LMFDB - Embedded newform 8048.2.a.u.1.17

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:	$0$
Dimension:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+1.08298 q^{3} +3.81229 q^{5} -4.19464 q^{7} -1.82716 q^{9} +O(q^{10})$	$q+1.08298 q^{3} +3.81229 q^{5} -4.19464 q^{7} -1.82716 q^{9} -2.27881 q^{11} -3.02088 q^{13} +4.12863 q^{15} -3.00674 q^{17} -3.39721 q^{19} -4.54271 q^{21} +0.954786 q^{23} +9.53358 q^{25} -5.22771 q^{27} +10.1698 q^{29} +9.84510 q^{31} -2.46790 q^{33} -15.9912 q^{35} +4.35659 q^{37} -3.27154 q^{39} +4.92893 q^{41} -5.37975 q^{43} -6.96566 q^{45} +11.4956 q^{47} +10.5950 q^{49} -3.25624 q^{51} +4.15359 q^{53} -8.68750 q^{55} -3.67910 q^{57} +0.911624 q^{59} -6.19541 q^{61} +7.66427 q^{63} -11.5165 q^{65} -4.89771 q^{67} +1.03401 q^{69} +8.71056 q^{71} +6.88092 q^{73} +10.3247 q^{75} +9.55880 q^{77} -7.64692 q^{79} -0.180022 q^{81} +4.88914 q^{83} -11.4626 q^{85} +11.0137 q^{87} -1.32368 q^{89} +12.6715 q^{91} +10.6620 q^{93} -12.9512 q^{95} +15.8885 q^{97} +4.16375 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	1.08298	0.625258	0.312629	−	0.949875i	$-0.398790\pi$
$3$	1.08298	0.625258	0.312629	+	0.949875i	$0.398790\pi$
$4$	0	0
$5$	3.81229	1.70491	0.852455	−	0.522801i	$-0.175113\pi$
$5$	3.81229	1.70491	0.852455	+	0.522801i	$0.175113\pi$
$6$	0	0
$7$	−4.19464	−1.58543	−0.792713	−	0.609595i	$-0.791332\pi$
$7$	−4.19464	−1.58543	−0.792713	+	0.609595i	$0.791332\pi$
$8$	0	0
$9$	−1.82716	−0.609053
$10$	0	0
$11$	−2.27881	−0.687088	−0.343544	−	0.939137i	$-0.611627\pi$
$11$	−2.27881	−0.687088	−0.343544	+	0.939137i	$0.611627\pi$
$12$	0	0
$13$	−3.02088	−0.837840	−0.418920	−	0.908023i	$-0.637591\pi$
$13$	−3.02088	−0.837840	−0.418920	+	0.908023i	$0.637591\pi$
$14$	0	0
$15$	4.12863	1.06601
$16$	0	0
$17$	−3.00674	−0.729242	−0.364621	−	0.931156i	$-0.618802\pi$
$17$	−3.00674	−0.729242	−0.364621	+	0.931156i	$0.618802\pi$
$18$	0	0
$19$	−3.39721	−0.779373	−0.389687	−	0.920948i	$-0.627417\pi$
$19$	−3.39721	−0.779373	−0.389687	+	0.920948i	$0.627417\pi$
$20$	0	0
$21$	−4.54271	−0.991300
$22$	0	0
$23$	0.954786	0.199087	0.0995433	−	0.995033i	$-0.468262\pi$
$23$	0.954786	0.199087	0.0995433	+	0.995033i	$0.468262\pi$
$24$	0	0
$25$	9.53358	1.90672
$26$	0	0
$27$	−5.22771	−1.00607
$28$	0	0
$29$	10.1698	1.88848	0.944242	−	0.329251i	$-0.106796\pi$
$29$	10.1698	1.88848	0.944242	+	0.329251i	$0.106796\pi$
$30$	0	0
$31$	9.84510	1.76823	0.884116	−	0.467267i	$-0.154761\pi$
$31$	9.84510	1.76823	0.884116	+	0.467267i	$0.154761\pi$
$32$	0	0
$33$	−2.46790	−0.429607
$34$	0	0
$35$	−15.9912	−2.70301
$36$	0	0
$37$	4.35659	0.716219	0.358110	−	0.933680i	$-0.383421\pi$
$37$	4.35659	0.716219	0.358110	+	0.933680i	$0.383421\pi$
$38$	0	0
$39$	−3.27154	−0.523866
$40$	0	0
$41$	4.92893	0.769770	0.384885	−	0.922964i	$-0.374241\pi$
$41$	4.92893	0.769770	0.384885	+	0.922964i	$0.374241\pi$
$42$	0	0
$43$	−5.37975	−0.820404	−0.410202	−	0.911995i	$-0.634542\pi$
$43$	−5.37975	−0.820404	−0.410202	+	0.911995i	$0.634542\pi$
$44$	0	0
$45$	−6.96566	−1.03838
$46$	0	0
$47$	11.4956	1.67681	0.838403	−	0.545051i	$-0.183490\pi$
$47$	11.4956	1.67681	0.838403	+	0.545051i	$0.183490\pi$
$48$	0	0
$49$	10.5950	1.51357
$50$	0	0
$51$	−3.25624	−0.455965
$52$	0	0
$53$	4.15359	0.570540	0.285270	−	0.958447i	$-0.407917\pi$
$53$	4.15359	0.570540	0.285270	+	0.958447i	$0.407917\pi$
$54$	0	0
$55$	−8.68750	−1.17142
$56$	0	0
$57$	−3.67910	−0.487309
$58$	0	0
$59$	0.911624	0.118683	0.0593417	−	0.998238i	$-0.481100\pi$
$59$	0.911624	0.118683	0.0593417	+	0.998238i	$0.481100\pi$
$60$	0	0
$61$	−6.19541	−0.793241	−0.396621	−	0.917983i	$-0.629817\pi$
$61$	−6.19541	−0.793241	−0.396621	+	0.917983i	$0.629817\pi$
$62$	0	0
$63$	7.66427	0.965607
$64$	0	0
$65$	−11.5165	−1.42844
$66$	0	0
$67$	−4.89771	−0.598351	−0.299175	−	0.954198i	$-0.596712\pi$
$67$	−4.89771	−0.598351	−0.299175	+	0.954198i	$0.596712\pi$
$68$	0	0
$69$	1.03401	0.124481
$70$	0	0
$71$	8.71056	1.03375	0.516877	−	0.856060i	$-0.327095\pi$
$71$	8.71056	1.03375	0.516877	+	0.856060i	$0.327095\pi$
$72$	0	0
$73$	6.88092	0.805351	0.402675	−	0.915343i	$-0.368080\pi$
$73$	6.88092	0.805351	0.402675	+	0.915343i	$0.368080\pi$
$74$	0	0
$75$	10.3247	1.19219
$76$	0	0
$77$	9.55880	1.08933
$78$	0	0
$79$	−7.64692	−0.860345	−0.430173	−	0.902747i	$-0.641547\pi$
$79$	−7.64692	−0.860345	−0.430173	+	0.902747i	$0.641547\pi$
$80$	0	0
$81$	−0.180022	−0.0200025
$82$	0	0
$83$	4.88914	0.536653	0.268326	−	0.963328i	$-0.413529\pi$
