LMFDB - Embedded newform 8512.2.a.ch.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 10x^{5} + 31x^{4} + 12x^{3} - 45x^{2} - 15x + 8 $ x^7 - 3x^6 - 10x^5 + 31x^4 + 12x^3 - 45x^2 - 15x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.811529$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.811529 q^{3} -0.636007 q^{5} -1.00000 q^{7} -2.34142 q^{9} +O(q^{10})$	$q+0.811529 q^{3} -0.636007 q^{5} -1.00000 q^{7} -2.34142 q^{9} -1.29803 q^{11} -1.02111 q^{13} -0.516138 q^{15} -7.09589 q^{17} +1.00000 q^{19} -0.811529 q^{21} -3.70216 q^{23} -4.59550 q^{25} -4.33472 q^{27} -4.94639 q^{29} +2.96414 q^{31} -1.05339 q^{33} +0.636007 q^{35} -2.78424 q^{37} -0.828662 q^{39} +5.64336 q^{41} -1.08208 q^{43} +1.48916 q^{45} +11.2285 q^{47} +1.00000 q^{49} -5.75852 q^{51} +6.23941 q^{53} +0.825557 q^{55} +0.811529 q^{57} -1.27447 q^{59} +11.3744 q^{61} +2.34142 q^{63} +0.649434 q^{65} -7.11692 q^{67} -3.00441 q^{69} +4.45671 q^{71} -2.12332 q^{73} -3.72938 q^{75} +1.29803 q^{77} +13.1745 q^{79} +3.50652 q^{81} -2.08289 q^{83} +4.51303 q^{85} -4.01413 q^{87} +7.92565 q^{89} +1.02111 q^{91} +2.40548 q^{93} -0.636007 q^{95} +9.66418 q^{97} +3.03924 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9	$ 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 3 q^{11} + 16 q^{13} + 8 q^{15} + 8 q^{17} + 7 q^{19} + 3 q^{21} + 10 q^{23} + 8 q^{25} - 6 q^{27} + 11 q^{29} + 14 q^{31} + 3 q^{33} - 5 q^{35} + 13 q^{37} - 6 q^{39} + 11 q^{41} - 11 q^{43} - 4 q^{45} + 7 q^{47} + 7 q^{49} - 24 q^{51} + 9 q^{53} - 8 q^{55} - 3 q^{57} - 23 q^{59} + 23 q^{61} - 8 q^{63} + 40 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} + 4 q^{73} - 48 q^{75} + 3 q^{77} + 29 q^{79} + 23 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} + 15 q^{89} - 16 q^{91} - 42 q^{93} + 5 q^{95} - 13 q^{97} - 47 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 3 * q^11 + 16 * q^13 + 8 * q^15 + 8 * q^17 + 7 * q^19 + 3 * q^21 + 10 * q^23 + 8 * q^25 - 6 * q^27 + 11 * q^29 + 14 * q^31 + 3 * q^33 - 5 * q^35 + 13 * q^37 - 6 * q^39 + 11 * q^41 - 11 * q^43 - 4 * q^45 + 7 * q^47 + 7 * q^49 - 24 * q^51 + 9 * q^53 - 8 * q^55 - 3 * q^57 - 23 * q^59 + 23 * q^61 - 8 * q^63 + 40 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 + 4 * q^73 - 48 * q^75 + 3 * q^77 + 29 * q^79 + 23 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 + 15 * q^89 - 16 * q^91 - 42 * q^93 + 5 * q^95 - 13 * q^97 - 47 * 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
$3$	0.811529	0.468536	0.234268	−	0.972172i	$-0.424731\pi$
$3$	0.811529	0.468536	0.234268	+	0.972172i	$0.424731\pi$
$4$	0	0
$5$	−0.636007	−0.284431	−0.142215	−	0.989836i	$-0.545423\pi$
$5$	−0.636007	−0.284431	−0.142215	+	0.989836i	$0.545423\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.34142	−0.780474
$10$	0	0
$11$	−1.29803	−0.391371	−0.195686	−	0.980667i	$-0.562693\pi$
$11$	−1.29803	−0.391371	−0.195686	+	0.980667i	$0.562693\pi$
$12$	0	0
$13$	−1.02111	−0.283206	−0.141603	−	0.989924i	$-0.545226\pi$
$13$	−1.02111	−0.283206	−0.141603	+	0.989924i	$0.545226\pi$
$14$	0	0
$15$	−0.516138	−0.133266
$16$	0	0
$17$	−7.09589	−1.72101	−0.860503	−	0.509445i	$-0.829851\pi$
$17$	−7.09589	−1.72101	−0.860503	+	0.509445i	$0.829851\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.811529	−0.177090
$22$	0	0
$23$	−3.70216	−0.771953	−0.385977	−	0.922509i	$-0.626135\pi$
$23$	−3.70216	−0.771953	−0.385977	+	0.922509i	$0.626135\pi$
$24$	0	0
$25$	−4.59550	−0.919099
$26$	0	0
$27$	−4.33472	−0.834217
$28$	0	0
$29$	−4.94639	−0.918521	−0.459260	−	0.888302i	$-0.651886\pi$
$29$	−4.94639	−0.918521	−0.459260	+	0.888302i	$0.651886\pi$
$30$	0	0
$31$	2.96414	0.532375	0.266187	−	0.963921i	$-0.414236\pi$
$31$	2.96414	0.532375	0.266187	+	0.963921i	$0.414236\pi$
$32$	0	0
$33$	−1.05339	−0.183372
$34$	0	0
$35$	0.636007	0.107505
$36$	0	0
$37$	−2.78424	−0.457726	−0.228863	−	0.973459i	$-0.573501\pi$
$37$	−2.78424	−0.457726	−0.228863	+	0.973459i	$0.573501\pi$
$38$	0	0
$39$	−0.828662	−0.132692
$40$	0	0
$41$	5.64336	0.881345	0.440673	−	0.897668i	$-0.354740\pi$
$41$	5.64336	0.881345	0.440673	+	0.897668i	$0.354740\pi$
$42$	0	0
$43$	−1.08208	−0.165016	−0.0825081	−	0.996590i	$-0.526293\pi$
$43$	−1.08208	−0.165016	−0.0825081	+	0.996590i	$0.526293\pi$
$44$	0	0
$45$	1.48916	0.221991
$46$	0	0
$47$	11.2285	1.63785	0.818924	−	0.573902i	$-0.194571\pi$
$47$	11.2285	1.63785	0.818924	+	0.573902i	$0.194571\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.75852	−0.806354
$52$	0	0
$53$	6.23941	0.857049	0.428524	−	0.903530i	$-0.359034\pi$
$53$	6.23941	0.857049	0.428524	+	0.903530i	$0.359034\pi$
$54$	0	0
$55$	0.825557	0.111318
$56$	0	0
$57$	0.811529	0.107490
$58$	0	0
$59$	−1.27447	−0.165921	−0.0829607	−	0.996553i	$-0.526438\pi$
$59$	−1.27447	−0.165921	−0.0829607	+	0.996553i	$0.526438\pi$
