LMFDB - Embedded newform 8048.2.a.y.1.2

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:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.95302 q^{3} +0.296089 q^{5} -1.50033 q^{7} +5.72032 q^{9} +O(q^{10})$	$q-2.95302 q^{3} +0.296089 q^{5} -1.50033 q^{7} +5.72032 q^{9} -1.08044 q^{11} -3.03263 q^{13} -0.874355 q^{15} -1.39026 q^{17} -6.49823 q^{19} +4.43051 q^{21} -6.01954 q^{23} -4.91233 q^{25} -8.03314 q^{27} +2.16074 q^{29} -4.62376 q^{31} +3.19055 q^{33} -0.444232 q^{35} +3.14887 q^{37} +8.95542 q^{39} -3.26900 q^{41} +3.55964 q^{43} +1.69372 q^{45} -0.543223 q^{47} -4.74900 q^{49} +4.10545 q^{51} -9.68893 q^{53} -0.319905 q^{55} +19.1894 q^{57} -8.24918 q^{59} +6.05574 q^{61} -8.58238 q^{63} -0.897928 q^{65} +8.90650 q^{67} +17.7758 q^{69} -11.1885 q^{71} +2.05063 q^{73} +14.5062 q^{75} +1.62102 q^{77} -9.79408 q^{79} +6.56107 q^{81} +6.12511 q^{83} -0.411639 q^{85} -6.38070 q^{87} -9.53552 q^{89} +4.54996 q^{91} +13.6540 q^{93} -1.92405 q^{95} +0.238166 q^{97} -6.18045 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.95302	−1.70493	−0.852463	−	0.522788i	$-0.824892\pi$
$3$	−2.95302	−1.70493	−0.852463	+	0.522788i	$0.824892\pi$
$4$	0	0
$5$	0.296089	0.132415	0.0662074	−	0.997806i	$-0.478910\pi$
$5$	0.296089	0.132415	0.0662074	+	0.997806i	$0.478910\pi$
$6$	0	0
$7$	−1.50033	−0.567073	−0.283536	−	0.958962i	$-0.591508\pi$
$7$	−1.50033	−0.567073	−0.283536	+	0.958962i	$0.591508\pi$
$8$	0	0
$9$	5.72032	1.90677
$10$	0	0
$11$	−1.08044	−0.325764	−0.162882	−	0.986646i	$-0.552079\pi$
$11$	−1.08044	−0.325764	−0.162882	+	0.986646i	$0.552079\pi$
$12$	0	0
$13$	−3.03263	−0.841101	−0.420550	−	0.907269i	$-0.638163\pi$
$13$	−3.03263	−0.841101	−0.420550	+	0.907269i	$0.638163\pi$
$14$	0	0
$15$	−0.874355	−0.225758
$16$	0	0
$17$	−1.39026	−0.337186	−0.168593	−	0.985686i	$-0.553922\pi$
$17$	−1.39026	−0.337186	−0.168593	+	0.985686i	$0.553922\pi$
$18$	0	0
$19$	−6.49823	−1.49080	−0.745398	−	0.666620i	$-0.767741\pi$
$19$	−6.49823	−1.49080	−0.745398	+	0.666620i	$0.767741\pi$
$20$	0	0
$21$	4.43051	0.966817
$22$	0	0
$23$	−6.01954	−1.25516	−0.627581	−	0.778551i	$-0.715955\pi$
$23$	−6.01954	−1.25516	−0.627581	+	0.778551i	$0.715955\pi$
$24$	0	0
$25$	−4.91233	−0.982466
$26$	0	0
$27$	−8.03314	−1.54598
$28$	0	0
$29$	2.16074	0.401239	0.200619	−	0.979669i	$-0.435705\pi$
$29$	2.16074	0.401239	0.200619	+	0.979669i	$0.435705\pi$
$30$	0	0
$31$	−4.62376	−0.830451	−0.415226	−	0.909718i	$-0.636297\pi$
$31$	−4.62376	−0.830451	−0.415226	+	0.909718i	$0.636297\pi$
$32$	0	0
$33$	3.19055	0.555404
$34$	0	0
$35$	−0.444232	−0.0750888
$36$	0	0
$37$	3.14887	0.517672	0.258836	−	0.965921i	$-0.416661\pi$
$37$	3.14887	0.517672	0.258836	+	0.965921i	$0.416661\pi$
$38$	0	0
$39$	8.95542	1.43401
$40$	0	0
$41$	−3.26900	−0.510532	−0.255266	−	0.966871i	$-0.582163\pi$
$41$	−3.26900	−0.510532	−0.255266	+	0.966871i	$0.582163\pi$
$42$	0	0
$43$	3.55964	0.542840	0.271420	−	0.962461i	$-0.412507\pi$
$43$	3.55964	0.542840	0.271420	+	0.962461i	$0.412507\pi$
$44$	0	0
$45$	1.69372	0.252485
$46$	0	0
$47$	−0.543223	−0.0792373	−0.0396186	−	0.999215i	$-0.512614\pi$
$47$	−0.543223	−0.0792373	−0.0396186	+	0.999215i	$0.512614\pi$
$48$	0	0
$49$	−4.74900	−0.678429
$50$	0	0
$51$	4.10545	0.574878
$52$	0	0
$53$	−9.68893	−1.33088	−0.665439	−	0.746452i	$-0.731756\pi$
$53$	−9.68893	−1.33088	−0.665439	+	0.746452i	$0.731756\pi$
$54$	0	0
$55$	−0.319905	−0.0431360
$56$	0	0
$57$	19.1894	2.54170
$58$	0	0
$59$	−8.24918	−1.07395	−0.536976	−	0.843598i	$-0.680433\pi$
$59$	−8.24918	−1.07395	−0.536976	+	0.843598i	$0.680433\pi$
$60$	0	0
$61$	6.05574	0.775358	0.387679	−	0.921794i	$-0.373277\pi$
$61$	6.05574	0.775358	0.387679	+	0.921794i	$0.373277\pi$
$62$	0	0
$63$	−8.58238	−1.08128
$64$	0	0
$65$	−0.897928	−0.111374
$66$	0	0
$67$	8.90650	1.08810	0.544051	−	0.839052i	$-0.316890\pi$
$67$	8.90650	1.08810	0.544051	+	0.839052i	$0.316890\pi$
$68$	0	0
$69$	17.7758	2.13996
$70$	0	0
$71$	−11.1885	−1.32784	−0.663918	−	0.747806i	$-0.731107\pi$
$71$	−11.1885	−1.32784	−0.663918	+	0.747806i	$0.731107\pi$
$72$	0	0
$73$	2.05063	0.240008	0.120004	−	0.992773i	$-0.461709\pi$
$73$	2.05063	0.240008	0.120004	+	0.992773i	$0.461709\pi$
$74$	0	0
$75$	14.5062	1.67503
$76$	0	0
$77$	1.62102	0.184732
$78$	0	0
$79$	−9.79408	−1.10192	−0.550960	−	0.834532i	$-0.685738\pi$
$79$	−9.79408	−1.10192	−0.550960	+	0.834532i	$0.685738\pi$
$80$	0	0
$81$	6.56107	0.729007
$82$	0	0
$83$	6.12511	0.672318	0.336159	−	0.941805i	$-0.390872\pi$