$83$	4.88914	0.536653	0.268326	+	0.963328i	$0.413529\pi$
$84$	0	0
$85$	−11.4626	−1.24329
$86$	0	0
$87$	11.0137	1.18079
$88$	0	0
$89$	−1.32368	−0.140310	−0.0701549	−	0.997536i	$-0.522349\pi$
$89$	−1.32368	−0.140310	−0.0701549	+	0.997536i	$0.522349\pi$
$90$	0	0
$91$	12.6715	1.32833
$92$	0	0
$93$	10.6620	1.10560
$94$	0	0
$95$	−12.9512	−1.32876
$96$	0	0
$97$	15.8885	1.61324	0.806618	−	0.591074i	$-0.201296\pi$
$97$	15.8885	1.61324	0.806618	+	0.591074i	$0.201296\pi$
$98$	0	0
$99$	4.16375	0.418472
$100$	0	0
$101$	−13.1510	−1.30858	−0.654288	−	0.756246i	$-0.727031\pi$
$101$	−13.1510	−1.30858	−0.654288	+	0.756246i	$0.727031\pi$
$102$	0	0
$103$	11.8316	1.16580	0.582900	−	0.812544i	$-0.301918\pi$
$103$	11.8316	1.16580	0.582900	+	0.812544i	$0.301918\pi$
$104$	0	0
$105$	−17.3181	−1.69008
$106$	0	0
$107$	−7.92281	−0.765927	−0.382964	−	0.923763i	$-0.625097\pi$
$107$	−7.92281	−0.765927	−0.382964	+	0.923763i	$0.625097\pi$
$108$	0	0
$109$	19.3910	1.85732	0.928660	−	0.370932i	$-0.120962\pi$
$109$	19.3910	1.85732	0.928660	+	0.370932i	$0.120962\pi$
$110$	0	0
$111$	4.71810	0.447822
$112$	0	0
$113$	5.57593	0.524540	0.262270	−	0.964995i	$-0.415529\pi$
$113$	5.57593	0.524540	0.262270	+	0.964995i	$0.415529\pi$
$114$	0	0
$115$	3.63993	0.339425
$116$	0	0
$117$	5.51962	0.510289
$118$	0	0
$119$	12.6122	1.15616
$120$	0	0
$121$	−5.80702	−0.527911
$122$	0	0
$123$	5.33793	0.481305
$124$	0	0
$125$	17.2833	1.54587
$126$	0	0
$127$	−19.9227	−1.76785	−0.883926	−	0.467626i	$-0.845109\pi$
$127$	−19.9227	−1.76785	−0.883926	+	0.467626i	$0.845109\pi$
$128$	0	0
$129$	−5.82615	−0.512964
$130$	0	0
$131$	16.5839	1.44894	0.724470	−	0.689306i	$-0.242084\pi$
$131$	16.5839	1.44894	0.724470	+	0.689306i	$0.242084\pi$
$132$	0	0
$133$	14.2501	1.23564
$134$	0	0
$135$	−19.9296	−1.71526
$136$	0	0
$137$	−9.03859	−0.772218	−0.386109	−	0.922453i	$-0.626181\pi$
$137$	−9.03859	−0.772218	−0.386109	+	0.922453i	$0.626181\pi$
$138$	0	0
$139$	−4.05614	−0.344037	−0.172019	−	0.985094i	$-0.555029\pi$
$139$	−4.05614	−0.344037	−0.172019	+	0.985094i	$0.555029\pi$
$140$	0	0
$141$	12.4495	1.04844
$142$	0	0
$143$	6.88401	0.575670
$144$	0	0
$145$	38.7703	3.21970
$146$	0	0
$147$	11.4742	0.946374
$148$	0	0
$149$	−15.8314	−1.29696	−0.648481	−	0.761231i	$-0.724595\pi$
$149$	−15.8314	−1.29696	−0.648481	+	0.761231i	$0.724595\pi$
$150$	0	0
$151$	11.6088	0.944712	0.472356	−	0.881408i	$-0.343404\pi$
$151$	11.6088	0.944712	0.472356	+	0.881408i	$0.343404\pi$
$152$	0	0
$153$	5.49379	0.444147
$154$	0	0
$155$	37.5324	3.01468
$156$	0	0
$157$	−3.16626	−0.252695	−0.126347	−	0.991986i	$-0.540325\pi$
$157$	−3.16626	−0.252695	−0.126347	+	0.991986i	$0.540325\pi$
$158$	0	0
$159$	4.49825	0.356734
$160$	0	0
$161$	−4.00499	−0.315637
$162$	0	0
$163$	13.4813	1.05594	0.527968	−	0.849264i	$-0.322954\pi$
$163$	13.4813	1.05594	0.527968	+	0.849264i	$0.322954\pi$
$164$	0	0
$165$	−9.40838	−0.732441
$166$	0	0
$167$	19.1923	1.48515	0.742573	−	0.669765i	$-0.233605\pi$
$167$	19.1923	1.48515	0.742573	+	0.669765i	$0.233605\pi$
$168$	0	0
$169$	−3.87431	−0.298024
$170$	0	0
$171$	6.20724	0.474679
$172$	0	0
$173$	14.4933	1.10191	0.550954	−	0.834536i	$-0.314264\pi$
$173$	14.4933	1.10191	0.550954	+	0.834536i	$0.314264\pi$
$174$	0	0
$175$	−39.9900	−3.02296
$176$	0	0
$177$	0.987269	0.0742077
$178$	0	0
$179$	22.7078	1.69726	0.848631	−	0.528985i	$-0.177427\pi$
$179$	22.7078	1.69726	0.848631	+	0.528985i	$0.177427\pi$
$180$	0	0
$181$	−8.18801	−0.608610	−0.304305	−	0.952575i	$-0.598424\pi$
$181$	−8.18801	−0.608610	−0.304305	+	0.952575i	$0.598424\pi$
$182$	0	0
$183$	−6.70950	−0.495980
$184$	0	0
$185$	16.6086	1.22109
$186$	0	0
$187$	6.85180	0.501053
$188$	0	0
$189$	21.9284	1.59505
$190$	0	0
$191$	−6.52751	−0.472314	−0.236157	−	0.971715i	$-0.575888\pi$
$191$	−6.52751	−0.472314	−0.236157	+	0.971715i	$0.575888\pi$
$192$	0	0
$193$	−3.05002	−0.219545	−0.109772	−	0.993957i	$-0.535012\pi$
$193$	−3.05002	−0.219545	−0.109772	+	0.993957i	$0.535012\pi$
$194$	0	0
$195$	−12.4721	−0.893145
$196$	0	0
$197$	6.84031	0.487352	0.243676	−	0.969857i	$-0.421647\pi$
$197$	6.84031	0.487352	0.243676	+	0.969857i	$0.421647\pi$
$198$	0	0
$199$	10.0866	0.715018	0.357509	−	0.933910i	$-0.383626\pi$
$199$	10.0866	0.715018	0.357509	+	0.933910i	$0.383626\pi$
$200$	0	0
$201$	−5.30411	−0.374123
$202$	0	0
$203$	−42.6587	−2.99405
$204$	0	0
$205$	18.7905	1.31239
$206$	0	0
$207$	−1.74454	−0.121254
$208$	0	0
$209$	7.74160	0.535498
$210$	0	0
