LMFDB - Embedded newform 8037.2.a.w.1.15

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

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-0.356944 q^{2} -1.87259 q^{4} +2.18998 q^{5} +5.04345 q^{7} +1.38230 q^{8} +O(q^{10})$	\(q-0.356944 q^{2} -1.87259 q^{4} +2.18998 q^{5} +5.04345 q^{7} +1.38230 q^{8} -0.781702 q^{10} -2.10048 q^{11} -3.68499 q^{13} -1.80023 q^{14} +3.25178 q^{16} -2.46234 q^{17} -1.00000 q^{19} -4.10094 q^{20} +0.749753 q^{22} +0.00459055 q^{23} -0.203973 q^{25} +1.31534 q^{26} -9.44433 q^{28} +8.46356 q^{29} -1.44490 q^{31} -3.92530 q^{32} +0.878917 q^{34} +11.0451 q^{35} -7.25432 q^{37} +0.356944 q^{38} +3.02721 q^{40} -10.4872 q^{41} -4.18889 q^{43} +3.93333 q^{44} -0.00163857 q^{46} +1.00000 q^{47} +18.4364 q^{49} +0.0728071 q^{50} +6.90049 q^{52} +2.86960 q^{53} -4.60001 q^{55} +6.97156 q^{56} -3.02102 q^{58} +4.00929 q^{59} +13.4280 q^{61} +0.515748 q^{62} -5.10244 q^{64} -8.07008 q^{65} +3.67576 q^{67} +4.61095 q^{68} -3.94248 q^{70} +5.86714 q^{71} +7.03548 q^{73} +2.58939 q^{74} +1.87259 q^{76} -10.5937 q^{77} -2.16270 q^{79} +7.12134 q^{80} +3.74335 q^{82} +16.8237 q^{83} -5.39248 q^{85} +1.49520 q^{86} -2.90349 q^{88} +16.7089 q^{89} -18.5851 q^{91} -0.00859622 q^{92} -0.356944 q^{94} -2.18998 q^{95} +9.91565 q^{97} -6.58078 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	$ 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) $ 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * 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 $

$2$	−0.356944	−0.252398	−0.126199	−	0.992005i	$-0.540278\pi$
$2$	−0.356944	−0.252398	−0.126199	+	0.992005i	$0.540278\pi$
$3$	0	0
$3$	0	0
$4$	−1.87259	−0.936295
$5$	2.18998	0.979390	0.489695	−	0.871894i	$-0.337108\pi$
$5$	2.18998	0.979390	0.489695	+	0.871894i	$0.337108\pi$
$6$	0	0
$7$	5.04345	1.90625	0.953123	−	0.302582i	$-0.0978487\pi$
$7$	5.04345	1.90625	0.953123	+	0.302582i	$0.0978487\pi$
$8$	1.38230	0.488717
$9$	0	0
$10$	−0.781702	−0.247196
$11$	−2.10048	−0.633318	−0.316659	−	0.948539i	$-0.602561\pi$
$11$	−2.10048	−0.633318	−0.316659	+	0.948539i	$0.602561\pi$
$12$	0	0
$13$	−3.68499	−1.02203	−0.511017	−	0.859571i	$-0.670731\pi$
$13$	−3.68499	−1.02203	−0.511017	+	0.859571i	$0.670731\pi$
$14$	−1.80023	−0.481132
$15$	0	0
$16$	3.25178	0.812944
$17$	−2.46234	−0.597204	−0.298602	−	0.954378i	$-0.596520\pi$
$17$	−2.46234	−0.597204	−0.298602	+	0.954378i	$0.596520\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.10094	−0.916999
$21$	0	0
$22$	0.749753	0.159848
$23$	0.00459055	0.000957196 0	0.000478598	−	1.00000i	$-0.499848\pi$
$23$	0.00459055	0.000957196 0	0.000478598		1.00000i	$0.499848\pi$
$24$	0	0
$25$	−0.203973	−0.0407946
$26$	1.31534	0.257959
$27$	0	0
$28$	−9.44433	−1.78481
$29$	8.46356	1.57164	0.785821	−	0.618453i	$-0.212240\pi$
$29$	8.46356	1.57164	0.785821	+	0.618453i	$0.212240\pi$
$30$	0	0
$31$	−1.44490	−0.259511	−0.129756	−	0.991546i	$-0.541419\pi$
$31$	−1.44490	−0.259511	−0.129756	+	0.991546i	$0.541419\pi$
$32$	−3.92530	−0.693902
$33$	0	0
$34$	0.878917	0.150733
$35$	11.0451	1.86696
$36$	0	0
$37$	−7.25432	−1.19260	−0.596301	−	0.802761i	$-0.703364\pi$
$37$	−7.25432	−1.19260	−0.596301	+	0.802761i	$0.703364\pi$
$38$	0.356944	0.0579040
$39$	0	0
$40$	3.02721	0.478644
$41$	−10.4872	−1.63783	−0.818913	−	0.573917i	$-0.805423\pi$
$41$	−10.4872	−1.63783	−0.818913	+	0.573917i	$0.805423\pi$
$42$	0	0
$43$	−4.18889	−0.638800	−0.319400	−	0.947620i	$-0.603481\pi$
$43$	−4.18889	−0.638800	−0.319400	+	0.947620i	$0.603481\pi$
$44$	3.93333	0.592972
$45$	0	0
$46$	−0.00163857	−0.000241594 0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	18.4364	2.63378
$50$	0.0728071	0.0102965
$51$	0	0
$52$	6.90049	0.956925
$53$	2.86960	0.394170	0.197085	−	0.980386i	$-0.436853\pi$
$53$	2.86960	0.394170	0.197085	+	0.980386i	$0.436853\pi$
$54$	0	0
$55$	−4.60001	−0.620265
$56$	6.97156	0.931614
$57$	0	0
$58$	−3.02102	−0.396679
$59$	4.00929	0.521966	0.260983	−	0.965343i	$-0.415953\pi$
$59$	4.00929	0.521966	0.260983	+	0.965343i	$0.415953\pi$
$60$	0	0
$61$	13.4280	1.71928	0.859638	−	0.510904i	$-0.170689\pi$
$61$	13.4280	1.71928	0.859638	+	0.510904i	$0.170689\pi$
$62$	0.515748	0.0655001
$63$	0	0
$64$	−5.10244	−0.637805
$65$	−8.07008	−1.00097
$66$	0	0
$67$	3.67576	0.449065	0.224533	−	0.974467i	$-0.427914\pi$
$67$	3.67576	0.449065	0.224533	+	0.974467i	$0.427914\pi$
$68$	4.61095	0.559160
$69$	0	0
$70$	−3.94248	−0.471216
$71$	5.86714	0.696301	0.348150	−	0.937439i	$-0.386810\pi$
$71$	5.86714	0.696301	0.348150	+	0.937439i	$0.386810\pi$
$72$	0	0
$73$	7.03548	0.823440	0.411720	−	0.911310i	$-0.364928\pi$
$73$	7.03548	0.823440	0.411720	+	0.911310i	$0.364928\pi$
$74$	2.58939	0.301010
$75$	0	0
$76$	1.87259	0.214801
$77$	−10.5937	−1.20726
$78$	0	0
$79$	−2.16270	−0.243323	−0.121661	−	0.992572i	$-0.538822\pi$
$79$	−2.16270	−0.243323	−0.121661	+	0.992572i	$0.538822\pi$
$80$	7.12134	0.796190
$81$	0	0
$82$	3.74335	0.413384
$83$	16.8237	1.84664	0.923322	−	0.384026i	$-0.125463\pi$
