LMFDB - Embedded newform 6498.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +3.64575 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +3.64575 q^{7} +1.00000 q^{8} +1.64575 q^{10} +4.64575 q^{11} -2.00000 q^{13} +3.64575 q^{14} +1.00000 q^{16} +1.64575 q^{20} +4.64575 q^{22} +1.64575 q^{23} -2.29150 q^{25} -2.00000 q^{26} +3.64575 q^{28} +1.64575 q^{29} +5.64575 q^{31} +1.00000 q^{32} +6.00000 q^{35} -0.354249 q^{37} +1.64575 q^{40} -0.291503 q^{41} +11.2915 q^{43} +4.64575 q^{44} +1.64575 q^{46} -4.35425 q^{47} +6.29150 q^{49} -2.29150 q^{50} -2.00000 q^{52} -12.5830 q^{53} +7.64575 q^{55} +3.64575 q^{56} +1.64575 q^{58} -7.93725 q^{59} +0.937254 q^{61} +5.64575 q^{62} +1.00000 q^{64} -3.29150 q^{65} -0.645751 q^{67} +6.00000 q^{70} -2.70850 q^{71} +1.70850 q^{73} -0.354249 q^{74} +16.9373 q^{77} +4.00000 q^{79} +1.64575 q^{80} -0.291503 q^{82} -7.93725 q^{83} +11.2915 q^{86} +4.64575 q^{88} -7.29150 q^{91} +1.64575 q^{92} -4.35425 q^{94} +3.70850 q^{97} +6.29150 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 6 q^{25} - 4 q^{26} + 2 q^{28} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 12 q^{35} - 6 q^{37} - 2 q^{40} + 10 q^{41} + 12 q^{43} + 4 q^{44} - 2 q^{46} - 14 q^{47} + 2 q^{49} + 6 q^{50} - 4 q^{52} - 4 q^{53} + 10 q^{55} + 2 q^{56} - 2 q^{58} - 14 q^{61} + 6 q^{62} + 2 q^{64} + 4 q^{65} + 4 q^{67} + 12 q^{70} - 16 q^{71} + 14 q^{73} - 6 q^{74} + 18 q^{77} + 8 q^{79} - 2 q^{80} + 10 q^{82} + 12 q^{86} + 4 q^{88} - 4 q^{91} - 2 q^{92} - 14 q^{94} + 18 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 6 * q^25 - 4 * q^26 + 2 * q^28 - 2 * q^29 + 6 * q^31 + 2 * q^32 + 12 * q^35 - 6 * q^37 - 2 * q^40 + 10 * q^41 + 12 * q^43 + 4 * q^44 - 2 * q^46 - 14 * q^47 + 2 * q^49 + 6 * q^50 - 4 * q^52 - 4 * q^53 + 10 * q^55 + 2 * q^56 - 2 * q^58 - 14 * q^61 + 6 * q^62 + 2 * q^64 + 4 * q^65 + 4 * q^67 + 12 * q^70 - 16 * q^71 + 14 * q^73 - 6 * q^74 + 18 * q^77 + 8 * q^79 - 2 * q^80 + 10 * q^82 + 12 * q^86 + 4 * q^88 - 4 * q^91 - 2 * q^92 - 14 * q^94 + 18 * q^97 + 2 * 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
$5$	1.64575	0.736002	0.368001	−	0.929825i	$-0.380042\pi$
$5$	1.64575	0.736002	0.368001	+	0.929825i	$0.380042\pi$
$6$	0	0
$7$	3.64575	1.37796	0.688982	−	0.724778i	$-0.258058\pi$
$7$	3.64575	1.37796	0.688982	+	0.724778i	$0.258058\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.64575	0.520432
$11$	4.64575	1.40075	0.700373	−	0.713777i	$-0.253017\pi$
$11$	4.64575	1.40075	0.700373	+	0.713777i	$0.253017\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.64575	0.974368
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	1.64575	0.368001
$21$	0	0
$22$	4.64575	0.990478
$23$	1.64575	0.343163	0.171581	−	0.985170i	$-0.445112\pi$
$23$	1.64575	0.343163	0.171581	+	0.985170i	$0.445112\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	−2.00000	−0.392232
$27$	0	0
$28$	3.64575	0.688982
$29$	1.64575	0.305608	0.152804	−	0.988256i	$-0.451170\pi$
$29$	1.64575	0.305608	0.152804	+	0.988256i	$0.451170\pi$
$30$	0	0
$31$	5.64575	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$31$	5.64575	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−0.354249	−0.0582381	−0.0291191	−	0.999576i	$-0.509270\pi$
$37$	−0.354249	−0.0582381	−0.0291191	+	0.999576i	$0.509270\pi$
$38$	0	0
$39$	0	0
$40$	1.64575	0.260216
$41$	−0.291503	−0.0455251	−0.0227625	−	0.999741i	$-0.507246\pi$
$41$	−0.291503	−0.0455251	−0.0227625	+	0.999741i	$0.507246\pi$
$42$	0	0
$43$	11.2915	1.72194	0.860969	−	0.508657i	$-0.169858\pi$
$43$	11.2915	1.72194	0.860969	+	0.508657i	$0.169858\pi$
$44$	4.64575	0.700373
$45$	0	0
$46$	1.64575	0.242653
$47$	−4.35425	−0.635132	−0.317566	−	0.948236i	$-0.602866\pi$
$47$	−4.35425	−0.635132	−0.317566	+	0.948236i	$0.602866\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	−2.29150	−0.324067
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−12.5830	−1.72841	−0.864204	−	0.503141i	$-0.832178\pi$
$53$	−12.5830	−1.72841	−0.864204	+	0.503141i	$0.832178\pi$
$54$	0	0
$55$	7.64575	1.03095
$56$	3.64575	0.487184
$57$	0	0
$58$	1.64575	0.216098
$59$	−7.93725	−1.03334	−0.516671	−	0.856184i	$-0.672829\pi$
$59$	−7.93725	−1.03334	−0.516671	+	0.856184i	$0.672829\pi$
$60$	0	0
$61$	0.937254	0.120003	0.0600015	−	0.998198i	$-0.480889\pi$
$61$	0.937254	0.120003	0.0600015	+	0.998198i	$0.480889\pi$
$62$	5.64575	0.717011
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.29150	−0.408261
$66$	0	0
$67$	−0.645751	−0.0788911	−0.0394455	−	0.999222i	$-0.512559\pi$
$67$	−0.645751	−0.0788911	−0.0394455	+	0.999222i	$0.512559\pi$
$68$	0	0
$69$	0	0
$70$	6.00000	0.717137
$71$	−2.70850	−0.321440	−0.160720	−	0.987000i	$-0.551382\pi$
$71$	−2.70850	−0.321440	−0.160720	+	0.987000i	$0.551382\pi$
$72$	0	0
$73$	1.70850	0.199964	0.0999822	−	0.994989i	$-0.468121\pi$
$73$	1.70850	0.199964	0.0999822	+	0.994989i	$0.468121\pi$
$74$	−0.354249	−0.0411806
$75$	0	0
$76$	0	0
$77$	16.9373	1.93018
