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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [720,2,Mod(289,720)] 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("720.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 2 \beta q^{17} + 4 q^{19} + 4 \beta q^{23} - 5 q^{25} - 8 q^{31} + 4 \beta q^{47} + 7 q^{49} + 2 \beta q^{53} + 2 q^{61} + 16 q^{79} - 8 \beta q^{83} - 10 q^{85} + 4 \beta q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{19} - 10 q^{25} - 16 q^{31} + 14 q^{49} + 4 q^{61} + 32 q^{79} - 20 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
− 2.23607i
2.23607i
0 0 0 − 2.23607i 0 0 0 0 0
289.2 0 0 0 2.23607i 0 0 0 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.f.d 2
3.b odd 2 1 inner 720.2.f.d 2
4.b odd 2 1 45.2.b.a ✓ 2
5.b even 2 1 inner 720.2.f.d 2
5.c odd 4 2 3600.2.a.bs 2
8.b even 2 1 2880.2.f.j 2
8.d odd 2 1 2880.2.f.k 2
12.b even 2 1 45.2.b.a ✓ 2
15.d odd 2 1 CM 720.2.f.d 2
15.e even 4 2 3600.2.a.bs 2
20.d odd 2 1 45.2.b.a ✓ 2
20.e even 4 2 225.2.a.f 2
24.f even 2 1 2880.2.f.k 2
24.h odd 2 1 2880.2.f.j 2
28.d even 2 1 2205.2.d.a 2
36.f odd 6 2 405.2.j.c 4
36.h even 6 2 405.2.j.c 4
40.e odd 2 1 2880.2.f.k 2
40.f even 2 1 2880.2.f.j 2
60.h even 2 1 45.2.b.a ✓ 2
60.l odd 4 2 225.2.a.f 2
84.h odd 2 1 2205.2.d.a 2
120.i odd 2 1 2880.2.f.j 2
120.m even 2 1 2880.2.f.k 2
140.c even 2 1 2205.2.d.a 2
180.n even 6 2 405.2.j.c 4
180.p odd 6 2 405.2.j.c 4
420.o odd 2 1 2205.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a ✓ 2 4.b odd 2 1
45.2.b.a ✓ 2 12.b even 2 1
45.2.b.a ✓ 2 20.d odd 2 1
45.2.b.a ✓ 2 60.h even 2 1
225.2.a.f 2 20.e even 4 2
225.2.a.f 2 60.l odd 4 2
405.2.j.c 4 36.f odd 6 2
405.2.j.c 4 36.h even 6 2
405.2.j.c 4 180.n even 6 2
405.2.j.c 4 180.p odd 6 2
720.2.f.d 2 1.a even 1 1 trivial
720.2.f.d 2 3.b odd 2 1 inner
720.2.f.d 2 5.b even 2 1 inner
720.2.f.d 2 15.d odd 2 1 CM
2205.2.d.a 2 28.d even 2 1
2205.2.d.a 2 84.h odd 2 1
2205.2.d.a 2 140.c even 2 1
2205.2.d.a 2 420.o odd 2 1
2880.2.f.j 2 8.b even 2 1
2880.2.f.j 2 24.h odd 2 1
2880.2.f.j 2 40.f even 2 1
2880.2.f.j 2 120.i odd 2 1
2880.2.f.k 2 8.d odd 2 1
2880.2.f.k 2 24.f even 2 1
2880.2.f.k 2 40.e odd 2 1
2880.2.f.k 2 120.m even 2 1
3600.2.a.bs 2 5.c odd 4 2
3600.2.a.bs 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 80 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 80 \) Copy content Toggle raw display
$53$ \( T^{2} + 20 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 320 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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