LMFDB - Embedded newform 6498.2.a.bs.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:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.18479 q^{5} +0.532089 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.18479 q^{5} +0.532089 q^{7} +1.00000 q^{8} -1.18479 q^{10} -1.87939 q^{11} -3.87939 q^{13} +0.532089 q^{14} +1.00000 q^{16} -1.16250 q^{17} -1.18479 q^{20} -1.87939 q^{22} +6.70233 q^{23} -3.59627 q^{25} -3.87939 q^{26} +0.532089 q^{28} +4.02229 q^{29} -1.95811 q^{31} +1.00000 q^{32} -1.16250 q^{34} -0.630415 q^{35} +6.88713 q^{37} -1.18479 q^{40} +8.98545 q^{41} +2.42602 q^{43} -1.87939 q^{44} +6.70233 q^{46} -2.04189 q^{47} -6.71688 q^{49} -3.59627 q^{50} -3.87939 q^{52} -12.9709 q^{53} +2.22668 q^{55} +0.532089 q^{56} +4.02229 q^{58} +2.68004 q^{59} -11.3473 q^{61} -1.95811 q^{62} +1.00000 q^{64} +4.59627 q^{65} -11.1702 q^{67} -1.16250 q^{68} -0.630415 q^{70} -6.07192 q^{71} -0.327696 q^{73} +6.88713 q^{74} -1.00000 q^{77} -16.1334 q^{79} -1.18479 q^{80} +8.98545 q^{82} -11.5740 q^{83} +1.37733 q^{85} +2.42602 q^{86} -1.87939 q^{88} +3.55943 q^{89} -2.06418 q^{91} +6.70233 q^{92} -2.04189 q^{94} +5.87939 q^{97} -6.71688 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8} - 6 q^{13} - 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} + 6 q^{29} - 9 q^{31} + 3 q^{32} - 6 q^{34} - 9 q^{35} - 9 q^{37} + 9 q^{41} + 15 q^{43} - 6 q^{46} - 3 q^{47} - 12 q^{49} + 3 q^{50} - 6 q^{52} - 3 q^{53} - 3 q^{56} + 6 q^{58} - 12 q^{59} - 33 q^{61} - 9 q^{62} + 3 q^{64} - 12 q^{67} - 6 q^{68} - 9 q^{70} + 15 q^{71} + 3 q^{73} - 9 q^{74} - 3 q^{77} - 15 q^{79} + 9 q^{82} - 27 q^{83} - 27 q^{85} + 15 q^{86} - 15 q^{89} + 3 q^{91} - 6 q^{92} - 3 q^{94} + 12 q^{97} - 12 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8 - 6 * q^13 - 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 + 6 * q^29 - 9 * q^31 + 3 * q^32 - 6 * q^34 - 9 * q^35 - 9 * q^37 + 9 * q^41 + 15 * q^43 - 6 * q^46 - 3 * q^47 - 12 * q^49 + 3 * q^50 - 6 * q^52 - 3 * q^53 - 3 * q^56 + 6 * q^58 - 12 * q^59 - 33 * q^61 - 9 * q^62 + 3 * q^64 - 12 * q^67 - 6 * q^68 - 9 * q^70 + 15 * q^71 + 3 * q^73 - 9 * q^74 - 3 * q^77 - 15 * q^79 + 9 * q^82 - 27 * q^83 - 27 * q^85 + 15 * q^86 - 15 * q^89 + 3 * q^91 - 6 * q^92 - 3 * q^94 + 12 * q^97 - 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.18479	−0.529855	−0.264928	−	0.964268i	$-0.585348\pi$
$5$	−1.18479	−0.529855	−0.264928	+	0.964268i	$0.585348\pi$
$6$	0	0
$7$	0.532089	0.201111	0.100555	−	0.994931i	$-0.467938\pi$
$7$	0.532089	0.201111	0.100555	+	0.994931i	$0.467938\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.18479	−0.374664
$11$	−1.87939	−0.566656	−0.283328	−	0.959023i	$-0.591439\pi$
$11$	−1.87939	−0.566656	−0.283328	+	0.959023i	$0.591439\pi$
$12$	0	0
$13$	−3.87939	−1.07595	−0.537974	−	0.842961i	$-0.680810\pi$
$13$	−3.87939	−1.07595	−0.537974	+	0.842961i	$0.680810\pi$
$14$	0.532089	0.142207
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.16250	−0.281949	−0.140974	−	0.990013i	$-0.545023\pi$
$17$	−1.16250	−0.281949	−0.140974	+	0.990013i	$0.545023\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−1.18479	−0.264928
$21$	0	0
$22$	−1.87939	−0.400686
$23$	6.70233	1.39753	0.698767	−	0.715350i	$-0.253733\pi$
$23$	6.70233	1.39753	0.698767	+	0.715350i	$0.253733\pi$
$24$	0	0
$25$	−3.59627	−0.719253
$26$	−3.87939	−0.760810
$27$	0	0
$28$	0.532089	0.100555
$29$	4.02229	0.746920	0.373460	−	0.927646i	$-0.378171\pi$
$29$	4.02229	0.746920	0.373460	+	0.927646i	$0.378171\pi$
$30$	0	0
$31$	−1.95811	−0.351687	−0.175844	−	0.984418i	$-0.556265\pi$
$31$	−1.95811	−0.351687	−0.175844	+	0.984418i	$0.556265\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.16250	−0.199368
$35$	−0.630415	−0.106560
$36$	0	0
$37$	6.88713	1.13224	0.566118	−	0.824324i	$-0.308445\pi$
$37$	6.88713	1.13224	0.566118	+	0.824324i	$0.308445\pi$
$38$	0	0
$39$	0	0
$40$	−1.18479	−0.187332
$41$	8.98545	1.40329	0.701646	−	0.712526i	$-0.252449\pi$
$41$	8.98545	1.40329	0.701646	+	0.712526i	$0.252449\pi$
$42$	0	0
$43$	2.42602	0.369965	0.184982	−	0.982742i	$-0.440777\pi$
$43$	2.42602	0.369965	0.184982	+	0.982742i	$0.440777\pi$
$44$	−1.87939	−0.283328
$45$	0	0
$46$	6.70233	0.988205
$47$	−2.04189	−0.297840	−0.148920	−	0.988849i	$-0.547580\pi$
$47$	−2.04189	−0.297840	−0.148920	+	0.988849i	$0.547580\pi$
$48$	0	0
$49$	−6.71688	−0.959554
$50$	−3.59627	−0.508589
$51$	0	0
$52$	−3.87939	−0.537974
$53$	−12.9709	−1.78169	−0.890845	−	0.454307i	$-0.849887\pi$
$53$	−12.9709	−1.78169	−0.890845	+	0.454307i	$0.849887\pi$
$54$	0	0
$55$	2.22668	0.300246
$56$	0.532089	0.0711034
$57$	0	0
$58$	4.02229	0.528152
$59$	2.68004	0.348912	0.174456	−	0.984665i	$-0.444183\pi$
$59$	2.68004	0.348912	0.174456	+	0.984665i	$0.444183\pi$
$60$	0	0
$61$	−11.3473	−1.45287	−0.726436	−	0.687234i	$-0.758825\pi$
$61$	−11.3473	−1.45287	−0.726436	+	0.687234i	$0.758825\pi$
$62$	−1.95811	−0.248680
$63$	0	0
$64$	1.00000	0.125000
$65$	4.59627	0.570097
$66$	0	0
$67$	−11.1702	−1.36466	−0.682331	−	0.731043i	$-0.739034\pi$
