Properties

Label 324.3.f.h
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( 10 - 5 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( 10 - 5 \zeta_{6} ) q^{7} -8 q^{8} -14 \zeta_{6} q^{10} + ( 10 - 5 \zeta_{6} ) q^{11} + ( -20 + 20 \zeta_{6} ) q^{13} + ( 10 - 20 \zeta_{6} ) q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 8 q^{17} + ( 6 - 12 \zeta_{6} ) q^{19} -28 q^{20} + ( 10 - 20 \zeta_{6} ) q^{22} + ( 2 + 2 \zeta_{6} ) q^{23} -24 \zeta_{6} q^{25} + 40 \zeta_{6} q^{26} + ( -20 - 20 \zeta_{6} ) q^{28} + 10 \zeta_{6} q^{29} + ( -31 - 31 \zeta_{6} ) q^{31} + 32 \zeta_{6} q^{32} + ( 16 - 16 \zeta_{6} ) q^{34} + ( 35 - 70 \zeta_{6} ) q^{35} -10 q^{37} + ( -12 - 12 \zeta_{6} ) q^{38} + ( -56 + 56 \zeta_{6} ) q^{40} + ( -50 + 50 \zeta_{6} ) q^{41} + ( -20 + 10 \zeta_{6} ) q^{43} + ( -20 - 20 \zeta_{6} ) q^{44} + ( 8 - 4 \zeta_{6} ) q^{46} + ( 100 - 50 \zeta_{6} ) q^{47} + ( 26 - 26 \zeta_{6} ) q^{49} -48 q^{50} + 80 q^{52} + 47 q^{53} + ( 35 - 70 \zeta_{6} ) q^{55} + ( -80 + 40 \zeta_{6} ) q^{56} + 20 q^{58} + ( 20 + 20 \zeta_{6} ) q^{59} + 64 \zeta_{6} q^{61} + ( -124 + 62 \zeta_{6} ) q^{62} + 64 q^{64} + 140 \zeta_{6} q^{65} + ( 50 + 50 \zeta_{6} ) q^{67} -32 \zeta_{6} q^{68} + ( -70 - 70 \zeta_{6} ) q^{70} -55 q^{73} + ( -20 + 20 \zeta_{6} ) q^{74} + ( -48 + 24 \zeta_{6} ) q^{76} + ( 75 - 75 \zeta_{6} ) q^{77} + ( -8 + 4 \zeta_{6} ) q^{79} + 112 \zeta_{6} q^{80} + 100 \zeta_{6} q^{82} + ( 34 - 17 \zeta_{6} ) q^{83} + ( 56 - 56 \zeta_{6} ) q^{85} + ( -20 + 40 \zeta_{6} ) q^{86} + ( -80 + 40 \zeta_{6} ) q^{88} -10 q^{89} + ( -100 + 200 \zeta_{6} ) q^{91} + ( 8 - 16 \zeta_{6} ) q^{92} + ( 100 - 200 \zeta_{6} ) q^{94} + ( -42 - 42 \zeta_{6} ) q^{95} + 25 \zeta_{6} q^{97} -52 \zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 7 q^{5} + 15 q^{7} - 16 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} - 4 q^{4} + 7 q^{5} + 15 q^{7} - 16 q^{8} - 14 q^{10} + 15 q^{11} - 20 q^{13} - 16 q^{16} + 16 q^{17} - 56 q^{20} + 6 q^{23} - 24 q^{25} + 40 q^{26} - 60 q^{28} + 10 q^{29} - 93 q^{31} + 32 q^{32} + 16 q^{34} - 20 q^{37} - 36 q^{38} - 56 q^{40} - 50 q^{41} - 30 q^{43} - 60 q^{44} + 12 q^{46} + 150 q^{47} + 26 q^{49} - 96 q^{50} + 160 q^{52} + 94 q^{53} - 120 q^{56} + 40 q^{58} + 60 q^{59} + 64 q^{61} - 186 q^{62} + 128 q^{64} + 140 q^{65} + 150 q^{67} - 32 q^{68} - 210 q^{70} - 110 q^{73} - 20 q^{74} - 72 q^{76} + 75 q^{77} - 12 q^{79} + 112 q^{80} + 100 q^{82} + 51 q^{83} + 56 q^{85} - 120 q^{88} - 20 q^{89} - 126 q^{95} + 25 q^{97} - 52 q^{98} + O(q^{100}) \)

Character values

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

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

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 3.50000 + 6.06218i 0 7.50000 + 4.33013i −8.00000 0 −7.00000 + 12.1244i
271.1 1.00000 1.73205i 0 −2.00000 3.46410i 3.50000 6.06218i 0 7.50000 4.33013i −8.00000 0 −7.00000 12.1244i
\(n\): 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.h 2
3.b odd 2 1 324.3.f.c 2
4.b odd 2 1 324.3.f.b 2
9.c even 3 1 108.3.d.b yes 2
9.c even 3 1 324.3.f.b 2
9.d odd 6 1 108.3.d.a 2
9.d odd 6 1 324.3.f.i 2
12.b even 2 1 324.3.f.i 2
36.f odd 6 1 108.3.d.b yes 2
36.f odd 6 1 inner 324.3.f.h 2
36.h even 6 1 108.3.d.a 2
36.h even 6 1 324.3.f.c 2
72.j odd 6 1 1728.3.g.a 2
72.l even 6 1 1728.3.g.a 2
72.n even 6 1 1728.3.g.f 2
72.p odd 6 1 1728.3.g.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 9.d odd 6 1
108.3.d.a 2 36.h even 6 1
108.3.d.b yes 2 9.c even 3 1
108.3.d.b yes 2 36.f odd 6 1
324.3.f.b 2 4.b odd 2 1
324.3.f.b 2 9.c even 3 1
324.3.f.c 2 3.b odd 2 1
324.3.f.c 2 36.h even 6 1
324.3.f.h 2 1.a even 1 1 trivial
324.3.f.h 2 36.f odd 6 1 inner
324.3.f.i 2 9.d odd 6 1
324.3.f.i 2 12.b even 2 1
1728.3.g.a 2 72.j odd 6 1
1728.3.g.a 2 72.l even 6 1
1728.3.g.f 2 72.n even 6 1
1728.3.g.f 2 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 \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{2} - 7 T_{5} + 49 \)
\( T_{7}^{2} - 15 T_{7} + 75 \)
\( T_{11}^{2} - 15 T_{11} + 75 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 49 - 7 T + T^{2} \)
$7$ \( 75 - 15 T + T^{2} \)
$11$ \( 75 - 15 T + T^{2} \)
$13$ \( 400 + 20 T + T^{2} \)
$17$ \( ( -8 + T )^{2} \)
$19$ \( 108 + T^{2} \)
$23$ \( 12 - 6 T + T^{2} \)
$29$ \( 100 - 10 T + T^{2} \)
$31$ \( 2883 + 93 T + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( 2500 + 50 T + T^{2} \)
$43$ \( 300 + 30 T + T^{2} \)
$47$ \( 7500 - 150 T + T^{2} \)
$53$ \( ( -47 + T )^{2} \)
$59$ \( 1200 - 60 T + T^{2} \)
$61$ \( 4096 - 64 T + T^{2} \)
$67$ \( 7500 - 150 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 55 + T )^{2} \)
$79$ \( 48 + 12 T + T^{2} \)
$83$ \( 867 - 51 T + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 625 - 25 T + T^{2} \)
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