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Properties

Label 324.6.e.e
Level 324
Weight 6
Character orbit 324.e
Analytic conductor 51.964
Analytic rank 0
Dimension 4
CM no
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -29 + 29 \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -29 + 29 \beta_{1} ) q^{7} + ( \beta_{2} - \beta_{3} ) q^{11} -329 \beta_{1} q^{13} -25 \beta_{3} q^{17} + 1799 q^{19} -41 \beta_{2} q^{23} + ( -4651 + 4651 \beta_{1} ) q^{25} + ( 16 \beta_{2} - 16 \beta_{3} ) q^{29} -5228 \beta_{1} q^{31} + 29 \beta_{3} q^{35} + 8783 q^{37} + 176 \beta_{2} q^{41} + ( -19976 + 19976 \beta_{1} ) q^{43} + ( -123 \beta_{2} + 123 \beta_{3} ) q^{47} + 15966 \beta_{1} q^{49} + 334 \beta_{3} q^{53} + 7776 q^{55} -65 \beta_{2} q^{59} + ( 1069 - 1069 \beta_{1} ) q^{61} + ( 329 \beta_{2} - 329 \beta_{3} ) q^{65} + 62077 \beta_{1} q^{67} -526 \beta_{3} q^{71} -48079 q^{73} -29 \beta_{2} q^{77} + ( -49979 + 49979 \beta_{1} ) q^{79} + ( -654 \beta_{2} + 654 \beta_{3} ) q^{83} + 194400 \beta_{1} q^{85} -997 \beta_{3} q^{89} + 9541 q^{91} -1799 \beta_{2} q^{95} + ( -12917 + 12917 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 58q^{7} + O(q^{10}) \) \( 4q - 58q^{7} - 658q^{13} + 7196q^{19} - 9302q^{25} - 10456q^{31} + 35132q^{37} - 39952q^{43} + 31932q^{49} + 31104q^{55} + 2138q^{61} + 124154q^{67} - 192316q^{73} - 99958q^{79} + 388800q^{85} + 38164q^{91} - 25834q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\( 18 \nu^{3} + 36 \nu \)
\(\beta_{3}\)\(=\)\( -18 \nu^{3} + 72 \nu \)

Display \(\nu^j\) in terms of \(\beta_i\)

\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/108\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2}\)\()/54\)

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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
1.22474 − 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 − 0.707107i
0 0 0 −44.0908 + 76.3675i 0 −14.5000 − 25.1147i 0 0 0
109.2 0 0 0 44.0908 − 76.3675i 0 −14.5000 − 25.1147i 0 0 0
217.1 0 0 0 −44.0908 − 76.3675i 0 −14.5000 + 25.1147i 0 0 0
217.2 0 0 0 44.0908 + 76.3675i 0 −14.5000 + 25.1147i 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 inner
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.e 4
3.b odd 2 1 inner 324.6.e.e 4
9.c even 3 1 108.6.a.d ✓ 2
9.c even 3 1 inner 324.6.e.e 4
9.d odd 6 1 108.6.a.d ✓ 2
9.d odd 6 1 inner 324.6.e.e 4
36.f odd 6 1 432.6.a.p 2
36.h even 6 1 432.6.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.d ✓ 2 9.c even 3 1
108.6.a.d ✓ 2 9.d odd 6 1
324.6.e.e 4 1.a even 1 1 trivial
324.6.e.e 4 3.b odd 2 1 inner
324.6.e.e 4 9.c even 3 1 inner
324.6.e.e 4 9.d odd 6 1 inner
432.6.a.p 2 36.f odd 6 1
432.6.a.p 2 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 \(S_{6}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} + 7776 T_{5}^{2} + 60466176 \)
\( T_{7}^{2} + 29 T_{7} + 841 \)
\( T_{11}^{4} + 7776 T_{11}^{2} + 60466176 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 1526 T^{2} - 7436949 T^{4} + 14902343750 T^{6} + 95367431640625 T^{8} \)
$7$ \( ( 1 + 29 T - 15966 T^{2} + 487403 T^{3} + 282475249 T^{4} )^{2} \)
$11$ \( 1 - 314326 T^{2} + 72863409675 T^{4} - 8152806925133926 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 + 329 T - 263052 T^{2} + 122155397 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 - 2020286 T^{2} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 - 1799 T + 2476099 T^{2} )^{4} \)
$23$ \( 1 + 198770 T^{2} - 41387001700749 T^{4} + 8234347633937011730 T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( 1 - 39031642 T^{2} + 1102761843915963 T^{4} - \)\(16\!\cdots\!42\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \)
$31$ \( ( 1 + 5228 T - 1297167 T^{2} + 149673201428 T^{3} + 819628286980801 T^{4} )^{2} \)
$37$ \( ( 1 - 8783 T + 69343957 T^{2} )^{4} \)
$41$ \( 1 + 9156974 T^{2} - 13338809137315725 T^{4} + \)\(12\!\cdots\!74\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \)
$43$ \( ( 1 + 19976 T + 252032133 T^{2} + 2936640657368 T^{3} + 21611482313284249 T^{4} )^{2} \)
$47$ \( 1 - 341046910 T^{2} + 63713862584718051 T^{4} - \)\(17\!\cdots\!90\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 - 31068470 T^{2} + 174887470365513049 T^{4} )^{2} \)
$59$ \( 1 - 1396994998 T^{2} + 1440478271136378603 T^{4} - \)\(71\!\cdots\!98\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( ( 1 - 1069 T - 843453540 T^{2} - 902873445769 T^{3} + 713342911662882601 T^{4} )^{2} \)
$67$ \( ( 1 - 62077 T + 2503428822 T^{2} - 83811716267239 T^{3} + 1822837804551761449 T^{4} )^{2} \)
$71$ \( ( 1 + 1457026126 T^{2} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 + 48079 T + 2073071593 T^{2} )^{4} \)
$79$ \( ( 1 + 49979 T - 579155958 T^{2} + 153788201765621 T^{3} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 4552161670 T^{2} + 5206134682611335451 T^{4} - \)\(70\!\cdots\!30\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 + 3438704914 T^{2} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( ( 1 + 12917 T - 8420491368 T^{2} + 110922674099669 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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