LMFDB - Embedded newform 5120.2.a.s.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5120,2,Mod(1,5120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,-8,0,4,0,8,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.8834058349$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.602887$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.01919 q^{3} -1.00000 q^{5} +4.02840 q^{7} -1.96126 q^{9} -0.914766 q^{11} -6.95074 q^{13} +1.01919 q^{15} +2.70862 q^{17} +0.619987 q^{19} -4.10569 q^{21} +3.60080 q^{23} +1.00000 q^{25} +5.05645 q^{27} +2.84146 q^{29} +4.30994 q^{31} +0.932316 q^{33} -4.02840 q^{35} +1.05212 q^{37} +7.08410 q^{39} -0.603979 q^{41} -7.11363 q^{43} +1.96126 q^{45} -10.8177 q^{47} +9.22800 q^{49} -2.76059 q^{51} +5.76178 q^{53} +0.914766 q^{55} -0.631882 q^{57} -1.73729 q^{59} -9.88410 q^{61} -7.90074 q^{63} +6.95074 q^{65} +7.41357 q^{67} -3.66988 q^{69} -13.7940 q^{71} -1.30876 q^{73} -1.01919 q^{75} -3.68504 q^{77} -0.611127 q^{79} +0.730326 q^{81} -1.83100 q^{83} -2.70862 q^{85} -2.89597 q^{87} +10.9236 q^{89} -28.0004 q^{91} -4.39263 q^{93} -0.619987 q^{95} -12.7571 q^{97} +1.79409 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 8 q^{5} + 4 q^{7} + 8 q^{9} - 8 q^{11} + 4 q^{15} - 16 q^{19} + 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} - 8 q^{45} + 20 q^{47} + 8 q^{49} - 24 q^{51} + 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^5 + 4 * q^7 + 8 * q^9 - 8 * q^11 + 4 * q^15 - 16 * q^19 + 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 - 8 * q^45 + 20 * q^47 + 8 * q^49 - 24 * q^51 + 8 * q^55 - 16 * q^59 + 20 * q^63 - 36 * q^67 + 16 * q^69 - 8 * q^71 - 4 * q^75 - 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 + 16 * q^95 - 40 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.01919	−0.588427	−0.294214	−	0.955740i	$-0.595058\pi$
$3$	−1.01919	−0.588427	−0.294214	+	0.955740i	$0.595058\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.02840	1.52259	0.761296	−	0.648405i	$-0.224563\pi$
$7$	4.02840	1.52259	0.761296	+	0.648405i	$0.224563\pi$
$8$	0	0
$9$	−1.96126	−0.653754
$10$	0	0
$11$	−0.914766	−0.275812	−0.137906	−	0.990445i	$-0.544037\pi$
$11$	−0.914766	−0.275812	−0.137906	+	0.990445i	$0.544037\pi$
$12$	0	0
$13$	−6.95074	−1.92779	−0.963895	−	0.266283i	$-0.914204\pi$
$13$	−6.95074	−1.92779	−0.963895	+	0.266283i	$0.914204\pi$
$14$	0	0
$15$	1.01919	0.263153
$16$	0	0
$17$	2.70862	0.656938	0.328469	−	0.944515i	$-0.393467\pi$
$17$	2.70862	0.656938	0.328469	+	0.944515i	$0.393467\pi$
$18$	0	0
$19$	0.619987	0.142235	0.0711174	−	0.997468i	$-0.477344\pi$
$19$	0.619987	0.142235	0.0711174	+	0.997468i	$0.477344\pi$
$20$	0	0
$21$	−4.10569	−0.895934
$22$	0	0
$23$	3.60080	0.750819	0.375410	−	0.926859i	$-0.377502\pi$
$23$	3.60080	0.750819	0.375410	+	0.926859i	$0.377502\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.05645	0.973113
$28$	0	0
$29$	2.84146	0.527645	0.263823	−	0.964571i	$-0.415017\pi$
$29$	2.84146	0.527645	0.263823	+	0.964571i	$0.415017\pi$
$30$	0	0
$31$	4.30994	0.774087	0.387044	−	0.922061i	$-0.373496\pi$
$31$	4.30994	0.774087	0.387044	+	0.922061i	$0.373496\pi$
$32$	0	0
$33$	0.932316	0.162295
$34$	0	0
$35$	−4.02840	−0.680924
$36$	0	0
$37$	1.05212	0.172967	0.0864837	−	0.996253i	$-0.472437\pi$
$37$	1.05212	0.172967	0.0864837	+	0.996253i	$0.472437\pi$
$38$	0	0
$39$	7.08410	1.13436
$40$	0	0
$41$	−0.603979	−0.0943256	−0.0471628	−	0.998887i	$-0.515018\pi$
$41$	−0.603979	−0.0943256	−0.0471628	+	0.998887i	$0.515018\pi$
$42$	0	0
$43$	−7.11363	−1.08482	−0.542409	−	0.840114i	$-0.682488\pi$
$43$	−7.11363	−1.08482	−0.542409	+	0.840114i	$0.682488\pi$
$44$	0	0
$45$	1.96126	0.292367
$46$	0	0
$47$	−10.8177	−1.57793	−0.788963	−	0.614440i	$-0.789382\pi$
$47$	−10.8177	−1.57793	−0.788963	+	0.614440i	$0.789382\pi$
$48$	0	0
$49$	9.22800	1.31829
$50$	0	0
$51$	−2.76059	−0.386560
$52$	0	0
$53$	5.76178	0.791442	0.395721	−	0.918371i	$-0.370495\pi$
$53$	5.76178	0.791442	0.395721	+	0.918371i	$0.370495\pi$
$54$	0	0
$55$	0.914766	0.123347
$56$	0	0
$57$	−0.631882	−0.0836948
$58$	0	0
$59$	−1.73729	−0.226177	−0.113088	−	0.993585i	$-0.536074\pi$
$59$	−1.73729	−0.226177	−0.113088	+	0.993585i	$0.536074\pi$
$60$	0	0
$61$	−9.88410	−1.26553	−0.632765	−	0.774344i	$-0.718080\pi$
$61$	−9.88410	−1.26553	−0.632765	+	0.774344i	$0.718080\pi$
$62$	0	0
$63$	−7.90074	−0.995400
$64$	0	0
$65$	6.95074	0.862134
$66$	0	0
$67$	7.41357	0.905712	0.452856	−	0.891584i	$-0.350405\pi$
$67$	7.41357	0.905712	0.452856	+	0.891584i	$0.350405\pi$
$68$	0	0
$69$	−3.66988	−0.441802
$70$	0	0
$71$	−13.7940	−1.63704	−0.818522	−	0.574475i	$-0.805206\pi$
$71$	−13.7940	−1.63704	−0.818522	+	0.574475i	$0.805206\pi$
$72$	0	0
$73$	−1.30876	−0.153179	−0.0765895	−	0.997063i	$-0.524403\pi$
$73$	−1.30876	−0.153179	−0.0765895	+	0.997063i	$0.524403\pi$
$74$	0	0
