Properties

Label 324.3.f.b
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( -10 + 5 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + ( 7 - 7 \zeta_{6} ) q^{5} + ( -10 + 5 \zeta_{6} ) q^{7} -8 q^{8} + ( -14 + 14 \zeta_{6} ) q^{10} + ( -10 + 5 \zeta_{6} ) q^{11} + ( -20 + 20 \zeta_{6} ) q^{13} + ( 20 - 10 \zeta_{6} ) q^{14} + 16 q^{16} + 8 q^{17} + ( -6 + 12 \zeta_{6} ) q^{19} + ( 28 - 28 \zeta_{6} ) q^{20} + ( 20 - 10 \zeta_{6} ) q^{22} + ( -2 - 2 \zeta_{6} ) q^{23} -24 \zeta_{6} q^{25} + ( 40 - 40 \zeta_{6} ) q^{26} + ( -40 + 20 \zeta_{6} ) q^{28} + 10 \zeta_{6} q^{29} + ( 31 + 31 \zeta_{6} ) q^{31} -32 q^{32} -16 q^{34} + ( -35 + 70 \zeta_{6} ) q^{35} -10 q^{37} + ( 12 - 24 \zeta_{6} ) q^{38} + ( -56 + 56 \zeta_{6} ) q^{40} + ( -50 + 50 \zeta_{6} ) q^{41} + ( 20 - 10 \zeta_{6} ) q^{43} + ( -40 + 20 \zeta_{6} ) q^{44} + ( 4 + 4 \zeta_{6} ) q^{46} + ( -100 + 50 \zeta_{6} ) q^{47} + ( 26 - 26 \zeta_{6} ) q^{49} + 48 \zeta_{6} q^{50} + ( -80 + 80 \zeta_{6} ) q^{52} + 47 q^{53} + ( -35 + 70 \zeta_{6} ) q^{55} + ( 80 - 40 \zeta_{6} ) q^{56} -20 \zeta_{6} q^{58} + ( -20 - 20 \zeta_{6} ) q^{59} + 64 \zeta_{6} q^{61} + ( -62 - 62 \zeta_{6} ) q^{62} + 64 q^{64} + 140 \zeta_{6} q^{65} + ( -50 - 50 \zeta_{6} ) q^{67} + 32 q^{68} + ( 70 - 140 \zeta_{6} ) q^{70} -55 q^{73} + 20 q^{74} + ( -24 + 48 \zeta_{6} ) q^{76} + ( 75 - 75 \zeta_{6} ) q^{77} + ( 8 - 4 \zeta_{6} ) q^{79} + ( 112 - 112 \zeta_{6} ) q^{80} + ( 100 - 100 \zeta_{6} ) q^{82} + ( -34 + 17 \zeta_{6} ) q^{83} + ( 56 - 56 \zeta_{6} ) q^{85} + ( -40 + 20 \zeta_{6} ) q^{86} + ( 80 - 40 \zeta_{6} ) q^{88} -10 q^{89} + ( 100 - 200 \zeta_{6} ) q^{91} + ( -8 - 8 \zeta_{6} ) q^{92} + ( 200 - 100 \zeta_{6} ) q^{94} + ( 42 + 42 \zeta_{6} ) q^{95} + 25 \zeta_{6} q^{97} + ( -52 + 52 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 15 q^{7} - 16 q^{8} + O(q^{10}) \) \( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 15 q^{7} - 16 q^{8} - 14 q^{10} - 15 q^{11} - 20 q^{13} + 30 q^{14} + 32 q^{16} + 16 q^{17} + 28 q^{20} + 30 q^{22} - 6 q^{23} - 24 q^{25} + 40 q^{26} - 60 q^{28} + 10 q^{29} + 93 q^{31} - 64 q^{32} - 32 q^{34} - 20 q^{37} - 56 q^{40} - 50 q^{41} + 30 q^{43} - 60 q^{44} + 12 q^{46} - 150 q^{47} + 26 q^{49} + 48 q^{50} - 80 q^{52} + 94 q^{53} + 120 q^{56} - 20 q^{58} - 60 q^{59} + 64 q^{61} - 186 q^{62} + 128 q^{64} + 140 q^{65} - 150 q^{67} + 64 q^{68} - 110 q^{73} + 40 q^{74} + 75 q^{77} + 12 q^{79} + 112 q^{80} + 100 q^{82} - 51 q^{83} + 56 q^{85} - 60 q^{86} + 120 q^{88} - 20 q^{89} - 24 q^{92} + 300 q^{94} + 126 q^{95} + 25 q^{97} - 52 q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 3.50000 + 6.06218i 0 −7.50000 4.33013i −8.00000 0 −7.00000 12.1244i
271.1 −2.00000 0 4.00000 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.b 2
3.b odd 2 1 324.3.f.i 2
4.b odd 2 1 324.3.f.h 2
9.c even 3 1 108.3.d.b yes 2
9.c even 3 1 324.3.f.h 2
9.d odd 6 1 108.3.d.a 2
9.d odd 6 1 324.3.f.c 2
12.b even 2 1 324.3.f.c 2
36.f odd 6 1 108.3.d.b yes 2
36.f odd 6 1 inner 324.3.f.b 2
36.h even 6 1 108.3.d.a 2
36.h even 6 1 324.3.f.i 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 1.a even 1 1 trivial
324.3.f.b 2 36.f odd 6 1 inner
324.3.f.c 2 9.d odd 6 1
324.3.f.c 2 12.b even 2 1
324.3.f.h 2 4.b odd 2 1
324.3.f.h 2 9.c even 3 1
324.3.f.i 2 3.b odd 2 1
324.3.f.i 2 36.h even 6 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

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