$83$	6.12511	0.672318	0.336159	+	0.941805i	$0.390872\pi$
$84$	0	0
$85$	−0.411639	−0.0446485
$86$	0	0
$87$	−6.38070	−0.684083
$88$	0	0
$89$	−9.53552	−1.01076	−0.505382	−	0.862896i	$-0.668648\pi$
$89$	−9.53552	−1.01076	−0.505382	+	0.862896i	$0.668648\pi$
$90$	0	0
$91$	4.54996	0.476965
$92$	0	0
$93$	13.6540	1.41586
$94$	0	0
$95$	−1.92405	−0.197404
$96$	0	0
$97$	0.238166	0.0241821	0.0120910	−	0.999927i	$-0.496151\pi$
$97$	0.238166	0.0241821	0.0120910	+	0.999927i	$0.496151\pi$
$98$	0	0
$99$	−6.18045	−0.621158
$100$	0	0
$101$	15.2200	1.51445	0.757224	−	0.653155i	$-0.226555\pi$
$101$	15.2200	1.51445	0.757224	+	0.653155i	$0.226555\pi$
$102$	0	0
$103$	−19.7565	−1.94667	−0.973334	−	0.229394i	$-0.926326\pi$
$103$	−19.7565	−1.94667	−0.973334	+	0.229394i	$0.926326\pi$
$104$	0	0
$105$	1.31182	0.128021
$106$	0	0
$107$	9.93824	0.960766	0.480383	−	0.877059i	$-0.340498\pi$
$107$	9.93824	0.960766	0.480383	+	0.877059i	$0.340498\pi$
$108$	0	0
$109$	1.83402	0.175667	0.0878335	−	0.996135i	$-0.472006\pi$
$109$	1.83402	0.175667	0.0878335	+	0.996135i	$0.472006\pi$
$110$	0	0
$111$	−9.29868	−0.882592
$112$	0	0
$113$	1.71329	0.161173	0.0805863	−	0.996748i	$-0.474321\pi$
$113$	1.71329	0.161173	0.0805863	+	0.996748i	$0.474321\pi$
$114$	0	0
$115$	−1.78232	−0.166202
$116$	0	0
$117$	−17.3476	−1.60379
$118$	0	0
$119$	2.08585	0.191209
$120$	0	0
$121$	−9.83265	−0.893878
$122$	0	0
$123$	9.65341	0.870418
$124$	0	0
$125$	−2.93493	−0.262508
$126$	0	0
$127$	7.44687	0.660803	0.330402	−	0.943840i	$-0.392816\pi$
$127$	7.44687	0.660803	0.330402	+	0.943840i	$0.392816\pi$
$128$	0	0
$129$	−10.5117	−0.925502
$130$	0	0
$131$	−3.14320	−0.274622	−0.137311	−	0.990528i	$-0.543846\pi$
$131$	−3.14320	−0.274622	−0.137311	+	0.990528i	$0.543846\pi$
$132$	0	0
$133$	9.74951	0.845390
$134$	0	0
$135$	−2.37852	−0.204711
$136$	0	0
$137$	−12.2105	−1.04321	−0.521606	−	0.853186i	$-0.674667\pi$
$137$	−12.2105	−1.04321	−0.521606	+	0.853186i	$0.674667\pi$
$138$	0	0
$139$	−9.72188	−0.824599	−0.412300	−	0.911048i	$-0.635274\pi$
$139$	−9.72188	−0.824599	−0.412300	+	0.911048i	$0.635274\pi$
$140$	0	0
$141$	1.60415	0.135094
$142$	0	0
$143$	3.27657	0.274001
$144$	0	0
$145$	0.639770	0.0531300
$146$	0	0
$147$	14.0239	1.15667
$148$	0	0
$149$	−6.67596	−0.546916	−0.273458	−	0.961884i	$-0.588167\pi$
$149$	−6.67596	−0.546916	−0.273458	+	0.961884i	$0.588167\pi$
$150$	0	0
$151$	18.2994	1.48918	0.744592	−	0.667519i	$-0.232644\pi$
$151$	18.2994	1.48918	0.744592	+	0.667519i	$0.232644\pi$
$152$	0	0
$153$	−7.95270	−0.642938
$154$	0	0
$155$	−1.36904	−0.109964
$156$	0	0
$157$	−18.4756	−1.47451	−0.737257	−	0.675612i	$-0.763879\pi$
$157$	−18.4756	−1.47451	−0.737257	+	0.675612i	$0.763879\pi$
$158$	0	0
$159$	28.6116	2.26905
$160$	0	0
$161$	9.03132	0.711768
$162$	0	0
$163$	−3.04796	−0.238735	−0.119367	−	0.992850i	$-0.538087\pi$
$163$	−3.04796	−0.238735	−0.119367	+	0.992850i	$0.538087\pi$
$164$	0	0
$165$	0.944686	0.0735437
$166$	0	0
$167$	−18.6519	−1.44333	−0.721664	−	0.692243i	$-0.756623\pi$
$167$	−18.6519	−1.44333	−0.721664	+	0.692243i	$0.756623\pi$
$168$	0	0
$169$	−3.80314	−0.292549
$170$	0	0
$171$	−37.1719	−2.84261
$172$	0	0
$173$	13.9721	1.06228	0.531141	−	0.847283i	$-0.321763\pi$
$173$	13.9721	1.06228	0.531141	+	0.847283i	$0.321763\pi$
$174$	0	0
$175$	7.37013	0.557130
$176$	0	0
$177$	24.3600	1.83101
$178$	0	0
$179$	−12.1379	−0.907228	−0.453614	−	0.891198i	$-0.649866\pi$
$179$	−12.1379	−0.907228	−0.453614	+	0.891198i	$0.649866\pi$
$180$	0	0
$181$	−12.9560	−0.963014	−0.481507	−	0.876442i	$-0.659910\pi$
$181$	−12.9560	−0.963014	−0.481507	+	0.876442i	$0.659910\pi$
$182$	0	0
$183$	−17.8827	−1.32193
$184$	0	0
$185$	0.932346	0.0685475
$186$	0	0
$187$	1.50208	0.109843
$188$	0	0
$189$	12.0524	0.876682
$190$	0	0
$191$	20.2407	1.46456	0.732281	−	0.681002i	$-0.238456\pi$
$191$	20.2407	1.46456	0.732281	+	0.681002i	$0.238456\pi$
$192$	0	0
$193$	−16.4980	−1.18755	−0.593775	−	0.804631i	$-0.702363\pi$
$193$	−16.4980	−1.18755	−0.593775	+	0.804631i	$0.702363\pi$
$194$	0	0
$195$	2.65160	0.189885
$196$	0	0
$197$	−1.82574	−0.130079	−0.0650395	−	0.997883i	$-0.520717\pi$
$197$	−1.82574	−0.130079	−0.0650395	+	0.997883i	$0.520717\pi$
$198$	0	0
$199$	−19.9397	−1.41349	−0.706744	−	0.707470i	$-0.749837\pi$
$199$	−19.9397	−1.41349	−0.706744	+	0.707470i	$0.749837\pi$
$200$	0	0
$201$	−26.3011	−1.85513
$202$	0	0
$203$	−3.24183	−0.227532
$204$	0	0
$205$	−0.967913	−0.0676020
$206$	0	0
$207$	−34.4337	−2.39331
