Properties

Label 1620.1.p.d.919.2
Level $1620$
Weight $1$
Character 1620.919
Analytic conductor $0.808$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -15
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] = [1620,1,Mod(379,1620)] 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("1620.379"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,0,0,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.808485320465\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1166400.1

Embedding invariants

Embedding label 919.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1620.919
Dual form 1620.1.p.d.379.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(0.866025 + 0.500000i) q^{5} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{17} -1.73205i q^{19} +1.00000i q^{20} +(-0.866025 + 1.50000i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-1.50000 - 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(0.500000 - 0.866025i) q^{34} +(0.866025 - 1.50000i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(-1.50000 + 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{49} +1.00000i q^{50} +1.00000i q^{53} +(-0.500000 - 0.866025i) q^{61} +(-0.866025 - 1.50000i) q^{62} -1.00000 q^{64} +(0.866025 - 0.500000i) q^{68} +(1.50000 - 0.866025i) q^{76} +(1.50000 - 0.866025i) q^{79} +(-0.866025 + 0.500000i) q^{80} +(0.866025 + 1.50000i) q^{83} +(0.500000 - 0.866025i) q^{85} -1.73205 q^{92} +(0.866025 - 1.50000i) q^{95} +(0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{10} - 2 q^{16} + 2 q^{25} - 6 q^{31} + 2 q^{34} - 2 q^{40} - 6 q^{46} + 2 q^{49} - 2 q^{61} - 4 q^{64} + 6 q^{76} + 6 q^{79} + 2 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(6\) 0 0
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0 0
\(10\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) − 1.00000i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(18\) 0 0
\(19\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(20\) 1.00000i 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) −1.50000 − 0.866025i −1.50000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(32\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(33\) 0 0
\(34\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0.866025 − 1.50000i 0.866025 − 1.50000i
\(39\) 0 0
\(40\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(62\) −0.866025 − 1.50000i −0.866025 − 1.50000i
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(69\) 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
\(76\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.50000 − 0.866025i 1.50000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(80\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(81\) 0 0
\(82\) 0 0
\(83\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.73205 −1.73205
\(93\) 0 0
\(94\) 0 0
\(95\) 0.866025 − 1.50000i 0.866025 − 1.50000i
\(96\) 0 0
\(97\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(99\) 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 1620.1.p.d.919.2 4
3.2 odd 2 inner 1620.1.p.d.919.1 4
4.3 odd 2 1620.1.p.a.919.1 4
5.4 even 2 inner 1620.1.p.d.919.1 4
9.2 odd 6 1620.1.p.a.379.2 4
9.4 even 3 540.1.f.a.379.2 yes 4
9.5 odd 6 540.1.f.a.379.3 yes 4
9.7 even 3 1620.1.p.a.379.1 4
12.11 even 2 1620.1.p.a.919.2 4
15.14 odd 2 CM 1620.1.p.d.919.2 4
20.19 odd 2 1620.1.p.a.919.2 4
36.7 odd 6 inner 1620.1.p.d.379.1 4
36.11 even 6 inner 1620.1.p.d.379.2 4
36.23 even 6 540.1.f.a.379.1 ✓ 4
36.31 odd 6 540.1.f.a.379.4 yes 4
45.4 even 6 540.1.f.a.379.3 yes 4
45.13 odd 12 2700.1.c.b.1351.2 2
45.14 odd 6 540.1.f.a.379.2 yes 4
45.22 odd 12 2700.1.c.a.1351.1 2
45.23 even 12 2700.1.c.a.1351.1 2
45.29 odd 6 1620.1.p.a.379.1 4
45.32 even 12 2700.1.c.b.1351.2 2
45.34 even 6 1620.1.p.a.379.2 4
60.59 even 2 1620.1.p.a.919.1 4
180.23 odd 12 2700.1.c.a.1351.2 2
180.59 even 6 540.1.f.a.379.4 yes 4
180.67 even 12 2700.1.c.a.1351.2 2
180.79 odd 6 inner 1620.1.p.d.379.2 4
180.103 even 12 2700.1.c.b.1351.1 2
180.119 even 6 inner 1620.1.p.d.379.1 4
180.139 odd 6 540.1.f.a.379.1 ✓ 4
180.167 odd 12 2700.1.c.b.1351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
540.1.f.a.379.1 ✓ 4 36.23 even 6
540.1.f.a.379.1 ✓ 4 180.139 odd 6
540.1.f.a.379.2 yes 4 9.4 even 3
540.1.f.a.379.2 yes 4 45.14 odd 6
540.1.f.a.379.3 yes 4 9.5 odd 6
540.1.f.a.379.3 yes 4 45.4 even 6
540.1.f.a.379.4 yes 4 36.31 odd 6
540.1.f.a.379.4 yes 4 180.59 even 6
1620.1.p.a.379.1 4 9.7 even 3
1620.1.p.a.379.1 4 45.29 odd 6
1620.1.p.a.379.2 4 9.2 odd 6
1620.1.p.a.379.2 4 45.34 even 6
1620.1.p.a.919.1 4 4.3 odd 2
1620.1.p.a.919.1 4 60.59 even 2
1620.1.p.a.919.2 4 12.11 even 2
1620.1.p.a.919.2 4 20.19 odd 2
1620.1.p.d.379.1 4 36.7 odd 6 inner
1620.1.p.d.379.1 4 180.119 even 6 inner
1620.1.p.d.379.2 4 36.11 even 6 inner
1620.1.p.d.379.2 4 180.79 odd 6 inner
1620.1.p.d.919.1 4 3.2 odd 2 inner
1620.1.p.d.919.1 4 5.4 even 2 inner
1620.1.p.d.919.2 4 1.1 even 1 trivial
1620.1.p.d.919.2 4 15.14 odd 2 CM
2700.1.c.a.1351.1 2 45.22 odd 12
2700.1.c.a.1351.1 2 45.23 even 12
2700.1.c.a.1351.2 2 180.23 odd 12
2700.1.c.a.1351.2 2 180.67 even 12
2700.1.c.b.1351.1 2 180.103 even 12
2700.1.c.b.1351.1 2 180.167 odd 12
2700.1.c.b.1351.2 2 45.13 odd 12
2700.1.c.b.1351.2 2 45.32 even 12