LMFDB - Embedded newform 6480.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.42801 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.42801 q^{7} -2.67282 q^{11} +4.67282 q^{13} +2.67282 q^{17} -4.67282 q^{19} +5.91764 q^{23} +1.00000 q^{25} -9.48963 q^{29} -6.96080 q^{31} -1.42801 q^{35} -1.81681 q^{37} -1.47116 q^{41} -0.471163 q^{43} -6.95684 q^{47} -4.96080 q^{49} -1.14399 q^{53} -2.67282 q^{55} +1.14399 q^{59} -2.52884 q^{61} +4.67282 q^{65} +6.59046 q^{67} +12.8745 q^{71} -1.71203 q^{73} +3.81681 q^{77} +0.287973 q^{79} +4.28402 q^{83} +2.67282 q^{85} -3.00000 q^{89} -6.67282 q^{91} -4.67282 q^{95} +7.83528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 5 * q^7	$ 3 q + 3 q^{5} - 5 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} - 4 q^{19} - 3 q^{23} + 3 q^{25} - 7 q^{29} - 8 q^{31} - 5 q^{35} + 6 q^{37} - 13 q^{41} - 10 q^{43} - 13 q^{47} - 2 q^{49} - 2 q^{53} + 2 q^{55} + 2 q^{59} + q^{61} + 4 q^{65} - 11 q^{67} + 10 q^{71} - 8 q^{73} - 2 q^{79} + 15 q^{83} - 2 q^{85} - 9 q^{89} - 10 q^{91} - 4 q^{95} - 18 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 5 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 - 4 * q^19 - 3 * q^23 + 3 * q^25 - 7 * q^29 - 8 * q^31 - 5 * q^35 + 6 * q^37 - 13 * q^41 - 10 * q^43 - 13 * q^47 - 2 * q^49 - 2 * q^53 + 2 * q^55 + 2 * q^59 + q^61 + 4 * q^65 - 11 * q^67 + 10 * q^71 - 8 * q^73 - 2 * q^79 + 15 * q^83 - 2 * q^85 - 9 * q^89 - 10 * q^91 - 4 * q^95 - 18 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.42801	−0.539736	−0.269868	−	0.962897i	$-0.586980\pi$
$7$	−1.42801	−0.539736	−0.269868	+	0.962897i	$0.586980\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.67282	−0.805887	−0.402943	−	0.915225i	$-0.632013\pi$
$11$	−2.67282	−0.805887	−0.402943	+	0.915225i	$0.632013\pi$
$12$	0	0
$13$	4.67282	1.29601	0.648004	−	0.761637i	$-0.275604\pi$
$13$	4.67282	1.29601	0.648004	+	0.761637i	$0.275604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.67282	0.648255	0.324127	−	0.946013i	$-0.394929\pi$
$17$	2.67282	0.648255	0.324127	+	0.946013i	$0.394929\pi$
$18$	0	0
$19$	−4.67282	−1.07202	−0.536010	−	0.844212i	$-0.680069\pi$
$19$	−4.67282	−1.07202	−0.536010	+	0.844212i	$0.680069\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.91764	1.23391	0.616957	−	0.786997i	$-0.288365\pi$
$23$	5.91764	1.23391	0.616957	+	0.786997i	$0.288365\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.48963	−1.76218	−0.881090	−	0.472948i	$-0.843190\pi$
$29$	−9.48963	−1.76218	−0.881090	+	0.472948i	$0.843190\pi$
$30$	0	0
$31$	−6.96080	−1.25020	−0.625098	−	0.780546i	$-0.714941\pi$
$31$	−6.96080	−1.25020	−0.625098	+	0.780546i	$0.714941\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.42801	−0.241377
$36$	0	0
$37$	−1.81681	−0.298682	−0.149341	−	0.988786i	$-0.547715\pi$
$37$	−1.81681	−0.298682	−0.149341	+	0.988786i	$0.547715\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.47116	−0.229757	−0.114879	−	0.993380i	$-0.536648\pi$
$41$	−1.47116	−0.229757	−0.114879	+	0.993380i	$0.536648\pi$
$42$	0	0
$43$	−0.471163	−0.0718517	−0.0359258	−	0.999354i	$-0.511438\pi$
$43$	−0.471163	−0.0718517	−0.0359258	+	0.999354i	$0.511438\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.95684	−1.01476	−0.507380	−	0.861722i	$-0.669386\pi$
$47$	−6.95684	−1.01476	−0.507380	+	0.861722i	$0.669386\pi$
$48$	0	0
$49$	−4.96080	−0.708685
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.14399	−0.157139	−0.0785693	−	0.996909i	$-0.525035\pi$
$53$	−1.14399	−0.157139	−0.0785693	+	0.996909i	$0.525035\pi$
$54$	0	0
$55$	−2.67282	−0.360403
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.14399	0.148934	0.0744672	−	0.997223i	$-0.476274\pi$
$59$	1.14399	0.148934	0.0744672	+	0.997223i	$0.476274\pi$
$60$	0	0
$61$	−2.52884	−0.323784	−0.161892	−	0.986808i	$-0.551760\pi$
$61$	−2.52884	−0.323784	−0.161892	+	0.986808i	$0.551760\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.67282	0.579592
$66$	0	0
$67$	6.59046	0.805153	0.402577	−	0.915386i	$-0.368115\pi$
$67$	6.59046	0.805153	0.402577	+	0.915386i	$0.368115\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.8745	1.52792	0.763960	−	0.645263i	$-0.223252\pi$
$71$	12.8745	1.52792	0.763960	+	0.645263i	$0.223252\pi$
$72$	0	0
$73$	−1.71203	−0.200378	−0.100189	−	0.994968i	$-0.531945\pi$
$73$	−1.71203	−0.200378	−0.100189	+	0.994968i	$0.531945\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.81681	0.434966
$78$	0	0
$79$	0.287973	0.0323995	0.0161998	−	0.999869i	$-0.494843\pi$
$79$	0.287973	0.0323995	0.0161998	+	0.999869i	$0.494843\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.28402	0.470232	0.235116	−	0.971967i	$-0.424453\pi$
