Properties

Label 1296.2.s.a
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -3 - 3 \zeta_{6} ) q^{7} + 5 \zeta_{6} q^{13} + ( -3 + 6 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( 12 - 6 \zeta_{6} ) q^{31} + 11 q^{37} + ( 6 + 6 \zeta_{6} ) q^{43} + 20 \zeta_{6} q^{49} + ( 1 - \zeta_{6} ) q^{61} + ( -18 + 9 \zeta_{6} ) q^{67} + 7 q^{73} + ( -3 - 3 \zeta_{6} ) q^{79} + ( 15 - 30 \zeta_{6} ) q^{91} + ( -19 + 19 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9q^{7} + O(q^{10}) \) \( 2q - 9q^{7} + 5q^{13} - 5q^{25} + 18q^{31} + 22q^{37} + 18q^{43} + 20q^{49} + q^{61} - 27q^{67} + 14q^{73} - 9q^{79} - 19q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −4.50000 2.59808i 0 0 0
863.1 0 0 0 0 0 −4.50000 + 2.59808i 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 \(\Q(\sqrt{-3}) \)
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 1296.2.s.a 2
3.b odd 2 1 CM 1296.2.s.a 2
4.b odd 2 1 1296.2.s.f 2
9.c even 3 1 432.2.c.a 2
9.c even 3 1 1296.2.s.f 2
9.d odd 6 1 432.2.c.a 2
9.d odd 6 1 1296.2.s.f 2
12.b even 2 1 1296.2.s.f 2
36.f odd 6 1 432.2.c.a 2
36.f odd 6 1 inner 1296.2.s.a 2
36.h even 6 1 432.2.c.a 2
36.h even 6 1 inner 1296.2.s.a 2
72.j odd 6 1 1728.2.c.b 2
72.l even 6 1 1728.2.c.b 2
72.n even 6 1 1728.2.c.b 2
72.p odd 6 1 1728.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.a 2 9.c even 3 1
432.2.c.a 2 9.d odd 6 1
432.2.c.a 2 36.f odd 6 1
432.2.c.a 2 36.h even 6 1
1296.2.s.a 2 1.a even 1 1 trivial
1296.2.s.a 2 3.b odd 2 1 CM
1296.2.s.a 2 36.f odd 6 1 inner
1296.2.s.a 2 36.h even 6 1 inner
1296.2.s.f 2 4.b odd 2 1
1296.2.s.f 2 9.c even 3 1
1296.2.s.f 2 9.d odd 6 1
1296.2.s.f 2 12.b even 2 1
1728.2.c.b 2 72.j odd 6 1
1728.2.c.b 2 72.l even 6 1
1728.2.c.b 2 72.n even 6 1
1728.2.c.b 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 9 T_{7} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 27 + 9 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 - 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 - 18 T + T^{2} \)
$37$ \( ( -11 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 - 18 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 243 + 27 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( 27 + 9 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 361 + 19 T + T^{2} \)
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