LMFDB - Embedded newform 8037.2.a.t.1.7

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:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2679)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.40720 q^{2} -0.0197875 q^{4} +4.17114 q^{5} -0.442129 q^{7} +2.84225 q^{8} +O(q^{10})$	\(q-1.40720 q^{2} -0.0197875 q^{4} +4.17114 q^{5} -0.442129 q^{7} +2.84225 q^{8} -5.86964 q^{10} +6.07689 q^{11} -5.08769 q^{13} +0.622164 q^{14} -3.96003 q^{16} +0.258151 q^{17} +1.00000 q^{19} -0.0825365 q^{20} -8.55141 q^{22} +2.95028 q^{23} +12.3984 q^{25} +7.15940 q^{26} +0.00874862 q^{28} -6.10998 q^{29} -9.36643 q^{31} -0.111931 q^{32} -0.363269 q^{34} -1.84418 q^{35} +9.75143 q^{37} -1.40720 q^{38} +11.8554 q^{40} +6.18056 q^{41} +1.35586 q^{43} -0.120246 q^{44} -4.15163 q^{46} +1.00000 q^{47} -6.80452 q^{49} -17.4471 q^{50} +0.100673 q^{52} -7.64224 q^{53} +25.3476 q^{55} -1.25664 q^{56} +8.59797 q^{58} +7.29658 q^{59} -10.5358 q^{61} +13.1804 q^{62} +8.07758 q^{64} -21.2215 q^{65} -12.8049 q^{67} -0.00510815 q^{68} +2.59513 q^{70} +7.82269 q^{71} +10.4207 q^{73} -13.7222 q^{74} -0.0197875 q^{76} -2.68677 q^{77} +9.60890 q^{79} -16.5179 q^{80} -8.69728 q^{82} +2.77631 q^{83} +1.07678 q^{85} -1.90797 q^{86} +17.2720 q^{88} +3.06153 q^{89} +2.24942 q^{91} -0.0583786 q^{92} -1.40720 q^{94} +4.17114 q^{95} +18.0978 q^{97} +9.57533 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	$ 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) $ 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * 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 $

$2$	−1.40720	−0.995041	−0.497520	−	0.867452i	$-0.665756\pi$
$2$	−1.40720	−0.995041	−0.497520	+	0.867452i	$0.665756\pi$
$3$	0	0
$3$	0	0
$4$	−0.0197875	−0.00989374
$5$	4.17114	1.86539	0.932696	−	0.360663i	$-0.117449\pi$
$5$	4.17114	1.86539	0.932696	+	0.360663i	$0.117449\pi$
$6$	0	0
$7$	−0.442129	−0.167109	−0.0835545	−	0.996503i	$-0.526627\pi$
$7$	−0.442129	−0.167109	−0.0835545	+	0.996503i	$0.526627\pi$
$8$	2.84225	1.00489
$9$	0	0
$10$	−5.86964	−1.85614
$11$	6.07689	1.83225	0.916126	−	0.400890i	$-0.131299\pi$
$11$	6.07689	1.83225	0.916126	+	0.400890i	$0.131299\pi$
$12$	0	0
$13$	−5.08769	−1.41107	−0.705536	−	0.708674i	$-0.749294\pi$
$13$	−5.08769	−1.41107	−0.705536	+	0.708674i	$0.749294\pi$
$14$	0.622164	0.166280
$15$	0	0
$16$	−3.96003	−0.990008
$17$	0.258151	0.0626107	0.0313054	−	0.999510i	$-0.490034\pi$
$17$	0.258151	0.0626107	0.0313054	+	0.999510i	$0.490034\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.0825365	−0.0184557
$21$	0	0
$22$	−8.55141	−1.82317
$23$	2.95028	0.615175	0.307588	−	0.951520i	$-0.400478\pi$
$23$	2.95028	0.615175	0.307588	+	0.951520i	$0.400478\pi$
$24$	0	0
$25$	12.3984	2.47969
$26$	7.15940	1.40407
$27$	0	0
$28$	0.00874862	0.00165333
$29$	−6.10998	−1.13460	−0.567298	−	0.823513i	$-0.692011\pi$
$29$	−6.10998	−1.13460	−0.567298	+	0.823513i	$0.692011\pi$
$30$	0	0
$31$	−9.36643	−1.68226	−0.841130	−	0.540832i	$-0.818109\pi$
$31$	−9.36643	−1.68226	−0.841130	+	0.540832i	$0.818109\pi$
$32$	−0.111931	−0.0197868
$33$	0	0
$34$	−0.363269	−0.0623002
$35$	−1.84418	−0.311724
$36$	0	0
$37$	9.75143	1.60312	0.801562	−	0.597911i	$-0.204002\pi$
$37$	9.75143	1.60312	0.801562	+	0.597911i	$0.204002\pi$
$38$	−1.40720	−0.228278
$39$	0	0
$40$	11.8554	1.87451
$41$	6.18056	0.965241	0.482621	−	0.875830i	$-0.339685\pi$
$41$	6.18056	0.965241	0.482621	+	0.875830i	$0.339685\pi$
$42$	0	0
$43$	1.35586	0.206767	0.103383	−	0.994642i	$-0.467033\pi$
$43$	1.35586	0.206767	0.103383	+	0.994642i	$0.467033\pi$
$44$	−0.120246	−0.0181278
$45$	0	0
$46$	−4.15163	−0.612124
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−6.80452	−0.972075
$50$	−17.4471	−2.46739
$51$	0	0
$52$	0.100673	0.0139608
$53$	−7.64224	−1.04974	−0.524872	−	0.851181i	$-0.675887\pi$
$53$	−7.64224	−1.04974	−0.524872	+	0.851181i	$0.675887\pi$
$54$	0	0
$55$	25.3476	3.41787
$56$	−1.25664	−0.167925
$57$	0	0
$58$	8.59797	1.12897
$59$	7.29658	0.949934	0.474967	−	0.880004i	$-0.342460\pi$
$59$	7.29658	0.949934	0.474967	+	0.880004i	$0.342460\pi$
$60$	0	0
$61$	−10.5358	−1.34897	−0.674484	−	0.738290i	$-0.735634\pi$
$61$	−10.5358	−1.34897	−0.674484	+	0.738290i	$0.735634\pi$
$62$	13.1804	1.67392
$63$	0	0
$64$	8.07758	1.00970
$65$	−21.2215	−2.63220
$66$	0	0
$67$	−12.8049	−1.56437	−0.782183	−	0.623048i	$-0.785894\pi$
$67$	−12.8049	−1.56437	−0.782183	+	0.623048i	$0.785894\pi$
$68$	−0.00510815	−0.000619454 0
$69$	0	0
$70$	2.59513	0.310178
$71$	7.82269	0.928383	0.464191	−	0.885735i	$-0.346345\pi$
$71$	7.82269	0.928383	0.464191	+	0.885735i	$0.346345\pi$
$72$	0	0
$73$	10.4207	1.21965	0.609824	−	0.792537i	$-0.291240\pi$
$73$	10.4207	1.21965	0.609824	+	0.792537i	$0.291240\pi$
$74$	−13.7222	−1.59517
$75$	0	0
$76$	−0.0197875	−0.00226978
$77$	−2.68677	−0.306186
$78$	0	0
$79$	9.60890	1.08109	0.540543	−	0.841316i	$-0.318219\pi$
$79$	9.60890	1.08109	0.540543	+	0.841316i	$0.318219\pi$
$80$	−16.5179	−1.84675
$81$	0	0
$82$	−8.69728	−0.960454
