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Properties

Label 720.2.a.c.1.1
Level $720$
Weight $2$
Character 720.1
Self dual yes
Analytic conductor $5.749$
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] = [720,2,Mod(1,720)] 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("720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-1,0,0,0,0,0,-4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.74922894553\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -4.00000 q^{11} -2.00000 q^{13} -2.00000 q^{17} -4.00000 q^{19} +1.00000 q^{25} +2.00000 q^{29} -10.0000 q^{37} -10.0000 q^{41} -4.00000 q^{43} +8.00000 q^{47} -7.00000 q^{49} +10.0000 q^{53} +4.00000 q^{55} -4.00000 q^{59} -2.00000 q^{61} +2.00000 q^{65} -12.0000 q^{67} -8.00000 q^{71} +10.0000 q^{73} +12.0000 q^{83} +2.00000 q^{85} +6.00000 q^{89} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −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 \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 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 720.2.a.c.1.1 1
3.2 odd 2 240.2.a.d.1.1 1
4.3 odd 2 45.2.a.a.1.1 1
5.2 odd 4 3600.2.f.e.2449.2 2
5.3 odd 4 3600.2.f.e.2449.1 2
5.4 even 2 3600.2.a.u.1.1 1
8.3 odd 2 2880.2.a.y.1.1 1
8.5 even 2 2880.2.a.bc.1.1 1
12.11 even 2 15.2.a.a.1.1 ✓ 1
15.2 even 4 1200.2.f.h.49.1 2
15.8 even 4 1200.2.f.h.49.2 2
15.14 odd 2 1200.2.a.e.1.1 1
20.3 even 4 225.2.b.b.199.1 2
20.7 even 4 225.2.b.b.199.2 2
20.19 odd 2 225.2.a.b.1.1 1
24.5 odd 2 960.2.a.a.1.1 1
24.11 even 2 960.2.a.l.1.1 1
28.27 even 2 2205.2.a.i.1.1 1
36.7 odd 6 405.2.e.c.271.1 2
36.11 even 6 405.2.e.f.271.1 2
36.23 even 6 405.2.e.f.136.1 2
36.31 odd 6 405.2.e.c.136.1 2
44.43 even 2 5445.2.a.c.1.1 1
48.5 odd 4 3840.2.k.r.1921.1 2
48.11 even 4 3840.2.k.m.1921.2 2
48.29 odd 4 3840.2.k.r.1921.2 2
48.35 even 4 3840.2.k.m.1921.1 2
52.51 odd 2 7605.2.a.g.1.1 1
60.23 odd 4 75.2.b.b.49.2 2
60.47 odd 4 75.2.b.b.49.1 2
60.59 even 2 75.2.a.b.1.1 1
84.11 even 6 735.2.i.e.226.1 2
84.23 even 6 735.2.i.e.361.1 2
84.47 odd 6 735.2.i.d.361.1 2
84.59 odd 6 735.2.i.d.226.1 2
84.83 odd 2 735.2.a.c.1.1 1
120.29 odd 2 4800.2.a.bz.1.1 1
120.53 even 4 4800.2.f.c.3649.1 2
120.59 even 2 4800.2.a.t.1.1 1
120.77 even 4 4800.2.f.c.3649.2 2
120.83 odd 4 4800.2.f.bf.3649.2 2
120.107 odd 4 4800.2.f.bf.3649.1 2
132.131 odd 2 1815.2.a.d.1.1 1
156.155 even 2 2535.2.a.j.1.1 1
204.203 even 2 4335.2.a.c.1.1 1
228.227 odd 2 5415.2.a.j.1.1 1
276.275 odd 2 7935.2.a.d.1.1 1
420.419 odd 2 3675.2.a.j.1.1 1
660.659 odd 2 9075.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 ✓ 1 12.11 even 2
45.2.a.a.1.1 1 4.3 odd 2
75.2.a.b.1.1 1 60.59 even 2
75.2.b.b.49.1 2 60.47 odd 4
75.2.b.b.49.2 2 60.23 odd 4
225.2.a.b.1.1 1 20.19 odd 2
225.2.b.b.199.1 2 20.3 even 4
225.2.b.b.199.2 2 20.7 even 4
240.2.a.d.1.1 1 3.2 odd 2
405.2.e.c.136.1 2 36.31 odd 6
405.2.e.c.271.1 2 36.7 odd 6
405.2.e.f.136.1 2 36.23 even 6
405.2.e.f.271.1 2 36.11 even 6
720.2.a.c.1.1 1 1.1 even 1 trivial
735.2.a.c.1.1 1 84.83 odd 2
735.2.i.d.226.1 2 84.59 odd 6
735.2.i.d.361.1 2 84.47 odd 6
735.2.i.e.226.1 2 84.11 even 6
735.2.i.e.361.1 2 84.23 even 6
960.2.a.a.1.1 1 24.5 odd 2
960.2.a.l.1.1 1 24.11 even 2
1200.2.a.e.1.1 1 15.14 odd 2
1200.2.f.h.49.1 2 15.2 even 4
1200.2.f.h.49.2 2 15.8 even 4
1815.2.a.d.1.1 1 132.131 odd 2
2205.2.a.i.1.1 1 28.27 even 2
2535.2.a.j.1.1 1 156.155 even 2
2880.2.a.y.1.1 1 8.3 odd 2
2880.2.a.bc.1.1 1 8.5 even 2
3600.2.a.u.1.1 1 5.4 even 2
3600.2.f.e.2449.1 2 5.3 odd 4
3600.2.f.e.2449.2 2 5.2 odd 4
3675.2.a.j.1.1 1 420.419 odd 2
3840.2.k.m.1921.1 2 48.35 even 4
3840.2.k.m.1921.2 2 48.11 even 4
3840.2.k.r.1921.1 2 48.5 odd 4
3840.2.k.r.1921.2 2 48.29 odd 4
4335.2.a.c.1.1 1 204.203 even 2
4800.2.a.t.1.1 1 120.59 even 2
4800.2.a.bz.1.1 1 120.29 odd 2
4800.2.f.c.3649.1 2 120.53 even 4
4800.2.f.c.3649.2 2 120.77 even 4
4800.2.f.bf.3649.1 2 120.107 odd 4
4800.2.f.bf.3649.2 2 120.83 odd 4
5415.2.a.j.1.1 1 228.227 odd 2
5445.2.a.c.1.1 1 44.43 even 2
7605.2.a.g.1.1 1 52.51 odd 2
7935.2.a.d.1.1 1 276.275 odd 2
9075.2.a.g.1.1 1 660.659 odd 2