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Properties

Label 120.1.i.a.29.1
Level $120$
Weight $1$
Character 120.29
Analytic conductor $0.060$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -15, -24, 40
Inner twists $8$

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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] = [120,1,Mod(29,120)] 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("120.29"); S:= CuspForms(chi, 1); 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, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 120.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.0598878015160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.3240000.1

Embedding invariants

Embedding label 29.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 120.29
Dual form 120.1.i.a.29.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{10} +1.00000i q^{12} +1.00000 q^{15} +1.00000 q^{16} +1.00000i q^{18} -1.00000i q^{20} +1.00000 q^{24} -1.00000 q^{25} +1.00000i q^{27} -1.00000i q^{30} -2.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} -1.00000 q^{40} -1.00000i q^{45} -1.00000i q^{48} +1.00000 q^{49} +1.00000i q^{50} -2.00000i q^{53} +1.00000 q^{54} -1.00000 q^{60} +2.00000i q^{62} -1.00000 q^{64} -1.00000i q^{72} +1.00000i q^{75} +2.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +2.00000i q^{83} -1.00000 q^{90} +2.00000i q^{93} -1.00000 q^{96} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{10} + 2 q^{15} + 2 q^{16} + 2 q^{24} - 2 q^{25} - 4 q^{31} + 2 q^{36} - 2 q^{40} + 2 q^{49} + 2 q^{54} - 2 q^{60} - 2 q^{64} + 4 q^{79} + 2 q^{81} - 2 q^{90} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 1.00000i
\(3\) − 1.00000i − 1.00000i
\(4\) −1.00000 −1.00000
\(5\) 1.00000i 1.00000i
\(6\) −1.00000 −1.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 1.00000i
\(9\) −1.00000 −1.00000
\(10\) 1.00000 1.00000
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 1.00000i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 1.00000
\(16\) 1.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 1.00000i − 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 1.00000
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000i 1.00000i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) − 1.00000i − 1.00000i
\(31\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −1.00000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 1.00000i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 1.00000i − 1.00000i
\(49\) 1.00000 1.00000
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −1.00000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.00000i 2.00000i
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 1.00000i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(80\) 1.00000i 1.00000i
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −1.00000 −1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) 2.00000i 2.00000i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 1.00000i − 1.00000i
\(99\) 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 120.1.i.a.29.1 ✓ 2
3.2 odd 2 inner 120.1.i.a.29.2 yes 2
4.3 odd 2 480.1.i.a.209.2 2
5.2 odd 4 600.1.n.b.101.1 1
5.3 odd 4 600.1.n.a.101.1 1
5.4 even 2 inner 120.1.i.a.29.2 yes 2
8.3 odd 2 480.1.i.a.209.1 2
8.5 even 2 inner 120.1.i.a.29.2 yes 2
9.2 odd 6 3240.1.bh.h.1349.2 4
9.4 even 3 3240.1.bh.h.269.2 4
9.5 odd 6 3240.1.bh.h.269.1 4
9.7 even 3 3240.1.bh.h.1349.1 4
12.11 even 2 480.1.i.a.209.1 2
15.2 even 4 600.1.n.a.101.1 1
15.8 even 4 600.1.n.b.101.1 1
15.14 odd 2 CM 120.1.i.a.29.1 ✓ 2
16.3 odd 4 3840.1.c.b.3329.1 1
16.5 even 4 3840.1.c.a.3329.1 1
