LMFDB - Embedded newform 6498.2.a.ce.1.5

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			1.5
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.83582 q^{5} +2.85095 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.83582 q^{5} +2.85095 q^{7} +1.00000 q^{8} +3.83582 q^{10} -3.66344 q^{11} +4.49378 q^{13} +2.85095 q^{14} +1.00000 q^{16} +6.37317 q^{17} +3.83582 q^{20} -3.66344 q^{22} +1.66783 q^{23} +9.71353 q^{25} +4.49378 q^{26} +2.85095 q^{28} +4.70533 q^{29} -9.51272 q^{31} +1.00000 q^{32} +6.37317 q^{34} +10.9357 q^{35} +5.40622 q^{37} +3.83582 q^{40} -4.30599 q^{41} -6.30284 q^{43} -3.66344 q^{44} +1.66783 q^{46} -2.14181 q^{47} +1.12793 q^{49} +9.71353 q^{50} +4.49378 q^{52} -7.03364 q^{53} -14.0523 q^{55} +2.85095 q^{56} +4.70533 q^{58} -0.605411 q^{59} -10.3689 q^{61} -9.51272 q^{62} +1.00000 q^{64} +17.2373 q^{65} -6.59685 q^{67} +6.37317 q^{68} +10.9357 q^{70} -7.25819 q^{71} +7.24562 q^{73} +5.40622 q^{74} -10.4443 q^{77} -11.6170 q^{79} +3.83582 q^{80} -4.30599 q^{82} +8.26499 q^{83} +24.4463 q^{85} -6.30284 q^{86} -3.66344 q^{88} -4.86111 q^{89} +12.8116 q^{91} +1.66783 q^{92} -2.14181 q^{94} +2.58165 q^{97} +1.12793 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.83582	1.71543	0.857716	−	0.514124i	$-0.171883\pi$
$5$	3.83582	1.71543	0.857716	+	0.514124i	$0.171883\pi$
$6$	0	0
$7$	2.85095	1.07756	0.538779	−	0.842447i	$-0.318886\pi$
$7$	2.85095	1.07756	0.538779	+	0.842447i	$0.318886\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.83582	1.21299
$11$	−3.66344	−1.10457	−0.552285	−	0.833655i	$-0.686244\pi$
$11$	−3.66344	−1.10457	−0.552285	+	0.833655i	$0.686244\pi$
$12$	0	0
$13$	4.49378	1.24635	0.623175	−	0.782082i	$-0.285842\pi$
$13$	4.49378	1.24635	0.623175	+	0.782082i	$0.285842\pi$
$14$	2.85095	0.761949
$15$	0	0
$16$	1.00000	0.250000
$17$	6.37317	1.54572	0.772860	−	0.634577i	$-0.218826\pi$
$17$	6.37317	1.54572	0.772860	+	0.634577i	$0.218826\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.83582	0.857716
$21$	0	0
$22$	−3.66344	−0.781049
$23$	1.66783	0.347767	0.173884	−	0.984766i	$-0.444368\pi$
$23$	1.66783	0.347767	0.173884	+	0.984766i	$0.444368\pi$
$24$	0	0
$25$	9.71353	1.94271
$26$	4.49378	0.881303
$27$	0	0
$28$	2.85095	0.538779
$29$	4.70533	0.873758	0.436879	−	0.899520i	$-0.356084\pi$
$29$	4.70533	0.873758	0.436879	+	0.899520i	$0.356084\pi$
$30$	0	0
$31$	−9.51272	−1.70854	−0.854268	−	0.519833i	$-0.825994\pi$
$31$	−9.51272	−1.70854	−0.854268	+	0.519833i	$0.825994\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.37317	1.09299
$35$	10.9357	1.84848
$36$	0	0
$37$	5.40622	0.888777	0.444389	−	0.895834i	$-0.353421\pi$
$37$	5.40622	0.888777	0.444389	+	0.895834i	$0.353421\pi$
$38$	0	0
$39$	0	0
$40$	3.83582	0.606497
$41$	−4.30599	−0.672482	−0.336241	−	0.941776i	$-0.609156\pi$
$41$	−4.30599	−0.672482	−0.336241	+	0.941776i	$0.609156\pi$
$42$	0	0
$43$	−6.30284	−0.961175	−0.480587	−	0.876947i	$-0.659576\pi$
$43$	−6.30284	−0.961175	−0.480587	+	0.876947i	$0.659576\pi$
$44$	−3.66344	−0.552285
$45$	0	0
$46$	1.66783	0.245909
$47$	−2.14181	−0.312415	−0.156208	−	0.987724i	$-0.549927\pi$
$47$	−2.14181	−0.312415	−0.156208	+	0.987724i	$0.549927\pi$
$48$	0	0
$49$	1.12793	0.161132
$50$	9.71353	1.37370
$51$	0	0
$52$	4.49378	0.623175
$53$	−7.03364	−0.966144	−0.483072	−	0.875581i	$-0.660479\pi$
$53$	−7.03364	−0.966144	−0.483072	+	0.875581i	$0.660479\pi$
$54$	0	0
$55$	−14.0523	−1.89481
$56$	2.85095	0.380974
$57$	0	0
$58$	4.70533	0.617840
$59$	−0.605411	−0.0788178	−0.0394089	−	0.999223i	$-0.512547\pi$
$59$	−0.605411	−0.0788178	−0.0394089	+	0.999223i	$0.512547\pi$
$60$	0	0
$61$	−10.3689	−1.32761	−0.663803	−	0.747908i	$-0.731058\pi$
$61$	−10.3689	−1.32761	−0.663803	+	0.747908i	$0.731058\pi$
$62$	−9.51272	−1.20812
$63$	0	0
$64$	1.00000	0.125000
$65$	17.2373	2.13803
$66$	0	0
$67$	−6.59685	−0.805933	−0.402967	−	0.915215i	$-0.632021\pi$
$67$	−6.59685	−0.805933	−0.402967	+	0.915215i	$0.632021\pi$
$68$	6.37317	0.772860
$69$	0	0
$70$	10.9357	1.30707
$71$	−7.25819	−0.861388	−0.430694	−	0.902498i	$-0.641731\pi$
$71$	−7.25819	−0.861388	−0.430694	+	0.902498i	$0.641731\pi$
$72$	0	0
$73$	7.24562	0.848036	0.424018	−	0.905654i	$-0.360619\pi$
$73$	7.24562	0.848036	0.424018	+	0.905654i	$0.360619\pi$
$74$	5.40622	0.628460
$75$	0	0
$76$	0	0
$77$	−10.4443	−1.19024
$78$	0	0
$79$	−11.6170	−1.30702	−0.653509	−	0.756919i	$-0.726704\pi$
$79$	−11.6170	−1.30702	−0.653509	+	0.756919i	$0.726704\pi$
$80$	3.83582	0.428858
$81$	0	0
$82$	−4.30599	−0.475517
$83$	8.26499	0.907201	0.453600	−	0.891205i	$-0.350139\pi$
$83$	8.26499	0.907201	0.453600	+	0.891205i	$0.350139\pi$
$84$	0	0
$85$	24.4463	2.65158
$86$	−6.30284	−0.679653
$87$	0	0
$88$	−3.66344	−0.390524
