LMFDB - Embedded newform 6498.2.a.bt.1.3

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			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.87939 q^{5} +1.22668 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.87939 q^{5} +1.22668 q^{7} +1.00000 q^{8} +3.87939 q^{10} +2.12061 q^{11} -0.490200 q^{13} +1.22668 q^{14} +1.00000 q^{16} +5.53209 q^{17} +3.87939 q^{20} +2.12061 q^{22} +8.94356 q^{23} +10.0496 q^{25} -0.490200 q^{26} +1.22668 q^{28} -8.47565 q^{29} -2.41147 q^{31} +1.00000 q^{32} +5.53209 q^{34} +4.75877 q^{35} -1.69459 q^{37} +3.87939 q^{40} +1.59627 q^{41} -6.63816 q^{43} +2.12061 q^{44} +8.94356 q^{46} -2.17024 q^{47} -5.49525 q^{49} +10.0496 q^{50} -0.490200 q^{52} +8.88713 q^{53} +8.22668 q^{55} +1.22668 q^{56} -8.47565 q^{58} -11.4192 q^{59} +0.0418891 q^{61} -2.41147 q^{62} +1.00000 q^{64} -1.90167 q^{65} -4.47565 q^{67} +5.53209 q^{68} +4.75877 q^{70} +2.63816 q^{71} -15.3628 q^{73} -1.69459 q^{74} +2.60132 q^{77} +4.66044 q^{79} +3.87939 q^{80} +1.59627 q^{82} -12.4807 q^{83} +21.4611 q^{85} -6.63816 q^{86} +2.12061 q^{88} +8.45605 q^{89} -0.601319 q^{91} +8.94356 q^{92} -2.17024 q^{94} +14.3773 q^{97} -5.49525 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.87939	1.73491	0.867457	−	0.497512i	$-0.165753\pi$
$5$	3.87939	1.73491	0.867457	+	0.497512i	$0.165753\pi$
$6$	0	0
$7$	1.22668	0.463642	0.231821	−	0.972758i	$-0.425532\pi$
$7$	1.22668	0.463642	0.231821	+	0.972758i	$0.425532\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.87939	1.22677
$11$	2.12061	0.639389	0.319695	−	0.947521i	$-0.396420\pi$
$11$	2.12061	0.639389	0.319695	+	0.947521i	$0.396420\pi$
$12$	0	0
$13$	−0.490200	−0.135957	−0.0679785	−	0.997687i	$-0.521655\pi$
$13$	−0.490200	−0.135957	−0.0679785	+	0.997687i	$0.521655\pi$
$14$	1.22668	0.327844
$15$	0	0
$16$	1.00000	0.250000
$17$	5.53209	1.34173	0.670864	−	0.741580i	$-0.265923\pi$
$17$	5.53209	1.34173	0.670864	+	0.741580i	$0.265923\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.87939	0.867457
$21$	0	0
$22$	2.12061	0.452117
$23$	8.94356	1.86486	0.932431	−	0.361348i	$-0.117683\pi$
$23$	8.94356	1.86486	0.932431	+	0.361348i	$0.117683\pi$
$24$	0	0
$25$	10.0496	2.00993
$26$	−0.490200	−0.0961361
$27$	0	0
$28$	1.22668	0.231821
$29$	−8.47565	−1.57389	−0.786945	−	0.617024i	$-0.788338\pi$
$29$	−8.47565	−1.57389	−0.786945	+	0.617024i	$0.788338\pi$
$30$	0	0
$31$	−2.41147	−0.433114	−0.216557	−	0.976270i	$-0.569483\pi$
$31$	−2.41147	−0.433114	−0.216557	+	0.976270i	$0.569483\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.53209	0.948745
$35$	4.75877	0.804379
$36$	0	0
$37$	−1.69459	−0.278589	−0.139295	−	0.990251i	$-0.544484\pi$
$37$	−1.69459	−0.278589	−0.139295	+	0.990251i	$0.544484\pi$
$38$	0	0
$39$	0	0
$40$	3.87939	0.613385
$41$	1.59627	0.249295	0.124647	−	0.992201i	$-0.460220\pi$
$41$	1.59627	0.249295	0.124647	+	0.992201i	$0.460220\pi$
$42$	0	0
$43$	−6.63816	−1.01231	−0.506155	−	0.862443i	$-0.668933\pi$
$43$	−6.63816	−1.01231	−0.506155	+	0.862443i	$0.668933\pi$
$44$	2.12061	0.319695
$45$	0	0
$46$	8.94356	1.31866
$47$	−2.17024	−0.316563	−0.158281	−	0.987394i	$-0.550595\pi$
$47$	−2.17024	−0.316563	−0.158281	+	0.987394i	$0.550595\pi$
$48$	0	0
$49$	−5.49525	−0.785036
$50$	10.0496	1.42123
$51$	0	0
$52$	−0.490200	−0.0679785
$53$	8.88713	1.22074	0.610370	−	0.792116i	$-0.291021\pi$
$53$	8.88713	1.22074	0.610370	+	0.792116i	$0.291021\pi$
$54$	0	0
$55$	8.22668	1.10929
$56$	1.22668	0.163922
$57$	0	0
$58$	−8.47565	−1.11291
$59$	−11.4192	−1.48666	−0.743328	−	0.668928i	$-0.766754\pi$
$59$	−11.4192	−1.48666	−0.743328	+	0.668928i	$0.766754\pi$
$60$	0	0
$61$	0.0418891	0.00536335	0.00268167	−	0.999996i	$-0.499146\pi$
$61$	0.0418891	0.00536335	0.00268167	+	0.999996i	$0.499146\pi$
$62$	−2.41147	−0.306258
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.90167	−0.235874
$66$	0	0
$67$	−4.47565	−0.546788	−0.273394	−	0.961902i	$-0.588146\pi$
$67$	−4.47565	−0.546788	−0.273394	+	0.961902i	$0.588146\pi$
$68$	5.53209	0.670864
$69$	0	0
$70$	4.75877	0.568782
$71$	2.63816	0.313091	0.156546	−	0.987671i	$-0.449964\pi$
$71$	2.63816	0.313091	0.156546	+	0.987671i	$0.449964\pi$
$72$	0	0
$73$	−15.3628	−1.79808	−0.899039	−	0.437869i	$-0.855733\pi$
$73$	−15.3628	−1.79808	−0.899039	+	0.437869i	$0.855733\pi$
$74$	−1.69459	−0.196992
$75$	0	0
$76$	0	0
$77$	2.60132	0.296448
$78$	0	0
$79$	4.66044	0.524341	0.262170	−	0.965022i	$-0.415562\pi$
$79$	4.66044	0.524341	0.262170	+	0.965022i	$0.415562\pi$
$80$	3.87939	0.433728
$81$	0	0
$82$	1.59627	0.176278
$83$	−12.4807	−1.36994	−0.684968	−	0.728573i	$-0.740184\pi$
$83$	−12.4807	−1.36994	−0.684968	+	0.728573i	$0.740184\pi$
$84$	0	0
$85$	21.4611	2.32778
$86$	−6.63816	−0.715811
$87$	0	0
$88$	2.12061	0.226058
$89$	8.45605	0.896340	0.448170	−	0.893948i	$-0.352076\pi$