$83$	16.8237	1.84664	0.923322	+	0.384026i	$0.125463\pi$
$84$	0	0
$85$	−5.39248	−0.584896
$86$	1.49520	0.161232
$87$	0	0
$88$	−2.90349	−0.309513
$89$	16.7089	1.77114	0.885572	−	0.464502i	$-0.153767\pi$
$89$	16.7089	1.77114	0.885572	+	0.464502i	$0.153767\pi$
$90$	0	0
$91$	−18.5851	−1.94825
$92$	−0.00859622	−0.000896218 0
$93$	0	0
$94$	−0.356944	−0.0368160
$95$	−2.18998	−0.224688
$96$	0	0
$97$	9.91565	1.00678	0.503391	−	0.864059i	$-0.332086\pi$
$97$	9.91565	1.00678	0.503391	+	0.864059i	$0.332086\pi$
$98$	−6.58078	−0.664759
$99$	0	0
$100$	0.381958	0.0381958
$101$	9.85094	0.980206	0.490103	−	0.871665i	$-0.336959\pi$
$101$	9.85094	0.980206	0.490103	+	0.871665i	$0.336959\pi$
$102$	0	0
$103$	−0.852306	−0.0839802	−0.0419901	−	0.999118i	$-0.513370\pi$
$103$	−0.852306	−0.0839802	−0.0419901	+	0.999118i	$0.513370\pi$
$104$	−5.09376	−0.499485
$105$	0	0
$106$	−1.02429	−0.0994876
$107$	13.9030	1.34405	0.672026	−	0.740527i	$-0.265424\pi$
$107$	13.9030	1.34405	0.672026	+	0.740527i	$0.265424\pi$
$108$	0	0
$109$	14.5333	1.39204	0.696020	−	0.718022i	$-0.254952\pi$
$109$	14.5333	1.39204	0.696020	+	0.718022i	$0.254952\pi$
$110$	1.64195	0.156554
$111$	0	0
$112$	16.4002	1.54967
$113$	−0.200784	−0.0188882	−0.00944409	−	0.999955i	$-0.503006\pi$
$113$	−0.200784	−0.0188882	−0.00944409	+	0.999955i	$0.503006\pi$
$114$	0	0
$115$	0.0100532	0.000937469 0
$116$	−15.8488	−1.47152
$117$	0	0
$118$	−1.43109	−0.131743
$119$	−12.4187	−1.13842
$120$	0	0
$121$	−6.58800	−0.598909
$122$	−4.79304	−0.433941
$123$	0	0
$124$	2.70570	0.242979
$125$	−11.3966	−1.01934
$126$	0	0
$127$	18.9727	1.68355	0.841776	−	0.539827i	$-0.181510\pi$
$127$	18.9727	1.68355	0.841776	+	0.539827i	$0.181510\pi$
$128$	9.67189	0.854882
$129$	0	0
$130$	2.88057	0.252642
$131$	−18.2657	−1.59588	−0.797940	−	0.602737i	$-0.794077\pi$
$131$	−18.2657	−1.59588	−0.797940	+	0.602737i	$0.794077\pi$
$132$	0	0
$133$	−5.04345	−0.437323
$134$	−1.31204	−0.113343
$135$	0	0
$136$	−3.40369	−0.291864
$137$	−5.08297	−0.434267	−0.217134	−	0.976142i	$-0.569671\pi$
$137$	−5.08297	−0.434267	−0.217134	+	0.976142i	$0.569671\pi$
$138$	0	0
$139$	9.87450	0.837544	0.418772	−	0.908091i	$-0.362461\pi$
$139$	9.87450	0.837544	0.418772	+	0.908091i	$0.362461\pi$
$140$	−20.6829	−1.74803
$141$	0	0
$142$	−2.09424	−0.175745
$143$	7.74025	0.647272
$144$	0	0
$145$	18.5350	1.53925
$146$	−2.51127	−0.207835
$147$	0	0
$148$	13.5844	1.11663
$149$	1.17613	0.0963526	0.0481763	−	0.998839i	$-0.484659\pi$
$149$	1.17613	0.0963526	0.0481763	+	0.998839i	$0.484659\pi$
$150$	0	0
$151$	6.03168	0.490851	0.245426	−	0.969415i	$-0.421072\pi$
$151$	6.03168	0.490851	0.245426	+	0.969415i	$0.421072\pi$
$152$	−1.38230	−0.112119
$153$	0	0
$154$	3.78135	0.304710
$155$	−3.16430	−0.254163
$156$	0	0
$157$	−20.9511	−1.67208	−0.836040	−	0.548669i	$-0.815135\pi$
$157$	−20.9511	−1.67208	−0.836040	+	0.548669i	$0.815135\pi$
$158$	0.771964	0.0614142
$159$	0	0
$160$	−8.59635	−0.679601
$161$	0.0231522	0.00182465
$162$	0	0
$163$	−6.20166	−0.485751	−0.242876	−	0.970057i	$-0.578091\pi$
$163$	−6.20166	−0.485751	−0.242876	+	0.970057i	$0.578091\pi$
$164$	19.6382	1.53349
$165$	0	0
$166$	−6.00513	−0.466089
$167$	1.79968	0.139263	0.0696316	−	0.997573i	$-0.477818\pi$
$167$	1.79968	0.139263	0.0696316	+	0.997573i	$0.477818\pi$
$168$	0	0
$169$	0.579181	0.0445524
$170$	1.92481	0.147626
$171$	0	0
$172$	7.84408	0.598105
$173$	16.7632	1.27449	0.637243	−	0.770663i	$-0.280075\pi$
$173$	16.7632	1.27449	0.637243	+	0.770663i	$0.280075\pi$
$174$	0	0
$175$	−1.02873	−0.0777646
$176$	−6.83029	−0.514852
$177$	0	0
$178$	−5.96416	−0.447033
$179$	11.5023	0.859723	0.429861	−	0.902895i	$-0.358562\pi$
$179$	11.5023	0.859723	0.429861	+	0.902895i	$0.358562\pi$
$180$	0	0
$181$	−6.23444	−0.463402	−0.231701	−	0.972787i	$-0.574429\pi$
$181$	−6.23444	−0.463402	−0.231701	+	0.972787i	$0.574429\pi$
$182$	6.63384	0.491733
$183$	0	0
$184$	0.00634551	0.000467798 0
$185$	−15.8868	−1.16802
$186$	0	0
$187$	5.17208	0.378220
$188$	−1.87259	−0.136573
$189$	0	0
$190$	0.781702	0.0567106
$191$	−15.2233	−1.10152	−0.550761	−	0.834663i	$-0.685662\pi$
$191$	−15.2233	−1.10152	−0.550761	+	0.834663i	$0.685662\pi$
$192$	0	0
$193$	4.71700	0.339537	0.169769	−	0.985484i	$-0.445698\pi$
$193$	4.71700	0.339537	0.169769	+	0.985484i	$0.445698\pi$
$194$	−3.53933	−0.254109
$195$	0	0
$196$	−34.5239	−2.46599
$197$	8.26748	0.589033	0.294517	−	0.955646i	$-0.404841\pi$
$197$	8.26748	0.589033	0.294517	+	0.955646i	$0.404841\pi$
$198$	0	0
$199$	−14.4445	−1.02394	−0.511971	−	0.859003i	$-0.671085\pi$
$199$	−14.4445	−1.02394	−0.511971	+	0.859003i	$0.671085\pi$
$200$	−0.281952	−0.0199370
$201$	0	0
$202$	−3.51624	−0.247402
$203$	42.6856	2.99594
$204$	0	0
$205$	−22.9668	−1.60407
$206$	0.304226	0.0211964
$207$	0	0
$208$	−11.9828	−0.830856
$209$	2.10048	0.145293
$210$	0	0
$211$	9.82707	0.676523	0.338262	−	0.941052i	$-0.390161\pi$
$211$	9.82707	0.676523	0.338262	+	0.941052i	$0.390161\pi$
$212$	−5.37359	−0.369059
