Properties

Label 120.2.a.a.1.1
Level $120$
Weight $2$
Character 120.1
Self dual yes
Analytic conductor $0.958$
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] = [120,2,Mod(1,120)] 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("120.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,-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: \(0.958204824255\)
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 \(\chi\) \(=\) 120.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -6.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} +4.00000 q^{19} +4.00000 q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -4.00000 q^{35} -6.00000 q^{37} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +9.00000 q^{49} -2.00000 q^{51} +10.0000 q^{53} +4.00000 q^{57} +6.00000 q^{61} +4.00000 q^{63} +6.00000 q^{65} -4.00000 q^{67} -8.00000 q^{69} -14.0000 q^{73} +1.00000 q^{75} +16.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +2.00000 q^{85} -6.00000 q^{87} +2.00000 q^{89} -24.0000 q^{91} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 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
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 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
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 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 120.2.a.a.1.1 ✓ 1
3.2 odd 2 360.2.a.e.1.1 1
4.3 odd 2 240.2.a.a.1.1 1
5.2 odd 4 600.2.f.c.49.1 2
5.3 odd 4 600.2.f.c.49.2 2
5.4 even 2 600.2.a.a.1.1 1
7.6 odd 2 5880.2.a.p.1.1 1
8.3 odd 2 960.2.a.n.1.1 1
8.5 even 2 960.2.a.g.1.1 1
9.2 odd 6 3240.2.q.a.1081.1 2
9.4 even 3 3240.2.q.m.2161.1 2
9.5 odd 6 3240.2.q.a.2161.1 2
9.7 even 3 3240.2.q.m.1081.1 2
12.11 even 2 720.2.a.f.1.1 1
15.2 even 4 1800.2.f.g.649.2 2
15.8 even 4 1800.2.f.g.649.1 2
15.14 odd 2 1800.2.a.c.1.1 1
16.3 odd 4 3840.2.k.z.1921.1 2
16.5 even 4 3840.2.k.a.1921.1 2
16.11 odd 4 3840.2.k.z.1921.2 2
16.13 even 4 3840.2.k.a.1921.2 2
20.3 even 4 1200.2.f.f.49.1 2
20.7 even 4 1200.2.f.f.49.2 2
20.19 odd 2 1200.2.a.r.1.1 1
24.5 odd 2 2880.2.a.r.1.1 1
24.11 even 2 2880.2.a.b.1.1 1
40.3 even 4 4800.2.f.n.3649.2 2
40.13 odd 4 4800.2.f.u.3649.1 2
40.19 odd 2 4800.2.a.bh.1.1 1
40.27 even 4 4800.2.f.n.3649.1 2
40.29 even 2 4800.2.a.bl.1.1 1
40.37 odd 4 4800.2.f.u.3649.2 2
60.23 odd 4 3600.2.f.l.2449.2 2
60.47 odd 4 3600.2.f.l.2449.1 2
60.59 even 2 3600.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
240.2.a.a.1.1 1 4.3 odd 2
360.2.a.e.1.1 1 3.2 odd 2
600.2.a.a.1.1 1 5.4 even 2
600.2.f.c.49.1 2 5.2 odd 4
600.2.f.c.49.2 2 5.3 odd 4
720.2.a.f.1.1 1 12.11 even 2
960.2.a.g.1.1 1 8.5 even 2
960.2.a.n.1.1 1 8.3 odd 2
1200.2.a.r.1.1 1 20.19 odd 2
1200.2.f.f.49.1 2 20.3 even 4
1200.2.f.f.49.2 2 20.7 even 4
1800.2.a.c.1.1 1 15.14 odd 2
1800.2.f.g.649.1 2 15.8 even 4
1800.2.f.g.649.2 2 15.2 even 4
2880.2.a.b.1.1 1 24.11 even 2
2880.2.a.r.1.1 1 24.5 odd 2
3240.2.q.a.1081.1 2 9.2 odd 6
3240.2.q.a.2161.1 2 9.5 odd 6
3240.2.q.m.1081.1 2 9.7 even 3
3240.2.q.m.2161.1 2 9.4 even 3
3600.2.a.bo.1.1 1 60.59 even 2
3600.2.f.l.2449.1 2 60.47 odd 4
3600.2.f.l.2449.2 2 60.23 odd 4
3840.2.k.a.1921.1 2 16.5 even 4
3840.2.k.a.1921.2 2 16.13 even 4
3840.2.k.z.1921.1 2 16.3 odd 4
3840.2.k.z.1921.2 2 16.11 odd 4
4800.2.a.bh.1.1 1 40.19 odd 2
4800.2.a.bl.1.1 1 40.29 even 2
4800.2.f.n.3649.1 2 40.27 even 4
4800.2.f.n.3649.2 2 40.3 even 4
4800.2.f.u.3649.1 2 40.13 odd 4
4800.2.f.u.3649.2 2 40.37 odd 4
5880.2.a.p.1.1 1 7.6 odd 2