$211$	−15.6262	−1.07575	−0.537875	−	0.843025i	$-0.680773\pi$
$211$	−15.6262	−1.07575	−0.537875	+	0.843025i	$0.680773\pi$
$212$	0	0
$213$	9.43335	0.646362
$214$	0	0
$215$	−20.5092	−1.39871
$216$	0	0
$217$	−41.2967	−2.80340
$218$	0	0
$219$	7.45189	0.503552
$220$	0	0
$221$	9.08300	0.610989
$222$	0	0
$223$	−10.6175	−0.711001	−0.355501	−	0.934676i	$-0.615690\pi$
$223$	−10.6175	−0.711001	−0.355501	+	0.934676i	$0.615690\pi$
$224$	0	0
$225$	−17.4194	−1.16129
$226$	0	0
$227$	19.7566	1.31129	0.655646	−	0.755068i	$-0.272396\pi$
$227$	19.7566	1.31129	0.655646	+	0.755068i	$0.272396\pi$
$228$	0	0
$229$	−17.7048	−1.16997	−0.584984	−	0.811045i	$-0.698899\pi$
$229$	−17.7048	−1.16997	−0.584984	+	0.811045i	$0.698899\pi$
$230$	0	0
$231$	10.3520	0.681110
$232$	0	0
$233$	3.89176	0.254958	0.127479	−	0.991841i	$-0.459312\pi$
$233$	3.89176	0.254958	0.127479	+	0.991841i	$0.459312\pi$
$234$	0	0
$235$	43.8246	2.85880
$236$	0	0
$237$	−8.28145	−0.537938
$238$	0	0
$239$	7.84648	0.507546	0.253773	−	0.967264i	$-0.418328\pi$
$239$	7.84648	0.507546	0.253773	+	0.967264i	$0.418328\pi$
$240$	0	0
$241$	17.1726	1.10618	0.553091	−	0.833121i	$-0.313448\pi$
$241$	17.1726	1.10618	0.553091	+	0.833121i	$0.313448\pi$
$242$	0	0
$243$	15.4882	0.993566
$244$	0	0
$245$	40.3913	2.58051
$246$	0	0
$247$	10.2625	0.652990
$248$	0	0
$249$	5.29483	0.335547
$250$	0	0
$251$	8.20804	0.518087	0.259043	−	0.965866i	$-0.416593\pi$
$251$	8.20804	0.518087	0.259043	+	0.965866i	$0.416593\pi$
$252$	0	0
$253$	−2.17578	−0.136790
$254$	0	0
$255$	−12.4137	−0.777378
$256$	0	0
$257$	27.0717	1.68869	0.844343	−	0.535803i	$-0.179991\pi$
$257$	27.0717	1.68869	0.844343	+	0.535803i	$0.179991\pi$
$258$	0	0
$259$	−18.2743	−1.13551
$260$	0	0
$261$	−18.5818	−1.15019
$262$	0	0
$263$	−9.05682	−0.558467	−0.279234	−	0.960223i	$-0.590080\pi$
$263$	−9.05682	−0.558467	−0.279234	+	0.960223i	$0.590080\pi$
$264$	0	0
$265$	15.8347	0.972719
$266$	0	0
$267$	−1.43352	−0.0877299
$268$	0	0
$269$	−0.962596	−0.0586905	−0.0293452	−	0.999569i	$-0.509342\pi$
$269$	−0.962596	−0.0586905	−0.0293452	+	0.999569i	$0.509342\pi$
$270$	0	0
$271$	−5.58876	−0.339493	−0.169746	−	0.985488i	$-0.554295\pi$
$271$	−5.58876	−0.339493	−0.169746	+	0.985488i	$0.554295\pi$
$272$	0	0
$273$	13.7230	0.830551
$274$	0	0
$275$	−21.7252	−1.31008
$276$	0	0
$277$	−26.6415	−1.60073	−0.800365	−	0.599513i	$-0.795361\pi$
$277$	−26.6415	−1.60073	−0.800365	+	0.599513i	$0.795361\pi$
$278$	0	0
$279$	−17.9886	−1.07695
$280$	0	0
$281$	−27.2137	−1.62343	−0.811716	−	0.584053i	$-0.801466\pi$
$281$	−27.2137	−1.62343	−0.811716	+	0.584053i	$0.801466\pi$
$282$	0	0
$283$	−4.53131	−0.269358	−0.134679	−	0.990889i	$-0.543000\pi$
$283$	−4.53131	−0.269358	−0.134679	+	0.990889i	$0.543000\pi$
$284$	0	0
$285$	−14.0258	−0.830818
$286$	0	0
$287$	−20.6751	−1.22041
$288$	0	0
$289$	−7.95949	−0.468206
$290$	0	0
$291$	17.2069	1.00869
$292$	0	0
$293$	6.39350	0.373512	0.186756	−	0.982406i	$-0.440203\pi$
$293$	6.39350	0.373512	0.186756	+	0.982406i	$0.440203\pi$
$294$	0	0
$295$	3.47538	0.202344
$296$	0	0
$297$	11.9130	0.691260
$298$	0	0
$299$	−2.88429	−0.166803
$300$	0	0
$301$	22.5661	1.30069
$302$	0	0
$303$	−14.2423	−0.818197
$304$	0	0
$305$	−23.6187	−1.35240
$306$	0	0
$307$	22.6042	1.29009	0.645045	−	0.764145i	$-0.276839\pi$
$307$	22.6042	1.29009	0.645045	+	0.764145i	$0.276839\pi$
$308$	0	0
$309$	12.8134	0.728926
$310$	0	0
$311$	3.75471	0.212910	0.106455	−	0.994318i	$-0.466050\pi$
$311$	3.75471	0.212910	0.106455	+	0.994318i	$0.466050\pi$
$312$	0	0
$313$	16.0058	0.904700	0.452350	−	0.891840i	$-0.350586\pi$
$313$	16.0058	0.904700	0.452350	+	0.891840i	$0.350586\pi$
$314$	0	0
$315$	29.2185	1.64627
$316$	0	0
$317$	10.7068	0.601353	0.300677	−	0.953726i	$-0.402788\pi$
$317$	10.7068	0.601353	0.300677	+	0.953726i	$0.402788\pi$
$318$	0	0
$319$	−23.1751	−1.29755
$320$	0	0
$321$	−8.58023	−0.478902
$322$	0	0
$323$	10.2145	0.568352
$324$	0	0
$325$	−28.7998	−1.59752
$326$	0	0
$327$	21.0000	1.16130
$328$	0	0
$329$	−48.2199	−2.65845
$330$	0	0
$331$	−6.94286	−0.381614	−0.190807	−	0.981628i	$-0.561110\pi$
$331$	−6.94286	−0.381614	−0.190807	+	0.981628i	$0.561110\pi$
$332$	0	0
$333$	−7.96018	−0.436215
$334$	0	0
$335$	−18.6715	−1.02013
$336$	0	0
$337$	21.8099	1.18806	0.594030	−	0.804443i	$-0.297536\pi$
$337$	21.8099	1.18806	0.594030	+	0.804443i	$0.297536\pi$
$338$	0	0
$339$	6.03862	0.327973
$340$	0	0
$341$	−22.4351	−1.21493
$342$	0	0
$343$	−15.0798	−0.814233
$344$	0	0
$345$	3.94196	0.212228
