Properties

Label 720.1.bu.a.79.2
Level $720$
Weight $1$
Character 720.79
Analytic conductor $0.359$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
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] = [720,1,Mod(79,720)] 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("720.79"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 720.bu (of order \(6\), degree \(2\), 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: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.10497600.1

Embedding invariants

Embedding label 79.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 720.79
Dual form 720.1.bu.a.319.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.866025 + 1.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.866025 + 0.500000i) q^{15} +1.73205i q^{21} +(-0.866025 - 1.50000i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000i q^{27} +(0.500000 - 0.866025i) q^{29} -1.73205 q^{35} +(-0.500000 - 0.866025i) q^{41} +1.00000 q^{45} +(0.866025 - 1.50000i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(-0.500000 + 0.866025i) q^{61} +(0.866025 + 1.50000i) q^{63} +(0.866025 + 1.50000i) q^{67} +(-1.50000 - 0.866025i) q^{69} +1.00000i q^{75} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 1.50000i) q^{83} -1.00000i q^{87} -1.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} + 2 q^{9} - 2 q^{25} + 2 q^{29} - 2 q^{41} + 4 q^{45} - 4 q^{49} - 2 q^{61} - 6 q^{69} - 2 q^{81} - 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(6\) 0 0
\(7\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(10\) 0 0
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(16\) 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
\(21\) 1.73205i 1.73205i
\(22\) 0 0
\(23\) −0.866025 − 1.50000i −0.866025 − 1.50000i −0.866025 − 0.500000i \(-0.833333\pi\)
− 1.00000i \(-0.5\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) − 1.00000i − 1.00000i
\(28\) 0 0
\(29\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
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
\(35\) −1.73205 −1.73205
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(44\) 0 0
\(45\) 1.00000 1.00000
\(46\) 0 0
\(47\) 0.866025 − 1.50000i 0.866025 − 1.50000i − 1.00000i \(-0.5\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(48\) 0 0
\(49\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(50\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −1.50000 − 0.866025i −1.50000 − 0.866025i
\(70\) 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
\(75\) 1.00000i 1.00000i
\(76\) 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
\(83\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.00000i − 1.00000i
\(88\) 0 0
\(89\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 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 720.1.bu.a.79.2 yes 4
3.2 odd 2 2160.1.bu.a.559.1 4
4.3 odd 2 inner 720.1.bu.a.79.1 ✓ 4
5.2 odd 4 3600.1.cc.a.1951.1 2
5.3 odd 4 3600.1.cc.b.1951.1 2
5.4 even 2 inner 720.1.bu.a.79.1 ✓ 4
8.3 odd 2 2880.1.bu.c.2239.2 4
8.5 even 2 2880.1.bu.c.2239.1 4
9.4 even 3 inner 720.1.bu.a.319.2 yes 4
9.5 odd 6 2160.1.bu.a.1279.1 4
12.11 even 2 2160.1.bu.a.559.2 4
15.14 odd 2 2160.1.bu.a.559.2 4
20.3 even 4 3600.1.cc.a.1951.1 2
20.7 even 4 3600.1.cc.b.1951.1 2
20.19 odd 2 CM 720.1.bu.a.79.2 yes 4
36.23 even 6 2160.1.bu.a.1279.2 4
36.31 odd 6 inner 720.1.bu.a.319.1 yes 4
40.19 odd 2 2880.1.bu.c.2239.1 4
40.29 even 2 2880.1.bu.c.2239.2 4
45.4 even 6 inner 720.1.bu.a.319.1 yes 4
45.13 odd 12 3600.1.cc.a.751.1 2
45.14 odd 6 2160.1.bu.a.1279.2 4
45.22 odd 12 3600.1.cc.b.751.1 2
60.59 even 2 2160.1.bu.a.559.1 4
72.13 even 6 2880.1.bu.c.319.1 4
72.67 odd 6 2880.1.bu.c.319.2 4
180.59 even 6 2160.1.bu.a.1279.1 4
180.67 even 12 3600.1.cc.a.751.1 2
180.103 even 12 3600.1.cc.b.751.1 2
180.139 odd 6 inner 720.1.bu.a.319.2 yes 4
360.139 odd 6 2880.1.bu.c.319.1 4
360.229 even 6 2880.1.bu.c.319.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.1.bu.a.79.1 ✓ 4 4.3 odd 2 inner
720.1.bu.a.79.1 ✓ 4 5.4 even 2 inner
720.1.bu.a.79.2 yes 4 1.1 even 1 trivial
720.1.bu.a.79.2 yes 4 20.19 odd 2 CM
720.1.bu.a.319.1 yes 4 36.31 odd 6 inner
720.1.bu.a.319.1 yes 4 45.4 even 6 inner
720.1.bu.a.319.2 yes 4 9.4 even 3 inner
720.1.bu.a.319.2 yes 4 180.139 odd 6 inner
2160.1.bu.a.559.1 4 3.2 odd 2
2160.1.bu.a.559.1 4 60.59 even 2
2160.1.bu.a.559.2 4 12.11 even 2
2160.1.bu.a.559.2 4 15.14 odd 2
2160.1.bu.a.1279.1 4 9.5 odd 6
2160.1.bu.a.1279.1 4 180.59 even 6
2160.1.bu.a.1279.2 4 36.23 even 6
2160.1.bu.a.1279.2 4 45.14 odd 6
2880.1.bu.c.319.1 4 72.13 even 6
2880.1.bu.c.319.1 4 360.139 odd 6
2880.1.bu.c.319.2 4 72.67 odd 6
2880.1.bu.c.319.2 4 360.229 even 6
2880.1.bu.c.2239.1 4 8.5 even 2
2880.1.bu.c.2239.1 4 40.19 odd 2
2880.1.bu.c.2239.2 4 8.3 odd 2
2880.1.bu.c.2239.2 4 40.29 even 2
3600.1.cc.a.751.1 2 45.13 odd 12
3600.1.cc.a.751.1 2 180.67 even 12
3600.1.cc.a.1951.1 2 5.2 odd 4
3600.1.cc.a.1951.1 2 20.3 even 4
3600.1.cc.b.751.1 2 45.22 odd 12
3600.1.cc.b.751.1 2 180.103 even 12
3600.1.cc.b.1951.1 2 5.3 odd 4
3600.1.cc.b.1951.1 2 20.7 even 4