LMFDB - Embedded newform 6498.2.a.bg.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} -1.64575 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} -1.64575 q^{7} +1.00000 q^{8} -3.64575 q^{10} -0.645751 q^{11} -2.00000 q^{13} -1.64575 q^{14} +1.00000 q^{16} -3.64575 q^{20} -0.645751 q^{22} -3.64575 q^{23} +8.29150 q^{25} -2.00000 q^{26} -1.64575 q^{28} -3.64575 q^{29} +0.354249 q^{31} +1.00000 q^{32} +6.00000 q^{35} -5.64575 q^{37} -3.64575 q^{40} +10.2915 q^{41} +0.708497 q^{43} -0.645751 q^{44} -3.64575 q^{46} -9.64575 q^{47} -4.29150 q^{49} +8.29150 q^{50} -2.00000 q^{52} +8.58301 q^{53} +2.35425 q^{55} -1.64575 q^{56} -3.64575 q^{58} +7.93725 q^{59} -14.9373 q^{61} +0.354249 q^{62} +1.00000 q^{64} +7.29150 q^{65} +4.64575 q^{67} +6.00000 q^{70} -13.2915 q^{71} +12.2915 q^{73} -5.64575 q^{74} +1.06275 q^{77} +4.00000 q^{79} -3.64575 q^{80} +10.2915 q^{82} +7.93725 q^{83} +0.708497 q^{86} -0.645751 q^{88} +3.29150 q^{91} -3.64575 q^{92} -9.64575 q^{94} +14.2915 q^{97} -4.29150 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.64575	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$6$	0	0
$7$	−1.64575	−0.622036	−0.311018	−	0.950404i	$-0.600670\pi$
$7$	−1.64575	−0.622036	−0.311018	+	0.950404i	$0.600670\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.64575	−1.15289
$11$	−0.645751	−0.194701	−0.0973507	−	0.995250i	$-0.531037\pi$
$11$	−0.645751	−0.194701	−0.0973507	+	0.995250i	$0.531037\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.64575	−0.439846
$15$	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
$20$	−3.64575	−0.815215
$21$	0	0
$22$	−0.645751	−0.137675
$23$	−3.64575	−0.760192	−0.380096	−	0.924947i	$-0.624109\pi$
$23$	−3.64575	−0.760192	−0.380096	+	0.924947i	$0.624109\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.64575	−0.311018
$29$	−3.64575	−0.676999	−0.338500	−	0.940967i	$-0.609919\pi$
$29$	−3.64575	−0.676999	−0.338500	+	0.940967i	$0.609919\pi$
$30$	0	0
$31$	0.354249	0.0636249	0.0318125	−	0.999494i	$-0.489872\pi$
$31$	0.354249	0.0636249	0.0318125	+	0.999494i	$0.489872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−5.64575	−0.928156	−0.464078	−	0.885794i	$-0.653614\pi$
$37$	−5.64575	−0.928156	−0.464078	+	0.885794i	$0.653614\pi$
$38$	0	0
$39$	0	0
$40$	−3.64575	−0.576444
$41$	10.2915	1.60726	0.803631	−	0.595127i	$-0.202898\pi$
$41$	10.2915	1.60726	0.803631	+	0.595127i	$0.202898\pi$
$42$	0	0
$43$	0.708497	0.108045	0.0540224	−	0.998540i	$-0.482796\pi$
$43$	0.708497	0.108045	0.0540224	+	0.998540i	$0.482796\pi$
$44$	−0.645751	−0.0973507
$45$	0	0
$46$	−3.64575	−0.537537
$47$	−9.64575	−1.40698	−0.703489	−	0.710706i	$-0.748375\pi$
$47$	−9.64575	−1.40698	−0.703489	+	0.710706i	$0.748375\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	8.29150	1.17260
$51$	0	0
$52$	−2.00000	−0.277350
$53$	8.58301	1.17897	0.589483	−	0.807781i	$-0.299331\pi$
$53$	8.58301	1.17897	0.589483	+	0.807781i	$0.299331\pi$
$54$	0	0
$55$	2.35425	0.317447
$56$	−1.64575	−0.219923
$57$	0	0
$58$	−3.64575	−0.478711
$59$	7.93725	1.03334	0.516671	−	0.856184i	$-0.327171\pi$
$59$	7.93725	1.03334	0.516671	+	0.856184i	$0.327171\pi$
$60$	0	0
$61$	−14.9373	−1.91252	−0.956260	−	0.292519i	$-0.905507\pi$
$61$	−14.9373	−1.91252	−0.956260	+	0.292519i	$0.905507\pi$
$62$	0.354249	0.0449896
$63$	0	0
$64$	1.00000	0.125000
$65$	7.29150	0.904400
$66$	0	0
$67$	4.64575	0.567569	0.283784	−	0.958888i	$-0.408410\pi$
$67$	4.64575	0.567569	0.283784	+	0.958888i	$0.408410\pi$
$68$	0	0
$69$	0	0
$70$	6.00000	0.717137
$71$	−13.2915	−1.57741	−0.788706	−	0.614771i	$-0.789248\pi$
$71$	−13.2915	−1.57741	−0.788706	+	0.614771i	$0.789248\pi$
$72$	0	0
$73$	12.2915	1.43861	0.719306	−	0.694694i	$-0.244460\pi$
$73$	12.2915	1.43861	0.719306	+	0.694694i	$0.244460\pi$
$74$	−5.64575	−0.656305
$75$	0	0
$76$	0	0
$77$	1.06275	0.121111
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−3.64575	−0.407607
$81$	0	0
$82$	10.2915	1.13651
$83$	7.93725	0.871227	0.435613	−	0.900134i	$-0.356531\pi$
$83$	7.93725	0.871227	0.435613	+	0.900134i	$0.356531\pi$
$84$	0	0
$85$	0	0
$86$	0.708497	0.0763992
$87$	0	0
$88$	−0.645751	−0.0688373
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	3.29150	0.345043
$92$	−3.64575	−0.380096
$93$	0	0
$94$	−9.64575	−0.994883
$95$	0	0
$96$	0	0
$97$	14.2915	1.45108	0.725541	−	0.688179i	$-0.241590\pi$
$97$	14.2915	1.45108	0.725541	+	0.688179i	$0.241590\pi$
$98$	−4.29150	−0.433507
$99$	0	0
$100$	8.29150	0.829150
$101$	8.35425	0.831279	0.415639	−	0.909529i	$-0.363558\pi$
$101$	8.35425	0.831279	0.415639	+	0.909529i	$0.363558\pi$
$102$	0	0
$103$	2.70850	0.266876	0.133438	−	0.991057i	$-0.457398\pi$
$103$	2.70850	0.266876	0.133438	+	0.991057i	$0.457398\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	8.58301	0.833655