$83$	4.28402	0.470232	0.235116	+	0.971967i	$0.424453\pi$
$84$	0	0
$85$	2.67282	0.289908
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−6.67282	−0.699502
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.67282	−0.479422
$96$	0	0
$97$	7.83528	0.795552	0.397776	−	0.917483i	$-0.369782\pi$
$97$	7.83528	0.795552	0.397776	+	0.917483i	$0.369782\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.20166	−0.418081	−0.209040	−	0.977907i	$-0.567034\pi$
$101$	−4.20166	−0.418081	−0.209040	+	0.977907i	$0.567034\pi$
$102$	0	0
$103$	1.81681	0.179016	0.0895078	−	0.995986i	$-0.471471\pi$
$103$	1.81681	0.179016	0.0895078	+	0.995986i	$0.471471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.9176	−1.15212	−0.576061	−	0.817407i	$-0.695411\pi$
$107$	−11.9176	−1.15212	−0.576061	+	0.817407i	$0.695411\pi$
$108$	0	0
$109$	−16.6521	−1.59498	−0.797491	−	0.603331i	$-0.793840\pi$
$109$	−16.6521	−1.59498	−0.797491	+	0.603331i	$0.793840\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	20.1233	1.89304	0.946518	−	0.322650i	$-0.104574\pi$
$113$	20.1233	1.89304	0.946518	+	0.322650i	$0.104574\pi$
$114$	0	0
$115$	5.91764	0.551823
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.81681	−0.349886
$120$	0	0
$121$	−3.85601	−0.350547
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.18714	−0.194078	−0.0970388	−	0.995281i	$-0.530937\pi$
$127$	−2.18714	−0.194078	−0.0970388	+	0.995281i	$0.530937\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	6.67282	0.578607
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.2017	−0.871587	−0.435793	−	0.900047i	$-0.643532\pi$
$137$	−10.2017	−0.871587	−0.435793	+	0.900047i	$0.643532\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.4896	−1.04444
$144$	0	0
$145$	−9.48963	−0.788071
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.0761	−1.64470	−0.822351	−	0.568981i	$-0.807338\pi$
$149$	−20.0761	−1.64470	−0.822351	+	0.568981i	$0.807338\pi$
$150$	0	0
$151$	−3.03920	−0.247327	−0.123663	−	0.992324i	$-0.539464\pi$
$151$	−3.03920	−0.247327	−0.123663	+	0.992324i	$0.539464\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.96080	−0.559105
$156$	0	0
$157$	0.201661	0.0160943	0.00804714	−	0.999968i	$-0.497438\pi$
$157$	0.201661	0.0160943	0.00804714	+	0.999968i	$0.497438\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.45043	−0.665987
$162$	0	0
$163$	−17.8168	−1.39552	−0.697760	−	0.716331i	$-0.745820\pi$
$163$	−17.8168	−1.39552	−0.697760	+	0.716331i	$0.745820\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.1008	−1.09116	−0.545578	−	0.838060i	$-0.683690\pi$
$167$	−14.1008	−1.09116	−0.545578	+	0.838060i	$0.683690\pi$
$168$	0	0
$169$	8.83528	0.679637
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.36638	0.331970	0.165985	−	0.986128i	$-0.446920\pi$
$173$	4.36638	0.331970	0.165985	+	0.986128i	$0.446920\pi$
$174$	0	0
$175$	−1.42801	−0.107947
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.1625	1.13330	0.566648	−	0.823960i	$-0.308240\pi$
$179$	15.1625	1.13330	0.566648	+	0.823960i	$0.308240\pi$
$180$	0	0
$181$	3.20166	0.237978	0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166	0.237978	0.118989	+	0.992896i	$0.462035\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.81681	−0.133575
$186$	0	0
$187$	−7.14399	−0.522420
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.83754	0.205317	0.102659	−	0.994717i	$-0.467265\pi$
$191$	2.83754	0.205317	0.102659	+	0.994717i	$0.467265\pi$
$192$	0	0
$193$	−18.7882	−1.35240	−0.676201	−	0.736717i	$-0.736375\pi$
$193$	−18.7882	−1.35240	−0.676201	+	0.736717i	$0.736375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.83528	0.415747	0.207873	−	0.978156i	$-0.433346\pi$
$197$	5.83528	0.415747	0.207873	+	0.978156i	$0.433346\pi$
$198$	0	0
$199$	−13.0761	−0.926943	−0.463472	−	0.886112i	$-0.653396\pi$
$199$	−13.0761	−0.926943	−0.463472	+	0.886112i	$0.653396\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.5513	0.951112
$204$	0	0
$205$	−1.47116	−0.102750
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.4896	0.863926
$210$	0	0
$211$	−8.38485	−0.577237	−0.288618	−	0.957444i	$-0.593196\pi$
$211$	−8.38485	−0.577237	−0.288618	+	0.957444i	$0.593196\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.471163	−0.0321330
$216$	0	0
$217$	9.94006	0.674776
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.4896	0.840144
$222$	0	0
$223$	−9.16641	−0.613828	−0.306914	−	0.951737i	$-0.599296\pi$
$223$	−9.16641	−0.613828	−0.306914	+	0.951737i	$0.599296\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.67282	0.177402	0.0887008	−	0.996058i	$-0.471729\pi$
