LMFDB - Embedded newform 8037.2.a.u.1.19

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:	$34$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.0794800 q^{2} -1.99368 q^{4} -2.46391 q^{5} -4.68518 q^{7} -0.317418 q^{8} +O(q^{10})$	\(q+0.0794800 q^{2} -1.99368 q^{4} -2.46391 q^{5} -4.68518 q^{7} -0.317418 q^{8} -0.195831 q^{10} -2.05142 q^{11} -0.761055 q^{13} -0.372378 q^{14} +3.96214 q^{16} +2.06104 q^{17} +1.00000 q^{19} +4.91225 q^{20} -0.163047 q^{22} -5.89990 q^{23} +1.07084 q^{25} -0.0604887 q^{26} +9.34077 q^{28} +6.84444 q^{29} +4.16385 q^{31} +0.949747 q^{32} +0.163812 q^{34} +11.5438 q^{35} +1.78852 q^{37} +0.0794800 q^{38} +0.782088 q^{40} -3.33076 q^{41} -10.7948 q^{43} +4.08988 q^{44} -0.468924 q^{46} +1.00000 q^{47} +14.9509 q^{49} +0.0851101 q^{50} +1.51730 q^{52} +4.19202 q^{53} +5.05451 q^{55} +1.48716 q^{56} +0.543996 q^{58} +7.20201 q^{59} -1.72741 q^{61} +0.330943 q^{62} -7.84879 q^{64} +1.87517 q^{65} +11.8729 q^{67} -4.10906 q^{68} +0.917505 q^{70} +4.40240 q^{71} +8.19240 q^{73} +0.142151 q^{74} -1.99368 q^{76} +9.61127 q^{77} -5.07491 q^{79} -9.76234 q^{80} -0.264729 q^{82} +11.3546 q^{83} -5.07822 q^{85} -0.857974 q^{86} +0.651157 q^{88} -11.4776 q^{89} +3.56568 q^{91} +11.7625 q^{92} +0.0794800 q^{94} -2.46391 q^{95} +2.25878 q^{97} +1.18830 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8}+O(q^{10}) $ 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8	$ 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8} - 18 q^{11} - 6 q^{13} - 12 q^{14} + 21 q^{16} - 36 q^{17} + 34 q^{19} - 40 q^{20} + 12 q^{22} - 38 q^{23} + 32 q^{25} - 15 q^{26} + 28 q^{28} - 14 q^{29} - 6 q^{31} - 35 q^{32} + 10 q^{34} - 46 q^{35} - 2 q^{37} - 5 q^{38} + 31 q^{40} - 18 q^{41} - 6 q^{43} - 42 q^{44} - 14 q^{46} + 34 q^{47} + 44 q^{49} - 9 q^{50} + 2 q^{52} - 32 q^{53} + 8 q^{55} + 4 q^{56} + 8 q^{58} - 62 q^{59} - 10 q^{61} - 30 q^{62} - 37 q^{64} - 8 q^{65} - 92 q^{68} - 62 q^{70} - 4 q^{71} - 8 q^{73} - 34 q^{74} + 31 q^{76} - 52 q^{77} + 40 q^{79} - 48 q^{80} - 2 q^{82} - 110 q^{83} - 12 q^{85} - 16 q^{86} - 44 q^{88} - 2 q^{89} - 28 q^{91} - 60 q^{92} - 5 q^{94} - 14 q^{95} + 2 q^{97} - 45 q^{98}+O(q^{100}) $ 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8 - 18 * q^11 - 6 * q^13 - 12 * q^14 + 21 * q^16 - 36 * q^17 + 34 * q^19 - 40 * q^20 + 12 * q^22 - 38 * q^23 + 32 * q^25 - 15 * q^26 + 28 * q^28 - 14 * q^29 - 6 * q^31 - 35 * q^32 + 10 * q^34 - 46 * q^35 - 2 * q^37 - 5 * q^38 + 31 * q^40 - 18 * q^41 - 6 * q^43 - 42 * q^44 - 14 * q^46 + 34 * q^47 + 44 * q^49 - 9 * q^50 + 2 * q^52 - 32 * q^53 + 8 * q^55 + 4 * q^56 + 8 * q^58 - 62 * q^59 - 10 * q^61 - 30 * q^62 - 37 * q^64 - 8 * q^65 - 92 * q^68 - 62 * q^70 - 4 * q^71 - 8 * q^73 - 34 * q^74 + 31 * q^76 - 52 * q^77 + 40 * q^79 - 48 * q^80 - 2 * q^82 - 110 * q^83 - 12 * q^85 - 16 * q^86 - 44 * q^88 - 2 * q^89 - 28 * q^91 - 60 * q^92 - 5 * q^94 - 14 * q^95 + 2 * q^97 - 45 * 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 $

$2$	0.0794800	0.0562009	0.0281004	−	0.999605i	$-0.491054\pi$
$2$	0.0794800	0.0562009	0.0281004	+	0.999605i	$0.491054\pi$
$3$	0	0
$3$	0	0
$4$	−1.99368	−0.996841
$5$	−2.46391	−1.10189	−0.550946	−	0.834541i	$-0.685733\pi$
$5$	−2.46391	−1.10189	−0.550946	+	0.834541i	$0.685733\pi$
$6$	0	0
$7$	−4.68518	−1.77083	−0.885416	−	0.464799i	$-0.846126\pi$
$7$	−4.68518	−1.77083	−0.885416	+	0.464799i	$0.846126\pi$
$8$	−0.317418	−0.112224
$9$	0	0
$10$	−0.195831	−0.0619273
$11$	−2.05142	−0.618526	−0.309263	−	0.950977i	$-0.600082\pi$
$11$	−2.05142	−0.618526	−0.309263	+	0.950977i	$0.600082\pi$
$12$	0	0
$13$	−0.761055	−0.211079	−0.105539	−	0.994415i	$-0.533657\pi$
$13$	−0.761055	−0.211079	−0.105539	+	0.994415i	$0.533657\pi$
$14$	−0.372378	−0.0995223
$15$	0	0
$16$	3.96214	0.990534
$17$	2.06104	0.499876	0.249938	−	0.968262i	$-0.419590\pi$
$17$	2.06104	0.499876	0.249938	+	0.968262i	$0.419590\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	4.91225	1.09841
$21$	0	0
$22$	−0.163047	−0.0347617
$23$	−5.89990	−1.23021	−0.615107	−	0.788443i	$-0.710887\pi$
$23$	−5.89990	−1.23021	−0.615107	+	0.788443i	$0.710887\pi$
$24$	0	0
$25$	1.07084	0.214167
$26$	−0.0604887	−0.0118628
$27$	0	0
$28$	9.34077	1.76524
$29$	6.84444	1.27098	0.635490	−	0.772109i	$-0.280798\pi$
$29$	6.84444	1.27098	0.635490	+	0.772109i	$0.280798\pi$
$30$	0	0
$31$	4.16385	0.747849	0.373925	−	0.927459i	$-0.378012\pi$
$31$	4.16385	0.747849	0.373925	+	0.927459i	$0.378012\pi$
$32$	0.949747	0.167893
$33$	0	0
$34$	0.163812	0.0280935
$35$	11.5438	1.95127
$36$	0	0
$37$	1.78852	0.294030	0.147015	−	0.989134i	$-0.453033\pi$
$37$	1.78852	0.294030	0.147015	+	0.989134i	$0.453033\pi$
$38$	0.0794800	0.0128934
$39$	0	0
$40$	0.782088	0.123659
$41$	−3.33076	−0.520178	−0.260089	−	0.965585i	$-0.583752\pi$
$41$	−3.33076	−0.520178	−0.260089	+	0.965585i	$0.583752\pi$
$42$	0	0
$43$	−10.7948	−1.64620	−0.823099	−	0.567898i	$-0.807757\pi$
$43$	−10.7948	−1.64620	−0.823099	+	0.567898i	$0.807757\pi$
$44$	4.08988	0.616572
$45$	0	0
$46$	−0.468924	−0.0691391
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	14.9509	2.13585
$50$	0.0851101	0.0120364
$51$	0	0
$52$	1.51730	0.210412
$53$	4.19202	0.575819	0.287909	−	0.957658i	$-0.407040\pi$