$89$	−4.86111	−0.515277	−0.257638	−	0.966241i	$-0.582944\pi$
$89$	−4.86111	−0.515277	−0.257638	+	0.966241i	$0.582944\pi$
$90$	0	0
$91$	12.8116	1.34302
$92$	1.66783	0.173884
$93$	0	0
$94$	−2.14181	−0.220911
$95$	0	0
$96$	0	0
$97$	2.58165	0.262126	0.131063	−	0.991374i	$-0.458161\pi$
$97$	2.58165	0.262126	0.131063	+	0.991374i	$0.458161\pi$
$98$	1.12793	0.113938
$99$	0	0
$100$	9.71353	0.971353
$101$	19.0054	1.89111	0.945554	−	0.325464i	$-0.105520\pi$
$101$	19.0054	1.89111	0.945554	+	0.325464i	$0.105520\pi$
$102$	0	0
$103$	6.79180	0.669216	0.334608	−	0.942357i	$-0.391396\pi$
$103$	6.79180	0.669216	0.334608	+	0.942357i	$0.391396\pi$
$104$	4.49378	0.440651
$105$	0	0
$106$	−7.03364	−0.683167
$107$	−11.1290	−1.07588	−0.537942	−	0.842982i	$-0.680798\pi$
$107$	−11.1290	−1.07588	−0.537942	+	0.842982i	$0.680798\pi$
$108$	0	0
$109$	−6.90226	−0.661116	−0.330558	−	0.943786i	$-0.607237\pi$
$109$	−6.90226	−0.661116	−0.330558	+	0.943786i	$0.607237\pi$
$110$	−14.0523	−1.33984
$111$	0	0
$112$	2.85095	0.269390
$113$	−0.692489	−0.0651439	−0.0325719	−	0.999469i	$-0.510370\pi$
$113$	−0.692489	−0.0651439	−0.0325719	+	0.999469i	$0.510370\pi$
$114$	0	0
$115$	6.39751	0.596571
$116$	4.70533	0.436879
$117$	0	0
$118$	−0.605411	−0.0557326
$119$	18.1696	1.66560
$120$	0	0
$121$	2.42082	0.220075
$122$	−10.3689	−0.938759
$123$	0	0
$124$	−9.51272	−0.854268
$125$	18.0803	1.61715
$126$	0	0
$127$	−2.13435	−0.189393	−0.0946967	−	0.995506i	$-0.530188\pi$
$127$	−2.13435	−0.189393	−0.0946967	+	0.995506i	$0.530188\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	17.2373	1.51182
$131$	4.43800	0.387750	0.193875	−	0.981026i	$-0.437894\pi$
$131$	4.43800	0.387750	0.193875	+	0.981026i	$0.437894\pi$
$132$	0	0
$133$	0	0
$134$	−6.59685	−0.569881
$135$	0	0
$136$	6.37317	0.546494
$137$	10.5134	0.898219	0.449109	−	0.893477i	$-0.351741\pi$
$137$	10.5134	0.898219	0.449109	+	0.893477i	$0.351741\pi$
$138$	0	0
$139$	7.15473	0.606857	0.303428	−	0.952854i	$-0.401869\pi$
$139$	7.15473	0.606857	0.303428	+	0.952854i	$0.401869\pi$
$140$	10.9357	0.924239
$141$	0	0
$142$	−7.25819	−0.609094
$143$	−16.4627	−1.37668
$144$	0	0
$145$	18.0488	1.49887
$146$	7.24562	0.599652
$147$	0	0
$148$	5.40622	0.444389
$149$	−19.6469	−1.60953	−0.804767	−	0.593591i	$-0.797710\pi$
$149$	−19.6469	−1.60953	−0.804767	+	0.593591i	$0.797710\pi$
$150$	0	0
$151$	15.6402	1.27278	0.636390	−	0.771367i	$-0.280427\pi$
$151$	15.6402	1.27278	0.636390	+	0.771367i	$0.280427\pi$
$152$	0	0
$153$	0	0
$154$	−10.4443	−0.841626
$155$	−36.4891	−2.93088
$156$	0	0
$157$	−4.49797	−0.358977	−0.179488	−	0.983760i	$-0.557444\pi$
$157$	−4.49797	−0.358977	−0.179488	+	0.983760i	$0.557444\pi$
$158$	−11.6170	−0.924202
$159$	0	0
$160$	3.83582	0.303248
$161$	4.75491	0.374739
$162$	0	0
$163$	−0.753285	−0.0590018	−0.0295009	−	0.999565i	$-0.509392\pi$
$163$	−0.753285	−0.0590018	−0.0295009	+	0.999565i	$0.509392\pi$
$164$	−4.30599	−0.336241
$165$	0	0
$166$	8.26499	0.641488
$167$	−24.7123	−1.91229	−0.956146	−	0.292890i	$-0.905383\pi$
$167$	−24.7123	−1.91229	−0.956146	+	0.292890i	$0.905383\pi$
$168$	0	0
$169$	7.19406	0.553389
$170$	24.4463	1.87495
$171$	0	0
$172$	−6.30284	−0.480587
$173$	1.98010	0.150544	0.0752721	−	0.997163i	$-0.476017\pi$
$173$	1.98010	0.150544	0.0752721	+	0.997163i	$0.476017\pi$
$174$	0	0
$175$	27.6928	2.09338
$176$	−3.66344	−0.276142
$177$	0	0
$178$	−4.86111	−0.364356
$179$	−0.0791904	−0.00591897	−0.00295948	−	0.999996i	$-0.500942\pi$
$179$	−0.0791904	−0.00591897	−0.00295948	+	0.999996i	$0.500942\pi$
$180$	0	0
$181$	−15.8219	−1.17603	−0.588015	−	0.808850i	$-0.700090\pi$
$181$	−15.8219	−1.17603	−0.588015	+	0.808850i	$0.700090\pi$
$182$	12.8116	0.949655
$183$	0	0
$184$	1.66783	0.122954
$185$	20.7373	1.52464
$186$	0	0
$187$	−23.3477	−1.70736
$188$	−2.14181	−0.156208
$189$	0	0
$190$	0	0
$191$	−13.2510	−0.958810	−0.479405	−	0.877594i	$-0.659148\pi$
$191$	−13.2510	−0.958810	−0.479405	+	0.877594i	$0.659148\pi$
$192$	0	0
$193$	0.177125	0.0127497	0.00637487	−	0.999980i	$-0.497971\pi$
$193$	0.177125	0.0127497	0.00637487	+	0.999980i	$0.497971\pi$
$194$	2.58165	0.185351
$195$	0	0
$196$	1.12793	0.0805661
$197$	0.892456	0.0635849	0.0317924	−	0.999494i	$-0.489878\pi$
$197$	0.892456	0.0635849	0.0317924	+	0.999494i	$0.489878\pi$
$198$	0	0
$199$	−14.1219	−1.00107	−0.500536	−	0.865716i	$-0.666864\pi$
$199$	−14.1219	−1.00107	−0.500536	+	0.865716i	$0.666864\pi$
$200$	9.71353	0.686851
$201$	0	0
$202$	19.0054	1.33722
$203$	13.4147	0.941526
$204$	0	0
$205$	−16.5170	−1.15360
$206$	6.79180	0.473207
$207$	0	0
$208$	4.49378	0.311588
$209$	0	0
$210$	0	0
$211$	12.2518	0.843446	0.421723	−	0.906725i	$-0.361425\pi$
$211$	12.2518	0.843446	0.421723	+	0.906725i	$0.361425\pi$
$212$	−7.03364	−0.483072