$75$	−1.01919	−0.117685
$76$	0	0
$77$	−3.68504	−0.419950
$78$	0	0
$79$	−0.611127	−0.0687571	−0.0343786	−	0.999409i	$-0.510945\pi$
$79$	−0.611127	−0.0687571	−0.0343786	+	0.999409i	$0.510945\pi$
$80$	0	0
$81$	0.730326	0.0811473
$82$	0	0
$83$	−1.83100	−0.200978	−0.100489	−	0.994938i	$-0.532041\pi$
$83$	−1.83100	−0.200978	−0.100489	+	0.994938i	$0.532041\pi$
$84$	0	0
$85$	−2.70862	−0.293792
$86$	0	0
$87$	−2.89597	−0.310481
$88$	0	0
$89$	10.9236	1.15790	0.578950	−	0.815363i	$-0.303463\pi$
$89$	10.9236	1.15790	0.578950	+	0.815363i	$0.303463\pi$
$90$	0	0
$91$	−28.0004	−2.93524
$92$	0	0
$93$	−4.39263	−0.455494
$94$	0	0
$95$	−0.619987	−0.0636093
$96$	0	0
$97$	−12.7571	−1.29528	−0.647642	−	0.761945i	$-0.724245\pi$
$97$	−12.7571	−1.29528	−0.647642	+	0.761945i	$0.724245\pi$
$98$	0	0
$99$	1.79409	0.180313
$100$	0	0
$101$	12.1595	1.20991	0.604956	−	0.796259i	$-0.293191\pi$
$101$	12.1595	1.20991	0.604956	+	0.796259i	$0.293191\pi$
$102$	0	0
$103$	12.0328	1.18563	0.592815	−	0.805338i	$-0.298016\pi$
$103$	12.0328	1.18563	0.592815	+	0.805338i	$0.298016\pi$
$104$	0	0
$105$	4.10569	0.400674
$106$	0	0
$107$	−3.35605	−0.324442	−0.162221	−	0.986754i	$-0.551866\pi$
$107$	−3.35605	−0.324442	−0.162221	+	0.986754i	$0.551866\pi$
$108$	0	0
$109$	4.58882	0.439529	0.219765	−	0.975553i	$-0.429471\pi$
$109$	4.58882	0.439529	0.219765	+	0.975553i	$0.429471\pi$
$110$	0	0
$111$	−1.07230	−0.101779
$112$	0	0
$113$	−17.3173	−1.62907	−0.814536	−	0.580114i	$-0.803008\pi$
$113$	−17.3173	−1.62907	−0.814536	+	0.580114i	$0.803008\pi$
$114$	0	0
$115$	−3.60080	−0.335776
$116$	0	0
$117$	13.6322	1.26030
$118$	0	0
$119$	10.9114	1.00025
$120$	0	0
$121$	−10.1632	−0.923928
$122$	0	0
$123$	0.615566	0.0555038
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.5438	−1.37929	−0.689645	−	0.724147i	$-0.742234\pi$
$127$	−15.5438	−1.37929	−0.689645	+	0.724147i	$0.742234\pi$
$128$	0	0
$129$	7.25011	0.638337
$130$	0	0
$131$	−15.9482	−1.39340	−0.696698	−	0.717364i	$-0.745348\pi$
$131$	−15.9482	−1.39340	−0.696698	+	0.717364i	$0.745348\pi$
$132$	0	0
$133$	2.49756	0.216566
$134$	0	0
$135$	−5.05645	−0.435190
$136$	0	0
$137$	3.67273	0.313782	0.156891	−	0.987616i	$-0.449853\pi$
$137$	3.67273	0.313782	0.156891	+	0.987616i	$0.449853\pi$
$138$	0	0
$139$	−7.40414	−0.628011	−0.314006	−	0.949421i	$-0.601671\pi$
$139$	−7.40414	−0.628011	−0.314006	+	0.949421i	$0.601671\pi$
$140$	0	0
$141$	11.0253	0.928495
$142$	0	0
$143$	6.35830	0.531708
$144$	0	0
$145$	−2.84146	−0.235970
$146$	0	0
$147$	−9.40505	−0.775715
$148$	0	0
$149$	−4.65829	−0.381622	−0.190811	−	0.981627i	$-0.561112\pi$
$149$	−4.65829	−0.381622	−0.190811	+	0.981627i	$0.561112\pi$
$150$	0	0
$151$	6.93206	0.564123	0.282061	−	0.959396i	$-0.408982\pi$
$151$	6.93206	0.564123	0.282061	+	0.959396i	$0.408982\pi$
$152$	0	0
$153$	−5.31232	−0.429475
$154$	0	0
$155$	−4.30994	−0.346182
$156$	0	0
$157$	7.99926	0.638411	0.319205	−	0.947686i	$-0.396584\pi$
$157$	7.99926	0.638411	0.319205	+	0.947686i	$0.396584\pi$
$158$	0	0
$159$	−5.87233	−0.465706
$160$	0	0
$161$	14.5055	1.14319
$162$	0	0
$163$	−15.5204	−1.21565	−0.607824	−	0.794072i	$-0.707957\pi$
$163$	−15.5204	−1.21565	−0.607824	+	0.794072i	$0.707957\pi$
$164$	0	0
$165$	−0.932316	−0.0725807
$166$	0	0
$167$	11.7686	0.910685	0.455343	−	0.890316i	$-0.349517\pi$
$167$	11.7686	0.910685	0.455343	+	0.890316i	$0.349517\pi$
$168$	0	0
$169$	35.3128	2.71637
$170$	0	0
$171$	−1.21596	−0.0929865
$172$	0	0
$173$	1.98309	0.150771	0.0753856	−	0.997154i	$-0.475981\pi$
$173$	1.98309	0.150771	0.0753856	+	0.997154i	$0.475981\pi$
$174$	0	0
$175$	4.02840	0.304518
$176$	0	0
$177$	1.77063	0.133088
$178$	0	0
$179$	−13.6632	−1.02123	−0.510616	−	0.859809i	$-0.670583\pi$
$179$	−13.6632	−1.02123	−0.510616	+	0.859809i	$0.670583\pi$
$180$	0	0
$181$	0.416973	0.0309933	0.0154967	−	0.999880i	$-0.495067\pi$
$181$	0.416973	0.0309933	0.0154967	+	0.999880i	$0.495067\pi$
$182$	0	0
$183$	10.0737	0.744672
$184$	0	0
$185$	−1.05212	−0.0773533
$186$	0	0
$187$	−2.47776	−0.181192
$188$	0	0
$189$	20.3694	1.48165
$190$	0	0
$191$	−16.9352	−1.22539	−0.612694	−	0.790320i	$-0.709914\pi$
$191$	−16.9352	−1.22539	−0.612694	+	0.790320i	$0.709914\pi$
$192$	0	0
$193$	−16.5927	−1.19437	−0.597185	−	0.802103i	$-0.703714\pi$
$193$	−16.5927	−1.19437	−0.597185	+	0.802103i	$0.703714\pi$
$194$	0	0
$195$	−7.08410	−0.507303
$196$	0	0
$197$	3.37137	0.240200	0.120100	−	0.992762i	$-0.461678\pi$
$197$	3.37137	0.240200	0.120100	+	0.992762i	$0.461678\pi$
$198$	0	0
$199$	10.1411	0.718883	0.359442	−	0.933168i	$-0.382967\pi$
$199$	10.1411	0.718883	0.359442	+	0.933168i	$0.382967\pi$
$200$	0	0
$201$	−7.55581	−0.532946
$202$	0	0
$203$	11.4465	0.803389
$204$	0	0
$205$	0.603979	0.0421837
$206$	0	0
$207$	−7.06211	−0.490851
$208$	0	0
$209$	−0.567143	−0.0392301
$210$	0	0
$211$	3.97636	0.273744	0.136872	−	0.990589i	$-0.456295\pi$
