LMFDB - Embedded newform 6498.2.a.bt.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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.65270 q^{5} +0.184793 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.65270 q^{5} +0.184793 q^{7} +1.00000 q^{8} +1.65270 q^{10} +4.34730 q^{11} +6.47565 q^{13} +0.184793 q^{14} +1.00000 q^{16} +2.12061 q^{17} +1.65270 q^{20} +4.34730 q^{22} -0.106067 q^{23} -2.26857 q^{25} +6.47565 q^{26} +0.184793 q^{28} +3.98545 q^{29} +3.22668 q^{31} +1.00000 q^{32} +2.12061 q^{34} +0.305407 q^{35} -4.06418 q^{37} +1.65270 q^{40} -8.63816 q^{41} +0.0418891 q^{43} +4.34730 q^{44} -0.106067 q^{46} +7.92127 q^{47} -6.96585 q^{49} -2.26857 q^{50} +6.47565 q^{52} -9.21213 q^{53} +7.18479 q^{55} +0.184793 q^{56} +3.98545 q^{58} +10.0915 q^{59} +3.59627 q^{61} +3.22668 q^{62} +1.00000 q^{64} +10.7023 q^{65} +7.98545 q^{67} +2.12061 q^{68} +0.305407 q^{70} -4.04189 q^{71} +15.1976 q^{73} -4.06418 q^{74} +0.803348 q^{77} -12.3969 q^{79} +1.65270 q^{80} -8.63816 q^{82} -8.45605 q^{83} +3.50475 q^{85} +0.0418891 q^{86} +4.34730 q^{88} -17.9368 q^{89} +1.19665 q^{91} -0.106067 q^{92} +7.92127 q^{94} -10.6878 q^{97} -6.96585 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{11} - 3 q^{14} + 3 q^{16} + 12 q^{17} + 6 q^{20} + 12 q^{22} + 12 q^{23} + 3 q^{25} - 3 q^{28} - 6 q^{29} + 3 q^{31} + 3 q^{32} + 12 q^{34} + 3 q^{35} - 3 q^{37} + 6 q^{40} - 9 q^{41} - 3 q^{43} + 12 q^{44} + 12 q^{46} + 15 q^{47} + 3 q^{50} - 3 q^{53} + 18 q^{55} - 3 q^{56} - 6 q^{58} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 6 q^{65} + 6 q^{67} + 12 q^{68} + 3 q^{70} - 9 q^{71} + 3 q^{73} - 3 q^{74} - 21 q^{77} - 9 q^{79} + 6 q^{80} - 9 q^{82} - 3 q^{83} + 27 q^{85} - 3 q^{86} + 12 q^{88} + 3 q^{89} + 27 q^{91} + 12 q^{92} + 15 q^{94} + 12 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8 + 6 * q^10 + 12 * q^11 - 3 * q^14 + 3 * q^16 + 12 * q^17 + 6 * q^20 + 12 * q^22 + 12 * q^23 + 3 * q^25 - 3 * q^28 - 6 * q^29 + 3 * q^31 + 3 * q^32 + 12 * q^34 + 3 * q^35 - 3 * q^37 + 6 * q^40 - 9 * q^41 - 3 * q^43 + 12 * q^44 + 12 * q^46 + 15 * q^47 + 3 * q^50 - 3 * q^53 + 18 * q^55 - 3 * q^56 - 6 * q^58 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 + 6 * q^67 + 12 * q^68 + 3 * q^70 - 9 * q^71 + 3 * q^73 - 3 * q^74 - 21 * q^77 - 9 * q^79 + 6 * q^80 - 9 * q^82 - 3 * q^83 + 27 * q^85 - 3 * q^86 + 12 * q^88 + 3 * q^89 + 27 * q^91 + 12 * q^92 + 15 * q^94 + 12 * 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
$5$	1.65270	0.739112	0.369556	−	0.929209i	$-0.379510\pi$
$5$	1.65270	0.739112	0.369556	+	0.929209i	$0.379510\pi$
$6$	0	0
$7$	0.184793	0.0698450	0.0349225	−	0.999390i	$-0.488882\pi$
$7$	0.184793	0.0698450	0.0349225	+	0.999390i	$0.488882\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.65270	0.522631
$11$	4.34730	1.31076	0.655380	−	0.755300i	$-0.272509\pi$
$11$	4.34730	1.31076	0.655380	+	0.755300i	$0.272509\pi$
$12$	0	0
$13$	6.47565	1.79602	0.898011	−	0.439972i	$-0.145012\pi$
$13$	6.47565	1.79602	0.898011	+	0.439972i	$0.145012\pi$
$14$	0.184793	0.0493879
$15$	0	0
$16$	1.00000	0.250000
$17$	2.12061	0.514325	0.257162	−	0.966368i	$-0.417212\pi$
$17$	2.12061	0.514325	0.257162	+	0.966368i	$0.417212\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	1.65270	0.369556
$21$	0	0
$22$	4.34730	0.926847
$23$	−0.106067	−0.0221165	−0.0110582	−	0.999939i	$-0.503520\pi$
$23$	−0.106067	−0.0221165	−0.0110582	+	0.999939i	$0.503520\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	6.47565	1.26998
$27$	0	0
$28$	0.184793	0.0349225
$29$	3.98545	0.740080	0.370040	−	0.929016i	$-0.379344\pi$
$29$	3.98545	0.740080	0.370040	+	0.929016i	$0.379344\pi$
$30$	0	0
$31$	3.22668	0.579529	0.289765	−	0.957098i	$-0.406423\pi$
$31$	3.22668	0.579529	0.289765	+	0.957098i	$0.406423\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.12061	0.363682
$35$	0.305407	0.0516233
$36$	0	0
$37$	−4.06418	−0.668147	−0.334073	−	0.942547i	$-0.608423\pi$
$37$	−4.06418	−0.668147	−0.334073	+	0.942547i	$0.608423\pi$
$38$	0	0
$39$	0	0
$40$	1.65270	0.261315
$41$	−8.63816	−1.34905	−0.674527	−	0.738251i	$-0.735652\pi$
$41$	−8.63816	−1.34905	−0.674527	+	0.738251i	$0.735652\pi$
$42$	0	0
$43$	0.0418891	0.00638802	0.00319401	−	0.999995i	$-0.498983\pi$
$43$	0.0418891	0.00638802	0.00319401	+	0.999995i	$0.498983\pi$
$44$	4.34730	0.655380
$45$	0	0
$46$	−0.106067	−0.0156387
$47$	7.92127	1.15544	0.577718	−	0.816236i	$-0.303943\pi$
$47$	7.92127	1.15544	0.577718	+	0.816236i	$0.303943\pi$
$48$	0	0
$49$	−6.96585	−0.995122
$50$	−2.26857	−0.320824
$51$	0	0
$52$	6.47565	0.898011
$53$	−9.21213	−1.26538	−0.632692	−	0.774404i	$-0.718050\pi$
$53$	−9.21213	−1.26538	−0.632692	+	0.774404i	$0.718050\pi$
$54$	0	0
$55$	7.18479	0.968797
$56$	0.184793	0.0246939
$57$	0	0
$58$	3.98545	0.523315
$59$	10.0915	1.31380	0.656902	−	0.753976i	$-0.271867\pi$
$59$	10.0915	1.31380	0.656902	+	0.753976i	$0.271867\pi$
$60$	0	0
$61$	3.59627	0.460455	0.230227	−	0.973137i	$-0.426053\pi$
$61$	3.59627	0.460455	0.230227	+	0.973137i	$0.426053\pi$
$62$	3.22668	0.409789
$63$	0	0
$64$	1.00000	0.125000