$213$	0	0
$214$	−4.96259	−0.339236
$215$	−9.17360	−0.625634
$216$	0	0
$217$	−7.28728	−0.494693
$218$	−5.18759	−0.351348
$219$	0	0
$220$	8.61394	0.580751
$221$	9.07370	0.610363
$222$	0	0
$223$	6.67689	0.447118	0.223559	−	0.974690i	$-0.428233\pi$
$223$	6.67689	0.447118	0.223559	+	0.974690i	$0.428233\pi$
$224$	−19.7971	−1.32275
$225$	0	0
$226$	0.0716688	0.00476733
$227$	24.9595	1.65662	0.828310	−	0.560270i	$-0.189303\pi$
$227$	24.9595	1.65662	0.828310	+	0.560270i	$0.189303\pi$
$228$	0	0
$229$	−17.8037	−1.17650	−0.588252	−	0.808677i	$-0.700184\pi$
$229$	−17.8037	−1.17650	−0.588252	+	0.808677i	$0.700184\pi$
$230$	−0.00358844	−0.000236615 0
$231$	0	0
$232$	11.6992	0.768088
$233$	13.9252	0.912271	0.456136	−	0.889910i	$-0.349233\pi$
$233$	13.9252	0.912271	0.456136	+	0.889910i	$0.349233\pi$
$234$	0	0
$235$	2.18998	0.142859
$236$	−7.50777	−0.488714
$237$	0	0
$238$	4.43278	0.287334
$239$	12.5964	0.814795	0.407397	−	0.913251i	$-0.366436\pi$
$239$	12.5964	0.814795	0.407397	+	0.913251i	$0.366436\pi$
$240$	0	0
$241$	18.8765	1.21594	0.607971	−	0.793959i	$-0.291984\pi$
$241$	18.8765	1.21594	0.607971	+	0.793959i	$0.291984\pi$
$242$	2.35155	0.151163
$243$	0	0
$244$	−25.1451	−1.60975
$245$	40.3755	2.57949
$246$	0	0
$247$	3.68499	0.234471
$248$	−1.99728	−0.126827
$249$	0	0
$250$	4.06796	0.257280
$251$	−12.0187	−0.758616	−0.379308	−	0.925270i	$-0.623838\pi$
$251$	−12.0187	−0.758616	−0.379308	+	0.925270i	$0.623838\pi$
$252$	0	0
$253$	−0.00964235	−0.000606209 0
$254$	−6.77219	−0.424925
$255$	0	0
$256$	6.75256	0.422035
$257$	0.0191270	0.00119311	0.000596554	−	1.00000i	$-0.499810\pi$
$257$	0.0191270	0.00119311	0.000596554		1.00000i	$0.499810\pi$
$258$	0	0
$259$	−36.5868	−2.27339
$260$	15.1119	0.937203
$261$	0	0
$262$	6.51983	0.402796
$263$	5.05694	0.311824	0.155912	−	0.987771i	$-0.450168\pi$
$263$	5.05694	0.311824	0.155912	+	0.987771i	$0.450168\pi$
$264$	0	0
$265$	6.28438	0.386046
$266$	1.80023	0.110379
$267$	0	0
$268$	−6.88319	−0.420458
$269$	−19.9911	−1.21888	−0.609440	−	0.792832i	$-0.708606\pi$
$269$	−19.9911	−1.21888	−0.609440	+	0.792832i	$0.708606\pi$
$270$	0	0
$271$	−6.57992	−0.399702	−0.199851	−	0.979826i	$-0.564046\pi$
$271$	−6.57992	−0.399702	−0.199851	+	0.979826i	$0.564046\pi$
$272$	−8.00697	−0.485494
$273$	0	0
$274$	1.81434	0.109608
$275$	0.428441	0.0258360
$276$	0	0
$277$	−4.67520	−0.280905	−0.140453	−	0.990087i	$-0.544856\pi$
$277$	−4.67520	−0.280905	−0.140453	+	0.990087i	$0.544856\pi$
$278$	−3.52464	−0.211394
$279$	0	0
$280$	15.2676	0.912414
$281$	32.7196	1.95189	0.975944	−	0.218020i	$-0.0699597\pi$
$281$	32.7196	1.95189	0.975944	+	0.218020i	$0.0699597\pi$
$282$	0	0
$283$	−13.8971	−0.826097	−0.413048	−	0.910709i	$-0.635536\pi$
$283$	−13.8971	−0.826097	−0.413048	+	0.910709i	$0.635536\pi$
$284$	−10.9867	−0.651943
$285$	0	0
$286$	−2.76284	−0.163370
$287$	−52.8917	−3.12210
$288$	0	0
$289$	−10.9369	−0.643347
$290$	−6.61598	−0.388504
$291$	0	0
$292$	−13.1746	−0.770984
$293$	0.763995	0.0446330	0.0223165	−	0.999751i	$-0.492896\pi$
$293$	0.763995	0.0446330	0.0223165	+	0.999751i	$0.492896\pi$
$294$	0	0
$295$	8.78029	0.511208
$296$	−10.0276	−0.582845
$297$	0	0
$298$	−0.419814	−0.0243192
$299$	−0.0169162	−0.000978286 0
$300$	0	0
$301$	−21.1265	−1.21771
$302$	−2.15297	−0.123890
$303$	0	0
$304$	−3.25178	−0.186502
$305$	29.4070	1.68384
$306$	0	0
$307$	−18.8404	−1.07528	−0.537639	−	0.843175i	$-0.680684\pi$
$307$	−18.8404	−1.07528	−0.537639	+	0.843175i	$0.680684\pi$
$308$	19.8376	1.13035
$309$	0	0
$310$	1.12948	0.0641501
$311$	1.12812	0.0639700	0.0319850	−	0.999488i	$-0.489817\pi$
$311$	1.12812	0.0639700	0.0319850	+	0.999488i	$0.489817\pi$
$312$	0	0
$313$	−23.9277	−1.35248	−0.676238	−	0.736683i	$-0.736391\pi$
$313$	−23.9277	−1.35248	−0.676238	+	0.736683i	$0.736391\pi$
$314$	7.47837	0.422029
$315$	0	0
$316$	4.04986	0.227822
$317$	15.5844	0.875306	0.437653	−	0.899144i	$-0.355810\pi$
$317$	15.5844	0.875306	0.437653	+	0.899144i	$0.355810\pi$
$318$	0	0
$319$	−17.7775	−0.995349
$320$	−11.1743	−0.624660
$321$	0	0
$322$	−0.00826406	−0.000460538 0
$323$	2.46234	0.137008
$324$	0	0
$325$	0.751640	0.0416935
$326$	2.21365	0.122602
$327$	0	0
$328$	−14.4965	−0.800433
$329$	5.04345	0.278055
$330$	0	0
$331$	−27.5394	−1.51370	−0.756851	−	0.653588i	$-0.773263\pi$
$331$	−27.5394	−1.51370	−0.756851	+	0.653588i	$0.773263\pi$
$332$	−31.5040	−1.72900
$333$	0	0
$334$	−0.642385	−0.0351497
$335$	8.04985	0.439810
$336$	0	0
$337$	−14.3758	−0.783098	−0.391549	−	0.920157i	$-0.628061\pi$
$337$	−14.3758	−0.783098	−0.391549	+	0.920157i	$0.628061\pi$
$338$	−0.206735	−0.0112449
$339$	0	0
$340$	10.0979	0.547636
$341$	3.03498	0.164353
$342$	0	0
$343$	57.6791	3.11438
$344$	−5.79030	−0.312192
$345$	0	0
$346$	−5.98355	−0.321677
$347$	24.2961	1.30428	0.652142	−	0.758097i	$-0.273871\pi$
$347$	24.2961	1.30428	0.652142	+	0.758097i	$0.273871\pi$
$348$	0	0
$349$	28.3940	1.51990	0.759949	−	0.649983i	$-0.225224\pi$
$349$	28.3940	1.51990	0.759949	+	0.649983i	$0.225224\pi$