$89$	8.45605	0.896340	0.448170	+	0.893948i	$0.352076\pi$
$90$	0	0
$91$	−0.601319	−0.0630354
$92$	8.94356	0.932431
$93$	0	0
$94$	−2.17024	−0.223844
$95$	0	0
$96$	0	0
$97$	14.3773	1.45980	0.729898	−	0.683556i	$-0.239567\pi$
$97$	14.3773	1.45980	0.729898	+	0.683556i	$0.239567\pi$
$98$	−5.49525	−0.555104
$99$	0	0
$100$	10.0496	1.00496
$101$	−5.69459	−0.566633	−0.283317	−	0.959026i	$-0.591435\pi$
$101$	−5.69459	−0.566633	−0.283317	+	0.959026i	$0.591435\pi$
$102$	0	0
$103$	0.313148	0.0308554	0.0154277	−	0.999881i	$-0.495089\pi$
$103$	0.313148	0.0308554	0.0154277	+	0.999881i	$0.495089\pi$
$104$	−0.490200	−0.0480680
$105$	0	0
$106$	8.88713	0.863194
$107$	−16.8280	−1.62682	−0.813412	−	0.581688i	$-0.802393\pi$
$107$	−16.8280	−1.62682	−0.813412	+	0.581688i	$0.802393\pi$
$108$	0	0
$109$	10.4115	0.997238	0.498619	−	0.866821i	$-0.333841\pi$
$109$	10.4115	0.997238	0.498619	+	0.866821i	$0.333841\pi$
$110$	8.22668	0.784383
$111$	0	0
$112$	1.22668	0.115911
$113$	−17.3824	−1.63520	−0.817598	−	0.575789i	$-0.804695\pi$
$113$	−17.3824	−1.63520	−0.817598	+	0.575789i	$0.804695\pi$
$114$	0	0
$115$	34.6955	3.23537
$116$	−8.47565	−0.786945
$117$	0	0
$118$	−11.4192	−1.05122
$119$	6.78611	0.622082
$120$	0	0
$121$	−6.50299	−0.591181
$122$	0.0418891	0.00379246
$123$	0	0
$124$	−2.41147	−0.216557
$125$	19.5895	1.75213
$126$	0	0
$127$	1.63041	0.144676	0.0723380	−	0.997380i	$-0.476954\pi$
$127$	1.63041	0.144676	0.0723380	+	0.997380i	$0.476954\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.90167	−0.166788
$131$	−5.96316	−0.521004	−0.260502	−	0.965473i	$-0.583888\pi$
$131$	−5.96316	−0.521004	−0.260502	+	0.965473i	$0.583888\pi$
$132$	0	0
$133$	0	0
$134$	−4.47565	−0.386637
$135$	0	0
$136$	5.53209	0.474373
$137$	0.0196004	0.00167457	0.000837286	−	1.00000i	$-0.499733\pi$
$137$	0.0196004	0.00167457	0.000837286		1.00000i	$0.499733\pi$
$138$	0	0
$139$	4.49525	0.381282	0.190641	−	0.981660i	$-0.438943\pi$
$139$	4.49525	0.381282	0.190641	+	0.981660i	$0.438943\pi$
$140$	4.75877	0.402190
$141$	0	0
$142$	2.63816	0.221389
$143$	−1.03952	−0.0869294
$144$	0	0
$145$	−32.8803	−2.73056
$146$	−15.3628	−1.27143
$147$	0	0
$148$	−1.69459	−0.139295
$149$	−12.0300	−0.985538	−0.492769	−	0.870160i	$-0.664015\pi$
$149$	−12.0300	−0.985538	−0.492769	+	0.870160i	$0.664015\pi$
$150$	0	0
$151$	−0.0591253	−0.00481155	−0.00240578	−	0.999997i	$-0.500766\pi$
$151$	−0.0591253	−0.00481155	−0.00240578	+	0.999997i	$0.500766\pi$
$152$	0	0
$153$	0	0
$154$	2.60132	0.209620
$155$	−9.35504	−0.751415
$156$	0	0
$157$	12.9581	1.03417	0.517085	−	0.855934i	$-0.327017\pi$
$157$	12.9581	1.03417	0.517085	+	0.855934i	$0.327017\pi$
$158$	4.66044	0.370765
$159$	0	0
$160$	3.87939	0.306692
$161$	10.9709	0.864628
$162$	0	0
$163$	15.0719	1.18052	0.590262	−	0.807212i	$-0.299024\pi$
$163$	15.0719	1.18052	0.590262	+	0.807212i	$0.299024\pi$
$164$	1.59627	0.124647
$165$	0	0
$166$	−12.4807	−0.968691
$167$	−1.94087	−0.150189	−0.0750947	−	0.997176i	$-0.523926\pi$
$167$	−1.94087	−0.150189	−0.0750947	+	0.997176i	$0.523926\pi$
$168$	0	0
$169$	−12.7597	−0.981516
$170$	21.4611	1.64599
$171$	0	0
$172$	−6.63816	−0.506155
$173$	16.0719	1.22193	0.610963	−	0.791659i	$-0.290783\pi$
$173$	16.0719	1.22193	0.610963	+	0.791659i	$0.290783\pi$
$174$	0	0
$175$	12.3277	0.931886
$176$	2.12061	0.159847
$177$	0	0
$178$	8.45605	0.633808
$179$	14.7743	1.10428	0.552140	−	0.833752i	$-0.313811\pi$
$179$	14.7743	1.10428	0.552140	+	0.833752i	$0.313811\pi$
$180$	0	0
$181$	18.2267	1.35478	0.677389	−	0.735625i	$-0.263111\pi$
$181$	18.2267	1.35478	0.677389	+	0.735625i	$0.263111\pi$
$182$	−0.601319	−0.0445727
$183$	0	0
$184$	8.94356	0.659328
$185$	−6.57398	−0.483328
$186$	0	0
$187$	11.7314	0.857887
$188$	−2.17024	−0.158281
$189$	0	0
$190$	0	0
$191$	−1.29086	−0.0934033	−0.0467017	−	0.998909i	$-0.514871\pi$
$191$	−1.29086	−0.0934033	−0.0467017	+	0.998909i	$0.514871\pi$
$192$	0	0
$193$	9.12061	0.656516	0.328258	−	0.944588i	$-0.393538\pi$
$193$	9.12061	0.656516	0.328258	+	0.944588i	$0.393538\pi$
$194$	14.3773	1.03223
$195$	0	0
$196$	−5.49525	−0.392518
$197$	−18.6236	−1.32688	−0.663439	−	0.748231i	$-0.730904\pi$
$197$	−18.6236	−1.32688	−0.663439	+	0.748231i	$0.730904\pi$
$198$	0	0
$199$	8.43107	0.597663	0.298832	−	0.954306i	$-0.403403\pi$
$199$	8.43107	0.597663	0.298832	+	0.954306i	$0.403403\pi$
$200$	10.0496	0.710616
$201$	0	0
$202$	−5.69459	−0.400670
$203$	−10.3969	−0.729721
$204$	0	0
$205$	6.19253	0.432505
$206$	0.313148	0.0218181
$207$	0	0
$208$	−0.490200	−0.0339892
$209$	0	0
$210$	0	0
$211$	1.46791	0.101055	0.0505276	−	0.998723i	$-0.483910\pi$
$211$	1.46791	0.101055	0.0505276	+	0.998723i	$0.483910\pi$
$212$	8.88713	0.610370
$213$	0	0
$214$	−16.8280	−1.15034
$215$	−25.7520	−1.75627
$216$	0	0
$217$	−2.95811	−0.200810
$218$	10.4115	0.705154
$219$	0	0
$220$	8.22668	0.554643
