Properties

Label 1296.2.s.d
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{7} - 7 \zeta_{6} q^{13} + ( - 10 \zeta_{6} + 5) q^{19} + (5 \zeta_{6} - 5) q^{25} + ( - 6 \zeta_{6} + 12) q^{31} - q^{37} + (6 \zeta_{6} + 6) q^{43} - 4 \zeta_{6} q^{49} + ( - 13 \zeta_{6} + 13) q^{61} + ( - 7 \zeta_{6} + 14) q^{67} - 17 q^{73} + ( - 7 \zeta_{6} - 7) q^{79} + ( - 14 \zeta_{6} + 7) q^{91} + ( - 5 \zeta_{6} + 5) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{7} - 7 q^{13} - 5 q^{25} + 18 q^{31} - 2 q^{37} + 18 q^{43} - 4 q^{49} + 13 q^{61} + 21 q^{67} - 34 q^{73} - 21 q^{79} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 1.50000 + 0.866025i 0 0 0
863.1 0 0 0 0 0 1.50000 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 \(\Q(\sqrt{-3}) \)
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.d 2
3.b odd 2 1 CM 1296.2.s.d 2
4.b odd 2 1 1296.2.s.c 2
9.c even 3 1 432.2.c.b 2
9.c even 3 1 1296.2.s.c 2
9.d odd 6 1 432.2.c.b 2
9.d odd 6 1 1296.2.s.c 2
12.b even 2 1 1296.2.s.c 2
36.f odd 6 1 432.2.c.b 2
36.f odd 6 1 inner 1296.2.s.d 2
36.h even 6 1 432.2.c.b 2
36.h even 6 1 inner 1296.2.s.d 2
72.j odd 6 1 1728.2.c.a 2
72.l even 6 1 1728.2.c.a 2
72.n even 6 1 1728.2.c.a 2
72.p odd 6 1 1728.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.b 2 9.c even 3 1
432.2.c.b 2 9.d odd 6 1
432.2.c.b 2 36.f odd 6 1
432.2.c.b 2 36.h even 6 1
1296.2.s.c 2 4.b odd 2 1
1296.2.s.c 2 9.c even 3 1
1296.2.s.c 2 9.d odd 6 1
1296.2.s.c 2 12.b even 2 1
1296.2.s.d 2 1.a even 1 1 trivial
1296.2.s.d 2 3.b odd 2 1 CM
1296.2.s.d 2 36.f odd 6 1 inner
1296.2.s.d 2 36.h even 6 1 inner
1728.2.c.a 2 72.j odd 6 1
1728.2.c.a 2 72.l even 6 1
1728.2.c.a 2 72.n even 6 1
1728.2.c.a 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 \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 75 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 21T + 147 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 17)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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