LMFDB - Embedded newform 6498.2.a.ce.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:	$6$
Coefficient field:	6.6.53327808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 18x^{4} + 87x^{2} - 127 $ x^6 - 18x^4 + 87x^2 - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 342)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.85095$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.83582 q^{5} -0.850952 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.83582 q^{5} -0.850952 q^{7} +1.00000 q^{8} -1.83582 q^{10} +3.29386 q^{11} -0.493780 q^{13} -0.850952 q^{14} +1.00000 q^{16} +1.38560 q^{17} -1.83582 q^{20} +3.29386 q^{22} +3.63757 q^{23} -1.62976 q^{25} -0.493780 q^{26} -0.850952 q^{28} -2.25197 q^{29} +6.81813 q^{31} +1.00000 q^{32} +1.38560 q^{34} +1.56220 q^{35} +4.72214 q^{37} -1.83582 q^{40} -6.27573 q^{41} +2.38662 q^{43} +3.29386 q^{44} +3.63757 q^{46} +1.56009 q^{47} -6.27588 q^{49} -1.62976 q^{50} -0.493780 q^{52} -13.3069 q^{53} -6.04694 q^{55} -0.850952 q^{56} -2.25197 q^{58} +2.64999 q^{59} +8.97974 q^{61} +6.81813 q^{62} +1.00000 q^{64} +0.906493 q^{65} -8.56659 q^{67} +1.38560 q^{68} +1.56220 q^{70} +10.8048 q^{71} -4.09767 q^{73} +4.72214 q^{74} -2.80291 q^{77} -0.511323 q^{79} -1.83582 q^{80} -6.27573 q^{82} +13.2526 q^{83} -2.54372 q^{85} +2.38662 q^{86} +3.29386 q^{88} +7.76784 q^{89} +0.420183 q^{91} +3.63757 q^{92} +1.56009 q^{94} +17.3893 q^{97} -6.27588 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 6 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 12 q^{13} + 6 q^{14} + 6 q^{16} + 12 q^{17} + 6 q^{20} + 6 q^{22} + 18 q^{23} + 18 q^{25} + 12 q^{26} + 6 q^{28} - 6 q^{29} - 6 q^{31} + 6 q^{32} + 12 q^{34} + 12 q^{35} + 12 q^{37} + 6 q^{40} - 18 q^{43} + 6 q^{44} + 18 q^{46} + 30 q^{47} + 18 q^{50} + 12 q^{52} - 18 q^{53} - 6 q^{55} + 6 q^{56} - 6 q^{58} - 6 q^{59} - 6 q^{62} + 6 q^{64} - 12 q^{65} + 18 q^{67} + 12 q^{68} + 12 q^{70} + 24 q^{71} - 6 q^{73} + 12 q^{74} + 30 q^{77} - 18 q^{79} + 6 q^{80} + 42 q^{83} - 12 q^{85} - 18 q^{86} + 6 q^{88} - 18 q^{89} + 18 q^{91} + 18 q^{92} + 30 q^{94} + 24 q^{97}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 12 * q^13 + 6 * q^14 + 6 * q^16 + 12 * q^17 + 6 * q^20 + 6 * q^22 + 18 * q^23 + 18 * q^25 + 12 * q^26 + 6 * q^28 - 6 * q^29 - 6 * q^31 + 6 * q^32 + 12 * q^34 + 12 * q^35 + 12 * q^37 + 6 * q^40 - 18 * q^43 + 6 * q^44 + 18 * q^46 + 30 * q^47 + 18 * q^50 + 12 * q^52 - 18 * q^53 - 6 * q^55 + 6 * q^56 - 6 * q^58 - 6 * q^59 - 6 * q^62 + 6 * q^64 - 12 * q^65 + 18 * q^67 + 12 * q^68 + 12 * q^70 + 24 * q^71 - 6 * q^73 + 12 * q^74 + 30 * q^77 - 18 * q^79 + 6 * q^80 + 42 * q^83 - 12 * q^85 - 18 * q^86 + 6 * q^88 - 18 * q^89 + 18 * q^91 + 18 * q^92 + 30 * q^94 + 24 * q^97

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
$5$	−1.83582	−0.821005	−0.410502	−	0.911860i	$-0.634647\pi$
$5$	−1.83582	−0.821005	−0.410502	+	0.911860i	$0.634647\pi$
$6$	0	0
$7$	−0.850952	−0.321629	−0.160815	−	0.986985i	$-0.551412\pi$
$7$	−0.850952	−0.321629	−0.160815	+	0.986985i	$0.551412\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.83582	−0.580538
$11$	3.29386	0.993136	0.496568	−	0.867998i	$-0.334593\pi$
$11$	3.29386	0.993136	0.496568	+	0.867998i	$0.334593\pi$
$12$	0	0
$13$	−0.493780	−0.136950	−0.0684750	−	0.997653i	$-0.521813\pi$
$13$	−0.493780	−0.136950	−0.0684750	+	0.997653i	$0.521813\pi$
$14$	−0.850952	−0.227426
$15$	0	0
$16$	1.00000	0.250000
$17$	1.38560	0.336059	0.168029	−	0.985782i	$-0.446260\pi$
$17$	1.38560	0.336059	0.168029	+	0.985782i	$0.446260\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−1.83582	−0.410502
$21$	0	0
$22$	3.29386	0.702253
$23$	3.63757	0.758487	0.379243	−	0.925297i	$-0.376184\pi$
$23$	3.63757	0.758487	0.379243	+	0.925297i	$0.376184\pi$
$24$	0	0
$25$	−1.62976	−0.325951
$26$	−0.493780	−0.0968383
$27$	0	0
$28$	−0.850952	−0.160815
$29$	−2.25197	−0.418180	−0.209090	−	0.977896i	$-0.567050\pi$
$29$	−2.25197	−0.418180	−0.209090	+	0.977896i	$0.567050\pi$
$30$	0	0
$31$	6.81813	1.22457	0.612286	−	0.790636i	$-0.290250\pi$
$31$	6.81813	1.22457	0.612286	+	0.790636i	$0.290250\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.38560	0.237629
$35$	1.56220	0.264059
$36$	0	0
$37$	4.72214	0.776314	0.388157	−	0.921593i	$-0.373112\pi$
$37$	4.72214	0.776314	0.388157	+	0.921593i	$0.373112\pi$
$38$	0	0
$39$	0	0
$40$	−1.83582	−0.290269
$41$	−6.27573	−0.980104	−0.490052	−	0.871693i	$-0.663022\pi$
$41$	−6.27573	−0.980104	−0.490052	+	0.871693i	$0.663022\pi$
$42$	0	0
$43$	2.38662	0.363956	0.181978	−	0.983303i	$-0.441750\pi$
$43$	2.38662	0.363956	0.181978	+	0.983303i	$0.441750\pi$
$44$	3.29386	0.496568
$45$	0	0
$46$	3.63757	0.536331
$47$	1.56009	0.227563	0.113781	−	0.993506i	$-0.463704\pi$
$47$	1.56009	0.227563	0.113781	+	0.993506i	$0.463704\pi$
$48$	0	0
$49$	−6.27588	−0.896554
$50$	−1.62976	−0.230482
$51$	0	0
$52$	−0.493780	−0.0684750
$53$	−13.3069	−1.82784	−0.913918	−	0.405898i	$-0.866959\pi$
$53$	−13.3069	−1.82784	−0.913918	+	0.405898i	$0.866959\pi$
$54$	0	0
$55$	−6.04694	−0.815369
$56$	−0.850952	−0.113713
$57$	0	0
$58$	−2.25197	−0.295698
$59$	2.64999	0.344999	0.172500	−	0.985010i	$-0.444816\pi$