$221$	−2.71183	−0.182417
$222$	0	0
$223$	−26.2686	−1.75907	−0.879537	−	0.475831i	$-0.842147\pi$
$223$	−26.2686	−1.75907	−0.879537	+	0.475831i	$0.842147\pi$
$224$	1.22668	0.0819611
$225$	0	0
$226$	−17.3824	−1.15626
$227$	18.3678	1.21912	0.609558	−	0.792742i	$-0.291347\pi$
$227$	18.3678	1.21912	0.609558	+	0.792742i	$0.291347\pi$
$228$	0	0
$229$	3.87164	0.255845	0.127923	−	0.991784i	$-0.459169\pi$
$229$	3.87164	0.255845	0.127923	+	0.991784i	$0.459169\pi$
$230$	34.6955	2.28776
$231$	0	0
$232$	−8.47565	−0.556454
$233$	−16.4466	−1.07745	−0.538725	−	0.842482i	$-0.681094\pi$
$233$	−16.4466	−1.07745	−0.538725	+	0.842482i	$0.681094\pi$
$234$	0	0
$235$	−8.41921	−0.549209
$236$	−11.4192	−0.743328
$237$	0	0
$238$	6.78611	0.439878
$239$	−3.30541	−0.213809	−0.106905	−	0.994269i	$-0.534094\pi$
$239$	−3.30541	−0.213809	−0.106905	+	0.994269i	$0.534094\pi$
$240$	0	0
$241$	−3.01548	−0.194244	−0.0971221	−	0.995272i	$-0.530964\pi$
$241$	−3.01548	−0.194244	−0.0971221	+	0.995272i	$0.530964\pi$
$242$	−6.50299	−0.418028
$243$	0	0
$244$	0.0418891	0.00268167
$245$	−21.3182	−1.36197
$246$	0	0
$247$	0	0
$248$	−2.41147	−0.153129
$249$	0	0
$250$	19.5895	1.23895
$251$	3.53478	0.223113	0.111557	−	0.993758i	$-0.464416\pi$
$251$	3.53478	0.223113	0.111557	+	0.993758i	$0.464416\pi$
$252$	0	0
$253$	18.9659	1.19237
$254$	1.63041	0.102301
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.70233	−0.293324	−0.146662	−	0.989187i	$-0.546853\pi$
$257$	−4.70233	−0.293324	−0.146662	+	0.989187i	$0.546853\pi$
$258$	0	0
$259$	−2.07873	−0.129166
$260$	−1.90167	−0.117937
$261$	0	0
$262$	−5.96316	−0.368405
$263$	−21.0624	−1.29876	−0.649382	−	0.760462i	$-0.724972\pi$
$263$	−21.0624	−1.29876	−0.649382	+	0.760462i	$0.724972\pi$
$264$	0	0
$265$	34.4766	2.11788
$266$	0	0
$267$	0	0
$268$	−4.47565	−0.273394
$269$	10.0787	0.614511	0.307255	−	0.951627i	$-0.400589\pi$
$269$	10.0787	0.614511	0.307255	+	0.951627i	$0.400589\pi$
$270$	0	0
$271$	−9.76382	−0.593110	−0.296555	−	0.955016i	$-0.595838\pi$
$271$	−9.76382	−0.593110	−0.296555	+	0.955016i	$0.595838\pi$
$272$	5.53209	0.335432
$273$	0	0
$274$	0.0196004	0.00118410
$275$	21.3114	1.28513
$276$	0	0
$277$	16.8571	1.01284	0.506422	−	0.862286i	$-0.330968\pi$
$277$	16.8571	1.01284	0.506422	+	0.862286i	$0.330968\pi$
$278$	4.49525	0.269607
$279$	0	0
$280$	4.75877	0.284391
$281$	−11.7246	−0.699432	−0.349716	−	0.936856i	$-0.613722\pi$
$281$	−11.7246	−0.699432	−0.349716	+	0.936856i	$0.613722\pi$
$282$	0	0
$283$	−21.5449	−1.28071	−0.640355	−	0.768079i	$-0.721213\pi$
$283$	−21.5449	−1.28071	−0.640355	+	0.768079i	$0.721213\pi$
$284$	2.63816	0.156546
$285$	0	0
$286$	−1.03952	−0.0614684
$287$	1.95811	0.115584
$288$	0	0
$289$	13.6040	0.800236
$290$	−32.8803	−1.93080
$291$	0	0
$292$	−15.3628	−0.899039
$293$	−11.1206	−0.649673	−0.324837	−	0.945770i	$-0.605309\pi$
$293$	−11.1206	−0.649673	−0.324837	+	0.945770i	$0.605309\pi$
$294$	0	0
$295$	−44.2995	−2.57922
$296$	−1.69459	−0.0984962
$297$	0	0
$298$	−12.0300	−0.696881
$299$	−4.38413	−0.253541
$300$	0	0
$301$	−8.14290	−0.469349
$302$	−0.0591253	−0.00340228
$303$	0	0
$304$	0	0
$305$	0.162504	0.00930494
$306$	0	0
$307$	−25.0401	−1.42912	−0.714558	−	0.699576i	$-0.753372\pi$
$307$	−25.0401	−1.42912	−0.714558	+	0.699576i	$0.753372\pi$
$308$	2.60132	0.148224
$309$	0	0
$310$	−9.35504	−0.531330
$311$	−0.0564370	−0.00320025	−0.00160012	−	0.999999i	$-0.500509\pi$
$311$	−0.0564370	−0.00320025	−0.00160012	+	0.999999i	$0.500509\pi$
$312$	0	0
$313$	13.7638	0.777977	0.388989	−	0.921243i	$-0.372825\pi$
$313$	13.7638	0.777977	0.388989	+	0.921243i	$0.372825\pi$
$314$	12.9581	0.731269
$315$	0	0
$316$	4.66044	0.262170
$317$	10.9040	0.612432	0.306216	−	0.951962i	$-0.400937\pi$
$317$	10.9040	0.612432	0.306216	+	0.951962i	$0.400937\pi$
$318$	0	0
$319$	−17.9736	−1.00633
$320$	3.87939	0.216864
$321$	0	0
$322$	10.9709	0.611385
$323$	0	0
$324$	0	0
$325$	−4.92633	−0.273263
$326$	15.0719	0.834756
$327$	0	0
$328$	1.59627	0.0881391
$329$	−2.66220	−0.146772
$330$	0	0
$331$	−8.59121	−0.472216	−0.236108	−	0.971727i	$-0.575872\pi$
$331$	−8.59121	−0.472216	−0.236108	+	0.971727i	$0.575872\pi$
$332$	−12.4807	−0.684968
$333$	0	0
$334$	−1.94087	−0.106200
$335$	−17.3628	−0.948630
$336$	0	0
$337$	2.31046	0.125859	0.0629294	−	0.998018i	$-0.479956\pi$
$337$	2.31046	0.125859	0.0629294	+	0.998018i	$0.479956\pi$
$338$	−12.7597	−0.694036
$339$	0	0
$340$	21.4611	1.16389
$341$	−5.11381	−0.276928
$342$	0	0
$343$	−15.3277	−0.827618
$344$	−6.63816	−0.357905
$345$	0	0
$346$	16.0719	0.864032
$347$	3.28581	0.176391	0.0881957	−	0.996103i	$-0.471890\pi$
$347$	3.28581	0.176391	0.0881957	+	0.996103i	$0.471890\pi$
$348$	0	0
$349$	29.3482	1.57097	0.785487	−	0.618878i	$-0.212412\pi$
$349$	29.3482	1.57097	0.785487	+	0.618878i	$0.212412\pi$
$350$	12.3277	0.658943
