LMFDB - Embedded newform 8048.2.a.v.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.447624 q^{3} +0.901345 q^{5} +1.57888 q^{7} -2.79963 q^{9} +O(q^{10})$	$q+0.447624 q^{3} +0.901345 q^{5} +1.57888 q^{7} -2.79963 q^{9} -2.73519 q^{11} -0.127802 q^{13} +0.403464 q^{15} -0.965178 q^{17} -1.08247 q^{19} +0.706743 q^{21} -0.734871 q^{23} -4.18758 q^{25} -2.59606 q^{27} +9.81943 q^{29} +6.56783 q^{31} -1.22434 q^{33} +1.42311 q^{35} -2.91899 q^{37} -0.0572072 q^{39} +1.43349 q^{41} +2.61043 q^{43} -2.52343 q^{45} -2.70258 q^{47} -4.50715 q^{49} -0.432037 q^{51} +0.901890 q^{53} -2.46535 q^{55} -0.484540 q^{57} -6.31776 q^{59} -5.52676 q^{61} -4.42028 q^{63} -0.115194 q^{65} +3.88478 q^{67} -0.328946 q^{69} +9.68884 q^{71} -1.43220 q^{73} -1.87446 q^{75} -4.31853 q^{77} +9.80388 q^{79} +7.23684 q^{81} -4.88746 q^{83} -0.869958 q^{85} +4.39541 q^{87} -16.4486 q^{89} -0.201784 q^{91} +2.93992 q^{93} -0.975680 q^{95} -8.63007 q^{97} +7.65752 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	0.447624	0.258436	0.129218	−	0.991616i	$-0.458753\pi$
$3$	0.447624	0.258436	0.129218	+	0.991616i	$0.458753\pi$
$4$	0	0
$5$	0.901345	0.403094	0.201547	−	0.979479i	$-0.435403\pi$
$5$	0.901345	0.403094	0.201547	+	0.979479i	$0.435403\pi$
$6$	0	0
$7$	1.57888	0.596760	0.298380	−	0.954447i	$-0.403554\pi$
$7$	1.57888	0.596760	0.298380	+	0.954447i	$0.403554\pi$
$8$	0	0
$9$	−2.79963	−0.933211
$10$	0	0
$11$	−2.73519	−0.824690	−0.412345	−	0.911028i	$-0.635290\pi$
$11$	−2.73519	−0.824690	−0.412345	+	0.911028i	$0.635290\pi$
$12$	0	0
$13$	−0.127802	−0.0354459	−0.0177229	−	0.999843i	$-0.505642\pi$
$13$	−0.127802	−0.0354459	−0.0177229	+	0.999843i	$0.505642\pi$
$14$	0	0
$15$	0.403464	0.104174
$16$	0	0
$17$	−0.965178	−0.234090	−0.117045	−	0.993127i	$-0.537342\pi$
$17$	−0.965178	−0.234090	−0.117045	+	0.993127i	$0.537342\pi$
$18$	0	0
$19$	−1.08247	−0.248336	−0.124168	−	0.992261i	$-0.539626\pi$
$19$	−1.08247	−0.248336	−0.124168	+	0.992261i	$0.539626\pi$
$20$	0	0
$21$	0.706743	0.154224
$22$	0	0
$23$	−0.734871	−0.153231	−0.0766156	−	0.997061i	$-0.524411\pi$
$23$	−0.734871	−0.153231	−0.0766156	+	0.997061i	$0.524411\pi$
$24$	0	0
$25$	−4.18758	−0.837516
$26$	0	0
$27$	−2.59606	−0.499611
$28$	0	0
$29$	9.81943	1.82342	0.911711	−	0.410833i	$-0.134762\pi$
$29$	9.81943	1.82342	0.911711	+	0.410833i	$0.134762\pi$
$30$	0	0
$31$	6.56783	1.17962	0.589809	−	0.807543i	$-0.299203\pi$
$31$	6.56783	1.17962	0.589809	+	0.807543i	$0.299203\pi$
$32$	0	0
$33$	−1.22434	−0.213130
$34$	0	0
$35$	1.42311	0.240550
$36$	0	0
$37$	−2.91899	−0.479879	−0.239939	−	0.970788i	$-0.577128\pi$
$37$	−2.91899	−0.479879	−0.239939	+	0.970788i	$0.577128\pi$
$38$	0	0
$39$	−0.0572072	−0.00916049
$40$	0	0
$41$	1.43349	0.223873	0.111936	−	0.993715i	$-0.464295\pi$
$41$	1.43349	0.223873	0.111936	+	0.993715i	$0.464295\pi$
$42$	0	0
$43$	2.61043	0.398087	0.199044	−	0.979991i	$-0.436216\pi$
$43$	2.61043	0.398087	0.199044	+	0.979991i	$0.436216\pi$
$44$	0	0
$45$	−2.52343	−0.376171
$46$	0	0
$47$	−2.70258	−0.394212	−0.197106	−	0.980382i	$-0.563154\pi$
$47$	−2.70258	−0.394212	−0.197106	+	0.980382i	$0.563154\pi$
$48$	0	0
$49$	−4.50715	−0.643878
$50$	0	0
$51$	−0.432037	−0.0604973
$52$	0	0
$53$	0.901890	0.123884	0.0619421	−	0.998080i	$-0.480271\pi$
$53$	0.901890	0.123884	0.0619421	+	0.998080i	$0.480271\pi$
$54$	0	0
$55$	−2.46535	−0.332427
$56$	0	0
$57$	−0.484540	−0.0641789
$58$	0	0
$59$	−6.31776	−0.822502	−0.411251	−	0.911522i	$-0.634908\pi$
$59$	−6.31776	−0.822502	−0.411251	+	0.911522i	$0.634908\pi$
$60$	0	0
$61$	−5.52676	−0.707629	−0.353815	−	0.935316i	$-0.615116\pi$
$61$	−5.52676	−0.707629	−0.353815	+	0.935316i	$0.615116\pi$
$62$	0	0
$63$	−4.42028	−0.556902
$64$	0	0
$65$	−0.115194	−0.0142880
$66$	0	0
$67$	3.88478	0.474602	0.237301	−	0.971436i	$-0.423737\pi$
$67$	3.88478	0.474602	0.237301	+	0.971436i	$0.423737\pi$
$68$	0	0
$69$	−0.328946	−0.0396004
$70$	0	0
$71$	9.68884	1.14985	0.574927	−	0.818205i	$-0.305031\pi$
$71$	9.68884	1.14985	0.574927	+	0.818205i	$0.305031\pi$
$72$	0	0
$73$	−1.43220	−0.167627	−0.0838134	−	0.996481i	$-0.526710\pi$
$73$	−1.43220	−0.167627	−0.0838134	+	0.996481i	$0.526710\pi$
$74$	0	0
$75$	−1.87446	−0.216444
$76$	0	0
$77$	−4.31853	−0.492142
$78$	0	0
$79$	9.80388	1.10302	0.551511	−	0.834167i	$-0.314051\pi$
$79$	9.80388	1.10302	0.551511	+	0.834167i	$0.314051\pi$
$80$	0	0
$81$	7.23684	0.804094
$82$	0	0
$83$	−4.88746	−0.536469	−0.268234	−	0.963354i	$-0.586440\pi$
$83$	−4.88746	−0.536469	−0.268234	+	0.963354i	$0.586440\pi$
