LMFDB - Embedded newform 8048.2.a.w.1.12

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:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.454325 q^{3} +0.622756 q^{5} +4.14589 q^{7} -2.79359 q^{9} +O(q^{10})$	$q-0.454325 q^{3} +0.622756 q^{5} +4.14589 q^{7} -2.79359 q^{9} -1.04776 q^{11} +4.67266 q^{13} -0.282934 q^{15} +1.91981 q^{17} -6.57918 q^{19} -1.88358 q^{21} +2.26582 q^{23} -4.61217 q^{25} +2.63217 q^{27} -9.03724 q^{29} +1.46769 q^{31} +0.476022 q^{33} +2.58188 q^{35} +6.96520 q^{37} -2.12291 q^{39} -0.613168 q^{41} +3.34066 q^{43} -1.73973 q^{45} +2.36839 q^{47} +10.1884 q^{49} -0.872217 q^{51} +0.528504 q^{53} -0.652497 q^{55} +2.98909 q^{57} +11.9591 q^{59} +4.74317 q^{61} -11.5819 q^{63} +2.90993 q^{65} +16.2911 q^{67} -1.02942 q^{69} -8.46958 q^{71} +6.36556 q^{73} +2.09543 q^{75} -4.34389 q^{77} +2.85384 q^{79} +7.18490 q^{81} +1.00931 q^{83} +1.19557 q^{85} +4.10585 q^{87} -13.9668 q^{89} +19.3724 q^{91} -0.666806 q^{93} -4.09723 q^{95} +5.06676 q^{97} +2.92700 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	−0.454325	−0.262305	−0.131152	−	0.991362i	$-0.541868\pi$
$3$	−0.454325	−0.262305	−0.131152	+	0.991362i	$0.541868\pi$
$4$	0	0
$5$	0.622756	0.278505	0.139253	−	0.990257i	$-0.455530\pi$
$5$	0.622756	0.278505	0.139253	+	0.990257i	$0.455530\pi$
$6$	0	0
$7$	4.14589	1.56700	0.783500	−	0.621391i	$-0.213432\pi$
$7$	4.14589	1.56700	0.783500	+	0.621391i	$0.213432\pi$
$8$	0	0
$9$	−2.79359	−0.931196
$10$	0	0
$11$	−1.04776	−0.315911	−0.157955	−	0.987446i	$-0.550490\pi$
$11$	−1.04776	−0.315911	−0.157955	+	0.987446i	$0.550490\pi$
$12$	0	0
$13$	4.67266	1.29596	0.647982	−	0.761656i	$-0.275613\pi$
$13$	4.67266	1.29596	0.647982	+	0.761656i	$0.275613\pi$
$14$	0	0
$15$	−0.282934	−0.0730532
$16$	0	0
$17$	1.91981	0.465622	0.232811	−	0.972522i	$-0.425208\pi$
$17$	1.91981	0.465622	0.232811	+	0.972522i	$0.425208\pi$
$18$	0	0
$19$	−6.57918	−1.50937	−0.754684	−	0.656088i	$-0.772210\pi$
$19$	−6.57918	−1.50937	−0.754684	+	0.656088i	$0.772210\pi$
$20$	0	0
$21$	−1.88358	−0.411032
$22$	0	0
$23$	2.26582	0.472456	0.236228	−	0.971698i	$-0.424089\pi$
$23$	2.26582	0.472456	0.236228	+	0.971698i	$0.424089\pi$
$24$	0	0
$25$	−4.61217	−0.922435
$26$	0	0
$27$	2.63217	0.506562
$28$	0	0
$29$	−9.03724	−1.67817	−0.839087	−	0.543997i	$-0.816910\pi$
$29$	−9.03724	−1.67817	−0.839087	+	0.543997i	$0.816910\pi$
$30$	0	0
$31$	1.46769	0.263604	0.131802	−	0.991276i	$-0.457924\pi$
$31$	1.46769	0.263604	0.131802	+	0.991276i	$0.457924\pi$
$32$	0	0
$33$	0.476022	0.0828648
$34$	0	0
$35$	2.58188	0.436418
$36$	0	0
$37$	6.96520	1.14507	0.572536	−	0.819880i	$-0.305960\pi$
$37$	6.96520	1.14507	0.572536	+	0.819880i	$0.305960\pi$
$38$	0	0
$39$	−2.12291	−0.339937
$40$	0	0
$41$	−0.613168	−0.0957608	−0.0478804	−	0.998853i	$-0.515247\pi$
$41$	−0.613168	−0.0957608	−0.0478804	+	0.998853i	$0.515247\pi$
$42$	0	0
$43$	3.34066	0.509446	0.254723	−	0.967014i	$-0.418016\pi$
$43$	3.34066	0.509446	0.254723	+	0.967014i	$0.418016\pi$
$44$	0	0
$45$	−1.73973	−0.259343
$46$	0	0
$47$	2.36839	0.345465	0.172733	−	0.984969i	$-0.444740\pi$
$47$	2.36839	0.345465	0.172733	+	0.984969i	$0.444740\pi$
$48$	0	0
$49$	10.1884	1.45549
$50$	0	0
$51$	−0.872217	−0.122135
$52$	0	0
$53$	0.528504	0.0725956	0.0362978	−	0.999341i	$-0.488444\pi$
$53$	0.528504	0.0725956	0.0362978	+	0.999341i	$0.488444\pi$
$54$	0	0
$55$	−0.652497	−0.0879827
$56$	0	0
$57$	2.98909	0.395914
$58$	0	0
$59$	11.9591	1.55695	0.778473	−	0.627678i	$-0.215994\pi$
$59$	11.9591	1.55695	0.778473	+	0.627678i	$0.215994\pi$
$60$	0	0
$61$	4.74317	0.607301	0.303650	−	0.952784i	$-0.401795\pi$
$61$	4.74317	0.607301	0.303650	+	0.952784i	$0.401795\pi$
$62$	0	0
$63$	−11.5819	−1.45919
$64$	0	0
$65$	2.90993	0.360933
$66$	0	0
$67$	16.2911	1.99028	0.995138	−	0.0984906i	$-0.0314014\pi$
$67$	16.2911	1.99028	0.995138	+	0.0984906i	$0.0314014\pi$
$68$	0	0
$69$	−1.02942	−0.123927
$70$	0	0
$71$	−8.46958	−1.00515	−0.502577	−	0.864532i	$-0.667615\pi$
$71$	−8.46958	−1.00515	−0.502577	+	0.864532i	$0.667615\pi$
$72$	0	0
$73$	6.36556	0.745033	0.372516	−	0.928026i	$-0.378495\pi$
$73$	6.36556	0.745033	0.372516	+	0.928026i	$0.378495\pi$
$74$	0	0
$75$	2.09543	0.241959
$76$	0	0
$77$	−4.34389	−0.495032
$78$	0	0
$79$	2.85384	0.321083	0.160541	−	0.987029i	$-0.448676\pi$
$79$	2.85384	0.321083	0.160541	+	0.987029i	$0.448676\pi$
$80$	0	0
$81$	7.18490	0.798323
$82$	0	0
$83$	1.00931	0.110786	0.0553932	−	0.998465i	$-0.482359\pi$
