Properties

Label 324.3.g.a
Level 324
Weight 3
Character orbit 324.g
Analytic conductor 8.828
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
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.

\(f(q)\) \(=\) \( q + ( -11 + 11 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -11 + 11 \zeta_{6} ) q^{7} -23 \zeta_{6} q^{13} -37 q^{19} + ( -25 + 25 \zeta_{6} ) q^{25} + 46 \zeta_{6} q^{31} -73 q^{37} + ( 22 - 22 \zeta_{6} ) q^{43} -72 \zeta_{6} q^{49} + ( -47 + 47 \zeta_{6} ) q^{61} + 13 \zeta_{6} q^{67} + 143 q^{73} + ( -11 + 11 \zeta_{6} ) q^{79} + 253 q^{91} + ( 169 - 169 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 11q^{7} + O(q^{10}) \) \( 2q - 11q^{7} - 23q^{13} - 74q^{19} - 25q^{25} + 46q^{31} - 146q^{37} + 22q^{43} - 72q^{49} - 47q^{61} + 13q^{67} + 286q^{73} - 11q^{79} + 506q^{91} + 169q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −5.50000 9.52628i 0 0 0
269.1 0 0 0 0 0 −5.50000 + 9.52628i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d 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.g.a 2
3.b odd 2 1 CM 324.3.g.a 2
4.b odd 2 1 1296.3.q.c 2
9.c even 3 1 108.3.c.a 1
9.c even 3 1 inner 324.3.g.a 2
9.d odd 6 1 108.3.c.a 1
9.d odd 6 1 inner 324.3.g.a 2
12.b even 2 1 1296.3.q.c 2
36.f odd 6 1 432.3.e.a 1
36.f odd 6 1 1296.3.q.c 2
36.h even 6 1 432.3.e.a 1
36.h even 6 1 1296.3.q.c 2
45.h odd 6 1 2700.3.g.b 1
45.j even 6 1 2700.3.g.b 1
45.k odd 12 2 2700.3.b.d 2
45.l even 12 2 2700.3.b.d 2
72.j odd 6 1 1728.3.e.c 1
72.l even 6 1 1728.3.e.b 1
72.n even 6 1 1728.3.e.c 1
72.p odd 6 1 1728.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 9.c even 3 1
108.3.c.a 1 9.d odd 6 1
324.3.g.a 2 1.a even 1 1 trivial
324.3.g.a 2 3.b odd 2 1 CM
324.3.g.a 2 9.c even 3 1 inner
324.3.g.a 2 9.d odd 6 1 inner
432.3.e.a 1 36.f odd 6 1
432.3.e.a 1 36.h even 6 1
1296.3.q.c 2 4.b odd 2 1
1296.3.q.c 2 12.b even 2 1
1296.3.q.c 2 36.f odd 6 1
1296.3.q.c 2 36.h even 6 1
1728.3.e.b 1 72.l even 6 1
1728.3.e.b 1 72.p odd 6 1
1728.3.e.c 1 72.j odd 6 1
1728.3.e.c 1 72.n even 6 1
2700.3.b.d 2 45.k odd 12 2
2700.3.b.d 2 45.l even 12 2
2700.3.g.b 1 45.h odd 6 1
2700.3.g.b 1 45.j even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} + 11 T_{7} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} ) \)
$7$ \( ( 1 - 2 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} ) \)
$11$ \( ( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} ) \)
$13$ \( ( 1 + T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \)
$17$ \( ( 1 - 17 T )^{2}( 1 + 17 T )^{2} \)
$19$ \( ( 1 + 37 T + 361 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} ) \)
$29$ \( ( 1 - 29 T + 841 T^{2} )( 1 + 29 T + 841 T^{2} ) \)
$31$ \( ( 1 - 59 T + 961 T^{2} )( 1 + 13 T + 961 T^{2} ) \)
$37$ \( ( 1 + 73 T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T + 1681 T^{2} )( 1 + 41 T + 1681 T^{2} ) \)
$43$ \( ( 1 - 83 T + 1849 T^{2} )( 1 + 61 T + 1849 T^{2} ) \)
$47$ \( ( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} ) \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( ( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} ) \)
$61$ \( ( 1 - 74 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} ) \)
$67$ \( ( 1 - 122 T + 4489 T^{2} )( 1 + 109 T + 4489 T^{2} ) \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 - 143 T + 5329 T^{2} )^{2} \)
$79$ \( ( 1 - 131 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} ) \)
$83$ \( ( 1 - 83 T + 6889 T^{2} )( 1 + 83 T + 6889 T^{2} ) \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 - 167 T + 9409 T^{2} )( 1 - 2 T + 9409 T^{2} ) \)
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