$84$	0	0
$85$	−0.869958	−0.0943602
$86$	0	0
$87$	4.39541	0.471237
$88$	0	0
$89$	−16.4486	−1.74355	−0.871776	−	0.489904i	$-0.837032\pi$
$89$	−16.4486	−1.74355	−0.871776	+	0.489904i	$0.837032\pi$
$90$	0	0
$91$	−0.201784	−0.0211527
$92$	0	0
$93$	2.93992	0.304855
$94$	0	0
$95$	−0.975680	−0.100103
$96$	0	0
$97$	−8.63007	−0.876250	−0.438125	−	0.898914i	$-0.644357\pi$
$97$	−8.63007	−0.876250	−0.438125	+	0.898914i	$0.644357\pi$
$98$	0	0
$99$	7.65752	0.769610
$100$	0	0
$101$	8.46581	0.842380	0.421190	−	0.906972i	$-0.361613\pi$
$101$	8.46581	0.842380	0.421190	+	0.906972i	$0.361613\pi$
$102$	0	0
$103$	−2.82426	−0.278282	−0.139141	−	0.990273i	$-0.544434\pi$
$103$	−2.82426	−0.278282	−0.139141	+	0.990273i	$0.544434\pi$
$104$	0	0
$105$	0.637019	0.0621667
$106$	0	0
$107$	−15.1261	−1.46230	−0.731148	−	0.682219i	$-0.761015\pi$
$107$	−15.1261	−1.46230	−0.731148	+	0.682219i	$0.761015\pi$
$108$	0	0
$109$	−14.9868	−1.43547	−0.717737	−	0.696315i	$-0.754822\pi$
$109$	−14.9868	−1.43547	−0.717737	+	0.696315i	$0.754822\pi$
$110$	0	0
$111$	−1.30661	−0.124018
$112$	0	0
$113$	−20.0027	−1.88170	−0.940848	−	0.338830i	$-0.889969\pi$
$113$	−20.0027	−1.88170	−0.940848	+	0.338830i	$0.889969\pi$
$114$	0	0
$115$	−0.662372	−0.0617665
$116$	0	0
$117$	0.357799	0.0330785
$118$	0	0
$119$	−1.52390	−0.139696
$120$	0	0
$121$	−3.51875	−0.319886
$122$	0	0
$123$	0.641663	0.0578568
$124$	0	0
$125$	−8.28117	−0.740691
$126$	0	0
$127$	−1.02653	−0.0910896	−0.0455448	−	0.998962i	$-0.514502\pi$
$127$	−1.02653	−0.0910896	−0.0455448	+	0.998962i	$0.514502\pi$
$128$	0	0
$129$	1.16849	0.102880
$130$	0	0
$131$	9.10716	0.795696	0.397848	−	0.917451i	$-0.369757\pi$
$131$	9.10716	0.795696	0.397848	+	0.917451i	$0.369757\pi$
$132$	0	0
$133$	−1.70909	−0.148197
$134$	0	0
$135$	−2.33994	−0.201390
$136$	0	0
$137$	−21.7857	−1.86128	−0.930640	−	0.365935i	$-0.880749\pi$
$137$	−21.7857	−1.86128	−0.930640	+	0.365935i	$0.880749\pi$
$138$	0	0
$139$	0.628032	0.0532689	0.0266345	−	0.999645i	$-0.491521\pi$
$139$	0.628032	0.0532689	0.0266345	+	0.999645i	$0.491521\pi$
$140$	0	0
$141$	−1.20974	−0.101879
$142$	0	0
$143$	0.349562	0.0292319
$144$	0	0
$145$	8.85069	0.735009
$146$	0	0
$147$	−2.01751	−0.166401
$148$	0	0
$149$	10.6024	0.868581	0.434291	−	0.900773i	$-0.356999\pi$
$149$	10.6024	0.868581	0.434291	+	0.900773i	$0.356999\pi$
$150$	0	0
$151$	−17.4339	−1.41875	−0.709377	−	0.704829i	$-0.751024\pi$
$151$	−17.4339	−1.41875	−0.709377	+	0.704829i	$0.751024\pi$
$152$	0	0
$153$	2.70214	0.218455
$154$	0	0
$155$	5.91988	0.475496
$156$	0	0
$157$	0.640002	0.0510778	0.0255389	−	0.999674i	$-0.491870\pi$
$157$	0.640002	0.0510778	0.0255389	+	0.999674i	$0.491870\pi$
$158$	0	0
$159$	0.403708	0.0320161
$160$	0	0
$161$	−1.16027	−0.0914421
$162$	0	0
$163$	−19.4870	−1.52634	−0.763170	−	0.646198i	$-0.776358\pi$
$163$	−19.4870	−1.52634	−0.763170	+	0.646198i	$0.776358\pi$
$164$	0	0
$165$	−1.10355	−0.0859111
$166$	0	0
$167$	−5.58619	−0.432272	−0.216136	−	0.976363i	$-0.569346\pi$
$167$	−5.58619	−0.432272	−0.216136	+	0.976363i	$0.569346\pi$
$168$	0	0
$169$	−12.9837	−0.998744
$170$	0	0
$171$	3.03052	0.231750
$172$	0	0
$173$	−21.2916	−1.61877	−0.809383	−	0.587281i	$-0.800198\pi$
$173$	−21.2916	−1.61877	−0.809383	+	0.587281i	$0.800198\pi$
$174$	0	0
$175$	−6.61167	−0.499795
$176$	0	0
$177$	−2.82798	−0.212564
$178$	0	0
$179$	−16.1793	−1.20930	−0.604650	−	0.796491i	$-0.706687\pi$
$179$	−16.1793	−1.20930	−0.604650	+	0.796491i	$0.706687\pi$
$180$	0	0
$181$	−21.8341	−1.62291	−0.811456	−	0.584413i	$-0.801325\pi$
$181$	−21.8341	−1.62291	−0.811456	+	0.584413i	$0.801325\pi$
$182$	0	0
$183$	−2.47391	−0.182877
$184$	0	0
$185$	−2.63101	−0.193436
$186$	0	0
$187$	2.63994	0.193052
$188$	0	0
$189$	−4.09885	−0.298148
$190$	0	0
$191$	17.0262	1.23197	0.615987	−	0.787757i	$-0.288758\pi$
$191$	17.0262	1.23197	0.615987	+	0.787757i	$0.288758\pi$
$192$	0	0
$193$	−2.80147	−0.201654	−0.100827	−	0.994904i	$-0.532149\pi$
$193$	−2.80147	−0.201654	−0.100827	+	0.994904i	$0.532149\pi$
$194$	0	0
$195$	−0.0515634	−0.00369253
$196$	0	0
$197$	8.21467	0.585271	0.292635	−	0.956224i	$-0.405468\pi$
$197$	8.21467	0.585271	0.292635	+	0.956224i	$0.405468\pi$
$198$	0	0
$199$	25.1008	1.77935	0.889674	−	0.456597i	$-0.150932\pi$
$199$	25.1008	1.77935	0.889674	+	0.456597i	$0.150932\pi$
$200$	0	0
$201$	1.73892	0.122654
$202$	0	0
$203$	15.5037	1.08814
$204$	0	0
$205$	1.29206	0.0902417
$206$	0	0
$207$	2.05737	0.142997
$208$	0	0
$209$	2.96076	0.204800
$210$	0	0