$350$	0.367199	0.0196276
$351$	0	0
$352$	8.24501	0.439460
$353$	34.7129	1.84758	0.923791	−	0.382897i	$-0.125074\pi$
$353$	34.7129	1.84758	0.923791	+	0.382897i	$0.125074\pi$
$354$	0	0
$355$	12.8489	0.681950
$356$	−31.2890	−1.65831
$357$	0	0
$358$	−4.10568	−0.216992
$359$	−35.1036	−1.85269	−0.926347	−	0.376670i	$-0.877069\pi$
$359$	−35.1036	−1.85269	−0.926347	+	0.376670i	$0.877069\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.22535	0.116962
$363$	0	0
$364$	34.8023	1.82414
$365$	15.4076	0.806470
$366$	0	0
$367$	−10.2760	−0.536401	−0.268200	−	0.963363i	$-0.586429\pi$
$367$	−10.2760	−0.536401	−0.268200	+	0.963363i	$0.586429\pi$
$368$	0.0149275	0.000778147 0
$369$	0	0
$370$	5.67071	0.294806
$371$	14.4727	0.751385
$372$	0	0
$373$	−10.7026	−0.554158	−0.277079	−	0.960847i	$-0.589366\pi$
$373$	−10.7026	−0.554158	−0.277079	+	0.960847i	$0.589366\pi$
$374$	−1.84615	−0.0954619
$375$	0	0
$376$	1.38230	0.0712866
$377$	−31.1882	−1.60627
$378$	0	0
$379$	−5.81782	−0.298842	−0.149421	−	0.988774i	$-0.547741\pi$
$379$	−5.81782	−0.298842	−0.149421	+	0.988774i	$0.547741\pi$
$380$	4.10094	0.210374
$381$	0	0
$382$	5.43388	0.278022
$383$	−17.1101	−0.874288	−0.437144	−	0.899392i	$-0.644010\pi$
$383$	−17.1101	−0.874288	−0.437144	+	0.899392i	$0.644010\pi$
$384$	0	0
$385$	−23.1999	−1.18238
$386$	−1.68371	−0.0856984
$387$	0	0
$388$	−18.5680	−0.942645
$389$	15.4391	0.782796	0.391398	−	0.920222i	$-0.371992\pi$
$389$	15.4391	0.782796	0.391398	+	0.920222i	$0.371992\pi$
$390$	0	0
$391$	−0.0113035	−0.000571642 0
$392$	25.4847	1.28717
$393$	0	0
$394$	−2.95103	−0.148671
$395$	−4.73628	−0.238308
$396$	0	0
$397$	8.02094	0.402559	0.201280	−	0.979534i	$-0.435490\pi$
$397$	8.02094	0.402559	0.201280	+	0.979534i	$0.435490\pi$
$398$	5.15588	0.258441
$399$	0	0
$400$	−0.663276	−0.0331638
$401$	22.9531	1.14622	0.573111	−	0.819477i	$-0.305736\pi$
$401$	22.9531	1.14622	0.573111	+	0.819477i	$0.305736\pi$
$402$	0	0
$403$	5.32444	0.265229
$404$	−18.4468	−0.917762
$405$	0	0
$406$	−15.2364	−0.756168
$407$	15.2375	0.755296
$408$	0	0
$409$	−8.90018	−0.440086	−0.220043	−	0.975490i	$-0.570620\pi$
$409$	−8.90018	−0.440086	−0.220043	+	0.975490i	$0.570620\pi$
$410$	8.19787	0.404864
$411$	0	0
$412$	1.59602	0.0786303
$413$	20.2207	0.994995
$414$	0	0
$415$	36.8437	1.80859
$416$	14.4647	0.709191
$417$	0	0
$418$	−0.749753	−0.0366716
$419$	−17.0109	−0.831039	−0.415519	−	0.909584i	$-0.636400\pi$
$419$	−17.0109	−0.831039	−0.415519	+	0.909584i	$0.636400\pi$
$420$	0	0
$421$	4.38372	0.213649	0.106825	−	0.994278i	$-0.465932\pi$
$421$	4.38372	0.213649	0.106825	+	0.994278i	$0.465932\pi$
$422$	−3.50772	−0.170753
$423$	0	0
$424$	3.96665	0.192637
$425$	0.502251	0.0243627
$426$	0	0
$427$	67.7234	3.27736
$428$	−26.0346	−1.25843
$429$	0	0
$430$	3.27446	0.157909
$431$	30.2311	1.45618	0.728091	−	0.685480i	$-0.240408\pi$
$431$	30.2311	1.45618	0.728091	+	0.685480i	$0.240408\pi$
$432$	0	0
$433$	−2.71008	−0.130238	−0.0651191	−	0.997877i	$-0.520743\pi$
$433$	−2.71008	−0.130238	−0.0651191	+	0.997877i	$0.520743\pi$
$434$	2.60115	0.124859
$435$	0	0
$436$	−27.2150	−1.30336
$437$	−0.00459055	−0.000219596 0
$438$	0	0
$439$	2.60980	0.124559	0.0622794	−	0.998059i	$-0.480163\pi$
$439$	2.60980	0.124559	0.0622794	+	0.998059i	$0.480163\pi$
$440$	−6.35859	−0.303134
$441$	0	0
$442$	−3.23880	−0.154054
$443$	−6.42607	−0.305312	−0.152656	−	0.988279i	$-0.548783\pi$
$443$	−6.42607	−0.305312	−0.152656	+	0.988279i	$0.548783\pi$
$444$	0	0
$445$	36.5923	1.73464
$446$	−2.38328	−0.112851
$447$	0	0
$448$	−25.7339	−1.21581
$449$	15.7190	0.741826	0.370913	−	0.928668i	$-0.379045\pi$
$449$	15.7190	0.741826	0.370913	+	0.928668i	$0.379045\pi$
$450$	0	0
$451$	22.0281	1.03726
$452$	0.375987	0.0176849
$453$	0	0
$454$	−8.90915	−0.418127
$455$	−40.7011	−1.90809
$456$	0	0
$457$	−30.7393	−1.43792	−0.718962	−	0.695050i	$-0.755382\pi$
$457$	−30.7393	−1.43792	−0.718962	+	0.695050i	$0.755382\pi$
$458$	6.35495	0.296947
$459$	0	0
$460$	−0.0188256	−0.000877747 0
$461$	17.6105	0.820202	0.410101	−	0.912040i	$-0.365493\pi$
$461$	17.6105	0.820202	0.410101	+	0.912040i	$0.365493\pi$
$462$	0	0
$463$	22.2251	1.03289	0.516444	−	0.856321i	$-0.327255\pi$
$463$	22.2251	1.03289	0.516444	+	0.856321i	$0.327255\pi$
$464$	27.5216	1.27766
$465$	0	0
$466$	−4.97053	−0.230255
$467$	9.78270	0.452689	0.226345	−	0.974047i	$-0.427322\pi$
$467$	9.78270	0.452689	0.226345	+	0.974047i	$0.427322\pi$
$468$	0	0
$469$	18.5385	0.856029
$470$	−0.781702	−0.0360572
$471$	0	0
$472$	5.54204	0.255093
$473$	8.79867	0.404563
$474$	0	0
$475$	0.203973	0.00935893
$476$	23.2551	1.06590
$477$	0	0
$478$	−4.49622	−0.205652
$479$	−16.8545	−0.770100	−0.385050	−	0.922896i	$-0.625816\pi$
$479$	−16.8545	−0.770100	−0.385050	+	0.922896i	$0.625816\pi$
$480$	0	0
$481$	26.7321	1.21888
$482$	−6.73786	−0.306901
$483$	0	0
$484$	12.3366	0.560755
$485$	21.7151	0.986032
$486$	0	0
$487$	−25.0040	−1.13304	−0.566520	−	0.824048i	$-0.691711\pi$
