Properties

Label 1296.2.a.o
Level $1296$
Weight $2$
Character orbit 1296.a
Self dual yes
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3486121020\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 2 q^{7} - 2 \beta q^{11} - q^{13} - 3 \beta q^{17} - 2 q^{19} + 2 \beta q^{23} - 2 q^{25} - \beta q^{29} - 8 q^{31} - 2 \beta q^{35} - 7 q^{37} + 4 \beta q^{41} - 2 q^{43} + 4 \beta q^{47} - 3 q^{49} - 6 q^{55} + 8 \beta q^{59} - 7 q^{61} - \beta q^{65} + 10 q^{67} - 6 \beta q^{71} - 7 q^{73} + 4 \beta q^{77} - 2 q^{79} - 8 \beta q^{83} - 9 q^{85} + 3 \beta q^{89} + 2 q^{91} - 2 \beta q^{95} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{13} - 4 q^{19} - 4 q^{25} - 16 q^{31} - 14 q^{37} - 4 q^{43} - 6 q^{49} - 12 q^{55} - 14 q^{61} + 20 q^{67} - 14 q^{73} - 4 q^{79} - 18 q^{85} + 4 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 0 0 −1.73205 0 −2.00000 0 0 0
1.2 0 0 0 1.73205 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.a.o 2
3.b odd 2 1 inner 1296.2.a.o 2
4.b odd 2 1 81.2.a.a 2
8.b even 2 1 5184.2.a.bq 2
8.d odd 2 1 5184.2.a.br 2
9.c even 3 2 1296.2.i.s 4
9.d odd 6 2 1296.2.i.s 4
12.b even 2 1 81.2.a.a 2
20.d odd 2 1 2025.2.a.j 2
20.e even 4 2 2025.2.b.k 4
24.f even 2 1 5184.2.a.br 2
24.h odd 2 1 5184.2.a.bq 2
28.d even 2 1 3969.2.a.i 2
36.f odd 6 2 81.2.c.b 4
36.h even 6 2 81.2.c.b 4
44.c even 2 1 9801.2.a.v 2
60.h even 2 1 2025.2.a.j 2
60.l odd 4 2 2025.2.b.k 4
84.h odd 2 1 3969.2.a.i 2
108.j odd 18 6 729.2.e.o 12
108.l even 18 6 729.2.e.o 12
132.d odd 2 1 9801.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 4.b odd 2 1
81.2.a.a 2 12.b even 2 1
81.2.c.b 4 36.f odd 6 2
81.2.c.b 4 36.h even 6 2
729.2.e.o 12 108.j odd 18 6
729.2.e.o 12 108.l even 18 6
1296.2.a.o 2 1.a even 1 1 trivial
1296.2.a.o 2 3.b odd 2 1 inner
1296.2.i.s 4 9.c even 3 2
1296.2.i.s 4 9.d odd 6 2
2025.2.a.j 2 20.d odd 2 1
2025.2.a.j 2 60.h even 2 1
2025.2.b.k 4 20.e even 4 2
2025.2.b.k 4 60.l odd 4 2
3969.2.a.i 2 28.d even 2 1
3969.2.a.i 2 84.h odd 2 1
5184.2.a.bq 2 8.b even 2 1
5184.2.a.bq 2 24.h odd 2 1
5184.2.a.br 2 8.d odd 2 1
5184.2.a.br 2 24.f even 2 1
9801.2.a.v 2 44.c even 2 1
9801.2.a.v 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1296))\):

\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) 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
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 27 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 3 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 48 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 192 \) Copy content Toggle raw display
$61$ \( (T + 7)^{2} \) Copy content Toggle raw display
$67$ \( (T - 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 108 \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 27 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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