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Properties

Label 1224.2.f.c
Level $1224$
Weight $2$
Character orbit 1224.f
Analytic conductor $9.774$
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] = [1224,2,Mod(613,1224)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1224.613"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,1,0,1,0,0,-12,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.77368920740\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4469724736.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{7} - \beta_{2} + \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{8} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots - 1) q^{10}+ \cdots + (\beta_{7} + 3 \beta_{5} + 3 \beta_{4} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{4} - 12 q^{7} - 5 q^{8} - 8 q^{10} - 6 q^{14} + 9 q^{16} - 8 q^{17} + 16 q^{20} + 4 q^{22} + 16 q^{23} - 8 q^{25} + 4 q^{26} - 20 q^{28} + 24 q^{31} - 9 q^{32} - q^{34} - 18 q^{38} + 20 q^{40}+ \cdots + 57 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 \) : Copy content Toggle raw display

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
613.1
−1.37971 − 0.310478i
−1.37971 + 0.310478i
−0.185533 − 1.40199i
−0.185533 + 1.40199i
0.733159 − 1.20933i
0.733159 + 1.20933i
1.33209 − 0.474920i
1.33209 + 0.474920i
−1.37971 − 0.310478i 0 1.80721 + 0.856739i − 2.33443i 0 −1.57978 −2.22743 − 1.74315i 0 −0.724789 + 3.22084i
613.2 −1.37971 + 0.310478i 0 1.80721 − 0.856739i 2.33443i 0 −1.57978 −2.22743 + 1.74315i 0 −0.724789 − 3.22084i
613.3 −0.185533 − 1.40199i 0 −1.93115 + 0.520231i − 3.84444i 0 −1.15650 1.08765 + 2.61094i 0 −5.38987 + 0.713272i
613.4 −0.185533 + 1.40199i 0 −1.93115 − 0.520231i 3.84444i 0 −1.15650 1.08765 − 2.61094i 0 −5.38987 − 0.713272i
613.5 0.733159 − 1.20933i 0 −0.924955 − 1.77326i 1.12786i 0 1.74755 −2.82260 − 0.181508i 0 1.36396 + 0.826905i
613.6 0.733159 + 1.20933i 0 −0.924955 + 1.77326i − 1.12786i 0 1.74755 −2.82260 + 0.181508i 0 1.36396 − 0.826905i
613.7 1.33209 − 0.474920i 0 1.54890 − 1.26527i 1.58069i 0 −5.01127 1.46237 − 2.42105i 0 0.750703 + 2.10562i
613.8 1.33209 + 0.474920i 0 1.54890 + 1.26527i − 1.58069i 0 −5.01127 1.46237 + 2.42105i 0 0.750703 − 2.10562i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 613.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.f.c 8
3.b odd 2 1 136.2.c.b ✓ 8
4.b odd 2 1 4896.2.f.d 8
8.b even 2 1 inner 1224.2.f.c 8
8.d odd 2 1 4896.2.f.d 8
12.b even 2 1 544.2.c.b 8
24.f even 2 1 544.2.c.b 8
24.h odd 2 1 136.2.c.b ✓ 8
48.i odd 4 2 4352.2.a.bf 8
48.k even 4 2 4352.2.a.bb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.c.b ✓ 8 3.b odd 2 1
136.2.c.b ✓ 8 24.h odd 2 1
544.2.c.b 8 12.b even 2 1
544.2.c.b 8 24.f even 2 1
1224.2.f.c 8 1.a even 1 1 trivial
1224.2.f.c 8 8.b even 2 1 inner
4352.2.a.bb 8 48.k even 4 2
4352.2.a.bf 8 48.i odd 4 2
4896.2.f.d 8 4.b odd 2 1
4896.2.f.d 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{8} + 24T_{5}^{6} + 160T_{5}^{4} + 368T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{23}^{4} - 8T_{23}^{3} - 50T_{23}^{2} + 278T_{23} + 944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} + 2 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 44 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 80 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots + 944)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + \cdots - 160)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 160 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$41$ \( (T^{4} - 56 T^{2} + \cdots - 80)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 188 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{3} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 112 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$59$ \( T^{8} + 300 T^{6} + \cdots + 10240000 \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{8} + 112 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$71$ \( (T^{4} - 18 T^{3} + \cdots - 400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} + \cdots + 304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 316 T^{6} + \cdots + 36966400 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots - 2456)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots - 688)^{2} \) Copy content Toggle raw display
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