$487$	−25.0040	−1.13304	−0.566520	+	0.824048i	$0.691711\pi$
$488$	18.5615	0.840238
$489$	0	0
$490$	−14.4118	−0.651058
$491$	−17.4254	−0.786396	−0.393198	−	0.919454i	$-0.628631\pi$
$491$	−17.4254	−0.786396	−0.393198	+	0.919454i	$0.628631\pi$
$492$	0	0
$493$	−20.8401	−0.938592
$494$	−1.31534	−0.0591798
$495$	0	0
$496$	−4.69849	−0.210968
$497$	29.5906	1.32732
$498$	0	0
$499$	3.60895	0.161559	0.0807793	−	0.996732i	$-0.474259\pi$
$499$	3.60895	0.161559	0.0807793	+	0.996732i	$0.474259\pi$
$500$	21.3412	0.954407
$501$	0	0
$502$	4.29002	0.191473
$503$	42.8376	1.91003	0.955017	−	0.296550i	$-0.0958361\pi$
$503$	42.8376	1.91003	0.955017	+	0.296550i	$0.0958361\pi$
$504$	0	0
$505$	21.5734	0.960004
$506$	0.00344178	0.000153006 0
$507$	0	0
$508$	−35.5280	−1.57630
$509$	10.8122	0.479241	0.239621	−	0.970867i	$-0.422977\pi$
$509$	10.8122	0.479241	0.239621	+	0.970867i	$0.422977\pi$
$510$	0	0
$511$	35.4831	1.56968
$512$	−21.7541	−0.961403
$513$	0	0
$514$	−0.00682726	−0.000301138 0
$515$	−1.86654	−0.0822494
$516$	0	0
$517$	−2.10048	−0.0923789
$518$	13.0595	0.573800
$519$	0	0
$520$	−11.1553	−0.489190
$521$	33.2280	1.45574	0.727872	−	0.685713i	$-0.240510\pi$
$521$	33.2280	1.45574	0.727872	+	0.685713i	$0.240510\pi$
$522$	0	0
$523$	−31.4716	−1.37616	−0.688078	−	0.725637i	$-0.741545\pi$
$523$	−31.4716	−1.37616	−0.688078	+	0.725637i	$0.741545\pi$
$524$	34.2041	1.49421
$525$	0	0
$526$	−1.80505	−0.0787038
$527$	3.55783	0.154981
$528$	0	0
$529$	−23.0000	−0.999999
$530$	−2.24317	−0.0974372
$531$	0	0
$532$	9.44433	0.409463
$533$	38.6453	1.67391
$534$	0	0
$535$	30.4473	1.31635
$536$	5.08100	0.219466
$537$	0	0
$538$	7.13571	0.307642
$539$	−38.7253	−1.66802
$540$	0	0
$541$	10.5458	0.453398	0.226699	−	0.973965i	$-0.427207\pi$
$541$	10.5458	0.453398	0.226699	+	0.973965i	$0.427207\pi$
$542$	2.34867	0.100884
$543$	0	0
$544$	9.66542	0.414401
$545$	31.8278	1.36335
$546$	0	0
$547$	−23.2934	−0.995952	−0.497976	−	0.867191i	$-0.665923\pi$
$547$	−23.2934	−0.995952	−0.497976	+	0.867191i	$0.665923\pi$
$548$	9.51832	0.406602
$549$	0	0
$550$	−0.152930	−0.00652094
$551$	−8.46356	−0.360560
$552$	0	0
$553$	−10.9075	−0.463834
$554$	1.66878	0.0708999
$555$	0	0
$556$	−18.4909	−0.784188
$557$	−29.4004	−1.24574	−0.622868	−	0.782327i	$-0.714032\pi$
$557$	−29.4004	−1.24574	−0.622868	+	0.782327i	$0.714032\pi$
$558$	0	0
$559$	15.4360	0.652875
$560$	35.9161	1.51773
$561$	0	0
$562$	−11.6791	−0.492652
$563$	−26.6002	−1.12106	−0.560531	−	0.828133i	$-0.689403\pi$
$563$	−26.6002	−1.12106	−0.560531	+	0.828133i	$0.689403\pi$
$564$	0	0
$565$	−0.439714	−0.0184989
$566$	4.96049	0.208505
$567$	0	0
$568$	8.11014	0.340294
$569$	−27.2552	−1.14260	−0.571298	−	0.820743i	$-0.693560\pi$
$569$	−27.2552	−1.14260	−0.571298	+	0.820743i	$0.693560\pi$
$570$	0	0
$571$	2.02795	0.0848670	0.0424335	−	0.999099i	$-0.486489\pi$
$571$	2.02795	0.0848670	0.0424335	+	0.999099i	$0.486489\pi$
$572$	−14.4943	−0.606038
$573$	0	0
$574$	18.8794	0.788011
$575$	−0.000936349 0	−3.90485e−5 0
$576$	0	0
$577$	29.1652	1.21416	0.607081	−	0.794640i	$-0.292341\pi$
$577$	29.1652	1.21416	0.607081	+	0.794640i	$0.292341\pi$
$578$	3.90386	0.162379
$579$	0	0
$580$	−34.7086	−1.44119
$581$	84.8497	3.52016
$582$	0	0
$583$	−6.02753	−0.249635
$584$	9.72514	0.402429
$585$	0	0
$586$	−0.272703	−0.0112653
$587$	−1.47214	−0.0607617	−0.0303809	−	0.999538i	$-0.509672\pi$
$587$	−1.47214	−0.0607617	−0.0303809	+	0.999538i	$0.509672\pi$
$588$	0	0
$589$	1.44490	0.0595360
$590$	−3.13407	−0.129028
$591$	0	0
$592$	−23.5894	−0.969520
$593$	15.6530	0.642793	0.321396	−	0.946945i	$-0.395848\pi$
$593$	15.6530	0.642793	0.321396	+	0.946945i	$0.395848\pi$
$594$	0	0
$595$	−27.1967	−1.11496
$596$	−2.20242	−0.0902145
$597$	0	0
$598$	0.00603812	0.000246917 0
$599$	31.2207	1.27564	0.637822	−	0.770184i	$-0.279836\pi$
$599$	31.2207	1.27564	0.637822	+	0.770184i	$0.279836\pi$
$600$	0	0
$601$	−15.1708	−0.618832	−0.309416	−	0.950927i	$-0.600134\pi$
$601$	−15.1708	−0.618832	−0.309416	+	0.950927i	$0.600134\pi$
$602$	7.54098	0.307347
$603$	0	0
$604$	−11.2949	−0.459582
$605$	−14.4276	−0.586565
$606$	0	0
$607$	45.1446	1.83236	0.916181	−	0.400766i	$-0.131256\pi$
$607$	45.1446	1.83236	0.916181	+	0.400766i	$0.131256\pi$
$608$	3.92530	0.159192
$609$	0	0
$610$	−10.4967	−0.424998
$611$	−3.68499	−0.149079
$612$	0	0
$613$	−2.66902	−0.107801	−0.0539003	−	0.998546i	$-0.517165\pi$
$613$	−2.66902	−0.107801	−0.0539003	+	0.998546i	$0.517165\pi$
$614$	6.72497	0.271398
$615$	0	0
$616$	−14.6436	−0.590008
$617$	−24.6541	−0.992538	−0.496269	−	0.868169i	$-0.665297\pi$
$617$	−24.6541	−0.992538	−0.496269	+	0.868169i	$0.665297\pi$
$618$	0	0
$619$	6.04699	0.243049	0.121524	−	0.992588i	$-0.461222\pi$
$619$	6.04699	0.243049	0.121524	+	0.992588i	$0.461222\pi$
$620$	5.92544	0.237972
$621$	0	0
$622$	−0.402677	−0.0161459
$623$	84.2708	3.37624
$624$	0	0
$625$	−23.9385	−0.957541
$626$	8.54087	0.341362
$627$	0	0
$628$	39.2328	1.56556
$629$	17.8626	0.712227
$630$	0	0
$631$	24.7027	0.983400	0.491700	−	0.870765i	$-0.336376\pi$
