LMFDB - Embedded newform 8028.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
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} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 38x^3 - 46x^2 - 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 892)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.206062$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.98220 q^{5} +2.92860 q^{7} +O(q^{10})$	$q+2.98220 q^{5} +2.92860 q^{7} -5.20687 q^{11} -3.57206 q^{13} -1.15097 q^{17} +0.707705 q^{19} +8.41293 q^{23} +3.89352 q^{25} +6.28563 q^{29} -3.16210 q^{31} +8.73368 q^{35} -5.59973 q^{37} +2.13466 q^{41} -3.38331 q^{43} -3.36596 q^{47} +1.57671 q^{49} +12.1384 q^{53} -15.5279 q^{55} +5.38548 q^{59} -0.765535 q^{61} -10.6526 q^{65} +1.36966 q^{67} +10.8021 q^{71} +6.71064 q^{73} -15.2488 q^{77} +8.16352 q^{79} +15.4302 q^{83} -3.43242 q^{85} -12.0436 q^{89} -10.4611 q^{91} +2.11052 q^{95} -0.0407865 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} - q^{7}+O(q^{10}) $ 7 * q + 7 * q^5 - q^7	$ 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	2.98220	1.33368	0.666840	−	0.745201i	$-0.267646\pi$
$5$	2.98220	1.33368	0.666840	+	0.745201i	$0.267646\pi$
$6$	0	0
$7$	2.92860	1.10691	0.553454	−	0.832880i	$-0.313310\pi$
$7$	2.92860	1.10691	0.553454	+	0.832880i	$0.313310\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.20687	−1.56993	−0.784965	−	0.619540i	$-0.787319\pi$
$11$	−5.20687	−1.56993	−0.784965	+	0.619540i	$0.787319\pi$
$12$	0	0
$13$	−3.57206	−0.990710	−0.495355	−	0.868691i	$-0.664962\pi$
$13$	−3.57206	−0.990710	−0.495355	+	0.868691i	$0.664962\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.15097	−0.279151	−0.139575	−	0.990211i	$-0.544574\pi$
$17$	−1.15097	−0.279151	−0.139575	+	0.990211i	$0.544574\pi$
$18$	0	0
$19$	0.707705	0.162359	0.0811793	−	0.996700i	$-0.474131\pi$
$19$	0.707705	0.162359	0.0811793	+	0.996700i	$0.474131\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.41293	1.75422	0.877109	−	0.480292i	$-0.159469\pi$
$23$	8.41293	1.75422	0.877109	+	0.480292i	$0.159469\pi$
$24$	0	0
$25$	3.89352	0.778703
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.28563	1.16721	0.583606	−	0.812037i	$-0.301641\pi$
$29$	6.28563	1.16721	0.583606	+	0.812037i	$0.301641\pi$
$30$	0	0
$31$	−3.16210	−0.567931	−0.283965	−	0.958835i	$-0.591650\pi$
$31$	−3.16210	−0.567931	−0.283965	+	0.958835i	$0.591650\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.73368	1.47626
$36$	0	0
$37$	−5.59973	−0.920589	−0.460295	−	0.887766i	$-0.652256\pi$
$37$	−5.59973	−0.920589	−0.460295	+	0.887766i	$0.652256\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.13466	0.333378	0.166689	−	0.986009i	$-0.446692\pi$
$41$	2.13466	0.333378	0.166689	+	0.986009i	$0.446692\pi$
$42$	0	0
$43$	−3.38331	−0.515950	−0.257975	−	0.966152i	$-0.583055\pi$
$43$	−3.38331	−0.515950	−0.257975	+	0.966152i	$0.583055\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.36596	−0.490976	−0.245488	−	0.969400i	$-0.578948\pi$
$47$	−3.36596	−0.490976	−0.245488	+	0.969400i	$0.578948\pi$
$48$	0	0
$49$	1.57671	0.225244
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.1384	1.66734	0.833671	−	0.552261i	$-0.186235\pi$
$53$	12.1384	1.66734	0.833671	+	0.552261i	$0.186235\pi$
$54$	0	0
$55$	−15.5279	−2.09378
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.38548	0.701130	0.350565	−	0.936538i	$-0.385990\pi$
$59$	5.38548	0.701130	0.350565	+	0.936538i	$0.385990\pi$
$60$	0	0
$61$	−0.765535	−0.0980167	−0.0490083	−	0.998798i	$-0.515606\pi$
$61$	−0.765535	−0.0980167	−0.0490083	+	0.998798i	$0.515606\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.6526	−1.32129
$66$	0	0
$67$	1.36966	0.167331	0.0836654	−	0.996494i	$-0.473337\pi$
$67$	1.36966	0.167331	0.0836654	+	0.996494i	$0.473337\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8021	1.28197	0.640985	−	0.767553i	$-0.278526\pi$
$71$	10.8021	1.28197	0.640985	+	0.767553i	$0.278526\pi$
$72$	0	0
$73$	6.71064	0.785421	0.392710	−	0.919662i	$-0.371537\pi$
$73$	6.71064	0.785421	0.392710	+	0.919662i	$0.371537\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.2488	−1.73777
$78$	0	0
$79$	8.16352	0.918468	0.459234	−	0.888315i	$-0.348124\pi$
$79$	8.16352	0.918468	0.459234	+	0.888315i	$0.348124\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.4302	1.69368	0.846840	−	0.531847i	$-0.178502\pi$
