LMFDB - Embedded newform 8037.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.93999$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93999 q^{2} +1.76355 q^{4} -1.45338 q^{5} -4.17438 q^{7} +0.458705 q^{8} +O(q^{10})$	\(q-1.93999 q^{2} +1.76355 q^{4} -1.45338 q^{5} -4.17438 q^{7} +0.458705 q^{8} +2.81953 q^{10} +3.52893 q^{11} -3.57964 q^{13} +8.09824 q^{14} -4.41699 q^{16} -3.85424 q^{17} +1.00000 q^{19} -2.56310 q^{20} -6.84609 q^{22} +2.10804 q^{23} -2.88770 q^{25} +6.94445 q^{26} -7.36174 q^{28} -4.78245 q^{29} +3.51534 q^{31} +7.65149 q^{32} +7.47718 q^{34} +6.06694 q^{35} +1.52551 q^{37} -1.93999 q^{38} -0.666671 q^{40} +1.32045 q^{41} +7.20851 q^{43} +6.22346 q^{44} -4.08957 q^{46} -1.00000 q^{47} +10.4254 q^{49} +5.60210 q^{50} -6.31288 q^{52} +4.61989 q^{53} -5.12887 q^{55} -1.91481 q^{56} +9.27790 q^{58} +8.71732 q^{59} +5.42152 q^{61} -6.81971 q^{62} -6.00983 q^{64} +5.20256 q^{65} -7.86699 q^{67} -6.79715 q^{68} -11.7698 q^{70} +16.1949 q^{71} -8.47421 q^{73} -2.95947 q^{74} +1.76355 q^{76} -14.7311 q^{77} -9.31782 q^{79} +6.41954 q^{80} -2.56166 q^{82} +3.21925 q^{83} +5.60165 q^{85} -13.9844 q^{86} +1.61874 q^{88} -15.8126 q^{89} +14.9428 q^{91} +3.71764 q^{92} +1.93999 q^{94} -1.45338 q^{95} -10.8533 q^{97} -20.2252 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

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 $

$2$	−1.93999	−1.37178	−0.685889	−	0.727706i	$-0.740587\pi$
$2$	−1.93999	−1.37178	−0.685889	+	0.727706i	$0.740587\pi$
$3$	0	0
$3$	0	0
$4$	1.76355	0.881776
$5$	−1.45338	−0.649969	−0.324985	−	0.945719i	$-0.605359\pi$
$5$	−1.45338	−0.649969	−0.324985	+	0.945719i	$0.605359\pi$
$6$	0	0
$7$	−4.17438	−1.57777	−0.788884	−	0.614543i	$-0.789341\pi$
$7$	−4.17438	−1.57777	−0.788884	+	0.614543i	$0.789341\pi$
$8$	0.458705	0.162177
$9$	0	0
$10$	2.81953	0.891614
$11$	3.52893	1.06401	0.532007	−	0.846740i	$-0.321438\pi$
$11$	3.52893	1.06401	0.532007	+	0.846740i	$0.321438\pi$
$12$	0	0
$13$	−3.57964	−0.992813	−0.496406	−	0.868090i	$-0.665347\pi$
$13$	−3.57964	−0.992813	−0.496406	+	0.868090i	$0.665347\pi$
$14$	8.09824	2.16435
$15$	0	0
$16$	−4.41699	−1.10425
$17$	−3.85424	−0.934790	−0.467395	−	0.884049i	$-0.654807\pi$
$17$	−3.85424	−0.934790	−0.467395	+	0.884049i	$0.654807\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.56310	−0.573127
$21$	0	0
$22$	−6.84609	−1.45959
$23$	2.10804	0.439557	0.219778	−	0.975550i	$-0.429467\pi$
$23$	2.10804	0.439557	0.219778	+	0.975550i	$0.429467\pi$
$24$	0	0
$25$	−2.88770	−0.577540
$26$	6.94445	1.36192
$27$	0	0
$28$	−7.36174	−1.39124
$29$	−4.78245	−0.888079	−0.444039	−	0.896007i	$-0.646455\pi$
$29$	−4.78245	−0.888079	−0.444039	+	0.896007i	$0.646455\pi$
$30$	0	0
$31$	3.51534	0.631373	0.315687	−	0.948864i	$-0.397765\pi$
$31$	3.51534	0.631373	0.315687	+	0.948864i	$0.397765\pi$
$32$	7.65149	1.35261
$33$	0	0
$34$	7.47718	1.28232
$35$	6.06694	1.02550
$36$	0	0
$37$	1.52551	0.250792	0.125396	−	0.992107i	$-0.459980\pi$
$37$	1.52551	0.250792	0.125396	+	0.992107i	$0.459980\pi$
$38$	−1.93999	−0.314708
$39$	0	0
$40$	−0.666671	−0.105410
$41$	1.32045	0.206220	0.103110	−	0.994670i	$-0.467121\pi$
$41$	1.32045	0.206220	0.103110	+	0.994670i	$0.467121\pi$
$42$	0	0
$43$	7.20851	1.09929	0.549643	−	0.835399i	$-0.314764\pi$
$43$	7.20851	1.09929	0.549643	+	0.835399i	$0.314764\pi$
$44$	6.22346	0.938222
$45$	0	0
$46$	−4.08957	−0.602974
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	10.4254	1.48935
$50$	5.60210	0.792257
$51$	0	0
$52$	−6.31288	−0.875439
$53$	4.61989	0.634591	0.317296	−	0.948327i	$-0.397225\pi$
$53$	4.61989	0.634591	0.317296	+	0.948327i	$0.397225\pi$
$54$	0	0
$55$	−5.12887	−0.691576
$56$	−1.91481	−0.255877
$57$	0	0
$58$	9.27790	1.21825
$59$	8.71732	1.13490	0.567449	−	0.823408i	$-0.307930\pi$
$59$	8.71732	1.13490	0.567449	+	0.823408i	$0.307930\pi$
$60$	0	0
$61$	5.42152	0.694154	0.347077	−	0.937837i	$-0.387174\pi$
$61$	5.42152	0.694154	0.347077	+	0.937837i	$0.387174\pi$
$62$	−6.81971	−0.866104
$63$	0	0
$64$	−6.00983	−0.751228
$65$	5.20256	0.645298
$66$	0	0
$67$	−7.86699	−0.961106	−0.480553	−	0.876966i	$-0.659564\pi$
$67$	−7.86699	−0.961106	−0.480553	+	0.876966i	$0.659564\pi$
$68$	−6.79715	−0.824276
$69$	0	0
$70$	−11.7698	−1.40676
$71$	16.1949	1.92198	0.960989	−	0.276587i	$-0.0892035\pi$
$71$	16.1949	1.92198	0.960989	+	0.276587i	$0.0892035\pi$
$72$	0	0
$73$	−8.47421	−0.991832	−0.495916	−	0.868371i	$-0.665168\pi$
$73$	−8.47421	−0.991832	−0.495916	+	0.868371i	$0.665168\pi$
$74$	−2.95947	−0.344031
$75$	0	0
$76$	1.76355	0.202293
$77$	−14.7311	−1.67877
$78$	0	0
$79$	−9.31782	−1.04834	−0.524168	−	0.851615i	$-0.675624\pi$
$79$	−9.31782	−1.04834	−0.524168	+	0.851615i	$0.675624\pi$
$80$	6.41954	0.717726
$81$	0	0
$82$	−2.56166	−0.282888
$83$	3.21925	0.353358	0.176679	−	0.984268i	$-0.443464\pi$