$213$	0	0
$214$	−11.1290	−0.760764
$215$	−24.1766	−1.64883
$216$	0	0
$217$	−27.1203	−1.84105
$218$	−6.90226	−0.467480
$219$	0	0
$220$	−14.0523	−0.947407
$221$	28.6396	1.92651
$222$	0	0
$223$	−26.0379	−1.74362	−0.871812	−	0.489841i	$-0.837055\pi$
$223$	−26.0379	−1.74362	−0.871812	+	0.489841i	$0.837055\pi$
$224$	2.85095	0.190487
$225$	0	0
$226$	−0.692489	−0.0460637
$227$	−1.86936	−0.124074	−0.0620370	−	0.998074i	$-0.519760\pi$
$227$	−1.86936	−0.124074	−0.0620370	+	0.998074i	$0.519760\pi$
$228$	0	0
$229$	−5.82691	−0.385053	−0.192527	−	0.981292i	$-0.561668\pi$
$229$	−5.82691	−0.385053	−0.192527	+	0.981292i	$0.561668\pi$
$230$	6.39751	0.421839
$231$	0	0
$232$	4.70533	0.308920
$233$	5.57651	0.365329	0.182665	−	0.983175i	$-0.441528\pi$
$233$	5.57651	0.365329	0.182665	+	0.983175i	$0.441528\pi$
$234$	0	0
$235$	−8.21561	−0.535927
$236$	−0.605411	−0.0394089
$237$	0	0
$238$	18.1696	1.17776
$239$	17.3437	1.12187	0.560935	−	0.827860i	$-0.310442\pi$
$239$	17.3437	1.12187	0.560935	+	0.827860i	$0.310442\pi$
$240$	0	0
$241$	15.3731	0.990268	0.495134	−	0.868817i	$-0.335119\pi$
$241$	15.3731	0.990268	0.495134	+	0.868817i	$0.335119\pi$
$242$	2.42082	0.155616
$243$	0	0
$244$	−10.3689	−0.663803
$245$	4.32652	0.276411
$246$	0	0
$247$	0	0
$248$	−9.51272	−0.604058
$249$	0	0
$250$	18.0803	1.14350
$251$	15.4887	0.977635	0.488818	−	0.872386i	$-0.337428\pi$
$251$	15.4887	0.977635	0.488818	+	0.872386i	$0.337428\pi$
$252$	0	0
$253$	−6.11001	−0.384133
$254$	−2.13435	−0.133921
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.1168	0.880584	0.440292	−	0.897855i	$-0.354875\pi$
$257$	14.1168	0.880584	0.440292	+	0.897855i	$0.354875\pi$
$258$	0	0
$259$	15.4129	0.957709
$260$	17.2373	1.06901
$261$	0	0
$262$	4.43800	0.274181
$263$	16.1817	0.997808	0.498904	−	0.866657i	$-0.333736\pi$
$263$	16.1817	0.997808	0.498904	+	0.866657i	$0.333736\pi$
$264$	0	0
$265$	−26.9798	−1.65735
$266$	0	0
$267$	0	0
$268$	−6.59685	−0.402967
$269$	−30.3525	−1.85063	−0.925314	−	0.379203i	$-0.876198\pi$
$269$	−30.3525	−1.85063	−0.925314	+	0.379203i	$0.876198\pi$
$270$	0	0
$271$	24.0290	1.45966	0.729829	−	0.683630i	$-0.239600\pi$
$271$	24.0290	1.45966	0.729829	+	0.683630i	$0.239600\pi$
$272$	6.37317	0.386430
$273$	0	0
$274$	10.5134	0.635136
$275$	−35.5850	−2.14586
$276$	0	0
$277$	−21.9370	−1.31807	−0.659033	−	0.752114i	$-0.729034\pi$
$277$	−21.9370	−1.31807	−0.659033	+	0.752114i	$0.729034\pi$
$278$	7.15473	0.429112
$279$	0	0
$280$	10.9357	0.653536
$281$	21.7983	1.30037	0.650187	−	0.759774i	$-0.274690\pi$
$281$	21.7983	1.30037	0.650187	+	0.759774i	$0.274690\pi$
$282$	0	0
$283$	−27.7599	−1.65015	−0.825076	−	0.565021i	$-0.808868\pi$
$283$	−27.7599	−1.65015	−0.825076	+	0.565021i	$0.808868\pi$
$284$	−7.25819	−0.430694
$285$	0	0
$286$	−16.4627	−0.973461
$287$	−12.2762	−0.724639
$288$	0	0
$289$	23.6172	1.38925
$290$	18.0488	1.05986
$291$	0	0
$292$	7.24562	0.424018
$293$	32.0006	1.86949	0.934746	−	0.355315i	$-0.115627\pi$
$293$	32.0006	1.86949	0.934746	+	0.355315i	$0.115627\pi$
$294$	0	0
$295$	−2.32225	−0.135207
$296$	5.40622	0.314230
$297$	0	0
$298$	−19.6469	−1.13811
$299$	7.49487	0.433440
$300$	0	0
$301$	−17.9691	−1.03572
$302$	15.6402	0.899992
$303$	0	0
$304$	0	0
$305$	−39.7734	−2.27742
$306$	0	0
$307$	17.0453	0.972828	0.486414	−	0.873729i	$-0.338305\pi$
$307$	17.0453	0.972828	0.486414	+	0.873729i	$0.338305\pi$
$308$	−10.4443	−0.595119
$309$	0	0
$310$	−36.4891	−2.07244
$311$	19.4288	1.10171	0.550853	−	0.834602i	$-0.314303\pi$
$311$	19.4288	1.10171	0.550853	+	0.834602i	$0.314303\pi$
$312$	0	0
$313$	−31.1894	−1.76293	−0.881464	−	0.472250i	$-0.843442\pi$
$313$	−31.1894	−1.76293	−0.881464	+	0.472250i	$0.843442\pi$
$314$	−4.49797	−0.253835
$315$	0	0
$316$	−11.6170	−0.653509
$317$	−0.479164	−0.0269125	−0.0134563	−	0.999909i	$-0.504283\pi$
$317$	−0.479164	−0.0269125	−0.0134563	+	0.999909i	$0.504283\pi$
$318$	0	0
$319$	−17.2377	−0.965127
$320$	3.83582	0.214429
$321$	0	0
$322$	4.75491	0.264981
$323$	0	0
$324$	0	0
$325$	43.6505	2.42129
$326$	−0.753285	−0.0417206
$327$	0	0
$328$	−4.30599	−0.237758
$329$	−6.10620	−0.336646
$330$	0	0
$331$	20.5670	1.13046	0.565232	−	0.824932i	$-0.308787\pi$
$331$	20.5670	1.13046	0.565232	+	0.824932i	$0.308787\pi$
$332$	8.26499	0.453600
$333$	0	0
$334$	−24.7123	−1.35219
$335$	−25.3043	−1.38252
$336$	0	0
$337$	−4.92815	−0.268454	−0.134227	−	0.990951i	$-0.542855\pi$
$337$	−4.92815	−0.268454	−0.134227	+	0.990951i	$0.542855\pi$
$338$	7.19406	0.391305
$339$	0	0
$340$	24.4463	1.32579
$341$	34.8493	1.88720
$342$	0	0
$343$	−16.7410	−0.903929
$344$	−6.30284	−0.339827
$345$	0	0
$346$	1.98010	0.106451
$347$	25.2211	1.35394	0.676971	−	0.736010i	$-0.263292\pi$
$347$	25.2211	1.35394	0.676971	+	0.736010i	$0.263292\pi$
