Properties

Label 5184.2.f.e
Level $5184$
Weight $2$
Character orbit 5184.f
Analytic conductor $41.394$
Analytic rank $0$
Dimension $16$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + ( \beta_{9} + \beta_{11} ) q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} + ( \beta_{9} + \beta_{11} ) q^{7} + \beta_{14} q^{11} + ( \beta_{5} - \beta_{9} - \beta_{11} ) q^{13} + ( -\beta_{12} - \beta_{14} ) q^{17} + ( 2 + \beta_{2} + \beta_{7} ) q^{19} + ( -2 \beta_{3} - \beta_{4} ) q^{23} + ( 1 - 2 \beta_{7} - \beta_{13} ) q^{25} + ( \beta_{1} + \beta_{3} - \beta_{8} ) q^{29} + ( 2 \beta_{5} + \beta_{9} - \beta_{10} ) q^{31} + ( -\beta_{6} + 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{35} + ( 4 \beta_{5} - \beta_{11} ) q^{37} + ( -2 \beta_{12} + \beta_{15} ) q^{41} + ( 2 + \beta_{2} + 2 \beta_{7} + \beta_{13} ) q^{43} + ( -\beta_{1} - 2 \beta_{3} ) q^{47} + ( -3 - \beta_{7} + \beta_{13} ) q^{49} + ( -2 \beta_{1} - \beta_{8} ) q^{53} + ( -2 \beta_{5} - \beta_{9} - \beta_{11} ) q^{55} + ( \beta_{6} - 2 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 4 \beta_{5} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{61} + ( \beta_{6} - 3 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{65} + ( 8 + \beta_{2} - \beta_{13} ) q^{67} + ( 2 \beta_{1} + \beta_{4} ) q^{71} + ( 2 + 2 \beta_{2} - \beta_{13} ) q^{73} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} ) q^{77} + ( 4 \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} + ( 2 \beta_{6} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{83} + ( 8 \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{85} + ( -2 \beta_{6} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{89} + ( 10 - \beta_{2} - \beta_{13} ) q^{91} + ( \beta_{1} + \beta_{4} + 2 \beta_{8} ) q^{95} + ( -2 - 2 \beta_{2} - 3 \beta_{7} + \beta_{13} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 114 \nu^{15} - 597 \nu^{13} + 3136 \nu^{11} + 2960 \nu^{9} - 10680 \nu^{7} + 28832 \nu^{5} - 12686 \nu^{3} + 3579 \nu \)\()/528\)
\(\beta_{2}\)\(=\)\((\)\( -105 \nu^{14} + 791 \nu^{12} - 4704 \nu^{10} + 8232 \nu^{8} - 10072 \nu^{6} + 2688 \nu^{4} - 441 \nu^{2} + 2183 \)\()/1056\)
\(\beta_{3}\)\(=\)\((\)\( -199 \nu^{15} + 1637 \nu^{13} - 10112 \nu^{11} + 22888 \nu^{9} - 36424 \nu^{7} + 27200 \nu^{5} - 13719 \nu^{3} + 3893 \nu \)\()/528\)
\(\beta_{4}\)\(=\)\((\)\( 489 \nu^{15} - 3983 \nu^{13} + 24512 \nu^{11} - 54248 \nu^{9} + 85144 \nu^{7} - 62432 \nu^{5} + 31481 \nu^{3} - 8927 \nu \)\()/1056\)
\(\beta_{5}\)\(=\)\((\)\( -59 \nu^{14} + 476 \nu^{12} - 2916 \nu^{10} + 6280 \nu^{8} - 9538 \nu^{6} + 6192 \nu^{4} - 2351 \nu^{2} + 260 \)\()/198\)
\(\beta_{6}\)\(=\)\((\)\( 323 \nu^{15} - 2478 \nu^{13} + 15104 \nu^{11} - 29336 \nu^{9} + 46256 \nu^{7} - 26432 \nu^{5} + 17619 \nu^{3} - 590 \nu \)\()/528\)
\(\beta_{7}\)\(=\)\((\)\( 15 \nu^{14} - 112 \nu^{12} + 672 \nu^{10} - 1176 \nu^{8} + 1616 \nu^{6} - 384 \nu^{4} + 63 \nu^{2} + 80 \)\()/48\)
\(\beta_{8}\)\(=\)\((\)\( 337 \nu^{15} - 3022 \nu^{13} + 19040 \nu^{11} - 50392 \nu^{9} + 83984 \nu^{7} - 79520 \nu^{5} + 38481 \nu^{3} - 10894 \nu \)\()/528\)
\(\beta_{9}\)\(=\)\((\)\( -196 \nu^{14} + 1533 \nu^{12} - 9344 \nu^{10} + 18816 \nu^{8} - 28616 \nu^{6} + 16352 \nu^{4} - 8708 \nu^{2} + 893 \)\()/528\)