$60$	0	0
$61$	11.3744	1.45634	0.728172	−	0.685395i	$-0.240370\pi$
$61$	11.3744	1.45634	0.728172	+	0.685395i	$0.240370\pi$
$62$	0	0
$63$	2.34142	0.294991
$64$	0	0
$65$	0.649434	0.0805524
$66$	0	0
$67$	−7.11692	−0.869470	−0.434735	−	0.900558i	$-0.643158\pi$
$67$	−7.11692	−0.869470	−0.434735	+	0.900558i	$0.643158\pi$
$68$	0	0
$69$	−3.00441	−0.361688
$70$	0	0
$71$	4.45671	0.528914	0.264457	−	0.964397i	$-0.414807\pi$
$71$	4.45671	0.528914	0.264457	+	0.964397i	$0.414807\pi$
$72$	0	0
$73$	−2.12332	−0.248515	−0.124258	−	0.992250i	$-0.539655\pi$
$73$	−2.12332	−0.248515	−0.124258	+	0.992250i	$0.539655\pi$
$74$	0	0
$75$	−3.72938	−0.430631
$76$	0	0
$77$	1.29803	0.147925
$78$	0	0
$79$	13.1745	1.48225	0.741125	−	0.671367i	$-0.234293\pi$
$79$	13.1745	1.48225	0.741125	+	0.671367i	$0.234293\pi$
$80$	0	0
$81$	3.50652	0.389613
$82$	0	0
$83$	−2.08289	−0.228627	−0.114313	−	0.993445i	$-0.536467\pi$
$83$	−2.08289	−0.228627	−0.114313	+	0.993445i	$0.536467\pi$
$84$	0	0
$85$	4.51303	0.489507
$86$	0	0
$87$	−4.01413	−0.430360
$88$	0	0
$89$	7.92565	0.840117	0.420059	−	0.907497i	$-0.362009\pi$
$89$	7.92565	0.840117	0.420059	+	0.907497i	$0.362009\pi$
$90$	0	0
$91$	1.02111	0.107042
$92$	0	0
$93$	2.40548	0.249437
$94$	0	0
$95$	−0.636007	−0.0652529
$96$	0	0
$97$	9.66418	0.981249	0.490624	−	0.871371i	$-0.336769\pi$
$97$	9.66418	0.981249	0.490624	+	0.871371i	$0.336769\pi$
$98$	0	0
$99$	3.03924	0.305455
$100$	0	0
$101$	16.7633	1.66801	0.834003	−	0.551759i	$-0.186043\pi$
$101$	16.7633	1.66801	0.834003	+	0.551759i	$0.186043\pi$
$102$	0	0
$103$	2.23385	0.220108	0.110054	−	0.993926i	$-0.464898\pi$
$103$	2.23385	0.220108	0.110054	+	0.993926i	$0.464898\pi$
$104$	0	0
$105$	0.516138	0.0503699
$106$	0	0
$107$	11.6185	1.12320	0.561601	−	0.827408i	$-0.310186\pi$
$107$	11.6185	1.12320	0.561601	+	0.827408i	$0.310186\pi$
$108$	0	0
$109$	−5.14717	−0.493010	−0.246505	−	0.969142i	$-0.579282\pi$
$109$	−5.14717	−0.493010	−0.246505	+	0.969142i	$0.579282\pi$
$110$	0	0
$111$	−2.25949	−0.214461
$112$	0	0
$113$	−0.0669556	−0.00629865	−0.00314933	−	0.999995i	$-0.501002\pi$
$113$	−0.0669556	−0.00629865	−0.00314933	+	0.999995i	$0.501002\pi$
$114$	0	0
$115$	2.35460	0.219567
$116$	0	0
$117$	2.39085	0.221035
$118$	0	0
$119$	7.09589	0.650479
$120$	0	0
$121$	−9.31511	−0.846828
$122$	0	0
$123$	4.57975	0.412942
$124$	0	0
$125$	6.10280	0.545851
$126$	0	0
$127$	1.92435	0.170758	0.0853791	−	0.996349i	$-0.472790\pi$
$127$	1.92435	0.170758	0.0853791	+	0.996349i	$0.472790\pi$
$128$	0	0
$129$	−0.878142	−0.0773161
$130$	0	0
$131$	−7.18486	−0.627744	−0.313872	−	0.949465i	$-0.601626\pi$
$131$	−7.18486	−0.627744	−0.313872	+	0.949465i	$0.601626\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	2.75691	0.237277
$136$	0	0
$137$	7.55147	0.645166	0.322583	−	0.946541i	$-0.395449\pi$
$137$	7.55147	0.645166	0.322583	+	0.946541i	$0.395449\pi$
$138$	0	0
$139$	−10.6473	−0.903088	−0.451544	−	0.892249i	$-0.649127\pi$
$139$	−10.6473	−0.903088	−0.451544	+	0.892249i	$0.649127\pi$
$140$	0	0
$141$	9.11226	0.767391
$142$	0	0
$143$	1.32544	0.110839
$144$	0	0
$145$	3.14593	0.261256
$146$	0	0
$147$	0.811529	0.0669338
$148$	0	0
$149$	−16.1264	−1.32113	−0.660564	−	0.750769i	$-0.729683\pi$
$149$	−16.1264	−1.32113	−0.660564	+	0.750769i	$0.729683\pi$
$150$	0	0
$151$	12.0872	0.983640	0.491820	−	0.870697i	$-0.336332\pi$
$151$	12.0872	0.983640	0.491820	+	0.870697i	$0.336332\pi$
$152$	0	0
$153$	16.6145	1.34320
$154$	0	0
$155$	−1.88521	−0.151424
$156$	0	0
$157$	5.07289	0.404861	0.202430	−	0.979297i	$-0.435116\pi$
$157$	5.07289	0.404861	0.202430	+	0.979297i	$0.435116\pi$
$158$	0	0
$159$	5.06346	0.401558
$160$	0	0
$161$	3.70216	0.291771
$162$	0	0
$163$	−10.1395	−0.794189	−0.397094	−	0.917778i	$-0.629981\pi$
$163$	−10.1395	−0.794189	−0.397094	+	0.917778i	$0.629981\pi$
$164$	0	0
$165$	0.669963	0.0521566
$166$	0	0
$167$	−12.0873	−0.935344	−0.467672	−	0.883902i	$-0.654907\pi$
$167$	−12.0873	−0.935344	−0.467672	+	0.883902i	$0.654907\pi$
$168$	0	0
$169$	−11.9573	−0.919795
$170$	0	0
$171$	−2.34142	−0.179053
$172$	0	0
$173$	1.18756	0.0902887	0.0451444	−	0.998980i	$-0.485625\pi$
$173$	1.18756	0.0902887	0.0451444	+	0.998980i	$0.485625\pi$
$174$	0	0
$175$	4.59550	0.347387
$176$	0	0
$177$	−1.03427	−0.0777401
$178$	0	0
$179$	−10.0156	−0.748603	−0.374301	−	0.927307i	$-0.622117\pi$
$179$	−10.0156	−0.748603	−0.374301	+	0.927307i	$0.622117\pi$
$180$	0	0
$181$	−10.8750	−0.808332	−0.404166	−	0.914686i	$-0.632438\pi$
$181$	−10.8750	−0.808332	−0.404166	+	0.914686i	$0.632438\pi$
$182$	0	0
$183$	9.23066	0.682350
$184$	0	0
$185$	1.77080	0.130191
$186$	0	0
$187$	9.21069	0.673553
$188$	0	0
$189$	4.33472	0.315304
$190$	0	0
$191$	−1.08397	−0.0784332	−0.0392166	−	0.999231i	$-0.512486\pi$
$191$	−1.08397	−0.0784332	−0.0392166	+	0.999231i	$0.512486\pi$
$192$	0	0
