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Properties

Label 1444.1.l.a
Level $1444$
Weight $1$
Character orbit 1444.l
Analytic conductor $0.721$
Analytic rank $0$
Dimension $12$
Projective image $D_{5}$
CM discriminant -4
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,1,Mod(99,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.l (of order \(18\), degree \(6\), not 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: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.6053445140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.2085136.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{3}) q^{2} + (\beta_{11} - \beta_{5}) q^{4} + (\beta_{9} - \beta_{7} + \beta_1) q^{5} - \beta_{8} q^{8} + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{3}) q^{2} + (\beta_{11} - \beta_{5}) q^{4} + (\beta_{9} - \beta_{7} + \beta_1) q^{5} - \beta_{8} q^{8} + \beta_{5} q^{9} + ( - \beta_{5} + \beta_{4}) q^{10} + (\beta_{10} + \beta_{4}) q^{13} + \beta_{3} q^{16} - \beta_{7} q^{17} + q^{18} + ( - \beta_{2} - 1) q^{20} + ( - \beta_{11} + \beta_{10} + \beta_{4}) q^{25} + (\beta_{6} - \beta_{2}) q^{26} - \beta_{4} q^{29} - \beta_{11} q^{32} - \beta_{10} q^{34} + (\beta_{9} - \beta_{3}) q^{36} + ( - \beta_{2} - 1) q^{37} + ( - \beta_{9} + \beta_{7} + \beta_{3}) q^{40} + ( - \beta_{3} - \beta_1) q^{41} + ( - \beta_{8} - \beta_{6} + \beta_{2} + 1) q^{45} - \beta_{8} q^{49} + (\beta_{8} + \beta_{6} - \beta_{2} - 1) q^{50} + (\beta_{7} - \beta_1) q^{52} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{53} + \beta_{2} q^{58} - \beta_{10} q^{61} + (\beta_{8} - 1) q^{64} + \beta_{8} q^{65} - \beta_{6} q^{68} + (\beta_{11} - \beta_{5}) q^{72} + ( - \beta_{3} - \beta_1) q^{73} + ( - \beta_{9} + \beta_{7} + \beta_{3}) q^{74} + ( - \beta_{11} + \beta_{10} + \beta_{5}) q^{80} - \beta_{9} q^{81} + (\beta_{11} - \beta_{10} - \beta_{4}) q^{82} - \beta_{5} q^{85} + (\beta_{11} - \beta_{10} - \beta_{4}) q^{89} + (\beta_{9} - \beta_{7} + \beta_1) q^{90} - \beta_{7} q^{97} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{8} + 12 q^{18} - 6 q^{20} + 3 q^{26} - 6 q^{37} + 3 q^{45} - 6 q^{49} - 3 q^{50} - 6 q^{58} - 6 q^{64} + 6 q^{65} + 3 q^{68}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 21 ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 55\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 55\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 89\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{9} + 17\nu^{6} - 85\nu^{3} + 1 ) / 34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{10} - 17\nu^{7} + 68\nu^{4} + 16\nu ) / 17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{9} + 17\nu^{6} - 68\nu^{3} + 1 ) / 17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{10} + 51\nu^{7} - 221\nu^{4} + 3\nu ) / 34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -12\nu^{11} + 51\nu^{8} - 221\nu^{5} + 3\nu^{2} ) / 34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\nu^{11} - 85\nu^{8} + 357\nu^{5} + 84\nu^{2} ) / 34 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} - 3\beta_{7} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - 5\beta_{10} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{8} - 8\beta_{6} + 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 13\beta_{7} + 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -13\beta_{11} - 21\beta_{10} - 21\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 34\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 34\beta_{3} - 55\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -55\beta_{5} - 89\beta_{4} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(-1\) \(-\beta_{5} + \beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.107320 + 0.608645i
