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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,5,Mod(449,450)] 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("450.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,32,0,0,0,0,0,0,0,0,0,0,0,256,0,0,556] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.5164833877\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + 8 q^{4} - 83 \beta_1 q^{7} - 16 \beta_{3} q^{8} + 57 \beta_{2} q^{11} + 41 \beta_1 q^{13} + 166 \beta_{2} q^{14} + 64 q^{16} + 363 \beta_{3} q^{17} + 139 q^{19} - 228 \beta_1 q^{22}+ \cdots + 8976 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4} + 256 q^{16} + 556 q^{19} - 4204 q^{31} - 5808 q^{34} - 2544 q^{46} - 17952 q^{49} + 23300 q^{61} + 2048 q^{64} + 4448 q^{76} - 37184 q^{79} + 13612 q^{91} - 33360 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.707107 + 0.707107i
0.707107 − 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
−2.82843 0 8.00000 0 0 − 83.0000i −22.6274 0 0
449.2 −2.82843 0 8.00000 0 0 83.0000i −22.6274 0 0
449.3 2.82843 0 8.00000 0 0 − 83.0000i 22.6274 0 0
449.4 2.82843 0 8.00000 0 0 83.0000i 22.6274 0 0
\(n\): 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
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.5.b.b 4
3.b odd 2 1 inner 450.5.b.b 4
5.b even 2 1 inner 450.5.b.b 4
5.c odd 4 1 450.5.d.a ✓ 2
5.c odd 4 1 450.5.d.d yes 2
15.d odd 2 1 inner 450.5.b.b 4
15.e even 4 1 450.5.d.a ✓ 2
15.e even 4 1 450.5.d.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.5.b.b 4 1.a even 1 1 trivial
450.5.b.b 4 3.b odd 2 1 inner
450.5.b.b 4 5.b even 2 1 inner
450.5.b.b 4 15.d odd 2 1 inner
450.5.d.a ✓ 2 5.c odd 4 1
450.5.d.a ✓ 2 15.e even 4 1
450.5.d.d yes 2 5.c odd 4 1
450.5.d.d yes 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 6889 \) acting on \(S_{5}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6889)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 6498)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 263538)^{2} \) Copy content Toggle raw display
$19$ \( (T - 139)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 50562)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 455058)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1051)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2795584)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 691488)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6325225)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8694450)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 152352)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 563922)^{2} \) Copy content Toggle raw display
$61$ \( (T - 5825)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 52693081)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 47706912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 20720704)^{2} \) Copy content Toggle raw display
$79$ \( (T + 9296)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 63686898)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36911232)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3214849)^{2} \) Copy content Toggle raw display
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