Properties

Label 324.3.d.d
Level $324$
Weight $3$
Character orbit 324.d
Self dual yes
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( 4 + \beta ) q^{5} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + ( 4 + \beta ) q^{5} + 8 q^{8} + ( 8 + 2 \beta ) q^{10} + ( 5 - 4 \beta ) q^{13} + 16 q^{16} + ( -8 - 5 \beta ) q^{17} + ( 16 + 4 \beta ) q^{20} + ( 18 + 8 \beta ) q^{25} + ( 10 - 8 \beta ) q^{26} + ( -20 + 7 \beta ) q^{29} + 32 q^{32} + ( -16 - 10 \beta ) q^{34} + ( 35 - 4 \beta ) q^{37} + ( 32 + 8 \beta ) q^{40} -80 q^{41} + 49 q^{49} + ( 36 + 16 \beta ) q^{50} + ( 20 - 16 \beta ) q^{52} -56 q^{53} + ( -40 + 14 \beta ) q^{58} + ( 11 + 20 \beta ) q^{61} + 64 q^{64} + ( -88 - 11 \beta ) q^{65} + ( -32 - 20 \beta ) q^{68} + ( -55 - 16 \beta ) q^{73} + ( 70 - 8 \beta ) q^{74} + ( 64 + 16 \beta ) q^{80} -160 q^{82} + ( -167 - 28 \beta ) q^{85} + ( -80 + 13 \beta ) q^{89} -130 q^{97} + 98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + O(q^{10}) \) \( 2 q + 4 q^{2} + 8 q^{4} + 8 q^{5} + 16 q^{8} + 16 q^{10} + 10 q^{13} + 32 q^{16} - 16 q^{17} + 32 q^{20} + 36 q^{25} + 20 q^{26} - 40 q^{29} + 64 q^{32} - 32 q^{34} + 70 q^{37} + 64 q^{40} - 160 q^{41} + 98 q^{49} + 72 q^{50} + 40 q^{52} - 112 q^{53} - 80 q^{58} + 22 q^{61} + 128 q^{64} - 176 q^{65} - 64 q^{68} - 110 q^{73} + 140 q^{74} + 128 q^{80} - 320 q^{82} - 334 q^{85} - 160 q^{89} - 260 q^{97} + 196 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.73205
1.73205
2.00000 0 4.00000 −1.19615 0 0 8.00000 0 −2.39230
163.2 2.00000 0 4.00000 9.19615 0 0 8.00000 0 18.3923
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.d yes 2
3.b odd 2 1 324.3.d.a 2
4.b odd 2 1 CM 324.3.d.d yes 2
9.c even 3 2 324.3.f.k 4
9.d odd 6 2 324.3.f.n 4
12.b even 2 1 324.3.d.a 2
36.f odd 6 2 324.3.f.k 4
36.h even 6 2 324.3.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.a 2 3.b odd 2 1
324.3.d.a 2 12.b even 2 1
324.3.d.d yes 2 1.a even 1 1 trivial
324.3.d.d yes 2 4.b odd 2 1 CM
324.3.f.k 4 9.c even 3 2
324.3.f.k 4 36.f odd 6 2
324.3.f.n 4 9.d odd 6 2
324.3.f.n 4 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 8 T_{5} - 11 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -11 - 8 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( -407 - 10 T + T^{2} \)
$17$ \( -611 + 16 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( -923 + 40 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 793 - 70 T + T^{2} \)
$41$ \( ( 80 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 56 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( -10679 - 22 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -3887 + 110 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 1837 + 160 T + T^{2} \)
$97$ \( ( 130 + T )^{2} \)
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