$631$	24.7027	0.983400	0.491700	+	0.870765i	$0.336376\pi$
$632$	−2.98950	−0.118916
$633$	0	0
$634$	−5.56276	−0.220925
$635$	41.5498	1.64885
$636$	0	0
$637$	−67.9381	−2.69181
$638$	6.34558	0.251224
$639$	0	0
$640$	21.1813	0.837264
$641$	−14.1627	−0.559393	−0.279696	−	0.960088i	$-0.590234\pi$
$641$	−14.1627	−0.559393	−0.279696	+	0.960088i	$0.590234\pi$
$642$	0	0
$643$	19.4976	0.768909	0.384454	−	0.923144i	$-0.374390\pi$
$643$	19.4976	0.768909	0.384454	+	0.923144i	$0.374390\pi$
$644$	−0.0433547	−0.00170841
$645$	0	0
$646$	−0.878917	−0.0345805
$647$	−11.9870	−0.471259	−0.235629	−	0.971843i	$-0.575715\pi$
$647$	−11.9870	−0.471259	−0.235629	+	0.971843i	$0.575715\pi$
$648$	0	0
$649$	−8.42143	−0.330570
$650$	−0.268294	−0.0105233
$651$	0	0
$652$	11.6132	0.454807
$653$	−43.4881	−1.70182	−0.850910	−	0.525312i	$-0.823949\pi$
$653$	−43.4881	−1.70182	−0.850910	+	0.525312i	$0.823949\pi$
$654$	0	0
$655$	−40.0015	−1.56299
$656$	−34.1021	−1.33146
$657$	0	0
$658$	−1.80023	−0.0701804
$659$	−6.22192	−0.242372	−0.121186	−	0.992630i	$-0.538670\pi$
$659$	−6.22192	−0.242372	−0.121186	+	0.992630i	$0.538670\pi$
$660$	0	0
$661$	25.2217	0.981011	0.490505	−	0.871438i	$-0.336812\pi$
$661$	25.2217	0.981011	0.490505	+	0.871438i	$0.336812\pi$
$662$	9.83003	0.382055
$663$	0	0
$664$	23.2554	0.902486
$665$	−11.0451	−0.428310
$666$	0	0
$667$	0.0388524	0.00150437
$668$	−3.37006	−0.130392
$669$	0	0
$670$	−2.87335	−0.111007
$671$	−28.2051	−1.08885
$672$	0	0
$673$	15.3602	0.592093	0.296046	−	0.955174i	$-0.404332\pi$
$673$	15.3602	0.592093	0.296046	+	0.955174i	$0.404332\pi$
$674$	5.13135	0.197652
$675$	0	0
$676$	−1.08457	−0.0417142
$677$	20.7693	0.798228	0.399114	−	0.916901i	$-0.369318\pi$
$677$	20.7693	0.798228	0.399114	+	0.916901i	$0.369318\pi$
$678$	0	0
$679$	50.0091	1.91917
$680$	−7.45402	−0.285848
$681$	0	0
$682$	−1.08332	−0.0414824
$683$	14.0606	0.538013	0.269006	−	0.963138i	$-0.413305\pi$
$683$	14.0606	0.538013	0.269006	+	0.963138i	$0.413305\pi$
$684$	0	0
$685$	−11.1316	−0.425317
$686$	−20.5882	−0.786062
$687$	0	0
$688$	−13.6213	−0.519309
$689$	−10.5745	−0.402855
$690$	0	0
$691$	31.0495	1.18118	0.590590	−	0.806972i	$-0.298895\pi$
$691$	31.0495	1.18118	0.590590	+	0.806972i	$0.298895\pi$
$692$	−31.3907	−1.19330
$693$	0	0
$694$	−8.67236	−0.329198
$695$	21.6250	0.820282
$696$	0	0
$697$	25.8230	0.978117
$698$	−10.1351	−0.383619
$699$	0	0
$700$	1.92639	0.0728107
$701$	−48.5778	−1.83476	−0.917378	−	0.398016i	$-0.869699\pi$
$701$	−48.5778	−1.83476	−0.917378	+	0.398016i	$0.869699\pi$
$702$	0	0
$703$	7.25432	0.273602
$704$	10.7176	0.403933
$705$	0	0
$706$	−12.3906	−0.466326
$707$	49.6828	1.86851
$708$	0	0
$709$	16.2554	0.610483	0.305241	−	0.952275i	$-0.401263\pi$
$709$	16.2554	0.610483	0.305241	+	0.952275i	$0.401263\pi$
$710$	−4.58635	−0.172123
$711$	0	0
$712$	23.0968	0.865587
$713$	−0.00663288	−0.000248403 0
$714$	0	0
$715$	16.9510	0.633932
$716$	−21.5391	−0.804954
$717$	0	0
$718$	12.5300	0.467616
$719$	−15.8415	−0.590787	−0.295393	−	0.955376i	$-0.595451\pi$
$719$	−15.8415	−0.590787	−0.295393	+	0.955376i	$0.595451\pi$
$720$	0	0
$721$	−4.29857	−0.160087
$722$	−0.356944	−0.0132841
$723$	0	0
$724$	11.6746	0.433881
$725$	−1.72634	−0.0641146
$726$	0	0
$727$	38.8969	1.44261	0.721303	−	0.692620i	$-0.243544\pi$
$727$	38.8969	1.44261	0.721303	+	0.692620i	$0.243544\pi$
$728$	−25.6902	−0.952141
$729$	0	0
$730$	−5.49965	−0.203551
$731$	10.3145	0.381494
$732$	0	0
$733$	18.7837	0.693793	0.346897	−	0.937903i	$-0.387235\pi$
$733$	18.7837	0.693793	0.346897	+	0.937903i	$0.387235\pi$
$734$	3.66795	0.135386
$735$	0	0
$736$	−0.0180193	−0.000664200 0
$737$	−7.72084	−0.284401
$738$	0	0
$739$	−2.14772	−0.0790052	−0.0395026	−	0.999219i	$-0.512577\pi$
$739$	−2.14772	−0.0790052	−0.0395026	+	0.999219i	$0.512577\pi$
$740$	29.7495	1.09361
$741$	0	0
$742$	−5.16595	−0.189648
$743$	20.2410	0.742569	0.371284	−	0.928519i	$-0.378918\pi$
$743$	20.2410	0.742569	0.371284	+	0.928519i	$0.378918\pi$
$744$	0	0
$745$	2.57571	0.0943668
$746$	3.82022	0.139868
$747$	0	0
$748$	−9.68519	−0.354126
$749$	70.1191	2.56210
$750$	0	0
$751$	−31.4690	−1.14832	−0.574159	−	0.818744i	$-0.694671\pi$
$751$	−31.4690	−1.14832	−0.574159	+	0.818744i	$0.694671\pi$
$752$	3.25178	0.118580
$753$	0	0
$754$	11.1324	0.405419
$755$	13.2093	0.480735
$756$	0	0
$757$	−20.5581	−0.747198	−0.373599	−	0.927590i	$-0.621876\pi$
$757$	−20.5581	−0.747198	−0.373599	+	0.927590i	$0.621876\pi$
$758$	2.07664	0.0754269
$759$	0	0
$760$	−3.02721	−0.109809
$761$	51.8731	1.88040	0.940200	−	0.340623i	$-0.110638\pi$
$761$	51.8731	1.88040	0.940200	+	0.340623i	$0.110638\pi$
$762$	0	0
$763$	73.2982	2.65357
$764$	28.5071	1.03135
$765$	0	0
$766$	6.10737	0.220668
$767$	−14.7742	−0.533466
$768$	0	0
$769$	−35.0239	−1.26299	−0.631497	−	0.775379i	$-0.717559\pi$
$769$	−35.0239	−1.26299	−0.631497	+	0.775379i	$0.717559\pi$
$770$	8.28109	0.298430
$771$	0	0
$772$	−8.83301	−0.317907
$773$	46.6995	1.67966	0.839832	−	0.542847i	$-0.182654\pi$