$53$	4.19202	0.575819	0.287909	+	0.957658i	$0.407040\pi$
$54$	0	0
$55$	5.05451	0.681549
$56$	1.48716	0.198730
$57$	0	0
$58$	0.543996	0.0714302
$59$	7.20201	0.937622	0.468811	−	0.883299i	$-0.344683\pi$
$59$	7.20201	0.937622	0.468811	+	0.883299i	$0.344683\pi$
$60$	0	0
$61$	−1.72741	−0.221172	−0.110586	−	0.993867i	$-0.535273\pi$
$61$	−1.72741	−0.221172	−0.110586	+	0.993867i	$0.535273\pi$
$62$	0.330943	0.0420298
$63$	0	0
$64$	−7.84879	−0.981099
$65$	1.87517	0.232586
$66$	0	0
$67$	11.8729	1.45050	0.725252	−	0.688484i	$-0.241723\pi$
$67$	11.8729	1.45050	0.725252	+	0.688484i	$0.241723\pi$
$68$	−4.10906	−0.498297
$69$	0	0
$70$	0.917505	0.109663
$71$	4.40240	0.522468	0.261234	−	0.965275i	$-0.415871\pi$
$71$	4.40240	0.522468	0.261234	+	0.965275i	$0.415871\pi$
$72$	0	0
$73$	8.19240	0.958848	0.479424	−	0.877583i	$-0.340846\pi$
$73$	8.19240	0.958848	0.479424	+	0.877583i	$0.340846\pi$
$74$	0.142151	0.0165247
$75$	0	0
$76$	−1.99368	−0.228691
$77$	9.61127	1.09531
$78$	0	0
$79$	−5.07491	−0.570972	−0.285486	−	0.958383i	$-0.592155\pi$
$79$	−5.07491	−0.570972	−0.285486	+	0.958383i	$0.592155\pi$
$80$	−9.76234	−1.09146
$81$	0	0
$82$	−0.264729	−0.0292344
$83$	11.3546	1.24633	0.623165	−	0.782090i	$-0.285846\pi$
$83$	11.3546	1.24633	0.623165	+	0.782090i	$0.285846\pi$
$84$	0	0
$85$	−5.07822	−0.550810
$86$	−0.857974	−0.0925177
$87$	0	0
$88$	0.651157	0.0694136
$89$	−11.4776	−1.21663	−0.608313	−	0.793697i	$-0.708154\pi$
$89$	−11.4776	−1.21663	−0.608313	+	0.793697i	$0.708154\pi$
$90$	0	0
$91$	3.56568	0.373785
$92$	11.7625	1.22633
$93$	0	0
$94$	0.0794800	0.00819774
$95$	−2.46391	−0.252792
$96$	0	0
$97$	2.25878	0.229345	0.114672	−	0.993403i	$-0.463418\pi$
$97$	2.25878	0.229345	0.114672	+	0.993403i	$0.463418\pi$
$98$	1.18830	0.120036
$99$	0	0
$100$	−2.13491	−0.213491
$101$	4.20870	0.418781	0.209391	−	0.977832i	$-0.432852\pi$
$101$	4.20870	0.418781	0.209391	+	0.977832i	$0.432852\pi$
$102$	0	0
$103$	7.76344	0.764954	0.382477	−	0.923965i	$-0.375071\pi$
$103$	7.76344	0.764954	0.382477	+	0.923965i	$0.375071\pi$
$104$	0.241573	0.0236881
$105$	0	0
$106$	0.333182	0.0323615
$107$	1.61655	0.156278	0.0781391	−	0.996942i	$-0.475102\pi$
$107$	1.61655	0.156278	0.0781391	+	0.996942i	$0.475102\pi$
$108$	0	0
$109$	2.32295	0.222499	0.111249	−	0.993793i	$-0.464515\pi$
$109$	2.32295	0.222499	0.111249	+	0.993793i	$0.464515\pi$
$110$	0.401732	0.0383037
$111$	0	0
$112$	−18.5633	−1.75407
$113$	13.3430	1.25520	0.627601	−	0.778535i	$-0.284037\pi$
$113$	13.3430	1.25520	0.627601	+	0.778535i	$0.284037\pi$
$114$	0	0
$115$	14.5368	1.35556
$116$	−13.6456	−1.26697
$117$	0	0
$118$	0.572416	0.0526952
$119$	−9.65636	−0.885197
$120$	0	0
$121$	−6.79168	−0.617425
$122$	−0.137294	−0.0124300
$123$	0	0
$124$	−8.30139	−0.745487
$125$	9.68109	0.865903
$126$	0	0
$127$	11.6923	1.03753	0.518763	−	0.854918i	$-0.326393\pi$
$127$	11.6923	1.03753	0.518763	+	0.854918i	$0.326393\pi$
$128$	−2.52332	−0.223032
$129$	0	0
$130$	0.149038	0.0130715
$131$	−3.12489	−0.273023	−0.136511	−	0.990639i	$-0.543589\pi$
$131$	−3.12489	−0.273023	−0.136511	+	0.990639i	$0.543589\pi$
$132$	0	0
$133$	−4.68518	−0.406257
$134$	0.943657	0.0815195
$135$	0	0
$136$	−0.654212	−0.0560982
$137$	−17.6732	−1.50992	−0.754960	−	0.655771i	$-0.772344\pi$
$137$	−17.6732	−1.50992	−0.754960	+	0.655771i	$0.772344\pi$
$138$	0	0
$139$	−9.17971	−0.778613	−0.389306	−	0.921108i	$-0.627285\pi$
$139$	−9.17971	−0.778613	−0.389306	+	0.921108i	$0.627285\pi$
$140$	−23.0148	−1.94510
$141$	0	0
$142$	0.349902	0.0293632
$143$	1.56124	0.130558
$144$	0	0
$145$	−16.8641	−1.40048
$146$	0.651132	0.0538881
$147$	0	0
$148$	−3.56573	−0.293102
$149$	−10.1429	−0.830943	−0.415471	−	0.909606i	$-0.636383\pi$
$149$	−10.1429	−0.830943	−0.415471	+	0.909606i	$0.636383\pi$
$150$	0	0
$151$	−16.1122	−1.31119	−0.655595	−	0.755113i	$-0.727582\pi$
$151$	−16.1122	−1.31119	−0.655595	+	0.755113i	$0.727582\pi$
$152$	−0.317418	−0.0257460
$153$	0	0
$154$	0.763904	0.0615571
$155$	−10.2593	−0.824050
$156$	0	0
$157$	14.4071	1.14981	0.574905	−	0.818221i	$-0.305039\pi$
$157$	14.4071	1.14981	0.574905	+	0.818221i	$0.305039\pi$
$158$	−0.403354	−0.0320891
$159$	0	0
$160$	−2.34009	−0.185000
$161$	27.6421	2.17850
$162$	0	0
$163$	19.5093	1.52808	0.764041	−	0.645167i	$-0.223212\pi$
$163$	19.5093	1.52808	0.764041	+	0.645167i	$0.223212\pi$
$164$	6.64048	0.518535
$165$	0	0
$166$	0.902464	0.0700448
$167$	−24.0163	−1.85844	−0.929219	−	0.369529i	$-0.879519\pi$
$167$	−24.0163	−1.85844	−0.929219	+	0.369529i	$0.879519\pi$
$168$	0	0
$169$	−12.4208	−0.955446
$170$	−0.403617	−0.0309560
$171$	0	0
$172$	21.5215	1.64100
$173$	14.3146	1.08832	0.544159	−	0.838982i	$-0.316849\pi$
$173$	14.3146	1.08832	0.544159	+	0.838982i	$0.316849\pi$
$174$	0	0
$175$	−5.01706	−0.379254
$176$	−8.12800	−0.612671
$177$	0	0
$178$	−0.912242	−0.0683754
$179$	−16.3416	−1.22143	−0.610713	−	0.791852i	$-0.709117\pi$
$179$	−16.3416	−1.22143	−0.610713	+	0.791852i	$0.709117\pi$
$180$	0	0
$181$	20.4718	1.52166	0.760830	−	0.648951i	$-0.224792\pi$
$181$	20.4718	1.52166	0.760830	+	0.648951i	$0.224792\pi$
$182$	0.283400	0.0210070
$183$	0	0
$184$	1.87274	0.138060
$185$	−4.40674	−0.323990
$186$	0	0
$187$	−4.22806	−0.309186
$188$	−1.99368	−0.145404
$189$	0	0
$190$	−0.195831	−0.0142071
$191$	−19.3370	−1.39917	−0.699587	−	0.714547i	$-0.746633\pi$