\(\beta_{10}\)\(=\)\((\)\( 9 \nu^{14} - 64 \nu^{12} + 384 \nu^{10} - 600 \nu^{8} + 944 \nu^{6} - 336 \nu^{4} + 441 \nu^{2} - 40 \)\()/24\)
\(\beta_{11}\)\(=\)\((\)\( -2009 \nu^{14} + 16409 \nu^{12} - 100704 \nu^{10} + 221992 \nu^{8} - 335752 \nu^{6} + 221376 \nu^{4} - 80105 \nu^{2} + 8969 \)\()/3168\)
\(\beta_{12}\)\(=\)\((\)\( -833 \nu^{15} + 6573 \nu^{13} - 40064 \nu^{11} + 81992 \nu^{9} - 122696 \nu^{7} + 70112 \nu^{5} - 27681 \nu^{3} + 1565 \nu \)\()/528\)
\(\beta_{13}\)\(=\)\((\)\( -80 \nu^{14} + 599 \nu^{12} - 3584 \nu^{10} + 6272 \nu^{8} - 8336 \nu^{6} + 2048 \nu^{4} - 336 \nu^{2} - 201 \)\()/88\)
\(\beta_{14}\)\(=\)\((\)\( -1931 \nu^{15} + 15267 \nu^{13} - 93056 \nu^{11} + 191096 \nu^{9} - 284984 \nu^{7} + 162848 \nu^{5} - 63867 \nu^{3} + 3635 \nu \)\()/1056\)
\(\beta_{15}\)\(=\)\((\)\( 1970 \nu^{15} - 15519 \nu^{13} + 94592 \nu^{11} - 192992 \nu^{9} + 289688 \nu^{7} - 165536 \nu^{5} + 71106 \nu^{3} - 3695 \nu \)\()/528\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{14} + 2 \beta_{12} + 2 \beta_{8} + \beta_{6} - 4 \beta_{4} + 2 \beta_{3} + \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13} + 6 \beta_{11} - 3 \beta_{10} - 9 \beta_{9} + \beta_{7} - 6 \beta_{5} - 4 \beta_{2} + 10\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} - 6 \beta_{14} + 3 \beta_{12} + 2 \beta_{6}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{13} + 36 \beta_{11} - 15 \beta_{10} - 33 \beta_{9} - 13 \beta_{7} - 54 \beta_{5} + 22 \beta_{2} - 40\)\()/12\)
\(\nu^{5}\)\(=\)\((\)\(-26 \beta_{15} - 66 \beta_{14} + 24 \beta_{12} - 50 \beta_{8} + 23 \beta_{6} + 112 \beta_{4} + 28 \beta_{3} - 43 \beta_{1}\)\()/12\)
\(\nu^{6}\)\(=\)\(-7 \beta_{13} - 14 \beta_{7} + 20 \beta_{2} - 34\)
\(\nu^{7}\)\(=\)\((\)\(150 \beta_{15} + 364 \beta_{14} - 118 \beta_{12} - 268 \beta_{8} - 131 \beta_{6} + 626 \beta_{4} + 182 \beta_{3} - 233 \beta_{1}\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(-239 \beta_{13} - 1140 \beta_{11} + 423 \beta_{10} + 867 \beta_{9} - 485 \beta_{7} + 1878 \beta_{5} + 662 \beta_{2} - 1106\)\()/12\)
\(\nu^{9}\)\(=\)\((\)\(421 \beta_{15} + 1008 \beta_{14} - 315 \beta_{12} - 367 \beta_{6}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(1337 \beta_{13} - 6342 \beta_{11} + 2331 \beta_{10} + 4761 \beta_{9} + 2723 \beta_{7} + 10500 \beta_{5} - 3668 \beta_{2} + 6098\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(4690 \beta_{15} + 11184 \beta_{14} - 3456 \beta_{12} + 8146 \beta_{8} - 4087 \beta_{6} - 19358 \beta_{4} - 5924 \beta_{3} + 7097 \beta_{1}\)\()/12\)
\(\nu^{12}\)\(=\)\(1240 \beta_{13} + 2528 \beta_{7} - 3392 \beta_{2} + 5631\)
\(\nu^{13}\)\(=\)\((\)\(-26064 \beta_{15} - 62078 \beta_{14} + 19118 \beta_{12} + 45182 \beta_{8} + 22711 \beta_{6} - 107500 \beta_{4} - 33010 \beta_{3} + 39367 \beta_{1}\)\()/12\)
\(\nu^{14}\)\(=\)\((\)\(41335 \beta_{13} + 195642 \beta_{11} - 71637 \beta_{10} - 146127 \beta_{9} + 84295 \beta_{7} - 324522 \beta_{5} - 112972 \beta_{2} + 187462\)\()/12\)
\(\nu^{15}\)\(=\)\((\)\(-72374 \beta_{15} - 172314 \beta_{14} + 53013 \beta_{12} + 63062 \beta_{6}\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2591.1
1.11871 + 0.645885i
1.11871 0.645885i
0.367543 + 0.212201i
0.367543 0.212201i
0.670418 0.387066i
0.670418 + 0.387066i
−2.04058 + 1.17813i
−2.04058 1.17813i
2.04058 1.17813i
2.04058 + 1.17813i
−0.670418 + 0.387066i
−0.670418 0.387066i
−0.367543 0.212201i
