Properties

Label 52.1.j.a
Level 52
Weight 1
Character orbit 52.j
Analytic conductor 0.026
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 52.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0259513806569\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 6.0.10816.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - q^{5} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} -\zeta_{6} q^{13} -\zeta_{6} q^{16} -\zeta_{6}^{2} q^{17} + q^{18} -\zeta_{6}^{2} q^{20} + \zeta_{6}^{2} q^{26} + \zeta_{6} q^{29} + \zeta_{6}^{2} q^{32} - q^{34} -\zeta_{6} q^{36} + \zeta_{6} q^{37} - q^{40} + \zeta_{6} q^{41} -\zeta_{6}^{2} q^{45} -\zeta_{6} q^{49} + q^{52} - q^{53} -\zeta_{6}^{2} q^{58} -\zeta_{6}^{2} q^{61} + q^{64} + \zeta_{6} q^{65} + \zeta_{6} q^{68} + \zeta_{6}^{2} q^{72} - q^{73} -\zeta_{6}^{2} q^{74} + \zeta_{6} q^{80} -\zeta_{6} q^{81} -\zeta_{6}^{2} q^{82} + \zeta_{6}^{2} q^{85} -2 \zeta_{6} q^{89} - q^{90} + 2 \zeta_{6}^{2} q^{97} + \zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 2q^{5} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 2q^{5} + 2q^{8} - q^{9} + q^{10} - q^{13} - q^{16} + q^{17} + 2q^{18} + q^{20} - q^{26} + q^{29} - q^{32} - 2q^{34} - q^{36} + q^{37} - 2q^{40} + q^{41} + q^{45} - q^{49} + 2q^{52} - 2q^{53} + q^{58} + q^{61} + 2q^{64} + q^{65} + q^{68} - q^{72} - 2q^{73} + q^{74} + q^{80} - q^{81} + q^{82} - q^{85} - 2q^{89} - 2q^{90} - 2q^{97} - q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
35.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 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 \(\Q(\sqrt{-1}) \)
13.c even 3 1 inner
52.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.1.j.a 2
3.b odd 2 1 468.1.br.a 2
4.b odd 2 1 CM 52.1.j.a 2
5.b even 2 1 1300.1.bc.a 2
5.c odd 4 2 1300.1.w.a 4
7.b odd 2 1 2548.1.bn.a 2
7.c even 3 1 2548.1.q.b 2
7.c even 3 1 2548.1.bi.b 2
7.d odd 6 1 2548.1.q.a 2
7.d odd 6 1 2548.1.bi.a 2
8.b even 2 1 832.1.bb.a 2
8.d odd 2 1 832.1.bb.a 2
12.b even 2 1 468.1.br.a 2
13.b even 2 1 676.1.j.a 2
13.c even 3 1 inner 52.1.j.a 2
13.c even 3 1 676.1.c.b 1
13.d odd 4 2 676.1.i.a 4
13.e even 6 1 676.1.c.a 1
13.e even 6 1 676.1.j.a 2
13.f odd 12 2 676.1.b.a 2
13.f odd 12 2 676.1.i.a 4
16.e even 4 2 3328.1.v.b 4
16.f odd 4 2 3328.1.v.b 4
20.d odd 2 1 1300.1.bc.a 2
20.e even 4 2 1300.1.w.a 4
28.d even 2 1 2548.1.bn.a 2
28.f even 6 1 2548.1.q.a 2
28.f even 6 1 2548.1.bi.a 2
28.g odd 6 1 2548.1.q.b 2
28.g odd 6 1 2548.1.bi.b 2