$193$	12.1528	0.874776	0.437388	−	0.899273i	$-0.355904\pi$
$193$	12.1528	0.874776	0.437388	+	0.899273i	$0.355904\pi$
$194$	0	0
$195$	0.527035	0.0377417
$196$	0	0
$197$	1.77068	0.126155	0.0630777	−	0.998009i	$-0.479908\pi$
$197$	1.77068	0.126155	0.0630777	+	0.998009i	$0.479908\pi$
$198$	0	0
$199$	−1.57152	−0.111402	−0.0557010	−	0.998447i	$-0.517739\pi$
$199$	−1.57152	−0.111402	−0.0557010	+	0.998447i	$0.517739\pi$
$200$	0	0
$201$	−5.77558	−0.407378
$202$	0	0
$203$	4.94639	0.347168
$204$	0	0
$205$	−3.58922	−0.250682
$206$	0	0
$207$	8.66831	0.602489
$208$	0	0
$209$	−1.29803	−0.0897868
$210$	0	0
$211$	−27.8073	−1.91434	−0.957168	−	0.289533i	$-0.906500\pi$
$211$	−27.8073	−1.91434	−0.957168	+	0.289533i	$0.906500\pi$
$212$	0	0
$213$	3.61675	0.247815
$214$	0	0
$215$	0.688213	0.0469357
$216$	0	0
$217$	−2.96414	−0.201219
$218$	0	0
$219$	−1.72313	−0.116438
$220$	0	0
$221$	7.24570	0.487399
$222$	0	0
$223$	4.55286	0.304882	0.152441	−	0.988313i	$-0.451287\pi$
$223$	4.55286	0.304882	0.152441	+	0.988313i	$0.451287\pi$
$224$	0	0
$225$	10.7600	0.717333
$226$	0	0
$227$	18.4223	1.22273	0.611364	−	0.791349i	$-0.290621\pi$
$227$	18.4223	1.22273	0.611364	+	0.791349i	$0.290621\pi$
$228$	0	0
$229$	−13.0123	−0.859880	−0.429940	−	0.902857i	$-0.641465\pi$
$229$	−13.0123	−0.859880	−0.429940	+	0.902857i	$0.641465\pi$
$230$	0	0
$231$	1.05339	0.0693080
$232$	0	0
$233$	17.4557	1.14356	0.571780	−	0.820407i	$-0.306253\pi$
$233$	17.4557	1.14356	0.571780	+	0.820407i	$0.306253\pi$
$234$	0	0
$235$	−7.14141	−0.465854
$236$	0	0
$237$	10.6915	0.694488
$238$	0	0
$239$	−2.15898	−0.139653	−0.0698265	−	0.997559i	$-0.522245\pi$
$239$	−2.15898	−0.139653	−0.0698265	+	0.997559i	$0.522245\pi$
$240$	0	0
$241$	5.95311	0.383473	0.191737	−	0.981446i	$-0.438588\pi$
$241$	5.95311	0.383473	0.191737	+	0.981446i	$0.438588\pi$
$242$	0	0
$243$	15.8498	1.01676
$244$	0	0
$245$	−0.636007	−0.0406330
$246$	0	0
$247$	−1.02111	−0.0649718
$248$	0	0
$249$	−1.69032	−0.107120
$250$	0	0
$251$	−11.3262	−0.714904	−0.357452	−	0.933932i	$-0.616354\pi$
$251$	−11.3262	−0.714904	−0.357452	+	0.933932i	$0.616354\pi$
$252$	0	0
$253$	4.80552	0.302120
$254$	0	0
$255$	3.66246	0.229352
$256$	0	0
$257$	22.5520	1.40675	0.703377	−	0.710817i	$-0.251675\pi$
$257$	22.5520	1.40675	0.703377	+	0.710817i	$0.251675\pi$
$258$	0	0
$259$	2.78424	0.173004
$260$	0	0
$261$	11.5816	0.716881
$262$	0	0
$263$	25.1703	1.55207	0.776033	−	0.630693i	$-0.217229\pi$
$263$	25.1703	1.55207	0.776033	+	0.630693i	$0.217229\pi$
$264$	0	0
$265$	−3.96831	−0.243771
$266$	0	0
$267$	6.43189	0.393626
$268$	0	0
$269$	−12.0045	−0.731926	−0.365963	−	0.930629i	$-0.619260\pi$
$269$	−12.0045	−0.731926	−0.365963	+	0.930629i	$0.619260\pi$
$270$	0	0
$271$	−6.81151	−0.413770	−0.206885	−	0.978365i	$-0.566333\pi$
$271$	−6.81151	−0.413770	−0.206885	+	0.978365i	$0.566333\pi$
$272$	0	0
$273$	0.828662	0.0501529
$274$	0	0
$275$	5.96510	0.359709
$276$	0	0
$277$	17.9562	1.07888	0.539441	−	0.842024i	$-0.318636\pi$
$277$	17.9562	1.07888	0.539441	+	0.842024i	$0.318636\pi$
$278$	0	0
$279$	−6.94029	−0.415504
$280$	0	0
$281$	2.17514	0.129758	0.0648791	−	0.997893i	$-0.479334\pi$
$281$	2.17514	0.129758	0.0648791	+	0.997893i	$0.479334\pi$
$282$	0	0
$283$	−25.3285	−1.50562	−0.752812	−	0.658235i	$-0.771303\pi$
$283$	−25.3285	−1.50562	−0.752812	+	0.658235i	$0.771303\pi$
$284$	0	0
$285$	−0.516138	−0.0305734
$286$	0	0
$287$	−5.64336	−0.333117
$288$	0	0
$289$	33.3516	1.96186
$290$	0	0
$291$	7.84276	0.459751
$292$	0	0
$293$	13.4142	0.783667	0.391833	−	0.920036i	$-0.371841\pi$
$293$	13.4142	0.783667	0.391833	+	0.920036i	$0.371841\pi$
$294$	0	0
$295$	0.810569	0.0471931
$296$	0	0
$297$	5.62660	0.326489
$298$	0	0
$299$	3.78032	0.218621
$300$	0	0
$301$	1.08208	0.0623703
$302$	0	0
$303$	13.6039	0.781522
$304$	0	0
$305$	−7.23420	−0.414229
$306$	0	0
$307$	−2.45091	−0.139881	−0.0699405	−	0.997551i	$-0.522281\pi$
$307$	−2.45091	−0.139881	−0.0699405	+	0.997551i	$0.522281\pi$
$308$	0	0
$309$	1.81283	0.103128
$310$	0	0
$311$	9.17314	0.520161	0.260080	−	0.965587i	$-0.416251\pi$
$311$	9.17314	0.520161	0.260080	+	0.965587i	$0.416251\pi$
$312$	0	0
$313$	−33.2472	−1.87924	−0.939622	−	0.342215i	$-0.888823\pi$
$313$	−33.2472	−1.87924	−0.939622	+	0.342215i	$0.888823\pi$
$314$	0	0
$315$	−1.48916	−0.0839046
$316$	0	0
$317$	11.0831	0.622487	0.311243	−	0.950330i	$-0.399255\pi$
$317$	11.0831	0.622487	0.311243	+	0.950330i	$0.399255\pi$
$318$	0	0
$319$	6.42057	0.359483
$320$	0	0
$321$	9.42874	0.526261
$322$	0	0
$323$	−7.09589	−0.394826
$324$	0	0
$325$	4.69252	0.260294
$326$	0	0
$327$	−4.17708	−0.230993
$328$	0	0
$329$	−11.2285	−0.619048
$330$	0	0
$331$	14.0938	0.774667	0.387334	−	0.921940i	$-0.373396\pi$
$331$	14.0938	0.774667	0.387334	+	0.921940i	$0.373396\pi$
$332$	0	0
$333$	6.51908	0.357243
$334$	0	0
