Properties

Label 324.3.f.m
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Defining polynomial: \(x^{4} - x^{3} + 4 x^{2} + 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{7} + ( 5 + 9 \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} + ( -3 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{7} + ( 5 + 9 \beta_{2} - \beta_{3} ) q^{8} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( 7 - 7 \beta_{2} ) q^{11} -16 \beta_{2} q^{13} + ( 10 + 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{14} + ( 11 - 5 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} ) q^{16} + ( 4 - 8 \beta_{3} ) q^{17} + ( 18 - 24 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{19} + ( 9 - 5 \beta_{1} - 4 \beta_{2} ) q^{20} + ( -7 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} ) q^{22} + ( 20 + 10 \beta_{2} ) q^{23} + ( 12 + 12 \beta_{2} ) q^{25} + ( -16 - 16 \beta_{2} + 16 \beta_{3} ) q^{26} + ( 27 - 10 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{28} + ( -14 + 28 \beta_{1} - 14 \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{31} + ( -18 - 11 \beta_{1} - 9 \beta_{2} - 11 \beta_{3} ) q^{32} + ( -20 - 4 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -13 - 26 \beta_{2} ) q^{35} + 26 q^{37} + ( -30 - 18 \beta_{1} + 48 \beta_{2} ) q^{38} + ( -9 \beta_{1} + 11 \beta_{2} + 9 \beta_{3} ) q^{40} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 6 - 4 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 7 + 35 \beta_{2} + 21 \beta_{3} ) q^{44} + ( 30 - 20 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{46} + ( -2 + 2 \beta_{2} ) q^{47} + 10 \beta_{2} q^{49} + ( 24 - 12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{50} + ( 16 + 16 \beta_{1} + 32 \beta_{2} + 32 \beta_{3} ) q^{52} + ( 19 - 38 \beta_{3} ) q^{53} + ( -21 + 28 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} ) q^{55} + ( 7 - 27 \beta_{1} + 20 \beta_{2} ) q^{56} + ( 14 \beta_{1} - 98 \beta_{2} - 14 \beta_{3} ) q^{58} + ( -88 - 44 \beta_{2} ) q^{59} + ( -8 - 8 \beta_{2} ) q^{61} + ( 5 + 13 \beta_{2} + 3 \beta_{3} ) q^{62} + ( -71 + 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{64} + ( -16 + 32 \beta_{1} - 16 \beta_{2} ) q^{65} + ( -20 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} ) q^{67} + ( -72 + 20 \beta_{1} - 36 \beta_{2} + 20 \beta_{3} ) q^{68} + ( -39 + 13 \beta_{1} - 26 \beta_{2} + 26 \beta_{3} ) q^{70} + ( -36 - 72 \beta_{2} ) q^{71} -19 q^{73} + ( 26 - 26 \beta_{1} ) q^{74} + ( 30 \beta_{1} + 102 \beta_{2} - 30 \beta_{3} ) q^{76} + ( -42 + 42 \beta_{1} - 21 \beta_{2} + 42 \beta_{3} ) q^{77} + ( 24 - 16 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} ) q^{79} + ( 29 + 65 \beta_{2} + 7 \beta_{3} ) q^{80} + ( -10 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 67 - 67 \beta_{2} ) q^{83} + 52 \beta_{2} q^{85} + ( -20 - 6 \beta_{1} - 10 \beta_{2} - 6 \beta_{3} ) q^{86} + ( 105 - 7 \beta_{1} + 98 \beta_{2} - 14 \beta_{3} ) q^{88} + ( 22 - 44 \beta_{3} ) q^{89} + ( -48 + 64 \beta_{1} - 32 \beta_{2} + 32 \beta_{3} ) q^{91} + ( -10 - 30 \beta_{1} + 40 \beta_{2} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 156 + 78 \beta_{2} ) q^{95} + ( -119 - 119 \beta_{2} ) q^{97} + ( 10 + 10 \beta_{2} - 10 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 5 q^{4} + O(q^{10}) \) \( 4 q + 3 q^{2} - 5 q^{4} + 26 q^{10} + 42 q^{11} + 32 q^{13} + 39 