$346$	0	0
$347$	3.80449	0.204236	0.102118	−	0.994772i	$-0.467438\pi$
$347$	3.80449	0.204236	0.102118	+	0.994772i	$0.467438\pi$
$348$	0	0
$349$	−13.3809	−0.716264	−0.358132	−	0.933671i	$-0.616586\pi$
$349$	−13.3809	−0.716264	−0.358132	+	0.933671i	$0.616586\pi$
$350$	0	0
$351$	15.7923	0.842928
$352$	0	0
$353$	2.58912	0.137805	0.0689026	−	0.997623i	$-0.478050\pi$
$353$	2.58912	0.137805	0.0689026	+	0.997623i	$0.478050\pi$
$354$	0	0
$355$	33.2072	1.76246
$356$	0	0
$357$	13.6588	0.722898
$358$	0	0
$359$	20.5137	1.08267	0.541337	−	0.840806i	$-0.317918\pi$
$359$	20.5137	1.08267	0.541337	+	0.840806i	$0.317918\pi$
$360$	0	0
$361$	−7.45897	−0.392577
$362$	0	0
$363$	−6.28887	−0.330080
$364$	0	0
$365$	26.2321	1.37305
$366$	0	0
$367$	−0.171790	−0.00896738	−0.00448369	−	0.999990i	$-0.501427\pi$
$367$	−0.171790	−0.00896738	−0.00448369	+	0.999990i	$0.501427\pi$
$368$	0	0
$369$	−9.00594	−0.468831
$370$	0	0
$371$	−17.4228	−0.904548
$372$	0	0
$373$	−18.8055	−0.973710	−0.486855	−	0.873483i	$-0.661856\pi$
$373$	−18.8055	−0.973710	−0.486855	+	0.873483i	$0.661856\pi$
$374$	0	0
$375$	18.7175	0.966567
$376$	0	0
$377$	−30.7217	−1.58225
$378$	0	0
$379$	−6.87651	−0.353223	−0.176611	−	0.984281i	$-0.556514\pi$
$379$	−6.87651	−0.353223	−0.176611	+	0.984281i	$0.556514\pi$
$380$	0	0
$381$	−21.5758	−1.10536
$382$	0	0
$383$	11.9529	0.610765	0.305382	−	0.952230i	$-0.401216\pi$
$383$	11.9529	0.610765	0.305382	+	0.952230i	$0.401216\pi$
$384$	0	0
$385$	36.4409	1.85720
$386$	0	0
$387$	9.82965	0.499669
$388$	0	0
$389$	11.9805	0.607433	0.303717	−	0.952762i	$-0.401772\pi$
$389$	11.9805	0.607433	0.303717	+	0.952762i	$0.401772\pi$
$390$	0	0
$391$	−2.87080	−0.145182
$392$	0	0
$393$	17.9600	0.905961
$394$	0	0
$395$	−29.1523	−1.46681
$396$	0	0
$397$	−15.5259	−0.779223	−0.389612	−	0.920979i	$-0.627391\pi$
$397$	−15.5259	−0.779223	−0.389612	+	0.920979i	$0.627391\pi$
$398$	0	0
$399$	15.4325	0.772592
$400$	0	0
$401$	−13.8963	−0.693947	−0.346973	−	0.937875i	$-0.612791\pi$
$401$	−13.8963	−0.693947	−0.346973	+	0.937875i	$0.612791\pi$
$402$	0	0
$403$	−29.7408	−1.48150
$404$	0	0
$405$	−0.686297	−0.0341024
$406$	0	0
$407$	−9.92785	−0.492105
$408$	0	0
$409$	−33.7567	−1.66916	−0.834580	−	0.550886i	$-0.814290\pi$
$409$	−33.7567	−1.66916	−0.834580	+	0.550886i	$0.814290\pi$
$410$	0	0
$411$	−9.78859	−0.482836
$412$	0	0
$413$	−3.82394	−0.188164
$414$	0	0
$415$	18.6388	0.914945
$416$	0	0
$417$	−4.39271	−0.215112
$418$	0	0
$419$	8.92862	0.436192	0.218096	−	0.975927i	$-0.430015\pi$
$419$	8.92862	0.436192	0.218096	+	0.975927i	$0.430015\pi$
$420$	0	0
$421$	−5.56415	−0.271180	−0.135590	−	0.990765i	$-0.543293\pi$
$421$	−5.56415	−0.271180	−0.135590	+	0.990765i	$0.543293\pi$
$422$	0	0
$423$	−21.0043	−1.02126
$424$	0	0
$425$	−28.6650	−1.39046
$426$	0	0
$427$	25.9875	1.25763
$428$	0	0
$429$	7.45523	0.359942
$430$	0	0
$431$	35.4344	1.70682	0.853408	−	0.521244i	$-0.174532\pi$
$431$	35.4344	1.70682	0.853408	+	0.521244i	$0.174532\pi$
$432$	0	0
$433$	−16.5957	−0.797538	−0.398769	−	0.917051i	$-0.630563\pi$
$433$	−16.5957	−0.797538	−0.398769	+	0.917051i	$0.630563\pi$
$434$	0	0
$435$	41.9874	2.01314
$436$	0	0
$437$	−3.24361	−0.155163
$438$	0	0
$439$	−40.7116	−1.94306	−0.971529	−	0.236919i	$-0.923862\pi$
$439$	−40.7116	−1.94306	−0.971529	+	0.236919i	$0.923862\pi$
$440$	0	0
$441$	−19.3588	−0.921846
$442$	0	0
$443$	14.1236	0.671031	0.335516	−	0.942035i	$-0.391089\pi$
$443$	14.1236	0.671031	0.335516	+	0.942035i	$0.391089\pi$
$444$	0	0
$445$	−5.04626	−0.239216
$446$	0	0
$447$	−17.1451	−0.810936
$448$	0	0
$449$	−16.3908	−0.773530	−0.386765	−	0.922178i	$-0.626407\pi$
$449$	−16.3908	−0.773530	−0.386765	+	0.922178i	$0.626407\pi$
$450$	0	0
$451$	−11.2321	−0.528900
$452$	0	0
$453$	12.5721	0.590689
$454$	0	0
$455$	48.3074	2.26469
$456$	0	0
$457$	28.6905	1.34208	0.671042	−	0.741420i	$-0.265847\pi$
$457$	28.6905	1.34208	0.671042	+	0.741420i	$0.265847\pi$
$458$	0	0
$459$	15.7184	0.733671
$460$	0	0
$461$	21.7560	1.01328	0.506638	−	0.862159i	$-0.330888\pi$
$461$	21.7560	1.01328	0.506638	+	0.862159i	$0.330888\pi$
$462$	0	0
$463$	15.1752	0.705249	0.352625	−	0.935765i	$-0.385289\pi$
$463$	15.1752	0.705249	0.352625	+	0.935765i	$0.385289\pi$
$464$	0	0
$465$	40.6468	1.88495
$466$	0	0
$467$	−13.9542	−0.645721	−0.322861	−	0.946447i	$-0.604645\pi$
$467$	−13.9542	−0.645721	−0.322861	+	0.946447i	$0.604645\pi$
$468$	0	0
$469$	20.5441	0.948640
$470$	0	0
$471$	−3.42899	−0.157999
$472$	0	0