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.64575	0.184001
$81$	0	0
$82$	−0.291503	−0.0321911
$83$	−7.93725	−0.871227	−0.435613	−	0.900134i	$-0.643469\pi$
$83$	−7.93725	−0.871227	−0.435613	+	0.900134i	$0.643469\pi$
$84$	0	0
$85$	0	0
$86$	11.2915	1.21759
$87$	0	0
$88$	4.64575	0.495239
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−7.29150	−0.764357
$92$	1.64575	0.171581
$93$	0	0
$94$	−4.35425	−0.449106
$95$	0	0
$96$	0	0
$97$	3.70850	0.376541	0.188270	−	0.982117i	$-0.439712\pi$
$97$	3.70850	0.376541	0.188270	+	0.982117i	$0.439712\pi$
$98$	6.29150	0.635538
$99$	0	0
$100$	−2.29150	−0.229150
$101$	13.6458	1.35780	0.678902	−	0.734229i	$-0.262456\pi$
$101$	13.6458	1.35780	0.678902	+	0.734229i	$0.262456\pi$
$102$	0	0
$103$	13.2915	1.30965	0.654825	−	0.755780i	$-0.272742\pi$
$103$	13.2915	1.30965	0.654825	+	0.755780i	$0.272742\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−12.5830	−1.22217
$107$	15.2915	1.47829	0.739143	−	0.673549i	$-0.235231\pi$
$107$	15.2915	1.47829	0.739143	+	0.673549i	$0.235231\pi$
$108$	0	0
$109$	−14.5830	−1.39680	−0.698399	−	0.715708i	$-0.746104\pi$
$109$	−14.5830	−1.39680	−0.698399	+	0.715708i	$0.746104\pi$
$110$	7.64575	0.728994
$111$	0	0
$112$	3.64575	0.344491
$113$	−15.5830	−1.46593	−0.732963	−	0.680269i	$-0.761863\pi$
$113$	−15.5830	−1.46593	−0.732963	+	0.680269i	$0.761863\pi$
$114$	0	0
$115$	2.70850	0.252569
$116$	1.64575	0.152804
$117$	0	0
$118$	−7.93725	−0.730683
$119$	0	0
$120$	0	0
$121$	10.5830	0.962091
$122$	0.937254	0.0848550
$123$	0	0
$124$	5.64575	0.507003
$125$	−12.0000	−1.07331
$126$	0	0
$127$	13.2915	1.17943	0.589715	−	0.807611i	$-0.299240\pi$
$127$	13.2915	1.17943	0.589715	+	0.807611i	$0.299240\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.29150	−0.288684
$131$	−1.93725	−0.169259	−0.0846293	−	0.996413i	$-0.526971\pi$
$131$	−1.93725	−0.169259	−0.0846293	+	0.996413i	$0.526971\pi$
$132$	0	0
$133$	0	0
$134$	−0.645751	−0.0557844
$135$	0	0
$136$	0	0
$137$	−15.5830	−1.33135	−0.665673	−	0.746244i	$-0.731855\pi$
$137$	−15.5830	−1.33135	−0.665673	+	0.746244i	$0.731855\pi$
$138$	0	0
$139$	18.6458	1.58151	0.790756	−	0.612131i	$-0.209688\pi$
$139$	18.6458	1.58151	0.790756	+	0.612131i	$0.209688\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−2.70850	−0.227292
$143$	−9.29150	−0.776994
$144$	0	0
$145$	2.70850	0.224928
$146$	1.70850	0.141396
$147$	0	0
$148$	−0.354249	−0.0291191
$149$	−10.9373	−0.896015	−0.448007	−	0.894030i	$-0.647866\pi$
$149$	−10.9373	−0.896015	−0.448007	+	0.894030i	$0.647866\pi$
$150$	0	0
$151$	−12.9373	−1.05282	−0.526409	−	0.850231i	$-0.676462\pi$
$151$	−12.9373	−1.05282	−0.526409	+	0.850231i	$0.676462\pi$
$152$	0	0
$153$	0	0
$154$	16.9373	1.36484
$155$	9.29150	0.746311
$156$	0	0
$157$	−10.5830	−0.844616	−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830	−0.844616	−0.422308	+	0.906452i	$0.638780\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	1.64575	0.130108
$161$	6.00000	0.472866
$162$	0	0
$163$	3.93725	0.308390	0.154195	−	0.988040i	$-0.450722\pi$
$163$	3.93725	0.308390	0.154195	+	0.988040i	$0.450722\pi$
$164$	−0.291503	−0.0227625
$165$	0	0
$166$	−7.93725	−0.616050
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	11.2915	0.860969
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−8.35425	−0.631522
$176$	4.64575	0.350187
$177$	0	0
$178$	0	0
$179$	4.06275	0.303664	0.151832	−	0.988406i	$-0.451483\pi$
$179$	4.06275	0.303664	0.151832	+	0.988406i	$0.451483\pi$
$180$	0	0
$181$	−22.2288	−1.65225	−0.826125	−	0.563487i	$-0.809460\pi$
$181$	−22.2288	−1.65225	−0.826125	+	0.563487i	$0.809460\pi$
$182$	−7.29150	−0.540482
$183$	0	0
$184$	1.64575	0.121326
$185$	−0.583005	−0.0428634
$186$	0	0
$187$	0	0
$188$	−4.35425	−0.317566
$189$	0	0
$190$	0	0
$191$	−6.58301	−0.476330	−0.238165	−	0.971225i	$-0.576546\pi$
$191$	−6.58301	−0.476330	−0.238165	+	0.971225i	$0.576546\pi$
$192$	0	0
$193$	−14.5830	−1.04971	−0.524854	−	0.851192i	$-0.675880\pi$
$193$	−14.5830	−1.04971	−0.524854	+	0.851192i	$0.675880\pi$
$194$	3.70850	0.266255
$195$	0	0
$196$	6.29150	0.449393
$197$	7.64575	0.544737	0.272369	−	0.962193i	$-0.412193\pi$
$197$	7.64575	0.544737	0.272369	+	0.962193i	$0.412193\pi$
$198$	0	0
$199$	−19.8745	−1.40887	−0.704433	−	0.709770i	$-0.748799\pi$
$199$	−19.8745	−1.40887	−0.704433	+	0.709770i	$0.748799\pi$
$200$	−2.29150	−0.162034
$201$	0	0
$202$	13.6458	0.960112
$203$	6.00000	0.421117
$204$	0	0
$205$	−0.479741	−0.0335066
$206$	13.2915	0.926063
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	13.2915	0.915025	0.457512	−	0.889203i	$-0.348741\pi$
$211$	13.2915	0.915025	0.457512	+	0.889203i	$0.348741\pi$
$212$	−12.5830	−0.864204
$213$	0	0
$214$	15.2915	1.04531
$215$	18.5830	1.26735
$216$	0	0
$217$	20.5830	1.39727
$218$	−14.5830	−0.987686
$219$	0	0
$220$	7.64575	0.515476
$221$	0	0
$222$	0	0
$223$	18.8118	1.25973	0.629864	−	0.776705i	$-0.283110\pi$
$223$	18.8118	1.25973	0.629864	+	0.776705i	$0.283110\pi$
$224$	3.64575	0.243592
$225$	0	0
$226$	−15.5830	−1.03657