$211$	−1.99433	−0.137295	−0.0686476	−	0.997641i	$-0.521868\pi$
$211$	−1.99433	−0.137295	−0.0686476	+	0.997641i	$0.521868\pi$
$212$	0	0
$213$	4.33696	0.297163
$214$	0	0
$215$	2.35290	0.160466
$216$	0	0
$217$	10.3698	0.703948
$218$	0	0
$219$	−0.641089	−0.0433208
$220$	0	0
$221$	0.123352	0.00829753
$222$	0	0
$223$	22.9920	1.53966	0.769830	−	0.638249i	$-0.220341\pi$
$223$	22.9920	1.53966	0.769830	+	0.638249i	$0.220341\pi$
$224$	0	0
$225$	11.7237	0.781579
$226$	0	0
$227$	−24.2522	−1.60967	−0.804836	−	0.593497i	$-0.797747\pi$
$227$	−24.2522	−1.60967	−0.804836	+	0.593497i	$0.797747\pi$
$228$	0	0
$229$	5.04283	0.333240	0.166620	−	0.986021i	$-0.446715\pi$
$229$	5.04283	0.333240	0.166620	+	0.986021i	$0.446715\pi$
$230$	0	0
$231$	−1.93308	−0.127187
$232$	0	0
$233$	−4.35483	−0.285294	−0.142647	−	0.989774i	$-0.545561\pi$
$233$	−4.35483	−0.285294	−0.142647	+	0.989774i	$0.545561\pi$
$234$	0	0
$235$	−2.43596	−0.158904
$236$	0	0
$237$	4.38845	0.285061
$238$	0	0
$239$	12.4760	0.807004	0.403502	−	0.914979i	$-0.367793\pi$
$239$	12.4760	0.807004	0.403502	+	0.914979i	$0.367793\pi$
$240$	0	0
$241$	8.29161	0.534110	0.267055	−	0.963681i	$-0.413950\pi$
$241$	8.29161	0.534110	0.267055	+	0.963681i	$0.413950\pi$
$242$	0	0
$243$	11.0275	0.707418
$244$	0	0
$245$	−4.06249	−0.259543
$246$	0	0
$247$	0.138342	0.00880249
$248$	0	0
$249$	−2.18775	−0.138643
$250$	0	0
$251$	17.5576	1.10822	0.554112	−	0.832442i	$-0.313058\pi$
$251$	17.5576	1.10822	0.554112	+	0.832442i	$0.313058\pi$
$252$	0	0
$253$	2.01001	0.126368
$254$	0	0
$255$	−0.389414	−0.0243861
$256$	0	0
$257$	4.60355	0.287162	0.143581	−	0.989639i	$-0.454138\pi$
$257$	4.60355	0.287162	0.143581	+	0.989639i	$0.454138\pi$
$258$	0	0
$259$	−4.60873	−0.286372
$260$	0	0
$261$	−27.4908	−1.70164
$262$	0	0
$263$	−13.8831	−0.856068	−0.428034	−	0.903763i	$-0.640794\pi$
$263$	−13.8831	−0.856068	−0.428034	+	0.903763i	$0.640794\pi$
$264$	0	0
$265$	0.812914	0.0499369
$266$	0	0
$267$	−7.36281	−0.450597
$268$	0	0
$269$	−2.71959	−0.165816	−0.0829082	−	0.996557i	$-0.526421\pi$
$269$	−2.71959	−0.165816	−0.0829082	+	0.996557i	$0.526421\pi$
$270$	0	0
$271$	−6.83383	−0.415126	−0.207563	−	0.978222i	$-0.566553\pi$
$271$	−6.83383	−0.415126	−0.207563	+	0.978222i	$0.566553\pi$
$272$	0	0
$273$	−0.0903232	−0.00546661
$274$	0	0
$275$	11.4538	0.690691
$276$	0	0
$277$	−16.4311	−0.987249	−0.493625	−	0.869675i	$-0.664328\pi$
$277$	−16.4311	−0.987249	−0.493625	+	0.869675i	$0.664328\pi$
$278$	0	0
$279$	−18.3875	−1.10083
$280$	0	0
$281$	15.7034	0.936787	0.468394	−	0.883520i	$-0.344833\pi$
$281$	15.7034	0.936787	0.468394	+	0.883520i	$0.344833\pi$
$282$	0	0
$283$	−26.5926	−1.58076	−0.790382	−	0.612615i	$-0.790118\pi$
$283$	−26.5926	−1.58076	−0.790382	+	0.612615i	$0.790118\pi$
$284$	0	0
$285$	−0.436738	−0.0258701
$286$	0	0
$287$	2.26330	0.133598
$288$	0	0
$289$	−16.0684	−0.945202
$290$	0	0
$291$	−3.86302	−0.226455
$292$	0	0
$293$	−21.3129	−1.24511	−0.622557	−	0.782574i	$-0.713906\pi$
$293$	−21.3129	−1.24511	−0.622557	+	0.782574i	$0.713906\pi$
$294$	0	0
$295$	−5.69448	−0.331545
$296$	0	0
$297$	7.10070	0.412024
$298$	0	0
$299$	0.0939179	0.00543141
$300$	0	0
$301$	4.12155	0.237562
$302$	0	0
$303$	3.78950	0.217701
$304$	0	0
$305$	−4.98152	−0.285241
$306$	0	0
$307$	28.3391	1.61740	0.808699	−	0.588223i	$-0.200172\pi$
$307$	28.3391	1.61740	0.808699	+	0.588223i	$0.200172\pi$
$308$	0	0
$309$	−1.26420	−0.0719181
$310$	0	0
$311$	−12.2327	−0.693655	−0.346827	−	0.937929i	$-0.612741\pi$
$311$	−12.2327	−0.693655	−0.346827	+	0.937929i	$0.612741\pi$
$312$	0	0
$313$	24.4674	1.38298	0.691489	−	0.722387i	$-0.256955\pi$
$313$	24.4674	1.38298	0.691489	+	0.722387i	$0.256955\pi$
$314$	0	0
$315$	−3.98419	−0.224484
$316$	0	0
$317$	13.7231	0.770768	0.385384	−	0.922756i	$-0.374069\pi$
$317$	13.7231	0.770768	0.385384	+	0.922756i	$0.374069\pi$
$318$	0	0
$319$	−26.8580	−1.50376
$320$	0	0
$321$	−6.77081	−0.377910
$322$	0	0
$323$	1.04478	0.0581330
$324$	0	0
$325$	0.535181	0.0296865
$326$	0	0
$327$	−6.70844	−0.370978
$328$	0	0
$329$	−4.26704	−0.235250
$330$	0	0
$331$	32.3649	1.77894	0.889469	−	0.456996i	$-0.151074\pi$
$331$	32.3649	1.77894	0.889469	+	0.456996i	$0.151074\pi$
$332$	0	0
$333$	8.17210	0.447828
$334$	0	0
$335$	3.50153	0.191309
$336$	0	0
$337$	−0.611067	−0.0332869	−0.0166435	−	0.999861i	$-0.505298\pi$
$337$	−0.611067	−0.0332869	−0.0166435	+	0.999861i	$0.505298\pi$
$338$	0	0
$339$	−8.95369	−0.486297
$340$	0	0
$341$	−17.9643	−0.972819
$342$	0	0
$343$	−18.1684	−0.981000