$83$	2.77631	0.304739	0.152370	−	0.988324i	$-0.451310\pi$
$83$	2.77631	0.304739	0.152370	+	0.988324i	$0.451310\pi$
$84$	0	0
$85$	1.07678	0.116794
$86$	−1.90797	−0.205741
$87$	0	0
$88$	17.2720	1.84120
$89$	3.06153	0.324522	0.162261	−	0.986748i	$-0.448121\pi$
$89$	3.06153	0.324522	0.162261	+	0.986748i	$0.448121\pi$
$90$	0	0
$91$	2.24942	0.235803
$92$	−0.0583786	−0.00608639
$93$	0	0
$94$	−1.40720	−0.145142
$95$	4.17114	0.427950
$96$	0	0
$97$	18.0978	1.83755	0.918776	−	0.394779i	$-0.129179\pi$
$97$	18.0978	1.83755	0.918776	+	0.394779i	$0.129179\pi$
$98$	9.57533	0.967254
$99$	0	0
$100$	−0.245334	−0.0245334
$101$	15.0802	1.50053	0.750267	−	0.661136i	$-0.229925\pi$
$101$	15.0802	1.50053	0.750267	+	0.661136i	$0.229925\pi$
$102$	0	0
$103$	3.77721	0.372180	0.186090	−	0.982533i	$-0.440418\pi$
$103$	3.77721	0.372180	0.186090	+	0.982533i	$0.440418\pi$
$104$	−14.4605	−1.41797
$105$	0	0
$106$	10.7542	1.04454
$107$	8.47342	0.819156	0.409578	−	0.912275i	$-0.365676\pi$
$107$	8.47342	0.819156	0.409578	+	0.912275i	$0.365676\pi$
$108$	0	0
$109$	−6.64031	−0.636027	−0.318013	−	0.948086i	$-0.603016\pi$
$109$	−6.64031	−0.636027	−0.318013	+	0.948086i	$0.603016\pi$
$110$	−35.6691	−3.40092
$111$	0	0
$112$	1.75084	0.165439
$113$	−4.70834	−0.442923	−0.221462	−	0.975169i	$-0.571083\pi$
$113$	−4.70834	−0.442923	−0.221462	+	0.975169i	$0.571083\pi$
$114$	0	0
$115$	12.3060	1.14754
$116$	0.120901	0.0112254
$117$	0	0
$118$	−10.2678	−0.945224
$119$	−0.114136	−0.0104628
$120$	0	0
$121$	25.9286	2.35715
$122$	14.8259	1.34228
$123$	0	0
$124$	0.185338	0.0166439
$125$	30.8600	2.76020
$126$	0	0
$127$	−0.105631	−0.00937321	−0.00468660	−	0.999989i	$-0.501492\pi$
$127$	−0.105631	−0.00937321	−0.00468660	+	0.999989i	$0.501492\pi$
$128$	−11.1429	−0.984903
$129$	0	0
$130$	29.8629	2.61915
$131$	−1.29461	−0.113110	−0.0565552	−	0.998399i	$-0.518012\pi$
$131$	−1.29461	−0.113110	−0.0565552	+	0.998399i	$0.518012\pi$
$132$	0	0
$133$	−0.442129	−0.0383374
$134$	18.0191	1.55661
$135$	0	0
$136$	0.733727	0.0629166
$137$	13.2024	1.12795	0.563977	−	0.825790i	$-0.309271\pi$
$137$	13.2024	1.12795	0.563977	+	0.825790i	$0.309271\pi$
$138$	0	0
$139$	7.87318	0.667794	0.333897	−	0.942610i	$-0.391636\pi$
$139$	7.87318	0.667794	0.333897	+	0.942610i	$0.391636\pi$
$140$	0.0364918	0.00308412
$141$	0	0
$142$	−11.0081	−0.923779
$143$	−30.9174	−2.58544
$144$	0	0
$145$	−25.4856	−2.11647
$146$	−14.6640	−1.21360
$147$	0	0
$148$	−0.192956	−0.0158609
$149$	13.4787	1.10422	0.552109	−	0.833772i	$-0.313823\pi$
$149$	13.4787	1.10422	0.552109	+	0.833772i	$0.313823\pi$
$150$	0	0
$151$	−18.5130	−1.50657	−0.753285	−	0.657694i	$-0.771532\pi$
$151$	−18.5130	−1.50657	−0.753285	+	0.657694i	$0.771532\pi$
$152$	2.84225	0.230537
$153$	0	0
$154$	3.78082	0.304667
$155$	−39.0687	−3.13808
$156$	0	0
$157$	6.98249	0.557263	0.278632	−	0.960398i	$-0.410119\pi$
$157$	6.98249	0.557263	0.278632	+	0.960398i	$0.410119\pi$
$158$	−13.5216	−1.07572
$159$	0	0
$160$	−0.466880	−0.0369101
$161$	−1.30440	−0.102801
$162$	0	0
$163$	−6.85803	−0.537162	−0.268581	−	0.963257i	$-0.586555\pi$
$163$	−6.85803	−0.537162	−0.268581	+	0.963257i	$0.586555\pi$
$164$	−0.122298	−0.00954985
$165$	0	0
$166$	−3.90682	−0.303228
$167$	−9.63232	−0.745371	−0.372685	−	0.927958i	$-0.621563\pi$
$167$	−9.63232	−0.745371	−0.372685	+	0.927958i	$0.621563\pi$
$168$	0	0
$169$	12.8846	0.991125
$170$	−1.51525	−0.116214
$171$	0	0
$172$	−0.0268291	−0.00204570
$173$	21.3176	1.62075	0.810374	−	0.585913i	$-0.199264\pi$
$173$	21.3176	1.62075	0.810374	+	0.585913i	$0.199264\pi$
$174$	0	0
$175$	−5.48171	−0.414378
$176$	−24.0647	−1.81395
$177$	0	0
$178$	−4.30819	−0.322912
$179$	7.06542	0.528095	0.264047	−	0.964510i	$-0.414943\pi$
$179$	7.06542	0.528095	0.264047	+	0.964510i	$0.414943\pi$
$180$	0	0
$181$	7.62537	0.566789	0.283395	−	0.959003i	$-0.408539\pi$
$181$	7.62537	0.566789	0.283395	+	0.959003i	$0.408539\pi$
$182$	−3.16538	−0.234633
$183$	0	0
$184$	8.38541	0.618181
$185$	40.6746	2.99046
$186$	0	0
$187$	1.56875	0.114719
$188$	−0.0197875	−0.00144315
$189$	0	0
$190$	−5.86964	−0.425828
$191$	−5.67892	−0.410912	−0.205456	−	0.978666i	$-0.565868\pi$
$191$	−5.67892	−0.410912	−0.205456	+	0.978666i	$0.565868\pi$
$192$	0	0
$193$	4.47657	0.322231	0.161115	−	0.986936i	$-0.448491\pi$
$193$	4.47657	0.322231	0.161115	+	0.986936i	$0.448491\pi$
$194$	−25.4672	−1.82844
$195$	0	0
$196$	0.134644	0.00961746
$197$	−4.36396	−0.310920	−0.155460	−	0.987842i	$-0.549686\pi$
$197$	−4.36396	−0.310920	−0.155460	+	0.987842i	$0.549686\pi$
$198$	0	0
$199$	10.7043	0.758808	0.379404	−	0.925231i	$-0.376129\pi$
$199$	10.7043	0.758808	0.379404	+	0.925231i	$0.376129\pi$
$200$	35.2394	2.49180
$201$	0	0
$202$	−21.2208	−1.49309
$203$	2.70140	0.189601
$204$	0	0
$205$	25.7800	1.80055
$206$	−5.31529	−0.370334
$207$	0	0
$208$	20.1474	1.39697
$209$	6.07689	0.420348
$210$	0	0
$211$	9.15604	0.630328	0.315164	−	0.949037i	$-0.397940\pi$
$211$	9.15604	0.630328	0.315164	+	0.949037i	$0.397940\pi$