39.i odd 6 1 468.1.br.a 2
52.b odd 2 1 676.1.j.a 2
52.f even 4 2 676.1.i.a 4
52.i odd 6 1 676.1.c.a 1
52.i odd 6 1 676.1.j.a 2
52.j odd 6 1 inner 52.1.j.a 2
52.j odd 6 1 676.1.c.b 1
52.l even 12 2 676.1.b.a 2
52.l even 12 2 676.1.i.a 4
65.n even 6 1 1300.1.bc.a 2
65.q odd 12 2 1300.1.w.a 4
91.g even 3 1 2548.1.bi.b 2
91.h even 3 1 2548.1.q.b 2
91.m odd 6 1 2548.1.bi.a 2
91.n odd 6 1 2548.1.bn.a 2
91.v odd 6 1 2548.1.q.a 2
104.n odd 6 1 832.1.bb.a 2
104.r even 6 1 832.1.bb.a 2
156.p even 6 1 468.1.br.a 2
208.bg odd 12 2 3328.1.v.b 4
208.bj even 12 2 3328.1.v.b 4
260.v odd 6 1 1300.1.bc.a 2
260.bj even 12 2 1300.1.w.a 4
364.q odd 6 1 2548.1.bi.b 2
364.v even 6 1 2548.1.bn.a 2
364.ba even 6 1 2548.1.q.a 2
364.bi odd 6 1 2548.1.q.b 2
364.br even 6 1 2548.1.bi.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 1.a even 1 1 trivial
52.1.j.a 2 4.b odd 2 1 CM
52.1.j.a 2 13.c even 3 1 inner
52.1.j.a 2 52.j odd 6 1 inner
468.1.br.a 2 3.b odd 2 1
468.1.br.a 2 12.b even 2 1
468.1.br.a 2 39.i odd 6 1
468.1.br.a 2 156.p even 6 1
676.1.b.a 2 13.f odd 12 2
676.1.b.a 2 52.l even 12 2
676.1.c.a 1 13.e even 6 1
676.1.c.a 1 52.i odd 6 1
676.1.c.b 1 13.c even 3 1
676.1.c.b 1 52.j odd 6 1
676.1.i.a 4 13.d odd 4 2
676.1.i.a 4 13.f odd 12 2
676.1.i.a 4 52.f even 4 2
676.1.i.a 4 52.l even 12 2
676.1.j.a 2 13.b even 2 1
676.1.j.a 2 13.e even 6 1
676.1.j.a 2 52.b odd 2 1
676.1.j.a 2 52.i odd 6 1
832.1.bb.a 2 8.b even 2 1
832.1.bb.a 2 8.d odd 2 1
832.1.bb.a 2 104.n odd 6 1
832.1.bb.a 2 104.r even 6 1
1300.1.w.a 4 5.c odd 4 2
1300.1.w.a 4 20.e even 4 2
1300.1.w.a 4 65.q odd 12 2
1300.1.w.a 4 260.bj even 12 2
1300.1.bc.a 2 5.b even 2 1
1300.1.bc.a 2 20.d odd 2 1
1300.1.bc.a 2 65.n even 6 1
1300.1.bc.a 2 260.v 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.v odd 6 1
2548.1.q.a 2 364.ba even 6 1
2548.1.q.b 2 7.c even 3 1
2548.1.q.b 2 28.g odd 6 1
2548.1.q.b 2 91.h even 3 1
2548.1.q.b 2 364.bi odd 6 1
2548.1.bi.a 2 7.d odd 6 1
2548.1.bi.a 2 28.f even 6 1
2548.1.bi.a 2 91.m odd 6 1
2548.1.bi.a 2 364.br even 6 1
2548.1.bi.b 2 7.c even 3 1
2548.1.bi.b 2 28.g odd 6 1
2548.1.bi.b 2 91.g even 3 1
2548.1.bi.b 2 364.q odd 6 1
2548.1.bn.a 2 7.b odd 2 1
2548.1.bn.a 2 28.d even 2 1
2548.1.bn.a 2 91.n odd 6 1
2548.1.bn.a 2 364.v even 6 1
3328.1.v.b 4 16.e even 4 2
3328.1.v.b 4 16.f odd 4 2
3328.1.v.b 4 208.bg odd 12 2
3328.1.v.b 4 208.bj even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

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