$59$	2.64999	0.344999	0.172500	+	0.985010i	$0.444816\pi$
$60$	0	0
$61$	8.97974	1.14974	0.574869	−	0.818245i	$-0.305053\pi$
$61$	8.97974	1.14974	0.574869	+	0.818245i	$0.305053\pi$
$62$	6.81813	0.865903
$63$	0	0
$64$	1.00000	0.125000
$65$	0.906493	0.112437
$66$	0	0
$67$	−8.56659	−1.04658	−0.523288	−	0.852156i	$-0.675295\pi$
$67$	−8.56659	−1.04658	−0.523288	+	0.852156i	$0.675295\pi$
$68$	1.38560	0.168029
$69$	0	0
$70$	1.56220	0.186718
$71$	10.8048	1.28230	0.641148	−	0.767417i	$-0.278458\pi$
$71$	10.8048	1.28230	0.641148	+	0.767417i	$0.278458\pi$
$72$	0	0
$73$	−4.09767	−0.479596	−0.239798	−	0.970823i	$-0.577081\pi$
$73$	−4.09767	−0.479596	−0.239798	+	0.970823i	$0.577081\pi$
$74$	4.72214	0.548937
$75$	0	0
$76$	0	0
$77$	−2.80291	−0.319422
$78$	0	0
$79$	−0.511323	−0.0575283	−0.0287642	−	0.999586i	$-0.509157\pi$
$79$	−0.511323	−0.0575283	−0.0287642	+	0.999586i	$0.509157\pi$
$80$	−1.83582	−0.205251
$81$	0	0
$82$	−6.27573	−0.693038
$83$	13.2526	1.45466	0.727328	−	0.686290i	$-0.240762\pi$
$83$	13.2526	1.45466	0.727328	+	0.686290i	$0.240762\pi$
$84$	0	0
$85$	−2.54372	−0.275906
$86$	2.38662	0.257356
$87$	0	0
$88$	3.29386	0.351127
$89$	7.76784	0.823389	0.411695	−	0.911322i	$-0.364937\pi$
$89$	7.76784	0.823389	0.411695	+	0.911322i	$0.364937\pi$
$90$	0	0
$91$	0.420183	0.0440472
$92$	3.63757	0.379243
$93$	0	0
$94$	1.56009	0.160911
$95$	0	0
$96$	0	0
$97$	17.3893	1.76561	0.882806	−	0.469738i	$-0.155652\pi$
$97$	17.3893	1.76561	0.882806	+	0.469738i	$0.155652\pi$
$98$	−6.27588	−0.633960
$99$	0	0
$100$	−1.62976	−0.162976
$101$	−6.01491	−0.598505	−0.299253	−	0.954174i	$-0.596737\pi$
$101$	−6.01491	−0.598505	−0.299253	+	0.954174i	$0.596737\pi$
$102$	0	0
$103$	−0.165503	−0.0163075	−0.00815376	−	0.999967i	$-0.502595\pi$
$103$	−0.165503	−0.0163075	−0.00815376	+	0.999967i	$0.502595\pi$
$104$	−0.493780	−0.0484192
$105$	0	0
$106$	−13.3069	−1.29248
$107$	12.8432	1.24160	0.620800	−	0.783969i	$-0.286808\pi$
$107$	12.8432	1.24160	0.620800	+	0.783969i	$0.286808\pi$
$108$	0	0
$109$	−8.87200	−0.849783	−0.424892	−	0.905244i	$-0.639688\pi$
$109$	−8.87200	−0.849783	−0.424892	+	0.905244i	$0.639688\pi$
$110$	−6.04694	−0.576553
$111$	0	0
$112$	−0.850952	−0.0804074
$113$	12.3830	1.16489	0.582446	−	0.812870i	$-0.302096\pi$
$113$	12.3830	1.16489	0.582446	+	0.812870i	$0.302096\pi$
$114$	0	0
$115$	−6.67794	−0.622721
$116$	−2.25197	−0.209090
$117$	0	0
$118$	2.64999	0.243951
$119$	−1.17908	−0.108086
$120$	0	0
$121$	−0.150495	−0.0136814
$122$	8.97974	0.812988
$123$	0	0
$124$	6.81813	0.612286
$125$	12.1711	1.08861
$126$	0	0
$127$	11.7803	1.04533	0.522664	−	0.852539i	$-0.324938\pi$
$127$	11.7803	1.04533	0.522664	+	0.852539i	$0.324938\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.906493	0.0795047
$131$	18.7991	1.64249	0.821243	−	0.570579i	$-0.193281\pi$
$131$	18.7991	1.64249	0.821243	+	0.570579i	$0.193281\pi$
$132$	0	0
$133$	0	0
$134$	−8.56659	−0.740041
$135$	0	0
$136$	1.38560	0.118815
$137$	13.5312	1.15605	0.578024	−	0.816020i	$-0.303824\pi$
$137$	13.5312	1.15605	0.578024	+	0.816020i	$0.303824\pi$
$138$	0	0
$139$	18.9445	1.60685	0.803427	−	0.595403i	$-0.203008\pi$
$139$	18.9445	1.60685	0.803427	+	0.595403i	$0.203008\pi$
$140$	1.56220	0.132030
$141$	0	0
$142$	10.8048	0.906721
$143$	−1.62644	−0.136010
$144$	0	0
$145$	4.13422	0.343328
$146$	−4.09767	−0.339125
$147$	0	0
$148$	4.72214	0.388157
$149$	15.1935	1.24470	0.622350	−	0.782739i	$-0.286178\pi$
$149$	15.1935	1.24470	0.622350	+	0.782739i	$0.286178\pi$
$150$	0	0
$151$	−3.70848	−0.301792	−0.150896	−	0.988550i	$-0.548216\pi$
$151$	−3.70848	−0.301792	−0.150896	+	0.988550i	$0.548216\pi$
$152$	0	0
$153$	0	0
$154$	−2.80291	−0.225865
$155$	−12.5169	−1.00538
$156$	0	0
$157$	−15.7587	−1.25768	−0.628842	−	0.777533i	$-0.716471\pi$
$157$	−15.7587	−1.25768	−0.628842	+	0.777533i	$0.716471\pi$
$158$	−0.511323	−0.0406787
$159$	0	0
$160$	−1.83582	−0.145135
$161$	−3.09540	−0.243952
$162$	0	0
$163$	−8.39467	−0.657521	−0.328761	−	0.944413i	$-0.606631\pi$
$163$	−8.39467	−0.657521	−0.328761	+	0.944413i	$0.606631\pi$
$164$	−6.27573	−0.490052
$165$	0	0
$166$	13.2526	1.02860
$167$	−3.84036	−0.297176	−0.148588	−	0.988899i	$-0.547473\pi$
$167$	−3.84036	−0.297176	−0.148588	+	0.988899i	$0.547473\pi$
$168$	0	0
$169$	−12.7562	−0.981245
$170$	−2.54372	−0.195095
$171$	0	0
$172$	2.38662	0.181978
$173$	4.84285	0.368195	0.184097	−	0.982908i	$-0.441064\pi$
$173$	4.84285	0.368195	0.184097	+	0.982908i	$0.441064\pi$
$174$	0	0
$175$	1.38684	0.104836
$176$	3.29386	0.248284
$177$	0	0
$178$	7.76784	0.582224
$179$	24.5771	1.83698	0.918491	−	0.395443i	$-0.129409\pi$
$179$	24.5771	1.83698	0.918491	+	0.395443i	$0.129409\pi$
$180$	0	0
$181$	−12.1200	−0.900869	−0.450434	−	0.892809i	$-0.648731\pi$
$181$	−12.1200	−0.900869	−0.450434	+	0.892809i	$0.648731\pi$
$182$	0.420183	0.0311461
$183$	0	0
$184$	3.63757	0.268166
$185$	−8.66900	−0.637358
$186$	0	0
$187$	4.56399	0.333752
$188$	1.56009	0.113781
$189$	0	0
$190$	0	0
$191$	−20.2083	−1.46222	−0.731112	−	0.682258i	$-0.760998\pi$
$191$	−20.2083	−1.46222	−0.731112	+	0.682258i	$0.760998\pi$
$192$	0	0
$193$	−14.6305	−1.05313	−0.526563	−	0.850136i	$-0.676519\pi$
