LMFDB - Embedded newform 6498.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +2.34730 q^{10} -2.71688 q^{11} -5.41147 q^{13} +3.57398 q^{14} +1.00000 q^{16} +3.87939 q^{17} +2.34730 q^{20} -2.71688 q^{22} +8.23442 q^{23} +0.509800 q^{25} -5.41147 q^{26} +3.57398 q^{28} +3.53209 q^{29} -6.53209 q^{31} +1.00000 q^{32} +3.87939 q^{34} +8.38919 q^{35} -0.389185 q^{37} +2.34730 q^{40} -1.94356 q^{41} +5.02229 q^{43} -2.71688 q^{44} +8.23442 q^{46} -2.98545 q^{47} +5.77332 q^{49} +0.509800 q^{50} -5.41147 q^{52} +8.30541 q^{53} -6.37733 q^{55} +3.57398 q^{56} +3.53209 q^{58} +2.73143 q^{59} +6.29086 q^{61} -6.53209 q^{62} +1.00000 q^{64} -12.7023 q^{65} +14.9368 q^{67} +3.87939 q^{68} +8.38919 q^{70} +9.02229 q^{71} +10.2909 q^{73} -0.389185 q^{74} -9.71007 q^{77} -13.0077 q^{79} +2.34730 q^{80} -1.94356 q^{82} +8.17024 q^{83} +9.10607 q^{85} +5.02229 q^{86} -2.71688 q^{88} +11.7246 q^{89} -19.3405 q^{91} +8.23442 q^{92} -2.98545 q^{94} +8.60401 q^{97} +5.77332 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34} + 21 q^{35} + 3 q^{37} + 6 q^{40} + 9 q^{41} + 9 q^{43} - 6 q^{46} + 9 q^{47} + 24 q^{49} + 3 q^{50} - 6 q^{52} + 27 q^{53} + 12 q^{55} + 3 q^{56} + 6 q^{58} + 18 q^{59} + 3 q^{61} - 15 q^{62} + 3 q^{64} - 12 q^{65} - 12 q^{67} + 6 q^{68} + 21 q^{70} + 21 q^{71} + 15 q^{73} + 3 q^{74} + 21 q^{77} - 15 q^{79} + 6 q^{80} + 9 q^{82} + 3 q^{83} + 15 q^{85} + 9 q^{86} + 3 q^{89} - 15 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.34730	1.04974	0.524871	−	0.851182i	$-0.324113\pi$
$5$	2.34730	1.04974	0.524871	+	0.851182i	$0.324113\pi$
$6$	0	0
$7$	3.57398	1.35084	0.675418	−	0.737435i	$-0.263963\pi$
$7$	3.57398	1.35084	0.675418	+	0.737435i	$0.263963\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.34730	0.742280
$11$	−2.71688	−0.819171	−0.409585	−	0.912272i	$-0.634327\pi$
$11$	−2.71688	−0.819171	−0.409585	+	0.912272i	$0.634327\pi$
$12$	0	0
$13$	−5.41147	−1.50087	−0.750436	−	0.660943i	$-0.770157\pi$
$13$	−5.41147	−1.50087	−0.750436	+	0.660943i	$0.770157\pi$
$14$	3.57398	0.955186
$15$	0	0
$16$	1.00000	0.250000
$17$	3.87939	0.940889	0.470445	−	0.882430i	$-0.344094\pi$
$17$	3.87939	0.940889	0.470445	+	0.882430i	$0.344094\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	2.34730	0.524871
$21$	0	0
$22$	−2.71688	−0.579241
$23$	8.23442	1.71700	0.858498	−	0.512817i	$-0.171398\pi$
$23$	8.23442	1.71700	0.858498	+	0.512817i	$0.171398\pi$
$24$	0	0
$25$	0.509800	0.101960
$26$	−5.41147	−1.06128
$27$	0	0
$28$	3.57398	0.675418
$29$	3.53209	0.655892	0.327946	−	0.944696i	$-0.393644\pi$
$29$	3.53209	0.655892	0.327946	+	0.944696i	$0.393644\pi$
$30$	0	0
$31$	−6.53209	−1.17320	−0.586599	−	0.809878i	$-0.699533\pi$
$31$	−6.53209	−1.17320	−0.586599	+	0.809878i	$0.699533\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.87939	0.665309
$35$	8.38919	1.41803
$36$	0	0
$37$	−0.389185	−0.0639817	−0.0319908	−	0.999488i	$-0.510185\pi$
$37$	−0.389185	−0.0639817	−0.0319908	+	0.999488i	$0.510185\pi$
$38$	0	0
$39$	0	0
$40$	2.34730	0.371140
$41$	−1.94356	−0.303534	−0.151767	−	0.988416i	$-0.548496\pi$
$41$	−1.94356	−0.303534	−0.151767	+	0.988416i	$0.548496\pi$
$42$	0	0
$43$	5.02229	0.765892	0.382946	−	0.923771i	$-0.374910\pi$
$43$	5.02229	0.765892	0.382946	+	0.923771i	$0.374910\pi$
$44$	−2.71688	−0.409585
$45$	0	0
$46$	8.23442	1.21410
$47$	−2.98545	−0.435473	−0.217736	−	0.976008i	$-0.569867\pi$
$47$	−2.98545	−0.435473	−0.217736	+	0.976008i	$0.569867\pi$
$48$	0	0
$49$	5.77332	0.824760
$50$	0.509800	0.0720966
$51$	0	0
$52$	−5.41147	−0.750436
$53$	8.30541	1.14084	0.570418	−	0.821355i	$-0.306781\pi$
$53$	8.30541	1.14084	0.570418	+	0.821355i	$0.306781\pi$
$54$	0	0
$55$	−6.37733	−0.859918
$56$	3.57398	0.477593
$57$	0	0
$58$	3.53209	0.463786
$59$	2.73143	0.355602	0.177801	−	0.984066i	$-0.443102\pi$
$59$	2.73143	0.355602	0.177801	+	0.984066i	$0.443102\pi$
$60$	0	0
$61$	6.29086	0.805462	0.402731	−	0.915318i	$-0.368061\pi$
$61$	6.29086	0.805462	0.402731	+	0.915318i	$0.368061\pi$
$62$	−6.53209	−0.829576
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.7023	−1.57553
$66$	0	0
$67$	14.9368	1.82482	0.912408	−	0.409283i	$-0.134221\pi$
$67$	14.9368	1.82482	0.912408	+	0.409283i	$0.134221\pi$
$68$	3.87939	0.470445
$69$	0	0
$70$	8.38919	1.00270
$71$	9.02229	1.07075	0.535374	−	0.844615i	$-0.320171\pi$
$71$	9.02229	1.07075	0.535374	+	0.844615i	$0.320171\pi$
$72$	0	0
$73$	10.2909	1.20445	0.602227	−	0.798325i	$-0.294280\pi$
$73$	10.2909	1.20445	0.602227	+	0.798325i	$0.294280\pi$
$74$	−0.389185	−0.0452419
$75$	0	0
$76$	0	0
$77$	−9.71007	−1.10657
$78$	0	0
$79$	−13.0077	−1.46349	−0.731743	−	0.681581i	$-0.761293\pi$
$79$	−13.0077	−1.46349	−0.731743	+	0.681581i	$0.761293\pi$
$80$	2.34730	0.262436
$81$	0	0
$82$	−1.94356	−0.214631
$83$	8.17024	0.896801	0.448400	−	0.893833i	$-0.351994\pi$
$83$	8.17024	0.896801	0.448400	+	0.893833i	$0.351994\pi$