$344$	0	0
$345$	−0.296493	−0.0159627
$346$	0	0
$347$	23.0394	1.23682	0.618409	−	0.785857i	$-0.287778\pi$
$347$	23.0394	1.23682	0.618409	+	0.785857i	$0.287778\pi$
$348$	0	0
$349$	−24.5411	−1.31365	−0.656827	−	0.754041i	$-0.728102\pi$
$349$	−24.5411	−1.31365	−0.656827	+	0.754041i	$0.728102\pi$
$350$	0	0
$351$	0.331781	0.0177092
$352$	0	0
$353$	34.1596	1.81813	0.909067	−	0.416651i	$-0.136796\pi$
$353$	34.1596	1.81813	0.909067	+	0.416651i	$0.136796\pi$
$354$	0	0
$355$	8.73298	0.463498
$356$	0	0
$357$	−0.682133	−0.0361023
$358$	0	0
$359$	11.9085	0.628507	0.314254	−	0.949339i	$-0.398246\pi$
$359$	11.9085	0.628507	0.314254	+	0.949339i	$0.398246\pi$
$360$	0	0
$361$	−17.8283	−0.938329
$362$	0	0
$363$	−1.57508	−0.0826700
$364$	0	0
$365$	−1.29091	−0.0675693
$366$	0	0
$367$	−5.95411	−0.310802	−0.155401	−	0.987851i	$-0.549667\pi$
$367$	−5.95411	−0.310802	−0.155401	+	0.987851i	$0.549667\pi$
$368$	0	0
$369$	−4.01323	−0.208921
$370$	0	0
$371$	1.42397	0.0739290
$372$	0	0
$373$	13.5045	0.699234	0.349617	−	0.936893i	$-0.386312\pi$
$373$	13.5045	0.699234	0.349617	+	0.936893i	$0.386312\pi$
$374$	0	0
$375$	−3.70685	−0.191421
$376$	0	0
$377$	−1.25494	−0.0646328
$378$	0	0
$379$	−22.3264	−1.14683	−0.573415	−	0.819265i	$-0.694382\pi$
$379$	−22.3264	−1.14683	−0.573415	+	0.819265i	$0.694382\pi$
$380$	0	0
$381$	−0.459499	−0.0235408
$382$	0	0
$383$	19.1315	0.977573	0.488786	−	0.872403i	$-0.337440\pi$
$383$	19.1315	0.977573	0.488786	+	0.872403i	$0.337440\pi$
$384$	0	0
$385$	−3.89248	−0.198379
$386$	0	0
$387$	−7.30825	−0.371499
$388$	0	0
$389$	5.32461	0.269969	0.134984	−	0.990848i	$-0.456902\pi$
$389$	5.32461	0.269969	0.134984	+	0.990848i	$0.456902\pi$
$390$	0	0
$391$	0.709281	0.0358699
$392$	0	0
$393$	4.07658	0.205636
$394$	0	0
$395$	8.83667	0.444621
$396$	0	0
$397$	−13.7991	−0.692558	−0.346279	−	0.938132i	$-0.612555\pi$
$397$	−13.7991	−0.692558	−0.346279	+	0.938132i	$0.612555\pi$
$398$	0	0
$399$	−0.765029	−0.0382994
$400$	0	0
$401$	−28.8332	−1.43986	−0.719931	−	0.694045i	$-0.755827\pi$
$401$	−28.8332	−1.43986	−0.719931	+	0.694045i	$0.755827\pi$
$402$	0	0
$403$	−0.839382	−0.0418126
$404$	0	0
$405$	6.52289	0.324125
$406$	0	0
$407$	7.98398	0.395751
$408$	0	0
$409$	−6.76400	−0.334458	−0.167229	−	0.985918i	$-0.553482\pi$
$409$	−6.76400	−0.334458	−0.167229	+	0.985918i	$0.553482\pi$
$410$	0	0
$411$	−9.75182	−0.481022
$412$	0	0
$413$	−9.97496	−0.490836
$414$	0	0
$415$	−4.40529	−0.216247
$416$	0	0
$417$	0.281122	0.0137666
$418$	0	0
$419$	35.4135	1.73006	0.865032	−	0.501717i	$-0.167298\pi$
$419$	35.4135	1.73006	0.865032	+	0.501717i	$0.167298\pi$
$420$	0	0
$421$	0.422696	0.0206010	0.0103005	−	0.999947i	$-0.496721\pi$
$421$	0.422696	0.0206010	0.0103005	+	0.999947i	$0.496721\pi$
$422$	0	0
$423$	7.56623	0.367883
$424$	0	0
$425$	4.04176	0.196054
$426$	0	0
$427$	−8.72608	−0.422284
$428$	0	0
$429$	0.156473	0.00755456
$430$	0	0
$431$	−13.6898	−0.659415	−0.329707	−	0.944083i	$-0.606950\pi$
$431$	−13.6898	−0.659415	−0.329707	+	0.944083i	$0.606950\pi$
$432$	0	0
$433$	18.9972	0.912946	0.456473	−	0.889737i	$-0.349112\pi$
$433$	18.9972	0.912946	0.456473	+	0.889737i	$0.349112\pi$
$434$	0	0
$435$	3.96178	0.189953
$436$	0	0
$437$	0.795476	0.0380528
$438$	0	0
$439$	−20.6855	−0.987265	−0.493633	−	0.869671i	$-0.664331\pi$
$439$	−20.6855	−0.987265	−0.493633	+	0.869671i	$0.664331\pi$
$440$	0	0
$441$	12.6184	0.600874
$442$	0	0
$443$	−28.7062	−1.36387	−0.681937	−	0.731411i	$-0.738862\pi$
$443$	−28.7062	−1.36387	−0.681937	+	0.731411i	$0.738862\pi$
$444$	0	0
$445$	−14.8259	−0.702815
$446$	0	0
$447$	4.74588	0.224473
$448$	0	0
$449$	1.92732	0.0909558	0.0454779	−	0.998965i	$-0.485519\pi$
$449$	1.92732	0.0909558	0.0454779	+	0.998965i	$0.485519\pi$
$450$	0	0
$451$	−3.92085	−0.184626
$452$	0	0
$453$	−7.80385	−0.366657
$454$	0	0
$455$	−0.181877	−0.00852650
$456$	0	0
$457$	−33.7613	−1.57929	−0.789645	−	0.613564i	$-0.789735\pi$
$457$	−33.7613	−1.57929	−0.789645	+	0.613564i	$0.789735\pi$
$458$	0	0
$459$	2.50566	0.116954
$460$	0	0
$461$	24.9746	1.16318	0.581592	−	0.813480i	$-0.302430\pi$
$461$	24.9746	1.16318	0.581592	+	0.813480i	$0.302430\pi$
$462$	0	0
$463$	6.36011	0.295579	0.147790	−	0.989019i	$-0.452784\pi$
$463$	6.36011	0.295579	0.147790	+	0.989019i	$0.452784\pi$
$464$	0	0
$465$	2.64988	0.122885
$466$	0	0
$467$	24.4726	1.13245	0.566227	−	0.824249i	$-0.308402\pi$
$467$	24.4726	1.13245	0.566227	+	0.824249i	$0.308402\pi$
$468$	0	0
$469$	6.13360	0.283223
$470$	0	0
$471$	0.286480	0.0132003