$107$	4.70850	0.455188	0.227594	−	0.973756i	$-0.426914\pi$
$107$	4.70850	0.455188	0.227594	+	0.973756i	$0.426914\pi$
$108$	0	0
$109$	6.58301	0.630538	0.315269	−	0.949002i	$-0.397905\pi$
$109$	6.58301	0.630538	0.315269	+	0.949002i	$0.397905\pi$
$110$	2.35425	0.224469
$111$	0	0
$112$	−1.64575	−0.155509
$113$	5.58301	0.525205	0.262602	−	0.964904i	$-0.415419\pi$
$113$	5.58301	0.525205	0.262602	+	0.964904i	$0.415419\pi$
$114$	0	0
$115$	13.2915	1.23944
$116$	−3.64575	−0.338500
$117$	0	0
$118$	7.93725	0.730683
$119$	0	0
$120$	0	0
$121$	−10.5830	−0.962091
$122$	−14.9373	−1.35236
$123$	0	0
$124$	0.354249	0.0318125
$125$	−12.0000	−1.07331
$126$	0	0
$127$	2.70850	0.240340	0.120170	−	0.992753i	$-0.461656\pi$
$127$	2.70850	0.240340	0.120170	+	0.992753i	$0.461656\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.29150	0.639507
$131$	13.9373	1.21770	0.608852	−	0.793284i	$-0.291630\pi$
$131$	13.9373	1.21770	0.608852	+	0.793284i	$0.291630\pi$
$132$	0	0
$133$	0	0
$134$	4.64575	0.401332
$135$	0	0
$136$	0	0
$137$	5.58301	0.476988	0.238494	−	0.971144i	$-0.423346\pi$
$137$	5.58301	0.476988	0.238494	+	0.971144i	$0.423346\pi$
$138$	0	0
$139$	13.3542	1.13269	0.566346	−	0.824167i	$-0.308356\pi$
$139$	13.3542	1.13269	0.566346	+	0.824167i	$0.308356\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−13.2915	−1.11540
$143$	1.29150	0.108001
$144$	0	0
$145$	13.2915	1.10380
$146$	12.2915	1.01725
$147$	0	0
$148$	−5.64575	−0.464078
$149$	4.93725	0.404476	0.202238	−	0.979336i	$-0.435179\pi$
$149$	4.93725	0.404476	0.202238	+	0.979336i	$0.435179\pi$
$150$	0	0
$151$	2.93725	0.239030	0.119515	−	0.992832i	$-0.461866\pi$
$151$	2.93725	0.239030	0.119515	+	0.992832i	$0.461866\pi$
$152$	0	0
$153$	0	0
$154$	1.06275	0.0856385
$155$	−1.29150	−0.103736
$156$	0	0
$157$	10.5830	0.844616	0.422308	−	0.906452i	$-0.361220\pi$
$157$	10.5830	0.844616	0.422308	+	0.906452i	$0.361220\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−3.64575	−0.288222
$161$	6.00000	0.472866
$162$	0	0
$163$	−11.9373	−0.934998	−0.467499	−	0.883994i	$-0.654845\pi$
$163$	−11.9373	−0.934998	−0.467499	+	0.883994i	$0.654845\pi$
$164$	10.2915	0.803631
$165$	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
$172$	0.708497	0.0540224
$173$	−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
$175$	−13.6458	−1.03152
$176$	−0.645751	−0.0486753
$177$	0	0
$178$	0	0
$179$	19.9373	1.49018	0.745090	−	0.666964i	$-0.232406\pi$
$179$	19.9373	1.49018	0.745090	+	0.666964i	$0.232406\pi$
$180$	0	0
$181$	4.22876	0.314321	0.157160	−	0.987573i	$-0.449766\pi$
$181$	4.22876	0.314321	0.157160	+	0.987573i	$0.449766\pi$
$182$	3.29150	0.243982
$183$	0	0
$184$	−3.64575	−0.268768
$185$	20.5830	1.51329
$186$	0	0
$187$	0	0
$188$	−9.64575	−0.703489
$189$	0	0
$190$	0	0
$191$	14.5830	1.05519	0.527595	−	0.849496i	$-0.323094\pi$
$191$	14.5830	1.05519	0.527595	+	0.849496i	$0.323094\pi$
$192$	0	0
$193$	6.58301	0.473855	0.236928	−	0.971527i	$-0.423860\pi$
$193$	6.58301	0.473855	0.236928	+	0.971527i	$0.423860\pi$
$194$	14.2915	1.02607
$195$	0	0
$196$	−4.29150	−0.306536
$197$	2.35425	0.167733	0.0838666	−	0.996477i	$-0.473273\pi$
$197$	2.35425	0.167733	0.0838666	+	0.996477i	$0.473273\pi$
$198$	0	0
$199$	11.8745	0.841762	0.420881	−	0.907116i	$-0.361721\pi$
$199$	11.8745	0.841762	0.420881	+	0.907116i	$0.361721\pi$
$200$	8.29150	0.586298
$201$	0	0
$202$	8.35425	0.587803
$203$	6.00000	0.421117
$204$	0	0
$205$	−37.5203	−2.62053
$206$	2.70850	0.188710
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	2.70850	0.186461	0.0932303	−	0.995645i	$-0.470281\pi$
$211$	2.70850	0.186461	0.0932303	+	0.995645i	$0.470281\pi$
$212$	8.58301	0.589483
$213$	0	0
$214$	4.70850	0.321866
$215$	−2.58301	−0.176159
$216$	0	0
$217$	−0.583005	−0.0395770
$218$	6.58301	0.445857
$219$	0	0
$220$	2.35425	0.158723
$221$	0	0
$222$	0	0
$223$	−28.8118	−1.92938	−0.964689	−	0.263391i	$-0.915159\pi$
$223$	−28.8118	−1.92938	−0.964689	+	0.263391i	$0.915159\pi$
$224$	−1.64575	−0.109961
$225$	0	0
$226$	5.58301	0.371376
$227$	−12.6458	−0.839328	−0.419664	−	0.907680i	$-0.637852\pi$
$227$	−12.6458	−0.839328	−0.419664	+	0.907680i	$0.637852\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	13.2915	0.876416
$231$	0	0
$232$	−3.64575	−0.239355
$233$	12.8745	0.843437	0.421719	−	0.906727i	$-0.361427\pi$
$233$	12.8745	0.843437	0.421719	+	0.906727i	$0.361427\pi$
$234$	0	0
$235$	35.1660	2.29398
$236$	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
$241$	−13.5830	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$241$	−13.5830	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$242$	−10.5830	−0.680301
$243$	0	0
$244$	−14.9373	−0.956260