$67$	−11.1702	−1.36466	−0.682331	+	0.731043i	$0.739034\pi$
$68$	−1.16250	−0.140974
$69$	0	0
$70$	−0.630415	−0.0753490
$71$	−6.07192	−0.720604	−0.360302	−	0.932836i	$-0.617326\pi$
$71$	−6.07192	−0.720604	−0.360302	+	0.932836i	$0.617326\pi$
$72$	0	0
$73$	−0.327696	−0.0383539	−0.0191770	−	0.999816i	$-0.506105\pi$
$73$	−0.327696	−0.0383539	−0.0191770	+	0.999816i	$0.506105\pi$
$74$	6.88713	0.800612
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−16.1334	−1.81515	−0.907575	−	0.419890i	$-0.862069\pi$
$79$	−16.1334	−1.81515	−0.907575	+	0.419890i	$0.862069\pi$
$80$	−1.18479	−0.132464
$81$	0	0
$82$	8.98545	0.992277
$83$	−11.5740	−1.27041	−0.635205	−	0.772344i	$-0.719084\pi$
$83$	−11.5740	−1.27041	−0.635205	+	0.772344i	$0.719084\pi$
$84$	0	0
$85$	1.37733	0.149392
$86$	2.42602	0.261605
$87$	0	0
$88$	−1.87939	−0.200343
$89$	3.55943	0.377299	0.188649	−	0.982044i	$-0.439589\pi$
$89$	3.55943	0.377299	0.188649	+	0.982044i	$0.439589\pi$
$90$	0	0
$91$	−2.06418	−0.216385
$92$	6.70233	0.698767
$93$	0	0
$94$	−2.04189	−0.210605
$95$	0	0
$96$	0	0
$97$	5.87939	0.596961	0.298481	−	0.954416i	$-0.403520\pi$
$97$	5.87939	0.596961	0.298481	+	0.954416i	$0.403520\pi$
$98$	−6.71688	−0.678507
$99$	0	0
$100$	−3.59627	−0.359627
$101$	−12.6459	−1.25831	−0.629157	−	0.777278i	$-0.716600\pi$
$101$	−12.6459	−1.25831	−0.629157	+	0.777278i	$0.716600\pi$
$102$	0	0
$103$	2.96316	0.291969	0.145985	−	0.989287i	$-0.453365\pi$
$103$	2.96316	0.291969	0.145985	+	0.989287i	$0.453365\pi$
$104$	−3.87939	−0.380405
$105$	0	0
$106$	−12.9709	−1.25985
$107$	−19.1138	−1.84780	−0.923901	−	0.382632i	$-0.875018\pi$
$107$	−19.1138	−1.84780	−0.923901	+	0.382632i	$0.875018\pi$
$108$	0	0
$109$	−16.4115	−1.57193	−0.785967	−	0.618268i	$-0.787834\pi$
$109$	−16.4115	−1.57193	−0.785967	+	0.618268i	$0.787834\pi$
$110$	2.22668	0.212306
$111$	0	0
$112$	0.532089	0.0502777
$113$	5.73648	0.539643	0.269821	−	0.962910i	$-0.413035\pi$
$113$	5.73648	0.539643	0.269821	+	0.962910i	$0.413035\pi$
$114$	0	0
$115$	−7.94087	−0.740490
$116$	4.02229	0.373460
$117$	0	0
$118$	2.68004	0.246718
$119$	−0.618555	−0.0567029
$120$	0	0
$121$	−7.46791	−0.678901
$122$	−11.3473	−1.02734
$123$	0	0
$124$	−1.95811	−0.175844
$125$	10.1848	0.910956
$126$	0	0
$127$	−1.88713	−0.167455	−0.0837277	−	0.996489i	$-0.526683\pi$
$127$	−1.88713	−0.167455	−0.0837277	+	0.996489i	$0.526683\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.59627	0.403119
$131$	−10.7023	−0.935067	−0.467534	−	0.883975i	$-0.654857\pi$
$131$	−10.7023	−0.935067	−0.467534	+	0.883975i	$0.654857\pi$
$132$	0	0
$133$	0	0
$134$	−11.1702	−0.964962
$135$	0	0
$136$	−1.16250	−0.0996839
$137$	11.2567	0.961726	0.480863	−	0.876796i	$-0.340323\pi$
$137$	11.2567	0.961726	0.480863	+	0.876796i	$0.340323\pi$
$138$	0	0
$139$	−12.1702	−1.03227	−0.516133	−	0.856508i	$-0.672629\pi$
$139$	−12.1702	−1.03227	−0.516133	+	0.856508i	$0.672629\pi$
$140$	−0.630415	−0.0532798
$141$	0	0
$142$	−6.07192	−0.509544
$143$	7.29086	0.609692
$144$	0	0
$145$	−4.76558	−0.395760
$146$	−0.327696	−0.0271203
$147$	0	0
$148$	6.88713	0.566118
$149$	16.3550	1.33986	0.669928	−	0.742426i	$-0.266325\pi$
$149$	16.3550	1.33986	0.669928	+	0.742426i	$0.266325\pi$
$150$	0	0
$151$	20.5226	1.67010	0.835052	−	0.550170i	$-0.185437\pi$
$151$	20.5226	1.67010	0.835052	+	0.550170i	$0.185437\pi$
$152$	0	0
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	2.31996	0.186343
$156$	0	0
$157$	−3.19934	−0.255335	−0.127668	−	0.991817i	$-0.540749\pi$
$157$	−3.19934	−0.255335	−0.127668	+	0.991817i	$0.540749\pi$
$158$	−16.1334	−1.28351
$159$	0	0
$160$	−1.18479	−0.0936661
$161$	3.56624	0.281059
$162$	0	0
$163$	−24.0077	−1.88043	−0.940216	−	0.340580i	$-0.889377\pi$
$163$	−24.0077	−1.88043	−0.940216	+	0.340580i	$0.889377\pi$
$164$	8.98545	0.701646
$165$	0	0
$166$	−11.5740	−0.898315
$167$	−3.66044	−0.283254	−0.141627	−	0.989920i	$-0.545233\pi$
$167$	−3.66044	−0.283254	−0.141627	+	0.989920i	$0.545233\pi$
$168$	0	0
$169$	2.04963	0.157664
$170$	1.37733	0.105636
$171$	0	0
$172$	2.42602	0.184982
$173$	5.33275	0.405441	0.202721	−	0.979237i	$-0.435022\pi$
$173$	5.33275	0.405441	0.202721	+	0.979237i	$0.435022\pi$
$174$	0	0
$175$	−1.91353	−0.144650
$176$	−1.87939	−0.141664
$177$	0	0
$178$	3.55943	0.266791
$179$	25.2567	1.88778	0.943888	−	0.330267i	$-0.107139\pi$
$179$	25.2567	1.88778	0.943888	+	0.330267i	$0.107139\pi$
$180$	0	0
$181$	−17.3063	−1.28637	−0.643185	−	0.765711i	$-0.722387\pi$
$181$	−17.3063	−1.28637	−0.643185	+	0.765711i	$0.722387\pi$
$182$	−2.06418	−0.153007
$183$	0	0
$184$	6.70233	0.494103
$185$	−8.15982	−0.599922
$186$	0	0
$187$	2.18479	0.159768
$188$	−2.04189	−0.148920
$189$	0	0
$190$	0	0
$191$	15.1780	1.09824	0.549120	−	0.835743i	$-0.314963\pi$
$191$	15.1780	1.09824	0.549120	+	0.835743i	$0.314963\pi$
$192$	0	0
$193$	13.0077	0.936318	0.468159	−	0.883644i	$-0.344918\pi$
$193$	13.0077	0.936318	0.468159	+	0.883644i	$0.344918\pi$
$194$	5.87939	0.422115
$195$	0	0
$196$	−6.71688	−0.479777
$197$	8.94862	0.637562	0.318781	−	0.947828i	$-0.396726\pi$
$197$	8.94862	0.637562	0.318781	+	0.947828i	$0.396726\pi$
$198$	0	0
