Properties

Label 5184.2.d.r
Level $5184$
Weight $2$
Character orbit 5184.d
Analytic conductor $41.394$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{8} - 24 x^{7} + 96 x^{5} + 304 x^{4} + 384 x^{3} + 288 x^{2} + 144 x + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + \beta_{4} q^{7} +O(q^{10})\) \( q + \beta_{8} q^{5} + \beta_{4} q^{7} + ( \beta_{7} + \beta_{9} ) q^{11} + \beta_{10} q^{13} + \beta_{1} q^{17} + \beta_{7} q^{19} + ( -\beta_{3} + \beta_{4} - \beta_{6} ) q^{23} + 2 q^{25} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{29} + ( \beta_{3} - \beta_{4} - \beta_{6} ) q^{31} + ( \beta_{5} + \beta_{7} + \beta_{9} ) q^{35} + ( -\beta_{8} - \beta_{10} ) q^{37} + ( 3 + \beta_{1} ) q^{41} + ( -\beta_{5} + \beta_{7} - \beta_{9} ) q^{43} + ( -\beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{47} + ( 4 + \beta_{1} - \beta_{2} ) q^{49} + ( 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{53} + ( -\beta_{4} + \beta_{6} ) q^{55} + ( \beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{59} + ( -\beta_{8} - \beta_{10} + \beta_{11} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{65} + ( 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{67} + ( -2 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{71} + ( -\beta_{1} + \beta_{2} ) q^{73} + ( 6 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{77} + ( \beta_{3} + \beta_{4} + \beta_{6} ) q^{79} + ( -\beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{83} + ( \beta_{10} - \beta_{11} ) q^{85} + ( 4 - \beta_{1} + \beta_{2} ) q^{89} + ( 3 \beta_{7} + 3 \beta_{9} ) q^{91} -\beta_{3} q^{95} + ( 1 - 3 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q + 24 q^{25} + 36 q^{41} + 48 q^{49} + 12 q^{65} + 48 q^{89} + 12 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 16 x^{8} - 24 x^{7} + 96 x^{5} + 304 x^{4} + 384 x^{3} + 288 x^{2} + 144 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 \)\()/51972\)
\(\beta_{2}\)\(=\)\((\)\( 1969 \nu^{11} - 2148 \nu^{10} + 4654 \nu^{9} - 11338 \nu^{8} - 17186 \nu^{7} - 23628 \nu^{6} - 17534 \nu^{5} + 254444 \nu^{4} + 441804 \nu^{3} + 378072 \nu^{2} + 214812 \nu - 442176 \)\()/51972\)
\(\beta_{3}\)\(=\)\((\)\( -34 \nu^{11} - 4 \nu^{10} + 17 \nu^{9} - 12 \nu^{8} + 546 \nu^{7} + 896 \nu^{6} - 196 \nu^{5} - 3468 \nu^{4} - 10454 \nu^{3} - 13152 \nu^{2} - 7260 \nu - 2736 \)\()/366\)
\(\beta_{4}\)\(=\)\((\)\( -11251 \nu^{11} - 3656 \nu^{10} + 8584 \nu^{9} - 6576 \nu^{8} + 184040 \nu^{7} + 325576 \nu^{6} - 51410 \nu^{5} - 1183104 \nu^{4} - 3657720 \nu^{3} - 4638708 \nu^{2} - 2561328 \nu - 967896 \)\()/103944\)
\(\beta_{5}\)\(=\)\((\)\( 5765 \nu^{11} - 3533 \nu^{10} + 2602 \nu^{9} - 2490 \nu^{8} - 88123 \nu^{7} - 86504 \nu^{6} + 51058 \nu^{5} + 522216 \nu^{4} + 1425948 \nu^{3} + 1357440 \nu^{2} + 954990 \nu + 355320 \)\()/51972\)
\(\beta_{6}\)\(=\)\((\)\( 6604 \nu^{11} + 4175 \nu^{10} - 8182 \nu^{9} + 6486 \nu^{8} - 110771 \nu^{7} - 217840 \nu^{6} + 19196 \nu^{5} + 732168 \nu^{4} + 2347908 \nu^{3} + 3011652 \nu^{2} + 1663614 \nu + 631152 \)\()/51972\)
\(\beta_{7}\)\(=\)\((\)\( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 \)\()/12993\)