$83$	15.4302	1.69368	0.846840	+	0.531847i	$0.178502\pi$
$84$	0	0
$85$	−3.43242	−0.372298
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.0436	−1.27662	−0.638309	−	0.769780i	$-0.720366\pi$
$89$	−12.0436	−1.27662	−0.638309	+	0.769780i	$0.720366\pi$
$90$	0	0
$91$	−10.4611	−1.09662
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.11052	0.216535
$96$	0	0
$97$	−0.0407865	−0.00414125	−0.00207062	−	0.999998i	$-0.500659\pi$
$97$	−0.0407865	−0.00414125	−0.00207062	+	0.999998i	$0.500659\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.0646	1.10097	0.550485	−	0.834845i	$-0.314443\pi$
$101$	11.0646	1.10097	0.550485	+	0.834845i	$0.314443\pi$
$102$	0	0
$103$	14.3445	1.41340	0.706701	−	0.707512i	$-0.250183\pi$
$103$	14.3445	1.41340	0.706701	+	0.707512i	$0.250183\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1834	1.46784	0.733919	−	0.679237i	$-0.237689\pi$
$107$	15.1834	1.46784	0.733919	+	0.679237i	$0.237689\pi$
$108$	0	0
$109$	17.1765	1.64521	0.822607	−	0.568610i	$-0.192519\pi$
$109$	17.1765	1.64521	0.822607	+	0.568610i	$0.192519\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.0562	1.04008	0.520040	−	0.854142i	$-0.325917\pi$
$113$	11.0562	1.04008	0.520040	+	0.854142i	$0.325917\pi$
$114$	0	0
$115$	25.0890	2.33956
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.37073	−0.308994
$120$	0	0
$121$	16.1115	1.46468
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.29975	−0.295139
$126$	0	0
$127$	−0.638436	−0.0566520	−0.0283260	−	0.999599i	$-0.509018\pi$
$127$	−0.638436	−0.0566520	−0.0283260	+	0.999599i	$0.509018\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.4273	−1.17315	−0.586575	−	0.809895i	$-0.699524\pi$
$131$	−13.4273	−1.17315	−0.586575	+	0.809895i	$0.699524\pi$
$132$	0	0
$133$	2.07259	0.179716
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.24320	−0.704264	−0.352132	−	0.935950i	$-0.614543\pi$
$137$	−8.24320	−0.704264	−0.352132	+	0.935950i	$0.614543\pi$
$138$	0	0
$139$	−9.35366	−0.793367	−0.396683	−	0.917955i	$-0.629839\pi$
$139$	−9.35366	−0.793367	−0.396683	+	0.917955i	$0.629839\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.5992	1.55535
$144$	0	0
$145$	18.7450	1.55669
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.9112	1.30350	0.651750	−	0.758434i	$-0.274035\pi$
$149$	15.9112	1.30350	0.651750	+	0.758434i	$0.274035\pi$
$150$	0	0
$151$	6.97525	0.567638	0.283819	−	0.958878i	$-0.408399\pi$
$151$	6.97525	0.567638	0.283819	+	0.958878i	$0.408399\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.43003	−0.757438
$156$	0	0
$157$	−11.3578	−0.906452	−0.453226	−	0.891396i	$-0.649727\pi$
$157$	−11.3578	−0.906452	−0.453226	+	0.891396i	$0.649727\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.6381	1.94176
$162$	0	0
$163$	0.0452103	0.00354114	0.00177057	−	0.999998i	$-0.499436\pi$
$163$	0.0452103	0.00354114	0.00177057	+	0.999998i	$0.499436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1875	−1.02048	−0.510239	−	0.860033i	$-0.670443\pi$
$167$	−13.1875	−1.02048	−0.510239	+	0.860033i	$0.670443\pi$
$168$	0	0
$169$	−0.240413	−0.0184933
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.82763	−0.214980	−0.107490	−	0.994206i	$-0.534281\pi$
$173$	−2.82763	−0.214980	−0.107490	+	0.994206i	$0.534281\pi$
$174$	0	0
$175$	11.4026	0.861953
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.9016	1.48751	0.743756	−	0.668451i	$-0.233042\pi$
$179$	19.9016	1.48751	0.743756	+	0.668451i	$0.233042\pi$
$180$	0	0
$181$	−5.70084	−0.423740	−0.211870	−	0.977298i	$-0.567955\pi$
$181$	−5.70084	−0.423740	−0.211870	+	0.977298i	$0.567955\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.6995	−1.22777
$186$	0	0
$187$	5.99294	0.438247
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.0766069	0.00554308	0.00277154	−	0.999996i	$-0.499118\pi$
$191$	0.0766069	0.00554308	0.00277154	+	0.999996i	$0.499118\pi$
$192$	0	0
$193$	10.9229	0.786248	0.393124	−	0.919485i	$-0.371394\pi$
$193$	10.9229	0.786248	0.393124	+	0.919485i	$0.371394\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.3812	0.882125	0.441063	−	0.897476i	$-0.354602\pi$
$197$	12.3812	0.882125	0.441063	+	0.897476i	$0.354602\pi$
$198$	0	0
$199$	4.81989	0.341673	0.170836	−	0.985299i	$-0.445353\pi$
$199$	4.81989	0.341673	0.170836	+	0.985299i	$0.445353\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.4081	1.29200
$204$	0	0
$205$	6.36599	0.444620