$211$	3.97636	0.273744	0.136872	+	0.990589i	$0.456295\pi$
$212$	0	0
$213$	14.0586	0.963281
$214$	0	0
$215$	7.11363	0.485146
$216$	0	0
$217$	17.3621	1.17862
$218$	0	0
$219$	1.33387	0.0901347
$220$	0	0
$221$	−18.8270	−1.26644
$222$	0	0
$223$	14.0502	0.940871	0.470436	−	0.882434i	$-0.344097\pi$
$223$	14.0502	0.940871	0.470436	+	0.882434i	$0.344097\pi$
$224$	0	0
$225$	−1.96126	−0.130751
$226$	0	0
$227$	−18.8790	−1.25305	−0.626523	−	0.779403i	$-0.715522\pi$
$227$	−18.8790	−1.25305	−0.626523	+	0.779403i	$0.715522\pi$
$228$	0	0
$229$	−12.4251	−0.821075	−0.410538	−	0.911844i	$-0.634659\pi$
$229$	−12.4251	−0.821075	−0.410538	+	0.911844i	$0.634659\pi$
$230$	0	0
$231$	3.75574	0.247110
$232$	0	0
$233$	−15.1472	−0.992329	−0.496165	−	0.868229i	$-0.665259\pi$
$233$	−15.1472	−0.992329	−0.496165	+	0.868229i	$0.665259\pi$
$234$	0	0
$235$	10.8177	0.705670
$236$	0	0
$237$	0.622852	0.0404586
$238$	0	0
$239$	−17.9151	−1.15883	−0.579414	−	0.815033i	$-0.696719\pi$
$239$	−17.9151	−1.15883	−0.579414	+	0.815033i	$0.696719\pi$
$240$	0	0
$241$	−25.6594	−1.65287	−0.826433	−	0.563035i	$-0.809634\pi$
$241$	−25.6594	−1.65287	−0.826433	+	0.563035i	$0.809634\pi$
$242$	0	0
$243$	−15.9137	−1.02086
$244$	0	0
$245$	−9.22800	−0.589556
$246$	0	0
$247$	−4.30937	−0.274199
$248$	0	0
$249$	1.86613	0.118261
$250$	0	0
$251$	8.41733	0.531297	0.265649	−	0.964070i	$-0.414414\pi$
$251$	8.41733	0.531297	0.265649	+	0.964070i	$0.414414\pi$
$252$	0	0
$253$	−3.29389	−0.207085
$254$	0	0
$255$	2.76059	0.172875
$256$	0	0
$257$	−4.17369	−0.260348	−0.130174	−	0.991491i	$-0.541554\pi$
$257$	−4.17369	−0.260348	−0.130174	+	0.991491i	$0.541554\pi$
$258$	0	0
$259$	4.23836	0.263359
$260$	0	0
$261$	−5.57284	−0.344950
$262$	0	0
$263$	9.14469	0.563885	0.281943	−	0.959431i	$-0.409021\pi$
$263$	9.14469	0.563885	0.281943	+	0.959431i	$0.409021\pi$
$264$	0	0
$265$	−5.76178	−0.353944
$266$	0	0
$267$	−11.1332	−0.681339
$268$	0	0
$269$	11.8798	0.724325	0.362162	−	0.932115i	$-0.382039\pi$
$269$	11.8798	0.724325	0.362162	+	0.932115i	$0.382039\pi$
$270$	0	0
$271$	−18.7794	−1.14077	−0.570383	−	0.821379i	$-0.693205\pi$
$271$	−18.7794	−1.14077	−0.570383	+	0.821379i	$0.693205\pi$
$272$	0	0
$273$	28.5376	1.72717
$274$	0	0
$275$	−0.914766	−0.0551625
$276$	0	0
$277$	−5.00868	−0.300942	−0.150471	−	0.988614i	$-0.548079\pi$
$277$	−5.00868	−0.300942	−0.150471	+	0.988614i	$0.548079\pi$
$278$	0	0
$279$	−8.45291	−0.506062
$280$	0	0
$281$	−2.31811	−0.138287	−0.0691433	−	0.997607i	$-0.522027\pi$
$281$	−2.31811	−0.138287	−0.0691433	+	0.997607i	$0.522027\pi$
$282$	0	0
$283$	2.30796	0.137194	0.0685970	−	0.997644i	$-0.478148\pi$
$283$	2.30796	0.137194	0.0685970	+	0.997644i	$0.478148\pi$
$284$	0	0
$285$	0.631882	0.0374295
$286$	0	0
$287$	−2.43307	−0.143619
$288$	0	0
$289$	−9.66335	−0.568433
$290$	0	0
$291$	13.0018	0.762180
$292$	0	0
$293$	16.3750	0.956636	0.478318	−	0.878187i	$-0.341247\pi$
$293$	16.3750	0.956636	0.478318	+	0.878187i	$0.341247\pi$
$294$	0	0
$295$	1.73729	0.101149
$296$	0	0
$297$	−4.62546	−0.268397
$298$	0	0
$299$	−25.0283	−1.44742
$300$	0	0
$301$	−28.6566	−1.65174
$302$	0	0
$303$	−12.3927	−0.711945
$304$	0	0
$305$	9.88410	0.565962
$306$	0	0
$307$	−16.5627	−0.945282	−0.472641	−	0.881255i	$-0.656699\pi$
$307$	−16.5627	−0.945282	−0.472641	+	0.881255i	$0.656699\pi$
$308$	0	0
$309$	−12.2637	−0.697657
$310$	0	0
$311$	11.2068	0.635477	0.317739	−	0.948178i	$-0.397077\pi$
$311$	11.2068	0.635477	0.317739	+	0.948178i	$0.397077\pi$
$312$	0	0
$313$	−7.50635	−0.424284	−0.212142	−	0.977239i	$-0.568044\pi$
$313$	−7.50635	−0.424284	−0.212142	+	0.977239i	$0.568044\pi$
$314$	0	0
$315$	7.90074	0.445156
$316$	0	0
$317$	23.0310	1.29355	0.646776	−	0.762680i	$-0.276117\pi$
$317$	23.0310	1.29355	0.646776	+	0.762680i	$0.276117\pi$
$318$	0	0
$319$	−2.59927	−0.145531
$320$	0	0
$321$	3.42044	0.190910
$322$	0	0
$323$	1.67931	0.0934394
$324$	0	0
$325$	−6.95074	−0.385558
$326$	0	0
$327$	−4.67686	−0.258631
$328$	0	0
$329$	−43.5781	−2.40254
$330$	0	0
$331$	−27.4139	−1.50680	−0.753402	−	0.657560i	$-0.771589\pi$
$331$	−27.4139	−1.50680	−0.753402	+	0.657560i	$0.771589\pi$
$332$	0	0
$333$	−2.06348	−0.113078
$334$	0	0
$335$	−7.41357	−0.405047
$336$	0	0
$337$	7.82991	0.426522	0.213261	−	0.976995i	$-0.431592\pi$
$337$	7.82991	0.426522	0.213261	+	0.976995i	$0.431592\pi$
$338$	0	0
$339$	17.6495	0.958590
$340$	0	0
$341$	−3.94258	−0.213503
$342$	0	0
$343$	8.97529	0.484620
$344$	0	0
$345$	3.66988	0.197580
$346$	0	0
$347$	−12.6113	−0.677009	−0.338505	−	0.940965i	$-0.609921\pi$
$347$	−12.6113	−0.677009	−0.338505	+	0.940965i	$0.609921\pi$
$348$	0	0
$349$	−9.46211	−0.506495	−0.253247	−	0.967402i	$-0.581499\pi$
$349$	−9.46211	−0.506495	−0.253247	+	0.967402i	$0.581499\pi$
$350$	0	0
$351$	−35.1461	−1.87596
$352$	0	0
$353$	−2.05215	−0.109225	−0.0546126	−	0.998508i	$-0.517392\pi$