$193$	−14.6305	−1.05313	−0.526563	+	0.850136i	$0.676519\pi$
$194$	17.3893	1.24848
$195$	0	0
$196$	−6.27588	−0.448277
$197$	18.9555	1.35052	0.675261	−	0.737579i	$-0.264031\pi$
$197$	18.9555	1.35052	0.675261	+	0.737579i	$0.264031\pi$
$198$	0	0
$199$	7.88063	0.558643	0.279322	−	0.960198i	$-0.409890\pi$
$199$	7.88063	0.558643	0.279322	+	0.960198i	$0.409890\pi$
$200$	−1.62976	−0.115241
$201$	0	0
$202$	−6.01491	−0.423207
$203$	1.91632	0.134499
$204$	0	0
$205$	11.5211	0.804670
$206$	−0.165503	−0.0115312
$207$	0	0
$208$	−0.493780	−0.0342375
$209$	0	0
$210$	0	0
$211$	0.699545	0.0481587	0.0240793	−	0.999710i	$-0.492335\pi$
$211$	0.699545	0.0481587	0.0240793	+	0.999710i	$0.492335\pi$
$212$	−13.3069	−0.913918
$213$	0	0
$214$	12.8432	0.877944
$215$	−4.38141	−0.298810
$216$	0	0
$217$	−5.80190	−0.393859
$218$	−8.87200	−0.600887
$219$	0	0
$220$	−6.04694	−0.407685
$221$	−0.684185	−0.0460232
$222$	0	0
$223$	−15.1411	−1.01392	−0.506960	−	0.861969i	$-0.669231\pi$
$223$	−15.1411	−1.01392	−0.506960	+	0.861969i	$0.669231\pi$
$224$	−0.850952	−0.0568566
$225$	0	0
$226$	12.3830	0.823702
$227$	17.0328	1.13051	0.565253	−	0.824917i	$-0.308778\pi$
$227$	17.0328	1.13051	0.565253	+	0.824917i	$0.308778\pi$
$228$	0	0
$229$	12.9202	0.853790	0.426895	−	0.904301i	$-0.359607\pi$
$229$	12.9202	0.853790	0.426895	+	0.904301i	$0.359607\pi$
$230$	−6.67794	−0.440330
$231$	0	0
$232$	−2.25197	−0.147849
$233$	26.0558	1.70697	0.853485	−	0.521117i	$-0.174485\pi$
$233$	26.0558	1.70697	0.853485	+	0.521117i	$0.174485\pi$
$234$	0	0
$235$	−2.86405	−0.186830
$236$	2.64999	0.172500
$237$	0	0
$238$	−1.17908	−0.0764286
$239$	26.1157	1.68928	0.844641	−	0.535333i	$-0.179814\pi$
$239$	26.1157	1.68928	0.844641	+	0.535333i	$0.179814\pi$
$240$	0	0
$241$	4.92282	0.317106	0.158553	−	0.987350i	$-0.449317\pi$
$241$	4.92282	0.317106	0.158553	+	0.987350i	$0.449317\pi$
$242$	−0.150495	−0.00967418
$243$	0	0
$244$	8.97974	0.574869
$245$	11.5214	0.736075
$246$	0	0
$247$	0	0
$248$	6.81813	0.432952
$249$	0	0
$250$	12.1711	0.769765
$251$	−6.87784	−0.434125	−0.217063	−	0.976158i	$-0.569648\pi$
$251$	−6.87784	−0.434125	−0.217063	+	0.976158i	$0.569648\pi$
$252$	0	0
$253$	11.9817	0.753280
$254$	11.7803	0.739159
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.85002	−0.115401	−0.0577006	−	0.998334i	$-0.518377\pi$
$257$	−1.85002	−0.115401	−0.0577006	+	0.998334i	$0.518377\pi$
$258$	0	0
$259$	−4.01831	−0.249686
$260$	0.906493	0.0562183
$261$	0	0
$262$	18.7991	1.16141
$263$	−9.28510	−0.572544	−0.286272	−	0.958148i	$-0.592416\pi$
$263$	−9.28510	−0.572544	−0.286272	+	0.958148i	$0.592416\pi$
$264$	0	0
$265$	24.4290	1.50066
$266$	0	0
$267$	0	0
$268$	−8.56659	−0.523288
$269$	−21.9007	−1.33531	−0.667653	−	0.744472i	$-0.732701\pi$
$269$	−21.9007	−1.33531	−0.667653	+	0.744472i	$0.732701\pi$
$270$	0	0
$271$	5.51949	0.335285	0.167643	−	0.985848i	$-0.446385\pi$
$271$	5.51949	0.335285	0.167643	+	0.985848i	$0.446385\pi$
$272$	1.38560	0.0840146
$273$	0	0
$274$	13.5312	0.817450
$275$	−5.36819	−0.323714
$276$	0	0
$277$	32.4896	1.95211	0.976056	−	0.217519i	$-0.0697965\pi$
$277$	32.4896	1.95211	0.976056	+	0.217519i	$0.0697965\pi$
$278$	18.9445	1.13622
$279$	0	0
$280$	1.56220	0.0933591
$281$	−27.6408	−1.64891	−0.824456	−	0.565926i	$-0.808519\pi$
$281$	−27.6408	−1.64891	−0.824456	+	0.565926i	$0.808519\pi$
$282$	0	0
$283$	−11.1914	−0.665262	−0.332631	−	0.943057i	$-0.607936\pi$
$283$	−11.1914	−0.665262	−0.332631	+	0.943057i	$0.607936\pi$
$284$	10.8048	0.641148
$285$	0	0
$286$	−1.62644	−0.0961736
$287$	5.34034	0.315230
$288$	0	0
$289$	−15.0801	−0.887065
$290$	4.13422	0.242770
$291$	0	0
$292$	−4.09767	−0.239798
$293$	−12.6060	−0.736450	−0.368225	−	0.929737i	$-0.620034\pi$
$293$	−12.6060	−0.736450	−0.368225	+	0.929737i	$0.620034\pi$
$294$	0	0
$295$	−4.86491	−0.283246
$296$	4.72214	0.274469
$297$	0	0
$298$	15.1935	0.880136
$299$	−1.79616	−0.103875
$300$	0	0
$301$	−2.03090	−0.117059
$302$	−3.70848	−0.213399
$303$	0	0
$304$	0	0
$305$	−16.4852	−0.943941
$306$	0	0
$307$	−29.7399	−1.69735	−0.848673	−	0.528918i	$-0.822598\pi$
$307$	−29.7399	−1.69735	−0.848673	+	0.528918i	$0.822598\pi$
$308$	−2.80291	−0.159711
$309$	0	0
$310$	−12.5169	−0.710911
$311$	23.5772	1.33694	0.668470	−	0.743739i	$-0.266949\pi$
$311$	23.5772	1.33694	0.668470	+	0.743739i	$0.266949\pi$
$312$	0	0
$313$	22.7907	1.28821	0.644104	−	0.764938i	$-0.277231\pi$
$313$	22.7907	1.28821	0.644104	+	0.764938i	$0.277231\pi$
$314$	−15.7587	−0.889317
$315$	0	0
$316$	−0.511323	−0.0287642
$317$	12.7514	0.716188	0.358094	−	0.933686i	$-0.383427\pi$
$317$	12.7514	0.716188	0.358094	+	0.933686i	$0.383427\pi$
$318$	0	0
$319$	−7.41767	−0.415310
$320$	−1.83582	−0.102626
$321$	0	0
$322$	−3.09540	−0.172500
$323$	0	0
$324$	0	0
$325$	0.804742	0.0446390
$326$	−8.39467	−0.464938
$327$	0	0
$328$	−6.27573	−0.346519
$329$	−1.32756	−0.0731909
$330$	0	0
$331$	−3.16768	−0.174112	−0.0870558	−	0.996203i	$-0.527746\pi$
$331$	−3.16768	−0.174112	−0.0870558	+	0.996203i	$0.527746\pi$
$332$	13.2526	0.727328
$333$	0	0
$334$	−3.84036	−0.210135
$335$	15.7267	0.859243
$336$	0	0
$337$	9.87946	0.538168	0.269084	−	0.963117i	$-0.413279\pi$
