Properties

Label 882.5.b.c
Level $882$
Weight $5$
Character orbit 882.b
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-53})\)
Defining polynomial: \(x^{4} - 52 x^{2} + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{3} q^{5} + 16 \beta_{1} q^{8} +O(q^{10})\) \( q -2 \beta_{1} q^{2} -8 q^{4} + \beta_{3} q^{5} + 16 \beta_{1} q^{8} + 2 \beta_{2} q^{10} + \beta_{1} q^{11} -2 \beta_{2} q^{13} + 64 q^{16} + 9 \beta_{3} q^{17} -11 \beta_{2} q^{19} -8 \beta_{3} q^{20} + 4 q^{22} -45 \beta_{1} q^{23} -223 q^{25} + 8 \beta_{3} q^{26} + 355 \beta_{1} q^{29} -31 \beta_{2} q^{31} -128 \beta_{1} q^{32} + 18 \beta_{2} q^{34} + 1560 q^{37} + 44 \beta_{3} q^{38} -16 \beta_{2} q^{40} -45 \beta_{3} q^{41} + 328 q^{43} -8 \beta_{1} q^{44} -180 q^{46} + 142 \beta_{3} q^{47} + 446 \beta_{1} q^{50} + 16 \beta_{2} q^{52} -2837 \beta_{1} q^{53} -\beta_{2} q^{55} + 1420 q^{58} -26 \beta_{3} q^{59} + 29 \beta_{2} q^{61} + 124 \beta_{3} q^{62} -512 q^{64} -1696 \beta_{1} q^{65} -1130 q^{67} -72 \beta_{3} q^{68} -299 \beta_{1} q^{71} -73 \beta_{2} q^{73} -3120 \beta_{1} q^{74} + 88 \beta_{2} q^{76} + 8650 q^{79} + 64 \beta_{3} q^{80} -90 \beta_{2} q^{82} -102 \beta_{3} q^{83} -7632 q^{85} -656 \beta_{1} q^{86} -32 q^{88} -457 \beta_{3} q^{89} + 360 \beta_{1} q^{92} + 284 \beta_{2} q^{94} -9328 \beta_{1} q^{95} -319 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 32q^{4} + O(q^{10}) \) \( 4q - 32q^{4} + 256q^{16} + 16q^{22} - 892q^{25} + 6240q^{37} + 1312q^{43} - 720q^{46} + 5680q^{58} - 2048q^{64} - 4520q^{67} + 34600q^{79} - 30528q^{85} - 128q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 52 x^{2} + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 25 \nu \)\()/27\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 316 \nu \)\()/27\)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} - 104 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 104\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(25 \beta_{2} + 316 \beta_{1}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−5.14782 + 0.707107i
5.14782 + 0.707107i
5.14782 0.707107i
−5.14782 0.707107i
2.82843i 0 −8.00000 29.1204i 0 0 22.6274i 0 −82.3650
197.2 2.82843i 0 −8.00000 29.1204i 0 0 22.6274i 0 82.3650
197.3 2.82843i 0 −8.00000 29.1204i 0 0 22.6274i 0 82.3650
197.4 2.82843i 0 −8.00000 29.1204i 0 0 22.6274i 0 −82.3650
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.5.b.c 4
3.b odd 2 1 inner 882.5.b.c 4
7.b odd 2 1 inner 882.5.b.c 4
21.c even 2 1 inner 882.5.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.5.b.c 4 1.a even 1 1 trivial
882.5.b.c 4 3.b odd 2 1 inner
882.5.b.c 4 7.b odd 2 1 inner
882.5.b.c 4 21.c 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 848 \)
\( T_{13}^{2} - 6784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 848 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 2 + T^{2} )^{2} \)
$13$ \( ( -6784 + T^{2} )^{2} \)
$17$ \( ( 68688 + T^{2} )^{2} \)
$19$ \( ( -205216 + T^{2} )^{2} \)
$23$ \( ( 4050 + T^{2} )^{2} \)
$29$ \( ( 252050 + T^{2} )^{2} \)
$31$ \( ( -1629856 + T^{2} )^{2} \)
$37$ \( ( -1560 + T )^{4} \)
$41$ \( ( 1717200 + T^{2} )^{2} \)
$43$ \( ( -328 + T )^{4} \)
$47$ \( ( 17099072 + T^{2} )^{2} \)
$53$ \( ( 16097138 + T^{2} )^{2} \)
$59$ \( ( 573248 + T^{2} )^{2} \)
$61$ \( ( -1426336 + T^{2} )^{2} \)
$67$ \( ( 1130 + T )^{4} \)
$71$ \( ( 178802 + T^{2} )^{2} \)
$73$ \( ( -9037984 + T^{2} )^{2} \)
$79$ \( ( -8650 + T )^{4} \)
$83$ \( ( 8822592 + T^{2} )^{2} \)
$89$ \( ( 177103952 + T^{2} )^{2} \)
$97$ \( ( -172586656 + T^{2} )^{2} \)
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