$245$	15.6458	0.999570
$246$	0	0
$247$	0	0
$248$	0.354249	0.0224948
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−2.77124	−0.174919	−0.0874597	−	0.996168i	$-0.527875\pi$
$251$	−2.77124	−0.174919	−0.0874597	+	0.996168i	$0.527875\pi$
$252$	0	0
$253$	2.35425	0.148010
$254$	2.70850	0.169946
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.70850	0.106573	0.0532866	−	0.998579i	$-0.483030\pi$
$257$	1.70850	0.106573	0.0532866	+	0.998579i	$0.483030\pi$
$258$	0	0
$259$	9.29150	0.577346
$260$	7.29150	0.452200
$261$	0	0
$262$	13.9373	0.861046
$263$	−4.93725	−0.304444	−0.152222	−	0.988346i	$-0.548643\pi$
$263$	−4.93725	−0.304444	−0.152222	+	0.988346i	$0.548643\pi$
$264$	0	0
$265$	−31.2915	−1.92222
$266$	0	0
$267$	0	0
$268$	4.64575	0.283784
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	17.6458	1.07190	0.535952	−	0.844249i	$-0.319953\pi$
$271$	17.6458	1.07190	0.535952	+	0.844249i	$0.319953\pi$
$272$	0	0
$273$	0	0
$274$	5.58301	0.337282
$275$	−5.35425	−0.322873
$276$	0	0
$277$	9.52026	0.572017	0.286008	−	0.958227i	$-0.407671\pi$
$277$	9.52026	0.572017	0.286008	+	0.958227i	$0.407671\pi$
$278$	13.3542	0.800935
$279$	0	0
$280$	6.00000	0.358569
$281$	−27.4575	−1.63798	−0.818989	−	0.573809i	$-0.805465\pi$
$281$	−27.4575	−1.63798	−0.818989	+	0.573809i	$0.805465\pi$
$282$	0	0
$283$	25.3542	1.50715	0.753577	−	0.657360i	$-0.228327\pi$
$283$	25.3542	1.50715	0.753577	+	0.657360i	$0.228327\pi$
$284$	−13.2915	−0.788706
$285$	0	0
$286$	1.29150	0.0763682
$287$	−16.9373	−0.999774
$288$	0	0
$289$	−17.0000	−1.00000
$290$	13.2915	0.780504
$291$	0	0
$292$	12.2915	0.719306
$293$	13.0627	0.763134	0.381567	−	0.924341i	$-0.375385\pi$
$293$	13.0627	0.763134	0.381567	+	0.924341i	$0.375385\pi$
$294$	0	0
$295$	−28.9373	−1.68479
$296$	−5.64575	−0.328153
$297$	0	0
$298$	4.93725	0.286007
$299$	7.29150	0.421678
$300$	0	0
$301$	−1.16601	−0.0672077
$302$	2.93725	0.169020
$303$	0	0
$304$	0	0
$305$	54.4575	3.11823
$306$	0	0
$307$	4.64575	0.265147	0.132574	−	0.991173i	$-0.457676\pi$
$307$	4.64575	0.265147	0.132574	+	0.991173i	$0.457676\pi$
$308$	1.06275	0.0605556
$309$	0	0
$310$	−1.29150	−0.0733524
$311$	−8.35425	−0.473726	−0.236863	−	0.971543i	$-0.576119\pi$
$311$	−8.35425	−0.473726	−0.236863	+	0.971543i	$0.576119\pi$
$312$	0	0
$313$	−22.8745	−1.29294	−0.646472	−	0.762938i	$-0.723756\pi$
$313$	−22.8745	−1.29294	−0.646472	+	0.762938i	$0.723756\pi$
$314$	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
$319$	2.35425	0.131813
$320$	−3.64575	−0.203804
$321$	0	0
$322$	6.00000	0.334367
$323$	0	0
$324$	0	0
$325$	−16.5830	−0.919860
$326$	−11.9373	−0.661143
$327$	0	0
$328$	10.2915	0.568253
$329$	15.8745	0.875190
$330$	0	0
$331$	27.8118	1.52867	0.764336	−	0.644818i	$-0.223067\pi$
$331$	27.8118	1.52867	0.764336	+	0.644818i	$0.223067\pi$
$332$	7.93725	0.435613
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−16.9373	−0.925381
$336$	0	0
$337$	20.2915	1.10535	0.552674	−	0.833397i	$-0.313607\pi$
$337$	20.2915	1.10535	0.552674	+	0.833397i	$0.313607\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	0	0
$341$	−0.228757	−0.0123879
$342$	0	0
$343$	18.5830	1.00339
$344$	0.708497	0.0381996
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−3.22876	−0.173329	−0.0866644	−	0.996238i	$-0.527621\pi$
$347$	−3.22876	−0.173329	−0.0866644	+	0.996238i	$0.527621\pi$
$348$	0	0
$349$	−21.1660	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$349$	−21.1660	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$350$	−13.6458	−0.729396
$351$	0	0
$352$	−0.645751	−0.0344187
$353$	18.8745	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$353$	18.8745	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$354$	0	0
$355$	48.4575	2.57186
$356$	0	0
$357$	0	0
$358$	19.9373	1.05372
$359$	−10.9373	−0.577246	−0.288623	−	0.957443i	$-0.593197\pi$
$359$	−10.9373	−0.577246	−0.288623	+	0.957443i	$0.593197\pi$
$360$	0	0
$361$	0	0
$362$	4.22876	0.222259
$363$	0	0
$364$	3.29150	0.172522
$365$	−44.8118	−2.34555
$366$	0	0
$367$	−10.2288	−0.533937	−0.266968	−	0.963705i	$-0.586022\pi$
$367$	−10.2288	−0.533937	−0.266968	+	0.963705i	$0.586022\pi$
$368$	−3.64575	−0.190048
$369$	0	0
$370$	20.5830	1.07006
$371$	−14.1255	−0.733359
$372$	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
$376$	−9.64575	−0.497442
$377$	7.29150	0.375531
$378$	0	0
$379$	−21.2915	−1.09367	−0.546836	−	0.837240i	$-0.684168\pi$
$379$	−21.2915	−1.09367	−0.546836	+	0.837240i	$0.684168\pi$
$380$	0	0
$381$	0	0
$382$	14.5830	0.746131
$383$	−31.5203	−1.61061	−0.805305	−	0.592861i	$-0.797998\pi$
$383$	−31.5203	−1.61061	−0.805305	+	0.592861i	$0.797998\pi$
$384$	0	0
$385$	−3.87451	−0.197463
$386$	6.58301	0.335066
$387$	0	0
$388$	14.2915	0.725541
