$	0	0
$835$	−16.1147	−0.557674
$836$	0	0
$837$	−18.8335	−0.650983
$838$	0	0
$839$	2.76020	0.0952925	0.0476463	−	0.998864i	$-0.484828\pi$
$839$	2.76020	0.0952925	0.0476463	+	0.998864i	$0.484828\pi$
$840$	0	0
$841$	10.8462	0.374008
$842$	0	0
$843$	35.1445	1.21044
$844$	0	0
$845$	−25.1207	−0.864178
$846$	0	0
$847$	−51.5943	−1.77280
$848$	0	0
$849$	−40.4033	−1.38664
$850$	0	0
$851$	75.5918	2.59125
$852$	0	0
$853$	−3.61671	−0.123834	−0.0619169	−	0.998081i	$-0.519721\pi$
$853$	−3.61671	−0.123834	−0.0619169	+	0.998081i	$0.519721\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.3716	−0.422607	−0.211303	−	0.977421i	$-0.567771\pi$
$857$	−12.3716	−0.422607	−0.211303	+	0.977421i	$0.567771\pi$
$858$	0	0
$859$	34.2618	1.16900	0.584498	−	0.811395i	$-0.301291\pi$
$859$	34.2618	1.16900	0.584498	+	0.811395i	$0.301291\pi$
$860$	0	0
$861$	62.4703	2.12898
$862$	0	0
$863$	−46.0895	−1.56890	−0.784452	−	0.620189i	$-0.787056\pi$
$863$	−46.0895	−1.56890	−0.784452	+	0.620189i	$0.787056\pi$
$864$	0	0
$865$	−1.14221	−0.0388365
$866$	0	0
$867$	−10.4386	−0.354515
$868$	0	0
$869$	−4.33705	−0.147124
$870$	0	0
$871$	1.01038	0.0342355
$872$	0	0
$873$	−23.7192	−0.802774
$874$	0	0
$875$	−58.0440	−1.96225
$876$	0	0
$877$	−36.9338	−1.24716	−0.623582	−	0.781758i	$-0.714323\pi$
$877$	−36.9338	−1.24716	−0.623582	+	0.781758i	$0.714323\pi$
$878$	0	0
$879$	−47.5524	−1.60390
$880$	0	0
$881$	−3.86180	−0.130107	−0.0650537	−	0.997882i	$-0.520722\pi$
$881$	−3.86180	−0.130107	−0.0650537	+	0.997882i	$0.520722\pi$
$882$	0	0
$883$	−12.9747	−0.436634	−0.218317	−	0.975878i	$-0.570057\pi$
$883$	−12.9747	−0.436634	−0.218317	+	0.975878i	$0.570057\pi$
$884$	0	0
$885$	−47.3570	−1.59189
$886$	0	0
$887$	14.7714	0.495974	0.247987	−	0.968763i	$-0.420231\pi$
$887$	14.7714	0.495974	0.247987	+	0.968763i	$0.420231\pi$
$888$	0	0
$889$	7.36069	0.246870
$890$	0	0
$891$	−6.82339	−0.228592
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−5.51163	−0.184234
$896$	0	0
$897$	−16.2613	−0.542950
$898$	0	0
$899$	61.2660	2.04334
$900$	0	0
$901$	−5.83849	−0.194508
$902$	0	0
$903$	41.9399	1.39567
$904$	0	0
$905$	42.7386	1.42068
$906$	0	0
$907$	7.73977	0.256995	0.128497	−	0.991710i	$-0.458985\pi$
$907$	7.73977	0.256995	0.128497	+	0.991710i	$0.458985\pi$
$908$	0	0
$909$	−3.55591	−0.117942
$910$	0	0
$911$	48.5235	1.60766	0.803828	−	0.594862i	$-0.202793\pi$
$911$	48.5235	1.60766	0.803828	+	0.594862i	$0.202793\pi$
$912$	0	0
$913$	−0.127606	−0.00422316
$914$	0	0
$915$	62.5276	2.06710
$916$	0	0
$917$	4.31939	0.142639
$918$	0	0
$919$	−7.35044	−0.242469	−0.121234	−	0.992624i	$-0.538685\pi$
$919$	−7.35044	−0.242469	−0.121234	+	0.992624i	$0.538685\pi$
$920$	0	0
$921$	−0.538911	−0.0177577
$922$	0	0
$923$	−6.58240	−0.216662
$924$	0	0
$925$	7.33221	0.241082
$926$	0	0
$927$	−3.97990	−0.130717
$928$	0	0
$929$	46.6758	1.53138	0.765691	−	0.643208i	$-0.222397\pi$
$929$	46.6758	1.53138	0.765691	+	0.643208i	$0.222397\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	61.0444	1.99851
$934$	0	0
$935$	−5.94897	−0.194552
$936$	0	0
$937$	−4.78745	−0.156399	−0.0781995	−	0.996938i	$-0.524917\pi$
$937$	−4.78745	−0.156399	−0.0781995	+	0.996938i	$0.524917\pi$