$83$	3.21925	0.353358	0.176679	+	0.984268i	$0.443464\pi$
$84$	0	0
$85$	5.60165	0.607585
$86$	−13.9844	−1.50798
$87$	0	0
$88$	1.61874	0.172558
$89$	−15.8126	−1.67614	−0.838069	−	0.545565i	$-0.816315\pi$
$89$	−15.8126	−1.67614	−0.838069	+	0.545565i	$0.816315\pi$
$90$	0	0
$91$	14.9428	1.56643
$92$	3.71764	0.387591
$93$	0	0
$94$	1.93999	0.200094
$95$	−1.45338	−0.149113
$96$	0	0
$97$	−10.8533	−1.10199	−0.550994	−	0.834509i	$-0.685751\pi$
$97$	−10.8533	−1.10199	−0.550994	+	0.834509i	$0.685751\pi$
$98$	−20.2252	−2.04306
$99$	0	0
$100$	−5.09261	−0.509261
$101$	−0.587586	−0.0584670	−0.0292335	−	0.999573i	$-0.509307\pi$
$101$	−0.587586	−0.0584670	−0.0292335	+	0.999573i	$0.509307\pi$
$102$	0	0
$103$	−5.13018	−0.505492	−0.252746	−	0.967533i	$-0.581334\pi$
$103$	−5.13018	−0.505492	−0.252746	+	0.967533i	$0.581334\pi$
$104$	−1.64200	−0.161011
$105$	0	0
$106$	−8.96253	−0.870518
$107$	20.4805	1.97992	0.989962	−	0.141333i	$-0.0451389\pi$
$107$	20.4805	1.97992	0.989962	+	0.141333i	$0.0451389\pi$
$108$	0	0
$109$	5.17345	0.495526	0.247763	−	0.968821i	$-0.420305\pi$
$109$	5.17345	0.495526	0.247763	+	0.968821i	$0.420305\pi$
$110$	9.94994	0.948689
$111$	0	0
$112$	18.4382	1.74224
$113$	−2.46314	−0.231713	−0.115857	−	0.993266i	$-0.536961\pi$
$113$	−2.46314	−0.231713	−0.115857	+	0.993266i	$0.536961\pi$
$114$	0	0
$115$	−3.06377	−0.285698
$116$	−8.43410	−0.783087
$117$	0	0
$118$	−16.9115	−1.55683
$119$	16.0891	1.47488
$120$	0	0
$121$	1.45338	0.132125
$122$	−10.5177	−0.952226
$123$	0	0
$124$	6.19948	0.556730
$125$	11.4638	1.02535
$126$	0	0
$127$	−2.05534	−0.182382	−0.0911908	−	0.995833i	$-0.529067\pi$
$127$	−2.05534	−0.182382	−0.0911908	+	0.995833i	$0.529067\pi$
$128$	−3.64400	−0.322087
$129$	0	0
$130$	−10.0929	−0.885206
$131$	10.9837	0.959650	0.479825	−	0.877364i	$-0.340700\pi$
$131$	10.9837	0.959650	0.479825	+	0.877364i	$0.340700\pi$
$132$	0	0
$133$	−4.17438	−0.361965
$134$	15.2619	1.31842
$135$	0	0
$136$	−1.76796	−0.151601
$137$	−13.9553	−1.19228	−0.596139	−	0.802881i	$-0.703299\pi$
$137$	−13.9553	−1.19228	−0.596139	+	0.802881i	$0.703299\pi$
$138$	0	0
$139$	5.03818	0.427333	0.213666	−	0.976907i	$-0.431459\pi$
$139$	5.03818	0.427333	0.213666	+	0.976907i	$0.431459\pi$
$140$	10.6994	0.904262
$141$	0	0
$142$	−31.4179	−2.63653
$143$	−12.6323	−1.05637
$144$	0	0
$145$	6.95070	0.577224
$146$	16.4399	1.36057
$147$	0	0
$148$	2.69032	0.221143
$149$	17.7925	1.45762	0.728808	−	0.684718i	$-0.240075\pi$
$149$	17.7925	1.45762	0.728808	+	0.684718i	$0.240075\pi$
$150$	0	0
$151$	9.04775	0.736295	0.368148	−	0.929767i	$-0.379992\pi$
$151$	9.04775	0.736295	0.368148	+	0.929767i	$0.379992\pi$
$152$	0.458705	0.0372059
$153$	0	0
$154$	28.5782	2.30290
$155$	−5.10910	−0.410373
$156$	0	0
$157$	12.1102	0.966498	0.483249	−	0.875483i	$-0.339457\pi$
$157$	12.1102	0.966498	0.483249	+	0.875483i	$0.339457\pi$
$158$	18.0765	1.43809
$159$	0	0
$160$	−11.1205	−0.879152
$161$	−8.79975	−0.693518
$162$	0	0
$163$	6.45692	0.505745	0.252873	−	0.967500i	$-0.418625\pi$
$163$	6.45692	0.505745	0.252873	+	0.967500i	$0.418625\pi$
$164$	2.32869	0.181840
$165$	0	0
$166$	−6.24530	−0.484730
$167$	2.28309	0.176671	0.0883355	−	0.996091i	$-0.471845\pi$
$167$	2.28309	0.176671	0.0883355	+	0.996091i	$0.471845\pi$
$168$	0	0
$169$	−0.186193	−0.0143225
$170$	−10.8671	−0.833472
$171$	0	0
$172$	12.7126	0.969325
$173$	0.469862	0.0357230	0.0178615	−	0.999840i	$-0.494314\pi$
$173$	0.469862	0.0357230	0.0178615	+	0.999840i	$0.494314\pi$
$174$	0	0
$175$	12.0544	0.911224
$176$	−15.5873	−1.17493
$177$	0	0
$178$	30.6763	2.29929
$179$	−21.9645	−1.64170	−0.820852	−	0.571141i	$-0.806501\pi$
$179$	−21.9645	−1.64170	−0.820852	+	0.571141i	$0.806501\pi$
$180$	0	0
$181$	18.2881	1.35934	0.679672	−	0.733517i	$-0.262122\pi$
$181$	18.2881	1.35934	0.679672	+	0.733517i	$0.262122\pi$
$182$	−28.9888	−2.14879
$183$	0	0
$184$	0.966968	0.0712858
$185$	−2.21714	−0.163007
$186$	0	0
$187$	−13.6014	−0.994630
$188$	−1.76355	−0.128620
$189$	0	0
$190$	2.81953	0.204550
$191$	13.6340	0.986523	0.493261	−	0.869881i	$-0.335805\pi$
$191$	13.6340	0.986523	0.493261	+	0.869881i	$0.335805\pi$
$192$	0	0
$193$	21.0290	1.51370	0.756851	−	0.653588i	$-0.226737\pi$
$193$	21.0290	1.51370	0.756851	+	0.653588i	$0.226737\pi$
$194$	21.0553	1.51168
$195$	0	0
$196$	18.3858	1.31327
$197$	−19.1731	−1.36603	−0.683013	−	0.730406i	$-0.739331\pi$
$197$	−19.1731	−1.36603	−0.683013	+	0.730406i	$0.739331\pi$
$198$	0	0
$199$	5.10720	0.362040	0.181020	−	0.983479i	$-0.442060\pi$
$199$	5.10720	0.362040	0.181020	+	0.983479i	$0.442060\pi$
$200$	−1.32460	−0.0936636
$201$	0	0
$202$	1.13991	0.0802038
$203$	19.9638	1.40118
$204$	0	0
$205$	−1.91911	−0.134037
$206$	9.95248	0.693422
$207$	0	0
$208$	15.8112	1.09631
$209$	3.52893	0.244102
$210$	0	0
$211$	−5.55541	−0.382450	−0.191225	−	0.981546i	$-0.561246\pi$