$335$	4.52641	0.247304
$336$	0	0
$337$	−12.7704	−0.695647	−0.347824	−	0.937560i	$-0.613079\pi$
$337$	−12.7704	−0.695647	−0.347824	+	0.937560i	$0.613079\pi$
$338$	0	0
$339$	−0.0543364	−0.00295115
$340$	0	0
$341$	−3.84754	−0.208356
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.91082	0.102875
$346$	0	0
$347$	12.8281	0.688649	0.344325	−	0.938851i	$-0.388108\pi$
$347$	12.8281	0.688649	0.344325	+	0.938851i	$0.388108\pi$
$348$	0	0
$349$	−2.68384	−0.143663	−0.0718313	−	0.997417i	$-0.522884\pi$
$349$	−2.68384	−0.143663	−0.0718313	+	0.997417i	$0.522884\pi$
$350$	0	0
$351$	4.42623	0.236255
$352$	0	0
$353$	6.87677	0.366013	0.183007	−	0.983112i	$-0.441417\pi$
$353$	6.87677	0.366013	0.183007	+	0.983112i	$0.441417\pi$
$354$	0	0
$355$	−2.83450	−0.150439
$356$	0	0
$357$	5.75852	0.304773
$358$	0	0
$359$	−24.8326	−1.31061	−0.655306	−	0.755363i	$-0.727461\pi$
$359$	−24.8326	−1.31061	−0.655306	+	0.755363i	$0.727461\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.55948	−0.396770
$364$	0	0
$365$	1.35044	0.0706854
$366$	0	0
$367$	−26.1933	−1.36728	−0.683638	−	0.729821i	$-0.739603\pi$
$367$	−26.1933	−1.36728	−0.683638	+	0.729821i	$0.739603\pi$
$368$	0	0
$369$	−13.2135	−0.687867
$370$	0	0
$371$	−6.23941	−0.323934
$372$	0	0
$373$	−14.5606	−0.753921	−0.376960	−	0.926229i	$-0.623031\pi$
$373$	−14.5606	−0.753921	−0.376960	+	0.926229i	$0.623031\pi$
$374$	0	0
$375$	4.95260	0.255751
$376$	0	0
$377$	5.05082	0.260130
$378$	0	0
$379$	11.0307	0.566609	0.283304	−	0.959030i	$-0.408569\pi$
$379$	11.0307	0.566609	0.283304	+	0.959030i	$0.408569\pi$
$380$	0	0
$381$	1.56166	0.0800064
$382$	0	0
$383$	20.8823	1.06703	0.533517	−	0.845789i	$-0.320870\pi$
$383$	20.8823	1.06703	0.533517	+	0.845789i	$0.320870\pi$
$384$	0	0
$385$	−0.825557	−0.0420743
$386$	0	0
$387$	2.53361	0.128791
$388$	0	0
$389$	−0.741281	−0.0375844	−0.0187922	−	0.999823i	$-0.505982\pi$
$389$	−0.741281	−0.0375844	−0.0187922	+	0.999823i	$0.505982\pi$
$390$	0	0
$391$	26.2701	1.32854
$392$	0	0
$393$	−5.83072	−0.294121
$394$	0	0
$395$	−8.37909	−0.421598
$396$	0	0
$397$	21.8743	1.09784	0.548919	−	0.835876i	$-0.315040\pi$
$397$	21.8743	1.09784	0.548919	+	0.835876i	$0.315040\pi$
$398$	0	0
$399$	−0.811529	−0.0406272
$400$	0	0
$401$	35.0381	1.74972	0.874859	−	0.484378i	$-0.160954\pi$
$401$	35.0381	1.74972	0.874859	+	0.484378i	$0.160954\pi$
$402$	0	0
$403$	−3.02672	−0.150772
$404$	0	0
$405$	−2.23017	−0.110818
$406$	0	0
$407$	3.61403	0.179141
$408$	0	0
$409$	−1.79179	−0.0885981	−0.0442991	−	0.999018i	$-0.514105\pi$
$409$	−1.79179	−0.0885981	−0.0442991	+	0.999018i	$0.514105\pi$
$410$	0	0
$411$	6.12823	0.302284
$412$	0	0
$413$	1.27447	0.0627124
$414$	0	0
$415$	1.32473	0.0650285
$416$	0	0
$417$	−8.64055	−0.423130
$418$	0	0
$419$	−28.7315	−1.40363	−0.701814	−	0.712361i	$-0.747626\pi$
$419$	−28.7315	−1.40363	−0.701814	+	0.712361i	$0.747626\pi$
$420$	0	0
$421$	−5.93235	−0.289125	−0.144563	−	0.989496i	$-0.546178\pi$
$421$	−5.93235	−0.289125	−0.144563	+	0.989496i	$0.546178\pi$
$422$	0	0
$423$	−26.2907	−1.27830
$424$	0	0
$425$	32.6091	1.58177
$426$	0	0
$427$	−11.3744	−0.550446
$428$	0	0
$429$	1.07563	0.0519319
$430$	0	0
$431$	−12.9682	−0.624655	−0.312328	−	0.949974i	$-0.601109\pi$
$431$	−12.9682	−0.624655	−0.312328	+	0.949974i	$0.601109\pi$
$432$	0	0
$433$	32.4728	1.56054	0.780272	−	0.625441i	$-0.215081\pi$
$433$	32.4728	1.56054	0.780272	+	0.625441i	$0.215081\pi$
$434$	0	0
$435$	2.55302	0.122408
$436$	0	0
$437$	−3.70216	−0.177098
$438$	0	0
$439$	−15.9348	−0.760526	−0.380263	−	0.924878i	$-0.624166\pi$
$439$	−15.9348	−0.760526	−0.380263	+	0.924878i	$0.624166\pi$
$440$	0	0
$441$	−2.34142	−0.111496
$442$	0	0
$443$	−31.1810	−1.48145	−0.740727	−	0.671806i	$-0.765519\pi$
$443$	−31.1810	−1.48145	−0.740727	+	0.671806i	$0.765519\pi$
$444$	0	0
$445$	−5.04077	−0.238955
$446$	0	0
$447$	−13.0871	−0.618997
$448$	0	0
$449$	14.3006	0.674886	0.337443	−	0.941346i	$-0.390438\pi$
$449$	14.3006	0.674886	0.337443	+	0.941346i	$0.390438\pi$
$450$	0	0
$451$	−7.32527	−0.344933
$452$	0	0
$453$	9.80909	0.460871
$454$	0	0
$455$	−0.649434	−0.0304460
$456$	0	0
$457$	12.8833	0.602655	0.301328	−	0.953521i	$-0.402570\pi$
$457$	12.8833	0.602655	0.301328	+	0.953521i	$0.402570\pi$
$458$	0	0
$459$	30.7587	1.43569
$460$	0	0
$461$	2.22705	0.103724	0.0518621	−	0.998654i	$-0.483484\pi$
$461$	2.22705	0.103724	0.0518621	+	0.998654i	$0.483484\pi$
$462$	0	0
$463$	5.26683	0.244770	0.122385	−	0.992483i	$-0.460946\pi$
$463$	5.26683	0.244770	0.122385	+	0.992483i	$0.460946\pi$
$464$	0	0
$465$	−1.52990	−0.0709475
$466$	0	0
$467$	−0.391021	−0.0180943	−0.00904715	−	0.999959i	$-0.502880\pi$
$467$	−0.391021	−0.0180943	−0.00904715	+	0.999959i	$0.502880\pi$
$468$	0	0
$469$	7.11692	0.328629
$470$	0	0
$471$	4.11679	0.189692
$472$	0	0
$473$	1.40458	0.0645827
$474$	0	0