−0.367543 + 0.212201i
−1.11871 0.645885i
−1.11871 + 0.645885i
0 0 0 −4.04682 0 3.89623i 0 0 0
2591.2 0 0 0 −4.04682 0 3.89623i 0 0 0
2591.3 0 0 0 −2.24254 0 3.77162i 0 0 0
2591.4 0 0 0 −2.24254 0 3.77162i 0 0 0
2591.5 0 0 0 −1.59733 0 1.16418i 0 0 0
2591.6 0 0 0 −1.59733 0 1.16418i 0 0 0
2591.7 0 0 0 −0.206954 0 3.03957i 0 0 0
2591.8 0 0 0 −0.206954 0 3.03957i 0 0 0
2591.9 0 0 0 0.206954 0 3.03957i 0 0 0
2591.10 0 0 0 0.206954 0 3.03957i 0 0 0
2591.11 0 0 0 1.59733 0 1.16418i 0 0 0
2591.12 0 0 0 1.59733 0 1.16418i 0 0 0
2591.13 0 0 0 2.24254 0 3.77162i 0 0 0
2591.14 0 0 0 2.24254 0 3.77162i 0 0 0
2591.15 0 0 0 4.04682 0 3.89623i 0 0 0
2591.16 0 0 0 4.04682 0 3.89623i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2591.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.f.e yes 16
3.b odd 2 1 inner 5184.2.f.e yes 16
4.b odd 2 1 5184.2.f.b 16
8.b even 2 1 5184.2.f.b 16
8.d odd 2 1 inner 5184.2.f.e yes 16
12.b even 2 1 5184.2.f.b 16
24.f even 2 1 inner 5184.2.f.e yes 16
24.h odd 2 1 5184.2.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5184.2.f.b 16 4.b odd 2 1
5184.2.f.b 16 8.b even 2 1
5184.2.f.b 16 12.b even 2 1
5184.2.f.b 16 24.h odd 2 1
5184.2.f.e yes 16 1.a even 1 1 trivial
5184.2.f.e yes 16 3.b odd 2 1 inner
5184.2.f.e yes 16 8.d odd 2 1 inner
5184.2.f.e yes 16 24.f even 2 1 inner

Hecke kernels

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

\( T_{5}^{8} - 24 T_{5}^{6} + 138 T_{5}^{4} - 216 T_{5}^{2} + 9 \)
\( T_{19}^{4} - 6 T_{19}^{3} - 6 T_{19}^{2} + 36 T_{19} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 9 - 216 T^{2} + 138 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( ( 2704 + 2656 T^{2} + 540 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$11$ \( ( 144 + 720 T^{2} + 300 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$13$ \( ( 81 + 3024 T^{2} + 702 T^{4} + 48 T^{6} + T^{8} )^{2} \)
$17$ \( ( 297 + 36 T^{2} + T^{4} )^{4} \)
$19$ \( ( 36 + 36 T - 6 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$23$ \( ( 11664 - 11664 T^{2} + 2700 T^{4} - 108 T^{6} + T^{8} )^{2} \)
$29$ \( ( 123201 - 42768 T^{2} + 4482 T^{4} - 144 T^{6} + T^{8} )^{2} \)
$31$ \( ( 135424 + 97408 T^{2} + 6192 T^{4} + 136 T^{6} + T^{8} )^{2} \)
$37$ \( ( 13689 + 15336 T^{2} + 3510 T^{4} + 120 T^{6} + T^{8} )^{2} \)
$41$ \( ( 11664 + 15552 T^{2} + 3672 T^{4} + 144 T^{6} + T^{8} )^{2} \)
$43$ \( ( -1404 + 756 T - 90 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$47$ \( ( 186624 - 62208 T^{2} + 5616 T^{4} - 144 T^{6} + T^{8} )^{2} \)
$53$ \( ( 992016 - 237312 T^{2} + 11640 T^{4} - 192 T^{6} + T^{8} )^{2} \)
$59$ \( ( 186624 + 248832 T^{2} + 15984 T^{4} + 288 T^{6} + T^{8} )^{2} \)
$61$ \( ( 42849 + 53568 T^{2} + 9774 T^{4} + 192 T^{6} + T^{8} )^{2} \)
$67$ \( ( 468 - 828 T + 282 T^{2} - 30 T^{3} + T^{4} )^{4} \)
$71$ \( ( 11664 - 11664 T^{2} + 2700 T^{4} - 180 T^{6} + T^{8} )^{2} \)
$73$ \( ( -347 + 476 T - 90 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$79$ \( ( 1336336 + 406912 T^{2} + 32316 T^{4} + 352 T^{6} + T^{8} )^{2} \)
$83$ \( ( 26378496 + 2539008 T^{2} + 67872 T^{4} + 480 T^{6} + T^{8} )^{2} \)
$89$ \( ( 20820969 + 2608848 T^{2} + 58266 T^{4} + 432 T^{6} + T^{8} )^{2} \)
$97$ \( ( -176 + 352 T - 108 T^{2} + 4 T^{3} + T^{4} )^{4} \)
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