$211$	−5.55541	−0.382450	−0.191225	+	0.981546i	$0.561246\pi$
$212$	8.14742	0.559567
$213$	0	0
$214$	−39.7319	−2.71602
$215$	−10.4767	−0.714503
$216$	0	0
$217$	−14.6743	−0.996160
$218$	−10.0364	−0.679752
$219$	0	0
$220$	−9.04503	−0.609816
$221$	13.7968	0.928072
$222$	0	0
$223$	0.525240	0.0351727	0.0175863	−	0.999845i	$-0.494402\pi$
$223$	0.525240	0.0351727	0.0175863	+	0.999845i	$0.494402\pi$
$224$	−31.9402	−2.13410
$225$	0	0
$226$	4.77847	0.317859
$227$	8.44069	0.560228	0.280114	−	0.959967i	$-0.409628\pi$
$227$	8.44069	0.560228	0.280114	+	0.959967i	$0.409628\pi$
$228$	0	0
$229$	−9.27034	−0.612602	−0.306301	−	0.951935i	$-0.599091\pi$
$229$	−9.27034	−0.612602	−0.306301	+	0.951935i	$0.599091\pi$
$230$	5.94368	0.391915
$231$	0	0
$232$	−2.19373	−0.144026
$233$	12.9187	0.846330	0.423165	−	0.906053i	$-0.360919\pi$
$233$	12.9187	0.846330	0.423165	+	0.906053i	$0.360919\pi$
$234$	0	0
$235$	1.45338	0.0948078
$236$	15.3735	1.00073
$237$	0	0
$238$	−31.2126	−2.02321
$239$	−18.1991	−1.17720	−0.588601	−	0.808423i	$-0.700321\pi$
$239$	−18.1991	−1.17720	−0.588601	+	0.808423i	$0.700321\pi$
$240$	0	0
$241$	−22.3645	−1.44062	−0.720312	−	0.693650i	$-0.756001\pi$
$241$	−22.3645	−1.44062	−0.720312	+	0.693650i	$0.756001\pi$
$242$	−2.81954	−0.181247
$243$	0	0
$244$	9.56113	0.612089
$245$	−15.1521	−0.968031
$246$	0	0
$247$	−3.57964	−0.227767
$248$	1.61250	0.102394
$249$	0	0
$250$	−22.2396	−1.40656
$251$	16.0757	1.01469	0.507345	−	0.861743i	$-0.330627\pi$
$251$	16.0757	1.01469	0.507345	+	0.861743i	$0.330627\pi$
$252$	0	0
$253$	7.43913	0.467694
$254$	3.98733	0.250187
$255$	0	0
$256$	19.0890	1.19306
$257$	12.3251	0.768819	0.384410	−	0.923163i	$-0.374405\pi$
$257$	12.3251	0.768819	0.384410	+	0.923163i	$0.374405\pi$
$258$	0	0
$259$	−6.36806	−0.395692
$260$	9.17498	0.569008
$261$	0	0
$262$	−21.3082	−1.31643
$263$	24.4653	1.50860	0.754298	−	0.656533i	$-0.227978\pi$
$263$	24.4653	1.50860	0.754298	+	0.656533i	$0.227978\pi$
$264$	0	0
$265$	−6.71444	−0.412465
$266$	8.09824	0.496535
$267$	0	0
$268$	−13.8739	−0.847480
$269$	−14.2527	−0.869002	−0.434501	−	0.900671i	$-0.643075\pi$
$269$	−14.2527	−0.869002	−0.434501	+	0.900671i	$0.643075\pi$
$270$	0	0
$271$	11.1230	0.675677	0.337838	−	0.941204i	$-0.390304\pi$
$271$	11.1230	0.675677	0.337838	+	0.941204i	$0.390304\pi$
$272$	17.0241	1.03224
$273$	0	0
$274$	27.0730	1.63554
$275$	−10.1905	−0.614511
$276$	0	0
$277$	12.4429	0.747623	0.373812	−	0.927505i	$-0.378051\pi$
$277$	12.4429	0.747623	0.373812	+	0.927505i	$0.378051\pi$
$278$	−9.77400	−0.586206
$279$	0	0
$280$	2.78294	0.166312
$281$	−0.144023	−0.00859172	−0.00429586	−	0.999991i	$-0.501367\pi$
$281$	−0.144023	−0.00859172	−0.00429586	+	0.999991i	$0.501367\pi$
$282$	0	0
$283$	−6.50189	−0.386497	−0.193249	−	0.981150i	$-0.561902\pi$
$283$	−6.50189	−0.386497	−0.193249	+	0.981150i	$0.561902\pi$
$284$	28.5605	1.69475
$285$	0	0
$286$	24.5065	1.44910
$287$	−5.51207	−0.325367
$288$	0	0
$289$	−2.14485	−0.126167
$290$	−13.4843	−0.791823
$291$	0	0
$292$	−14.9447	−0.874574
$293$	−8.89949	−0.519914	−0.259957	−	0.965620i	$-0.583708\pi$
$293$	−8.89949	−0.519914	−0.259957	+	0.965620i	$0.583708\pi$
$294$	0	0
$295$	−12.6695	−0.737649
$296$	0.699759	0.0406727
$297$	0	0
$298$	−34.5172	−1.99953
$299$	−7.54602	−0.436397
$300$	0	0
$301$	−30.0910	−1.73442
$302$	−17.5525	−1.01003
$303$	0	0
$304$	−4.41699	−0.253332
$305$	−7.87950	−0.451179
$306$	0	0
$307$	−19.4627	−1.11079	−0.555397	−	0.831586i	$-0.687434\pi$
$307$	−19.4627	−1.11079	−0.555397	+	0.831586i	$0.687434\pi$
$308$	−25.9791	−1.48030
$309$	0	0
$310$	9.91160	0.562941
$311$	28.1838	1.59816	0.799078	−	0.601227i	$-0.205321\pi$
$311$	28.1838	1.59816	0.799078	+	0.601227i	$0.205321\pi$
$312$	0	0
$313$	−31.1743	−1.76208	−0.881038	−	0.473046i	$-0.843154\pi$
$313$	−31.1743	−1.76208	−0.881038	+	0.473046i	$0.843154\pi$
$314$	−23.4936	−1.32582
$315$	0	0
$316$	−16.4325	−0.924398
$317$	−1.21663	−0.0683330	−0.0341665	−	0.999416i	$-0.510878\pi$
$317$	−1.21663	−0.0683330	−0.0341665	+	0.999416i	$0.510878\pi$
$318$	0	0
$319$	−16.8770	−0.944928
$320$	8.73453	0.488275
$321$	0	0
$322$	17.0714	0.951353
$323$	−3.85424	−0.214456
$324$	0	0
$325$	10.3369	0.573389
$326$	−12.5264	−0.693770
$327$	0	0
$328$	0.605698	0.0334441
$329$	4.17438	0.230141
$330$	0	0
$331$	17.5229	0.963148	0.481574	−	0.876405i	$-0.340065\pi$
$331$	17.5229	0.963148	0.481574	+	0.876405i	$0.340065\pi$
$332$	5.67731	0.311583
$333$	0	0
$334$	−4.42917	−0.242354
$335$	11.4337	0.624689
$336$	0	0
$337$	−5.93760	−0.323442	−0.161721	−	0.986837i	$-0.551704\pi$
$337$	−5.93760	−0.323442	−0.161721	+	0.986837i	$0.551704\pi$
$338$	0.361211	0.0196473
$339$	0	0
$340$	9.87881	0.535754
$341$	12.4054	0.671790
$342$	0	0
$343$	−14.2991	−0.772079
$344$	3.30658	0.178279
$345$	0	0
$346$	−0.911527	−0.0490040
$347$	13.7315	0.737144	0.368572	−	0.929599i	$-0.379847\pi$
