Properties

Label 3920.1.bt.a.79.1
Level $3920$
Weight $1$
Character 3920.79
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
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] = [3920,1,Mod(79,3920)] 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("3920.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3920, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.bt (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,0,0,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3920.79
Dual form 3920.1.bt.a.1439.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{29} +2.00000 q^{41} +(0.500000 + 0.866025i) q^{45} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 - 0.866025i) q^{81} +(1.00000 - 1.73205i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + q^{9} - q^{25} + 4 q^{29} + 4 q^{41} + q^{45} - 2 q^{61} - q^{81} + 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 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
\(45\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(82\) 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
\(89\) 1.00000 − 1.73205i 1.00000 − 1.73205i 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(90\) 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 3920.1.bt.a.79.1 2
4.3 odd 2 CM 3920.1.bt.a.79.1 2
5.4 even 2 RM 3920.1.bt.a.79.1 2
7.2 even 3 3920.1.j.a.3039.1 1
7.3 odd 6 3920.1.bt.b.1439.1 2
7.4 even 3 inner 3920.1.bt.a.1439.1 2
7.5 odd 6 80.1.h.a.79.1 ✓ 1
7.6 odd 2 3920.1.bt.b.79.1 2
20.19 odd 2 CM 3920.1.bt.a.79.1 2
21.5 even 6 720.1.j.a.559.1 1
28.3 even 6 3920.1.bt.b.1439.1 2
28.11 odd 6 inner 3920.1.bt.a.1439.1 2
28.19 even 6 80.1.h.a.79.1 ✓ 1
28.23 odd 6 3920.1.j.a.3039.1 1
28.27 even 2 3920.1.bt.b.79.1 2
35.4 even 6 inner 3920.1.bt.a.1439.1 2
35.9 even 6 3920.1.j.a.3039.1 1
35.12 even 12 400.1.b.a.351.1 1
35.19 odd 6 80.1.h.a.79.1 ✓ 1
35.24 odd 6 3920.1.bt.b.1439.1 2
35.33 even 12 400.1.b.a.351.1 1
35.34 odd 2 3920.1.bt.b.79.1 2
56.5 odd 6 320.1.h.a.319.1 1
56.19 even 6 320.1.h.a.319.1 1
84.47 odd 6 720.1.j.a.559.1 1
105.47 odd 12 3600.1.e.a.3151.1 1
105.68 odd 12 3600.1.e.a.3151.1 1
105.89 even 6 720.1.j.a.559.1 1
112.5 odd 12 1280.1.e.a.639.1 2
112.19 even 12 1280.1.e.a.639.2 2
112.61 odd 12 1280.1.e.a.639.2 2
112.75 even 12 1280.1.e.a.639.1 2
140.19 even 6 80.1.h.a.79.1 ✓ 1
140.39 odd 6 inner 3920.1.bt.a.1439.1 2
140.47 odd 12 400.1.b.a.351.1 1
140.59 even 6 3920.1.bt.b.1439.1 2
140.79 odd 6 3920.1.j.a.3039.1 1
140.103 odd 12 400.1.b.a.351.1 1
140.139 even 2 3920.1.bt.b.79.1 2
168.5 even 6 2880.1.j.a.1279.1 1
168.131 odd 6 2880.1.j.a.1279.1 1
280.19 even 6 320.1.h.a.319.1 1
280.117 even 12 1600.1.b.a.1151.1 1
280.173 even 12 1600.1.b.a.1151.1 1
280.187 odd 12 1600.1.b.a.1151.1 1
280.229 odd 6 320.1.h.a.319.1 1
280.243 odd 12 1600.1.b.a.1151.1 1
420.47 even 12 3600.1.e.a.3151.1 1
420.299 odd 6 720.1.j.a.559.1 1
420.383 even 12 3600.1.e.a.3151.1 1
560.19 even 12 1280.1.e.a.639.2 2
560.229 odd 12 1280.1.e.a.639.1 2
560.299 even 12 1280.1.e.a.639.1 2
560.509 odd 12 1280.1.e.a.639.2 2
840.299 odd 6 2880.1.j.a.1279.1 1
840.509 even 6 2880.1.j.a.1279.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.1.h.a.79.1 ✓ 1 7.5 odd 6
80.1.h.a.79.1 ✓ 1 28.19 even 6
80.1.h.a.79.1 ✓ 1 35.19 odd 6
80.1.h.a.79.1 ✓ 1 140.19 even 6
320.1.h.a.319.1 1 56.5 odd 6
320.1.h.a.319.1 1 56.19 even 6
320.1.h.a.319.1 1 280.19 even 6
320.1.h.a.319.1 1 280.229 odd 6
400.1.b.a.351.1 1 35.12 even 12
400.1.b.a.351.1 1 35.33 even 12
400.1.b.a.351.1 1 140.47 odd 12
400.1.b.a.351.1 1 140.103 odd 12
720.1.j.a.559.1 1 21.5 even 6
720.1.j.a.559.1 1 84.47 odd 6
720.1.j.a.559.1 1 105.89 even 6
720.1.j.a.559.1 1 420.299 odd 6
1280.1.e.a.639.1 2 112.5 odd 12
1280.1.e.a.639.1 2 112.75 even 12
1280.1.e.a.639.1 2 560.229 odd 12
1280.1.e.a.639.1 2 560.299 even 12
1280.1.e.a.639.2 2 112.19 even 12
1280.1.e.a.639.2 2 112.61 odd 12
1280.1.e.a.639.2 2 560.19 even 12
1280.1.e.a.639.2 2 560.509 odd 12
1600.1.b.a.1151.1 1 280.117 even 12
1600.1.b.a.1151.1 1 280.173 even 12
1600.1.b.a.1151.1 1 280.187 odd 12
1600.1.b.a.1151.1 1 280.243 odd 12
2880.1.j.a.1279.1 1 168.5 even 6
2880.1.j.a.1279.1 1 168.131 odd 6
2880.1.j.a.1279.1 1 840.299 odd 6
2880.1.j.a.1279.1 1 840.509 even 6
3600.1.e.a.3151.1 1 105.47 odd 12
3600.1.e.a.3151.1 1 105.68 odd 12
3600.1.e.a.3151.1 1 420.47 even 12
3600.1.e.a.3151.1 1 420.383 even 12
3920.1.j.a.3039.1 1 7.2 even 3
3920.1.j.a.3039.1 1 28.23 odd 6
3920.1.j.a.3039.1 1 35.9 even 6
3920.1.j.a.3039.1 1 140.79 odd 6
3920.1.bt.a.79.1 2 1.1 even 1 trivial
3920.1.bt.a.79.1 2 4.3 odd 2 CM
3920.1.bt.a.79.1 2 5.4 even 2 RM
3920.1.bt.a.79.1 2 20.19 odd 2 CM
3920.1.bt.a.1439.1 2 7.4 even 3 inner
3920.1.bt.a.1439.1 2 28.11 odd 6 inner
3920.1.bt.a.1439.1 2 35.4 even 6 inner
3920.1.bt.a.1439.1 2 140.39 odd 6 inner
3920.1.bt.b.79.1 2 7.6 odd 2
3920.1.bt.b.79.1 2 28.27 even 2
3920.1.bt.b.79.1 2 35.34 odd 2
3920.1.bt.b.79.1 2 140.139 even 2
3920.1.bt.b.1439.1 2 7.3 odd 6
3920.1.bt.b.1439.1 2 28.3 even 6
3920.1.bt.b.1439.1 2 35.24 odd 6
3920.1.bt.b.1439.1 2 140.59 even 6