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.68493	−0.254892
$210$	0	0
$211$	8.58524	0.591032	0.295516	−	0.955338i	$-0.404508\pi$
$211$	8.58524	0.591032	0.295516	+	0.955338i	$0.404508\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−10.0897	−0.688112
$216$	0	0
$217$	−9.26055	−0.628647
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.11132	0.276557
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.9840	−0.729036	−0.364518	−	0.931196i	$-0.618766\pi$
$227$	−10.9840	−0.729036	−0.364518	+	0.931196i	$0.618766\pi$
$228$	0	0
$229$	19.3515	1.27878	0.639391	−	0.768882i	$-0.279186\pi$
$229$	19.3515	1.27878	0.639391	+	0.768882i	$0.279186\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.61226	−0.367671	−0.183836	−	0.982957i	$-0.558851\pi$
$233$	−5.61226	−0.367671	−0.183836	+	0.982957i	$0.558851\pi$
$234$	0	0
$235$	−10.0380	−0.654805
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.3688	−1.70566	−0.852828	−	0.522192i	$-0.825114\pi$
$239$	−26.3688	−1.70566	−0.852828	+	0.522192i	$0.825114\pi$
$240$	0	0
$241$	2.24294	0.144480	0.0722401	−	0.997387i	$-0.476985\pi$
$241$	2.24294	0.144480	0.0722401	+	0.997387i	$0.476985\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.70206	0.300404
$246$	0	0
$247$	−2.52796	−0.160850
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.1319	0.702642	0.351321	−	0.936255i	$-0.385733\pi$
$251$	11.1319	0.702642	0.351321	+	0.936255i	$0.385733\pi$
$252$	0	0
$253$	−43.8050	−2.75400
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.11244	−0.131770	−0.0658852	−	0.997827i	$-0.520987\pi$
$257$	−2.11244	−0.131770	−0.0658852	+	0.997827i	$0.520987\pi$
$258$	0	0
$259$	−16.3994	−1.01901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.75335	0.416430	0.208215	−	0.978083i	$-0.433235\pi$
$263$	6.75335	0.416430	0.208215	+	0.978083i	$0.433235\pi$
$264$	0	0
$265$	36.1992	2.22370
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.9757	−0.852113	−0.426057	−	0.904696i	$-0.640098\pi$
$269$	−13.9757	−0.852113	−0.426057	+	0.904696i	$0.640098\pi$
$270$	0	0
$271$	−18.3289	−1.11340	−0.556702	−	0.830712i	$-0.687933\pi$
$271$	−18.3289	−1.11340	−0.556702	+	0.830712i	$0.687933\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.2730	−1.22251
$276$	0	0
$277$	−24.3461	−1.46281	−0.731407	−	0.681941i	$-0.761136\pi$
$277$	−24.3461	−1.46281	−0.731407	+	0.681941i	$0.761136\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.4030	−1.45576	−0.727880	−	0.685705i	$-0.759494\pi$
$281$	−24.4030	−1.45576	−0.727880	+	0.685705i	$0.759494\pi$
$282$	0	0
$283$	23.6103	1.40349	0.701744	−	0.712429i	$-0.252405\pi$
$283$	23.6103	1.40349	0.701744	+	0.712429i	$0.252405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.25158	0.369019
$288$	0	0
$289$	−15.6753	−0.922075
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.3426	−1.88948	−0.944738	−	0.327826i	$-0.893684\pi$
$293$	−32.3426	−1.88948	−0.944738	+	0.327826i	$0.893684\pi$
$294$	0	0
$295$	16.0606	0.935083
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−30.0515	−1.73792
$300$	0	0
$301$	−9.90837	−0.571109
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.28298	−0.130723
$306$	0	0
$307$	17.2032	0.981841	0.490921	−	0.871204i	$-0.336660\pi$
$307$	17.2032	0.981841	0.490921	+	0.871204i	$0.336660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.31619	0.414863	0.207431	−	0.978250i	$-0.433490\pi$
$311$	7.31619	0.414863	0.207431	+	0.978250i	$0.433490\pi$
$312$	0	0
$313$	−31.9735	−1.80725	−0.903624	−	0.428326i	$-0.859103\pi$
$313$	−31.9735	−1.80725	−0.903624	+	0.428326i	$0.859103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.1720	1.24530	0.622652	−	0.782499i	$-0.286055\pi$
$317$	22.1720	1.24530	0.622652	+	0.782499i	$0.286055\pi$
$318$	0	0
$319$	−32.7285	−1.83244
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.814546	−0.0453225
$324$	0	0
$325$	−13.9079	−0.771469
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.85757	−0.543465
$330$	0	0
$331$	31.1443	1.71185	0.855923	−	0.517103i	$-0.172990\pi$
$331$	31.1443	1.71185	0.855923	+	0.517103i	$0.172990\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.08461	0.223166
$336$	0	0
$337$	−10.5025	−0.572105	−0.286053	−	0.958214i	$-0.592343\pi$
$337$	−10.5025	−0.572105	−0.286053	+	0.958214i	$0.592343\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.4647	0.891611
$342$	0	0
$343$	−15.8827	−0.857583