q^{14} + 7 q^{16} + 39 q^{20} + 21 q^{22} + 60 q^{23} + 24 q^{25} + 78 q^{28} - 87 q^{32} - 52 q^{34} + 104 q^{37} - 234 q^{38} - 13 q^{40} + 60 q^{46} - 12 q^{47} - 20 q^{49} + 36 q^{50} + 80 q^{52} - 39 q^{56} + 182 q^{58} - 264 q^{59} - 16 q^{61} - 230 q^{64} - 156 q^{68} - 39 q^{70} - 76 q^{73} + 78 q^{74} - 234 q^{76} - 52 q^{82} + 402 q^{83} - 104 q^{85} - 78 q^{86} + 189 q^{88} - 150 q^{92} - 6 q^{94} + 468 q^{95} - 238 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 4 x^{2} + 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 4 \nu^{2} - 4 \nu - 3 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 7 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{3} - 7\)

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
1.15139 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 1.12824i
−0.151388 + 1.99426i 0 −3.95416 0.603814i −1.80278 3.12250i 0 −5.40833 3.12250i 1.80278 7.79423i 0 6.50000 3.12250i
55.2 1.65139 1.12824i 0 1.45416 3.72631i 1.80278 + 3.12250i 0 5.40833 + 3.12250i −1.80278 7.79423i 0 6.50000 + 3.12250i
271.1 −0.151388 1.99426i 0 −3.95416 + 0.603814i −1.80278 + 3.12250i 0 −5.40833 + 3.12250i 1.80278 + 7.79423i 0 6.50000 + 3.12250i
271.2 1.65139 + 1.12824i 0 1.45416 + 3.72631i 1.80278 3.12250i 0 5.40833 3.12250i −1.80278 + 7.79423i 0 6.50000 3.12250i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.m 4
3.b odd 2 1 324.3.f.l 4
4.b odd 2 1 324.3.f.l 4
9.c even 3 1 108.3.d.c 4
9.c even 3 1 324.3.f.l 4
9.d odd 6 1 108.3.d.c 4
9.d odd 6 1 inner 324.3.f.m 4
12.b even 2 1 inner 324.3.f.m 4
36.f odd 6 1 108.3.d.c 4
36.f odd 6 1 inner 324.3.f.m 4
36.h even 6 1 108.3.d.c 4
36.h even 6 1 324.3.f.l 4
72.j odd 6 1 1728.3.g.i 4
72.l even 6 1 1728.3.g.i 4
72.n even 6 1 1728.3.g.i 4
72.p odd 6 1 1728.3.g.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.c 4 9.c even 3 1
108.3.d.c 4 9.d odd 6 1
108.3.d.c 4 36.f odd 6 1
108.3.d.c 4 36.h even 6 1
324.3.f.l 4 3.b odd 2 1
324.3.f.l 4 4.b odd 2 1
324.3.f.l 4 9.c even 3 1
324.3.f.l 4 36.h even 6 1
324.3.f.m 4 1.a even 1 1 trivial
324.3.f.m 4 9.d odd 6 1 inner
324.3.f.m 4 12.b even 2 1 inner
324.3.f.m 4 36.f odd 6 1 inner
1728.3.g.i 4 72.j odd 6 1
1728.3.g.i 4 72.l even 6 1
1728.3.g.i 4 72.n even 6 1
1728.3.g.i 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{4} + 13 T_{5}^{2} + 169 \)
\( T_{7}^{4} - 39 T_{7}^{2} + 1521 \)
\( T_{11}^{2} - 21 T_{11} + 147 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 12 T + 7 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 169 + 13 T^{2} + T^{4} \)
$7$ \( 1521 - 39 T^{2} + T^{4} \)
$11$ \( ( 147 - 21 T + T^{2} )^{2} \)
$13$ \( ( 256 - 16 T + T^{2} )^{2} \)
$17$ \( ( -208 + T^{2} )^{2} \)
$19$ \( ( 1404 + T^{2} )^{2} \)
$23$ \( ( 300 - 30 T + T^{2} )^{2} \)
$29$ \( 6492304 + 2548 T^{2} + T^{4} \)
$31$ \( 1521 - 39 T^{2} + T^{4} \)
$37$ \( ( -26 + T )^{4} \)
$41$ \( 2704 + 52 T^{2} + T^{4} \)
$43$ \( 24336 - 156 T^{2} + T^{4} \)
$47$ \( ( 12 + 6 T + T^{2} )^{2} \)
$53$ \( ( -4693 + T^{2} )^{2} \)
$59$ \( ( 5808 + 132 T + T^{2} )^{2} \)
$61$ \( ( 64 + 8 T + T^{2} )^{2} \)
$67$ \( 15210000 - 3900 T^{2} + T^{4} \)
$71$ \( ( 3888 + T^{2} )^{2} \)
$73$ \( ( 19 + T )^{4} \)
$79$ \( 6230016 - 2496 T^{2} + T^{4} \)
$83$ \( ( 13467 - 201 T + T^{2} )^{2} \)
$89$ \( ( -6292 + T^{2} )^{2} \)
$97$ \( ( 14161 + 119 T + T^{2} )^{2} \)
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