Properties

Label 324.3.d.c
Level 324
Weight 3
Character orbit 324.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -4 q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -4 q^{5} + ( -2 + 4 \zeta_{6} ) q^{7} -8 q^{8} -8 \zeta_{6} q^{10} + ( 7 - 14 \zeta_{6} ) q^{11} -22 q^{13} + ( -8 + 4 \zeta_{6} ) q^{14} -16 \zeta_{6} q^{16} + 11 q^{17} + ( -9 + 18 \zeta_{6} ) q^{19} + ( 16 - 16 \zeta_{6} ) q^{20} + ( 28 - 14 \zeta_{6} ) q^{22} + ( 14 - 28 \zeta_{6} ) q^{23} -9 q^{25} -44 \zeta_{6} q^{26} + ( -8 - 8 \zeta_{6} ) q^{28} -34 q^{29} + ( -4 + 8 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} + 22 \zeta_{6} q^{34} + ( 8 - 16 \zeta_{6} ) q^{35} -16 q^{37} + ( -36 + 18 \zeta_{6} ) q^{38} + 32 q^{40} -13 q^{41} + ( -29 + 58 \zeta_{6} ) q^{43} + ( 28 + 28 \zeta_{6} ) q^{44} + ( 56 - 28 \zeta_{6} ) q^{46} + ( -2 + 4 \zeta_{6} ) q^{47} + 37 q^{49} -18 \zeta_{6} q^{50} + ( 88 - 88 \zeta_{6} ) q^{52} -52 q^{53} + ( -28 + 56 \zeta_{6} ) q^{55} + ( 16 - 32 \zeta_{6} ) q^{56} -68 \zeta_{6} q^{58} + ( -31 + 62 \zeta_{6} ) q^{59} -16 q^{61} + ( -16 + 8 \zeta_{6} ) q^{62} + 64 q^{64} + 88 q^{65} + ( -67 + 134 \zeta_{6} ) q^{67} + ( -44 + 44 \zeta_{6} ) q^{68} + ( 32 - 16 \zeta_{6} ) q^{70} -25 q^{73} -32 \zeta_{6} q^{74} + ( -36 - 36 \zeta_{6} ) q^{76} + 42 q^{77} + ( 16 - 32 \zeta_{6} ) q^{79} + 64 \zeta_{6} q^{80} -26 \zeta_{6} q^{82} + ( -20 + 40 \zeta_{6} ) q^{83} -44 q^{85} + ( -116 + 58 \zeta_{6} ) q^{86} + ( -56 + 112 \zeta_{6} ) q^{88} + 2 q^{89} + ( 44 - 88 \zeta_{6} ) q^{91} + ( 56 + 56 \zeta_{6} ) q^{92} + ( -8 + 4 \zeta_{6} ) q^{94} + ( 36 - 72 \zeta_{6} ) q^{95} -43 q^{97} + 74 \zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 8q^{5} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 8q^{5} - 16q^{8} - 8q^{10} - 44q^{13} - 12q^{14} - 16q^{16} + 22q^{17} + 16q^{20} + 42q^{22} - 18q^{25} - 44q^{26} - 24q^{28} - 68q^{29} + 32q^{32} + 22q^{34} - 32q^{37} - 54q^{38} + 64q^{40} - 26q^{41} + 84q^{44} + 84q^{46} + 74q^{49} - 18q^{50} + 88q^{52} - 104q^{53} - 68q^{58} - 32q^{61} - 24q^{62} + 128q^{64} + 176q^{65} - 44q^{68} + 48q^{70} - 50q^{73} - 32q^{74} - 108q^{76} + 84q^{77} + 64q^{80} - 26q^{82} - 88q^{85} - 174q^{86} + 4q^{89} + 168q^{92} - 12q^{94} - 86q^{97} + 74q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i −4.00000 0 3.46410i −8.00000 0 −4.00000 + 6.92820i
163.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.00000 0 3.46410i −8.00000 0 −4.00000 6.92820i
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.c 2
3.b odd 2 1 324.3.d.b 2
4.b odd 2 1 inner 324.3.d.c 2
9.c even 3 1 108.3.f.a 2
9.c even 3 1 108.3.f.b 2
9.d odd 6 1 36.3.f.a 2
9.d odd 6 1 36.3.f.b yes 2
12.b even 2 1 324.3.d.b 2
36.f odd 6 1 108.3.f.a 2
36.f odd 6 1 108.3.f.b 2
36.h even 6 1 36.3.f.a 2
36.h even 6 1 36.3.f.b yes 2
72.j odd 6 1 576.3.o.a 2
72.j odd 6 1 576.3.o.b 2
72.l even 6 1 576.3.o.a 2
72.l even 6 1 576.3.o.b 2
72.n even 6 1 1728.3.o.a 2
72.n even 6 1 1728.3.o.b 2
72.p odd 6 1 1728.3.o.a 2
72.p odd 6 1 1728.3.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 9.d odd 6 1
36.3.f.a 2 36.h even 6 1
36.3.f.b yes 2 9.d odd 6 1
36.3.f.b yes 2 36.h even 6 1
108.3.f.a 2 9.c even 3 1
108.3.f.a 2 36.f odd 6 1
108.3.f.b 2 9.c even 3 1
108.3.f.b 2 36.f odd 6 1
324.3.d.b 2 3.b odd 2 1
324.3.d.b 2 12.b even 2 1
324.3.d.c 2 1.a even 1 1 trivial
324.3.d.c 2 4.b odd 2 1 inner
576.3.o.a 2 72.j odd 6 1
576.3.o.a 2 72.l even 6 1
576.3.o.b 2 72.j odd 6 1
576.3.o.b 2 72.l even 6 1
1728.3.o.a 2 72.n even 6 1
1728.3.o.a 2 72.p odd 6 1
1728.3.o.b 2 72.n even 6 1
1728.3.o.b 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 4 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ 1
$5$ \( ( 1 + 4 T + 25 T^{2} )^{2} \)
$7$ \( 1 - 86 T^{2} + 2401 T^{4} \)
$11$ \( 1 - 95 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 + 22 T + 169 T^{2} )^{2} \)
$17$ \( ( 1 - 11 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 479 T^{2} + 130321 T^{4} \)
$23$ \( 1 - 470 T^{2} + 279841 T^{4} \)
$29$ \( ( 1 + 34 T + 841 T^{2} )^{2} \)
$31$ \( 1 - 1874 T^{2} + 923521 T^{4} \)
$37$ \( ( 1 + 16 T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 + 13 T + 1681 T^{2} )^{2} \)
$43$ \( 1 - 1175 T^{2} + 3418801 T^{4} \)
$47$ \( 1 - 4406 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 + 52 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 4079 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 16 T + 3721 T^{2} )^{2} \)
$67$ \( ( 1 - 67 T + 4489 T^{2} )( 1 + 67 T + 4489 T^{2} ) \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 + 25 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 11714 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 12578 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 - 2 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 + 43 T + 9409 T^{2} )^{2} \)
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