$389$	−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
$392$	−4.29150	−0.216754
$393$	0	0
$394$	2.35425	0.118605
$395$	−14.5830	−0.733751
$396$	0	0
$397$	36.9373	1.85383	0.926914	−	0.375274i	$-0.122451\pi$
$397$	36.9373	1.85383	0.926914	+	0.375274i	$0.122451\pi$
$398$	11.8745	0.595215
$399$	0	0
$400$	8.29150	0.414575
$401$	6.41699	0.320449	0.160225	−	0.987081i	$-0.448778\pi$
$401$	6.41699	0.320449	0.160225	+	0.987081i	$0.448778\pi$
$402$	0	0
$403$	−0.708497	−0.0352928
$404$	8.35425	0.415639
$405$	0	0
$406$	6.00000	0.297775
$407$	3.64575	0.180713
$408$	0	0
$409$	−13.5830	−0.671636	−0.335818	−	0.941927i	$-0.609013\pi$
$409$	−13.5830	−0.671636	−0.335818	+	0.941927i	$0.609013\pi$
$410$	−37.5203	−1.85299
$411$	0	0
$412$	2.70850	0.133438
$413$	−13.0627	−0.642776
$414$	0	0
$415$	−28.9373	−1.42047
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	−31.7490	−1.55104	−0.775520	−	0.631322i	$-0.782512\pi$
$419$	−31.7490	−1.55104	−0.775520	+	0.631322i	$0.782512\pi$
$420$	0	0
$421$	−22.8118	−1.11178	−0.555889	−	0.831257i	$-0.687622\pi$
$421$	−22.8118	−1.11178	−0.555889	+	0.831257i	$0.687622\pi$
$422$	2.70850	0.131848
$423$	0	0
$424$	8.58301	0.416828
$425$	0	0
$426$	0	0
$427$	24.5830	1.18966
$428$	4.70850	0.227594
$429$	0	0
$430$	−2.58301	−0.124564
$431$	−3.87451	−0.186628	−0.0933142	−	0.995637i	$-0.529746\pi$
$431$	−3.87451	−0.186628	−0.0933142	+	0.995637i	$0.529746\pi$
$432$	0	0
$433$	13.8745	0.666766	0.333383	−	0.942791i	$-0.391810\pi$
$433$	13.8745	0.666766	0.333383	+	0.942791i	$0.391810\pi$
$434$	−0.583005	−0.0279851
$435$	0	0
$436$	6.58301	0.315269
$437$	0	0
$438$	0	0
$439$	36.8118	1.75693	0.878465	−	0.477807i	$-0.158568\pi$
$439$	36.8118	1.75693	0.878465	+	0.477807i	$0.158568\pi$
$440$	2.35425	0.112234
$441$	0	0
$442$	0	0
$443$	−5.35425	−0.254388	−0.127194	−	0.991878i	$-0.540597\pi$
$443$	−5.35425	−0.254388	−0.127194	+	0.991878i	$0.540597\pi$
$444$	0	0
$445$	0	0
$446$	−28.8118	−1.36428
$447$	0	0
$448$	−1.64575	−0.0777544
$449$	−13.7085	−0.646944	−0.323472	−	0.946238i	$-0.604850\pi$
$449$	−13.7085	−0.646944	−0.323472	+	0.946238i	$0.604850\pi$
$450$	0	0
$451$	−6.64575	−0.312936
$452$	5.58301	0.262602
$453$	0	0
$454$	−12.6458	−0.593495
$455$	−12.0000	−0.562569
$456$	0	0
$457$	1.12549	0.0526483	0.0263242	−	0.999653i	$-0.491620\pi$
$457$	1.12549	0.0526483	0.0263242	+	0.999653i	$0.491620\pi$
$458$	20.0000	0.934539
$459$	0	0
$460$	13.2915	0.619720
$461$	23.1660	1.07895	0.539474	−	0.842002i	$-0.318623\pi$
$461$	23.1660	1.07895	0.539474	+	0.842002i	$0.318623\pi$
$462$	0	0
$463$	14.4575	0.671898	0.335949	−	0.941880i	$-0.390943\pi$
$463$	14.4575	0.671898	0.335949	+	0.941880i	$0.390943\pi$
$464$	−3.64575	−0.169250
$465$	0	0
$466$	12.8745	0.596400
$467$	24.6458	1.14047	0.570235	−	0.821482i	$-0.306852\pi$
$467$	24.6458	1.14047	0.570235	+	0.821482i	$0.306852\pi$
$468$	0	0
$469$	−7.64575	−0.353048
$470$	35.1660	1.62209
$471$	0	0
$472$	7.93725	0.365342
$473$	−0.457513	−0.0210365
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	14.5830	0.666315	0.333157	−	0.942871i	$-0.391886\pi$
$479$	14.5830	0.666315	0.333157	+	0.942871i	$0.391886\pi$
$480$	0	0
$481$	11.2915	0.514848
$482$	−13.5830	−0.618689
$483$	0	0
$484$	−10.5830	−0.481046
$485$	−52.1033	−2.36589
$486$	0	0
$487$	22.2288	1.00728	0.503641	−	0.863913i	$-0.331994\pi$
$487$	22.2288	1.00728	0.503641	+	0.863913i	$0.331994\pi$
$488$	−14.9373	−0.676178
$489$	0	0
$490$	15.6458	0.706803
$491$	−28.7085	−1.29560	−0.647798	−	0.761812i	$-0.724310\pi$
$491$	−28.7085	−1.29560	−0.647798	+	0.761812i	$0.724310\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0.354249	0.0159062
$497$	21.8745	0.981206
$498$	0	0
$499$	−31.2288	−1.39799	−0.698996	−	0.715126i	$-0.746369\pi$
$499$	−31.2288	−1.39799	−0.698996	+	0.715126i	$0.746369\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−2.77124	−0.123687
$503$	25.0627	1.11749	0.558746	−	0.829339i	$-0.311283\pi$
$503$	25.0627	1.11749	0.558746	+	0.829339i	$0.311283\pi$
$504$	0	0
$505$	−30.4575	−1.35534
$506$	2.35425	0.104659
$507$	0	0
$508$	2.70850	0.120170
$509$	31.7490	1.40725	0.703625	−	0.710571i	$-0.251563\pi$
$509$	31.7490	1.40725	0.703625	+	0.710571i	$0.251563\pi$
$510$	0	0
$511$	−20.2288	−0.894868
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.70850	0.0753586
$515$	−9.87451	−0.435123
$516$	0	0
$517$	6.22876	0.273940
$518$	9.29150	0.408245
$519$	0	0
$520$	7.29150	0.319754
$521$	−22.2915	−0.976608	−0.488304	−	0.872673i	$-0.662384\pi$
$521$	−22.2915	−0.976608	−0.488304	+	0.872673i	$0.662384\pi$
$522$	0	0
$523$	−29.8745	−1.30632	−0.653161	−	0.757219i	$-0.726557\pi$
$523$	−29.8745	−1.30632	−0.653161	+	0.757219i	$0.726557\pi$
$524$	13.9373	0.608852
$525$	0	0
$526$	−4.93725	−0.215275