$199$	−4.50475	−0.319333	−0.159667	−	0.987171i	$-0.551042\pi$
$199$	−4.50475	−0.319333	−0.159667	+	0.987171i	$0.551042\pi$
$200$	−3.59627	−0.254294
$201$	0	0
$202$	−12.6459	−0.889762
$203$	2.14022	0.150214
$204$	0	0
$205$	−10.6459	−0.743542
$206$	2.96316	0.206453
$207$	0	0
$208$	−3.87939	−0.268987
$209$	0	0
$210$	0	0
$211$	−6.40373	−0.440851	−0.220426	−	0.975404i	$-0.570745\pi$
$211$	−6.40373	−0.440851	−0.220426	+	0.975404i	$0.570745\pi$
$212$	−12.9709	−0.890845
$213$	0	0
$214$	−19.1138	−1.30659
$215$	−2.87433	−0.196028
$216$	0	0
$217$	−1.04189	−0.0707280
$218$	−16.4115	−1.11153
$219$	0	0
$220$	2.22668	0.150123
$221$	4.50980	0.303362
$222$	0	0
$223$	3.24897	0.217567	0.108784	−	0.994065i	$-0.465304\pi$
$223$	3.24897	0.217567	0.108784	+	0.994065i	$0.465304\pi$
$224$	0.532089	0.0355517
$225$	0	0
$226$	5.73648	0.381585
$227$	11.7023	0.776711	0.388356	−	0.921510i	$-0.373043\pi$
$227$	11.7023	0.776711	0.388356	+	0.921510i	$0.373043\pi$
$228$	0	0
$229$	−22.9067	−1.51372	−0.756860	−	0.653578i	$-0.773267\pi$
$229$	−22.9067	−1.51372	−0.756860	+	0.653578i	$0.773267\pi$
$230$	−7.94087	−0.523606
$231$	0	0
$232$	4.02229	0.264076
$233$	3.19934	0.209596	0.104798	−	0.994494i	$-0.466580\pi$
$233$	3.19934	0.209596	0.104798	+	0.994494i	$0.466580\pi$
$234$	0	0
$235$	2.41921	0.157812
$236$	2.68004	0.174456
$237$	0	0
$238$	−0.618555	−0.0400950
$239$	−6.17293	−0.399294	−0.199647	−	0.979868i	$-0.563980\pi$
$239$	−6.17293	−0.399294	−0.199647	+	0.979868i	$0.563980\pi$
$240$	0	0
$241$	1.53714	0.0990160	0.0495080	−	0.998774i	$-0.484235\pi$
$241$	1.53714	0.0990160	0.0495080	+	0.998774i	$0.484235\pi$
$242$	−7.46791	−0.480056
$243$	0	0
$244$	−11.3473	−0.726436
$245$	7.95811	0.508425
$246$	0	0
$247$	0	0
$248$	−1.95811	−0.124340
$249$	0	0
$250$	10.1848	0.644143
$251$	−17.9436	−1.13259	−0.566294	−	0.824203i	$-0.691623\pi$
$251$	−17.9436	−1.13259	−0.566294	+	0.824203i	$0.691623\pi$
$252$	0	0
$253$	−12.5963	−0.791920
$254$	−1.88713	−0.118409
$255$	0	0
$256$	1.00000	0.0625000
$257$	−29.4124	−1.83470	−0.917348	−	0.398087i	$-0.869674\pi$
$257$	−29.4124	−1.83470	−0.917348	+	0.398087i	$0.869674\pi$
$258$	0	0
$259$	3.66456	0.227705
$260$	4.59627	0.285048
$261$	0	0
$262$	−10.7023	−0.661192
$263$	−10.6827	−0.658726	−0.329363	−	0.944203i	$-0.606834\pi$
$263$	−10.6827	−0.658726	−0.329363	+	0.944203i	$0.606834\pi$
$264$	0	0
$265$	15.3678	0.944038
$266$	0	0
$267$	0	0
$268$	−11.1702	−0.682331
$269$	20.5767	1.25458	0.627291	−	0.778785i	$-0.284164\pi$
$269$	20.5767	1.25458	0.627291	+	0.778785i	$0.284164\pi$
$270$	0	0
$271$	1.87670	0.114001	0.0570006	−	0.998374i	$-0.481846\pi$
$271$	1.87670	0.114001	0.0570006	+	0.998374i	$0.481846\pi$
$272$	−1.16250	−0.0704871
$273$	0	0
$274$	11.2567	0.680043
$275$	6.75877	0.407569
$276$	0	0
$277$	4.87258	0.292765	0.146382	−	0.989228i	$-0.453237\pi$
$277$	4.87258	0.292765	0.146382	+	0.989228i	$0.453237\pi$
$278$	−12.1702	−0.729923
$279$	0	0
$280$	−0.630415	−0.0376745
$281$	0.403733	0.0240847	0.0120424	−	0.999927i	$-0.496167\pi$
$281$	0.403733	0.0240847	0.0120424	+	0.999927i	$0.496167\pi$
$282$	0	0
$283$	17.5895	1.04558	0.522792	−	0.852460i	$-0.324890\pi$
$283$	17.5895	1.04558	0.522792	+	0.852460i	$0.324890\pi$
$284$	−6.07192	−0.360302
$285$	0	0
$286$	7.29086	0.431118
$287$	4.78106	0.282217
$288$	0	0
$289$	−15.6486	−0.920505
$290$	−4.76558	−0.279844
$291$	0	0
$292$	−0.327696	−0.0191770
$293$	13.1753	0.769709	0.384855	−	0.922977i	$-0.374252\pi$
$293$	13.1753	0.769709	0.384855	+	0.922977i	$0.374252\pi$
$294$	0	0
$295$	−3.17530	−0.184873
$296$	6.88713	0.400306
$297$	0	0
$298$	16.3550	0.947422
$299$	−26.0009	−1.50367
$300$	0	0
$301$	1.29086	0.0744039
$302$	20.5226	1.18094
$303$	0	0
$304$	0	0
$305$	13.4442	0.769812
$306$	0	0
$307$	16.9017	0.964629	0.482315	−	0.875998i	$-0.339796\pi$
$307$	16.9017	0.964629	0.482315	+	0.875998i	$0.339796\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	2.31996	0.131765
$311$	2.38144	0.135039	0.0675197	−	0.997718i	$-0.478491\pi$
$311$	2.38144	0.135039	0.0675197	+	0.997718i	$0.478491\pi$
$312$	0	0
$313$	−29.8631	−1.68796	−0.843981	−	0.536374i	$-0.819794\pi$
$313$	−29.8631	−1.68796	−0.843981	+	0.536374i	$0.819794\pi$
$314$	−3.19934	−0.180549
$315$	0	0
$316$	−16.1334	−0.907575
$317$	7.02229	0.394411	0.197206	−	0.980362i	$-0.436813\pi$
$317$	7.02229	0.394411	0.197206	+	0.980362i	$0.436813\pi$
$318$	0	0
$319$	−7.55943	−0.423247
$320$	−1.18479	−0.0662319
$321$	0	0
$322$	3.56624	0.198739
$323$	0	0
$324$	0	0
$325$	13.9513	0.773879
$326$	−24.0077	−1.32967
$327$	0	0
$328$	8.98545	0.496139
$329$	−1.08647	−0.0598988
$330$	0	0
$331$	27.1985	1.49497	0.747483	−	0.664281i	$-0.231262\pi$
$331$	27.1985	1.49497	0.747483	+	0.664281i	$0.231262\pi$
$332$	−11.5740	−0.635205
$333$	0	0
$334$	−3.66044	−0.200291
$335$	13.2344	0.723074
$336$	0	0
$337$	−11.2172	−0.611039	−0.305520	−	0.952186i	$-0.598830\pi$
$337$	−11.2172	−0.611039	−0.305520	+	0.952186i	$0.598830\pi$
$338$	2.04963	0.111485
$339$	0	0
$340$	1.37733	0.0746960
$341$	3.68004	0.199286
$342$	0	0
$343$	−7.29860	−0.394087
$344$	2.42602	0.130802