$227$	−7.35425	−0.488119	−0.244059	−	0.969760i	$-0.578479\pi$
$227$	−7.35425	−0.488119	−0.244059	+	0.969760i	$0.578479\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	2.70850	0.178593
$231$	0	0
$232$	1.64575	0.108049
$233$	−18.8745	−1.23651	−0.618255	−	0.785978i	$-0.712160\pi$
$233$	−18.8745	−1.23651	−0.618255	+	0.785978i	$0.712160\pi$
$234$	0	0
$235$	−7.16601	−0.467459
$236$	−7.93725	−0.516671
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	7.58301	0.488464	0.244232	−	0.969717i	$-0.421464\pi$
$241$	7.58301	0.488464	0.244232	+	0.969717i	$0.421464\pi$
$242$	10.5830	0.680301
$243$	0	0
$244$	0.937254	0.0600015
$245$	10.3542	0.661509
$246$	0	0
$247$	0	0
$248$	5.64575	0.358506
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−29.2288	−1.84490	−0.922451	−	0.386113i	$-0.873817\pi$
$251$	−29.2288	−1.84490	−0.922451	+	0.386113i	$0.873817\pi$
$252$	0	0
$253$	7.64575	0.480684
$254$	13.2915	0.833983
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.2915	0.766723	0.383361	−	0.923598i	$-0.374766\pi$
$257$	12.2915	0.766723	0.383361	+	0.923598i	$0.374766\pi$
$258$	0	0
$259$	−1.29150	−0.0802501
$260$	−3.29150	−0.204130
$261$	0	0
$262$	−1.93725	−0.119684
$263$	10.9373	0.674420	0.337210	−	0.941429i	$-0.390517\pi$
$263$	10.9373	0.674420	0.337210	+	0.941429i	$0.390517\pi$
$264$	0	0
$265$	−20.7085	−1.27211
$266$	0	0
$267$	0	0
$268$	−0.645751	−0.0394455
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	12.3542	0.750467	0.375234	−	0.926930i	$-0.377562\pi$
$271$	12.3542	0.750467	0.375234	+	0.926930i	$0.377562\pi$
$272$	0	0
$273$	0	0
$274$	−15.5830	−0.941404
$275$	−10.6458	−0.641963
$276$	0	0
$277$	−27.5203	−1.65353	−0.826766	−	0.562546i	$-0.809822\pi$
$277$	−27.5203	−1.65353	−0.826766	+	0.562546i	$0.809822\pi$
$278$	18.6458	1.11830
$279$	0	0
$280$	6.00000	0.358569
$281$	25.4575	1.51867	0.759334	−	0.650701i	$-0.225525\pi$
$281$	25.4575	1.51867	0.759334	+	0.650701i	$0.225525\pi$
$282$	0	0
$283$	30.6458	1.82170	0.910850	−	0.412737i	$-0.135427\pi$
$283$	30.6458	1.82170	0.910850	+	0.412737i	$0.135427\pi$
$284$	−2.70850	−0.160720
$285$	0	0
$286$	−9.29150	−0.549418
$287$	−1.06275	−0.0627319
$288$	0	0
$289$	−17.0000	−1.00000
$290$	2.70850	0.159048
$291$	0	0
$292$	1.70850	0.0999822
$293$	28.9373	1.69053	0.845266	−	0.534345i	$-0.179442\pi$
$293$	28.9373	1.69053	0.845266	+	0.534345i	$0.179442\pi$
$294$	0	0
$295$	−13.0627	−0.760542
$296$	−0.354249	−0.0205903
$297$	0	0
$298$	−10.9373	−0.633578
$299$	−3.29150	−0.190353
$300$	0	0
$301$	41.1660	2.37277
$302$	−12.9373	−0.744455
$303$	0	0
$304$	0	0
$305$	1.54249	0.0883225
$306$	0	0
$307$	−0.645751	−0.0368550	−0.0184275	−	0.999830i	$-0.505866\pi$
$307$	−0.645751	−0.0368550	−0.0184275	+	0.999830i	$0.505866\pi$
$308$	16.9373	0.965090
$309$	0	0
$310$	9.29150	0.527722
$311$	−13.6458	−0.773780	−0.386890	−	0.922126i	$-0.626451\pi$
$311$	−13.6458	−0.773780	−0.386890	+	0.922126i	$0.626451\pi$
$312$	0	0
$313$	8.87451	0.501617	0.250808	−	0.968037i	$-0.419304\pi$
$313$	8.87451	0.501617	0.250808	+	0.968037i	$0.419304\pi$
$314$	−10.5830	−0.597234
$315$	0	0
$316$	4.00000	0.225018
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	7.64575	0.428080
$320$	1.64575	0.0920003
$321$	0	0
$322$	6.00000	0.334367
$323$	0	0
$324$	0	0
$325$	4.58301	0.254219
$326$	3.93725	0.218064
$327$	0	0
$328$	−0.291503	−0.0160955
$329$	−15.8745	−0.875190
$330$	0	0
$331$	−19.8118	−1.08895	−0.544476	−	0.838776i	$-0.683272\pi$
$331$	−19.8118	−1.08895	−0.544476	+	0.838776i	$0.683272\pi$
$332$	−7.93725	−0.435613
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−1.06275	−0.0580640
$336$	0	0
$337$	9.70850	0.528856	0.264428	−	0.964405i	$-0.414817\pi$
$337$	9.70850	0.528856	0.264428	+	0.964405i	$0.414817\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	0	0
$341$	26.2288	1.42037
$342$	0	0
$343$	−2.58301	−0.139469
$344$	11.2915	0.608797
$345$	0	0
$346$	−6.00000	−0.322562
$347$	23.2288	1.24698	0.623492	−	0.781829i	$-0.285713\pi$
$347$	23.2288	1.24698	0.623492	+	0.781829i	$0.285713\pi$
$348$	0	0
$349$	21.1660	1.13299	0.566495	−	0.824065i	$-0.308299\pi$
$349$	21.1660	1.13299	0.566495	+	0.824065i	$0.308299\pi$
$350$	−8.35425	−0.446553
$351$	0	0
$352$	4.64575	0.247619
$353$	−12.8745	−0.685241	−0.342620	−	0.939474i	$-0.611314\pi$
$353$	−12.8745	−0.685241	−0.342620	+	0.939474i	$0.611314\pi$
$354$	0	0
$355$	−4.45751	−0.236580
$356$	0	0
$357$	0	0
$358$	4.06275	0.214723
$359$	4.93725	0.260578	0.130289	−	0.991476i	$-0.458409\pi$
$359$	4.93725	0.260578	0.130289	+	0.991476i	$0.458409\pi$
$360$	0	0
$361$	0	0
$362$	−22.2288	−1.16832
$363$	0	0
$364$	−7.29150	−0.382179
$365$	2.81176	0.147174
$366$	0	0
$367$	16.2288	0.847134	0.423567	−	0.905865i	$-0.360778\pi$
$367$	16.2288	0.847134	0.423567	+	0.905865i	$0.360778\pi$
$368$	1.64575	0.0857907
$369$	0	0
$370$	−0.583005	−0.0303090
$371$	−45.8745	−2.38169
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	−4.35425	−0.224553
$377$	−3.29150	−0.169521
$378$	0	0
