Properties

Label 324.3.f.p
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
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{5} - \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( 2 - \beta_{1} - 2 \beta_{5} ) q^{7} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{5} - \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{5} + ( 2 - \beta_{1} - 2 \beta_{5} ) 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} + ( -\beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{11} + ( 1 - \beta_{1} - 4 \beta_{5} - 2 \beta_{7} ) q^{13} + ( \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{14} + ( -14 - 2 \beta_{5} - 2 \beta_{7} ) q^{16} + ( -10 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{17} + ( 3 - 6 \beta_{1} ) q^{19} + ( -8 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} ) q^{20} + ( -10 - 10 \beta_{1} - 4 \beta_{5} + 2 \beta_{7} ) q^{22} + ( -15 \beta_{2} + 9 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} ) q^{23} + ( -3 \beta_{1} - 4 \beta_{5} - 8 \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} - 12 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{29} + ( -8 - 8 \beta_{1} + 8 \beta_{7} ) q^{31} + ( 16 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{32} + ( 32 + 6 \beta_{1} - 6 \beta_{5} - 8 \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} - 6 \beta_{3} ) q^{38} + ( 28 + 8 \beta_{1} - 4 \beta_{5} + 4 \beta_{7} ) q^{40} + ( 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{6} ) q^{41} + ( -40 + 20 \beta_{1} + 4 \beta_{5} ) 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} + ( 11 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + \beta_{6} ) q^{47} + ( 14 - 14 \beta_{1} - 8 \beta_{5} - 4 \beta_{7} ) q^{49} + ( 7 \beta_{2} - 15 \beta_{3} - 8 \beta_{4} + 16 \beta_{6} ) q^{50} + ( -29 - 31 \beta_{1} - 4 \beta_{5} - 3 \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} + ( 32 \beta_{2} - 36 \beta_{3} - 6 \beta_{4} + 4 \beta_{6} ) q^{56} + ( -24 - 8 \beta_{1} - 8 \beta_{7} ) q^{58} + ( -15 \beta_{2} + 33 \beta_{3} + 5 \beta_{4} - 14 \beta_{6} ) q^{59} + ( -5 \beta_{1} + 2 \beta_{5} + 4 \beta_{7} ) q^{61} + ( 16 \beta_{2} - 16 \beta_{6} ) q^{62} + ( -44 + 12 \beta_{5} + 12 \beta_{7} ) q^{64} + ( -23 \beta_{2} + 45 \beta_{3} + 15 \beta_{4} - 11 \beta_{6} ) q^{65} + ( 19 + 19 \beta_{1} - 16 \beta_{7} ) q^{67} + ( -32 \beta_{2} - 8 \beta_{3} - 12 \beta_{4} + 16 \beta_{6} ) q^{68} + ( 4 + 70 \beta_{1} + 6 \beta_{5} + 20 \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} + ( 9 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} ) q^{74} + ( 3 - 6 \beta_{1} + 3 \beta_{5} - 3 \beta_{7} ) q^{76} + ( 5 \beta_{2} - 27 \beta_{3} + 5 \beta_{4} - 16 \beta_{6} ) q^{77} + ( 110 - 55 \beta_{1} - 14 \beta_{5} ) 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} + ( 2 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} - 2 \beta_{6} ) q^{83} + ( -52 + 52 \beta_{1} + 24 \beta_{5} + 12 \beta_{7} ) q^{85} + ( 16 \beta_{2} + 24 \beta_{3} + 8 \beta_{4} ) q^{86} + ( 