\(\beta_{8}\)\(=\)\((\)\( -118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 4320 \)\()/852\)
\(\beta_{9}\)\(=\)\((\)\( 26035 \nu^{11} - 16590 \nu^{10} + 14290 \nu^{9} - 9780 \nu^{8} - 416670 \nu^{7} - 347068 \nu^{6} + 155770 \nu^{5} + 2327880 \nu^{4} + 6525980 \nu^{3} + 6240620 \nu^{2} + 4402140 \nu + 1639800 \)\()/103944\)
\(\beta_{10}\)\(=\)\((\)\( -17920 \nu^{11} + 13314 \nu^{10} - 5985 \nu^{9} + 1146 \nu^{8} + 286651 \nu^{7} + 215040 \nu^{6} - 211740 \nu^{5} - 1622514 \nu^{4} - 4157302 \nu^{3} - 3407556 \nu^{2} - 1863546 \nu - 639072 \)\()/51972\)
\(\beta_{11}\)\(=\)\((\)\( -29743 \nu^{11} + 22494 \nu^{10} - 12243 \nu^{9} + 5262 \nu^{8} + 472510 \nu^{7} + 356916 \nu^{6} - 347886 \nu^{5} - 2656440 \nu^{4} - 6971578 \nu^{3} - 5717928 \nu^{2} - 3128892 \nu - 1079712 \)\()/51972\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{9} - 7 \beta_{7} + 2 \beta_{5} + 6 \beta_{4} - 9 \beta_{3}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{11} + 4 \beta_{10} + 6 \beta_{8} + 4 \beta_{6} + 8 \beta_{4} - 3 \beta_{3}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{11} + 3 \beta_{10} + 30 \beta_{8} + 2 \beta_{2} + 7 \beta_{1} + 32\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{11} + 16 \beta_{10} + 6 \beta_{9} + 39 \beta_{8} - 57 \beta_{7} - 16 \beta_{6} + 54 \beta_{5} - 38 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 54 \beta_{1} + 120\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(16 \beta_{9} - 65 \beta_{7} + 40 \beta_{5}\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-47 \beta_{11} + 35 \beta_{10} + 24 \beta_{9} + 108 \beta_{8} - 156 \beta_{7} + 35 \beta_{6} + 129 \beta_{5} + 94 \beta_{4} - 60 \beta_{3} - 12 \beta_{2} - 129 \beta_{1} - 336\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-249 \beta_{11} + 144 \beta_{10} + 669 \beta_{8} - 35 \beta_{2} - 214 \beta_{1} - 704\)\()/6\)
\(\nu^{9}\)\(=\)\((\)\(-236 \beta_{11} + 164 \beta_{10} + 570 \beta_{8} - 164 \beta_{6} - 472 \beta_{4} + 321 \beta_{3}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(164 \beta_{9} - 790 \beta_{7} - 393 \beta_{6} + 557 \beta_{5} - 1278 \beta_{4} + 954 \beta_{3}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(1193 \beta_{11} - 800 \beta_{10} + 786 \beta_{9} - 2949 \beta_{8} - 4227 \beta_{7} - 800 \beta_{6} + 3186 \beta_{5} - 2386 \beta_{4} + 1671 \beta_{3} - 393 \beta_{2} - 3186 \beta_{1} - 9240\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
−0.673288 + 0.180407i
0.583700 + 2.17840i
−1.50511 + 0.403293i
−0.403293 1.50511i
2.17840 0.583700i
−0.180407 0.673288i
−0.673288 0.180407i
0.583700 2.17840i
−1.50511 0.403293i
−0.403293 + 1.50511i
2.17840 + 0.583700i
−0.180407 + 0.673288i
0 0 0 1.73205i 0 −4.35461 0 0 0
2593.2 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.3 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.4 0 0 0 1.73205i 0 0.990721 0 0 0
2593.5 0 0 0 1.73205i 0 3.61328 0 0 0
2593.6 0 0 0 1.73205i 0 4.35461 0 0 0
2593.7 0 0 0 1.73205i 0 −4.35461 0 0 0
2593.8 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.9 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.10 0 0 0 1.73205i 0 0.990721 0 0 0