$379$	−10.7085	−0.550059	−0.275029	−	0.961436i	$-0.588688\pi$
$379$	−10.7085	−0.550059	−0.275029	+	0.961436i	$0.588688\pi$
$380$	0	0
$381$	0	0
$382$	−6.58301	−0.336816
$383$	5.52026	0.282072	0.141036	−	0.990004i	$-0.454957\pi$
$383$	5.52026	0.282072	0.141036	+	0.990004i	$0.454957\pi$
$384$	0	0
$385$	27.8745	1.42062
$386$	−14.5830	−0.742255
$387$	0	0
$388$	3.70850	0.188270
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	6.29150	0.317769
$393$	0	0
$394$	7.64575	0.385187
$395$	6.58301	0.331227
$396$	0	0
$397$	21.0627	1.05711	0.528554	−	0.848899i	$-0.322734\pi$
$397$	21.0627	1.05711	0.528554	+	0.848899i	$0.322734\pi$
$398$	−19.8745	−0.996219
$399$	0	0
$400$	−2.29150	−0.114575
$401$	27.5830	1.37743	0.688715	−	0.725032i	$-0.258175\pi$
$401$	27.5830	1.37743	0.688715	+	0.725032i	$0.258175\pi$
$402$	0	0
$403$	−11.2915	−0.562470
$404$	13.6458	0.678902
$405$	0	0
$406$	6.00000	0.297775
$407$	−1.64575	−0.0815769
$408$	0	0
$409$	7.58301	0.374955	0.187478	−	0.982269i	$-0.439969\pi$
$409$	7.58301	0.374955	0.187478	+	0.982269i	$0.439969\pi$
$410$	−0.479741	−0.0236927
$411$	0	0
$412$	13.2915	0.654825
$413$	−28.9373	−1.42391
$414$	0	0
$415$	−13.0627	−0.641225
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	31.7490	1.55104	0.775520	−	0.631322i	$-0.217488\pi$
$419$	31.7490	1.55104	0.775520	+	0.631322i	$0.217488\pi$
$420$	0	0
$421$	24.8118	1.20925	0.604626	−	0.796510i	$-0.293323\pi$
$421$	24.8118	1.20925	0.604626	+	0.796510i	$0.293323\pi$
$422$	13.2915	0.647020
$423$	0	0
$424$	−12.5830	−0.611085
$425$	0	0
$426$	0	0
$427$	3.41699	0.165360
$428$	15.2915	0.739143
$429$	0	0
$430$	18.5830	0.896152
$431$	27.8745	1.34267	0.671334	−	0.741155i	$-0.265722\pi$
$431$	27.8745	1.34267	0.671334	+	0.741155i	$0.265722\pi$
$432$	0	0
$433$	−17.8745	−0.858994	−0.429497	−	0.903068i	$-0.641309\pi$
$433$	−17.8745	−0.858994	−0.429497	+	0.903068i	$0.641309\pi$
$434$	20.5830	0.988016
$435$	0	0
$436$	−14.5830	−0.698399
$437$	0	0
$438$	0	0
$439$	−10.8118	−0.516017	−0.258009	−	0.966143i	$-0.583066\pi$
$439$	−10.8118	−0.516017	−0.258009	+	0.966143i	$0.583066\pi$
$440$	7.64575	0.364497
$441$	0	0
$442$	0	0
$443$	−10.6458	−0.505795	−0.252897	−	0.967493i	$-0.581384\pi$
$443$	−10.6458	−0.505795	−0.252897	+	0.967493i	$0.581384\pi$
$444$	0	0
$445$	0	0
$446$	18.8118	0.890763
$447$	0	0
$448$	3.64575	0.172246
$449$	−24.2915	−1.14639	−0.573193	−	0.819420i	$-0.694296\pi$
$449$	−24.2915	−1.14639	−0.573193	+	0.819420i	$0.694296\pi$
$450$	0	0
$451$	−1.35425	−0.0637691
$452$	−15.5830	−0.732963
$453$	0	0
$454$	−7.35425	−0.345152
$455$	−12.0000	−0.562569
$456$	0	0
$457$	32.8745	1.53780	0.768902	−	0.639366i	$-0.220803\pi$
$457$	32.8745	1.53780	0.768902	+	0.639366i	$0.220803\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	2.70850	0.126284
$461$	−19.1660	−0.892650	−0.446325	−	0.894871i	$-0.647267\pi$
$461$	−19.1660	−0.892650	−0.446325	+	0.894871i	$0.647267\pi$
$462$	0	0
$463$	−38.4575	−1.78727	−0.893636	−	0.448792i	$-0.851854\pi$
$463$	−38.4575	−1.78727	−0.893636	+	0.448792i	$0.851854\pi$
$464$	1.64575	0.0764021
$465$	0	0
$466$	−18.8745	−0.874345
$467$	19.3542	0.895608	0.447804	−	0.894132i	$-0.352206\pi$
$467$	19.3542	0.895608	0.447804	+	0.894132i	$0.352206\pi$
$468$	0	0
$469$	−2.35425	−0.108709
$470$	−7.16601	−0.330543
$471$	0	0
$472$	−7.93725	−0.365342
$473$	52.4575	2.41200
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	−6.58301	−0.300785	−0.150393	−	0.988626i	$-0.548054\pi$
$479$	−6.58301	−0.300785	−0.150393	+	0.988626i	$0.548054\pi$
$480$	0	0
$481$	0.708497	0.0323047
$482$	7.58301	0.345396
$483$	0	0
$484$	10.5830	0.481046
$485$	6.10326	0.277135
$486$	0	0
$487$	−4.22876	−0.191623	−0.0958116	−	0.995399i	$-0.530545\pi$
$487$	−4.22876	−0.191623	−0.0958116	+	0.995399i	$0.530545\pi$
$488$	0.937254	0.0424275
$489$	0	0
$490$	10.3542	0.467757
$491$	−39.2915	−1.77320	−0.886600	−	0.462536i	$-0.846939\pi$
$491$	−39.2915	−1.77320	−0.886600	+	0.462536i	$0.846939\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	5.64575	0.253502
$497$	−9.87451	−0.442932
$498$	0	0
$499$	−4.77124	−0.213590	−0.106795	−	0.994281i	$-0.534059\pi$
$499$	−4.77124	−0.213590	−0.106795	+	0.994281i	$0.534059\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−29.2288	−1.30454
$503$	40.9373	1.82530	0.912651	−	0.408740i	$-0.134032\pi$
$503$	40.9373	1.82530	0.912651	+	0.408740i	$0.134032\pi$
$504$	0	0
$505$	22.4575	0.999346
$506$	7.64575	0.339895
$507$	0	0
$508$	13.2915	0.589715
$509$	−31.7490	−1.40725	−0.703625	−	0.710571i	$-0.748437\pi$
$509$	−31.7490	−1.40725	−0.703625	+	0.710571i	$0.748437\pi$
$510$	0	0
$511$	6.22876	0.275544
$512$	1.00000	0.0441942
$513$	0	0
$514$	12.2915	0.542155
$515$	21.8745	0.963906
$516$	0	0
$517$	−20.2288	−0.889660
$518$	−1.29150	−0.0567454
$519$	0	0
$520$	−3.29150	−0.144342
$521$	−11.7085	−0.512959	−0.256479	−	0.966550i	$-0.582563\pi$
$521$	−11.7085	−0.512959	−0.256479	+	0.966550i	$0.582563\pi$
$522$	0	0
$523$	1.87451	0.0819665	0.0409833	−	0.999160i	$-0.486951\pi$