$83$	1.00931	0.110786	0.0553932	+	0.998465i	$0.482359\pi$
$84$	0	0
$85$	1.19557	0.129678
$86$	0	0
$87$	4.10585	0.440193
$88$	0	0
$89$	−13.9668	−1.48048	−0.740240	−	0.672342i	$-0.765288\pi$
$89$	−13.9668	−1.48048	−0.740240	+	0.672342i	$0.765288\pi$
$90$	0	0
$91$	19.3724	2.03078
$92$	0	0
$93$	−0.666806	−0.0691446
$94$	0	0
$95$	−4.09723	−0.420367
$96$	0	0
$97$	5.06676	0.514451	0.257226	−	0.966351i	$-0.417192\pi$
$97$	5.06676	0.514451	0.257226	+	0.966351i	$0.417192\pi$
$98$	0	0
$99$	2.92700	0.294175
$100$	0	0
$101$	6.33120	0.629978	0.314989	−	0.949095i	$-0.397999\pi$
$101$	6.33120	0.629978	0.314989	+	0.949095i	$0.397999\pi$
$102$	0	0
$103$	−2.20033	−0.216805	−0.108403	−	0.994107i	$-0.534574\pi$
$103$	−2.20033	−0.216805	−0.108403	+	0.994107i	$0.534574\pi$
$104$	0	0
$105$	−1.17301	−0.114474
$106$	0	0
$107$	17.7928	1.72009	0.860047	−	0.510214i	$-0.170434\pi$
$107$	17.7928	1.72009	0.860047	+	0.510214i	$0.170434\pi$
$108$	0	0
$109$	−2.58716	−0.247805	−0.123902	−	0.992294i	$-0.539541\pi$
$109$	−2.58716	−0.247805	−0.123902	+	0.992294i	$0.539541\pi$
$110$	0	0
$111$	−3.16446	−0.300358
$112$	0	0
$113$	−12.1305	−1.14114	−0.570572	−	0.821248i	$-0.693278\pi$
$113$	−12.1305	−1.14114	−0.570572	+	0.821248i	$0.693278\pi$
$114$	0	0
$115$	1.41105	0.131581
$116$	0	0
$117$	−13.0535	−1.20680
$118$	0	0
$119$	7.95932	0.729630
$120$	0	0
$121$	−9.90221	−0.900201
$122$	0	0
$123$	0.278578	0.0251185
$124$	0	0
$125$	−5.98604	−0.535408
$126$	0	0
$127$	5.65530	0.501827	0.250913	−	0.968010i	$-0.419269\pi$
$127$	5.65530	0.501827	0.250913	+	0.968010i	$0.419269\pi$
$128$	0	0
$129$	−1.51775	−0.133630
$130$	0	0
$131$	8.13561	0.710811	0.355406	−	0.934712i	$-0.384343\pi$
$131$	8.13561	0.710811	0.355406	+	0.934712i	$0.384343\pi$
$132$	0	0
$133$	−27.2766	−2.36518
$134$	0	0
$135$	1.63920	0.141080
$136$	0	0
$137$	6.87315	0.587213	0.293606	−	0.955926i	$-0.405144\pi$
$137$	6.87315	0.587213	0.293606	+	0.955926i	$0.405144\pi$
$138$	0	0
$139$	22.1165	1.87589	0.937947	−	0.346778i	$-0.112724\pi$
$139$	22.1165	1.87589	0.937947	+	0.346778i	$0.112724\pi$
$140$	0	0
$141$	−1.07602	−0.0906172
$142$	0	0
$143$	−4.89582	−0.409409
$144$	0	0
$145$	−5.62800	−0.467380
$146$	0	0
$147$	−4.62886	−0.381782
$148$	0	0
$149$	11.3728	0.931699	0.465849	−	0.884864i	$-0.345749\pi$
$149$	11.3728	0.931699	0.465849	+	0.884864i	$0.345749\pi$
$150$	0	0
$151$	5.78479	0.470759	0.235380	−	0.971904i	$-0.424367\pi$
$151$	5.78479	0.470759	0.235380	+	0.971904i	$0.424367\pi$
$152$	0	0
$153$	−5.36316	−0.433585
$154$	0	0
$155$	0.914011	0.0734151
$156$	0	0
$157$	−11.2534	−0.898122	−0.449061	−	0.893501i	$-0.648241\pi$
$157$	−11.2534	−0.898122	−0.449061	+	0.893501i	$0.648241\pi$
$158$	0	0
$159$	−0.240113	−0.0190422
$160$	0	0
$161$	9.39384	0.740338
$162$	0	0
$163$	−5.33382	−0.417777	−0.208888	−	0.977939i	$-0.566985\pi$
$163$	−5.33382	−0.417777	−0.208888	+	0.977939i	$0.566985\pi$
$164$	0	0
$165$	0.296446	0.0230783
$166$	0	0
$167$	8.90420	0.689028	0.344514	−	0.938781i	$-0.388044\pi$
$167$	8.90420	0.689028	0.344514	+	0.938781i	$0.388044\pi$
$168$	0	0
$169$	8.83379	0.679522
$170$	0	0
$171$	18.3795	1.40552
$172$	0	0
$173$	−18.7460	−1.42523	−0.712617	−	0.701554i	$-0.752490\pi$
$173$	−18.7460	−1.42523	−0.712617	+	0.701554i	$0.752490\pi$
$174$	0	0
$175$	−19.1216	−1.44546
$176$	0	0
$177$	−5.43334	−0.408394
$178$	0	0
$179$	2.93684	0.219510	0.109755	−	0.993959i	$-0.464993\pi$
$179$	2.93684	0.219510	0.109755	+	0.993959i	$0.464993\pi$
$180$	0	0
$181$	−5.66934	−0.421399	−0.210699	−	0.977551i	$-0.567574\pi$
$181$	−5.66934	−0.421399	−0.210699	+	0.977551i	$0.567574\pi$
$182$	0	0
$183$	−2.15494	−0.159298
$184$	0	0
$185$	4.33762	0.318908
$186$	0	0
$187$	−2.01149	−0.147095
$188$	0	0
$189$	10.9127	0.793783
$190$	0	0
$191$	−1.75891	−0.127270	−0.0636351	−	0.997973i	$-0.520269\pi$
$191$	−1.75891	−0.127270	−0.0636351	+	0.997973i	$0.520269\pi$
$192$	0	0
$193$	−6.02765	−0.433880	−0.216940	−	0.976185i	$-0.569608\pi$
$193$	−6.02765	−0.433880	−0.216940	+	0.976185i	$0.569608\pi$
$194$	0	0
$195$	−1.32205	−0.0946743
$196$	0	0
$197$	−9.52626	−0.678718	−0.339359	−	0.940657i	$-0.610210\pi$
$197$	−9.52626	−0.678718	−0.339359	+	0.940657i	$0.610210\pi$
$198$	0	0
$199$	11.3572	0.805092	0.402546	−	0.915400i	$-0.368125\pi$
$199$	11.3572	0.805092	0.402546	+	0.915400i	$0.368125\pi$
$200$	0	0
$201$	−7.40146	−0.522059
$202$	0	0
$203$	−37.4675	−2.62970
$204$	0	0
$205$	−0.381854	−0.0266699
$206$	0	0
$207$	−6.32976	−0.439949
$208$	0	0
$209$	6.89338	0.476825