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.1308	0.919629	0.459815	−	0.888015i	$-0.347916\pi$
$347$	17.1308	0.919629	0.459815	+	0.888015i	$0.347916\pi$
$348$	0	0
$349$	−0.502793	−0.0269139	−0.0134570	−	0.999909i	$-0.504284\pi$
$349$	−0.502793	−0.0269139	−0.0134570	+	0.999909i	$0.504284\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.6891	−0.888272	−0.444136	−	0.895959i	$-0.646489\pi$
$353$	−16.6891	−0.888272	−0.444136	+	0.895959i	$0.646489\pi$
$354$	0	0
$355$	32.2140	1.70974
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.3022	1.44095	0.720477	−	0.693479i	$-0.243923\pi$
$359$	27.3022	1.44095	0.720477	+	0.693479i	$0.243923\pi$
$360$	0	0
$361$	−18.4992	−0.973640
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.0125	1.04750
$366$	0	0
$367$	−5.92501	−0.309283	−0.154641	−	0.987971i	$-0.549422\pi$
$367$	−5.92501	−0.309283	−0.154641	+	0.987971i	$0.549422\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.5486	1.84559
$372$	0	0
$373$	−18.9096	−0.979103	−0.489551	−	0.871974i	$-0.662839\pi$
$373$	−18.9096	−0.979103	−0.489551	+	0.871974i	$0.662839\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.4526	−1.15637
$378$	0	0
$379$	−8.13485	−0.417859	−0.208930	−	0.977931i	$-0.566998\pi$
$379$	−8.13485	−0.417859	−0.208930	+	0.977931i	$0.566998\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.4946	−1.66040	−0.830199	−	0.557467i	$-0.811773\pi$
$383$	−32.4946	−1.66040	−0.830199	+	0.557467i	$0.811773\pi$
$384$	0	0
$385$	−45.4751	−2.31763
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.3087	1.58741	0.793706	−	0.608301i	$-0.208149\pi$
$389$	31.3087	1.58741	0.793706	+	0.608301i	$0.208149\pi$
$390$	0	0
$391$	−9.68301	−0.489691
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.3453	1.22494
$396$	0	0
$397$	−14.9439	−0.750012	−0.375006	−	0.927022i	$-0.622359\pi$
$397$	−14.9439	−0.750012	−0.375006	+	0.927022i	$0.622359\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.4212	−0.770098	−0.385049	−	0.922896i	$-0.625815\pi$
$401$	−15.4212	−0.770098	−0.385049	+	0.922896i	$0.625815\pi$
$402$	0	0
$403$	11.2952	0.562655
$404$	0	0
$405$	0	0
$406$	0	0
$407$	29.1570	1.44526
$408$	0	0
$409$	−20.1215	−0.994942	−0.497471	−	0.867481i	$-0.665738\pi$
$409$	−20.1215	−0.994942	−0.497471	+	0.867481i	$0.665738\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.7719	0.776086
$414$	0	0
$415$	46.0158	2.25883
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.75221	0.427573	0.213787	−	0.976880i	$-0.431420\pi$
$419$	8.75221	0.427573	0.213787	+	0.976880i	$0.431420\pi$
$420$	0	0
$421$	17.0547	0.831197	0.415599	−	0.909548i	$-0.363572\pi$
$421$	17.0547	0.831197	0.415599	+	0.909548i	$0.363572\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.48131	−0.217376
$426$	0	0
$427$	−2.24195	−0.108495
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.05579	−0.243529	−0.121764	−	0.992559i	$-0.538855\pi$
$431$	−5.05579	−0.243529	−0.121764	+	0.992559i	$0.538855\pi$
$432$	0	0
$433$	10.2145	0.490875	0.245438	−	0.969412i	$-0.421068\pi$
$433$	10.2145	0.490875	0.245438	+	0.969412i	$0.421068\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.95387	0.284812
$438$	0	0
$439$	−30.4380	−1.45273	−0.726363	−	0.687312i	$-0.758791\pi$
$439$	−30.4380	−1.45273	−0.726363	+	0.687312i	$0.758791\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0371	1.14204	0.571019	−	0.820937i	$-0.306548\pi$
$443$	24.0371	1.14204	0.571019	+	0.820937i	$0.306548\pi$
$444$	0	0
$445$	−35.9164	−1.70260
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.2255	0.529765	0.264883	−	0.964281i	$-0.414667\pi$
$449$	11.2255	0.529765	0.264883	+	0.964281i	$0.414667\pi$
$450$	0	0
$451$	−11.1149	−0.523381
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−31.1972	−1.46255
$456$	0	0
$457$	25.3943	1.18790	0.593948	−	0.804504i	$-0.297569\pi$
$457$	25.3943	1.18790	0.593948	+	0.804504i	$0.297569\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−37.1611	−1.73077	−0.865383	−	0.501110i	$-0.832925\pi$
$461$	−37.1611	−1.73077	−0.865383	+	0.501110i	$0.832925\pi$
$462$	0	0
$463$	−14.8349	−0.689437	−0.344718	−	0.938706i	$-0.612026\pi$
$463$	−14.8349	−0.689437	−0.344718	+	0.938706i	$0.612026\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.93220	−0.320784	−0.160392	−	0.987053i	$-0.551276\pi$
$467$	−6.93220	−0.320784	−0.160392	+	0.987053i	$0.551276\pi$
$468$	0	0
$469$	4.01120	0.185220