$227$	2.67282	0.177402	0.0887008	+	0.996058i	$0.471729\pi$
$228$	0	0
$229$	2.54731	0.168331	0.0841654	−	0.996452i	$-0.473178\pi$
$229$	2.54731	0.168331	0.0841654	+	0.996452i	$0.473178\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.22013	−0.407494	−0.203747	−	0.979024i	$-0.565312\pi$
$233$	−6.22013	−0.407494	−0.203747	+	0.979024i	$0.565312\pi$
$234$	0	0
$235$	−6.95684	−0.453814
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.12325	−0.525450	−0.262725	−	0.964871i	$-0.584621\pi$
$239$	−8.12325	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$240$	0	0
$241$	−26.3641	−1.69826	−0.849131	−	0.528182i	$-0.822874\pi$
$241$	−26.3641	−1.69826	−0.849131	+	0.528182i	$0.822874\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.96080	−0.316934
$246$	0	0
$247$	−21.8353	−1.38935
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.549569	0.0346885	0.0173443	−	0.999850i	$-0.494479\pi$
$251$	0.549569	0.0346885	0.0173443	+	0.999850i	$0.494479\pi$
$252$	0	0
$253$	−15.8168	−0.994394
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	2.59442	0.161209
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.8952	−0.733490	−0.366745	−	0.930321i	$-0.619528\pi$
$263$	−11.8952	−0.733490	−0.366745	+	0.930321i	$0.619528\pi$
$264$	0	0
$265$	−1.14399	−0.0702745
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.5737	1.74217	0.871084	−	0.491134i	$-0.163417\pi$
$269$	28.5737	1.74217	0.871084	+	0.491134i	$0.163417\pi$
$270$	0	0
$271$	23.3641	1.41927	0.709635	−	0.704570i	$-0.248860\pi$
$271$	23.3641	1.41927	0.709635	+	0.704570i	$0.248860\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.67282	−0.161177
$276$	0	0
$277$	−15.0761	−0.905838	−0.452919	−	0.891552i	$-0.649617\pi$
$277$	−15.0761	−0.905838	−0.452919	+	0.891552i	$0.649617\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.65209	0.396831	0.198415	−	0.980118i	$-0.436421\pi$
$281$	6.65209	0.396831	0.198415	+	0.980118i	$0.436421\pi$
$282$	0	0
$283$	−26.8969	−1.59886	−0.799428	−	0.600762i	$-0.794864\pi$
$283$	−26.8969	−1.59886	−0.799428	+	0.600762i	$0.794864\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.10083	0.124008
$288$	0	0
$289$	−9.85601	−0.579765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.3849	0.723531	0.361765	−	0.932269i	$-0.382174\pi$
$293$	12.3849	0.723531	0.361765	+	0.932269i	$0.382174\pi$
$294$	0	0
$295$	1.14399	0.0666055
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.6521	1.59916
$300$	0	0
$301$	0.672824	0.0387809
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.52884	−0.144801
$306$	0	0
$307$	2.49359	0.142317	0.0711583	−	0.997465i	$-0.477330\pi$
$307$	2.49359	0.142317	0.0711583	+	0.997465i	$0.477330\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.2201	1.37340	0.686699	−	0.726942i	$-0.259059\pi$
$311$	24.2201	1.37340	0.686699	+	0.726942i	$0.259059\pi$
$312$	0	0
$313$	−35.0841	−1.98307	−0.991534	−	0.129848i	$-0.958551\pi$
$313$	−35.0841	−1.98307	−0.991534	+	0.129848i	$0.958551\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.4712	−0.588119	−0.294060	−	0.955787i	$-0.595006\pi$
$317$	−10.4712	−0.588119	−0.294060	+	0.955787i	$0.595006\pi$
$318$	0	0
$319$	25.3641	1.42012
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.4896	−0.694942
$324$	0	0
$325$	4.67282	0.259202
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.93442	0.547702
$330$	0	0
$331$	−16.7776	−0.922181	−0.461090	−	0.887353i	$-0.652542\pi$
$331$	−16.7776	−0.922181	−0.461090	+	0.887353i	$0.652542\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.59046	0.360076
$336$	0	0
$337$	−27.1809	−1.48064	−0.740320	−	0.672255i	$-0.765326\pi$
$337$	−27.1809	−1.48064	−0.740320	+	0.672255i	$0.765326\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.6050	1.00752
$342$	0	0
$343$	17.0801	0.922239
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.5658	1.26508	0.632539	−	0.774529i	$-0.282013\pi$
$347$	23.5658	1.26508	0.632539	+	0.774529i	$0.282013\pi$
$348$	0	0
$349$	−10.7120	−0.573402	−0.286701	−	0.958020i	$-0.592559\pi$
$349$	−10.7120	−0.573402	−0.286701	+	0.958020i	$0.592559\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.2672	1.45129	0.725644	−	0.688070i	$-0.241542\pi$
$353$	27.2672	1.45129	0.725644	+	0.688070i	$0.241542\pi$
$354$	0	0
$355$	12.8745	0.683307
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.6807	0.563707	0.281854	−	0.959457i	$-0.409051\pi$
$359$	10.6807	0.563707	0.281854	+	0.959457i	$0.409051\pi$
$360$	0	0
$361$	2.83528	0.149225
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.71203	−0.0896116
$366$	0	0
$367$	8.47116	0.442191	0.221096	−	0.975252i	$-0.429037\pi$
