Properties

Label 1296.2.c.a
Level $1296$
Weight $2$
Character orbit 1296.c
Analytic conductor $10.349$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{7} -3 q^{11} -5 q^{13} + ( -4 + 8 \zeta_{6} ) q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} -9 q^{23} + 2 q^{25} + ( -1 + 2 \zeta_{6} ) q^{29} + ( -3 + 6 \zeta_{6} ) q^{31} -3 q^{35} + 2 q^{37} + ( -3 + 6 \zeta_{6} ) q^{41} + ( -3 + 6 \zeta_{6} ) q^{43} -3 q^{47} + 4 q^{49} + ( -3 + 6 \zeta_{6} ) q^{55} + 3 q^{59} - q^{61} + ( -5 + 10 \zeta_{6} ) q^{65} + ( 5 - 10 \zeta_{6} ) q^{67} -12 q^{71} -2 q^{73} + ( -3 + 6 \zeta_{6} ) q^{77} + ( 5 - 10 \zeta_{6} ) q^{79} -15 q^{83} + 12 q^{85} + ( 4 - 8 \zeta_{6} ) q^{89} + ( -5 + 10 \zeta_{6} ) q^{91} -6 q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 6q^{11} - 10q^{13} - 18q^{23} + 4q^{25} - 6q^{35} + 4q^{37} - 6q^{47} + 8q^{49} + 6q^{59} - 2q^{61} - 24q^{71} - 4q^{73} - 30q^{83} + 24q^{85} - 12q^{95} - 10q^{97} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 1.73205i 0 1.73205i 0 0 0
1295.2 0 0 0 1.73205i 0 1.73205i 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
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.a 2
3.b odd 2 1 1296.2.c.c 2
4.b odd 2 1 1296.2.c.c 2
8.b even 2 1 5184.2.c.d 2
8.d odd 2 1 5184.2.c.b 2
9.c even 3 1 144.2.s.b 2
9.c even 3 1 432.2.s.b 2
9.d odd 6 1 144.2.s.c yes 2
9.d odd 6 1 432.2.s.a 2
12.b even 2 1 inner 1296.2.c.a 2
24.f even 2 1 5184.2.c.d 2
24.h odd 2 1 5184.2.c.b 2
36.f odd 6 1 144.2.s.c yes 2
36.f odd 6 1 432.2.s.a 2
36.h even 6 1 144.2.s.b 2
36.h even 6 1 432.2.s.b 2
72.j odd 6 1 576.2.s.c 2
72.j odd 6 1 1728.2.s.c 2
72.l even 6 1 576.2.s.b 2
72.l even 6 1 1728.2.s.d 2
72.n even 6 1 576.2.s.b 2
72.n even 6 1 1728.2.s.d 2
72.p odd 6 1 576.2.s.c 2
72.p odd 6 1 1728.2.s.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.s.b 2 9.c even 3 1
144.2.s.b 2 36.h even 6 1
144.2.s.c yes 2 9.d odd 6 1
144.2.s.c yes 2 36.f odd 6 1
432.2.s.a 2 9.d odd 6 1
432.2.s.a 2 36.f odd 6 1
432.2.s.b 2 9.c even 3 1
432.2.s.b 2 36.h even 6 1
576.2.s.b 2 72.l even 6 1
576.2.s.b 2 72.n even 6 1
576.2.s.c 2 72.j odd 6 1
576.2.s.c 2 72.p odd 6 1
1296.2.c.a 2 1.a even 1 1 trivial
1296.2.c.a 2 12.b even 2 1 inner
1296.2.c.c 2 3.b odd 2 1
1296.2.c.c 2 4.b odd 2 1
1728.2.s.c 2 72.j odd 6 1
1728.2.s.c 2 72.p odd 6 1
1728.2.s.d 2 72.l even 6 1
1728.2.s.d 2 72.n even 6 1
5184.2.c.b 2 8.d odd 2 1
5184.2.c.b 2 24.h odd 2 1
5184.2.c.d 2 8.b even 2 1
5184.2.c.d 2 24.f even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 7 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 14 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( ( 1 + 9 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 55 T^{2} + 841 T^{4} \)
$31$ \( 1 - 35 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 55 T^{2} + 1681 T^{4} \)
$43$ \( 1 - 59 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 3 T + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 - 3 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + T + 61 T^{2} )^{2} \)
$67$ \( 1 - 59 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 83 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 15 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 130 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{2} \)
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