$345$	0	0
$346$	5.33275	0.286690
$347$	−1.93582	−0.103920	−0.0519602	−	0.998649i	$-0.516547\pi$
$347$	−1.93582	−0.103920	−0.0519602	+	0.998649i	$0.516547\pi$
$348$	0	0
$349$	14.3969	0.770650	0.385325	−	0.922781i	$-0.374089\pi$
$349$	14.3969	0.770650	0.385325	+	0.922781i	$0.374089\pi$
$350$	−1.91353	−0.102283
$351$	0	0
$352$	−1.87939	−0.100172
$353$	−15.7237	−0.836887	−0.418444	−	0.908243i	$-0.637424\pi$
$353$	−15.7237	−0.836887	−0.418444	+	0.908243i	$0.637424\pi$
$354$	0	0
$355$	7.19396	0.381816
$356$	3.55943	0.188649
$357$	0	0
$358$	25.2567	1.33486
$359$	−15.5175	−0.818984	−0.409492	−	0.912314i	$-0.634294\pi$
$359$	−15.5175	−0.818984	−0.409492	+	0.912314i	$0.634294\pi$
$360$	0	0
$361$	0	0
$362$	−17.3063	−0.909601
$363$	0	0
$364$	−2.06418	−0.108192
$365$	0.388252	0.0203220
$366$	0	0
$367$	−15.7442	−0.821842	−0.410921	−	0.911671i	$-0.634793\pi$
$367$	−15.7442	−0.821842	−0.410921	+	0.911671i	$0.634793\pi$
$368$	6.70233	0.349383
$369$	0	0
$370$	−8.15982	−0.424209
$371$	−6.90167	−0.358317
$372$	0	0
$373$	−29.6287	−1.53411	−0.767057	−	0.641579i	$-0.778280\pi$
$373$	−29.6287	−1.53411	−0.767057	+	0.641579i	$0.778280\pi$
$374$	2.18479	0.112973
$375$	0	0
$376$	−2.04189	−0.105302
$377$	−15.6040	−0.803647
$378$	0	0
$379$	4.72462	0.242688	0.121344	−	0.992611i	$-0.461280\pi$
$379$	4.72462	0.242688	0.121344	+	0.992611i	$0.461280\pi$
$380$	0	0
$381$	0	0
$382$	15.1780	0.776573
$383$	5.24123	0.267814	0.133907	−	0.990994i	$-0.457248\pi$
$383$	5.24123	0.267814	0.133907	+	0.990994i	$0.457248\pi$
$384$	0	0
$385$	1.18479	0.0603826
$386$	13.0077	0.662077
$387$	0	0
$388$	5.87939	0.298481
$389$	15.0770	0.764433	0.382216	−	0.924073i	$-0.375161\pi$
$389$	15.0770	0.764433	0.382216	+	0.924073i	$0.375161\pi$
$390$	0	0
$391$	−7.79149	−0.394033
$392$	−6.71688	−0.339254
$393$	0	0
$394$	8.94862	0.450825
$395$	19.1147	0.961767
$396$	0	0
$397$	−14.5449	−0.729987	−0.364993	−	0.931010i	$-0.618929\pi$
$397$	−14.5449	−0.729987	−0.364993	+	0.931010i	$0.618929\pi$
$398$	−4.50475	−0.225803
$399$	0	0
$400$	−3.59627	−0.179813
$401$	−2.85473	−0.142559	−0.0712793	−	0.997456i	$-0.522708\pi$
$401$	−2.85473	−0.142559	−0.0712793	+	0.997456i	$0.522708\pi$
$402$	0	0
$403$	7.59627	0.378397
$404$	−12.6459	−0.629157
$405$	0	0
$406$	2.14022	0.106217
$407$	−12.9436	−0.641589
$408$	0	0
$409$	34.2276	1.69245	0.846223	−	0.532828i	$-0.178871\pi$
$409$	34.2276	1.69245	0.846223	+	0.532828i	$0.178871\pi$
$410$	−10.6459	−0.525763
$411$	0	0
$412$	2.96316	0.145985
$413$	1.42602	0.0701700
$414$	0	0
$415$	13.7128	0.673133
$416$	−3.87939	−0.190203
$417$	0	0
$418$	0	0
$419$	7.91891	0.386864	0.193432	−	0.981114i	$-0.438038\pi$
$419$	7.91891	0.386864	0.193432	+	0.981114i	$0.438038\pi$
$420$	0	0
$421$	16.2098	0.790016	0.395008	−	0.918678i	$-0.370742\pi$
$421$	16.2098	0.790016	0.395008	+	0.918678i	$0.370742\pi$
$422$	−6.40373	−0.311729
$423$	0	0
$424$	−12.9709	−0.629923
$425$	4.18067	0.202792
$426$	0	0
$427$	−6.03777	−0.292188
$428$	−19.1138	−0.923901
$429$	0	0
$430$	−2.87433	−0.138613
$431$	−25.4320	−1.22502	−0.612508	−	0.790464i	$-0.709839\pi$
$431$	−25.4320	−1.22502	−0.612508	+	0.790464i	$0.709839\pi$
$432$	0	0
$433$	11.8479	0.569375	0.284687	−	0.958620i	$-0.408110\pi$
$433$	11.8479	0.569375	0.284687	+	0.958620i	$0.408110\pi$
$434$	−1.04189	−0.0500123
$435$	0	0
$436$	−16.4115	−0.785967
$437$	0	0
$438$	0	0
$439$	25.8598	1.23422	0.617110	−	0.786877i	$-0.288303\pi$
$439$	25.8598	1.23422	0.617110	+	0.786877i	$0.288303\pi$
$440$	2.22668	0.106153
$441$	0	0
$442$	4.50980	0.214509
$443$	−0.923029	−0.0438544	−0.0219272	−	0.999760i	$-0.506980\pi$
$443$	−0.923029	−0.0438544	−0.0219272	+	0.999760i	$0.506980\pi$
$444$	0	0
$445$	−4.21719	−0.199914
$446$	3.24897	0.153843
$447$	0	0
$448$	0.532089	0.0251388
$449$	−9.34461	−0.440999	−0.220500	−	0.975387i	$-0.570769\pi$
$449$	−9.34461	−0.440999	−0.220500	+	0.975387i	$0.570769\pi$
$450$	0	0
$451$	−16.8871	−0.795184
$452$	5.73648	0.269821
$453$	0	0
$454$	11.7023	0.549218
$455$	2.44562	0.114653
$456$	0	0
$457$	30.9009	1.44548	0.722740	−	0.691120i	$-0.242882\pi$
$457$	30.9009	1.44548	0.722740	+	0.691120i	$0.242882\pi$
$458$	−22.9067	−1.07036
$459$	0	0
$460$	−7.94087	−0.370245
$461$	29.6878	1.38270	0.691349	−	0.722521i	$-0.257017\pi$
$461$	29.6878	1.38270	0.691349	+	0.722521i	$0.257017\pi$
$462$	0	0
$463$	−11.8307	−0.549819	−0.274909	−	0.961470i	$-0.588648\pi$
$463$	−11.8307	−0.549819	−0.274909	+	0.961470i	$0.588648\pi$
$464$	4.02229	0.186730
$465$	0	0
$466$	3.19934	0.148207
$467$	−38.3182	−1.77315	−0.886577	−	0.462580i	$-0.846924\pi$
$467$	−38.3182	−1.77315	−0.886577	+	0.462580i	$0.846924\pi$
$468$	0	0
$469$	−5.94356	−0.274448
$470$	2.41921	0.111590
$471$	0	0
$472$	2.68004	0.123359
$473$	−4.55943	−0.209643
$474$	0	0
$475$	0	0
$476$	−0.618555	−0.0283514
$477$	0	0
$478$	−6.17293	−0.282343
$479$	−37.2841	−1.70355	−0.851776	−	0.523906i	$-0.824474\pi$
$479$	−37.2841	−1.70355	−0.851776	+	0.523906i	$0.824474\pi$
$480$	0	0
$481$	−26.7178	−1.21823
$482$	1.53714	0.0700149
$483$	0	0
$484$	−7.46791	−0.339451
$485$	−6.96585	−0.316303