$473$	12.2594	0.563689
$474$	0	0
$475$	−32.3876	−1.48604
$476$	0	0
$477$	−7.58927	−0.347489
$478$	0	0
$479$	31.0093	1.41685	0.708427	−	0.705785i	$-0.249405\pi$
$479$	31.0093	1.41685	0.708427	+	0.705785i	$0.249405\pi$
$480$	0	0
$481$	−13.1607	−0.600077
$482$	0	0
$483$	−4.33731	−0.197355
$484$	0	0
$485$	60.5717	2.75042
$486$	0	0
$487$	11.7183	0.531005	0.265503	−	0.964110i	$-0.414462\pi$
$487$	11.7183	0.531005	0.265503	+	0.964110i	$0.414462\pi$
$488$	0	0
$489$	14.5999	0.660232
$490$	0	0
$491$	−4.00695	−0.180831	−0.0904156	−	0.995904i	$-0.528820\pi$
$491$	−4.00695	−0.180831	−0.0904156	+	0.995904i	$0.528820\pi$
$492$	0	0
$493$	−30.5780	−1.37716
$494$	0	0
$495$	15.8734	0.713458
$496$	0	0
$497$	−36.5377	−1.63894
$498$	0	0
$499$	−37.1067	−1.66112	−0.830561	−	0.556927i	$-0.811980\pi$
$499$	−37.1067	−1.66112	−0.830561	+	0.556927i	$0.811980\pi$
$500$	0	0
$501$	20.7849	0.928600
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−50.1355	−2.23100
$506$	0	0
$507$	−4.19579	−0.186342
$508$	0	0
$509$	15.7904	0.699895	0.349948	−	0.936769i	$-0.386199\pi$
$509$	15.7904	0.699895	0.349948	+	0.936769i	$0.386199\pi$
$510$	0	0
$511$	−28.8630	−1.27682
$512$	0	0
$513$	17.7596	0.784106
$514$	0	0
$515$	45.1055	1.98759
$516$	0	0
$517$	−26.1963	−1.15211
$518$	0	0
$519$	15.6960	0.688977
$520$	0	0
$521$	−12.3114	−0.539373	−0.269687	−	0.962948i	$-0.586920\pi$
$521$	−12.3114	−0.539373	−0.269687	+	0.962948i	$0.586920\pi$
$522$	0	0
$523$	3.22860	0.141177	0.0705884	−	0.997506i	$-0.477512\pi$
$523$	3.22860	0.141177	0.0705884	+	0.997506i	$0.477512\pi$
$524$	0	0
$525$	−43.3083	−1.89013
$526$	0	0
$527$	−29.6017	−1.28947
$528$	0	0
$529$	−22.0884	−0.960364
$530$	0	0
$531$	−1.66568	−0.0722844
$532$	0	0
$533$	−14.8897	−0.644945
$534$	0	0
$535$	−30.2041	−1.30584
$536$	0	0
$537$	24.5921	1.06123
$538$	0	0
$539$	−24.1441	−1.03996
$540$	0	0
$541$	−29.3113	−1.26019	−0.630096	−	0.776517i	$-0.716985\pi$
$541$	−29.3113	−1.26019	−0.630096	+	0.776517i	$0.716985\pi$
$542$	0	0
$543$	−8.86744	−0.380538
$544$	0	0
$545$	73.9241	3.16656
$546$	0	0
$547$	−33.6999	−1.44090	−0.720452	−	0.693505i	$-0.756066\pi$
$547$	−33.6999	−1.44090	−0.720452	+	0.693505i	$0.756066\pi$
$548$	0	0
$549$	11.3200	0.483126
$550$	0	0
$551$	−34.5489	−1.47183
$552$	0	0
$553$	32.0761	1.36401
$554$	0	0
$555$	17.9868	0.763496
$556$	0	0
$557$	29.9607	1.26948	0.634739	−	0.772727i	$-0.281108\pi$
$557$	29.9607	1.26948	0.634739	+	0.772727i	$0.281108\pi$
$558$	0	0
$559$	16.2516	0.687367
$560$	0	0
$561$	7.42035	0.313288
$562$	0	0
$563$	19.4400	0.819297	0.409648	−	0.912243i	$-0.365651\pi$
$563$	19.4400	0.819297	0.409648	+	0.912243i	$0.365651\pi$
$564$	0	0
$565$	21.2571	0.894293
$566$	0	0
$567$	0.755128	0.0317124
$568$	0	0
$569$	−2.10666	−0.0883159	−0.0441580	−	0.999025i	$-0.514060\pi$
$569$	−2.10666	−0.0883159	−0.0441580	+	0.999025i	$0.514060\pi$
$570$	0	0
$571$	4.21722	0.176485	0.0882426	−	0.996099i	$-0.471875\pi$
$571$	4.21722	0.176485	0.0882426	+	0.996099i	$0.471875\pi$
$572$	0	0
$573$	−7.06915	−0.295318
$574$	0	0
$575$	9.10253	0.379602
$576$	0	0
$577$	11.5233	0.479723	0.239861	−	0.970807i	$-0.422898\pi$
$577$	11.5233	0.479723	0.239861	+	0.970807i	$0.422898\pi$
$578$	0	0
$579$	−3.30310	−0.137272
$580$	0	0
$581$	−20.5082	−0.850823
$582$	0	0
$583$	−9.46526	−0.392011
$584$	0	0
$585$	21.0424	0.869996
$586$	0	0
$587$	−0.879492	−0.0363005	−0.0181503	−	0.999835i	$-0.505778\pi$
$587$	−0.879492	−0.0363005	−0.0181503	+	0.999835i	$0.505778\pi$
$588$	0	0
$589$	−33.4459	−1.37811
$590$	0	0
$591$	7.40791	0.304721
$592$	0	0
$593$	39.0485	1.60353	0.801765	−	0.597639i	$-0.203894\pi$
$593$	39.0485	1.60353	0.801765	+	0.597639i	$0.203894\pi$
$594$	0	0
$595$	48.0814	1.97115
$596$	0	0
$597$	10.9235	0.447070
$598$	0	0
$599$	−48.1521	−1.96744	−0.983721	−	0.179703i	$-0.942486\pi$
$599$	−48.1521	−1.96744	−0.983721	+	0.179703i	$0.942486\pi$
$600$	0	0
$601$	−9.66650	−0.394305	−0.197152	−	0.980373i	$-0.563169\pi$
$601$	−9.66650	−0.394305	−0.197152	+	0.980373i	$0.563169\pi$
$602$	0	0
$603$	8.94889	0.364427
$604$	0	0
$605$	−22.1381	−0.900040
$606$	0	0
$607$	−38.4722	−1.56154	−0.780768	−	0.624821i	$-0.785172\pi$
$607$	−38.4722	−1.56154	−0.780768	+	0.624821i	$0.785172\pi$
$608$	0	0
$609$	−46.1984	−1.87205
$610$	0	0
$611$	−34.7268	−1.40490
$612$	0	0
$613$	8.45580	0.341527	0.170763	−	0.985312i	$-0.445377\pi$
$613$	8.45580	0.341527	0.170763	+	0.985312i	$0.445377\pi$
$614$	0	0
$615$	20.3498	0.820581
$616$	0	0
