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Properties

Label 1156.2.h.g
Level $1156$
Weight $2$
Character orbit 1156.h
Analytic conductor $9.231$
Analytic rank $0$
Dimension $16$
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] = [1156,2,Mod(733,1156)] 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("1156.733"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.10646188457767892278050816.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3791x^{8} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{15} - \beta_{13}) q^{5} + ( - 3 \beta_{2} - \beta_1) q^{7} + ( - \beta_{14} - 3 \beta_{12}) q^{9} + (2 \beta_{11} + \beta_{9}) q^{11} + (\beta_{10} + \beta_{8}) q^{13} + (2 \beta_{5} + 7 \beta_{3}) q^{15}+ \cdots + ( - 8 \beta_{7} + 5 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 168 q^{33} - 24 q^{35} + 16 q^{67} + 48 q^{69}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3791x^{8} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 2286\nu ) / 3905 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 6191\nu^{2} ) / 19525 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 1505 ) / 781 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{10} - 17621\nu^{2} ) / 19525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 6191\nu^{3} ) / 19525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{11} + 17621\nu^{3} ) / 97625 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{12} - 4416\nu^{4} ) / 44375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -41\nu^{13} - 136681\nu^{5} ) / 2440625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41\nu^{12} + 136681\nu^{4} ) / 488125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{13} - 4416\nu^{5} ) / 44375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -96\nu^{14} - 379561\nu^{6} ) / 12203125 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -301\nu^{15} - 1062966\nu^{7} ) / 61015625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 301\nu^{14} + 1062966\nu^{6} ) / 12203125 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -96\nu^{15} - 379561\nu^{7} ) / 12203125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} + 6\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{10} - 41\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -41\beta_{11} + 55\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -96\beta_{14} - 301\beta_{12} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -301\beta_{15} + 480\beta_{13} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 781\beta_{4} - 1505 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3905\beta_{2} - 2286\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -6191\beta_{5} - 17621\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 30955\beta_{7} - 17621\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 48576\beta_{10} + 136681\beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 136681\beta_{11} - 242880\beta_{9} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 379561\beta_{14} + 1062966\beta_{12} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1062966\beta_{15} - 1897805\beta_{13} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(-\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
733.1
1.06818 − 2.57881i
0.685496 − 1.65493i
−0.685496 + 1.65493i
−1.06818 + 2.57881i
1.06818 + 2.57881i
0.685496 + 1.65493i
−0.685496 − 1.65493i
−1.06818 − 2.57881i
−2.57881 + 1.06818i
−1.65493 + 0.685496i
1.65493 − 0.685496i
2.57881 − 1.06818i
−2.57881 − 1.06818i
−1.65493 − 0.685496i
1.65493 + 0.685496i
2.57881 + 1.06818i
0 −2.57881 + 1.06818i 0 1.45086 + 3.50269i 0 0.0798707 − 0.192825i 0 3.38795 − 3.38795i 0
733.2 0 −1.65493 + 0.685496i 0 0.302813 + 0.731055i 0 −1.83355 + 4.42657i 0 0.147582 − 0.147582i 0
733.3 0 1.65493 − 0.685496i 0 −0.302813 − 0.731055i 0 1.83355 − 4.42657i 0 0.147582 − 0.147582i 0
733.4 0 2.57881 − 1.06818i 0 −1.45086 − 3.50269i 0 −0.0798707 + 0.192825i 0 3.38795 − 3.38795i 0
757.1 0 −2.57881 − 1.06818i 0 1.45086 − 3.50269i 0 0.0798707 + 0.192825i 0 3.38795 + 3.38795i 0
757.2 0 −1.65493 − 0.685496i 0 0.302813 − 0.731055i 0 −1.83355 − 4.42657i 0 0.147582 + 0.147582i 0
757.3 0 1.65493 + 0.685496i 0 −0.302813 + 0.731055i 0 1.83355 + 4.42657i 0 0.147582 + 0.147582i 0
757.4 0 2.57881 + 1.06818i 0 −1.45086 + 3.50269i 0 −0.0798707 − 0.192825i 0 3.38795 + 3.38795i 0
977.1 0 −1.06818 + 2.57881i 0 −3.50269 − 1.45086i 0 −0.192825 + 0.0798707i 0 −3.38795 − 3.38795i 0
977.2 0 −0.685496 + 1.65493i 0 −0.731055 − 0.302813i 0 4.42657 − 1.83355i 0 −0.147582 − 0.147582i 0
977.3 0 0.685496 − 1.65493i 0 0.731055 + 0.302813i 0 −4.42657 + 1.83355i 0 −0.147582 − 0.147582i 0
977.4 0 1.06818 − 2.57881i 0 3.50269 + 1.45086i 0 0.192825 − 0.0798707i 0 −3.38795 − 3.38795i 0
1001.1 0 −1.06818 − 2.57881i 0 −3.50269 + 1.45086i 0 −0.192825 − 0.0798707i 0 −3.38795 + 3.38795i 0
1001.2 0 −0.685496 − 1.65493i 0 −0.731055 + 0.302813i 0 4.42657 + 1.83355i 0 −0.147582 + 0.147582i 0
1001.3 0 0.685496 + 1.65493i 0 0.731055 − 0.302813i 0 −4.42657 − 1.83355i 0 −0.147582 + 0.147582i 0
1001.4 0 1.06818 + 2.57881i 0 3.50269 − 1.45086i 0 0.192825 + 0.0798707i 0 −3.38795 + 3.38795i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 733.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.h.g 16
17.b even 2 1 inner 1156.2.h.g 16
17.c even 4 2 inner 1156.2.h.g 16
17.d even 8 4 inner 1156.2.h.g 16
17.e odd 16 1 1156.2.a.b ✓ 2
17.e odd 16 1 1156.2.a.d yes 2
17.e odd 16 2 1156.2.b.b 4
17.e odd 16 4 1156.2.e.e 8
68.i even 16 1 4624.2.a.k 2
68.i even 16 1 4624.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1156.2.a.b ✓ 2 17.e odd 16 1
1156.2.a.d yes 2 17.e odd 16 1
1156.2.b.b 4 17.e odd 16 2
1156.2.e.e 8 17.e odd 16 4
1156.2.h.g 16 1.a even 1 1 trivial
1156.2.h.g 16 17.b even 2 1 inner
1156.2.h.g 16 17.c even 4 2 inner
1156.2.h.g 16 17.d even 8 4 inner
4624.2.a.k 2 68.i even 16 1
4624.2.a.u 2 68.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 3791T_{3}^{8} + 390625 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3791 T^{8} + 390625 \) Copy content Toggle raw display
$5$ \( T^{16} + 42687 T^{8} + 6561 \) Copy content Toggle raw display
$7$ \( T^{16} + 277727 T^{8} + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} + 194481)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 11 T^{2} + 25)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 527 T^{4} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2562890625 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 45767944570401 \) Copy content Toggle raw display
$31$ \( (T^{8} + 390625)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 25600000000 \) Copy content Toggle raw display
$41$ \( (T^{8} + 1679616)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 114 T^{2} + 225)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 2151 T^{4} + 50625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 7056)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 20)^{8} \) Copy content Toggle raw display
$71$ \( (T^{8} + 1679616)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( (T^{8} + 256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 2151 T^{4} + 50625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 84)^{8} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 86\!\cdots\!41 \) Copy content Toggle raw display
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