$486$	0	0
$487$	32.9786	1.49441	0.747203	−	0.664596i	$-0.231397\pi$
$487$	32.9786	1.49441	0.747203	+	0.664596i	$0.231397\pi$
$488$	−11.3473	−0.513668
$489$	0	0
$490$	7.95811	0.359511
$491$	−17.6810	−0.797931	−0.398966	−	0.916966i	$-0.630631\pi$
$491$	−17.6810	−0.797931	−0.398966	+	0.916966i	$0.630631\pi$
$492$	0	0
$493$	−4.67593	−0.210593
$494$	0	0
$495$	0	0
$496$	−1.95811	−0.0879218
$497$	−3.23080	−0.144921
$498$	0	0
$499$	27.1239	1.21423	0.607117	−	0.794613i	$-0.292326\pi$
$499$	27.1239	1.21423	0.607117	+	0.794613i	$0.292326\pi$
$500$	10.1848	0.455478
$501$	0	0
$502$	−17.9436	−0.800860
$503$	−0.571290	−0.0254725	−0.0127363	−	0.999919i	$-0.504054\pi$
$503$	−0.571290	−0.0254725	−0.0127363	+	0.999919i	$0.504054\pi$
$504$	0	0
$505$	14.9828	0.666724
$506$	−12.5963	−0.559972
$507$	0	0
$508$	−1.88713	−0.0837277
$509$	−40.5509	−1.79739	−0.898693	−	0.438579i	$-0.855482\pi$
$509$	−40.5509	−1.79739	−0.898693	+	0.438579i	$0.855482\pi$
$510$	0	0
$511$	−0.174363	−0.00771338
$512$	1.00000	0.0441942
$513$	0	0
$514$	−29.4124	−1.29733
$515$	−3.51073	−0.154701
$516$	0	0
$517$	3.83750	0.168773
$518$	3.66456	0.161012
$519$	0	0
$520$	4.59627	0.201560
$521$	−18.8999	−0.828020	−0.414010	−	0.910272i	$-0.635872\pi$
$521$	−18.8999	−0.828020	−0.414010	+	0.910272i	$0.635872\pi$
$522$	0	0
$523$	−4.74186	−0.207347	−0.103673	−	0.994611i	$-0.533060\pi$
$523$	−4.74186	−0.207347	−0.103673	+	0.994611i	$0.533060\pi$
$524$	−10.7023	−0.467534
$525$	0	0
$526$	−10.6827	−0.465789
$527$	2.27631	0.0991577
$528$	0	0
$529$	21.9213	0.953099
$530$	15.3678	0.667536
$531$	0	0
$532$	0	0
$533$	−34.8580	−1.50987
$534$	0	0
$535$	22.6459	0.979067
$536$	−11.1702	−0.482481
$537$	0	0
$538$	20.5767	0.887123
$539$	12.6236	0.543737
$540$	0	0
$541$	−28.3610	−1.21934	−0.609668	−	0.792657i	$-0.708697\pi$
$541$	−28.3610	−1.21934	−0.609668	+	0.792657i	$0.708697\pi$
$542$	1.87670	0.0806110
$543$	0	0
$544$	−1.16250	−0.0498419
$545$	19.4442	0.832898
$546$	0	0
$547$	−6.88032	−0.294181	−0.147091	−	0.989123i	$-0.546991\pi$
$547$	−6.88032	−0.294181	−0.147091	+	0.989123i	$0.546991\pi$
$548$	11.2567	0.480863
$549$	0	0
$550$	6.75877	0.288195
$551$	0	0
$552$	0	0
$553$	−8.58441	−0.365046
$554$	4.87258	0.207016
$555$	0	0
$556$	−12.1702	−0.516133
$557$	14.1625	0.600085	0.300042	−	0.953926i	$-0.402999\pi$
$557$	14.1625	0.600085	0.300042	+	0.953926i	$0.402999\pi$
$558$	0	0
$559$	−9.41147	−0.398063
$560$	−0.630415	−0.0266399
$561$	0	0
$562$	0.403733	0.0170305
$563$	36.2550	1.52796	0.763982	−	0.645237i	$-0.223242\pi$
$563$	36.2550	1.52796	0.763982	+	0.645237i	$0.223242\pi$
$564$	0	0
$565$	−6.79654	−0.285933
$566$	17.5895	0.739340
$567$	0	0
$568$	−6.07192	−0.254772
$569$	30.2918	1.26990	0.634949	−	0.772554i	$-0.281021\pi$
$569$	30.2918	1.26990	0.634949	+	0.772554i	$0.281021\pi$
$570$	0	0
$571$	3.39094	0.141906	0.0709532	−	0.997480i	$-0.477396\pi$
$571$	3.39094	0.141906	0.0709532	+	0.997480i	$0.477396\pi$
$572$	7.29086	0.304846
$573$	0	0
$574$	4.78106	0.199558
$575$	−24.1034	−1.00518
$576$	0	0
$577$	−22.3027	−0.928474	−0.464237	−	0.885711i	$-0.653671\pi$
$577$	−22.3027	−0.928474	−0.464237	+	0.885711i	$0.653671\pi$
$578$	−15.6486	−0.650895
$579$	0	0
$580$	−4.76558	−0.197880
$581$	−6.15839	−0.255493
$582$	0	0
$583$	24.3773	1.00961
$584$	−0.327696	−0.0135602
$585$	0	0
$586$	13.1753	0.544267
$587$	28.7502	1.18665	0.593324	−	0.804964i	$-0.297815\pi$
$587$	28.7502	1.18665	0.593324	+	0.804964i	$0.297815\pi$
$588$	0	0
$589$	0	0
$590$	−3.17530	−0.130725
$591$	0	0
$592$	6.88713	0.283059
$593$	−11.7760	−0.483583	−0.241791	−	0.970328i	$-0.577735\pi$
$593$	−11.7760	−0.483583	−0.241791	+	0.970328i	$0.577735\pi$
$594$	0	0
$595$	0.732860	0.0300443
$596$	16.3550	0.669928
$597$	0	0
$598$	−26.0009	−1.06326
$599$	21.6313	0.883833	0.441916	−	0.897056i	$-0.354299\pi$
$599$	21.6313	0.883833	0.441916	+	0.897056i	$0.354299\pi$
$600$	0	0
$601$	−26.1935	−1.06845	−0.534227	−	0.845341i	$-0.679397\pi$
$601$	−26.1935	−1.06845	−0.534227	+	0.845341i	$0.679397\pi$
$602$	1.29086	0.0526115
$603$	0	0
$604$	20.5226	0.835052
$605$	8.84793	0.359719
$606$	0	0
$607$	−13.2249	−0.536783	−0.268392	−	0.963310i	$-0.586492\pi$
$607$	−13.2249	−0.536783	−0.268392	+	0.963310i	$0.586492\pi$
$608$	0	0
$609$	0	0
$610$	13.4442	0.544339
$611$	7.92127	0.320460
$612$	0	0
$613$	−5.62267	−0.227098	−0.113549	−	0.993532i	$-0.536222\pi$
$613$	−5.62267	−0.227098	−0.113549	+	0.993532i	$0.536222\pi$
$614$	16.9017	0.682096
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−9.47834	−0.381584	−0.190792	−	0.981631i	$-0.561106\pi$
$617$	−9.47834	−0.381584	−0.190792	+	0.981631i	$0.561106\pi$
$618$	0	0
$619$	−14.7493	−0.592823	−0.296412	−	0.955060i	$-0.595790\pi$
$619$	−14.7493	−0.592823	−0.296412	+	0.955060i	$0.595790\pi$
$620$	2.31996	0.0931716
$621$	0	0
$622$	2.38144	0.0954872
$623$	1.89393	0.0758788
$624$	0	0
$625$	5.91447	0.236579
$626$	−29.8631	−1.19357
$627$	0	0
$628$	−3.19934	−0.127668
$629$	−8.00631	−0.319232
$630$	0	0
$631$	25.0009	0.995271	0.497636	−	0.867386i	$-0.334202\pi$
$631$	25.0009	0.995271	0.497636	+	0.867386i	$0.334202\pi$
