Properties

Label 2548.1.bi.b
Level $2548$
Weight $1$
Character orbit 2548.bi
Analytic conductor $1.272$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2548.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.27161765219\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.676.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.25969216.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\zeta_{6}^{2} q^{5} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + q^{2} + q^{4} -\zeta_{6}^{2} q^{5} + q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{10} -\zeta_{6} q^{13} + q^{16} - q^{17} -\zeta_{6} q^{18} -\zeta_{6}^{2} q^{20} -\zeta_{6} q^{26} + \zeta_{6} q^{29} + q^{32} - q^{34} -\zeta_{6} q^{36} - q^{37} -\zeta_{6}^{2} q^{40} + \zeta_{6} q^{41} - q^{45} -\zeta_{6} q^{52} + \zeta_{6} q^{53} + \zeta_{6} q^{58} + \zeta_{6} q^{61} + q^{64} - q^{65} - q^{68} -\zeta_{6} q^{72} + \zeta_{6} q^{73} - q^{74} -\zeta_{6}^{2} q^{80} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} + \zeta_{6}^{2} q^{85} + 2 q^{89} - q^{90} + 2 \zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + q^{5} + 2q^{8} - q^{9} + q^{10} - q^{13} + 2q^{16} - 2q^{17} - q^{18} + q^{20} - q^{26} + q^{29} + 2q^{32} - 2q^{34} - q^{36} - 2q^{37} + q^{40} + q^{41} - 2q^{45} - q^{52} + q^{53} + q^{58} + q^{61} + 2q^{64} - 2q^{65} - 2q^{68} - q^{72} + q^{73} - 2q^{74} + q^{80} - q^{81} + q^{82} - q^{85} + 4q^{89} - 2q^{90} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(\zeta_{6}^{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
471.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 0 1.00000 0.500000 0.866025i 0 0 1.00000 −0.500000 0.866025i 0.500000 0.866025i
1439.1 1.00000 0 1.00000 0.500000 + 0.866025i 0 0 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
91.h even 3 1 inner
364.bi odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.1.bi.b 2
4.b odd 2 1 CM 2548.1.bi.b 2
7.b odd 2 1 2548.1.bi.a 2
7.c even 3 1 52.1.j.a 2
7.c even 3 1 2548.1.q.b 2
7.d odd 6 1 2548.1.q.a 2
7.d odd 6 1 2548.1.bn.a 2
13.c even 3 1 2548.1.q.b 2
21.h odd 6 1 468.1.br.a 2
28.d even 2 1 2548.1.bi.a 2
28.f even 6 1 2548.1.q.a 2
28.f even 6 1 2548.1.bn.a 2
28.g odd 6 1 52.1.j.a 2
28.g odd 6 1 2548.1.q.b 2
35.j even 6 1 1300.1.bc.a 2
35.l odd 12 2 1300.1.w.a 4
52.j odd 6 1 2548.1.q.b 2
56.k odd 6 1 832.1.bb.a 2
56.p even 6 1 832.1.bb.a 2
84.n even 6 1 468.1.br.a 2
91.g even 3 1 52.1.j.a 2
91.h even 3 1 676.1.c.b 1
91.h even 3 1 inner 2548.1.bi.b 2
91.k even 6 1 676.1.c.a 1
91.m odd 6 1 2548.1.bn.a 2
91.n odd 6 1 2548.1.q.a 2
91.r even 6 1 676.1.j.a 2
91.u even 6 1 676.1.j.a 2
91.v odd 6 1 2548.1.bi.a 2
91.x odd 12 2 676.1.b.a 2
91.z odd 12 2 676.1.i.a 4
91.bd odd 12 2 676.1.i.a 4
112.u odd 12 2 3328.1.v.b 4
112.w even 12 2 3328.1.v.b 4