$617$	−14.4715	−0.582599	−0.291299	−	0.956632i	$-0.594088\pi$
$617$	−14.4715	−0.582599	−0.291299	+	0.956632i	$0.594088\pi$
$618$	0	0
$619$	−17.3060	−0.695585	−0.347793	−	0.937571i	$-0.613069\pi$
$619$	−17.3060	−0.695585	−0.347793	+	0.937571i	$0.613069\pi$
$620$	0	0
$621$	−4.99134	−0.200296
$622$	0	0
$623$	5.55237	0.222451
$624$	0	0
$625$	18.2213	0.728851
$626$	0	0
$627$	8.38399	0.334824
$628$	0	0
$629$	−13.0992	−0.522298
$630$	0	0
$631$	−9.14821	−0.364184	−0.182092	−	0.983281i	$-0.558287\pi$
$631$	−9.14821	−0.364184	−0.182092	+	0.983281i	$0.558287\pi$
$632$	0	0
$633$	−16.9228	−0.672621
$634$	0	0
$635$	−75.9511	−3.01403
$636$	0	0
$637$	−32.0062	−1.26813
$638$	0	0
$639$	−15.9156	−0.629610
$640$	0	0
$641$	1.33878	0.0528786	0.0264393	−	0.999650i	$-0.491583\pi$
$641$	1.33878	0.0528786	0.0264393	+	0.999650i	$0.491583\pi$
$642$	0	0
$643$	13.0037	0.512817	0.256408	−	0.966569i	$-0.417461\pi$
$643$	13.0037	0.512817	0.256408	+	0.966569i	$0.417461\pi$
$644$	0	0
$645$	−22.2110	−0.874557
$646$	0	0
$647$	−33.7529	−1.32696	−0.663481	−	0.748193i	$-0.730922\pi$
$647$	−33.7529	−1.32696	−0.663481	+	0.748193i	$0.730922\pi$
$648$	0	0
$649$	−2.07742	−0.0815459
$650$	0	0
$651$	−44.7234	−1.75285
$652$	0	0
$653$	6.76583	0.264767	0.132384	−	0.991199i	$-0.457737\pi$
$653$	6.76583	0.264767	0.132384	+	0.991199i	$0.457737\pi$
$654$	0	0
$655$	63.2226	2.47031
$656$	0	0
$657$	−12.5725	−0.490501
$658$	0	0
$659$	1.75198	0.0682476	0.0341238	−	0.999418i	$-0.489136\pi$
$659$	1.75198	0.0682476	0.0341238	+	0.999418i	$0.489136\pi$
$660$	0	0
$661$	−44.1272	−1.71635	−0.858174	−	0.513359i	$-0.828401\pi$
$661$	−44.1272	−1.71635	−0.858174	+	0.513359i	$0.828401\pi$
$662$	0	0
$663$	9.83669	0.382026
$664$	0	0
$665$	54.3255	2.10665
$666$	0	0
$667$	9.70999	0.375972
$668$	0	0
$669$	−11.4985	−0.444559
$670$	0	0
$671$	14.1182	0.545026
$672$	0	0
$673$	−24.4965	−0.944269	−0.472134	−	0.881527i	$-0.656516\pi$
$673$	−24.4965	−0.944269	−0.472134	+	0.881527i	$0.656516\pi$
$674$	0	0
$675$	−49.8388	−1.91830
$676$	0	0
$677$	−23.8523	−0.916719	−0.458360	−	0.888767i	$-0.651563\pi$
$677$	−23.8523	−0.916719	−0.458360	+	0.888767i	$0.651563\pi$
$678$	0	0
$679$	−66.6467	−2.55766
$680$	0	0
$681$	21.3960	0.819896
$682$	0	0
$683$	34.9397	1.33693	0.668466	−	0.743743i	$-0.266951\pi$
$683$	34.9397	1.33693	0.668466	+	0.743743i	$0.266951\pi$
$684$	0	0
$685$	−34.4577	−1.31656
$686$	0	0
$687$	−19.1740	−0.731532
$688$	0	0
$689$	−12.5475	−0.478021
$690$	0	0
$691$	−14.1394	−0.537890	−0.268945	−	0.963156i	$-0.586675\pi$
$691$	−14.1394	−0.537890	−0.268945	+	0.963156i	$0.586675\pi$
$692$	0	0
$693$	−17.4654	−0.663457
$694$	0	0
$695$	−15.4632	−0.586552
$696$	0	0
$697$	−14.8200	−0.561349
$698$	0	0
$699$	4.21469	0.159414
$700$	0	0
$701$	17.7005	0.668539	0.334270	−	0.942478i	$-0.391510\pi$
$701$	17.7005	0.668539	0.334270	+	0.942478i	$0.391510\pi$
$702$	0	0
$703$	−14.8003	−0.558202
$704$	0	0
$705$	47.4611	1.78749
$706$	0	0
$707$	55.1638	2.07465
$708$	0	0
$709$	22.1227	0.830834	0.415417	−	0.909631i	$-0.363636\pi$
$709$	22.1227	0.830834	0.415417	+	0.909631i	$0.363636\pi$
$710$	0	0
$711$	13.9721	0.523995
$712$	0	0
$713$	9.39997	0.352032
$714$	0	0
$715$	26.2439	0.981465
$716$	0	0
$717$	8.49756	0.317347
$718$	0	0
$719$	8.65699	0.322851	0.161426	−	0.986885i	$-0.448391\pi$
$719$	8.65699	0.322851	0.161426	+	0.986885i	$0.448391\pi$
$720$	0	0
$721$	−49.6293	−1.84829
$722$	0	0
$723$	18.5975	0.691649
$724$	0	0
$725$	96.9546	3.60080
$726$	0	0
$727$	27.2630	1.01113	0.505565	−	0.862788i	$-0.331284\pi$
$727$	27.2630	1.01113	0.505565	+	0.862788i	$0.331284\pi$
$728$	0	0
$729$	17.3134	0.641238
$730$	0	0
$731$	16.1755	0.598273
$732$	0	0
$733$	29.5285	1.09066	0.545330	−	0.838222i	$-0.316404\pi$
$733$	29.5285	1.09066	0.545330	+	0.838222i	$0.316404\pi$
$734$	0	0
$735$	43.7429	1.61348
$736$	0	0
$737$	11.1610	0.411119
$738$	0	0
$739$	7.30761	0.268815	0.134407	−	0.990926i	$-0.457087\pi$
$739$	7.30761	0.268815	0.134407	+	0.990926i	$0.457087\pi$
$740$	0	0
$741$	11.1141	0.408287
$742$	0	0
$743$	50.7217	1.86080	0.930400	−	0.366546i	$-0.119460\pi$
$743$	50.7217	1.86080	0.930400	+	0.366546i	$0.119460\pi$
$744$	0	0
$745$	−60.3541	−2.21120
$746$	0	0
$747$	−8.93323	−0.326850
$748$	0	0
$749$	33.2334	1.21432
$750$	0	0
$751$	−28.0748	−1.02446	−0.512231	−	0.858848i	$-0.671181\pi$
$751$	−28.0748	−1.02446	−0.512231	+	0.858848i	$0.671181\pi$
$752$	0	0
$753$	8.88913	0.323938
$754$	0	0
$755$	44.2562	1.61065
$756$	0	0