$632$	−16.1334	−0.641753
$633$	0	0
$634$	7.02229	0.278891
$635$	2.23585	0.0887271
$636$	0	0
$637$	26.0574	1.03243
$638$	−7.55943	−0.299281
$639$	0	0
$640$	−1.18479	−0.0468330
$641$	43.7306	1.72726	0.863628	−	0.504130i	$-0.168187\pi$
$641$	43.7306	1.72726	0.863628	+	0.504130i	$0.168187\pi$
$642$	0	0
$643$	−33.4534	−1.31927	−0.659636	−	0.751585i	$-0.729290\pi$
$643$	−33.4534	−1.31927	−0.659636	+	0.751585i	$0.729290\pi$
$644$	3.56624	0.140529
$645$	0	0
$646$	0	0
$647$	32.1266	1.26303	0.631514	−	0.775365i	$-0.282434\pi$
$647$	32.1266	1.26303	0.631514	+	0.775365i	$0.282434\pi$
$648$	0	0
$649$	−5.03684	−0.197713
$650$	13.9513	0.547215
$651$	0	0
$652$	−24.0077	−0.940216
$653$	−2.89630	−0.113341	−0.0566704	−	0.998393i	$-0.518048\pi$
$653$	−2.89630	−0.113341	−0.0566704	+	0.998393i	$0.518048\pi$
$654$	0	0
$655$	12.6800	0.495450
$656$	8.98545	0.350823
$657$	0	0
$658$	−1.08647	−0.0423549
$659$	−18.1935	−0.708717	−0.354358	−	0.935110i	$-0.615301\pi$
$659$	−18.1935	−0.708717	−0.354358	+	0.935110i	$0.615301\pi$
$660$	0	0
$661$	21.4192	0.833111	0.416555	−	0.909110i	$-0.363237\pi$
$661$	21.4192	0.833111	0.416555	+	0.909110i	$0.363237\pi$
$662$	27.1985	1.05710
$663$	0	0
$664$	−11.5740	−0.449157
$665$	0	0
$666$	0	0
$667$	26.9587	1.04385
$668$	−3.66044	−0.141627
$669$	0	0
$670$	13.2344	0.511290
$671$	21.3259	0.823279
$672$	0	0
$673$	40.6914	1.56854	0.784269	−	0.620421i	$-0.213038\pi$
$673$	40.6914	1.56854	0.784269	+	0.620421i	$0.213038\pi$
$674$	−11.2172	−0.432070
$675$	0	0
$676$	2.04963	0.0788319
$677$	0.136096	0.00523061	0.00261530	−	0.999997i	$-0.499168\pi$
$677$	0.136096	0.00523061	0.00261530	+	0.999997i	$0.499168\pi$
$678$	0	0
$679$	3.12836	0.120055
$680$	1.37733	0.0528180
$681$	0	0
$682$	3.68004	0.140916
$683$	43.6459	1.67006	0.835032	−	0.550202i	$-0.185449\pi$
$683$	43.6459	1.67006	0.835032	+	0.550202i	$0.185449\pi$
$684$	0	0
$685$	−13.3369	−0.509576
$686$	−7.29860	−0.278662
$687$	0	0
$688$	2.42602	0.0924912
$689$	50.3191	1.91701
$690$	0	0
$691$	15.8675	0.603629	0.301815	−	0.953367i	$-0.402408\pi$
$691$	15.8675	0.603629	0.301815	+	0.953367i	$0.402408\pi$
$692$	5.33275	0.202721
$693$	0	0
$694$	−1.93582	−0.0734828
$695$	14.4192	0.546952
$696$	0	0
$697$	−10.4456	−0.395656
$698$	14.3969	0.544932
$699$	0	0
$700$	−1.91353	−0.0723248
$701$	−19.2098	−0.725543	−0.362771	−	0.931878i	$-0.618169\pi$
$701$	−19.2098	−0.725543	−0.362771	+	0.931878i	$0.618169\pi$
$702$	0	0
$703$	0	0
$704$	−1.87939	−0.0708320
$705$	0	0
$706$	−15.7237	−0.591769
$707$	−6.72874	−0.253060
$708$	0	0
$709$	33.0259	1.24031	0.620157	−	0.784478i	$-0.287069\pi$
$709$	33.0259	1.24031	0.620157	+	0.784478i	$0.287069\pi$
$710$	7.19396	0.269985
$711$	0	0
$712$	3.55943	0.133395
$713$	−13.1239	−0.491494
$714$	0	0
$715$	−8.63816	−0.323049
$716$	25.2567	0.943888
$717$	0	0
$718$	−15.5175	−0.579109
$719$	−39.4448	−1.47104	−0.735521	−	0.677501i	$-0.763063\pi$
$719$	−39.4448	−1.47104	−0.735521	+	0.677501i	$0.763063\pi$
$720$	0	0
$721$	1.57667	0.0587181
$722$	0	0
$723$	0	0
$724$	−17.3063	−0.643185
$725$	−14.4652	−0.537225
$726$	0	0
$727$	18.2517	0.676917	0.338458	−	0.940981i	$-0.390095\pi$
$727$	18.2517	0.676917	0.338458	+	0.940981i	$0.390095\pi$
$728$	−2.06418	−0.0765035
$729$	0	0
$730$	0.388252	0.0143698
$731$	−2.82026	−0.104311
$732$	0	0
$733$	15.3928	0.568546	0.284273	−	0.958743i	$-0.408248\pi$
$733$	15.3928	0.568546	0.284273	+	0.958743i	$0.408248\pi$
$734$	−15.7442	−0.581130
$735$	0	0
$736$	6.70233	0.247051
$737$	20.9932	0.773294
$738$	0	0
$739$	15.1771	0.558297	0.279148	−	0.960248i	$-0.409948\pi$
$739$	15.1771	0.558297	0.279148	+	0.960248i	$0.409948\pi$
$740$	−8.15982	−0.299961
$741$	0	0
$742$	−6.90167	−0.253368
$743$	9.86659	0.361970	0.180985	−	0.983486i	$-0.442071\pi$
$743$	9.86659	0.361970	0.180985	+	0.983486i	$0.442071\pi$
$744$	0	0
$745$	−19.3773	−0.709930
$746$	−29.6287	−1.08478
$747$	0	0
$748$	2.18479	0.0798839
$749$	−10.1702	−0.371613
$750$	0	0
$751$	−9.75877	−0.356103	−0.178051	−	0.984021i	$-0.556979\pi$
$751$	−9.75877	−0.356103	−0.178051	+	0.984021i	$0.556979\pi$
$752$	−2.04189	−0.0744600
$753$	0	0
$754$	−15.6040	−0.568264
$755$	−24.3150	−0.884914
$756$	0	0
$757$	−43.5039	−1.58118	−0.790589	−	0.612348i	$-0.790225\pi$
$757$	−43.5039	−1.58118	−0.790589	+	0.612348i	$0.790225\pi$
$758$	4.72462	0.171606
$759$	0	0
$760$	0	0
$761$	−44.7137	−1.62087	−0.810435	−	0.585828i	$-0.800769\pi$
$761$	−44.7137	−1.62087	−0.810435	+	0.585828i	$0.800769\pi$
$762$	0	0
$763$	−8.73236	−0.316133
$764$	15.1780	0.549120
$765$	0	0
$766$	5.24123	0.189373
$767$	−10.3969	−0.375411
$768$	0	0
$769$	22.1019	0.797017	0.398508	−	0.917165i	$-0.369528\pi$
$769$	22.1019	0.797017	0.398508	+	0.917165i	$0.369528\pi$
$770$	1.18479	0.0426970
$771$	0	0
$772$	13.0077	0.468159
$773$	−42.1789	−1.51707	−0.758535	−	0.651632i	$-0.774085\pi$
$773$	−42.1789	−1.51707	−0.758535	+	0.651632i	$0.774085\pi$
$774$	0	0
$775$	7.04189	0.252952
$776$	5.87939	0.211058
$777$	0	0
$778$	15.0770	0.540536
$779$	0	0
$780$	0	0
$781$	11.4115	0.408335
$782$	−7.79149	−0.278623
$783$	0	0