$367$	8.47116	0.442191	0.221096	+	0.975252i	$0.429037\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.63362	0.0848133
$372$	0	0
$373$	10.1233	0.524162	0.262081	−	0.965046i	$-0.415591\pi$
$373$	10.1233	0.524162	0.262081	+	0.965046i	$0.415591\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−44.3434	−2.28380
$378$	0	0
$379$	−11.9216	−0.612371	−0.306186	−	0.951972i	$-0.599053\pi$
$379$	−11.9216	−0.612371	−0.306186	+	0.951972i	$0.599053\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.81681	−0.501616	−0.250808	−	0.968037i	$-0.580696\pi$
$383$	−9.81681	−0.501616	−0.250808	+	0.968037i	$0.580696\pi$
$384$	0	0
$385$	3.81681	0.194523
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.22013	0.467479	0.233740	−	0.972299i	$-0.424904\pi$
$389$	9.22013	0.467479	0.233740	+	0.972299i	$0.424904\pi$
$390$	0	0
$391$	15.8168	0.799890
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.287973	0.0144895
$396$	0	0
$397$	−22.9793	−1.15330	−0.576648	−	0.816993i	$-0.695640\pi$
$397$	−22.9793	−1.15330	−0.576648	+	0.816993i	$0.695640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.0656	−0.552589	−0.276294	−	0.961073i	$-0.589106\pi$
$401$	−11.0656	−0.552589	−0.276294	+	0.961073i	$0.589106\pi$
$402$	0	0
$403$	−32.5266	−1.62026
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.85601	0.240704
$408$	0	0
$409$	−17.6336	−0.871926	−0.435963	−	0.899964i	$-0.643592\pi$
$409$	−17.6336	−0.871926	−0.435963	+	0.899964i	$0.643592\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.63362	−0.0803852
$414$	0	0
$415$	4.28402	0.210294
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−37.0347	−1.80926	−0.904631	−	0.426195i	$-0.859854\pi$
$419$	−37.0347	−1.80926	−0.904631	+	0.426195i	$0.859854\pi$
$420$	0	0
$421$	5.05767	0.246496	0.123248	−	0.992376i	$-0.460669\pi$
$421$	5.05767	0.246496	0.123248	+	0.992376i	$0.460669\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.67282	0.129651
$426$	0	0
$427$	3.61120	0.174758
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.23030	0.251935	0.125967	−	0.992034i	$-0.459797\pi$
$431$	5.23030	0.251935	0.125967	+	0.992034i	$0.459797\pi$
$432$	0	0
$433$	34.3434	1.65044	0.825219	−	0.564813i	$-0.191052\pi$
$433$	34.3434	1.65044	0.825219	+	0.564813i	$0.191052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.6521	−1.32278
$438$	0	0
$439$	19.5473	0.932942	0.466471	−	0.884536i	$-0.345525\pi$
$439$	19.5473	0.932942	0.466471	+	0.884536i	$0.345525\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.5042	−0.499067	−0.249534	−	0.968366i	$-0.580277\pi$
$443$	−10.5042	−0.499067	−0.249534	+	0.968366i	$0.580277\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.8560	1.07864	0.539321	−	0.842100i	$-0.318681\pi$
$449$	22.8560	1.07864	0.539321	+	0.842100i	$0.318681\pi$
$450$	0	0
$451$	3.93216	0.185158
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.67282	−0.312827
$456$	0	0
$457$	−14.0264	−0.656126	−0.328063	−	0.944656i	$-0.606396\pi$
$457$	−14.0264	−0.656126	−0.328063	+	0.944656i	$0.606396\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.1025	−1.12257	−0.561283	−	0.827624i	$-0.689692\pi$
$461$	−24.1025	−1.12257	−0.561283	+	0.827624i	$0.689692\pi$
$462$	0	0
$463$	33.9401	1.57733	0.788664	−	0.614824i	$-0.210773\pi$
$463$	33.9401	1.57733	0.788664	+	0.614824i	$0.210773\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.3720	1.26663	0.633313	−	0.773896i	$-0.281695\pi$
$467$	27.3720	1.26663	0.633313	+	0.773896i	$0.281695\pi$
$468$	0	0
$469$	−9.41123	−0.434570
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.25934	0.0579043
$474$	0	0
$475$	−4.67282	−0.214404
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.23030	−0.238978	−0.119489	−	0.992835i	$-0.538126\pi$
$479$	−5.23030	−0.238978	−0.119489	+	0.992835i	$0.538126\pi$
$480$	0	0
$481$	−8.48963	−0.387094
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.83528	0.355782
$486$	0	0
$487$	24.0185	1.08838	0.544190	−	0.838962i	$-0.316837\pi$
$487$	24.0185	1.08838	0.544190	+	0.838962i	$0.316837\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.7776	−0.666904	−0.333452	−	0.942767i	$-0.608214\pi$
$491$	−14.7776	−0.666904	−0.333452	+	0.942767i	$0.608214\pi$
$492$	0	0
$493$	−25.3641	−1.14234
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.3849	−0.824673
$498$	0	0
$499$	−24.8560	−1.11271	−0.556354	−	0.830945i	$-0.687800\pi$
$499$	−24.8560	−1.11271	−0.556354	+	0.830945i	$0.687800\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.9154	−1.73515	−0.867576	−	0.497305i	$-0.834323\pi$
$503$	−38.9154	−1.73515	−0.867576	+	0.497305i	$0.834323\pi$
$504$	0	0
$505$	−4.20166	−0.186971
