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Properties

Label 2280.1.el.a
Level $2280$
Weight $1$
Character orbit 2280.el
Analytic conductor $1.138$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2280,1,Mod(149,2280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2280, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 9, 9, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2280.149");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2280.el (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.13786822880\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{36}^{7} q^{2} + \zeta_{36}^{17} q^{3} + \zeta_{36}^{14} q^{4} - \zeta_{36}^{5} q^{5} + \zeta_{36}^{6} q^{6} + \zeta_{36}^{3} q^{8} - \zeta_{36}^{16} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{7} q^{2} + \zeta_{36}^{17} q^{3} + \zeta_{36}^{14} q^{4} - \zeta_{36}^{5} q^{5} + \zeta_{36}^{6} q^{6} + \zeta_{36}^{3} q^{8} - \zeta_{36}^{16} q^{9} + \zeta_{36}^{12} q^{10} - \zeta_{36}^{13} q^{12} + \zeta_{36}^{4} q^{15} - \zeta_{36}^{10} q^{16} + (\zeta_{36}^{13} - \zeta_{36}^{9}) q^{17} - \zeta_{36}^{5} q^{18} + \zeta_{36}^{8} q^{19} + \zeta_{36} q^{20} + (\zeta_{36}^{7} + \zeta_{36}) q^{23} - \zeta_{36}^{2} q^{24} + \zeta_{36}^{10} q^{25} + \zeta_{36}^{15} q^{27} - \zeta_{36}^{11} q^{30} + ( - \zeta_{36}^{4} + \zeta_{36}^{2}) q^{31} + \zeta_{36}^{17} q^{32} + (\zeta_{36}^{16} + \zeta_{36}^{2}) q^{34} + \zeta_{36}^{12} q^{36} - \zeta_{36}^{15} q^{38} - \zeta_{36}^{8} q^{40} - \zeta_{36}^{3} q^{45} + ( - \zeta_{36}^{14} - \zeta_{36}^{8}) q^{46} + ( - \zeta_{36}^{11} + \zeta_{36}^{3}) q^{47} + \zeta_{36}^{9} q^{48} - \zeta_{36}^{6} q^{49} - \zeta_{36}^{17} q^{50} + ( - \zeta_{36}^{12} + \zeta_{36}^{8}) q^{51} + ( - \zeta_{36}^{15} - \zeta_{36}^{11}) q^{53} + \zeta_{36}^{4} q^{54} - \zeta_{36}^{7} q^{57} - q^{60} + ( - \zeta_{36}^{16} - \zeta_{36}^{10}) q^{61} + (\zeta_{36}^{11} - \zeta_{36}^{9}) q^{62} + \zeta_{36}^{6} q^{64} + ( - \zeta_{36}^{9} + \zeta_{36}^{5}) q^{68} + ( - \zeta_{36}^{6} - 1) q^{69} + \zeta_{36} q^{72} - \zeta_{36}^{9} q^{75} - \zeta_{36}^{4} q^{76} - \zeta_{36}^{8} q^{79} + \zeta_{36}^{15} q^{80} - \zeta_{36}^{14} q^{81} + ( - \zeta_{36}^{13} + \zeta_{36}^{11}) q^{83} + (\zeta_{36}^{14} + 1) q^{85} + \zeta_{36}^{10} q^{90} + (\zeta_{36}^{15} - \zeta_{36}^{3}) q^{92} + (\zeta_{36}^{3} - \zeta_{36}) q^{93} + ( - \zeta_{36}^{10} - 1) q^{94} - \zeta_{36}^{13} q^{95} - \zeta_{36}^{16} q^{96} + \zeta_{36}^{13} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{6} - 6 q^{10} - 6 q^{36} - 6 q^{49} + 6 q^{51} - 12 q^{60} + 6 q^{64} - 18 q^{69} + 12 q^{85} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times\).