$527$	0	0
$528$	0	0
$529$	−9.70850	−0.422109
$530$	−31.2915	−1.35922
$531$	0	0
$532$	0	0
$533$	−20.5830	−0.891549
$534$	0	0
$535$	−17.1660	−0.742151
$536$	4.64575	0.200666
$537$	0	0
$538$	0	0
$539$	2.77124	0.119366
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	17.6458	0.757950
$543$	0	0
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−0.708497	−0.0302932	−0.0151466	−	0.999885i	$-0.504821\pi$
$547$	−0.708497	−0.0302932	−0.0151466	+	0.999885i	$0.504821\pi$
$548$	5.58301	0.238494
$549$	0	0
$550$	−5.35425	−0.228306
$551$	0	0
$552$	0	0
$553$	−6.58301	−0.279938
$554$	9.52026	0.404477
$555$	0	0
$556$	13.3542	0.566346
$557$	26.5830	1.12636	0.563179	−	0.826335i	$-0.309578\pi$
$557$	26.5830	1.12636	0.563179	+	0.826335i	$0.309578\pi$
$558$	0	0
$559$	−1.41699	−0.0599325
$560$	6.00000	0.253546
$561$	0	0
$562$	−27.4575	−1.15823
$563$	25.9373	1.09312	0.546562	−	0.837418i	$-0.315936\pi$
$563$	25.9373	1.09312	0.546562	+	0.837418i	$0.315936\pi$
$564$	0	0
$565$	−20.3542	−0.856310
$566$	25.3542	1.06572
$567$	0	0
$568$	−13.2915	−0.557699
$569$	−14.5830	−0.611351	−0.305676	−	0.952136i	$-0.598882\pi$
$569$	−14.5830	−0.611351	−0.305676	+	0.952136i	$0.598882\pi$
$570$	0	0
$571$	−39.8118	−1.66607	−0.833035	−	0.553220i	$-0.813399\pi$
$571$	−39.8118	−1.66607	−0.833035	+	0.553220i	$0.813399\pi$
$572$	1.29150	0.0540004
$573$	0	0
$574$	−16.9373	−0.706947
$575$	−30.2288	−1.26063
$576$	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
$580$	13.2915	0.551900
$581$	−13.0627	−0.541934
$582$	0	0
$583$	−5.54249	−0.229546
$584$	12.2915	0.508626
$585$	0	0
$586$	13.0627	0.539617
$587$	45.8745	1.89344	0.946722	−	0.322053i	$-0.104373\pi$
$587$	45.8745	1.89344	0.946722	+	0.322053i	$0.104373\pi$
$588$	0	0
$589$	0	0
$590$	−28.9373	−1.19133
$591$	0	0
$592$	−5.64575	−0.232039
$593$	−40.2915	−1.65457	−0.827287	−	0.561780i	$-0.810117\pi$
$593$	−40.2915	−1.65457	−0.827287	+	0.561780i	$0.810117\pi$
$594$	0	0
$595$	0	0
$596$	4.93725	0.202238
$597$	0	0
$598$	7.29150	0.298172
$599$	−1.06275	−0.0434226	−0.0217113	−	0.999764i	$-0.506911\pi$
$599$	−1.06275	−0.0434226	−0.0217113	+	0.999764i	$0.506911\pi$
$600$	0	0
$601$	−31.5830	−1.28830	−0.644149	−	0.764900i	$-0.722788\pi$
$601$	−31.5830	−1.28830	−0.644149	+	0.764900i	$0.722788\pi$
$602$	−1.16601	−0.0475230
$603$	0	0
$604$	2.93725	0.119515
$605$	38.5830	1.56862
$606$	0	0
$607$	8.93725	0.362752	0.181376	−	0.983414i	$-0.441945\pi$
$607$	8.93725	0.362752	0.181376	+	0.983414i	$0.441945\pi$
$608$	0	0
$609$	0	0
$610$	54.4575	2.20492
$611$	19.2915	0.780451
$612$	0	0
$613$	28.5830	1.15446	0.577228	−	0.816583i	$-0.304134\pi$
$613$	28.5830	1.15446	0.577228	+	0.816583i	$0.304134\pi$
$614$	4.64575	0.187487
$615$	0	0
$616$	1.06275	0.0428193
$617$	−0.874508	−0.0352064	−0.0176032	−	0.999845i	$-0.505604\pi$
$617$	−0.874508	−0.0352064	−0.0176032	+	0.999845i	$0.505604\pi$
$618$	0	0
$619$	44.4575	1.78690	0.893449	−	0.449164i	$-0.148278\pi$
$619$	44.4575	1.78690	0.893449	+	0.449164i	$0.148278\pi$
$620$	−1.29150	−0.0518680
$621$	0	0
$622$	−8.35425	−0.334975
$623$	0	0
$624$	0	0
$625$	2.29150	0.0916601
$626$	−22.8745	−0.914249
$627$	0	0
$628$	10.5830	0.422308
$629$	0	0
$630$	0	0
$631$	22.8118	0.908122	0.454061	−	0.890971i	$-0.349975\pi$
$631$	22.8118	0.908122	0.454061	+	0.890971i	$0.349975\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−6.00000	−0.238290
$635$	−9.87451	−0.391858
$636$	0	0
$637$	8.58301	0.340071
$638$	2.35425	0.0932056
$639$	0	0
$640$	−3.64575	−0.144111
$641$	−18.8745	−0.745498	−0.372749	−	0.927932i	$-0.621585\pi$
$641$	−18.8745	−0.745498	−0.372749	+	0.927932i	$0.621585\pi$
$642$	0	0
$643$	30.5203	1.20360	0.601801	−	0.798646i	$-0.294450\pi$
$643$	30.5203	1.20360	0.601801	+	0.798646i	$0.294450\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	−30.4575	−1.19741	−0.598704	−	0.800970i	$-0.704317\pi$
$647$	−30.4575	−1.19741	−0.598704	+	0.800970i	$0.704317\pi$
$648$	0	0
$649$	−5.12549	−0.201193
$650$	−16.5830	−0.650439
$651$	0	0
$652$	−11.9373	−0.467499
$653$	−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
$655$	−50.8118	−1.98538
$656$	10.2915	0.401816
$657$	0	0
$658$	15.8745	0.618853
$659$	2.58301	0.100620	0.0503098	−	0.998734i	$-0.483979\pi$
$659$	2.58301	0.100620	0.0503098	+	0.998734i	$0.483979\pi$
$660$	0	0
$661$	10.2288	0.397853	0.198926	−	0.980014i	$-0.436255\pi$
$661$	10.2288	0.397853	0.198926	+	0.980014i	$0.436255\pi$
$662$	27.8118	1.08093
$663$	0	0
$664$	7.93725	0.308025
$665$	0	0
$666$	0	0
$667$	13.2915	0.514649
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−16.9373	−0.654343
$671$	9.64575	0.372370
$672$	0	0
$673$	−17.8745	−0.689012	−0.344506	−	0.938784i	$-0.611954\pi$
$673$	−17.8745	−0.689012	−0.344506	+	0.938784i	$0.611954\pi$