20 - 100 \beta_{1} + 16 \beta_{5} + 12 \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} + ( -40 \beta_{2} + 12 \beta_{3} - 20 \beta_{4} + 12 \beta_{6} ) q^{92} + ( -10 + 62 \beta_{1} + 20 \beta_{5} + 2 \beta_{7} ) q^{94} + ( -9 \beta_{2} + 15 \beta_{3} + 3 \beta_{4} - 6 \beta_{6} ) q^{95} + ( 31 \beta_{1} + 16 \beta_{5} + 32 \beta_{7} ) q^{97} + ( 8 \beta_{2} - 26 \beta_{3} - 16 \beta_{4} + 8 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + 12q^{7} + O(q^{10}) \) \( 8q + 8q^{4} + 12q^{7} + 16q^{10} + 4q^{13} - 112q^{16} - 120q^{22} - 12q^{25} - 108q^{28} - 96q^{31} + 280q^{34} - 56q^{37} + 256q^{40} - 240q^{43} + 288q^{46} + 56q^{49} - 356q^{52} - 224q^{58} - 20q^{61} - 352q^{64} + 228q^{67} + 312q^{70} + 232q^{73} + 660q^{79} + 448q^{82} - 208q^{85} - 240q^{88} + 168q^{94} + 124q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

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

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
1.14412 1.98168i
0.437016 0.756934i
−0.437016 + 0.756934i
−1.14412 + 1.98168i
−1.58114 1.22474i 0 1.00000 + 3.87298i 3.70246 + 6.41285i 0 8.20820 + 4.73901i 3.16228 7.34847i 0 2.00000 14.6742i
55.2 −1.58114 + 1.22474i 0 1.00000 3.87298i −0.540182 0.935622i 0 −5.20820 3.00696i 3.16228 + 7.34847i 0 2.00000 + 0.817763i
55.3 1.58114 1.22474i 0 1.00000 3.87298i 0.540182 + 0.935622i 0 −5.20820 3.00696i −3.16228 7.34847i 0 2.00000 + 0.817763i
55.4 1.58114 + 1.22474i 0 1.00000 + 3.87298i −3.70246 6.41285i 0 8.20820 + 4.73901i −3.16228 + 7.34847i 0 2.00000 14.6742i
271.1 −1.58114 1.22474i 0 1.00000 + 3.87298i −0.540182 + 0.935622i 0 −5.20820 + 3.00696i 3.16228 7.34847i 0 2.00000 0.817763i
271.2 −1.58114 + 1.22474i 0 1.00000 3.87298i 3.70246 6.41285i 0 8.20820 4.73901i 3.16228 + 7.34847i 0 2.00000 + 14.6742i
271.3 1.58114 1.22474i 0 1.00000 3.87298i −3.70246 + 6.41285i 0 8.20820 4.73901i −3.16228 7.34847i 0 2.00000 + 14.6742i
271.4 1.58114 + 1.22474i 0 1.00000 + 3.87298i 0.540182 0.935622i 0 −5.20820 + 3.00696i −3.16228 + 7.34847i 0 2.00000 0.817763i
\(n\): 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.p 8
3.b odd 2 1 inner 324.3.f.p 8
4.b odd 2 1 324.3.f.o 8
9.c even 3 1 108.3.d.d 8
9.c even 3 1 324.3.f.o 8
9.d odd 6 1 108.3.d.d 8
9.d odd 6 1 324.3.f.o 8
12.b even 2 1 324.3.f.o 8
36.f odd 6 1 108.3.d.d 8
36.f odd 6 1 inner 324.3.f.p 8
36.h even 6 1 108.3.d.d 8
36.h even 6 1 inner 324.3.f.p 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 4.b odd 2 1
324.3.f.o 8 9.c even 3 1
324.3.f.o 8 9.d odd 6 1
324.3.f.o 8 12.b even 2 1
324.3.f.p 8 1.a even 1 1 trivial
324.3.f.p 8 3.b odd 2 1 inner
324.3.f.p 8 36.f odd 6 1 inner
324.3.f.p 8 36.h even 6 1 inner
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

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 16 T^{4} )^{2} \)
$3$ 1
$5$ \( 1 - 44 T^{2} + 922 T^{4} + 10384 T^{6} - 572429 T^{8} + 6490000 T^{10} + 360156250 T^{12} - 10742187500 T^{14} + 152587890625 T^{16} \)
