Properties

Label 1296.1.q.a
Level 1296
Weight 1
Character orbit 1296.q
Analytic conductor 0.647
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1296.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.108.1
Artin image $C_6\times S_3$
Artin field Galois closure of 12.0.25389989167104.5

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{7} -\zeta_{6}^{2} q^{13} + q^{19} -\zeta_{6} q^{25} -2 \zeta_{6}^{2} q^{31} - q^{37} + 2 \zeta_{6} q^{43} + \zeta_{6} q^{61} + \zeta_{6}^{2} q^{67} - q^{73} -\zeta_{6} q^{79} - q^{91} + \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} + q^{13} + 2q^{19} - q^{25} + 2q^{31} - 2q^{37} + 2q^{43} + q^{61} - q^{67} - 2q^{73} - q^{79} - 2q^{91} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −0.500000 + 0.866025i 0 0 0
1025.1 0 0 0 0 0 −0.500000 0.866025i 0 0 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 \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.1.q.a 2
3.b odd 2 1 CM 1296.1.q.a 2
4.b odd 2 1 324.1.g.a 2
9.c even 3 1 432.1.e.a 1
9.c even 3 1 inner 1296.1.q.a 2
9.d odd 6 1 432.1.e.a 1
9.d odd 6 1 inner 1296.1.q.a 2
12.b even 2 1 324.1.g.a 2
36.f odd 6 1 108.1.c.a 1
36.f odd 6 1 324.1.g.a 2
36.h even 6 1 108.1.c.a 1
36.h even 6 1 324.1.g.a 2
72.j odd 6 1 1728.1.e.b 1
72.l even 6 1 1728.1.e.a 1
72.n even 6 1 1728.1.e.b 1
72.p odd 6 1 1728.1.e.a 1
108.j odd 18 6 2916.1.k.c 6
108.l even 18 6 2916.1.k.c 6
180.n even 6 1 2700.1.g.b 1
180.p odd 6 1 2700.1.g.b 1
180.v odd 12 2 2700.1.b.b 2
180.x even 12 2 2700.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 36.f odd 6 1
108.1.c.a 1 36.h even 6 1
324.1.g.a 2 4.b odd 2 1
324.1.g.a 2 12.b even 2 1
324.1.g.a 2 36.f odd 6 1
324.1.g.a 2 36.h even 6 1
432.1.e.a 1 9.c even 3 1
432.1.e.a 1 9.d odd 6 1
1296.1.q.a 2 1.a even 1 1 trivial
1296.1.q.a 2 3.b odd 2 1 CM
1296.1.q.a 2 9.c even 3 1 inner
1296.1.q.a 2 9.d odd 6 1 inner
1728.1.e.a 1 72.l even 6 1
1728.1.e.a 1 72.p odd 6 1
1728.1.e.b 1 72.j odd 6 1
1728.1.e.b 1 72.n even 6 1
2700.1.b.b 2 180.v odd 12 2
2700.1.b.b 2 180.x even 12 2
2700.1.g.b 1 180.n even 6 1
2700.1.g.b 1 180.p odd 6 1
2916.1.k.c 6 108.j odd 18 6
2916.1.k.c 6 108.l even 18 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1296, [\chi])\).

Hecke characteristic polynomials

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