Properties

Label 1296.2.i.p
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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

Hecke kernels

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

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

Hecke characteristic polynomials

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