Properties

Label 360.2.a.c.1.1
Level $360$
Weight $2$
Character 360.1
Self dual yes
Analytic conductor $2.875$
Analytic rank $0$
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] = [360,2,Mod(1,360)] 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("360.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-1,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 360.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.00000 q^{7} +2.00000 q^{11} +4.00000 q^{13} -2.00000 q^{17} +4.00000 q^{19} +8.00000 q^{23} +1.00000 q^{25} -10.0000 q^{29} +4.00000 q^{31} -2.00000 q^{35} -8.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} +6.00000 q^{53} -2.00000 q^{55} -14.0000 q^{59} -14.0000 q^{61} -4.00000 q^{65} -4.00000 q^{67} +12.0000 q^{71} +6.00000 q^{73} +4.00000 q^{77} -12.0000 q^{79} +4.00000 q^{83} +2.00000 q^{85} -12.0000 q^{89} +8.00000 q^{91} -4.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 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 \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 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
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 360.2.a.c.1.1 ✓ 1
3.2 odd 2 360.2.a.d.1.1 yes 1
4.3 odd 2 720.2.a.a.1.1 1
5.2 odd 4 1800.2.f.h.649.2 2
5.3 odd 4 1800.2.f.h.649.1 2
5.4 even 2 1800.2.a.i.1.1 1
8.3 odd 2 2880.2.a.w.1.1 1
8.5 even 2 2880.2.a.bd.1.1 1
9.2 odd 6 3240.2.q.d.1081.1 2
9.4 even 3 3240.2.q.n.2161.1 2
9.5 odd 6 3240.2.q.d.2161.1 2
9.7 even 3 3240.2.q.n.1081.1 2
12.11 even 2 720.2.a.i.1.1 1
15.2 even 4 1800.2.f.d.649.2 2
15.8 even 4 1800.2.f.d.649.1 2
15.14 odd 2 1800.2.a.f.1.1 1
20.3 even 4 3600.2.f.g.2449.2 2
20.7 even 4 3600.2.f.g.2449.1 2
20.19 odd 2 3600.2.a.bd.1.1 1
24.5 odd 2 2880.2.a.n.1.1 1
24.11 even 2 2880.2.a.e.1.1 1
60.23 odd 4 3600.2.f.q.2449.2 2
60.47 odd 4 3600.2.f.q.2449.1 2
60.59 even 2 3600.2.a.bh.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
360.2.a.d.1.1 yes 1 3.2 odd 2
720.2.a.a.1.1 1 4.3 odd 2
720.2.a.i.1.1 1 12.11 even 2
1800.2.a.f.1.1 1 15.14 odd 2
1800.2.a.i.1.1 1 5.4 even 2
1800.2.f.d.649.1 2 15.8 even 4
1800.2.f.d.649.2 2 15.2 even 4
1800.2.f.h.649.1 2 5.3 odd 4
1800.2.f.h.649.2 2 5.2 odd 4
2880.2.a.e.1.1 1 24.11 even 2
2880.2.a.n.1.1 1 24.5 odd 2
2880.2.a.w.1.1 1 8.3 odd 2
2880.2.a.bd.1.1 1 8.5 even 2
3240.2.q.d.1081.1 2 9.2 odd 6
3240.2.q.d.2161.1 2 9.5 odd 6
3240.2.q.n.1081.1 2 9.7 even 3
3240.2.q.n.2161.1 2 9.4 even 3
3600.2.a.bd.1.1 1 20.19 odd 2
3600.2.a.bh.1.1 1 60.59 even 2
3600.2.f.g.2449.1 2 20.7 even 4
3600.2.f.g.2449.2 2 20.3 even 4
3600.2.f.q.2449.1 2 60.47 odd 4
3600.2.f.q.2449.2 2 60.23 odd 4