Properties

Label 324.3.f.n
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{12}^{2} ) q^{2} -4 \zeta_{12}^{2} q^{4} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{5} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{12}^{2} ) q^{2} -4 \zeta_{12}^{2} q^{4} + ( -3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{5} -8 q^{8} + ( 8 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{10} + ( -12 \zeta_{12} - 5 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{13} + ( -16 + 16 \zeta_{12}^{2} ) q^{16} + ( 8 - 30 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{17} + ( 16 - 12 \zeta_{12} - 16 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{20} + ( -18 + 24 \zeta_{12} + 18 \zeta_{12}^{2} - 48 \zeta_{12}^{3} ) q^{25} + ( -10 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{26} + ( -20 - 21 \zeta_{12} + 20 \zeta_{12}^{2} + 42 \zeta_{12}^{3} ) q^{29} + 32 \zeta_{12}^{2} q^{32} + ( 16 - 30 \zeta_{12} - 16 \zeta_{12}^{2} + 60 \zeta_{12}^{3} ) q^{34} + ( 35 + 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{37} + ( 24 \zeta_{12} - 32 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{40} -80 \zeta_{12}^{2} q^{41} -49 \zeta_{12}^{2} q^{49} + ( -48 \zeta_{12} + 36 \zeta_{12}^{2} - 48 \zeta_{12}^{3} ) q^{50} + ( -20 - 48 \zeta_{12} + 20 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{52} + 56 q^{53} + ( 42 \zeta_{12} + 40 \zeta_{12}^{2} + 42 \zeta_{12}^{3} ) q^{58} + ( -11 + 60 \zeta_{12} + 11 \zeta_{12}^{2} - 120 \zeta_{12}^{3} ) q^{61} + 64 q^{64} + ( -88 + 33 \zeta_{12} + 88 \zeta_{12}^{2} - 66 \zeta_{12}^{3} ) q^{65} + ( 60 \zeta_{12} - 32 \zeta_{12}^{2} + 60 \zeta_{12}^{3} ) q^{68} + ( -55 + 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{73} + ( 70 + 24 \zeta_{12} - 70 \zeta_{12}^{2} - 48 \zeta_{12}^{3} ) q^{74} + ( -64 + 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{80} -160 q^{82} + ( -84 \zeta_{12} + 167 \zeta_{12}^{2} - 84 \zeta_{12}^{3} ) q^{85} + ( 80 + 78 \zeta_{12} - 39 \zeta_{12}^{3} ) q^{89} + ( 130 - 130 \zeta_{12}^{2} ) q^{97} -98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8} + O(q^{10}) \) \( 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8} + 32 q^{10} - 10 q^{13} - 32 q^{16} + 32 q^{17} + 32 q^{20} - 36 q^{25} - 40 q^{26} - 40 q^{29} + 64 q^{32} + 32 q^{34} + 140 q^{37} - 64 q^{40} - 160 q^{41} - 98 q^{49} + 72 q^{50} - 40 q^{52} + 224 q^{53} + 80 q^{58} - 22 q^{61} + 256 q^{64} - 176 q^{65} - 64 q^{68} - 220 q^{73} + 140 q^{74} - 256 q^{80} - 640 q^{82} + 334 q^{85} + 320 q^{89} + 260 q^{97} - 392 q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.00000 1.73205i 0 −2.00000 3.46410i −0.598076 1.03590i 0 0 −8.00000 0 −2.39230
55.2 1.00000 1.73205i 0 −2.00000 3.46410i 4.59808 + 7.96410i 0 0 −8.00000 0 18.3923
271.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −0.598076 + 1.03590i 0 0 −8.00000 0 −2.39230
271.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.59808 7.96410i 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 \(\Q(\sqrt{-1}) \)
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} - 8 T_{5}^{3} + 75 T_{5}^{2} + 88 T_{5} + 121 \)
\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 165649 - 4070 T + 507 T^{2} + 10 T^{3} + T^{4} \)
$17$ \( ( -611 - 16 T + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 851929 - 36920 T + 2523 T^{2} + 40 T^{3} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 793 - 70 T + T^{2} )^{2} \)
$41$ \( ( 6400 + 80 T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -56 + T )^{4} \)
$59$ \( T^{4} \)
$61$ \( 114041041 - 234938 T + 11163 T^{2} + 22 T^{3} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -3887 + 110 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 1837 - 160 T + T^{2} )^{2} \)
$97$ \( ( 16900 - 130 T + T^{2} )^{2} \)
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