$347$	13.7315	0.737144	0.368572	+	0.929599i	$0.379847\pi$
$348$	0	0
$349$	1.48849	0.0796771	0.0398386	−	0.999206i	$-0.487316\pi$
$349$	1.48849	0.0796771	0.0398386	+	0.999206i	$0.487316\pi$
$350$	−23.3853	−1.25000
$351$	0	0
$352$	27.0016	1.43919
$353$	−22.5072	−1.19794	−0.598969	−	0.800772i	$-0.704423\pi$
$353$	−22.5072	−1.19794	−0.598969	+	0.800772i	$0.704423\pi$
$354$	0	0
$355$	−23.5372	−1.24923
$356$	−27.8864	−1.47798
$357$	0	0
$358$	42.6109	2.25206
$359$	−5.24285	−0.276707	−0.138354	−	0.990383i	$-0.544181\pi$
$359$	−5.24285	−0.276707	−0.138354	+	0.990383i	$0.544181\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−35.4787	−1.86472
$363$	0	0
$364$	26.3524	1.38124
$365$	12.3162	0.644660
$366$	0	0
$367$	−31.0699	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$367$	−31.0699	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$368$	−9.31118	−0.485379
$369$	0	0
$370$	4.30122	0.223610
$371$	−19.2852	−1.00124
$372$	0	0
$373$	−34.6922	−1.79629	−0.898147	−	0.439696i	$-0.855086\pi$
$373$	−34.6922	−1.79629	−0.898147	+	0.439696i	$0.855086\pi$
$374$	26.3865	1.36441
$375$	0	0
$376$	−0.458705	−0.0236559
$377$	17.1194	0.881696
$378$	0	0
$379$	25.7134	1.32081	0.660404	−	0.750911i	$-0.270385\pi$
$379$	25.7134	1.32081	0.660404	+	0.750911i	$0.270385\pi$
$380$	−2.56310	−0.131484
$381$	0	0
$382$	−26.4498	−1.35329
$383$	1.28054	0.0654327	0.0327164	−	0.999465i	$-0.489584\pi$
$383$	1.28054	0.0654327	0.0327164	+	0.999465i	$0.489584\pi$
$384$	0	0
$385$	21.4098	1.09115
$386$	−40.7960	−2.07646
$387$	0	0
$388$	−19.1404	−0.971706
$389$	−2.39063	−0.121210	−0.0606049	−	0.998162i	$-0.519303\pi$
$389$	−2.39063	−0.121210	−0.0606049	+	0.998162i	$0.519303\pi$
$390$	0	0
$391$	−8.12488	−0.410893
$392$	4.78220	0.241538
$393$	0	0
$394$	37.1956	1.87389
$395$	13.5423	0.681386
$396$	0	0
$397$	−16.9533	−0.850860	−0.425430	−	0.904991i	$-0.639877\pi$
$397$	−16.9533	−0.850860	−0.425430	+	0.904991i	$0.639877\pi$
$398$	−9.90790	−0.496638
$399$	0	0
$400$	12.7549	0.637747
$401$	−14.1496	−0.706598	−0.353299	−	0.935510i	$-0.614940\pi$
$401$	−14.1496	−0.706598	−0.353299	+	0.935510i	$0.614940\pi$
$402$	0	0
$403$	−12.5836	−0.626835
$404$	−1.03624	−0.0515548
$405$	0	0
$406$	−38.7295	−1.92211
$407$	5.38342	0.266846
$408$	0	0
$409$	−7.51986	−0.371833	−0.185917	−	0.982566i	$-0.559525\pi$
$409$	−7.51986	−0.371833	−0.185917	+	0.982566i	$0.559525\pi$
$410$	3.72305	0.183868
$411$	0	0
$412$	−9.04734	−0.445730
$413$	−36.3894	−1.79061
$414$	0	0
$415$	−4.67878	−0.229672
$416$	−27.3896	−1.34288
$417$	0	0
$418$	−6.84609	−0.334853
$419$	−38.7799	−1.89452	−0.947261	−	0.320464i	$-0.896161\pi$
$419$	−38.7799	−1.89452	−0.947261	+	0.320464i	$0.896161\pi$
$420$	0	0
$421$	20.4566	0.996996	0.498498	−	0.866891i	$-0.333885\pi$
$421$	20.4566	0.996996	0.498498	+	0.866891i	$0.333885\pi$
$422$	10.7774	0.524637
$423$	0	0
$424$	2.11917	0.102916
$425$	11.1299	0.539879
$426$	0	0
$427$	−22.6315	−1.09521
$428$	36.1184	1.74585
$429$	0	0
$430$	20.3246	0.980139
$431$	−18.4202	−0.887272	−0.443636	−	0.896207i	$-0.646312\pi$
$431$	−18.4202	−0.887272	−0.443636	+	0.896207i	$0.646312\pi$
$432$	0	0
$433$	−10.3924	−0.499426	−0.249713	−	0.968320i	$-0.580336\pi$
$433$	−10.3924	−0.499426	−0.249713	+	0.968320i	$0.580336\pi$
$434$	28.4681	1.36651
$435$	0	0
$436$	9.12364	0.436943
$437$	2.10804	0.100841
$438$	0	0
$439$	15.3018	0.730314	0.365157	−	0.930946i	$-0.381015\pi$
$439$	15.3018	0.730314	0.365157	+	0.930946i	$0.381015\pi$
$440$	−2.35264	−0.112158
$441$	0	0
$442$	−26.7656	−1.27311
$443$	−36.6998	−1.74366	−0.871831	−	0.489807i	$-0.837067\pi$
$443$	−36.6998	−1.74366	−0.871831	+	0.489807i	$0.837067\pi$
$444$	0	0
$445$	22.9817	1.08944
$446$	−1.01896	−0.0482491
$447$	0	0
$448$	25.0873	1.18526
$449$	−27.7462	−1.30942	−0.654711	−	0.755879i	$-0.727210\pi$
$449$	−27.7462	−1.30942	−0.654711	+	0.755879i	$0.727210\pi$
$450$	0	0
$451$	4.65979	0.219421
$452$	−4.34388	−0.204319
$453$	0	0
$454$	−16.3748	−0.768509
$455$	−21.7174	−1.01813
$456$	0	0
$457$	−20.0997	−0.940224	−0.470112	−	0.882607i	$-0.655787\pi$
$457$	−20.0997	−0.940224	−0.470112	+	0.882607i	$0.655787\pi$
$458$	17.9844	0.840354
$459$	0	0
$460$	−5.40312	−0.251922
$461$	27.8024	1.29489	0.647443	−	0.762114i	$-0.275838\pi$
$461$	27.8024	1.29489	0.647443	+	0.762114i	$0.275838\pi$
$462$	0	0
$463$	9.63521	0.447786	0.223893	−	0.974614i	$-0.428123\pi$
$463$	9.63521	0.447786	0.223893	+	0.974614i	$0.428123\pi$
$464$	21.1240	0.980658
$465$	0	0
$466$	−25.0621	−1.16098
$467$	11.8665	0.549117	0.274558	−	0.961570i	$-0.411468\pi$
$467$	11.8665	0.549117	0.274558	+	0.961570i	$0.411468\pi$
$468$	0	0
$469$	32.8398	1.51640
$470$	−2.81953	−0.130055
$471$	0	0
$472$	3.99868	0.184054
$473$	25.4383	1.16966
$474$	0	0
$475$	−2.88770	−0.132497
$476$	28.3739	1.30052
$477$	0	0
$478$	35.3061	1.61486
