Properties

Label 1296.2.c.f
Level $1296$
Weight $2$
Character orbit 1296.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{5} -3 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{12}^{2} ) q^{5} -3 \zeta_{12}^{3} q^{7} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{11} + q^{13} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} + 6 \zeta_{12}^{3} q^{19} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{23} + 2 q^{25} + ( -5 + 10 \zeta_{12}^{2} ) q^{29} + 3 \zeta_{12}^{3} q^{31} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} -4 q^{37} + ( -3 + 6 \zeta_{12}^{2} ) q^{41} -3 \zeta_{12}^{3} q^{43} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{47} -2 q^{49} + ( -6 + 12 \zeta_{12}^{2} ) q^{53} -9 \zeta_{12}^{3} q^{55} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} -7 q^{61} + ( -1 + 2 \zeta_{12}^{2} ) q^{65} -9 \zeta_{12}^{3} q^{67} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( -9 + 18 \zeta_{12}^{2} ) q^{77} + 15 \zeta_{12}^{3} q^{79} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} -6 q^{85} + ( 2 - 4 \zeta_{12}^{2} ) q^{89} -3 \zeta_{12}^{3} q^{91} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{95} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} + 8q^{25} - 16q^{37} - 8q^{49} - 28q^{61} + 16q^{73} - 24q^{85} + 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1295.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 1.73205i 0 3.00000i 0 0 0
1295.2 0 0 0 1.73205i 0 3.00000i 0 0 0
1295.3 0 0 0 1.73205i 0 3.00000i 0 0 0
1295.4 0 0 0 1.73205i 0 3.00000i 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
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.c.f 4
3.b odd 2 1 inner 1296.2.c.f 4
4.b odd 2 1 inner 1296.2.c.f 4
8.b even 2 1 5184.2.c.f 4
8.d odd 2 1 5184.2.c.f 4
9.c even 3 1 144.2.s.e 4
9.c even 3 1 432.2.s.e 4
9.d odd 6 1 144.2.s.e 4
9.d odd 6 1 432.2.s.e 4
12.b even 2 1 inner 1296.2.c.f 4
24.f even 2 1 5184.2.c.f 4
24.h odd 2 1 5184.2.c.f 4
36.f odd 6 1 144.2.s.e 4
36.f odd 6 1 432.2.s.e 4
36.h even 6 1 144.2.s.e 4
36.h even 6 1 432.2.s.e 4
72.j odd 6 1 576.2.s.e 4
72.j odd 6 1 1728.2.s.e 4
72.l even 6 1 576.2.s.e 4
72.l even 6 1 1728.2.s.e 4
72.n even 6 1 576.2.s.e 4
72.n even 6 1 1728.2.s.e 4
72.p odd 6 1 576.2.s.e 4
72.p odd 6 1 1728.2.s.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.e 4 9.c even 3 1
144.2.s.e 4 9.d odd 6 1
144.2.s.e 4 36.f odd 6 1
144.2.s.e 4 36.h even 6 1
432.2.s.e 4 9.c even 3 1
432.2.s.e 4 9.d odd 6 1
432.2.s.e 4 36.f odd 6 1
432.2.s.e 4 36.h even 6 1
576.2.s.e 4 72.j odd 6 1
576.2.s.e 4 72.l even 6 1
576.2.s.e 4 72.n even 6 1
576.2.s.e 4 72.p odd 6 1
1296.2.c.f 4 1.a even 1 1 trivial
1296.2.c.f 4 3.b odd 2 1 inner
1296.2.c.f 4 4.b odd 2 1 inner
1296.2.c.f 4 12.b even 2 1 inner
1728.2.s.e 4 72.j odd 6 1
1728.2.s.e 4 72.l even 6 1
1728.2.s.e 4 72.n even 6 1
1728.2.s.e 4 72.p odd 6 1
5184.2.c.f 4 8.b even 2 1
5184.2.c.f 4 8.d odd 2 1
5184.2.c.f 4 24.f even 2 1
5184.2.c.f 4 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 3 \)
\( T_{7}^{2} + 9 \)
\( T_{11}^{2} - 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 3 + T^{2} )^{2} \)
$7$ \( ( 9 + T^{2} )^{2} \)
$11$ \( ( -27 + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( ( -27 + T^{2} )^{2} \)
$29$ \( ( 75 + T^{2} )^{2} \)
$31$ \( ( 9 + T^{2} )^{2} \)
$37$ \( ( 4 + T )^{4} \)
$41$ \( ( 27 + T^{2} )^{2} \)
$43$ \( ( 9 + T^{2} )^{2} \)
$47$ \( ( -27 + T^{2} )^{2} \)
$53$ \( ( 108 + T^{2} )^{2} \)
$59$ \( ( -27 + T^{2} )^{2} \)
$61$ \( ( 7 + T )^{4} \)
$67$ \( ( 81 + T^{2} )^{2} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( ( 225 + T^{2} )^{2} \)
$83$ \( ( -27 + T^{2} )^{2} \)
$89$ \( ( 12 + T^{2} )^{2} \)
$97$ \( ( -1 + T )^{4} \)
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