$351$	0	0
$352$	2.12061	0.113029
$353$	7.73917	0.411914	0.205957	−	0.978561i	$-0.433969\pi$
$353$	7.73917	0.411914	0.205957	+	0.978561i	$0.433969\pi$
$354$	0	0
$355$	10.2344	0.543187
$356$	8.45605	0.448170
$357$	0	0
$358$	14.7743	0.780843
$359$	−2.86753	−0.151342	−0.0756711	−	0.997133i	$-0.524110\pi$
$359$	−2.86753	−0.151342	−0.0756711	+	0.997133i	$0.524110\pi$
$360$	0	0
$361$	0	0
$362$	18.2267	0.957973
$363$	0	0
$364$	−0.601319	−0.0315177
$365$	−59.5981	−3.11951
$366$	0	0
$367$	−22.9659	−1.19881	−0.599404	−	0.800447i	$-0.704596\pi$
$367$	−22.9659	−1.19881	−0.599404	+	0.800447i	$0.704596\pi$
$368$	8.94356	0.466215
$369$	0	0
$370$	−6.57398	−0.341765
$371$	10.9017	0.565987
$372$	0	0
$373$	23.1908	1.20077	0.600387	−	0.799710i	$-0.295013\pi$
$373$	23.1908	1.20077	0.600387	+	0.799710i	$0.295013\pi$
$374$	11.7314	0.606618
$375$	0	0
$376$	−2.17024	−0.111922
$377$	4.15476	0.213981
$378$	0	0
$379$	33.5185	1.72173	0.860864	−	0.508835i	$-0.169924\pi$
$379$	33.5185	1.72173	0.860864	+	0.508835i	$0.169924\pi$
$380$	0	0
$381$	0	0
$382$	−1.29086	−0.0660461
$383$	6.72967	0.343870	0.171935	−	0.985108i	$-0.444998\pi$
$383$	6.72967	0.343870	0.171935	+	0.985108i	$0.444998\pi$
$384$	0	0
$385$	10.0915	0.514311
$386$	9.12061	0.464227
$387$	0	0
$388$	14.3773	0.729898
$389$	19.0223	0.964468	0.482234	−	0.876042i	$-0.339825\pi$
$389$	19.0223	0.964468	0.482234	+	0.876042i	$0.339825\pi$
$390$	0	0
$391$	49.4766	2.50214
$392$	−5.49525	−0.277552
$393$	0	0
$394$	−18.6236	−0.938244
$395$	18.0797	0.909686
$396$	0	0
$397$	−26.9145	−1.35080	−0.675399	−	0.737452i	$-0.736029\pi$
$397$	−26.9145	−1.35080	−0.675399	+	0.737452i	$0.736029\pi$
$398$	8.43107	0.422612
$399$	0	0
$400$	10.0496	0.502481
$401$	−5.70140	−0.284714	−0.142357	−	0.989815i	$-0.545468\pi$
$401$	−5.70140	−0.284714	−0.142357	+	0.989815i	$0.545468\pi$
$402$	0	0
$403$	1.18210	0.0588848
$404$	−5.69459	−0.283317
$405$	0	0
$406$	−10.3969	−0.515991
$407$	−3.59358	−0.178127
$408$	0	0
$409$	0.807467	0.0399267	0.0199633	−	0.999801i	$-0.493645\pi$
$409$	0.807467	0.0399267	0.0199633	+	0.999801i	$0.493645\pi$
$410$	6.19253	0.305827
$411$	0	0
$412$	0.313148	0.0154277
$413$	−14.0077	−0.689276
$414$	0	0
$415$	−48.4175	−2.37672
$416$	−0.490200	−0.0240340
$417$	0	0
$418$	0	0
$419$	19.1215	0.934149	0.467074	−	0.884218i	$-0.345308\pi$
$419$	19.1215	0.934149	0.467074	+	0.884218i	$0.345308\pi$
$420$	0	0
$421$	28.0232	1.36577	0.682884	−	0.730527i	$-0.260725\pi$
$421$	28.0232	1.36577	0.682884	+	0.730527i	$0.260725\pi$
$422$	1.46791	0.0714568
$423$	0	0
$424$	8.88713	0.431597
$425$	55.5954	2.69678
$426$	0	0
$427$	0.0513845	0.00248667
$428$	−16.8280	−0.813412
$429$	0	0
$430$	−25.7520	−1.24187
$431$	−9.03684	−0.435289	−0.217645	−	0.976028i	$-0.569837\pi$
$431$	−9.03684	−0.435289	−0.217645	+	0.976028i	$0.569837\pi$
$432$	0	0
$433$	−1.46286	−0.0703005	−0.0351503	−	0.999382i	$-0.511191\pi$
$433$	−1.46286	−0.0703005	−0.0351503	+	0.999382i	$0.511191\pi$
$434$	−2.95811	−0.141994
$435$	0	0
$436$	10.4115	0.498619
$437$	0	0
$438$	0	0
$439$	5.41653	0.258517	0.129258	−	0.991611i	$-0.458740\pi$
$439$	5.41653	0.258517	0.129258	+	0.991611i	$0.458740\pi$
$440$	8.22668	0.392192
$441$	0	0
$442$	−2.71183	−0.128989
$443$	28.4662	1.35247	0.676234	−	0.736687i	$-0.263611\pi$
$443$	28.4662	1.35247	0.676234	+	0.736687i	$0.263611\pi$
$444$	0	0
$445$	32.8043	1.55507
$446$	−26.2686	−1.24385
$447$	0	0
$448$	1.22668	0.0579553
$449$	−22.7648	−1.07434	−0.537168	−	0.843476i	$-0.680506\pi$
$449$	−22.7648	−1.07434	−0.537168	+	0.843476i	$0.680506\pi$
$450$	0	0
$451$	3.38507	0.159397
$452$	−17.3824	−0.817598
$453$	0	0
$454$	18.3678	0.862045
$455$	−2.33275	−0.109361
$456$	0	0
$457$	−1.13011	−0.0528643	−0.0264322	−	0.999651i	$-0.508415\pi$
$457$	−1.13011	−0.0528643	−0.0264322	+	0.999651i	$0.508415\pi$
$458$	3.87164	0.180910
$459$	0	0
$460$	34.6955	1.61769
$461$	12.9831	0.604683	0.302341	−	0.953200i	$-0.402232\pi$
$461$	12.9831	0.604683	0.302341	+	0.953200i	$0.402232\pi$
$462$	0	0
$463$	−30.6245	−1.42324	−0.711622	−	0.702563i	$-0.752039\pi$
$463$	−30.6245	−1.42324	−0.711622	+	0.702563i	$0.752039\pi$
$464$	−8.47565	−0.393472
$465$	0	0
$466$	−16.4466	−0.761872
$467$	−11.0574	−0.511674	−0.255837	−	0.966720i	$-0.582351\pi$
$467$	−11.0574	−0.511674	−0.255837	+	0.966720i	$0.582351\pi$
$468$	0	0
$469$	−5.49020	−0.253514
$470$	−8.41921	−0.388349
$471$	0	0
$472$	−11.4192	−0.525612
$473$	−14.0770	−0.647260
$474$	0	0
$475$	0	0
$476$	6.78611	0.311041
$477$	0	0
$478$	−3.30541	−0.151186
$479$	17.6382	0.805908	0.402954	−	0.915220i	$-0.367983\pi$
$479$	17.6382	0.805908	0.402954	+	0.915220i	$0.367983\pi$
$480$	0	0
$481$	0.830689	0.0378762
$482$	−3.01548	−0.137351
$483$	0	0
$484$	−6.50299	−0.295591
$485$	55.7752	2.53262
$486$	0	0
$487$	15.2935	0.693017	0.346508	−	0.938047i	$-0.387367\pi$