$337$	9.87946	0.538168	0.269084	+	0.963117i	$0.413279\pi$
$338$	−12.7562	−0.693845
$339$	0	0
$340$	−2.54372	−0.137953
$341$	22.4580	1.21617
$342$	0	0
$343$	11.2971	0.609988
$344$	2.38662	0.128678
$345$	0	0
$346$	4.84285	0.260353
$347$	14.4794	0.777297	0.388648	−	0.921386i	$-0.372942\pi$
$347$	14.4794	0.777297	0.388648	+	0.921386i	$0.372942\pi$
$348$	0	0
$349$	3.60760	0.193110	0.0965552	−	0.995328i	$-0.469218\pi$
$349$	3.60760	0.193110	0.0965552	+	0.995328i	$0.469218\pi$
$350$	1.38684	0.0741299
$351$	0	0
$352$	3.29386	0.175563
$353$	4.02595	0.214280	0.107140	−	0.994244i	$-0.465831\pi$
$353$	4.02595	0.214280	0.107140	+	0.994244i	$0.465831\pi$
$354$	0	0
$355$	−19.8357	−1.05277
$356$	7.76784	0.411695
$357$	0	0
$358$	24.5771	1.29894
$359$	−4.95939	−0.261746	−0.130873	−	0.991399i	$-0.541778\pi$
$359$	−4.95939	−0.261746	−0.130873	+	0.991399i	$0.541778\pi$
$360$	0	0
$361$	0	0
$362$	−12.1200	−0.637011
$363$	0	0
$364$	0.420183	0.0220236
$365$	7.52259	0.393750
$366$	0	0
$367$	24.8168	1.29543	0.647714	−	0.761883i	$-0.275725\pi$
$367$	24.8168	1.29543	0.647714	+	0.761883i	$0.275725\pi$
$368$	3.63757	0.189622
$369$	0	0
$370$	−8.66900	−0.450680
$371$	11.3235	0.587886
$372$	0	0
$373$	−0.543724	−0.0281530	−0.0140765	−	0.999901i	$-0.504481\pi$
$373$	−0.543724	−0.0281530	−0.0140765	+	0.999901i	$0.504481\pi$
$374$	4.56399	0.235998
$375$	0	0
$376$	1.56009	0.0804556
$377$	1.11198	0.0572698
$378$	0	0
$379$	−2.73521	−0.140498	−0.0702492	−	0.997529i	$-0.522379\pi$
$379$	−2.73521	−0.140498	−0.0702492	+	0.997529i	$0.522379\pi$
$380$	0	0
$381$	0	0
$382$	−20.2083	−1.03395
$383$	−34.2196	−1.74854	−0.874270	−	0.485440i	$-0.838659\pi$
$383$	−34.2196	−1.74854	−0.874270	+	0.485440i	$0.838659\pi$
$384$	0	0
$385$	5.14565	0.262247
$386$	−14.6305	−0.744672
$387$	0	0
$388$	17.3893	0.882806
$389$	−20.6645	−1.04773	−0.523867	−	0.851800i	$-0.675511\pi$
$389$	−20.6645	−1.04773	−0.523867	+	0.851800i	$0.675511\pi$
$390$	0	0
$391$	5.04024	0.254896
$392$	−6.27588	−0.316980
$393$	0	0
$394$	18.9555	0.954963
$395$	0.938698	0.0472310
$396$	0	0
$397$	−3.94846	−0.198167	−0.0990837	−	0.995079i	$-0.531591\pi$
$397$	−3.94846	−0.198167	−0.0990837	+	0.995079i	$0.531591\pi$
$398$	7.88063	0.395020
$399$	0	0
$400$	−1.62976	−0.0814878
$401$	−19.2120	−0.959402	−0.479701	−	0.877432i	$-0.659255\pi$
$401$	−19.2120	−0.959402	−0.479701	+	0.877432i	$0.659255\pi$
$402$	0	0
$403$	−3.36666	−0.167705
$404$	−6.01491	−0.299253
$405$	0	0
$406$	1.91632	0.0951052
$407$	15.5540	0.770986
$408$	0	0
$409$	−2.39954	−0.118650	−0.0593248	−	0.998239i	$-0.518895\pi$
$409$	−2.39954	−0.118650	−0.0593248	+	0.998239i	$0.518895\pi$
$410$	11.5211	0.568988
$411$	0	0
$412$	−0.165503	−0.00815376
$413$	−2.25501	−0.110962
$414$	0	0
$415$	−24.3293	−1.19428
$416$	−0.493780	−0.0242096
$417$	0	0
$418$	0	0
$419$	−29.0694	−1.42013	−0.710066	−	0.704135i	$-0.751335\pi$
$419$	−29.0694	−1.42013	−0.710066	+	0.704135i	$0.751335\pi$
$420$	0	0
$421$	21.3663	1.04133	0.520665	−	0.853761i	$-0.325684\pi$
$421$	21.3663	1.04133	0.520665	+	0.853761i	$0.325684\pi$
$422$	0.699545	0.0340533
$423$	0	0
$424$	−13.3069	−0.646238
$425$	−2.25820	−0.109539
$426$	0	0
$427$	−7.64133	−0.369790
$428$	12.8432	0.620800
$429$	0	0
$430$	−4.38141	−0.211290
$431$	−20.3463	−0.980048	−0.490024	−	0.871709i	$-0.663012\pi$
$431$	−20.3463	−0.980048	−0.490024	+	0.871709i	$0.663012\pi$
$432$	0	0
$433$	−34.7782	−1.67133	−0.835667	−	0.549236i	$-0.814919\pi$
$433$	−34.7782	−1.67133	−0.835667	+	0.549236i	$0.814919\pi$
$434$	−5.80190	−0.278500
$435$	0	0
$436$	−8.87200	−0.424892
$437$	0	0
$438$	0	0
$439$	21.5384	1.02797	0.513987	−	0.857798i	$-0.328168\pi$
$439$	21.5384	1.02797	0.513987	+	0.857798i	$0.328168\pi$
$440$	−6.04694	−0.288277
$441$	0	0
$442$	−0.684185	−0.0325433
$443$	15.1746	0.720969	0.360485	−	0.932765i	$-0.382611\pi$
$443$	15.1746	0.720969	0.360485	+	0.932765i	$0.382611\pi$
$444$	0	0
$445$	−14.2604	−0.676006
$446$	−15.1411	−0.716950
$447$	0	0
$448$	−0.850952	−0.0402037
$449$	−16.5316	−0.780175	−0.390088	−	0.920778i	$-0.627555\pi$
$449$	−16.5316	−0.780175	−0.390088	+	0.920778i	$0.627555\pi$
$450$	0	0
$451$	−20.6714	−0.973377
$452$	12.3830	0.582446
$453$	0	0
$454$	17.0328	0.799389
$455$	−0.771382	−0.0361629
$456$	0	0
$457$	0.605508	0.0283245	0.0141622	−	0.999900i	$-0.495492\pi$
$457$	0.605508	0.0283245	0.0141622	+	0.999900i	$0.495492\pi$
$458$	12.9202	0.603721
$459$	0	0
$460$	−6.67794	−0.311361
$461$	14.2973	0.665892	0.332946	−	0.942946i	$-0.391957\pi$
$461$	14.2973	0.665892	0.332946	+	0.942946i	$0.391957\pi$
$462$	0	0
$463$	−10.7275	−0.498548	−0.249274	−	0.968433i	$-0.580192\pi$
$463$	−10.7275	−0.498548	−0.249274	+	0.968433i	$0.580192\pi$
$464$	−2.25197	−0.104545
$465$	0	0
$466$	26.0558	1.20701
$467$	18.6617	0.863561	0.431780	−	0.901979i	$-0.357886\pi$
$467$	18.6617	0.863561	0.431780	+	0.901979i	$0.357886\pi$
$468$	0	0
$469$	7.28975	0.336610
$470$	−2.86405	−0.132109
$471$	0	0
$472$	2.64999	0.121976
$473$	7.86119	0.361458
$474$	0	0
$475$	0	0
$476$	−1.17908	−0.0540432
$477$	0	0
$478$	26.1157	1.19450
$479$	33.2187	1.51780	0.758900	−	0.651207i	$-0.225737\pi$
$479$	33.2187	1.51780	0.758900	+	0.651207i	$0.225737\pi$