$212$	0.151221	0.0103859
$213$	0	0
$214$	−11.9238	−0.815094
$215$	5.65549	0.385701
$216$	0	0
$217$	4.14117	0.281121
$218$	9.34425	0.632872
$219$	0	0
$220$	−0.501565	−0.0338155
$221$	−1.31339	−0.0883482
$222$	0	0
$223$	−4.77148	−0.319522	−0.159761	−	0.987156i	$-0.551072\pi$
$223$	−4.77148	−0.319522	−0.159761	+	0.987156i	$0.551072\pi$
$224$	0.0494878	0.00330655
$225$	0	0
$226$	6.62557	0.440727
$227$	−5.94976	−0.394900	−0.197450	−	0.980313i	$-0.563266\pi$
$227$	−5.94976	−0.394900	−0.197450	+	0.980313i	$0.563266\pi$
$228$	0	0
$229$	−2.21705	−0.146507	−0.0732535	−	0.997313i	$-0.523338\pi$
$229$	−2.21705	−0.146507	−0.0732535	+	0.997313i	$0.523338\pi$
$230$	−17.3170	−1.14185
$231$	0	0
$232$	−17.3661	−1.14014
$233$	−13.7358	−0.899859	−0.449929	−	0.893064i	$-0.648551\pi$
$233$	−13.7358	−0.899859	−0.449929	+	0.893064i	$0.648551\pi$
$234$	0	0
$235$	4.17114	0.272095
$236$	−0.144381	−0.00939841
$237$	0	0
$238$	0.160612	0.0104109
$239$	−1.30704	−0.0845457	−0.0422728	−	0.999106i	$-0.513460\pi$
$239$	−1.30704	−0.0845457	−0.0422728	+	0.999106i	$0.513460\pi$
$240$	0	0
$241$	13.2253	0.851916	0.425958	−	0.904743i	$-0.359937\pi$
$241$	13.2253	0.851916	0.425958	+	0.904743i	$0.359937\pi$
$242$	−36.4868	−2.34546
$243$	0	0
$244$	0.208476	0.0133463
$245$	−28.3826	−1.81330
$246$	0	0
$247$	−5.08769	−0.323722
$248$	−26.6217	−1.69048
$249$	0	0
$250$	−43.4262	−2.74651
$251$	5.24222	0.330886	0.165443	−	0.986219i	$-0.447095\pi$
$251$	5.24222	0.330886	0.165443	+	0.986219i	$0.447095\pi$
$252$	0	0
$253$	17.9285	1.12716
$254$	0.148644	0.00932673
$255$	0	0
$256$	−0.474853	−0.0296783
$257$	−12.6062	−0.786352	−0.393176	−	0.919463i	$-0.628624\pi$
$257$	−12.6062	−0.786352	−0.393176	+	0.919463i	$0.628624\pi$
$258$	0	0
$259$	−4.31139	−0.267897
$260$	0.419920	0.0260423
$261$	0	0
$262$	1.82177	0.112549
$263$	−5.60476	−0.345604	−0.172802	−	0.984957i	$-0.555282\pi$
$263$	−5.60476	−0.345604	−0.172802	+	0.984957i	$0.555282\pi$
$264$	0	0
$265$	−31.8769	−1.95818
$266$	0.622164	0.0381473
$267$	0	0
$268$	0.253377	0.0154774
$269$	−30.3737	−1.85192	−0.925960	−	0.377622i	$-0.876742\pi$
$269$	−30.3737	−1.85192	−0.925960	+	0.377622i	$0.876742\pi$
$270$	0	0
$271$	14.5629	0.884633	0.442316	−	0.896859i	$-0.354157\pi$
$271$	14.5629	0.884633	0.442316	+	0.896859i	$0.354157\pi$
$272$	−1.02228	−0.0619851
$273$	0	0
$274$	−18.5784	−1.12236
$275$	75.3440	4.54342
$276$	0	0
$277$	1.50941	0.0906914	0.0453457	−	0.998971i	$-0.485561\pi$
$277$	1.50941	0.0906914	0.0453457	+	0.998971i	$0.485561\pi$
$278$	−11.0791	−0.664483
$279$	0	0
$280$	−5.24162	−0.313247
$281$	0.263521	0.0157203	0.00786016	−	0.999969i	$-0.497498\pi$
$281$	0.263521	0.0157203	0.00786016	+	0.999969i	$0.497498\pi$
$282$	0	0
$283$	−18.1804	−1.08071	−0.540357	−	0.841436i	$-0.681711\pi$
$283$	−18.1804	−1.08071	−0.540357	+	0.841436i	$0.681711\pi$
$284$	−0.154791	−0.00918518
$285$	0	0
$286$	43.5069	2.57262
$287$	−2.73260	−0.161300
$288$	0	0
$289$	−16.9334	−0.996080
$290$	35.8634	2.10597
$291$	0	0
$292$	−0.206199	−0.0120669
$293$	29.1901	1.70530	0.852651	−	0.522480i	$-0.174993\pi$
$293$	29.1901	1.70530	0.852651	+	0.522480i	$0.174993\pi$
$294$	0	0
$295$	30.4351	1.77200
$296$	27.7160	1.61096
$297$	0	0
$298$	−18.9672	−1.09874
$299$	−15.0101	−0.868057
$300$	0	0
$301$	−0.599465	−0.0345526
$302$	26.0516	1.49910
$303$	0	0
$304$	−3.96003	−0.227123
$305$	−43.9462	−2.51635
$306$	0	0
$307$	−33.4126	−1.90696	−0.953478	−	0.301461i	$-0.902526\pi$
$307$	−33.4126	−1.90696	−0.953478	+	0.301461i	$0.902526\pi$
$308$	0.0531644	0.00302932
$309$	0	0
$310$	54.9775	3.12251
$311$	21.5388	1.22136	0.610678	−	0.791879i	$-0.290897\pi$
$311$	21.5388	1.22136	0.610678	+	0.791879i	$0.290897\pi$
$312$	0	0
$313$	−22.4744	−1.27033	−0.635164	−	0.772377i	$-0.719067\pi$
$313$	−22.4744	−1.27033	−0.635164	+	0.772377i	$0.719067\pi$
$314$	−9.82576	−0.554500
$315$	0	0
$316$	−0.190136	−0.0106960
$317$	25.6423	1.44022	0.720108	−	0.693862i	$-0.244092\pi$
$317$	25.6423	1.44022	0.720108	+	0.693862i	$0.244092\pi$
$318$	0	0
$319$	−37.1297	−2.07887
$320$	33.6927	1.88348
$321$	0	0
$322$	1.83555	0.102291
$323$	0.258151	0.0143639
$324$	0	0
$325$	−63.0795	−3.49902
$326$	9.65062	0.534498
$327$	0	0
$328$	17.5667	0.969957
$329$	−0.442129	−0.0243753
$330$	0	0
$331$	−7.15158	−0.393086	−0.196543	−	0.980495i	$-0.562972\pi$
$331$	−7.15158	−0.393086	−0.196543	+	0.980495i	$0.562972\pi$
$332$	−0.0549361	−0.00301501
$333$	0	0
$334$	13.5546	0.741675
$335$	−53.4111	−2.91816
$336$	0	0
$337$	−10.4295	−0.568131	−0.284065	−	0.958805i	$-0.591683\pi$
$337$	−10.4295	−0.568131	−0.284065	+	0.958805i	$0.591683\pi$
$338$	−18.1312	−0.986210
$339$	0	0
$340$	−0.0213068	−0.00115553
$341$	−56.9188	−3.08233
$342$	0	0
$343$	6.10338	0.329551
$344$	3.85369	0.207777
$345$	0	0
$346$	−29.9982	−1.61271
$347$	4.19124	0.224997	0.112499	−	0.993652i	$-0.464115\pi$
$347$	4.19124	0.224997	0.112499	+	0.993652i	$0.464115\pi$
$348$	0	0
$349$	−5.04331	−0.269962	−0.134981	−	0.990848i	$-0.543097\pi$