$65$	10.7023	1.32746
$66$	0	0
$67$	7.98545	0.975578	0.487789	−	0.872961i	$-0.337803\pi$
$67$	7.98545	0.975578	0.487789	+	0.872961i	$0.337803\pi$
$68$	2.12061	0.257162
$69$	0	0
$70$	0.305407	0.0365032
$71$	−4.04189	−0.479684	−0.239842	−	0.970812i	$-0.577096\pi$
$71$	−4.04189	−0.479684	−0.239842	+	0.970812i	$0.577096\pi$
$72$	0	0
$73$	15.1976	1.77874	0.889371	−	0.457185i	$-0.151142\pi$
$73$	15.1976	1.77874	0.889371	+	0.457185i	$0.151142\pi$
$74$	−4.06418	−0.472451
$75$	0	0
$76$	0	0
$77$	0.803348	0.0915500
$78$	0	0
$79$	−12.3969	−1.39476	−0.697382	−	0.716700i	$-0.745652\pi$
$79$	−12.3969	−1.39476	−0.697382	+	0.716700i	$0.745652\pi$
$80$	1.65270	0.184778
$81$	0	0
$82$	−8.63816	−0.953925
$83$	−8.45605	−0.928172	−0.464086	−	0.885790i	$-0.653617\pi$
$83$	−8.45605	−0.928172	−0.464086	+	0.885790i	$0.653617\pi$
$84$	0	0
$85$	3.50475	0.380143
$86$	0.0418891	0.00451701
$87$	0	0
$88$	4.34730	0.463423
$89$	−17.9368	−1.90129	−0.950646	−	0.310277i	$-0.899578\pi$
$89$	−17.9368	−1.90129	−0.950646	+	0.310277i	$0.899578\pi$
$90$	0	0
$91$	1.19665	0.125443
$92$	−0.106067	−0.0110582
$93$	0	0
$94$	7.92127	0.817017
$95$	0	0
$96$	0	0
$97$	−10.6878	−1.08518	−0.542590	−	0.839998i	$-0.682556\pi$
$97$	−10.6878	−1.08518	−0.542590	+	0.839998i	$0.682556\pi$
$98$	−6.96585	−0.703657
$99$	0	0
$100$	−2.26857	−0.226857
$101$	−8.06418	−0.802416	−0.401208	−	0.915987i	$-0.631409\pi$
$101$	−8.06418	−0.802416	−0.401208	+	0.915987i	$0.631409\pi$
$102$	0	0
$103$	−17.9290	−1.76660	−0.883299	−	0.468810i	$-0.844683\pi$
$103$	−17.9290	−1.76660	−0.883299	+	0.468810i	$0.844683\pi$
$104$	6.47565	0.634990
$105$	0	0
$106$	−9.21213	−0.894762
$107$	−13.9881	−1.35228	−0.676142	−	0.736771i	$-0.736350\pi$
$107$	−13.9881	−1.35228	−0.676142	+	0.736771i	$0.736350\pi$
$108$	0	0
$109$	4.77332	0.457201	0.228600	−	0.973520i	$-0.426585\pi$
$109$	4.77332	0.457201	0.228600	+	0.973520i	$0.426585\pi$
$110$	7.18479	0.685043
$111$	0	0
$112$	0.184793	0.0174613
$113$	−0.753718	−0.0709038	−0.0354519	−	0.999371i	$-0.511287\pi$
$113$	−0.753718	−0.0709038	−0.0354519	+	0.999371i	$0.511287\pi$
$114$	0	0
$115$	−0.175297	−0.0163465
$116$	3.98545	0.370040
$117$	0	0
$118$	10.0915	0.929000
$119$	0.391874	0.0359230
$120$	0	0
$121$	7.89899	0.718090
$122$	3.59627	0.325591
$123$	0	0
$124$	3.22668	0.289765
$125$	−12.0128	−1.07446
$126$	0	0
$127$	10.8229	0.960381	0.480191	−	0.877164i	$-0.340567\pi$
$127$	10.8229	0.960381	0.480191	+	0.877164i	$0.340567\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.7023	0.938657
$131$	−10.8452	−0.947553	−0.473776	−	0.880645i	$-0.657109\pi$
$131$	−10.8452	−0.947553	−0.473776	+	0.880645i	$0.657109\pi$
$132$	0	0
$133$	0	0
$134$	7.98545	0.689838
$135$	0	0
$136$	2.12061	0.181841
$137$	13.9513	1.19194	0.595970	−	0.803007i	$-0.296768\pi$
$137$	13.9513	1.19194	0.595970	+	0.803007i	$0.296768\pi$
$138$	0	0
$139$	5.96585	0.506017	0.253008	−	0.967464i	$-0.418580\pi$
$139$	5.96585	0.506017	0.253008	+	0.967464i	$0.418580\pi$
$140$	0.305407	0.0258116
$141$	0	0
$142$	−4.04189	−0.339188
$143$	28.1516	2.35415
$144$	0	0
$145$	6.58677	0.547002
$146$	15.1976	1.25776
$147$	0	0
$148$	−4.06418	−0.334073
$149$	14.2199	1.16494	0.582469	−	0.812853i	$-0.302087\pi$
$149$	14.2199	1.16494	0.582469	+	0.812853i	$0.302087\pi$
$150$	0	0
$151$	15.2003	1.23698	0.618490	−	0.785792i	$-0.287745\pi$
$151$	15.2003	1.23698	0.618490	+	0.785792i	$0.287745\pi$
$152$	0	0
$153$	0	0
$154$	0.803348	0.0647356
$155$	5.33275	0.428337
$156$	0	0
$157$	9.40373	0.750500	0.375250	−	0.926924i	$-0.377557\pi$
$157$	9.40373	0.750500	0.375250	+	0.926924i	$0.377557\pi$
$158$	−12.3969	−0.986246
$159$	0	0
$160$	1.65270	0.130658
$161$	−0.0196004	−0.00154472
$162$	0	0
$163$	−7.62361	−0.597127	−0.298564	−	0.954390i	$-0.596507\pi$
$163$	−7.62361	−0.597127	−0.298564	+	0.954390i	$0.596507\pi$
$164$	−8.63816	−0.674527
$165$	0	0
$166$	−8.45605	−0.656317
$167$	−17.2003	−1.33100	−0.665499	−	0.746399i	$-0.731781\pi$
$167$	−17.2003	−1.33100	−0.665499	+	0.746399i	$0.731781\pi$
$168$	0	0
$169$	28.9341	2.22570
$170$	3.50475	0.268802
$171$	0	0
$172$	0.0418891	0.00319401
$173$	−6.62361	−0.503584	−0.251792	−	0.967781i	$-0.581020\pi$
$173$	−6.62361	−0.503584	−0.251792	+	0.967781i	$0.581020\pi$
$174$	0	0
$175$	−0.419215	−0.0316897
$176$	4.34730	0.327690
$177$	0	0
$178$	−17.9368	−1.34442
$179$	−21.4243	−1.60132	−0.800662	−	0.599116i	$-0.795519\pi$
$179$	−21.4243	−1.60132	−0.800662	+	0.599116i	$0.795519\pi$
$180$	0	0
$181$	17.1848	1.27734	0.638668	−	0.769483i	$-0.279486\pi$
$181$	17.1848	1.27734	0.638668	+	0.769483i	$0.279486\pi$
$182$	1.19665	0.0887018
$183$	0	0
$184$	−0.106067	−0.00781935
$185$	−6.71688	−0.493835
$186$	0	0
$187$	9.21894	0.674156
$188$	7.92127	0.577718
$189$	0	0
$190$	0	0
$191$	6.57398	0.475676	0.237838	−	0.971305i	$-0.423561\pi$
$191$	6.57398	0.475676	0.237838	+	0.971305i	$0.423561\pi$
$192$	0	0
$193$	11.3473	0.816796	0.408398	−	0.912804i	$-0.366088\pi$
$193$	11.3473	0.816796	0.408398	+	0.912804i	$0.366088\pi$
$194$	−10.6878	−0.767338
$195$	0	0
$196$	−6.96585	−0.497561
$197$	−6.44831	−0.459423	−0.229712	−	0.973259i	$-0.573778\pi$
