Properties

Label 324.3.c.a
Level $324$
Weight $3$
Character orbit 324.c
Analytic conductor $8.828$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -1 + \beta_{2} ) q^{7} -\beta_{3} q^{11} + ( -1 + 2 \beta_{2} ) q^{13} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -7 + 5 \beta_{2} ) q^{19} + ( 6 \beta_{1} + \beta_{3} ) q^{23} + ( 7 + 3 \beta_{2} ) q^{25} + ( -5 \beta_{1} - 3 \beta_{3} ) q^{29} + ( 8 + 6 \beta_{2} ) q^{31} + ( -6 \beta_{1} + \beta_{3} ) q^{35} + ( 14 + \beta_{2} ) q^{37} + ( 12 \beta_{1} - \beta_{3} ) q^{41} + ( -13 + 9 \beta_{2} ) q^{43} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -21 - 2 \beta_{2} ) q^{49} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 9 + 3 \beta_{2} ) q^{55} + ( -14 \beta_{1} + 2 \beta_{3} ) q^{59} + ( -16 - 9 \beta_{2} ) q^{61} + ( -11 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 5 + 3 \beta_{2} ) q^{67} + ( -2 \beta_{1} - 7 \beta_{3} ) q^{71} + ( -64 - 9 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{77} + ( 77 - 7 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{83} + ( 72 - 21 \beta_{2} ) q^{85} + ( 19 \beta_{1} - \beta_{3} ) q^{89} + ( 55 - 3 \beta_{2} ) q^{91} + ( -32 \beta_{1} + 5 \beta_{3} ) q^{95} + ( -40 - 22 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} + O(q^{10}) \) \( 4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( 3 \nu^{2} + 6 \)
\(\beta_{3}\)\(=\)\( 9 \nu^{3} + 33 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 11 \beta_{1}\)\()/9\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.93185i
0.517638i
0.517638i
1.93185i
0 0 0 5.79555i 0 −6.19615 0 0 0
161.2 0 0 0 1.55291i 0 4.19615 0 0 0
161.3 0 0 0 1.55291i 0 4.19615 0 0 0
161.4 0 0 0 5.79555i 0 −6.19615 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
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 324.3.c.a 4
3.b odd 2 1 inner 324.3.c.a 4
4.b odd 2 1 1296.3.e.f 4
9.c even 3 2 324.3.g.d 8
9.d odd 6 2 324.3.g.d 8
12.b even 2 1 1296.3.e.f 4
36.f odd 6 2 1296.3.q.n 8
36.h even 6 2 1296.3.q.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.c.a 4 1.a even 1 1 trivial
324.3.c.a 4 3.b odd 2 1 inner
324.3.g.d 8 9.c even 3 2
324.3.g.d 8 9.d odd 6 2
1296.3.e.f 4 4.b odd 2 1
1296.3.e.f 4 12.b even 2 1
1296.3.q.n 8 36.f odd 6 2
1296.3.q.n 8 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 36 T_{5}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 81 + 36 T^{2} + T^{4} \)
$7$ \( ( -26 + 2 T + T^{2} )^{2} \)
$11$ \( 324 + 252 T^{2} + T^{4} \)
$13$ \( ( -107 + 2 T + T^{2} )^{2} \)
$17$ \( 558009 + 1548 T^{2} + T^{4} \)
$19$ \( ( -626 + 14 T + T^{2} )^{2} \)
$23$ \( 715716 + 1764 T^{2} + T^{4} \)
$29$ \( 1996569 + 3708 T^{2} + T^{4} \)
$31$ \( ( -908 - 16 T + T^{2} )^{2} \)
$37$ \( ( 169 - 28 T + T^{2} )^{2} \)
$41$ \( 39204 + 5004 T^{2} + T^{4} \)
$43$ \( ( -2018 + 26 T + T^{2} )^{2} \)
$47$ \( 944784 + 3888 T^{2} + T^{4} \)
$53$ \( 1127844 + 3276 T^{2} + T^{4} \)
$59$ \( 685584 + 7056 T^{2} + T^{4} \)
$61$ \( ( -1931 + 32 T + T^{2} )^{2} \)
$67$ \( ( -218 - 10 T + T^{2} )^{2} \)
$71$ \( 171396 + 12996 T^{2} + T^{4} \)
$73$ \( ( 1909 + 128 T + T^{2} )^{2} \)
$79$ \( ( 4606 - 154 T + T^{2} )^{2} \)
$83$ \( 3779136 + 27216 T^{2} + T^{4} \)
$89$ \( 2313441 + 12564 T^{2} + T^{4} \)
$97$ \( ( -11468 + 80 T + T^{2} )^{2} \)
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