$208$	0	0
$209$	7.02093	0.485648
$210$	0	0
$211$	−26.4549	−1.82123	−0.910615	−	0.413257i	$-0.864391\pi$
$211$	−26.4549	−1.82123	−0.910615	+	0.413257i	$0.864391\pi$
$212$	0	0
$213$	33.0400	2.26386
$214$	0	0
$215$	1.05397	0.0718801
$216$	0	0
$217$	6.93718	0.470926
$218$	0	0
$219$	−6.05554	−0.409195
$220$	0	0
$221$	4.21613	0.283608
$222$	0	0
$223$	9.54654	0.639283	0.319642	−	0.947538i	$-0.396437\pi$
$223$	9.54654	0.639283	0.319642	+	0.947538i	$0.396437\pi$
$224$	0	0
$225$	−28.1001	−1.87334
$226$	0	0
$227$	20.7580	1.37775	0.688877	−	0.724878i	$-0.258104\pi$
$227$	20.7580	1.37775	0.688877	+	0.724878i	$0.258104\pi$
$228$	0	0
$229$	−19.5329	−1.29077	−0.645385	−	0.763857i	$-0.723303\pi$
$229$	−19.5329	−1.29077	−0.645385	+	0.763857i	$0.723303\pi$
$230$	0	0
$231$	−4.78689	−0.314954
$232$	0	0
$233$	−8.82104	−0.577886	−0.288943	−	0.957346i	$-0.593304\pi$
$233$	−8.82104	−0.577886	−0.288943	+	0.957346i	$0.593304\pi$
$234$	0	0
$235$	−0.160842	−0.0104922
$236$	0	0
$237$	28.9221	1.87869
$238$	0	0
$239$	11.7112	0.757534	0.378767	−	0.925492i	$-0.376348\pi$
$239$	11.7112	0.757534	0.378767	+	0.925492i	$0.376348\pi$
$240$	0	0
$241$	11.0233	0.710073	0.355036	−	0.934853i	$-0.384468\pi$
$241$	11.0233	0.710073	0.355036	+	0.934853i	$0.384468\pi$
$242$	0	0
$243$	4.72448	0.303075
$244$	0	0
$245$	−1.40613	−0.0898341
$246$	0	0
$247$	19.7067	1.25391
$248$	0	0
$249$	−18.0876	−1.14625
$250$	0	0
$251$	−22.8586	−1.44282	−0.721410	−	0.692508i	$-0.756506\pi$
$251$	−22.8586	−1.44282	−0.721410	+	0.692508i	$0.756506\pi$
$252$	0	0
$253$	6.50374	0.408887
$254$	0	0
$255$	1.21558	0.0761224
$256$	0	0
$257$	24.6709	1.53893	0.769463	−	0.638691i	$-0.220524\pi$
$257$	24.6709	1.53893	0.769463	+	0.638691i	$0.220524\pi$
$258$	0	0
$259$	−4.72436	−0.293557
$260$	0	0
$261$	12.3601	0.765071
$262$	0	0
$263$	10.2114	0.629663	0.314831	−	0.949148i	$-0.398052\pi$
$263$	10.2114	0.629663	0.314831	+	0.949148i	$0.398052\pi$
$264$	0	0
$265$	−2.86878	−0.176228
$266$	0	0
$267$	28.1586	1.72328
$268$	0	0
$269$	−19.9445	−1.21604	−0.608019	−	0.793923i	$-0.708036\pi$
$269$	−19.9445	−1.21604	−0.608019	+	0.793923i	$0.708036\pi$
$270$	0	0
$271$	11.9680	0.727003	0.363502	−	0.931594i	$-0.381581\pi$
$271$	11.9680	0.727003	0.363502	+	0.931594i	$0.381581\pi$
$272$	0	0
$273$	−13.4361	−0.813190
$274$	0	0
$275$	5.30747	0.320052
$276$	0	0
$277$	1.16091	0.0697526	0.0348763	−	0.999392i	$-0.488896\pi$
$277$	1.16091	0.0697526	0.0348763	+	0.999392i	$0.488896\pi$
$278$	0	0
$279$	−26.4494	−1.58348
$280$	0	0
$281$	22.2179	1.32541	0.662705	−	0.748881i	$-0.269408\pi$
$281$	22.2179	1.32541	0.662705	+	0.748881i	$0.269408\pi$
$282$	0	0
$283$	2.99211	0.177862	0.0889312	−	0.996038i	$-0.471655\pi$
$283$	2.99211	0.177862	0.0889312	+	0.996038i	$0.471655\pi$
$284$	0	0
$285$	5.68176	0.336559
$286$	0	0
$287$	4.90458	0.289508
$288$	0	0
$289$	−15.0672	−0.886305
$290$	0	0
$291$	−0.703308	−0.0412287
$292$	0	0
$293$	8.82647	0.515648	0.257824	−	0.966192i	$-0.416995\pi$
$293$	8.82647	0.515648	0.257824	+	0.966192i	$0.416995\pi$
$294$	0	0
$295$	−2.44249	−0.142207
$296$	0	0
$297$	8.67931	0.503625
$298$	0	0
$299$	18.2551	1.05572
$300$	0	0
$301$	−5.34064	−0.307830
$302$	0	0
$303$	−44.9450	−2.58202
$304$	0	0
$305$	1.79304	0.102669
$306$	0	0
$307$	16.8775	0.963252	0.481626	−	0.876377i	$-0.340046\pi$
$307$	16.8775	0.963252	0.481626	+	0.876377i	$0.340046\pi$
$308$	0	0
$309$	58.3414	3.31892
$310$	0	0
$311$	6.49297	0.368183	0.184091	−	0.982909i	$-0.441066\pi$
$311$	6.49297	0.368183	0.184091	+	0.982909i	$0.441066\pi$
$312$	0	0
$313$	2.98804	0.168894	0.0844470	−	0.996428i	$-0.473088\pi$
$313$	2.98804	0.168894	0.0844470	+	0.996428i	$0.473088\pi$
$314$	0	0
$315$	−2.54115	−0.143177
$316$	0	0
$317$	9.60100	0.539246	0.269623	−	0.962966i	$-0.413101\pi$
$317$	9.60100	0.539246	0.269623	+	0.962966i	$0.413101\pi$
$318$	0	0
$319$	−2.33454	−0.130709
$320$	0	0
$321$	−29.3478	−1.63803
$322$	0	0
$323$	9.03420	0.502676
$324$	0	0
$325$	14.8973	0.826353
$326$	0	0
$327$	−5.41588	−0.299499
$328$	0	0
$329$	0.815016	0.0449333
$330$	0	0
$331$	17.9426	0.986215	0.493108	−	0.869968i	$-0.335861\pi$
$331$	17.9426	0.986215	0.493108	+	0.869968i	$0.335861\pi$
$332$	0	0
$333$	18.0126	0.987082
$334$	0	0
$335$	2.63711	0.144081
$336$	0	0
$337$	−4.63987	−0.252750	−0.126375	−	0.991983i	$-0.540334\pi$
$337$	−4.63987	−0.252750	−0.126375	+	0.991983i	$0.540334\pi$
$338$	0	0
$339$	−5.05937	−0.274787
$340$	0	0
$341$	4.99568	0.270531
$342$	0	0