−0.280969 − 1.59345i
−0.580762 + 0.211380i
1.52045 − 0.553400i
0.107320 − 0.608645i
−0.280969 + 1.59345i
−0.580762 − 0.211380i
1.52045 + 0.553400i
−1.23949 − 1.04005i
0.473442 + 0.397265i
−1.23949 + 1.04005i
0.473442 − 0.397265i
−0.939693 − 0.342020i 0 0.766044 + 0.642788i −1.23949 + 1.04005i 0 0 −0.500000 − 0.866025i −0.939693 + 0.342020i 1.52045 − 0.553400i
99.2 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.473442 − 0.397265i 0 0 −0.500000 − 0.866025i −0.939693 + 0.342020i −0.580762 + 0.211380i
415.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.280969 + 1.59345i 0 0 −0.500000 + 0.866025i 0.766044 − 0.642788i −1.23949 + 1.04005i
415.2 0.766044 + 0.642788i 0 0.173648 + 0.984808i 0.107320 − 0.608645i 0 0 −0.500000 + 0.866025i 0.766044 − 0.642788i 0.473442 − 0.397265i
423.1 −0.939693 + 0.342020i 0 0.766044 − 0.642788i −1.23949 − 1.04005i 0 0 −0.500000 + 0.866025i −0.939693 − 0.342020i 1.52045 + 0.553400i
423.2 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.473442 + 0.397265i 0 0 −0.500000 + 0.866025i −0.939693 − 0.342020i −0.580762 − 0.211380i
595.1 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.280969 − 1.59345i 0 0 −0.500000 − 0.866025i 0.766044 + 0.642788i −1.23949 − 1.04005i
595.2 0.766044 − 0.642788i 0 0.173648 − 0.984808i 0.107320 + 0.608645i 0 0 −0.500000 − 0.866025i 0.766044 + 0.642788i 0.473442 + 0.397265i
967.1 0.173648 − 0.984808i 0 −0.939693 − 0.342020i −0.580762 + 0.211380i 0 0 −0.500000 + 0.866025i 0.173648 + 0.984808i 0.107320 + 0.608645i
967.2 0.173648 − 0.984808i 0 −0.939693 − 0.342020i 1.52045 − 0.553400i 0 0 −0.500000 + 0.866025i 0.173648 + 0.984808i −0.280969 − 1.59345i
1111.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −0.580762 − 0.211380i 0 0 −0.500000 − 0.866025i 0.173648 − 0.984808i 0.107320 − 0.608645i
1111.2 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 1.52045 + 0.553400i 0 0 −0.500000 − 0.866025i 0.173648 − 0.984808i −0.280969 + 1.59345i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
19.c even 3 2 inner
19.e even 9 3 inner
76.g odd 6 2 inner
76.l odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.l.a 12
4.b odd 2 1 CM 1444.1.l.a 12
19.b odd 2 1 1444.1.l.b 12
19.c even 3 2 inner 1444.1.l.a 12
19.d odd 6 2 1444.1.l.b 12
19.e even 9 1 1444.1.b.b yes 2
19.e even 9 2 1444.1.g.a 4
19.e even 9 3 inner 1444.1.l.a 12
19.f odd 18 1 1444.1.b.a ✓ 2
19.f odd 18 2 1444.1.g.b 4
19.f odd 18 3 1444.1.l.b 12
76.d even 2 1 1444.1.l.b 12
76.f even 6 2 1444.1.l.b 12
76.g odd 6 2 inner 1444.1.l.a 12
76.k even 18 1 1444.1.b.a ✓ 2
76.k even 18 2 1444.1.g.b 4
76.k even 18 3 1444.1.l.b 12
76.l odd 18 1 1444.1.b.b yes 2
76.l odd 18 2 1444.1.g.a 4
76.l odd 18 3 inner 1444.1.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a ✓ 2 19.f odd 18 1
1444.1.b.a ✓ 2 76.k even 18 1
1444.1.b.b yes 2 19.e even 9 1
1444.1.b.b yes 2 76.l odd 18 1
1444.1.g.a 4 19.e even 9 2
1444.1.g.a 4 76.l odd 18 2
1444.1.g.b 4 19.f odd 18 2
1444.1.g.b 4 76.k even 18 2
1444.1.l.a 12 1.a even 1 1 trivial
1444.1.l.a 12 4.b odd 2 1 CM
1444.1.l.a 12 19.c even 3 2 inner
1444.1.l.a 12 19.e even 9 3 inner
1444.1.l.a 12 76.g odd 6 2 inner
1444.1.l.a 12 76.l odd 18 3 inner
1444.1.l.b 12 19.b odd 2 1
1444.1.l.b 12 19.d odd 6 2
1444.1.l.b 12 19.f odd 18 3
1444.1.l.b 12 76.d even 2 1
1444.1.l.b 12 76.f even 6 2
1444.1.l.b 12 76.k even 18 3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{12} - 4T_{13}^{9} + 17T_{13}^{6} + 4T_{13}^{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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