$487$	15.2935	0.693017	0.346508	+	0.938047i	$0.387367\pi$
$488$	0.0418891	0.00189623
$489$	0	0
$490$	−21.3182	−0.963058
$491$	1.77837	0.0802568	0.0401284	−	0.999195i	$-0.487223\pi$
$491$	1.77837	0.0802568	0.0401284	+	0.999195i	$0.487223\pi$
$492$	0	0
$493$	−46.8881	−2.11173
$494$	0	0
$495$	0	0
$496$	−2.41147	−0.108278
$497$	3.23618	0.145162
$498$	0	0
$499$	−0.107822	−0.00482675	−0.00241338	−	0.999997i	$-0.500768\pi$
$499$	−0.107822	−0.00482675	−0.00241338	+	0.999997i	$0.500768\pi$
$500$	19.5895	0.876067
$501$	0	0
$502$	3.53478	0.157765
$503$	23.6901	1.05629	0.528146	−	0.849154i	$-0.322887\pi$
$503$	23.6901	1.05629	0.528146	+	0.849154i	$0.322887\pi$
$504$	0	0
$505$	−22.0915	−0.983060
$506$	18.9659	0.843135
$507$	0	0
$508$	1.63041	0.0723380
$509$	−3.57398	−0.158414	−0.0792069	−	0.996858i	$-0.525239\pi$
$509$	−3.57398	−0.158414	−0.0792069	+	0.996858i	$0.525239\pi$
$510$	0	0
$511$	−18.8452	−0.833664
$512$	1.00000	0.0441942
$513$	0	0
$514$	−4.70233	−0.207411
$515$	1.21482	0.0535315
$516$	0	0
$517$	−4.60225	−0.202407
$518$	−2.07873	−0.0913340
$519$	0	0
$520$	−1.90167	−0.0833939
$521$	34.1215	1.49489	0.747446	−	0.664322i	$-0.231280\pi$
$521$	34.1215	1.49489	0.747446	+	0.664322i	$0.231280\pi$
$522$	0	0
$523$	−17.0669	−0.746282	−0.373141	−	0.927775i	$-0.621719\pi$
$523$	−17.0669	−0.746282	−0.373141	+	0.927775i	$0.621719\pi$
$524$	−5.96316	−0.260502
$525$	0	0
$526$	−21.0624	−0.918365
$527$	−13.3405	−0.581121
$528$	0	0
$529$	56.9873	2.47771
$530$	34.4766	1.49757
$531$	0	0
$532$	0	0
$533$	−0.782490	−0.0338934
$534$	0	0
$535$	−65.2823	−2.82240
$536$	−4.47565	−0.193319
$537$	0	0
$538$	10.0787	0.434525
$539$	−11.6533	−0.501944
$540$	0	0
$541$	26.3746	1.13393	0.566967	−	0.823740i	$-0.308117\pi$
$541$	26.3746	1.13393	0.566967	+	0.823740i	$0.308117\pi$
$542$	−9.76382	−0.419392
$543$	0	0
$544$	5.53209	0.237186
$545$	40.3901	1.73012
$546$	0	0
$547$	34.8958	1.49204	0.746018	−	0.665925i	$-0.231963\pi$
$547$	34.8958	1.49204	0.746018	+	0.665925i	$0.231963\pi$
$548$	0.0196004	0.000837286 0
$549$	0	0
$550$	21.3114	0.908721
$551$	0	0
$552$	0	0
$553$	5.71688	0.243107
$554$	16.8571	0.716189
$555$	0	0
$556$	4.49525	0.190641
$557$	32.5921	1.38097	0.690487	−	0.723345i	$-0.257396\pi$
$557$	32.5921	1.38097	0.690487	+	0.723345i	$0.257396\pi$
$558$	0	0
$559$	3.25402	0.137630
$560$	4.75877	0.201095
$561$	0	0
$562$	−11.7246	−0.494573
$563$	−4.79561	−0.202111	−0.101055	−	0.994881i	$-0.532222\pi$
$563$	−4.79561	−0.202111	−0.101055	+	0.994881i	$0.532222\pi$
$564$	0	0
$565$	−67.4329	−2.83693
$566$	−21.5449	−0.905599
$567$	0	0
$568$	2.63816	0.110695
$569$	−32.7006	−1.37088	−0.685440	−	0.728129i	$-0.740390\pi$
$569$	−32.7006	−1.37088	−0.685440	+	0.728129i	$0.740390\pi$
$570$	0	0
$571$	−10.9531	−0.458371	−0.229186	−	0.973383i	$-0.573606\pi$
$571$	−10.9531	−0.458371	−0.229186	+	0.973383i	$0.573606\pi$
$572$	−1.03952	−0.0434647
$573$	0	0
$574$	1.95811	0.0817300
$575$	89.8795	3.74823
$576$	0	0
$577$	−18.3027	−0.761952	−0.380976	−	0.924585i	$-0.624412\pi$
$577$	−18.3027	−0.761952	−0.380976	+	0.924585i	$0.624412\pi$
$578$	13.6040	0.565852
$579$	0	0
$580$	−32.8803	−1.36528
$581$	−15.3099	−0.635160
$582$	0	0
$583$	18.8462	0.780529
$584$	−15.3628	−0.635716
$585$	0	0
$586$	−11.1206	−0.459388
$587$	5.68416	0.234611	0.117305	−	0.993096i	$-0.462574\pi$
$587$	5.68416	0.234611	0.117305	+	0.993096i	$0.462574\pi$
$588$	0	0
$589$	0	0
$590$	−44.2995	−1.82378
$591$	0	0
$592$	−1.69459	−0.0696473
$593$	10.5098	0.431586	0.215793	−	0.976439i	$-0.430766\pi$
$593$	10.5098	0.431586	0.215793	+	0.976439i	$0.430766\pi$
$594$	0	0
$595$	26.3259	1.07926
$596$	−12.0300	−0.492769
$597$	0	0
$598$	−4.38413	−0.179281
$599$	−44.7110	−1.82684	−0.913421	−	0.407016i	$-0.866569\pi$
$599$	−44.7110	−1.82684	−0.913421	+	0.407016i	$0.866569\pi$
$600$	0	0
$601$	−32.8726	−1.34090	−0.670450	−	0.741955i	$-0.733899\pi$
$601$	−32.8726	−1.34090	−0.670450	+	0.741955i	$0.733899\pi$
$602$	−8.14290	−0.331880
$603$	0	0
$604$	−0.0591253	−0.00240578
$605$	−25.2276	−1.02565
$606$	0	0
$607$	−29.9486	−1.21558	−0.607788	−	0.794099i	$-0.707943\pi$
$607$	−29.9486	−1.21558	−0.607788	+	0.794099i	$0.707943\pi$
$608$	0	0
$609$	0	0
$610$	0.162504	0.00657959
$611$	1.06385	0.0430389
$612$	0	0
$613$	5.84018	0.235883	0.117941	−	0.993021i	$-0.462370\pi$
$613$	5.84018	0.235883	0.117941	+	0.993021i	$0.462370\pi$
$614$	−25.0401	−1.01054
$615$	0	0
$616$	2.60132	0.104810
$617$	12.5716	0.506114	0.253057	−	0.967451i	$-0.418564\pi$
$617$	12.5716	0.506114	0.253057	+	0.967451i	$0.418564\pi$
$618$	0	0
$619$	8.28581	0.333035	0.166517	−	0.986039i	$-0.446748\pi$
$619$	8.28581	0.333035	0.166517	+	0.986039i	$0.446748\pi$
$620$	−9.35504	−0.375707
$621$	0	0
$622$	−0.0564370	−0.00226292
$623$	10.3729	0.415581
$624$	0	0
$625$	25.7469	1.02988