$523$	1.87451	0.0819665	0.0409833	+	0.999160i	$0.486951\pi$
$524$	−1.93725	−0.0846293
$525$	0	0
$526$	10.9373	0.476887
$527$	0	0
$528$	0	0
$529$	−20.2915	−0.882239
$530$	−20.7085	−0.899520
$531$	0	0
$532$	0	0
$533$	0.583005	0.0252528
$534$	0	0
$535$	25.1660	1.08802
$536$	−0.645751	−0.0278922
$537$	0	0
$538$	0	0
$539$	29.2288	1.25897
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	12.3542	0.530660
$543$	0	0
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−11.2915	−0.482790	−0.241395	−	0.970427i	$-0.577605\pi$
$547$	−11.2915	−0.482790	−0.241395	+	0.970427i	$0.577605\pi$
$548$	−15.5830	−0.665673
$549$	0	0
$550$	−10.6458	−0.453936
$551$	0	0
$552$	0	0
$553$	14.5830	0.620132
$554$	−27.5203	−1.16922
$555$	0	0
$556$	18.6458	0.790756
$557$	5.41699	0.229525	0.114763	−	0.993393i	$-0.463389\pi$
$557$	5.41699	0.229525	0.114763	+	0.993393i	$0.463389\pi$
$558$	0	0
$559$	−22.5830	−0.955159
$560$	6.00000	0.253546
$561$	0	0
$562$	25.4575	1.07386
$563$	10.0627	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$563$	10.0627	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$564$	0	0
$565$	−25.6458	−1.07892
$566$	30.6458	1.28814
$567$	0	0
$568$	−2.70850	−0.113646
$569$	6.58301	0.275974	0.137987	−	0.990434i	$-0.455937\pi$
$569$	6.58301	0.275974	0.137987	+	0.990434i	$0.455937\pi$
$570$	0	0
$571$	7.81176	0.326912	0.163456	−	0.986551i	$-0.447736\pi$
$571$	7.81176	0.326912	0.163456	+	0.986551i	$0.447736\pi$
$572$	−9.29150	−0.388497
$573$	0	0
$574$	−1.06275	−0.0443582
$575$	−3.77124	−0.157272
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	2.70850	0.112464
$581$	−28.9373	−1.20052
$582$	0	0
$583$	−58.4575	−2.42106
$584$	1.70850	0.0706981
$585$	0	0
$586$	28.9373	1.19539
$587$	14.1255	0.583021	0.291511	−	0.956568i	$-0.405842\pi$
$587$	14.1255	0.583021	0.291511	+	0.956568i	$0.405842\pi$
$588$	0	0
$589$	0	0
$590$	−13.0627	−0.537785
$591$	0	0
$592$	−0.354249	−0.0145595
$593$	−29.7085	−1.21998	−0.609991	−	0.792408i	$-0.708827\pi$
$593$	−29.7085	−1.21998	−0.609991	+	0.792408i	$0.708827\pi$
$594$	0	0
$595$	0	0
$596$	−10.9373	−0.448007
$597$	0	0
$598$	−3.29150	−0.134600
$599$	−16.9373	−0.692037	−0.346019	−	0.938228i	$-0.612467\pi$
$599$	−16.9373	−0.692037	−0.346019	+	0.938228i	$0.612467\pi$
$600$	0	0
$601$	−10.4170	−0.424918	−0.212459	−	0.977170i	$-0.568147\pi$
$601$	−10.4170	−0.424918	−0.212459	+	0.977170i	$0.568147\pi$
$602$	41.1660	1.67780
$603$	0	0
$604$	−12.9373	−0.526409
$605$	17.4170	0.708102
$606$	0	0
$607$	−6.93725	−0.281574	−0.140787	−	0.990040i	$-0.544963\pi$
$607$	−6.93725	−0.281574	−0.140787	+	0.990040i	$0.544963\pi$
$608$	0	0
$609$	0	0
$610$	1.54249	0.0624535
$611$	8.70850	0.352308
$612$	0	0
$613$	7.41699	0.299570	0.149785	−	0.988719i	$-0.452142\pi$
$613$	7.41699	0.299570	0.149785	+	0.988719i	$0.452142\pi$
$614$	−0.645751	−0.0260604
$615$	0	0
$616$	16.9373	0.682421
$617$	30.8745	1.24296	0.621480	−	0.783430i	$-0.286532\pi$
$617$	30.8745	1.24296	0.621480	+	0.783430i	$0.286532\pi$
$618$	0	0
$619$	−8.45751	−0.339936	−0.169968	−	0.985450i	$-0.554366\pi$
$619$	−8.45751	−0.339936	−0.169968	+	0.985450i	$0.554366\pi$
$620$	9.29150	0.373156
$621$	0	0
$622$	−13.6458	−0.547145
$623$	0	0
$624$	0	0
$625$	−8.29150	−0.331660
$626$	8.87451	0.354697
$627$	0	0
$628$	−10.5830	−0.422308
$629$	0	0
$630$	0	0
$631$	−24.8118	−0.987741	−0.493870	−	0.869536i	$-0.664418\pi$
$631$	−24.8118	−0.987741	−0.493870	+	0.869536i	$0.664418\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−6.00000	−0.238290
$635$	21.8745	0.868063
$636$	0	0
$637$	−12.5830	−0.498557
$638$	7.64575	0.302698
$639$	0	0
$640$	1.64575	0.0650540
$641$	12.8745	0.508512	0.254256	−	0.967137i	$-0.418169\pi$
$641$	12.8745	0.508512	0.254256	+	0.967137i	$0.418169\pi$
$642$	0	0
$643$	−6.52026	−0.257134	−0.128567	−	0.991701i	$-0.541038\pi$
$643$	−6.52026	−0.257134	−0.128567	+	0.991701i	$0.541038\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	22.4575	0.882896	0.441448	−	0.897287i	$-0.354465\pi$
$647$	22.4575	0.882896	0.441448	+	0.897287i	$0.354465\pi$
$648$	0	0
$649$	−36.8745	−1.44745
$650$	4.58301	0.179760
$651$	0	0
$652$	3.93725	0.154195
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	−3.18824	−0.124575
$656$	−0.291503	−0.0113813
$657$	0	0
$658$	−15.8745	−0.618853
$659$	−18.5830	−0.723891	−0.361946	−	0.932199i	$-0.617887\pi$
$659$	−18.5830	−0.723891	−0.361946	+	0.932199i	$0.617887\pi$
$660$	0	0
$661$	−16.2288	−0.631225	−0.315613	−	0.948888i	$-0.602210\pi$
$661$	−16.2288	−0.631225	−0.315613	+	0.948888i	$0.602210\pi$
$662$	−19.8118	−0.770006
$663$	0	0
$664$	−7.93725	−0.308025
$665$	0	0
$666$	0	0
$667$	2.70850	0.104873
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−1.06275	−0.0410575
$671$	4.35425	0.168094
$672$	0	0
$673$	13.8745	0.534823	0.267411	−	0.963582i	$-0.413832\pi$
$673$	13.8745	0.534823	0.267411	+	0.963582i	$0.413832\pi$
$674$	9.70850	0.373957
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−11.4170	−0.438791	−0.219395	−	0.975636i	$-0.570408\pi$
