Properties

Label 324.3.f.e
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 8 - 8 \zeta_{6} ) q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + ( 8 - 8 \zeta_{6} ) q^{5} + 8 q^{8} -16 q^{10} + ( 10 - 10 \zeta_{6} ) q^{13} -16 \zeta_{6} q^{16} + 16 q^{17} + 32 \zeta_{6} q^{20} -39 \zeta_{6} q^{25} -20 q^{26} -40 \zeta_{6} q^{29} + ( -32 + 32 \zeta_{6} ) q^{32} -32 \zeta_{6} q^{34} -70 q^{37} + ( 64 - 64 \zeta_{6} ) q^{40} + ( 80 - 80 \zeta_{6} ) q^{41} + ( -49 + 49 \zeta_{6} ) q^{49} + ( -78 + 78 \zeta_{6} ) q^{50} + 40 \zeta_{6} q^{52} -56 q^{53} + ( -80 + 80 \zeta_{6} ) q^{58} + 22 \zeta_{6} q^{61} + 64 q^{64} -80 \zeta_{6} q^{65} + ( -64 + 64 \zeta_{6} ) q^{68} + 110 q^{73} + 140 \zeta_{6} q^{74} -128 q^{80} -160 q^{82} + ( 128 - 128 \zeta_{6} ) q^{85} + 160 q^{89} + 130 \zeta_{6} q^{97} + 98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 8q^{5} + 16q^{8} - 32q^{10} + 10q^{13} - 16q^{16} + 32q^{17} + 32q^{20} - 39q^{25} - 40q^{26} - 40q^{29} - 32q^{32} - 32q^{34} - 140q^{37} + 64q^{40} + 80q^{41} - 49q^{49} - 78q^{50} + 40q^{52} - 112q^{53} - 80q^{58} + 22q^{61} + 128q^{64} - 80q^{65} - 64q^{68} + 220q^{73} + 140q^{74} - 256q^{80} - 320q^{82} + 128q^{85} + 320q^{89} + 130q^{97} + 196q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i 4.00000 + 6.92820i 0 0 8.00000 0 −16.0000
271.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i 4.00000 6.92820i 0 0 8.00000 0 −16.0000
\(n\): 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.e 2
3.b odd 2 1 324.3.f.f 2
4.b odd 2 1 CM 324.3.f.e 2
9.c even 3 1 36.3.d.b yes 1
9.c even 3 1 inner 324.3.f.e 2
9.d odd 6 1 36.3.d.a 1
9.d odd 6 1 324.3.f.f 2
12.b even 2 1 324.3.f.f 2
36.f odd 6 1 36.3.d.b yes 1
36.f odd 6 1 inner 324.3.f.e 2
36.h even 6 1 36.3.d.a 1
36.h even 6 1 324.3.f.f 2
45.h odd 6 1 900.3.c.c 1
45.j even 6 1 900.3.c.b 1
45.k odd 12 2 900.3.f.a 2
45.l even 12 2 900.3.f.b 2
72.j odd 6 1 576.3.g.a 1
72.l even 6 1 576.3.g.a 1
72.n even 6 1 576.3.g.c 1
72.p odd 6 1 576.3.g.c 1
144.u even 12 2 2304.3.b.d 2
144.v odd 12 2 2304.3.b.e 2
144.w odd 12 2 2304.3.b.d 2
144.x even 12 2 2304.3.b.e 2
180.n even 6 1 900.3.c.c 1
180.p odd 6 1 900.3.c.b 1
180.v odd 12 2 900.3.f.b 2
180.x even 12 2 900.3.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a 1 9.d odd 6 1
36.3.d.a 1 36.h even 6 1
36.3.d.b yes 1 9.c even 3 1
36.3.d.b yes 1 36.f odd 6 1
324.3.f.e 2 1.a even 1 1 trivial
324.3.f.e 2 4.b odd 2 1 CM
324.3.f.e 2 9.c even 3 1 inner
324.3.f.e 2 36.f odd 6 1 inner
324.3.f.f 2 3.b odd 2 1
324.3.f.f 2 9.d odd 6 1
324.3.f.f 2 12.b even 2 1
324.3.f.f 2 36.h even 6 1
576.3.g.a 1 72.j odd 6 1
576.3.g.a 1 72.l even 6 1
576.3.g.c 1 72.n even 6 1
576.3.g.c 1 72.p odd 6 1
900.3.c.b 1 45.j even 6 1
900.3.c.b 1 180.p odd 6 1
900.3.c.c 1 45.h odd 6 1
900.3.c.c 1 180.n even 6 1
900.3.f.a 2 45.k odd 12 2
900.3.f.a 2 180.x even 12 2
900.3.f.b 2 45.l even 12 2
900.3.f.b 2 180.v odd 12 2
2304.3.b.d 2 144.u even 12 2
2304.3.b.d 2 144.w odd 12 2
2304.3.b.e 2 144.v odd 12 2
2304.3.b.e 2 144.x even 12 2

Hecke kernels

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

\( T_{5}^{2} - 8 T_{5} + 64 \)
\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 64 - 8 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 100 - 10 T + T^{2} \)
$17$ \( ( -16 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1600 + 40 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 70 + T )^{2} \)
$41$ \( 6400 - 80 T + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 56 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 484 - 22 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -110 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -160 + T )^{2} \)
$97$ \( 16900 - 130 T + T^{2} \)
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