$7$ \( ( 1 - 6 T + 53 T^{2} - 246 T^{3} - 132 T^{4} - 12054 T^{5} + 127253 T^{6} - 705894 T^{7} + 5764801 T^{8} )^{2} \)
$11$ \( 1 + 124 T^{2} + 250 T^{4} - 1755344 T^{6} - 210683021 T^{8} - 25699991504 T^{10} + 53589720250 T^{12} + 389165118713404 T^{14} + 45949729863572161 T^{16} \)
$13$ \( ( 1 - 2 T - 155 T^{2} + 358 T^{3} - 3956 T^{4} + 60502 T^{5} - 4426955 T^{6} - 9653618 T^{7} + 815730721 T^{8} )^{2} \)
$17$ \( ( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2} \)
$19$ \( ( 1 - 695 T^{2} + 130321 T^{4} )^{4} \)
$23$ \( 1 + 604 T^{2} - 268070 T^{4} + 44215216 T^{6} + 220143229459 T^{8} + 12373230260656 T^{10} - 20992825824277670 T^{12} + 13236433156940273884 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \)
$29$ \( 1 - 2468 T^{2} + 3338026 T^{4} - 3303260048 T^{6} + 2770686029299 T^{8} - 2336333070009488 T^{10} + 1669835532870554986 T^{12} - \)\(87\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \)
$31$ \( ( 1 + 48 T + 1922 T^{2} + 55392 T^{3} + 1146243 T^{4} + 53231712 T^{5} + 1775007362 T^{6} + 42600176688 T^{7} + 852891037441 T^{8} )^{2} \)
$37$ \( ( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4} \)
$41$ \( 1 - 3140 T^{2} + 4692298 T^{4} + 1520450800 T^{6} - 9144788425517 T^{8} + 4296430573058800 T^{10} + 37467648682754010058 T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( ( 1 + 120 T + 9458 T^{2} + 558960 T^{3} + 27153363 T^{4} + 1033517040 T^{5} + 32335019858 T^{6} + 758563565880 T^{7} + 11688200277601 T^{8} )^{2} \)
$47$ \( 1 + 4444 T^{2} + 5312410 T^{4} + 20786205616 T^{6} + 88238287231699 T^{8} + 101430052606488496 T^{10} + \)\(12\!\cdots\!10\)\( T^{12} + \)\(51\!\cdots\!04\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2} \)
$59$ \( 1 + 6076 T^{2} + 12975610 T^{4} - 1777570256 T^{6} - 20949143941901 T^{8} - 21539460494814416 T^{10} + \)\(19\!\cdots\!10\)\( T^{12} + \)\(10\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \)
$61$ \( ( 1 + 10 T - 7187 T^{2} - 1550 T^{3} + 38882428 T^{4} - 5767550 T^{5} - 99510059267 T^{6} + 515203743610 T^{7} + 191707312997281 T^{8} )^{2} \)
$67$ \( ( 1 - 114 T + 10553 T^{2} - 709194 T^{3} + 37996068 T^{4} - 3183571866 T^{5} + 212654779913 T^{6} - 10312255567266 T^{7} + 406067677556641 T^{8} )^{2} \)
$71$ \( ( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2} \)
$73$ \( ( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4} \)
$79$ \( ( 1 - 330 T + 54917 T^{2} - 6143610 T^{3} + 534190908 T^{4} - 38342270010 T^{5} + 2139021598277 T^{6} - 80218860321930 T^{7} + 1517108809906561 T^{8} )^{2} \)
$83$ \( 1 + 26116 T^{2} + 416905450 T^{4} + 4445553374224 T^{6} + 35678198362101619 T^{8} + \)\(21\!\cdots\!04\)\( T^{10} + \)\(93\!\cdots\!50\)\( T^{12} + \)\(27\!\cdots\!76\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \)
$89$ \( ( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2} \)
$97$ \( ( 1 - 62 T - 4415 T^{2} + 654658 T^{3} - 56486396 T^{4} + 6159677122 T^{5} - 390856775615 T^{6} - 51644264305598 T^{7} + 7837433594376961 T^{8} )^{2} \)
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