$84$	0	0
$85$	9.10607	0.987692
$86$	5.02229	0.541567
$87$	0	0
$88$	−2.71688	−0.289621
$89$	11.7246	1.24281	0.621404	−	0.783491i	$-0.286563\pi$
$89$	11.7246	1.24281	0.621404	+	0.783491i	$0.286563\pi$
$90$	0	0
$91$	−19.3405	−2.02743
$92$	8.23442	0.858498
$93$	0	0
$94$	−2.98545	−0.307926
$95$	0	0
$96$	0	0
$97$	8.60401	0.873605	0.436802	−	0.899558i	$-0.356111\pi$
$97$	8.60401	0.873605	0.436802	+	0.899558i	$0.356111\pi$
$98$	5.77332	0.583193
$99$	0	0
$100$	0.509800	0.0509800
$101$	1.28581	0.127943	0.0639713	−	0.997952i	$-0.479623\pi$
$101$	1.28581	0.127943	0.0639713	+	0.997952i	$0.479623\pi$
$102$	0	0
$103$	−0.736482	−0.0725677	−0.0362839	−	0.999342i	$-0.511552\pi$
$103$	−0.736482	−0.0725677	−0.0362839	+	0.999342i	$0.511552\pi$
$104$	−5.41147	−0.530639
$105$	0	0
$106$	8.30541	0.806692
$107$	10.0273	0.969380	0.484690	−	0.874686i	$-0.338932\pi$
$107$	10.0273	0.969380	0.484690	+	0.874686i	$0.338932\pi$
$108$	0	0
$109$	−13.9213	−1.33342	−0.666708	−	0.745319i	$-0.732297\pi$
$109$	−13.9213	−1.33342	−0.666708	+	0.745319i	$0.732297\pi$
$110$	−6.37733	−0.608054
$111$	0	0
$112$	3.57398	0.337709
$113$	−0.226682	−0.0213244	−0.0106622	−	0.999943i	$-0.503394\pi$
$113$	−0.226682	−0.0213244	−0.0106622	+	0.999943i	$0.503394\pi$
$114$	0	0
$115$	19.3286	1.80240
$116$	3.53209	0.327946
$117$	0	0
$118$	2.73143	0.251448
$119$	13.8648	1.27099
$120$	0	0
$121$	−3.61856	−0.328960
$122$	6.29086	0.569548
$123$	0	0
$124$	−6.53209	−0.586599
$125$	−10.5398	−0.942711
$126$	0	0
$127$	−8.69459	−0.771520	−0.385760	−	0.922599i	$-0.626061\pi$
$127$	−8.69459	−0.771520	−0.385760	+	0.922599i	$0.626061\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−12.7023	−1.11407
$131$	12.6432	1.10464	0.552321	−	0.833631i	$-0.313742\pi$
$131$	12.6432	1.10464	0.552321	+	0.833631i	$0.313742\pi$
$132$	0	0
$133$	0	0
$134$	14.9368	1.29034
$135$	0	0
$136$	3.87939	0.332655
$137$	−21.7939	−1.86197	−0.930987	−	0.365052i	$-0.881051\pi$
$137$	−21.7939	−1.86197	−0.930987	+	0.365052i	$0.881051\pi$
$138$	0	0
$139$	−11.2267	−0.952235	−0.476117	−	0.879382i	$-0.657956\pi$
$139$	−11.2267	−0.952235	−0.476117	+	0.879382i	$0.657956\pi$
$140$	8.38919	0.709016
$141$	0	0
$142$	9.02229	0.757134
$143$	14.7023	1.22947
$144$	0	0
$145$	8.29086	0.688518
$146$	10.2909	0.851678
$147$	0	0
$148$	−0.389185	−0.0319908
$149$	21.9786	1.80056	0.900280	−	0.435311i	$-0.143361\pi$
$149$	21.9786	1.80056	0.900280	+	0.435311i	$0.143361\pi$
$150$	0	0
$151$	−2.36184	−0.192204	−0.0961021	−	0.995371i	$-0.530638\pi$
$151$	−2.36184	−0.192204	−0.0961021	+	0.995371i	$0.530638\pi$
$152$	0	0
$153$	0	0
$154$	−9.71007	−0.782460
$155$	−15.3327	−1.23156
$156$	0	0
$157$	−14.3550	−1.14566	−0.572828	−	0.819675i	$-0.694154\pi$
$157$	−14.3550	−1.14566	−0.572828	+	0.819675i	$0.694154\pi$
$158$	−13.0077	−1.03484
$159$	0	0
$160$	2.34730	0.185570
$161$	29.4296	2.31938
$162$	0	0
$163$	−17.1411	−1.34260	−0.671299	−	0.741186i	$-0.734263\pi$
$163$	−17.1411	−1.34260	−0.671299	+	0.741186i	$0.734263\pi$
$164$	−1.94356	−0.151767
$165$	0	0
$166$	8.17024	0.634134
$167$	−14.6186	−1.13122	−0.565609	−	0.824674i	$-0.691359\pi$
$167$	−14.6186	−1.13122	−0.565609	+	0.824674i	$0.691359\pi$
$168$	0	0
$169$	16.2841	1.25262
$170$	9.10607	0.698403
$171$	0	0
$172$	5.02229	0.382946
$173$	−20.6682	−1.57137	−0.785687	−	0.618625i	$-0.787690\pi$
$173$	−20.6682	−1.57137	−0.785687	+	0.618625i	$0.787690\pi$
$174$	0	0
$175$	1.82201	0.137731
$176$	−2.71688	−0.204793
$177$	0	0
$178$	11.7246	0.878798
$179$	5.69459	0.425634	0.212817	−	0.977092i	$-0.431736\pi$
$179$	5.69459	0.425634	0.212817	+	0.977092i	$0.431736\pi$
$180$	0	0
$181$	22.2199	1.65159	0.825795	−	0.563970i	$-0.190727\pi$
$181$	22.2199	1.65159	0.825795	+	0.563970i	$0.190727\pi$
$182$	−19.3405	−1.43361
$183$	0	0
$184$	8.23442	0.607050
$185$	−0.913534	−0.0671643
$186$	0	0
$187$	−10.5398	−0.770749
$188$	−2.98545	−0.217736
$189$	0	0
$190$	0	0
$191$	−6.04694	−0.437541	−0.218771	−	0.975776i	$-0.570205\pi$
$191$	−6.04694	−0.437541	−0.218771	+	0.975776i	$0.570205\pi$
$192$	0	0
$193$	−6.29860	−0.453383	−0.226692	−	0.973967i	$-0.572791\pi$
$193$	−6.29860	−0.453383	−0.226692	+	0.973967i	$0.572791\pi$
$194$	8.60401	0.617732
$195$	0	0
$196$	5.77332	0.412380
$197$	−19.9368	−1.42044	−0.710218	−	0.703982i	$-0.751403\pi$
$197$	−19.9368	−1.42044	−0.710218	+	0.703982i	$0.751403\pi$
$198$	0	0
$199$	12.8821	0.913186	0.456593	−	0.889676i	$-0.349070\pi$
$199$	12.8821	0.913186	0.456593	+	0.889676i	$0.349070\pi$
$200$	0.509800	0.0360483
$201$	0	0
$202$	1.28581	0.0904691
$203$	12.6236	0.886004
$204$	0	0
$205$	−4.56212	−0.318632
$206$	−0.736482	−0.0513131
$207$	0	0
$208$	−5.41147	−0.375218
$209$	0	0
$210$	0	0
$211$	−2.31315	−0.159244	−0.0796218	−	0.996825i	$-0.525371\pi$
$211$	−2.31315	−0.159244	−0.0796218	+	0.996825i	$0.525371\pi$
$212$	8.30541	0.570418
$213$	0	0
$214$	10.0273	0.685455
$215$	11.7888	0.803989