$674$	20.2915	0.781599
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−32.5830	−1.25227	−0.626133	−	0.779716i	$-0.715363\pi$
$677$	−32.5830	−1.25227	−0.626133	+	0.779716i	$0.715363\pi$
$678$	0	0
$679$	−23.5203	−0.902625
$680$	0	0
$681$	0	0
$682$	−0.228757	−0.00875954
$683$	−26.5830	−1.01717	−0.508585	−	0.861012i	$-0.669831\pi$
$683$	−26.5830	−1.01717	−0.508585	+	0.861012i	$0.669831\pi$
$684$	0	0
$685$	−20.3542	−0.777696
$686$	18.5830	0.709502
$687$	0	0
$688$	0.708497	0.0270112
$689$	−17.1660	−0.653973
$690$	0	0
$691$	−18.5830	−0.706931	−0.353465	−	0.935448i	$-0.614997\pi$
$691$	−18.5830	−0.706931	−0.353465	+	0.935448i	$0.614997\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−3.22876	−0.122562
$695$	−48.6863	−1.84678
$696$	0	0
$697$	0	0
$698$	−21.1660	−0.801145
$699$	0	0
$700$	−13.6458	−0.515761
$701$	−15.6458	−0.590932	−0.295466	−	0.955353i	$-0.595475\pi$
$701$	−15.6458	−0.590932	−0.295466	+	0.955353i	$0.595475\pi$
$702$	0	0
$703$	0	0
$704$	−0.645751	−0.0243377
$705$	0	0
$706$	18.8745	0.710351
$707$	−13.7490	−0.517085
$708$	0	0
$709$	−1.64575	−0.0618075	−0.0309037	−	0.999522i	$-0.509839\pi$
$709$	−1.64575	−0.0618075	−0.0309037	+	0.999522i	$0.509839\pi$
$710$	48.4575	1.81858
$711$	0	0
$712$	0	0
$713$	−1.29150	−0.0483672
$714$	0	0
$715$	−4.70850	−0.176088
$716$	19.9373	0.745090
$717$	0	0
$718$	−10.9373	−0.408175
$719$	−13.2915	−0.495689	−0.247845	−	0.968800i	$-0.579722\pi$
$719$	−13.2915	−0.495689	−0.247845	+	0.968800i	$0.579722\pi$
$720$	0	0
$721$	−4.45751	−0.166006
$722$	0	0
$723$	0	0
$724$	4.22876	0.157160
$725$	−30.2288	−1.12267
$726$	0	0
$727$	22.5830	0.837557	0.418779	−	0.908088i	$-0.362458\pi$
$727$	22.5830	0.837557	0.418779	+	0.908088i	$0.362458\pi$
$728$	3.29150	0.121991
$729$	0	0
$730$	−44.8118	−1.65856
$731$	0	0
$732$	0	0
$733$	42.1033	1.55512	0.777560	−	0.628809i	$-0.216457\pi$
$733$	42.1033	1.55512	0.777560	+	0.628809i	$0.216457\pi$
$734$	−10.2288	−0.377550
$735$	0	0
$736$	−3.64575	−0.134384
$737$	−3.00000	−0.110506
$738$	0	0
$739$	13.8118	0.508074	0.254037	−	0.967195i	$-0.418242\pi$
$739$	13.8118	0.508074	0.254037	+	0.967195i	$0.418242\pi$
$740$	20.5830	0.756646
$741$	0	0
$742$	−14.1255	−0.518563
$743$	−10.4797	−0.384464	−0.192232	−	0.981349i	$-0.561573\pi$
$743$	−10.4797	−0.384464	−0.192232	+	0.981349i	$0.561573\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	4.00000	0.146450
$747$	0	0
$748$	0	0
$749$	−7.74902	−0.283143
$750$	0	0
$751$	−23.8745	−0.871193	−0.435597	−	0.900142i	$-0.643463\pi$
$751$	−23.8745	−0.871193	−0.435597	+	0.900142i	$0.643463\pi$
$752$	−9.64575	−0.351744
$753$	0	0
$754$	7.29150	0.265541
$755$	−10.7085	−0.389722
$756$	0	0
$757$	16.5830	0.602720	0.301360	−	0.953511i	$-0.402559\pi$
$757$	16.5830	0.602720	0.301360	+	0.953511i	$0.402559\pi$
$758$	−21.2915	−0.773342
$759$	0	0
$760$	0	0
$761$	−11.1255	−0.403299	−0.201649	−	0.979458i	$-0.564630\pi$
$761$	−11.1255	−0.403299	−0.201649	+	0.979458i	$0.564630\pi$
$762$	0	0
$763$	−10.8340	−0.392217
$764$	14.5830	0.527595
$765$	0	0
$766$	−31.5203	−1.13887
$767$	−15.8745	−0.573195
$768$	0	0
$769$	24.7085	0.891011	0.445506	−	0.895279i	$-0.353024\pi$
$769$	24.7085	0.891011	0.445506	+	0.895279i	$0.353024\pi$
$770$	−3.87451	−0.139628
$771$	0	0
$772$	6.58301	0.236928
$773$	10.9373	0.393386	0.196693	−	0.980465i	$-0.436980\pi$
$773$	10.9373	0.393386	0.196693	+	0.980465i	$0.436980\pi$
$774$	0	0
$775$	2.93725	0.105509
$776$	14.2915	0.513035
$777$	0	0
$778$	−12.0000	−0.430221
$779$	0	0
$780$	0	0
$781$	8.58301	0.307124
$782$	0	0
$783$	0	0
$784$	−4.29150	−0.153268
$785$	−38.5830	−1.37709
$786$	0	0
$787$	5.47974	0.195332	0.0976658	−	0.995219i	$-0.468862\pi$
$787$	5.47974	0.195332	0.0976658	+	0.995219i	$0.468862\pi$
$788$	2.35425	0.0838666
$789$	0	0
$790$	−14.5830	−0.518840
$791$	−9.18824	−0.326696
$792$	0	0
$793$	29.8745	1.06087
$794$	36.9373	1.31085
$795$	0	0
$796$	11.8745	0.420881
$797$	2.81176	0.0995977	0.0497989	−	0.998759i	$-0.484142\pi$
$797$	2.81176	0.0995977	0.0497989	+	0.998759i	$0.484142\pi$
$798$	0	0
$799$	0	0
$800$	8.29150	0.293149
$801$	0	0
$802$	6.41699	0.226592
$803$	−7.93725	−0.280100
$804$	0	0
$805$	−21.8745	−0.770975
$806$	−0.708497	−0.0249558
$807$	0	0
$808$	8.35425	0.293901
$809$	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
$811$	20.7085	0.727174	0.363587	−	0.931560i	$-0.381552\pi$
$811$	20.7085	0.727174	0.363587	+	0.931560i	$0.381552\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	3.64575	0.127784
$815$	43.5203	1.52445
$816$	0	0
$817$	0	0
$818$	−13.5830	−0.474919
$819$	0	0
$820$	−37.5203	−1.31026
$821$	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
$823$	−0.125492	−0.00437438	−0.00218719	−	0.999998i	$-0.500696\pi$