$197$	−6.44831	−0.459423	−0.229712	+	0.973259i	$0.573778\pi$
$198$	0	0
$199$	16.7246	1.18558	0.592789	−	0.805358i	$-0.298027\pi$
$199$	16.7246	1.18558	0.592789	+	0.805358i	$0.298027\pi$
$200$	−2.26857	−0.160412
$201$	0	0
$202$	−8.06418	−0.567394
$203$	0.736482	0.0516909
$204$	0	0
$205$	−14.2763	−0.997101
$206$	−17.9290	−1.24917
$207$	0	0
$208$	6.47565	0.449006
$209$	0	0
$210$	0	0
$211$	4.87939	0.335911	0.167955	−	0.985795i	$-0.446284\pi$
$211$	4.87939	0.335911	0.167955	+	0.985795i	$0.446284\pi$
$212$	−9.21213	−0.632692
$213$	0	0
$214$	−13.9881	−0.956210
$215$	0.0692302	0.00472146
$216$	0	0
$217$	0.596267	0.0404772
$218$	4.77332	0.323290
$219$	0	0
$220$	7.18479	0.484399
$221$	13.7324	0.923739
$222$	0	0
$223$	−28.7811	−1.92732	−0.963661	−	0.267128i	$-0.913925\pi$
$223$	−28.7811	−1.92732	−0.963661	+	0.267128i	$0.913925\pi$
$224$	0.184793	0.0123470
$225$	0	0
$226$	−0.753718	−0.0501366
$227$	−3.75608	−0.249300	−0.124650	−	0.992201i	$-0.539781\pi$
$227$	−3.75608	−0.249300	−0.124650	+	0.992201i	$0.539781\pi$
$228$	0	0
$229$	17.5175	1.15759	0.578796	−	0.815472i	$-0.303523\pi$
$229$	17.5175	1.15759	0.578796	+	0.815472i	$0.303523\pi$
$230$	−0.175297	−0.0115587
$231$	0	0
$232$	3.98545	0.261658
$233$	7.00505	0.458916	0.229458	−	0.973319i	$-0.426305\pi$
$233$	7.00505	0.458916	0.229458	+	0.973319i	$0.426305\pi$
$234$	0	0
$235$	13.0915	0.853997
$236$	10.0915	0.656902
$237$	0	0
$238$	0.391874	0.0254014
$239$	−0.935822	−0.0605333	−0.0302667	−	0.999542i	$-0.509636\pi$
$239$	−0.935822	−0.0605333	−0.0302667	+	0.999542i	$0.509636\pi$
$240$	0	0
$241$	28.7297	1.85064	0.925321	−	0.379186i	$-0.123796\pi$
$241$	28.7297	1.85064	0.925321	+	0.379186i	$0.123796\pi$
$242$	7.89899	0.507766
$243$	0	0
$244$	3.59627	0.230227
$245$	−11.5125	−0.735506
$246$	0	0
$247$	0	0
$248$	3.22668	0.204894
$249$	0	0
$250$	−12.0128	−0.759756
$251$	−24.1857	−1.52659	−0.763295	−	0.646050i	$-0.776420\pi$
$251$	−24.1857	−1.52659	−0.763295	+	0.646050i	$0.776420\pi$
$252$	0	0
$253$	−0.461104	−0.0289894
$254$	10.8229	0.679092
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.80066	0.548970	0.274485	−	0.961591i	$-0.411493\pi$
$257$	8.80066	0.548970	0.274485	+	0.961591i	$0.411493\pi$
$258$	0	0
$259$	−0.751030	−0.0466667
$260$	10.7023	0.663731
$261$	0	0
$262$	−10.8452	−0.670021
$263$	−1.30810	−0.0806606	−0.0403303	−	0.999186i	$-0.512841\pi$
$263$	−1.30810	−0.0806606	−0.0403303	+	0.999186i	$0.512841\pi$
$264$	0	0
$265$	−15.2249	−0.935260
$266$	0	0
$267$	0	0
$268$	7.98545	0.487789
$269$	8.75103	0.533560	0.266780	−	0.963757i	$-0.414040\pi$
$269$	8.75103	0.533560	0.266780	+	0.963757i	$0.414040\pi$
$270$	0	0
$271$	−13.7469	−0.835065	−0.417533	−	0.908662i	$-0.637105\pi$
$271$	−13.7469	−0.835065	−0.417533	+	0.908662i	$0.637105\pi$
$272$	2.12061	0.128581
$273$	0	0
$274$	13.9513	0.842829
$275$	−9.86215	−0.594710
$276$	0	0
$277$	25.0077	1.50257	0.751285	−	0.659978i	$-0.229434\pi$
$277$	25.0077	1.50257	0.751285	+	0.659978i	$0.229434\pi$
$278$	5.96585	0.357808
$279$	0	0
$280$	0.305407	0.0182516
$281$	12.1557	0.725148	0.362574	−	0.931955i	$-0.381898\pi$
$281$	12.1557	0.725148	0.362574	+	0.931955i	$0.381898\pi$
$282$	0	0
$283$	−10.6973	−0.635887	−0.317944	−	0.948110i	$-0.602992\pi$
$283$	−10.6973	−0.635887	−0.317944	+	0.948110i	$0.602992\pi$
$284$	−4.04189	−0.239842
$285$	0	0
$286$	28.1516	1.66464
$287$	−1.59627	−0.0942246
$288$	0	0
$289$	−12.5030	−0.735470
$290$	6.58677	0.386789
$291$	0	0
$292$	15.1976	0.889371
$293$	−13.3473	−0.779757	−0.389879	−	0.920866i	$-0.627483\pi$
$293$	−13.3473	−0.779757	−0.389879	+	0.920866i	$0.627483\pi$
$294$	0	0
$295$	16.6783	0.971048
$296$	−4.06418	−0.236226
$297$	0	0
$298$	14.2199	0.823735
$299$	−0.686852	−0.0397217
$300$	0	0
$301$	0.00774079	0.000446172 0
$302$	15.2003	0.874677
$303$	0	0
$304$	0	0
$305$	5.94356	0.340327
$306$	0	0
$307$	−15.6631	−0.893942	−0.446971	−	0.894548i	$-0.647497\pi$
$307$	−15.6631	−0.893942	−0.446971	+	0.894548i	$0.647497\pi$
$308$	0.803348	0.0457750
$309$	0	0
$310$	5.33275	0.302880
$311$	−9.10607	−0.516358	−0.258179	−	0.966097i	$-0.583122\pi$
$311$	−9.10607	−0.516358	−0.258179	+	0.966097i	$0.583122\pi$
$312$	0	0
$313$	17.7469	1.00311	0.501557	−	0.865124i	$-0.332761\pi$
$313$	17.7469	1.00311	0.501557	+	0.865124i	$0.332761\pi$
$314$	9.40373	0.530683
$315$	0	0
$316$	−12.3969	−0.697382
$317$	31.0455	1.74369	0.871845	−	0.489782i	$-0.162924\pi$
$317$	31.0455	1.74369	0.871845	+	0.489782i	$0.162924\pi$
$318$	0	0
$319$	17.3259	0.970066
$320$	1.65270	0.0923889
$321$	0	0
$322$	−0.0196004	−0.00109229
$323$	0	0
$324$	0	0
$325$	−14.6905	−0.814881
$326$	−7.62361	−0.422233
$327$	0	0
$328$	−8.63816	−0.476962
$329$	1.46379	0.0807015
$330$	0	0
$331$	10.0797	0.554028	0.277014	−	0.960866i	$-0.410655\pi$
$331$	10.0797	0.554028	0.277014	+	0.960866i	$0.410655\pi$
$332$	−8.45605	−0.464086
$333$	0	0
$334$	−17.2003	−0.941157
$335$	13.1976	0.721061
$336$	0	0
$337$	8.37733	0.456342	0.228171	−	0.973621i	$-0.426725\pi$
$337$	8.37733	0.456342	0.228171	+	0.973621i	$0.426725\pi$
$338$	28.9341	1.57381
$339$	0	0