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.02073	0.0895674	0.0447837	−	0.998997i	$-0.485740\pi$
$509$	2.02073	0.0895674	0.0447837	+	0.998997i	$0.485740\pi$
$510$	0	0
$511$	2.44479	0.108151
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.81681	0.0800582
$516$	0	0
$517$	18.5944	0.817782
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−23.0290	−1.00892	−0.504460	−	0.863435i	$-0.668308\pi$
$521$	−23.0290	−1.00892	−0.504460	+	0.863435i	$0.668308\pi$
$522$	0	0
$523$	41.1170	1.79792	0.898961	−	0.438028i	$-0.144323\pi$
$523$	41.1170	1.79792	0.898961	+	0.438028i	$0.144323\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.6050	−0.810446
$528$	0	0
$529$	12.0185	0.522542
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.87448	−0.297767
$534$	0	0
$535$	−11.9176	−0.515245
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.2593	0.571120
$540$	0	0
$541$	5.20957	0.223977	0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957	0.223977	0.111988	+	0.993710i	$0.464278\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.6521	−0.713297
$546$	0	0
$547$	−40.0409	−1.71203	−0.856013	−	0.516955i	$-0.827065\pi$
$547$	−40.0409	−1.71203	−0.856013	+	0.516955i	$0.827065\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.3434	1.88909
$552$	0	0
$553$	−0.411227	−0.0174872
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.4033	−0.610288	−0.305144	−	0.952306i	$-0.598705\pi$
$557$	−14.4033	−0.610288	−0.305144	+	0.952306i	$0.598705\pi$
$558$	0	0
$559$	−2.20166	−0.0931203
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.3681	−1.23772	−0.618858	−	0.785503i	$-0.712405\pi$
$563$	−29.3681	−1.23772	−0.618858	+	0.785503i	$0.712405\pi$
$564$	0	0
$565$	20.1233	0.846592
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.8066	1.96224	0.981118	−	0.193409i	$-0.0619543\pi$
$569$	46.8066	1.96224	0.981118	+	0.193409i	$0.0619543\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.91764	0.246783
$576$	0	0
$577$	28.2386	1.17559	0.587794	−	0.809010i	$-0.299996\pi$
$577$	28.2386	1.17559	0.587794	+	0.809010i	$0.299996\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.11761	−0.253801
$582$	0	0
$583$	3.05767	0.126636
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.0824	−0.746339	−0.373169	−	0.927763i	$-0.621729\pi$
$587$	−18.0824	−0.746339	−0.373169	+	0.927763i	$0.621729\pi$
$588$	0	0
$589$	32.5266	1.34023
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.73840	−0.317778	−0.158889	−	0.987296i	$-0.550791\pi$
$593$	−7.73840	−0.317778	−0.158889	+	0.987296i	$0.550791\pi$
$594$	0	0
$595$	−3.81681	−0.156474
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.9216	1.14085	0.570423	−	0.821351i	$-0.306779\pi$
$599$	27.9216	1.14085	0.570423	+	0.821351i	$0.306779\pi$
$600$	0	0
$601$	38.4403	1.56801	0.784006	−	0.620754i	$-0.213173\pi$
$601$	38.4403	1.56801	0.784006	+	0.620754i	$0.213173\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.85601	−0.156769
$606$	0	0
$607$	0.639834	0.0259701	0.0129850	−	0.999916i	$-0.495867\pi$
$607$	0.639834	0.0259701	0.0129850	+	0.999916i	$0.495867\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.5081	−1.31514
$612$	0	0
$613$	42.7467	1.72652	0.863262	−	0.504757i	$-0.168418\pi$
$613$	42.7467	1.72652	0.863262	+	0.504757i	$0.168418\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.1025	0.849556	0.424778	−	0.905298i	$-0.360352\pi$
$617$	21.1025	0.849556	0.424778	+	0.905298i	$0.360352\pi$
$618$	0	0
$619$	13.6521	0.548724	0.274362	−	0.961626i	$-0.411533\pi$
$619$	13.6521	0.548724	0.274362	+	0.961626i	$0.411533\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.28402	0.171636
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.85601	−0.193622
$630$	0	0
$631$	−33.2593	−1.32403	−0.662017	−	0.749489i	$-0.730299\pi$
$631$	−33.2593	−1.32403	−0.662017	+	0.749489i	$0.730299\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.18714	−0.0867941
$636$	0	0
$637$	−23.1809	−0.918462
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.2857	1.03822	0.519112	−	0.854706i	$-0.326263\pi$
$641$	26.2857	1.03822	0.519112	+	0.854706i	$0.326263\pi$
$642$	0	0
$643$	−20.5826	−0.811697	−0.405848	−	0.913940i	$-0.633024\pi$
$643$	−20.5826	−0.811697	−0.405848	+	0.913940i	$0.633024\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.2527	−0.914159	−0.457079	−	0.889426i	$-0.651104\pi$
$647$	−23.2527	−0.914159	−0.457079	+	0.889426i	$0.651104\pi$
$648$	0	0
$649$	−3.05767	−0.120024
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.7591	0.734102	0.367051	−	0.930201i	$-0.380367\pi$
$653$	18.7591	0.734102	0.367051	+	0.930201i	$0.380367\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.280067	0.0109099	0.00545494	−	0.999985i	$-0.498264\pi$