$823$	−0.125492	−0.00437438	−0.00218719	+	0.999998i	$0.500696\pi$
$824$	2.70850	0.0943550
$825$	0	0
$826$	−13.0627	−0.454511
$827$	−47.3542	−1.64667	−0.823334	−	0.567557i	$-0.807889\pi$
$827$	−47.3542	−1.64667	−0.823334	+	0.567557i	$0.807889\pi$
$828$	0	0
$829$	−25.1660	−0.874052	−0.437026	−	0.899449i	$-0.643968\pi$
$829$	−25.1660	−0.874052	−0.437026	+	0.899449i	$0.643968\pi$
$830$	−28.9373	−1.00443
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	43.7490	1.51400
$836$	0	0
$837$	0	0
$838$	−31.7490	−1.09675
$839$	−4.47974	−0.154658	−0.0773289	−	0.997006i	$-0.524639\pi$
$839$	−4.47974	−0.154658	−0.0773289	+	0.997006i	$0.524639\pi$
$840$	0	0
$841$	−15.7085	−0.541672
$842$	−22.8118	−0.786145
$843$	0	0
$844$	2.70850	0.0932303
$845$	32.8118	1.12876
$846$	0	0
$847$	17.4170	0.598455
$848$	8.58301	0.294742
$849$	0	0
$850$	0	0
$851$	20.5830	0.705576
$852$	0	0
$853$	−12.5830	−0.430834	−0.215417	−	0.976522i	$-0.569111\pi$
$853$	−12.5830	−0.430834	−0.215417	+	0.976522i	$0.569111\pi$
$854$	24.5830	0.841213
$855$	0	0
$856$	4.70850	0.160933
$857$	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	$-0.572460\pi$
$859$	−13.2288	−0.451359	−0.225680	+	0.974202i	$0.572460\pi$
$860$	−2.58301	−0.0880797
$861$	0	0
$862$	−3.87451	−0.131966
$863$	−46.9373	−1.59776	−0.798881	−	0.601489i	$-0.794575\pi$
$863$	−46.9373	−1.59776	−0.798881	+	0.601489i	$0.794575\pi$
$864$	0	0
$865$	21.8745	0.743756
$866$	13.8745	0.471475
$867$	0	0
$868$	−0.583005	−0.0197885
$869$	−2.58301	−0.0876225
$870$	0	0
$871$	−9.29150	−0.314831
$872$	6.58301	0.222929
$873$	0	0
$874$	0	0
$875$	19.7490	0.667639
$876$	0	0
$877$	36.3542	1.22760	0.613798	−	0.789463i	$-0.289641\pi$
$877$	36.3542	1.22760	0.613798	+	0.789463i	$0.289641\pi$
$878$	36.8118	1.24234
$879$	0	0
$880$	2.35425	0.0793617
$881$	5.12549	0.172682	0.0863411	−	0.996266i	$-0.472483\pi$
$881$	5.12549	0.172682	0.0863411	+	0.996266i	$0.472483\pi$
$882$	0	0
$883$	40.3948	1.35939	0.679696	−	0.733494i	$-0.262112\pi$
$883$	40.3948	1.35939	0.679696	+	0.733494i	$0.262112\pi$
$884$	0	0
$885$	0	0
$886$	−5.35425	−0.179880
$887$	−38.5830	−1.29549	−0.647745	−	0.761857i	$-0.724288\pi$
$887$	−38.5830	−1.29549	−0.647745	+	0.761857i	$0.724288\pi$
$888$	0	0
$889$	−4.45751	−0.149500
$890$	0	0
$891$	0	0
$892$	−28.8118	−0.964689
$893$	0	0
$894$	0	0
$895$	−72.6863	−2.42963
$896$	−1.64575	−0.0549807
$897$	0	0
$898$	−13.7085	−0.457458
$899$	−1.29150	−0.0430740
$900$	0	0
$901$	0	0
$902$	−6.64575	−0.221279
$903$	0	0
$904$	5.58301	0.185688
$905$	−15.4170	−0.512478
$906$	0	0
$907$	−24.0627	−0.798990	−0.399495	−	0.916735i	$-0.630815\pi$
$907$	−24.0627	−0.798990	−0.399495	+	0.916735i	$0.630815\pi$
$908$	−12.6458	−0.419664
$909$	0	0
$910$	−12.0000	−0.397796
$911$	1.06275	0.0352103	0.0176052	−	0.999845i	$-0.494396\pi$
$911$	1.06275	0.0352103	0.0176052	+	0.999845i	$0.494396\pi$
$912$	0	0
$913$	−5.12549	−0.169629
$914$	1.12549	0.0372280
$915$	0	0
$916$	20.0000	0.660819
$917$	−22.9373	−0.757455
$918$	0	0
$919$	11.8745	0.391704	0.195852	−	0.980633i	$-0.437253\pi$
$919$	11.8745	0.391704	0.195852	+	0.980633i	$0.437253\pi$
$920$	13.2915	0.438208
$921$	0	0
$922$	23.1660	0.762932
$923$	26.5830	0.874990
$924$	0	0
$925$	−46.8118	−1.53916
$926$	14.4575	0.475103
$927$	0	0
$928$	−3.64575	−0.119678
$929$	−11.5830	−0.380026	−0.190013	−	0.981782i	$-0.560853\pi$
$929$	−11.5830	−0.380026	−0.190013	+	0.981782i	$0.560853\pi$
$930$	0	0
$931$	0	0
$932$	12.8745	0.421719
$933$	0	0
$934$	24.6458	0.806434
$935$	0	0
$936$	0	0
$937$	38.8745	1.26997	0.634987	−	0.772522i	$-0.281005\pi$
$937$	38.8745	1.26997	0.634987	+	0.772522i	$0.281005\pi$
$938$	−7.64575	−0.249643
$939$	0	0
$940$	35.1660	1.14699
$941$	59.1660	1.92876	0.964378	−	0.264527i	$-0.0852157\pi$
$941$	59.1660	1.92876	0.964378	+	0.264527i	$0.0852157\pi$
$942$	0	0
$943$	−37.5203	−1.22183
$944$	7.93725	0.258336
$945$	0	0
$946$	−0.457513	−0.0148750
$947$	−55.7490	−1.81160	−0.905800	−	0.423706i	$-0.860729\pi$
$947$	−55.7490	−1.81160	−0.905800	+	0.423706i	$0.860729\pi$
$948$	0	0
$949$	−24.5830	−0.797998
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.4575	1.27815	0.639077	−	0.769143i	$-0.279316\pi$
$953$	39.4575	1.27815	0.639077	+	0.769143i	$0.279316\pi$
$954$	0	0
$955$	−53.1660	−1.72041
$956$	12.0000	0.388108
$957$	0	0
$958$	14.5830	0.471156
$959$	−9.18824	−0.296704
$960$	0	0
$961$	−30.8745	−0.995952
$962$	11.2915	0.364053
$963$	0	0
$964$	−13.5830	−0.437479
$965$	−24.0000	−0.772587
$966$	0	0
$967$	−2.70850	−0.0870994	−0.0435497	−	0.999051i	$-0.513867\pi$
$967$	−2.70850	−0.0870994	−0.0435497	+	0.999051i	$0.513867\pi$
$968$	−10.5830	−0.340151
$969$	0	0
$970$	−52.1033	−1.67293
$971$	14.3948	0.461950	0.230975	−	0.972960i	$-0.425808\pi$
