Properties

Label 57.1.h.a
Level 57
Weight 1
Character orbit 57.h
Analytic conductor 0.028
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 57.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0284467057201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1083.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.9747.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} - q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} - q^{7} -\zeta_{6} q^{9} + q^{12} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} + q^{19} -\zeta_{6}^{2} q^{21} -\zeta_{6} q^{25} + q^{27} + \zeta_{6} q^{28} - q^{31} + \zeta_{6}^{2} q^{36} - q^{37} - q^{39} -\zeta_{6}^{2} q^{43} -\zeta_{6} q^{48} -\zeta_{6}^{2} q^{52} + \zeta_{6}^{2} q^{57} + \zeta_{6} q^{61} + \zeta_{6} q^{63} + q^{64} + \zeta_{6} q^{67} -\zeta_{6}^{2} q^{73} + q^{75} -\zeta_{6} q^{76} -\zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} - q^{84} -\zeta_{6} q^{91} -\zeta_{6}^{2} q^{93} + 2 \zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{4} - 2q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{4} - 2q^{7} - q^{9} + 2q^{12} + q^{13} - q^{16} + 2q^{19} + q^{21} - q^{25} + 2q^{27} + q^{28} - 2q^{31} - q^{36} - 2q^{37} - 2q^{39} + q^{43} - q^{48} + q^{52} - q^{57} + q^{61} + q^{63} + 2q^{64} + q^{67} + q^{73} + 2q^{75} - q^{76} + q^{79} - q^{81} - 2q^{84} - q^{91} + q^{93} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i −0.500000 0.866025i 0 0 −1.00000 0 −0.500000 0.866025i 0
26.1 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 0 −1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.1.h.a 2
3.b odd 2 1 CM 57.1.h.a 2
4.b odd 2 1 912.1.bl.a 2
5.b even 2 1 1425.1.t.a 2
5.c odd 4 2 1425.1.o.a 4
7.b odd 2 1 2793.1.bf.a 2
7.c even 3 1 2793.1.n.a 2
7.c even 3 1 2793.1.bi.b 2
7.d odd 6 1 2793.1.n.b 2
7.d odd 6 1 2793.1.bi.a 2
8.b even 2 1 3648.1.bl.b 2
8.d odd 2 1 3648.1.bl.a 2
9.c even 3 1 1539.1.j.a 2
9.c even 3 1 1539.1.n.a 2
9.d odd 6 1 1539.1.j.a 2
9.d odd 6 1 1539.1.n.a 2
12.b even 2 1 912.1.bl.a 2
15.d odd 2 1 1425.1.t.a 2
15.e even 4 2 1425.1.o.a 4
19.b odd 2 1 1083.1.h.a 2
19.c even 3 1 inner 57.1.h.a 2
19.c even 3 1 1083.1.b.b 1
19.d odd 6 1 1083.1.b.a 1
19.d odd 6 1 1083.1.h.a 2
19.e even 9 6 1083.1.l.a 6
19.f odd 18 6 1083.1.l.b 6
21.c even 2 1 2793.1.bf.a 2
21.g even 6 1 2793.1.n.b 2
21.g even 6 1 2793.1.bi.a 2
21.h odd 6 1 2793.1.n.a 2
21.h odd 6 1 2793.1.bi.b 2
24.f even 2 1 3648.1.bl.a 2
24.h odd 2 1 3648.1.bl.b 2
57.d even 2 1 1083.1.h.a 2
57.f even 6 1 1083.1.b.a 1
57.f even 6 1 1083.1.h.a 2
57.h odd 6 1 inner 57.1.h.a 2
57.h odd 6 1 1083.1.b.b 1