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17.6164	0.810005
$474$	0	0
$475$	2.75546	0.126429
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.9787	−1.91805	−0.959027	−	0.283315i	$-0.908566\pi$
$479$	−41.9787	−1.91805	−0.959027	+	0.283315i	$0.908566\pi$
$480$	0	0
$481$	20.0025	0.912037
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.121634	−0.00552310
$486$	0	0
$487$	−20.5694	−0.932090	−0.466045	−	0.884761i	$-0.654321\pi$
$487$	−20.5694	−0.932090	−0.466045	+	0.884761i	$0.654321\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.74689	0.214224	0.107112	−	0.994247i	$-0.465840\pi$
$491$	4.74689	0.214224	0.107112	+	0.994247i	$0.465840\pi$
$492$	0	0
$493$	−7.23456	−0.325828
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.6350	1.41902
$498$	0	0
$499$	21.1148	0.945229	0.472614	−	0.881269i	$-0.343310\pi$
$499$	21.1148	0.945229	0.472614	+	0.881269i	$0.343310\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	40.9178	1.82443	0.912217	−	0.409708i	$-0.134369\pi$
$503$	40.9178	1.82443	0.912217	+	0.409708i	$0.134369\pi$
$504$	0	0
$505$	32.9969	1.46834
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.1732	0.495244	0.247622	−	0.968857i	$-0.420351\pi$
$509$	11.1732	0.495244	0.247622	+	0.968857i	$0.420351\pi$
$510$	0	0
$511$	19.6528	0.869388
$512$	0	0
$513$	0	0
$514$	0	0
$515$	42.7781	1.88503
$516$	0	0
$517$	17.5261	0.770798
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.72325	0.206929	0.103465	−	0.994633i	$-0.467007\pi$
$521$	4.72325	0.206929	0.103465	+	0.994633i	$0.467007\pi$
$522$	0	0
$523$	11.1788	0.488813	0.244406	−	0.969673i	$-0.421407\pi$
$523$	11.1788	0.488813	0.244406	+	0.969673i	$0.421407\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.63948	0.158538
$528$	0	0
$529$	47.7774	2.07728
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.62514	−0.330281
$534$	0	0
$535$	45.2800	1.95763
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8.20971	−0.353617
$540$	0	0
$541$	−7.99659	−0.343800	−0.171900	−	0.985114i	$-0.554991\pi$
$541$	−7.99659	−0.343800	−0.171900	+	0.985114i	$0.554991\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	51.2239	2.19419
$546$	0	0
$547$	3.53823	0.151284	0.0756419	−	0.997135i	$-0.475899\pi$
$547$	3.53823	0.151284	0.0756419	+	0.997135i	$0.475899\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.44837	0.189507
$552$	0	0
$553$	23.9077	1.01666
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.8276	−1.43332	−0.716662	−	0.697421i	$-0.754331\pi$
$557$	−33.8276	−1.43332	−0.716662	+	0.697421i	$0.754331\pi$
$558$	0	0
$559$	12.0854	0.511157
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.13647	−0.385056	−0.192528	−	0.981291i	$-0.561669\pi$
$563$	−9.13647	−0.385056	−0.192528	+	0.981291i	$0.561669\pi$
$564$	0	0
$565$	32.9718	1.38713
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.9698	−1.17255	−0.586277	−	0.810111i	$-0.699407\pi$
$569$	−27.9698	−1.17255	−0.586277	+	0.810111i	$0.699407\pi$
$570$	0	0
$571$	24.0470	1.00634	0.503168	−	0.864189i	$-0.332168\pi$
$571$	24.0470	1.00634	0.503168	+	0.864189i	$0.332168\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	32.7559	1.36601
$576$	0	0
$577$	−39.9871	−1.66469	−0.832343	−	0.554261i	$-0.813001\pi$
$577$	−39.9871	−1.66469	−0.832343	+	0.554261i	$0.813001\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.1888	1.87475
$582$	0	0
$583$	−63.2032	−2.61761
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1739	0.791393	0.395696	−	0.918381i	$-0.370503\pi$
$587$	19.1739	0.791393	0.395696	+	0.918381i	$0.370503\pi$
$588$	0	0
$589$	−2.23784	−0.0922085
$590$	0	0
$591$	0	0
$592$	0	0
$593$	35.9817	1.47759	0.738795	−	0.673930i	$-0.235395\pi$
$593$	35.9817	1.47759	0.738795	+	0.673930i	$0.235395\pi$
$594$	0	0
$595$	−10.0522	−0.412099
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.5944	0.514592	0.257296	−	0.966333i	$-0.417168\pi$
$599$	12.5944	0.514592	0.257296	+	0.966333i	$0.417168\pi$
$600$	0	0
$601$	0.111962	0.00456703	0.00228351	−	0.999997i	$-0.499273\pi$
$601$	0.111962	0.00456703	0.00228351	+	0.999997i	$0.499273\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	48.0476	1.95341
$606$	0	0
$607$	−37.6891	−1.52975	−0.764876	−	0.644177i	$-0.777200\pi$
$607$	−37.6891	−1.52975	−0.764876	+	0.644177i	$0.777200\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0234	0.486415
$612$	0	0
$613$	26.9089	1.08684	0.543419	−	0.839462i	$-0.317129\pi$