$472$	0	0
$473$	−7.14002	−0.328299
$474$	0	0
$475$	4.53293	0.207985
$476$	0	0
$477$	−2.52496	−0.115610
$478$	0	0
$479$	18.9355	0.865186	0.432593	−	0.901589i	$-0.357599\pi$
$479$	18.9355	0.865186	0.432593	+	0.901589i	$0.357599\pi$
$480$	0	0
$481$	0.373052	0.0170097
$482$	0	0
$483$	−0.519365	−0.0236319
$484$	0	0
$485$	−7.77866	−0.353211
$486$	0	0
$487$	33.8295	1.53296	0.766481	−	0.642267i	$-0.222006\pi$
$487$	33.8295	1.53296	0.766481	+	0.642267i	$0.222006\pi$
$488$	0	0
$489$	−8.72285	−0.394461
$490$	0	0
$491$	−12.1072	−0.546390	−0.273195	−	0.961959i	$-0.588080\pi$
$491$	−12.1072	−0.546390	−0.273195	+	0.961959i	$0.588080\pi$
$492$	0	0
$493$	−9.47750	−0.426845
$494$	0	0
$495$	6.90207	0.310225
$496$	0	0
$497$	15.2975	0.686186
$498$	0	0
$499$	36.3596	1.62768	0.813839	−	0.581090i	$-0.197374\pi$
$499$	36.3596	1.62768	0.813839	+	0.581090i	$0.197374\pi$
$500$	0	0
$501$	−2.50051	−0.111715
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	7.63062	0.339558
$506$	0	0
$507$	−5.81180	−0.258111
$508$	0	0
$509$	41.0203	1.81819	0.909096	−	0.416587i	$-0.136774\pi$
$509$	41.0203	1.81819	0.909096	+	0.416587i	$0.136774\pi$
$510$	0	0
$511$	−2.26127	−0.100033
$512$	0	0
$513$	2.81016	0.124071
$514$	0	0
$515$	−2.54563	−0.112174
$516$	0	0
$517$	7.39207	0.325103
$518$	0	0
$519$	−9.53061	−0.418347
$520$	0	0
$521$	7.83402	0.343215	0.171607	−	0.985165i	$-0.445104\pi$
$521$	7.83402	0.343215	0.171607	+	0.985165i	$0.445104\pi$
$522$	0	0
$523$	−40.8245	−1.78513	−0.892565	−	0.450919i	$-0.851096\pi$
$523$	−40.8245	−1.78513	−0.892565	+	0.450919i	$0.851096\pi$
$524$	0	0
$525$	−2.95954	−0.129165
$526$	0	0
$527$	−6.33913	−0.276137
$528$	0	0
$529$	−22.4600	−0.976520
$530$	0	0
$531$	17.6874	0.767568
$532$	0	0
$533$	−0.183202	−0.00793537
$534$	0	0
$535$	−13.6338	−0.589442
$536$	0	0
$537$	−7.24226	−0.312526
$538$	0	0
$539$	12.3279	0.531000
$540$	0	0
$541$	−23.3073	−1.00206	−0.501030	−	0.865430i	$-0.667045\pi$
$541$	−23.3073	−1.00206	−0.501030	+	0.865430i	$0.667045\pi$
$542$	0	0
$543$	−9.77345	−0.419419
$544$	0	0
$545$	−13.5083	−0.578630
$546$	0	0
$547$	1.00198	0.0428416	0.0214208	−	0.999771i	$-0.493181\pi$
$547$	1.00198	0.0428416	0.0214208	+	0.999771i	$0.493181\pi$
$548$	0	0
$549$	15.4729	0.660367
$550$	0	0
$551$	−10.6292	−0.452821
$552$	0	0
$553$	15.4791	0.658239
$554$	0	0
$555$	−1.17771	−0.0499908
$556$	0	0
$557$	−43.2243	−1.83147	−0.915737	−	0.401778i	$-0.868392\pi$
$557$	−43.2243	−1.83147	−0.915737	+	0.401778i	$0.868392\pi$
$558$	0	0
$559$	−0.333618	−0.0141105
$560$	0	0
$561$	1.18170	0.0498915
$562$	0	0
$563$	−43.0714	−1.81524	−0.907621	−	0.419790i	$-0.862104\pi$
$563$	−43.0714	−1.81524	−0.907621	+	0.419790i	$0.862104\pi$
$564$	0	0
$565$	−18.0293	−0.758499
$566$	0	0
$567$	11.4261	0.479850
$568$	0	0
$569$	45.1586	1.89315	0.946573	−	0.322491i	$-0.104520\pi$
$569$	45.1586	1.89315	0.946573	+	0.322491i	$0.104520\pi$
$570$	0	0
$571$	−3.24438	−0.135773	−0.0678866	−	0.997693i	$-0.521626\pi$
$571$	−3.24438	−0.135773	−0.0678866	+	0.997693i	$0.521626\pi$
$572$	0	0
$573$	7.62134	0.318386
$574$	0	0
$575$	3.07733	0.128333
$576$	0	0
$577$	−1.72220	−0.0716960	−0.0358480	−	0.999357i	$-0.511413\pi$
$577$	−1.72220	−0.0716960	−0.0358480	+	0.999357i	$0.511413\pi$
$578$	0	0
$579$	−1.25400	−0.0521146
$580$	0	0
$581$	−7.71670	−0.320143
$582$	0	0
$583$	−2.46684	−0.102166
$584$	0	0
$585$	0.322500	0.0133337
$586$	0	0
$587$	17.8085	0.735036	0.367518	−	0.930017i	$-0.380208\pi$
$587$	17.8085	0.735036	0.367518	+	0.930017i	$0.380208\pi$
$588$	0	0
$589$	−7.10949	−0.292941
$590$	0	0
$591$	3.67708	0.151255
$592$	0	0
$593$	15.2632	0.626783	0.313391	−	0.949624i	$-0.398535\pi$
$593$	15.2632	0.626783	0.313391	+	0.949624i	$0.398535\pi$
$594$	0	0
$595$	−1.37356	−0.0563104
$596$	0	0
$597$	11.2357	0.459847
$598$	0	0
$599$	−20.2179	−0.826082	−0.413041	−	0.910712i	$-0.635533\pi$
$599$	−20.2179	−0.826082	−0.413041	+	0.910712i	$0.635533\pi$
$600$	0	0
$601$	−23.4812	−0.957818	−0.478909	−	0.877865i	$-0.658968\pi$
$601$	−23.4812	−0.957818	−0.478909	+	0.877865i	$0.658968\pi$
$602$	0	0
$603$	−10.8760	−0.442904
$604$	0	0
$605$	−3.17160	−0.128944
$606$	0	0
$607$	−9.16706	−0.372079	−0.186040	−	0.982542i	$-0.559565\pi$
$607$	−9.16706	−0.372079	−0.186040	+	0.982542i	$0.559565\pi$
$608$	0	0
$609$	6.93981	0.281215
$610$	0	0
$611$	0.345395	0.0139732
$612$	0	0
$613$	39.4458	1.59320	0.796601	−	0.604505i	$-0.206629\pi$
$613$	39.4458	1.59320	0.796601	+	0.604505i	$0.206629\pi$
$614$	0	0
$615$	0.578359	0.0233217
$616$	0	0
