Properties

Label 324.2.h.e
Level $324$
Weight $2$
Character orbit 324.h
Analytic conductor $2.587$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -1 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{10} + ( -2 - 3 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{13} -4 \zeta_{24}^{4} q^{16} + ( -\zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{20} + ( -3 \zeta_{24}^{2} + 9 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{25} + ( -\zeta_{24} + \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{26} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{29} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{32} + ( -3 \zeta_{24}^{2} - 5 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{34} + ( -1 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{37} + ( -2 + 6 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 12 \zeta_{24}^{6} ) q^{40} + ( -\zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{41} + ( -7 + 7 \zeta_{24}^{4} ) q^{49} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{50} + ( 6 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{52} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{53} + ( 7 - 3 \zeta_{24}^{2} - 7 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{58} + ( 6 \zeta_{24}^{2} - 5 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{61} -8 q^{64} + ( 14 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 14 \zeta_{24}^{7} ) q^{65} + ( 6 \zeta_{24} + 8 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{68} + ( 8 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{73} + ( 7 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{74} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{80} + 2 q^{82} + ( 11 + 6 \zeta_{24}^{2} - 11 \zeta_{24}^{4} - 12 \zeta_{24}^{6} ) q^{85} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{89} -8 \zeta_{24}^{4} q^{97} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + O(q^{10}) \) \( 8 q + 8 q^{4} - 8 q^{10} - 8 q^{13} - 16 q^{16} + 36 q^{25} - 20 q^{34} - 8 q^{37} - 8 q^{40} - 28 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 64 q^{64} + 64 q^{73} + 16 q^{82} + 44 q^{85} - 32 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{24}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
−1.22474 0.707107i 0 1.00000 + 1.73205i −2.56961 + 1.48356i 0 0 2.82843i 0 4.19615
107.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.79435 2.19067i 0 0 2.82843i 0 −6.19615
107.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.79435 + 2.19067i 0 0 2.82843i 0 −6.19615
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.56961 1.48356i 0 0 2.82843i 0 4.19615
215.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −2.56961 1.48356i 0 0 2.82843i 0 4.19615
215.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.79435 + 2.19067i 0 0 2.82843i 0 −6.19615
215.3 1.22474 0.707107i 0 1.00000 1.73205i −3.79435 2.19067i 0 0 2.82843i 0 −6.19615
215.4 1.22474 0.707107i 0 1.00000 1.73205i 2.56961 + 1.48356i 0 0 2.82843i 0 4.19615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.h.e 8
3.b odd 2 1 inner 324.2.h.e 8
4.b odd 2 1 CM 324.2.h.e 8
9.c even 3 1 324.2.b.a 4
9.c even 3 1 inner 324.2.h.e 8
9.d odd 6 1 324.2.b.a 4
9.d odd 6 1 inner 324.2.h.e 8
12.b even 2 1 inner 324.2.h.e 8
36.f odd 6 1 324.2.b.a 4
36.f odd 6 1 inner 324.2.h.e 8
36.h even 6 1 324.2.b.a 4
36.h even 6 1 inner 324.2.h.e 8
72.j odd 6 1 5184.2.c.e 4
72.l even 6 1 5184.2.c.e 4
72.n even 6 1 5184.2.c.e 4
72.p odd 6 1 5184.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.b.a 4 9.c even 3 1
324.2.b.a 4 9.d odd 6 1
324.2.b.a 4 36.f odd 6 1
324.2.b.a 4 36.h even 6 1
324.2.h.e 8 1.a even 1 1 trivial
324.2.h.e 8 3.b odd 2 1 inner
324.2.h.e 8 4.b odd 2 1 CM
324.2.h.e 8 9.c even 3 1 inner
324.2.h.e 8 9.d odd 6 1 inner
324.2.h.e 8 12.b even 2 1 inner
324.2.h.e 8 36.f odd 6 1 inner
324.2.h.e 8 36.h even 6 1 inner
5184.2.c.e 4 72.j odd 6 1
5184.2.c.e 4 72.l even 6 1
5184.2.c.e 4 72.n even 6 1
5184.2.c.e 4 72.p odd 6 1

Hecke kernels

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

\( T_{5}^{8} - 28 T_{5}^{6} + 615 T_{5}^{4} - 4732 T_{5}^{2} + 28561 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 28561 - 4732 T^{2} + 615 T^{4} - 28 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$17$ \( ( 1 + 52 T^{2} + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( 14641 - 9196 T^{2} + 5655 T^{4} - 76 T^{6} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( -107 + 2 T + T^{2} )^{4} \)
$41$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( 50 + T^{2} )^{4} \)
$59$ \( T^{8} \)
$61$ \( ( 6889 - 830 T + 183 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 37 - 16 T + T^{2} )^{4} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 5041 + 196 T^{2} + T^{4} )^{2} \)
$97$ \( ( 64 + 8 T + T^{2} )^{4} \)
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