Properties

Label 324.2.a.c
Level $324$
Weight $2$
Character orbit 324.a
Self dual yes
Analytic conductor $2.587$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{5} - q^{7} + 3 q^{11} - q^{13} + 6 q^{17} - 4 q^{19} - 3 q^{23} + 4 q^{25} + 3 q^{29} + 5 q^{31} - 3 q^{35} + 2 q^{37} + 3 q^{41} - q^{43} - 9 q^{47} - 6 q^{49} - 6 q^{53} + 9 q^{55} - 3 q^{59} - 13 q^{61} - 3 q^{65} - 7 q^{67} - 12 q^{71} - 10 q^{73} - 3 q^{77} + 11 q^{79} - 9 q^{83} + 18 q^{85} + 6 q^{89} + q^{91} - 12 q^{95} + 11 q^{97}+O(q^{100}) \) 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
0
0 0 0 3.00000 0 −1.00000 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

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.a.c 1
3.b odd 2 1 324.2.a.a 1
4.b odd 2 1 1296.2.a.k 1
5.b even 2 1 8100.2.a.j 1
5.c odd 4 2 8100.2.d.h 2
8.b even 2 1 5184.2.a.e 1
8.d odd 2 1 5184.2.a.f 1
9.c even 3 2 36.2.e.a 2
9.d odd 6 2 108.2.e.a 2
12.b even 2 1 1296.2.a.b 1
15.d odd 2 1 8100.2.a.g 1
15.e even 4 2 8100.2.d.c 2
24.f even 2 1 5184.2.a.bb 1
24.h odd 2 1 5184.2.a.ba 1
36.f odd 6 2 144.2.i.a 2
36.h even 6 2 432.2.i.c 2
45.h odd 6 2 2700.2.i.b 2
45.j even 6 2 900.2.i.b 2
45.k odd 12 4 900.2.s.b 4
45.l even 12 4 2700.2.s.b 4
63.g even 3 2 1764.2.l.c 2
63.h even 3 2 1764.2.i.a 2
63.i even 6 2 5292.2.i.a 2
63.j odd 6 2 5292.2.i.c 2
63.k odd 6 2 1764.2.l.a 2
63.l odd 6 2 1764.2.j.b 2
63.n odd 6 2 5292.2.l.a 2
63.o even 6 2 5292.2.j.a 2
63.s even 6 2 5292.2.l.c 2
63.t odd 6 2 1764.2.i.c 2
72.j odd 6 2 1728.2.i.d 2
72.l even 6 2 1728.2.i.c 2
72.n even 6 2 576.2.i.f 2
72.p odd 6 2 576.2.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 9.c even 3 2
108.2.e.a 2 9.d odd 6 2
144.2.i.a 2 36.f odd 6 2
324.2.a.a 1 3.b odd 2 1
324.2.a.c 1 1.a even 1 1 trivial
432.2.i.c 2 36.h even 6 2
576.2.i.e 2 72.p odd 6 2
576.2.i.f 2 72.n even 6 2
900.2.i.b 2 45.j even 6 2
900.2.s.b 4 45.k odd 12 4
1296.2.a.b 1 12.b even 2 1
1296.2.a.k 1 4.b odd 2 1
1728.2.i.c 2 72.l even 6 2
1728.2.i.d 2 72.j odd 6 2
1764.2.i.a 2 63.h even 3 2
1764.2.i.c 2 63.t odd 6 2
1764.2.j.b 2 63.l odd 6 2
1764.2.l.a 2 63.k odd 6 2
1764.2.l.c 2 63.g even 3 2
2700.2.i.b 2 45.h odd 6 2
2700.2.s.b 4 45.l even 12 4
5184.2.a.e 1 8.b even 2 1
5184.2.a.f 1 8.d odd 2 1
5184.2.a.ba 1 24.h odd 2 1
5184.2.a.bb 1 24.f even 2 1
5292.2.i.a 2 63.i even 6 2
5292.2.i.c 2 63.j odd 6 2
5292.2.j.a 2 63.o even 6 2
5292.2.l.a 2 63.n odd 6 2
5292.2.l.c 2 63.s even 6 2
8100.2.a.g 1 15.d odd 2 1
8100.2.a.j 1 5.b even 2 1
8100.2.d.c 2 15.e even 4 2
8100.2.d.h 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(324))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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