16.11 odd 4 3840.1.c.c.3329.1 1
16.13 even 4 3840.1.c.d.3329.1 1
20.3 even 4 2400.1.n.a.401.1 1
20.7 even 4 2400.1.n.b.401.1 1
20.19 odd 2 480.1.i.a.209.1 2
24.5 odd 2 CM 120.1.i.a.29.1 ✓ 2
24.11 even 2 480.1.i.a.209.2 2
40.3 even 4 2400.1.n.b.401.1 1
40.13 odd 4 600.1.n.b.101.1 1
40.19 odd 2 480.1.i.a.209.2 2
40.27 even 4 2400.1.n.a.401.1 1
40.29 even 2 RM 120.1.i.a.29.1 ✓ 2
40.37 odd 4 600.1.n.a.101.1 1
45.4 even 6 3240.1.bh.h.269.1 4
45.14 odd 6 3240.1.bh.h.269.2 4
45.29 odd 6 3240.1.bh.h.1349.1 4
45.34 even 6 3240.1.bh.h.1349.2 4
48.5 odd 4 3840.1.c.d.3329.1 1
48.11 even 4 3840.1.c.b.3329.1 1
48.29 odd 4 3840.1.c.a.3329.1 1
48.35 even 4 3840.1.c.c.3329.1 1
60.23 odd 4 2400.1.n.b.401.1 1
60.47 odd 4 2400.1.n.a.401.1 1
60.59 even 2 480.1.i.a.209.2 2
72.5 odd 6 3240.1.bh.h.269.2 4
72.13 even 6 3240.1.bh.h.269.1 4
72.29 odd 6 3240.1.bh.h.1349.1 4
72.61 even 6 3240.1.bh.h.1349.2 4
80.19 odd 4 3840.1.c.c.3329.1 1
80.29 even 4 3840.1.c.a.3329.1 1
80.59 odd 4 3840.1.c.b.3329.1 1
80.69 even 4 3840.1.c.d.3329.1 1
120.29 odd 2 inner 120.1.i.a.29.2 yes 2
120.53 even 4 600.1.n.a.101.1 1
120.59 even 2 480.1.i.a.209.1 2
120.77 even 4 600.1.n.b.101.1 1
120.83 odd 4 2400.1.n.a.401.1 1
120.107 odd 4 2400.1.n.b.401.1 1
240.29 odd 4 3840.1.c.d.3329.1 1
240.59 even 4 3840.1.c.c.3329.1 1
240.149 odd 4 3840.1.c.a.3329.1 1
240.179 even 4 3840.1.c.b.3329.1 1
360.29 odd 6 3240.1.bh.h.1349.2 4
360.149 odd 6 3240.1.bh.h.269.1 4
360.229 even 6 3240.1.bh.h.269.2 4
360.349 even 6 3240.1.bh.h.1349.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.1.i.a.29.1 ✓ 2 1.1 even 1 trivial
120.1.i.a.29.1 ✓ 2 15.14 odd 2 CM
120.1.i.a.29.1 ✓ 2 24.5 odd 2 CM
120.1.i.a.29.1 ✓ 2 40.29 even 2 RM
120.1.i.a.29.2 yes 2 3.2 odd 2 inner
120.1.i.a.29.2 yes 2 5.4 even 2 inner
120.1.i.a.29.2 yes 2 8.5 even 2 inner
120.1.i.a.29.2 yes 2 120.29 odd 2 inner
480.1.i.a.209.1 2 8.3 odd 2
480.1.i.a.209.1 2 12.11 even 2
480.1.i.a.209.1 2 20.19 odd 2
480.1.i.a.209.1 2 120.59 even 2
480.1.i.a.209.2 2 4.3 odd 2
480.1.i.a.209.2 2 24.11 even 2
480.1.i.a.209.2 2 40.19 odd 2
480.1.i.a.209.2 2 60.59 even 2
600.1.n.a.101.1 1 5.3 odd 4
600.1.n.a.101.1 1 15.2 even 4
600.1.n.a.101.1 1 40.37 odd 4
600.1.n.a.101.1 1 120.53 even 4
600.1.n.b.101.1 1 5.2 odd 4
600.1.n.b.101.1 1 15.8 even 4
600.1.n.b.101.1 1 40.13 odd 4
600.1.n.b.101.1 1 120.77 even 4
2400.1.n.a.401.1 1 20.3 even 4
2400.1.n.a.401.1 1 40.27 even 4
2400.1.n.a.401.1 1 60.47 odd 4
2400.1.n.a.401.1 1 120.83 odd 4
2400.1.n.b.401.1 1 20.7 even 4
2400.1.n.b.401.1 1 40.3 even 4
2400.1.n.b.401.1 1 60.23 odd 4
2400.1.n.b.401.1 1 120.107 odd 4
3240.1.bh.h.269.1 4 9.5 odd 6
3240.1.bh.h.269.1 4 45.4 even 6
3240.1.bh.h.269.1 4 72.13 even 6
3240.1.bh.h.269.1 4 360.149 odd 6
3240.1.bh.h.269.2 4 9.4 even 3
3240.1.bh.h.269.2 4 45.14 odd 6
3240.1.bh.h.269.2 4 72.5 odd 6
3240.1.bh.h.269.2 4 360.229 even 6
3240.1.bh.h.1349.1 4 9.7 even 3
3240.1.bh.h.1349.1 4 45.29 odd 6
3240.1.bh.h.1349.1 4 72.29 odd 6
3240.1.bh.h.1349.1 4 360.349 even 6
3240.1.bh.h.1349.2 4 9.2 odd 6
3240.1.bh.h.1349.2 4 45.34 even 6
3240.1.bh.h.1349.2 4 72.61 even 6
3240.1.bh.h.1349.2 4 360.29 odd 6
3840.1.c.a.3329.1 1 16.5 even 4
3840.1.c.a.3329.1 1 48.29 odd 4
3840.1.c.a.3329.1 1 80.29 even 4
3840.1.c.a.3329.1 1 240.149 odd 4
3840.1.c.b.3329.1 1 16.3 odd 4
3840.1.c.b.3329.1 1 48.11 even 4
3840.1.c.b.3329.1 1 80.59 odd 4
3840.1.c.b.3329.1 1 240.179 even 4
3840.1.c.c.3329.1 1 16.11 odd 4
3840.1.c.c.3329.1 1 48.35 even 4
3840.1.c.c.3329.1 1 80.19 odd 4
3840.1.c.c.3329.1 1 240.59 even 4
3840.1.c.d.3329.1 1 16.13 even 4
3840.1.c.d.3329.1 1 48.5 odd 4
3840.1.c.d.3329.1 1 80.69 even 4
3840.1.c.d.3329.1 1 240.29 odd 4