$216$	0	0
$217$	−23.3455	−1.58480
$218$	−13.9213	−0.942868
$219$	0	0
$220$	−6.37733	−0.429959
$221$	−20.9932	−1.41215
$222$	0	0
$223$	−4.07098	−0.272613	−0.136307	−	0.990667i	$-0.543523\pi$
$223$	−4.07098	−0.272613	−0.136307	+	0.990667i	$0.543523\pi$
$224$	3.57398	0.238796
$225$	0	0
$226$	−0.226682	−0.0150786
$227$	−0.440570	−0.0292417	−0.0146208	−	0.999893i	$-0.504654\pi$
$227$	−0.440570	−0.0292417	−0.0146208	+	0.999893i	$0.504654\pi$
$228$	0	0
$229$	−3.38919	−0.223964	−0.111982	−	0.993710i	$-0.535720\pi$
$229$	−3.38919	−0.223964	−0.111982	+	0.993710i	$0.535720\pi$
$230$	19.3286	1.27449
$231$	0	0
$232$	3.53209	0.231893
$233$	−2.11381	−0.138480	−0.0692401	−	0.997600i	$-0.522057\pi$
$233$	−2.11381	−0.138480	−0.0692401	+	0.997600i	$0.522057\pi$
$234$	0	0
$235$	−7.00774	−0.457135
$236$	2.73143	0.177801
$237$	0	0
$238$	13.8648	0.898724
$239$	4.02910	0.260621	0.130310	−	0.991473i	$-0.458403\pi$
$239$	4.02910	0.260621	0.130310	+	0.991473i	$0.458403\pi$
$240$	0	0
$241$	10.1771	0.655562	0.327781	−	0.944754i	$-0.393699\pi$
$241$	10.1771	0.655562	0.327781	+	0.944754i	$0.393699\pi$
$242$	−3.61856	−0.232610
$243$	0	0
$244$	6.29086	0.402731
$245$	13.5517	0.865786
$246$	0	0
$247$	0	0
$248$	−6.53209	−0.414788
$249$	0	0
$250$	−10.5398	−0.666597
$251$	−2.90941	−0.183641	−0.0918203	−	0.995776i	$-0.529269\pi$
$251$	−2.90941	−0.183641	−0.0918203	+	0.995776i	$0.529269\pi$
$252$	0	0
$253$	−22.3719	−1.40651
$254$	−8.69459	−0.545547
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.55943	−0.159653	−0.0798264	−	0.996809i	$-0.525437\pi$
$257$	−2.55943	−0.159653	−0.0798264	+	0.996809i	$0.525437\pi$
$258$	0	0
$259$	−1.39094	−0.0864288
$260$	−12.7023	−0.787765
$261$	0	0
$262$	12.6432	0.781100
$263$	−15.9290	−0.982225	−0.491113	−	0.871096i	$-0.663410\pi$
$263$	−15.9290	−0.982225	−0.491113	+	0.871096i	$0.663410\pi$
$264$	0	0
$265$	19.4953	1.19758
$266$	0	0
$267$	0	0
$268$	14.9368	0.912408
$269$	16.8949	1.03010	0.515049	−	0.857161i	$-0.327774\pi$
$269$	16.8949	1.03010	0.515049	+	0.857161i	$0.327774\pi$
$270$	0	0
$271$	−0.622674	−0.0378248	−0.0189124	−	0.999821i	$-0.506020\pi$
$271$	−0.622674	−0.0378248	−0.0189124	+	0.999821i	$0.506020\pi$
$272$	3.87939	0.235222
$273$	0	0
$274$	−21.7939	−1.31661
$275$	−1.38507	−0.0835227
$276$	0	0
$277$	−27.4766	−1.65091	−0.825454	−	0.564469i	$-0.809081\pi$
$277$	−27.4766	−1.65091	−0.825454	+	0.564469i	$0.809081\pi$
$278$	−11.2267	−0.673332
$279$	0	0
$280$	8.38919	0.501350
$281$	−2.48070	−0.147986	−0.0739932	−	0.997259i	$-0.523574\pi$
$281$	−2.48070	−0.147986	−0.0739932	+	0.997259i	$0.523574\pi$
$282$	0	0
$283$	2.28312	0.135717	0.0678587	−	0.997695i	$-0.478383\pi$
$283$	2.28312	0.135717	0.0678587	+	0.997695i	$0.478383\pi$
$284$	9.02229	0.535374
$285$	0	0
$286$	14.7023	0.869367
$287$	−6.94625	−0.410024
$288$	0	0
$289$	−1.95037	−0.114728
$290$	8.29086	0.486856
$291$	0	0
$292$	10.2909	0.602227
$293$	−2.39599	−0.139975	−0.0699877	−	0.997548i	$-0.522296\pi$
$293$	−2.39599	−0.139975	−0.0699877	+	0.997548i	$0.522296\pi$
$294$	0	0
$295$	6.41147	0.373290
$296$	−0.389185	−0.0226209
$297$	0	0
$298$	21.9786	1.27319
$299$	−44.5604	−2.57699
$300$	0	0
$301$	17.9495	1.03459
$302$	−2.36184	−0.135909
$303$	0	0
$304$	0	0
$305$	14.7665	0.845528
$306$	0	0
$307$	5.23349	0.298691	0.149345	−	0.988785i	$-0.452283\pi$
$307$	5.23349	0.298691	0.149345	+	0.988785i	$0.452283\pi$
$308$	−9.71007	−0.553283
$309$	0	0
$310$	−15.3327	−0.870842
$311$	10.9932	0.623367	0.311683	−	0.950186i	$-0.399107\pi$
$311$	10.9932	0.623367	0.311683	+	0.950186i	$0.399107\pi$
$312$	0	0
$313$	−30.4516	−1.72123	−0.860613	−	0.509259i	$-0.829920\pi$
$313$	−30.4516	−1.72123	−0.860613	+	0.509259i	$0.829920\pi$
$314$	−14.3550	−0.810102
$315$	0	0
$316$	−13.0077	−0.731743
$317$	11.3396	0.636893	0.318446	−	0.947941i	$-0.396839\pi$
$317$	11.3396	0.636893	0.318446	+	0.947941i	$0.396839\pi$
$318$	0	0
$319$	−9.59627	−0.537288
$320$	2.34730	0.131218
$321$	0	0
$322$	29.4296	1.64005
$323$	0	0
$324$	0	0
$325$	−2.75877	−0.153029
$326$	−17.1411	−0.949360
$327$	0	0
$328$	−1.94356	−0.107315
$329$	−10.6699	−0.588253
$330$	0	0
$331$	−12.9317	−0.710791	−0.355395	−	0.934716i	$-0.615654\pi$
$331$	−12.9317	−0.710791	−0.355395	+	0.934716i	$0.615654\pi$
$332$	8.17024	0.448400
$333$	0	0
$334$	−14.6186	−0.799892
$335$	35.0610	1.91559
$336$	0	0
$337$	−10.7469	−0.585422	−0.292711	−	0.956201i	$-0.594557\pi$
$337$	−10.7469	−0.585422	−0.292711	+	0.956201i	$0.594557\pi$
$338$	16.2841	0.885736
$339$	0	0
$340$	9.10607	0.493846
$341$	17.7469	0.961049
$342$	0	0
$343$	−4.38413	−0.236721
$344$	5.02229	0.270784
$345$	0	0
$346$	−20.6682	−1.11113
$347$	−22.8033	−1.22415	−0.612074	−	0.790801i	$-0.709665\pi$
$347$	−22.8033	−1.22415	−0.612074	+	0.790801i	$0.709665\pi$
$348$	0	0
$349$	−25.3628	−1.35764	−0.678819	−	0.734305i	$-0.737508\pi$
$349$	−25.3628	−1.35764	−0.678819	+	0.734305i	$0.737508\pi$