$343$	17.6274	0.951791
$344$	0	0
$345$	5.26322	0.283362
$346$	0	0
$347$	27.4649	1.47439	0.737197	−	0.675678i	$-0.236149\pi$
$347$	27.4649	1.47439	0.737197	+	0.675678i	$0.236149\pi$
$348$	0	0
$349$	25.2889	1.35369	0.676843	−	0.736127i	$-0.263348\pi$
$349$	25.2889	1.35369	0.676843	+	0.736127i	$0.263348\pi$
$350$	0	0
$351$	24.3616	1.30032
$352$	0	0
$353$	−21.0998	−1.12303	−0.561514	−	0.827467i	$-0.689781\pi$
$353$	−21.0998	−1.12303	−0.561514	+	0.827467i	$0.689781\pi$
$354$	0	0
$355$	−3.31280	−0.175825
$356$	0	0
$357$	−6.15954	−0.325997
$358$	0	0
$359$	1.18153	0.0623589	0.0311794	−	0.999514i	$-0.490074\pi$
$359$	1.18153	0.0623589	0.0311794	+	0.999514i	$0.490074\pi$
$360$	0	0
$361$	23.2270	1.22247
$362$	0	0
$363$	29.0360	1.52400
$364$	0	0
$365$	0.607167	0.0317806
$366$	0	0
$367$	−6.20462	−0.323879	−0.161939	−	0.986801i	$-0.551775\pi$
$367$	−6.20462	−0.323879	−0.161939	+	0.986801i	$0.551775\pi$
$368$	0	0
$369$	−18.6997	−0.973467
$370$	0	0
$371$	14.5366	0.754704
$372$	0	0
$373$	−12.1178	−0.627437	−0.313719	−	0.949516i	$-0.601575\pi$
$373$	−12.1178	−0.627437	−0.313719	+	0.949516i	$0.601575\pi$
$374$	0	0
$375$	8.66690	0.447557
$376$	0	0
$377$	−6.55272	−0.337482
$378$	0	0
$379$	4.44159	0.228149	0.114075	−	0.993472i	$-0.463610\pi$
$379$	4.44159	0.228149	0.114075	+	0.993472i	$0.463610\pi$
$380$	0	0
$381$	−21.9907	−1.12662
$382$	0	0
$383$	2.94227	0.150343	0.0751714	−	0.997171i	$-0.476050\pi$
$383$	2.94227	0.150343	0.0751714	+	0.997171i	$0.476050\pi$
$384$	0	0
$385$	0.479965	0.0244613
$386$	0	0
$387$	20.3623	1.03507
$388$	0	0
$389$	6.09252	0.308903	0.154451	−	0.988000i	$-0.450639\pi$
$389$	6.09252	0.308903	0.154451	+	0.988000i	$0.450639\pi$
$390$	0	0
$391$	8.36870	0.423224
$392$	0	0
$393$	9.28192	0.468211
$394$	0	0
$395$	−2.89992	−0.145911
$396$	0	0
$397$	−21.6982	−1.08900	−0.544500	−	0.838761i	$-0.683281\pi$
$397$	−21.6982	−1.08900	−0.544500	+	0.838761i	$0.683281\pi$
$398$	0	0
$399$	−28.7905	−1.44133
$400$	0	0
$401$	0.701886	0.0350505	0.0175253	−	0.999846i	$-0.494421\pi$
$401$	0.701886	0.0350505	0.0175253	+	0.999846i	$0.494421\pi$
$402$	0	0
$403$	14.0222	0.698493
$404$	0	0
$405$	1.94266	0.0965314
$406$	0	0
$407$	−3.40216	−0.168639
$408$	0	0
$409$	5.98144	0.295763	0.147882	−	0.989005i	$-0.452755\pi$
$409$	5.98144	0.295763	0.147882	+	0.989005i	$0.452755\pi$
$410$	0	0
$411$	36.0578	1.77860
$412$	0	0
$413$	12.3765	0.609008
$414$	0	0
$415$	1.81358	0.0890250
$416$	0	0
$417$	28.7089	1.40588
$418$	0	0
$419$	−23.6263	−1.15422	−0.577111	−	0.816666i	$-0.695820\pi$
$419$	−23.6263	−1.15422	−0.577111	+	0.816666i	$0.695820\pi$
$420$	0	0
$421$	11.2415	0.547878	0.273939	−	0.961747i	$-0.411673\pi$
$421$	11.2415	0.547878	0.273939	+	0.961747i	$0.411673\pi$
$422$	0	0
$423$	−3.10741	−0.151087
$424$	0	0
$425$	6.82939	0.331274
$426$	0	0
$427$	−9.08563	−0.439684
$428$	0	0
$429$	−9.67577	−0.467151
$430$	0	0
$431$	18.7276	0.902074	0.451037	−	0.892505i	$-0.351054\pi$
$431$	18.7276	0.902074	0.451037	+	0.892505i	$0.351054\pi$
$432$	0	0
$433$	−1.71078	−0.0822147	−0.0411073	−	0.999155i	$-0.513089\pi$
$433$	−1.71078	−0.0822147	−0.0411073	+	0.999155i	$0.513089\pi$
$434$	0	0
$435$	−1.88925	−0.0905827
$436$	0	0
$437$	39.1164	1.87119
$438$	0	0
$439$	−15.4963	−0.739597	−0.369799	−	0.929112i	$-0.620573\pi$
$439$	−15.4963	−0.739597	−0.369799	+	0.929112i	$0.620573\pi$
$440$	0	0
$441$	−27.1658	−1.29361
$442$	0	0
$443$	25.3106	1.20254	0.601271	−	0.799045i	$-0.294661\pi$
$443$	25.3106	1.20254	0.601271	+	0.799045i	$0.294661\pi$
$444$	0	0
$445$	−2.82336	−0.133840
$446$	0	0
$447$	19.7142	0.932451
$448$	0	0
$449$	22.6041	1.06675	0.533377	−	0.845878i	$-0.320923\pi$
$449$	22.6041	1.06675	0.533377	+	0.845878i	$0.320923\pi$
$450$	0	0
$451$	3.53195	0.166313
$452$	0	0
$453$	−54.0385	−2.53895
$454$	0	0
$455$	1.34719	0.0631573
$456$	0	0
$457$	−26.0870	−1.22030	−0.610150	−	0.792286i	$-0.708891\pi$
$457$	−26.0870	−1.22030	−0.610150	+	0.792286i	$0.708891\pi$
$458$	0	0
$459$	11.1681	0.521283
$460$	0	0
$461$	−20.2284	−0.942129	−0.471064	−	0.882099i	$-0.656130\pi$
$461$	−20.2284	−0.942129	−0.471064	+	0.882099i	$0.656130\pi$
$462$	0	0
$463$	32.6328	1.51657	0.758286	−	0.651922i	$-0.226037\pi$
$463$	32.6328	1.51657	0.758286	+	0.651922i	$0.226037\pi$
$464$	0	0
$465$	4.04281	0.187481
$466$	0	0
$467$	11.1582	0.516340	0.258170	−	0.966099i	$-0.416880\pi$
$467$	11.1582	0.516340	0.258170	+	0.966099i	$0.416880\pi$
$468$	0	0
$469$	−13.3627	−0.617033
$470$	0	0
$471$	54.5588	2.51394