$938$	0	0
$939$	−3.94471	−0.128731
$940$	0	0
$941$	−4.27902	−0.139492	−0.0697460	−	0.997565i	$-0.522219\pi$
$941$	−4.27902	−0.139492	−0.0697460	+	0.997565i	$0.522219\pi$
$942$	0	0
$943$	50.7342	1.65213
$944$	0	0
$945$	19.1898	0.624245
$946$	0	0
$947$	−53.5099	−1.73884	−0.869419	−	0.494075i	$-0.835507\pi$
$947$	−53.5099	−1.73884	−0.869419	+	0.494075i	$0.835507\pi$
$948$	0	0
$949$	5.42160	0.175993
$950$	0	0
$951$	−38.5327	−1.24951
$952$	0	0
$953$	−43.2210	−1.40007	−0.700033	−	0.714111i	$-0.746831\pi$
$953$	−43.2210	−1.40007	−0.700033	+	0.714111i	$0.746831\pi$
$954$	0	0
$955$	13.9064	0.450002
$956$	0	0
$957$	9.01672	0.291469
$958$	0	0
$959$	4.02017	0.129818
$960$	0	0
$961$	63.2002	2.03872
$962$	0	0
$963$	31.8800	1.02732
$964$	0	0
$965$	−32.4907	−1.04591
$966$	0	0
$967$	−23.9680	−0.770758	−0.385379	−	0.922758i	$-0.625929\pi$
$967$	−23.9680	−0.770758	−0.385379	+	0.922758i	$0.625929\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.8145	0.571696	0.285848	−	0.958275i	$-0.407725\pi$
$971$	17.8145	0.571696	0.285848	+	0.958275i	$0.407725\pi$
$972$	0	0
$973$	94.8346	3.04026
$974$	0	0
$975$	−1.57731	−0.0505143
$976$	0	0
$977$	−19.0026	−0.607947	−0.303973	−	0.952681i	$-0.598313\pi$
$977$	−19.0026	−0.607947	−0.303973	+	0.952681i	$0.598313\pi$
$978$	0	0
$979$	4.74884	0.151774
$980$	0	0
$981$	−0.663739	−0.0211916
$982$	0	0
$983$	4.08920	0.130425	0.0652126	−	0.997871i	$-0.479227\pi$
$983$	4.08920	0.130425	0.0652126	+	0.997871i	$0.479227\pi$
$984$	0	0
$985$	39.9572	1.27314
$986$	0	0
$987$	−44.3184	−1.41067
$988$	0	0
$989$	34.0608	1.08307
$990$	0	0
$991$	43.6388	1.38623	0.693115	−	0.720827i	$-0.256238\pi$
$991$	43.6388	1.38623	0.693115	+	0.720827i	$0.256238\pi$
$992$	0	0
$993$	27.0951	0.859837
$994$	0	0
$995$	29.0206	0.920014
$996$	0	0
$997$	−46.7001	−1.47901	−0.739504	−	0.673152i	$-0.764940\pi$
$997$	−46.7001	−1.47901	−0.739504	+	0.673152i	$0.764940\pi$
$998$	0	0
$999$	−16.3654	−0.517778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.v.1.3	✓	8
4.3	odd	2		5776.2.a.cc.1.6		8
19.18	odd	2		2888.2.a.w.1.6	yes	8
76.75	even	2		5776.2.a.ca.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.3	✓	8	1.1	even	1	trivial
2888.2.a.w.1.6	yes	8	19.18	odd	2
5776.2.a.ca.1.3		8	76.75	even	2
5776.2.a.cc.1.6		8	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26814 q^{3} +2.03239 q^{5} +4.86584 q^{7} +2.14447 q^{9} +0.629774 q^{11} -0.799892 q^{13} -4.60975 q^{15} -4.64783 q^{17} -11.0364 q^{21} -8.96303 q^{23} -0.869391 q^{25} +1.94047 q^{27} -6.31239 q^{29} -9.70568 q^{31} -1.42842 q^{33} +9.88928 q^{35} -8.43373 q^{37} +1.81427 q^{39} -5.66039 q^{41} -3.80014 q^{43} +4.35840 q^{45} +4.01565 q^{47} +16.6764 q^{49} +10.5419 q^{51} +1.25618 q^{53} +1.27995 q^{55} +10.2732 q^{59} -13.5642 q^{61} +10.4346 q^{63} -1.62569 q^{65} -1.26315 q^{67} +20.3294 q^{69} +8.22911 q^{71} -6.77791 q^{73} +1.97190 q^{75} +3.06438 q^{77} -6.88668 q^{79} -10.8347 q^{81} -0.202623 q^{83} -9.44620 q^{85} +14.3174 q^{87} +7.54055 q^{89} -3.89214 q^{91} +22.0139 q^{93} -11.0607 q^{97} +1.35053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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