\(n\) \(457\) \(761\) \(1141\) \(1711\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\) \(\zeta_{36}^{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.342020 + 0.939693i
0.342020 − 0.939693i
0.642788 − 0.766044i
−0.642788 + 0.766044i
−0.342020 − 0.939693i
0.342020 + 0.939693i
0.984808 − 0.173648i
−0.984808 + 0.173648i
0.642788 + 0.766044i
−0.642788 − 0.766044i
0.984808 + 0.173648i
−0.984808 − 0.173648i
−0.642788 − 0.766044i 0.342020 + 0.939693i −0.173648 + 0.984808i 0.984808 + 0.173648i 0.500000 − 0.866025i 0 0.866025 − 0.500000i −0.766044 + 0.642788i −0.500000 − 0.866025i
149.2 0.642788 + 0.766044i −0.342020 − 0.939693i −0.173648 + 0.984808i −0.984808 − 0.173648i 0.500000 − 0.866025i 0 −0.866025 + 0.500000i −0.766044 + 0.642788i −0.500000 − 0.866025i
389.1 −0.984808 − 0.173648i −0.642788 − 0.766044i 0.939693 + 0.342020i 0.342020 − 0.939693i 0.500000 + 0.866025i 0 −0.866025 − 0.500000i −0.173648 + 0.984808i −0.500000 + 0.866025i
389.2 0.984808 + 0.173648i 0.642788 + 0.766044i 0.939693 + 0.342020i −0.342020 + 0.939693i 0.500000 + 0.866025i 0 0.866025 + 0.500000i −0.173648 + 0.984808i −0.500000 + 0.866025i
1469.1 −0.642788 + 0.766044i 0.342020 − 0.939693i −0.173648 − 0.984808i 0.984808 − 0.173648i 0.500000 + 0.866025i 0 0.866025 + 0.500000i −0.766044 − 0.642788i −0.500000 + 0.866025i
1469.2 0.642788 − 0.766044i −0.342020 + 0.939693i −0.173648 − 0.984808i −0.984808 + 0.173648i 0.500000 + 0.866025i 0 −0.866025 − 0.500000i −0.766044 − 0.642788i −0.500000 + 0.866025i
1829.1 −0.342020 + 0.939693i −0.984808 − 0.173648i −0.766044 − 0.642788i −0.642788 + 0.766044i 0.500000 − 0.866025i 0 0.866025 − 0.500000i 0.939693 + 0.342020i −0.500000 − 0.866025i
1829.2 0.342020 − 0.939693i 0.984808 + 0.173648i −0.766044 − 0.642788i 0.642788 − 0.766044i 0.500000 − 0.866025i 0 −0.866025 + 0.500000i 0.939693 + 0.342020i −0.500000 − 0.866025i
2069.1 −0.984808 + 0.173648i −0.642788 + 0.766044i 0.939693 − 0.342020i 0.342020 + 0.939693i 0.500000 − 0.866025i 0 −0.866025 + 0.500000i −0.173648 − 0.984808i −0.500000 − 0.866025i
2069.2 0.984808 − 0.173648i 0.642788 − 0.766044i 0.939693 − 0.342020i −0.342020 − 0.939693i 0.500000 − 0.866025i 0 0.866025 − 0.500000i −0.173648 − 0.984808i −0.500000 − 0.866025i
2189.1 −0.342020 − 0.939693i −0.984808 + 0.173648i −0.766044 + 0.642788i −0.642788 − 0.766044i 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.939693 − 0.342020i −0.500000 + 0.866025i
2189.2 0.342020 + 0.939693i 0.984808 − 0.173648i −0.766044 + 0.642788i 0.642788 + 0.766044i 0.500000 + 0.866025i 0 −0.866025 − 0.500000i 0.939693 − 0.342020i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
152.t even 18 1 inner
456.bh odd 18 1 inner
760.cj even 18 1 inner
2280.el odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2280.1.el.a ✓ 12
3.b odd 2 1 inner 2280.1.el.a ✓ 12
5.b even 2 1 inner 2280.1.el.a ✓ 12
8.b even 2 1 2280.1.el.b yes 12
15.d odd 2 1 CM 2280.1.el.a ✓ 12
19.e even 9 1 2280.1.el.b yes 12
24.h odd 2 1 2280.1.el.b yes 12
40.f even 2 1 2280.1.el.b yes 12
57.l odd 18 1 2280.1.el.b yes 12
95.p even 18 1 2280.1.el.b yes 12
120.i odd 2 1 2280.1.el.b yes 12
152.t even 18 1 inner 2280.1.el.a ✓ 12
285.bd odd 18 1 2280.1.el.b yes 12
456.bh odd 18 1 inner 2280.1.el.a ✓ 12
760.cj even 18 1 inner 2280.1.el.a ✓ 12
2280.el odd 18 1 inner 2280.1.el.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.1.el.a ✓ 12 1.a even 1 1 trivial
2280.1.el.a ✓ 12 3.b odd 2 1 inner
2280.1.el.a ✓ 12 5.b even 2 1 inner
2280.1.el.a ✓ 12 15.d odd 2 1 CM
2280.1.el.a ✓ 12 152.t even 18 1 inner
2280.1.el.a ✓ 12 456.bh odd 18 1 inner
2280.1.el.a ✓ 12 760.cj even 18 1 inner
2280.1.el.a ✓ 12 2280.el odd 18 1 inner
2280.1.el.b yes 12 8.b even 2 1
2280.1.el.b yes 12 19.e even 9 1
2280.1.el.b yes 12 24.h odd 2 1
2280.1.el.b yes 12 40.f even 2 1
2280.1.el.b yes 12 57.l odd 18 1
2280.1.el.b yes 12 95.p even 18 1
2280.1.el.b yes 12 120.i odd 2 1
2280.1.el.b yes 12 285.bd odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{61}^{6} + 9T_{61}^{3} + 27 \) acting on \(S_{1}^{\mathrm{new}}(2280, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} + 3 T^{4} - 2 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} - 3 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{3} + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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