$626$	13.7638	0.550113
$627$	0	0
$628$	12.9581	0.517085
$629$	−9.37464	−0.373791
$630$	0	0
$631$	6.82800	0.271818	0.135909	−	0.990721i	$-0.456604\pi$
$631$	6.82800	0.271818	0.135909	+	0.990721i	$0.456604\pi$
$632$	4.66044	0.185383
$633$	0	0
$634$	10.9040	0.433055
$635$	6.32501	0.251000
$636$	0	0
$637$	2.69377	0.106731
$638$	−17.9736	−0.711581
$639$	0	0
$640$	3.87939	0.153346
$641$	38.7110	1.52899	0.764496	−	0.644628i	$-0.222988\pi$
$641$	38.7110	1.52899	0.764496	+	0.644628i	$0.222988\pi$
$642$	0	0
$643$	−12.6013	−0.496948	−0.248474	−	0.968639i	$-0.579929\pi$
$643$	−12.6013	−0.496948	−0.248474	+	0.968639i	$0.579929\pi$
$644$	10.9709	0.432314
$645$	0	0
$646$	0	0
$647$	−8.41241	−0.330726	−0.165363	−	0.986233i	$-0.552880\pi$
$647$	−8.41241	−0.330726	−0.165363	+	0.986233i	$0.552880\pi$
$648$	0	0
$649$	−24.2158	−0.950552
$650$	−4.92633	−0.193226
$651$	0	0
$652$	15.0719	0.590262
$653$	−26.3800	−1.03233	−0.516165	−	0.856489i	$-0.672641\pi$
$653$	−26.3800	−1.03233	−0.516165	+	0.856489i	$0.672641\pi$
$654$	0	0
$655$	−23.1334	−0.903897
$656$	1.59627	0.0623237
$657$	0	0
$658$	−2.66220	−0.103783
$659$	32.0069	1.24681	0.623406	−	0.781898i	$-0.285748\pi$
$659$	32.0069	1.24681	0.623406	+	0.781898i	$0.285748\pi$
$660$	0	0
$661$	40.5280	1.57636	0.788178	−	0.615448i	$-0.211025\pi$
$661$	40.5280	1.57636	0.788178	+	0.615448i	$0.211025\pi$
$662$	−8.59121	−0.333907
$663$	0	0
$664$	−12.4807	−0.484345
$665$	0	0
$666$	0	0
$667$	−75.8025	−2.93509
$668$	−1.94087	−0.0750947
$669$	0	0
$670$	−17.3628	−0.670783
$671$	0.0888306	0.00342927
$672$	0	0
$673$	4.56118	0.175821	0.0879104	−	0.996128i	$-0.471981\pi$
$673$	4.56118	0.175821	0.0879104	+	0.996128i	$0.471981\pi$
$674$	2.31046	0.0889956
$675$	0	0
$676$	−12.7597	−0.490758
$677$	33.4415	1.28526	0.642631	−	0.766176i	$-0.277843\pi$
$677$	33.4415	1.28526	0.642631	+	0.766176i	$0.277843\pi$
$678$	0	0
$679$	17.6364	0.676823
$680$	21.4611	0.822996
$681$	0	0
$682$	−5.11381	−0.195818
$683$	−26.4825	−1.01332	−0.506662	−	0.862145i	$-0.669121\pi$
$683$	−26.4825	−1.01332	−0.506662	+	0.862145i	$0.669121\pi$
$684$	0	0
$685$	0.0760373	0.00290524
$686$	−15.3277	−0.585214
$687$	0	0
$688$	−6.63816	−0.253077
$689$	−4.35647	−0.165968
$690$	0	0
$691$	33.4005	1.27062	0.635308	−	0.772259i	$-0.280873\pi$
$691$	33.4005	1.27062	0.635308	+	0.772259i	$0.280873\pi$
$692$	16.0719	0.610963
$693$	0	0
$694$	3.28581	0.124728
$695$	17.4388	0.661492
$696$	0	0
$697$	8.83069	0.334486
$698$	29.3482	1.11085
$699$	0	0
$700$	12.3277	0.465943
$701$	−18.8794	−0.713065	−0.356532	−	0.934283i	$-0.616041\pi$
$701$	−18.8794	−0.713065	−0.356532	+	0.934283i	$0.616041\pi$
$702$	0	0
$703$	0	0
$704$	2.12061	0.0799237
$705$	0	0
$706$	7.73917	0.291268
$707$	−6.98545	−0.262715
$708$	0	0
$709$	−0.213067	−0.00800191	−0.00400096	−	0.999992i	$-0.501274\pi$
$709$	−0.213067	−0.00800191	−0.00400096	+	0.999992i	$0.501274\pi$
$710$	10.2344	0.384091
$711$	0	0
$712$	8.45605	0.316904
$713$	−21.5672	−0.807697
$714$	0	0
$715$	−4.03272	−0.150815
$716$	14.7743	0.552140
$717$	0	0
$718$	−2.86753	−0.107015
$719$	12.9659	0.483545	0.241772	−	0.970333i	$-0.422271\pi$
$719$	12.9659	0.483545	0.241772	+	0.970333i	$0.422271\pi$
$720$	0	0
$721$	0.384133	0.0143059
$722$	0	0
$723$	0	0
$724$	18.2267	0.677389
$725$	−85.1772	−3.16340
$726$	0	0
$727$	25.7246	0.954073	0.477037	−	0.878883i	$-0.341711\pi$
$727$	25.7246	0.954073	0.477037	+	0.878883i	$0.341711\pi$
$728$	−0.601319	−0.0222864
$729$	0	0
$730$	−59.5981	−2.20583
$731$	−36.7229	−1.35824
$732$	0	0
$733$	8.51991	0.314690	0.157345	−	0.987544i	$-0.449707\pi$
$733$	8.51991	0.314690	0.157345	+	0.987544i	$0.449707\pi$
$734$	−22.9659	−0.847685
$735$	0	0
$736$	8.94356	0.329664
$737$	−9.49113	−0.349610
$738$	0	0
$739$	−13.3054	−0.489447	−0.244724	−	0.969593i	$-0.578697\pi$
$739$	−13.3054	−0.489447	−0.244724	+	0.969593i	$0.578697\pi$
$740$	−6.57398	−0.241664
$741$	0	0
$742$	10.9017	0.400213
$743$	−26.9564	−0.988933	−0.494466	−	0.869197i	$-0.664637\pi$
$743$	−26.9564	−0.988933	−0.494466	+	0.869197i	$0.664637\pi$
$744$	0	0
$745$	−46.6691	−1.70982
$746$	23.1908	0.849075
$747$	0	0
$748$	11.7314	0.428944
$749$	−20.6426	−0.754264
$750$	0	0
$751$	−18.4397	−0.672876	−0.336438	−	0.941706i	$-0.609222\pi$
$751$	−18.4397	−0.672876	−0.336438	+	0.941706i	$0.609222\pi$
$752$	−2.17024	−0.0791407
$753$	0	0
$754$	4.15476	0.151308
$755$	−0.229370	−0.00834763
$756$	0	0
$757$	29.9162	1.08732	0.543662	−	0.839304i	$-0.317037\pi$
$757$	29.9162	1.08732	0.543662	+	0.839304i	$0.317037\pi$
$758$	33.5185	1.21745
$759$	0	0
$760$	0	0
$761$	28.6691	1.03925	0.519627	−	0.854393i	$-0.326071\pi$
$761$	28.6691	1.03925	0.519627	+	0.854393i	$0.326071\pi$
$762$	0	0
$763$	12.7716	0.462362
$764$	−1.29086	−0.0467017
$765$	0	0
$766$	6.72967	0.243153
$767$	5.59770	0.202121
$768$	0	0