$348$	0	0
$349$	−12.5143	−0.669877	−0.334938	−	0.942240i	$-0.608715\pi$
$349$	−12.5143	−0.669877	−0.334938	+	0.942240i	$0.608715\pi$
$350$	27.6928	1.48024
$351$	0	0
$352$	−3.66344	−0.195262
$353$	8.41193	0.447722	0.223861	−	0.974621i	$-0.428134\pi$
$353$	8.41193	0.447722	0.223861	+	0.974621i	$0.428134\pi$
$354$	0	0
$355$	−27.8411	−1.47765
$356$	−4.86111	−0.257638
$357$	0	0
$358$	−0.0791904	−0.00418534
$359$	13.7877	0.727688	0.363844	−	0.931460i	$-0.381464\pi$
$359$	13.7877	0.727688	0.363844	+	0.931460i	$0.381464\pi$
$360$	0	0
$361$	0	0
$362$	−15.8219	−0.831578
$363$	0	0
$364$	12.8116	0.671508
$365$	27.7929	1.45475
$366$	0	0
$367$	4.10000	0.214018	0.107009	−	0.994258i	$-0.465873\pi$
$367$	4.10000	0.214018	0.107009	+	0.994258i	$0.465873\pi$
$368$	1.66783	0.0869418
$369$	0	0
$370$	20.7373	1.07808
$371$	−20.0526	−1.04108
$372$	0	0
$373$	26.4463	1.36934	0.684670	−	0.728854i	$-0.259947\pi$
$373$	26.4463	1.36934	0.684670	+	0.728854i	$0.259947\pi$
$374$	−23.3477	−1.20728
$375$	0	0
$376$	−2.14181	−0.110455
$377$	21.1447	1.08901
$378$	0	0
$379$	1.56826	0.0805564	0.0402782	−	0.999189i	$-0.487176\pi$
$379$	1.56826	0.0805564	0.0402782	+	0.999189i	$0.487176\pi$
$380$	0	0
$381$	0	0
$382$	−13.2510	−0.677981
$383$	21.9392	1.12104	0.560520	−	0.828141i	$-0.310601\pi$
$383$	21.9392	1.12104	0.560520	+	0.828141i	$0.310601\pi$
$384$	0	0
$385$	−40.0625	−2.04177
$386$	0.177125	0.00901543
$387$	0	0
$388$	2.58165	0.131063
$389$	14.7774	0.749244	0.374622	−	0.927178i	$-0.377772\pi$
$389$	14.7774	0.749244	0.374622	+	0.927178i	$0.377772\pi$
$390$	0	0
$391$	10.6294	0.537551
$392$	1.12793	0.0569688
$393$	0	0
$394$	0.892456	0.0449613
$395$	−44.5609	−2.24210
$396$	0	0
$397$	−5.47169	−0.274616	−0.137308	−	0.990528i	$-0.543845\pi$
$397$	−5.47169	−0.274616	−0.137308	+	0.990528i	$0.543845\pi$
$398$	−14.1219	−0.707865
$399$	0	0
$400$	9.71353	0.485677
$401$	−11.7257	−0.585553	−0.292776	−	0.956181i	$-0.594579\pi$
$401$	−11.7257	−0.585553	−0.292776	+	0.956181i	$0.594579\pi$
$402$	0	0
$403$	−42.7481	−2.12943
$404$	19.0054	0.945554
$405$	0	0
$406$	13.4147	0.665759
$407$	−19.8054	−0.981716
$408$	0	0
$409$	−17.2072	−0.850839	−0.425420	−	0.904996i	$-0.639873\pi$
$409$	−17.2072	−0.850839	−0.425420	+	0.904996i	$0.639873\pi$
$410$	−16.5170	−0.815717
$411$	0	0
$412$	6.79180	0.334608
$413$	−1.72600	−0.0849308
$414$	0	0
$415$	31.7030	1.55624
$416$	4.49378	0.220326
$417$	0	0
$418$	0	0
$419$	31.9505	1.56089	0.780443	−	0.625227i	$-0.214994\pi$
$419$	31.9505	1.56089	0.780443	+	0.625227i	$0.214994\pi$
$420$	0	0
$421$	−31.0906	−1.51526	−0.757631	−	0.652683i	$-0.773643\pi$
$421$	−31.0906	−1.51526	−0.757631	+	0.652683i	$0.773643\pi$
$422$	12.2518	0.596406
$423$	0	0
$424$	−7.03364	−0.341584
$425$	61.9060	3.00288
$426$	0	0
$427$	−29.5613	−1.43057
$428$	−11.1290	−0.537942
$429$	0	0
$430$	−24.1766	−1.16590
$431$	15.0956	0.727130	0.363565	−	0.931569i	$-0.381559\pi$
$431$	15.0956	0.727130	0.363565	+	0.931569i	$0.381559\pi$
$432$	0	0
$433$	−16.0311	−0.770406	−0.385203	−	0.922832i	$-0.625869\pi$
$433$	−16.0311	−0.770406	−0.385203	+	0.922832i	$0.625869\pi$
$434$	−27.1203	−1.30182
$435$	0	0
$436$	−6.90226	−0.330558
$437$	0	0
$438$	0	0
$439$	−6.26212	−0.298875	−0.149437	−	0.988771i	$-0.547746\pi$
$439$	−6.26212	−0.298875	−0.149437	+	0.988771i	$0.547746\pi$
$440$	−14.0523	−0.669918
$441$	0	0
$442$	28.6396	1.36225
$443$	−19.2192	−0.913133	−0.456566	−	0.889689i	$-0.650921\pi$
$443$	−19.2192	−0.913133	−0.456566	+	0.889689i	$0.650921\pi$
$444$	0	0
$445$	−18.6464	−0.883922
$446$	−26.0379	−1.23293
$447$	0	0
$448$	2.85095	0.134695
$449$	32.3059	1.52461	0.762304	−	0.647219i	$-0.224068\pi$
$449$	32.3059	1.52461	0.762304	+	0.647219i	$0.224068\pi$
$450$	0	0
$451$	15.7747	0.742804
$452$	−0.692489	−0.0325719
$453$	0	0
$454$	−1.86936	−0.0877336
$455$	49.1428	2.30385
$456$	0	0
$457$	1.80865	0.0846053	0.0423026	−	0.999105i	$-0.486531\pi$
$457$	1.80865	0.0846053	0.0423026	+	0.999105i	$0.486531\pi$
$458$	−5.82691	−0.272274
$459$	0	0
$460$	6.39751	0.298285
$461$	−35.5344	−1.65500	−0.827501	−	0.561464i	$-0.810238\pi$
$461$	−35.5344	−1.65500	−0.827501	+	0.561464i	$0.810238\pi$
$462$	0	0
$463$	−1.19885	−0.0557155	−0.0278577	−	0.999612i	$-0.508869\pi$
$463$	−1.19885	−0.0557155	−0.0278577	+	0.999612i	$0.508869\pi$
$464$	4.70533	0.218440
$465$	0	0
$466$	5.57651	0.258327
$467$	0.570033	0.0263780	0.0131890	−	0.999913i	$-0.495802\pi$
$467$	0.570033	0.0263780	0.0131890	+	0.999913i	$0.495802\pi$
$468$	0	0
$469$	−18.8073	−0.868440
$470$	−8.21561	−0.378958
$471$	0	0
$472$	−0.605411	−0.0278663
$473$	23.0901	1.06168
$474$	0	0
$475$	0	0
$476$	18.1696	0.832802
$477$	0	0
$478$	17.3437	0.793282
$479$	−25.3025	−1.15610	−0.578050	−	0.816002i	$-0.696186\pi$
