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Properties

Label 36.19.d.a
Level $36$
Weight $19$
Character orbit 36.d
Self dual yes
Analytic conductor $73.939$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,19,Mod(19,36)] 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("36.19"); S:= CuspForms(chi, 19); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 19, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-512] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.9390095755\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 
\(f(q)\) \(=\) \( q - 512 q^{2} + 262144 q^{4} + 3443528 q^{5} - 134217728 q^{8} - 1763086336 q^{10} + 8483805110 q^{13} + 68719476736 q^{16} + 226387416944 q^{17} + 902700204032 q^{20} + 8043187821159 q^{25} - 4343708216320 q^{26}+ \cdots - 83\!\cdots\!88 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−512.000 0 262144. 3.44353e6 0 0 −1.34218e8 0 −1.76309e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.19.d.a ✓ 1
3.b odd 2 1 36.19.d.b yes 1
4.b odd 2 1 CM 36.19.d.a ✓ 1
12.b even 2 1 36.19.d.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.19.d.a ✓ 1 1.a even 1 1 trivial
36.19.d.a ✓ 1 4.b odd 2 1 CM
36.19.d.b yes 1 3.b odd 2 1
36.19.d.b yes 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3443528 \) acting on \(S_{19}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 512 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3443528 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 8483805110 \) Copy content Toggle raw display
$17$ \( T - 226387416944 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 15557481145960 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 256224119739770 \) Copy content Toggle raw display
$41$ \( T + 267604647192880 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6310582201707784 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 23\!\cdots\!82 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 11\!\cdots\!90 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 41\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T + 48\!\cdots\!70 \) Copy content Toggle raw display
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