Properties

Label 1296.2.s.i
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_{3} - \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{13} - 2 \beta_{3} q^{17} + ( - 8 \beta_{2} + 4) q^{19} + 4 \beta_{2} q^{25} + (2 \beta_{3} - 2 \beta_1) q^{29} + ( - 3 \beta_{2} - 3) q^{31} + ( - \beta_{3} + 2 \beta_1) q^{35} + 8 q^{37} + ( - 6 \beta_{2} + 12) q^{43} + (2 \beta_{3} + 2 \beta_1) q^{47} + (4 \beta_{2} - 4) q^{49} + 3 \beta_{3} q^{53} + ( - 18 \beta_{2} + 9) q^{55} + (4 \beta_{3} - 2 \beta_1) q^{59} + 4 \beta_{2} q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{65} + ( - 2 \beta_{2} - 2) q^{67} + (2 \beta_{3} - 4 \beta_1) q^{71} + q^{73} - 3 \beta_1 q^{77} + ( - 2 \beta_{2} + 4) q^{79} + (\beta_{3} + \beta_1) q^{83} + ( - 18 \beta_{2} + 18) q^{85} + 2 \beta_{3} q^{89} + ( - 4 \beta_{2} + 2) q^{91} + ( - 8 \beta_{3} + 4 \beta_1) q^{95} + 5 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{7} + 4 q^{13} + 8 q^{25} - 18 q^{31} + 32 q^{37} + 36 q^{43} - 8 q^{49} + 8 q^{61} - 12 q^{67} + 4 q^{73} + 12 q^{79} + 36 q^{85} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −2.59808 + 1.50000i 0 1.50000 + 0.866025i 0 0 0
431.2 0 0 0 2.59808 1.50000i 0 1.50000 + 0.866025i 0 0 0
863.1 0 0 0 −2.59808 1.50000i 0 1.50000 0.866025i 0 0 0
863.2 0 0 0 2.59808 + 1.50000i 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 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 1296.2.s.i 4
3.b odd 2 1 inner 1296.2.s.i 4
4.b odd 2 1 1296.2.s.g 4
9.c even 3 1 432.2.c.c 4
9.c even 3 1 1296.2.s.g 4
9.d odd 6 1 432.2.c.c 4
9.d odd 6 1 1296.2.s.g 4
12.b even 2 1 1296.2.s.g 4
36.f odd 6 1 432.2.c.c 4
36.f odd 6 1 inner 1296.2.s.i 4
36.h even 6 1 432.2.c.c 4
36.h even 6 1 inner 1296.2.s.i 4
72.j odd 6 1 1728.2.c.e 4
72.l even 6 1 1728.2.c.e 4
72.n even 6 1 1728.2.c.e 4
72.p odd 6 1 1728.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.c 4 9.c even 3 1
432.2.c.c 4 9.d odd 6 1
432.2.c.c 4 36.f odd 6 1
432.2.c.c 4 36.h even 6 1
1296.2.s.g 4 4.b odd 2 1
1296.2.s.g 4 9.c even 3 1
1296.2.s.g 4 9.d odd 6 1
1296.2.s.g 4 12.b even 2 1
1296.2.s.i 4 1.a even 1 1 trivial
1296.2.s.i 4 3.b odd 2 1 inner
1296.2.s.i 4 36.f odd 6 1 inner
1296.2.s.i 4 36.h even 6 1 inner
1728.2.c.e 4 72.j odd 6 1
1728.2.c.e 4 72.l even 6 1
1728.2.c.e 4 72.n even 6 1
1728.2.c.e 4 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 \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{4} - 9T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$31$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T - 8)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 18 T + 108)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
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