Properties

Label 324.3.f.a
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $2$
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.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 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 8 - 4 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 8 - 4 \zeta_{6} ) q^{7} -8 q^{8} + ( 4 - 4 \zeta_{6} ) q^{10} + ( 8 - 4 \zeta_{6} ) q^{11} + ( -2 + 2 \zeta_{6} ) q^{13} + ( -16 + 8 \zeta_{6} ) q^{14} + 16 q^{16} -10 q^{17} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -8 + 8 \zeta_{6} ) q^{20} + ( -16 + 8 \zeta_{6} ) q^{22} + ( 16 + 16 \zeta_{6} ) q^{23} + 21 \zeta_{6} q^{25} + ( 4 - 4 \zeta_{6} ) q^{26} + ( 32 - 16 \zeta_{6} ) q^{28} -26 \zeta_{6} q^{29} + ( 4 + 4 \zeta_{6} ) q^{31} -32 q^{32} + 20 q^{34} + ( -8 + 16 \zeta_{6} ) q^{35} + 26 q^{37} + ( -24 + 48 \zeta_{6} ) q^{38} + ( 16 - 16 \zeta_{6} ) q^{40} + ( 58 - 58 \zeta_{6} ) q^{41} + ( 56 - 28 \zeta_{6} ) q^{43} + ( 32 - 16 \zeta_{6} ) q^{44} + ( -32 - 32 \zeta_{6} ) q^{46} + ( 80 - 40 \zeta_{6} ) q^{47} + ( -1 + \zeta_{6} ) q^{49} -42 \zeta_{6} q^{50} + ( -8 + 8 \zeta_{6} ) q^{52} + 74 q^{53} + ( -8 + 16 \zeta_{6} ) q^{55} + ( -64 + 32 \zeta_{6} ) q^{56} + 52 \zeta_{6} q^{58} + ( 52 + 52 \zeta_{6} ) q^{59} -26 \zeta_{6} q^{61} + ( -8 - 8 \zeta_{6} ) q^{62} + 64 q^{64} -4 \zeta_{6} q^{65} + ( 4 + 4 \zeta_{6} ) q^{67} -40 q^{68} + ( 16 - 32 \zeta_{6} ) q^{70} -46 q^{73} -52 q^{74} + ( 48 - 96 \zeta_{6} ) q^{76} + ( 48 - 48 \zeta_{6} ) q^{77} + ( -136 + 68 \zeta_{6} ) q^{79} + ( -32 + 32 \zeta_{6} ) q^{80} + ( -116 + 116 \zeta_{6} ) q^{82} + ( 56 - 28 \zeta_{6} ) q^{83} + ( 20 - 20 \zeta_{6} ) q^{85} + ( -112 + 56 \zeta_{6} ) q^{86} + ( -64 + 32 \zeta_{6} ) q^{88} -82 q^{89} + ( -8 + 16 \zeta_{6} ) q^{91} + ( 64 + 64 \zeta_{6} ) q^{92} + ( -160 + 80 \zeta_{6} ) q^{94} + ( 24 + 24 \zeta_{6} ) q^{95} -2 \zeta_{6} q^{97} + ( 2 - 2 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 2q^{5} + 12q^{7} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 2q^{5} + 12q^{7} - 16q^{8} + 4q^{10} + 12q^{11} - 2q^{13} - 24q^{14} + 32q^{16} - 20q^{17} - 8q^{20} - 24q^{22} + 48q^{23} + 21q^{25} + 4q^{26} + 48q^{28} - 26q^{29} + 12q^{31} - 64q^{32} + 40q^{34} + 52q^{37} + 16q^{40} + 58q^{41} + 84q^{43} + 48q^{44} - 96q^{46} + 120q^{47} - q^{49} - 42q^{50} - 8q^{52} + 148q^{53} - 96q^{56} + 52q^{58} + 156q^{59} - 26q^{61} - 24q^{62} + 128q^{64} - 4q^{65} + 12q^{67} - 80q^{68} - 92q^{73} - 104q^{74} + 48q^{77} - 204q^{79} - 32q^{80} - 116q^{82} + 84q^{83} + 20q^{85} - 168q^{86} - 96q^{88} - 164q^{89} + 192q^{92} - 240q^{94} + 72q^{95} - 2q^{97} + 2q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 −1.00000 1.73205i 0 6.00000 + 3.46410i −8.00000 0 2.00000 + 3.46410i