$350$	1.82201	0.0973908
$351$	0	0
$352$	−2.71688	−0.144810
$353$	12.1925	0.648943	0.324472	−	0.945895i	$-0.394814\pi$
$353$	12.1925	0.648943	0.324472	+	0.945895i	$0.394814\pi$
$354$	0	0
$355$	21.1780	1.12401
$356$	11.7246	0.621404
$357$	0	0
$358$	5.69459	0.300969
$359$	−29.1634	−1.53919	−0.769594	−	0.638534i	$-0.779541\pi$
$359$	−29.1634	−1.53919	−0.769594	+	0.638534i	$0.779541\pi$
$360$	0	0
$361$	0	0
$362$	22.2199	1.16785
$363$	0	0
$364$	−19.3405	−1.01372
$365$	24.1557	1.26437
$366$	0	0
$367$	13.9923	0.730390	0.365195	−	0.930931i	$-0.381002\pi$
$367$	13.9923	0.730390	0.365195	+	0.930931i	$0.381002\pi$
$368$	8.23442	0.429249
$369$	0	0
$370$	−0.913534	−0.0474923
$371$	29.6833	1.54108
$372$	0	0
$373$	6.82026	0.353140	0.176570	−	0.984288i	$-0.443500\pi$
$373$	6.82026	0.353140	0.176570	+	0.984288i	$0.443500\pi$
$374$	−10.5398	−0.545002
$375$	0	0
$376$	−2.98545	−0.153963
$377$	−19.1138	−0.984411
$378$	0	0
$379$	−31.9341	−1.64034	−0.820171	−	0.572118i	$-0.806122\pi$
$379$	−31.9341	−1.64034	−0.820171	+	0.572118i	$0.806122\pi$
$380$	0	0
$381$	0	0
$382$	−6.04694	−0.309388
$383$	−36.8188	−1.88135	−0.940677	−	0.339303i	$-0.889809\pi$
$383$	−36.8188	−1.88135	−0.940677	+	0.339303i	$0.889809\pi$
$384$	0	0
$385$	−22.7924	−1.16161
$386$	−6.29860	−0.320590
$387$	0	0
$388$	8.60401	0.436802
$389$	3.35504	0.170107	0.0850536	−	0.996376i	$-0.472894\pi$
$389$	3.35504	0.170107	0.0850536	+	0.996376i	$0.472894\pi$
$390$	0	0
$391$	31.9445	1.61550
$392$	5.77332	0.291597
$393$	0	0
$394$	−19.9368	−1.00440
$395$	−30.5330	−1.53628
$396$	0	0
$397$	6.88444	0.345520	0.172760	−	0.984964i	$-0.444731\pi$
$397$	6.88444	0.345520	0.172760	+	0.984964i	$0.444731\pi$
$398$	12.8821	0.645720
$399$	0	0
$400$	0.509800	0.0254900
$401$	−5.03003	−0.251188	−0.125594	−	0.992082i	$-0.540084\pi$
$401$	−5.03003	−0.251188	−0.125594	+	0.992082i	$0.540084\pi$
$402$	0	0
$403$	35.3482	1.76082
$404$	1.28581	0.0639713
$405$	0	0
$406$	12.6236	0.626499
$407$	1.05737	0.0524119
$408$	0	0
$409$	10.4088	0.514681	0.257341	−	0.966321i	$-0.417154\pi$
$409$	10.4088	0.514681	0.257341	+	0.966321i	$0.417154\pi$
$410$	−4.56212	−0.225307
$411$	0	0
$412$	−0.736482	−0.0362839
$413$	9.76207	0.480360
$414$	0	0
$415$	19.1780	0.941410
$416$	−5.41147	−0.265319
$417$	0	0
$418$	0	0
$419$	6.22256	0.303992	0.151996	−	0.988381i	$-0.451430\pi$
$419$	6.22256	0.303992	0.151996	+	0.988381i	$0.451430\pi$
$420$	0	0
$421$	26.6195	1.29735	0.648677	−	0.761064i	$-0.275323\pi$
$421$	26.6195	1.29735	0.648677	+	0.761064i	$0.275323\pi$
$422$	−2.31315	−0.112602
$423$	0	0
$424$	8.30541	0.403346
$425$	1.97771	0.0959331
$426$	0	0
$427$	22.4834	1.08805
$428$	10.0273	0.484690
$429$	0	0
$430$	11.7888	0.568506
$431$	−13.7324	−0.661465	−0.330732	−	0.943725i	$-0.607296\pi$
$431$	−13.7324	−0.661465	−0.330732	+	0.943725i	$0.607296\pi$
$432$	0	0
$433$	24.6946	1.18675	0.593373	−	0.804927i	$-0.297796\pi$
$433$	24.6946	1.18675	0.593373	+	0.804927i	$0.297796\pi$
$434$	−23.3455	−1.12062
$435$	0	0
$436$	−13.9213	−0.666708
$437$	0	0
$438$	0	0
$439$	−18.1411	−0.865830	−0.432915	−	0.901435i	$-0.642515\pi$
$439$	−18.1411	−0.865830	−0.432915	+	0.901435i	$0.642515\pi$
$440$	−6.37733	−0.304027
$441$	0	0
$442$	−20.9932	−0.998544
$443$	12.8625	0.611115	0.305557	−	0.952174i	$-0.401157\pi$
$443$	12.8625	0.611115	0.305557	+	0.952174i	$0.401157\pi$
$444$	0	0
$445$	27.5212	1.30463
$446$	−4.07098	−0.192767
$447$	0	0
$448$	3.57398	0.168855
$449$	−21.9317	−1.03502	−0.517511	−	0.855677i	$-0.673141\pi$
$449$	−21.9317	−1.03502	−0.517511	+	0.855677i	$0.673141\pi$
$450$	0	0
$451$	5.28043	0.248646
$452$	−0.226682	−0.0106622
$453$	0	0
$454$	−0.440570	−0.0206770
$455$	−45.3979	−2.12828
$456$	0	0
$457$	30.8452	1.44288	0.721440	−	0.692477i	$-0.243481\pi$
$457$	30.8452	1.44288	0.721440	+	0.692477i	$0.243481\pi$
$458$	−3.38919	−0.158366
$459$	0	0
$460$	19.3286	0.901202
$461$	12.3200	0.573798	0.286899	−	0.957961i	$-0.407376\pi$
$461$	12.3200	0.573798	0.286899	+	0.957961i	$0.407376\pi$
$462$	0	0
$463$	12.2094	0.567421	0.283711	−	0.958910i	$-0.408435\pi$
$463$	12.2094	0.567421	0.283711	+	0.958910i	$0.408435\pi$
$464$	3.53209	0.163973
$465$	0	0
$466$	−2.11381	−0.0979202
$467$	26.7401	1.23738	0.618692	−	0.785633i	$-0.287663\pi$
$467$	26.7401	1.23738	0.618692	+	0.785633i	$0.287663\pi$
$468$	0	0
$469$	53.3836	2.46503
$470$	−7.00774	−0.323243
$471$	0	0
$472$	2.73143	0.125724
$473$	−13.6450	−0.627396
$474$	0	0
$475$	0	0
$476$	13.8648	0.635494
$477$	0	0
$478$	4.02910	0.184287
$479$	−7.73648	−0.353489	−0.176744	−	0.984257i	$-0.556557\pi$
$479$	−7.73648	−0.353489	−0.176744	+	0.984257i	$0.556557\pi$
$480$	0	0
$481$	2.10607	0.0960284
$482$	10.1771	0.463552
$483$	0	0
$484$	−3.61856	−0.164480
$485$	20.1962	0.917060
$486$	0	0
$487$	−13.3919	−0.606844	−0.303422	−	0.952856i	$-0.598129\pi$
$487$	−13.3919	−0.606844	−0.303422	+	0.952856i	$0.598129\pi$