$480$	0	0
$481$	−2.33170	−0.106316
$482$	4.92282	0.224228
$483$	0	0
$484$	−0.150495	−0.00684068
$485$	−31.9236	−1.44958
$486$	0	0
$487$	−30.6089	−1.38702	−0.693511	−	0.720446i	$-0.743937\pi$
$487$	−30.6089	−1.38702	−0.693511	+	0.720446i	$0.743937\pi$
$488$	8.97974	0.406494
$489$	0	0
$490$	11.5214	0.520484
$491$	4.18063	0.188669	0.0943347	−	0.995541i	$-0.469928\pi$
$491$	4.18063	0.188669	0.0943347	+	0.995541i	$0.469928\pi$
$492$	0	0
$493$	−3.12034	−0.140533
$494$	0	0
$495$	0	0
$496$	6.81813	0.306143
$497$	−9.19438	−0.412424
$498$	0	0
$499$	−13.0004	−0.581980	−0.290990	−	0.956726i	$-0.593985\pi$
$499$	−13.0004	−0.581980	−0.290990	+	0.956726i	$0.593985\pi$
$500$	12.1711	0.544306
$501$	0	0
$502$	−6.87784	−0.306973
$503$	10.8250	0.482664	0.241332	−	0.970443i	$-0.422416\pi$
$503$	10.8250	0.482664	0.241332	+	0.970443i	$0.422416\pi$
$504$	0	0
$505$	11.0423	0.491376
$506$	11.9817	0.532650
$507$	0	0
$508$	11.7803	0.522664
$509$	11.9879	0.531353	0.265676	−	0.964062i	$-0.414405\pi$
$509$	11.9879	0.531353	0.265676	+	0.964062i	$0.414405\pi$
$510$	0	0
$511$	3.48692	0.154252
$512$	1.00000	0.0441942
$513$	0	0
$514$	−1.85002	−0.0816009
$515$	0.303834	0.0133885
$516$	0	0
$517$	5.13872	0.226001
$518$	−4.01831	−0.176554
$519$	0	0
$520$	0.906493	0.0397524
$521$	39.5844	1.73422	0.867112	−	0.498114i	$-0.165974\pi$
$521$	39.5844	1.73422	0.867112	+	0.498114i	$0.165974\pi$
$522$	0	0
$523$	13.6366	0.596286	0.298143	−	0.954521i	$-0.403633\pi$
$523$	13.6366	0.596286	0.298143	+	0.954521i	$0.403633\pi$
$524$	18.7991	0.821243
$525$	0	0
$526$	−9.28510	−0.404850
$527$	9.44723	0.411528
$528$	0	0
$529$	−9.76805	−0.424698
$530$	24.4290	1.06113
$531$	0	0
$532$	0	0
$533$	3.09883	0.134225
$534$	0	0
$535$	−23.5779	−1.01936
$536$	−8.56659	−0.370020
$537$	0	0
$538$	−21.9007	−0.944204
$539$	−20.6719	−0.890400
$540$	0	0
$541$	−43.9351	−1.88892	−0.944458	−	0.328631i	$-0.893413\pi$
$541$	−43.9351	−1.88892	−0.944458	+	0.328631i	$0.893413\pi$
$542$	5.51949	0.237083
$543$	0	0
$544$	1.38560	0.0594073
$545$	16.2874	0.697676
$546$	0	0
$547$	−5.60273	−0.239555	−0.119778	−	0.992801i	$-0.538218\pi$
$547$	−5.60273	−0.239555	−0.119778	+	0.992801i	$0.538218\pi$
$548$	13.5312	0.578024
$549$	0	0
$550$	−5.36819	−0.228900
$551$	0	0
$552$	0	0
$553$	0.435111	0.0185028
$554$	32.4896	1.38035
$555$	0	0
$556$	18.9445	0.803427
$557$	−37.4888	−1.58845	−0.794225	−	0.607623i	$-0.792123\pi$
$557$	−37.4888	−1.58845	−0.794225	+	0.607623i	$0.792123\pi$
$558$	0	0
$559$	−1.17847	−0.0498438
$560$	1.56220	0.0660148
$561$	0	0
$562$	−27.6408	−1.16596
$563$	1.57318	0.0663015	0.0331507	−	0.999450i	$-0.489446\pi$
$563$	1.57318	0.0663015	0.0331507	+	0.999450i	$0.489446\pi$
$564$	0	0
$565$	−22.7329	−0.956381
$566$	−11.1914	−0.470411
$567$	0	0
$568$	10.8048	0.453360
$569$	−23.7454	−0.995457	−0.497729	−	0.867333i	$-0.665832\pi$
$569$	−23.7454	−0.995457	−0.497729	+	0.867333i	$0.665832\pi$
$570$	0	0
$571$	−17.0880	−0.715110	−0.357555	−	0.933892i	$-0.616389\pi$
$571$	−17.0880	−0.715110	−0.357555	+	0.933892i	$0.616389\pi$
$572$	−1.62644	−0.0680050
$573$	0	0
$574$	5.34034	0.222902
$575$	−5.92836	−0.247230
$576$	0	0
$577$	18.0438	0.751175	0.375587	−	0.926787i	$-0.377441\pi$
$577$	18.0438	0.751175	0.375587	+	0.926787i	$0.377441\pi$
$578$	−15.0801	−0.627249
$579$	0	0
$580$	4.13422	0.171664
$581$	−11.2773	−0.467860
$582$	0	0
$583$	−43.8309	−1.81529
$584$	−4.09767	−0.169563
$585$	0	0
$586$	−12.6060	−0.520749
$587$	1.57749	0.0651100	0.0325550	−	0.999470i	$-0.489636\pi$
$587$	1.57749	0.0651100	0.0325550	+	0.999470i	$0.489636\pi$
$588$	0	0
$589$	0	0
$590$	−4.86491	−0.200285
$591$	0	0
$592$	4.72214	0.194079
$593$	−2.31500	−0.0950657	−0.0475329	−	0.998870i	$-0.515136\pi$
$593$	−2.31500	−0.0950657	−0.0475329	+	0.998870i	$0.515136\pi$
$594$	0	0
$595$	2.16459	0.0887394
$596$	15.1935	0.622350
$597$	0	0
$598$	−1.79616	−0.0734506
$599$	4.61960	0.188752	0.0943758	−	0.995537i	$-0.469914\pi$
$599$	4.61960	0.188752	0.0943758	+	0.995537i	$0.469914\pi$
$600$	0	0
$601$	37.2151	1.51804	0.759019	−	0.651069i	$-0.225679\pi$
$601$	37.2151	1.51804	0.759019	+	0.651069i	$0.225679\pi$
$602$	−2.03090	−0.0827733
$603$	0	0
$604$	−3.70848	−0.150896
$605$	0.276282	0.0112325
$606$	0	0
$607$	−38.7882	−1.57437	−0.787183	−	0.616720i	$-0.788461\pi$
$607$	−38.7882	−1.57437	−0.787183	+	0.616720i	$0.788461\pi$
$608$	0	0
$609$	0	0
$610$	−16.4852	−0.667467
$611$	−0.770343	−0.0311647
$612$	0	0
$613$	−33.0790	−1.33605	−0.668024	−	0.744139i	$-0.732860\pi$
$613$	−33.0790	−1.33605	−0.668024	+	0.744139i	$0.732860\pi$
$614$	−29.7399	−1.20021
$615$	0	0
$616$	−2.80291	−0.112933
$617$	−1.71327	−0.0689737	−0.0344869	−	0.999405i	$-0.510980\pi$
$617$	−1.71327	−0.0689737	−0.0344869	+	0.999405i	$0.510980\pi$
$618$	0	0
$619$	28.0387	1.12697	0.563486	−	0.826126i	$-0.309460\pi$
$619$	28.0387	1.12697	0.563486	+	0.826126i	$0.309460\pi$
$620$	−12.5169	−0.502690
$621$	0	0
$622$	23.5772	0.945359
$623$	−6.61005	−0.264826
$624$	0	0
$625$	−14.1951	−0.567805
$626$	22.7907	0.910900
$627$	0	0
$628$	−15.7587	−0.628842
$629$	6.54301	0.260887
$630$	0	0
$631$	6.56631	0.261401	0.130700	−	0.991422i	$-0.458277\pi$
