Properties

Label 324.3.f.o
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.207360000.1
Defining polynomial: \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{7} ) q^{4} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{1} + \beta_{7} ) q^{4} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} ) q^{5} + ( -1 - \beta_{1} - 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{8} + ( -2 + 8 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{10} + ( 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{11} + ( \beta_{1} + 2 \beta_{5} + 4 \beta_{7} ) q^{13} + ( 3 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{14} + ( 14 \beta_{1} + 2 \beta_{5} ) q^{16} + ( 10 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} ) q^{17} + ( 3 - 6 \beta_{1} ) q^{19} + ( 8 \beta_{3} + 4 \beta_{4} ) q^{20} + ( 20 - 10 \beta_{1} + 6 \beta_{5} + 4 \beta_{7} ) q^{22} + ( -7 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} ) q^{23} + ( -3 + 3 \beta_{1} + 8 \beta_{5} + 4 \beta_{7} ) q^{25} + ( -4 \beta_{2} + 7 \beta_{3} + 8 \beta_{4} - 4 \beta_{6} ) q^{26} + ( 2 - 31 \beta_{1} - 3 \beta_{5} - 2 \beta_{7} ) q^{28} + ( -4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{29} + ( 16 - 8 \beta_{1} + 8 \beta_{5} ) q^{31} + ( 12 \beta_{3} + 4 \beta_{6} ) q^{32} + ( 6 - 38 \beta_{1} + 8 \beta_{5} + 2 \beta_{7} ) q^{34} + ( 4 \beta_{2} - 24 \beta_{3} - 10 \beta_{4} + 3 \beta_{6} ) q^{35} + ( -7 - 2 \beta_{5} + 2 \beta_{7} ) q^{37} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -36 + 28 \beta_{1} + 8 \beta_{5} + 4 \beta_{7} ) q^{40} + ( 16 \beta_{2} - 8 \beta_{6} ) q^{41} + ( 20 + 20 \beta_{1} + 4 \beta_{7} ) q^{43} + ( -24 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} + 4 \beta_{6} ) q^{44} + ( 22 + 28 \beta_{1} - 10 \beta_{5} - 6 \beta_{7} ) q^{46} + ( 3 \beta_{2} - 21 \beta_{3} - \beta_{4} + 10 \beta_{6} ) q^{47} + ( 14 \beta_{1} + 4 \beta_{5} + 8 \beta_{7} ) q^{49} + ( -\beta_{2} + 8 \beta_{4} + 8 \beta_{6} ) q^{50} + ( -31 + 60 \beta_{1} + 3 \beta_{5} - \beta_{7} ) q^{52} + ( -12 \beta_{2} + 28 \beta_{3} - 8 \beta_{4} - 10 \beta_{6} ) q^{53} + ( -20 + 40 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} ) q^{55} + ( -30 \beta_{3} - 4 \beta_{4} - 2 \beta_{6} ) q^{56} + ( 32 - 24 \beta_{1} - 8 \beta_{5} ) q^{58} + ( -19 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} - 5 \beta_{6} ) q^{59} + ( -5 + 5 \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{61} + ( -16 \beta_{2} + 16 \beta_{6} ) q^{62} + ( -44 + 12 \beta_{5} + 12 \beta_{7} ) q^{64} + ( 11 \beta_{2} + 19 \beta_{3} + 11 \beta_{4} + 4 \beta_{6} ) q^{65} + ( -38 + 19 \beta_{1} - 16 \beta_{5} ) q^{67} + ( -8 \beta_{2} - 36 \beta_{3} + 4 \beta_{4} + 12 \beta_{6} ) q^{68} + ( 70 - 74 \beta_{1} - 20 \beta_{5} - 14 \beta_{7} ) q^{70} + ( 24 \beta_{2} - 48 \beta_{3} - 12 \beta_{4} - 6 \beta_{6} ) q^{71} + ( 29 + 4 \beta_{5} - 4 \beta_{7} ) q^{73} + ( 5 \beta_{2} - \beta_{3} + 4 \beta_{4} - 8 \beta_{6} ) q^{74} + ( 3 + 3 \beta_{1} - 6 \beta_{5} - 3 \beta_{7} ) q^{76} + ( 43 \beta_{2} - 33 \beta_{3} - 11 \beta_{4} - 5 \beta_{6} ) q^{77} + ( -55 - 55 \beta_{1} - 14 \beta_{7} ) q^{79} + ( 32 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{6} ) q^{80} + ( 64 - 16 \beta_{1} + 16 \beta_{5} ) q^{82} + ( -6 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} ) q^{83} + ( -52 \beta_{1} - 12 \beta_{5} - 24 \beta_{7} ) q^{85} + ( -24 \beta_{2} + 48 \beta_{3} + 8 \beta_{4} - 8 \beta_{6} ) q^{86} + ( -100 + 80 \beta_{1} - 12 \beta_{5} + 4 \beta_{7} ) q^{88} + ( 10 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{6} ) q^{89} + ( 61 - 122 \beta_{1} - 8 \beta_{5} - 8 \beta_{7} ) q^{91} + ( -16 \beta_{2} + 48 \beta_{3} - 12 \beta_{4} - 8 \beta_{6} ) q^{92} + ( -52 - 10 \beta_{1} - 18 \beta_{5} - 20 \beta_{7} ) q^{94} + ( -9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} ) q^{95} + ( 31 - 31 \beta_{1} - 32 \beta_{5} - 16 \beta_{7} ) q^{97} + ( -8 \beta_{2} + 26 \beta_{3} + 16 \beta_{4} - 8 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} - 12 q^{7} + O(q^{10}) \) \( 8 q - 4 q^{4} - 12 q^{7} + 16 q^{10} + 4 q^{13} + 56 q^{16} + 120 q^{22} - 12 q^{25} - 108 q^{28} + 96 q^{31} - 104 q^{34} - 56 q^{37} - 176 q^{40} + 240 q^{43} + 288 q^{46} + 56 q^{49} - 8 q^{52} + 160 q^{58} - 20 q^{61} - 352 q^{64} - 228 q^{67} + 264 q^{70} + 232 q^{73} + 36 q^{76} - 660 q^{79} + 448 q^{82} - 208 q^{85} - 480 q^{88} - 456 q^{94} + 124 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 6 x^{6} + 32 x^{4} + 24 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{6} - 16 \nu^{4} - 96 \nu^{2} - 8 \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} - 24 \nu^{3} + 24 \nu \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( 5 \nu^{7} + 32 \nu^{5} + 160 \nu^{3} + 184 \nu \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} - 16 \nu^{5} - 64 \nu^{3} + 264 \nu \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{6} - 48 \nu^{4} - 224 \nu^{2} - 312 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 16 \nu^{5} - 80 \nu^{3} + 24 \nu \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{6} - 48 \nu^{4} - 224 \nu^{2} + 120 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{6} - \beta_{4} + 3 \beta_{3} - \beta_{2}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + \beta_{5} - 9 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{6} + 5 \beta_{4} - 3 \beta_{3} - 7 \beta_{2}\)\()/3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} - 4 \beta_{5} + 14 \beta_{1} - 14\)
\(\nu^{5}\)\(=\)\((\)\(-22 \beta_{6} - 4 \beta_{4} - 12 \beta_{3} + 56 \beta_{2}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-32 \beta_{7} + 32 \beta_{5} + 216\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(136 \beta_{6} - 116 \beta_{4} + 156 \beta_{3} - 116 \beta_{2}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.437016 0.756934i
−1.14412 + 1.98168i
1.14412 1.98168i
−0.437016 + 0.756934i
0.437016 + 0.756934i
−1.14412 1.98168i
1.14412 + 1.98168i
−0.437016 0.756934i
−1.85123 0.756934i 0 2.85410 + 2.80252i −3.70246 6.41285i 0 −8.20820 4.73901i −3.16228 7.34847i 0 2.00000 + 14.6742i
55.2 −0.270091 + 1.98168i 0 −3.85410 1.07047i −0.540182 0.935622i 0 5.20820 + 3.00696i 3.16228 7.34847i 0 2.00000 0.817763i
55.3 0.270091 1.98168i 0 −3.85410 1.07047i 0.540182 + 0.935622i 0 5.20820 + 3.00696i −3.16228 + 7.34847i 0 2.00000 0.817763i
55.4 1.85123 + 0.756934i 0 2.85410 + 2.80252i 3.70246 + 6.41285i 0 −8.20820 4.73901i 3.16228 + 7.34847i 0 2.00000 + 14.6742i
271.1 −1.85123 + 0.756934i 0 2.85410 2.80252i −3.70246 + 6.41285i 0 −8.20820 + 4.73901i −3.16228 + 7.34847i 0 2.00000 14.6742i