$784$	−6.71688	−0.239889
$785$	3.79055	0.135291
$786$	0	0
$787$	−28.4793	−1.01518	−0.507588	−	0.861600i	$-0.669463\pi$
$787$	−28.4793	−1.01518	−0.507588	+	0.861600i	$0.669463\pi$
$788$	8.94862	0.318781
$789$	0	0
$790$	19.1147	0.680072
$791$	3.05232	0.108528
$792$	0	0
$793$	44.0205	1.56321
$794$	−14.5449	−0.516179
$795$	0	0
$796$	−4.50475	−0.159667
$797$	−16.6081	−0.588290	−0.294145	−	0.955761i	$-0.595035\pi$
$797$	−16.6081	−0.588290	−0.294145	+	0.955761i	$0.595035\pi$
$798$	0	0
$799$	2.37370	0.0839756
$800$	−3.59627	−0.127147
$801$	0	0
$802$	−2.85473	−0.100804
$803$	0.615867	0.0217335
$804$	0	0
$805$	−4.22525	−0.148921
$806$	7.59627	0.267567
$807$	0	0
$808$	−12.6459	−0.444881
$809$	13.6783	0.480903	0.240452	−	0.970661i	$-0.422704\pi$
$809$	13.6783	0.480903	0.240452	+	0.970661i	$0.422704\pi$
$810$	0	0
$811$	−0.618231	−0.0217090	−0.0108545	−	0.999941i	$-0.503455\pi$
$811$	−0.618231	−0.0217090	−0.0108545	+	0.999941i	$0.503455\pi$
$812$	2.14022	0.0751068
$813$	0	0
$814$	−12.9436	−0.453672
$815$	28.4442	0.996357
$816$	0	0
$817$	0	0
$818$	34.2276	1.19674
$819$	0	0
$820$	−10.6459	−0.371771
$821$	−6.11205	−0.213312	−0.106656	−	0.994296i	$-0.534014\pi$
$821$	−6.11205	−0.213312	−0.106656	+	0.994296i	$0.534014\pi$
$822$	0	0
$823$	30.6546	1.06855	0.534276	−	0.845310i	$-0.320584\pi$
$823$	30.6546	1.06855	0.534276	+	0.845310i	$0.320584\pi$
$824$	2.96316	0.103227
$825$	0	0
$826$	1.42602	0.0496177
$827$	50.7134	1.76348	0.881738	−	0.471739i	$-0.156373\pi$
$827$	50.7134	1.76348	0.881738	+	0.471739i	$0.156373\pi$
$828$	0	0
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	13.7128	0.475977
$831$	0	0
$832$	−3.87939	−0.134493
$833$	7.80840	0.270545
$834$	0	0
$835$	4.33687	0.150083
$836$	0	0
$837$	0	0
$838$	7.91891	0.273554
$839$	21.4979	0.742191	0.371096	−	0.928595i	$-0.378982\pi$
$839$	21.4979	0.742191	0.371096	+	0.928595i	$0.378982\pi$
$840$	0	0
$841$	−12.8212	−0.442110
$842$	16.2098	0.558626
$843$	0	0
$844$	−6.40373	−0.220426
$845$	−2.42839	−0.0835390
$846$	0	0
$847$	−3.97359	−0.136534
$848$	−12.9709	−0.445423
$849$	0	0
$850$	4.18067	0.143396
$851$	46.1598	1.58234
$852$	0	0
$853$	−46.2276	−1.58280	−0.791402	−	0.611296i	$-0.790648\pi$
$853$	−46.2276	−1.58280	−0.791402	+	0.611296i	$0.790648\pi$
$854$	−6.03777	−0.206608
$855$	0	0
$856$	−19.1138	−0.653296
$857$	43.6623	1.49148	0.745738	−	0.666239i	$-0.232097\pi$
$857$	43.6623	1.49148	0.745738	+	0.666239i	$0.232097\pi$
$858$	0	0
$859$	20.1652	0.688027	0.344014	−	0.938965i	$-0.388213\pi$
$859$	20.1652	0.688027	0.344014	+	0.938965i	$0.388213\pi$
$860$	−2.87433	−0.0980139
$861$	0	0
$862$	−25.4320	−0.866218
$863$	32.6932	1.11289	0.556444	−	0.830885i	$-0.312165\pi$
$863$	32.6932	1.11289	0.556444	+	0.830885i	$0.312165\pi$
$864$	0	0
$865$	−6.31820	−0.214825
$866$	11.8479	0.402609
$867$	0	0
$868$	−1.04189	−0.0353640
$869$	30.3209	1.02857
$870$	0	0
$871$	43.3337	1.46831
$872$	−16.4115	−0.555763
$873$	0	0
$874$	0	0
$875$	5.41921	0.183203
$876$	0	0
$877$	30.2968	1.02305	0.511526	−	0.859268i	$-0.329080\pi$
$877$	30.2968	1.02305	0.511526	+	0.859268i	$0.329080\pi$
$878$	25.8598	0.872725
$879$	0	0
$880$	2.22668	0.0750614
$881$	−23.5577	−0.793678	−0.396839	−	0.917888i	$-0.629893\pi$
$881$	−23.5577	−0.793678	−0.396839	+	0.917888i	$0.629893\pi$
$882$	0	0
$883$	36.8289	1.23939	0.619696	−	0.784842i	$-0.287256\pi$
$883$	36.8289	1.23939	0.619696	+	0.784842i	$0.287256\pi$
$884$	4.50980	0.151681
$885$	0	0
$886$	−0.923029	−0.0310098
$887$	−4.30810	−0.144652	−0.0723258	−	0.997381i	$-0.523042\pi$
$887$	−4.30810	−0.144652	−0.0723258	+	0.997381i	$0.523042\pi$
$888$	0	0
$889$	−1.00412	−0.0336771
$890$	−4.21719	−0.141360
$891$	0	0
$892$	3.24897	0.108784
$893$	0	0
$894$	0	0
$895$	−29.9240	−1.00025
$896$	0.532089	0.0177758
$897$	0	0
$898$	−9.34461	−0.311834
$899$	−7.87609	−0.262682
$900$	0	0
$901$	15.0787	0.502345
$902$	−16.8871	−0.562280
$903$	0	0
$904$	5.73648	0.190793
$905$	20.5044	0.681590
$906$	0	0
$907$	23.3405	0.775008	0.387504	−	0.921868i	$-0.373337\pi$
$907$	23.3405	0.775008	0.387504	+	0.921868i	$0.373337\pi$
$908$	11.7023	0.388356
$909$	0	0
$910$	2.44562	0.0810716
$911$	−22.5631	−0.747547	−0.373774	−	0.927520i	$-0.621936\pi$
$911$	−22.5631	−0.747547	−0.373774	+	0.927520i	$0.621936\pi$
$912$	0	0
$913$	21.7520	0.719885
$914$	30.9009	1.02211
$915$	0	0
$916$	−22.9067	−0.756860
$917$	−5.69459	−0.188052
$918$	0	0
$919$	4.25578	0.140385	0.0701926	−	0.997533i	$-0.477639\pi$
$919$	4.25578	0.140385	0.0701926	+	0.997533i	$0.477639\pi$
$920$	−7.94087	−0.261803
$921$	0	0
$922$	29.6878	0.977715
$923$	23.5553	0.775333
$924$	0	0
$925$	−24.7679	−0.814365
$926$	−11.8307	−0.388781
$927$	0	0
$928$	4.02229	0.132038
$929$	−59.5245	−1.95293	−0.976467	−	0.215666i	$-0.930808\pi$
$929$	−59.5245	−1.95293	−0.976467	+	0.215666i	$0.930808\pi$
$930$	0	0
$931$	0	0
$932$	3.19934	0.104798
$933$	0	0
$934$	−38.3182	−1.25381
$935$	−2.58853	−0.0846538
$936$	0	0
$937$	33.5090	1.09469	0.547345	−	0.836907i	$-0.315638\pi$
$937$	33.5090	1.09469	0.547345	+	0.836907i	$0.315638\pi$
$938$	−5.94356	−0.194064
