Properties

Label 324.4.e.d
Level $324$
Weight $4$
Character orbit 324.e
Analytic conductor $19.117$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1166188419\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (17 \zeta_{6} - 17) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (17 \zeta_{6} - 17) q^{7} - 89 \zeta_{6} q^{13} + 107 q^{19} + ( - 125 \zeta_{6} + 125) q^{25} - 308 \zeta_{6} q^{31} - 433 q^{37} + ( - 520 \zeta_{6} + 520) q^{43} + 54 \zeta_{6} q^{49} + ( - 901 \zeta_{6} + 901) q^{61} - 1007 \zeta_{6} q^{67} - 271 q^{73} + (503 \zeta_{6} - 503) q^{79} + 1513 q^{91} + (1853 \zeta_{6} - 1853) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 17 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 17 q^{7} - 89 q^{13} + 214 q^{19} + 125 q^{25} - 308 q^{31} - 866 q^{37} + 520 q^{43} + 54 q^{49} + 901 q^{61} - 1007 q^{67} - 542 q^{73} - 503 q^{79} + 3026 q^{91} - 1853 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −8.50000 14.7224i 0 0 0
217.1 0 0 0 0 0 −8.50000 + 14.7224i 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 324.4.e.d 2
3.b odd 2 1 CM 324.4.e.d 2
9.c even 3 1 108.4.a.c 1
9.c even 3 1 inner 324.4.e.d 2
9.d odd 6 1 108.4.a.c 1
9.d odd 6 1 inner 324.4.e.d 2
36.f odd 6 1 432.4.a.g 1
36.h even 6 1 432.4.a.g 1
72.j odd 6 1 1728.4.a.q 1
72.l even 6 1 1728.4.a.p 1
72.n even 6 1 1728.4.a.q 1
72.p odd 6 1 1728.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.c 1 9.c even 3 1
108.4.a.c 1 9.d odd 6 1
324.4.e.d 2 1.a even 1 1 trivial
324.4.e.d 2 3.b odd 2 1 CM
324.4.e.d 2 9.c even 3 1 inner
324.4.e.d 2 9.d odd 6 1 inner
432.4.a.g 1 36.f odd 6 1
432.4.a.g 1 36.h even 6 1
1728.4.a.p 1 72.l even 6 1
1728.4.a.p 1 72.p odd 6 1
1728.4.a.q 1 72.j odd 6 1
1728.4.a.q 1 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} + 17T_{7} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 89T + 7921 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 107)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 308T + 94864 \) Copy content Toggle raw display
$37$ \( (T + 433)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 520T + 270400 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 901T + 811801 \) Copy content Toggle raw display
$67$ \( T^{2} + 1007 T + 1014049 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 271)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 503T + 253009 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1853 T + 3433609 \) Copy content Toggle raw display
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