$191$	−19.3370	−1.39917	−0.699587	+	0.714547i	$0.746633\pi$
$192$	0	0
$193$	−14.1327	−1.01730	−0.508648	−	0.860974i	$-0.669855\pi$
$193$	−14.1327	−1.01730	−0.508648	+	0.860974i	$0.669855\pi$
$194$	0.179528	0.0128894
$195$	0	0
$196$	−29.8074	−2.12910
$197$	−9.04099	−0.644144	−0.322072	−	0.946715i	$-0.604379\pi$
$197$	−9.04099	−0.644144	−0.322072	+	0.946715i	$0.604379\pi$
$198$	0	0
$199$	10.7437	0.761600	0.380800	−	0.924658i	$-0.375649\pi$
$199$	10.7437	0.761600	0.380800	+	0.924658i	$0.375649\pi$
$200$	−0.339903	−0.0240348
$201$	0	0
$202$	0.334507	0.0235359
$203$	−32.0674	−2.25069
$204$	0	0
$205$	8.20669	0.573180
$206$	0.617038	0.0429911
$207$	0	0
$208$	−3.01540	−0.209081
$209$	−2.05142	−0.141900
$210$	0	0
$211$	15.1225	1.04108	0.520538	−	0.853838i	$-0.325731\pi$
$211$	15.1225	1.04108	0.520538	+	0.853838i	$0.325731\pi$
$212$	−8.35756	−0.574000
$213$	0	0
$214$	0.128484	0.00878296
$215$	26.5975	1.81393
$216$	0	0
$217$	−19.5084	−1.32432
$218$	0.184628	0.0125046
$219$	0	0
$220$	−10.0771	−0.679397
$221$	−1.56857	−0.105513
$222$	0	0
$223$	−7.78062	−0.521029	−0.260514	−	0.965470i	$-0.583892\pi$
$223$	−7.78062	−0.521029	−0.260514	+	0.965470i	$0.583892\pi$
$224$	−4.44973	−0.297310
$225$	0	0
$226$	1.06050	0.0705435
$227$	−26.1193	−1.73360	−0.866801	−	0.498655i	$-0.833828\pi$
$227$	−26.1193	−1.73360	−0.866801	+	0.498655i	$0.833828\pi$
$228$	0	0
$229$	−20.1108	−1.32896	−0.664478	−	0.747308i	$-0.731346\pi$
$229$	−20.1108	−1.32896	−0.664478	+	0.747308i	$0.731346\pi$
$230$	1.15539	0.0761839
$231$	0	0
$232$	−2.17255	−0.142635
$233$	−13.0501	−0.854939	−0.427470	−	0.904030i	$-0.640595\pi$
$233$	−13.0501	−0.854939	−0.427470	+	0.904030i	$0.640595\pi$
$234$	0	0
$235$	−2.46391	−0.160728
$236$	−14.3585	−0.934660
$237$	0	0
$238$	−0.767487	−0.0497488
$239$	9.96750	0.644744	0.322372	−	0.946613i	$-0.395520\pi$
$239$	9.96750	0.644744	0.322372	+	0.946613i	$0.395520\pi$
$240$	0	0
$241$	20.8348	1.34209	0.671044	−	0.741418i	$-0.265846\pi$
$241$	20.8348	1.34209	0.671044	+	0.741418i	$0.265846\pi$
$242$	−0.539803	−0.0346998
$243$	0	0
$244$	3.44390	0.220473
$245$	−36.8377	−2.35347
$246$	0	0
$247$	−0.761055	−0.0484248
$248$	−1.32168	−0.0839268
$249$	0	0
$250$	0.769453	0.0486645
$251$	−18.6193	−1.17524	−0.587618	−	0.809138i	$-0.699934\pi$
$251$	−18.6193	−1.17524	−0.587618	+	0.809138i	$0.699934\pi$
$252$	0	0
$253$	12.1032	0.760920
$254$	0.929306	0.0583098
$255$	0	0
$256$	15.4970	0.968564
$257$	8.03005	0.500901	0.250450	−	0.968129i	$-0.419421\pi$
$257$	8.03005	0.500901	0.250450	+	0.968129i	$0.419421\pi$
$258$	0	0
$259$	−8.37952	−0.520678
$260$	−3.73849	−0.231851
$261$	0	0
$262$	−0.248366	−0.0153441
$263$	−20.5320	−1.26606	−0.633029	−	0.774128i	$-0.718189\pi$
$263$	−20.5320	−1.26606	−0.633029	+	0.774128i	$0.718189\pi$
$264$	0	0
$265$	−10.3288	−0.634490
$266$	−0.372378	−0.0228320
$267$	0	0
$268$	−23.6708	−1.44592
$269$	29.2173	1.78141	0.890706	−	0.454580i	$-0.150211\pi$
$269$	29.2173	1.78141	0.890706	+	0.454580i	$0.150211\pi$
$270$	0	0
$271$	10.0395	0.609859	0.304929	−	0.952375i	$-0.401367\pi$
$271$	10.0395	0.609859	0.304929	+	0.952375i	$0.401367\pi$
$272$	8.16613	0.495145
$273$	0	0
$274$	−1.40466	−0.0848588
$275$	−2.19674	−0.132468
$276$	0	0
$277$	10.6578	0.640365	0.320183	−	0.947356i	$-0.396256\pi$
$277$	10.6578	0.640365	0.320183	+	0.947356i	$0.396256\pi$
$278$	−0.729603	−0.0437587
$279$	0	0
$280$	−3.66423	−0.218979
$281$	19.2456	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$281$	19.2456	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$282$	0	0
$283$	3.96013	0.235405	0.117703	−	0.993049i	$-0.462447\pi$
$283$	3.96013	0.235405	0.117703	+	0.993049i	$0.462447\pi$
$284$	−8.77698	−0.520818
$285$	0	0
$286$	0.124088	0.00733745
$287$	15.6052	0.921147
$288$	0	0
$289$	−12.7521	−0.750124
$290$	−1.34036	−0.0787084
$291$	0	0
$292$	−16.3330	−0.955819
$293$	4.80117	0.280488	0.140244	−	0.990117i	$-0.455211\pi$
$293$	4.80117	0.280488	0.140244	+	0.990117i	$0.455211\pi$
$294$	0	0
$295$	−17.7451	−1.03316
$296$	−0.567707	−0.0329973
$297$	0	0
$298$	−0.806161	−0.0466997
$299$	4.49015	0.259672
$300$	0	0
$301$	50.5758	2.91514
$302$	−1.28059	−0.0736899
$303$	0	0
$304$	3.96214	0.227244
$305$	4.25617	0.243707
$306$	0	0
$307$	−23.8461	−1.36097	−0.680485	−	0.732762i	$-0.738231\pi$
$307$	−23.8461	−1.36097	−0.680485	+	0.732762i	$0.738231\pi$
$308$	−19.1618	−1.09185
$309$	0	0
$310$	−0.815412	−0.0463123
$311$	−22.4784	−1.27463	−0.637317	−	0.770602i	$-0.719956\pi$
$311$	−22.4784	−1.27463	−0.637317	+	0.770602i	$0.719956\pi$
$312$	0	0
$313$	−9.92439	−0.560960	−0.280480	−	0.959860i	$-0.590494\pi$
$313$	−9.92439	−0.560960	−0.280480	+	0.959860i	$0.590494\pi$
$314$	1.14507	0.0646202
$315$	0	0
$316$	10.1178	0.569168
$317$	−26.1858	−1.47074	−0.735370	−	0.677666i	$-0.762992\pi$
$317$	−26.1858	−1.47074	−0.735370	+	0.677666i	$0.762992\pi$
$318$	0	0
$319$	−14.0408	−0.786135
$320$	19.3387	1.08107
$321$	0	0
$322$	2.19700	0.122434
$323$	2.06104	0.114679
$324$	0	0
$325$	−0.814966	−0.0452062
$326$	1.55060	0.0858796
$327$	0	0
$328$	1.05724	0.0583765
$329$	−4.68518	−0.258302
$330$	0	0
$331$	14.8820	0.817991	0.408995	−	0.912536i	$-0.365879\pi$
$331$	14.8820	0.817991	0.408995	+	0.912536i	$0.365879\pi$
$332$	−22.6375	−1.24239