$349$	−5.04331	−0.269962	−0.134981	+	0.990848i	$0.543097\pi$
$350$	7.71386	0.412323
$351$	0	0
$352$	−0.680192	−0.0362543
$353$	18.6388	0.992045	0.496023	−	0.868310i	$-0.334793\pi$
$353$	18.6388	0.992045	0.496023	+	0.868310i	$0.334793\pi$
$354$	0	0
$355$	32.6296	1.73180
$356$	−0.0605800	−0.00321073
$357$	0	0
$358$	−9.94246	−0.525476
$359$	−28.8673	−1.52356	−0.761780	−	0.647836i	$-0.775674\pi$
$359$	−28.8673	−1.52356	−0.761780	+	0.647836i	$0.775674\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.7304	−0.563979
$363$	0	0
$364$	−0.0445103	−0.00233297
$365$	43.4661	2.27512
$366$	0	0
$367$	24.1903	1.26273	0.631363	−	0.775487i	$-0.282496\pi$
$367$	24.1903	1.26273	0.631363	+	0.775487i	$0.282496\pi$
$368$	−11.6832	−0.609029
$369$	0	0
$370$	−57.2373	−2.97563
$371$	3.37886	0.175421
$372$	0	0
$373$	−19.9409	−1.03250	−0.516251	−	0.856437i	$-0.672673\pi$
$373$	−19.9409	−1.03250	−0.516251	+	0.856437i	$0.672673\pi$
$374$	−2.20755	−0.114150
$375$	0	0
$376$	2.84225	0.146578
$377$	31.0857	1.60100
$378$	0	0
$379$	38.2761	1.96611	0.983054	−	0.183314i	$-0.0586825\pi$
$379$	38.2761	1.96611	0.983054	+	0.183314i	$0.0586825\pi$
$380$	−0.0825365	−0.00423403
$381$	0	0
$382$	7.99137	0.408874
$383$	26.0112	1.32911	0.664554	−	0.747240i	$-0.268622\pi$
$383$	26.0112	1.32911	0.664554	+	0.747240i	$0.268622\pi$
$384$	0	0
$385$	−11.2069	−0.571157
$386$	−6.29944	−0.320633
$387$	0	0
$388$	−0.358110	−0.0181803
$389$	15.8505	0.803651	0.401826	−	0.915716i	$-0.368376\pi$
$389$	15.8505	0.803651	0.401826	+	0.915716i	$0.368376\pi$
$390$	0	0
$391$	0.761615	0.0385165
$392$	−19.3401	−0.976824
$393$	0	0
$394$	6.14097	0.309378
$395$	40.0801	2.01665
$396$	0	0
$397$	2.21190	0.111012	0.0555060	−	0.998458i	$-0.482323\pi$
$397$	2.21190	0.111012	0.0555060	+	0.998458i	$0.482323\pi$
$398$	−15.0631	−0.755045
$399$	0	0
$400$	−49.0983	−2.45491
$401$	−14.3354	−0.715874	−0.357937	−	0.933746i	$-0.616520\pi$
$401$	−14.3354	−0.715874	−0.357937	+	0.933746i	$0.616520\pi$
$402$	0	0
$403$	47.6535	2.37379
$404$	−0.298399	−0.0148459
$405$	0	0
$406$	−3.80141	−0.188661
$407$	59.2584	2.93733
$408$	0	0
$409$	−15.1522	−0.749228	−0.374614	−	0.927181i	$-0.622225\pi$
$409$	−15.1522	−0.749228	−0.374614	+	0.927181i	$0.622225\pi$
$410$	−36.2776	−1.79162
$411$	0	0
$412$	−0.0747415	−0.00368225
$413$	−3.22603	−0.158743
$414$	0	0
$415$	11.5804	0.568458
$416$	0.569470	0.0279205
$417$	0	0
$418$	−8.55141	−0.418263
$419$	−9.51794	−0.464982	−0.232491	−	0.972599i	$-0.574688\pi$
$419$	−9.51794	−0.464982	−0.232491	+	0.972599i	$0.574688\pi$
$420$	0	0
$421$	−10.3441	−0.504142	−0.252071	−	0.967709i	$-0.581112\pi$
$421$	−10.3441	−0.504142	−0.252071	+	0.967709i	$0.581112\pi$
$422$	−12.8844	−0.627202
$423$	0	0
$424$	−21.7211	−1.05487
$425$	3.20067	0.155255
$426$	0	0
$427$	4.65817	0.225424
$428$	−0.167668	−0.00810452
$429$	0	0
$430$	−7.95841	−0.383789
$431$	28.0810	1.35261	0.676307	−	0.736619i	$-0.263579\pi$
$431$	28.0810	1.35261	0.676307	+	0.736619i	$0.263579\pi$
$432$	0	0
$433$	−15.0932	−0.725335	−0.362667	−	0.931919i	$-0.618134\pi$
$433$	−15.0932	−0.725335	−0.362667	+	0.931919i	$0.618134\pi$
$434$	−5.82745	−0.279727
$435$	0	0
$436$	0.131395	0.00629269
$437$	2.95028	0.141131
$438$	0	0
$439$	−24.7472	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$439$	−24.7472	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$440$	72.0441	3.43457
$441$	0	0
$442$	1.84820	0.0879101
$443$	2.30866	0.109688	0.0548439	−	0.998495i	$-0.482534\pi$
$443$	2.30866	0.109688	0.0548439	+	0.998495i	$0.482534\pi$
$444$	0	0
$445$	12.7701	0.605360
$446$	6.71443	0.317937
$447$	0	0
$448$	−3.57133	−0.168729
$449$	−34.1482	−1.61155	−0.805777	−	0.592220i	$-0.798252\pi$
$449$	−34.1482	−1.61155	−0.805777	+	0.592220i	$0.798252\pi$
$450$	0	0
$451$	37.5586	1.76857
$452$	0.0931662	0.00438217
$453$	0	0
$454$	8.37251	0.392941
$455$	9.38264	0.439865
$456$	0	0
$457$	25.7787	1.20588	0.602938	−	0.797788i	$-0.293997\pi$
$457$	25.7787	1.20588	0.602938	+	0.797788i	$0.293997\pi$
$458$	3.11984	0.145780
$459$	0	0
$460$	−0.243505	−0.0113535
$461$	−14.2995	−0.665992	−0.332996	−	0.942928i	$-0.608060\pi$
$461$	−14.2995	−0.665992	−0.332996	+	0.942928i	$0.608060\pi$
$462$	0	0
$463$	15.7000	0.729639	0.364819	−	0.931078i	$-0.381131\pi$
$463$	15.7000	0.729639	0.364819	+	0.931078i	$0.381131\pi$
$464$	24.1957	1.12326
$465$	0	0
$466$	19.3290	0.895396
$467$	4.48943	0.207746	0.103873	−	0.994591i	$-0.466876\pi$
$467$	4.48943	0.207746	0.103873	+	0.994591i	$0.466876\pi$
$468$	0	0
$469$	5.66141	0.261420
$470$	−5.86964	−0.270746
$471$	0	0
$472$	20.7387	0.954575
$473$	8.23942	0.378849
$474$	0	0
$475$	12.3984	0.568880
$476$	0.00225846	0.000103516 0
$477$	0	0
$478$	1.83927	0.0841264
$479$	−11.0567	−0.505193	−0.252597	−	0.967572i	$-0.581285\pi$
$479$	−11.0567	−0.505193	−0.252597	+	0.967572i	$0.581285\pi$
$480$	0	0
$481$	−49.6123	−2.26213
$482$	−18.6106	−0.847691
$483$	0	0
$484$	−0.513063	−0.0233210
$485$	75.4885	3.42776
$486$	0	0
$487$	−7.36450	−0.333717	−0.166859	−	0.985981i	$-0.553362\pi$