$353$	−2.05215	−0.109225	−0.0546126	+	0.998508i	$0.517392\pi$
$354$	0	0
$355$	13.7940	0.732108
$356$	0	0
$357$	−11.1208	−0.588573
$358$	0	0
$359$	−9.52634	−0.502781	−0.251391	−	0.967886i	$-0.580888\pi$
$359$	−9.52634	−0.502781	−0.251391	+	0.967886i	$0.580888\pi$
$360$	0	0
$361$	−18.6156	−0.979769
$362$	0	0
$363$	10.3582	0.543664
$364$	0	0
$365$	1.30876	0.0685038
$366$	0	0
$367$	3.39736	0.177341	0.0886703	−	0.996061i	$-0.471738\pi$
$367$	3.39736	0.177341	0.0886703	+	0.996061i	$0.471738\pi$
$368$	0	0
$369$	1.18456	0.0616657
$370$	0	0
$371$	23.2108	1.20504
$372$	0	0
$373$	−31.8049	−1.64679	−0.823397	−	0.567465i	$-0.807924\pi$
$373$	−31.8049	−1.64679	−0.823397	+	0.567465i	$0.807924\pi$
$374$	0	0
$375$	1.01919	0.0526305
$376$	0	0
$377$	−19.7502	−1.01719
$378$	0	0
$379$	−21.1876	−1.08833	−0.544166	−	0.838978i	$-0.683154\pi$
$379$	−21.1876	−1.08833	−0.544166	+	0.838978i	$0.683154\pi$
$380$	0	0
$381$	15.8420	0.811612
$382$	0	0
$383$	26.1197	1.33466	0.667328	−	0.744764i	$-0.267438\pi$
$383$	26.1197	1.33466	0.667328	+	0.744764i	$0.267438\pi$
$384$	0	0
$385$	3.68504	0.187807
$386$	0	0
$387$	13.9517	0.709204
$388$	0	0
$389$	2.94714	0.149426	0.0747131	−	0.997205i	$-0.476196\pi$
$389$	2.94714	0.149426	0.0747131	+	0.997205i	$0.476196\pi$
$390$	0	0
$391$	9.75322	0.493241
$392$	0	0
$393$	16.2541	0.819912
$394$	0	0
$395$	0.611127	0.0307491
$396$	0	0
$397$	−2.02632	−0.101698	−0.0508490	−	0.998706i	$-0.516193\pi$
$397$	−2.02632	−0.101698	−0.0508490	+	0.998706i	$0.516193\pi$
$398$	0	0
$399$	−2.54547	−0.127433
$400$	0	0
$401$	29.9853	1.49739	0.748697	−	0.662912i	$-0.230680\pi$
$401$	29.9853	1.49739	0.748697	+	0.662912i	$0.230680\pi$
$402$	0	0
$403$	−29.9573	−1.49228
$404$	0	0
$405$	−0.730326	−0.0362902
$406$	0	0
$407$	−0.962443	−0.0477065
$408$	0	0
$409$	4.17833	0.206605	0.103302	−	0.994650i	$-0.467059\pi$
$409$	4.17833	0.206605	0.103302	+	0.994650i	$0.467059\pi$
$410$	0	0
$411$	−3.74319	−0.184638
$412$	0	0
$413$	−6.99852	−0.344375
$414$	0	0
$415$	1.83100	0.0898801
$416$	0	0
$417$	7.54620	0.369539
$418$	0	0
$419$	−34.6012	−1.69038	−0.845189	−	0.534468i	$-0.820512\pi$
$419$	−34.6012	−1.69038	−0.845189	+	0.534468i	$0.820512\pi$
$420$	0	0
$421$	−36.2063	−1.76458	−0.882292	−	0.470702i	$-0.844001\pi$
$421$	−36.2063	−1.76458	−0.882292	+	0.470702i	$0.844001\pi$
$422$	0	0
$423$	21.2164	1.03158
$424$	0	0
$425$	2.70862	0.131388
$426$	0	0
$427$	−39.8171	−1.92689
$428$	0	0
$429$	−6.48029	−0.312871
$430$	0	0
$431$	−17.6126	−0.848367	−0.424184	−	0.905576i	$-0.639439\pi$
$431$	−17.6126	−0.848367	−0.424184	+	0.905576i	$0.639439\pi$
$432$	0	0
$433$	−27.0568	−1.30027	−0.650133	−	0.759820i	$-0.725287\pi$
$433$	−27.0568	−1.30027	−0.650133	+	0.759820i	$0.725287\pi$
$434$	0	0
$435$	2.89597	0.138851
$436$	0	0
$437$	2.23245	0.106793
$438$	0	0
$439$	−22.9965	−1.09756	−0.548780	−	0.835967i	$-0.684908\pi$
$439$	−22.9965	−1.09756	−0.548780	+	0.835967i	$0.684908\pi$
$440$	0	0
$441$	−18.0985	−0.861834
$442$	0	0
$443$	19.4758	0.925325	0.462662	−	0.886535i	$-0.346894\pi$
$443$	19.4758	0.925325	0.462662	+	0.886535i	$0.346894\pi$
$444$	0	0
$445$	−10.9236	−0.517828
$446$	0	0
$447$	4.74766	0.224557
$448$	0	0
$449$	6.88838	0.325083	0.162541	−	0.986702i	$-0.448031\pi$
$449$	6.88838	0.325083	0.162541	+	0.986702i	$0.448031\pi$
$450$	0	0
$451$	0.552499	0.0260162
$452$	0	0
$453$	−7.06505	−0.331945
$454$	0	0
$455$	28.0004	1.31268
$456$	0	0
$457$	2.52622	0.118171	0.0590857	−	0.998253i	$-0.481181\pi$
$457$	2.52622	0.118171	0.0590857	+	0.998253i	$0.481181\pi$
$458$	0	0
$459$	13.6960	0.639275
$460$	0	0
$461$	−13.0603	−0.608279	−0.304139	−	0.952628i	$-0.598369\pi$
$461$	−13.0603	−0.608279	−0.304139	+	0.952628i	$0.598369\pi$
$462$	0	0
$463$	11.2676	0.523652	0.261826	−	0.965115i	$-0.415675\pi$
$463$	11.2676	0.523652	0.261826	+	0.965115i	$0.415675\pi$
$464$	0	0
$465$	4.39263	0.203703
$466$	0	0
$467$	36.5279	1.69031	0.845154	−	0.534522i	$-0.179508\pi$
$467$	36.5279	1.69031	0.845154	+	0.534522i	$0.179508\pi$
$468$	0	0
$469$	29.8648	1.37903
$470$	0	0
$471$	−8.15273	−0.375658
$472$	0	0
$473$	6.50731	0.299206
$474$	0	0
$475$	0.619987	0.0284470
$476$	0	0
$477$	−11.3004	−0.517408
$478$	0	0
$479$	−15.7261	−0.718545	−0.359273	−	0.933233i	$-0.616975\pi$
$479$	−15.7261	−0.718545	−0.359273	+	0.933233i	$0.616975\pi$
$480$	0	0
$481$	−7.31301	−0.333445
$482$	0	0
$483$	−14.7838	−0.672685
$484$	0	0
$485$	12.7571	0.579268
$486$	0	0
$487$	−35.3717	−1.60284	−0.801422	−	0.598100i	$-0.795923\pi$
$487$	−35.3717	−1.60284	−0.801422	+	0.598100i	$0.795923\pi$
$488$	0	0
$489$	15.8181	0.715320
$490$	0	0
$491$	11.2529	0.507838	0.253919	−	0.967225i	$-0.418280\pi$
$491$	11.2529	0.507838	0.253919	+	0.967225i	$0.418280\pi$
$492$	0	0
$493$	7.69644	0.346630
$494$	0	0
$495$	−1.79409	−0.0806385
$496$	0	0
$497$	−55.5677	−2.49255
$498$	0	0