$333$	0	0
$334$	−1.90882	−0.104446
$335$	−29.2537	−1.59830
$336$	0	0
$337$	18.1287	0.987535	0.493767	−	0.869594i	$-0.335619\pi$
$337$	18.1287	0.987535	0.493767	+	0.869594i	$0.335619\pi$
$338$	−0.987205	−0.0536969
$339$	0	0
$340$	10.1244	0.549070
$341$	−8.54180	−0.462564
$342$	0	0
$343$	−37.2515	−2.01139
$344$	3.42648	0.184743
$345$	0	0
$346$	1.13772	0.0611644
$347$	−23.5518	−1.26433	−0.632165	−	0.774834i	$-0.717833\pi$
$347$	−23.5518	−1.26433	−0.632165	+	0.774834i	$0.717833\pi$
$348$	0	0
$349$	−3.04303	−0.162890	−0.0814449	−	0.996678i	$-0.525953\pi$
$349$	−3.04303	−0.162890	−0.0814449	+	0.996678i	$0.525953\pi$
$350$	−0.398756	−0.0213144
$351$	0	0
$352$	−1.94833	−0.103846
$353$	−27.0954	−1.44214	−0.721071	−	0.692861i	$-0.756350\pi$
$353$	−27.0954	−1.44214	−0.721071	+	0.692861i	$0.756350\pi$
$354$	0	0
$355$	−10.8471	−0.575704
$356$	22.8828	1.21278
$357$	0	0
$358$	−1.29883	−0.0686451
$359$	26.4512	1.39604	0.698020	−	0.716078i	$-0.254065\pi$
$359$	26.4512	1.39604	0.698020	+	0.716078i	$0.254065\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.62710	0.0855186
$363$	0	0
$364$	−7.10884	−0.372604
$365$	−20.1853	−1.05655
$366$	0	0
$367$	−9.99411	−0.521688	−0.260844	−	0.965381i	$-0.584001\pi$
$367$	−9.99411	−0.521688	−0.260844	+	0.965381i	$0.584001\pi$
$368$	−23.3762	−1.21857
$369$	0	0
$370$	−0.350247	−0.0182085
$371$	−19.6404	−1.01968
$372$	0	0
$373$	−21.0242	−1.08859	−0.544296	−	0.838893i	$-0.683203\pi$
$373$	−21.0242	−1.08859	−0.544296	+	0.838893i	$0.683203\pi$
$374$	−0.336046	−0.0173765
$375$	0	0
$376$	−0.317418	−0.0163696
$377$	−5.20899	−0.268277
$378$	0	0
$379$	1.97801	0.101603	0.0508017	−	0.998709i	$-0.483822\pi$
$379$	1.97801	0.101603	0.0508017	+	0.998709i	$0.483822\pi$
$380$	4.91225	0.251993
$381$	0	0
$382$	−1.53690	−0.0786348
$383$	−0.0668538	−0.00341607	−0.00170804	−	0.999999i	$-0.500544\pi$
$383$	−0.0668538	−0.00341607	−0.00170804	+	0.999999i	$0.500544\pi$
$384$	0	0
$385$	−23.6813	−1.20691
$386$	−1.12327	−0.0571729
$387$	0	0
$388$	−4.50329	−0.228620
$389$	26.1399	1.32535	0.662673	−	0.748909i	$-0.269422\pi$
$389$	26.1399	1.32535	0.662673	+	0.748909i	$0.269422\pi$
$390$	0	0
$391$	−12.1599	−0.614955
$392$	−4.74569	−0.239694
$393$	0	0
$394$	−0.718578	−0.0362014
$395$	12.5041	0.629150
$396$	0	0
$397$	19.9828	1.00291	0.501454	−	0.865184i	$-0.332799\pi$
$397$	19.9828	1.00291	0.501454	+	0.865184i	$0.332799\pi$
$398$	0.853908	0.0428025
$399$	0	0
$400$	4.24280	0.212140
$401$	−30.2308	−1.50965	−0.754827	−	0.655924i	$-0.772279\pi$
$401$	−30.2308	−1.50965	−0.754827	+	0.655924i	$0.772279\pi$
$402$	0	0
$403$	−3.16892	−0.157855
$404$	−8.39081	−0.417458
$405$	0	0
$406$	−2.54872	−0.126491
$407$	−3.66900	−0.181865
$408$	0	0
$409$	12.9313	0.639410	0.319705	−	0.947517i	$-0.396416\pi$
$409$	12.9313	0.639410	0.319705	+	0.947517i	$0.396416\pi$
$410$	0.652267	0.0322132
$411$	0	0
$412$	−15.4778	−0.762538
$413$	−33.7427	−1.66037
$414$	0	0
$415$	−27.9767	−1.37332
$416$	−0.722809	−0.0354386
$417$	0	0
$418$	−0.163047	−0.00797488
$419$	19.9689	0.975543	0.487771	−	0.872971i	$-0.337810\pi$
$419$	19.9689	0.975543	0.487771	+	0.872971i	$0.337810\pi$
$420$	0	0
$421$	39.4659	1.92345	0.961724	−	0.274020i	$-0.0883535\pi$
$421$	39.4659	1.92345	0.961724	+	0.274020i	$0.0883535\pi$
$422$	1.20194	0.0585094
$423$	0	0
$424$	−1.33062	−0.0646208
$425$	2.20704	0.107057
$426$	0	0
$427$	8.09321	0.391658
$428$	−3.22289	−0.155785
$429$	0	0
$430$	2.11397	0.101945
$431$	20.4516	0.985117	0.492559	−	0.870279i	$-0.336062\pi$
$431$	20.4516	0.985117	0.492559	+	0.870279i	$0.336062\pi$
$432$	0	0
$433$	−6.06085	−0.291266	−0.145633	−	0.989339i	$-0.546522\pi$
$433$	−6.06085	−0.291266	−0.145633	+	0.989339i	$0.546522\pi$
$434$	−1.55053	−0.0744276
$435$	0	0
$436$	−4.63123	−0.221796
$437$	−5.89990	−0.282231
$438$	0	0
$439$	33.7605	1.61130	0.805649	−	0.592393i	$-0.201817\pi$
$439$	33.7605	1.61130	0.805649	+	0.592393i	$0.201817\pi$
$440$	−1.60439	−0.0764863
$441$	0	0
$442$	−0.124670	−0.00592993
$443$	−24.8410	−1.18023	−0.590117	−	0.807318i	$-0.700918\pi$
$443$	−24.8410	−1.18023	−0.590117	+	0.807318i	$0.700918\pi$
$444$	0	0
$445$	28.2798	1.34059
$446$	−0.618403	−0.0292823
$447$	0	0
$448$	36.7730	1.73736
$449$	−17.7522	−0.837780	−0.418890	−	0.908037i	$-0.637581\pi$
$449$	−17.7522	−0.837780	−0.418890	+	0.908037i	$0.637581\pi$
$450$	0	0
$451$	6.83279	0.321743
$452$	−26.6017	−1.25124
$453$	0	0
$454$	−2.07597	−0.0974299
$455$	−8.78550	−0.411871
$456$	0	0
$457$	18.5669	0.868525	0.434262	−	0.900786i	$-0.357009\pi$
$457$	18.5669	0.868525	0.434262	+	0.900786i	$0.357009\pi$
$458$	−1.59840	−0.0746885
$459$	0	0
$460$	−28.9818	−1.35128
$461$	−2.76395	−0.128730	−0.0643649	−	0.997926i	$-0.520502\pi$
$461$	−2.76395	−0.128730	−0.0643649	+	0.997926i	$0.520502\pi$
$462$	0	0
$463$	27.4063	1.27368	0.636839	−	0.770996i	$-0.280241\pi$
$463$	27.4063	1.27368	0.636839	+	0.770996i	$0.280241\pi$
$464$	27.1186	1.25895
$465$	0	0
$466$	−1.03722	−0.0480483
$467$	−24.0041	−1.11078	−0.555389	−	0.831591i	$-0.687431\pi$
$467$	−24.0041	−1.11078	−0.555389	+	0.831591i	$0.687431\pi$
$468$	0	0
$469$	−55.6266	−2.56860
$470$	−0.195831	−0.00903303
$471$	0	0
$472$	−2.28605	−0.105224
$473$	22.1447	1.01822
$474$	0	0
$475$	1.07084	0.0491334
$476$	19.2517	0.882401
$477$	0	0
