Properties

Label 324.2.e.c
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( 4 - 4 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 4 - 4 \zeta_{6} ) q^{7} -2 \zeta_{6} q^{13} + 8 q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{31} -10 q^{37} + ( -8 + 8 \zeta_{6} ) q^{43} -9 \zeta_{6} q^{49} + ( -14 + 14 \zeta_{6} ) q^{61} + 16 \zeta_{6} q^{67} -10 q^{73} + ( 4 - 4 \zeta_{6} ) q^{79} -8 q^{91} + ( -14 + 14 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{7} + O(q^{10}) \) \( 2q + 4q^{7} - 2q^{13} + 16q^{19} + 5q^{25} + 4q^{31} - 20q^{37} - 8q^{43} - 9q^{49} - 14q^{61} + 16q^{67} - 20q^{73} + 4q^{79} - 16q^{91} - 14q^{97} + O(q^{100}) \)

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 2.00000 + 3.46410i 0 0 0
217.1 0 0 0 0 0 2.00000 3.46410i 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.c 2
3.b odd 2 1 CM 324.2.e.c 2
4.b odd 2 1 1296.2.i.h 2
9.c even 3 1 36.2.a.a 1
9.c even 3 1 inner 324.2.e.c 2
9.d odd 6 1 36.2.a.a 1
9.d odd 6 1 inner 324.2.e.c 2
12.b even 2 1 1296.2.i.h 2
36.f odd 6 1 144.2.a.a 1
36.f odd 6 1 1296.2.i.h 2
36.h even 6 1 144.2.a.a 1
36.h even 6 1 1296.2.i.h 2
45.h odd 6 1 900.2.a.g 1
45.j even 6 1 900.2.a.g 1
45.k odd 12 2 900.2.d.b 2
45.l even 12 2 900.2.d.b 2
63.g even 3 1 1764.2.k.h 2
63.h even 3 1 1764.2.k.h 2
63.i even 6 1 1764.2.k.g 2
63.j odd 6 1 1764.2.k.h 2
63.k odd 6 1 1764.2.k.g 2
63.l odd 6 1 1764.2.a.e 1
63.n odd 6 1 1764.2.k.h 2
63.o even 6 1 1764.2.a.e 1
63.s even 6 1 1764.2.k.g 2
63.t odd 6 1 1764.2.k.g 2
72.j odd 6 1 576.2.a.e 1
72.l even 6 1 576.2.a.f 1
72.n even 6 1 576.2.a.e 1
72.p odd 6 1 576.2.a.f 1
99.g even 6 1 4356.2.a.g 1
99.h odd 6 1 4356.2.a.g 1
117.n odd 6 1 6084.2.a.i 1
117.t even 6 1 6084.2.a.i 1
117.y odd 12 2 6084.2.b.f 2
117.z even 12 2 6084.2.b.f 2
144.u even 12 2 2304.2.d.a 2
144.v odd 12 2 2304.2.d.a 2
144.w odd 12 2 2304.2.d.q 2
144.x even 12 2 2304.2.d.q 2
180.n even 6 1 3600.2.a.e 1
180.p odd 6 1 3600.2.a.e 1
180.v odd 12 2 3600.2.f.m 2
180.x even 12 2 3600.2.f.m 2
252.s odd 6 1 7056.2.a.bb 1
252.bi even 6 1 7056.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 9.c even 3 1
36.2.a.a 1 9.d odd 6 1
144.2.a.a 1 36.f odd 6 1
144.2.a.a 1 36.h even 6 1
324.2.e.c 2 1.a even 1 1 trivial
324.2.e.c 2 3.b odd 2 1 CM
324.2.e.c 2 9.c even 3 1 inner
324.2.e.c 2 9.d odd 6 1 inner
576.2.a.e 1 72.j odd 6 1
576.2.a.e 1 72.n even 6 1
576.2.a.f 1 72.l even 6 1
576.2.a.f 1 72.p odd 6 1
900.2.a.g 1 45.h odd 6 1
900.2.a.g 1 45.j even 6 1
900.2.d.b 2 45.k odd 12 2
900.2.d.b 2 45.l even 12 2
1296.2.i.h 2 4.b odd 2 1
1296.2.i.h 2 12.b even 2 1
1296.2.i.h 2 36.f odd 6 1
1296.2.i.h 2 36.h even 6 1
1764.2.a.e 1 63.l odd 6 1
1764.2.a.e 1 63.o even 6 1
1764.2.k.g 2 63.i even 6 1
1764.2.k.g 2 63.k odd 6 1
1764.2.k.g 2 63.s even 6 1
1764.2.k.g 2 63.t odd 6 1
1764.2.k.h 2 63.g even 3 1
1764.2.k.h 2 63.h even 3 1
1764.2.k.h 2 63.j odd 6 1
1764.2.k.h 2 63.n odd 6 1
2304.2.d.a 2 144.u even 12 2
2304.2.d.a 2 144.v odd 12 2
2304.2.d.q 2 144.w odd 12 2
2304.2.d.q 2 144.x even 12 2
3600.2.a.e 1 180.n even 6 1
3600.2.a.e 1 180.p odd 6 1
3600.2.f.m 2 180.v odd 12 2
3600.2.f.m 2 180.x even 12 2
4356.2.a.g 1 99.g even 6 1
4356.2.a.g 1 99.h odd 6 1
6084.2.a.i 1 117.n odd 6 1
6084.2.a.i 1 117.t even 6 1
6084.2.b.f 2 117.y odd 12 2
6084.2.b.f 2 117.z even 12 2
7056.2.a.bb 1 252.s odd 6 1
7056.2.a.bb 1 252.bi even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} - 4 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 196 + 14 T + T^{2} \)
$67$ \( 256 - 16 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 196 + 14 T + T^{2} \)
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