140.p odd 6 1 1300.1.bc.a 2
140.w even 12 2 1300.1.w.a 4
273.bm odd 6 1 468.1.br.a 2
364.q odd 6 1 52.1.j.a 2
364.s odd 6 1 676.1.j.a 2
364.v even 6 1 2548.1.q.a 2
364.ba even 6 1 2548.1.bi.a 2
364.bi odd 6 1 676.1.c.b 1
364.bi odd 6 1 inner 2548.1.bi.b 2
364.bk odd 6 1 676.1.c.a 1
364.bl odd 6 1 676.1.j.a 2
364.br even 6 1 2548.1.bn.a 2
364.bt even 12 2 676.1.i.a 4
364.ca even 12 2 676.1.b.a 2
364.ce even 12 2 676.1.i.a 4
455.bm even 6 1 1300.1.bc.a 2
455.cx odd 12 2 1300.1.w.a 4
728.bg even 6 1 832.1.bb.a 2
728.di odd 6 1 832.1.bb.a 2
1092.dc even 6 1 468.1.br.a 2
1456.fh even 12 2 3328.1.v.b 4
1456.fk odd 12 2 3328.1.v.b 4
1820.dv odd 6 1 1300.1.bc.a 2
1820.es even 12 2 1300.1.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 7.c even 3 1
52.1.j.a 2 28.g odd 6 1
52.1.j.a 2 91.g even 3 1
52.1.j.a 2 364.q odd 6 1
468.1.br.a 2 21.h odd 6 1
468.1.br.a 2 84.n even 6 1
468.1.br.a 2 273.bm odd 6 1
468.1.br.a 2 1092.dc even 6 1
676.1.b.a 2 91.x odd 12 2
676.1.b.a 2 364.ca even 12 2
676.1.c.a 1 91.k even 6 1
676.1.c.a 1 364.bk odd 6 1
676.1.c.b 1 91.h even 3 1
676.1.c.b 1 364.bi odd 6 1
676.1.i.a 4 91.z odd 12 2
676.1.i.a 4 91.bd odd 12 2
676.1.i.a 4 364.bt even 12 2
676.1.i.a 4 364.ce even 12 2
676.1.j.a 2 91.r even 6 1
676.1.j.a 2 91.u even 6 1
676.1.j.a 2 364.s odd 6 1
676.1.j.a 2 364.bl odd 6 1
832.1.bb.a 2 56.k odd 6 1
832.1.bb.a 2 56.p even 6 1
832.1.bb.a 2 728.bg even 6 1
832.1.bb.a 2 728.di odd 6 1
1300.1.w.a 4 35.l odd 12 2
1300.1.w.a 4 140.w even 12 2
1300.1.w.a 4 455.cx odd 12 2
1300.1.w.a 4 1820.es even 12 2
1300.1.bc.a 2 35.j even 6 1
1300.1.bc.a 2 140.p odd 6 1
1300.1.bc.a 2 455.bm even 6 1
1300.1.bc.a 2 1820.dv odd 6 1
2548.1.q.a 2 7.d odd 6 1
2548.1.q.a 2 28.f even 6 1
2548.1.q.a 2 91.n odd 6 1
2548.1.q.a 2 364.v even 6 1
2548.1.q.b 2 7.c even 3 1
2548.1.q.b 2 13.c even 3 1
2548.1.q.b 2 28.g odd 6 1
2548.1.q.b 2 52.j odd 6 1
2548.1.bi.a 2 7.b odd 2 1
2548.1.bi.a 2 28.d even 2 1
2548.1.bi.a 2 91.v odd 6 1
2548.1.bi.a 2 364.ba even 6 1
2548.1.bi.b 2 1.a even 1 1 trivial
2548.1.bi.b 2 4.b odd 2 1 CM
2548.1.bi.b 2 91.h even 3 1 inner
2548.1.bi.b 2 364.bi odd 6 1 inner
2548.1.bn.a 2 7.d odd 6 1
2548.1.bn.a 2 28.f even 6 1
2548.1.bn.a 2 91.m odd 6 1
2548.1.bn.a 2 364.br even 6 1
3328.1.v.b 4 112.u odd 12 2
3328.1.v.b 4 112.w even 12 2
3328.1.v.b 4 1456.fh even 12 2
3328.1.v.b 4 1456.fk odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2548, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1 + T + T^{2} \)
$17$ \( ( 1 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1 - T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( 1 - T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 1 - T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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