$769$	−9.76146	−0.352007	−0.176004	−	0.984390i	$-0.556317\pi$
$769$	−9.76146	−0.352007	−0.176004	+	0.984390i	$0.556317\pi$
$770$	10.0915	0.363673
$771$	0	0
$772$	9.12061	0.328258
$773$	−3.89662	−0.140152	−0.0700759	−	0.997542i	$-0.522324\pi$
$773$	−3.89662	−0.140152	−0.0700759	+	0.997542i	$0.522324\pi$
$774$	0	0
$775$	−24.2344	−0.870526
$776$	14.3773	0.516116
$777$	0	0
$778$	19.0223	0.681982
$779$	0	0
$780$	0	0
$781$	5.59451	0.200187
$782$	49.4766	1.76928
$783$	0	0
$784$	−5.49525	−0.196259
$785$	50.2695	1.79420
$786$	0	0
$787$	2.94087	0.104831	0.0524154	−	0.998625i	$-0.483308\pi$
$787$	2.94087	0.104831	0.0524154	+	0.998625i	$0.483308\pi$
$788$	−18.6236	−0.663439
$789$	0	0
$790$	18.0797	0.643245
$791$	−21.3226	−0.758146
$792$	0	0
$793$	−0.0205340	−0.000729184 0
$794$	−26.9145	−0.955159
$795$	0	0
$796$	8.43107	0.298832
$797$	8.26950	0.292921	0.146460	−	0.989217i	$-0.453212\pi$
$797$	8.26950	0.292921	0.146460	+	0.989217i	$0.453212\pi$
$798$	0	0
$799$	−12.0060	−0.424741
$800$	10.0496	0.355308
$801$	0	0
$802$	−5.70140	−0.201323
$803$	−32.5785	−1.14967
$804$	0	0
$805$	42.5604	1.50006
$806$	1.18210	0.0416378
$807$	0	0
$808$	−5.69459	−0.200335
$809$	2.09059	0.0735011	0.0367505	−	0.999324i	$-0.488299\pi$
$809$	2.09059	0.0735011	0.0367505	+	0.999324i	$0.488299\pi$
$810$	0	0
$811$	−0.864837	−0.0303685	−0.0151843	−	0.999885i	$-0.504833\pi$
$811$	−0.864837	−0.0303685	−0.0151843	+	0.999885i	$0.504833\pi$
$812$	−10.3969	−0.364861
$813$	0	0
$814$	−3.59358	−0.125955
$815$	58.4698	2.04811
$816$	0	0
$817$	0	0
$818$	0.807467	0.0282324
$819$	0	0
$820$	6.19253	0.216253
$821$	−22.7656	−0.794524	−0.397262	−	0.917705i	$-0.630040\pi$
$821$	−22.7656	−0.794524	−0.397262	+	0.917705i	$0.630040\pi$
$822$	0	0
$823$	−33.1162	−1.15436	−0.577179	−	0.816618i	$-0.695846\pi$
$823$	−33.1162	−1.15436	−0.577179	+	0.816618i	$0.695846\pi$
$824$	0.313148	0.0109090
$825$	0	0
$826$	−14.0077	−0.487392
$827$	−26.5149	−0.922012	−0.461006	−	0.887397i	$-0.652511\pi$
$827$	−26.5149	−0.922012	−0.461006	+	0.887397i	$0.652511\pi$
$828$	0	0
$829$	3.15745	0.109663	0.0548314	−	0.998496i	$-0.482538\pi$
$829$	3.15745	0.109663	0.0548314	+	0.998496i	$0.482538\pi$
$830$	−48.4175	−1.68059
$831$	0	0
$832$	−0.490200	−0.0169946
$833$	−30.4002	−1.05331
$834$	0	0
$835$	−7.52940	−0.260566
$836$	0	0
$837$	0	0
$838$	19.1215	0.660543
$839$	−42.1189	−1.45410	−0.727052	−	0.686582i	$-0.759110\pi$
$839$	−42.1189	−1.45410	−0.727052	+	0.686582i	$0.759110\pi$
$840$	0	0
$841$	42.8367	1.47713
$842$	28.0232	0.965744
$843$	0	0
$844$	1.46791	0.0505276
$845$	−49.4998	−1.70285
$846$	0	0
$847$	−7.97710	−0.274096
$848$	8.88713	0.305185
$849$	0	0
$850$	55.5954	1.90691
$851$	−15.1557	−0.519531
$852$	0	0
$853$	56.5235	1.93533	0.967664	−	0.252241i	$-0.0811677\pi$
$853$	56.5235	1.93533	0.967664	+	0.252241i	$0.0811677\pi$
$854$	0.0513845	0.00175834
$855$	0	0
$856$	−16.8280	−0.575169
$857$	13.1916	0.450616	0.225308	−	0.974288i	$-0.427661\pi$
$857$	13.1916	0.450616	0.225308	+	0.974288i	$0.427661\pi$
$858$	0	0
$859$	−21.7510	−0.742136	−0.371068	−	0.928606i	$-0.621008\pi$
$859$	−21.7510	−0.742136	−0.371068	+	0.928606i	$0.621008\pi$
$860$	−25.7520	−0.878135
$861$	0	0
$862$	−9.03684	−0.307796
$863$	−46.9100	−1.59684	−0.798418	−	0.602104i	$-0.794329\pi$
$863$	−46.9100	−1.59684	−0.798418	+	0.602104i	$0.794329\pi$
$864$	0	0
$865$	62.3492	2.11994
$866$	−1.46286	−0.0497100
$867$	0	0
$868$	−2.95811	−0.100405
$869$	9.88301	0.335258
$870$	0	0
$871$	2.19396	0.0743396
$872$	10.4115	0.352577
$873$	0	0
$874$	0	0
$875$	24.0300	0.812363
$876$	0	0
$877$	2.67467	0.0903171	0.0451586	−	0.998980i	$-0.485621\pi$
$877$	2.67467	0.0903171	0.0451586	+	0.998980i	$0.485621\pi$
$878$	5.41653	0.182799
$879$	0	0
$880$	8.22668	0.277321
$881$	27.1097	0.913349	0.456674	−	0.889634i	$-0.349040\pi$
$881$	27.1097	0.913349	0.456674	+	0.889634i	$0.349040\pi$
$882$	0	0
$883$	−43.4243	−1.46134	−0.730671	−	0.682729i	$-0.760793\pi$
$883$	−43.4243	−1.46134	−0.730671	+	0.682729i	$0.760793\pi$
$884$	−2.71183	−0.0912087
$885$	0	0
$886$	28.4662	0.956339
$887$	−21.4371	−0.719786	−0.359893	−	0.932994i	$-0.617187\pi$
$887$	−21.4371	−0.719786	−0.359893	+	0.932994i	$0.617187\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	32.8043	1.09960
$891$	0	0
$892$	−26.2686	−0.879537
$893$	0	0
$894$	0	0
$895$	57.3150	1.91583
$896$	1.22668	0.0409806
$897$	0	0
$898$	−22.7648	−0.759670
$899$	20.4388	0.681673
$900$	0	0
$901$	49.1644	1.63790
$902$	3.38507	0.112710
$903$	0	0
$904$	−17.3824	−0.578129
$905$	70.7083	2.35042
$906$	0	0
$907$	16.3791	0.543858	0.271929	−	0.962317i	$-0.412338\pi$
$907$	16.3791	0.543858	0.271929	+	0.962317i	$0.412338\pi$
$908$	18.3678	0.609558
$909$	0	0
$910$	−2.33275	−0.0773299
$911$	28.4502	0.942596	0.471298	−	0.881974i	$-0.343786\pi$
$911$	28.4502	0.942596	0.471298	+	0.881974i	$0.343786\pi$