$487$	−7.36450	−0.333717	−0.166859	+	0.985981i	$0.553362\pi$
$488$	−29.9452	−1.35556
$489$	0	0
$490$	39.9401	1.80431
$491$	−23.9984	−1.08303	−0.541515	−	0.840691i	$-0.682149\pi$
$491$	−23.9984	−1.08303	−0.541515	+	0.840691i	$0.682149\pi$
$492$	0	0
$493$	−1.57730	−0.0710378
$494$	7.15940	0.322117
$495$	0	0
$496$	37.0914	1.66545
$497$	−3.45864	−0.155141
$498$	0	0
$499$	3.63672	0.162802	0.0814010	−	0.996681i	$-0.474061\pi$
$499$	3.63672	0.162802	0.0814010	+	0.996681i	$0.474061\pi$
$500$	−0.610642	−0.0273087
$501$	0	0
$502$	−7.37685	−0.329245
$503$	−1.53424	−0.0684082	−0.0342041	−	0.999415i	$-0.510890\pi$
$503$	−1.53424	−0.0684082	−0.0342041	+	0.999415i	$0.510890\pi$
$504$	0	0
$505$	62.9016	2.79908
$506$	−25.2290	−1.12157
$507$	0	0
$508$	0.00209017	9.27361e−5 0
$509$	−39.9960	−1.77279	−0.886395	−	0.462930i	$-0.846798\pi$
$509$	−39.9960	−1.77279	−0.886395	+	0.462930i	$0.846798\pi$
$510$	0	0
$511$	−4.60728	−0.203814
$512$	22.9540	1.01443
$513$	0	0
$514$	17.7394	0.782452
$515$	15.7553	0.694261
$516$	0	0
$517$	6.07689	0.267261
$518$	6.06698	0.266568
$519$	0	0
$520$	−60.3167	−2.64506
$521$	−38.7500	−1.69767	−0.848833	−	0.528661i	$-0.822694\pi$
$521$	−38.7500	−1.69767	−0.848833	+	0.528661i	$0.822694\pi$
$522$	0	0
$523$	1.63054	0.0712987	0.0356494	−	0.999364i	$-0.488650\pi$
$523$	1.63054	0.0712987	0.0356494	+	0.999364i	$0.488650\pi$
$524$	0.0256170	0.00111909
$525$	0	0
$526$	7.88702	0.343891
$527$	−2.41795	−0.105328
$528$	0	0
$529$	−14.2959	−0.621560
$530$	44.8572	1.94847
$531$	0	0
$532$	0.00874862	0.000379301 0
$533$	−31.4448	−1.36203
$534$	0	0
$535$	35.3438	1.52805
$536$	−36.3947	−1.57201
$537$	0	0
$538$	42.7419	1.84274
$539$	−41.3504	−1.78109
$540$	0	0
$541$	15.8387	0.680959	0.340480	−	0.940252i	$-0.389411\pi$
$541$	15.8387	0.680959	0.340480	+	0.940252i	$0.389411\pi$
$542$	−20.4929	−0.880246
$543$	0	0
$544$	−0.0288950	−0.00123886
$545$	−27.6977	−1.18644
$546$	0	0
$547$	42.6176	1.82220	0.911099	−	0.412188i	$-0.135235\pi$
$547$	42.6176	1.82220	0.911099	+	0.412188i	$0.135235\pi$
$548$	−0.261242	−0.0111597
$549$	0	0
$550$	−106.024	−4.52088
$551$	−6.10998	−0.260294
$552$	0	0
$553$	−4.24837	−0.180659
$554$	−2.12404	−0.0902416
$555$	0	0
$556$	−0.155790	−0.00660699
$557$	−25.5998	−1.08470	−0.542349	−	0.840153i	$-0.682465\pi$
$557$	−25.5998	−1.08470	−0.542349	+	0.840153i	$0.682465\pi$
$558$	0	0
$559$	−6.89820	−0.291763
$560$	7.30303	0.308609
$561$	0	0
$562$	−0.370826	−0.0156424
$563$	21.8475	0.920764	0.460382	−	0.887721i	$-0.347713\pi$
$563$	21.8475	0.920764	0.460382	+	0.887721i	$0.347713\pi$
$564$	0	0
$565$	−19.6392	−0.826226
$566$	25.5835	1.07536
$567$	0	0
$568$	22.2340	0.932918
$569$	−34.4593	−1.44461	−0.722305	−	0.691574i	$-0.756917\pi$
$569$	−34.4593	−1.44461	−0.722305	+	0.691574i	$0.756917\pi$
$570$	0	0
$571$	44.5956	1.86627	0.933133	−	0.359531i	$-0.117063\pi$
$571$	44.5956	1.86627	0.933133	+	0.359531i	$0.117063\pi$
$572$	0.611777	0.0255797
$573$	0	0
$574$	3.84532	0.160501
$575$	36.5788	1.52544
$576$	0	0
$577$	42.0439	1.75031	0.875155	−	0.483842i	$-0.160759\pi$
$577$	42.0439	1.75031	0.875155	+	0.483842i	$0.160759\pi$
$578$	23.8286	0.991140
$579$	0	0
$580$	0.504297	0.0209398
$581$	−1.22748	−0.0509247
$582$	0	0
$583$	−46.4411	−1.92339
$584$	29.6181	1.22561
$585$	0	0
$586$	−41.0763	−1.69685
$587$	37.4645	1.54633	0.773164	−	0.634207i	$-0.218673\pi$
$587$	37.4645	1.54633	0.773164	+	0.634207i	$0.218673\pi$
$588$	0	0
$589$	−9.36643	−0.385937
$590$	−42.8283	−1.76321
$591$	0	0
$592$	−38.6160	−1.58711
$593$	−11.2086	−0.460282	−0.230141	−	0.973157i	$-0.573919\pi$
$593$	−11.2086	−0.460282	−0.230141	+	0.973157i	$0.573919\pi$
$594$	0	0
$595$	−0.476077	−0.0195172
$596$	−0.266710	−0.0109249
$597$	0	0
$598$	21.1222	0.863752
$599$	42.1994	1.72422	0.862110	−	0.506722i	$-0.169143\pi$
$599$	42.1994	1.72422	0.862110	+	0.506722i	$0.169143\pi$
$600$	0	0
$601$	9.60342	0.391732	0.195866	−	0.980631i	$-0.437248\pi$
$601$	9.60342	0.391732	0.195866	+	0.980631i	$0.437248\pi$
$602$	0.843567	0.0343812
$603$	0	0
$604$	0.366327	0.0149056
$605$	108.152	4.39701
$606$	0	0
$607$	−26.6706	−1.08253	−0.541263	−	0.840853i	$-0.682054\pi$
$607$	−26.6706	−1.08253	−0.541263	+	0.840853i	$0.682054\pi$
$608$	−0.111931	−0.00453939
$609$	0	0
$610$	61.8411	2.50387
$611$	−5.08769	−0.205826
$612$	0	0
$613$	−3.59511	−0.145205	−0.0726026	−	0.997361i	$-0.523130\pi$
$613$	−3.59511	−0.145205	−0.0726026	+	0.997361i	$0.523130\pi$
$614$	47.0182	1.89750
$615$	0	0
$616$	−7.63646	−0.307682
$617$	26.2336	1.05613	0.528063	−	0.849205i	$-0.322919\pi$
$617$	26.2336	1.05613	0.528063	+	0.849205i	$0.322919\pi$
$618$	0	0
$619$	40.1722	1.61466	0.807328	−	0.590103i	$-0.200913\pi$
$619$	40.1722	1.61466	0.807328	+	0.590103i	$0.200913\pi$
$620$	0.773072	0.0310473
$621$	0	0
$622$	−30.3095	−1.21530
$623$	−1.35359	−0.0542305
$624$	0	0
$625$	66.7292	2.66917
$626$	31.6260	1.26403
$627$	0	0
$628$	−0.138166	−0.00551342
$629$	2.51734	0.100373
$630$	0	0
$631$	−0.983196	−0.0391404	−0.0195702	−	0.999808i	$-0.506230\pi$