$472$	0	0
$473$	−3.84597	−0.176838
$474$	0	0
$475$	31.9215	1.46466
$476$	0	0
$477$	−55.4238	−2.53768
$478$	0	0
$479$	−39.8555	−1.82105	−0.910523	−	0.413458i	$-0.864321\pi$
$479$	−39.8555	−1.82105	−0.910523	+	0.413458i	$0.864321\pi$
$480$	0	0
$481$	−9.54938	−0.435414
$482$	0	0
$483$	−26.6697	−1.21351
$484$	0	0
$485$	0.0705182	0.00320207
$486$	0	0
$487$	4.95238	0.224414	0.112207	−	0.993685i	$-0.464208\pi$
$487$	4.95238	0.224414	0.112207	+	0.993685i	$0.464208\pi$
$488$	0	0
$489$	9.00069	0.407025
$490$	0	0
$491$	−29.0669	−1.31177	−0.655886	−	0.754860i	$-0.727705\pi$
$491$	−29.0669	−1.31177	−0.655886	+	0.754860i	$0.727705\pi$
$492$	0	0
$493$	−3.00398	−0.135292
$494$	0	0
$495$	−1.82996	−0.0822506
$496$	0	0
$497$	16.7865	0.752979
$498$	0	0
$499$	41.1744	1.84322	0.921611	−	0.388116i	$-0.126874\pi$
$499$	41.1744	1.84322	0.921611	+	0.388116i	$0.126874\pi$
$500$	0	0
$501$	55.0795	2.46077
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	4.50648	0.200536
$506$	0	0
$507$	11.2307	0.498775
$508$	0	0
$509$	10.2005	0.452129	0.226064	−	0.974112i	$-0.427414\pi$
$509$	10.2005	0.452129	0.226064	+	0.974112i	$0.427414\pi$
$510$	0	0
$511$	−3.07662	−0.136102
$512$	0	0
$513$	52.2012	2.30474
$514$	0	0
$515$	−5.84968	−0.257768
$516$	0	0
$517$	0.586919	0.0258127
$518$	0	0
$519$	−41.2600	−1.81111
$520$	0	0
$521$	4.56885	0.200165	0.100083	−	0.994979i	$-0.468089\pi$
$521$	4.56885	0.200165	0.100083	+	0.994979i	$0.468089\pi$
$522$	0	0
$523$	−10.9773	−0.480006	−0.240003	−	0.970772i	$-0.577148\pi$
$523$	−10.9773	−0.480006	−0.240003	+	0.970772i	$0.577148\pi$
$524$	0	0
$525$	−21.7641	−0.949865
$526$	0	0
$527$	6.42820	0.280017
$528$	0	0
$529$	13.2349	0.575431
$530$	0	0
$531$	−47.1879	−2.04778
$532$	0	0
$533$	9.91367	0.429409
$534$	0	0
$535$	2.94260	0.127220
$536$	0	0
$537$	35.8434	1.54676
$538$	0	0
$539$	5.13100	0.221008
$540$	0	0
$541$	7.63103	0.328084	0.164042	−	0.986453i	$-0.447547\pi$
$541$	7.63103	0.328084	0.164042	+	0.986453i	$0.447547\pi$
$542$	0	0
$543$	38.2594	1.64187
$544$	0	0
$545$	0.543032	0.0232609
$546$	0	0
$547$	14.6432	0.626098	0.313049	−	0.949737i	$-0.398650\pi$
$547$	14.6432	0.626098	0.313049	+	0.949737i	$0.398650\pi$
$548$	0	0
$549$	34.6408	1.47843
$550$	0	0
$551$	−14.0410	−0.598165
$552$	0	0
$553$	14.6944	0.624869
$554$	0	0
$555$	−2.75324	−0.116868
$556$	0	0
$557$	11.3161	0.479479	0.239739	−	0.970837i	$-0.422938\pi$
$557$	11.3161	0.479479	0.239739	+	0.970837i	$0.422938\pi$
$558$	0	0
$559$	−10.7951	−0.456583
$560$	0	0
$561$	−4.43568	−0.187275
$562$	0	0
$563$	30.5090	1.28580	0.642900	−	0.765950i	$-0.277731\pi$
$563$	30.5090	1.28580	0.642900	+	0.765950i	$0.277731\pi$
$564$	0	0
$565$	0.507285	0.0213416
$566$	0	0
$567$	−9.84378	−0.413400
$568$	0	0
$569$	−23.3792	−0.980109	−0.490054	−	0.871692i	$-0.663023\pi$
$569$	−23.3792	−0.980109	−0.490054	+	0.871692i	$0.663023\pi$
$570$	0	0
$571$	17.0333	0.712821	0.356410	−	0.934330i	$-0.384001\pi$
$571$	17.0333	0.712821	0.356410	+	0.934330i	$0.384001\pi$
$572$	0	0
$573$	−59.7710	−2.49697
$574$	0	0
$575$	29.5700	1.23315
$576$	0	0
$577$	−23.2105	−0.966265	−0.483132	−	0.875547i	$-0.660501\pi$
$577$	−23.2105	−0.966265	−0.483132	+	0.875547i	$0.660501\pi$
$578$	0	0
$579$	48.7188	2.02468
$580$	0	0
$581$	−9.18971	−0.381253
$582$	0	0
$583$	10.4683	0.433552
$584$	0	0
$585$	−5.13643	−0.212365
$586$	0	0
$587$	−31.4570	−1.29837	−0.649184	−	0.760632i	$-0.724889\pi$
$587$	−31.4570	−1.29837	−0.649184	+	0.760632i	$0.724889\pi$
$588$	0	0
$589$	30.0462	1.23803
$590$	0	0
$591$	5.39146	0.221775
$592$	0	0
$593$	25.5103	1.04758	0.523791	−	0.851847i	$-0.324517\pi$
$593$	25.5103	1.04758	0.523791	+	0.851847i	$0.324517\pi$
$594$	0	0
$595$	0.617595	0.0253189
$596$	0	0
$597$	58.8823	2.40989
$598$	0	0
$599$	−14.3951	−0.588167	−0.294084	−	0.955780i	$-0.595014\pi$
$599$	−14.3951	−0.588167	−0.294084	+	0.955780i	$0.595014\pi$
$600$	0	0
$601$	22.7514	0.928048	0.464024	−	0.885823i	$-0.346405\pi$
$601$	22.7514	0.928048	0.464024	+	0.885823i	$0.346405\pi$
$602$	0	0
$603$	50.9480	2.07476
$604$	0	0
$605$	−2.91134	−0.118363
$606$	0	0
$607$	4.75246	0.192897	0.0964483	−	0.995338i	$-0.469252\pi$
$607$	4.75246	0.192897	0.0964483	+	0.995338i	$0.469252\pi$
$608$	0	0
$609$	9.57317	0.387924
$610$	0	0
$611$	1.64740	0.0666465
$612$	0	0
$613$	19.6437	0.793401	0.396701	−	0.917948i	$-0.370155\pi$
$613$	19.6437	0.793401	0.396701	+	0.917948i	$0.370155\pi$
$614$	0	0
$615$	2.85826	0.115256
$616$	0	0