$479$	−21.9546	−1.00313	−0.501566	−	0.865120i	$-0.667242\pi$
$479$	−21.9546	−1.00313	−0.501566	+	0.865120i	$0.667242\pi$
$480$	0	0
$481$	−5.46077	−0.248990
$482$	43.3869	1.97622
$483$	0	0
$484$	2.56311	0.116505
$485$	15.7739	0.716258
$486$	0	0
$487$	6.84320	0.310095	0.155048	−	0.987907i	$-0.450447\pi$
$487$	6.84320	0.310095	0.155048	+	0.987907i	$0.450447\pi$
$488$	2.48688	0.112576
$489$	0	0
$490$	29.3949	1.32792
$491$	22.4293	1.01222	0.506109	−	0.862469i	$-0.331083\pi$
$491$	22.4293	1.01222	0.506109	+	0.862469i	$0.331083\pi$
$492$	0	0
$493$	18.4327	0.830167
$494$	6.94445	0.312446
$495$	0	0
$496$	−15.5272	−0.697192
$497$	−67.6035	−3.03243
$498$	0	0
$499$	11.0424	0.494327	0.247164	−	0.968974i	$-0.420501\pi$
$499$	11.0424	0.494327	0.247164	+	0.968974i	$0.420501\pi$
$500$	20.2170	0.904131
$501$	0	0
$502$	−31.1867	−1.39193
$503$	−19.9880	−0.891221	−0.445610	−	0.895227i	$-0.647013\pi$
$503$	−19.9880	−0.891221	−0.445610	+	0.895227i	$0.647013\pi$
$504$	0	0
$505$	0.853984	0.0380018
$506$	−14.4318	−0.641573
$507$	0	0
$508$	−3.62469	−0.160820
$509$	22.9209	1.01595	0.507976	−	0.861371i	$-0.330394\pi$
$509$	22.9209	1.01595	0.507976	+	0.861371i	$0.330394\pi$
$510$	0	0
$511$	35.3746	1.56488
$512$	−29.7444	−1.31453
$513$	0	0
$514$	−23.9106	−1.05465
$515$	7.45608	0.328554
$516$	0	0
$517$	−3.52893	−0.155202
$518$	12.3540	0.542801
$519$	0	0
$520$	2.38644	0.104652
$521$	−39.3293	−1.72305	−0.861523	−	0.507718i	$-0.830489\pi$
$521$	−39.3293	−1.72305	−0.861523	+	0.507718i	$0.830489\pi$
$522$	0	0
$523$	18.0097	0.787507	0.393754	−	0.919216i	$-0.371176\pi$
$523$	18.0097	0.787507	0.393754	+	0.919216i	$0.371176\pi$
$524$	19.3703	0.846197
$525$	0	0
$526$	−47.4624	−2.06946
$527$	−13.5489	−0.590201
$528$	0	0
$529$	−18.5562	−0.806790
$530$	13.0259	0.565810
$531$	0	0
$532$	−7.36174	−0.319172
$533$	−4.72674	−0.204738
$534$	0	0
$535$	−29.7658	−1.28689
$536$	−3.60863	−0.155869
$537$	0	0
$538$	27.6501	1.19208
$539$	36.7907	1.58469
$540$	0	0
$541$	27.5315	1.18367	0.591835	−	0.806059i	$-0.298404\pi$
$541$	27.5315	1.18367	0.591835	+	0.806059i	$0.298404\pi$
$542$	−21.5786	−0.926879
$543$	0	0
$544$	−29.4907	−1.26440
$545$	−7.51896	−0.322077
$546$	0	0
$547$	−13.4068	−0.573232	−0.286616	−	0.958046i	$-0.592530\pi$
$547$	−13.4068	−0.573232	−0.286616	+	0.958046i	$0.592530\pi$
$548$	−24.6108	−1.05132
$549$	0	0
$550$	19.7695	0.842972
$551$	−4.78245	−0.203739
$552$	0	0
$553$	38.8961	1.65403
$554$	−24.1391	−1.02557
$555$	0	0
$556$	8.88509	0.376812
$557$	40.4014	1.71186	0.855931	−	0.517089i	$-0.172984\pi$
$557$	40.4014	1.71186	0.855931	+	0.517089i	$0.172984\pi$
$558$	0	0
$559$	−25.8038	−1.09139
$560$	−26.7976	−1.13241
$561$	0	0
$562$	0.279404	0.0117859
$563$	10.8231	0.456139	0.228070	−	0.973645i	$-0.426759\pi$
$563$	10.8231	0.456139	0.228070	+	0.973645i	$0.426759\pi$
$564$	0	0
$565$	3.57987	0.150606
$566$	12.6136	0.530188
$567$	0	0
$568$	7.42867	0.311700
$569$	26.0708	1.09295	0.546473	−	0.837477i	$-0.315970\pi$
$569$	26.0708	1.09295	0.546473	+	0.837477i	$0.315970\pi$
$570$	0	0
$571$	21.7045	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$571$	21.7045	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$572$	−22.2777	−0.931479
$573$	0	0
$574$	10.6933	0.446331
$575$	−6.08738	−0.253861
$576$	0	0
$577$	−47.9578	−1.99651	−0.998254	−	0.0590661i	$-0.981188\pi$
$577$	−47.9578	−1.99651	−0.998254	+	0.0590661i	$0.981188\pi$
$578$	4.16098	0.173074
$579$	0	0
$580$	12.2579	0.508982
$581$	−13.4384	−0.557517
$582$	0	0
$583$	16.3033	0.675214
$584$	−3.88716	−0.160852
$585$	0	0
$586$	17.2649	0.713207
$587$	27.6370	1.14070	0.570350	−	0.821402i	$-0.306808\pi$
$587$	27.6370	1.14070	0.570350	+	0.821402i	$0.306808\pi$
$588$	0	0
$589$	3.51534	0.144847
$590$	24.5788	1.01189
$591$	0	0
$592$	−6.73816	−0.276937
$593$	−29.7458	−1.22151	−0.610757	−	0.791818i	$-0.709135\pi$
$593$	−29.7458	−1.22151	−0.610757	+	0.791818i	$0.709135\pi$
$594$	0	0
$595$	−23.3834	−0.958627
$596$	31.3780	1.28529
$597$	0	0
$598$	14.6392	0.598641
$599$	23.0641	0.942372	0.471186	−	0.882034i	$-0.343826\pi$
$599$	23.0641	0.942372	0.471186	+	0.882034i	$0.343826\pi$
$600$	0	0
$601$	−23.8230	−0.971761	−0.485881	−	0.874025i	$-0.661501\pi$
$601$	−23.8230	−0.971761	−0.485881	+	0.874025i	$0.661501\pi$
$602$	58.3762	2.37924
$603$	0	0
$604$	15.9562	0.649248
$605$	−2.11231	−0.0858774
$606$	0	0
$607$	−4.73994	−0.192388	−0.0961942	−	0.995363i	$-0.530667\pi$
$607$	−4.73994	−0.192388	−0.0961942	+	0.995363i	$0.530667\pi$
$608$	7.65149	0.310309
$609$	0	0
$610$	15.2861	0.618918
$611$	3.57964	0.144817
$612$	0	0
$613$	−0.258042	−0.0104222	−0.00521111	−	0.999986i	$-0.501659\pi$
$613$	−0.258042	−0.0104222	−0.00521111	+	0.999986i	$0.501659\pi$
$614$	37.7573	1.52376
$615$	0	0
$616$	−6.75724	−0.272257
$617$	5.21546	0.209966	0.104983	−	0.994474i	$-0.466521\pi$