$613$	26.9089	1.08684	0.543419	+	0.839462i	$0.317129\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.7381	1.55954	0.779768	−	0.626069i	$-0.215337\pi$
$617$	38.7381	1.55954	0.779768	+	0.626069i	$0.215337\pi$
$618$	0	0
$619$	−31.6913	−1.27378	−0.636891	−	0.770954i	$-0.719780\pi$
$619$	−31.6913	−1.27378	−0.636891	+	0.770954i	$0.719780\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−35.2709	−1.41310
$624$	0	0
$625$	−29.3081	−1.17232
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.44511	0.256983
$630$	0	0
$631$	11.0219	0.438775	0.219388	−	0.975638i	$-0.429594\pi$
$631$	11.0219	0.438775	0.219388	+	0.975638i	$0.429594\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.90394	−0.0755557
$636$	0	0
$637$	−5.63209	−0.223152
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.1024	−1.14948	−0.574738	−	0.818337i	$-0.694896\pi$
$641$	−29.1024	−1.14948	−0.574738	+	0.818337i	$0.694896\pi$
$642$	0	0
$643$	−17.4304	−0.687387	−0.343694	−	0.939082i	$-0.611678\pi$
$643$	−17.4304	−0.687387	−0.343694	+	0.939082i	$0.611678\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.8861	1.21426	0.607129	−	0.794604i	$-0.292321\pi$
$647$	30.8861	1.21426	0.607129	+	0.794604i	$0.292321\pi$
$648$	0	0
$649$	−28.0415	−1.10072
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28.3230	−1.10837	−0.554183	−	0.832395i	$-0.686969\pi$
$653$	−28.3230	−1.10837	−0.554183	+	0.832395i	$0.686969\pi$
$654$	0	0
$655$	−40.0429	−1.56461
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.4176	−0.873265	−0.436632	−	0.899640i	$-0.643829\pi$
$659$	−22.4176	−0.873265	−0.436632	+	0.899640i	$0.643829\pi$
$660$	0	0
$661$	36.9684	1.43790	0.718952	−	0.695059i	$-0.244622\pi$
$661$	36.9684	1.43790	0.718952	+	0.695059i	$0.244622\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.18086	0.239684
$666$	0	0
$667$	52.8806	2.04754
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.98604	0.153879
$672$	0	0
$673$	7.13722	0.275120	0.137560	−	0.990493i	$-0.456074\pi$
$673$	7.13722	0.275120	0.137560	+	0.990493i	$0.456074\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.3307	−1.81907	−0.909533	−	0.415631i	$-0.863561\pi$
$677$	−47.3307	−1.81907	−0.909533	+	0.415631i	$0.863561\pi$
$678$	0	0
$679$	−0.119448	−0.00458398
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.8928	1.52646	0.763228	−	0.646129i	$-0.223613\pi$
$683$	39.8928	1.52646	0.763228	+	0.646129i	$0.223613\pi$
$684$	0	0
$685$	−24.5829	−0.939263
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.3592	−1.65185
$690$	0	0
$691$	44.5988	1.69662	0.848308	−	0.529502i	$-0.177621\pi$
$691$	44.5988	1.69662	0.848308	+	0.529502i	$0.177621\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27.8945	−1.05810
$696$	0	0
$697$	−2.45693	−0.0930629
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.5468	−1.04043	−0.520214	−	0.854036i	$-0.674148\pi$
$701$	−27.5468	−1.04043	−0.520214	+	0.854036i	$0.674148\pi$
$702$	0	0
$703$	−3.96295	−0.149466
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.4038	1.21867
$708$	0	0
$709$	40.0715	1.50492	0.752459	−	0.658639i	$-0.228867\pi$
$709$	40.0715	1.50492	0.752459	+	0.658639i	$0.228867\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−26.6026	−0.996274
$714$	0	0
$715$	55.4666	2.07433
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.1706	−0.752237	−0.376118	−	0.926572i	$-0.622741\pi$
$719$	−20.1706	−0.752237	−0.376118	+	0.926572i	$0.622741\pi$
$720$	0	0
$721$	42.0092	1.56451
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.4732	0.908912
$726$	0	0
$727$	16.8007	0.623103	0.311551	−	0.950229i	$-0.399151\pi$
$727$	16.8007	0.623103	0.311551	+	0.950229i	$0.399151\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.89408	0.144028
$732$	0	0
$733$	17.1422	0.633161	0.316580	−	0.948566i	$-0.397465\pi$
$733$	17.1422	0.633161	0.316580	+	0.948566i	$0.397465\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.13165	−0.262698
$738$	0	0
$739$	23.3169	0.857727	0.428864	−	0.903369i	$-0.358914\pi$
$739$	23.3169	0.857727	0.428864	+	0.903369i	$0.358914\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.3512	1.33360	0.666798	−	0.745239i	$-0.267665\pi$
$743$	36.3512	1.33360	0.666798	+	0.745239i	$0.267665\pi$
$744$	0	0
$745$	47.4505	1.73845
$746$	0	0
$747$	0	0
$748$	0	0
$749$	44.4662	1.62476
$750$	0	0
$751$	−3.85110	−0.140529	−0.0702643	−	0.997528i	$-0.522384\pi$