$939$	0	0
$940$	2.41921	0.0789061
$941$	5.52435	0.180089	0.0900443	−	0.995938i	$-0.471299\pi$
$941$	5.52435	0.180089	0.0900443	+	0.995938i	$0.471299\pi$
$942$	0	0
$943$	60.2235	1.96115
$944$	2.68004	0.0872280
$945$	0	0
$946$	−4.55943	−0.148240
$947$	14.9050	0.484347	0.242173	−	0.970233i	$-0.422140\pi$
$947$	14.9050	0.484347	0.242173	+	0.970233i	$0.422140\pi$
$948$	0	0
$949$	1.27126	0.0412668
$950$	0	0
$951$	0	0
$952$	−0.618555	−0.0200475
$953$	46.3661	1.50194	0.750972	−	0.660334i	$-0.229585\pi$
$953$	46.3661	1.50194	0.750972	+	0.660334i	$0.229585\pi$
$954$	0	0
$955$	−17.9828	−0.581909
$956$	−6.17293	−0.199647
$957$	0	0
$958$	−37.2841	−1.20459
$959$	5.98957	0.193413
$960$	0	0
$961$	−27.1658	−0.876316
$962$	−26.7178	−0.861417
$963$	0	0
$964$	1.53714	0.0495080
$965$	−15.4115	−0.496113
$966$	0	0
$967$	−23.2431	−0.747448	−0.373724	−	0.927540i	$-0.621919\pi$
$967$	−23.2431	−0.747448	−0.373724	+	0.927540i	$0.621919\pi$
$968$	−7.46791	−0.240028
$969$	0	0
$970$	−6.96585	−0.223660
$971$	−4.93045	−0.158226	−0.0791128	−	0.996866i	$-0.525209\pi$
$971$	−4.93045	−0.158226	−0.0791128	+	0.996866i	$0.525209\pi$
$972$	0	0
$973$	−6.47565	−0.207600
$974$	32.9786	1.05670
$975$	0	0
$976$	−11.3473	−0.363218
$977$	−32.7442	−1.04758	−0.523790	−	0.851847i	$-0.675482\pi$
$977$	−32.7442	−1.04758	−0.523790	+	0.851847i	$0.675482\pi$
$978$	0	0
$979$	−6.68954	−0.213799
$980$	7.95811	0.254213
$981$	0	0
$982$	−17.6810	−0.564223
$983$	25.4406	0.811428	0.405714	−	0.914000i	$-0.367023\pi$
$983$	25.4406	0.811428	0.405714	+	0.914000i	$0.367023\pi$
$984$	0	0
$985$	−10.6023	−0.337816
$986$	−4.67593	−0.148912
$987$	0	0
$988$	0	0
$989$	16.2600	0.517038
$990$	0	0
$991$	10.1533	0.322531	0.161266	−	0.986911i	$-0.448442\pi$
$991$	10.1533	0.322531	0.161266	+	0.986911i	$0.448442\pi$
$992$	−1.95811	−0.0621701
$993$	0	0
$994$	−3.23080	−0.102475
$995$	5.33719	0.169200
$996$	0	0
$997$	55.7698	1.76625	0.883124	−	0.469140i	$-0.155436\pi$
$997$	55.7698	1.76625	0.883124	+	0.469140i	$0.155436\pi$
$998$	27.1239	0.858592
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bs.1.2		3
3.2	odd	2		2166.2.a.o.1.2		3
19.9	even	9		342.2.u.a.271.1		6
19.17	even	9		342.2.u.a.289.1		6
19.18	odd	2		6498.2.a.bn.1.2		3
57.17	odd	18		114.2.i.d.61.1	yes	6
57.47	odd	18		114.2.i.d.43.1	✓	6
57.56	even	2		2166.2.a.u.1.2		3
228.47	even	18		912.2.bo.f.385.1		6
228.131	even	18		912.2.bo.f.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.d.43.1	✓	6	57.47	odd	18
114.2.i.d.61.1	yes	6	57.17	odd	18
342.2.u.a.271.1		6	19.9	even	9
342.2.u.a.289.1		6	19.17	even	9
912.2.bo.f.289.1		6	228.131	even	18
912.2.bo.f.385.1		6	228.47	even	18
2166.2.a.o.1.2		3	3.2	odd	2
2166.2.a.u.1.2		3	57.56	even	2
6498.2.a.bn.1.2		3	19.18	odd	2
6498.2.a.bs.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.18479 q^{5} +0.532089 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.18479 q^{5} +0.532089 q^{7} +1.00000 q^{8} -1.18479 q^{10} -1.87939 q^{11} -3.87939 q^{13} +0.532089 q^{14} +1.00000 q^{16} -1.16250 q^{17} -1.18479 q^{20} -1.87939 q^{22} +6.70233 q^{23} -3.59627 q^{25} -3.87939 q^{26} +0.532089 q^{28} +4.02229 q^{29} -1.95811 q^{31} +1.00000 q^{32} -1.16250 q^{34} -0.630415 q^{35} +6.88713 q^{37} -1.18479 q^{40} +8.98545 q^{41} +2.42602 q^{43} -1.87939 q^{44} +6.70233 q^{46} -2.04189 q^{47} -6.71688 q^{49} -3.59627 q^{50} -3.87939 q^{52} -12.9709 q^{53} +2.22668 q^{55} +0.532089 q^{56} +4.02229 q^{58} +2.68004 q^{59} -11.3473 q^{61} -1.95811 q^{62} +1.00000 q^{64} +4.59627 q^{65} -11.1702 q^{67} -1.16250 q^{68} -0.630415 q^{70} -6.07192 q^{71} -0.327696 q^{73} +6.88713 q^{74} -1.00000 q^{77} -16.1334 q^{79} -1.18479 q^{80} +8.98545 q^{82} -11.5740 q^{83} +1.37733 q^{85} +2.42602 q^{86} -1.87939 q^{88} +3.55943 q^{89} -2.06418 q^{91} +6.70233 q^{92} -2.04189 q^{94} +5.87939 q^{97} -6.71688 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{7} + 3 q^{8} - 6 q^{13} - 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} + 6 q^{29} - 9 q^{31} + 3 q^{32} - 6 q^{34} - 9 q^{35} - 9 q^{37} + 9 q^{41} + 15 q^{43} - 6 q^{46} - 3 q^{47} - 12 q^{49} + 3 q^{50} - 6 q^{52} - 3 q^{53} - 3 q^{56} + 6 q^{58} - 12 q^{59} - 33 q^{61} - 9 q^{62} + 3 q^{64} - 12 q^{67} - 6 q^{68} - 9 q^{70} + 15 q^{71} + 3 q^{73} - 9 q^{74} - 3 q^{77} - 15 q^{79} + 9 q^{82} - 27 q^{83} - 27 q^{85} + 15 q^{86} - 15 q^{89} + 3 q^{91} - 6 q^{92} - 3 q^{94} + 12 q^{97} - 12 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^7 + 3 * q^8 - 6 * q^13 - 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 + 6 * q^29 - 9 * q^31 + 3 * q^32 - 6 * q^34 - 9 * q^35 - 9 * q^37 + 9 * q^41 + 15 * q^43 - 6 * q^46 - 3 * q^47 - 12 * q^49 + 3 * q^50 - 6 * q^52 - 3 * q^53 - 3 * q^56 + 6 * q^58 - 12 * q^59 - 33 * q^61 - 9 * q^62 + 3 * q^64 - 12 * q^67 - 6 * q^68 - 9 * q^70 + 15 * q^71 + 3 * q^73 - 9 * q^74 - 3 * q^77 - 15 * q^79 + 9 * q^82 - 27 * q^83 - 27 * q^85 + 15 * q^86 - 15 * q^89 + 3 * q^91 - 6 * q^92 - 3 * q^94 + 12 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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