Properties

Label 5184.2.f.b
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: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + (\beta_{11} + \beta_{9}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + (\beta_{11} + \beta_{9}) q^{7} + \beta_{14} q^{11} + (\beta_{11} + \beta_{9} - \beta_{5}) q^{13} + (\beta_{14} + \beta_{12}) q^{17} + ( - \beta_{7} - \beta_{2} - 2) q^{19} + (\beta_{4} + 2 \beta_{3}) q^{23} + ( - \beta_{13} - 2 \beta_{7} + 1) q^{25} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{29} + ( - \beta_{10} + \beta_{9} + 2 \beta_{5}) q^{31} + (2 \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{6}) q^{35} + (\beta_{11} - 4 \beta_{5}) q^{37} + ( - \beta_{15} + 2 \beta_{12}) q^{41} + ( - \beta_{13} - 2 \beta_{7} - \beta_{2} - 2) q^{43} + (2 \beta_{3} + \beta_1) q^{47} + (\beta_{13} - \beta_{7} - 3) q^{49} + ( - \beta_{8} - 2 \beta_1) q^{53} + ( - \beta_{11} - \beta_{9} - 2 \beta_{5}) q^{55} + ( - 2 \beta_{15} - 2 \beta_{14} + \beta_{6}) q^{59} + (\beta_{11} - \beta_{10} + \beta_{9} - 4 \beta_{5}) q^{61} + (2 \beta_{15} - \beta_{14} + 3 \beta_{12} - \beta_{6}) q^{65} + (\beta_{13} - \beta_{2} - 8) q^{67} + ( - \beta_{4} - 2 \beta_1) q^{71} + ( - \beta_{13} + 2 \beta_{2} + 2) q^{73} + (2 \beta_{8} - 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{77} + ( - \beta_{11} + \beta_{10} + 4 \beta_{9}) q^{79} + (2 \beta_{15} + 2 \beta_{14} - 2 \beta_{12} + 2 \beta_{6}) q^{83} + ( - \beta_{11} + \beta_{10} + \beta_{9} - 8 \beta_{5}) q^{85} + (\beta_{15} - \beta_{14} - \beta_{12} + 2 \beta_{6}) q^{89} + (\beta_{13} + \beta_{2} - 10) q^{91} + ( - 2 \beta_{8} - \beta_{4} - \beta_1) q^{95} + (\beta_{13} - 3 \beta_{7} - 2 \beta_{2} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{19} + 16 q^{25} - 24 q^{43} - 48 q^{49} - 120 q^{67} + 16 q^{73} - 168 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 114\nu^{15} - 597\nu^{13} + 3136\nu^{11} + 2960\nu^{9} - 10680\nu^{7} + 28832\nu^{5} - 12686\nu^{3} + 3579\nu ) / 528 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -105\nu^{14} + 791\nu^{12} - 4704\nu^{10} + 8232\nu^{8} - 10072\nu^{6} + 2688\nu^{4} - 441\nu^{2} + 2183 ) / 1056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 199 \nu^{15} + 1637 \nu^{13} - 10112 \nu^{11} + 22888 \nu^{9} - 36424 \nu^{7} + 27200 \nu^{5} - 13719 \nu^{3} + 3893 \nu ) / 528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 489 \nu^{15} - 3983 \nu^{13} + 24512 \nu^{11} - 54248 \nu^{9} + 85144 \nu^{7} - 62432 \nu^{5} + 31481 \nu^{3} - 8927 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -59\nu^{14} + 476\nu^{12} - 2916\nu^{10} + 6280\nu^{8} - 9538\nu^{6} + 6192\nu^{4} - 2351\nu^{2} + 260 ) / 198 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 323 \nu^{15} - 2478 \nu^{13} + 15104 \nu^{11} - 29336 \nu^{9} + 46256 \nu^{7} - 26432 \nu^{5} + 17619 \nu^{3} - 590 \nu ) / 528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\nu^{14} - 112\nu^{12} + 672\nu^{10} - 1176\nu^{8} + 1616\nu^{6} - 384\nu^{4} + 63\nu^{2} + 80 ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 337 \nu^{15} - 3022 \nu^{13} + 19040 \nu^{11} - 50392 \nu^{9} + 83984 \nu^{7} - 79520 \nu^{5} + 38481 \nu^{3} - 10894 \nu ) / 528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -196\nu^{14} + 1533\nu^{12} - 9344\nu^{10} + 18816\nu^{8} - 28616\nu^{6} + 16352\nu^{4} - 8708\nu^{2} + 893 ) / 528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9\nu^{14} - 64\nu^{12} + 384\nu^{10} - 