$617$	5.21546	0.209966	0.104983	+	0.994474i	$0.466521\pi$
$618$	0	0
$619$	−5.39607	−0.216886	−0.108443	−	0.994103i	$-0.534587\pi$
$619$	−5.39607	−0.216886	−0.108443	+	0.994103i	$0.534587\pi$
$620$	−9.01017	−0.361857
$621$	0	0
$622$	−54.6762	−2.19232
$623$	66.0080	2.64455
$624$	0	0
$625$	−2.22269	−0.0889074
$626$	60.4777	2.41718
$627$	0	0
$628$	21.3569	0.852235
$629$	−5.87968	−0.234438
$630$	0	0
$631$	48.2396	1.92039	0.960193	−	0.279337i	$-0.0901145\pi$
$631$	48.2396	1.92039	0.960193	+	0.279337i	$0.0901145\pi$
$632$	−4.27413	−0.170016
$633$	0	0
$634$	2.36025	0.0937377
$635$	2.98717	0.118542
$636$	0	0
$637$	−37.3193	−1.47864
$638$	32.7411	1.29623
$639$	0	0
$640$	5.29609	0.209346
$641$	−43.9111	−1.73438	−0.867192	−	0.497974i	$-0.834077\pi$
$641$	−43.9111	−1.73438	−0.867192	+	0.497974i	$0.834077\pi$
$642$	0	0
$643$	29.7505	1.17324	0.586622	−	0.809861i	$-0.300458\pi$
$643$	29.7505	1.17324	0.586622	+	0.809861i	$0.300458\pi$
$644$	−15.5188	−0.611528
$645$	0	0
$646$	7.47718	0.294186
$647$	−30.2161	−1.18792	−0.593959	−	0.804495i	$-0.702436\pi$
$647$	−30.2161	−1.18792	−0.593959	+	0.804495i	$0.702436\pi$
$648$	0	0
$649$	30.7629	1.20755
$650$	−20.0535	−0.786563
$651$	0	0
$652$	11.3871	0.445954
$653$	−29.4055	−1.15072	−0.575362	−	0.817899i	$-0.695139\pi$
$653$	−29.4055	−1.15072	−0.575362	+	0.817899i	$0.695139\pi$
$654$	0	0
$655$	−15.9634	−0.623743
$656$	−5.83242	−0.227718
$657$	0	0
$658$	−8.09824	−0.315702
$659$	−1.18612	−0.0462049	−0.0231024	−	0.999733i	$-0.507354\pi$
$659$	−1.18612	−0.0462049	−0.0231024	+	0.999733i	$0.507354\pi$
$660$	0	0
$661$	−49.2323	−1.91492	−0.957458	−	0.288573i	$-0.906819\pi$
$661$	−49.2323	−1.91492	−0.957458	+	0.288573i	$0.906819\pi$
$662$	−33.9943	−1.32123
$663$	0	0
$664$	1.47669	0.0573065
$665$	6.06694	0.235266
$666$	0	0
$667$	−10.0816	−0.390361
$668$	4.02635	0.155784
$669$	0	0
$670$	−22.1812	−0.856935
$671$	19.1322	0.738590
$672$	0	0
$673$	−22.0935	−0.851642	−0.425821	−	0.904807i	$-0.640015\pi$
$673$	−22.0935	−0.851642	−0.425821	+	0.904807i	$0.640015\pi$
$674$	11.5189	0.443690
$675$	0	0
$676$	−0.328361	−0.0126293
$677$	−13.2043	−0.507481	−0.253740	−	0.967272i	$-0.581661\pi$
$677$	−13.2043	−0.507481	−0.253740	+	0.967272i	$0.581661\pi$
$678$	0	0
$679$	45.3059	1.73868
$680$	2.56951	0.0985361
$681$	0	0
$682$	−24.0663	−0.921547
$683$	−26.7155	−1.02224	−0.511120	−	0.859510i	$-0.670769\pi$
$683$	−26.7155	−1.02224	−0.511120	+	0.859510i	$0.670769\pi$
$684$	0	0
$685$	20.2822	0.774944
$686$	27.7401	1.05912
$687$	0	0
$688$	−31.8399	−1.21388
$689$	−16.5375	−0.630030
$690$	0	0
$691$	−32.7474	−1.24577	−0.622885	−	0.782313i	$-0.714040\pi$
$691$	−32.7474	−1.24577	−0.622885	+	0.782313i	$0.714040\pi$
$692$	0.828627	0.0314997
$693$	0	0
$694$	−26.6389	−1.01120
$695$	−7.32236	−0.277753
$696$	0	0
$697$	−5.08933	−0.192772
$698$	−2.88766	−0.109299
$699$	0	0
$700$	21.2585	0.803495
$701$	16.5943	0.626760	0.313380	−	0.949628i	$-0.398539\pi$
$701$	16.5943	0.626760	0.313380	+	0.949628i	$0.398539\pi$
$702$	0	0
$703$	1.52551	0.0575357
$704$	−21.2083	−0.799317
$705$	0	0
$706$	43.6637	1.64331
$707$	2.45281	0.0922474
$708$	0	0
$709$	−48.4996	−1.82144	−0.910721	−	0.413022i	$-0.864473\pi$
$709$	−48.4996	−1.82144	−0.910721	+	0.413022i	$0.864473\pi$
$710$	45.6619	1.71366
$711$	0	0
$712$	−7.25334	−0.271830
$713$	7.41047	0.277524
$714$	0	0
$715$	18.3595	0.686606
$716$	−38.7356	−1.44762
$717$	0	0
$718$	10.1711	0.379581
$719$	−8.89573	−0.331755	−0.165877	−	0.986146i	$-0.553046\pi$
$719$	−8.89573	−0.331755	−0.165877	+	0.986146i	$0.553046\pi$
$720$	0	0
$721$	21.4153	0.797548
$722$	−1.93999	−0.0721989
$723$	0	0
$724$	32.2520	1.19864
$725$	13.8103	0.512901
$726$	0	0
$727$	−46.9746	−1.74219	−0.871096	−	0.491112i	$-0.836591\pi$
$727$	−46.9746	−1.74219	−0.871096	+	0.491112i	$0.836591\pi$
$728$	6.85432	0.254038
$729$	0	0
$730$	−23.8933	−0.884331
$731$	−27.7833	−1.02760
$732$	0	0
$733$	−0.955564	−0.0352946	−0.0176473	−	0.999844i	$-0.505618\pi$
$733$	−0.955564	−0.0352946	−0.0176473	+	0.999844i	$0.505618\pi$
$734$	60.2752	2.22480
$735$	0	0
$736$	16.1296	0.594547
$737$	−27.7621	−1.02263
$738$	0	0
$739$	−46.4831	−1.70991	−0.854955	−	0.518703i	$-0.826415\pi$
$739$	−46.4831	−1.70991	−0.854955	+	0.518703i	$0.826415\pi$
$740$	−3.91004	−0.143736
$741$	0	0
$742$	37.4130	1.37348
$743$	−4.76761	−0.174907	−0.0874534	−	0.996169i	$-0.527873\pi$
$743$	−4.76761	−0.174907	−0.0874534	+	0.996169i	$0.527873\pi$
$744$	0	0
$745$	−25.8591	−0.947406
$746$	67.3024	2.46412
$747$	0	0
$748$	−23.9867	−0.877041
$749$	−85.4934	−3.12386
$750$	0	0
$751$	21.7978	0.795414	0.397707	−	0.917513i	$-0.369806\pi$
$751$	21.7978	0.795414	0.397707	+	0.917513i	$0.369806\pi$
$752$	4.41699	0.161071
$753$	0	0
$754$	−33.2115	−1.20949
$755$	−13.1498	−0.478569
$756$	0	0
$757$	4.70994	0.171186	0.0855929	−	0.996330i	$-0.472722\pi$