$659$	0.280067	0.0109099	0.00545494	+	0.999985i	$0.498264\pi$
$660$	0	0
$661$	−39.7859	−1.54749	−0.773746	−	0.633496i	$-0.781619\pi$
$661$	−39.7859	−1.54749	−0.773746	+	0.633496i	$0.781619\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.67282	0.258761
$666$	0	0
$667$	−56.1562	−2.17438
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.75914	0.260934
$672$	0	0
$673$	33.5288	1.29244	0.646221	−	0.763150i	$-0.276348\pi$
$673$	33.5288	1.29244	0.646221	+	0.763150i	$0.276348\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.4874	1.05643	0.528213	−	0.849112i	$-0.322862\pi$
$677$	27.4874	1.05643	0.528213	+	0.849112i	$0.322862\pi$
$678$	0	0
$679$	−11.1888	−0.429388
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.5865	1.32342	0.661708	−	0.749762i	$-0.269832\pi$
$683$	34.5865	1.32342	0.661708	+	0.749762i	$0.269832\pi$
$684$	0	0
$685$	−10.2017	−0.389785
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.34565	−0.203653
$690$	0	0
$691$	−40.7282	−1.54938	−0.774688	−	0.632344i	$-0.782093\pi$
$691$	−40.7282	−1.54938	−0.774688	+	0.632344i	$0.782093\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−3.93216	−0.148941
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.4712	−0.735416	−0.367708	−	0.929941i	$-0.619857\pi$
$701$	−19.4712	−0.735416	−0.367708	+	0.929941i	$0.619857\pi$
$702$	0	0
$703$	8.48963	0.320193
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	15.0863	0.566578	0.283289	−	0.959035i	$-0.408574\pi$
$709$	15.0863	0.566578	0.283289	+	0.959035i	$0.408574\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−41.1915	−1.54263
$714$	0	0
$715$	−12.4896	−0.467086
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.43196	0.127990	0.0639952	−	0.997950i	$-0.479616\pi$
$719$	3.43196	0.127990	0.0639952	+	0.997950i	$0.479616\pi$
$720$	0	0
$721$	−2.59442	−0.0966211
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.48963	−0.352436
$726$	0	0
$727$	35.7714	1.32669	0.663344	−	0.748315i	$-0.269137\pi$
$727$	35.7714	1.32669	0.663344	+	0.748315i	$0.269137\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.25934	−0.0465782
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.6151	−0.648862
$738$	0	0
$739$	−6.08631	−0.223889	−0.111944	−	0.993714i	$-0.535708\pi$
$739$	−6.08631	−0.223889	−0.111944	+	0.993714i	$0.535708\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.5019	0.935574	0.467787	−	0.883841i	$-0.345051\pi$
$743$	25.5019	0.935574	0.467787	+	0.883841i	$0.345051\pi$
$744$	0	0
$745$	−20.0761	−0.735533
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.0185	0.621841
$750$	0	0
$751$	18.3928	0.671161	0.335581	−	0.942011i	$-0.391068\pi$
$751$	18.3928	0.671161	0.335581	+	0.942011i	$0.391068\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.03920	−0.110608
$756$	0	0
$757$	−41.8986	−1.52283	−0.761415	−	0.648264i	$-0.775495\pi$
$757$	−41.8986	−1.52283	−0.761415	+	0.648264i	$0.775495\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.97136	−0.288962	−0.144481	−	0.989508i	$-0.546151\pi$
$761$	−7.97136	−0.288962	−0.144481	+	0.989508i	$0.546151\pi$
$762$	0	0
$763$	23.7793	0.860868
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.34565	0.193020
$768$	0	0
$769$	6.02864	0.217398	0.108699	−	0.994075i	$-0.465331\pi$
$769$	6.02864	0.217398	0.108699	+	0.994075i	$0.465331\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.4033	1.59708	0.798538	−	0.601944i	$-0.205607\pi$
$773$	44.4033	1.59708	0.798538	+	0.601944i	$0.205607\pi$
$774$	0	0
$775$	−6.96080	−0.250039
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.87448	0.246304
$780$	0	0
$781$	−34.4112	−1.23133
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.201661	0.00719758
$786$	0	0
$787$	−16.8375	−0.600194	−0.300097	−	0.953909i	$-0.597019\pi$
$787$	−16.8375	−0.600194	−0.300097	+	0.953909i	$0.597019\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−28.7361	−1.02174
$792$	0	0
$793$	−11.8168	−0.419627
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.3720	−1.18210	−0.591049	−	0.806636i	$-0.701286\pi$
$797$	−33.3720	−1.18210	−0.591049	+	0.806636i	$0.701286\pi$
$798$	0	0
$799$	−18.5944	−0.657823
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.57595	0.161482
$804$	0	0
$805$	−8.45043	−0.297839
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.1809	1.02595	0.512973	−	0.858404i	$-0.328544\pi$
$809$	29.1809	1.02595	0.512973	+	0.858404i	$0.328544\pi$
$810$	0	0
$811$	15.5552	0.546217	0.273109	−	0.961983i	$-0.411948\pi$
$811$	15.5552	0.546217	0.273109	+	0.961983i	$0.411948\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.8168	−0.624096