$475$	−4.59550	−0.210856
$476$	0	0
$477$	−14.6091	−0.668904
$478$	0	0
$479$	28.1075	1.28426	0.642132	−	0.766594i	$-0.278050\pi$
$479$	28.1075	1.28426	0.642132	+	0.766594i	$0.278050\pi$
$480$	0	0
$481$	2.84302	0.129631
$482$	0	0
$483$	3.00441	0.136705
$484$	0	0
$485$	−6.14648	−0.279097
$486$	0	0
$487$	32.7247	1.48290	0.741448	−	0.671010i	$-0.234139\pi$
$487$	32.7247	1.48290	0.741448	+	0.671010i	$0.234139\pi$
$488$	0	0
$489$	−8.22851	−0.372106
$490$	0	0
$491$	−13.1414	−0.593063	−0.296531	−	0.955023i	$-0.595830\pi$
$491$	−13.1414	−0.593063	−0.296531	+	0.955023i	$0.595830\pi$
$492$	0	0
$493$	35.0990	1.58078
$494$	0	0
$495$	−1.93298	−0.0868809
$496$	0	0
$497$	−4.45671	−0.199911
$498$	0	0
$499$	−4.07868	−0.182587	−0.0912935	−	0.995824i	$-0.529100\pi$
$499$	−4.07868	−0.182587	−0.0912935	+	0.995824i	$0.529100\pi$
$500$	0	0
$501$	−9.80920	−0.438243
$502$	0	0
$503$	−27.6725	−1.23386	−0.616928	−	0.787020i	$-0.711623\pi$
$503$	−27.6725	−1.23386	−0.616928	+	0.787020i	$0.711623\pi$
$504$	0	0
$505$	−10.6615	−0.474433
$506$	0	0
$507$	−9.70371	−0.430957
$508$	0	0
$509$	27.7347	1.22932	0.614660	−	0.788792i	$-0.289293\pi$
$509$	27.7347	1.22932	0.614660	+	0.788792i	$0.289293\pi$
$510$	0	0
$511$	2.12332	0.0939300
$512$	0	0
$513$	−4.33472	−0.191382
$514$	0	0
$515$	−1.42074	−0.0626054
$516$	0	0
$517$	−14.5750	−0.641007
$518$	0	0
$519$	0.963741	0.0423036
$520$	0	0
$521$	11.7538	0.514943	0.257471	−	0.966286i	$-0.417111\pi$
$521$	11.7538	0.514943	0.257471	+	0.966286i	$0.417111\pi$
$522$	0	0
$523$	−36.8157	−1.60984	−0.804919	−	0.593385i	$-0.797791\pi$
$523$	−36.8157	−1.60984	−0.804919	+	0.593385i	$0.797791\pi$
$524$	0	0
$525$	3.72938	0.162763
$526$	0	0
$527$	−21.0332	−0.916220
$528$	0	0
$529$	−9.29404	−0.404089
$530$	0	0
$531$	2.98406	0.129497
$532$	0	0
$533$	−5.76251	−0.249602
$534$	0	0
$535$	−7.38944	−0.319473
$536$	0	0
$537$	−8.12796	−0.350748
$538$	0	0
$539$	−1.29803	−0.0559102
$540$	0	0
$541$	−7.04710	−0.302979	−0.151489	−	0.988459i	$-0.548407\pi$
$541$	−7.04710	−0.302979	−0.151489	+	0.988459i	$0.548407\pi$
$542$	0	0
$543$	−8.82537	−0.378733
$544$	0	0
$545$	3.27363	0.140227
$546$	0	0
$547$	−8.94352	−0.382397	−0.191199	−	0.981551i	$-0.561237\pi$
$547$	−8.94352	−0.382397	−0.191199	+	0.981551i	$0.561237\pi$
$548$	0	0
$549$	−26.6323	−1.13664
$550$	0	0
$551$	−4.94639	−0.210723
$552$	0	0
$553$	−13.1745	−0.560238
$554$	0	0
$555$	1.43705	0.0609994
$556$	0	0
$557$	45.2610	1.91777	0.958885	−	0.283794i	$-0.0915931\pi$
$557$	45.2610	1.91777	0.958885	+	0.283794i	$0.0915931\pi$
$558$	0	0
$559$	1.10493	0.0467335
$560$	0	0
$561$	7.47474	0.315584
$562$	0	0
$563$	32.3507	1.36342	0.681710	−	0.731623i	$-0.261237\pi$
$563$	32.3507	1.36342	0.681710	+	0.731623i	$0.261237\pi$
$564$	0	0
$565$	0.0425842	0.00179153
$566$	0	0
$567$	−3.50652	−0.147260
$568$	0	0
$569$	−8.97303	−0.376169	−0.188085	−	0.982153i	$-0.560228\pi$
$569$	−8.97303	−0.376169	−0.188085	+	0.982153i	$0.560228\pi$
$570$	0	0
$571$	17.1457	0.717525	0.358763	−	0.933429i	$-0.383199\pi$
$571$	17.1457	0.717525	0.358763	+	0.933429i	$0.383199\pi$
$572$	0	0
$573$	−0.879672	−0.0367488
$574$	0	0
$575$	17.0132	0.709501
$576$	0	0
$577$	14.8515	0.618276	0.309138	−	0.951017i	$-0.399959\pi$
$577$	14.8515	0.618276	0.309138	+	0.951017i	$0.399959\pi$
$578$	0	0
$579$	9.86233	0.409864
$580$	0	0
$581$	2.08289	0.0864128
$582$	0	0
$583$	−8.09896	−0.335424
$584$	0	0
$585$	−1.52060	−0.0628691
$586$	0	0
$587$	12.7544	0.526432	0.263216	−	0.964737i	$-0.415217\pi$
$587$	12.7544	0.526432	0.263216	+	0.964737i	$0.415217\pi$
$588$	0	0
$589$	2.96414	0.122135
$590$	0	0
$591$	1.43695	0.0591084
$592$	0	0
$593$	−38.3977	−1.57681	−0.788403	−	0.615159i	$-0.789092\pi$
$593$	−38.3977	−1.57681	−0.788403	+	0.615159i	$0.789092\pi$
$594$	0	0
$595$	−4.51303	−0.185016
$596$	0	0
$597$	−1.27533	−0.0521959
$598$	0	0
$599$	4.16331	0.170108	0.0850541	−	0.996376i	$-0.472894\pi$
$599$	4.16331	0.170108	0.0850541	+	0.996376i	$0.472894\pi$
$600$	0	0
$601$	9.28102	0.378581	0.189290	−	0.981921i	$-0.439381\pi$
$601$	9.28102	0.378581	0.189290	+	0.981921i	$0.439381\pi$
$602$	0	0
$603$	16.6637	0.678599
$604$	0	0
$605$	5.92447	0.240864
$606$	0	0
$607$	18.2012	0.738764	0.369382	−	0.929278i	$-0.379569\pi$
$607$	18.2012	0.738764	0.369382	+	0.929278i	$0.379569\pi$
$608$	0	0
$609$	4.01413	0.162661
$610$	0	0
$611$	−11.4656	−0.463848
$612$	0	0
$613$	−15.1116	−0.610351	−0.305175	−	0.952296i	$-0.598715\pi$
$613$	−15.1116	−0.610351	−0.305175	+	0.952296i	$0.598715\pi$
$614$	0	0
$615$	−2.91275	−0.117454
$616$	0	0
$617$	18.5113	0.745236	0.372618	−	0.927985i	$-0.378460\pi$
$617$	18.5113	0.745236	0.372618	+	0.927985i	$0.378460\pi$
$618$	0	0
$619$	7.80912	0.313875	0.156938	−	0.987609i	$-0.449838\pi$
$619$	7.80912	0.313875	0.156938	+	0.987609i	$0.449838\pi$
$620$	0	0
$621$	16.0478	0.643976
$622$	0	0