$631$	6.56631	0.261401	0.130700	+	0.991422i	$0.458277\pi$
$632$	−0.511323	−0.0203393
$633$	0	0
$634$	12.7514	0.506421
$635$	−21.6264	−0.858219
$636$	0	0
$637$	3.09891	0.122783
$638$	−7.41767	−0.293668
$639$	0	0
$640$	−1.83582	−0.0725673
$641$	12.0337	0.475304	0.237652	−	0.971350i	$-0.423622\pi$
$641$	12.0337	0.475304	0.237652	+	0.971350i	$0.423622\pi$
$642$	0	0
$643$	−29.2059	−1.15177	−0.575883	−	0.817532i	$-0.695342\pi$
$643$	−29.2059	−1.15177	−0.575883	+	0.817532i	$0.695342\pi$
$644$	−3.09540	−0.121976
$645$	0	0
$646$	0	0
$647$	8.90424	0.350062	0.175031	−	0.984563i	$-0.443997\pi$
$647$	8.90424	0.350062	0.175031	+	0.984563i	$0.443997\pi$
$648$	0	0
$649$	8.72869	0.342631
$650$	0.804742	0.0315646
$651$	0	0
$652$	−8.39467	−0.328761
$653$	30.9043	1.20938	0.604690	−	0.796461i	$-0.293297\pi$
$653$	30.9043	1.20938	0.604690	+	0.796461i	$0.293297\pi$
$654$	0	0
$655$	−34.5118	−1.34849
$656$	−6.27573	−0.245026
$657$	0	0
$658$	−1.32756	−0.0517538
$659$	7.35811	0.286631	0.143316	−	0.989677i	$-0.454224\pi$
$659$	7.35811	0.286631	0.143316	+	0.989677i	$0.454224\pi$
$660$	0	0
$661$	−40.0889	−1.55928	−0.779639	−	0.626229i	$-0.784598\pi$
$661$	−40.0889	−1.55928	−0.779639	+	0.626229i	$0.784598\pi$
$662$	−3.16768	−0.123116
$663$	0	0
$664$	13.2526	0.514299
$665$	0	0
$666$	0	0
$667$	−8.19171	−0.317184
$668$	−3.84036	−0.148588
$669$	0	0
$670$	15.7267	0.607577
$671$	29.5780	1.14185
$672$	0	0
$673$	−34.5259	−1.33088	−0.665438	−	0.746453i	$-0.731755\pi$
$673$	−34.5259	−1.33088	−0.665438	+	0.746453i	$0.731755\pi$
$674$	9.87946	0.380543
$675$	0	0
$676$	−12.7562	−0.490622
$677$	−4.58550	−0.176235	−0.0881176	−	0.996110i	$-0.528085\pi$
$677$	−4.58550	−0.176235	−0.0881176	+	0.996110i	$0.528085\pi$
$678$	0	0
$679$	−14.7974	−0.567873
$680$	−2.54372	−0.0975474
$681$	0	0
$682$	22.4580	0.859959
$683$	1.71380	0.0655766	0.0327883	−	0.999462i	$-0.489561\pi$
$683$	1.71380	0.0655766	0.0327883	+	0.999462i	$0.489561\pi$
$684$	0	0
$685$	−24.8409	−0.949121
$686$	11.2971	0.431327
$687$	0	0
$688$	2.38662	0.0909891
$689$	6.57066	0.250322
$690$	0	0
$691$	−39.4163	−1.49947	−0.749734	−	0.661740i	$-0.769818\pi$
$691$	−39.4163	−1.49947	−0.749734	+	0.661740i	$0.769818\pi$
$692$	4.84285	0.184097
$693$	0	0
$694$	14.4794	0.549632
$695$	−34.7788	−1.31923
$696$	0	0
$697$	−8.69568	−0.329372
$698$	3.60760	0.136550
$699$	0	0
$700$	1.38684	0.0524178
$701$	−12.9098	−0.487597	−0.243799	−	0.969826i	$-0.578394\pi$
$701$	−12.9098	−0.487597	−0.243799	+	0.969826i	$0.578394\pi$
$702$	0	0
$703$	0	0
$704$	3.29386	0.124142
$705$	0	0
$706$	4.02595	0.151519
$707$	5.11839	0.192497
$708$	0	0
$709$	5.27461	0.198092	0.0990461	−	0.995083i	$-0.468421\pi$
$709$	5.27461	0.198092	0.0990461	+	0.995083i	$0.468421\pi$
$710$	−19.8357	−0.744422
$711$	0	0
$712$	7.76784	0.291112
$713$	24.8015	0.928822
$714$	0	0
$715$	2.98586	0.111665
$716$	24.5771	0.918491
$717$	0	0
$718$	−4.95939	−0.185083
$719$	6.36492	0.237371	0.118686	−	0.992932i	$-0.462132\pi$
$719$	6.36492	0.237371	0.118686	+	0.992932i	$0.462132\pi$
$720$	0	0
$721$	0.140835	0.00524498
$722$	0	0
$723$	0	0
$724$	−12.1200	−0.450434
$725$	3.67016	0.136306
$726$	0	0
$727$	43.0363	1.59613	0.798064	−	0.602572i	$-0.205857\pi$
$727$	43.0363	1.59613	0.798064	+	0.602572i	$0.205857\pi$
$728$	0.420183	0.0155730
$729$	0	0
$730$	7.52259	0.278424
$731$	3.30691	0.122311
$732$	0	0
$733$	36.5209	1.34893	0.674464	−	0.738307i	$-0.264375\pi$
$733$	36.5209	1.34893	0.674464	+	0.738307i	$0.264375\pi$
$734$	24.8168	0.916006
$735$	0	0
$736$	3.63757	0.134083
$737$	−28.2171	−1.03939
$738$	0	0
$739$	32.7826	1.20593	0.602964	−	0.797769i	$-0.293986\pi$
$739$	32.7826	1.20593	0.602964	+	0.797769i	$0.293986\pi$
$740$	−8.66900	−0.318679
$741$	0	0
$742$	11.3235	0.415698
$743$	−39.1320	−1.43561	−0.717807	−	0.696242i	$-0.754854\pi$
$743$	−39.1320	−1.43561	−0.717807	+	0.696242i	$0.754854\pi$
$744$	0	0
$745$	−27.8926	−1.02190
$746$	−0.543724	−0.0199072
$747$	0	0
$748$	4.56399	0.166876
$749$	−10.9290	−0.399335
$750$	0	0
$751$	−25.9562	−0.947155	−0.473577	−	0.880752i	$-0.657038\pi$
$751$	−25.9562	−0.947155	−0.473577	+	0.880752i	$0.657038\pi$
$752$	1.56009	0.0568907
$753$	0	0
$754$	1.11198	0.0404959
$755$	6.80812	0.247773
$756$	0	0
$757$	11.1516	0.405312	0.202656	−	0.979250i	$-0.435043\pi$
$757$	11.1516	0.405312	0.202656	+	0.979250i	$0.435043\pi$
$758$	−2.73521	−0.0993474
$759$	0	0
$760$	0	0
$761$	42.5374	1.54198	0.770991	−	0.636846i	$-0.219761\pi$
$761$	42.5374	1.54198	0.770991	+	0.636846i	$0.219761\pi$
$762$	0	0
$763$	7.54964	0.273315
$764$	−20.2083	−0.731112
$765$	0	0
$766$	−34.2196	−1.23640
$767$	−1.30851	−0.0472476
$768$	0	0
$769$	3.16667	0.114193	0.0570966	−	0.998369i	$-0.481816\pi$
$769$	3.16667	0.114193	0.0570966	+	0.998369i	$0.481816\pi$
$770$	5.14565	0.185436
$771$	0	0
$772$	−14.6305	−0.526563
$773$	−33.9016	−1.21936	−0.609678	−	0.792649i	$-0.708701\pi$
$773$	−33.9016	−1.21936	−0.609678	+	0.792649i	$0.708701\pi$
$774$	0	0
$775$	−11.1119	−0.399151
$776$	17.3893	0.624238
$777$	0	0
$778$	−20.6645	−0.740859
$779$	0	0
$780$	0	0
$781$	35.5896	1.27349
$782$	5.04024	0.180239
$783$	0	0
$784$	−6.27588	−0.224139
$785$	28.9303	1.03257