$479$	−25.3025	−1.15610	−0.578050	+	0.816002i	$0.696186\pi$
$480$	0	0
$481$	24.2944	1.10773
$482$	15.3731	0.700225
$483$	0	0
$484$	2.42082	0.110037
$485$	9.90273	0.449660
$486$	0	0
$487$	−24.1268	−1.09329	−0.546644	−	0.837365i	$-0.684095\pi$
$487$	−24.1268	−1.09329	−0.546644	+	0.837365i	$0.684095\pi$
$488$	−10.3689	−0.469379
$489$	0	0
$490$	4.32652	0.195452
$491$	−13.0719	−0.589926	−0.294963	−	0.955509i	$-0.595307\pi$
$491$	−13.0719	−0.589926	−0.294963	+	0.955509i	$0.595307\pi$
$492$	0	0
$493$	29.9879	1.35059
$494$	0	0
$495$	0	0
$496$	−9.51272	−0.427134
$497$	−20.6927	−0.928196
$498$	0	0
$499$	22.9167	1.02589	0.512945	−	0.858421i	$-0.328554\pi$
$499$	22.9167	1.02589	0.512945	+	0.858421i	$0.328554\pi$
$500$	18.0803	0.808575
$501$	0	0
$502$	15.4887	0.691292
$503$	29.0431	1.29497	0.647484	−	0.762079i	$-0.275821\pi$
$503$	29.0431	1.29497	0.647484	+	0.762079i	$0.275821\pi$
$504$	0	0
$505$	72.9014	3.24407
$506$	−6.11001	−0.271623
$507$	0	0
$508$	−2.13435	−0.0946967
$509$	−0.850010	−0.0376760	−0.0188380	−	0.999823i	$-0.505997\pi$
$509$	−0.850010	−0.0376760	−0.0188380	+	0.999823i	$0.505997\pi$
$510$	0	0
$511$	20.6569	0.913808
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.1168	0.622667
$515$	26.0521	1.14799
$516$	0	0
$517$	7.84640	0.345084
$518$	15.4129	0.677203
$519$	0	0
$520$	17.2373	0.755908
$521$	0.0478945	0.00209830	0.00104915	−	0.999999i	$-0.499666\pi$
$521$	0.0478945	0.00209830	0.00104915	+	0.999999i	$0.499666\pi$
$522$	0	0
$523$	23.3302	1.02016	0.510079	−	0.860127i	$-0.329616\pi$
$523$	23.3302	1.02016	0.510079	+	0.860127i	$0.329616\pi$
$524$	4.43800	0.193875
$525$	0	0
$526$	16.1817	0.705557
$527$	−60.6262	−2.64092
$528$	0	0
$529$	−20.2183	−0.879058
$530$	−26.9798	−1.17193
$531$	0	0
$532$	0	0
$533$	−19.3502	−0.838149
$534$	0	0
$535$	−42.6890	−1.84560
$536$	−6.59685	−0.284940
$537$	0	0
$538$	−30.3525	−1.30859
$539$	−4.13209	−0.177982
$540$	0	0
$541$	27.4526	1.18028	0.590140	−	0.807301i	$-0.299072\pi$
$541$	27.4526	1.18028	0.590140	+	0.807301i	$0.299072\pi$
$542$	24.0290	1.03213
$543$	0	0
$544$	6.37317	0.273247
$545$	−26.4758	−1.13410
$546$	0	0
$547$	12.6979	0.542922	0.271461	−	0.962450i	$-0.412493\pi$
$547$	12.6979	0.542922	0.271461	+	0.962450i	$0.412493\pi$
$548$	10.5134	0.449109
$549$	0	0
$550$	−35.5850	−1.51735
$551$	0	0
$552$	0	0
$553$	−33.1196	−1.40839
$554$	−21.9370	−0.932014
$555$	0	0
$556$	7.15473	0.303428
$557$	−1.20768	−0.0511712	−0.0255856	−	0.999673i	$-0.508145\pi$
$557$	−1.20768	−0.0511712	−0.0255856	+	0.999673i	$0.508145\pi$
$558$	0	0
$559$	−28.3236	−1.19796
$560$	10.9357	0.462120
$561$	0	0
$562$	21.7983	0.919504
$563$	38.3294	1.61539	0.807696	−	0.589599i	$-0.200714\pi$
$563$	38.3294	1.61539	0.807696	+	0.589599i	$0.200714\pi$
$564$	0	0
$565$	−2.65626	−0.111750
$566$	−27.7599	−1.16683
$567$	0	0
$568$	−7.25819	−0.304547
$569$	6.62650	0.277798	0.138899	−	0.990307i	$-0.455644\pi$
$569$	6.62650	0.277798	0.138899	+	0.990307i	$0.455644\pi$
$570$	0	0
$571$	33.0352	1.38248	0.691239	−	0.722626i	$-0.257065\pi$
$571$	33.0352	1.38248	0.691239	+	0.722626i	$0.257065\pi$
$572$	−16.4627	−0.688341
$573$	0	0
$574$	−12.2762	−0.512397
$575$	16.2006	0.675610
$576$	0	0
$577$	−10.6784	−0.444547	−0.222273	−	0.974984i	$-0.571348\pi$
$577$	−10.6784	−0.444547	−0.222273	+	0.974984i	$0.571348\pi$
$578$	23.6172	0.982348
$579$	0	0
$580$	18.0488	0.749437
$581$	23.5631	0.977562
$582$	0	0
$583$	25.7673	1.06717
$584$	7.24562	0.299826
$585$	0	0
$586$	32.0006	1.32193
$587$	−11.6530	−0.480972	−0.240486	−	0.970653i	$-0.577307\pi$
$587$	−11.6530	−0.480972	−0.240486	+	0.970653i	$0.577307\pi$
$588$	0	0
$589$	0	0
$590$	−2.32225	−0.0956055
$591$	0	0
$592$	5.40622	0.222194
$593$	−40.7748	−1.67442	−0.837209	−	0.546882i	$-0.815814\pi$
$593$	−40.7748	−1.67442	−0.837209	+	0.546882i	$0.815814\pi$
$594$	0	0
$595$	69.6953	2.85723
$596$	−19.6469	−0.804767
$597$	0	0
$598$	7.49487	0.306488
$599$	10.2087	0.417117	0.208559	−	0.978010i	$-0.433123\pi$
$599$	10.2087	0.417117	0.208559	+	0.978010i	$0.433123\pi$
$600$	0	0
$601$	−28.1083	−1.14656	−0.573280	−	0.819360i	$-0.694329\pi$
$601$	−28.1083	−1.14656	−0.573280	+	0.819360i	$0.694329\pi$
$602$	−17.9691	−0.732366
$603$	0	0
$604$	15.6402	0.636390
$605$	9.28584	0.377523
$606$	0	0
$607$	−1.50296	−0.0610032	−0.0305016	−	0.999535i	$-0.509710\pi$
$607$	−1.50296	−0.0610032	−0.0305016	+	0.999535i	$0.509710\pi$
$608$	0	0
$609$	0	0
$610$	−39.7734	−1.61038
$611$	−9.62483	−0.389379
$612$	0	0
$613$	−17.6412	−0.712520	−0.356260	−	0.934387i	$-0.615948\pi$
$613$	−17.6412	−0.712520	−0.356260	+	0.934387i	$0.615948\pi$
$614$	17.0453	0.687893
$615$	0	0
$616$	−10.4443	−0.420813
$617$	−10.4852	−0.422120	−0.211060	−	0.977473i	$-0.567692\pi$