$340$	3.50475	0.190072
$341$	14.0273	0.759623
$342$	0	0
$343$	−2.58079	−0.139349
$344$	0.0418891	0.00225851
$345$	0	0
$346$	−6.62361	−0.356087
$347$	−13.0155	−0.698708	−0.349354	−	0.936991i	$-0.613599\pi$
$347$	−13.0155	−0.698708	−0.349354	+	0.936991i	$0.613599\pi$
$348$	0	0
$349$	−6.70739	−0.359038	−0.179519	−	0.983754i	$-0.557454\pi$
$349$	−6.70739	−0.359038	−0.179519	+	0.983754i	$0.557454\pi$
$350$	−0.419215	−0.0224080
$351$	0	0
$352$	4.34730	0.231712
$353$	−10.6459	−0.566624	−0.283312	−	0.959028i	$-0.591433\pi$
$353$	−10.6459	−0.566624	−0.283312	+	0.959028i	$0.591433\pi$
$354$	0	0
$355$	−6.68004	−0.354540
$356$	−17.9368	−0.950646
$357$	0	0
$358$	−21.4243	−1.13231
$359$	29.1634	1.53919	0.769594	−	0.638534i	$-0.220459\pi$
$359$	29.1634	1.53919	0.769594	+	0.638534i	$0.220459\pi$
$360$	0	0
$361$	0	0
$362$	17.1848	0.903213
$363$	0	0
$364$	1.19665	0.0627216
$365$	25.1171	1.31469
$366$	0	0
$367$	−3.53890	−0.184729	−0.0923644	−	0.995725i	$-0.529442\pi$
$367$	−3.53890	−0.184729	−0.0923644	+	0.995725i	$0.529442\pi$
$368$	−0.106067	−0.00552912
$369$	0	0
$370$	−6.71688	−0.349194
$371$	−1.70233	−0.0883808
$372$	0	0
$373$	−10.2094	−0.528625	−0.264313	−	0.964437i	$-0.585145\pi$
$373$	−10.2094	−0.528625	−0.264313	+	0.964437i	$0.585145\pi$
$374$	9.21894	0.476700
$375$	0	0
$376$	7.92127	0.408509
$377$	25.8084	1.32920
$378$	0	0
$379$	−12.6287	−0.648691	−0.324345	−	0.945939i	$-0.605144\pi$
$379$	−12.6287	−0.648691	−0.324345	+	0.945939i	$0.605144\pi$
$380$	0	0
$381$	0	0
$382$	6.57398	0.336354
$383$	−8.71419	−0.445274	−0.222637	−	0.974901i	$-0.571467\pi$
$383$	−8.71419	−0.445274	−0.222637	+	0.974901i	$0.571467\pi$
$384$	0	0
$385$	1.32770	0.0676657
$386$	11.3473	0.577562
$387$	0	0
$388$	−10.6878	−0.542590
$389$	8.64496	0.438317	0.219159	−	0.975689i	$-0.429669\pi$
$389$	8.64496	0.438317	0.219159	+	0.975689i	$0.429669\pi$
$390$	0	0
$391$	−0.224927	−0.0113750
$392$	−6.96585	−0.351829
$393$	0	0
$394$	−6.44831	−0.324861
$395$	−20.4884	−1.03089
$396$	0	0
$397$	−6.87433	−0.345013	−0.172506	−	0.985008i	$-0.555187\pi$
$397$	−6.87433	−0.345013	−0.172506	+	0.985008i	$0.555187\pi$
$398$	16.7246	0.838330
$399$	0	0
$400$	−2.26857	−0.113429
$401$	−29.4388	−1.47010	−0.735052	−	0.678011i	$-0.762842\pi$
$401$	−29.4388	−1.47010	−0.735052	+	0.678011i	$0.762842\pi$
$402$	0	0
$403$	20.8949	1.04085
$404$	−8.06418	−0.401208
$405$	0	0
$406$	0.736482	0.0365510
$407$	−17.6682	−0.875779
$408$	0	0
$409$	21.2763	1.05205	0.526023	−	0.850470i	$-0.323683\pi$
$409$	21.2763	1.05205	0.526023	+	0.850470i	$0.323683\pi$
$410$	−14.2763	−0.705057
$411$	0	0
$412$	−17.9290	−0.883299
$413$	1.86484	0.0917626
$414$	0	0
$415$	−13.9753	−0.686023
$416$	6.47565	0.317495
$417$	0	0
$418$	0	0
$419$	−15.8922	−0.776384	−0.388192	−	0.921579i	$-0.626900\pi$
$419$	−15.8922	−0.776384	−0.388192	+	0.921579i	$0.626900\pi$
$420$	0	0
$421$	−19.5945	−0.954978	−0.477489	−	0.878638i	$-0.658453\pi$
$421$	−19.5945	−0.954978	−0.477489	+	0.878638i	$0.658453\pi$
$422$	4.87939	0.237525
$423$	0	0
$424$	−9.21213	−0.447381
$425$	−4.81076	−0.233356
$426$	0	0
$427$	0.664563	0.0321605
$428$	−13.9881	−0.676142
$429$	0	0
$430$	0.0692302	0.00333858
$431$	−4.15476	−0.200128	−0.100064	−	0.994981i	$-0.531905\pi$
$431$	−4.15476	−0.200128	−0.100064	+	0.994981i	$0.531905\pi$
$432$	0	0
$433$	3.56212	0.171184	0.0855922	−	0.996330i	$-0.472722\pi$
$433$	3.56212	0.171184	0.0855922	+	0.996330i	$0.472722\pi$
$434$	0.596267	0.0286217
$435$	0	0
$436$	4.77332	0.228600
$437$	0	0
$438$	0	0
$439$	8.21482	0.392072	0.196036	−	0.980597i	$-0.437193\pi$
$439$	8.21482	0.392072	0.196036	+	0.980597i	$0.437193\pi$
$440$	7.18479	0.342522
$441$	0	0
$442$	13.7324	0.653182
$443$	18.9463	0.900164	0.450082	−	0.892987i	$-0.351395\pi$
$443$	18.9463	0.900164	0.450082	+	0.892987i	$0.351395\pi$
$444$	0	0
$445$	−29.6441	−1.40527
$446$	−28.7811	−1.36282
$447$	0	0
$448$	0.184793	0.00873063
$449$	10.4926	0.495175	0.247587	−	0.968866i	$-0.420362\pi$
$449$	10.4926	0.495175	0.247587	+	0.968866i	$0.420362\pi$
$450$	0	0
$451$	−37.5526	−1.76828
$452$	−0.753718	−0.0354519
$453$	0	0
$454$	−3.75608	−0.176282
$455$	1.97771	0.0927165
$456$	0	0
$457$	−0.415593	−0.0194406	−0.00972031	−	0.999953i	$-0.503094\pi$
$457$	−0.415593	−0.0194406	−0.00972031	+	0.999953i	$0.503094\pi$
$458$	17.5175	0.818541
$459$	0	0
$460$	−0.175297	−0.00817327
$461$	−25.2576	−1.17637	−0.588183	−	0.808728i	$-0.700156\pi$
$461$	−25.2576	−1.17637	−0.588183	+	0.808728i	$0.700156\pi$
$462$	0	0
$463$	18.7912	0.873299	0.436650	−	0.899632i	$-0.356165\pi$
$463$	18.7912	0.873299	0.436650	+	0.899632i	$0.356165\pi$
$464$	3.98545	0.185020
$465$	0	0
$466$	7.00505	0.324503
$467$	17.1334	0.792840	0.396420	−	0.918069i	$-0.370252\pi$
$467$	17.1334	0.792840	0.396420	+	0.918069i	$0.370252\pi$
$468$	0	0
$469$	1.47565	0.0681393
$470$	13.0915	0.603867
$471$	0	0
$472$	10.0915	0.464500
$473$	0.182104	0.00837316
$474$	0	0
$475$	0	0
$476$	0.391874	0.0179615
$477$	0	0
$478$	−0.935822	−0.0428035
$479$	10.9581	0.500689	0.250344	−	0.968157i	$-0.419456\pi$
$479$	10.9581	0.500689	0.250344	+	0.968157i	$0.419456\pi$
