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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1521,2,Mod(1351,1521)] 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("1521.1351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-12,0,0,0,0,0,-20,0,0,0,0,0,-4,0,0,0,0,0,40,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,20,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{4} - \beta_1 q^{5} - 2 \beta_{2} q^{7} + \beta_1 q^{8} - 5 q^{10} + 2 \beta_1 q^{11} - 2 \beta_{3} q^{14} - q^{16} - \beta_{3} q^{17} - 2 \beta_{2} q^{19} + 3 \beta_1 q^{20} + 10 q^{22}+ \cdots + 5 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 20 q^{10} - 4 q^{16} + 40 q^{22} + 20 q^{40} + 8 q^{43} - 20 q^{49} + 40 q^{55} - 28 q^{61} + 52 q^{64} + 32 q^{79} - 20 q^{82} - 40 q^{88} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 10\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.93649 + 1.11803i
−1.93649 + 1.11803i
−1.93649 − 1.11803i
1.93649 − 1.11803i
− 2.23607i 0 −3.00000 − 2.23607i 0 − 3.46410i 2.23607i 0 −5.00000
1351.2 − 2.23607i 0 −3.00000 − 2.23607i 0 3.46410i 2.23607i 0 −5.00000
1351.3 2.23607i 0 −3.00000 2.23607i 0 − 3.46410i − 2.23607i 0 −5.00000
1351.4 2.23607i 0 −3.00000 2.23607i 0 3.46410i − 2.23607i 0 −5.00000
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.g 4
3.b odd 2 1 inner 1521.2.b.g 4
13.b even 2 1 inner 1521.2.b.g 4
13.c even 3 1 117.2.q.d ✓ 4
13.d odd 4 2 1521.2.a.u 4
13.e even 6 1 117.2.q.d ✓ 4
39.d odd 2 1 inner 1521.2.b.g 4
39.f even 4 2 1521.2.a.u 4
39.h odd 6 1 117.2.q.d ✓ 4
39.i odd 6 1 117.2.q.d ✓ 4
52.i odd 6 1 1872.2.by.l 4
52.j odd 6 1 1872.2.by.l 4
156.p even 6 1 1872.2.by.l 4
156.r even 6 1 1872.2.by.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.d ✓ 4 13.c even 3 1
117.2.q.d ✓ 4 13.e even 6 1
117.2.q.d ✓ 4 39.h odd 6 1
117.2.q.d ✓ 4 39.i odd 6 1
1521.2.a.u 4 13.d odd 4 2
1521.2.a.u 4 39.f even 4 2
1521.2.b.g 4 1.a even 1 1 trivial
1521.2.b.g 4 3.b odd 2 1 inner
1521.2.b.g 4 13.b even 2 1 inner
1521.2.b.g 4 39.d odd 2 1 inner
1872.2.by.l 4 52.i odd 6 1
1872.2.by.l 4 52.j odd 6 1
1872.2.by.l 4 156.p even 6 1
1872.2.by.l 4 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 15)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 135)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 243)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
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