$623$	−7.92565	−0.317535
$624$	0	0
$625$	19.0961	0.763842
$626$	0	0
$627$	−1.05339	−0.0420684
$628$	0	0
$629$	19.7567	0.787750
$630$	0	0
$631$	−34.9920	−1.39301	−0.696505	−	0.717551i	$-0.745263\pi$
$631$	−34.9920	−1.39301	−0.696505	+	0.717551i	$0.745263\pi$
$632$	0	0
$633$	−22.5665	−0.896936
$634$	0	0
$635$	−1.22390	−0.0485689
$636$	0	0
$637$	−1.02111	−0.0404580
$638$	0	0
$639$	−10.4350	−0.412804
$640$	0	0
$641$	35.7796	1.41321	0.706605	−	0.707608i	$-0.250226\pi$
$641$	35.7796	1.41321	0.706605	+	0.707608i	$0.250226\pi$
$642$	0	0
$643$	−17.3962	−0.686038	−0.343019	−	0.939328i	$-0.611450\pi$
$643$	−17.3962	−0.686038	−0.343019	+	0.939328i	$0.611450\pi$
$644$	0	0
$645$	0.558504	0.0219911
$646$	0	0
$647$	−30.7502	−1.20892	−0.604458	−	0.796637i	$-0.706610\pi$
$647$	−30.7502	−1.20892	−0.604458	+	0.796637i	$0.706610\pi$
$648$	0	0
$649$	1.65430	0.0649369
$650$	0	0
$651$	−2.40548	−0.0942783
$652$	0	0
$653$	−44.6013	−1.74538	−0.872691	−	0.488272i	$-0.837627\pi$
$653$	−44.6013	−1.74538	−0.872691	+	0.488272i	$0.837627\pi$
$654$	0	0
$655$	4.56962	0.178550
$656$	0	0
$657$	4.97158	0.193960
$658$	0	0
$659$	−36.9039	−1.43757	−0.718785	−	0.695232i	$-0.755302\pi$
$659$	−36.9039	−1.43757	−0.718785	+	0.695232i	$0.755302\pi$
$660$	0	0
$661$	22.3132	0.867882	0.433941	−	0.900941i	$-0.357123\pi$
$661$	22.3132	0.867882	0.433941	+	0.900941i	$0.357123\pi$
$662$	0	0
$663$	5.88009	0.228364
$664$	0	0
$665$	0.636007	0.0246633
$666$	0	0
$667$	18.3123	0.709055
$668$	0	0
$669$	3.69477	0.142848
$670$	0	0
$671$	−14.7643	−0.569971
$672$	0	0
$673$	28.2034	1.08716	0.543581	−	0.839356i	$-0.317068\pi$
$673$	28.2034	1.08716	0.543581	+	0.839356i	$0.317068\pi$
$674$	0	0
$675$	19.9202	0.766728
$676$	0	0
$677$	27.3009	1.04926	0.524629	−	0.851331i	$-0.324204\pi$
$677$	27.3009	1.04926	0.524629	+	0.851331i	$0.324204\pi$
$678$	0	0
$679$	−9.66418	−0.370877
$680$	0	0
$681$	14.9502	0.572893
$682$	0	0
$683$	27.9117	1.06801	0.534005	−	0.845481i	$-0.320686\pi$
$683$	27.9117	1.06801	0.534005	+	0.845481i	$0.320686\pi$
$684$	0	0
$685$	−4.80279	−0.183505
$686$	0	0
$687$	−10.5599	−0.402885
$688$	0	0
$689$	−6.37114	−0.242721
$690$	0	0
$691$	22.6534	0.861778	0.430889	−	0.902405i	$-0.358200\pi$
$691$	22.6534	0.861778	0.430889	+	0.902405i	$0.358200\pi$
$692$	0	0
$693$	−3.03924	−0.115451
$694$	0	0
$695$	6.77173	0.256866
$696$	0	0
$697$	−40.0447	−1.51680
$698$	0	0
$699$	14.1658	0.535800
$700$	0	0
$701$	9.84112	0.371694	0.185847	−	0.982579i	$-0.440497\pi$
$701$	9.84112	0.371694	0.185847	+	0.982579i	$0.440497\pi$
$702$	0	0
$703$	−2.78424	−0.105010
$704$	0	0
$705$	−5.79546	−0.218270
$706$	0	0
$707$	−16.7633	−0.630447
$708$	0	0
$709$	−28.3147	−1.06338	−0.531691	−	0.846938i	$-0.678443\pi$
$709$	−28.3147	−1.06338	−0.531691	+	0.846938i	$0.678443\pi$
$710$	0	0
$711$	−30.8471	−1.15686
$712$	0	0
$713$	−10.9737	−0.410968
$714$	0	0
$715$	−0.842987	−0.0315259
$716$	0	0
$717$	−1.75208	−0.0654325
$718$	0	0
$719$	−20.6519	−0.770185	−0.385092	−	0.922878i	$-0.625830\pi$
$719$	−20.6519	−0.770185	−0.385092	+	0.922878i	$0.625830\pi$
$720$	0	0
$721$	−2.23385	−0.0831929
$722$	0	0
$723$	4.83112	0.179671
$724$	0	0
$725$	22.7311	0.844212
$726$	0	0
$727$	−15.2781	−0.566635	−0.283317	−	0.959026i	$-0.591435\pi$
$727$	−15.2781	−0.566635	−0.283317	+	0.959026i	$0.591435\pi$
$728$	0	0
$729$	2.34300	0.0867779
$730$	0	0
$731$	7.67835	0.283994
$732$	0	0
$733$	−2.45927	−0.0908352	−0.0454176	−	0.998968i	$-0.514462\pi$
$733$	−2.45927	−0.0908352	−0.0454176	+	0.998968i	$0.514462\pi$
$734$	0	0
$735$	−0.516138	−0.0190380
$736$	0	0
$737$	9.23799	0.340286
$738$	0	0
$739$	40.5385	1.49123	0.745616	−	0.666376i	$-0.232155\pi$
$739$	40.5385	1.49123	0.745616	+	0.666376i	$0.232155\pi$
$740$	0	0
$741$	−0.828662	−0.0304417
$742$	0	0
$743$	−0.152709	−0.00560235	−0.00280117	−	0.999996i	$-0.500892\pi$
$743$	−0.152709	−0.00560235	−0.00280117	+	0.999996i	$0.500892\pi$
$744$	0	0
$745$	10.2565	0.375770
$746$	0	0
$747$	4.87692	0.178437
$748$	0	0
$749$	−11.6185	−0.424531
$750$	0	0
$751$	2.79407	0.101957	0.0509785	−	0.998700i	$-0.483766\pi$
$751$	2.79407	0.101957	0.0509785	+	0.998700i	$0.483766\pi$
$752$	0	0
$753$	−9.19154	−0.334958
$754$	0	0
$755$	−7.68752	−0.279778
$756$	0	0
$757$	−42.7727	−1.55460	−0.777301	−	0.629129i	$-0.783412\pi$
$757$	−42.7727	−1.55460	−0.777301	+	0.629129i	$0.783412\pi$
$758$	0	0
$759$	3.89982	0.141554
$760$	0	0
$761$	−22.9768	−0.832907	−0.416454	−	0.909157i	$-0.636727\pi$
$761$	−22.9768	−0.832907	−0.416454	+	0.909157i	$0.636727\pi$
$762$	0	0
$763$	5.14717	0.186340
$764$	0	0
$765$	−10.5669	−0.382047
$766$	0	0
$767$	1.30137	0.0469899
$768$	0	0
$769$	16.4778	0.594204	0.297102	−	0.954846i	$-0.403980\pi$
$769$	16.4778	0.594204	0.297102	+	0.954846i	$0.403980\pi$
$770$	0	0
$771$	18.3016	0.659115
$772$	0	0
$773$	38.2050	1.37414	0.687070	−	0.726592i	$-0.258897\pi$
