Properties

Label 324.2.e.b
Level $324$
Weight $2$
Character orbit 324.e
Analytic conductor $2.587$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -5 + 5 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -5 + 5 \zeta_{6} ) q^{7} + 7 \zeta_{6} q^{13} - q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} - q^{37} + ( -8 + 8 \zeta_{6} ) q^{43} -18 \zeta_{6} q^{49} + ( 13 - 13 \zeta_{6} ) q^{61} -11 \zeta_{6} q^{67} + 17 q^{73} + ( 13 - 13 \zeta_{6} ) q^{79} -35 q^{91} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 5q^{7} + O(q^{10}) \) \( 2q - 5q^{7} + 7q^{13} - 2q^{19} + 5q^{25} + 4q^{31} - 2q^{37} - 8q^{43} - 18q^{49} + 13q^{61} - 11q^{67} + 34q^{73} + 13q^{79} - 70q^{91} - 5q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −2.50000 4.33013i 0 0 0
217.1 0 0 0 0 0 −2.50000 + 4.33013i 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 324.2.e.b 2
3.b odd 2 1 CM 324.2.e.b 2
4.b odd 2 1 1296.2.i.j 2
9.c even 3 1 108.2.a.a 1
9.c even 3 1 inner 324.2.e.b 2
9.d odd 6 1 108.2.a.a 1
9.d odd 6 1 inner 324.2.e.b 2
12.b even 2 1 1296.2.i.j 2
36.f odd 6 1 432.2.a.d 1
36.f odd 6 1 1296.2.i.j 2
36.h even 6 1 432.2.a.d 1
36.h even 6 1 1296.2.i.j 2
45.h odd 6 1 2700.2.a.b 1
45.j even 6 1 2700.2.a.b 1
45.k odd 12 2 2700.2.d.g 2
45.l even 12 2 2700.2.d.g 2
63.l odd 6 1 5292.2.a.j 1
63.o even 6 1 5292.2.a.j 1
72.j odd 6 1 1728.2.a.p 1
72.l even 6 1 1728.2.a.m 1
72.n even 6 1 1728.2.a.p 1
72.p odd 6 1 1728.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 9.c even 3 1
108.2.a.a 1 9.d odd 6 1
324.2.e.b 2 1.a even 1 1 trivial
324.2.e.b 2 3.b odd 2 1 CM
324.2.e.b 2 9.c even 3 1 inner
324.2.e.b 2 9.d odd 6 1 inner
432.2.a.d 1 36.f odd 6 1
432.2.a.d 1 36.h even 6 1
1296.2.i.j 2 4.b odd 2 1
1296.2.i.j 2 12.b even 2 1
1296.2.i.j 2 36.f odd 6 1
1296.2.i.j 2 36.h even 6 1
1728.2.a.m 1 72.l even 6 1
1728.2.a.m 1 72.p odd 6 1
1728.2.a.p 1 72.j odd 6 1
1728.2.a.p 1 72.n even 6 1
2700.2.a.b 1 45.h odd 6 1
2700.2.a.b 1 45.j even 6 1
2700.2.d.g 2 45.k odd 12 2
2700.2.d.g 2 45.l even 12 2
5292.2.a.j 1 63.l odd 6 1
5292.2.a.j 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(324, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} + 5 T_{7} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 49 - 7 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -17 + T )^{2} \)
$79$ \( 169 - 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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