$617$	−17.0522	−0.686495	−0.343247	−	0.939245i	$-0.611527\pi$
$617$	−17.0522	−0.686495	−0.343247	+	0.939245i	$0.611527\pi$
$618$	0	0
$619$	−35.9531	−1.44508	−0.722538	−	0.691332i	$-0.757024\pi$
$619$	−35.9531	−1.44508	−0.722538	+	0.691332i	$0.757024\pi$
$620$	0	0
$621$	1.90776	0.0765560
$622$	0	0
$623$	−25.9704	−1.04048
$624$	0	0
$625$	13.4737	0.538948
$626$	0	0
$627$	1.32531	0.0529277
$628$	0	0
$629$	2.81734	0.112335
$630$	0	0
$631$	40.4393	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$631$	40.4393	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$632$	0	0
$633$	−0.892710	−0.0354820
$634$	0	0
$635$	−0.925255	−0.0367176
$636$	0	0
$637$	0.576022	0.0228228
$638$	0	0
$639$	−27.1252	−1.07306
$640$	0	0
$641$	24.0274	0.949025	0.474512	−	0.880249i	$-0.342624\pi$
$641$	24.0274	0.949025	0.474512	+	0.880249i	$0.342624\pi$
$642$	0	0
$643$	−5.72714	−0.225856	−0.112928	−	0.993603i	$-0.536023\pi$
$643$	−5.72714	−0.225856	−0.112928	+	0.993603i	$0.536023\pi$
$644$	0	0
$645$	1.05321	0.0414703
$646$	0	0
$647$	−13.3266	−0.523922	−0.261961	−	0.965079i	$-0.584369\pi$
$647$	−13.3266	−0.523922	−0.261961	+	0.965079i	$0.584369\pi$
$648$	0	0
$649$	17.2803	0.678309
$650$	0	0
$651$	4.64177	0.181925
$652$	0	0
$653$	3.34007	0.130707	0.0653535	−	0.997862i	$-0.479182\pi$
$653$	3.34007	0.130707	0.0653535	+	0.997862i	$0.479182\pi$
$654$	0	0
$655$	8.20869	0.320740
$656$	0	0
$657$	4.00965	0.156431
$658$	0	0
$659$	21.8770	0.852207	0.426103	−	0.904674i	$-0.359886\pi$
$659$	21.8770	0.852207	0.426103	+	0.904674i	$0.359886\pi$
$660$	0	0
$661$	7.79954	0.303367	0.151683	−	0.988429i	$-0.451531\pi$
$661$	7.79954	0.303367	0.151683	+	0.988429i	$0.451531\pi$
$662$	0	0
$663$	0.0552152	0.00214438
$664$	0	0
$665$	−1.54048	−0.0597372
$666$	0	0
$667$	−7.21601	−0.279405
$668$	0	0
$669$	10.2918	0.397903
$670$	0	0
$671$	15.1167	0.583575
$672$	0	0
$673$	17.6284	0.679524	0.339762	−	0.940511i	$-0.389653\pi$
$673$	17.6284	0.679524	0.339762	+	0.940511i	$0.389653\pi$
$674$	0	0
$675$	10.8712	0.418432
$676$	0	0
$677$	−27.5792	−1.05995	−0.529977	−	0.848012i	$-0.677800\pi$
$677$	−27.5792	−1.05995	−0.529977	+	0.848012i	$0.677800\pi$
$678$	0	0
$679$	−13.6258	−0.522911
$680$	0	0
$681$	−10.8558	−0.415997
$682$	0	0
$683$	0.636916	0.0243709	0.0121855	−	0.999926i	$-0.496121\pi$
$683$	0.636916	0.0243709	0.0121855	+	0.999926i	$0.496121\pi$
$684$	0	0
$685$	−19.6364	−0.750270
$686$	0	0
$687$	2.25729	0.0861211
$688$	0	0
$689$	−0.115263	−0.00439118
$690$	0	0
$691$	−51.5685	−1.96176	−0.980878	−	0.194623i	$-0.937652\pi$
$691$	−51.5685	−1.96176	−0.980878	+	0.194623i	$0.937652\pi$
$692$	0	0
$693$	12.0903	0.459272
$694$	0	0
$695$	0.566073	0.0214724
$696$	0	0
$697$	−1.38357	−0.0524064
$698$	0	0
$699$	−1.94933	−0.0737303
$700$	0	0
$701$	−32.0254	−1.20958	−0.604792	−	0.796384i	$-0.706744\pi$
$701$	−32.0254	−1.20958	−0.604792	+	0.796384i	$0.706744\pi$
$702$	0	0
$703$	3.15972	0.119171
$704$	0	0
$705$	−1.09039	−0.0410666
$706$	0	0
$707$	13.3665	0.502698
$708$	0	0
$709$	36.6315	1.37573	0.687863	−	0.725841i	$-0.258549\pi$
$709$	36.6315	1.37573	0.687863	+	0.725841i	$0.258549\pi$
$710$	0	0
$711$	−27.4473	−1.02935
$712$	0	0
$713$	−4.82651	−0.180754
$714$	0	0
$715$	0.315076	0.0117832
$716$	0	0
$717$	5.58455	0.208559
$718$	0	0
$719$	−17.4462	−0.650635	−0.325317	−	0.945605i	$-0.605471\pi$
$719$	−17.4462	−0.650635	−0.325317	+	0.945605i	$0.605471\pi$
$720$	0	0
$721$	−4.45915	−0.166068
$722$	0	0
$723$	3.71153	0.138033
$724$	0	0
$725$	−41.1196	−1.52714
$726$	0	0
$727$	−26.0541	−0.966294	−0.483147	−	0.875539i	$-0.660506\pi$
$727$	−26.0541	−0.966294	−0.483147	+	0.875539i	$0.660506\pi$
$728$	0	0
$729$	−16.7743	−0.621271
$730$	0	0
$731$	−2.51953	−0.0931883
$732$	0	0
$733$	−37.0798	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$733$	−37.0798	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$734$	0	0
$735$	−1.81847	−0.0670752
$736$	0	0
$737$	−10.6256	−0.391400
$738$	0	0
$739$	−37.5384	−1.38087	−0.690436	−	0.723393i	$-0.742581\pi$
$739$	−37.5384	−1.38087	−0.690436	+	0.723393i	$0.742581\pi$
$740$	0	0
$741$	0.0619252	0.00227488
$742$	0	0
$743$	−0.654911	−0.0240263	−0.0120132	−	0.999928i	$-0.503824\pi$
$743$	−0.654911	−0.0240263	−0.0120132	+	0.999928i	$0.503824\pi$
$744$	0	0
$745$	9.55640	0.350119
$746$	0	0
$747$	13.6831	0.500639
$748$	0	0
$749$	−23.8823	−0.872639
$750$	0	0
$751$	−17.0374	−0.621704	−0.310852	−	0.950458i	$-0.600614\pi$
$751$	−17.0374	−0.621704	−0.310852	+	0.950458i	$0.600614\pi$
$752$	0	0
