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Properties

Label 2535.2.a.j
Level $2535$
Weight $2$
Character orbit 2535.a
Self dual yes
Analytic conductor $20.242$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2535,2,Mod(1,2535)] 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("2535.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2535, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1,-1,-1,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2420769124\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3 q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} + q^{15} - q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + q^{20} + 4 q^{22} + 3 q^{24} + q^{25} - q^{27}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −1.00000 −1.00000 −1.00000 −1.00000 0 −3.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2535.2.a.j 1
3.b odd 2 1 7605.2.a.g 1
13.b even 2 1 15.2.a.a ✓ 1
39.d odd 2 1 45.2.a.a 1
52.b odd 2 1 240.2.a.d 1
65.d even 2 1 75.2.a.b 1
65.h odd 4 2 75.2.b.b 2
91.b odd 2 1 735.2.a.c 1
91.r even 6 2 735.2.i.e 2
91.s odd 6 2 735.2.i.d 2
104.e even 2 1 960.2.a.l 1
104.h odd 2 1 960.2.a.a 1
117.n odd 6 2 405.2.e.c 2
117.t even 6 2 405.2.e.f 2
143.d odd 2 1 1815.2.a.d 1
156.h even 2 1 720.2.a.c 1
195.e odd 2 1 225.2.a.b 1
195.s even 4 2 225.2.b.b 2
208.o odd 4 2 3840.2.k.r 2
208.p even 4 2 3840.2.k.m 2
221.b even 2 1 4335.2.a.c 1
247.d odd 2 1 5415.2.a.j 1
260.g odd 2 1 1200.2.a.e 1
260.p even 4 2 1200.2.f.h 2
273.g even 2 1 2205.2.a.i 1
299.c odd 2 1 7935.2.a.d 1
312.b odd 2 1 2880.2.a.y 1
312.h even 2 1 2880.2.a.bc 1
429.e even 2 1 5445.2.a.c 1
455.h odd 2 1 3675.2.a.j 1
520.b odd 2 1 4800.2.a.bz 1
520.p even 2 1 4800.2.a.t 1
520.bc even 4 2 4800.2.f.c 2
520.bg odd 4 2 4800.2.f.bf 2
715.c odd 2 1 9075.2.a.g 1
780.d even 2 1 3600.2.a.u 1
780.w odd 4 2 3600.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a ✓ 1 13.b even 2 1
45.2.a.a 1 39.d odd 2 1
75.2.a.b 1 65.d even 2 1
75.2.b.b 2 65.h odd 4 2
225.2.a.b 1 195.e odd 2 1
225.2.b.b 2 195.s even 4 2
240.2.a.d 1 52.b odd 2 1
405.2.e.c 2 117.n odd 6 2
405.2.e.f 2 117.t even 6 2
720.2.a.c 1 156.h even 2 1
735.2.a.c 1 91.b odd 2 1
735.2.i.d 2 91.s odd 6 2
735.2.i.e 2 91.r even 6 2
960.2.a.a 1 104.h odd 2 1
960.2.a.l 1 104.e even 2 1
1200.2.a.e 1 260.g odd 2 1
1200.2.f.h 2 260.p even 4 2
1815.2.a.d 1 143.d odd 2 1
2205.2.a.i 1 273.g even 2 1
2535.2.a.j 1 1.a even 1 1 trivial
2880.2.a.y 1 312.b odd 2 1
2880.2.a.bc 1 312.h even 2 1
3600.2.a.u 1 780.d even 2 1
3600.2.f.e 2 780.w odd 4 2
3675.2.a.j 1 455.h odd 2 1
3840.2.k.m 2 208.p even 4 2
3840.2.k.r 2 208.o odd 4 2
4335.2.a.c 1 221.b even 2 1
4800.2.a.t 1 520.p even 2 1
4800.2.a.bz 1 520.b odd 2 1
4800.2.f.c 2 520.bc even 4 2
4800.2.f.bf 2 520.bg odd 4 2
5415.2.a.j 1 247.d odd 2 1
5445.2.a.c 1 429.e even 2 1
7605.2.a.g 1 3.b odd 2 1
7935.2.a.d 1 299.c odd 2 1
9075.2.a.g 1 715.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2535))\):

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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