Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2704,2,Mod(1,2704)] 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("2704.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,3,0,1,0,1] 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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9} - 2 q^{11} + 3 q^{15} - 3 q^{17} + 6 q^{19} + 3 q^{21} + 4 q^{23} - 4 q^{25} + 9 q^{27} + 2 q^{29} + 4 q^{31} - 6 q^{33} + q^{35} - 3 q^{37} + 5 q^{43} + 6 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
0 3.00000 0 1.00000 0 1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( +1 \)

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 2704.2.a.n 1
4.b odd 2 1 338.2.a.a 1
12.b even 2 1 3042.2.a.l 1
13.b even 2 1 208.2.a.d 1
13.d odd 4 2 2704.2.f.j 2
20.d odd 2 1 8450.2.a.y 1
39.d odd 2 1 1872.2.a.m 1
52.b odd 2 1 26.2.a.b ✓ 1
52.f even 4 2 338.2.b.a 2
52.i odd 6 2 338.2.c.c 2
52.j odd 6 2 338.2.c.g 2
52.l even 12 4 338.2.e.d 4
65.d even 2 1 5200.2.a.c 1
104.e even 2 1 832.2.a.a 1
104.h odd 2 1 832.2.a.j 1
156.h even 2 1 234.2.a.b 1
156.l odd 4 2 3042.2.b.f 2
208.o odd 4 2 3328.2.b.g 2
208.p even 4 2 3328.2.b.k 2
260.g odd 2 1 650.2.a.g 1
260.p even 4 2 650.2.b.a 2
312.b odd 2 1 7488.2.a.v 1
312.h even 2 1 7488.2.a.w 1
364.h even 2 1 1274.2.a.o 1
364.x even 6 2 1274.2.f.a 2
364.bl odd 6 2 1274.2.f.l 2
468.x even 6 2 2106.2.e.t 2
468.bg odd 6 2 2106.2.e.h 2
572.b even 2 1 3146.2.a.a 1
780.d even 2 1 5850.2.a.bn 1
780.w odd 4 2 5850.2.e.v 2
884.h odd 2 1 7514.2.a.i 1
988.g even 2 1 9386.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b ✓ 1 52.b odd 2 1
208.2.a.d 1 13.b even 2 1
234.2.a.b 1 156.h even 2 1
338.2.a.a 1 4.b odd 2 1
338.2.b.a 2 52.f even 4 2
338.2.c.c 2 52.i odd 6 2
338.2.c.g 2 52.j odd 6 2
338.2.e.d 4 52.l even 12 4
650.2.a.g 1 260.g odd 2 1
650.2.b.a 2 260.p even 4 2
832.2.a.a 1 104.e even 2 1
832.2.a.j 1 104.h odd 2 1
1274.2.a.o 1 364.h even 2 1
1274.2.f.a 2 364.x even 6 2
1274.2.f.l 2 364.bl odd 6 2
1872.2.a.m 1 39.d odd 2 1
2106.2.e.h 2 468.bg odd 6 2
2106.2.e.t 2 468.x even 6 2
2704.2.a.n 1 1.a even 1 1 trivial
2704.2.f.j 2 13.d odd 4 2
3042.2.a.l 1 12.b even 2 1
3042.2.b.f 2 156.l odd 4 2
3146.2.a.a 1 572.b even 2 1
3328.2.b.g 2 208.o odd 4 2
3328.2.b.k 2 208.p even 4 2
5200.2.a.c 1 65.d even 2 1
5850.2.a.bn 1 780.d even 2 1
5850.2.e.v 2 780.w odd 4 2
7488.2.a.v 1 312.b odd 2 1
7488.2.a.w 1 312.h even 2 1
7514.2.a.i 1 884.h odd 2 1
8450.2.a.y 1 20.d odd 2 1
9386.2.a.f 1 988.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T - 13 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 5 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 14 \) Copy content Toggle raw display
show more
show less