Properties

Label 324.6.e.c
Level 324
Weight 6
Character orbit 324.e
Analytic conductor 51.964
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.9643576194\)
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: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 25 - 25 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 25 - 25 \zeta_{6} ) q^{7} + 427 \zeta_{6} q^{13} -1711 q^{19} + ( 3125 - 3125 \zeta_{6} ) q^{25} + 10324 \zeta_{6} q^{31} -6661 q^{37} + ( 3352 - 3352 \zeta_{6} ) q^{43} + 16182 \zeta_{6} q^{49} + ( -56927 + 56927 \zeta_{6} ) q^{61} + 37939 \zeta_{6} q^{67} + 79577 q^{73} + ( -90857 + 90857 \zeta_{6} ) q^{79} + 10675 q^{91} + ( -177725 + 177725 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 25q^{7} + O(q^{10}) \) \( 2q + 25q^{7} + 427q^{13} - 3422q^{19} + 3125q^{25} + 10324q^{31} - 13322q^{37} + 3352q^{43} + 16182q^{49} - 56927q^{61} + 37939q^{67} + 159154q^{73} - 90857q^{79} + 21350q^{91} - 177725q^{97} + O(q^{100}) \)

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 12.5000 + 21.6506i 0 0 0
217.1 0 0 0 0 0 12.5000 21.6506i 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 324.6.e.c 2
3.b odd 2 1 CM 324.6.e.c 2
9.c even 3 1 108.6.a.a 1
9.c even 3 1 inner 324.6.e.c 2
9.d odd 6 1 108.6.a.a 1
9.d odd 6 1 inner 324.6.e.c 2
36.f odd 6 1 432.6.a.e 1
36.h even 6 1 432.6.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.a 1 9.c even 3 1
108.6.a.a 1 9.d odd 6 1
324.6.e.c 2 1.a even 1 1 trivial
324.6.e.c 2 3.b odd 2 1 CM
324.6.e.c 2 9.c even 3 1 inner
324.6.e.c 2 9.d odd 6 1 inner
432.6.a.e 1 36.f odd 6 1
432.6.a.e 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(324, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} - 25 T_{7} + 625 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 3125 T^{2} + 9765625 T^{4} \)
$7$ \( ( 1 - 236 T + 16807 T^{2} )( 1 + 211 T + 16807 T^{2} ) \)
$11$ \( 1 - 161051 T^{2} + 25937424601 T^{4} \)
$13$ \( ( 1 - 1202 T + 371293 T^{2} )( 1 + 775 T + 371293 T^{2} ) \)
$17$ \( ( 1 + 1419857 T^{2} )^{2} \)
$19$ \( ( 1 + 1711 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 6436343 T^{2} + 41426511213649 T^{4} \)
$29$ \( 1 - 20511149 T^{2} + 420707233300201 T^{4} \)
$31$ \( ( 1 - 7601 T + 28629151 T^{2} )( 1 - 2723 T + 28629151 T^{2} ) \)
$37$ \( ( 1 + 6661 T + 69343957 T^{2} )^{2} \)
$41$ \( 1 - 115856201 T^{2} + 13422659310152401 T^{4} \)
$43$ \( ( 1 - 22475 T + 147008443 T^{2} )( 1 + 19123 T + 147008443 T^{2} ) \)
$47$ \( 1 - 229345007 T^{2} + 52599132235830049 T^{4} \)
$53$ \( ( 1 + 418195493 T^{2} )^{2} \)
$59$ \( 1 - 714924299 T^{2} + 511116753300641401 T^{4} \)
$61$ \( ( 1 + 18301 T + 844596301 T^{2} )( 1 + 38626 T + 844596301 T^{2} ) \)
$67$ \( ( 1 - 73475 T + 1350125107 T^{2} )( 1 + 35536 T + 1350125107 T^{2} ) \)
$71$ \( ( 1 + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 79577 T + 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 - 9707 T + 3077056399 T^{2} )( 1 + 100564 T + 3077056399 T^{2} ) \)
$83$ \( 1 - 3939040643 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 + 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 + 43339 T + 8587340257 T^{2} )( 1 + 134386 T + 8587340257 T^{2} ) \)
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