$816$	0	0
$817$	2.20166	0.0770264
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.7177	−0.827752	−0.413876	−	0.910333i	$-0.635825\pi$
$821$	−23.7177	−0.827752	−0.413876	+	0.910333i	$0.635825\pi$
$822$	0	0
$823$	−19.3681	−0.675129	−0.337564	−	0.941302i	$-0.609603\pi$
$823$	−19.3681	−0.675129	−0.337564	+	0.941302i	$0.609603\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.2241	−1.81601	−0.908005	−	0.418960i	$-0.862395\pi$
$827$	−52.2241	−1.81601	−0.908005	+	0.418960i	$0.862395\pi$
$828$	0	0
$829$	−26.6442	−0.925391	−0.462695	−	0.886517i	$-0.653118\pi$
$829$	−26.6442	−0.925391	−0.462695	+	0.886517i	$0.653118\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13.2593	−0.459409
$834$	0	0
$835$	−14.1008	−0.487979
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.0392	0.864449	0.432225	−	0.901766i	$-0.357729\pi$
$839$	25.0392	0.864449	0.432225	+	0.901766i	$0.357729\pi$
$840$	0	0
$841$	61.0532	2.10528
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.83528	0.303943
$846$	0	0
$847$	5.50641	0.189203
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.7512	−0.368547
$852$	0	0
$853$	−10.8745	−0.372335	−0.186168	−	0.982518i	$-0.559607\pi$
$853$	−10.8745	−0.372335	−0.186168	+	0.982518i	$0.559607\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.5944	−0.635173	−0.317587	−	0.948229i	$-0.602872\pi$
$857$	−18.5944	−0.635173	−0.317587	+	0.948229i	$0.602872\pi$
$858$	0	0
$859$	4.66492	0.159165	0.0795825	−	0.996828i	$-0.474641\pi$
$859$	4.66492	0.159165	0.0795825	+	0.996828i	$0.474641\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.0594	0.955152	0.477576	−	0.878590i	$-0.341516\pi$
$863$	28.0594	0.955152	0.477576	+	0.878590i	$0.341516\pi$
$864$	0	0
$865$	4.36638	0.148461
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.769701	−0.0261103
$870$	0	0
$871$	30.7961	1.04349
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.42801	−0.0482754
$876$	0	0
$877$	34.6683	1.17067	0.585333	−	0.810793i	$-0.300964\pi$
$877$	34.6683	1.17067	0.585333	+	0.810793i	$0.300964\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.29854	−0.178512	−0.0892561	−	0.996009i	$-0.528449\pi$
$881$	−5.29854	−0.178512	−0.0892561	+	0.996009i	$0.528449\pi$
$882$	0	0
$883$	10.3025	0.346706	0.173353	−	0.984860i	$-0.444540\pi$
$883$	10.3025	0.346706	0.173353	+	0.984860i	$0.444540\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.4504	−1.39177	−0.695885	−	0.718154i	$-0.744988\pi$
$887$	−41.4504	−1.39177	−0.695885	+	0.718154i	$0.744988\pi$
$888$	0	0
$889$	3.12325	0.104751
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.5081	1.08784
$894$	0	0
$895$	15.1625	0.506825
$896$	0	0
$897$	0	0
$898$	0	0
$899$	66.0554	2.20307
$900$	0	0
$901$	−3.05767	−0.101866
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.20166	0.106427
$906$	0	0
$907$	16.7921	0.557573	0.278787	−	0.960353i	$-0.410068\pi$
$907$	16.7921	0.557573	0.278787	+	0.960353i	$0.410068\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.3536	−1.23758	−0.618789	−	0.785557i	$-0.712377\pi$
$911$	−37.3536	−1.23758	−0.618789	+	0.785557i	$0.712377\pi$
$912$	0	0
$913$	−11.4504	−0.378954
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.56804	−0.282942
$918$	0	0
$919$	−37.1316	−1.22486	−0.612429	−	0.790526i	$-0.709807\pi$
$919$	−37.1316	−1.22486	−0.612429	+	0.790526i	$0.709807\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	60.1602	1.98020
$924$	0	0
$925$	−1.81681	−0.0597364
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47.9955	−1.57468	−0.787340	−	0.616519i	$-0.788542\pi$
$929$	−47.9955	−1.57468	−0.787340	+	0.616519i	$0.788542\pi$
$930$	0	0
$931$	23.1809	0.759724
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.14399	−0.233633
$936$	0	0
$937$	22.0079	0.718967	0.359483	−	0.933151i	$-0.382953\pi$
$937$	22.0079	0.718967	0.359483	+	0.933151i	$0.382953\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.5843	1.32301	0.661504	−	0.749941i	$-0.269918\pi$
$941$	40.5843	1.32301	0.661504	+	0.749941i	$0.269918\pi$
$942$	0	0
$943$	−8.70581	−0.283500
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.2320	−0.982408	−0.491204	−	0.871045i	$-0.663443\pi$
$947$	−30.2320	−0.982408	−0.491204	+	0.871045i	$0.663443\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.50811	0.0812455	0.0406227	−	0.999175i	$-0.487066\pi$
$953$	2.50811	0.0812455	0.0406227	+	0.999175i	$0.487066\pi$
$954$	0	0
$955$	2.83754	0.0918207
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.5680	0.470427
$960$	0	0
$961$	17.4527	0.562990
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.7882	−0.604813
$966$	0	0