$631$	−0.983196	−0.0391404	−0.0195702	+	0.999808i	$0.506230\pi$
$632$	27.3109	1.08637
$633$	0	0
$634$	−36.0839	−1.43307
$635$	−0.440601	−0.0174847
$636$	0	0
$637$	34.6193	1.37167
$638$	52.2489	2.06856
$639$	0	0
$640$	−46.4787	−1.83723
$641$	21.8488	0.862976	0.431488	−	0.902119i	$-0.357989\pi$
$641$	21.8488	0.862976	0.431488	+	0.902119i	$0.357989\pi$
$642$	0	0
$643$	−4.29728	−0.169468	−0.0847341	−	0.996404i	$-0.527004\pi$
$643$	−4.29728	−0.169468	−0.0847341	+	0.996404i	$0.527004\pi$
$644$	0.0258108	0.00101709
$645$	0	0
$646$	−0.363269	−0.0142926
$647$	−17.2843	−0.679515	−0.339757	−	0.940513i	$-0.610345\pi$
$647$	−17.2843	−0.679515	−0.339757	+	0.940513i	$0.610345\pi$
$648$	0	0
$649$	44.3406	1.74052
$650$	88.7655	3.48167
$651$	0	0
$652$	0.135703	0.00531455
$653$	−46.0723	−1.80295	−0.901473	−	0.432835i	$-0.857513\pi$
$653$	−46.0723	−1.80295	−0.901473	+	0.432835i	$0.857513\pi$
$654$	0	0
$655$	−5.40000	−0.210995
$656$	−24.4752	−0.955597
$657$	0	0
$658$	0.622164	0.0242545
$659$	−13.4140	−0.522534	−0.261267	−	0.965267i	$-0.584140\pi$
$659$	−13.4140	−0.522534	−0.261267	+	0.965267i	$0.584140\pi$
$660$	0	0
$661$	−37.5226	−1.45946	−0.729731	−	0.683734i	$-0.760355\pi$
$661$	−37.5226	−1.45946	−0.729731	+	0.683734i	$0.760355\pi$
$662$	10.0637	0.391137
$663$	0	0
$664$	7.89094	0.306228
$665$	−1.84418	−0.0715143
$666$	0	0
$667$	−18.0261	−0.697975
$668$	0.190599	0.00737451
$669$	0	0
$670$	75.1601	2.90369
$671$	−64.0247	−2.47165
$672$	0	0
$673$	37.6081	1.44968	0.724842	−	0.688915i	$-0.241913\pi$
$673$	37.6081	1.44968	0.724842	+	0.688915i	$0.241913\pi$
$674$	14.6764	0.565313
$675$	0	0
$676$	−0.254954	−0.00980594
$677$	−12.8435	−0.493616	−0.246808	−	0.969064i	$-0.579382\pi$
$677$	−12.8435	−0.493616	−0.246808	+	0.969064i	$0.579382\pi$
$678$	0	0
$679$	−8.00155	−0.307071
$680$	3.06048	0.117364
$681$	0	0
$682$	80.0962	3.06704
$683$	16.2649	0.622358	0.311179	−	0.950351i	$-0.399276\pi$
$683$	16.2649	0.622358	0.311179	+	0.950351i	$0.399276\pi$
$684$	0	0
$685$	55.0690	2.10408
$686$	−8.58867	−0.327917
$687$	0	0
$688$	−5.36925	−0.204701
$689$	38.8814	1.48126
$690$	0	0
$691$	−14.1598	−0.538663	−0.269331	−	0.963048i	$-0.586803\pi$
$691$	−14.1598	−0.538663	−0.269331	+	0.963048i	$0.586803\pi$
$692$	−0.421822	−0.0160353
$693$	0	0
$694$	−5.89791	−0.223881
$695$	32.8402	1.24570
$696$	0	0
$697$	1.59551	0.0604344
$698$	7.09694	0.268623
$699$	0	0
$700$	0.108469	0.00409975
$701$	27.2694	1.02995	0.514975	−	0.857205i	$-0.327801\pi$
$701$	27.2694	1.02995	0.514975	+	0.857205i	$0.327801\pi$
$702$	0	0
$703$	9.75143	0.367782
$704$	49.0866	1.85002
$705$	0	0
$706$	−26.2286	−0.987125
$707$	−6.66738	−0.250753
$708$	0	0
$709$	−39.8748	−1.49753	−0.748765	−	0.662836i	$-0.769353\pi$
$709$	−39.8748	−1.49753	−0.748765	+	0.662836i	$0.769353\pi$
$710$	−45.9163	−1.72321
$711$	0	0
$712$	8.70162	0.326107
$713$	−27.6336	−1.03489
$714$	0	0
$715$	−128.961	−4.82286
$716$	−0.139807	−0.00522483
$717$	0	0
$718$	40.6221	1.51600
$719$	16.8025	0.626627	0.313313	−	0.949650i	$-0.398561\pi$
$719$	16.8025	0.626627	0.313313	+	0.949650i	$0.398561\pi$
$720$	0	0
$721$	−1.67001	−0.0621945
$722$	−1.40720	−0.0523706
$723$	0	0
$724$	−0.150887	−0.00560767
$725$	−75.7543	−2.81344
$726$	0	0
$727$	39.2078	1.45414	0.727069	−	0.686564i	$-0.240882\pi$
$727$	39.2078	1.45414	0.727069	+	0.686564i	$0.240882\pi$
$728$	6.39339	0.236955
$729$	0	0
$730$	−61.1656	−2.26384
$731$	0.350016	0.0129458
$732$	0	0
$733$	−21.8749	−0.807968	−0.403984	−	0.914766i	$-0.632375\pi$
$733$	−21.8749	−0.807968	−0.403984	+	0.914766i	$0.632375\pi$
$734$	−34.0407	−1.25646
$735$	0	0
$736$	−0.330227	−0.0121723
$737$	−77.8140	−2.86631
$738$	0	0
$739$	−7.53478	−0.277172	−0.138586	−	0.990350i	$-0.544256\pi$
$739$	−7.53478	−0.277172	−0.138586	+	0.990350i	$0.544256\pi$
$740$	−0.804849	−0.0295868
$741$	0	0
$742$	−4.75473	−0.174552
$743$	0.985274	0.0361462	0.0180731	−	0.999837i	$-0.494247\pi$
$743$	0.985274	0.0361462	0.0180731	+	0.999837i	$0.494247\pi$
$744$	0	0
$745$	56.2216	2.05980
$746$	28.0609	1.02738
$747$	0	0
$748$	−0.0310417	−0.00113500
$749$	−3.74634	−0.136888
$750$	0	0
$751$	25.2741	0.922266	0.461133	−	0.887331i	$-0.347443\pi$
$751$	25.2741	0.922266	0.461133	+	0.887331i	$0.347443\pi$
$752$	−3.96003	−0.144408
$753$	0	0
$754$	−43.7438	−1.59306
$755$	−77.2206	−2.81034
$756$	0	0
$757$	8.38016	0.304582	0.152291	−	0.988336i	$-0.451335\pi$
$757$	8.38016	0.304582	0.152291	+	0.988336i	$0.451335\pi$
$758$	−53.8621	−1.95636
$759$	0	0
$760$	11.8554	0.430041
$761$	−47.3401	−1.71608	−0.858039	−	0.513585i	$-0.828317\pi$
$761$	−47.3401	−1.71608	−0.858039	+	0.513585i	$0.828317\pi$
$762$	0	0
$763$	2.93587	0.106286
$764$	0.112371	0.00406546
$765$	0	0
$766$	−36.6029	−1.32252
$767$	−37.1228	−1.34043
$768$	0	0
$769$	−10.6298	−0.383321	−0.191661	−	0.981461i	$-0.561387\pi$
$769$	−10.6298	−0.383321	−0.191661	+	0.981461i	$0.561387\pi$
$770$	15.7704	0.568324
$771$	0	0
$772$	−0.0885802	−0.00318807