$488$	6.29086	0.284774
$489$	0	0
$490$	13.5517	0.612203
$491$	20.2472	0.913744	0.456872	−	0.889532i	$-0.348970\pi$
$491$	20.2472	0.913744	0.456872	+	0.889532i	$0.348970\pi$
$492$	0	0
$493$	13.7023	0.617122
$494$	0	0
$495$	0	0
$496$	−6.53209	−0.293299
$497$	32.2455	1.44641
$498$	0	0
$499$	−4.73742	−0.212076	−0.106038	−	0.994362i	$-0.533816\pi$
$499$	−4.73742	−0.212076	−0.106038	+	0.994362i	$0.533816\pi$
$500$	−10.5398	−0.471356
$501$	0	0
$502$	−2.90941	−0.129854
$503$	−8.59896	−0.383408	−0.191704	−	0.981453i	$-0.561401\pi$
$503$	−8.59896	−0.383408	−0.191704	+	0.981453i	$0.561401\pi$
$504$	0	0
$505$	3.01817	0.134307
$506$	−22.3719	−0.994554
$507$	0	0
$508$	−8.69459	−0.385760
$509$	5.36278	0.237701	0.118850	−	0.992912i	$-0.462079\pi$
$509$	5.36278	0.237701	0.118850	+	0.992912i	$0.462079\pi$
$510$	0	0
$511$	36.7793	1.62702
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.55943	−0.112892
$515$	−1.72874	−0.0761774
$516$	0	0
$517$	8.11112	0.356727
$518$	−1.39094	−0.0611144
$519$	0	0
$520$	−12.7023	−0.557034
$521$	−23.7151	−1.03898	−0.519489	−	0.854477i	$-0.673878\pi$
$521$	−23.7151	−1.03898	−0.519489	+	0.854477i	$0.673878\pi$
$522$	0	0
$523$	31.3158	1.36935	0.684673	−	0.728850i	$-0.259945\pi$
$523$	31.3158	1.36935	0.684673	+	0.728850i	$0.259945\pi$
$524$	12.6432	0.552321
$525$	0	0
$526$	−15.9290	−0.694538
$527$	−25.3405	−1.10385
$528$	0	0
$529$	44.8057	1.94807
$530$	19.4953	0.846820
$531$	0	0
$532$	0	0
$533$	10.5175	0.455565
$534$	0	0
$535$	23.5371	1.01760
$536$	14.9368	0.645170
$537$	0	0
$538$	16.8949	0.728389
$539$	−15.6854	−0.675619
$540$	0	0
$541$	9.58946	0.412283	0.206142	−	0.978522i	$-0.433909\pi$
$541$	9.58946	0.412283	0.206142	+	0.978522i	$0.433909\pi$
$542$	−0.622674	−0.0267461
$543$	0	0
$544$	3.87939	0.166327
$545$	−32.6774	−1.39974
$546$	0	0
$547$	−10.2754	−0.439343	−0.219672	−	0.975574i	$-0.570499\pi$
$547$	−10.2754	−0.439343	−0.219672	+	0.975574i	$0.570499\pi$
$548$	−21.7939	−0.930987
$549$	0	0
$550$	−1.38507	−0.0590594
$551$	0	0
$552$	0	0
$553$	−46.4894	−1.97693
$554$	−27.4766	−1.16737
$555$	0	0
$556$	−11.2267	−0.476117
$557$	16.6774	0.706642	0.353321	−	0.935502i	$-0.385052\pi$
$557$	16.6774	0.706642	0.353321	+	0.935502i	$0.385052\pi$
$558$	0	0
$559$	−27.1780	−1.14951
$560$	8.38919	0.354508
$561$	0	0
$562$	−2.48070	−0.104642
$563$	3.43107	0.144603	0.0723013	−	0.997383i	$-0.476966\pi$
$563$	3.43107	0.144603	0.0723013	+	0.997383i	$0.476966\pi$
$564$	0	0
$565$	−0.532089	−0.0223851
$566$	2.28312	0.0959666
$567$	0	0
$568$	9.02229	0.378567
$569$	−43.5681	−1.82647	−0.913235	−	0.407433i	$-0.866424\pi$
$569$	−43.5681	−1.82647	−0.913235	+	0.407433i	$0.866424\pi$
$570$	0	0
$571$	−8.14115	−0.340696	−0.170348	−	0.985384i	$-0.554489\pi$
$571$	−8.14115	−0.340696	−0.170348	+	0.985384i	$0.554489\pi$
$572$	14.7023	0.614735
$573$	0	0
$574$	−6.94625	−0.289931
$575$	4.19791	0.175065
$576$	0	0
$577$	35.0496	1.45914	0.729568	−	0.683909i	$-0.239721\pi$
$577$	35.0496	1.45914	0.729568	+	0.683909i	$0.239721\pi$
$578$	−1.95037	−0.0811247
$579$	0	0
$580$	8.29086	0.344259
$581$	29.2003	1.21143
$582$	0	0
$583$	−22.5648	−0.934539
$584$	10.2909	0.425839
$585$	0	0
$586$	−2.39599	−0.0989775
$587$	−9.76651	−0.403107	−0.201554	−	0.979478i	$-0.564599\pi$
$587$	−9.76651	−0.403107	−0.201554	+	0.979478i	$0.564599\pi$
$588$	0	0
$589$	0	0
$590$	6.41147	0.263956
$591$	0	0
$592$	−0.389185	−0.0159954
$593$	32.6973	1.34272	0.671358	−	0.741133i	$-0.265711\pi$
$593$	32.6973	1.34272	0.671358	+	0.741133i	$0.265711\pi$
$594$	0	0
$595$	32.5449	1.33421
$596$	21.9786	0.900280
$597$	0	0
$598$	−44.5604	−1.82221
$599$	−8.84430	−0.361368	−0.180684	−	0.983541i	$-0.557831\pi$
$599$	−8.84430	−0.361368	−0.180684	+	0.983541i	$0.557831\pi$
$600$	0	0
$601$	21.8152	0.889861	0.444930	−	0.895565i	$-0.353228\pi$
$601$	21.8152	0.889861	0.444930	+	0.895565i	$0.353228\pi$
$602$	17.9495	0.731569
$603$	0	0
$604$	−2.36184	−0.0961021
$605$	−8.49382	−0.345323
$606$	0	0
$607$	26.5963	1.07951	0.539755	−	0.841822i	$-0.318517\pi$
$607$	26.5963	1.07951	0.539755	+	0.841822i	$0.318517\pi$
$608$	0	0
$609$	0	0
$610$	14.7665	0.597879
$611$	16.1557	0.653590
$612$	0	0
$613$	−15.1411	−0.611545	−0.305773	−	0.952105i	$-0.598915\pi$
$613$	−15.1411	−0.611545	−0.305773	+	0.952105i	$0.598915\pi$
$614$	5.23349	0.211206
$615$	0	0
$616$	−9.71007	−0.391230
$617$	29.3892	1.18316	0.591582	−	0.806245i	$-0.298504\pi$
$617$	29.3892	1.18316	0.591582	+	0.806245i	$0.298504\pi$
$618$	0	0
$619$	31.4439	1.26384	0.631918	−	0.775035i	$-0.282268\pi$
$619$	31.4439	1.26384	0.631918	+	0.775035i	$0.282268\pi$
$620$	−15.3327	−0.615778
$621$	0	0
$622$	10.9932	0.440787
$623$	41.9035	1.67883
$624$	0	0
$625$	−27.2891	−1.09156
$626$	−30.4516	−1.21709
$627$	0	0
$628$	−14.3550	−0.572828
$629$	−1.50980	−0.0601997
$630$	0	0
$631$	−29.8753	−1.18932	−0.594658	−	0.803979i	$-0.702712\pi$
