Properties

Label 1296.2.i.d
Level $1296$
Weight $2$
Character orbit 1296.i
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.i (of order \(3\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 324)
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.

\(f(q)\) \(=\) \( q -3 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} -3 q^{17} -2 q^{19} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -3 + 3 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} -6 q^{35} + 5 q^{37} + 6 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{49} -6 q^{53} + 18 q^{55} -12 \zeta_{6} q^{59} + ( -5 + 5 \zeta_{6} ) q^{61} + ( -15 + 15 \zeta_{6} ) q^{65} + 2 \zeta_{6} q^{67} -6 q^{71} - q^{73} + 12 \zeta_{6} q^{77} + ( -10 + 10 \zeta_{6} ) q^{79} + 9 \zeta_{6} q^{85} -3 q^{89} -10 q^{91} + 6 \zeta_{6} q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + 2 q^{7} + O(q^{10}) \) \( 2 q - 3 q^{5} + 2 q^{7} - 6 q^{11} - 5 q^{13} - 6 q^{17} - 4 q^{19} + 6 q^{23} - 4 q^{25} - 3 q^{29} - 4 q^{31} - 12 q^{35} + 10 q^{37} + 6 q^{41} - 10 q^{43} + 3 q^{49} - 12 q^{53} + 36 q^{55} - 12 q^{59} - 5 q^{61} - 15 q^{65} + 2 q^{67} - 12 q^{71} - 2 q^{73} + 12 q^{77} - 10 q^{79} + 9 q^{85} - 6 q^{89} - 20 q^{91} + 6 q^{95} + 10 q^{97} + O(q^{100}) \)

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 1.00000 + 1.73205i 0 0 0
865.1 0 0 0 −1.50000 2.59808i 0 1.00000 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
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.d 2
3.b odd 2 1 1296.2.i.p 2
4.b odd 2 1 324.2.e.a 2
9.c even 3 1 1296.2.a.j 1
9.c even 3 1 inner 1296.2.i.d 2
9.d odd 6 1 1296.2.a.a 1
9.d odd 6 1 1296.2.i.p 2
12.b even 2 1 324.2.e.d 2
36.f odd 6 1 324.2.a.d yes 1
36.f odd 6 1 324.2.e.a 2
36.h even 6 1 324.2.a.b 1
36.h even 6 1 324.2.e.d 2
72.j odd 6 1 5184.2.a.z 1
72.l even 6 1 5184.2.a.bc 1
72.n even 6 1 5184.2.a.d 1
72.p odd 6 1 5184.2.a.g 1
180.n even 6 1 8100.2.a.f 1
180.p odd 6 1 8100.2.a.a 1
180.v odd 12 2 8100.2.d.j 2
180.x even 12 2 8100.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.a.b 1 36.h even 6 1
324.2.a.d yes 1 36.f odd 6 1
324.2.e.a 2 4.b odd 2 1
324.2.e.a 2 36.f odd 6 1
324.2.e.d 2 12.b even 2 1
324.2.e.d 2 36.h even 6 1
1296.2.a.a 1 9.d odd 6 1
1296.2.a.j 1 9.c even 3 1
1296.2.i.d 2 1.a even 1 1 trivial
1296.2.i.d 2 9.c even 3 1 inner
1296.2.i.p 2 3.b odd 2 1
1296.2.i.p 2 9.d odd 6 1
5184.2.a.d 1 72.n even 6 1
5184.2.a.g 1 72.p odd 6 1
5184.2.a.z 1 72.j odd 6 1
5184.2.a.bc 1 72.l even 6 1
8100.2.a.a 1 180.p odd 6 1
8100.2.a.f 1 180.n even 6 1
8100.2.d.a 2 180.x even 12 2
8100.2.d.j 2 180.v odd 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 36 + 6 T + T^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( ( -5 + T )^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 100 + 10 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 25 + 5 T + T^{2} \)
$67$ \( 4 - 2 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( 100 + 10 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 3 + T )^{2} \)
$97$ \( 100 - 10 T + T^{2} \)
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