$478$	0.792217	0.0362352
$479$	5.71644	0.261191	0.130595	−	0.991436i	$-0.458311\pi$
$479$	5.71644	0.261191	0.130595	+	0.991436i	$0.458311\pi$
$480$	0	0
$481$	−1.36116	−0.0620635
$482$	1.65595	0.0754265
$483$	0	0
$484$	13.5405	0.615475
$485$	−5.56543	−0.252713
$486$	0	0
$487$	1.58892	0.0720010	0.0360005	−	0.999352i	$-0.488538\pi$
$487$	1.58892	0.0720010	0.0360005	+	0.999352i	$0.488538\pi$
$488$	0.548310	0.0248208
$489$	0	0
$490$	−2.92786	−0.132267
$491$	−21.0770	−0.951191	−0.475595	−	0.879664i	$-0.657767\pi$
$491$	−21.0770	−0.951191	−0.475595	+	0.879664i	$0.657767\pi$
$492$	0	0
$493$	14.1067	0.635333
$494$	−0.0604887	−0.00272151
$495$	0	0
$496$	16.4977	0.740770
$497$	−20.6260	−0.925204
$498$	0	0
$499$	2.52467	0.113020	0.0565099	−	0.998402i	$-0.482003\pi$
$499$	2.52467	0.113020	0.0565099	+	0.998402i	$0.482003\pi$
$500$	−19.3010	−0.863168
$501$	0	0
$502$	−1.47986	−0.0660493
$503$	18.5286	0.826150	0.413075	−	0.910697i	$-0.364455\pi$
$503$	18.5286	0.826150	0.413075	+	0.910697i	$0.364455\pi$
$504$	0	0
$505$	−10.3698	−0.461452
$506$	0.961960	0.0427644
$507$	0	0
$508$	−23.3108	−1.03425
$509$	32.8276	1.45506	0.727529	−	0.686077i	$-0.240669\pi$
$509$	32.8276	1.45506	0.727529	+	0.686077i	$0.240669\pi$
$510$	0	0
$511$	−38.3829	−1.69796
$512$	6.27833	0.277466
$513$	0	0
$514$	0.638229	0.0281511
$515$	−19.1284	−0.842897
$516$	0	0
$517$	−2.05142	−0.0902213
$518$	−0.666004	−0.0292626
$519$	0	0
$520$	−0.595212	−0.0261018
$521$	2.18186	0.0955891	0.0477946	−	0.998857i	$-0.484781\pi$
$521$	2.18186	0.0955891	0.0477946	+	0.998857i	$0.484781\pi$
$522$	0	0
$523$	8.11734	0.354947	0.177473	−	0.984126i	$-0.443208\pi$
$523$	8.11734	0.354947	0.177473	+	0.984126i	$0.443208\pi$
$524$	6.23003	0.272160
$525$	0	0
$526$	−1.63188	−0.0711536
$527$	8.58187	0.373832
$528$	0	0
$529$	11.8089	0.513429
$530$	−0.820929	−0.0356589
$531$	0	0
$532$	9.34077	0.404974
$533$	2.53489	0.109798
$534$	0	0
$535$	−3.98304	−0.172202
$536$	−3.76867	−0.162782
$537$	0	0
$538$	2.32219	0.100117
$539$	−30.6706	−1.32108
$540$	0	0
$541$	9.46894	0.407101	0.203551	−	0.979064i	$-0.434752\pi$
$541$	9.46894	0.407101	0.203551	+	0.979064i	$0.434752\pi$
$542$	0.797943	0.0342746
$543$	0	0
$544$	1.95747	0.0839257
$545$	−5.72354	−0.245170
$546$	0	0
$547$	−7.01695	−0.300023	−0.150011	−	0.988684i	$-0.547931\pi$
$547$	−7.01695	−0.300023	−0.150011	+	0.988684i	$0.547931\pi$
$548$	35.2347	1.50515
$549$	0	0
$550$	−0.174597	−0.00744482
$551$	6.84444	0.291583
$552$	0	0
$553$	23.7769	1.01110
$554$	0.847082	0.0359891
$555$	0	0
$556$	18.3014	0.776153
$557$	37.9503	1.60801	0.804004	−	0.594624i	$-0.202699\pi$
$557$	37.9503	1.60801	0.804004	+	0.594624i	$0.202699\pi$
$558$	0	0
$559$	8.21547	0.347477
$560$	45.7383	1.93280
$561$	0	0
$562$	1.52964	0.0645241
$563$	−15.2224	−0.641546	−0.320773	−	0.947156i	$-0.603943\pi$
$563$	−15.2224	−0.641546	−0.320773	+	0.947156i	$0.603943\pi$
$564$	0	0
$565$	−32.8759	−1.38310
$566$	0.314751	0.0132300
$567$	0	0
$568$	−1.39740	−0.0586336
$569$	41.9586	1.75900	0.879498	−	0.475902i	$-0.157878\pi$
$569$	41.9586	1.75900	0.879498	+	0.475902i	$0.157878\pi$
$570$	0	0
$571$	23.5393	0.985088	0.492544	−	0.870288i	$-0.336067\pi$
$571$	23.5393	0.985088	0.492544	+	0.870288i	$0.336067\pi$
$572$	−3.11262	−0.130145
$573$	0	0
$574$	1.24030	0.0517692
$575$	−6.31783	−0.263472
$576$	0	0
$577$	−8.33006	−0.346785	−0.173393	−	0.984853i	$-0.555473\pi$
$577$	−8.33006	−0.346785	−0.173393	+	0.984853i	$0.555473\pi$
$578$	−1.01354	−0.0421576
$579$	0	0
$580$	33.6216	1.39606
$581$	−53.1984	−2.20704
$582$	0	0
$583$	−8.59959	−0.356159
$584$	−2.60041	−0.107606
$585$	0	0
$586$	0.381597	0.0157636
$587$	−2.67028	−0.110214	−0.0551072	−	0.998480i	$-0.517550\pi$
$587$	−2.67028	−0.110214	−0.0551072	+	0.998480i	$0.517550\pi$
$588$	0	0
$589$	4.16385	0.171568
$590$	−1.41038	−0.0580644
$591$	0	0
$592$	7.08635	0.291247
$593$	24.0034	0.985703	0.492851	−	0.870113i	$-0.335955\pi$
$593$	24.0034	0.985703	0.492851	+	0.870113i	$0.335955\pi$
$594$	0	0
$595$	23.7924	0.975392
$596$	20.2218	0.828318
$597$	0	0
$598$	0.356877	0.0145938
$599$	−19.9631	−0.815670	−0.407835	−	0.913056i	$-0.633716\pi$
$599$	−19.9631	−0.815670	−0.407835	+	0.913056i	$0.633716\pi$
$600$	0	0
$601$	−30.8621	−1.25889	−0.629447	−	0.777044i	$-0.716718\pi$
$601$	−30.8621	−1.25889	−0.629447	+	0.777044i	$0.716718\pi$
$602$	4.01976	0.163833
$603$	0	0
$604$	32.1225	1.30705
$605$	16.7341	0.680337
$606$	0	0
$607$	29.6048	1.20162	0.600811	−	0.799391i	$-0.294844\pi$
$607$	29.6048	1.20162	0.600811	+	0.799391i	$0.294844\pi$
$608$	0.949747	0.0385173
$609$	0	0
$610$	0.338280	0.0136966
$611$	−0.761055	−0.0307890
$612$	0	0
$613$	35.0037	1.41379	0.706894	−	0.707319i	$-0.250096\pi$
$613$	35.0037	1.41379	0.706894	+	0.707319i	$0.250096\pi$
$614$	−1.89529	−0.0764877
$615$	0	0
$616$	−3.05079	−0.122920
$617$	9.83853	0.396084	0.198042	−	0.980193i	$-0.436542\pi$
$617$	9.83853	0.396084	0.198042	+	0.980193i	$0.436542\pi$
$618$	0	0
$619$	−15.5675	−0.625712	−0.312856	−	0.949801i	$-0.601286\pi$
$619$	−15.5675	−0.625712	−0.312856	+	0.949801i	$0.601286\pi$
$620$	20.4539	0.821447
$621$	0	0
$622$	−1.78658	−0.0716355
$623$	53.7748	2.15444
$624$	0	0
$625$	−29.2075	−1.16830
$626$	−0.788791	−0.0315264
$627$	0	0
$628$	−28.7231	−1.14618
$629$	3.68621	0.146979