$773$	38.2050	1.37414	0.687070	+	0.726592i	$0.258897\pi$
$774$	0	0
$775$	−13.6217	−0.489305
$776$	0	0
$777$	2.25949	0.0810588
$778$	0	0
$779$	5.64336	0.202195
$780$	0	0
$781$	−5.78495	−0.207002
$782$	0	0
$783$	21.4412	0.766245
$784$	0	0
$785$	−3.22639	−0.115155
$786$	0	0
$787$	37.2974	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$787$	37.2974	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$788$	0	0
$789$	20.4264	0.727199
$790$	0	0
$791$	0.0669556	0.00238067
$792$	0	0
$793$	−11.6145	−0.412445
$794$	0	0
$795$	−3.22039	−0.114216
$796$	0	0
$797$	23.7648	0.841791	0.420895	−	0.907109i	$-0.361716\pi$
$797$	23.7648	0.841791	0.420895	+	0.907109i	$0.361716\pi$
$798$	0	0
$799$	−79.6763	−2.81875
$800$	0	0
$801$	−18.5573	−0.655690
$802$	0	0
$803$	2.75613	0.0972618
$804$	0	0
$805$	−2.35460	−0.0829886
$806$	0	0
$807$	−9.74198	−0.342934
$808$	0	0
$809$	−1.83710	−0.0645890	−0.0322945	−	0.999478i	$-0.510281\pi$
$809$	−1.83710	−0.0645890	−0.0322945	+	0.999478i	$0.510281\pi$
$810$	0	0
$811$	32.3038	1.13434	0.567169	−	0.823601i	$-0.308039\pi$
$811$	32.3038	1.13434	0.567169	+	0.823601i	$0.308039\pi$
$812$	0	0
$813$	−5.52774	−0.193866
$814$	0	0
$815$	6.44880	0.225892
$816$	0	0
$817$	−1.08208	−0.0378573
$818$	0	0
$819$	−2.39085	−0.0835432
$820$	0	0
$821$	−17.6352	−0.615472	−0.307736	−	0.951472i	$-0.599571\pi$
$821$	−17.6352	−0.615472	−0.307736	+	0.951472i	$0.599571\pi$
$822$	0	0
$823$	52.6367	1.83480	0.917401	−	0.397965i	$-0.130283\pi$
$823$	52.6367	1.83480	0.917401	+	0.397965i	$0.130283\pi$
$824$	0	0
$825$	4.84085	0.168537
$826$	0	0
$827$	22.5231	0.783204	0.391602	−	0.920135i	$-0.371921\pi$
$827$	22.5231	0.783204	0.391602	+	0.920135i	$0.371921\pi$
$828$	0	0
$829$	26.2488	0.911659	0.455830	−	0.890067i	$-0.349343\pi$
$829$	26.2488	0.911659	0.455830	+	0.890067i	$0.349343\pi$
$830$	0	0
$831$	14.5719	0.505495
$832$	0	0
$833$	−7.09589	−0.245858
$834$	0	0
$835$	7.68761	0.266041
$836$	0	0
$837$	−12.8487	−0.444116
$838$	0	0
$839$	5.27136	0.181987	0.0909937	−	0.995851i	$-0.470996\pi$
$839$	5.27136	0.181987	0.0909937	+	0.995851i	$0.470996\pi$
$840$	0	0
$841$	−4.53326	−0.156319
$842$	0	0
$843$	1.76519	0.0607964
$844$	0	0
$845$	7.60494	0.261618
$846$	0	0
$847$	9.31511	0.320071
$848$	0	0
$849$	−20.5548	−0.705440
$850$	0	0
$851$	10.3077	0.353343
$852$	0	0
$853$	7.92638	0.271394	0.135697	−	0.990750i	$-0.456673\pi$
$853$	7.92638	0.271394	0.135697	+	0.990750i	$0.456673\pi$
$854$	0	0
$855$	1.48916	0.0509282
$856$	0	0
$857$	−52.3189	−1.78718	−0.893589	−	0.448886i	$-0.851821\pi$
$857$	−52.3189	−1.78718	−0.893589	+	0.448886i	$0.851821\pi$
$858$	0	0
$859$	−21.1473	−0.721535	−0.360768	−	0.932656i	$-0.617485\pi$
$859$	−21.1473	−0.721535	−0.360768	+	0.932656i	$0.617485\pi$
$860$	0	0
$861$	−4.57975	−0.156078
$862$	0	0
$863$	−33.6834	−1.14660	−0.573298	−	0.819347i	$-0.694336\pi$
$863$	−33.6834	−1.14660	−0.573298	+	0.819347i	$0.694336\pi$
$864$	0	0
$865$	−0.755298	−0.0256809
$866$	0	0
$867$	27.0658	0.919203
$868$	0	0
$869$	−17.1010	−0.580110
$870$	0	0
$871$	7.26717	0.246239
$872$	0	0
$873$	−22.6279	−0.765839
$874$	0	0
$875$	−6.10280	−0.206312
$876$	0	0
$877$	37.9082	1.28007	0.640035	−	0.768346i	$-0.278920\pi$
$877$	37.9082	1.28007	0.640035	+	0.768346i	$0.278920\pi$
$878$	0	0
$879$	10.8860	0.367176
$880$	0	0
$881$	25.4413	0.857138	0.428569	−	0.903509i	$-0.359018\pi$
$881$	25.4413	0.857138	0.428569	+	0.903509i	$0.359018\pi$
$882$	0	0
$883$	−44.7224	−1.50503	−0.752514	−	0.658576i	$-0.771159\pi$
$883$	−44.7224	−1.50503	−0.752514	+	0.658576i	$0.771159\pi$
$884$	0	0
$885$	0.657800	0.0221117
$886$	0	0
$887$	−53.9753	−1.81231	−0.906157	−	0.422942i	$-0.860998\pi$
$887$	−53.9753	−1.81231	−0.906157	+	0.422942i	$0.860998\pi$
$888$	0	0
$889$	−1.92435	−0.0645405
$890$	0	0
$891$	−4.55157	−0.152483
$892$	0	0
$893$	11.2285	0.375748
$894$	0	0
$895$	6.37000	0.212926
$896$	0	0
$897$	3.06784	0.102432
$898$	0	0
$899$	−14.6618	−0.488997
$900$	0	0
$901$	−44.2742	−1.47499
$902$	0	0
$903$	0.878142	0.0292227
$904$	0	0
$905$	6.91657	0.229915
$906$	0	0
$907$	33.2363	1.10359	0.551797	−	0.833978i	$-0.313942\pi$
$907$	33.2363	1.10359	0.551797	+	0.833978i	$0.313942\pi$
$908$	0	0
$909$	−39.2499	−1.30184
$910$	0	0
$911$	−47.2924	−1.56687	−0.783433	−	0.621476i	$-0.786533\pi$
$911$	−47.2924	−1.56687	−0.783433	+	0.621476i	$0.786533\pi$
$912$	0	0
$913$	2.70366	0.0894780
$914$	0	0
$915$	−5.87076	−0.194081
$916$	0	0
$917$	7.18486	0.237265
$918$	0	0
$919$	23.6716	0.780853	0.390426	−	0.920634i	$-0.372328\pi$
$919$	23.6716	0.780853	0.390426	+	0.920634i	$0.372328\pi$
$920$	0	0
$921$	−1.98899	−0.0655393
$922$	0	0
$923$	−4.55080	−0.149791
$924$	0	0
$925$	12.7950	0.420696
$926$	0	0
$927$	−5.23038	−0.171788
$928$	0	0
$929$	−26.6785	−0.875293	−0.437646	−	0.899147i	$-0.644188\pi$