$912$	0	0
$913$	−26.4668	−0.875922
$914$	−1.13011	−0.0373807
$915$	0	0
$916$	3.87164	0.127923
$917$	−7.31490	−0.241559
$918$	0	0
$919$	−51.2535	−1.69070	−0.845349	−	0.534215i	$-0.820607\pi$
$919$	−51.2535	−1.69070	−0.845349	+	0.534215i	$0.820607\pi$
$920$	34.6955	1.14388
$921$	0	0
$922$	12.9831	0.427575
$923$	−1.29322	−0.0425670
$924$	0	0
$925$	−17.0300	−0.559944
$926$	−30.6245	−1.00638
$927$	0	0
$928$	−8.47565	−0.278227
$929$	−51.5536	−1.69142	−0.845709	−	0.533645i	$-0.820822\pi$
$929$	−51.5536	−1.69142	−0.845709	+	0.533645i	$0.820822\pi$
$930$	0	0
$931$	0	0
$932$	−16.4466	−0.538725
$933$	0	0
$934$	−11.0574	−0.361808
$935$	45.5107	1.48836
$936$	0	0
$937$	−14.4638	−0.472511	−0.236256	−	0.971691i	$-0.575920\pi$
$937$	−14.4638	−0.472511	−0.236256	+	0.971691i	$0.575920\pi$
$938$	−5.49020	−0.179261
$939$	0	0
$940$	−8.41921	−0.274605
$941$	−13.1803	−0.429667	−0.214834	−	0.976651i	$-0.568921\pi$
$941$	−13.1803	−0.429667	−0.214834	+	0.976651i	$0.568921\pi$
$942$	0	0
$943$	14.2763	0.464901
$944$	−11.4192	−0.371664
$945$	0	0
$946$	−14.0770	−0.457682
$947$	42.8266	1.39168	0.695838	−	0.718199i	$-0.255033\pi$
$947$	42.8266	1.39168	0.695838	+	0.718199i	$0.255033\pi$
$948$	0	0
$949$	7.53083	0.244461
$950$	0	0
$951$	0	0
$952$	6.78611	0.219939
$953$	−32.2175	−1.04363	−0.521814	−	0.853059i	$-0.674744\pi$
$953$	−32.2175	−1.04363	−0.521814	+	0.853059i	$0.674744\pi$
$954$	0	0
$955$	−5.00774	−0.162047
$956$	−3.30541	−0.106905
$957$	0	0
$958$	17.6382	0.569863
$959$	0.0240434	0.000776402 0
$960$	0	0
$961$	−25.1848	−0.812413
$962$	0.830689	0.0267825
$963$	0	0
$964$	−3.01548	−0.0971221
$965$	35.3824	1.13900
$966$	0	0
$967$	34.5485	1.11100	0.555502	−	0.831515i	$-0.312526\pi$
$967$	34.5485	1.11100	0.555502	+	0.831515i	$0.312526\pi$
$968$	−6.50299	−0.209014
$969$	0	0
$970$	55.7752	1.79083
$971$	−1.18716	−0.0380977	−0.0190488	−	0.999819i	$-0.506064\pi$
$971$	−1.18716	−0.0380977	−0.0190488	+	0.999819i	$0.506064\pi$
$972$	0	0
$973$	5.51424	0.176779
$974$	15.2935	0.490037
$975$	0	0
$976$	0.0418891	0.00134084
$977$	20.8827	0.668096	0.334048	−	0.942556i	$-0.391585\pi$
$977$	20.8827	0.668096	0.334048	+	0.942556i	$0.391585\pi$
$978$	0	0
$979$	17.9320	0.573110
$980$	−21.3182	−0.680985
$981$	0	0
$982$	1.77837	0.0567501
$983$	−35.0966	−1.11941	−0.559703	−	0.828693i	$-0.689085\pi$
$983$	−35.0966	−1.11941	−0.559703	+	0.828693i	$0.689085\pi$
$984$	0	0
$985$	−72.2481	−2.30202
$986$	−46.8881	−1.49322
$987$	0	0
$988$	0	0
$989$	−59.3688	−1.88782
$990$	0	0
$991$	49.7357	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$991$	49.7357	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$992$	−2.41147	−0.0765644
$993$	0	0
$994$	3.23618	0.102645
$995$	32.7074	1.03689
$996$	0	0
$997$	41.8334	1.32488	0.662438	−	0.749117i	$-0.269522\pi$
$997$	41.8334	1.32488	0.662438	+	0.749117i	$0.269522\pi$
$998$	−0.107822	−0.00341303
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bt.1.3		3
3.2	odd	2		2166.2.a.n.1.1		3
19.2	odd	18		342.2.u.d.289.1		6
19.10	odd	18		342.2.u.d.271.1		6
19.18	odd	2		6498.2.a.bo.1.3		3
57.2	even	18		114.2.i.b.61.1	yes	6
57.29	even	18		114.2.i.b.43.1	✓	6
57.56	even	2		2166.2.a.t.1.1		3
228.59	odd	18		912.2.bo.c.289.1		6
228.143	odd	18		912.2.bo.c.385.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.b.43.1	✓	6	57.29	even	18
114.2.i.b.61.1	yes	6	57.2	even	18
342.2.u.d.271.1		6	19.10	odd	18
342.2.u.d.289.1		6	19.2	odd	18
912.2.bo.c.289.1		6	228.59	odd	18
912.2.bo.c.385.1		6	228.143	odd	18
2166.2.a.n.1.1		3	3.2	odd	2
2166.2.a.t.1.1		3	57.56	even	2
6498.2.a.bo.1.3		3	19.18	odd	2
6498.2.a.bt.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.87939 q^{5} +1.22668 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.87939 q^{5} +1.22668 q^{7} +1.00000 q^{8} +3.87939 q^{10} +2.12061 q^{11} -0.490200 q^{13} +1.22668 q^{14} +1.00000 q^{16} +5.53209 q^{17} +3.87939 q^{20} +2.12061 q^{22} +8.94356 q^{23} +10.0496 q^{25} -0.490200 q^{26} +1.22668 q^{28} -8.47565 q^{29} -2.41147 q^{31} +1.00000 q^{32} +5.53209 q^{34} +4.75877 q^{35} -1.69459 q^{37} +3.87939 q^{40} +1.59627 q^{41} -6.63816 q^{43} +2.12061 q^{44} +8.94356 q^{46} -2.17024 q^{47} -5.49525 q^{49} +10.0496 q^{50} -0.490200 q^{52} +8.88713 q^{53} +8.22668 q^{55} +1.22668 q^{56} -8.47565 q^{58} -11.4192 q^{59} +0.0418891 q^{61} -2.41147 q^{62} +1.00000 q^{64} -1.90167 q^{65} -4.47565 q^{67} +5.53209 q^{68} +4.75877 q^{70} +2.63816 q^{71} -15.3628 q^{73} -1.69459 q^{74} +2.60132 q^{77} +4.66044 q^{79} +3.87939 q^{80} +1.59627 q^{82} -12.4807 q^{83} +21.4611 q^{85} -6.63816 q^{86} +2.12061 q^{88} +8.45605 q^{89} -0.601319 q^{91} +8.94356 q^{92} -2.17024 q^{94} +14.3773 q^{97} -5.49525 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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