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Properties

Label 444.2.c.a
Level $444$
Weight $2$
Character orbit 444.c
Analytic conductor $3.545$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(371,444)] 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("444.371"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.386672896.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{7} - \beta_{2}) q^{3} + (\beta_{4} + \beta_{3}) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{4} + 1) q^{6} + ( - 2 \beta_{7} - \beta_{4} - \beta_{2} - 1) q^{7}+ \cdots + ( - 4 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 8 q^{6} + 4 q^{8} + 2 q^{9} + 4 q^{11} + 10 q^{12} - 16 q^{13} + 18 q^{14} - 8 q^{15} - 6 q^{16} - 10 q^{18} + 8 q^{20} - 22 q^{21} - 2 q^{22} + 8 q^{23} - 4 q^{24} + 8 q^{25}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} + 2\nu^{3} + 12\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 2\nu^{3} + 16\nu^{2} - 12\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} - 4\nu^{4} - 6\nu^{3} - 8\nu^{2} - 4\nu - 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{3} - 4\nu^{2} - 12\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} - 2\nu^{6} - \nu^{5} + 2\nu^{3} + 8\nu^{2} + 12\nu + 8 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} + 2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} - \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + \beta_{3} - 3\beta_{2} + 6\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
0.621372 + 1.27039i
0.621372 − 1.27039i
−1.19503 − 0.756243i
−1.19503 + 0.756243i
1.40961 + 0.114062i
1.40961 − 0.114062i
−0.835949 + 1.14070i
−0.835949 − 1.14070i
−1.27039 − 0.621372i 1.72779 + 0.121372i 1.22779 + 1.57877i − 2.00000i −2.11956 − 1.22779i 0.242745i −0.578773 − 2.76858i 2.97054 + 0.419412i −1.24274 + 2.54078i
371.2 −1.27039 + 0.621372i 1.72779 − 0.121372i 1.22779 − 1.57877i 2.00000i −2.11956 + 1.22779i − 0.242745i −0.578773 + 2.76858i 2.97054 − 0.419412i −1.24274 − 2.54078i
371.3 −0.756243 − 1.19503i −0.356193 + 1.69503i −0.856193 + 1.80747i 2.00000i 2.29498 − 0.856193i 3.39006i 2.80747 − 0.343707i −2.74625 − 1.20752i 2.39006 − 1.51249i
371.4 −0.756243 + 1.19503i −0.356193 − 1.69503i −0.856193 − 1.80747i − 2.00000i 2.29498 + 0.856193i − 3.39006i 2.80747 + 0.343707i −2.74625 + 1.20752i 2.39006 + 1.51249i
371.5 −0.114062 − 1.40961i −1.47398 + 0.909606i −1.97398 + 0.321565i − 2.00000i 1.45031 + 1.97398i 1.81921i 0.678435 + 2.74586i 1.34523 − 2.68148i −2.81921 + 0.228124i
371.6 −0.114062 + 1.40961i −1.47398 − 0.909606i −1.97398 − 0.321565i 2.00000i 1.45031 − 1.97398i − 1.81921i 0.678435 − 2.74586i 1.34523 + 2.68148i −2.81921 − 0.228124i
371.7 1.14070 − 0.835949i 1.10238 + 1.33595i 0.602380 − 1.90713i 2.00000i 2.37427 + 0.602380i 2.67190i −0.907128 − 2.67901i −0.569517 + 2.94545i 1.67190 + 2.28139i
371.8 1.14070 + 0.835949i 1.10238 − 1.33595i 0.602380 + 1.90713i − 2.00000i 2.37427 − 0.602380i − 2.67190i −0.907128 + 2.67901i −0.569517 − 2.94545i 1.67190 − 2.28139i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 371.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.c.a ✓ 8
3.b odd 2 1 444.2.c.b yes 8
4.b odd 2 1 444.2.c.b yes 8
12.b even 2 1 inner 444.2.c.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.c.a ✓ 8 1.a even 1 1 trivial
444.2.c.a ✓ 8 12.b even 2 1 inner
444.2.c.b yes 8 3.b odd 2 1
444.2.c.b yes 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 11T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 22 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 11 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 104 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$19$ \( T^{8} + 88 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} - 40 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 152 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$37$ \( (T - 1)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 70 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$43$ \( T^{8} + 176 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$47$ \( (T^{4} + 30 T^{3} + \cdots + 2512)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 278 T^{6} + \cdots + 9535744 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} + \cdots + 8672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots + 2272)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 208 T^{6} + \cdots + 891136 \) Copy content Toggle raw display
$71$ \( (T^{4} + 34 T^{3} + \cdots + 4304)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} + \cdots - 124)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 152 T^{6} + \cdots + 200704 \) Copy content Toggle raw display
$83$ \( (T^{4} - 10 T^{3} + \cdots - 2752)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 488 T^{6} + \cdots + 61465600 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 5600)^{2} \) Copy content Toggle raw display
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