$929$	−26.6785	−0.875293	−0.437646	+	0.899147i	$0.644188\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	7.44426	0.243714
$934$	0	0
$935$	−5.85806	−0.191579
$936$	0	0
$937$	24.9994	0.816696	0.408348	−	0.912826i	$-0.366105\pi$
$937$	24.9994	0.816696	0.408348	+	0.912826i	$0.366105\pi$
$938$	0	0
$939$	−26.9811	−0.880494
$940$	0	0
$941$	20.2696	0.660769	0.330385	−	0.943846i	$-0.392822\pi$
$941$	20.2696	0.660769	0.330385	+	0.943846i	$0.392822\pi$
$942$	0	0
$943$	−20.8926	−0.680357
$944$	0	0
$945$	−2.75691	−0.0896822
$946$	0	0
$947$	44.4560	1.44462	0.722312	−	0.691567i	$-0.243079\pi$
$947$	44.4560	1.44462	0.722312	+	0.691567i	$0.243079\pi$
$948$	0	0
$949$	2.16814	0.0703810
$950$	0	0
$951$	8.99422	0.291658
$952$	0	0
$953$	53.4069	1.73002	0.865010	−	0.501754i	$-0.167312\pi$
$953$	53.4069	1.73002	0.865010	+	0.501754i	$0.167312\pi$
$954$	0	0
$955$	0.689411	0.0223088
$956$	0	0
$957$	5.21048	0.168431
$958$	0	0
$959$	−7.55147	−0.243850
$960$	0	0
$961$	−22.2139	−0.716577
$962$	0	0
$963$	−27.2038	−0.876630
$964$	0	0
$965$	−7.72925	−0.248813
$966$	0	0
$967$	45.2384	1.45477	0.727384	−	0.686231i	$-0.240736\pi$
$967$	45.2384	1.45477	0.727384	+	0.686231i	$0.240736\pi$
$968$	0	0
$969$	−5.75852	−0.184990
$970$	0	0
$971$	40.3899	1.29617	0.648086	−	0.761567i	$-0.275570\pi$
$971$	40.3899	1.29617	0.648086	+	0.761567i	$0.275570\pi$
$972$	0	0
$973$	10.6473	0.341335
$974$	0	0
$975$	3.80811	0.121957
$976$	0	0
$977$	−4.45440	−0.142509	−0.0712544	−	0.997458i	$-0.522700\pi$
$977$	−4.45440	−0.142509	−0.0712544	+	0.997458i	$0.522700\pi$
$978$	0	0
$979$	−10.2878	−0.328798
$980$	0	0
$981$	12.0517	0.384781
$982$	0	0
$983$	−37.4751	−1.19527	−0.597634	−	0.801769i	$-0.703893\pi$
$983$	−37.4751	−1.19527	−0.597634	+	0.801769i	$0.703893\pi$
$984$	0	0
$985$	−1.12616	−0.0358825
$986$	0	0
$987$	−9.11226	−0.290047
$988$	0	0
$989$	4.00604	0.127385
$990$	0	0
$991$	6.64266	0.211011	0.105506	−	0.994419i	$-0.466354\pi$
$991$	6.64266	0.211011	0.105506	+	0.994419i	$0.466354\pi$
$992$	0	0
$993$	11.4375	0.362960
$994$	0	0
$995$	0.999496	0.0316862
$996$	0	0
$997$	0.970611	0.0307396	0.0153698	−	0.999882i	$-0.495107\pi$
$997$	0.970611	0.0307396	0.0153698	+	0.999882i	$0.495107\pi$
$998$	0	0
$999$	12.0689	0.381843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ch.1.5		7
4.3	odd	2		8512.2.a.ci.1.3		7
8.3	odd	2		4256.2.a.p.1.5	✓	7
8.5	even	2		4256.2.a.q.1.3	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.p.1.5	✓	7	8.3	odd	2
4256.2.a.q.1.3	yes	7	8.5	even	2
8512.2.a.ch.1.5		7	1.1	even	1	trivial
8512.2.a.ci.1.3		7	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+0.811529 q^{3} -0.636007 q^{5} -1.00000 q^{7} -2.34142 q^{9} +O(q^{10})\)	\(q+0.811529 q^{3} -0.636007 q^{5} -1.00000 q^{7} -2.34142 q^{9} -1.29803 q^{11} -1.02111 q^{13} -0.516138 q^{15} -7.09589 q^{17} +1.00000 q^{19} -0.811529 q^{21} -3.70216 q^{23} -4.59550 q^{25} -4.33472 q^{27} -4.94639 q^{29} +2.96414 q^{31} -1.05339 q^{33} +0.636007 q^{35} -2.78424 q^{37} -0.828662 q^{39} +5.64336 q^{41} -1.08208 q^{43} +1.48916 q^{45} +11.2285 q^{47} +1.00000 q^{49} -5.75852 q^{51} +6.23941 q^{53} +0.825557 q^{55} +0.811529 q^{57} -1.27447 q^{59} +11.3744 q^{61} +2.34142 q^{63} +0.649434 q^{65} -7.11692 q^{67} -3.00441 q^{69} +4.45671 q^{71} -2.12332 q^{73} -3.72938 q^{75} +1.29803 q^{77} +13.1745 q^{79} +3.50652 q^{81} -2.08289 q^{83} +4.51303 q^{85} -4.01413 q^{87} +7.92565 q^{89} +1.02111 q^{91} +2.40548 q^{93} -0.636007 q^{95} +9.66418 q^{97} +3.03924 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9	\( 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 3 q^{11} + 16 q^{13} + 8 q^{15} + 8 q^{17} + 7 q^{19} + 3 q^{21} + 10 q^{23} + 8 q^{25} - 6 q^{27} + 11 q^{29} + 14 q^{31} + 3 q^{33} - 5 q^{35} + 13 q^{37} - 6 q^{39} + 11 q^{41} - 11 q^{43} - 4 q^{45} + 7 q^{47} + 7 q^{49} - 24 q^{51} + 9 q^{53} - 8 q^{55} - 3 q^{57} - 23 q^{59} + 23 q^{61} - 8 q^{63} + 40 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} + 4 q^{73} - 48 q^{75} + 3 q^{77} + 29 q^{79} + 23 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} + 15 q^{89} - 16 q^{91} - 42 q^{93} + 5 q^{95} - 13 q^{97} - 47 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 3 * q^11 + 16 * q^13 + 8 * q^15 + 8 * q^17 + 7 * q^19 + 3 * q^21 + 10 * q^23 + 8 * q^25 - 6 * q^27 + 11 * q^29 + 14 * q^31 + 3 * q^33 - 5 * q^35 + 13 * q^37 - 6 * q^39 + 11 * q^41 - 11 * q^43 - 4 * q^45 + 7 * q^47 + 7 * q^49 - 24 * q^51 + 9 * q^53 - 8 * q^55 - 3 * q^57 - 23 * q^59 + 23 * q^61 - 8 * q^63 + 40 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 + 4 * q^73 - 48 * q^75 + 3 * q^77 + 29 * q^79 + 23 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 + 15 * q^89 - 16 * q^91 - 42 * q^93 + 5 * q^95 - 13 * q^97 - 47 * q^99

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