Properties

Label 324.3.c.b
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( 1 - \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 1 - \beta_{2} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{11} + ( -3 + \beta_{2} ) q^{13} + \beta_{3} q^{17} + \beta_{2} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + ( -8 + 3 \beta_{2} ) q^{25} + 7 \beta_{1} q^{29} + ( 5 - 3 \beta_{2} ) q^{31} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{35} + ( -14 - 4 \beta_{2} ) q^{37} + ( -7 \beta_{1} + \beta_{3} ) q^{41} + 23 q^{43} + ( -\beta_{1} + 2 \beta_{3} ) q^{47} + ( 26 - \beta_{2} ) q^{49} -8 \beta_{1} q^{53} + ( 21 + 3 \beta_{2} ) q^{55} + ( 9 \beta_{1} + \beta_{3} ) q^{59} + ( -19 - 3 \beta_{2} ) q^{61} + ( -9 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -61 + 6 \beta_{2} ) q^{67} -2 \beta_{3} q^{71} + ( 20 + 3 \beta_{2} ) q^{73} + ( 9 \beta_{1} - 8 \beta_{3} ) q^{77} + ( -39 - 5 \beta_{2} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} ) q^{83} + ( -12 + 6 \beta_{2} ) q^{85} -8 \beta_{3} q^{89} + ( -77 + 3 \beta_{2} ) q^{91} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{95} + ( 99 - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7} + O(q^{10}) \) \( 4 q + 2 q^{7} - 10 q^{13} + 2 q^{19} - 26 q^{25} + 14 q^{31} - 64 q^{37} + 92 q^{43} + 102 q^{49} + 90 q^{55} - 82 q^{61} - 232 q^{67} + 86 q^{73} - 166 q^{79} - 36 q^{85} - 302 q^{91} + 392 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + \nu^{2} - \nu + 3 \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + \nu^{2} + 5 \nu + 2 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{3} + 5 \nu^{2} - 5 \nu - 21 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 21\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} - 5 \beta_{1} + 36\)\()/9\)

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
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
0 0 0 7.57301i 0 9.11684 0 0 0
161.2 0 0 0 2.37686i 0 −8.11684 0 0 0
161.3 0 0 0 2.37686i 0 −8.11684 0 0 0
161.4 0 0 0 7.57301i 0 9.11684 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.b 4
3.b odd 2 1 inner 324.3.c.b 4
4.b odd 2 1 1296.3.e.e 4
9.c even 3 1 36.3.g.a 4
9.c even 3 1 108.3.g.a 4
9.d odd 6 1 36.3.g.a 4
9.d odd 6 1 108.3.g.a 4
12.b even 2 1 1296.3.e.e 4
36.f odd 6 1 144.3.q.b 4
36.f odd 6 1 432.3.q.b 4
36.h even 6 1 144.3.q.b 4
36.h even 6 1 432.3.q.b 4
45.h odd 6 1 900.3.p.a 4
45.h odd 6 1 2700.3.p.b 4
45.j even 6 1 900.3.p.a 4
45.j even 6 1 2700.3.p.b 4
45.k odd 12 2 900.3.u.a 8
45.k odd 12 2 2700.3.u.b 8
45.l even 12 2 900.3.u.a 8
45.l even 12 2 2700.3.u.b 8
72.j odd 6 1 576.3.q.d 4
72.j odd 6 1 1728.3.q.g 4
72.l even 6 1 576.3.q.g 4
72.l even 6 1 1728.3.q.h 4
72.n even 6 1 576.3.q.d 4
72.n even 6 1 1728.3.q.g 4
72.p odd 6 1 576.3.q.g 4
72.p odd 6 1 1728.3.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 9.c even 3 1
36.3.g.a 4 9.d odd 6 1
108.3.g.a 4 9.c even 3 1
108.3.g.a 4 9.d odd 6 1
144.3.q.b 4 36.f odd 6 1
144.3.q.b 4 36.h even 6 1
324.3.c.b 4 1.a even 1 1 trivial
324.3.c.b 4 3.b odd 2 1 inner
432.3.q.b 4 36.f odd 6 1
432.3.q.b 4 36.h even 6 1
576.3.q.d 4 72.j odd 6 1
576.3.q.d 4 72.n even 6 1
576.3.q.g 4 72.l even 6 1
576.3.q.g 4 72.p odd 6 1
900.3.p.a 4 45.h odd 6 1
900.3.p.a 4 45.j even 6 1
900.3.u.a 8 45.k odd 12 2
900.3.u.a 8 45.l even 12 2
1296.3.e.e 4 4.b odd 2 1
1296.3.e.e 4 12.b even 2 1
1728.3.q.g 4 72.j odd 6 1
1728.3.q.g 4 72.n even 6 1
1728.3.q.h 4 72.l even 6 1
1728.3.q.h 4 72.p odd 6 1
2700.3.p.b 4 45.h odd 6 1
2700.3.p.b 4 45.j even 6 1
2700.3.u.b 8 45.k odd 12 2
2700.3.u.b 8 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 63 T_{5}^{2} + 324 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 + 63 T^{2} + T^{4} \)
$7$ \( ( -74 - T + T^{2} )^{2} \)
$11$ \( 81 + 414 T^{2} + T^{4} \)
$13$ \( ( -68 + 5 T + T^{2} )^{2} \)
$17$ \( 20736 + 387 T^{2} + T^{4} \)
$19$ \( ( -74 - T + T^{2} )^{2} \)
$23$ \( 627264 + 1683 T^{2} + T^{4} \)
$29$ \( 777924 + 3087 T^{2} + T^{4} \)
$31$ \( ( -656 - 7 T + T^{2} )^{2} \)
$37$ \( ( -932 + 32 T + T^{2} )^{2} \)
$41$ \( 2424249 + 3222 T^{2} + T^{4} \)
$43$ \( ( -23 + T )^{4} \)
$47$ \( 104976 + 1539 T^{2} + T^{4} \)
$53$ \( 1327104 + 4032 T^{2} + T^{4} \)
$59$ \( 68121 + 5814 T^{2} + T^{4} \)
$61$ \( ( -248 + 41 T + T^{2} )^{2} \)
$67$ \( ( 691 + 116 T + T^{2} )^{2} \)
$71$ \( 331776 + 1548 T^{2} + T^{4} \)
$73$ \( ( -206 - 43 T + T^{2} )^{2} \)
$79$ \( ( -134 + 83 T + T^{2} )^{2} \)
$83$ \( 104976 + 1539 T^{2} + T^{4} \)
$89$ \( 84934656 + 24768 T^{2} + T^{4} \)
$97$ \( ( 9307 - 196 T + T^{2} )^{2} \)
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