$677$	−11.4170	−0.438791	−0.219395	+	0.975636i	$0.570408\pi$
$678$	0	0
$679$	13.5203	0.518860
$680$	0	0
$681$	0	0
$682$	26.2288	1.00435
$683$	−5.41699	−0.207276	−0.103638	−	0.994615i	$-0.533048\pi$
$683$	−5.41699	−0.207276	−0.103638	+	0.994615i	$0.533048\pi$
$684$	0	0
$685$	−25.6458	−0.979874
$686$	−2.58301	−0.0986196
$687$	0	0
$688$	11.2915	0.430485
$689$	25.1660	0.958749
$690$	0	0
$691$	2.58301	0.0982622	0.0491311	−	0.998792i	$-0.484355\pi$
$691$	2.58301	0.0982622	0.0491311	+	0.998792i	$0.484355\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	23.2288	0.881752
$695$	30.6863	1.16400
$696$	0	0
$697$	0	0
$698$	21.1660	0.801145
$699$	0	0
$700$	−8.35425	−0.315761
$701$	−10.3542	−0.391075	−0.195537	−	0.980696i	$-0.562645\pi$
$701$	−10.3542	−0.391075	−0.195537	+	0.980696i	$0.562645\pi$
$702$	0	0
$703$	0	0
$704$	4.64575	0.175093
$705$	0	0
$706$	−12.8745	−0.484538
$707$	49.7490	1.87100
$708$	0	0
$709$	3.64575	0.136919	0.0684595	−	0.997654i	$-0.478192\pi$
$709$	3.64575	0.136919	0.0684595	+	0.997654i	$0.478192\pi$
$710$	−4.45751	−0.167287
$711$	0	0
$712$	0	0
$713$	9.29150	0.347970
$714$	0	0
$715$	−15.2915	−0.571870
$716$	4.06275	0.151832
$717$	0	0
$718$	4.93725	0.184257
$719$	−2.70850	−0.101010	−0.0505050	−	0.998724i	$-0.516083\pi$
$719$	−2.70850	−0.101010	−0.0505050	+	0.998724i	$0.516083\pi$
$720$	0	0
$721$	48.4575	1.80465
$722$	0	0
$723$	0	0
$724$	−22.2288	−0.826125
$725$	−3.77124	−0.140060
$726$	0	0
$727$	1.41699	0.0525534	0.0262767	−	0.999655i	$-0.491635\pi$
$727$	1.41699	0.0525534	0.0262767	+	0.999655i	$0.491635\pi$
$728$	−7.29150	−0.270241
$729$	0	0
$730$	2.81176	0.104068
$731$	0	0
$732$	0	0
$733$	−16.1033	−0.594788	−0.297394	−	0.954755i	$-0.596117\pi$
$733$	−16.1033	−0.594788	−0.297394	+	0.954755i	$0.596117\pi$
$734$	16.2288	0.599014
$735$	0	0
$736$	1.64575	0.0606632
$737$	−3.00000	−0.110506
$738$	0	0
$739$	−33.8118	−1.24379	−0.621893	−	0.783102i	$-0.713636\pi$
$739$	−33.8118	−1.24379	−0.621893	+	0.783102i	$0.713636\pi$
$740$	−0.583005	−0.0214317
$741$	0	0
$742$	−45.8745	−1.68411
$743$	−47.5203	−1.74335	−0.871675	−	0.490085i	$-0.836966\pi$
$743$	−47.5203	−1.74335	−0.871675	+	0.490085i	$0.836966\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	4.00000	0.146450
$747$	0	0
$748$	0	0
$749$	55.7490	2.03702
$750$	0	0
$751$	7.87451	0.287345	0.143672	−	0.989625i	$-0.454109\pi$
$751$	7.87451	0.287345	0.143672	+	0.989625i	$0.454109\pi$
$752$	−4.35425	−0.158783
$753$	0	0
$754$	−3.29150	−0.119869
$755$	−21.2915	−0.774877
$756$	0	0
$757$	−4.58301	−0.166572	−0.0832861	−	0.996526i	$-0.526542\pi$
$757$	−4.58301	−0.166572	−0.0832861	+	0.996526i	$0.526542\pi$
$758$	−10.7085	−0.388950
$759$	0	0
$760$	0	0
$761$	−42.8745	−1.55420	−0.777100	−	0.629377i	$-0.783310\pi$
$761$	−42.8745	−1.55420	−0.777100	+	0.629377i	$0.783310\pi$
$762$	0	0
$763$	−53.1660	−1.92474
$764$	−6.58301	−0.238165
$765$	0	0
$766$	5.52026	0.199455
$767$	15.8745	0.573195
$768$	0	0
$769$	35.2915	1.27264	0.636322	−	0.771424i	$-0.280455\pi$
$769$	35.2915	1.27264	0.636322	+	0.771424i	$0.280455\pi$
$770$	27.8745	1.00453
$771$	0	0
$772$	−14.5830	−0.524854
$773$	−4.93725	−0.177581	−0.0887903	−	0.996050i	$-0.528300\pi$
$773$	−4.93725	−0.177581	−0.0887903	+	0.996050i	$0.528300\pi$
$774$	0	0
$775$	−12.9373	−0.464720
$776$	3.70850	0.133127
$777$	0	0
$778$	−12.0000	−0.430221
$779$	0	0
$780$	0	0
$781$	−12.5830	−0.450255
$782$	0	0
$783$	0	0
$784$	6.29150	0.224697
$785$	−17.4170	−0.621639
$786$	0	0
$787$	42.5203	1.51568	0.757842	−	0.652438i	$-0.226254\pi$
$787$	42.5203	1.51568	0.757842	+	0.652438i	$0.226254\pi$
$788$	7.64575	0.272369
$789$	0	0
$790$	6.58301	0.234213
$791$	−56.8118	−2.01999
$792$	0	0
$793$	−1.87451	−0.0665657
$794$	21.0627	0.747489
$795$	0	0
$796$	−19.8745	−0.704433
$797$	−44.8118	−1.58731	−0.793657	−	0.608365i	$-0.791826\pi$
$797$	−44.8118	−1.58731	−0.793657	+	0.608365i	$0.791826\pi$
$798$	0	0
$799$	0	0
$800$	−2.29150	−0.0810169
$801$	0	0
$802$	27.5830	0.973990
$803$	7.93725	0.280100
$804$	0	0
$805$	9.87451	0.348031
$806$	−11.2915	−0.397726
$807$	0	0
$808$	13.6458	0.480056
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	31.2915	1.09879	0.549397	−	0.835562i	$-0.314858\pi$
$811$	31.2915	1.09879	0.549397	+	0.835562i	$0.314858\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−1.64575	−0.0576836
$815$	6.47974	0.226975
$816$	0	0
$817$	0	0
$818$	7.58301	0.265134
$819$	0	0
$820$	−0.479741	−0.0167533
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−31.8745	−1.11108	−0.555538	−	0.831491i	$-0.687488\pi$
$823$	−31.8745	−1.11108	−0.555538	+	0.831491i	$0.687488\pi$
$824$	13.2915	0.463031
$825$	0	0
$826$	−28.9373	−1.00686
$827$	−52.6458	−1.83067	−0.915336	−	0.402691i	$-0.868075\pi$
$827$	−52.6458	−1.83067	−0.915336	+	0.402691i	$0.868075\pi$
$828$	0	0
$829$	17.1660	0.596200	0.298100	−	0.954535i	$-0.403647\pi$
$829$	17.1660	0.596200	0.298100	+	0.954535i	$0.403647\pi$
$830$	−13.0627	−0.453415