$786$	0	0
$787$	16.5243	0.589027	0.294513	−	0.955647i	$-0.404842\pi$
$787$	16.5243	0.589027	0.294513	+	0.955647i	$0.404842\pi$
$788$	18.9555	0.675261
$789$	0	0
$790$	0.938698	0.0333974
$791$	−10.5373	−0.374663
$792$	0	0
$793$	−4.43402	−0.157457
$794$	−3.94846	−0.140125
$795$	0	0
$796$	7.88063	0.279322
$797$	−44.2751	−1.56830	−0.784152	−	0.620569i	$-0.786902\pi$
$797$	−44.2751	−1.56830	−0.784152	+	0.620569i	$0.786902\pi$
$798$	0	0
$799$	2.16167	0.0764744
$800$	−1.62976	−0.0576206
$801$	0	0
$802$	−19.2120	−0.678399
$803$	−13.4971	−0.476304
$804$	0	0
$805$	5.68260	0.200286
$806$	−3.36666	−0.118585
$807$	0	0
$808$	−6.01491	−0.211604
$809$	0.878874	0.0308996	0.0154498	−	0.999881i	$-0.495082\pi$
$809$	0.878874	0.0308996	0.0154498	+	0.999881i	$0.495082\pi$
$810$	0	0
$811$	−22.6310	−0.794682	−0.397341	−	0.917671i	$-0.630067\pi$
$811$	−22.6310	−0.794682	−0.397341	+	0.917671i	$0.630067\pi$
$812$	1.91632	0.0672496
$813$	0	0
$814$	15.5540	0.545169
$815$	15.4111	0.539828
$816$	0	0
$817$	0	0
$818$	−2.39954	−0.0838980
$819$	0	0
$820$	11.5211	0.402335
$821$	−34.7357	−1.21228	−0.606142	−	0.795356i	$-0.707284\pi$
$821$	−34.7357	−1.21228	−0.606142	+	0.795356i	$0.707284\pi$
$822$	0	0
$823$	−1.95628	−0.0681917	−0.0340959	−	0.999419i	$-0.510855\pi$
$823$	−1.95628	−0.0681917	−0.0340959	+	0.999419i	$0.510855\pi$
$824$	−0.165503	−0.00576558
$825$	0	0
$826$	−2.25501	−0.0784619
$827$	6.91826	0.240571	0.120286	−	0.992739i	$-0.461619\pi$
$827$	6.91826	0.240571	0.120286	+	0.992739i	$0.461619\pi$
$828$	0	0
$829$	12.2863	0.426722	0.213361	−	0.976973i	$-0.431559\pi$
$829$	12.2863	0.426722	0.213361	+	0.976973i	$0.431559\pi$
$830$	−24.3293	−0.844483
$831$	0	0
$832$	−0.493780	−0.0171188
$833$	−8.69589	−0.301295
$834$	0	0
$835$	7.05022	0.243983
$836$	0	0
$837$	0	0
$838$	−29.0694	−1.00419
$839$	15.4977	0.535039	0.267520	−	0.963552i	$-0.413796\pi$
$839$	15.4977	0.535039	0.267520	+	0.963552i	$0.413796\pi$
$840$	0	0
$841$	−23.9286	−0.825125
$842$	21.3663	0.736331
$843$	0	0
$844$	0.699545	0.0240793
$845$	23.4181	0.805607
$846$	0	0
$847$	0.128064	0.00440033
$848$	−13.3069	−0.456959
$849$	0	0
$850$	−2.25820	−0.0774555
$851$	17.1771	0.588824
$852$	0	0
$853$	22.0678	0.755587	0.377793	−	0.925890i	$-0.376683\pi$
$853$	22.0678	0.755587	0.377793	+	0.925890i	$0.376683\pi$
$854$	−7.64133	−0.261481
$855$	0	0
$856$	12.8432	0.438972
$857$	27.2714	0.931574	0.465787	−	0.884897i	$-0.345771\pi$
$857$	27.2714	0.931574	0.465787	+	0.884897i	$0.345771\pi$
$858$	0	0
$859$	25.2049	0.859979	0.429989	−	0.902834i	$-0.358517\pi$
$859$	25.2049	0.859979	0.429989	+	0.902834i	$0.358517\pi$
$860$	−4.38141	−0.149405
$861$	0	0
$862$	−20.3463	−0.692999
$863$	27.7553	0.944800	0.472400	−	0.881384i	$-0.343388\pi$
$863$	27.7553	0.944800	0.472400	+	0.881384i	$0.343388\pi$
$864$	0	0
$865$	−8.89061	−0.302290
$866$	−34.7782	−1.18181
$867$	0	0
$868$	−5.80190	−0.196929
$869$	−1.68423	−0.0571334
$870$	0	0
$871$	4.23001	0.143329
$872$	−8.87200	−0.300444
$873$	0	0
$874$	0	0
$875$	−10.3570	−0.350130
$876$	0	0
$877$	−12.5360	−0.423311	−0.211655	−	0.977344i	$-0.567885\pi$
$877$	−12.5360	−0.423311	−0.211655	+	0.977344i	$0.567885\pi$
$878$	21.5384	0.726887
$879$	0	0
$880$	−6.04694	−0.203842
$881$	−56.4989	−1.90350	−0.951748	−	0.306881i	$-0.900715\pi$
$881$	−56.4989	−1.90350	−0.951748	+	0.306881i	$0.900715\pi$
$882$	0	0
$883$	18.4427	0.620645	0.310322	−	0.950631i	$-0.399563\pi$
$883$	18.4427	0.620645	0.310322	+	0.950631i	$0.399563\pi$
$884$	−0.684185	−0.0230116
$885$	0	0
$886$	15.1746	0.509802
$887$	−12.3548	−0.414834	−0.207417	−	0.978253i	$-0.566506\pi$
$887$	−12.3548	−0.414834	−0.207417	+	0.978253i	$0.566506\pi$
$888$	0	0
$889$	−10.0244	−0.336208
$890$	−14.2604	−0.478009
$891$	0	0
$892$	−15.1411	−0.506960
$893$	0	0
$894$	0	0
$895$	−45.1192	−1.50817
$896$	−0.850952	−0.0284283
$897$	0	0
$898$	−16.5316	−0.551667
$899$	−15.3542	−0.512092
$900$	0	0
$901$	−18.4380	−0.614260
$902$	−20.6714	−0.688281
$903$	0	0
$904$	12.3830	0.411851
$905$	22.2501	0.739618
$906$	0	0
$907$	9.09823	0.302102	0.151051	−	0.988526i	$-0.451734\pi$
$907$	9.09823	0.302102	0.151051	+	0.988526i	$0.451734\pi$
$908$	17.0328	0.565253
$909$	0	0
$910$	−0.771382	−0.0255711
$911$	5.50145	0.182271	0.0911355	−	0.995839i	$-0.470950\pi$
$911$	5.50145	0.182271	0.0911355	+	0.995839i	$0.470950\pi$
$912$	0	0
$913$	43.6520	1.44467
$914$	0.605508	0.0200284
$915$	0	0
$916$	12.9202	0.426895
$917$	−15.9971	−0.528272
$918$	0	0
$919$	−46.4390	−1.53188	−0.765941	−	0.642911i	$-0.777727\pi$
$919$	−46.4390	−1.53188	−0.765941	+	0.642911i	$0.777727\pi$
$920$	−6.67794	−0.220165
$921$	0	0
$922$	14.2973	0.470856
$923$	−5.33521	−0.175611
$924$	0	0
$925$	−7.69593	−0.253041
$926$	−10.7275	−0.352527
$927$	0	0
$928$	−2.25197	−0.0739245
$929$	−8.58255	−0.281584	−0.140792	−	0.990039i	$-0.544965\pi$
$929$	−8.58255	−0.281584	−0.140792	+	0.990039i	$0.544965\pi$
$930$	0	0
$931$	0	0
$932$	26.0558	0.853485
$933$	0	0
$934$	18.6617	0.610630
$935$	−8.37867	−0.274012
$936$	0	0
$937$	1.54442	0.0504540	0.0252270	−	0.999682i	$-0.491969\pi$
$937$	1.54442	0.0504540	0.0252270	+	0.999682i	$0.491969\pi$
$938$	7.28975	0.238019
$939$	0	0