$757$	4.70994	0.171186	0.0855929	+	0.996330i	$0.472722\pi$
$758$	−49.8836	−1.81185
$759$	0	0
$760$	−0.666671	−0.0241827
$761$	30.7869	1.11603	0.558013	−	0.829832i	$-0.311564\pi$
$761$	30.7869	1.11603	0.558013	+	0.829832i	$0.311564\pi$
$762$	0	0
$763$	−21.5959	−0.781825
$764$	24.0443	0.869892
$765$	0	0
$766$	−2.48424	−0.0897592
$767$	−31.2049	−1.12674
$768$	0	0
$769$	30.4527	1.09815	0.549077	−	0.835772i	$-0.314979\pi$
$769$	30.4527	1.09815	0.549077	+	0.835772i	$0.314979\pi$
$770$	−41.5348	−1.49681
$771$	0	0
$772$	37.0858	1.33475
$773$	−38.6312	−1.38947	−0.694733	−	0.719267i	$-0.744478\pi$
$773$	−38.6312	−1.38947	−0.694733	+	0.719267i	$0.744478\pi$
$774$	0	0
$775$	−10.1512	−0.364643
$776$	−4.97847	−0.178717
$777$	0	0
$778$	4.63780	0.166273
$779$	1.32045	0.0473101
$780$	0	0
$781$	57.1506	2.04501
$782$	15.7622	0.563654
$783$	0	0
$784$	−46.0491	−1.64461
$785$	−17.6006	−0.628194
$786$	0	0
$787$	−11.3158	−0.403366	−0.201683	−	0.979451i	$-0.564641\pi$
$787$	−11.3158	−0.403366	−0.201683	+	0.979451i	$0.564641\pi$
$788$	−33.8128	−1.20453
$789$	0	0
$790$	−26.2719	−0.934711
$791$	10.2821	0.365589
$792$	0	0
$793$	−19.4071	−0.689165
$794$	32.8891	1.16719
$795$	0	0
$796$	9.00681	0.319238
$797$	50.8196	1.80012	0.900061	−	0.435764i	$-0.143522\pi$
$797$	50.8196	1.80012	0.900061	+	0.435764i	$0.143522\pi$
$798$	0	0
$799$	3.85424	0.136353
$800$	−22.0952	−0.781184
$801$	0	0
$802$	27.4501	0.969296
$803$	−29.9049	−1.05532
$804$	0	0
$805$	12.7893	0.450765
$806$	24.4121	0.859879
$807$	0	0
$808$	−0.269529	−0.00948199
$809$	11.2204	0.394488	0.197244	−	0.980354i	$-0.436801\pi$
$809$	11.2204	0.394488	0.197244	+	0.980354i	$0.436801\pi$
$810$	0	0
$811$	−3.01587	−0.105901	−0.0529507	−	0.998597i	$-0.516863\pi$
$811$	−3.01587	−0.105901	−0.0529507	+	0.998597i	$0.516863\pi$
$812$	35.2072	1.23553
$813$	0	0
$814$	−10.4438	−0.366054
$815$	−9.38433	−0.328719
$816$	0	0
$817$	7.20851	0.252194
$818$	14.5884	0.510073
$819$	0	0
$820$	−3.38445	−0.118190
$821$	15.7491	0.549646	0.274823	−	0.961495i	$-0.411381\pi$
$821$	15.7491	0.549646	0.274823	+	0.961495i	$0.411381\pi$
$822$	0	0
$823$	15.9505	0.556000	0.278000	−	0.960581i	$-0.410328\pi$
$823$	15.9505	0.556000	0.278000	+	0.960581i	$0.410328\pi$
$824$	−2.35324	−0.0819790
$825$	0	0
$826$	70.5950	2.45631
$827$	−8.47694	−0.294772	−0.147386	−	0.989079i	$-0.547086\pi$
$827$	−8.47694	−0.294772	−0.147386	+	0.989079i	$0.547086\pi$
$828$	0	0
$829$	14.0677	0.488593	0.244296	−	0.969701i	$-0.421443\pi$
$829$	14.0677	0.488593	0.244296	+	0.969701i	$0.421443\pi$
$830$	9.07677	0.315059
$831$	0	0
$832$	21.5130	0.745829
$833$	−40.1821	−1.39223
$834$	0	0
$835$	−3.31819	−0.114831
$836$	6.22346	0.215243
$837$	0	0
$838$	75.2325	2.59886
$839$	−38.7149	−1.33659	−0.668294	−	0.743897i	$-0.732975\pi$
$839$	−38.7149	−1.33659	−0.668294	+	0.743897i	$0.732975\pi$
$840$	0	0
$841$	−6.12816	−0.211316
$842$	−39.6856	−1.36766
$843$	0	0
$844$	−9.79725	−0.337235
$845$	0.270608	0.00930919
$846$	0	0
$847$	−6.06696	−0.208463
$848$	−20.4060	−0.700745
$849$	0	0
$850$	−21.5918	−0.740594
$851$	3.21583	0.110237
$852$	0	0
$853$	23.5683	0.806963	0.403481	−	0.914988i	$-0.367800\pi$
$853$	23.5683	0.806963	0.403481	+	0.914988i	$0.367800\pi$
$854$	43.9048	1.50239
$855$	0	0
$856$	9.39451	0.321098
$857$	26.6279	0.909592	0.454796	−	0.890596i	$-0.349712\pi$
$857$	26.6279	0.909592	0.454796	+	0.890596i	$0.349712\pi$
$858$	0	0
$859$	−47.0788	−1.60631	−0.803155	−	0.595771i	$-0.796847\pi$
$859$	−47.0788	−1.60631	−0.803155	+	0.595771i	$0.796847\pi$
$860$	−18.4761	−0.630031
$861$	0	0
$862$	35.7350	1.21714
$863$	−27.1913	−0.925603	−0.462802	−	0.886462i	$-0.653156\pi$
$863$	−27.1913	−0.925603	−0.462802	+	0.886462i	$0.653156\pi$
$864$	0	0
$865$	−0.682886	−0.0232188
$866$	20.1611	0.685101
$867$	0	0
$868$	−25.8790	−0.878390
$869$	−32.8820	−1.11544
$870$	0	0
$871$	28.1610	0.954198
$872$	2.37309	0.0803628
$873$	0	0
$874$	−4.08957	−0.138332
$875$	−47.8542	−1.61777
$876$	0	0
$877$	−21.7670	−0.735019	−0.367509	−	0.930020i	$-0.619789\pi$
$877$	−21.7670	−0.735019	−0.367509	+	0.930020i	$0.619789\pi$
$878$	−29.6853	−1.00183
$879$	0	0
$880$	22.6541	0.763671
$881$	−1.77365	−0.0597558	−0.0298779	−	0.999554i	$-0.509512\pi$
$881$	−1.77365	−0.0597558	−0.0298779	+	0.999554i	$0.509512\pi$
$882$	0	0
$883$	−11.3776	−0.382885	−0.191443	−	0.981504i	$-0.561317\pi$
$883$	−11.3776	−0.382885	−0.191443	+	0.981504i	$0.561317\pi$
$884$	24.3313	0.818352
$885$	0	0
$886$	71.1972	2.39192
$887$	−29.2826	−0.983212	−0.491606	−	0.870818i	$-0.663590\pi$
$887$	−29.2826	−0.983212	−0.491606	+	0.870818i	$0.663590\pi$
$888$	0	0
$889$	8.57975	0.287756
$890$	−44.5842	−1.49447
$891$	0	0
$892$	0.926288	0.0310144
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	31.9227	1.06706
$896$	15.2114	0.508178
$897$	0	0
$898$	53.8272	1.79624