$630$	0	0
$631$	3.60741	0.143609	0.0718044	−	0.997419i	$-0.477124\pi$
$631$	3.60741	0.143609	0.0718044	+	0.997419i	$0.477124\pi$
$632$	1.61087	0.0640768
$633$	0	0
$634$	−2.08125	−0.0826568
$635$	−28.8088	−1.14324
$636$	0	0
$637$	−11.3785	−0.450832
$638$	−1.11596	−0.0441814
$639$	0	0
$640$	6.21721	0.245757
$641$	−9.60894	−0.379530	−0.189765	−	0.981830i	$-0.560773\pi$
$641$	−9.60894	−0.379530	−0.189765	+	0.981830i	$0.560773\pi$
$642$	0	0
$643$	−14.2021	−0.560075	−0.280038	−	0.959989i	$-0.590347\pi$
$643$	−14.2021	−0.560075	−0.280038	+	0.959989i	$0.590347\pi$
$644$	−55.1096	−2.17162
$645$	0	0
$646$	0.163812	0.00644508
$647$	36.3769	1.43013	0.715063	−	0.699060i	$-0.246398\pi$
$647$	36.3769	1.43013	0.715063	+	0.699060i	$0.246398\pi$
$648$	0	0
$649$	−14.7743	−0.579944
$650$	−0.0647735	−0.00254063
$651$	0	0
$652$	−38.8953	−1.52326
$653$	−1.87493	−0.0733719	−0.0366859	−	0.999327i	$-0.511680\pi$
$653$	−1.87493	−0.0733719	−0.0366859	+	0.999327i	$0.511680\pi$
$654$	0	0
$655$	7.69943	0.300841
$656$	−13.1969	−0.515254
$657$	0	0
$658$	−0.372378	−0.0145168
$659$	22.5266	0.877512	0.438756	−	0.898606i	$-0.355419\pi$
$659$	22.5266	0.877512	0.438756	+	0.898606i	$0.355419\pi$
$660$	0	0
$661$	−22.7799	−0.886036	−0.443018	−	0.896513i	$-0.646092\pi$
$661$	−22.7799	−0.886036	−0.443018	+	0.896513i	$0.646092\pi$
$662$	1.18282	0.0459718
$663$	0	0
$664$	−3.60416	−0.139868
$665$	11.5438	0.447651
$666$	0	0
$667$	−40.3815	−1.56358
$668$	47.8809	1.85257
$669$	0	0
$670$	−2.32508	−0.0898258
$671$	3.54363	0.136800
$672$	0	0
$673$	−13.2643	−0.511301	−0.255651	−	0.966769i	$-0.582290\pi$
$673$	−13.2643	−0.511301	−0.255651	+	0.966769i	$0.582290\pi$
$674$	1.44087	0.0555003
$675$	0	0
$676$	24.7631	0.952428
$677$	−21.6726	−0.832947	−0.416474	−	0.909148i	$-0.636734\pi$
$677$	−21.6726	−0.832947	−0.416474	+	0.909148i	$0.636734\pi$
$678$	0	0
$679$	−10.5828	−0.406131
$680$	1.61192	0.0618142
$681$	0	0
$682$	−0.678902	−0.0259965
$683$	−45.8407	−1.75404	−0.877022	−	0.480450i	$-0.840473\pi$
$683$	−45.8407	−1.75404	−0.877022	+	0.480450i	$0.840473\pi$
$684$	0	0
$685$	43.5450	1.66377
$686$	−2.96075	−0.113042
$687$	0	0
$688$	−42.7706	−1.63062
$689$	−3.19036	−0.121543
$690$	0	0
$691$	5.16284	0.196404	0.0982018	−	0.995167i	$-0.468691\pi$
$691$	5.16284	0.196404	0.0982018	+	0.995167i	$0.468691\pi$
$692$	−28.5387	−1.08488
$693$	0	0
$694$	−1.87190	−0.0710564
$695$	22.6179	0.857948
$696$	0	0
$697$	−6.86484	−0.260024
$698$	−0.241860	−0.00915454
$699$	0	0
$700$	10.0024	0.378057
$701$	−36.4823	−1.37792	−0.688958	−	0.724801i	$-0.741932\pi$
$701$	−36.4823	−1.37792	−0.688958	+	0.724801i	$0.741932\pi$
$702$	0	0
$703$	1.78852	0.0674552
$704$	16.1012	0.606835
$705$	0	0
$706$	−2.15354	−0.0810496
$707$	−19.7185	−0.741591
$708$	0	0
$709$	43.4962	1.63353	0.816767	−	0.576967i	$-0.195764\pi$
$709$	43.4962	1.63353	0.816767	+	0.576967i	$0.195764\pi$
$710$	−0.862127	−0.0323551
$711$	0	0
$712$	3.64321	0.136535
$713$	−24.5663	−0.920015
$714$	0	0
$715$	−3.84676	−0.143861
$716$	32.5799	1.21757
$717$	0	0
$718$	2.10234	0.0784587
$719$	28.7752	1.07313	0.536567	−	0.843858i	$-0.319721\pi$
$719$	28.7752	1.07313	0.536567	+	0.843858i	$0.319721\pi$
$720$	0	0
$721$	−36.3731	−1.35461
$722$	0.0794800	0.00295794
$723$	0	0
$724$	−40.8144	−1.51685
$725$	7.32928	0.272203
$726$	0	0
$727$	−4.11060	−0.152454	−0.0762268	−	0.997091i	$-0.524287\pi$
$727$	−4.11060	−0.152454	−0.0762268	+	0.997091i	$0.524287\pi$
$728$	−1.13181	−0.0419477
$729$	0	0
$730$	−1.60433	−0.0593789
$731$	−22.2486	−0.822895
$732$	0	0
$733$	27.2282	1.00570	0.502849	−	0.864375i	$-0.332285\pi$
$733$	27.2282	1.00570	0.502849	+	0.864375i	$0.332285\pi$
$734$	−0.794332	−0.0293193
$735$	0	0
$736$	−5.60341	−0.206545
$737$	−24.3563	−0.897174
$738$	0	0
$739$	2.68699	0.0988424	0.0494212	−	0.998778i	$-0.484262\pi$
$739$	2.68699	0.0988424	0.0494212	+	0.998778i	$0.484262\pi$
$740$	8.78564	0.322966
$741$	0	0
$742$	−1.56102	−0.0573068
$743$	−28.2321	−1.03573	−0.517867	−	0.855461i	$-0.673274\pi$
$743$	−28.2321	−1.03573	−0.517867	+	0.855461i	$0.673274\pi$
$744$	0	0
$745$	24.9913	0.915609
$746$	−1.67101	−0.0611798
$747$	0	0
$748$	8.42941	0.308210
$749$	−7.57385	−0.276742
$750$	0	0
$751$	−28.1013	−1.02543	−0.512716	−	0.858558i	$-0.671361\pi$
$751$	−28.1013	−1.02543	−0.512716	+	0.858558i	$0.671361\pi$
$752$	3.96214	0.144484
$753$	0	0
$754$	−0.414011	−0.0150774
$755$	39.6989	1.44479
$756$	0	0
$757$	−28.5398	−1.03730	−0.518648	−	0.854988i	$-0.673564\pi$
$757$	−28.5398	−1.03730	−0.518648	+	0.854988i	$0.673564\pi$
$758$	0.157212	0.00571020
$759$	0	0
$760$	0.782088	0.0283693
$761$	38.2157	1.38532	0.692659	−	0.721265i	$-0.256439\pi$
$761$	38.2157	1.38532	0.692659	+	0.721265i	$0.256439\pi$
$762$	0	0
$763$	−10.8835	−0.394008
$764$	38.5518	1.39476
$765$	0	0
$766$	−0.00531354	−0.000191986 0
$767$	−5.48113	−0.197912
$768$	0	0
$769$	14.9460	0.538966	0.269483	−	0.963005i	$-0.413147\pi$
$769$	14.9460	0.538966	0.269483	+	0.963005i	$0.413147\pi$
$770$	−1.88219	−0.0678293
$771$	0	0
$772$	28.1762	1.01408
$773$	−3.99746	−0.143779	−0.0718893	−	0.997413i	$-0.522903\pi$
$773$	−3.99746	−0.143779	−0.0718893	+	0.997413i	$0.522903\pi$
$774$	0	0
$775$	4.45880	0.160165
$776$	−0.716978	−0.0257380
$777$	0	0
$778$	2.07760	0.0744856