$480$	0	0
$481$	−26.3182	−1.20001
$482$	28.7297	1.30860
$483$	0	0
$484$	7.89899	0.359045
$485$	−17.6637	−0.802069
$486$	0	0
$487$	−16.8803	−0.764920	−0.382460	−	0.923972i	$-0.624923\pi$
$487$	−16.8803	−0.764920	−0.382460	+	0.923972i	$0.624923\pi$
$488$	3.59627	0.162795
$489$	0	0
$490$	−11.5125	−0.520081
$491$	11.2567	0.508008	0.254004	−	0.967203i	$-0.418252\pi$
$491$	11.2567	0.508008	0.254004	+	0.967203i	$0.418252\pi$
$492$	0	0
$493$	8.45161	0.380641
$494$	0	0
$495$	0	0
$496$	3.22668	0.144882
$497$	−0.746911	−0.0335035
$498$	0	0
$499$	−9.77063	−0.437393	−0.218697	−	0.975793i	$-0.570181\pi$
$499$	−9.77063	−0.437393	−0.218697	+	0.975793i	$0.570181\pi$
$500$	−12.0128	−0.537228
$501$	0	0
$502$	−24.1857	−1.07946
$503$	37.4374	1.66925	0.834625	−	0.550818i	$-0.185684\pi$
$503$	37.4374	1.66925	0.834625	+	0.550818i	$0.185684\pi$
$504$	0	0
$505$	−13.3277	−0.593075
$506$	−0.461104	−0.0204986
$507$	0	0
$508$	10.8229	0.480191
$509$	−3.71688	−0.164748	−0.0823739	−	0.996601i	$-0.526250\pi$
$509$	−3.71688	−0.164748	−0.0823739	+	0.996601i	$0.526250\pi$
$510$	0	0
$511$	2.80840	0.124236
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.80066	0.388180
$515$	−29.6313	−1.30571
$516$	0	0
$517$	34.4361	1.51450
$518$	−0.751030	−0.0329984
$519$	0	0
$520$	10.7023	0.469328
$521$	−0.892178	−0.0390870	−0.0195435	−	0.999809i	$-0.506221\pi$
$521$	−0.892178	−0.0390870	−0.0195435	+	0.999809i	$0.506221\pi$
$522$	0	0
$523$	14.0651	0.615024	0.307512	−	0.951544i	$-0.400504\pi$
$523$	14.0651	0.615024	0.307512	+	0.951544i	$0.400504\pi$
$524$	−10.8452	−0.473776
$525$	0	0
$526$	−1.30810	−0.0570357
$527$	6.84255	0.298066
$528$	0	0
$529$	−22.9887	−0.999511
$530$	−15.2249	−0.661329
$531$	0	0
$532$	0	0
$533$	−55.9377	−2.42293
$534$	0	0
$535$	−23.1183	−0.999489
$536$	7.98545	0.344919
$537$	0	0
$538$	8.75103	0.377284
$539$	−30.2826	−1.30436
$540$	0	0
$541$	25.6186	1.10143	0.550714	−	0.834694i	$-0.314356\pi$
$541$	25.6186	1.10143	0.550714	+	0.834694i	$0.314356\pi$
$542$	−13.7469	−0.590480
$543$	0	0
$544$	2.12061	0.0909206
$545$	7.88888	0.337923
$546$	0	0
$547$	−36.3164	−1.55278	−0.776390	−	0.630253i	$-0.782951\pi$
$547$	−36.3164	−1.55278	−0.776390	+	0.630253i	$0.782951\pi$
$548$	13.9513	0.595970
$549$	0	0
$550$	−9.86215	−0.420523
$551$	0	0
$552$	0	0
$553$	−2.29086	−0.0974172
$554$	25.0077	1.06248
$555$	0	0
$556$	5.96585	0.253008
$557$	−23.3191	−0.988063	−0.494032	−	0.869444i	$-0.664477\pi$
$557$	−23.3191	−0.988063	−0.494032	+	0.869444i	$0.664477\pi$
$558$	0	0
$559$	0.271259	0.0114730
$560$	0.305407	0.0129058
$561$	0	0
$562$	12.1557	0.512757
$563$	4.53983	0.191331	0.0956655	−	0.995414i	$-0.469502\pi$
$563$	4.53983	0.191331	0.0956655	+	0.995414i	$0.469502\pi$
$564$	0	0
$565$	−1.24567	−0.0524058
$566$	−10.6973	−0.449640
$567$	0	0
$568$	−4.04189	−0.169594
$569$	−6.26621	−0.262693	−0.131347	−	0.991337i	$-0.541930\pi$
$569$	−6.26621	−0.262693	−0.131347	+	0.991337i	$0.541930\pi$
$570$	0	0
$571$	1.03777	0.0434293	0.0217147	−	0.999764i	$-0.493087\pi$
$571$	1.03777	0.0434293	0.0217147	+	0.999764i	$0.493087\pi$
$572$	28.1516	1.17708
$573$	0	0
$574$	−1.59627	−0.0666269
$575$	0.240620	0.0100346
$576$	0	0
$577$	−40.2422	−1.67530	−0.837652	−	0.546205i	$-0.816072\pi$
$577$	−40.2422	−1.67530	−0.837652	+	0.546205i	$0.816072\pi$
$578$	−12.5030	−0.520056
$579$	0	0
$580$	6.58677	0.273501
$581$	−1.56262	−0.0648282
$582$	0	0
$583$	−40.0479	−1.65861
$584$	15.1976	0.628881
$585$	0	0
$586$	−13.3473	−0.551372
$587$	48.2354	1.99089	0.995443	−	0.0953574i	$-0.0303994\pi$
$587$	48.2354	1.99089	0.995443	+	0.0953574i	$0.0303994\pi$
$588$	0	0
$589$	0	0
$590$	16.6783	0.686634
$591$	0	0
$592$	−4.06418	−0.167037
$593$	17.4757	0.717639	0.358820	−	0.933407i	$-0.383179\pi$
$593$	17.4757	0.717639	0.358820	+	0.933407i	$0.383179\pi$
$594$	0	0
$595$	0.647651	0.0265511
$596$	14.2199	0.582469
$597$	0	0
$598$	−0.686852	−0.0280875
$599$	21.9050	0.895013	0.447506	−	0.894281i	$-0.352312\pi$
$599$	21.9050	0.895013	0.447506	+	0.894281i	$0.352312\pi$
$600$	0	0
$601$	−9.27807	−0.378460	−0.189230	−	0.981933i	$-0.560599\pi$
$601$	−9.27807	−0.378460	−0.189230	+	0.981933i	$0.560599\pi$
$602$	0.00774079	0.000315491 0
$603$	0	0
$604$	15.2003	0.618490
$605$	13.0547	0.530748
$606$	0	0
$607$	−29.3354	−1.19069	−0.595344	−	0.803471i	$-0.702984\pi$
$607$	−29.3354	−1.19069	−0.595344	+	0.803471i	$0.702984\pi$
$608$	0	0
$609$	0	0
$610$	5.94356	0.240648
$611$	51.2954	2.07519
$612$	0	0
$613$	−24.2499	−0.979444	−0.489722	−	0.871879i	$-0.662902\pi$
$613$	−24.2499	−0.979444	−0.489722	+	0.871879i	$0.662902\pi$
$614$	−15.6631	−0.632113
$615$	0	0
$616$	0.803348	0.0323678
$617$	−20.0310	−0.806416	−0.403208	−	0.915108i	$-0.632105\pi$
$617$	−20.0310	−0.806416	−0.403208	+	0.915108i	$0.632105\pi$
$618$	0	0
$619$	−8.01548	−0.322169	−0.161085	−	0.986941i	$-0.551499\pi$
$619$	−8.01548	−0.322169	−0.161085	+	0.986941i	$0.551499\pi$
$620$	5.33275	0.214168
$621$	0	0
$622$	−9.10607	−0.365120
$623$	−3.31458	−0.132796
$624$	0	0
$625$	−8.51073	−0.340429
$626$	17.7469	0.709309
$627$	0	0
$628$	9.40373	0.375250
$629$	−8.61856	−0.343644