$499$	−16.3856	−0.733520	−0.366760	−	0.930316i	$-0.619533\pi$
$499$	−16.3856	−0.733520	−0.366760	+	0.930316i	$0.619533\pi$
$500$	0	0
$501$	−11.9944	−0.535872
$502$	0	0
$503$	23.5051	1.04804	0.524020	−	0.851706i	$-0.324432\pi$
$503$	23.5051	1.04804	0.524020	+	0.851706i	$0.324432\pi$
$504$	0	0
$505$	−12.1595	−0.541089
$506$	0	0
$507$	−35.9903	−1.59839
$508$	0	0
$509$	−4.36117	−0.193305	−0.0966527	−	0.995318i	$-0.530814\pi$
$509$	−4.36117	−0.193305	−0.0966527	+	0.995318i	$0.530814\pi$
$510$	0	0
$511$	−5.27222	−0.233229
$512$	0	0
$513$	3.13493	0.138411
$514$	0	0
$515$	−12.0328	−0.530230
$516$	0	0
$517$	9.89568	0.435212
$518$	0	0
$519$	−2.02113	−0.0887178
$520$	0	0
$521$	11.5762	0.507161	0.253580	−	0.967314i	$-0.418392\pi$
$521$	11.5762	0.507161	0.253580	+	0.967314i	$0.418392\pi$
$522$	0	0
$523$	5.62716	0.246059	0.123029	−	0.992403i	$-0.460739\pi$
$523$	5.62716	0.246059	0.123029	+	0.992403i	$0.460739\pi$
$524$	0	0
$525$	−4.10569	−0.179187
$526$	0	0
$527$	11.6740	0.508527
$528$	0	0
$529$	−10.0342	−0.436271
$530$	0	0
$531$	3.40729	0.147864
$532$	0	0
$533$	4.19810	0.181840
$534$	0	0
$535$	3.35605	0.145095
$536$	0	0
$537$	13.9253	0.600921
$538$	0	0
$539$	−8.44146	−0.363600
$540$	0	0
$541$	24.3454	1.04669	0.523346	−	0.852120i	$-0.324684\pi$
$541$	24.3454	1.04669	0.523346	+	0.852120i	$0.324684\pi$
$542$	0	0
$543$	−0.424973	−0.0182373
$544$	0	0
$545$	−4.58882	−0.196564
$546$	0	0
$547$	−28.7948	−1.23118	−0.615588	−	0.788068i	$-0.711081\pi$
$547$	−28.7948	−1.23118	−0.615588	+	0.788068i	$0.711081\pi$
$548$	0	0
$549$	19.3853	0.827344
$550$	0	0
$551$	1.76167	0.0750495
$552$	0	0
$553$	−2.46186	−0.104689
$554$	0	0
$555$	1.07230	0.0455168
$556$	0	0
$557$	32.1068	1.36041	0.680205	−	0.733022i	$-0.261891\pi$
$557$	32.1068	1.36041	0.680205	+	0.733022i	$0.261891\pi$
$558$	0	0
$559$	49.4451	2.09130
$560$	0	0
$561$	2.52529	0.106618
$562$	0	0
$563$	−21.8005	−0.918781	−0.459391	−	0.888234i	$-0.651932\pi$
$563$	−21.8005	−0.918781	−0.459391	+	0.888234i	$0.651932\pi$
$564$	0	0
$565$	17.3173	0.728543
$566$	0	0
$567$	2.94204	0.123554
$568$	0	0
$569$	−22.6529	−0.949660	−0.474830	−	0.880078i	$-0.657490\pi$
$569$	−22.6529	−0.949660	−0.474830	+	0.880078i	$0.657490\pi$
$570$	0	0
$571$	19.0835	0.798621	0.399311	−	0.916816i	$-0.369250\pi$
$571$	19.0835	0.798621	0.399311	+	0.916816i	$0.369250\pi$
$572$	0	0
$573$	17.2601	0.721051
$574$	0	0
$575$	3.60080	0.150164
$576$	0	0
$577$	6.08684	0.253398	0.126699	−	0.991941i	$-0.459562\pi$
$577$	6.08684	0.253398	0.126699	+	0.991941i	$0.459562\pi$
$578$	0	0
$579$	16.9111	0.702800
$580$	0	0
$581$	−7.37599	−0.306007
$582$	0	0
$583$	−5.27068	−0.218289
$584$	0	0
$585$	−13.6322	−0.563623
$586$	0	0
$587$	30.3233	1.25158	0.625789	−	0.779993i	$-0.284777\pi$
$587$	30.3233	1.25158	0.625789	+	0.779993i	$0.284777\pi$
$588$	0	0
$589$	2.67210	0.110102
$590$	0	0
$591$	−3.43605	−0.141340
$592$	0	0
$593$	28.2005	1.15806	0.579028	−	0.815308i	$-0.303432\pi$
$593$	28.2005	1.15806	0.579028	+	0.815308i	$0.303432\pi$
$594$	0	0
$595$	−10.9114	−0.447325
$596$	0	0
$597$	−10.3357	−0.423010
$598$	0	0
$599$	38.0516	1.55475	0.777374	−	0.629039i	$-0.216551\pi$
$599$	38.0516	1.55475	0.777374	+	0.629039i	$0.216551\pi$
$600$	0	0
$601$	19.0716	0.777947	0.388974	−	0.921249i	$-0.372830\pi$
$601$	19.0716	0.777947	0.388974	+	0.921249i	$0.372830\pi$
$602$	0	0
$603$	−14.5400	−0.592113
$604$	0	0
$605$	10.1632	0.413193
$606$	0	0
$607$	5.73433	0.232749	0.116375	−	0.993205i	$-0.462873\pi$
$607$	5.73433	0.232749	0.116375	+	0.993205i	$0.462873\pi$
$608$	0	0
$609$	−11.6661	−0.472736
$610$	0	0
$611$	75.1912	3.04191
$612$	0	0
$613$	−7.59316	−0.306685	−0.153342	−	0.988173i	$-0.549004\pi$
$613$	−7.59316	−0.306685	−0.153342	+	0.988173i	$0.549004\pi$
$614$	0	0
$615$	−0.615566	−0.0248220
$616$	0	0
$617$	28.2915	1.13897	0.569487	−	0.822000i	$-0.307142\pi$
$617$	28.2915	1.13897	0.569487	+	0.822000i	$0.307142\pi$
$618$	0	0
$619$	26.8233	1.07812	0.539059	−	0.842268i	$-0.318780\pi$
$619$	26.8233	1.07812	0.539059	+	0.842268i	$0.318780\pi$
$620$	0	0
$621$	18.2073	0.730632
$622$	0	0
$623$	44.0046	1.76301
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.578024	0.0230841
$628$	0	0
$629$	2.84980	0.113629
$630$	0	0
$631$	−41.7662	−1.66269	−0.831344	−	0.555758i	$-0.812428\pi$
$631$	−41.7662	−1.66269	−0.831344	+	0.555758i	$0.812428\pi$
$632$	0	0
$633$	−4.05265	−0.161078
$634$	0	0
$635$	15.5438	0.616837
$636$	0	0
$637$	−64.1415	−2.54138
$638$	0	0
$639$	27.0536	1.07022
$640$	0	0
$641$	−2.85195	−0.112645	−0.0563227	−	0.998413i	$-0.517938\pi$
$641$	−2.85195	−0.112645	−0.0563227	+	0.998413i	$0.517938\pi$
$642$	0	0
$643$	−45.1023	−1.77866	−0.889330	−	0.457266i	$-0.848829\pi$
$643$	−45.1023	−1.77866	−0.889330	+	0.457266i	$0.848829\pi$
$644$	0	0
$645$	−7.25011	−0.285473
$646$	0	0
$647$	7.83402	0.307987	0.153994	−	0.988072i	$-0.450786\pi$