$779$	−3.33076	−0.119337
$780$	0	0
$781$	−9.03116	−0.323160
$782$	−0.966473	−0.0345610
$783$	0	0
$784$	59.2376	2.11563
$785$	−35.4977	−1.26697
$786$	0	0
$787$	23.1997	0.826981	0.413491	−	0.910508i	$-0.364309\pi$
$787$	23.1997	0.826981	0.413491	+	0.910508i	$0.364309\pi$
$788$	18.0249	0.642109
$789$	0	0
$790$	0.993826	0.0353587
$791$	−62.5143	−2.22275
$792$	0	0
$793$	1.31465	0.0466846
$794$	1.58823	0.0563643
$795$	0	0
$796$	−21.4195	−0.759194
$797$	−44.0173	−1.55917	−0.779586	−	0.626295i	$-0.784570\pi$
$797$	−44.0173	−1.55917	−0.779586	+	0.626295i	$0.784570\pi$
$798$	0	0
$799$	2.06104	0.0729144
$800$	1.01702	0.0359572
$801$	0	0
$802$	−2.40274	−0.0848438
$803$	−16.8060	−0.593072
$804$	0	0
$805$	−68.1076	−2.40048
$806$	−0.251866	−0.00887159
$807$	0	0
$808$	−1.33592	−0.0469974
$809$	−23.2335	−0.816845	−0.408423	−	0.912793i	$-0.633921\pi$
$809$	−23.2335	−0.816845	−0.408423	+	0.912793i	$0.633921\pi$
$810$	0	0
$811$	4.10021	0.143978	0.0719889	−	0.997405i	$-0.477065\pi$
$811$	4.10021	0.143978	0.0719889	+	0.997405i	$0.477065\pi$
$812$	63.9323	2.24358
$813$	0	0
$814$	−0.291612	−0.0102210
$815$	−48.0690	−1.68378
$816$	0	0
$817$	−10.7948	−0.377664
$818$	1.02778	0.0359354
$819$	0	0
$820$	−16.3615	−0.571369
$821$	−14.5887	−0.509149	−0.254574	−	0.967053i	$-0.581935\pi$
$821$	−14.5887	−0.509149	−0.254574	+	0.967053i	$0.581935\pi$
$822$	0	0
$823$	−26.2696	−0.915702	−0.457851	−	0.889029i	$-0.651381\pi$
$823$	−26.2696	−0.915702	−0.457851	+	0.889029i	$0.651381\pi$
$824$	−2.46425	−0.0858464
$825$	0	0
$826$	−2.68187	−0.0933143
$827$	−4.72631	−0.164350	−0.0821750	−	0.996618i	$-0.526187\pi$
$827$	−4.72631	−0.164350	−0.0821750	+	0.996618i	$0.526187\pi$
$828$	0	0
$829$	40.1129	1.39318	0.696589	−	0.717470i	$-0.254700\pi$
$829$	40.1129	1.39318	0.696589	+	0.717470i	$0.254700\pi$
$830$	−2.22359	−0.0771819
$831$	0	0
$832$	5.97336	0.207089
$833$	30.8145	1.06766
$834$	0	0
$835$	59.1740	2.04780
$836$	4.08988	0.141451
$837$	0	0
$838$	1.58713	0.0548263
$839$	−14.2836	−0.493124	−0.246562	−	0.969127i	$-0.579301\pi$
$839$	−14.2836	−0.493124	−0.246562	+	0.969127i	$0.579301\pi$
$840$	0	0
$841$	17.8463	0.615391
$842$	3.13675	0.108099
$843$	0	0
$844$	−30.1495	−1.03779
$845$	30.6037	1.05280
$846$	0	0
$847$	31.8203	1.09336
$848$	16.6094	0.570368
$849$	0	0
$850$	0.175416	0.00601670
$851$	−10.5521	−0.361720
$852$	0	0
$853$	16.0879	0.550838	0.275419	−	0.961324i	$-0.411183\pi$
$853$	16.0879	0.550838	0.275419	+	0.961324i	$0.411183\pi$
$854$	0.643248	0.0220115
$855$	0	0
$856$	−0.513123	−0.0175382
$857$	−14.1903	−0.484730	−0.242365	−	0.970185i	$-0.577923\pi$
$857$	−14.1903	−0.484730	−0.242365	+	0.970185i	$0.577923\pi$
$858$	0	0
$859$	−7.23337	−0.246799	−0.123400	−	0.992357i	$-0.539380\pi$
$859$	−7.23337	−0.246799	−0.123400	+	0.992357i	$0.539380\pi$
$860$	−53.0270	−1.80820
$861$	0	0
$862$	1.62549	0.0553644
$863$	−38.3633	−1.30590	−0.652951	−	0.757400i	$-0.726469\pi$
$863$	−38.3633	−1.30590	−0.652951	+	0.757400i	$0.726469\pi$
$864$	0	0
$865$	−35.2698	−1.19921
$866$	−0.481717	−0.0163694
$867$	0	0
$868$	38.8935	1.32013
$869$	10.4108	0.353161
$870$	0	0
$871$	−9.03592	−0.306170
$872$	−0.737347	−0.0249697
$873$	0	0
$874$	−0.468924	−0.0158616
$875$	−45.3577	−1.53337
$876$	0	0
$877$	1.64497	0.0555467	0.0277734	−	0.999614i	$-0.491158\pi$
$877$	1.64497	0.0555467	0.0277734	+	0.999614i	$0.491158\pi$
$878$	2.68328	0.0905563
$879$	0	0
$880$	20.0266	0.675098
$881$	−0.989141	−0.0333250	−0.0166625	−	0.999861i	$-0.505304\pi$
$881$	−0.989141	−0.0333250	−0.0166625	+	0.999861i	$0.505304\pi$
$882$	0	0
$883$	14.2502	0.479557	0.239779	−	0.970828i	$-0.422925\pi$
$883$	14.2502	0.479557	0.239779	+	0.970828i	$0.422925\pi$
$884$	3.12722	0.105180
$885$	0	0
$886$	−1.97437	−0.0663301
$887$	−37.5603	−1.26115	−0.630576	−	0.776128i	$-0.717181\pi$
$887$	−37.5603	−1.26115	−0.630576	+	0.776128i	$0.717181\pi$
$888$	0	0
$889$	−54.7807	−1.83728
$890$	2.24768	0.0753424
$891$	0	0
$892$	15.5121	0.519383
$893$	1.00000	0.0334637
$894$	0	0
$895$	40.2641	1.34588
$896$	11.8222	0.394952
$897$	0	0
$898$	−1.41095	−0.0470839
$899$	28.4992	0.950502
$900$	0	0
$901$	8.63993	0.287838
$902$	0.543070	0.0180823
$903$	0	0
$904$	−4.23530	−0.140864
$905$	−50.4407	−1.67671
$906$	0	0
$907$	30.6773	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$907$	30.6773	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$908$	52.0737	1.72813
$909$	0	0
$910$	−0.698272	−0.0231475
$911$	33.8509	1.12153	0.560765	−	0.827975i	$-0.310507\pi$
$911$	33.8509	1.12153	0.560765	+	0.827975i	$0.310507\pi$
$912$	0	0
$913$	−23.2931	−0.770888
$914$	1.47570	0.0488118
$915$	0	0
$916$	40.0945	1.32476
$917$	14.6407	0.483477
$918$	0	0
$919$	35.3546	1.16624	0.583121	−	0.812385i	$-0.301831\pi$
$919$	35.3546	1.16624	0.583121	+	0.812385i	$0.301831\pi$
$920$	−4.61424	−0.152127
$921$	0	0
$922$	−0.219679	−0.00723473
$923$	−3.35047	−0.110282
$924$	0	0
$925$	1.91521	0.0629717
$926$	2.17825	0.0715818
$927$	0	0
$928$	6.50048	0.213389
$929$	31.3707	1.02924	0.514620	−	0.857418i	$-0.327933\pi$
$929$	31.3707	1.02924	0.514620	+	0.857418i	$0.327933\pi$
$930$	0	0
$931$	14.9509	0.489997
$932$	26.0177	0.852239
$933$	0	0
$934$	−1.90785	−0.0624267
$935$	10.4175	0.340690
$936$	0	0
$937$	21.9779	0.717988	0.358994	−	0.933340i	$-0.383120\pi$
