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Properties

Label 2880.1.j.a.1279.1
Level $2880$
Weight $1$
Character 2880.1279
Self dual yes
Analytic conductor $1.437$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
Inner twists $4$

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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] = [2880,1,Mod(1279,2880)] 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("2880.1279"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2880.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4$
Artin field: Galois closure of \(\Q(\sqrt{6 +3 i})\)
Stark unit: Root of $x^{4} - 2788x^{3} - 3642x^{2} - 2788x + 1$

Embedding invariants

Embedding label 1279.1
Character \(\chi\) \(=\) 2880.1279

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +1.00000 q^{25} +2.00000 q^{29} +2.00000 q^{41} -1.00000 q^{49} +2.00000 q^{61} -2.00000 q^{89} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −1.00000
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 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 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 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 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 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
\(89\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 2880.1.j.a.1279.1 1
3.2 odd 2 320.1.h.a.319.1 1
4.3 odd 2 CM 2880.1.j.a.1279.1 1
5.4 even 2 RM 2880.1.j.a.1279.1 1
8.3 odd 2 720.1.j.a.559.1 1
8.5 even 2 720.1.j.a.559.1 1
12.11 even 2 320.1.h.a.319.1 1
15.2 even 4 1600.1.b.a.1151.1 1
15.8 even 4 1600.1.b.a.1151.1 1
15.14 odd 2 320.1.h.a.319.1 1
20.19 odd 2 CM 2880.1.j.a.1279.1 1
24.5 odd 2 80.1.h.a.79.1 ✓ 1
24.11 even 2 80.1.h.a.79.1 ✓ 1
40.3 even 4 3600.1.e.a.3151.1 1
40.13 odd 4 3600.1.e.a.3151.1 1
40.19 odd 2 720.1.j.a.559.1 1
40.27 even 4 3600.1.e.a.3151.1 1
40.29 even 2 720.1.j.a.559.1 1
40.37 odd 4 3600.1.e.a.3151.1 1
48.5 odd 4 1280.1.e.a.639.2 2
48.11 even 4 1280.1.e.a.639.2 2
48.29 odd 4 1280.1.e.a.639.1 2
48.35 even 4 1280.1.e.a.639.1 2
60.23 odd 4 1600.1.b.a.1151.1 1
60.47 odd 4 1600.1.b.a.1151.1 1
60.59 even 2 320.1.h.a.319.1 1
120.29 odd 2 80.1.h.a.79.1 ✓ 1
120.53 even 4 400.1.b.a.351.1 1
120.59 even 2 80.1.h.a.79.1 ✓ 1
120.77 even 4 400.1.b.a.351.1 1
120.83 odd 4 400.1.b.a.351.1 1
120.107 odd 4 400.1.b.a.351.1 1
168.5 even 6 3920.1.bt.a.1439.1 2
168.11 even 6 3920.1.bt.b.79.1 2
168.53 odd 6 3920.1.bt.b.79.1 2
168.59 odd 6 3920.1.bt.a.79.1 2
168.83 odd 2 3920.1.j.a.3039.1 1
168.101 even 6 3920.1.bt.a.79.1 2
168.107 even 6 3920.1.bt.b.1439.1 2
168.125 even 2 3920.1.j.a.3039.1 1
168.131 odd 6 3920.1.bt.a.1439.1 2
168.149 odd 6 3920.1.bt.b.1439.1 2
240.29 odd 4 1280.1.e.a.639.1 2
240.59 even 4 1280.1.e.a.639.2 2
240.149 odd 4 1280.1.e.a.639.2 2
240.179 even 4 1280.1.e.a.639.1 2
840.59 odd 6 3920.1.bt.a.79.1 2
840.149 odd 6 3920.1.bt.b.1439.1 2
840.179 even 6 3920.1.bt.b.79.1 2
840.269 even 6 3920.1.bt.a.79.1 2
840.299 odd 6 3920.1.bt.a.1439.1 2
840.389 odd 6 3920.1.bt.b.79.1 2
840.419 odd 2 3920.1.j.a.3039.1 1
840.509 even 6 3920.1.bt.a.1439.1 2
840.629 even 2 3920.1.j.a.3039.1 1
840.779 even 6 3920.1.bt.b.1439.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.1.h.a.79.1 ✓ 1 24.5 odd 2
80.1.h.a.79.1 ✓ 1 24.11 even 2
80.1.h.a.79.1 ✓ 1 120.29 odd 2
80.1.h.a.79.1 ✓ 1 120.59 even 2
320.1.h.a.319.1 1 3.2 odd 2
320.1.h.a.319.1 1 12.11 even 2
320.1.h.a.319.1 1 15.14 odd 2
320.1.h.a.319.1 1 60.59 even 2
400.1.b.a.351.1 1 120.53 even 4
400.1.b.a.351.1 1 120.77 even 4
400.1.b.a.351.1 1 120.83 odd 4
400.1.b.a.351.1 1 120.107 odd 4
720.1.j.a.559.1 1 8.3 odd 2
720.1.j.a.559.1 1 8.5 even 2
720.1.j.a.559.1 1 40.19 odd 2
720.1.j.a.559.1 1 40.29 even 2
1280.1.e.a.639.1 2 48.29 odd 4
1280.1.e.a.639.1 2 48.35 even 4
1280.1.e.a.639.1 2 240.29 odd 4
1280.1.e.a.639.1 2 240.179 even 4
1280.1.e.a.639.2 2 48.5 odd 4
1280.1.e.a.639.2 2 48.11 even 4
1280.1.e.a.639.2 2 240.59 even 4
1280.1.e.a.639.2 2 240.149 odd 4
1600.1.b.a.1151.1 1 15.2 even 4
1600.1.b.a.1151.1 1 15.8 even 4
1600.1.b.a.1151.1 1 60.23 odd 4
1600.1.b.a.1151.1 1 60.47 odd 4
2880.1.j.a.1279.1 1 1.1 even 1 trivial
2880.1.j.a.1279.1 1 4.3 odd 2 CM
2880.1.j.a.1279.1 1 5.4 even 2 RM
2880.1.j.a.1279.1 1 20.19 odd 2 CM
3600.1.e.a.3151.1 1 40.3 even 4
3600.1.e.a.3151.1 1 40.13 odd 4
3600.1.e.a.3151.1 1 40.27 even 4
3600.1.e.a.3151.1 1 40.37 odd 4
3920.1.j.a.3039.1 1 168.83 odd 2
3920.1.j.a.3039.1 1 168.125 even 2
3920.1.j.a.3039.1 1 840.419 odd 2
3920.1.j.a.3039.1 1 840.629 even 2
3920.1.bt.a.79.1 2 168.59 odd 6
3920.1.bt.a.79.1 2 168.101 even 6
3920.1.bt.a.79.1 2 840.59 odd 6
3920.1.bt.a.79.1 2 840.269 even 6
3920.1.bt.a.1439.1 2 168.5 even 6
3920.1.bt.a.1439.1 2 168.131 odd 6
3920.1.bt.a.1439.1 2 840.299 odd 6
3920.1.bt.a.1439.1 2 840.509 even 6
3920.1.bt.b.79.1 2 168.11 even 6
3920.1.bt.b.79.1 2 168.53 odd 6
3920.1.bt.b.79.1 2 840.179 even 6
3920.1.bt.b.79.1 2 840.389 odd 6
3920.1.bt.b.1439.1 2 168.107 even 6
3920.1.bt.b.1439.1 2 168.149 odd 6
3920.1.bt.b.1439.1 2 840.149 odd 6
3920.1.bt.b.1439.1 2 840.779 even 6