$773$	46.6995	1.67966	0.839832	+	0.542847i	$0.182654\pi$
$774$	0	0
$775$	0.294720	0.0105867
$776$	13.7064	0.492031
$777$	0	0
$778$	−5.51091	−0.197576
$779$	10.4872	0.375743
$780$	0	0
$781$	−12.3238	−0.440980
$782$	0.00403471	0.000144281 0
$783$	0	0
$784$	59.9512	2.14111
$785$	−45.8825	−1.63762
$786$	0	0
$787$	7.17267	0.255678	0.127839	−	0.991795i	$-0.459196\pi$
$787$	7.17267	0.255678	0.127839	+	0.991795i	$0.459196\pi$
$788$	−15.4816	−0.551509
$789$	0	0
$790$	1.69059	0.0601484
$791$	−1.01265	−0.0360055
$792$	0	0
$793$	−49.4820	−1.75716
$794$	−2.86303	−0.101605
$795$	0	0
$796$	27.0486	0.958713
$797$	−1.58839	−0.0562637	−0.0281319	−	0.999604i	$-0.508956\pi$
$797$	−1.58839	−0.0562637	−0.0281319	+	0.999604i	$0.508956\pi$
$798$	0	0
$799$	−2.46234	−0.0871112
$800$	0.800656	0.0283075
$801$	0	0
$802$	−8.19298	−0.289304
$803$	−14.7779	−0.521499
$804$	0	0
$805$	0.0507030	0.00178705
$806$	−1.90053	−0.0669433
$807$	0	0
$808$	13.6170	0.479043
$809$	32.2025	1.13218	0.566090	−	0.824343i	$-0.308455\pi$
$809$	32.2025	1.13218	0.566090	+	0.824343i	$0.308455\pi$
$810$	0	0
$811$	−5.91677	−0.207766	−0.103883	−	0.994590i	$-0.533127\pi$
$811$	−5.91677	−0.207766	−0.103883	+	0.994590i	$0.533127\pi$
$812$	−79.9326	−2.80508
$813$	0	0
$814$	−5.43895	−0.190635
$815$	−13.5815	−0.475740
$816$	0	0
$817$	4.18889	0.146551
$818$	3.17687	0.111077
$819$	0	0
$820$	43.0074	1.50188
$821$	8.43574	0.294409	0.147205	−	0.989106i	$-0.452972\pi$
$821$	8.43574	0.294409	0.147205	+	0.989106i	$0.452972\pi$
$822$	0	0
$823$	−29.5768	−1.03098	−0.515491	−	0.856895i	$-0.672390\pi$
$823$	−29.5768	−1.03098	−0.515491	+	0.856895i	$0.672390\pi$
$824$	−1.17814	−0.0410425
$825$	0	0
$826$	−7.21766	−0.251134
$827$	51.3500	1.78561	0.892807	−	0.450440i	$-0.148733\pi$
$827$	51.3500	1.78561	0.892807	+	0.450440i	$0.148733\pi$
$828$	0	0
$829$	−9.22327	−0.320337	−0.160169	−	0.987090i	$-0.551204\pi$
$829$	−9.22327	−0.320337	−0.160169	+	0.987090i	$0.551204\pi$
$830$	−13.1511	−0.456483
$831$	0	0
$832$	18.8025	0.651858
$833$	−45.3967	−1.57290
$834$	0	0
$835$	3.94126	0.136393
$836$	−3.93333	−0.136037
$837$	0	0
$838$	6.07196	0.209752
$839$	6.56807	0.226755	0.113377	−	0.993552i	$-0.463833\pi$
$839$	6.56807	0.226755	0.113377	+	0.993552i	$0.463833\pi$
$840$	0	0
$841$	42.6318	1.47006
$842$	−1.56474	−0.0539246
$843$	0	0
$844$	−18.4021	−0.633426
$845$	1.26840	0.0436342
$846$	0	0
$847$	−33.2263	−1.14167
$848$	9.33130	0.320438
$849$	0	0
$850$	−0.179276	−0.00614910
$851$	−0.0333013	−0.00114155
$852$	0	0
$853$	−46.9231	−1.60662	−0.803308	−	0.595564i	$-0.796929\pi$
$853$	−46.9231	−1.60662	−0.803308	+	0.595564i	$0.796929\pi$
$854$	−24.1735	−0.827199
$855$	0	0
$856$	19.2181	0.656861
$857$	−41.0000	−1.40053	−0.700267	−	0.713881i	$-0.746936\pi$
$857$	−41.0000	−1.40053	−0.700267	+	0.713881i	$0.746936\pi$
$858$	0	0
$859$	−25.2174	−0.860406	−0.430203	−	0.902732i	$-0.641558\pi$
$859$	−25.2174	−0.860406	−0.430203	+	0.902732i	$0.641558\pi$
$860$	17.1784	0.585779
$861$	0	0
$862$	−10.7908	−0.367537
$863$	−25.2797	−0.860532	−0.430266	−	0.902702i	$-0.641580\pi$
$863$	−25.2797	−0.860532	−0.430266	+	0.902702i	$0.641580\pi$
$864$	0	0
$865$	36.7112	1.24822
$866$	0.967348	0.0328718
$867$	0	0
$868$	13.6461	0.463178
$869$	4.54271	0.154101
$870$	0	0
$871$	−13.5451	−0.458960
$872$	20.0894	0.680313
$873$	0	0
$874$	0.00163857	5.54255e−5 0
$875$	−57.4783	−1.94312
$876$	0	0
$877$	26.4021	0.891535	0.445768	−	0.895149i	$-0.352931\pi$
$877$	26.4021	0.891535	0.445768	+	0.895149i	$0.352931\pi$
$878$	−0.931553	−0.0314384
$879$	0	0
$880$	−14.9582	−0.504241
$881$	−39.1114	−1.31770	−0.658849	−	0.752275i	$-0.728956\pi$
$881$	−39.1114	−1.31770	−0.658849	+	0.752275i	$0.728956\pi$
$882$	0	0
$883$	−5.29507	−0.178193	−0.0890967	−	0.996023i	$-0.528398\pi$
$883$	−5.29507	−0.178193	−0.0890967	+	0.996023i	$0.528398\pi$
$884$	−16.9913	−0.571480
$885$	0	0
$886$	2.29375	0.0770601
$887$	−18.9805	−0.637302	−0.318651	−	0.947872i	$-0.603230\pi$
$887$	−18.9805	−0.637302	−0.318651	+	0.947872i	$0.603230\pi$
$888$	0	0
$889$	95.6878	3.20927
$890$	−13.0614	−0.437819
$891$	0	0
$892$	−12.5031	−0.418634
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	25.1899	0.842004
$896$	48.7797	1.62962
$897$	0	0
$898$	−5.61081	−0.187235
$899$	−12.2290	−0.407859
$900$	0	0
$901$	−7.06592	−0.235400
$902$	−7.86282	−0.261803
$903$	0	0
$904$	−0.277544	−0.00923097
$905$	−13.6533	−0.453852
$906$	0	0
$907$	−47.3404	−1.57191	−0.785956	−	0.618282i	$-0.787829\pi$
$907$	−47.3404	−1.57191	−0.785956	+	0.618282i	$0.787829\pi$
$908$	−46.7389	−1.55109
$909$	0	0
$910$	14.5280	0.481599
$911$	−15.4060	−0.510422	−0.255211	−	0.966885i	$-0.582145\pi$
$911$	−15.4060	−0.510422	−0.255211	+	0.966885i	$0.582145\pi$
$912$	0	0
$913$	−35.3379	−1.16951
$914$	10.9722	0.362929
$915$	0	0
$916$	33.3391	1.10156
$917$	−92.1221	−3.04214
$918$	0	0
$919$	7.89906	0.260566	0.130283	−	0.991477i	$-0.458411\pi$
$919$	7.89906	0.260566	0.130283	+	0.991477i	$0.458411\pi$
