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Properties

Label 1156.2.b.e
Level $1156$
Weight $2$
Character orbit 1156.b
Analytic conductor $9.231$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(577,1156)] 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("1156.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.419904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 6x^{4} + 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + \beta_1) q^{3} + (2 \beta_{3} - \beta_1) q^{5} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{7} + ( - \beta_{4} - 2 \beta_{2}) q^{9} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{11} + ( - \beta_{4} - \beta_{2} + 2) q^{13}+ \cdots + (10 \beta_{5} + 8 \beta_{3} + 10 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{13} - 6 q^{15} - 24 q^{19} - 12 q^{21} - 6 q^{25} + 24 q^{33} - 42 q^{35} + 18 q^{43} - 6 q^{47} - 18 q^{49} + 12 q^{53} + 36 q^{55} + 30 q^{59} - 30 q^{67} - 30 q^{69} + 60 q^{77} + 30 q^{81}+ \cdots + 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} + 9x^{2} + 1 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
− 0.347296i
− 1.53209i
− 1.87939i
1.87939i
1.53209i
0.347296i
0 − 2.87939i 0 − 1.65270i 0 − 3.18479i 0 −5.29086 0
577.2 0 − 0.652704i 0 − 0.467911i 0 1.41147i 0 2.57398 0
577.3 0 − 0.532089i 0 3.87939i 0 4.22668i 0 2.71688 0
577.4 0 0.532089i 0 − 3.87939i 0 − 4.22668i 0 2.71688 0
577.5 0 0.652704i 0 0.467911i 0 − 1.41147i 0 2.57398 0
577.6 0 2.87939i 0 1.65270i 0 3.18479i 0 −5.29086 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 577.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.b.e 6
17.b even 2 1 inner 1156.2.b.e 6
17.c even 4 1 1156.2.a.e ✓ 3
17.c even 4 1 1156.2.a.f yes 3
17.d even 8 4 1156.2.e.g 12
17.e odd 16 8 1156.2.h.h 24
68.f odd 4 1 4624.2.a.be 3
68.f odd 4 1 4624.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1156.2.a.e ✓ 3 17.c even 4 1
1156.2.a.f yes 3 17.c even 4 1
1156.2.b.e 6 1.a even 1 1 trivial
1156.2.b.e 6 17.b even 2 1 inner
1156.2.e.g 12 17.d even 8 4
1156.2.h.h 24 17.e odd 16 8
4624.2.a.be 3 68.f odd 4 1
4624.2.a.bf 3 68.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 9T_{3}^{4} + 6T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 18 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 30 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{6} + 36 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( (T^{3} - 6 T^{2} + 3 T + 19)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} + \cdots - 37)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 126 T^{4} + \cdots + 45369 \) Copy content Toggle raw display
$29$ \( T^{6} + 90 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$31$ \( T^{6} + 69 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} + 81 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$41$ \( T^{6} + 162 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 3 T^{2} + \cdots - 219)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} + 24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 15 T^{2} + \cdots + 51)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 165 T^{4} + \cdots + 104329 \) Copy content Toggle raw display
$67$ \( (T^{3} + 15 T^{2} + \cdots + 19)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 477 T^{4} + \cdots + 3143529 \) Copy content Toggle raw display
$73$ \( T^{6} + 165 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$79$ \( T^{6} + 363 T^{4} + \cdots + 1371241 \) Copy content Toggle raw display
$83$ \( (T^{3} - 3 T^{2} + \cdots + 867)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 15 T^{2} + 54 T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 84 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
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