$647$	7.83402	0.307987	0.153994	+	0.988072i	$0.450786\pi$
$648$	0	0
$649$	1.58922	0.0623823
$650$	0	0
$651$	−17.6952	−0.693532
$652$	0	0
$653$	17.9353	0.701863	0.350932	−	0.936401i	$-0.385865\pi$
$653$	17.9353	0.701863	0.350932	+	0.936401i	$0.385865\pi$
$654$	0	0
$655$	15.9482	0.623146
$656$	0	0
$657$	2.56682	0.100141
$658$	0	0
$659$	−18.3415	−0.714485	−0.357243	−	0.934012i	$-0.616283\pi$
$659$	−18.3415	−0.714485	−0.357243	+	0.934012i	$0.616283\pi$
$660$	0	0
$661$	−9.69862	−0.377232	−0.188616	−	0.982051i	$-0.560400\pi$
$661$	−9.69862	−0.377232	−0.188616	+	0.982051i	$0.560400\pi$
$662$	0	0
$663$	19.1882	0.745206
$664$	0	0
$665$	−2.49756	−0.0968511
$666$	0	0
$667$	10.2315	0.396166
$668$	0	0
$669$	−14.3198	−0.553634
$670$	0	0
$671$	9.04164	0.349049
$672$	0	0
$673$	23.1277	0.891508	0.445754	−	0.895155i	$-0.352936\pi$
$673$	23.1277	0.891508	0.445754	+	0.895155i	$0.352936\pi$
$674$	0	0
$675$	5.05645	0.194623
$676$	0	0
$677$	28.6408	1.10076	0.550378	−	0.834916i	$-0.314484\pi$
$677$	28.6408	1.10076	0.550378	+	0.834916i	$0.314484\pi$
$678$	0	0
$679$	−51.3905	−1.97219
$680$	0	0
$681$	19.2412	0.737326
$682$	0	0
$683$	37.1878	1.42295	0.711475	−	0.702711i	$-0.248027\pi$
$683$	37.1878	1.42295	0.711475	+	0.702711i	$0.248027\pi$
$684$	0	0
$685$	−3.67273	−0.140328
$686$	0	0
$687$	12.6635	0.483143
$688$	0	0
$689$	−40.0487	−1.52573
$690$	0	0
$691$	6.94704	0.264278	0.132139	−	0.991231i	$-0.457815\pi$
$691$	6.94704	0.264278	0.132139	+	0.991231i	$0.457815\pi$
$692$	0	0
$693$	7.22733	0.274544
$694$	0	0
$695$	7.40414	0.280855
$696$	0	0
$697$	−1.63595	−0.0619661
$698$	0	0
$699$	15.4379	0.583913
$700$	0	0
$701$	−17.4794	−0.660189	−0.330094	−	0.943948i	$-0.607081\pi$
$701$	−17.4794	−0.660189	−0.330094	+	0.943948i	$0.607081\pi$
$702$	0	0
$703$	0.652300	0.0246020
$704$	0	0
$705$	−11.0253	−0.415235
$706$	0	0
$707$	48.9832	1.84220
$708$	0	0
$709$	37.6289	1.41318	0.706591	−	0.707622i	$-0.250232\pi$
$709$	37.6289	1.41318	0.706591	+	0.707622i	$0.250232\pi$
$710$	0	0
$711$	1.19858	0.0449502
$712$	0	0
$713$	15.5192	0.581200
$714$	0	0
$715$	−6.35830	−0.237787
$716$	0	0
$717$	18.2588	0.681886
$718$	0	0
$719$	−50.9765	−1.90110	−0.950551	−	0.310570i	$-0.899480\pi$
$719$	−50.9765	−1.90110	−0.950551	+	0.310570i	$0.899480\pi$
$720$	0	0
$721$	48.4731	1.80523
$722$	0	0
$723$	26.1517	0.972591
$724$	0	0
$725$	2.84146	0.105529
$726$	0	0
$727$	13.2824	0.492616	0.246308	−	0.969192i	$-0.420782\pi$
$727$	13.2824	0.492616	0.246308	+	0.969192i	$0.420782\pi$
$728$	0	0
$729$	14.0280	0.519556
$730$	0	0
$731$	−19.2682	−0.712659
$732$	0	0
$733$	30.1334	1.11300	0.556501	−	0.830847i	$-0.312143\pi$
$733$	30.1334	1.11300	0.556501	+	0.830847i	$0.312143\pi$
$734$	0	0
$735$	9.40505	0.346910
$736$	0	0
$737$	−6.78168	−0.249807
$738$	0	0
$739$	8.25676	0.303730	0.151865	−	0.988401i	$-0.451472\pi$
$739$	8.25676	0.303730	0.151865	+	0.988401i	$0.451472\pi$
$740$	0	0
$741$	4.39205	0.161346
$742$	0	0
$743$	3.25778	0.119516	0.0597582	−	0.998213i	$-0.480967\pi$
$743$	3.25778	0.119516	0.0597582	+	0.998213i	$0.480967\pi$
$744$	0	0
$745$	4.65829	0.170667
$746$	0	0
$747$	3.59106	0.131390
$748$	0	0
$749$	−13.5195	−0.493993
$750$	0	0
$751$	−20.7322	−0.756530	−0.378265	−	0.925697i	$-0.623479\pi$
$751$	−20.7322	−0.756530	−0.378265	+	0.925697i	$0.623479\pi$
$752$	0	0
$753$	−8.57882	−0.312630
$754$	0	0
$755$	−6.93206	−0.252283
$756$	0	0
$757$	32.9904	1.19906	0.599529	−	0.800353i	$-0.295355\pi$
$757$	32.9904	1.19906	0.599529	+	0.800353i	$0.295355\pi$
$758$	0	0
$759$	3.35709	0.121854
$760$	0	0
$761$	24.0242	0.870878	0.435439	−	0.900218i	$-0.356593\pi$
$761$	24.0242	0.870878	0.435439	+	0.900218i	$0.356593\pi$
$762$	0	0
$763$	18.4856	0.669224
$764$	0	0
$765$	5.31232	0.192067
$766$	0	0
$767$	12.0755	0.436021
$768$	0	0
$769$	−2.70862	−0.0976754	−0.0488377	−	0.998807i	$-0.515552\pi$
$769$	−2.70862	−0.0976754	−0.0488377	+	0.998807i	$0.515552\pi$
$770$	0	0
$771$	4.25377	0.153196
$772$	0	0
$773$	−32.4524	−1.16723	−0.583616	−	0.812030i	$-0.698363\pi$
$773$	−32.4524	−1.16723	−0.583616	+	0.812030i	$0.698363\pi$
$774$	0	0
$775$	4.30994	0.154817
$776$	0	0
$777$	−4.31967	−0.154967
$778$	0	0
$779$	−0.374459	−0.0134164
$780$	0	0
$781$	12.6183	0.451517
$782$	0	0
$783$	14.3677	0.513459
$784$	0	0
$785$	−7.99926	−0.285506
$786$	0	0
$787$	19.8827	0.708743	0.354371	−	0.935105i	$-0.384695\pi$
$787$	19.8827	0.708743	0.354371	+	0.935105i	$0.384695\pi$
$788$	0	0
$789$	−9.32013	−0.331805
$790$	0	0
$791$	−69.7609	−2.48041
$792$	0	0
$793$	68.7019	2.43967
$794$	0	0
$795$	5.87233	0.208270
$796$	0	0
$797$	50.0997	1.77462	0.887311	−	0.461171i	$-0.152571\pi$
$797$	50.0997	1.77462	0.887311	+	0.461171i	$0.152571\pi$
$798$	0	0
$799$	−29.3011	−1.03660
$800$	0	0
$801$	−21.4240	−0.756981
$802$	0	0
$803$	1.19721	0.0422487
$804$	0	0
$805$	−14.5055	−0.511251
$806$	0	0
$807$	−12.1077	−0.426212