$210$	0	0
$211$	−7.78349	−0.535837	−0.267919	−	0.963442i	$-0.586336\pi$
$211$	−7.78349	−0.535837	−0.267919	+	0.963442i	$0.586336\pi$
$212$	0	0
$213$	3.84794	0.263657
$214$	0	0
$215$	2.08042	0.141883
$216$	0	0
$217$	6.08487	0.413068
$218$	0	0
$219$	−2.89203	−0.195426
$220$	0	0
$221$	8.97062	0.603429
$222$	0	0
$223$	19.2921	1.29189	0.645947	−	0.763382i	$-0.276463\pi$
$223$	19.2921	1.29189	0.645947	+	0.763382i	$0.276463\pi$
$224$	0	0
$225$	12.8845	0.858968
$226$	0	0
$227$	−15.4660	−1.02651	−0.513256	−	0.858236i	$-0.671561\pi$
$227$	−15.4660	−1.02651	−0.513256	+	0.858236i	$0.671561\pi$
$228$	0	0
$229$	16.8349	1.11248	0.556241	−	0.831021i	$-0.312243\pi$
$229$	16.8349	1.11248	0.556241	+	0.831021i	$0.312243\pi$
$230$	0	0
$231$	1.97354	0.129849
$232$	0	0
$233$	9.10178	0.596277	0.298139	−	0.954523i	$-0.403634\pi$
$233$	9.10178	0.596277	0.298139	+	0.954523i	$0.403634\pi$
$234$	0	0
$235$	1.47493	0.0962139
$236$	0	0
$237$	−1.29657	−0.0842215
$238$	0	0
$239$	−1.18942	−0.0769369	−0.0384684	−	0.999260i	$-0.512248\pi$
$239$	−1.18942	−0.0769369	−0.0384684	+	0.999260i	$0.512248\pi$
$240$	0	0
$241$	18.0461	1.16245	0.581225	−	0.813743i	$-0.302573\pi$
$241$	18.0461	1.16245	0.581225	+	0.813743i	$0.302573\pi$
$242$	0	0
$243$	−11.1608	−0.715966
$244$	0	0
$245$	6.34492	0.405362
$246$	0	0
$247$	−30.7423	−1.95609
$248$	0	0
$249$	−0.458556	−0.0290598
$250$	0	0
$251$	22.0895	1.39428	0.697140	−	0.716935i	$-0.254456\pi$
$251$	22.0895	1.39428	0.697140	+	0.716935i	$0.254456\pi$
$252$	0	0
$253$	−2.37403	−0.149254
$254$	0	0
$255$	−0.543179	−0.0340152
$256$	0	0
$257$	−1.87083	−0.116699	−0.0583496	−	0.998296i	$-0.518584\pi$
$257$	−1.87083	−0.116699	−0.0583496	+	0.998296i	$0.518584\pi$
$258$	0	0
$259$	28.8770	1.79433
$260$	0	0
$261$	25.2463	1.56271
$262$	0	0
$263$	4.02177	0.247993	0.123997	−	0.992283i	$-0.460429\pi$
$263$	4.02177	0.247993	0.123997	+	0.992283i	$0.460429\pi$
$264$	0	0
$265$	0.329129	0.0202183
$266$	0	0
$267$	6.34548	0.388337
$268$	0	0
$269$	16.7112	1.01890	0.509451	−	0.860500i	$-0.329849\pi$
$269$	16.7112	1.01890	0.509451	+	0.860500i	$0.329849\pi$
$270$	0	0
$271$	−15.2794	−0.928160	−0.464080	−	0.885793i	$-0.653615\pi$
$271$	−15.2794	−0.928160	−0.464080	+	0.885793i	$0.653615\pi$
$272$	0	0
$273$	−8.80135	−0.532682
$274$	0	0
$275$	4.83244	0.291407
$276$	0	0
$277$	−18.3567	−1.10295	−0.551474	−	0.834192i	$-0.685935\pi$
$277$	−18.3567	−1.10295	−0.551474	+	0.834192i	$0.685935\pi$
$278$	0	0
$279$	−4.10011	−0.245467
$280$	0	0
$281$	−5.98387	−0.356968	−0.178484	−	0.983943i	$-0.557119\pi$
$281$	−5.98387	−0.356968	−0.178484	+	0.983943i	$0.557119\pi$
$282$	0	0
$283$	9.98825	0.593740	0.296870	−	0.954918i	$-0.404057\pi$
$283$	9.98825	0.593740	0.296870	+	0.954918i	$0.404057\pi$
$284$	0	0
$285$	1.86147	0.110264
$286$	0	0
$287$	−2.54213	−0.150057
$288$	0	0
$289$	−13.3143	−0.783196
$290$	0	0
$291$	−2.30195	−0.134943
$292$	0	0
$293$	14.6436	0.855491	0.427745	−	0.903899i	$-0.359308\pi$
$293$	14.6436	0.855491	0.427745	+	0.903899i	$0.359308\pi$
$294$	0	0
$295$	7.44763	0.433618
$296$	0	0
$297$	−2.75788	−0.160028
$298$	0	0
$299$	10.5874	0.612285
$300$	0	0
$301$	13.8500	0.798303
$302$	0	0
$303$	−2.87642	−0.165246
$304$	0	0
$305$	2.95384	0.169136
$306$	0	0
$307$	−7.14918	−0.408025	−0.204013	−	0.978968i	$-0.565398\pi$
$307$	−7.14918	−0.408025	−0.204013	+	0.978968i	$0.565398\pi$
$308$	0	0
$309$	0.999667	0.0568691
$310$	0	0
$311$	15.2038	0.862127	0.431064	−	0.902322i	$-0.358138\pi$
$311$	15.2038	0.862127	0.431064	+	0.902322i	$0.358138\pi$
$312$	0	0
$313$	−19.5332	−1.10408	−0.552042	−	0.833816i	$-0.686151\pi$
$313$	−19.5332	−1.10408	−0.552042	+	0.833816i	$0.686151\pi$
$314$	0	0
$315$	−7.21272	−0.406391
$316$	0	0
$317$	18.6502	1.04750	0.523749	−	0.851873i	$-0.324533\pi$
$317$	18.6502	1.04750	0.523749	+	0.851873i	$0.324533\pi$
$318$	0	0
$319$	9.46883	0.530153
$320$	0	0
$321$	−8.08371	−0.451189
$322$	0	0
$323$	−12.6308	−0.702795
$324$	0	0
$325$	−21.5511	−1.19544
$326$	0	0
$327$	1.17541	0.0650004
$328$	0	0
$329$	9.81910	0.541345
$330$	0	0
$331$	−16.9994	−0.934373	−0.467187	−	0.884159i	$-0.654732\pi$
$331$	−16.9994	−0.934373	−0.467187	+	0.884159i	$0.654732\pi$
$332$	0	0
$333$	−19.4579	−1.06629
$334$	0	0
$335$	10.1454	0.554302
$336$	0	0
$337$	25.9818	1.41532	0.707659	−	0.706554i	$-0.249751\pi$
$337$	25.9818	1.41532	0.707659	+	0.706554i	$0.249751\pi$
$338$	0	0
$339$	5.51120	0.299327
$340$	0	0
$341$	−1.53778	−0.0832753
$342$	0	0
$343$	13.2189	0.713756