271.2 −0.270091 1.98168i 0 −3.85410 + 1.07047i −0.540182 + 0.935622i 0 5.20820 3.00696i 3.16228 + 7.34847i 0 2.00000 + 0.817763i
271.3 0.270091 + 1.98168i 0 −3.85410 + 1.07047i 0.540182 0.935622i 0 5.20820 3.00696i −3.16228 7.34847i 0 2.00000 + 0.817763i
271.4 1.85123 0.756934i 0 2.85410 2.80252i 3.70246 6.41285i 0 −8.20820 + 4.73901i 3.16228 7.34847i 0 2.00000 14.6742i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
36.f odd 6 1 inner
36.h even 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.o 8
3.b odd 2 1 inner 324.3.f.o 8
4.b odd 2 1 324.3.f.p 8
9.c even 3 1 108.3.d.d 8
9.c even 3 1 324.3.f.p 8
9.d odd 6 1 108.3.d.d 8
9.d odd 6 1 324.3.f.p 8
12.b even 2 1 324.3.f.p 8
36.f odd 6 1 108.3.d.d 8
36.f odd 6 1 inner 324.3.f.o 8
36.h even 6 1 108.3.d.d 8
36.h even 6 1 inner 324.3.f.o 8
72.j odd 6 1 1728.3.g.l 8
72.l even 6 1 1728.3.g.l 8
72.n even 6 1 1728.3.g.l 8
72.p odd 6 1 1728.3.g.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d 8 9.c even 3 1
108.3.d.d 8 9.d odd 6 1
108.3.d.d 8 36.f odd 6 1
108.3.d.d 8 36.h even 6 1
324.3.f.o 8 1.a even 1 1 trivial
324.3.f.o 8 3.b odd 2 1 inner
324.3.f.o 8 36.f odd 6 1 inner
324.3.f.o 8 36.h even 6 1 inner
324.3.f.p 8 4.b odd 2 1
324.3.f.p 8 9.c even 3 1
324.3.f.p 8 9.d odd 6 1
324.3.f.p 8 12.b even 2 1
1728.3.g.l 8 72.j odd 6 1
1728.3.g.l 8 72.l even 6 1
1728.3.g.l 8 72.n even 6 1
1728.3.g.l 8 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 \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{8} + 56 T_{5}^{6} + 3072 T_{5}^{4} + 3584 T_{5}^{2} + 4096 \)
\( T_{7}^{4} + 6 T_{7}^{3} - 45 T_{7}^{2} - 342 T_{7} + 3249 \)
\( T_{11}^{8} - 360 T_{11}^{6} + 115200 T_{11}^{4} - 5184000 T_{11}^{2} + 207360000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 32 T^{2} - 12 T^{4} + 2 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 4096 + 3584 T^{2} + 3072 T^{4} + 56 T^{6} + T^{8} \)
$7$ \( ( 3249 - 342 T - 45 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$11$ \( 207360000 - 5184000 T^{2} + 115200 T^{4} - 360 T^{6} + T^{8} \)
$13$ \( ( 32041 + 358 T + 183 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$17$ \( ( 222784 - 1016 T^{2} + T^{4} )^{2} \)
$19$ \( ( 27 + T^{2} )^{4} \)
$23$ \( 306402103296 - 836946432 T^{2} + 1732608 T^{4} - 1512 T^{6} + T^{8} \)
$29$ \( 268435456 + 14680064 T^{2} + 786432 T^{4} + 896 T^{6} + T^{8} \)
$31$ \( ( 589824 + 36864 T - 48 T^{3} + T^{4} )^{2} \)
$37$ \( ( -131 + 14 T + T^{2} )^{4} \)
$41$ \( 68719476736 + 939524096 T^{2} + 12582912 T^{4} + 3584 T^{6} + T^{8} \)
$43$ \( ( 921600 - 115200 T + 5760 T^{2} - 120 T^{3} + T^{4} )^{2} \)
$47$ \( 20816369750016 - 20038482432 T^{2} + 14727168 T^{4} - 4392 T^{6} + T^{8} \)
$53$ \( ( 33547264 - 11744 T^{2} + T^{4} )^{2} \)
$59$ \( 34524743602176 - 46113090048 T^{2} + 55715328 T^{4} - 7848 T^{6} + T^{8} \)
$61$ \( ( 24025 - 1550 T + 255 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$67$ \( ( 7601049 - 314298 T + 1575 T^{2} + 114 T^{3} + T^{4} )^{2} \)
$71$ \( ( 90326016 + 19872 T^{2} + T^{4} )^{2} \)
$73$ \( ( 121 - 58 T + T^{2} )^{4} \)
$79$ \( ( 37638225 + 2024550 T + 42435 T^{2} + 330 T^{3} + T^{4} )^{2} \)
$83$ \( 53084160000 - 331776000 T^{2} + 1843200 T^{4} - 1440 T^{6} + T^{8} \)
$89$ \( ( 222784 - 1016 T^{2} + T^{4} )^{2} \)
$97$ \( ( 111492481 + 654658 T + 14403 T^{2} - 62 T^{3} + T^{4} )^{2} \)
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