$773$	46.7518	1.68154	0.840772	−	0.541390i	$-0.182102\pi$
$773$	46.7518	1.68154	0.840772	+	0.541390i	$0.182102\pi$
$774$	0	0
$775$	−116.129	−4.17148
$776$	51.4384	1.84653
$777$	0	0
$778$	−22.3048	−0.799666
$779$	6.18056	0.221442
$780$	0	0
$781$	47.5377	1.70103
$782$	−1.07175	−0.0383255
$783$	0	0
$784$	26.9461	0.962362
$785$	29.1250	1.03951
$786$	0	0
$787$	6.62552	0.236174	0.118087	−	0.993003i	$-0.462324\pi$
$787$	6.62552	0.236174	0.118087	+	0.993003i	$0.462324\pi$
$788$	0.0863519	0.00307616
$789$	0	0
$790$	−56.4007	−2.00665
$791$	2.08169	0.0740164
$792$	0	0
$793$	53.6028	1.90349
$794$	−3.11259	−0.110462
$795$	0	0
$796$	−0.211811	−0.00750745
$797$	22.5759	0.799680	0.399840	−	0.916585i	$-0.369066\pi$
$797$	22.5759	0.799680	0.399840	+	0.916585i	$0.369066\pi$
$798$	0	0
$799$	0.258151	0.00913271
$800$	−1.38777	−0.0490650
$801$	0	0
$802$	20.1727	0.712324
$803$	63.3253	2.23470
$804$	0	0
$805$	−5.44085	−0.191765
$806$	−67.0581	−2.36202
$807$	0	0
$808$	42.8615	1.50786
$809$	−24.6003	−0.864900	−0.432450	−	0.901658i	$-0.642351\pi$
$809$	−24.6003	−0.864900	−0.432450	+	0.901658i	$0.642351\pi$
$810$	0	0
$811$	−28.1986	−0.990186	−0.495093	−	0.868840i	$-0.664866\pi$
$811$	−28.1986	−0.990186	−0.495093	+	0.868840i	$0.664866\pi$
$812$	−0.0534539	−0.00187586
$813$	0	0
$814$	−83.3884	−2.92276
$815$	−28.6058	−1.00202
$816$	0	0
$817$	1.35586	0.0474356
$818$	21.3222	0.745512
$819$	0	0
$820$	−0.510122	−0.0178142
$821$	14.0893	0.491719	0.245860	−	0.969305i	$-0.420930\pi$
$821$	14.0893	0.491719	0.245860	+	0.969305i	$0.420930\pi$
$822$	0	0
$823$	3.51922	0.122672	0.0613362	−	0.998117i	$-0.480464\pi$
$823$	3.51922	0.122672	0.0613362	+	0.998117i	$0.480464\pi$
$824$	10.7358	0.373998
$825$	0	0
$826$	4.53967	0.157955
$827$	47.4013	1.64830	0.824152	−	0.566369i	$-0.191652\pi$
$827$	47.4013	1.64830	0.824152	+	0.566369i	$0.191652\pi$
$828$	0	0
$829$	13.8190	0.479954	0.239977	−	0.970779i	$-0.422860\pi$
$829$	13.8190	0.479954	0.239977	+	0.970779i	$0.422860\pi$
$830$	−16.2959	−0.565639
$831$	0	0
$832$	−41.0962	−1.42476
$833$	−1.75659	−0.0608623
$834$	0	0
$835$	−40.1778	−1.39041
$836$	−0.120246	−0.00415881
$837$	0	0
$838$	13.3936	0.462676
$839$	22.7777	0.786373	0.393186	−	0.919459i	$-0.371373\pi$
$839$	22.7777	0.786373	0.393186	+	0.919459i	$0.371373\pi$
$840$	0	0
$841$	8.33190	0.287307
$842$	14.5562	0.501641
$843$	0	0
$844$	−0.181175	−0.00623630
$845$	53.7436	1.84884
$846$	0	0
$847$	−11.4638	−0.393901
$848$	30.2635	1.03925
$849$	0	0
$850$	−4.50398	−0.154485
$851$	28.7694	0.986203
$852$	0	0
$853$	−47.6343	−1.63097	−0.815484	−	0.578780i	$-0.803529\pi$
$853$	−47.6343	−1.63097	−0.815484	+	0.578780i	$0.803529\pi$
$854$	−6.55497	−0.224307
$855$	0	0
$856$	24.0835	0.823158
$857$	23.6375	0.807443	0.403721	−	0.914882i	$-0.367716\pi$
$857$	23.6375	0.807443	0.403721	+	0.914882i	$0.367716\pi$
$858$	0	0
$859$	51.3736	1.75284	0.876422	−	0.481544i	$-0.159924\pi$
$859$	51.3736	1.75284	0.876422	+	0.481544i	$0.159924\pi$
$860$	−0.111908	−0.00381603
$861$	0	0
$862$	−39.5156	−1.34591
$863$	−27.2621	−0.928014	−0.464007	−	0.885832i	$-0.653589\pi$
$863$	−27.2621	−0.928014	−0.464007	+	0.885832i	$0.653589\pi$
$864$	0	0
$865$	88.9189	3.02333
$866$	21.2392	0.721738
$867$	0	0
$868$	−0.0819433	−0.00278134
$869$	58.3923	1.98082
$870$	0	0
$871$	65.1474	2.20743
$872$	−18.8734	−0.639134
$873$	0	0
$874$	−4.15163	−0.140431
$875$	−13.6441	−0.461254
$876$	0	0
$877$	48.9294	1.65223	0.826114	−	0.563502i	$-0.190546\pi$
$877$	48.9294	1.65223	0.826114	+	0.563502i	$0.190546\pi$
$878$	34.8243	1.17526
$879$	0	0
$880$	−100.377	−3.38372
$881$	−10.3722	−0.349449	−0.174725	−	0.984617i	$-0.555904\pi$
$881$	−10.3722	−0.349449	−0.174725	+	0.984617i	$0.555904\pi$
$882$	0	0
$883$	4.73192	0.159242	0.0796210	−	0.996825i	$-0.474629\pi$
$883$	4.73192	0.159242	0.0796210	+	0.996825i	$0.474629\pi$
$884$	0.0259887	0.000874095 0
$885$	0	0
$886$	−3.24875	−0.109144
$887$	−49.5719	−1.66446	−0.832230	−	0.554431i	$-0.812936\pi$
$887$	−49.5719	−1.66446	−0.832230	+	0.554431i	$0.812936\pi$
$888$	0	0
$889$	0.0467024	0.00156635
$890$	−17.9701	−0.602358
$891$	0	0
$892$	0.0944156	0.00316127
$893$	1.00000	0.0334637
$894$	0	0
$895$	29.4709	0.985104
$896$	4.92660	0.164586
$897$	0	0
$898$	48.0534	1.60356
$899$	57.2287	1.90869
$900$	0	0
$901$	−1.97285	−0.0657252
$902$	−52.8525	−1.75979
$903$	0	0
$904$	−13.3823	−0.445087
$905$	31.8065	1.05728
$906$	0	0
$907$	30.9399	1.02734	0.513672	−	0.857987i	$-0.328285\pi$
$907$	30.9399	1.02734	0.513672	+	0.857987i	$0.328285\pi$
$908$	0.117731	0.00390704
$909$	0	0
$910$	−13.2032	−0.437683
$911$	46.7228	1.54800	0.773998	−	0.633188i	$-0.218254\pi$
$911$	46.7228	1.54800	0.773998	+	0.633188i	$0.218254\pi$
$912$	0	0
$913$	16.8713	0.558359
$914$	−36.2758	−1.19990
$915$	0	0
$916$	0.0438699	0.00144950
$917$	0.572383	0.0189018
$918$	0	0
$919$	18.0914	0.596780	0.298390	−	0.954444i	$-0.403550\pi$
$919$	18.0914	0.596780	0.298390	+	0.954444i	$0.403550\pi$
$920$	34.9768	1.15315
$921$	0	0
$922$	20.1222	0.662689