$344$	0	0
$345$	−0.641077	−0.0345144
$346$	0	0
$347$	−22.8924	−1.22893	−0.614463	−	0.788945i	$-0.710627\pi$
$347$	−22.8924	−1.22893	−0.614463	+	0.788945i	$0.710627\pi$
$348$	0	0
$349$	5.87573	0.314521	0.157260	−	0.987557i	$-0.449734\pi$
$349$	5.87573	0.314521	0.157260	+	0.987557i	$0.449734\pi$
$350$	0	0
$351$	12.2993	0.656486
$352$	0	0
$353$	−3.39799	−0.180857	−0.0904285	−	0.995903i	$-0.528824\pi$
$353$	−3.39799	−0.180857	−0.0904285	+	0.995903i	$0.528824\pi$
$354$	0	0
$355$	−5.27449	−0.279941
$356$	0	0
$357$	−3.61612	−0.191385
$358$	0	0
$359$	15.6670	0.826875	0.413437	−	0.910533i	$-0.364328\pi$
$359$	15.6670	0.826875	0.413437	+	0.910533i	$0.364328\pi$
$360$	0	0
$361$	24.2857	1.27819
$362$	0	0
$363$	4.49882	0.236127
$364$	0	0
$365$	3.96419	0.207495
$366$	0	0
$367$	17.4997	0.913478	0.456739	−	0.889601i	$-0.349017\pi$
$367$	17.4997	0.913478	0.456739	+	0.889601i	$0.349017\pi$
$368$	0	0
$369$	1.71294	0.0891721
$370$	0	0
$371$	2.19112	0.113757
$372$	0	0
$373$	6.99394	0.362132	0.181066	−	0.983471i	$-0.442045\pi$
$373$	6.99394	0.362132	0.181066	+	0.983471i	$0.442045\pi$
$374$	0	0
$375$	2.71961	0.140440
$376$	0	0
$377$	−42.2280	−2.17485
$378$	0	0
$379$	36.3663	1.86801	0.934005	−	0.357260i	$-0.116289\pi$
$379$	36.3663	1.86801	0.934005	+	0.357260i	$0.116289\pi$
$380$	0	0
$381$	−2.56934	−0.131631
$382$	0	0
$383$	−8.74892	−0.447049	−0.223525	−	0.974698i	$-0.571756\pi$
$383$	−8.74892	−0.447049	−0.223525	+	0.974698i	$0.571756\pi$
$384$	0	0
$385$	−2.70518	−0.137869
$386$	0	0
$387$	−9.33244	−0.474395
$388$	0	0
$389$	−8.77216	−0.444766	−0.222383	−	0.974959i	$-0.571384\pi$
$389$	−8.77216	−0.444766	−0.222383	+	0.974959i	$0.571384\pi$
$390$	0	0
$391$	4.34994	0.219986
$392$	0	0
$393$	−3.69621	−0.186449
$394$	0	0
$395$	1.77725	0.0894232
$396$	0	0
$397$	18.2764	0.917268	0.458634	−	0.888625i	$-0.348339\pi$
$397$	18.2764	0.917268	0.458634	+	0.888625i	$0.348339\pi$
$398$	0	0
$399$	12.3924	0.620398
$400$	0	0
$401$	11.0917	0.553895	0.276948	−	0.960885i	$-0.410677\pi$
$401$	11.0917	0.553895	0.276948	+	0.960885i	$0.410677\pi$
$402$	0	0
$403$	6.85800	0.341621
$404$	0	0
$405$	4.47444	0.222337
$406$	0	0
$407$	−7.29783	−0.361740
$408$	0	0
$409$	−6.15940	−0.304563	−0.152281	−	0.988337i	$-0.548662\pi$
$409$	−6.15940	−0.304563	−0.152281	+	0.988337i	$0.548662\pi$
$410$	0	0
$411$	−3.12265	−0.154029
$412$	0	0
$413$	49.5813	2.43974
$414$	0	0
$415$	0.628556	0.0308546
$416$	0	0
$417$	−10.0481	−0.492056
$418$	0	0
$419$	9.96805	0.486971	0.243486	−	0.969904i	$-0.421709\pi$
$419$	9.96805	0.486971	0.243486	+	0.969904i	$0.421709\pi$
$420$	0	0
$421$	−21.9170	−1.06817	−0.534084	−	0.845431i	$-0.679344\pi$
$421$	−21.9170	−1.06817	−0.534084	+	0.845431i	$0.679344\pi$
$422$	0	0
$423$	−6.61631	−0.321696
$424$	0	0
$425$	−8.85449	−0.429506
$426$	0	0
$427$	19.6647	0.951641
$428$	0	0
$429$	2.22429	0.107390
$430$	0	0
$431$	−23.4394	−1.12904	−0.564518	−	0.825421i	$-0.690938\pi$
$431$	−23.4394	−1.12904	−0.564518	+	0.825421i	$0.690938\pi$
$432$	0	0
$433$	28.3972	1.36468	0.682342	−	0.731033i	$-0.260962\pi$
$433$	28.3972	1.36468	0.682342	+	0.731033i	$0.260962\pi$
$434$	0	0
$435$	2.55694	0.122596
$436$	0	0
$437$	−14.9072	−0.713110
$438$	0	0
$439$	−16.7175	−0.797883	−0.398941	−	0.916976i	$-0.630622\pi$
$439$	−16.7175	−0.797883	−0.398941	+	0.916976i	$0.630622\pi$
$440$	0	0
$441$	−28.4623	−1.35535
$442$	0	0
$443$	−4.58348	−0.217768	−0.108884	−	0.994054i	$-0.534728\pi$
$443$	−4.58348	−0.217768	−0.108884	+	0.994054i	$0.534728\pi$
$444$	0	0
$445$	−8.69793	−0.412322
$446$	0	0
$447$	−5.16696	−0.244389
$448$	0	0
$449$	16.3239	0.770373	0.385187	−	0.922839i	$-0.374137\pi$
$449$	16.3239	0.770373	0.385187	+	0.922839i	$0.374137\pi$
$450$	0	0
$451$	0.642451	0.0302518
$452$	0	0
$453$	−2.62817	−0.123482
$454$	0	0
$455$	12.0643	0.565582
$456$	0	0
$457$	−0.239524	−0.0112045	−0.00560223	−	0.999984i	$-0.501783\pi$
$457$	−0.239524	−0.0112045	−0.00560223	+	0.999984i	$0.501783\pi$
$458$	0	0
$459$	5.05327	0.235866
$460$	0	0
$461$	−29.8979	−1.39248	−0.696241	−	0.717808i	$-0.745146\pi$
$461$	−29.8979	−1.39248	−0.696241	+	0.717808i	$0.745146\pi$
$462$	0	0
$463$	−29.9674	−1.39271	−0.696353	−	0.717700i	$-0.745195\pi$
$463$	−29.9674	−1.39271	−0.696353	+	0.717700i	$0.745195\pi$
$464$	0	0
$465$	−0.415258	−0.0192571
$466$	0	0
$467$	38.2390	1.76949	0.884746	−	0.466074i	$-0.154332\pi$
$467$	38.2390	1.76949	0.884746	+	0.466074i	$0.154332\pi$
$468$	0	0
$469$	67.5412	3.11876
$470$	0	0
$471$	5.11272	0.235582