$967$	21.7529	0.699527	0.349763	−	0.936838i	$-0.386262\pi$
$967$	21.7529	0.699527	0.349763	+	0.936838i	$0.386262\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.7512	−0.730122	−0.365061	−	0.930984i	$-0.618952\pi$
$971$	−22.7512	−0.730122	−0.365061	+	0.930984i	$0.618952\pi$
$972$	0	0
$973$	11.4241	0.366238
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.2778	1.25661	0.628304	−	0.777968i	$-0.283749\pi$
$977$	39.2778	1.25661	0.628304	+	0.777968i	$0.283749\pi$
$978$	0	0
$979$	8.01847	0.256271
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.7899	−1.07773	−0.538865	−	0.842392i	$-0.681147\pi$
$983$	−33.7899	−1.07773	−0.538865	+	0.842392i	$0.681147\pi$
$984$	0	0
$985$	5.83528	0.185928
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.78817	−0.0886587
$990$	0	0
$991$	23.7983	0.755979	0.377990	−	0.925810i	$-0.376616\pi$
$991$	23.7983	0.755979	0.377990	+	0.925810i	$0.376616\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.0761	−0.414542
$996$	0	0
$997$	51.6785	1.63667	0.818337	−	0.574739i	$-0.194896\pi$
$997$	51.6785	1.63667	0.818337	+	0.574739i	$0.194896\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bv.1.2		3
3.2	odd	2		6480.2.a.bs.1.2		3
4.3	odd	2		405.2.a.j.1.2		3
9.2	odd	6		2160.2.q.k.1441.2		6
9.4	even	3		720.2.q.i.241.2		6
9.5	odd	6		2160.2.q.k.721.2		6
9.7	even	3		720.2.q.i.481.2		6
12.11	even	2		405.2.a.i.1.2		3
20.3	even	4		2025.2.b.l.649.3		6
20.7	even	4		2025.2.b.l.649.4		6
20.19	odd	2		2025.2.a.n.1.2		3
36.7	odd	6		45.2.e.b.31.2	yes	6
36.11	even	6		135.2.e.b.91.2		6
36.23	even	6		135.2.e.b.46.2		6
36.31	odd	6		45.2.e.b.16.2	✓	6
60.23	odd	4		2025.2.b.m.649.4		6
60.47	odd	4		2025.2.b.m.649.3		6
60.59	even	2		2025.2.a.o.1.2		3
180.7	even	12		225.2.k.b.49.4		12
180.23	odd	12		675.2.k.b.424.3		12
180.43	even	12		225.2.k.b.49.3		12
180.47	odd	12		675.2.k.b.199.3		12
180.59	even	6		675.2.e.b.451.2		6
180.67	even	12		225.2.k.b.124.3		12
180.79	odd	6		225.2.e.b.76.2		6
180.83	odd	12		675.2.k.b.199.4		12
180.103	even	12		225.2.k.b.124.4		12
180.119	even	6		675.2.e.b.226.2		6
180.139	odd	6		225.2.e.b.151.2		6
180.167	odd	12		675.2.k.b.424.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.2	✓	6	36.31	odd	6
45.2.e.b.31.2	yes	6	36.7	odd	6
135.2.e.b.46.2		6	36.23	even	6
135.2.e.b.91.2		6	36.11	even	6
225.2.e.b.76.2		6	180.79	odd	6
225.2.e.b.151.2		6	180.139	odd	6
225.2.k.b.49.3		12	180.43	even	12
225.2.k.b.49.4		12	180.7	even	12
225.2.k.b.124.3		12	180.67	even	12
225.2.k.b.124.4		12	180.103	even	12
405.2.a.i.1.2		3	12.11	even	2
405.2.a.j.1.2		3	4.3	odd	2
675.2.e.b.226.2		6	180.119	even	6
675.2.e.b.451.2		6	180.59	even	6
675.2.k.b.199.3		12	180.47	odd	12
675.2.k.b.199.4		12	180.83	odd	12
675.2.k.b.424.3		12	180.23	odd	12
675.2.k.b.424.4		12	180.167	odd	12
720.2.q.i.241.2		6	9.4	even	3
720.2.q.i.481.2		6	9.7	even	3
2025.2.a.n.1.2		3	20.19	odd	2
2025.2.a.o.1.2		3	60.59	even	2
2025.2.b.l.649.3		6	20.3	even	4
2025.2.b.l.649.4		6	20.7	even	4
2025.2.b.m.649.3		6	60.47	odd	4
2025.2.b.m.649.4		6	60.23	odd	4
2160.2.q.k.721.2		6	9.5	odd	6
2160.2.q.k.1441.2		6	9.2	odd	6
6480.2.a.bs.1.2		3	3.2	odd	2
6480.2.a.bv.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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.42801 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.42801 q^{7} -2.67282 q^{11} +4.67282 q^{13} +2.67282 q^{17} -4.67282 q^{19} +5.91764 q^{23} +1.00000 q^{25} -9.48963 q^{29} -6.96080 q^{31} -1.42801 q^{35} -1.81681 q^{37} -1.47116 q^{41} -0.471163 q^{43} -6.95684 q^{47} -4.96080 q^{49} -1.14399 q^{53} -2.67282 q^{55} +1.14399 q^{59} -2.52884 q^{61} +4.67282 q^{65} +6.59046 q^{67} +12.8745 q^{71} -1.71203 q^{73} +3.81681 q^{77} +0.287973 q^{79} +4.28402 q^{83} +2.67282 q^{85} -3.00000 q^{89} -6.67282 q^{91} -4.67282 q^{95} +7.83528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 5 * q^7	\( 3 q + 3 q^{5} - 5 q^{7} + 2 q^{11} + 4 q^{13} - 2 q^{17} - 4 q^{19} - 3 q^{23} + 3 q^{25} - 7 q^{29} - 8 q^{31} - 5 q^{35} + 6 q^{37} - 13 q^{41} - 10 q^{43} - 13 q^{47} - 2 q^{49} - 2 q^{53} + 2 q^{55} + 2 q^{59} + q^{61} + 4 q^{65} - 11 q^{67} + 10 q^{71} - 8 q^{73} - 2 q^{79} + 15 q^{83} - 2 q^{85} - 9 q^{89} - 10 q^{91} - 4 q^{95} - 18 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 5 * q^7 + 2 * q^11 + 4 * q^13 - 2 * q^17 - 4 * q^19 - 3 * q^23 + 3 * q^25 - 7 * q^29 - 8 * q^31 - 5 * q^35 + 6 * q^37 - 13 * q^41 - 10 * q^43 - 13 * q^47 - 2 * q^49 - 2 * q^53 + 2 * q^55 + 2 * q^59 + q^61 + 4 * q^65 - 11 * q^67 + 10 * q^71 - 8 * q^73 - 2 * q^79 + 15 * q^83 - 2 * q^85 - 9 * q^89 - 10 * q^91 - 4 * q^95 - 18 * q^97

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