$617$	−29.7864	−1.19916	−0.599578	−	0.800317i	$-0.704665\pi$
$617$	−29.7864	−1.19916	−0.599578	+	0.800317i	$0.704665\pi$
$618$	0	0
$619$	−24.8362	−0.998253	−0.499127	−	0.866529i	$-0.666346\pi$
$619$	−24.8362	−0.998253	−0.499127	+	0.866529i	$0.666346\pi$
$620$	0	0
$621$	48.3559	1.94045
$622$	0	0
$623$	14.3065	0.573176
$624$	0	0
$625$	23.6927	0.947706
$626$	0	0
$627$	−20.7329	−0.827994
$628$	0	0
$629$	−4.37774	−0.174552
$630$	0	0
$631$	−6.64629	−0.264585	−0.132292	−	0.991211i	$-0.542234\pi$
$631$	−6.64629	−0.264585	−0.132292	+	0.991211i	$0.542234\pi$
$632$	0	0
$633$	78.1217	3.10506
$634$	0	0
$635$	2.20493	0.0875002
$636$	0	0
$637$	14.4020	0.570627
$638$	0	0
$639$	−64.0020	−2.53188
$640$	0	0
$641$	−18.3035	−0.722945	−0.361473	−	0.932383i	$-0.617726\pi$
$641$	−18.3035	−0.722945	−0.361473	+	0.932383i	$0.617726\pi$
$642$	0	0
$643$	42.6824	1.68323	0.841615	−	0.540078i	$-0.181605\pi$
$643$	42.6824	1.68323	0.841615	+	0.540078i	$0.181605\pi$
$644$	0	0
$645$	−3.11239	−0.122550
$646$	0	0
$647$	−30.1951	−1.18709	−0.593546	−	0.804800i	$-0.702273\pi$
$647$	−30.1951	−1.18709	−0.593546	+	0.804800i	$0.702273\pi$
$648$	0	0
$649$	8.91272	0.349855
$650$	0	0
$651$	−20.4856	−0.802894
$652$	0	0
$653$	−0.771846	−0.0302047	−0.0151023	−	0.999886i	$-0.504807\pi$
$653$	−0.771846	−0.0302047	−0.0151023	+	0.999886i	$0.504807\pi$
$654$	0	0
$655$	−0.930665	−0.0363641
$656$	0	0
$657$	11.7302	0.457640
$658$	0	0
$659$	−31.7491	−1.23677	−0.618386	−	0.785875i	$-0.712213\pi$
$659$	−31.7491	−1.23677	−0.618386	+	0.785875i	$0.712213\pi$
$660$	0	0
$661$	−11.2498	−0.437566	−0.218783	−	0.975774i	$-0.570209\pi$
$661$	−11.2498	−0.437566	−0.218783	+	0.975774i	$0.570209\pi$
$662$	0	0
$663$	−12.4503	−0.483530
$664$	0	0
$665$	2.88672	0.111942
$666$	0	0
$667$	−13.0067	−0.503620
$668$	0	0
$669$	−28.1911	−1.08993
$670$	0	0
$671$	−6.54285	−0.252584
$672$	0	0
$673$	38.0896	1.46825	0.734123	−	0.679016i	$-0.237593\pi$
$673$	38.0896	1.46825	0.734123	+	0.679016i	$0.237593\pi$
$674$	0	0
$675$	39.4615	1.51887
$676$	0	0
$677$	−6.70050	−0.257521	−0.128760	−	0.991676i	$-0.541100\pi$
$677$	−6.70050	−0.257521	−0.128760	+	0.991676i	$0.541100\pi$
$678$	0	0
$679$	−0.357328	−0.0137130
$680$	0	0
$681$	−61.2987	−2.34897
$682$	0	0
$683$	−12.4423	−0.476091	−0.238045	−	0.971254i	$-0.576507\pi$
$683$	−12.4423	−0.476091	−0.238045	+	0.971254i	$0.576507\pi$
$684$	0	0
$685$	−3.61539	−0.138137
$686$	0	0
$687$	57.6810	2.20067
$688$	0	0
$689$	29.3830	1.11940
$690$	0	0
$691$	19.6632	0.748023	0.374012	−	0.927424i	$-0.377982\pi$
$691$	19.6632	0.748023	0.374012	+	0.927424i	$0.377982\pi$
$692$	0	0
$693$	9.27272	0.352242
$694$	0	0
$695$	−2.87854	−0.109189
$696$	0	0
$697$	4.54474	0.172144
$698$	0	0
$699$	26.0487	0.985252
$700$	0	0
$701$	12.0649	0.455686	0.227843	−	0.973698i	$-0.426833\pi$
$701$	12.0649	0.455686	0.227843	+	0.973698i	$0.426833\pi$
$702$	0	0
$703$	−20.4621	−0.771743
$704$	0	0
$705$	0.474970	0.0178884
$706$	0	0
$707$	−22.8351	−0.858802
$708$	0	0
$709$	35.1541	1.32024	0.660120	−	0.751160i	$-0.270505\pi$
$709$	35.1541	1.32024	0.660120	+	0.751160i	$0.270505\pi$
$710$	0	0
$711$	−56.0252	−2.10111
$712$	0	0
$713$	27.8329	1.04235
$714$	0	0
$715$	0.970155	0.0362818
$716$	0	0
$717$	−34.5834	−1.29154
$718$	0	0
$719$	8.27360	0.308553	0.154277	−	0.988028i	$-0.450695\pi$
$719$	8.27360	0.308553	0.154277	+	0.988028i	$0.450695\pi$
$720$	0	0
$721$	29.6414	1.10390
$722$	0	0
$723$	−32.5520	−1.21062
$724$	0	0
$725$	−10.6143	−0.394204
$726$	0	0
$727$	10.4091	0.386051	0.193026	−	0.981194i	$-0.438170\pi$
$727$	10.4091	0.386051	0.193026	+	0.981194i	$0.438170\pi$
$728$	0	0
$729$	−33.6347	−1.24573
$730$	0	0
$731$	−4.94881	−0.183038
$732$	0	0
$733$	−11.4218	−0.421874	−0.210937	−	0.977500i	$-0.567652\pi$
$733$	−11.4218	−0.421874	−0.210937	+	0.977500i	$0.567652\pi$
$734$	0	0
$735$	4.15231	0.153160
$736$	0	0
$737$	−9.62292	−0.354465
$738$	0	0
$739$	−1.64671	−0.0605753	−0.0302876	−	0.999541i	$-0.509642\pi$
$739$	−1.64671	−0.0605753	−0.0302876	+	0.999541i	$0.509642\pi$
$740$	0	0
$741$	−58.1944	−2.13782
$742$	0	0
$743$	−14.0207	−0.514368	−0.257184	−	0.966362i	$-0.582795\pi$
$743$	−14.0207	−0.514368	−0.257184	+	0.966362i	$0.582795\pi$
$744$	0	0
$745$	−1.97668	−0.0724198
$746$	0	0
$747$	35.0376	1.28196
$748$	0	0
$749$	−14.9107	−0.544824
$750$	0	0
$751$	30.3868	1.10883	0.554415	−	0.832240i	$-0.312942\pi$
$751$	30.3868	1.10883	0.554415	+	0.832240i	$0.312942\pi$
$752$	0	0
$753$	67.5017	2.45990
$754$	0	0
$755$	5.41825	0.197190