$472$	0	0
$473$	−3.50020	−0.160939
$474$	0	0
$475$	30.3443	1.39229
$476$	0	0
$477$	−1.47642	−0.0676008
$478$	0	0
$479$	−38.0256	−1.73743	−0.868716	−	0.495310i	$-0.835054\pi$
$479$	−38.0256	−1.73743	−0.868716	+	0.495310i	$0.835054\pi$
$480$	0	0
$481$	32.5460	1.48397
$482$	0	0
$483$	−4.26786	−0.194194
$484$	0	0
$485$	3.15535	0.143277
$486$	0	0
$487$	−10.6833	−0.484107	−0.242053	−	0.970263i	$-0.577821\pi$
$487$	−10.6833	−0.484107	−0.242053	+	0.970263i	$0.577821\pi$
$488$	0	0
$489$	2.42329	0.109585
$490$	0	0
$491$	−27.4935	−1.24077	−0.620383	−	0.784299i	$-0.713023\pi$
$491$	−27.4935	−1.24077	−0.620383	+	0.784299i	$0.713023\pi$
$492$	0	0
$493$	−17.3498	−0.781395
$494$	0	0
$495$	1.82281	0.0819292
$496$	0	0
$497$	−35.1140	−1.57508
$498$	0	0
$499$	25.4066	1.13736	0.568678	−	0.822560i	$-0.307455\pi$
$499$	25.4066	1.13736	0.568678	+	0.822560i	$0.307455\pi$
$500$	0	0
$501$	−4.04540	−0.180735
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	3.94280	0.175452
$506$	0	0
$507$	−4.01341	−0.178242
$508$	0	0
$509$	−0.623759	−0.0276476	−0.0138238	−	0.999904i	$-0.504400\pi$
$509$	−0.623759	−0.0276476	−0.0138238	+	0.999904i	$0.504400\pi$
$510$	0	0
$511$	26.3909	1.16747
$512$	0	0
$513$	−17.3175	−0.764588
$514$	0	0
$515$	−1.37027	−0.0603814
$516$	0	0
$517$	−2.48150	−0.109136
$518$	0	0
$519$	8.51678	0.373845
$520$	0	0
$521$	−23.9312	−1.04845	−0.524223	−	0.851581i	$-0.675644\pi$
$521$	−23.9312	−1.04845	−0.524223	+	0.851581i	$0.675644\pi$
$522$	0	0
$523$	−11.4777	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$523$	−11.4777	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$524$	0	0
$525$	8.68742	0.379150
$526$	0	0
$527$	2.81768	0.122740
$528$	0	0
$529$	−17.8661	−0.776786
$530$	0	0
$531$	−33.4089	−1.44982
$532$	0	0
$533$	−2.86513	−0.124103
$534$	0	0
$535$	11.0806	0.479055
$536$	0	0
$537$	−1.33428	−0.0575785
$538$	0	0
$539$	−10.6750	−0.459805
$540$	0	0
$541$	28.8461	1.24019	0.620095	−	0.784526i	$-0.287094\pi$
$541$	28.8461	1.24019	0.620095	+	0.784526i	$0.287094\pi$
$542$	0	0
$543$	2.57572	0.110535
$544$	0	0
$545$	−1.61117	−0.0690149
$546$	0	0
$547$	13.1604	0.562697	0.281348	−	0.959606i	$-0.409218\pi$
$547$	13.1604	0.562697	0.281348	+	0.959606i	$0.409218\pi$
$548$	0	0
$549$	−13.2505	−0.565516
$550$	0	0
$551$	59.4577	2.53298
$552$	0	0
$553$	11.8317	0.503137
$554$	0	0
$555$	−1.97069	−0.0836512
$556$	0	0
$557$	0.769673	0.0326121	0.0163060	−	0.999867i	$-0.494809\pi$
$557$	0.769673	0.0326121	0.0163060	+	0.999867i	$0.494809\pi$
$558$	0	0
$559$	15.6098	0.660224
$560$	0	0
$561$	0.913871	0.0385837
$562$	0	0
$563$	−2.69799	−0.113707	−0.0568534	−	0.998383i	$-0.518107\pi$
$563$	−2.69799	−0.113707	−0.0568534	+	0.998383i	$0.518107\pi$
$564$	0	0
$565$	−7.55436	−0.317814
$566$	0	0
$567$	29.7879	1.25097
$568$	0	0
$569$	−33.4495	−1.40228	−0.701138	−	0.713026i	$-0.747324\pi$
$569$	−33.4495	−1.40228	−0.701138	+	0.713026i	$0.747324\pi$
$570$	0	0
$571$	−17.5765	−0.735555	−0.367777	−	0.929914i	$-0.619881\pi$
$571$	−17.5765	−0.735555	−0.367777	+	0.929914i	$0.619881\pi$
$572$	0	0
$573$	0.799117	0.0333836
$574$	0	0
$575$	−10.4503	−0.435810
$576$	0	0
$577$	21.8017	0.907617	0.453808	−	0.891099i	$-0.350065\pi$
$577$	21.8017	0.907617	0.453808	+	0.891099i	$0.350065\pi$
$578$	0	0
$579$	2.73851	0.113809
$580$	0	0
$581$	4.18450	0.173602
$582$	0	0
$583$	−0.553744	−0.0229337
$584$	0	0
$585$	−8.12915	−0.336099
$586$	0	0
$587$	6.84732	0.282619	0.141310	−	0.989965i	$-0.454869\pi$
$587$	6.84732	0.282619	0.141310	+	0.989965i	$0.454869\pi$
$588$	0	0
$589$	−9.65617	−0.397876
$590$	0	0
$591$	4.32802	0.178031
$592$	0	0
$593$	3.06717	0.125954	0.0629768	−	0.998015i	$-0.479941\pi$
$593$	3.06717	0.125954	0.0629768	+	0.998015i	$0.479941\pi$
$594$	0	0
$595$	4.95672	0.203206
$596$	0	0
$597$	−5.15987	−0.211179
$598$	0	0
$599$	22.0535	0.901081	0.450540	−	0.892756i	$-0.351231\pi$
$599$	22.0535	0.901081	0.450540	+	0.892756i	$0.351231\pi$
$600$	0	0
$601$	16.6919	0.680877	0.340438	−	0.940267i	$-0.389425\pi$
$601$	16.6919	0.680877	0.340438	+	0.940267i	$0.389425\pi$
$602$	0	0
$603$	−45.5107	−1.85334
$604$	0	0
$605$	−6.16666	−0.250710
$606$	0	0
$607$	11.4832	0.466088	0.233044	−	0.972466i	$-0.425131\pi$
$607$	11.4832	0.466088	0.233044	+	0.972466i	$0.425131\pi$
$608$	0	0
$609$	17.0224	0.689783
$610$	0	0
$611$	11.0667	0.447711
$612$	0	0
$613$	30.2982	1.22373	0.611867	−	0.790961i	$-0.290419\pi$
$613$	30.2982	1.22373	0.611867	+	0.790961i	$0.290419\pi$
$614$	0	0