$757$	43.8199	1.59266	0.796332	−	0.604860i	$-0.206771\pi$
$757$	43.8199	1.59266	0.796332	+	0.604860i	$0.206771\pi$
$758$	0	0
$759$	−2.35632	−0.0855290
$760$	0	0
$761$	19.9806	0.724295	0.362147	−	0.932121i	$-0.382044\pi$
$761$	19.9806	0.724295	0.362147	+	0.932121i	$0.382044\pi$
$762$	0	0
$763$	−81.3383	−2.94464
$764$	0	0
$765$	20.9440	0.757230
$766$	0	0
$767$	−2.75390	−0.0994377
$768$	0	0
$769$	18.9990	0.685121	0.342560	−	0.939496i	$-0.388706\pi$
$769$	18.9990	0.685121	0.342560	+	0.939496i	$0.388706\pi$
$770$	0	0
$771$	29.3181	1.05586
$772$	0	0
$773$	50.3756	1.81188	0.905941	−	0.423403i	$-0.139165\pi$
$773$	50.3756	1.81188	0.905941	+	0.423403i	$0.139165\pi$
$774$	0	0
$775$	93.8591	3.37152
$776$	0	0
$777$	−19.7907	−0.709988
$778$	0	0
$779$	−16.7446	−0.599938
$780$	0	0
$781$	−19.8497	−0.710279
$782$	0	0
$783$	−53.1647	−1.89995
$784$	0	0
$785$	−12.0707	−0.430822
$786$	0	0
$787$	−21.7818	−0.776438	−0.388219	−	0.921567i	$-0.626910\pi$
$787$	−21.7818	−0.776438	−0.388219	+	0.921567i	$0.626910\pi$
$788$	0	0
$789$	−9.80834	−0.349186
$790$	0	0
$791$	−23.3890	−0.831619
$792$	0	0
$793$	18.7156	0.664610
$794$	0	0
$795$	17.1487	0.608200
$796$	0	0
$797$	−26.8983	−0.952785	−0.476393	−	0.879233i	$-0.658056\pi$
$797$	−26.8983	−0.952785	−0.476393	+	0.879233i	$0.658056\pi$
$798$	0	0
$799$	−34.5643	−1.22280
$800$	0	0
$801$	2.41857	0.0854561
$802$	0	0
$803$	−15.6803	−0.553347
$804$	0	0
$805$	−15.2682	−0.538133
$806$	0	0
$807$	−1.04247	−0.0366967
$808$	0	0
$809$	52.4610	1.84443	0.922215	−	0.386677i	$-0.126377\pi$
$809$	52.4610	1.84443	0.922215	+	0.386677i	$0.126377\pi$
$810$	0	0
$811$	30.3166	1.06456	0.532281	−	0.846568i	$-0.321335\pi$
$811$	30.3166	1.06456	0.532281	+	0.846568i	$0.321335\pi$
$812$	0	0
$813$	−6.05250	−0.212271
$814$	0	0
$815$	51.3946	1.80028
$816$	0	0
$817$	18.2761	0.639401
$818$	0	0
$819$	−23.1528	−0.809025
$820$	0	0
$821$	−41.5005	−1.44838	−0.724189	−	0.689602i	$-0.757786\pi$
$821$	−41.5005	−1.44838	−0.724189	+	0.689602i	$0.757786\pi$
$822$	0	0
$823$	11.9413	0.416248	0.208124	−	0.978102i	$-0.433264\pi$
$823$	11.9413	0.416248	0.208124	+	0.978102i	$0.433264\pi$
$824$	0	0
$825$	−23.5280	−0.819139
$826$	0	0
$827$	20.9505	0.728520	0.364260	−	0.931297i	$-0.381322\pi$
$827$	20.9505	0.728520	0.364260	+	0.931297i	$0.381322\pi$
$828$	0	0
$829$	55.7250	1.93541	0.967704	−	0.252088i	$-0.0811171\pi$
$829$	55.7250	1.93541	0.967704	+	0.252088i	$0.0811171\pi$
$830$	0	0
$831$	−28.8521	−1.00087
$832$	0	0
$833$	−31.8565	−1.10376
$834$	0	0
$835$	73.1668	2.53204
$836$	0	0
$837$	−51.4673	−1.77897
$838$	0	0
$839$	8.40377	0.290130	0.145065	−	0.989422i	$-0.453661\pi$
$839$	8.40377	0.290130	0.145065	+	0.989422i	$0.453661\pi$
$840$	0	0
$841$	74.4249	2.56637
$842$	0	0
$843$	−29.4718	−1.01506
$844$	0	0
$845$	−14.7700	−0.508103
$846$	0	0
$847$	24.3584	0.836963
$848$	0	0
$849$	−4.90731	−0.168418
$850$	0	0
$851$	4.15961	0.142590
$852$	0	0
$853$	25.7710	0.882382	0.441191	−	0.897413i	$-0.354556\pi$
$853$	25.7710	0.882382	0.441191	+	0.897413i	$0.354556\pi$
$854$	0	0
$855$	23.6638	0.809285
$856$	0	0
$857$	23.6912	0.809274	0.404637	−	0.914477i	$-0.367398\pi$
$857$	23.6912	0.809274	0.404637	+	0.914477i	$0.367398\pi$
$858$	0	0
$859$	20.0310	0.683447	0.341724	−	0.939800i	$-0.388989\pi$
$859$	20.0310	0.683447	0.341724	+	0.939800i	$0.388989\pi$
$860$	0	0
$861$	−22.3907	−0.763073
$862$	0	0
$863$	30.1680	1.02693	0.513465	−	0.858110i	$-0.328362\pi$
$863$	30.1680	1.02693	0.513465	+	0.858110i	$0.328362\pi$
$864$	0	0
$865$	55.2529	1.87865
$866$	0	0
$867$	−8.61996	−0.292749
$868$	0	0
$869$	17.4259	0.591133
$870$	0	0
$871$	14.7954	0.501322
$872$	0	0
$873$	−29.0308	−0.982545
$874$	0	0
$875$	−72.4974	−2.45086
$876$	0	0
$877$	−16.7333	−0.565044	−0.282522	−	0.959261i	$-0.591171\pi$
$877$	−16.7333	−0.565044	−0.282522	+	0.959261i	$0.591171\pi$
$878$	0	0
$879$	6.92402	0.233541
$880$	0	0
$881$	25.7558	0.867736	0.433868	−	0.900976i	$-0.357148\pi$
$881$	25.7558	0.867736	0.433868	+	0.900976i	$0.357148\pi$
$882$	0	0
$883$	25.8734	0.870710	0.435355	−	0.900259i	$-0.356623\pi$
$883$	25.8734	0.870710	0.435355	+	0.900259i	$0.356623\pi$
$884$	0	0
$885$	3.76376	0.126517
$886$	0	0
$887$	31.9623	1.07319	0.536595	−	0.843840i	$-0.319710\pi$
$887$	31.9623	1.07319	0.536595	+	0.843840i	$0.319710\pi$
$888$	0	0
$889$	83.5685	2.80280
$890$	0	0
$891$	0.410237	0.0137434
$892$	0	0
$893$	−39.0530	−1.30686
$894$	0	0
$895$	86.5689	2.89368
$896$	0	0
$897$	−3.12362	−0.104295
$898$	0	0