$756$	0	0
$757$	35.6066	1.29415	0.647073	−	0.762428i	$-0.275993\pi$
$757$	35.6066	1.29415	0.647073	+	0.762428i	$0.275993\pi$
$758$	0	0
$759$	−19.2057	−0.697122
$760$	0	0
$761$	19.5237	0.707735	0.353867	−	0.935296i	$-0.384866\pi$
$761$	19.5237	0.707735	0.353867	+	0.935296i	$0.384866\pi$
$762$	0	0
$763$	−2.75164	−0.0996159
$764$	0	0
$765$	−2.35470	−0.0851345
$766$	0	0
$767$	25.0167	0.903302
$768$	0	0
$769$	−22.7480	−0.820315	−0.410158	−	0.912015i	$-0.634526\pi$
$769$	−22.7480	−0.820315	−0.410158	+	0.912015i	$0.634526\pi$
$770$	0	0
$771$	−72.8535	−2.62375
$772$	0	0
$773$	−26.9261	−0.968466	−0.484233	−	0.874939i	$-0.660901\pi$
$773$	−26.9261	−0.968466	−0.484233	+	0.874939i	$0.660901\pi$
$774$	0	0
$775$	22.7134	0.815890
$776$	0	0
$777$	13.9511	0.500494
$778$	0	0
$779$	21.2427	0.761099
$780$	0	0
$781$	12.0885	0.432561
$782$	0	0
$783$	−17.3575	−0.620307
$784$	0	0
$785$	−5.47042	−0.195248
$786$	0	0
$787$	14.0145	0.499563	0.249781	−	0.968302i	$-0.419641\pi$
$787$	14.0145	0.499563	0.249781	+	0.968302i	$0.419641\pi$
$788$	0	0
$789$	−30.1545	−1.07353
$790$	0	0
$791$	−2.57050	−0.0913965
$792$	0	0
$793$	−18.3648	−0.652154
$794$	0	0
$795$	8.47157	0.300456
$796$	0	0
$797$	−11.0189	−0.390310	−0.195155	−	0.980772i	$-0.562521\pi$
$797$	−11.0189	−0.390310	−0.195155	+	0.980772i	$0.562521\pi$
$798$	0	0
$799$	0.755219	0.0267177
$800$	0	0
$801$	−54.5462	−1.92730
$802$	0	0
$803$	−2.21557	−0.0781859
$804$	0	0
$805$	2.67407	0.0942487
$806$	0	0
$807$	58.8965	2.07325
$808$	0	0
$809$	−30.3627	−1.06749	−0.533747	−	0.845644i	$-0.679217\pi$
$809$	−30.3627	−1.06749	−0.533747	+	0.845644i	$0.679217\pi$
$810$	0	0
$811$	26.4051	0.927208	0.463604	−	0.886043i	$-0.346556\pi$
$811$	26.4051	0.927208	0.463604	+	0.886043i	$0.346556\pi$
$812$	0	0
$813$	−35.3417	−1.23949
$814$	0	0
$815$	−0.902468	−0.0316121
$816$	0	0
$817$	−23.1314	−0.809264
$818$	0	0
$819$	26.0272	0.909464
$820$	0	0
$821$	24.9355	0.870256	0.435128	−	0.900369i	$-0.356703\pi$
$821$	24.9355	0.870256	0.435128	+	0.900369i	$0.356703\pi$
$822$	0	0
$823$	23.6557	0.824587	0.412294	−	0.911051i	$-0.364728\pi$
$823$	23.6557	0.824587	0.412294	+	0.911051i	$0.364728\pi$
$824$	0	0
$825$	−15.6730	−0.545666
$826$	0	0
$827$	−16.4449	−0.571844	−0.285922	−	0.958253i	$-0.592300\pi$
$827$	−16.4449	−0.571844	−0.285922	+	0.958253i	$0.592300\pi$
$828$	0	0
$829$	−54.8297	−1.90431	−0.952157	−	0.305608i	$-0.901140\pi$
$829$	−54.8297	−1.90431	−0.952157	+	0.305608i	$0.901140\pi$
$830$	0	0
$831$	−3.42820	−0.118923
$832$	0	0
$833$	6.60232	0.228757
$834$	0	0
$835$	−5.52262	−0.191118
$836$	0	0
$837$	37.1433	1.28386
$838$	0	0
$839$	−32.8538	−1.13424	−0.567119	−	0.823636i	$-0.691942\pi$
$839$	−32.8538	−1.13424	−0.567119	+	0.823636i	$0.691942\pi$
$840$	0	0
$841$	−24.3312	−0.839007
$842$	0	0
$843$	−65.6099	−2.25973
$844$	0	0
$845$	−1.12607	−0.0387379
$846$	0	0
$847$	14.7523	0.506893
$848$	0	0
$849$	−8.83576	−0.303242
$850$	0	0
$851$	−18.9548	−0.649762
$852$	0	0
$853$	30.6542	1.04958	0.524790	−	0.851231i	$-0.324144\pi$
$853$	30.6542	1.04958	0.524790	+	0.851231i	$0.324144\pi$
$854$	0	0
$855$	−11.0062	−0.376404
$856$	0	0
$857$	−25.5261	−0.871954	−0.435977	−	0.899958i	$-0.643597\pi$
$857$	−25.5261	−0.871954	−0.435977	+	0.899958i	$0.643597\pi$
$858$	0	0
$859$	25.2024	0.859895	0.429948	−	0.902854i	$-0.358532\pi$
$859$	25.2024	0.859895	0.429948	+	0.902854i	$0.358532\pi$
$860$	0	0
$861$	−14.4833	−0.493590
$862$	0	0
$863$	−34.7274	−1.18213	−0.591066	−	0.806623i	$-0.701293\pi$
$863$	−34.7274	−1.18213	−0.591066	+	0.806623i	$0.701293\pi$
$864$	0	0
$865$	4.13699	0.140662
$866$	0	0
$867$	44.4937	1.51108
$868$	0	0
$869$	10.5819	0.358966
$870$	0	0
$871$	−27.0101	−0.915204
$872$	0	0
$873$	1.36238	0.0461097
$874$	0	0
$875$	4.40337	0.148861
$876$	0	0
$877$	−13.3314	−0.450170	−0.225085	−	0.974339i	$-0.572266\pi$
$877$	−13.3314	−0.450170	−0.225085	+	0.974339i	$0.572266\pi$
$878$	0	0
$879$	−26.0647	−0.879141
$880$	0	0
$881$	−9.19968	−0.309945	−0.154973	−	0.987919i	$-0.549529\pi$
$881$	−9.19968	−0.309945	−0.154973	+	0.987919i	$0.549529\pi$
$882$	0	0
$883$	37.2819	1.25464	0.627318	−	0.778764i	$-0.284153\pi$
$883$	37.2819	1.25464	0.627318	+	0.778764i	$0.284153\pi$
$884$	0	0
$885$	7.21271	0.242453
$886$	0	0
$887$	40.8188	1.37056	0.685280	−	0.728280i	$-0.259680\pi$
$887$	40.8188	1.37056	0.685280	+	0.728280i	$0.259680\pi$
$888$	0	0
$889$	−11.1728	−0.374723
$890$	0	0
$891$	−7.08882	−0.237485
$892$	0	0
$893$	3.52999	0.118127
$894$	0	0
