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Properties

Label 567.2.a.a.1.1
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(1,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 567.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} -5.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +6.00000 q^{23} -4.00000 q^{25} +5.00000 q^{26} +1.00000 q^{28} -5.00000 q^{29} -6.00000 q^{31} -5.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} -3.00000 q^{37} +2.00000 q^{38} +3.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +2.00000 q^{44} -6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +5.00000 q^{52} -6.00000 q^{53} -2.00000 q^{55} -3.00000 q^{56} +5.00000 q^{58} +6.00000 q^{59} +7.00000 q^{61} +6.00000 q^{62} +7.00000 q^{64} -5.00000 q^{65} -2.00000 q^{67} -3.00000 q^{68} +1.00000 q^{70} -12.0000 q^{71} -15.0000 q^{73} +3.00000 q^{74} +2.00000 q^{76} +2.00000 q^{77} +14.0000 q^{79} -1.00000 q^{80} +10.0000 q^{82} +18.0000 q^{83} +3.00000 q^{85} +4.00000 q^{86} -6.00000 q^{88} -5.00000 q^{89} +5.00000 q^{91} -6.00000 q^{92} +6.00000 q^{94} -2.00000 q^{95} -18.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 5.00000 0.656532
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 − 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 10.0000 1.10432
\(83\) 18.0000 1.97576 0.987878 − 0.155230i \(-0.0496119\pi\)
0.987878 + 0.155230i \(0.0496119\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −5.00000 −0.529999 −0.264999 − 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.a.a.1.1 ✓ 1
3.2 odd 2 567.2.a.b.1.1 yes 1
4.3 odd 2 9072.2.a.q.1.1 1
7.6 odd 2 3969.2.a.b.1.1 1
9.2 odd 6 567.2.f.c.190.1 2
9.4 even 3 567.2.f.f.379.1 2
9.5 odd 6 567.2.f.c.379.1 2
9.7 even 3 567.2.f.f.190.1 2
12.11 even 2 9072.2.a.j.1.1 1
21.20 even 2 3969.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
567.2.a.b.1.1 yes 1 3.2 odd 2
567.2.f.c.190.1 2 9.2 odd 6
567.2.f.c.379.1 2 9.5 odd 6
567.2.f.f.190.1 2 9.7 even 3
567.2.f.f.379.1 2 9.4 even 3
3969.2.a.b.1.1 1 7.6 odd 2
3969.2.a.e.1.1 1 21.20 even 2
9072.2.a.j.1.1 1 12.11 even 2
9072.2.a.q.1.1 1 4.3 odd 2