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	−19.7490	−0.683443
$836$	0	0
$837$	0	0
$838$	31.7490	1.09675
$839$	−41.5203	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$839$	−41.5203	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$840$	0	0
$841$	−26.2915	−0.906604
$842$	24.8118	0.855070
$843$	0	0
$844$	13.2915	0.457512
$845$	−14.8118	−0.509540
$846$	0	0
$847$	38.5830	1.32573
$848$	−12.5830	−0.432102
$849$	0	0
$850$	0	0
$851$	−0.583005	−0.0199852
$852$	0	0
$853$	8.58301	0.293877	0.146938	−	0.989146i	$-0.453058\pi$
$853$	8.58301	0.293877	0.146938	+	0.989146i	$0.453058\pi$
$854$	3.41699	0.116927
$855$	0	0
$856$	15.2915	0.522653
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	13.2288	0.451359	0.225680	−	0.974202i	$-0.427540\pi$
$859$	13.2288	0.451359	0.225680	+	0.974202i	$0.427540\pi$
$860$	18.5830	0.633675
$861$	0	0
$862$	27.8745	0.949410
$863$	−31.0627	−1.05739	−0.528694	−	0.848812i	$-0.677318\pi$
$863$	−31.0627	−1.05739	−0.528694	+	0.848812i	$0.677318\pi$
$864$	0	0
$865$	−9.87451	−0.335743
$866$	−17.8745	−0.607401
$867$	0	0
$868$	20.5830	0.698633
$869$	18.5830	0.630385
$870$	0	0
$871$	1.29150	0.0437609
$872$	−14.5830	−0.493843
$873$	0	0
$874$	0	0
$875$	−43.7490	−1.47899
$876$	0	0
$877$	41.6458	1.40628	0.703139	−	0.711053i	$-0.251781\pi$
$877$	41.6458	1.40628	0.703139	+	0.711053i	$0.251781\pi$
$878$	−10.8118	−0.364879
$879$	0	0
$880$	7.64575	0.257738
$881$	36.8745	1.24233	0.621167	−	0.783678i	$-0.286659\pi$
$881$	36.8745	1.24233	0.621167	+	0.783678i	$0.286659\pi$
$882$	0	0
$883$	−28.3948	−0.955560	−0.477780	−	0.878480i	$-0.658558\pi$
$883$	−28.3948	−0.955560	−0.477780	+	0.878480i	$0.658558\pi$
$884$	0	0
$885$	0	0
$886$	−10.6458	−0.357651
$887$	−17.4170	−0.584805	−0.292403	−	0.956295i	$-0.594455\pi$
$887$	−17.4170	−0.584805	−0.292403	+	0.956295i	$0.594455\pi$
$888$	0	0
$889$	48.4575	1.62521
$890$	0	0
$891$	0	0
$892$	18.8118	0.629864
$893$	0	0
$894$	0	0
$895$	6.68627	0.223497
$896$	3.64575	0.121796
$897$	0	0
$898$	−24.2915	−0.810618
$899$	9.29150	0.309889
$900$	0	0
$901$	0	0
$902$	−1.35425	−0.0450915
$903$	0	0
$904$	−15.5830	−0.518283
$905$	−36.5830	−1.21606
$906$	0	0
$907$	−39.9373	−1.32609	−0.663047	−	0.748577i	$-0.730737\pi$
$907$	−39.9373	−1.32609	−0.663047	+	0.748577i	$0.730737\pi$
$908$	−7.35425	−0.244059
$909$	0	0
$910$	−12.0000	−0.397796
$911$	16.9373	0.561156	0.280578	−	0.959831i	$-0.409474\pi$
$911$	16.9373	0.561156	0.280578	+	0.959831i	$0.409474\pi$
$912$	0	0
$913$	−36.8745	−1.22037
$914$	32.8745	1.08739
$915$	0	0
$916$	20.0000	0.660819
$917$	−7.06275	−0.233232
$918$	0	0
$919$	−19.8745	−0.655600	−0.327800	−	0.944747i	$-0.606307\pi$
$919$	−19.8745	−0.655600	−0.327800	+	0.944747i	$0.606307\pi$
$920$	2.70850	0.0892965
$921$	0	0
$922$	−19.1660	−0.631199
$923$	5.41699	0.178303
$924$	0	0
$925$	0.811762	0.0266906
$926$	−38.4575	−1.26379
$927$	0	0
$928$	1.64575	0.0540244
$929$	9.58301	0.314408	0.157204	−	0.987566i	$-0.449752\pi$
$929$	9.58301	0.314408	0.157204	+	0.987566i	$0.449752\pi$
$930$	0	0
$931$	0	0
$932$	−18.8745	−0.618255
$933$	0	0
$934$	19.3542	0.633290
$935$	0	0
$936$	0	0
$937$	7.12549	0.232780	0.116390	−	0.993204i	$-0.462868\pi$
$937$	7.12549	0.232780	0.116390	+	0.993204i	$0.462868\pi$
$938$	−2.35425	−0.0768689
$939$	0	0
$940$	−7.16601	−0.233729
$941$	16.8340	0.548772	0.274386	−	0.961620i	$-0.411525\pi$
$941$	16.8340	0.548772	0.274386	+	0.961620i	$0.411525\pi$
$942$	0	0
$943$	−0.479741	−0.0156225
$944$	−7.93725	−0.258336
$945$	0	0
$946$	52.4575	1.70554
$947$	7.74902	0.251809	0.125905	−	0.992042i	$-0.459817\pi$
$947$	7.74902	0.251809	0.125905	+	0.992042i	$0.459817\pi$
$948$	0	0
$949$	−3.41699	−0.110920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.4575	−0.435932	−0.217966	−	0.975956i	$-0.569942\pi$
$953$	−13.4575	−0.435932	−0.217966	+	0.975956i	$0.569942\pi$
$954$	0	0
$955$	−10.8340	−0.350580
$956$	12.0000	0.388108
$957$	0	0
$958$	−6.58301	−0.212687
$959$	−56.8118	−1.83455
$960$	0	0
$961$	0.874508	0.0282099
$962$	0.708497	0.0228429
$963$	0	0
$964$	7.58301	0.244232
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−13.2915	−0.427426	−0.213713	−	0.976897i	$-0.568556\pi$
$967$	−13.2915	−0.427426	−0.213713	+	0.976897i	$0.568556\pi$
$968$	10.5830	0.340151
$969$	0	0
$970$	6.10326	0.195964
$971$	−54.3948	−1.74561	−0.872806	−	0.488068i	$-0.837702\pi$
$971$	−54.3948	−1.74561	−0.872806	+	0.488068i	$0.837702\pi$
$972$	0	0
$973$	67.9778	2.17927
$974$	−4.22876	−0.135498
$975$	0	0
$976$	0.937254	0.0300008
$977$	−7.45751	−0.238587	−0.119293	−	0.992859i	$-0.538063\pi$
$977$	−7.45751	−0.238587	−0.119293	+	0.992859i	$0.538063\pi$
$978$	0	0
$979$	0	0
$980$	10.3542	0.330754
$981$	0	0
$982$	−39.2915	−1.25384
$983$	31.7490	1.01264	0.506318	−	0.862347i	$-0.331006\pi$
$983$	31.7490	1.01264	0.506318	+	0.862347i	$0.331006\pi$
$984$	0	0
$985$	12.5830	0.400928
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.5830	0.590905
$990$	0	0
$991$	2.83399	0.0900246	0.0450123	−	0.998986i	$-0.485667\pi$