$630$	0	0
$631$	3.98814	0.158765	0.0793827	−	0.996844i	$-0.474705\pi$
$631$	3.98814	0.158765	0.0793827	+	0.996844i	$0.474705\pi$
$632$	−12.3969	−0.493123
$633$	0	0
$634$	31.0455	1.23297
$635$	17.8871	0.709829
$636$	0	0
$637$	−45.1084	−1.78726
$638$	17.3259	0.685941
$639$	0	0
$640$	1.65270	0.0653288
$641$	−27.9050	−1.10218	−0.551090	−	0.834446i	$-0.685788\pi$
$641$	−27.9050	−1.10218	−0.551090	+	0.834446i	$0.685788\pi$
$642$	0	0
$643$	−10.8033	−0.426042	−0.213021	−	0.977048i	$-0.568330\pi$
$643$	−10.8033	−0.426042	−0.213021	+	0.977048i	$0.568330\pi$
$644$	−0.0196004	−0.000772362 0
$645$	0	0
$646$	0	0
$647$	34.4662	1.35500	0.677502	−	0.735521i	$-0.263062\pi$
$647$	34.4662	1.35500	0.677502	+	0.735521i	$0.263062\pi$
$648$	0	0
$649$	43.8708	1.72208
$650$	−14.6905	−0.576208
$651$	0	0
$652$	−7.62361	−0.298564
$653$	22.9941	0.899830	0.449915	−	0.893071i	$-0.351454\pi$
$653$	22.9941	0.899830	0.449915	+	0.893071i	$0.351454\pi$
$654$	0	0
$655$	−17.9240	−0.700347
$656$	−8.63816	−0.337263
$657$	0	0
$658$	1.46379	0.0570646
$659$	−34.0374	−1.32591	−0.662955	−	0.748659i	$-0.730698\pi$
$659$	−34.0374	−1.32591	−0.662955	+	0.748659i	$0.730698\pi$
$660$	0	0
$661$	−8.56036	−0.332960	−0.166480	−	0.986045i	$-0.553240\pi$
$661$	−8.56036	−0.332960	−0.166480	+	0.986045i	$0.553240\pi$
$662$	10.0797	0.391757
$663$	0	0
$664$	−8.45605	−0.328158
$665$	0	0
$666$	0	0
$667$	−0.422724	−0.0163680
$668$	−17.2003	−0.665499
$669$	0	0
$670$	13.1976	0.509867
$671$	15.6340	0.603545
$672$	0	0
$673$	12.1402	0.467971	0.233985	−	0.972240i	$-0.424823\pi$
$673$	12.1402	0.467971	0.233985	+	0.972240i	$0.424823\pi$
$674$	8.37733	0.322683
$675$	0	0
$676$	28.9341	1.11285
$677$	1.55344	0.0597037	0.0298519	−	0.999554i	$-0.490496\pi$
$677$	1.55344	0.0597037	0.0298519	+	0.999554i	$0.490496\pi$
$678$	0	0
$679$	−1.97502	−0.0757944
$680$	3.50475	0.134401
$681$	0	0
$682$	14.0273	0.537135
$683$	−35.3892	−1.35413	−0.677065	−	0.735923i	$-0.736748\pi$
$683$	−35.3892	−1.35413	−0.677065	+	0.735923i	$0.736748\pi$
$684$	0	0
$685$	23.0574	0.880977
$686$	−2.58079	−0.0985348
$687$	0	0
$688$	0.0418891	0.00159701
$689$	−59.6546	−2.27266
$690$	0	0
$691$	−39.2823	−1.49437	−0.747185	−	0.664617i	$-0.768595\pi$
$691$	−39.2823	−1.49437	−0.747185	+	0.664617i	$0.768595\pi$
$692$	−6.62361	−0.251792
$693$	0	0
$694$	−13.0155	−0.494061
$695$	9.85978	0.374003
$696$	0	0
$697$	−18.3182	−0.693851
$698$	−6.70739	−0.253878
$699$	0	0
$700$	−0.419215	−0.0158448
$701$	−16.6527	−0.628964	−0.314482	−	0.949263i	$-0.601831\pi$
$701$	−16.6527	−0.628964	−0.314482	+	0.949263i	$0.601831\pi$
$702$	0	0
$703$	0	0
$704$	4.34730	0.163845
$705$	0	0
$706$	−10.6459	−0.400664
$707$	−1.49020	−0.0560447
$708$	0	0
$709$	43.5645	1.63610	0.818049	−	0.575148i	$-0.195056\pi$
$709$	43.5645	1.63610	0.818049	+	0.575148i	$0.195056\pi$
$710$	−6.68004	−0.250698
$711$	0	0
$712$	−17.9368	−0.672208
$713$	−0.342244	−0.0128171
$714$	0	0
$715$	46.5262	1.73998
$716$	−21.4243	−0.800662
$717$	0	0
$718$	29.1634	1.08837
$719$	−6.46110	−0.240959	−0.120479	−	0.992716i	$-0.538443\pi$
$719$	−6.46110	−0.240959	−0.120479	+	0.992716i	$0.538443\pi$
$720$	0	0
$721$	−3.31315	−0.123388
$722$	0	0
$723$	0	0
$724$	17.1848	0.638668
$725$	−9.04128	−0.335785
$726$	0	0
$727$	1.84430	0.0684014	0.0342007	−	0.999415i	$-0.489111\pi$
$727$	1.84430	0.0684014	0.0342007	+	0.999415i	$0.489111\pi$
$728$	1.19665	0.0443509
$729$	0	0
$730$	25.1171	0.929626
$731$	0.0888306	0.00328552
$732$	0	0
$733$	32.3587	1.19519	0.597597	−	0.801796i	$-0.296122\pi$
$733$	32.3587	1.19519	0.597597	+	0.801796i	$0.296122\pi$
$734$	−3.53890	−0.130623
$735$	0	0
$736$	−0.106067	−0.00390968
$737$	34.7151	1.27875
$738$	0	0
$739$	−10.9358	−0.402281	−0.201140	−	0.979562i	$-0.564465\pi$
$739$	−10.9358	−0.402281	−0.201140	+	0.979562i	$0.564465\pi$
$740$	−6.71688	−0.246917
$741$	0	0
$742$	−1.70233	−0.0624946
$743$	−10.4706	−0.384129	−0.192065	−	0.981382i	$-0.561518\pi$
$743$	−10.4706	−0.384129	−0.192065	+	0.981382i	$0.561518\pi$
$744$	0	0
$745$	23.5012	0.861019
$746$	−10.2094	−0.373794
$747$	0	0
$748$	9.21894	0.337078
$749$	−2.58490	−0.0944503
$750$	0	0
$751$	26.3797	0.962609	0.481304	−	0.876554i	$-0.340163\pi$
$751$	26.3797	0.962609	0.481304	+	0.876554i	$0.340163\pi$
$752$	7.92127	0.288859
$753$	0	0
$754$	25.8084	0.939887
$755$	25.1215	0.914267
$756$	0	0
$757$	22.8075	0.828951	0.414476	−	0.910060i	$-0.363965\pi$
$757$	22.8075	0.828951	0.414476	+	0.910060i	$0.363965\pi$
$758$	−12.6287	−0.458694
$759$	0	0
$760$	0	0
$761$	−41.5012	−1.50442	−0.752209	−	0.658924i	$-0.771012\pi$
$761$	−41.5012	−1.50442	−0.752209	+	0.658924i	$0.771012\pi$
$762$	0	0
$763$	0.882074	0.0319332
$764$	6.57398	0.237838
$765$	0	0
$766$	−8.71419	−0.314857
$767$	65.3492	2.35962
$768$	0	0
$769$	19.0009	0.685191	0.342596	−	0.939483i	$-0.388694\pi$
$769$	19.0009	0.685191	0.342596	+	0.939483i	$0.388694\pi$
$770$	1.32770	0.0478468
$771$	0	0
$772$	11.3473	0.408398
$773$	17.1438	0.616621	0.308310	−	0.951286i	$-0.400236\pi$
$773$	17.1438	0.616621	0.308310	+	0.951286i	$0.400236\pi$
$774$	0	0
$775$	−7.31996	−0.262941
$776$	−10.6878	−0.383669