$971$	14.3948	0.461950	0.230975	+	0.972960i	$0.425808\pi$
$972$	0	0
$973$	−21.9778	−0.704575
$974$	22.2288	0.712255
$975$	0	0
$976$	−14.9373	−0.478130
$977$	45.4575	1.45431	0.727157	−	0.686471i	$-0.240841\pi$
$977$	45.4575	1.45431	0.727157	+	0.686471i	$0.240841\pi$
$978$	0	0
$979$	0	0
$980$	15.6458	0.499785
$981$	0	0
$982$	−28.7085	−0.916125
$983$	−31.7490	−1.01264	−0.506318	−	0.862347i	$-0.668994\pi$
$983$	−31.7490	−1.01264	−0.506318	+	0.862347i	$0.668994\pi$
$984$	0	0
$985$	−8.58301	−0.273477
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.58301	−0.0821348
$990$	0	0
$991$	45.1660	1.43475	0.717373	−	0.696690i	$-0.245344\pi$
$991$	45.1660	1.43475	0.717373	+	0.696690i	$0.245344\pi$
$992$	0.354249	0.0112474
$993$	0	0
$994$	21.8745	0.693817
$995$	−43.2915	−1.37243
$996$	0	0
$997$	−10.2288	−0.323948	−0.161974	−	0.986795i	$-0.551786\pi$
$997$	−10.2288	−0.323948	−0.161974	+	0.986795i	$0.551786\pi$
$998$	−31.2288	−0.988529
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bg.1.1		2
3.2	odd	2		722.2.a.g.1.2		2
12.11	even	2		5776.2.a.z.1.1		2
19.8	odd	6		342.2.g.f.235.2		4
19.12	odd	6		342.2.g.f.163.2		4
19.18	odd	2		6498.2.a.ba.1.1		2
57.2	even	18		722.2.e.n.99.1		12
57.5	odd	18		722.2.e.o.595.1		12
57.8	even	6		38.2.c.b.7.2	✓	4
57.11	odd	6		722.2.c.j.653.1		4
57.14	even	18		722.2.e.n.595.2		12
57.17	odd	18		722.2.e.o.99.2		12
57.23	odd	18		722.2.e.o.415.1		12
57.26	odd	6		722.2.c.j.429.1		4
57.29	even	18		722.2.e.n.423.1		12
57.32	even	18		722.2.e.n.245.1		12
57.35	odd	18		722.2.e.o.389.2		12
57.41	even	18		722.2.e.n.389.1		12
57.44	odd	18		722.2.e.o.245.2		12
57.47	odd	18		722.2.e.o.423.2		12
57.50	even	6		38.2.c.b.11.2	yes	4
57.53	even	18		722.2.e.n.415.2		12
57.56	even	2		722.2.a.j.1.1		2
76.27	even	6		2736.2.s.v.577.2		4
76.31	even	6		2736.2.s.v.1873.2		4
228.107	odd	6		304.2.i.e.49.1		4
228.179	odd	6		304.2.i.e.273.1		4
228.227	odd	2		5776.2.a.ba.1.2		2
285.8	odd	12		950.2.j.g.349.1		8
285.107	odd	12		950.2.j.g.49.1		8
285.122	odd	12		950.2.j.g.349.4		8
285.164	even	6		950.2.e.k.201.1		4
285.179	even	6		950.2.e.k.501.1		4
285.278	odd	12		950.2.j.g.49.4		8
456.107	odd	6		1216.2.i.k.961.2		4
456.179	odd	6		1216.2.i.k.577.2		4
456.221	even	6		1216.2.i.l.961.1		4
456.293	even	6		1216.2.i.l.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.b.7.2	✓	4	57.8	even	6
38.2.c.b.11.2	yes	4	57.50	even	6
304.2.i.e.49.1		4	228.107	odd	6
304.2.i.e.273.1		4	228.179	odd	6
342.2.g.f.163.2		4	19.12	odd	6
342.2.g.f.235.2		4	19.8	odd	6
722.2.a.g.1.2		2	3.2	odd	2
722.2.a.j.1.1		2	57.56	even	2
722.2.c.j.429.1		4	57.26	odd	6
722.2.c.j.653.1		4	57.11	odd	6
722.2.e.n.99.1		12	57.2	even	18
722.2.e.n.245.1		12	57.32	even	18
722.2.e.n.389.1		12	57.41	even	18
722.2.e.n.415.2		12	57.53	even	18
722.2.e.n.423.1		12	57.29	even	18
722.2.e.n.595.2		12	57.14	even	18
722.2.e.o.99.2		12	57.17	odd	18
722.2.e.o.245.2		12	57.44	odd	18
722.2.e.o.389.2		12	57.35	odd	18
722.2.e.o.415.1		12	57.23	odd	18
722.2.e.o.423.2		12	57.47	odd	18
722.2.e.o.595.1		12	57.5	odd	18
950.2.e.k.201.1		4	285.164	even	6
950.2.e.k.501.1		4	285.179	even	6
950.2.j.g.49.1		8	285.107	odd	12
950.2.j.g.49.4		8	285.278	odd	12
950.2.j.g.349.1		8	285.8	odd	12
950.2.j.g.349.4		8	285.122	odd	12
1216.2.i.k.577.2		4	456.179	odd	6
1216.2.i.k.961.2		4	456.107	odd	6
1216.2.i.l.577.1		4	456.293	even	6
1216.2.i.l.961.1		4	456.221	even	6
2736.2.s.v.577.2		4	76.27	even	6
2736.2.s.v.1873.2		4	76.31	even	6
5776.2.a.z.1.1		2	12.11	even	2
5776.2.a.ba.1.2		2	228.227	odd	2
6498.2.a.ba.1.1		2	19.18	odd	2
6498.2.a.bg.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} -1.64575 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.64575 q^{5} -1.64575 q^{7} +1.00000 q^{8} -3.64575 q^{10} -0.645751 q^{11} -2.00000 q^{13} -1.64575 q^{14} +1.00000 q^{16} -3.64575 q^{20} -0.645751 q^{22} -3.64575 q^{23} +8.29150 q^{25} -2.00000 q^{26} -1.64575 q^{28} -3.64575 q^{29} +0.354249 q^{31} +1.00000 q^{32} +6.00000 q^{35} -5.64575 q^{37} -3.64575 q^{40} +10.2915 q^{41} +0.708497 q^{43} -0.645751 q^{44} -3.64575 q^{46} -9.64575 q^{47} -4.29150 q^{49} +8.29150 q^{50} -2.00000 q^{52} +8.58301 q^{53} +2.35425 q^{55} -1.64575 q^{56} -3.64575 q^{58} +7.93725 q^{59} -14.9373 q^{61} +0.354249 q^{62} +1.00000 q^{64} +7.29150 q^{65} +4.64575 q^{67} +6.00000 q^{70} -13.2915 q^{71} +12.2915 q^{73} -5.64575 q^{74} +1.06275 q^{77} +4.00000 q^{79} -3.64575 q^{80} +10.2915 q^{82} +7.93725 q^{83} +0.708497 q^{86} -0.645751 q^{88} +3.29150 q^{91} -3.64575 q^{92} -9.64575 q^{94} +14.2915 q^{97} -4.29150 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 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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