$631$	−29.8753	−1.18932	−0.594658	+	0.803979i	$0.702712\pi$
$632$	−13.0077	−0.517420
$633$	0	0
$634$	11.3396	0.450351
$635$	−20.4088	−0.809898
$636$	0	0
$637$	−31.2422	−1.23786
$638$	−9.59627	−0.379920
$639$	0	0
$640$	2.34730	0.0927850
$641$	7.18479	0.283782	0.141891	−	0.989882i	$-0.454682\pi$
$641$	7.18479	0.283782	0.141891	+	0.989882i	$0.454682\pi$
$642$	0	0
$643$	9.08378	0.358229	0.179115	−	0.983828i	$-0.442677\pi$
$643$	9.08378	0.358229	0.179115	+	0.983828i	$0.442677\pi$
$644$	29.4296	1.15969
$645$	0	0
$646$	0	0
$647$	−37.5749	−1.47722	−0.738611	−	0.674132i	$-0.764518\pi$
$647$	−37.5749	−1.47722	−0.738611	+	0.674132i	$0.764518\pi$
$648$	0	0
$649$	−7.42097	−0.291299
$650$	−2.75877	−0.108208
$651$	0	0
$652$	−17.1411	−0.671299
$653$	−1.74153	−0.0681515	−0.0340758	−	0.999419i	$-0.510849\pi$
$653$	−1.74153	−0.0681515	−0.0340758	+	0.999419i	$0.510849\pi$
$654$	0	0
$655$	29.6774	1.15959
$656$	−1.94356	−0.0758834
$657$	0	0
$658$	−10.6699	−0.415958
$659$	−5.20439	−0.202734	−0.101367	−	0.994849i	$-0.532322\pi$
$659$	−5.20439	−0.202734	−0.101367	+	0.994849i	$0.532322\pi$
$660$	0	0
$661$	24.8408	0.966195	0.483097	−	0.875567i	$-0.339512\pi$
$661$	24.8408	0.966195	0.483097	+	0.875567i	$0.339512\pi$
$662$	−12.9317	−0.502605
$663$	0	0
$664$	8.17024	0.317067
$665$	0	0
$666$	0	0
$667$	29.0847	1.12616
$668$	−14.6186	−0.565609
$669$	0	0
$670$	35.0610	1.35452
$671$	−17.0915	−0.659811
$672$	0	0
$673$	−8.99226	−0.346626	−0.173313	−	0.984867i	$-0.555447\pi$
$673$	−8.99226	−0.346626	−0.173313	+	0.984867i	$0.555447\pi$
$674$	−10.7469	−0.413956
$675$	0	0
$676$	16.2841	0.626310
$677$	6.41559	0.246571	0.123286	−	0.992371i	$-0.460657\pi$
$677$	6.41559	0.246571	0.123286	+	0.992371i	$0.460657\pi$
$678$	0	0
$679$	30.7505	1.18010
$680$	9.10607	0.349202
$681$	0	0
$682$	17.7469	0.679564
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	0	0
$685$	−51.1566	−1.95459
$686$	−4.38413	−0.167387
$687$	0	0
$688$	5.02229	0.191473
$689$	−44.9445	−1.71225
$690$	0	0
$691$	−44.3019	−1.68532	−0.842662	−	0.538443i	$-0.819013\pi$
$691$	−44.3019	−1.68532	−0.842662	+	0.538443i	$0.819013\pi$
$692$	−20.6682	−0.785687
$693$	0	0
$694$	−22.8033	−0.865603
$695$	−26.3523	−0.999602
$696$	0	0
$697$	−7.53983	−0.285591
$698$	−25.3628	−0.959995
$699$	0	0
$700$	1.82201	0.0688657
$701$	46.7229	1.76470	0.882349	−	0.470595i	$-0.155961\pi$
$701$	46.7229	1.76470	0.882349	+	0.470595i	$0.155961\pi$
$702$	0	0
$703$	0	0
$704$	−2.71688	−0.102396
$705$	0	0
$706$	12.1925	0.458872
$707$	4.59545	0.172830
$708$	0	0
$709$	−15.1949	−0.570656	−0.285328	−	0.958430i	$-0.592103\pi$
$709$	−15.1949	−0.570656	−0.285328	+	0.958430i	$0.592103\pi$
$710$	21.1780	0.794796
$711$	0	0
$712$	11.7246	0.439399
$713$	−53.7880	−2.01438
$714$	0	0
$715$	34.5107	1.29063
$716$	5.69459	0.212817
$717$	0	0
$718$	−29.1634	−1.08837
$719$	−23.3969	−0.872558	−0.436279	−	0.899811i	$-0.643704\pi$
$719$	−23.3969	−0.872558	−0.436279	+	0.899811i	$0.643704\pi$
$720$	0	0
$721$	−2.63217	−0.0980271
$722$	0	0
$723$	0	0
$724$	22.2199	0.825795
$725$	1.80066	0.0668748
$726$	0	0
$727$	34.0820	1.26403	0.632016	−	0.774955i	$-0.282228\pi$
$727$	34.0820	1.26403	0.632016	+	0.774955i	$0.282228\pi$
$728$	−19.3405	−0.716806
$729$	0	0
$730$	24.1557	0.894042
$731$	19.4834	0.720619
$732$	0	0
$733$	−30.0128	−1.10855	−0.554274	−	0.832334i	$-0.687004\pi$
$733$	−30.0128	−1.10855	−0.554274	+	0.832334i	$0.687004\pi$
$734$	13.9923	0.516464
$735$	0	0
$736$	8.23442	0.303525
$737$	−40.5814	−1.49483
$738$	0	0
$739$	23.8425	0.877062	0.438531	−	0.898716i	$-0.355499\pi$
$739$	23.8425	0.877062	0.438531	+	0.898716i	$0.355499\pi$
$740$	−0.913534	−0.0335822
$741$	0	0
$742$	29.6833	1.08971
$743$	30.9941	1.13706	0.568532	−	0.822661i	$-0.307512\pi$
$743$	30.9941	1.13706	0.568532	+	0.822661i	$0.307512\pi$
$744$	0	0
$745$	51.5904	1.89013
$746$	6.82026	0.249707
$747$	0	0
$748$	−10.5398	−0.385374
$749$	35.8375	1.30947
$750$	0	0
$751$	−22.0446	−0.804418	−0.402209	−	0.915548i	$-0.631757\pi$
$751$	−22.0446	−0.804418	−0.402209	+	0.915548i	$0.631757\pi$
$752$	−2.98545	−0.108868
$753$	0	0
$754$	−19.1138	−0.696084
$755$	−5.54395	−0.201765
$756$	0	0
$757$	−14.3250	−0.520651	−0.260326	−	0.965521i	$-0.583830\pi$
$757$	−14.3250	−0.520651	−0.260326	+	0.965521i	$0.583830\pi$
$758$	−31.9341	−1.15990
$759$	0	0
$760$	0	0
$761$	−17.2635	−0.625802	−0.312901	−	0.949786i	$-0.601301\pi$
$761$	−17.2635	−0.625802	−0.312901	+	0.949786i	$0.601301\pi$
$762$	0	0
$763$	−49.7543	−1.80123
$764$	−6.04694	−0.218771
$765$	0	0
$766$	−36.8188	−1.33032
$767$	−14.7811	−0.533713
$768$	0	0
$769$	18.3200	0.660634	0.330317	−	0.943870i	$-0.392844\pi$
$769$	18.3200	0.660634	0.330317	+	0.943870i	$0.392844\pi$
$770$	−22.7924	−0.821382
$771$	0	0
$772$	−6.29860	−0.226692
$773$	−41.1438	−1.47984	−0.739920	−	0.672694i	$-0.765137\pi$
$773$	−41.1438	−1.47984	−0.739920	+	0.672694i	$0.765137\pi$
$774$	0	0