$617$	−10.4852	−0.422120	−0.211060	+	0.977473i	$0.567692\pi$
$618$	0	0
$619$	20.1598	0.810290	0.405145	−	0.914252i	$-0.367221\pi$
$619$	20.1598	0.810290	0.405145	+	0.914252i	$0.367221\pi$
$620$	−36.4891	−1.46544
$621$	0	0
$622$	19.4288	0.779023
$623$	−13.8588	−0.555241
$624$	0	0
$625$	20.7851	0.831403
$626$	−31.1894	−1.24658
$627$	0	0
$628$	−4.49797	−0.179488
$629$	34.4547	1.37380
$630$	0	0
$631$	−23.2865	−0.927021	−0.463510	−	0.886092i	$-0.653410\pi$
$631$	−23.2865	−0.927021	−0.463510	+	0.886092i	$0.653410\pi$
$632$	−11.6170	−0.462101
$633$	0	0
$634$	−0.479164	−0.0190300
$635$	−8.18700	−0.324891
$636$	0	0
$637$	5.06865	0.200827
$638$	−17.2377	−0.682448
$639$	0	0
$640$	3.83582	0.151624
$641$	14.7601	0.582990	0.291495	−	0.956572i	$-0.405847\pi$
$641$	14.7601	0.582990	0.291495	+	0.956572i	$0.405847\pi$
$642$	0	0
$643$	−18.0176	−0.710546	−0.355273	−	0.934763i	$-0.615612\pi$
$643$	−18.0176	−0.710546	−0.355273	+	0.934763i	$0.615612\pi$
$644$	4.75491	0.187370
$645$	0	0
$646$	0	0
$647$	9.58832	0.376956	0.188478	−	0.982077i	$-0.439645\pi$
$647$	9.58832	0.376956	0.188478	+	0.982077i	$0.439645\pi$
$648$	0	0
$649$	2.21789	0.0870597
$650$	43.6505	1.71211
$651$	0	0
$652$	−0.753285	−0.0295009
$653$	−14.6239	−0.572277	−0.286139	−	0.958188i	$-0.592372\pi$
$653$	−14.6239	−0.572277	−0.286139	+	0.958188i	$0.592372\pi$
$654$	0	0
$655$	17.0234	0.665159
$656$	−4.30599	−0.168121
$657$	0	0
$658$	−6.10620	−0.238044
$659$	40.8303	1.59052	0.795261	−	0.606267i	$-0.207334\pi$
$659$	40.8303	1.59052	0.795261	+	0.606267i	$0.207334\pi$
$660$	0	0
$661$	27.0930	1.05380	0.526899	−	0.849928i	$-0.323355\pi$
$661$	27.0930	1.05380	0.526899	+	0.849928i	$0.323355\pi$
$662$	20.5670	0.799358
$663$	0	0
$664$	8.26499	0.320744
$665$	0	0
$666$	0	0
$667$	7.84771	0.303864
$668$	−24.7123	−0.956146
$669$	0	0
$670$	−25.3043	−0.977592
$671$	37.9860	1.46643
$672$	0	0
$673$	0.707094	0.0272565	0.0136282	−	0.999907i	$-0.495662\pi$
$673$	0.707094	0.0272565	0.0136282	+	0.999907i	$0.495662\pi$
$674$	−4.92815	−0.189825
$675$	0	0
$676$	7.19406	0.276695
$677$	−0.282024	−0.0108391	−0.00541954	−	0.999985i	$-0.501725\pi$
$677$	−0.282024	−0.0108391	−0.00541954	+	0.999985i	$0.501725\pi$
$678$	0	0
$679$	7.36015	0.282456
$680$	24.4463	0.937474
$681$	0	0
$682$	34.8493	1.33445
$683$	−29.7448	−1.13815	−0.569076	−	0.822285i	$-0.692699\pi$
$683$	−29.7448	−1.13815	−0.569076	+	0.822285i	$0.692699\pi$
$684$	0	0
$685$	40.3275	1.54083
$686$	−16.7410	−0.639174
$687$	0	0
$688$	−6.30284	−0.240294
$689$	−31.6076	−1.20415
$690$	0	0
$691$	−18.4180	−0.700654	−0.350327	−	0.936628i	$-0.613929\pi$
$691$	−18.4180	−0.700654	−0.350327	+	0.936628i	$0.613929\pi$
$692$	1.98010	0.0752721
$693$	0	0
$694$	25.2211	0.957382
$695$	27.4443	1.04102
$696$	0	0
$697$	−27.4428	−1.03947
$698$	−12.5143	−0.473674
$699$	0	0
$700$	27.6928	1.04669
$701$	12.6395	0.477387	0.238694	−	0.971095i	$-0.423281\pi$
$701$	12.6395	0.477387	0.238694	+	0.971095i	$0.423281\pi$
$702$	0	0
$703$	0	0
$704$	−3.66344	−0.138071
$705$	0	0
$706$	8.41193	0.316587
$707$	54.1835	2.03778
$708$	0	0
$709$	−14.4381	−0.542232	−0.271116	−	0.962547i	$-0.587393\pi$
$709$	−14.4381	−0.542232	−0.271116	+	0.962547i	$0.587393\pi$
$710$	−27.8411	−1.04486
$711$	0	0
$712$	−4.86111	−0.182178
$713$	−15.8656	−0.594172
$714$	0	0
$715$	−63.1480	−2.36160
$716$	−0.0791904	−0.00295948
$717$	0	0
$718$	13.7877	0.514553
$719$	−15.4000	−0.574323	−0.287162	−	0.957882i	$-0.592712\pi$
$719$	−15.4000	−0.574323	−0.287162	+	0.957882i	$0.592712\pi$
$720$	0	0
$721$	19.3631	0.721119
$722$	0	0
$723$	0	0
$724$	−15.8219	−0.588015
$725$	45.7054	1.69746
$726$	0	0
$727$	17.1230	0.635057	0.317528	−	0.948249i	$-0.397147\pi$
$727$	17.1230	0.635057	0.317528	+	0.948249i	$0.397147\pi$
$728$	12.8116	0.474828
$729$	0	0
$730$	27.7929	1.02866
$731$	−40.1691	−1.48571
$732$	0	0
$733$	24.3384	0.898960	0.449480	−	0.893290i	$-0.351609\pi$
$733$	24.3384	0.898960	0.449480	+	0.893290i	$0.351609\pi$
$734$	4.10000	0.151334
$735$	0	0
$736$	1.66783	0.0614771
$737$	24.1672	0.890210
$738$	0	0
$739$	−27.7083	−1.01927	−0.509633	−	0.860392i	$-0.670219\pi$
$739$	−27.7083	−1.01927	−0.509633	+	0.860392i	$0.670219\pi$
$740$	20.7373	0.762318
$741$	0	0
$742$	−20.0526	−0.736152
$743$	32.3382	1.18637	0.593186	−	0.805065i	$-0.297870\pi$
$743$	32.3382	1.18637	0.593186	+	0.805065i	$0.297870\pi$
$744$	0	0
$745$	−75.3619	−2.76105
$746$	26.4463	0.968269
$747$	0	0
$748$	−23.3477	−0.853678
$749$	−31.7283	−1.15933
$750$	0	0
$751$	1.56288	0.0570304	0.0285152	−	0.999593i	$-0.490922\pi$
$751$	1.56288	0.0570304	0.0285152	+	0.999593i	$0.490922\pi$
$752$	−2.14181	−0.0781038
$753$	0	0
$754$	21.1447	0.770046
$755$	59.9930	2.18337
$756$	0	0
$757$	11.6268	0.422582	0.211291	−	0.977423i	$-0.432233\pi$
