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Properties

Label 32.12.a.c
Level $32$
Weight $12$
Character orbit 32.a
Self dual yes
Analytic conductor $24.587$
Analytic rank $0$
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] = [32,12,Mod(1,32)] 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("32.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,14060] 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: \(24.5869817779\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{273}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(\beta = 32\sqrt{273}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 7030 q^{5} - 66 \beta q^{7} + 102405 q^{9} - 1371 \beta q^{11} - 1182066 q^{13} - 7030 \beta q^{15} + 5443874 q^{17} + 39675 \beta q^{19} + 18450432 q^{21} - 12486 \beta q^{23} + 592775 q^{25} + \cdots - 140397255 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14060 q^{5} + 204810 q^{9} - 2364132 q^{13} + 10887748 q^{17} + 36900864 q^{21} + 1185550 q^{25} + 328092988 q^{29} + 766531584 q^{33} + 1080453740 q^{37} + 559998644 q^{41} + 1439814300 q^{45} - 1519196462 q^{49}+ \cdots + 201406465188 q^{97}+O(q^{100}) \) 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
8.76136
−7.76136
0 −528.727 0 7030.00 0 −34896.0 0 102405. 0
1.2 0 528.727 0 7030.00 0 34896.0 0 102405. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.12.a.c ✓ 2
4.b odd 2 1 inner 32.12.a.c ✓ 2
8.b even 2 1 64.12.a.i 2
8.d odd 2 1 64.12.a.i 2
16.e even 4 2 256.12.b.i 4
16.f odd 4 2 256.12.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.12.a.c ✓ 2 1.a even 1 1 trivial
32.12.a.c ✓ 2 4.b odd 2 1 inner
64.12.a.i 2 8.b even 2 1
64.12.a.i 2 8.d odd 2 1
256.12.b.i 4 16.e even 4 2
256.12.b.i 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 279552 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(32))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 279552 \) Copy content Toggle raw display
$5$ \( (T - 7030)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1217728512 \) Copy content Toggle raw display
$11$ \( T^{2} - 525457400832 \) Copy content Toggle raw display
$13$ \( (T + 1182066)^{2} \) Copy content Toggle raw display
$17$ \( (T - 5443874)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 440044375680000 \) Copy content Toggle raw display
$23$ \( T^{2} - 43582211592192 \) Copy content Toggle raw display
$29$ \( (T - 164046494)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 39\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( (T - 540226870)^{2} \) Copy content Toggle raw display
$41$ \( (T - 279999322)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 91\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{2} - 35\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( (T - 4776519398)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 19\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( (T + 7479685282)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 11\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{2} - 72\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( (T + 21803168118)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 11\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{2} - 38\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( (T - 22195771450)^{2} \) Copy content Toggle raw display
$97$ \( (T - 100703232594)^{2} \) Copy content Toggle raw display
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