$775$	−3.33006	−0.119619
$776$	8.60401	0.308866
$777$	0	0
$778$	3.35504	0.120284
$779$	0	0
$780$	0	0
$781$	−24.5125	−0.877126
$782$	31.9445	1.14233
$783$	0	0
$784$	5.77332	0.206190
$785$	−33.6955	−1.20264
$786$	0	0
$787$	−13.6732	−0.487398	−0.243699	−	0.969851i	$-0.578361\pi$
$787$	−13.6732	−0.487398	−0.243699	+	0.969851i	$0.578361\pi$
$788$	−19.9368	−0.710218
$789$	0	0
$790$	−30.5330	−1.08632
$791$	−0.810155	−0.0288058
$792$	0	0
$793$	−34.0428	−1.20890
$794$	6.88444	0.244320
$795$	0	0
$796$	12.8821	0.456593
$797$	33.4935	1.18640	0.593200	−	0.805055i	$-0.297864\pi$
$797$	33.4935	1.18640	0.593200	+	0.805055i	$0.297864\pi$
$798$	0	0
$799$	−11.5817	−0.409732
$800$	0.509800	0.0180242
$801$	0	0
$802$	−5.03003	−0.177617
$803$	−27.9590	−0.986653
$804$	0	0
$805$	69.0801	2.43475
$806$	35.3482	1.24509
$807$	0	0
$808$	1.28581	0.0452345
$809$	−10.3841	−0.365087	−0.182543	−	0.983198i	$-0.558433\pi$
$809$	−10.3841	−0.365087	−0.182543	+	0.983198i	$0.558433\pi$
$810$	0	0
$811$	−30.9855	−1.08805	−0.544023	−	0.839070i	$-0.683100\pi$
$811$	−30.9855	−1.08805	−0.544023	+	0.839070i	$0.683100\pi$
$812$	12.6236	0.443002
$813$	0	0
$814$	1.05737	0.0370608
$815$	−40.2354	−1.40938
$816$	0	0
$817$	0	0
$818$	10.4088	0.363935
$819$	0	0
$820$	−4.56212	−0.159316
$821$	44.2719	1.54510	0.772549	−	0.634955i	$-0.218981\pi$
$821$	44.2719	1.54510	0.772549	+	0.634955i	$0.218981\pi$
$822$	0	0
$823$	−19.9659	−0.695966	−0.347983	−	0.937501i	$-0.613133\pi$
$823$	−19.9659	−0.695966	−0.347983	+	0.937501i	$0.613133\pi$
$824$	−0.736482	−0.0256566
$825$	0	0
$826$	9.76207	0.339666
$827$	8.42871	0.293095	0.146547	−	0.989204i	$-0.453184\pi$
$827$	8.42871	0.293095	0.146547	+	0.989204i	$0.453184\pi$
$828$	0	0
$829$	−4.24722	−0.147512	−0.0737559	−	0.997276i	$-0.523499\pi$
$829$	−4.24722	−0.147512	−0.0737559	+	0.997276i	$0.523499\pi$
$830$	19.1780	0.665678
$831$	0	0
$832$	−5.41147	−0.187609
$833$	22.3969	0.776007
$834$	0	0
$835$	−34.3141	−1.18749
$836$	0	0
$837$	0	0
$838$	6.22256	0.214955
$839$	6.04870	0.208824	0.104412	−	0.994534i	$-0.466704\pi$
$839$	6.04870	0.208824	0.104412	+	0.994534i	$0.466704\pi$
$840$	0	0
$841$	−16.5243	−0.569805
$842$	26.6195	0.917368
$843$	0	0
$844$	−2.31315	−0.0796218
$845$	38.2235	1.31493
$846$	0	0
$847$	−12.9326	−0.444371
$848$	8.30541	0.285209
$849$	0	0
$850$	1.97771	0.0678349
$851$	−3.20472	−0.109856
$852$	0	0
$853$	−25.7588	−0.881964	−0.440982	−	0.897516i	$-0.645370\pi$
$853$	−25.7588	−0.881964	−0.440982	+	0.897516i	$0.645370\pi$
$854$	22.4834	0.769366
$855$	0	0
$856$	10.0273	0.342727
$857$	−58.2116	−1.98847	−0.994236	−	0.107215i	$-0.965807\pi$
$857$	−58.2116	−1.98847	−0.994236	+	0.107215i	$0.965807\pi$
$858$	0	0
$859$	−10.1162	−0.345159	−0.172580	−	0.984996i	$-0.555210\pi$
$859$	−10.1162	−0.345159	−0.172580	+	0.984996i	$0.555210\pi$
$860$	11.7888	0.401995
$861$	0	0
$862$	−13.7324	−0.467726
$863$	−36.3414	−1.23708	−0.618538	−	0.785755i	$-0.712275\pi$
$863$	−36.3414	−1.23708	−0.618538	+	0.785755i	$0.712275\pi$
$864$	0	0
$865$	−48.5144	−1.64954
$866$	24.6946	0.839156
$867$	0	0
$868$	−23.3455	−0.792399
$869$	35.3405	1.19884
$870$	0	0
$871$	−80.8299	−2.73882
$872$	−13.9213	−0.471434
$873$	0	0
$874$	0	0
$875$	−37.6691	−1.27345
$876$	0	0
$877$	7.17942	0.242432	0.121216	−	0.992626i	$-0.461321\pi$
$877$	7.17942	0.242432	0.121216	+	0.992626i	$0.461321\pi$
$878$	−18.1411	−0.612234
$879$	0	0
$880$	−6.37733	−0.214980
$881$	−13.1712	−0.443748	−0.221874	−	0.975075i	$-0.571217\pi$
$881$	−13.1712	−0.443748	−0.221874	+	0.975075i	$0.571217\pi$
$882$	0	0
$883$	−12.5330	−0.421770	−0.210885	−	0.977511i	$-0.567635\pi$
$883$	−12.5330	−0.421770	−0.210885	+	0.977511i	$0.567635\pi$
$884$	−20.9932	−0.706077
$885$	0	0
$886$	12.8625	0.432123
$887$	−53.0015	−1.77962	−0.889809	−	0.456333i	$-0.849162\pi$
$887$	−53.0015	−1.77962	−0.889809	+	0.456333i	$0.849162\pi$
$888$	0	0
$889$	−31.0743	−1.04220
$890$	27.5212	0.922511
$891$	0	0
$892$	−4.07098	−0.136307
$893$	0	0
$894$	0	0
$895$	13.3669	0.446806
$896$	3.57398	0.119398
$897$	0	0
$898$	−21.9317	−0.731870
$899$	−23.0719	−0.769492
$900$	0	0
$901$	32.2199	1.07340
$902$	5.28043	0.175819
$903$	0	0
$904$	−0.226682	−0.00753932
$905$	52.1566	1.73375
$906$	0	0
$907$	−22.1634	−0.735925	−0.367962	−	0.929841i	$-0.619944\pi$
$907$	−22.1634	−0.735925	−0.367962	+	0.929841i	$0.619944\pi$
$908$	−0.440570	−0.0146208
$909$	0	0
$910$	−45.3979	−1.50492
$911$	37.1908	1.23219	0.616093	−	0.787674i	$-0.288715\pi$
$911$	37.1908	1.23219	0.616093	+	0.787674i	$0.288715\pi$
$912$	0	0
$913$	−22.1976	−0.734633
$914$	30.8452	1.02027
$915$	0	0
$916$	−3.38919	−0.111982
$917$	45.1865	1.49219
$918$	0	0
$919$	11.1557	0.367992	0.183996	−	0.982927i	$-0.441097\pi$
$919$	11.1557	0.367992	0.183996	+	0.982927i	$0.441097\pi$
$920$	19.3286	0.637246
$921$	0	0
$922$	12.3200	0.405736
$923$	−48.8239	−1.60706
$924$	0	0
$925$	−0.198407	−0.00652358