57.j even 18 6 1083.1.l.b 6
57.l odd 18 6 1083.1.l.a 6
76.g odd 6 1 912.1.bl.a 2
95.i even 6 1 1425.1.t.a 2
95.m odd 12 2 1425.1.o.a 4
133.g even 3 1 2793.1.n.a 2
133.h even 3 1 2793.1.bi.b 2
133.k odd 6 1 2793.1.n.b 2
133.m odd 6 1 2793.1.bf.a 2
133.t odd 6 1 2793.1.bi.a 2
152.k odd 6 1 3648.1.bl.a 2
152.p even 6 1 3648.1.bl.b 2
171.g even 3 1 1539.1.j.a 2
171.h even 3 1 1539.1.n.a 2
171.j odd 6 1 1539.1.n.a 2
171.n odd 6 1 1539.1.j.a 2
228.m even 6 1 912.1.bl.a 2
285.n odd 6 1 1425.1.t.a 2
285.v even 12 2 1425.1.o.a 4
399.n odd 6 1 2793.1.bi.b 2
399.p even 6 1 2793.1.bi.a 2
399.z even 6 1 2793.1.bf.a 2
399.bd even 6 1 2793.1.n.b 2
399.bi odd 6 1 2793.1.n.a 2
456.u even 6 1 3648.1.bl.a 2
456.x odd 6 1 3648.1.bl.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 1.a even 1 1 trivial
57.1.h.a 2 3.b odd 2 1 CM
57.1.h.a 2 19.c even 3 1 inner
57.1.h.a 2 57.h odd 6 1 inner
912.1.bl.a 2 4.b odd 2 1
912.1.bl.a 2 12.b even 2 1
912.1.bl.a 2 76.g odd 6 1
912.1.bl.a 2 228.m even 6 1
1083.1.b.a 1 19.d odd 6 1
1083.1.b.a 1 57.f even 6 1
1083.1.b.b 1 19.c even 3 1
1083.1.b.b 1 57.h odd 6 1
1083.1.h.a 2 19.b odd 2 1
1083.1.h.a 2 19.d odd 6 1
1083.1.h.a 2 57.d even 2 1
1083.1.h.a 2 57.f even 6 1
1083.1.l.a 6 19.e even 9 6
1083.1.l.a 6 57.l odd 18 6
1083.1.l.b 6 19.f odd 18 6
1083.1.l.b 6 57.j even 18 6
1425.1.o.a 4 5.c odd 4 2
1425.1.o.a 4 15.e even 4 2
1425.1.o.a 4 95.m odd 12 2
1425.1.o.a 4 285.v even 12 2
1425.1.t.a 2 5.b even 2 1
1425.1.t.a 2 15.d odd 2 1
1425.1.t.a 2 95.i even 6 1
1425.1.t.a 2 285.n odd 6 1
1539.1.j.a 2 9.c even 3 1
1539.1.j.a 2 9.d odd 6 1
1539.1.j.a 2 171.g even 3 1
1539.1.j.a 2 171.n odd 6 1
1539.1.n.a 2 9.c even 3 1
1539.1.n.a 2 9.d odd 6 1
1539.1.n.a 2 171.h even 3 1
1539.1.n.a 2 171.j odd 6 1
2793.1.n.a 2 7.c even 3 1
2793.1.n.a 2 21.h odd 6 1
2793.1.n.a 2 133.g even 3 1
2793.1.n.a 2 399.bi odd 6 1
2793.1.n.b 2 7.d odd 6 1
2793.1.n.b 2 21.g even 6 1
2793.1.n.b 2 133.k odd 6 1
2793.1.n.b 2 399.bd even 6 1
2793.1.bf.a 2 7.b odd 2 1
2793.1.bf.a 2 21.c even 2 1
2793.1.bf.a 2 133.m odd 6 1
2793.1.bf.a 2 399.z even 6 1
2793.1.bi.a 2 7.d odd 6 1
2793.1.bi.a 2 21.g even 6 1
2793.1.bi.a 2 133.t odd 6 1
2793.1.bi.a 2 399.p even 6 1
2793.1.bi.b 2 7.c even 3 1
2793.1.bi.b 2 21.h odd 6 1
2793.1.bi.b 2 133.h even 3 1
2793.1.bi.b 2 399.n odd 6 1
3648.1.bl.a 2 8.d odd 2 1
3648.1.bl.a 2 24.f even 2 1
3648.1.bl.a 2 152.k odd 6 1
3648.1.bl.a 2 456.u even 6 1
3648.1.bl.b 2 8.b even 2 1
3648.1.bl.b 2 24.h odd 2 1
3648.1.bl.b 2 152.p even 6 1
3648.1.bl.b 2 456.x odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

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