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Properties

Label 18.19.b.a
Level $18$
Weight $19$
Character orbit 18.b
Analytic conductor $36.970$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,19,Mod(17,18)] 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("18.17"); S:= CuspForms(chi, 19); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 19, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.9695047878\)
Analytic rank: \(0\)
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: 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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 256 \beta q^{2} - 131072 q^{4} + 459525 \beta q^{5} - 352276 q^{7} + 33554432 \beta q^{8} + 235276800 q^{10} + 664810332 \beta q^{11} + 4851691832 q^{13} + 90182656 \beta q^{14} + 17179869184 q^{16} + \cdots + 41\!\cdots\!88 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 262144 q^{4} - 704552 q^{7} + 470553600 q^{10} + 9703383664 q^{13} + 34359738368 q^{16} - 90924168992 q^{19} + 680765779968 q^{22} + 6784741628750 q^{25} + 92347039744 q^{28} - 37887193137992 q^{31}+ \cdots - 85\!\cdots\!92 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
− 1.41421i
− 362.039i 0 −131072. 649866.i 0 −352276. 4.74531e7i 0 2.35277e8
17.2 362.039i 0 −131072. − 649866.i 0 −352276. − 4.74531e7i 0 2.35277e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.19.b.a ✓ 2
3.b odd 2 1 inner 18.19.b.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.19.b.a ✓ 2 1.a even 1 1 trivial
18.19.b.a ✓ 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 422326451250 \) acting on \(S_{19}^{\mathrm{new}}(18, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 131072 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 422326451250 \) Copy content Toggle raw display
$7$ \( (T + 352276)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 88\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( (T - 4851691832)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 10\!\cdots\!42 \) Copy content Toggle raw display
$19$ \( (T + 45462084496)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 32\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{2} + 54\!\cdots\!18 \) Copy content Toggle raw display
$31$ \( (T + 18943596568996)^{2} \) Copy content Toggle raw display
$37$ \( (T + 166886471089186)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 18\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( (T + 730556755353160)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{2} + 20\!\cdots\!98 \) Copy content Toggle raw display
$59$ \( T^{2} + 76\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( (T + 69\!\cdots\!30)^{2} \) Copy content Toggle raw display
$67$ \( (T - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 42\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( (T - 95\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T - 22\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 21\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{2} + 98\!\cdots\!22 \) Copy content Toggle raw display
$97$ \( (T + 42\!\cdots\!96)^{2} \) Copy content Toggle raw display
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