$899$	−16.8119	−0.560709
$900$	0	0
$901$	−17.8062	−0.593209
$902$	−9.03993	−0.300997
$903$	0	0
$904$	−1.12986	−0.0375785
$905$	−26.5795	−0.883531
$906$	0	0
$907$	−36.1560	−1.20054	−0.600271	−	0.799797i	$-0.704941\pi$
$907$	−36.1560	−1.20054	−0.600271	+	0.799797i	$0.704941\pi$
$908$	14.8856	0.493996
$909$	0	0
$910$	42.1316	1.39665
$911$	40.1559	1.33042	0.665212	−	0.746655i	$-0.268341\pi$
$911$	40.1559	1.33042	0.665212	+	0.746655i	$0.268341\pi$
$912$	0	0
$913$	11.3605	0.375978
$914$	38.9932	1.28978
$915$	0	0
$916$	−16.3487	−0.540178
$917$	−45.8501	−1.51410
$918$	0	0
$919$	−31.4308	−1.03680	−0.518402	−	0.855137i	$-0.673473\pi$
$919$	−31.4308	−1.03680	−0.518402	+	0.855137i	$0.673473\pi$
$920$	−1.40537	−0.0463336
$921$	0	0
$922$	−53.9363	−1.77630
$923$	−57.9718	−1.90816
$924$	0	0
$925$	−4.40521	−0.144843
$926$	−18.6922	−0.614263
$927$	0	0
$928$	−36.5929	−1.20122
$929$	−6.86362	−0.225188	−0.112594	−	0.993641i	$-0.535916\pi$
$929$	−6.86362	−0.225188	−0.112594	+	0.993641i	$0.535916\pi$
$930$	0	0
$931$	10.4254	0.341680
$932$	22.7828	0.746274
$933$	0	0
$934$	−23.0209	−0.753266
$935$	19.7679	0.646479
$936$	0	0
$937$	26.2168	0.856467	0.428233	−	0.903668i	$-0.359136\pi$
$937$	26.2168	0.856467	0.428233	+	0.903668i	$0.359136\pi$
$938$	−63.7088	−2.08017
$939$	0	0
$940$	2.56310	0.0835992
$941$	−45.6994	−1.48976	−0.744878	−	0.667200i	$-0.767493\pi$
$941$	−45.6994	−1.48976	−0.744878	+	0.667200i	$0.767493\pi$
$942$	0	0
$943$	2.78356	0.0906453
$944$	−38.5043	−1.25321
$945$	0	0
$946$	−49.3501	−1.60451
$947$	−44.7307	−1.45355	−0.726777	−	0.686873i	$-0.758983\pi$
$947$	−44.7307	−1.45355	−0.726777	+	0.686873i	$0.758983\pi$
$948$	0	0
$949$	30.3346	0.984703
$950$	5.60210	0.181756
$951$	0	0
$952$	7.38013	0.239191
$953$	−5.58175	−0.180811	−0.0904053	−	0.995905i	$-0.528816\pi$
$953$	−5.58175	−0.180811	−0.0904053	+	0.995905i	$0.528816\pi$
$954$	0	0
$955$	−19.8153	−0.641209
$956$	−32.0951	−1.03803
$957$	0	0
$958$	42.5917	1.37607
$959$	58.2546	1.88114
$960$	0	0
$961$	−18.6424	−0.601368
$962$	10.5938	0.341559
$963$	0	0
$964$	−39.4410	−1.27031
$965$	−30.5631	−0.983860
$966$	0	0
$967$	−15.7106	−0.505220	−0.252610	−	0.967568i	$-0.581289\pi$
$967$	−15.7106	−0.505220	−0.252610	+	0.967568i	$0.581289\pi$
$968$	0.666672	0.0214277
$969$	0	0
$970$	−30.6013	−0.982547
$971$	45.8725	1.47212	0.736060	−	0.676916i	$-0.236684\pi$
$971$	45.8725	1.47212	0.736060	+	0.676916i	$0.236684\pi$
$972$	0	0
$973$	−21.0313	−0.674231
$974$	−13.2757	−0.425382
$975$	0	0
$976$	−23.9468	−0.766518
$977$	−15.0568	−0.481709	−0.240855	−	0.970561i	$-0.577428\pi$
$977$	−15.0568	−0.481709	−0.240855	+	0.970561i	$0.577428\pi$
$978$	0	0
$979$	−55.8018	−1.78343
$980$	−26.7215	−0.853587
$981$	0	0
$982$	−43.5125	−1.38854
$983$	48.2625	1.53933	0.769667	−	0.638446i	$-0.220422\pi$
$983$	48.2625	1.53933	0.769667	+	0.638446i	$0.220422\pi$
$984$	0	0
$985$	27.8657	0.887875
$986$	−35.7592	−1.13881
$987$	0	0
$988$	−6.31288	−0.200839
$989$	15.1958	0.483199
$990$	0	0
$991$	−11.9582	−0.379863	−0.189932	−	0.981797i	$-0.560827\pi$
$991$	−11.9582	−0.379863	−0.189932	+	0.981797i	$0.560827\pi$
$992$	26.8976	0.853999
$993$	0	0
$994$	131.150	4.15983
$995$	−7.42267	−0.235315
$996$	0	0
$997$	−43.3816	−1.37391	−0.686955	−	0.726700i	$-0.741053\pi$
$997$	−43.3816	−1.37391	−0.686955	+	0.726700i	$0.741053\pi$
$998$	−21.4222	−0.678108
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.2		12
3.2	odd	2		893.2.a.a.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.11	✓	12	3.2	odd	2
8037.2.a.m.1.2		12	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93999 q^{2} +1.76355 q^{4} -1.45338 q^{5} -4.17438 q^{7} +0.458705 q^{8} +O(q^{10})\)	\(q-1.93999 q^{2} +1.76355 q^{4} -1.45338 q^{5} -4.17438 q^{7} +0.458705 q^{8} +2.81953 q^{10} +3.52893 q^{11} -3.57964 q^{13} +8.09824 q^{14} -4.41699 q^{16} -3.85424 q^{17} +1.00000 q^{19} -2.56310 q^{20} -6.84609 q^{22} +2.10804 q^{23} -2.88770 q^{25} +6.94445 q^{26} -7.36174 q^{28} -4.78245 q^{29} +3.51534 q^{31} +7.65149 q^{32} +7.47718 q^{34} +6.06694 q^{35} +1.52551 q^{37} -1.93999 q^{38} -0.666671 q^{40} +1.32045 q^{41} +7.20851 q^{43} +6.22346 q^{44} -4.08957 q^{46} -1.00000 q^{47} +10.4254 q^{49} +5.60210 q^{50} -6.31288 q^{52} +4.61989 q^{53} -5.12887 q^{55} -1.91481 q^{56} +9.27790 q^{58} +8.71732 q^{59} +5.42152 q^{61} -6.81971 q^{62} -6.00983 q^{64} +5.20256 q^{65} -7.86699 q^{67} -6.79715 q^{68} -11.7698 q^{70} +16.1949 q^{71} -8.47421 q^{73} -2.95947 q^{74} +1.76355 q^{76} -14.7311 q^{77} -9.31782 q^{79} +6.41954 q^{80} -2.56166 q^{82} +3.21925 q^{83} +5.60165 q^{85} -13.9844 q^{86} +1.61874 q^{88} -15.8126 q^{89} +14.9428 q^{91} +3.71764 q^{92} +1.93999 q^{94} -1.45338 q^{95} -10.8533 q^{97} -20.2252 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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