600\nu^{8} + 944\nu^{6} - 336\nu^{4} + 441\nu^{2} - 40 ) / 24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2009 \nu^{14} + 16409 \nu^{12} - 100704 \nu^{10} + 221992 \nu^{8} - 335752 \nu^{6} + 221376 \nu^{4} - 80105 \nu^{2} + 8969 ) / 3168 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 833 \nu^{15} + 6573 \nu^{13} - 40064 \nu^{11} + 81992 \nu^{9} - 122696 \nu^{7} + 70112 \nu^{5} - 27681 \nu^{3} + 1565 \nu ) / 528 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -80\nu^{14} + 599\nu^{12} - 3584\nu^{10} + 6272\nu^{8} - 8336\nu^{6} + 2048\nu^{4} - 336\nu^{2} - 201 ) / 88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1931 \nu^{15} + 15267 \nu^{13} - 93056 \nu^{11} + 191096 \nu^{9} - 284984 \nu^{7} + 162848 \nu^{5} - 63867 \nu^{3} + 3635 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1970 \nu^{15} - 15519 \nu^{13} + 94592 \nu^{11} - 192992 \nu^{9} + 289688 \nu^{7} - 165536 \nu^{5} + 71106 \nu^{3} - 3695 \nu ) / 528 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{14} + 2\beta_{12} + 2\beta_{8} + \beta_{6} - 4\beta_{4} + 2\beta_{3} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + 6\beta_{11} - 3\beta_{10} - 9\beta_{9} + \beta_{7} - 6\beta_{5} - 4\beta_{2} + 10 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - 6\beta_{14} + 3\beta_{12} + 2\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{13} + 36\beta_{11} - 15\beta_{10} - 33\beta_{9} - 13\beta_{7} - 54\beta_{5} + 22\beta_{2} - 40 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -26\beta_{15} - 66\beta_{14} + 24\beta_{12} - 50\beta_{8} + 23\beta_{6} + 112\beta_{4} + 28\beta_{3} - 43\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{13} - 14\beta_{7} + 20\beta_{2} - 34 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 150 \beta_{15} + 364 \beta_{14} - 118 \beta_{12} - 268 \beta_{8} - 131 \beta_{6} + 626 \beta_{4} + 182 \beta_{3} - 233 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 239 \beta_{13} - 1140 \beta_{11} + 423 \beta_{10} + 867 \beta_{9} - 485 \beta_{7} + 1878 \beta_{5} + 662 \beta_{2} - 1106 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 421\beta_{15} + 1008\beta_{14} - 315\beta_{12} - 367\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1337 \beta_{13} - 6342 \beta_{11} + 2331 \beta_{10} + 4761 \beta_{9} + 2723 \beta_{7} + 10500 \beta_{5} - 3668 \beta_{2} + 6098 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4690 \beta_{15} + 11184 \beta_{14} - 3456 \beta_{12} + 8146 \beta_{8} - 4087 \beta_{6} - 19358 \beta_{4} - 5924 \beta_{3} + 7097 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1240\beta_{13} + 2528\beta_{7} - 3392\beta_{2} + 5631 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26064 \beta_{15} - 62078 \beta_{14} + 19118 \beta_{12} + 45182 \beta_{8} + 22711 \beta_{6} - 107500 \beta_{4} - 33010 \beta_{3} + 39367 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 41335 \beta_{13} + 195642 \beta_{11} - 71637 \beta_{10} - 146127 \beta_{9} + 84295 \beta_{7} - 324522 \beta_{5} - 112972 \beta_{2} + 187462 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -72374\beta_{15} - 172314\beta_{14} + 53013\beta_{12} + 63062\beta_{6} ) / 3 \) Copy content Toggle raw display

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

Hecke kernels

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

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

Hecke characteristic polynomials

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