$808$	0	0
$809$	−16.9217	−0.594935	−0.297467	−	0.954732i	$-0.596142\pi$
$809$	−16.9217	−0.594935	−0.297467	+	0.954732i	$0.596142\pi$
$810$	0	0
$811$	−28.8881	−1.01440	−0.507200	−	0.861829i	$-0.669319\pi$
$811$	−28.8881	−1.01440	−0.507200	+	0.861829i	$0.669319\pi$
$812$	0	0
$813$	19.1397	0.671257
$814$	0	0
$815$	15.5204	0.543655
$816$	0	0
$817$	−4.41036	−0.154299
$818$	0	0
$819$	54.9160	1.91892
$820$	0	0
$821$	45.9002	1.60193	0.800964	−	0.598713i	$-0.204321\pi$
$821$	45.9002	1.60193	0.800964	+	0.598713i	$0.204321\pi$
$822$	0	0
$823$	−6.50705	−0.226821	−0.113411	−	0.993548i	$-0.536178\pi$
$823$	−6.50705	−0.226821	−0.113411	+	0.993548i	$0.536178\pi$
$824$	0	0
$825$	0.932316	0.0324591
$826$	0	0
$827$	−38.2584	−1.33038	−0.665188	−	0.746676i	$-0.731649\pi$
$827$	−38.2584	−1.33038	−0.665188	+	0.746676i	$0.731649\pi$
$828$	0	0
$829$	−12.0804	−0.419571	−0.209786	−	0.977747i	$-0.567277\pi$
$829$	−12.0804	−0.419571	−0.209786	+	0.977747i	$0.567277\pi$
$830$	0	0
$831$	5.10477	0.177083
$832$	0	0
$833$	24.9952	0.866032
$834$	0	0
$835$	−11.7686	−0.407271
$836$	0	0
$837$	21.7930	0.753275
$838$	0	0
$839$	24.4138	0.842860	0.421430	−	0.906861i	$-0.361528\pi$
$839$	24.4138	0.842860	0.421430	+	0.906861i	$0.361528\pi$
$840$	0	0
$841$	−20.9261	−0.721590
$842$	0	0
$843$	2.36258	0.0813716
$844$	0	0
$845$	−35.3128	−1.21480
$846$	0	0
$847$	−40.9414	−1.40676
$848$	0	0
$849$	−2.35224	−0.0807287
$850$	0	0
$851$	3.78847	0.129867
$852$	0	0
$853$	11.6428	0.398642	0.199321	−	0.979934i	$-0.436126\pi$
$853$	11.6428	0.398642	0.199321	+	0.979934i	$0.436126\pi$
$854$	0	0
$855$	1.21596	0.0415848
$856$	0	0
$857$	−4.73909	−0.161884	−0.0809421	−	0.996719i	$-0.525793\pi$
$857$	−4.73909	−0.161884	−0.0809421	+	0.996719i	$0.525793\pi$
$858$	0	0
$859$	13.6489	0.465693	0.232847	−	0.972513i	$-0.425196\pi$
$859$	13.6489	0.465693	0.232847	+	0.972513i	$0.425196\pi$
$860$	0	0
$861$	2.47975	0.0845096
$862$	0	0
$863$	−3.80368	−0.129479	−0.0647393	−	0.997902i	$-0.520622\pi$
$863$	−3.80368	−0.129479	−0.0647393	+	0.997902i	$0.520622\pi$
$864$	0	0
$865$	−1.98309	−0.0674269
$866$	0	0
$867$	9.84875	0.334481
$868$	0	0
$869$	0.559038	0.0189641
$870$	0	0
$871$	−51.5299	−1.74602
$872$	0	0
$873$	25.0199	0.846796
$874$	0	0
$875$	−4.02840	−0.136185
$876$	0	0
$877$	−3.37880	−0.114094	−0.0570469	−	0.998371i	$-0.518168\pi$
$877$	−3.37880	−0.114094	−0.0570469	+	0.998371i	$0.518168\pi$
$878$	0	0
$879$	−16.6891	−0.562911
$880$	0	0
$881$	23.6195	0.795762	0.397881	−	0.917437i	$-0.369746\pi$
$881$	23.6195	0.795762	0.397881	+	0.917437i	$0.369746\pi$
$882$	0	0
$883$	13.6437	0.459146	0.229573	−	0.973291i	$-0.426267\pi$
$883$	13.6437	0.459146	0.229573	+	0.973291i	$0.426267\pi$
$884$	0	0
$885$	−1.77063	−0.0595189
$886$	0	0
$887$	−8.91140	−0.299215	−0.149608	−	0.988745i	$-0.547801\pi$
$887$	−8.91140	−0.299215	−0.149608	+	0.988745i	$0.547801\pi$
$888$	0	0
$889$	−62.6167	−2.10010
$890$	0	0
$891$	−0.668077	−0.0223814
$892$	0	0
$893$	−6.70685	−0.224436
$894$	0	0
$895$	13.6632	0.456709
$896$	0	0
$897$	25.5084	0.851702
$898$	0	0
$899$	12.2465	0.408444
$900$	0	0
$901$	15.6065	0.519928
$902$	0	0
$903$	29.2064	0.971927
$904$	0	0
$905$	−0.416973	−0.0138606
$906$	0	0
$907$	33.2162	1.10293	0.551463	−	0.834200i	$-0.314070\pi$
$907$	33.2162	1.10293	0.551463	+	0.834200i	$0.314070\pi$
$908$	0	0
$909$	−23.8479	−0.790984
$910$	0	0
$911$	1.77171	0.0586993	0.0293497	−	0.999569i	$-0.490656\pi$
$911$	1.77171	0.0586993	0.0293497	+	0.999569i	$0.490656\pi$
$912$	0	0
$913$	1.67493	0.0554322
$914$	0	0
$915$	−10.0737	−0.333027
$916$	0	0
$917$	−64.2455	−2.12157
$918$	0	0
$919$	−46.2001	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$919$	−46.2001	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$920$	0	0
$921$	16.8804	0.556229
$922$	0	0
$923$	95.8784	3.15588
$924$	0	0
$925$	1.05212	0.0345935
$926$	0	0
$927$	−23.5995	−0.775110
$928$	0	0
$929$	−35.5011	−1.16475	−0.582376	−	0.812920i	$-0.697877\pi$
$929$	−35.5011	−1.16475	−0.582376	+	0.812920i	$0.697877\pi$
$930$	0	0
$931$	5.72124	0.187506
$932$	0	0
$933$	−11.4218	−0.373932
$934$	0	0
$935$	2.47776	0.0810313
$936$	0	0
$937$	10.0385	0.327945	0.163972	−	0.986465i	$-0.447569\pi$
$937$	10.0385	0.327945	0.163972	+	0.986465i	$0.447569\pi$
$938$	0	0
$939$	7.65036	0.249660
$940$	0	0
$941$	−59.8731	−1.95181	−0.975904	−	0.218200i	$-0.929981\pi$
$941$	−59.8731	−1.95181	−0.975904	+	0.218200i	$0.929981\pi$
$942$	0	0
$943$	−2.17481	−0.0708215
$944$	0	0
$945$	−20.3694	−0.662616
$946$	0	0
$947$	51.3903	1.66996	0.834980	−	0.550280i	$-0.185479\pi$
$947$	51.3903	1.66996	0.834980	+	0.550280i	$0.185479\pi$
$948$	0	0
$949$	9.09687	0.295297
$950$	0	0
$951$	−23.4729	−0.761161
$952$	0	0
$953$	−32.7338	−1.06035	−0.530176	−	0.847888i	$-0.677874\pi$
$953$	−32.7338	−1.06035	−0.530176	+	0.847888i	$0.677874\pi$
$954$	0	0
$955$	16.9352	0.548010
