Properties

Label 1296.2.i.n
Level $1296$
Weight $2$
Character orbit 1296.i
Analytic conductor $10.349$
Analytic rank $0$
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.i (of order \(3\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + \zeta_{6} q^{13} -3 q^{17} + 4 q^{19} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -9 + 9 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} -12 q^{35} - q^{37} -6 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -6 q^{53} + ( 1 - \zeta_{6} ) q^{61} + ( -3 + 3 \zeta_{6} ) q^{65} -4 \zeta_{6} q^{67} + 12 q^{71} + 11 q^{73} + ( -16 + 16 \zeta_{6} ) q^{79} + ( -12 + 12 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} -3 q^{89} -4 q^{91} + 12 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + 3q^{5} - 4q^{7} + q^{13} - 6q^{17} + 8q^{19} - 4q^{25} - 9q^{29} - 4q^{31} - 24q^{35} - 2q^{37} - 6q^{41} + 8q^{43} - 12q^{47} - 9q^{49} - 12q^{53} + q^{61} - 3q^{65} - 4q^{67} + 24q^{71} + 22q^{73} - 16q^{79} - 12q^{83} - 9q^{85} - 6q^{89} - 8q^{91} + 12q^{95} - 2q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.50000 2.59808i 0 −2.00000 3.46410i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 0 −2.00000 + 3.46410i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.i.n 2
3.b odd 2 1 1296.2.i.b 2
4.b odd 2 1 162.2.c.d 2
9.c even 3 1 1296.2.a.c 1
9.c even 3 1 inner 1296.2.i.n 2
9.d odd 6 1 1296.2.a.l 1
9.d odd 6 1 1296.2.i.b 2
12.b even 2 1 162.2.c.a 2
36.f odd 6 1 162.2.a.a 1
36.f odd 6 1 162.2.c.d 2
36.h even 6 1 162.2.a.d yes 1
36.h even 6 1 162.2.c.a 2
72.j odd 6 1 5184.2.a.h 1
72.l even 6 1 5184.2.a.c 1
72.n even 6 1 5184.2.a.bd 1
72.p odd 6 1 5184.2.a.y 1
180.n even 6 1 4050.2.a.r 1
180.p odd 6 1 4050.2.a.bh 1
180.v odd 12 2 4050.2.c.n 2
180.x even 12 2 4050.2.c.g 2
252.s odd 6 1 7938.2.a.s 1
252.bi even 6 1 7938.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 36.f odd 6 1
162.2.a.d yes 1 36.h even 6 1
162.2.c.a 2 12.b even 2 1
162.2.c.a 2 36.h even 6 1
162.2.c.d 2 4.b odd 2 1
162.2.c.d 2 36.f odd 6 1
1296.2.a.c 1 9.c even 3 1
1296.2.a.l 1 9.d odd 6 1
1296.2.i.b 2 3.b odd 2 1
1296.2.i.b 2 9.d odd 6 1
1296.2.i.n 2 1.a even 1 1 trivial
1296.2.i.n 2 9.c even 3 1 inner
4050.2.a.r 1 180.n even 6 1
4050.2.a.bh 1 180.p odd 6 1
4050.2.c.g 2 180.x even 12 2
4050.2.c.n 2 180.v odd 12 2
5184.2.a.c 1 72.l even 6 1
5184.2.a.h 1 72.j odd 6 1
5184.2.a.y 1 72.p odd 6 1
5184.2.a.bd 1 72.n even 6 1
7938.2.a.n 1 252.bi even 6 1
7938.2.a.s 1 252.s odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{7}^{2} + 4 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 23 T^{2} + 529 T^{4} \)
$29$ \( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 + T + 37 T^{2} )^{2} \)
$41$ \( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )( 1 + 5 T + 43 T^{2} ) \)
$47$ \( 1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 + 3 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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