$757$	11.6268	0.422582	0.211291	+	0.977423i	$0.432233\pi$
$758$	1.56826	0.0569619
$759$	0	0
$760$	0	0
$761$	−5.00443	−0.181410	−0.0907051	−	0.995878i	$-0.528912\pi$
$761$	−5.00443	−0.181410	−0.0907051	+	0.995878i	$0.528912\pi$
$762$	0	0
$763$	−19.6780	−0.712391
$764$	−13.2510	−0.479405
$765$	0	0
$766$	21.9392	0.792695
$767$	−2.72058	−0.0982346
$768$	0	0
$769$	4.05968	0.146396	0.0731980	−	0.997317i	$-0.476680\pi$
$769$	4.05968	0.146396	0.0731980	+	0.997317i	$0.476680\pi$
$770$	−40.0625	−1.44375
$771$	0	0
$772$	0.177125	0.00637487
$773$	−7.51314	−0.270229	−0.135115	−	0.990830i	$-0.543140\pi$
$773$	−7.51314	−0.270229	−0.135115	+	0.990830i	$0.543140\pi$
$774$	0	0
$775$	−92.4021	−3.31918
$776$	2.58165	0.0926757
$777$	0	0
$778$	14.7774	0.529796
$779$	0	0
$780$	0	0
$781$	26.5900	0.951464
$782$	10.6294	0.380106
$783$	0	0
$784$	1.12793	0.0402830
$785$	−17.2534	−0.615800
$786$	0	0
$787$	1.92558	0.0686396	0.0343198	−	0.999411i	$-0.489074\pi$
$787$	1.92558	0.0686396	0.0343198	+	0.999411i	$0.489074\pi$
$788$	0.892456	0.0317924
$789$	0	0
$790$	−44.5609	−1.58540
$791$	−1.97425	−0.0701963
$792$	0	0
$793$	−46.5957	−1.65466
$794$	−5.47169	−0.194183
$795$	0	0
$796$	−14.1219	−0.500536
$797$	−10.0363	−0.355504	−0.177752	−	0.984075i	$-0.556882\pi$
$797$	−10.0363	−0.355504	−0.177752	+	0.984075i	$0.556882\pi$
$798$	0	0
$799$	−13.6501	−0.482906
$800$	9.71353	0.343425
$801$	0	0
$802$	−11.7257	−0.414048
$803$	−26.5439	−0.936715
$804$	0	0
$805$	18.2390	0.642840
$806$	−42.7481	−1.50574
$807$	0	0
$808$	19.0054	0.668608
$809$	−0.853288	−0.0300000	−0.0150000	−	0.999887i	$-0.504775\pi$
$809$	−0.853288	−0.0300000	−0.0150000	+	0.999887i	$0.504775\pi$
$810$	0	0
$811$	−23.2425	−0.816155	−0.408078	−	0.912947i	$-0.633801\pi$
$811$	−23.2425	−0.816155	−0.408078	+	0.912947i	$0.633801\pi$
$812$	13.4147	0.470763
$813$	0	0
$814$	−19.8054	−0.694178
$815$	−2.88947	−0.101214
$816$	0	0
$817$	0	0
$818$	−17.2072	−0.601634
$819$	0	0
$820$	−16.5170	−0.576799
$821$	−2.75810	−0.0962584	−0.0481292	−	0.998841i	$-0.515326\pi$
$821$	−2.75810	−0.0962584	−0.0481292	+	0.998841i	$0.515326\pi$
$822$	0	0
$823$	25.2427	0.879905	0.439953	−	0.898021i	$-0.354995\pi$
$823$	25.2427	0.879905	0.439953	+	0.898021i	$0.354995\pi$
$824$	6.79180	0.236604
$825$	0	0
$826$	−1.72600	−0.0600551
$827$	−35.4810	−1.23379	−0.616897	−	0.787044i	$-0.711611\pi$
$827$	−35.4810	−1.23379	−0.616897	+	0.787044i	$0.711611\pi$
$828$	0	0
$829$	8.13793	0.282642	0.141321	−	0.989964i	$-0.454865\pi$
$829$	8.13793	0.282642	0.141321	+	0.989964i	$0.454865\pi$
$830$	31.7030	1.10043
$831$	0	0
$832$	4.49378	0.155794
$833$	7.18846	0.249065
$834$	0	0
$835$	−94.7919	−3.28041
$836$	0	0
$837$	0	0
$838$	31.9505	1.10371
$839$	52.0164	1.79580	0.897902	−	0.440195i	$-0.145091\pi$
$839$	52.0164	1.79580	0.897902	+	0.440195i	$0.145091\pi$
$840$	0	0
$841$	−6.85984	−0.236546
$842$	−31.0906	−1.07145
$843$	0	0
$844$	12.2518	0.421723
$845$	27.5951	0.949302
$846$	0	0
$847$	6.90164	0.237143
$848$	−7.03364	−0.241536
$849$	0	0
$850$	61.9060	2.12336
$851$	9.01667	0.309087
$852$	0	0
$853$	−8.33274	−0.285307	−0.142654	−	0.989773i	$-0.545564\pi$
$853$	−8.33274	−0.285307	−0.142654	+	0.989773i	$0.545564\pi$
$854$	−29.5613	−1.01157
$855$	0	0
$856$	−11.1290	−0.380382
$857$	−6.43836	−0.219930	−0.109965	−	0.993935i	$-0.535074\pi$
$857$	−6.43836	−0.219930	−0.109965	+	0.993935i	$0.535074\pi$
$858$	0	0
$859$	26.8106	0.914766	0.457383	−	0.889270i	$-0.348787\pi$
$859$	26.8106	0.914766	0.457383	+	0.889270i	$0.348787\pi$
$860$	−24.1766	−0.824415
$861$	0	0
$862$	15.0956	0.514158
$863$	48.0256	1.63481	0.817405	−	0.576063i	$-0.195412\pi$
$863$	48.0256	1.63481	0.817405	+	0.576063i	$0.195412\pi$
$864$	0	0
$865$	7.59531	0.258248
$866$	−16.0311	−0.544760
$867$	0	0
$868$	−27.1203	−0.920523
$869$	42.5583	1.44369
$870$	0	0
$871$	−29.6448	−1.00448
$872$	−6.90226	−0.233740
$873$	0	0
$874$	0	0
$875$	51.5460	1.74257
$876$	0	0
$877$	−18.4991	−0.624670	−0.312335	−	0.949972i	$-0.601111\pi$
$877$	−18.4991	−0.624670	−0.312335	+	0.949972i	$0.601111\pi$
$878$	−6.26212	−0.211336
$879$	0	0
$880$	−14.0523	−0.473704
$881$	−36.2285	−1.22057	−0.610285	−	0.792182i	$-0.708945\pi$
$881$	−36.2285	−1.22057	−0.610285	+	0.792182i	$0.708945\pi$
$882$	0	0
$883$	42.6525	1.43537	0.717685	−	0.696368i	$-0.245202\pi$
$883$	42.6525	1.43537	0.717685	+	0.696368i	$0.245202\pi$
$884$	28.6396	0.963254
$885$	0	0
$886$	−19.2192	−0.645682
$887$	57.9561	1.94598	0.972988	−	0.230854i	$-0.0741520\pi$
$887$	57.9561	1.94598	0.972988	+	0.230854i	$0.0741520\pi$
$888$	0	0
$889$	−6.08494	−0.204082
$890$	−18.6464	−0.625027
$891$	0	0
$892$	−26.0379	−0.871812
$893$	0	0
$894$	0	0
$895$	−0.303760	−0.0101536
$896$	2.85095	0.0952436
$897$	0	0