$923$	−39.7995	−1.31001
$924$	0	0
$925$	120.903	3.97525
$926$	−22.0930	−0.726020
$927$	0	0
$928$	0.683895	0.0224500
$929$	1.41987	0.0465843	0.0232921	−	0.999729i	$-0.492585\pi$
$929$	1.41987	0.0465843	0.0232921	+	0.999729i	$0.492585\pi$
$930$	0	0
$931$	−6.80452	−0.223009
$932$	0.271796	0.00890297
$933$	0	0
$934$	−6.31752	−0.206716
$935$	6.54350	0.213995
$936$	0	0
$937$	−20.9246	−0.683578	−0.341789	−	0.939777i	$-0.611033\pi$
$937$	−20.9246	−0.683578	−0.341789	+	0.939777i	$0.611033\pi$
$938$	−7.96674	−0.260123
$939$	0	0
$940$	−0.0825365	−0.00269204
$941$	−29.7735	−0.970588	−0.485294	−	0.874351i	$-0.661287\pi$
$941$	−29.7735	−0.970588	−0.485294	+	0.874351i	$0.661287\pi$
$942$	0	0
$943$	18.2344	0.593792
$944$	−28.8947	−0.940443
$945$	0	0
$946$	−11.5945	−0.376970
$947$	36.6510	1.19100	0.595498	−	0.803357i	$-0.296955\pi$
$947$	36.6510	1.19100	0.595498	+	0.803357i	$0.296955\pi$
$948$	0	0
$949$	−53.0172	−1.72101
$950$	−17.4471	−0.566059
$951$	0	0
$952$	−0.324402	−0.0105139
$953$	−18.2791	−0.592118	−0.296059	−	0.955170i	$-0.595672\pi$
$953$	−18.2791	−0.592118	−0.296059	+	0.955170i	$0.595672\pi$
$954$	0	0
$955$	−23.6876	−0.766512
$956$	0.0258631	0.000836473 0
$957$	0	0
$958$	15.5590	0.502688
$959$	−5.83715	−0.188491
$960$	0	0
$961$	56.7301	1.83000
$962$	69.8144	2.25091
$963$	0	0
$964$	−0.261695	−0.00842864
$965$	18.6724	0.601087
$966$	0	0
$967$	−8.30195	−0.266973	−0.133486	−	0.991051i	$-0.542617\pi$
$967$	−8.30195	−0.266973	−0.133486	+	0.991051i	$0.542617\pi$
$968$	73.6955	2.36866
$969$	0	0
$970$	−106.227	−3.41076
$971$	9.58992	0.307755	0.153878	−	0.988090i	$-0.450824\pi$
$971$	9.58992	0.307755	0.153878	+	0.988090i	$0.450824\pi$
$972$	0	0
$973$	−3.48096	−0.111594
$974$	10.3633	0.332062
$975$	0	0
$976$	41.7220	1.33549
$977$	−30.9793	−0.991115	−0.495558	−	0.868575i	$-0.665036\pi$
$977$	−30.9793	−0.991115	−0.495558	+	0.868575i	$0.665036\pi$
$978$	0	0
$979$	18.6046	0.594606
$980$	0.561621	0.0179403
$981$	0	0
$982$	33.7705	1.07766
$983$	18.0888	0.576944	0.288472	−	0.957488i	$-0.406853\pi$
$983$	18.0888	0.576944	0.288472	+	0.957488i	$0.406853\pi$
$984$	0	0
$985$	−18.2027	−0.579987
$986$	2.21957	0.0706855
$987$	0	0
$988$	0.100673	0.00320282
$989$	4.00016	0.127198
$990$	0	0
$991$	−25.7958	−0.819431	−0.409715	−	0.912213i	$-0.634372\pi$
$991$	−25.7958	−0.819431	−0.409715	+	0.912213i	$0.634372\pi$
$992$	1.04839	0.0332865
$993$	0	0
$994$	4.86699	0.154372
$995$	44.6492	1.41547
$996$	0	0
$997$	−14.9047	−0.472038	−0.236019	−	0.971748i	$-0.575843\pi$
$997$	−14.9047	−0.472038	−0.236019	+	0.971748i	$0.575843\pi$
$998$	−5.11760	−0.161995
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.t.1.7		24
3.2	odd	2		2679.2.a.o.1.18	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2679.2.a.o.1.18	✓	24	3.2	odd	2
8037.2.a.t.1.7		24	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)

\(f(q)\)	\(=\)	\(q-1.40720 q^{2} -0.0197875 q^{4} +4.17114 q^{5} -0.442129 q^{7} +2.84225 q^{8} +O(q^{10})\)	\(q-1.40720 q^{2} -0.0197875 q^{4} +4.17114 q^{5} -0.442129 q^{7} +2.84225 q^{8} -5.86964 q^{10} +6.07689 q^{11} -5.08769 q^{13} +0.622164 q^{14} -3.96003 q^{16} +0.258151 q^{17} +1.00000 q^{19} -0.0825365 q^{20} -8.55141 q^{22} +2.95028 q^{23} +12.3984 q^{25} +7.15940 q^{26} +0.00874862 q^{28} -6.10998 q^{29} -9.36643 q^{31} -0.111931 q^{32} -0.363269 q^{34} -1.84418 q^{35} +9.75143 q^{37} -1.40720 q^{38} +11.8554 q^{40} +6.18056 q^{41} +1.35586 q^{43} -0.120246 q^{44} -4.15163 q^{46} +1.00000 q^{47} -6.80452 q^{49} -17.4471 q^{50} +0.100673 q^{52} -7.64224 q^{53} +25.3476 q^{55} -1.25664 q^{56} +8.59797 q^{58} +7.29658 q^{59} -10.5358 q^{61} +13.1804 q^{62} +8.07758 q^{64} -21.2215 q^{65} -12.8049 q^{67} -0.00510815 q^{68} +2.59513 q^{70} +7.82269 q^{71} +10.4207 q^{73} -13.7222 q^{74} -0.0197875 q^{76} -2.68677 q^{77} +9.60890 q^{79} -16.5179 q^{80} -8.69728 q^{82} +2.77631 q^{83} +1.07678 q^{85} -1.90797 q^{86} +17.2720 q^{88} +3.06153 q^{89} +2.24942 q^{91} -0.0583786 q^{92} -1.40720 q^{94} +4.17114 q^{95} +18.0978 q^{97} +9.57533 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8	\( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 32 * q^4 + 2 * q^5 + 6 * q^7 - 9 * q^8 + 7 * q^10 + 3 * q^11 + 11 * q^13 - 9 * q^14 + 40 * q^16 + 6 * q^17 + 24 * q^19 + 17 * q^20 + 15 * q^22 + 19 * q^23 + 54 * q^25 + q^26 + 26 * q^28 - 32 * q^29 + 12 * q^31 - 30 * q^32 + 38 * q^34 + 35 * q^35 + 14 * q^37 - 2 * q^38 + 47 * q^40 - 30 * q^41 + 10 * q^43 - q^44 - 3 * q^46 + 24 * q^47 + 56 * q^49 - 44 * q^50 - 10 * q^53 + 24 * q^55 + 6 * q^56 + 41 * q^58 - 11 * q^59 + 10 * q^61 - 2 * q^62 + 61 * q^64 - 35 * q^65 + 12 * q^67 + 22 * q^68 - 20 * q^70 - 44 * q^71 + 33 * q^73 - 36 * q^74 + 32 * q^76 + 23 * q^77 + 50 * q^79 + 13 * q^80 + 5 * q^82 + 72 * q^83 + 33 * q^85 - 44 * q^86 + 38 * q^88 - 48 * q^89 + 47 * q^91 + 29 * q^92 - 2 * q^94 + 2 * q^95 + 50 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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