$777$	0	0
$778$	8.64496	0.309937
$779$	0	0
$780$	0	0
$781$	−17.5713	−0.628750
$782$	−0.224927	−0.00804337
$783$	0	0
$784$	−6.96585	−0.248780
$785$	15.5416	0.554703
$786$	0	0
$787$	18.2003	0.648770	0.324385	−	0.945925i	$-0.394843\pi$
$787$	18.2003	0.648770	0.324385	+	0.945925i	$0.394843\pi$
$788$	−6.44831	−0.229712
$789$	0	0
$790$	−20.4884	−0.728946
$791$	−0.139281	−0.00495228
$792$	0	0
$793$	23.2882	0.826987
$794$	−6.87433	−0.243961
$795$	0	0
$796$	16.7246	0.592789
$797$	−26.4584	−0.937205	−0.468603	−	0.883409i	$-0.655242\pi$
$797$	−26.4584	−0.937205	−0.468603	+	0.883409i	$0.655242\pi$
$798$	0	0
$799$	16.7980	0.594270
$800$	−2.26857	−0.0802061
$801$	0	0
$802$	−29.4388	−1.03952
$803$	66.0684	2.33150
$804$	0	0
$805$	−0.0323936	−0.00114172
$806$	20.8949	0.735990
$807$	0	0
$808$	−8.06418	−0.283697
$809$	30.5672	1.07468	0.537342	−	0.843364i	$-0.319428\pi$
$809$	30.5672	1.07468	0.537342	+	0.843364i	$0.319428\pi$
$810$	0	0
$811$	6.85710	0.240785	0.120393	−	0.992726i	$-0.461585\pi$
$811$	6.85710	0.240785	0.120393	+	0.992726i	$0.461585\pi$
$812$	0.736482	0.0258454
$813$	0	0
$814$	−17.6682	−0.619270
$815$	−12.5996	−0.441343
$816$	0	0
$817$	0	0
$818$	21.2763	0.743909
$819$	0	0
$820$	−14.2763	−0.498550
$821$	−39.6800	−1.38484	−0.692422	−	0.721493i	$-0.743456\pi$
$821$	−39.6800	−1.38484	−0.692422	+	0.721493i	$0.743456\pi$
$822$	0	0
$823$	−46.7205	−1.62857	−0.814287	−	0.580462i	$-0.802872\pi$
$823$	−46.7205	−1.62857	−0.814287	+	0.580462i	$0.802872\pi$
$824$	−17.9290	−0.624587
$825$	0	0
$826$	1.86484	0.0648860
$827$	−41.9172	−1.45760	−0.728801	−	0.684725i	$-0.759922\pi$
$827$	−41.9172	−1.45760	−0.728801	+	0.684725i	$0.759922\pi$
$828$	0	0
$829$	0.502059	0.0174372	0.00871862	−	0.999962i	$-0.497225\pi$
$829$	0.502059	0.0174372	0.00871862	+	0.999962i	$0.497225\pi$
$830$	−13.9753	−0.485091
$831$	0	0
$832$	6.47565	0.224503
$833$	−14.7719	−0.511816
$834$	0	0
$835$	−28.4270	−0.983755
$836$	0	0
$837$	0	0
$838$	−15.8922	−0.548986
$839$	−31.4142	−1.08454	−0.542269	−	0.840205i	$-0.682435\pi$
$839$	−31.4142	−1.08454	−0.542269	+	0.840205i	$0.682435\pi$
$840$	0	0
$841$	−13.1162	−0.452282
$842$	−19.5945	−0.675271
$843$	0	0
$844$	4.87939	0.167955
$845$	47.8194	1.64504
$846$	0	0
$847$	1.45967	0.0501550
$848$	−9.21213	−0.316346
$849$	0	0
$850$	−4.81076	−0.165008
$851$	0.431074	0.0147770
$852$	0	0
$853$	18.8128	0.644139	0.322070	−	0.946716i	$-0.395621\pi$
$853$	18.8128	0.644139	0.322070	+	0.946716i	$0.395621\pi$
$854$	0.664563	0.0227409
$855$	0	0
$856$	−13.9881	−0.478105
$857$	29.9632	1.02352	0.511761	−	0.859128i	$-0.328993\pi$
$857$	29.9632	1.02352	0.511761	+	0.859128i	$0.328993\pi$
$858$	0	0
$859$	−33.1702	−1.13175	−0.565877	−	0.824490i	$-0.691462\pi$
$859$	−33.1702	−1.13175	−0.565877	+	0.824490i	$0.691462\pi$
$860$	0.0692302	0.00236073
$861$	0	0
$862$	−4.15476	−0.141512
$863$	−38.2475	−1.30196	−0.650981	−	0.759094i	$-0.725642\pi$
$863$	−38.2475	−1.30196	−0.650981	+	0.759094i	$0.725642\pi$
$864$	0	0
$865$	−10.9469	−0.372204
$866$	3.56212	0.121046
$867$	0	0
$868$	0.596267	0.0202386
$869$	−53.8931	−1.82820
$870$	0	0
$871$	51.7110	1.75216
$872$	4.77332	0.161645
$873$	0	0
$874$	0	0
$875$	−2.21987	−0.0750455
$876$	0	0
$877$	48.1671	1.62649	0.813243	−	0.581924i	$-0.197700\pi$
$877$	48.1671	1.62649	0.813243	+	0.581924i	$0.197700\pi$
$878$	8.21482	0.277237
$879$	0	0
$880$	7.18479	0.242199
$881$	−37.7083	−1.27043	−0.635213	−	0.772337i	$-0.719088\pi$
$881$	−37.7083	−1.27043	−0.635213	+	0.772337i	$0.719088\pi$
$882$	0	0
$883$	−30.3500	−1.02136	−0.510679	−	0.859771i	$-0.670606\pi$
$883$	−30.3500	−1.02136	−0.510679	+	0.859771i	$0.670606\pi$
$884$	13.7324	0.461869
$885$	0	0
$886$	18.9463	0.636512
$887$	−0.926651	−0.0311139	−0.0155569	−	0.999879i	$-0.504952\pi$
$887$	−0.926651	−0.0311139	−0.0155569	+	0.999879i	$0.504952\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	−29.6441	−0.993674
$891$	0	0
$892$	−28.7811	−0.963661
$893$	0	0
$894$	0	0
$895$	−35.4080	−1.18356
$896$	0.184793	0.00617349
$897$	0	0
$898$	10.4926	0.350141
$899$	12.8598	0.428898
$900$	0	0
$901$	−19.5354	−0.650818
$902$	−37.5526	−1.25037
$903$	0	0
$904$	−0.753718	−0.0250683
$905$	28.4014	0.944093
$906$	0	0
$907$	4.24535	0.140964	0.0704822	−	0.997513i	$-0.477546\pi$
$907$	4.24535	0.140964	0.0704822	+	0.997513i	$0.477546\pi$
$908$	−3.75608	−0.124650
$909$	0	0
$910$	1.97771	0.0655605
$911$	−56.5509	−1.87361	−0.936807	−	0.349847i	$-0.886234\pi$
$911$	−56.5509	−1.87361	−0.936807	+	0.349847i	$0.886234\pi$
$912$	0	0
$913$	−36.7610	−1.21661
$914$	−0.415593	−0.0137466
$915$	0	0
$916$	17.5175	0.578796
$917$	−2.00412	−0.0661818
$918$	0	0
$919$	58.9555	1.94476	0.972382	−	0.233396i	$-0.0749838\pi$
$919$	58.9555	1.94476	0.972382	+	0.233396i	$0.0749838\pi$
$920$	−0.175297	−0.00577937
$921$	0	0
$922$	−25.2576	−0.831816
$923$	−26.1739	−0.861523
$924$	0	0
$925$	9.21987	0.303148
$926$	18.7912	0.617516
$927$	0	0
$928$	3.98545	0.130829
$929$	12.4070	0.407061	0.203531	−	0.979069i	$-0.434758\pi$
$929$	12.4070	0.407061	0.203531	+	0.979069i	$0.434758\pi$
$930$	0	0
$931$	0	0
$932$	7.00505	0.229458