$991$	2.83399	0.0900246	0.0450123	+	0.998986i	$0.485667\pi$
$992$	5.64575	0.179253
$993$	0	0
$994$	−9.87451	−0.313200
$995$	−32.7085	−1.03693
$996$	0	0
$997$	16.2288	0.513970	0.256985	−	0.966415i	$-0.417271\pi$
$997$	16.2288	0.513970	0.256985	+	0.966415i	$0.417271\pi$
$998$	−4.77124	−0.151031
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bg.1.2		2
3.2	odd	2		722.2.a.g.1.1		2
12.11	even	2		5776.2.a.z.1.2		2
19.8	odd	6		342.2.g.f.235.1		4
19.12	odd	6		342.2.g.f.163.1		4
19.18	odd	2		6498.2.a.ba.1.2		2
57.2	even	18		722.2.e.n.99.2		12
57.5	odd	18		722.2.e.o.595.2		12
57.8	even	6		38.2.c.b.7.1	✓	4
57.11	odd	6		722.2.c.j.653.2		4
57.14	even	18		722.2.e.n.595.1		12
57.17	odd	18		722.2.e.o.99.1		12
57.23	odd	18		722.2.e.o.415.2		12
57.26	odd	6		722.2.c.j.429.2		4
57.29	even	18		722.2.e.n.423.2		12
57.32	even	18		722.2.e.n.245.2		12
57.35	odd	18		722.2.e.o.389.1		12
57.41	even	18		722.2.e.n.389.2		12
57.44	odd	18		722.2.e.o.245.1		12
57.47	odd	18		722.2.e.o.423.1		12
57.50	even	6		38.2.c.b.11.1	yes	4
57.53	even	18		722.2.e.n.415.1		12
57.56	even	2		722.2.a.j.1.2		2
76.27	even	6		2736.2.s.v.577.1		4
76.31	even	6		2736.2.s.v.1873.1		4
228.107	odd	6		304.2.i.e.49.2		4
228.179	odd	6		304.2.i.e.273.2		4
228.227	odd	2		5776.2.a.ba.1.1		2
285.8	odd	12		950.2.j.g.349.2		8
285.107	odd	12		950.2.j.g.49.2		8
285.122	odd	12		950.2.j.g.349.3		8
285.164	even	6		950.2.e.k.201.2		4
285.179	even	6		950.2.e.k.501.2		4
285.278	odd	12		950.2.j.g.49.3		8
456.107	odd	6		1216.2.i.k.961.1		4
456.179	odd	6		1216.2.i.k.577.1		4
456.221	even	6		1216.2.i.l.961.2		4
456.293	even	6		1216.2.i.l.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.b.7.1	✓	4	57.8	even	6
38.2.c.b.11.1	yes	4	57.50	even	6
304.2.i.e.49.2		4	228.107	odd	6
304.2.i.e.273.2		4	228.179	odd	6
342.2.g.f.163.1		4	19.12	odd	6
342.2.g.f.235.1		4	19.8	odd	6
722.2.a.g.1.1		2	3.2	odd	2
722.2.a.j.1.2		2	57.56	even	2
722.2.c.j.429.2		4	57.26	odd	6
722.2.c.j.653.2		4	57.11	odd	6
722.2.e.n.99.2		12	57.2	even	18
722.2.e.n.245.2		12	57.32	even	18
722.2.e.n.389.2		12	57.41	even	18
722.2.e.n.415.1		12	57.53	even	18
722.2.e.n.423.2		12	57.29	even	18
722.2.e.n.595.1		12	57.14	even	18
722.2.e.o.99.1		12	57.17	odd	18
722.2.e.o.245.1		12	57.44	odd	18
722.2.e.o.389.1		12	57.35	odd	18
722.2.e.o.415.2		12	57.23	odd	18
722.2.e.o.423.1		12	57.47	odd	18
722.2.e.o.595.2		12	57.5	odd	18
950.2.e.k.201.2		4	285.164	even	6
950.2.e.k.501.2		4	285.179	even	6
950.2.j.g.49.2		8	285.107	odd	12
950.2.j.g.49.3		8	285.278	odd	12
950.2.j.g.349.2		8	285.8	odd	12
950.2.j.g.349.3		8	285.122	odd	12
1216.2.i.k.577.1		4	456.179	odd	6
1216.2.i.k.961.1		4	456.107	odd	6
1216.2.i.l.577.2		4	456.293	even	6
1216.2.i.l.961.2		4	456.221	even	6
2736.2.s.v.577.1		4	76.27	even	6
2736.2.s.v.1873.1		4	76.31	even	6
5776.2.a.z.1.2		2	12.11	even	2
5776.2.a.ba.1.1		2	228.227	odd	2
6498.2.a.ba.1.2		2	19.18	odd	2
6498.2.a.bg.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +3.64575 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +3.64575 q^{7} +1.00000 q^{8} +1.64575 q^{10} +4.64575 q^{11} -2.00000 q^{13} +3.64575 q^{14} +1.00000 q^{16} +1.64575 q^{20} +4.64575 q^{22} +1.64575 q^{23} -2.29150 q^{25} -2.00000 q^{26} +3.64575 q^{28} +1.64575 q^{29} +5.64575 q^{31} +1.00000 q^{32} +6.00000 q^{35} -0.354249 q^{37} +1.64575 q^{40} -0.291503 q^{41} +11.2915 q^{43} +4.64575 q^{44} +1.64575 q^{46} -4.35425 q^{47} +6.29150 q^{49} -2.29150 q^{50} -2.00000 q^{52} -12.5830 q^{53} +7.64575 q^{55} +3.64575 q^{56} +1.64575 q^{58} -7.93725 q^{59} +0.937254 q^{61} +5.64575 q^{62} +1.00000 q^{64} -3.29150 q^{65} -0.645751 q^{67} +6.00000 q^{70} -2.70850 q^{71} +1.70850 q^{73} -0.354249 q^{74} +16.9373 q^{77} +4.00000 q^{79} +1.64575 q^{80} -0.291503 q^{82} -7.93725 q^{83} +11.2915 q^{86} +4.64575 q^{88} -7.29150 q^{91} +1.64575 q^{92} -4.35425 q^{94} +3.70850 q^{97} +6.29150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 6 q^{25} - 4 q^{26} + 2 q^{28} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 12 q^{35} - 6 q^{37} - 2 q^{40} + 10 q^{41} + 12 q^{43} + 4 q^{44} - 2 q^{46} - 14 q^{47} + 2 q^{49} + 6 q^{50} - 4 q^{52} - 4 q^{53} + 10 q^{55} + 2 q^{56} - 2 q^{58} - 14 q^{61} + 6 q^{62} + 2 q^{64} + 4 q^{65} + 4 q^{67} + 12 q^{70} - 16 q^{71} + 14 q^{73} - 6 q^{74} + 18 q^{77} + 8 q^{79} - 2 q^{80} + 10 q^{82} + 12 q^{86} + 4 q^{88} - 4 q^{91} - 2 q^{92} - 14 q^{94} + 18 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 6 * q^25 - 4 * q^26 + 2 * q^28 - 2 * q^29 + 6 * q^31 + 2 * q^32 + 12 * q^35 - 6 * q^37 - 2 * q^40 + 10 * q^41 + 12 * q^43 + 4 * q^44 - 2 * q^46 - 14 * q^47 + 2 * q^49 + 6 * q^50 - 4 * q^52 - 4 * q^53 + 10 * q^55 + 2 * q^56 - 2 * q^58 - 14 * q^61 + 6 * q^62 + 2 * q^64 + 4 * q^65 + 4 * q^67 + 12 * q^70 - 16 * q^71 + 14 * q^73 - 6 * q^74 + 18 * q^77 + 8 * q^79 - 2 * q^80 + 10 * q^82 + 12 * q^86 + 4 * q^88 - 4 * q^91 - 2 * q^92 - 14 * q^94 + 18 * q^97 + 2 * q^98

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