Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,2,Mod(1145,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.1145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2548.j (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.3458824350\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{5} - \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{11} + q^{13} - 2 q^{15} + ( - 2 \zeta_{6} + 2) q^{17} + \zeta_{6} q^{19} + 7 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + 4 q^{27} - 5 q^{29} + ( - 9 \zeta_{6} + 9) q^{31} - 8 \zeta_{6} q^{33} + 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{39} + 2 q^{41} + q^{43} + (\zeta_{6} - 1) q^{45} - 9 \zeta_{6} q^{47} - 4 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} - 4 q^{55} + 2 q^{57} - 14 \zeta_{6} q^{61} - \zeta_{6} q^{65} + (10 \zeta_{6} - 10) q^{67} + 14 q^{69} - 14 q^{71} + (3 \zeta_{6} - 3) q^{73} - 8 \zeta_{6} q^{75} - 5 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 5 q^{83} - 2 q^{85} + (10 \zeta_{6} - 10) q^{87} + 9 \zeta_{6} q^{89} - 18 \zeta_{6} q^{93} + ( - \zeta_{6} + 1) q^{95} - q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{5} - q^{9} + 4 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} + 8 q^{27} - 10 q^{29} + 9 q^{31} - 8 q^{33} + 2 q^{37} + 2 q^{39} + 4 q^{41} + 2 q^{43} - q^{45} - 9 q^{47} - 4 q^{51} - 3 q^{53} - 8 q^{55} + 4 q^{57} - 14 q^{61} - q^{65} - 10 q^{67} + 28 q^{69} - 28 q^{71} - 3 q^{73} - 8 q^{75} - 5 q^{79} + 11 q^{81} + 10 q^{83} - 4 q^{85} - 10 q^{87} + 9 q^{89} - 18 q^{93} + q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1145.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 1.73205i 0 −0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
1353.1 0 1.00000 + 1.73205i 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 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
7.c 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.j.j 2
7.b odd 2 1 2548.2.j.c 2
7.c even 3 1 364.2.a.a 1
7.c even 3 1 inner 2548.2.j.j 2
7.d odd 6 1 2548.2.a.i 1
7.d odd 6 1 2548.2.j.c 2
21.h odd 6 1 3276.2.a.b 1
28.g odd 6 1 1456.2.a.m 1
35.j even 6 1 9100.2.a.l 1
56.k odd 6 1 5824.2.a.d 1
56.p even 6 1 5824.2.a.bb 1
91.r even 6 1 4732.2.a.a 1
91.z odd 12 2 4732.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.a.a 1 7.c even 3 1
1456.2.a.m 1 28.g odd 6 1
2548.2.a.i 1 7.d odd 6 1
2548.2.j.c 2 7.b odd 2 1
2548.2.j.c 2 7.d odd 6 1
2548.2.j.j 2 1.a even 1 1 trivial
2548.2.j.j 2 7.c even 3 1 inner
3276.2.a.b 1 21.h odd 6 1
4732.2.a.a 1 91.r even 6 1
4732.2.g.a 2 91.z odd 12 2
5824.2.a.d 1 56.k odd 6 1
5824.2.a.bb 1 56.p even 6 1
9100.2.a.l 1 35.j even 6 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( (T - 5)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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