$751$	−3.85110	−0.140529	−0.0702643	+	0.997528i	$0.522384\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.8016	0.757047
$756$	0	0
$757$	−36.2643	−1.31805	−0.659024	−	0.752122i	$-0.729030\pi$
$757$	−36.2643	−1.31805	−0.659024	+	0.752122i	$0.729030\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.96295	−0.252407	−0.126203	−	0.992004i	$-0.540279\pi$
$761$	−6.96295	−0.252407	−0.126203	+	0.992004i	$0.540279\pi$
$762$	0	0
$763$	50.3032	1.82110
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.2372	−0.694617
$768$	0	0
$769$	5.23229	0.188681	0.0943405	−	0.995540i	$-0.469926\pi$
$769$	5.23229	0.188681	0.0943405	+	0.995540i	$0.469926\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.7051	0.672776	0.336388	−	0.941723i	$-0.390795\pi$
$773$	18.7051	0.672776	0.336388	+	0.941723i	$0.390795\pi$
$774$	0	0
$775$	−12.3117	−0.442250
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.51071	0.0541269
$780$	0	0
$781$	−56.2450	−2.01260
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33.8712	−1.20892
$786$	0	0
$787$	−21.6846	−0.772973	−0.386487	−	0.922295i	$-0.626311\pi$
$787$	−21.6846	−0.772973	−0.386487	+	0.922295i	$0.626311\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.3792	1.15127
$792$	0	0
$793$	2.73453	0.0971061
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.0182	0.886190	0.443095	−	0.896475i	$-0.353880\pi$
$797$	25.0182	0.886190	0.443095	+	0.896475i	$0.353880\pi$
$798$	0	0
$799$	3.87412	0.137056
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.9414	−1.23306
$804$	0	0
$805$	73.4758	2.58968
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.7092	−1.14999	−0.574997	−	0.818156i	$-0.694997\pi$
$809$	−32.7092	−1.14999	−0.574997	+	0.818156i	$0.694997\pi$
$810$	0	0
$811$	42.6994	1.49938	0.749689	−	0.661791i	$-0.230203\pi$
$811$	42.6994	1.49938	0.749689	+	0.661791i	$0.230203\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.134826	0.00472275
$816$	0	0
$817$	−2.39438	−0.0837689
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.5713	−0.438740	−0.219370	−	0.975642i	$-0.570400\pi$
$821$	−12.5713	−0.438740	−0.219370	+	0.975642i	$0.570400\pi$
$822$	0	0
$823$	−33.0342	−1.15150	−0.575750	−	0.817626i	$-0.695290\pi$
$823$	−33.0342	−1.15150	−0.575750	+	0.817626i	$0.695290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−31.5473	−1.09701	−0.548504	−	0.836148i	$-0.684802\pi$
$827$	−31.5473	−1.09701	−0.548504	+	0.836148i	$0.684802\pi$
$828$	0	0
$829$	45.6954	1.58707	0.793534	−	0.608526i	$-0.208239\pi$
$829$	45.6954	1.58707	0.793534	+	0.608526i	$0.208239\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.81474	−0.0628771
$834$	0	0
$835$	−39.3277	−1.36099
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.1321	−0.660512	−0.330256	−	0.943891i	$-0.607135\pi$
$839$	−19.1321	−0.660512	−0.330256	+	0.943891i	$0.607135\pi$
$840$	0	0
$841$	10.5092	0.362385
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.716960	−0.0246642
$846$	0	0
$847$	47.1841	1.62126
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−47.1101	−1.61491
$852$	0	0
$853$	−33.1471	−1.13494	−0.567468	−	0.823396i	$-0.692077\pi$
$853$	−33.1471	−1.13494	−0.567468	+	0.823396i	$0.692077\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.6068	0.806392	0.403196	−	0.915114i	$-0.367899\pi$
$857$	23.6068	0.806392	0.403196	+	0.915114i	$0.367899\pi$
$858$	0	0
$859$	10.3327	0.352549	0.176274	−	0.984341i	$-0.443595\pi$
$859$	10.3327	0.352549	0.176274	+	0.984341i	$0.443595\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.2884	−1.30335	−0.651676	−	0.758498i	$-0.725934\pi$
$863$	−38.2884	−1.30335	−0.651676	+	0.758498i	$0.725934\pi$
$864$	0	0
$865$	−8.43255	−0.286715
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−42.5064	−1.44193
$870$	0	0
$871$	−4.89251	−0.165776
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.66366	−0.326691
$876$	0	0
$877$	−15.3817	−0.519402	−0.259701	−	0.965689i	$-0.583624\pi$
$877$	−15.3817	−0.519402	−0.259701	+	0.965689i	$0.583624\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.2453	1.08637	0.543185	−	0.839613i	$-0.317218\pi$
$881$	32.2453	1.08637	0.543185	+	0.839613i	$0.317218\pi$
$882$	0	0
$883$	−32.0184	−1.07750	−0.538752	−	0.842464i	$-0.681104\pi$
$883$	−32.0184	−1.07750	−0.538752	+	0.842464i	$0.681104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.2511	0.579236	0.289618	−	0.957142i	$-0.406472\pi$