$898$	32.3059	1.07806
$899$	−44.7605	−1.49285
$900$	0	0
$901$	−44.8265	−1.49339
$902$	15.7747	0.525242
$903$	0	0
$904$	−0.692489	−0.0230318
$905$	−60.6898	−2.01740
$906$	0	0
$907$	−34.5867	−1.14843	−0.574216	−	0.818704i	$-0.694693\pi$
$907$	−34.5867	−1.14843	−0.574216	+	0.818704i	$0.694693\pi$
$908$	−1.86936	−0.0620370
$909$	0	0
$910$	49.1428	1.62907
$911$	−19.6453	−0.650877	−0.325439	−	0.945563i	$-0.605512\pi$
$911$	−19.6453	−0.650877	−0.325439	+	0.945563i	$0.605512\pi$
$912$	0	0
$913$	−30.2783	−1.00207
$914$	1.80865	0.0598250
$915$	0	0
$916$	−5.82691	−0.192527
$917$	12.6525	0.417823
$918$	0	0
$919$	20.3790	0.672240	0.336120	−	0.941819i	$-0.390885\pi$
$919$	20.3790	0.672240	0.336120	+	0.941819i	$0.390885\pi$
$920$	6.39751	0.210920
$921$	0	0
$922$	−35.5344	−1.17026
$923$	−32.6167	−1.07359
$924$	0	0
$925$	52.5135	1.72663
$926$	−1.19885	−0.0393968
$927$	0	0
$928$	4.70533	0.154460
$929$	−9.34915	−0.306736	−0.153368	−	0.988169i	$-0.549012\pi$
$929$	−9.34915	−0.306736	−0.153368	+	0.988169i	$0.549012\pi$
$930$	0	0
$931$	0	0
$932$	5.57651	0.182665
$933$	0	0
$934$	0.570033	0.0186520
$935$	−89.5578	−2.92885
$936$	0	0
$937$	−61.2077	−1.99957	−0.999784	−	0.0208001i	$-0.993379\pi$
$937$	−61.2077	−1.99957	−0.999784	+	0.0208001i	$0.993379\pi$
$938$	−18.8073	−0.614080
$939$	0	0
$940$	−8.21561	−0.267964
$941$	57.1078	1.86166	0.930831	−	0.365450i	$-0.119085\pi$
$941$	57.1078	1.86166	0.930831	+	0.365450i	$0.119085\pi$
$942$	0	0
$943$	−7.18167	−0.233867
$944$	−0.605411	−0.0197044
$945$	0	0
$946$	23.0901	0.750724
$947$	−2.97980	−0.0968303	−0.0484152	−	0.998827i	$-0.515417\pi$
$947$	−2.97980	−0.0968303	−0.0484152	+	0.998827i	$0.515417\pi$
$948$	0	0
$949$	32.5602	1.05695
$950$	0	0
$951$	0	0
$952$	18.1696	0.588880
$953$	−17.6165	−0.570653	−0.285327	−	0.958430i	$-0.592102\pi$
$953$	−17.6165	−0.570653	−0.285327	+	0.958430i	$0.592102\pi$
$954$	0	0
$955$	−50.8286	−1.64477
$956$	17.3437	0.560935
$957$	0	0
$958$	−25.3025	−0.817486
$959$	29.9731	0.967883
$960$	0	0
$961$	59.4919	1.91909
$962$	24.2944	0.783282
$963$	0	0
$964$	15.3731	0.495134
$965$	0.679420	0.0218713
$966$	0	0
$967$	−16.9691	−0.545689	−0.272844	−	0.962058i	$-0.587964\pi$
$967$	−16.9691	−0.545689	−0.272844	+	0.962058i	$0.587964\pi$
$968$	2.42082	0.0778081
$969$	0	0
$970$	9.90273	0.317958
$971$	−47.7242	−1.53154	−0.765771	−	0.643114i	$-0.777642\pi$
$971$	−47.7242	−1.53154	−0.765771	+	0.643114i	$0.777642\pi$
$972$	0	0
$973$	20.3978	0.653923
$974$	−24.1268	−0.773071
$975$	0	0
$976$	−10.3689	−0.331901
$977$	−55.6138	−1.77924	−0.889622	−	0.456698i	$-0.849032\pi$
$977$	−55.6138	−1.77924	−0.889622	+	0.456698i	$0.849032\pi$
$978$	0	0
$979$	17.8084	0.569159
$980$	4.32652	0.138206
$981$	0	0
$982$	−13.0719	−0.417140
$983$	−11.8952	−0.379399	−0.189700	−	0.981842i	$-0.560751\pi$
$983$	−11.8952	−0.379399	−0.189700	+	0.981842i	$0.560751\pi$
$984$	0	0
$985$	3.42330	0.109076
$986$	29.9879	0.955008
$987$	0	0
$988$	0	0
$989$	−10.5121	−0.334265
$990$	0	0
$991$	−12.1049	−0.384525	−0.192262	−	0.981344i	$-0.561582\pi$
$991$	−12.1049	−0.384525	−0.192262	+	0.981344i	$0.561582\pi$
$992$	−9.51272	−0.302029
$993$	0	0
$994$	−20.6927	−0.656334
$995$	−54.1690	−1.71727
$996$	0	0
$997$	−43.1379	−1.36619	−0.683096	−	0.730328i	$-0.739367\pi$
$997$	−43.1379	−1.36619	−0.683096	+	0.730328i	$0.739367\pi$
$998$	22.9167	0.725414
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.83582 q^{5} +2.85095 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.83582 q^{5} +2.85095 q^{7} +1.00000 q^{8} +3.83582 q^{10} -3.66344 q^{11} +4.49378 q^{13} +2.85095 q^{14} +1.00000 q^{16} +6.37317 q^{17} +3.83582 q^{20} -3.66344 q^{22} +1.66783 q^{23} +9.71353 q^{25} +4.49378 q^{26} +2.85095 q^{28} +4.70533 q^{29} -9.51272 q^{31} +1.00000 q^{32} +6.37317 q^{34} +10.9357 q^{35} +5.40622 q^{37} +3.83582 q^{40} -4.30599 q^{41} -6.30284 q^{43} -3.66344 q^{44} +1.66783 q^{46} -2.14181 q^{47} +1.12793 q^{49} +9.71353 q^{50} +4.49378 q^{52} -7.03364 q^{53} -14.0523 q^{55} +2.85095 q^{56} +4.70533 q^{58} -0.605411 q^{59} -10.3689 q^{61} -9.51272 q^{62} +1.00000 q^{64} +17.2373 q^{65} -6.59685 q^{67} +6.37317 q^{68} +10.9357 q^{70} -7.25819 q^{71} +7.24562 q^{73} +5.40622 q^{74} -10.4443 q^{77} -11.6170 q^{79} +3.83582 q^{80} -4.30599 q^{82} +8.26499 q^{83} +24.4463 q^{85} -6.30284 q^{86} -3.66344 q^{88} -4.86111 q^{89} +12.8116 q^{91} +1.66783 q^{92} -2.14181 q^{94} +2.58165 q^{97} +1.12793 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) 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

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Representations

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Database

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