$940$	−2.86405	−0.0934151
$941$	−19.5302	−0.636667	−0.318333	−	0.947979i	$-0.603123\pi$
$941$	−19.5302	−0.636667	−0.318333	+	0.947979i	$0.603123\pi$
$942$	0	0
$943$	−22.8284	−0.743396
$944$	2.64999	0.0862498
$945$	0	0
$946$	7.86119	0.255589
$947$	6.11227	0.198622	0.0993110	−	0.995056i	$-0.468336\pi$
$947$	6.11227	0.198622	0.0993110	+	0.995056i	$0.468336\pi$
$948$	0	0
$949$	2.02335	0.0656806
$950$	0	0
$951$	0	0
$952$	−1.17908	−0.0382143
$953$	40.1381	1.30020	0.650101	−	0.759848i	$-0.274727\pi$
$953$	40.1381	1.30020	0.650101	+	0.759848i	$0.274727\pi$
$954$	0	0
$955$	37.0989	1.20049
$956$	26.1157	0.844641
$957$	0	0
$958$	33.2187	1.07325
$959$	−11.5144	−0.371819
$960$	0	0
$961$	15.4869	0.499577
$962$	−2.33170	−0.0751770
$963$	0	0
$964$	4.92282	0.158553
$965$	26.8590	0.864621
$966$	0	0
$967$	−24.7369	−0.795484	−0.397742	−	0.917497i	$-0.630206\pi$
$967$	−24.7369	−0.795484	−0.397742	+	0.917497i	$0.630206\pi$
$968$	−0.150495	−0.00483709
$969$	0	0
$970$	−31.9236	−1.02500
$971$	41.6540	1.33674	0.668370	−	0.743829i	$-0.266992\pi$
$971$	41.6540	1.33674	0.668370	+	0.743829i	$0.266992\pi$
$972$	0	0
$973$	−16.1209	−0.516812
$974$	−30.6089	−0.980772
$975$	0	0
$976$	8.97974	0.287435
$977$	−37.1582	−1.18880	−0.594398	−	0.804171i	$-0.702609\pi$
$977$	−37.1582	−1.18880	−0.594398	+	0.804171i	$0.702609\pi$
$978$	0	0
$979$	25.5862	0.817737
$980$	11.5214	0.368038
$981$	0	0
$982$	4.18063	0.133409
$983$	−33.2236	−1.05967	−0.529834	−	0.848101i	$-0.677746\pi$
$983$	−33.2236	−1.05967	−0.529834	+	0.848101i	$0.677746\pi$
$984$	0	0
$985$	−34.7989	−1.10878
$986$	−3.12034	−0.0993719
$987$	0	0
$988$	0	0
$989$	8.68151	0.276056
$990$	0	0
$991$	−31.2160	−0.991609	−0.495804	−	0.868434i	$-0.665127\pi$
$991$	−31.2160	−0.991609	−0.495804	+	0.868434i	$0.665127\pi$
$992$	6.81813	0.216476
$993$	0	0
$994$	−9.19438	−0.291628
$995$	−14.4674	−0.458649
$996$	0	0
$997$	11.0798	0.350899	0.175450	−	0.984488i	$-0.443862\pi$
$997$	11.0798	0.350899	0.175450	+	0.984488i	$0.443862\pi$
$998$	−13.0004	−0.411522
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.ce.1.2		6
3.2	odd	2		6498.2.a.cb.1.5		6
19.4	even	9		342.2.u.f.73.1	✓	12
19.5	even	9		342.2.u.f.253.1	yes	12
19.18	odd	2		6498.2.a.cc.1.2		6
57.5	odd	18		342.2.u.g.253.2	yes	12
57.23	odd	18		342.2.u.g.73.2	yes	12
57.56	even	2		6498.2.a.cd.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.u.f.73.1	✓	12	19.4	even	9
342.2.u.f.253.1	yes	12	19.5	even	9
342.2.u.g.73.2	yes	12	57.23	odd	18
342.2.u.g.253.2	yes	12	57.5	odd	18
6498.2.a.cb.1.5		6	3.2	odd	2
6498.2.a.cc.1.2		6	19.18	odd	2
6498.2.a.cd.1.5		6	57.56	even	2
6498.2.a.ce.1.2		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.83582 q^{5} -0.850952 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.83582 q^{5} -0.850952 q^{7} +1.00000 q^{8} -1.83582 q^{10} +3.29386 q^{11} -0.493780 q^{13} -0.850952 q^{14} +1.00000 q^{16} +1.38560 q^{17} -1.83582 q^{20} +3.29386 q^{22} +3.63757 q^{23} -1.62976 q^{25} -0.493780 q^{26} -0.850952 q^{28} -2.25197 q^{29} +6.81813 q^{31} +1.00000 q^{32} +1.38560 q^{34} +1.56220 q^{35} +4.72214 q^{37} -1.83582 q^{40} -6.27573 q^{41} +2.38662 q^{43} +3.29386 q^{44} +3.63757 q^{46} +1.56009 q^{47} -6.27588 q^{49} -1.62976 q^{50} -0.493780 q^{52} -13.3069 q^{53} -6.04694 q^{55} -0.850952 q^{56} -2.25197 q^{58} +2.64999 q^{59} +8.97974 q^{61} +6.81813 q^{62} +1.00000 q^{64} +0.906493 q^{65} -8.56659 q^{67} +1.38560 q^{68} +1.56220 q^{70} +10.8048 q^{71} -4.09767 q^{73} +4.72214 q^{74} -2.80291 q^{77} -0.511323 q^{79} -1.83582 q^{80} -6.27573 q^{82} +13.2526 q^{83} -2.54372 q^{85} +2.38662 q^{86} +3.29386 q^{88} +7.76784 q^{89} +0.420183 q^{91} +3.63757 q^{92} +1.56009 q^{94} +17.3893 q^{97} -6.27588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 6 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} + 6 q^{11} + 12 q^{13} + 6 q^{14} + 6 q^{16} + 12 q^{17} + 6 q^{20} + 6 q^{22} + 18 q^{23} + 18 q^{25} + 12 q^{26} + 6 q^{28} - 6 q^{29} - 6 q^{31} + 6 q^{32} + 12 q^{34} + 12 q^{35} + 12 q^{37} + 6 q^{40} - 18 q^{43} + 6 q^{44} + 18 q^{46} + 30 q^{47} + 18 q^{50} + 12 q^{52} - 18 q^{53} - 6 q^{55} + 6 q^{56} - 6 q^{58} - 6 q^{59} - 6 q^{62} + 6 q^{64} - 12 q^{65} + 18 q^{67} + 12 q^{68} + 12 q^{70} + 24 q^{71} - 6 q^{73} + 12 q^{74} + 30 q^{77} - 18 q^{79} + 6 q^{80} + 42 q^{83} - 12 q^{85} - 18 q^{86} + 6 q^{88} - 18 q^{89} + 18 q^{91} + 18 q^{92} + 30 q^{94} + 24 q^{97}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 + 6 * q^11 + 12 * q^13 + 6 * q^14 + 6 * q^16 + 12 * q^17 + 6 * q^20 + 6 * q^22 + 18 * q^23 + 18 * q^25 + 12 * q^26 + 6 * q^28 - 6 * q^29 - 6 * q^31 + 6 * q^32 + 12 * q^34 + 12 * q^35 + 12 * q^37 + 6 * q^40 - 18 * q^43 + 6 * q^44 + 18 * q^46 + 30 * q^47 + 18 * q^50 + 12 * q^52 - 18 * q^53 - 6 * q^55 + 6 * q^56 - 6 * q^58 - 6 * q^59 - 6 * q^62 + 6 * q^64 - 12 * q^65 + 18 * q^67 + 12 * q^68 + 12 * q^70 + 24 * q^71 - 6 * q^73 + 12 * q^74 + 30 * q^77 - 18 * q^79 + 6 * q^80 + 42 * q^83 - 12 * q^85 - 18 * q^86 + 6 * q^88 - 18 * q^89 + 18 * q^91 + 18 * q^92 + 30 * q^94 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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