$933$	0	0
$934$	17.1334	0.560622
$935$	15.2362	0.498276
$936$	0	0
$937$	27.8016	0.908238	0.454119	−	0.890941i	$-0.349954\pi$
$937$	27.8016	0.908238	0.454119	+	0.890941i	$0.349954\pi$
$938$	1.47565	0.0481817
$939$	0	0
$940$	13.0915	0.426998
$941$	−19.9617	−0.650734	−0.325367	−	0.945588i	$-0.605488\pi$
$941$	−19.9617	−0.650734	−0.325367	+	0.945588i	$0.605488\pi$
$942$	0	0
$943$	0.916222	0.0298363
$944$	10.0915	0.328451
$945$	0	0
$946$	0.182104	0.00592072
$947$	−29.9992	−0.974842	−0.487421	−	0.873167i	$-0.662062\pi$
$947$	−29.9992	−0.974842	−0.487421	+	0.873167i	$0.662062\pi$
$948$	0	0
$949$	98.4143	3.19466
$950$	0	0
$951$	0	0
$952$	0.391874	0.0127007
$953$	22.9377	0.743025	0.371512	−	0.928428i	$-0.378839\pi$
$953$	22.9377	0.743025	0.371512	+	0.928428i	$0.378839\pi$
$954$	0	0
$955$	10.8648	0.351578
$956$	−0.935822	−0.0302667
$957$	0	0
$958$	10.9581	0.354040
$959$	2.57810	0.0832511
$960$	0	0
$961$	−20.5885	−0.664146
$962$	−26.3182	−0.848533
$963$	0	0
$964$	28.7297	0.925321
$965$	18.7537	0.603704
$966$	0	0
$967$	−37.8485	−1.21713	−0.608563	−	0.793505i	$-0.708254\pi$
$967$	−37.8485	−1.21713	−0.608563	+	0.793505i	$0.708254\pi$
$968$	7.89899	0.253883
$969$	0	0
$970$	−17.6637	−0.567149
$971$	−29.3364	−0.941449	−0.470724	−	0.882280i	$-0.656007\pi$
$971$	−29.3364	−0.941449	−0.470724	+	0.882280i	$0.656007\pi$
$972$	0	0
$973$	1.10244	0.0353428
$974$	−16.8803	−0.540880
$975$	0	0
$976$	3.59627	0.115114
$977$	14.1611	0.453053	0.226526	−	0.974005i	$-0.427263\pi$
$977$	14.1611	0.453053	0.226526	+	0.974005i	$0.427263\pi$
$978$	0	0
$979$	−77.9764	−2.49214
$980$	−11.5125	−0.367753
$981$	0	0
$982$	11.2567	0.359216
$983$	−34.7692	−1.10897	−0.554483	−	0.832195i	$-0.687084\pi$
$983$	−34.7692	−1.10897	−0.554483	+	0.832195i	$0.687084\pi$
$984$	0	0
$985$	−10.6571	−0.339565
$986$	8.45161	0.269154
$987$	0	0
$988$	0	0
$989$	−0.00444304	−0.000141280 0
$990$	0	0
$991$	5.48784	0.174327	0.0871634	−	0.996194i	$-0.472220\pi$
$991$	5.48784	0.174327	0.0871634	+	0.996194i	$0.472220\pi$
$992$	3.22668	0.102447
$993$	0	0
$994$	−0.746911	−0.0236906
$995$	27.6408	0.876274
$996$	0	0
$997$	−9.62454	−0.304812	−0.152406	−	0.988318i	$-0.548702\pi$
$997$	−9.62454	−0.304812	−0.152406	+	0.988318i	$0.548702\pi$
$998$	−9.77063	−0.309284
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.b.55.1	✓	6	57.32	even	18
114.2.i.b.85.1	yes	6	57.41	even	18
342.2.u.d.55.1		6	19.13	odd	18
342.2.u.d.199.1		6	19.3	odd	18
912.2.bo.c.625.1		6	228.203	odd	18
912.2.bo.c.769.1		6	228.155	odd	18
2166.2.a.n.1.2		3	3.2	odd	2
2166.2.a.t.1.2		3	57.56	even	2
6498.2.a.bo.1.2		3	19.18	odd	2
6498.2.a.bt.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.65270 q^{5} +0.184793 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.65270 q^{5} +0.184793 q^{7} +1.00000 q^{8} +1.65270 q^{10} +4.34730 q^{11} +6.47565 q^{13} +0.184793 q^{14} +1.00000 q^{16} +2.12061 q^{17} +1.65270 q^{20} +4.34730 q^{22} -0.106067 q^{23} -2.26857 q^{25} +6.47565 q^{26} +0.184793 q^{28} +3.98545 q^{29} +3.22668 q^{31} +1.00000 q^{32} +2.12061 q^{34} +0.305407 q^{35} -4.06418 q^{37} +1.65270 q^{40} -8.63816 q^{41} +0.0418891 q^{43} +4.34730 q^{44} -0.106067 q^{46} +7.92127 q^{47} -6.96585 q^{49} -2.26857 q^{50} +6.47565 q^{52} -9.21213 q^{53} +7.18479 q^{55} +0.184793 q^{56} +3.98545 q^{58} +10.0915 q^{59} +3.59627 q^{61} +3.22668 q^{62} +1.00000 q^{64} +10.7023 q^{65} +7.98545 q^{67} +2.12061 q^{68} +0.305407 q^{70} -4.04189 q^{71} +15.1976 q^{73} -4.06418 q^{74} +0.803348 q^{77} -12.3969 q^{79} +1.65270 q^{80} -8.63816 q^{82} -8.45605 q^{83} +3.50475 q^{85} +0.0418891 q^{86} +4.34730 q^{88} -17.9368 q^{89} +1.19665 q^{91} -0.106067 q^{92} +7.92127 q^{94} -10.6878 q^{97} -6.96585 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8} + 6 q^{10} + 12 q^{11} - 3 q^{14} + 3 q^{16} + 12 q^{17} + 6 q^{20} + 12 q^{22} + 12 q^{23} + 3 q^{25} - 3 q^{28} - 6 q^{29} + 3 q^{31} + 3 q^{32} + 12 q^{34} + 3 q^{35} - 3 q^{37} + 6 q^{40} - 9 q^{41} - 3 q^{43} + 12 q^{44} + 12 q^{46} + 15 q^{47} + 3 q^{50} - 3 q^{53} + 18 q^{55} - 3 q^{56} - 6 q^{58} - 3 q^{61} + 3 q^{62} + 3 q^{64} + 6 q^{65} + 6 q^{67} + 12 q^{68} + 3 q^{70} - 9 q^{71} + 3 q^{73} - 3 q^{74} - 21 q^{77} - 9 q^{79} + 6 q^{80} - 9 q^{82} - 3 q^{83} + 27 q^{85} - 3 q^{86} + 12 q^{88} + 3 q^{89} + 27 q^{91} + 12 q^{92} + 15 q^{94} + 12 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 3 * q^8 + 6 * q^10 + 12 * q^11 - 3 * q^14 + 3 * q^16 + 12 * q^17 + 6 * q^20 + 12 * q^22 + 12 * q^23 + 3 * q^25 - 3 * q^28 - 6 * q^29 + 3 * q^31 + 3 * q^32 + 12 * q^34 + 3 * q^35 - 3 * q^37 + 6 * q^40 - 9 * q^41 - 3 * q^43 + 12 * q^44 + 12 * q^46 + 15 * q^47 + 3 * q^50 - 3 * q^53 + 18 * q^55 - 3 * q^56 - 6 * q^58 - 3 * q^61 + 3 * q^62 + 3 * q^64 + 6 * q^65 + 6 * q^67 + 12 * q^68 + 3 * q^70 - 9 * q^71 + 3 * q^73 - 3 * q^74 - 21 * q^77 - 9 * q^79 + 6 * q^80 - 9 * q^82 - 3 * q^83 + 27 * q^85 - 3 * q^86 + 12 * q^88 + 3 * q^89 + 27 * q^91 + 12 * q^92 + 15 * q^94 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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