$887$	17.2511	0.579236	0.289618	+	0.957142i	$0.406472\pi$
$888$	0	0
$889$	−1.86972	−0.0627085
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.38211	−0.0797142
$894$	0	0
$895$	59.3504	1.98387
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.8758	−0.662896
$900$	0	0
$901$	−13.9709	−0.465440
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.0010	−0.565134
$906$	0	0
$907$	20.5674	0.682931	0.341465	−	0.939894i	$-0.389077\pi$
$907$	20.5674	0.682931	0.341465	+	0.939894i	$0.389077\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.6685	0.784172	0.392086	−	0.919929i	$-0.371754\pi$
$911$	23.6685	0.784172	0.392086	+	0.919929i	$0.371754\pi$
$912$	0	0
$913$	−80.3428	−2.65896
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−39.3232	−1.29857
$918$	0	0
$919$	48.1573	1.58856	0.794281	−	0.607551i	$-0.207848\pi$
$919$	48.1573	1.58856	0.794281	+	0.607551i	$0.207848\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−38.5856	−1.27006
$924$	0	0
$925$	−21.8026	−0.716866
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.51250	−0.279286	−0.139643	−	0.990202i	$-0.544595\pi$
$929$	−8.51250	−0.279286	−0.139643	+	0.990202i	$0.544595\pi$
$930$	0	0
$931$	1.11584	0.0365703
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.8721	0.584481
$936$	0	0
$937$	27.2488	0.890181	0.445090	−	0.895486i	$-0.353172\pi$
$937$	27.2488	0.890181	0.445090	+	0.895486i	$0.353172\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−55.6700	−1.81479	−0.907394	−	0.420280i	$-0.861932\pi$
$941$	−55.6700	−1.81479	−0.907394	+	0.420280i	$0.861932\pi$
$942$	0	0
$943$	17.9588	0.584818
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.0339	0.975971	0.487985	−	0.872852i	$-0.337732\pi$
$947$	30.0339	0.975971	0.487985	+	0.872852i	$0.337732\pi$
$948$	0	0
$949$	−23.9708	−0.778124
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.11916	−0.263005	−0.131503	−	0.991316i	$-0.541980\pi$
$953$	−8.11916	−0.263005	−0.131503	+	0.991316i	$0.541980\pi$
$954$	0	0
$955$	0.228457	0.00739270
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.1410	−0.779555
$960$	0	0
$961$	−21.0011	−0.677455
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.5743	1.04860
$966$	0	0
$967$	−2.71932	−0.0874474	−0.0437237	−	0.999044i	$-0.513922\pi$
$967$	−2.71932	−0.0874474	−0.0437237	+	0.999044i	$0.513922\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28.3198	−0.908826	−0.454413	−	0.890791i	$-0.650151\pi$
$971$	−28.3198	−0.908826	−0.454413	+	0.890791i	$0.650151\pi$
$972$	0	0
$973$	−27.3931	−0.878184
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.95029	0.0943881	0.0471941	−	0.998886i	$-0.484972\pi$
$977$	2.95029	0.0943881	0.0471941	+	0.998886i	$0.484972\pi$
$978$	0	0
$979$	62.7094	2.00420
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.5342	−1.67558	−0.837790	−	0.545993i	$-0.816152\pi$
$983$	−52.5342	−1.67558	−0.837790	+	0.545993i	$0.816152\pi$
$984$	0	0
$985$	36.9233	1.17647
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.4635	−0.905088
$990$	0	0
$991$	14.4527	0.459105	0.229552	−	0.973296i	$-0.426274\pi$
$991$	14.4527	0.459105	0.229552	+	0.973296i	$0.426274\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.3739	0.455682
$996$	0	0
$997$	−22.8137	−0.722518	−0.361259	−	0.932466i	$-0.617653\pi$
$997$	−22.8137	−0.722518	−0.361259	+	0.932466i	$0.617653\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.j.1.6		7
3.2	odd	2		892.2.a.d.1.4	✓	7
12.11	even	2		3568.2.a.m.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
892.2.a.d.1.4	✓	7	3.2	odd	2
3568.2.a.m.1.4		7	12.11	even	2
8028.2.a.j.1.6		7	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.98220 q^{5} +2.92860 q^{7} +O(q^{10})\)	\(q+2.98220 q^{5} +2.92860 q^{7} -5.20687 q^{11} -3.57206 q^{13} -1.15097 q^{17} +0.707705 q^{19} +8.41293 q^{23} +3.89352 q^{25} +6.28563 q^{29} -3.16210 q^{31} +8.73368 q^{35} -5.59973 q^{37} +2.13466 q^{41} -3.38331 q^{43} -3.36596 q^{47} +1.57671 q^{49} +12.1384 q^{53} -15.5279 q^{55} +5.38548 q^{59} -0.765535 q^{61} -10.6526 q^{65} +1.36966 q^{67} +10.8021 q^{71} +6.71064 q^{73} -15.2488 q^{77} +8.16352 q^{79} +15.4302 q^{83} -3.43242 q^{85} -12.0436 q^{89} -10.4611 q^{91} +2.11052 q^{95} -0.0407865 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - q^{7}+O(q^{10}) \) 7 * q + 7 * q^5 - q^7	\( 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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