$895$	−3.59389	−0.120131
$896$	0	0
$897$	−53.9075	−1.79992
$898$	0	0
$899$	−9.99073	−0.333209
$900$	0	0
$901$	13.4701	0.448754
$902$	0	0
$903$	15.7710	0.524827
$904$	0	0
$905$	−3.83613	−0.127517
$906$	0	0
$907$	−45.9131	−1.52452	−0.762259	−	0.647272i	$-0.775910\pi$
$907$	−45.9131	−1.52452	−0.762259	+	0.647272i	$0.775910\pi$
$908$	0	0
$909$	87.0633	2.88771
$910$	0	0
$911$	8.03072	0.266070	0.133035	−	0.991111i	$-0.457528\pi$
$911$	8.03072	0.266070	0.133035	+	0.991111i	$0.457528\pi$
$912$	0	0
$913$	−6.61780	−0.219017
$914$	0	0
$915$	−5.29487	−0.175043
$916$	0	0
$917$	4.71584	0.155731
$918$	0	0
$919$	0.446040	0.0147135	0.00735675	−	0.999973i	$-0.497658\pi$
$919$	0.446040	0.0147135	0.00735675	+	0.999973i	$0.497658\pi$
$920$	0	0
$921$	−49.8397	−1.64227
$922$	0	0
$923$	33.9307	1.11684
$924$	0	0
$925$	−15.4683	−0.508595
$926$	0	0
$927$	−113.014	−3.71185
$928$	0	0
$929$	−12.9374	−0.424461	−0.212230	−	0.977220i	$-0.568073\pi$
$929$	−12.9374	−0.424461	−0.212230	+	0.977220i	$0.568073\pi$
$930$	0	0
$931$	30.8601	1.01140
$932$	0	0
$933$	−19.1739	−0.627724
$934$	0	0
$935$	0.444750	0.0145449
$936$	0	0
$937$	−7.03852	−0.229938	−0.114969	−	0.993369i	$-0.536677\pi$
$937$	−7.03852	−0.229938	−0.114969	+	0.993369i	$0.536677\pi$
$938$	0	0
$939$	−8.82374	−0.287952
$940$	0	0
$941$	−31.5187	−1.02748	−0.513740	−	0.857946i	$-0.671740\pi$
$941$	−31.5187	−1.02748	−0.513740	+	0.857946i	$0.671740\pi$
$942$	0	0
$943$	19.6779	0.640800
$944$	0	0
$945$	3.56858	0.116086
$946$	0	0
$947$	−4.74004	−0.154031	−0.0770154	−	0.997030i	$-0.524539\pi$
$947$	−4.74004	−0.154031	−0.0770154	+	0.997030i	$0.524539\pi$
$948$	0	0
$949$	−6.21880	−0.201871
$950$	0	0
$951$	−28.3519	−0.919374
$952$	0	0
$953$	33.4794	1.08450	0.542252	−	0.840216i	$-0.317572\pi$
$953$	33.4794	1.08450	0.542252	+	0.840216i	$0.317572\pi$
$954$	0	0
$955$	5.99303	0.193930
$956$	0	0
$957$	6.89395	0.222850
$958$	0	0
$959$	18.3198	0.591577
$960$	0	0
$961$	−9.62087	−0.310351
$962$	0	0
$963$	56.8499	1.83196
$964$	0	0
$965$	−4.88486	−0.157249
$966$	0	0
$967$	−45.5868	−1.46597	−0.732986	−	0.680244i	$-0.761874\pi$
$967$	−45.5868	−1.46597	−0.732986	+	0.680244i	$0.761874\pi$
$968$	0	0
$969$	−26.6782	−0.857026
$970$	0	0
$971$	22.7805	0.731061	0.365530	−	0.930799i	$-0.380888\pi$
$971$	22.7805	0.731061	0.365530	+	0.930799i	$0.380888\pi$
$972$	0	0
$973$	14.5861	0.467608
$974$	0	0
$975$	−43.9920	−1.40887
$976$	0	0
$977$	53.1254	1.69963	0.849817	−	0.527079i	$-0.176713\pi$
$977$	53.1254	1.69963	0.849817	+	0.527079i	$0.176713\pi$
$978$	0	0
$979$	10.3025	0.329271
$980$	0	0
$981$	10.4912	0.334957
$982$	0	0
$983$	21.7755	0.694532	0.347266	−	0.937767i	$-0.387110\pi$
$983$	21.7755	0.694532	0.347266	+	0.937767i	$0.387110\pi$
$984$	0	0
$985$	−0.540582	−0.0172244
$986$	0	0
$987$	−2.40676	−0.0766079
$988$	0	0
$989$	−21.4274	−0.681352
$990$	0	0
$991$	−25.8165	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$991$	−25.8165	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$992$	0	0
$993$	−52.9849	−1.68142
$994$	0	0
$995$	−5.90392	−0.187167
$996$	0	0
$997$	24.7743	0.784611	0.392305	−	0.919835i	$-0.371678\pi$
$997$	24.7743	0.784611	0.392305	+	0.919835i	$0.371678\pi$
$998$	0	0
$999$	−25.2954	−0.800310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.2		33
4.3	odd	2		4024.2.a.f.1.32	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.32	✓	33	4.3	odd	2
8048.2.a.y.1.2		33	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)

\(f(q)\)	\(=\)	\(q-2.95302 q^{3} +0.296089 q^{5} -1.50033 q^{7} +5.72032 q^{9} +O(q^{10})\)	\(q-2.95302 q^{3} +0.296089 q^{5} -1.50033 q^{7} +5.72032 q^{9} -1.08044 q^{11} -3.03263 q^{13} -0.874355 q^{15} -1.39026 q^{17} -6.49823 q^{19} +4.43051 q^{21} -6.01954 q^{23} -4.91233 q^{25} -8.03314 q^{27} +2.16074 q^{29} -4.62376 q^{31} +3.19055 q^{33} -0.444232 q^{35} +3.14887 q^{37} +8.95542 q^{39} -3.26900 q^{41} +3.55964 q^{43} +1.69372 q^{45} -0.543223 q^{47} -4.74900 q^{49} +4.10545 q^{51} -9.68893 q^{53} -0.319905 q^{55} +19.1894 q^{57} -8.24918 q^{59} +6.05574 q^{61} -8.58238 q^{63} -0.897928 q^{65} +8.90650 q^{67} +17.7758 q^{69} -11.1885 q^{71} +2.05063 q^{73} +14.5062 q^{75} +1.62102 q^{77} -9.79408 q^{79} +6.56107 q^{81} +6.12511 q^{83} -0.411639 q^{85} -6.38070 q^{87} -9.53552 q^{89} +4.54996 q^{91} +13.6540 q^{93} -1.92405 q^{95} +0.238166 q^{97} -6.18045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

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