$899$	100.123	3.33928
$900$	0	0
$901$	−12.4888	−0.416062
$902$	0	0
$903$	24.4386	0.813266
$904$	0	0
$905$	−31.2151	−1.03762
$906$	0	0
$907$	11.1059	0.368764	0.184382	−	0.982855i	$-0.440972\pi$
$907$	11.1059	0.368764	0.184382	+	0.982855i	$0.440972\pi$
$908$	0	0
$909$	24.0290	0.796991
$910$	0	0
$911$	−24.2946	−0.804915	−0.402457	−	0.915439i	$-0.631844\pi$
$911$	−24.2946	−0.804915	−0.402457	+	0.915439i	$0.631844\pi$
$912$	0	0
$913$	−11.1414	−0.368728
$914$	0	0
$915$	−25.5786	−0.845602
$916$	0	0
$917$	−69.5634	−2.29719
$918$	0	0
$919$	−3.95529	−0.130473	−0.0652365	−	0.997870i	$-0.520780\pi$
$919$	−3.95529	−0.130473	−0.0652365	+	0.997870i	$0.520780\pi$
$920$	0	0
$921$	24.4799	0.806639
$922$	0	0
$923$	−26.3135	−0.866120
$924$	0	0
$925$	41.5339	1.36563
$926$	0	0
$927$	−21.6182	−0.710034
$928$	0	0
$929$	34.6286	1.13613	0.568064	−	0.822984i	$-0.307692\pi$
$929$	34.6286	1.13613	0.568064	+	0.822984i	$0.307692\pi$
$930$	0	0
$931$	−35.9935	−1.17964
$932$	0	0
$933$	4.06627	0.133124
$934$	0	0
$935$	26.1211	0.854251
$936$	0	0
$937$	−7.65993	−0.250239	−0.125120	−	0.992142i	$-0.539931\pi$
$937$	−7.65993	−0.250239	−0.125120	+	0.992142i	$0.539931\pi$
$938$	0	0
$939$	17.3339	0.565671
$940$	0	0
$941$	−16.4953	−0.537733	−0.268866	−	0.963177i	$-0.586649\pi$
$941$	−16.4953	−0.537733	−0.268866	+	0.963177i	$0.586649\pi$
$942$	0	0
$943$	4.70608	0.153251
$944$	0	0
$945$	83.5973	2.71942
$946$	0	0
$947$	−36.1892	−1.17599	−0.587996	−	0.808864i	$-0.700083\pi$
$947$	−36.1892	−1.17599	−0.587996	+	0.808864i	$0.700083\pi$
$948$	0	0
$949$	−20.7864	−0.674755
$950$	0	0
$951$	11.5952	0.376001
$952$	0	0
$953$	−18.2117	−0.589936	−0.294968	−	0.955507i	$-0.595309\pi$
$953$	−18.2117	−0.589936	−0.294968	+	0.955507i	$0.595309\pi$
$954$	0	0
$955$	−24.8848	−0.805252
$956$	0	0
$957$	−25.0981	−0.811306
$958$	0	0
$959$	37.9136	1.22429
$960$	0	0
$961$	65.9260	2.12665
$962$	0	0
$963$	14.4762	0.466490
$964$	0	0
$965$	−11.6276	−0.374304
$966$	0	0
$967$	−5.49491	−0.176704	−0.0883522	−	0.996089i	$-0.528160\pi$
$967$	−5.49491	−0.176704	−0.0883522	+	0.996089i	$0.528160\pi$
$968$	0	0
$969$	11.0621	0.355367
$970$	0	0
$971$	9.70979	0.311602	0.155801	−	0.987788i	$-0.450204\pi$
$971$	9.70979	0.311602	0.155801	+	0.987788i	$0.450204\pi$
$972$	0	0
$973$	17.0140	0.545445
$974$	0	0
$975$	−31.1895	−0.998864
$976$	0	0
$977$	30.8205	0.986035	0.493018	−	0.870019i	$-0.335894\pi$
$977$	30.8205	0.986035	0.493018	+	0.870019i	$0.335894\pi$
$978$	0	0
$979$	3.01642	0.0964052
$980$	0	0
$981$	−35.4304	−1.13121
$982$	0	0
$983$	9.78408	0.312064	0.156032	−	0.987752i	$-0.450130\pi$
$983$	9.78408	0.312064	0.156032	+	0.987752i	$0.450130\pi$
$984$	0	0
$985$	26.0773	0.830891
$986$	0	0
$987$	−52.2211	−1.66222
$988$	0	0
$989$	−5.13651	−0.163331
$990$	0	0
$991$	−26.7644	−0.850201	−0.425100	−	0.905146i	$-0.639761\pi$
$991$	−26.7644	−0.850201	−0.425100	+	0.905146i	$0.639761\pi$
$992$	0	0
$993$	−7.51897	−0.238607
$994$	0	0
$995$	38.4529	1.21904
$996$	0	0
$997$	−38.3627	−1.21496	−0.607479	−	0.794336i	$-0.707819\pi$
$997$	−38.3627	−1.21496	−0.607479	+	0.794336i	$0.707819\pi$
$998$	0	0
$999$	−22.7750	−0.720569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.17		26
4.3	odd	2		503.2.a.f.1.9	✓	26
12.11	even	2		4527.2.a.o.1.18		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.9	✓	26	4.3	odd	2
4527.2.a.o.1.18		26	12.11	even	2
8048.2.a.u.1.17		26	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)

\(f(q)\)	\(=\)	\(q+1.08298 q^{3} +3.81229 q^{5} -4.19464 q^{7} -1.82716 q^{9} +O(q^{10})\)	\(q+1.08298 q^{3} +3.81229 q^{5} -4.19464 q^{7} -1.82716 q^{9} -2.27881 q^{11} -3.02088 q^{13} +4.12863 q^{15} -3.00674 q^{17} -3.39721 q^{19} -4.54271 q^{21} +0.954786 q^{23} +9.53358 q^{25} -5.22771 q^{27} +10.1698 q^{29} +9.84510 q^{31} -2.46790 q^{33} -15.9912 q^{35} +4.35659 q^{37} -3.27154 q^{39} +4.92893 q^{41} -5.37975 q^{43} -6.96566 q^{45} +11.4956 q^{47} +10.5950 q^{49} -3.25624 q^{51} +4.15359 q^{53} -8.68750 q^{55} -3.67910 q^{57} +0.911624 q^{59} -6.19541 q^{61} +7.66427 q^{63} -11.5165 q^{65} -4.89771 q^{67} +1.03401 q^{69} +8.71056 q^{71} +6.88092 q^{73} +10.3247 q^{75} +9.55880 q^{77} -7.64692 q^{79} -0.180022 q^{81} +4.88914 q^{83} -11.4626 q^{85} +11.0137 q^{87} -1.32368 q^{89} +12.6715 q^{91} +10.6620 q^{93} -12.9512 q^{95} +15.8885 q^{97} +4.16375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

Introduction

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