↑

Properties

Label 525.2.g.b
Level $525$
Weight $2$
Character orbit 525.g
Analytic conductor $4.192$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [525,2,Mod(524,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.524"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} + q^{4} - 3 q^{6} + ( - \beta_{3} - \beta_1) q^{7} - \beta_1 q^{8} + 3 q^{9} - \beta_{2} q^{11} - \beta_1 q^{12} + ( - \beta_{2} - 3) q^{14} - 5 q^{16} - 3 \beta_{3} q^{17}+ \cdots - 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{6} + 12 q^{9} - 12 q^{14} - 20 q^{16} + 12 q^{21} + 12 q^{24} + 12 q^{36} + 24 q^{41} - 24 q^{46} - 4 q^{49} - 36 q^{54} + 12 q^{56} + 48 q^{59} + 4 q^{64} + 24 q^{69} - 32 q^{79}+ \cdots + 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−1.73205 1.73205 1.00000 0 −3.00000 1.73205 − 2.00000i 1.73205 3.00000 0
524.2 −1.73205 1.73205 1.00000 0 −3.00000 1.73205 + 2.00000i 1.73205 3.00000 0
524.3 1.73205 −1.73205 1.00000 0 −3.00000 −1.73205 − 2.00000i −1.73205 3.00000 0
524.4 1.73205 −1.73205 1.00000 0 −3.00000 −1.73205 + 2.00000i −1.73205 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.c even 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.g.b 4
3.b odd 2 1 525.2.g.c 4
5.b even 2 1 inner 525.2.g.b 4
5.c odd 4 1 105.2.b.a ✓ 2
5.c odd 4 1 525.2.b.a 2
7.b odd 2 1 525.2.g.c 4
15.d odd 2 1 525.2.g.c 4
15.e even 4 1 105.2.b.b yes 2
15.e even 4 1 525.2.b.b 2
20.e even 4 1 1680.2.f.b 2
21.c even 2 1 inner 525.2.g.b 4
35.c odd 2 1 525.2.g.c 4
35.f even 4 1 105.2.b.b yes 2
35.f even 4 1 525.2.b.b 2
35.k even 12 1 735.2.s.a 2
35.k even 12 1 735.2.s.f 2
35.l odd 12 1 735.2.s.b 2
35.l odd 12 1 735.2.s.d 2
60.l odd 4 1 1680.2.f.c 2
105.g even 2 1 inner 525.2.g.b 4
105.k odd 4 1 105.2.b.a ✓ 2
105.k odd 4 1 525.2.b.a 2
105.w odd 12 1 735.2.s.b 2
105.w odd 12 1 735.2.s.d 2
105.x even 12 1 735.2.s.a 2
105.x even 12 1 735.2.s.f 2
140.j odd 4 1 1680.2.f.c 2
420.w even 4 1 1680.2.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a ✓ 2 5.c odd 4 1
105.2.b.a ✓ 2 105.k odd 4 1
105.2.b.b yes 2 15.e even 4 1
105.2.b.b yes 2 35.f even 4 1
525.2.b.a 2 5.c odd 4 1
525.2.b.a 2 105.k odd 4 1
525.2.b.b 2 15.e even 4 1
525.2.b.b 2 35.f even 4 1
525.2.g.b 4 1.a even 1 1 trivial
525.2.g.b 4 5.b even 2 1 inner
525.2.g.b 4 21.c even 2 1 inner
525.2.g.b 4 105.g even 2 1 inner
525.2.g.c 4 3.b odd 2 1
525.2.g.c 4 7.b odd 2 1
525.2.g.c 4 15.d odd 2 1
525.2.g.c 4 35.c odd 2 1
735.2.s.a 2 35.k even 12 1
735.2.s.a 2 105.x even 12 1
735.2.s.b 2 35.l odd 12 1
735.2.s.b 2 105.w odd 12 1
735.2.s.d 2 35.l odd 12 1
735.2.s.d 2 105.w odd 12 1
735.2.s.f 2 35.k even 12 1
735.2.s.f 2 105.x even 12 1
1680.2.f.b 2 20.e even 4 1
1680.2.f.b 2 420.w even 4 1
1680.2.f.c 2 60.l odd 4 1
1680.2.f.c 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{41} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T - 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
show more
show less