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Properties

Label 2548.2.l.g
Level $2548$
Weight $2$
Character orbit 2548.l
Analytic conductor $20.346$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [2548,2,Mod(373,2548)] 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("2548.373"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2548, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2548.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,4,0,-3,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.3458824350\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} + (3 \zeta_{6} - 3) q^{5} + q^{9} + (\zeta_{6} + 3) q^{13} + (6 \zeta_{6} - 6) q^{15} + ( - 3 \zeta_{6} + 3) q^{17} - 2 q^{19} + 6 \zeta_{6} q^{23} - 4 \zeta_{6} q^{25} - 4 q^{27} + (9 \zeta_{6} - 9) q^{29} + \cdots + 14 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} - 3 q^{5} + 2 q^{9} + 7 q^{13} - 6 q^{15} + 3 q^{17} - 4 q^{19} + 6 q^{23} - 4 q^{25} - 8 q^{27} - 9 q^{29} + 2 q^{31} + 7 q^{37} + 14 q^{39} + 3 q^{41} + 4 q^{43} - 3 q^{45} - 6 q^{47} + 6 q^{51}+ \cdots + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1 + \zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 2.00000 0 −1.50000 + 2.59808i 0 0 0 1.00000 0
1537.1 0 2.00000 0 −1.50000 − 2.59808i 0 0 0 1.00000 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
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.l.g 2
7.b odd 2 1 2548.2.l.b 2
7.c even 3 1 2548.2.i.b 2
7.c even 3 1 2548.2.k.a 2
7.d odd 6 1 52.2.e.b ✓ 2
7.d odd 6 1 2548.2.i.g 2
13.c even 3 1 2548.2.i.b 2
21.g even 6 1 468.2.l.d 2
28.f even 6 1 208.2.i.a 2
35.i odd 6 1 1300.2.i.b 2
35.k even 12 2 1300.2.bb.d 4
56.j odd 6 1 832.2.i.c 2
56.m even 6 1 832.2.i.i 2
84.j odd 6 1 1872.2.t.m 2
91.g even 3 1 inner 2548.2.l.g 2
91.h even 3 1 2548.2.k.a 2
91.l odd 6 1 676.2.e.d 2
91.m odd 6 1 676.2.a.a 1
91.m odd 6 1 2548.2.l.b 2
91.n odd 6 1 2548.2.i.g 2
91.p odd 6 1 676.2.a.b 1
91.s odd 6 1 676.2.e.d 2
91.v odd 6 1 52.2.e.b ✓ 2
91.w even 12 2 676.2.d.a 2
91.ba even 12 2 676.2.h.d 4
91.bb even 12 2 676.2.h.d 4
273.r even 6 1 468.2.l.d 2
273.y even 6 1 6084.2.a.c 1
273.bf even 6 1 6084.2.a.o 1
273.ch odd 12 2 6084.2.b.k 2
364.ba even 6 1 208.2.i.a 2
364.bp even 6 1 2704.2.a.m 1
364.br even 6 1 2704.2.a.l 1
364.cg odd 12 2 2704.2.f.i 2
455.y odd 6 1 1300.2.i.b 2
455.cs even 12 2 1300.2.bb.d 4
728.bq odd 6 1 832.2.i.c 2
728.cl even 6 1 832.2.i.i 2
1092.bt odd 6 1 1872.2.t.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.e.b ✓ 2 7.d odd 6 1
52.2.e.b ✓ 2 91.v odd 6 1
208.2.i.a 2 28.f even 6 1
208.2.i.a 2 364.ba even 6 1
468.2.l.d 2 21.g even 6 1
468.2.l.d 2 273.r even 6 1
676.2.a.a 1 91.m odd 6 1
676.2.a.b 1 91.p odd 6 1
676.2.d.a 2 91.w even 12 2
676.2.e.d 2 91.l odd 6 1
676.2.e.d 2 91.s odd 6 1
676.2.h.d 4 91.ba even 12 2
676.2.h.d 4 91.bb even 12 2
832.2.i.c 2 56.j odd 6 1
832.2.i.c 2 728.bq odd 6 1
832.2.i.i 2 56.m even 6 1
832.2.i.i 2 728.cl even 6 1
1300.2.i.b 2 35.i odd 6 1
1300.2.i.b 2 455.y odd 6 1
1300.2.bb.d 4 35.k even 12 2
1300.2.bb.d 4 455.cs even 12 2
1872.2.t.m 2 84.j odd 6 1
1872.2.t.m 2 1092.bt odd 6 1
2548.2.i.b 2 7.c even 3 1
2548.2.i.b 2 13.c even 3 1
2548.2.i.g 2 7.d odd 6 1
2548.2.i.g 2 91.n odd 6 1
2548.2.k.a 2 7.c even 3 1
2548.2.k.a 2 91.h even 3 1
2548.2.l.b 2 7.b odd 2 1
2548.2.l.b 2 91.m odd 6 1
2548.2.l.g 2 1.a even 1 1 trivial
2548.2.l.g 2 91.g even 3 1 inner
2704.2.a.l 1 364.br even 6 1
2704.2.a.m 1 364.bp even 6 1
2704.2.f.i 2 364.cg odd 12 2
6084.2.a.c 1 273.y even 6 1
6084.2.a.o 1 273.bf even 6 1
6084.2.b.k 2 273.ch odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 5)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
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