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Properties

Label 1156.4.a.d
Level $1156$
Weight $4$
Character orbit 1156.a
Self dual yes
Analytic conductor $68.206$
Analytic rank $1$
Dimension $2$
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] = [1156,4,Mod(1,1156)] 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("1156.1"); S:= CuspForms(chi, 4); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.2062079666\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta q^{3} - \beta q^{5} - 20 \beta q^{7} + 5 q^{9} + 32 \beta q^{11} + 36 q^{13} - 8 q^{15} - 20 q^{19} - 160 q^{21} - 48 \beta q^{23} - 123 q^{25} - 88 \beta q^{27} + 85 \beta q^{29} + 96 \beta q^{31} + \cdots + 160 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9} + 72 q^{13} - 16 q^{15} - 40 q^{19} - 320 q^{21} - 246 q^{25} + 512 q^{33} + 80 q^{35} - 936 q^{43} + 752 q^{47} + 914 q^{49} - 456 q^{53} - 128 q^{55} + 744 q^{59} - 1864 q^{67} - 768 q^{69}+ \cdots + 1536 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −5.65685 0 1.41421 0 28.2843 0 5.00000 0
1.2 0 5.65685 0 −1.41421 0 −28.2843 0 5.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( +1 \)

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 1156.4.a.d 2
17.b even 2 1 inner 1156.4.a.d 2
17.c even 4 2 1156.4.b.a 2
17.d even 8 2 68.4.e.a ✓ 2
51.g odd 8 2 612.4.k.a 2
68.g odd 8 2 272.4.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.e.a ✓ 2 17.d even 8 2
272.4.o.b 2 68.g odd 8 2
612.4.k.a 2 51.g odd 8 2
1156.4.a.d 2 1.a even 1 1 trivial
1156.4.a.d 2 17.b even 2 1 inner
1156.4.b.a 2 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 32 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 32 \) Copy content Toggle raw display
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 800 \) Copy content Toggle raw display
$11$ \( T^{2} - 2048 \) Copy content Toggle raw display
$13$ \( (T - 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4608 \) Copy content Toggle raw display
$29$ \( T^{2} - 14450 \) Copy content Toggle raw display
$31$ \( T^{2} - 18432 \) Copy content Toggle raw display
$37$ \( T^{2} - 11250 \) Copy content Toggle raw display
$41$ \( T^{2} - 18050 \) Copy content Toggle raw display
$43$ \( (T + 468)^{2} \) Copy content Toggle raw display
$47$ \( (T - 376)^{2} \) Copy content Toggle raw display
$53$ \( (T + 228)^{2} \) Copy content Toggle raw display
$59$ \( (T - 372)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 417698 \) Copy content Toggle raw display
$67$ \( (T + 932)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 798848 \) Copy content Toggle raw display
$73$ \( T^{2} - 11858 \) Copy content Toggle raw display
$79$ \( T^{2} - 516128 \) Copy content Toggle raw display
$83$ \( (T + 92)^{2} \) Copy content Toggle raw display
$89$ \( (T + 1344)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1051250 \) Copy content Toggle raw display
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