$920$	0.0138966	0.000458156 0
$921$	0	0
$922$	−6.28596	−0.207017
$923$	−21.6204	−0.711643
$924$	0	0
$925$	1.47969	0.0486518
$926$	−7.93313	−0.260699
$927$	0	0
$928$	−33.2220	−1.09057
$929$	13.4890	0.442560	0.221280	−	0.975210i	$-0.428976\pi$
$929$	13.4890	0.442560	0.221280	+	0.975210i	$0.428976\pi$
$930$	0	0
$931$	−18.4364	−0.604230
$932$	−26.0762	−0.854155
$933$	0	0
$934$	−3.49188	−0.114258
$935$	11.3268	0.370425
$936$	0	0
$937$	14.1615	0.462637	0.231319	−	0.972878i	$-0.425696\pi$
$937$	14.1615	0.462637	0.231319	+	0.972878i	$0.425696\pi$
$938$	−6.61722	−0.216060
$939$	0	0
$940$	−4.10094	−0.133758
$941$	−46.0199	−1.50021	−0.750103	−	0.661321i	$-0.769996\pi$
$941$	−46.0199	−1.50021	−0.750103	+	0.661321i	$0.769996\pi$
$942$	0	0
$943$	−0.0481421	−0.00156772
$944$	13.0373	0.424329
$945$	0	0
$946$	−3.14063	−0.102111
$947$	−36.0353	−1.17099	−0.585495	−	0.810676i	$-0.699100\pi$
$947$	−36.0353	−1.17099	−0.585495	+	0.810676i	$0.699100\pi$
$948$	0	0
$949$	−25.9257	−0.841584
$950$	−0.0728071	−0.00236217
$951$	0	0
$952$	−17.1663	−0.556364
$953$	28.8819	0.935576	0.467788	−	0.883841i	$-0.345051\pi$
$953$	28.8819	0.935576	0.467788	+	0.883841i	$0.345051\pi$
$954$	0	0
$955$	−33.3388	−1.07882
$956$	−23.5879	−0.762889
$957$	0	0
$958$	6.01610	0.194371
$959$	−25.6357	−0.827820
$960$	0	0
$961$	−28.9123	−0.932654
$962$	−9.54188	−0.307642
$963$	0	0
$964$	−35.3480	−1.13848
$965$	10.3301	0.332539
$966$	0	0
$967$	−21.9419	−0.705602	−0.352801	−	0.935698i	$-0.614771\pi$
$967$	−21.9419	−0.705602	−0.352801	+	0.935698i	$0.614771\pi$
$968$	−9.10658	−0.292697
$969$	0	0
$970$	−7.75108	−0.248872
$971$	54.4976	1.74891	0.874455	−	0.485107i	$-0.161219\pi$
$971$	54.4976	1.74891	0.874455	+	0.485107i	$0.161219\pi$
$972$	0	0
$973$	49.8016	1.59656
$974$	8.92504	0.285977
$975$	0	0
$976$	43.6648	1.39768
$977$	−10.1590	−0.325016	−0.162508	−	0.986707i	$-0.551958\pi$
$977$	−10.1590	−0.325016	−0.162508	+	0.986707i	$0.551958\pi$
$978$	0	0
$979$	−35.0967	−1.12170
$980$	−75.6067	−2.41517
$981$	0	0
$982$	6.21988	0.198484
$983$	−49.0267	−1.56371	−0.781854	−	0.623462i	$-0.785726\pi$
$983$	−49.0267	−1.56371	−0.781854	+	0.623462i	$0.785726\pi$
$984$	0	0
$985$	18.1056	0.576893
$986$	7.43876	0.236899
$987$	0	0
$988$	−6.90049	−0.219534
$989$	−0.0192293	−0.000611457 0
$990$	0	0
$991$	53.9441	1.71359	0.856795	−	0.515657i	$-0.172452\pi$
$991$	53.9441	1.71359	0.856795	+	0.515657i	$0.172452\pi$
$992$	5.67166	0.180075
$993$	0	0
$994$	−10.5622	−0.335013
$995$	−31.6332	−1.00284
$996$	0	0
$997$	−3.15041	−0.0997745	−0.0498872	−	0.998755i	$-0.515886\pi$
$997$	−3.15041	−0.0997745	−0.0498872	+	0.998755i	$0.515886\pi$
$998$	−1.28819	−0.0407770
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.15	yes	34
3.2	odd	2		8037.2.a.v.1.20	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.20	✓	34	3.2	odd	2
8037.2.a.w.1.15	yes	34	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)

\(f(q)\)	\(=\)	\(q-0.356944 q^{2} -1.87259 q^{4} +2.18998 q^{5} +5.04345 q^{7} +1.38230 q^{8} +O(q^{10})\)	\(q-0.356944 q^{2} -1.87259 q^{4} +2.18998 q^{5} +5.04345 q^{7} +1.38230 q^{8} -0.781702 q^{10} -2.10048 q^{11} -3.68499 q^{13} -1.80023 q^{14} +3.25178 q^{16} -2.46234 q^{17} -1.00000 q^{19} -4.10094 q^{20} +0.749753 q^{22} +0.00459055 q^{23} -0.203973 q^{25} +1.31534 q^{26} -9.44433 q^{28} +8.46356 q^{29} -1.44490 q^{31} -3.92530 q^{32} +0.878917 q^{34} +11.0451 q^{35} -7.25432 q^{37} +0.356944 q^{38} +3.02721 q^{40} -10.4872 q^{41} -4.18889 q^{43} +3.93333 q^{44} -0.00163857 q^{46} +1.00000 q^{47} +18.4364 q^{49} +0.0728071 q^{50} +6.90049 q^{52} +2.86960 q^{53} -4.60001 q^{55} +6.97156 q^{56} -3.02102 q^{58} +4.00929 q^{59} +13.4280 q^{61} +0.515748 q^{62} -5.10244 q^{64} -8.07008 q^{65} +3.67576 q^{67} +4.61095 q^{68} -3.94248 q^{70} +5.86714 q^{71} +7.03548 q^{73} +2.58939 q^{74} +1.87259 q^{76} -10.5937 q^{77} -2.16270 q^{79} +7.12134 q^{80} +3.74335 q^{82} +16.8237 q^{83} -5.39248 q^{85} +1.49520 q^{86} -2.90349 q^{88} +16.7089 q^{89} -18.5851 q^{91} -0.00859622 q^{92} -0.356944 q^{94} -2.18998 q^{95} +9.91565 q^{97} -6.58078 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 34 * q + 5 * q^2 + 31 * q^4 + 6 * q^5 + 15 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^13 + 12 * q^14 + 21 * q^16 + 4 * q^17 - 34 * q^19 + 20 * q^20 - 8 * q^22 + 26 * q^23 + 32 * q^25 + 29 * q^26 - 4 * q^28 + 14 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 50 * q^35 - 10 * q^37 - 5 * q^38 + 17 * q^40 + 18 * q^41 + 6 * q^43 + 6 * q^44 + 18 * q^46 + 34 * q^47 + 28 * q^49 + 41 * q^50 + 10 * q^52 + 40 * q^53 - 8 * q^55 + 76 * q^56 + 4 * q^58 + 62 * q^59 - 2 * q^61 + 50 * q^62 + 11 * q^64 + 32 * q^65 + 20 * q^67 + 28 * q^68 + 22 * q^70 + 52 * q^71 - 8 * q^73 + 10 * q^74 - 31 * q^76 + 36 * q^77 - 12 * q^79 + 92 * q^80 + 10 * q^82 + 82 * q^83 - 4 * q^85 + 40 * q^86 - 16 * q^88 + 58 * q^89 + 100 * q^92 + 5 * q^94 - 6 * q^95 - 6 * q^97 + 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more