$615$	0.173486	0.00699563
$616$	0	0
$617$	−22.2232	−0.894673	−0.447337	−	0.894366i	$-0.647627\pi$
$617$	−22.2232	−0.894673	−0.447337	+	0.894366i	$0.647627\pi$
$618$	0	0
$619$	38.6718	1.55435	0.777176	−	0.629283i	$-0.216651\pi$
$619$	38.6718	1.55435	0.777176	+	0.629283i	$0.216651\pi$
$620$	0	0
$621$	5.96402	0.239328
$622$	0	0
$623$	−57.9050	−2.31991
$624$	0	0
$625$	19.3330	0.773321
$626$	0	0
$627$	−3.13184	−0.125074
$628$	0	0
$629$	13.3718	0.533170
$630$	0	0
$631$	−5.89984	−0.234869	−0.117435	−	0.993081i	$-0.537467\pi$
$631$	−5.89984	−0.234869	−0.117435	+	0.993081i	$0.537467\pi$
$632$	0	0
$633$	3.53623	0.140553
$634$	0	0
$635$	3.52187	0.139761
$636$	0	0
$637$	47.6072	1.88626
$638$	0	0
$639$	23.6605	0.935996
$640$	0	0
$641$	5.65655	0.223420	0.111710	−	0.993741i	$-0.464367\pi$
$641$	5.65655	0.223420	0.111710	+	0.993741i	$0.464367\pi$
$642$	0	0
$643$	28.5741	1.12685	0.563427	−	0.826166i	$-0.309483\pi$
$643$	28.5741	1.12685	0.563427	+	0.826166i	$0.309483\pi$
$644$	0	0
$645$	−0.945187	−0.0372167
$646$	0	0
$647$	−12.3042	−0.483728	−0.241864	−	0.970310i	$-0.577759\pi$
$647$	−12.3042	−0.483728	−0.241864	+	0.970310i	$0.577759\pi$
$648$	0	0
$649$	−12.5303	−0.491856
$650$	0	0
$651$	−2.76451	−0.108350
$652$	0	0
$653$	43.0548	1.68487	0.842433	−	0.538802i	$-0.181123\pi$
$653$	43.0548	1.68487	0.842433	+	0.538802i	$0.181123\pi$
$654$	0	0
$655$	5.06650	0.197965
$656$	0	0
$657$	−17.7828	−0.693772
$658$	0	0
$659$	−36.1040	−1.40641	−0.703206	−	0.710987i	$-0.748249\pi$
$659$	−36.1040	−1.40641	−0.703206	+	0.710987i	$0.748249\pi$
$660$	0	0
$661$	48.1029	1.87098	0.935492	−	0.353347i	$-0.114957\pi$
$661$	48.1029	1.87098	0.935492	+	0.353347i	$0.114957\pi$
$662$	0	0
$663$	−4.07558	−0.158282
$664$	0	0
$665$	−16.9867	−0.658715
$666$	0	0
$667$	−20.4767	−0.792863
$668$	0	0
$669$	−8.76488	−0.338870
$670$	0	0
$671$	−4.96969	−0.191853
$672$	0	0
$673$	2.64934	0.102125	0.0510623	−	0.998695i	$-0.483739\pi$
$673$	2.64934	0.102125	0.0510623	+	0.998695i	$0.483739\pi$
$674$	0	0
$675$	−12.1400	−0.467270
$676$	0	0
$677$	8.77489	0.337247	0.168623	−	0.985681i	$-0.446068\pi$
$677$	8.77489	0.337247	0.168623	+	0.985681i	$0.446068\pi$
$678$	0	0
$679$	21.0062	0.806145
$680$	0	0
$681$	7.02657	0.269259
$682$	0	0
$683$	29.3004	1.12115	0.560574	−	0.828104i	$-0.310581\pi$
$683$	29.3004	1.12115	0.560574	+	0.828104i	$0.310581\pi$
$684$	0	0
$685$	4.28030	0.163542
$686$	0	0
$687$	−7.64851	−0.291809
$688$	0	0
$689$	2.46952	0.0940813
$690$	0	0
$691$	−38.6326	−1.46965	−0.734826	−	0.678256i	$-0.762736\pi$
$691$	−38.6326	−1.46965	−0.734826	+	0.678256i	$0.762736\pi$
$692$	0	0
$693$	12.1350	0.460972
$694$	0	0
$695$	13.7732	0.522446
$696$	0	0
$697$	−1.17717	−0.0445883
$698$	0	0
$699$	−4.13517	−0.156406
$700$	0	0
$701$	43.3475	1.63721	0.818606	−	0.574355i	$-0.194747\pi$
$701$	43.3475	1.63721	0.818606	+	0.574355i	$0.194747\pi$
$702$	0	0
$703$	−45.8253	−1.72833
$704$	0	0
$705$	−0.670098	−0.0252374
$706$	0	0
$707$	26.2485	0.987176
$708$	0	0
$709$	−44.5209	−1.67202	−0.836008	−	0.548717i	$-0.815117\pi$
$709$	−44.5209	−1.67202	−0.836008	+	0.548717i	$0.815117\pi$
$710$	0	0
$711$	−7.97247	−0.298991
$712$	0	0
$713$	3.32551	0.124541
$714$	0	0
$715$	−3.04890	−0.114022
$716$	0	0
$717$	0.540381	0.0201809
$718$	0	0
$719$	17.5083	0.652948	0.326474	−	0.945206i	$-0.394140\pi$
$719$	17.5083	0.652948	0.326474	+	0.945206i	$0.394140\pi$
$720$	0	0
$721$	−9.12235	−0.339734
$722$	0	0
$723$	−8.19879	−0.304916
$724$	0	0
$725$	41.6813	1.54801
$726$	0	0
$727$	−11.1535	−0.413659	−0.206830	−	0.978377i	$-0.566315\pi$
$727$	−11.1535	−0.413659	−0.206830	+	0.978377i	$0.566315\pi$
$728$	0	0
$729$	−16.4841	−0.610522
$730$	0	0
$731$	6.41343	0.237209
$732$	0	0
$733$	−8.50192	−0.314026	−0.157013	−	0.987597i	$-0.550186\pi$
$733$	−8.50192	−0.314026	−0.157013	+	0.987597i	$0.550186\pi$
$734$	0	0
$735$	−2.88265	−0.106328
$736$	0	0
$737$	−17.0691	−0.628749
$738$	0	0
$739$	−11.7543	−0.432389	−0.216195	−	0.976350i	$-0.569365\pi$
$739$	−11.7543	−0.432389	−0.216195	+	0.976350i	$0.569365\pi$
$740$	0	0
$741$	13.9670	0.513091
$742$	0	0
$743$	−6.71422	−0.246321	−0.123160	−	0.992387i	$-0.539303\pi$
$743$	−6.71422	−0.246321	−0.123160	+	0.992387i	$0.539303\pi$
$744$	0	0
$745$	7.08250	0.259483
$746$	0	0
$747$	−2.81960	−0.103164
$748$	0	0
$749$	73.7671	2.69539
$750$	0	0
$751$	−20.0634	−0.732123	−0.366062	−	0.930591i	$-0.619294\pi$
$751$	−20.0634	−0.732123	−0.366062	+	0.930591i	$0.619294\pi$
$752$	0	0
$753$	−10.0358	−0.365726