$937$	21.9779	0.717988	0.358994	+	0.933340i	$0.383120\pi$
$938$	−4.42120	−0.144357
$939$	0	0
$940$	4.91225	0.160220
$941$	14.0721	0.458738	0.229369	−	0.973340i	$-0.426334\pi$
$941$	14.0721	0.458738	0.229369	+	0.973340i	$0.426334\pi$
$942$	0	0
$943$	19.6512	0.639930
$944$	28.5354	0.928747
$945$	0	0
$946$	1.76006	0.0572246
$947$	8.90246	0.289291	0.144646	−	0.989484i	$-0.453796\pi$
$947$	8.90246	0.289291	0.144646	+	0.989484i	$0.453796\pi$
$948$	0	0
$949$	−6.23487	−0.202392
$950$	0.0851101	0.00276134
$951$	0	0
$952$	3.06510	0.0993405
$953$	18.3910	0.595744	0.297872	−	0.954606i	$-0.403723\pi$
$953$	18.3910	0.595744	0.297872	+	0.954606i	$0.403723\pi$
$954$	0	0
$955$	47.6445	1.54174
$956$	−19.8720	−0.642708
$957$	0	0
$958$	0.454343	0.0146791
$959$	82.8020	2.67381
$960$	0	0
$961$	−13.6624	−0.440722
$962$	−0.108185	−0.00348802
$963$	0	0
$964$	−41.5380	−1.33785
$965$	34.8217	1.12095
$966$	0	0
$967$	52.0362	1.67337	0.836686	−	0.547684i	$-0.184490\pi$
$967$	52.0362	1.67337	0.836686	+	0.547684i	$0.184490\pi$
$968$	2.15580	0.0692901
$969$	0	0
$970$	−0.442340	−0.0142027
$971$	−33.0762	−1.06147	−0.530733	−	0.847539i	$-0.678083\pi$
$971$	−33.0762	−1.06147	−0.530733	+	0.847539i	$0.678083\pi$
$972$	0	0
$973$	43.0086	1.37879
$974$	0.126288	0.00404652
$975$	0	0
$976$	−6.84422	−0.219078
$977$	−4.97185	−0.159063	−0.0795317	−	0.996832i	$-0.525342\pi$
$977$	−4.97185	−0.159063	−0.0795317	+	0.996832i	$0.525342\pi$
$978$	0	0
$979$	23.5454	0.752515
$980$	73.4426	2.34604
$981$	0	0
$982$	−1.67520	−0.0534577
$983$	−42.5161	−1.35605	−0.678027	−	0.735037i	$-0.737165\pi$
$983$	−42.5161	−1.35605	−0.678027	+	0.735037i	$0.737165\pi$
$984$	0	0
$985$	22.2762	0.709777
$986$	1.12120	0.0357062
$987$	0	0
$988$	1.51730	0.0482718
$989$	63.6885	2.02518
$990$	0	0
$991$	−48.7330	−1.54806	−0.774028	−	0.633152i	$-0.781761\pi$
$991$	−48.7330	−1.54806	−0.774028	+	0.633152i	$0.781761\pi$
$992$	3.95460	0.125559
$993$	0	0
$994$	−1.63936	−0.0519972
$995$	−26.4714	−0.839201
$996$	0	0
$997$	−33.6612	−1.06606	−0.533030	−	0.846097i	$-0.678947\pi$
$997$	−33.6612	−1.06606	−0.533030	+	0.846097i	$0.678947\pi$
$998$	0.200661	0.00635181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.u.1.19	✓	34
3.2	odd	2		8037.2.a.x.1.16	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.u.1.19	✓	34	1.1	even	1	trivial
8037.2.a.x.1.16	yes	34	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0794800 q^{2} -1.99368 q^{4} -2.46391 q^{5} -4.68518 q^{7} -0.317418 q^{8} +O(q^{10})\)	\(q+0.0794800 q^{2} -1.99368 q^{4} -2.46391 q^{5} -4.68518 q^{7} -0.317418 q^{8} -0.195831 q^{10} -2.05142 q^{11} -0.761055 q^{13} -0.372378 q^{14} +3.96214 q^{16} +2.06104 q^{17} +1.00000 q^{19} +4.91225 q^{20} -0.163047 q^{22} -5.89990 q^{23} +1.07084 q^{25} -0.0604887 q^{26} +9.34077 q^{28} +6.84444 q^{29} +4.16385 q^{31} +0.949747 q^{32} +0.163812 q^{34} +11.5438 q^{35} +1.78852 q^{37} +0.0794800 q^{38} +0.782088 q^{40} -3.33076 q^{41} -10.7948 q^{43} +4.08988 q^{44} -0.468924 q^{46} +1.00000 q^{47} +14.9509 q^{49} +0.0851101 q^{50} +1.51730 q^{52} +4.19202 q^{53} +5.05451 q^{55} +1.48716 q^{56} +0.543996 q^{58} +7.20201 q^{59} -1.72741 q^{61} +0.330943 q^{62} -7.84879 q^{64} +1.87517 q^{65} +11.8729 q^{67} -4.10906 q^{68} +0.917505 q^{70} +4.40240 q^{71} +8.19240 q^{73} +0.142151 q^{74} -1.99368 q^{76} +9.61127 q^{77} -5.07491 q^{79} -9.76234 q^{80} -0.264729 q^{82} +11.3546 q^{83} -5.07822 q^{85} -0.857974 q^{86} +0.651157 q^{88} -11.4776 q^{89} +3.56568 q^{91} +11.7625 q^{92} +0.0794800 q^{94} -2.46391 q^{95} +2.25878 q^{97} +1.18830 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8}+O(q^{10}) \) 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8	\( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8} - 18 q^{11} - 6 q^{13} - 12 q^{14} + 21 q^{16} - 36 q^{17} + 34 q^{19} - 40 q^{20} + 12 q^{22} - 38 q^{23} + 32 q^{25} - 15 q^{26} + 28 q^{28} - 14 q^{29} - 6 q^{31} - 35 q^{32} + 10 q^{34} - 46 q^{35} - 2 q^{37} - 5 q^{38} + 31 q^{40} - 18 q^{41} - 6 q^{43} - 42 q^{44} - 14 q^{46} + 34 q^{47} + 44 q^{49} - 9 q^{50} + 2 q^{52} - 32 q^{53} + 8 q^{55} + 4 q^{56} + 8 q^{58} - 62 q^{59} - 10 q^{61} - 30 q^{62} - 37 q^{64} - 8 q^{65} - 92 q^{68} - 62 q^{70} - 4 q^{71} - 8 q^{73} - 34 q^{74} + 31 q^{76} - 52 q^{77} + 40 q^{79} - 48 q^{80} - 2 q^{82} - 110 q^{83} - 12 q^{85} - 16 q^{86} - 44 q^{88} - 2 q^{89} - 28 q^{91} - 60 q^{92} - 5 q^{94} - 14 q^{95} + 2 q^{97} - 45 q^{98}+O(q^{100}) \) 34 * q - 5 * q^2 + 31 * q^4 - 14 * q^5 - 15 * q^8 - 18 * q^11 - 6 * q^13 - 12 * q^14 + 21 * q^16 - 36 * q^17 + 34 * q^19 - 40 * q^20 + 12 * q^22 - 38 * q^23 + 32 * q^25 - 15 * q^26 + 28 * q^28 - 14 * q^29 - 6 * q^31 - 35 * q^32 + 10 * q^34 - 46 * q^35 - 2 * q^37 - 5 * q^38 + 31 * q^40 - 18 * q^41 - 6 * q^43 - 42 * q^44 - 14 * q^46 + 34 * q^47 + 44 * q^49 - 9 * q^50 + 2 * q^52 - 32 * q^53 + 8 * q^55 + 4 * q^56 + 8 * q^58 - 62 * q^59 - 10 * q^61 - 30 * q^62 - 37 * q^64 - 8 * q^65 - 92 * q^68 - 62 * q^70 - 4 * q^71 - 8 * q^73 - 34 * q^74 + 31 * q^76 - 52 * q^77 + 40 * q^79 - 48 * q^80 - 2 * q^82 - 110 * q^83 - 12 * q^85 - 16 * q^86 - 44 * q^88 - 2 * q^89 - 28 * q^91 - 60 * q^92 - 5 * q^94 - 14 * q^95 + 2 * q^97 - 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more