Properties

Label 1296.1.o.b
Level 1296
Weight 1
Character orbit 1296.o
Analytic conductor 0.647
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -3, -4, 12
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{12})\)
Artin image $C_3\times D_4$
Artin field Galois closure of 12.6.304679870005248.2

$q$-expansion

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

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{25} + 2 q^{37} + \zeta_{6}^{2} q^{49} -2 \zeta_{6} q^{61} -2 q^{73} + 2 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{13} + q^{25} + 4q^{37} - q^{49} - 2q^{61} - 4q^{73} + 2q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 0 0 0 0
703.1 0 0 0 0 0 0 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.1.o.b 2
3.b odd 2 1 CM 1296.1.o.b 2
4.b odd 2 1 CM 1296.1.o.b 2
9.c even 3 1 144.1.g.a 1
9.c even 3 1 inner 1296.1.o.b 2
9.d odd 6 1 144.1.g.a 1
9.d odd 6 1 inner 1296.1.o.b 2
12.b even 2 1 RM 1296.1.o.b 2
36.f odd 6 1 144.1.g.a 1
36.f odd 6 1 inner 1296.1.o.b 2
36.h even 6 1 144.1.g.a 1
36.h even 6 1 inner 1296.1.o.b 2
45.h odd 6 1 3600.1.e.b 1
45.j even 6 1 3600.1.e.b 1
45.k odd 12 2 3600.1.j.a 2
45.l even 12 2 3600.1.j.a 2
72.j odd 6 1 576.1.g.a 1
72.l even 6 1 576.1.g.a 1
72.n even 6 1 576.1.g.a 1
72.p odd 6 1 576.1.g.a 1
144.u even 12 2 2304.1.b.a 2
144.v odd 12 2 2304.1.b.a 2
144.w odd 12 2 2304.1.b.a 2
144.x even 12 2 2304.1.b.a 2
180.n even 6 1 3600.1.e.b 1
180.p odd 6 1 3600.1.e.b 1
180.v odd 12 2 3600.1.j.a 2
180.x even 12 2 3600.1.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 9.c even 3 1
144.1.g.a 1 9.d odd 6 1
144.1.g.a 1 36.f odd 6 1
144.1.g.a 1 36.h even 6 1
576.1.g.a 1 72.j odd 6 1
576.1.g.a 1 72.l even 6 1
576.1.g.a 1 72.n even 6 1
576.1.g.a 1 72.p odd 6 1
1296.1.o.b 2 1.a even 1 1 trivial
1296.1.o.b 2 3.b odd 2 1 CM
1296.1.o.b 2 4.b odd 2 1 CM
1296.1.o.b 2 9.c even 3 1 inner
1296.1.o.b 2 9.d odd 6 1 inner
1296.1.o.b 2 12.b even 2 1 RM
1296.1.o.b 2 36.f odd 6 1 inner
1296.1.o.b 2 36.h even 6 1 inner
2304.1.b.a 2 144.u even 12 2
2304.1.b.a 2 144.v odd 12 2
2304.1.b.a 2 144.w odd 12 2
2304.1.b.a 2 144.x even 12 2
3600.1.e.b 1 45.h odd 6 1
3600.1.e.b 1 45.j even 6 1
3600.1.e.b 1 180.n even 6 1
3600.1.e.b 1 180.p odd 6 1
3600.1.j.a 2 45.k odd 12 2
3600.1.j.a 2 45.l even 12 2
3600.1.j.a 2 180.v odd 12 2
3600.1.j.a 2 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \)
\( T_{7} \)

Hecke characteristic polynomials

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