2593.11 0 0 0 1.73205i 0 3.61328 0 0 0
2593.12 0 0 0 1.73205i 0 4.35461 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2593.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.d.r 12
3.b odd 2 1 5184.2.d.q 12
4.b odd 2 1 inner 5184.2.d.r 12
8.b even 2 1 inner 5184.2.d.r 12
8.d odd 2 1 inner 5184.2.d.r 12
9.c even 3 1 1728.2.r.e 12
9.c even 3 1 1728.2.r.f 12
9.d odd 6 1 576.2.r.e 12
9.d odd 6 1 576.2.r.f yes 12
12.b even 2 1 5184.2.d.q 12
24.f even 2 1 5184.2.d.q 12
24.h odd 2 1 5184.2.d.q 12
36.f odd 6 1 1728.2.r.e 12
36.f odd 6 1 1728.2.r.f 12
36.h even 6 1 576.2.r.e 12
36.h even 6 1 576.2.r.f yes 12
72.j odd 6 1 576.2.r.e 12
72.j odd 6 1 576.2.r.f yes 12
72.l even 6 1 576.2.r.e 12
72.l even 6 1 576.2.r.f yes 12
72.n even 6 1 1728.2.r.e 12
72.n even 6 1 1728.2.r.f 12
72.p odd 6 1 1728.2.r.e 12
72.p odd 6 1 1728.2.r.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 9.d odd 6 1
576.2.r.e 12 36.h even 6 1
576.2.r.e 12 72.j odd 6 1
576.2.r.e 12 72.l even 6 1
576.2.r.f yes 12 9.d odd 6 1
576.2.r.f yes 12 36.h even 6 1
576.2.r.f yes 12 72.j odd 6 1
576.2.r.f yes 12 72.l even 6 1
1728.2.r.e 12 9.c even 3 1
1728.2.r.e 12 36.f odd 6 1
1728.2.r.e 12 72.n even 6 1
1728.2.r.e 12 72.p odd 6 1
1728.2.r.f 12 9.c even 3 1
1728.2.r.f 12 36.f odd 6 1
1728.2.r.f 12 72.n even 6 1
1728.2.r.f 12 72.p odd 6 1
5184.2.d.q 12 3.b odd 2 1
5184.2.d.q 12 12.b even 2 1
5184.2.d.q 12 24.f even 2 1
5184.2.d.q 12 24.h odd 2 1
5184.2.d.r 12 1.a even 1 1 trivial
5184.2.d.r 12 4.b odd 2 1 inner
5184.2.d.r 12 8.b even 2 1 inner
5184.2.d.r 12 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5184, [\chi])\):

\( T_{5}^{2} + 3 \)
\( T_{7}^{6} - 33 T_{7}^{4} + 279 T_{7}^{2} - 243 \)
\( T_{17}^{3} - 24 T_{17} - 36 \)
\( T_{23}^{6} - 117 T_{23}^{4} + 4347 T_{23}^{2} - 49923 \)
\( T_{41}^{3} - 9 T_{41}^{2} + 3 T_{41} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 3 + T^{2} )^{6} \)
$7$ \( ( -243 + 279 T^{2} - 33 T^{4} + T^{6} )^{2} \)
$11$ \( ( 81 + 171 T^{2} + 39 T^{4} + T^{6} )^{2} \)
$13$ \( ( 243 + 675 T^{2} + 57 T^{4} + T^{6} )^{2} \)
$17$ \( ( -36 - 24 T + T^{3} )^{4} \)
$19$ \( ( 4 + T^{2} )^{6} \)
$23$ \( ( -49923 + 4347 T^{2} - 117 T^{4} + T^{6} )^{2} \)
$29$ \( ( 93987 + 7155 T^{2} + 153 T^{4} + T^{6} )^{2} \)
$31$ \( ( -9747 + 2619 T^{2} - 117 T^{4} + T^{6} )^{2} \)
$37$ \( ( 3888 + 1008 T^{2} + 60 T^{4} + T^{6} )^{2} \)
$41$ \( ( 9 + 3 T - 9 T^{2} + T^{3} )^{4} \)
$43$ \( ( 6889 + 1935 T^{2} + 123 T^{4} + T^{6} )^{2} \)
$47$ \( ( -717363 + 27135 T^{2} - 297 T^{4} + T^{6} )^{2} \)
$53$ \( ( 432 + 1728 T^{2} + 180 T^{4} + T^{6} )^{2} \)
$59$ \( ( 729 + 711 T^{2} + 147 T^{4} + T^{6} )^{2} \)
$61$ \( ( 4563 + 3267 T^{2} + 153 T^{4} + T^{6} )^{2} \)
$67$ \( ( 18769 + 8091 T^{2} + 231 T^{4} + T^{6} )^{2} \)
$71$ \( ( -124848 + 19872 T^{2} - 288 T^{4} + T^{6} )^{2} \)
$73$ \( ( -164 - 84 T + T^{3} )^{4} \)
$79$ \( ( -2187 + 891 T^{2} - 93 T^{4} + T^{6} )^{2} \)
$83$ \( ( 6561 + 3159 T^{2} + 171 T^{4} + T^{6} )^{2} \)
$89$ \( ( 108 - 36 T - 12 T^{2} + T^{3} )^{4} \)
$97$ \( ( 1187 - 213 T - 3 T^{2} + T^{3} )^{4} \)
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