$956$	0	0
$957$	2.64914	0.0856344
$958$	0	0
$959$	14.7952	0.477762
$960$	0	0
$961$	−12.4244	−0.400789
$962$	0	0
$963$	6.58209	0.212105
$964$	0	0
$965$	16.5927	0.534139
$966$	0	0
$967$	−49.7169	−1.59879	−0.799394	−	0.600807i	$-0.794846\pi$
$967$	−49.7169	−1.59879	−0.799394	+	0.600807i	$0.794846\pi$
$968$	0	0
$969$	−1.71153	−0.0549823
$970$	0	0
$971$	−35.0636	−1.12524	−0.562622	−	0.826715i	$-0.690207\pi$
$971$	−35.0636	−1.12524	−0.562622	+	0.826715i	$0.690207\pi$
$972$	0	0
$973$	−29.8269	−0.956205
$974$	0	0
$975$	7.08410	0.226873
$976$	0	0
$977$	49.4546	1.58219	0.791096	−	0.611692i	$-0.209511\pi$
$977$	49.4546	1.58219	0.791096	+	0.611692i	$0.209511\pi$
$978$	0	0
$979$	−9.99253	−0.319363
$980$	0	0
$981$	−8.99988	−0.287344
$982$	0	0
$983$	−23.9656	−0.764383	−0.382191	−	0.924083i	$-0.624830\pi$
$983$	−23.9656	−0.764383	−0.382191	+	0.924083i	$0.624830\pi$
$984$	0	0
$985$	−3.37137	−0.107421
$986$	0	0
$987$	44.4142	1.41372
$988$	0	0
$989$	−25.6148	−0.814503
$990$	0	0
$991$	28.8345	0.915957	0.457978	−	0.888963i	$-0.348574\pi$
$991$	28.8345	0.915957	0.457978	+	0.888963i	$0.348574\pi$
$992$	0	0
$993$	27.9399	0.886645
$994$	0	0
$995$	−10.1411	−0.321494
$996$	0	0
$997$	−8.53071	−0.270170	−0.135085	−	0.990834i	$-0.543131\pi$
$997$	−8.53071	−0.270170	−0.135085	+	0.990834i	$0.543131\pi$
$998$	0	0
$999$	5.31998	0.168317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.s.1.4		8
4.3	odd	2		5120.2.a.u.1.5		8
8.3	odd	2		5120.2.a.t.1.4		8
8.5	even	2		5120.2.a.v.1.5		8
32.3	odd	8		320.2.l.a.81.6		16
32.5	even	8		640.2.l.b.481.6		16
32.11	odd	8		320.2.l.a.241.6		16
32.13	even	8		640.2.l.b.161.6		16
32.19	odd	8		640.2.l.a.161.3		16
32.21	even	8		80.2.l.a.21.1	✓	16
32.27	odd	8		640.2.l.a.481.3		16
32.29	even	8		80.2.l.a.61.1	yes	16
96.11	even	8		2880.2.t.c.2161.1		16
96.29	odd	8		720.2.t.c.541.8		16
96.35	even	8		2880.2.t.c.721.4		16
96.53	odd	8		720.2.t.c.181.8		16
160.3	even	8		1600.2.q.h.849.3		16
160.29	even	8		400.2.l.h.301.8		16
160.43	even	8		1600.2.q.g.49.6		16
160.53	odd	8		400.2.q.h.149.4		16
160.67	even	8		1600.2.q.g.849.6		16
160.93	odd	8		400.2.q.g.349.5		16
160.99	odd	8		1600.2.l.i.401.3		16
160.107	even	8		1600.2.q.h.49.3		16
160.117	odd	8		400.2.q.g.149.5		16
160.139	odd	8		1600.2.l.i.1201.3		16
160.149	even	8		400.2.l.h.101.8		16
160.157	odd	8		400.2.q.h.349.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.1	✓	16	32.21	even	8
80.2.l.a.61.1	yes	16	32.29	even	8
320.2.l.a.81.6		16	32.3	odd	8
320.2.l.a.241.6		16	32.11	odd	8
400.2.l.h.101.8		16	160.149	even	8
400.2.l.h.301.8		16	160.29	even	8
400.2.q.g.149.5		16	160.117	odd	8
400.2.q.g.349.5		16	160.93	odd	8
400.2.q.h.149.4		16	160.53	odd	8
400.2.q.h.349.4		16	160.157	odd	8
640.2.l.a.161.3		16	32.19	odd	8
640.2.l.a.481.3		16	32.27	odd	8
640.2.l.b.161.6		16	32.13	even	8
640.2.l.b.481.6		16	32.5	even	8
720.2.t.c.181.8		16	96.53	odd	8
720.2.t.c.541.8		16	96.29	odd	8
1600.2.l.i.401.3		16	160.99	odd	8
1600.2.l.i.1201.3		16	160.139	odd	8
1600.2.q.g.49.6		16	160.43	even	8
1600.2.q.g.849.6		16	160.67	even	8
1600.2.q.h.49.3		16	160.107	even	8
1600.2.q.h.849.3		16	160.3	even	8
2880.2.t.c.721.4		16	96.35	even	8
2880.2.t.c.2161.1		16	96.11	even	8
5120.2.a.s.1.4		8	1.1	even	1	trivial
5120.2.a.t.1.4		8	8.3	odd	2
5120.2.a.u.1.5		8	4.3	odd	2
5120.2.a.v.1.5		8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q-1.01919 q^{3} -1.00000 q^{5} +4.02840 q^{7} -1.96126 q^{9} -0.914766 q^{11} -6.95074 q^{13} +1.01919 q^{15} +2.70862 q^{17} +0.619987 q^{19} -4.10569 q^{21} +3.60080 q^{23} +1.00000 q^{25} +5.05645 q^{27} +2.84146 q^{29} +4.30994 q^{31} +0.932316 q^{33} -4.02840 q^{35} +1.05212 q^{37} +7.08410 q^{39} -0.603979 q^{41} -7.11363 q^{43} +1.96126 q^{45} -10.8177 q^{47} +9.22800 q^{49} -2.76059 q^{51} +5.76178 q^{53} +0.914766 q^{55} -0.631882 q^{57} -1.73729 q^{59} -9.88410 q^{61} -7.90074 q^{63} +6.95074 q^{65} +7.41357 q^{67} -3.66988 q^{69} -13.7940 q^{71} -1.30876 q^{73} -1.01919 q^{75} -3.68504 q^{77} -0.611127 q^{79} +0.730326 q^{81} -1.83100 q^{83} -2.70862 q^{85} -2.89597 q^{87} +10.9236 q^{89} -28.0004 q^{91} -4.39263 q^{93} -0.619987 q^{95} -12.7571 q^{97} +1.79409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 8 q^{5} + 4 q^{7} + 8 q^{9} - 8 q^{11} + 4 q^{15} - 16 q^{19} + 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} - 8 q^{45} + 20 q^{47} + 8 q^{49} - 24 q^{51} + 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^5 + 4 * q^7 + 8 * q^9 - 8 * q^11 + 4 * q^15 - 16 * q^19 + 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 - 8 * q^45 + 20 * q^47 + 8 * q^49 - 24 * q^51 + 8 * q^55 - 16 * q^59 + 20 * q^63 - 36 * q^67 + 16 * q^69 - 8 * q^71 - 4 * q^75 - 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 + 16 * q^95 - 40 * q^99

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