$753$	7.85920	0.286405
$754$	0	0
$755$	−15.7140	−0.571891
$756$	0	0
$757$	45.7650	1.66336	0.831678	−	0.555258i	$-0.187381\pi$
$757$	45.7650	1.66336	0.831678	+	0.555258i	$0.187381\pi$
$758$	0	0
$759$	0.899728	0.0326581
$760$	0	0
$761$	3.97736	0.144179	0.0720896	−	0.997398i	$-0.477033\pi$
$761$	3.97736	0.144179	0.0720896	+	0.997398i	$0.477033\pi$
$762$	0	0
$763$	−23.6623	−0.856632
$764$	0	0
$765$	2.43556	0.0880580
$766$	0	0
$767$	0.807422	0.0291543
$768$	0	0
$769$	30.5810	1.10278	0.551389	−	0.834248i	$-0.314098\pi$
$769$	30.5810	1.10278	0.551389	+	0.834248i	$0.314098\pi$
$770$	0	0
$771$	2.06066	0.0742129
$772$	0	0
$773$	38.2675	1.37638	0.688192	−	0.725528i	$-0.258405\pi$
$773$	38.2675	1.37638	0.688192	+	0.725528i	$0.258405\pi$
$774$	0	0
$775$	−27.5033	−0.987948
$776$	0	0
$777$	−2.06298	−0.0740089
$778$	0	0
$779$	−1.55171	−0.0555957
$780$	0	0
$781$	−26.5008	−0.948273
$782$	0	0
$783$	−25.4918	−0.911001
$784$	0	0
$785$	0.576863	0.0205891
$786$	0	0
$787$	23.0437	0.821420	0.410710	−	0.911766i	$-0.365281\pi$
$787$	23.0437	0.821420	0.410710	+	0.911766i	$0.365281\pi$
$788$	0	0
$789$	−6.21440	−0.221239
$790$	0	0
$791$	−31.5818	−1.12292
$792$	0	0
$793$	0.706331	0.0250825
$794$	0	0
$795$	0.363880	0.0129055
$796$	0	0
$797$	54.3456	1.92502	0.962510	−	0.271247i	$-0.0874359\pi$
$797$	54.3456	1.92502	0.962510	+	0.271247i	$0.0874359\pi$
$798$	0	0
$799$	2.60847	0.0922811
$800$	0	0
$801$	46.0502	1.62710
$802$	0	0
$803$	3.91735	0.138240
$804$	0	0
$805$	−1.04580	−0.0368597
$806$	0	0
$807$	−1.21735	−0.0428529
$808$	0	0
$809$	28.0937	0.987720	0.493860	−	0.869541i	$-0.335585\pi$
$809$	28.0937	0.987720	0.493860	+	0.869541i	$0.335585\pi$
$810$	0	0
$811$	−13.6466	−0.479198	−0.239599	−	0.970872i	$-0.577016\pi$
$811$	−13.6466	−0.479198	−0.239599	+	0.970872i	$0.577016\pi$
$812$	0	0
$813$	−3.05899	−0.107283
$814$	0	0
$815$	−17.5645	−0.615258
$816$	0	0
$817$	−2.82572	−0.0988593
$818$	0	0
$819$	0.564920	0.0197399
$820$	0	0
$821$	−8.16976	−0.285126	−0.142563	−	0.989786i	$-0.545534\pi$
$821$	−8.16976	−0.285126	−0.142563	+	0.989786i	$0.545534\pi$
$822$	0	0
$823$	−3.93119	−0.137033	−0.0685163	−	0.997650i	$-0.521827\pi$
$823$	−3.93119	−0.137033	−0.0685163	+	0.997650i	$0.521827\pi$
$824$	0	0
$825$	5.12700	0.178499
$826$	0	0
$827$	−44.6674	−1.55324	−0.776618	−	0.629972i	$-0.783066\pi$
$827$	−44.6674	−1.55324	−0.776618	+	0.629972i	$0.783066\pi$
$828$	0	0
$829$	−38.5802	−1.33995	−0.669974	−	0.742385i	$-0.733695\pi$
$829$	−38.5802	−1.33995	−0.669974	+	0.742385i	$0.733695\pi$
$830$	0	0
$831$	−7.35496	−0.255141
$832$	0	0
$833$	4.35020	0.150726
$834$	0	0
$835$	−5.03508	−0.174246
$836$	0	0
$837$	−17.0505	−0.589350
$838$	0	0
$839$	−8.85031	−0.305547	−0.152773	−	0.988261i	$-0.548820\pi$
$839$	−8.85031	−0.305547	−0.152773	+	0.988261i	$0.548820\pi$
$840$	0	0
$841$	67.4211	2.32487
$842$	0	0
$843$	7.02923	0.242099
$844$	0	0
$845$	−11.7028	−0.402587
$846$	0	0
$847$	−5.55567	−0.190895
$848$	0	0
$849$	−11.9035	−0.408526
$850$	0	0
$851$	2.14508	0.0735324
$852$	0	0
$853$	−2.45889	−0.0841907	−0.0420953	−	0.999114i	$-0.513403\pi$
$853$	−2.45889	−0.0841907	−0.0420953	+	0.999114i	$0.513403\pi$
$854$	0	0
$855$	2.73154	0.0934169
$856$	0	0
$857$	−36.1084	−1.23344	−0.616719	−	0.787183i	$-0.711539\pi$
$857$	−36.1084	−1.23344	−0.616719	+	0.787183i	$0.711539\pi$
$858$	0	0
$859$	−31.4366	−1.07260	−0.536301	−	0.844027i	$-0.680179\pi$
$859$	−31.4366	−1.07260	−0.536301	+	0.844027i	$0.680179\pi$
$860$	0	0
$861$	1.01311	0.0345266
$862$	0	0
$863$	11.2863	0.384189	0.192094	−	0.981376i	$-0.438472\pi$
$863$	11.2863	0.384189	0.192094	+	0.981376i	$0.438472\pi$
$864$	0	0
$865$	−19.1910	−0.652514
$866$	0	0
$867$	−7.19262	−0.244274
$868$	0	0
$869$	−26.8155	−0.909652
$870$	0	0
$871$	−0.496483	−0.0168227
$872$	0	0
$873$	24.1610	0.817726
$874$	0	0
$875$	−13.0750	−0.442014
$876$	0	0
$877$	7.09812	0.239687	0.119843	−	0.992793i	$-0.461761\pi$
$877$	7.09812	0.239687	0.119843	+	0.992793i	$0.461761\pi$
$878$	0	0
$879$	−9.54018	−0.321782
$880$	0	0
$881$	−2.63729	−0.0888526	−0.0444263	−	0.999013i	$-0.514146\pi$
$881$	−2.63729	−0.0888526	−0.0444263	+	0.999013i	$0.514146\pi$
$882$	0	0
$883$	−6.24123	−0.210034	−0.105017	−	0.994470i	$-0.533490\pi$
$883$	−6.24123	−0.210034	−0.105017	+	0.994470i	$0.533490\pi$
$884$	0	0
$885$	−2.54898	−0.0856832
$886$	0	0
$887$	24.5186	0.823255	0.411627	−	0.911352i	$-0.364961\pi$
$887$	24.5186	0.823255	0.411627	+	0.911352i	$0.364961\pi$
$888$	0	0
$889$	−1.62076	−0.0543586
$890$	0	0