271.1 −2.00000 0 4.00000 −1.00000 + 1.73205i 0 6.00000 3.46410i −8.00000 0 2.00000 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a 2
3.b odd 2 1 324.3.f.j 2
4.b odd 2 1 324.3.f.g 2
9.c even 3 1 36.3.d.c 2
9.c even 3 1 324.3.f.g 2
9.d odd 6 1 12.3.d.a 2
9.d odd 6 1 324.3.f.d 2
12.b even 2 1 324.3.f.d 2
36.f odd 6 1 36.3.d.c 2
36.f odd 6 1 inner 324.3.f.a 2
36.h even 6 1 12.3.d.a 2
36.h even 6 1 324.3.f.j 2
45.h odd 6 1 300.3.c.b 2
45.j even 6 1 900.3.c.e 2
45.k odd 12 2 900.3.f.c 4
45.l even 12 2 300.3.f.a 4
63.o even 6 1 588.3.g.b 2
72.j odd 6 1 192.3.g.b 2
72.l even 6 1 192.3.g.b 2
72.n even 6 1 576.3.g.e 2
72.p odd 6 1 576.3.g.e 2
144.u even 12 2 768.3.b.c 4
144.v odd 12 2 2304.3.b.l 4
144.w odd 12 2 768.3.b.c 4
144.x even 12 2 2304.3.b.l 4
180.n even 6 1 300.3.c.b 2
180.p odd 6 1 900.3.c.e 2
180.v odd 12 2 300.3.f.a 4
180.x even 12 2 900.3.f.c 4
252.s odd 6 1 588.3.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.d.a 2 9.d odd 6 1
12.3.d.a 2 36.h even 6 1
36.3.d.c 2 9.c even 3 1
36.3.d.c 2 36.f odd 6 1
192.3.g.b 2 72.j odd 6 1
192.3.g.b 2 72.l even 6 1
300.3.c.b 2 45.h odd 6 1
300.3.c.b 2 180.n even 6 1
300.3.f.a 4 45.l even 12 2
300.3.f.a 4 180.v odd 12 2
324.3.f.a 2 1.a even 1 1 trivial
324.3.f.a 2 36.f odd 6 1 inner
324.3.f.d 2 9.d odd 6 1
324.3.f.d 2 12.b even 2 1
324.3.f.g 2 4.b odd 2 1
324.3.f.g 2 9.c even 3 1
324.3.f.j 2 3.b odd 2 1
324.3.f.j 2 36.h even 6 1
576.3.g.e 2 72.n even 6 1
576.3.g.e 2 72.p odd 6 1
588.3.g.b 2 63.o even 6 1
588.3.g.b 2 252.s odd 6 1
768.3.b.c 4 144.u even 12 2
768.3.b.c 4 144.w odd 12 2
900.3.c.e 2 45.j even 6 1
900.3.c.e 2 180.p odd 6 1
900.3.f.c 4 45.k odd 12 2
900.3.f.c 4 180.x even 12 2
2304.3.b.l 4 144.v odd 12 2
2304.3.b.l 4 144.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 \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7}^{2} - 12 T_{7} + 48 \)
\( T_{11}^{2} - 12 T_{11} + 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( 48 - 12 T + T^{2} \)
$11$ \( 48 - 12 T + T^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( ( 10 + T )^{2} \)
$19$ \( 432 + T^{2} \)
$23$ \( 768 - 48 T + T^{2} \)
$29$ \( 676 + 26 T + T^{2} \)
$31$ \( 48 - 12 T + T^{2} \)
$37$ \( ( -26 + T )^{2} \)
$41$ \( 3364 - 58 T + T^{2} \)
$43$ \( 2352 - 84 T + T^{2} \)
$47$ \( 4800 - 120 T + T^{2} \)
$53$ \( ( -74 + T )^{2} \)
$59$ \( 8112 - 156 T + T^{2} \)
$61$ \( 676 + 26 T + T^{2} \)
$67$ \( 48 - 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 46 + T )^{2} \)
$79$ \( 13872 + 204 T + T^{2} \)
$83$ \( 2352 - 84 T + T^{2} \)
$89$ \( ( 82 + T )^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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