$754$	0	0
$755$	3.60251	0.131109
$756$	0	0
$757$	8.97087	0.326052	0.163026	−	0.986622i	$-0.447875\pi$
$757$	8.97087	0.326052	0.163026	+	0.986622i	$0.447875\pi$
$758$	0	0
$759$	1.07858	0.0391500
$760$	0	0
$761$	−5.94861	−0.215637	−0.107819	−	0.994171i	$-0.534387\pi$
$761$	−5.94861	−0.215637	−0.107819	+	0.994171i	$0.534387\pi$
$762$	0	0
$763$	−10.7261	−0.388310
$764$	0	0
$765$	−3.33994	−0.120756
$766$	0	0
$767$	55.8810	2.01775
$768$	0	0
$769$	16.5388	0.596405	0.298203	−	0.954503i	$-0.403613\pi$
$769$	16.5388	0.596405	0.298203	+	0.954503i	$0.403613\pi$
$770$	0	0
$771$	0.849965	0.0306107
$772$	0	0
$773$	47.7218	1.71643	0.858217	−	0.513288i	$-0.171573\pi$
$773$	47.7218	1.71643	0.858217	+	0.513288i	$0.171573\pi$
$774$	0	0
$775$	−6.76922	−0.243158
$776$	0	0
$777$	−13.1195	−0.470661
$778$	0	0
$779$	4.03415	0.144538
$780$	0	0
$781$	8.87406	0.317539
$782$	0	0
$783$	−23.7876	−0.850099
$784$	0	0
$785$	−7.00815	−0.250132
$786$	0	0
$787$	48.2845	1.72116	0.860578	−	0.509318i	$-0.170102\pi$
$787$	48.2845	1.72116	0.860578	+	0.509318i	$0.170102\pi$
$788$	0	0
$789$	−1.82719	−0.0650498
$790$	0	0
$791$	−50.2919	−1.78817
$792$	0	0
$793$	22.1632	0.787040
$794$	0	0
$795$	−0.149532	−0.00530334
$796$	0	0
$797$	0.0420293	0.00148876	0.000744378	−	1.00000i	$-0.499763\pi$
$797$	0.0420293	0.00148876	0.000744378		1.00000i	$0.499763\pi$
$798$	0	0
$799$	4.54686	0.160856
$800$	0	0
$801$	39.0176	1.37862
$802$	0	0
$803$	−6.66956	−0.235364
$804$	0	0
$805$	5.85007	0.206188
$806$	0	0
$807$	−7.59233	−0.267263
$808$	0	0
$809$	28.8318	1.01367	0.506837	−	0.862042i	$-0.330815\pi$
$809$	28.8318	1.01367	0.506837	+	0.862042i	$0.330815\pi$
$810$	0	0
$811$	18.3160	0.643163	0.321582	−	0.946882i	$-0.395786\pi$
$811$	18.3160	0.643163	0.321582	+	0.946882i	$0.395786\pi$
$812$	0	0
$813$	6.94183	0.243461
$814$	0	0
$815$	−3.32167	−0.116353
$816$	0	0
$817$	−21.9788	−0.768942
$818$	0	0
$819$	−54.1184	−1.89105
$820$	0	0
$821$	−47.7932	−1.66799	−0.833996	−	0.551770i	$-0.813953\pi$
$821$	−47.7932	−1.66799	−0.833996	+	0.551770i	$0.813953\pi$
$822$	0	0
$823$	−28.5560	−0.995398	−0.497699	−	0.867350i	$-0.665822\pi$
$823$	−28.5560	−0.995398	−0.497699	+	0.867350i	$0.665822\pi$
$824$	0	0
$825$	−2.19550	−0.0764374
$826$	0	0
$827$	−37.9725	−1.32043	−0.660217	−	0.751075i	$-0.729536\pi$
$827$	−37.9725	−1.32043	−0.660217	+	0.751075i	$0.729536\pi$
$828$	0	0
$829$	19.2340	0.668025	0.334012	−	0.942569i	$-0.391597\pi$
$829$	19.2340	0.668025	0.334012	+	0.942569i	$0.391597\pi$
$830$	0	0
$831$	8.33992	0.289309
$832$	0	0
$833$	19.5599	0.677709
$834$	0	0
$835$	5.54515	0.191898
$836$	0	0
$837$	3.86320	0.133532
$838$	0	0
$839$	38.4653	1.32797	0.663985	−	0.747746i	$-0.268864\pi$
$839$	38.4653	1.32797	0.663985	+	0.747746i	$0.268864\pi$
$840$	0	0
$841$	52.6718	1.81627
$842$	0	0
$843$	2.71862	0.0936343
$844$	0	0
$845$	5.50130	0.189250
$846$	0	0
$847$	−41.0535	−1.41061
$848$	0	0
$849$	−4.53791	−0.155741
$850$	0	0
$851$	15.7819	0.540996
$852$	0	0
$853$	−17.8231	−0.610252	−0.305126	−	0.952312i	$-0.598699\pi$
$853$	−17.8231	−0.610252	−0.305126	+	0.952312i	$0.598699\pi$
$854$	0	0
$855$	11.4460	0.391444
$856$	0	0
$857$	−24.2111	−0.827036	−0.413518	−	0.910496i	$-0.635700\pi$
$857$	−24.2111	−0.827036	−0.413518	+	0.910496i	$0.635700\pi$
$858$	0	0
$859$	27.3204	0.932160	0.466080	−	0.884743i	$-0.345666\pi$
$859$	27.3204	0.932160	0.466080	+	0.884743i	$0.345666\pi$
$860$	0	0
$861$	1.15495	0.0393607
$862$	0	0
$863$	3.41306	0.116182	0.0580909	−	0.998311i	$-0.481499\pi$
$863$	3.41306	0.116182	0.0580909	+	0.998311i	$0.481499\pi$
$864$	0	0
$865$	−11.6742	−0.396935
$866$	0	0
$867$	6.04904	0.205436
$868$	0	0
$869$	−2.99014	−0.101433
$870$	0	0
$871$	76.1229	2.57933
$872$	0	0
$873$	−14.1544	−0.479055
$874$	0	0
$875$	−24.8175	−0.838985
$876$	0	0
$877$	43.5871	1.47183	0.735915	−	0.677074i	$-0.236752\pi$
$877$	43.5871	1.47183	0.735915	+	0.677074i	$0.236752\pi$
$878$	0	0
$879$	−6.65297	−0.224399
$880$	0	0
$881$	−22.7754	−0.767323	−0.383662	−	0.923474i	$-0.625337\pi$
$881$	−22.7754	−0.767323	−0.383662	+	0.923474i	$0.625337\pi$
$882$	0	0
$883$	−10.5889	−0.356345	−0.178173	−	0.983999i	$-0.557019\pi$
$883$	−10.5889	−0.356345	−0.178173	+	0.983999i	$0.557019\pi$
$884$	0	0
$885$	−3.38364	−0.113740
$886$	0	0
$887$	49.2237	1.65277	0.826386	−	0.563104i	$-0.190393\pi$
$887$	49.2237	1.65277	0.826386	+	0.563104i	$0.190393\pi$
$888$	0	0
$889$	23.4463	0.786363
$890$	0	0
$891$	−7.52803	−0.252199
$892$	0	0
$893$	−15.5821	−0.521435