$926$	12.2094	0.401227
$927$	0	0
$928$	3.53209	0.115946
$929$	12.4124	0.407238	0.203619	−	0.979050i	$-0.434730\pi$
$929$	12.4124	0.407238	0.203619	+	0.979050i	$0.434730\pi$
$930$	0	0
$931$	0	0
$932$	−2.11381	−0.0692401
$933$	0	0
$934$	26.7401	0.874963
$935$	−24.7401	−0.809088
$936$	0	0
$937$	14.1557	0.462446	0.231223	−	0.972901i	$-0.425727\pi$
$937$	14.1557	0.462446	0.231223	+	0.972901i	$0.425727\pi$
$938$	53.3836	1.74304
$939$	0	0
$940$	−7.00774	−0.228567
$941$	18.6509	0.608004	0.304002	−	0.952671i	$-0.401677\pi$
$941$	18.6509	0.608004	0.304002	+	0.952671i	$0.401677\pi$
$942$	0	0
$943$	−16.0041	−0.521166
$944$	2.73143	0.0889005
$945$	0	0
$946$	−13.6450	−0.443636
$947$	42.9350	1.39520	0.697600	−	0.716487i	$-0.254251\pi$
$947$	42.9350	1.39520	0.697600	+	0.716487i	$0.254251\pi$
$948$	0	0
$949$	−55.6887	−1.80773
$950$	0	0
$951$	0	0
$952$	13.8648	0.449362
$953$	37.3500	1.20988	0.604942	−	0.796269i	$-0.293196\pi$
$953$	37.3500	1.20988	0.604942	+	0.796269i	$0.293196\pi$
$954$	0	0
$955$	−14.1940	−0.459306
$956$	4.02910	0.130310
$957$	0	0
$958$	−7.73648	−0.249954
$959$	−77.8907	−2.51522
$960$	0	0
$961$	11.6682	0.376393
$962$	2.10607	0.0679023
$963$	0	0
$964$	10.1771	0.327781
$965$	−14.7847	−0.475936
$966$	0	0
$967$	−26.0297	−0.837059	−0.418529	−	0.908203i	$-0.637454\pi$
$967$	−26.0297	−0.837059	−0.418529	+	0.908203i	$0.637454\pi$
$968$	−3.61856	−0.116305
$969$	0	0
$970$	20.1962	0.648459
$971$	−6.69459	−0.214840	−0.107420	−	0.994214i	$-0.534259\pi$
$971$	−6.69459	−0.214840	−0.107420	+	0.994214i	$0.534259\pi$
$972$	0	0
$973$	−40.1239	−1.28631
$974$	−13.3919	−0.429103
$975$	0	0
$976$	6.29086	0.201366
$977$	−4.87939	−0.156105	−0.0780527	−	0.996949i	$-0.524870\pi$
$977$	−4.87939	−0.156105	−0.0780527	+	0.996949i	$0.524870\pi$
$978$	0	0
$979$	−31.8544	−1.01807
$980$	13.5517	0.432893
$981$	0	0
$982$	20.2472	0.646115
$983$	−6.22130	−0.198429	−0.0992144	−	0.995066i	$-0.531633\pi$
$983$	−6.22130	−0.198429	−0.0992144	+	0.995066i	$0.531633\pi$
$984$	0	0
$985$	−46.7975	−1.49109
$986$	13.7023	0.436371
$987$	0	0
$988$	0	0
$989$	41.3556	1.31503
$990$	0	0
$991$	9.32687	0.296278	0.148139	−	0.988967i	$-0.452672\pi$
$991$	9.32687	0.296278	0.148139	+	0.988967i	$0.452672\pi$
$992$	−6.53209	−0.207394
$993$	0	0
$994$	32.2455	1.02276
$995$	30.2380	0.958610
$996$	0	0
$997$	−23.5152	−0.744733	−0.372367	−	0.928086i	$-0.621454\pi$
$997$	−23.5152	−0.744733	−0.372367	+	0.928086i	$0.621454\pi$
$998$	−4.73742	−0.149960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bu.1.2		3
3.2	odd	2		2166.2.a.p.1.2		3
19.3	odd	18		342.2.u.b.199.1		6
19.13	odd	18		342.2.u.b.55.1		6
19.18	odd	2		6498.2.a.bp.1.2		3
57.32	even	18		114.2.i.c.55.1	✓	6
57.41	even	18		114.2.i.c.85.1	yes	6
57.56	even	2		2166.2.a.r.1.2		3
228.155	odd	18		912.2.bo.d.769.1		6
228.203	odd	18		912.2.bo.d.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.55.1	✓	6	57.32	even	18
114.2.i.c.85.1	yes	6	57.41	even	18
342.2.u.b.55.1		6	19.13	odd	18
342.2.u.b.199.1		6	19.3	odd	18
912.2.bo.d.625.1		6	228.203	odd	18
912.2.bo.d.769.1		6	228.155	odd	18
2166.2.a.p.1.2		3	3.2	odd	2
2166.2.a.r.1.2		3	57.56	even	2
6498.2.a.bp.1.2		3	19.18	odd	2
6498.2.a.bu.1.2		3	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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +2.34730 q^{10} -2.71688 q^{11} -5.41147 q^{13} +3.57398 q^{14} +1.00000 q^{16} +3.87939 q^{17} +2.34730 q^{20} -2.71688 q^{22} +8.23442 q^{23} +0.509800 q^{25} -5.41147 q^{26} +3.57398 q^{28} +3.53209 q^{29} -6.53209 q^{31} +1.00000 q^{32} +3.87939 q^{34} +8.38919 q^{35} -0.389185 q^{37} +2.34730 q^{40} -1.94356 q^{41} +5.02229 q^{43} -2.71688 q^{44} +8.23442 q^{46} -2.98545 q^{47} +5.77332 q^{49} +0.509800 q^{50} -5.41147 q^{52} +8.30541 q^{53} -6.37733 q^{55} +3.57398 q^{56} +3.53209 q^{58} +2.73143 q^{59} +6.29086 q^{61} -6.53209 q^{62} +1.00000 q^{64} -12.7023 q^{65} +14.9368 q^{67} +3.87939 q^{68} +8.38919 q^{70} +9.02229 q^{71} +10.2909 q^{73} -0.389185 q^{74} -9.71007 q^{77} -13.0077 q^{79} +2.34730 q^{80} -1.94356 q^{82} +8.17024 q^{83} +9.10607 q^{85} +5.02229 q^{86} -2.71688 q^{88} +11.7246 q^{89} -19.3405 q^{91} +8.23442 q^{92} -2.98545 q^{94} +8.60401 q^{97} +5.77332 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34} + 21 q^{35} + 3 q^{37} + 6 q^{40} + 9 q^{41} + 9 q^{43} - 6 q^{46} + 9 q^{47} + 24 q^{49} + 3 q^{50} - 6 q^{52} + 27 q^{53} + 12 q^{55} + 3 q^{56} + 6 q^{58} + 18 q^{59} + 3 q^{61} - 15 q^{62} + 3 q^{64} - 12 q^{65} - 12 q^{67} + 6 q^{68} + 21 q^{70} + 21 q^{71} + 15 q^{73} + 3 q^{74} + 21 q^{77} - 15 q^{79} + 6 q^{80} + 9 q^{82} + 3 q^{83} + 15 q^{85} + 9 q^{86} + 3 q^{89} - 15 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Introduction

L-functions

Modular forms

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Database

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