$891$	−19.7941	−0.663128
$892$	0	0
$893$	2.92547	0.0978970
$894$	0	0
$895$	−14.5831	−0.487461
$896$	0	0
$897$	0.0420399	0.00140367
$898$	0	0
$899$	64.4923	2.15094
$900$	0	0
$901$	−0.870485	−0.0290001
$902$	0	0
$903$	1.84491	0.0613946
$904$	0	0
$905$	−19.6800	−0.654185
$906$	0	0
$907$	−12.2547	−0.406911	−0.203456	−	0.979084i	$-0.565217\pi$
$907$	−12.2547	−0.406911	−0.203456	+	0.979084i	$0.565217\pi$
$908$	0	0
$909$	−23.7012	−0.786118
$910$	0	0
$911$	−9.26239	−0.306877	−0.153438	−	0.988158i	$-0.549035\pi$
$911$	−9.26239	−0.306877	−0.153438	+	0.988158i	$0.549035\pi$
$912$	0	0
$913$	13.3681	0.442421
$914$	0	0
$915$	−2.22985	−0.0737164
$916$	0	0
$917$	14.3791	0.474839
$918$	0	0
$919$	−51.4107	−1.69588	−0.847942	−	0.530090i	$-0.822158\pi$
$919$	−51.4107	−1.69588	−0.847942	+	0.530090i	$0.822158\pi$
$920$	0	0
$921$	12.6853	0.417994
$922$	0	0
$923$	−1.23825	−0.0407576
$924$	0	0
$925$	12.2235	0.401906
$926$	0	0
$927$	7.90688	0.259696
$928$	0	0
$929$	1.94174	0.0637065	0.0318532	−	0.999493i	$-0.489859\pi$
$929$	1.94174	0.0637065	0.0318532	+	0.999493i	$0.489859\pi$
$930$	0	0
$931$	4.87886	0.159898
$932$	0	0
$933$	−5.47567	−0.179265
$934$	0	0
$935$	2.37950	0.0778180
$936$	0	0
$937$	33.7004	1.10094	0.550472	−	0.834854i	$-0.314448\pi$
$937$	33.7004	1.10094	0.550472	+	0.834854i	$0.314448\pi$
$938$	0	0
$939$	10.9522	0.357411
$940$	0	0
$941$	19.5026	0.635766	0.317883	−	0.948130i	$-0.397028\pi$
$941$	19.5026	0.635766	0.317883	+	0.948130i	$0.397028\pi$
$942$	0	0
$943$	−1.05343	−0.0343043
$944$	0	0
$945$	−3.69448	−0.120181
$946$	0	0
$947$	33.4170	1.08591	0.542954	−	0.839763i	$-0.317306\pi$
$947$	33.4170	1.08591	0.542954	+	0.839763i	$0.317306\pi$
$948$	0	0
$949$	0.183038	0.00594168
$950$	0	0
$951$	6.14280	0.199194
$952$	0	0
$953$	35.7494	1.15804	0.579019	−	0.815314i	$-0.303436\pi$
$953$	35.7494	1.15804	0.579019	+	0.815314i	$0.303436\pi$
$954$	0	0
$955$	15.3465	0.496600
$956$	0	0
$957$	−12.0223	−0.388625
$958$	0	0
$959$	−34.3970	−1.11074
$960$	0	0
$961$	12.1364	0.391497
$962$	0	0
$963$	42.3475	1.36463
$964$	0	0
$965$	−2.52509	−0.0812855
$966$	0	0
$967$	−24.9239	−0.801499	−0.400749	−	0.916188i	$-0.631250\pi$
$967$	−24.9239	−0.801499	−0.400749	+	0.916188i	$0.631250\pi$
$968$	0	0
$969$	0.467668	0.0150237
$970$	0	0
$971$	−19.1722	−0.615266	−0.307633	−	0.951505i	$-0.599537\pi$
$971$	−19.1722	−0.615266	−0.307633	+	0.951505i	$0.599537\pi$
$972$	0	0
$973$	0.991585	0.0317887
$974$	0	0
$975$	0.239560	0.00767205
$976$	0	0
$977$	−15.2149	−0.486768	−0.243384	−	0.969930i	$-0.578257\pi$
$977$	−15.2149	−0.486768	−0.243384	+	0.969930i	$0.578257\pi$
$978$	0	0
$979$	44.9901	1.43789
$980$	0	0
$981$	41.9575	1.33960
$982$	0	0
$983$	20.8694	0.665632	0.332816	−	0.942992i	$-0.392001\pi$
$983$	20.8694	0.665632	0.332816	+	0.942992i	$0.392001\pi$
$984$	0	0
$985$	7.40425	0.235919
$986$	0	0
$987$	−1.91003	−0.0607970
$988$	0	0
$989$	−1.91833	−0.0609993
$990$	0	0
$991$	48.6095	1.54413	0.772066	−	0.635543i	$-0.219224\pi$
$991$	48.6095	1.54413	0.772066	+	0.635543i	$0.219224\pi$
$992$	0	0
$993$	14.4873	0.459741
$994$	0	0
$995$	22.6245	0.717244
$996$	0	0
$997$	−38.6835	−1.22512	−0.612559	−	0.790425i	$-0.709860\pi$
$997$	−38.6835	−1.22512	−0.612559	+	0.790425i	$0.709860\pi$
$998$	0	0
$999$	7.57786	0.239753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.16		28
4.3	odd	2		4024.2.a.d.1.13	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.13	✓	28	4.3	odd	2
8048.2.a.v.1.16		28	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.447624 q^{3} +0.901345 q^{5} +1.57888 q^{7} -2.79963 q^{9} +O(q^{10})\)	\(q+0.447624 q^{3} +0.901345 q^{5} +1.57888 q^{7} -2.79963 q^{9} -2.73519 q^{11} -0.127802 q^{13} +0.403464 q^{15} -0.965178 q^{17} -1.08247 q^{19} +0.706743 q^{21} -0.734871 q^{23} -4.18758 q^{25} -2.59606 q^{27} +9.81943 q^{29} +6.56783 q^{31} -1.22434 q^{33} +1.42311 q^{35} -2.91899 q^{37} -0.0572072 q^{39} +1.43349 q^{41} +2.61043 q^{43} -2.52343 q^{45} -2.70258 q^{47} -4.50715 q^{49} -0.432037 q^{51} +0.901890 q^{53} -2.46535 q^{55} -0.484540 q^{57} -6.31776 q^{59} -5.52676 q^{61} -4.42028 q^{63} -0.115194 q^{65} +3.88478 q^{67} -0.328946 q^{69} +9.68884 q^{71} -1.43220 q^{73} -1.87446 q^{75} -4.31853 q^{77} +9.80388 q^{79} +7.23684 q^{81} -4.88746 q^{83} -0.869958 q^{85} +4.39541 q^{87} -16.4486 q^{89} -0.201784 q^{91} +2.93992 q^{93} -0.975680 q^{95} -8.63007 q^{97} +7.65752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

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