$894$	0	0
$895$	1.82894	0.0611347
$896$	0	0
$897$	−4.81012	−0.160605
$898$	0	0
$899$	−13.2638	−0.442374
$900$	0	0
$901$	1.01463	0.0338021
$902$	0	0
$903$	−6.29242	−0.209399
$904$	0	0
$905$	−3.53062	−0.117362
$906$	0	0
$907$	−16.4067	−0.544777	−0.272388	−	0.962187i	$-0.587814\pi$
$907$	−16.4067	−0.544777	−0.272388	+	0.962187i	$0.587814\pi$
$908$	0	0
$909$	−17.6868	−0.586633
$910$	0	0
$911$	−36.2635	−1.20146	−0.600732	−	0.799451i	$-0.705124\pi$
$911$	−36.2635	−1.20146	−0.600732	+	0.799451i	$0.705124\pi$
$912$	0	0
$913$	−1.05751	−0.0349986
$914$	0	0
$915$	−1.34200	−0.0443653
$916$	0	0
$917$	33.7294	1.11384
$918$	0	0
$919$	27.6586	0.912371	0.456186	−	0.889885i	$-0.349215\pi$
$919$	27.6586	0.912371	0.456186	+	0.889885i	$0.349215\pi$
$920$	0	0
$921$	3.24805	0.107027
$922$	0	0
$923$	−39.5755	−1.30264
$924$	0	0
$925$	−32.1247	−1.05625
$926$	0	0
$927$	6.14683	0.201888
$928$	0	0
$929$	−12.6814	−0.416062	−0.208031	−	0.978122i	$-0.566706\pi$
$929$	−12.6814	−0.416062	−0.208031	+	0.978122i	$0.566706\pi$
$930$	0	0
$931$	−67.0316	−2.19687
$932$	0	0
$933$	−6.90746	−0.226140
$934$	0	0
$935$	−1.25267	−0.0409667
$936$	0	0
$937$	11.6391	0.380234	0.190117	−	0.981761i	$-0.439113\pi$
$937$	11.6391	0.380234	0.190117	+	0.981761i	$0.439113\pi$
$938$	0	0
$939$	8.87444	0.289606
$940$	0	0
$941$	−18.6504	−0.607986	−0.303993	−	0.952674i	$-0.598320\pi$
$941$	−18.6504	−0.607986	−0.303993	+	0.952674i	$0.598320\pi$
$942$	0	0
$943$	−1.38933	−0.0452427
$944$	0	0
$945$	6.79596	0.221073
$946$	0	0
$947$	−10.3051	−0.334871	−0.167436	−	0.985883i	$-0.553549\pi$
$947$	−10.3051	−0.334871	−0.167436	+	0.985883i	$0.553549\pi$
$948$	0	0
$949$	29.7441	0.965535
$950$	0	0
$951$	−8.47324	−0.274764
$952$	0	0
$953$	−54.0619	−1.75124	−0.875619	−	0.483003i	$-0.839546\pi$
$953$	−54.0619	−1.75124	−0.875619	+	0.483003i	$0.839546\pi$
$954$	0	0
$955$	−1.09537	−0.0354454
$956$	0	0
$957$	−4.30193	−0.139062
$958$	0	0
$959$	28.4954	0.920163
$960$	0	0
$961$	−28.8459	−0.930513
$962$	0	0
$963$	−49.7058	−1.60175
$964$	0	0
$965$	−3.75376	−0.120838
$966$	0	0
$967$	−32.6646	−1.05042	−0.525212	−	0.850972i	$-0.676014\pi$
$967$	−32.6646	−1.05042	−0.525212	+	0.850972i	$0.676014\pi$
$968$	0	0
$969$	5.73848	0.184346
$970$	0	0
$971$	52.7236	1.69198	0.845991	−	0.533198i	$-0.179010\pi$
$971$	52.7236	1.69198	0.845991	+	0.533198i	$0.179010\pi$
$972$	0	0
$973$	91.6926	2.93953
$974$	0	0
$975$	9.79122	0.313570
$976$	0	0
$977$	2.24928	0.0719610	0.0359805	−	0.999352i	$-0.488545\pi$
$977$	2.24928	0.0719610	0.0359805	+	0.999352i	$0.488545\pi$
$978$	0	0
$979$	14.6338	0.467700
$980$	0	0
$981$	7.22745	0.230755
$982$	0	0
$983$	−28.3617	−0.904598	−0.452299	−	0.891866i	$-0.649396\pi$
$983$	−28.3617	−0.904598	−0.452299	+	0.891866i	$0.649396\pi$
$984$	0	0
$985$	−5.93254	−0.189026
$986$	0	0
$987$	−4.46106	−0.141997
$988$	0	0
$989$	7.56933	0.240691
$990$	0	0
$991$	−39.6659	−1.26003	−0.630014	−	0.776584i	$-0.716951\pi$
$991$	−39.6659	−1.26003	−0.630014	+	0.776584i	$0.716951\pi$
$992$	0	0
$993$	7.72327	0.245090
$994$	0	0
$995$	7.07278	0.224222
$996$	0	0
$997$	36.7817	1.16489	0.582444	−	0.812871i	$-0.302096\pi$
$997$	36.7817	1.16489	0.582444	+	0.812871i	$0.302096\pi$
$998$	0	0
$999$	18.3336	0.580050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.12		29
4.3	odd	2		4024.2.a.e.1.18	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.18	✓	29	4.3	odd	2
8048.2.a.w.1.12		29	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)

\(f(q)\)	\(=\)	\(q-0.454325 q^{3} +0.622756 q^{5} +4.14589 q^{7} -2.79359 q^{9} +O(q^{10})\)	\(q-0.454325 q^{3} +0.622756 q^{5} +4.14589 q^{7} -2.79359 q^{9} -1.04776 q^{11} +4.67266 q^{13} -0.282934 q^{15} +1.91981 q^{17} -6.57918 q^{19} -1.88358 q^{21} +2.26582 q^{23} -4.61217 q^{25} +2.63217 q^{27} -9.03724 q^{29} +1.46769 q^{31} +0.476022 q^{33} +2.58188 q^{35} +6.96520 q^{37} -2.12291 q^{39} -0.613168 q^{41} +3.34066 q^{43} -1.73973 q^{45} +2.36839 q^{47} +10.1884 q^{49} -0.872217 q^{51} +0.528504 q^{53} -0.652497 q^{55} +2.98909 q^{57} +11.9591 q^{59} +4.74317 q^{61} -11.5819 q^{63} +2.90993 q^{65} +16.2911 q^{67} -1.02942 q^{69} -8.46958 q^{71} +6.36556 q^{73} +2.09543 q^{75} -4.34389 q^{77} +2.85384 q^{79} +7.18490 q^{81} +1.00931 q^{83} +1.19557 q^{85} +4.10585 q^{87} -13.9668 q^{89} +19.3724 q^{91} -0.666806 q^{93} -4.09723 q^{95} +5.06676 q^{97} +2.92700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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