Properties

Label 882.4.g.bg
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_{1} q^{5} -8 q^{8} +O(q^{10})\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_{1} q^{5} -8 q^{8} + ( 10 \beta_{1} + 10 \beta_{3} ) q^{10} + 40 \beta_{2} q^{11} + 45 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} -8 \beta_{1} q^{19} + 20 \beta_{3} q^{20} -80 q^{22} + ( -68 - 68 \beta_{2} ) q^{23} -75 \beta_{2} q^{25} -90 \beta_{1} q^{26} + 110 q^{29} + ( -84 \beta_{1} - 84 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + 2 \beta_{3} q^{34} + ( 20 + 20 \beta_{2} ) q^{37} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{38} -40 \beta_{1} q^{40} + 35 \beta_{3} q^{41} -340 q^{43} + ( -160 - 160 \beta_{2} ) q^{44} -136 \beta_{2} q^{46} -64 \beta_{1} q^{47} + 150 q^{50} + ( -180 \beta_{1} - 180 \beta_{3} ) q^{52} + 628 \beta_{2} q^{53} + 200 \beta_{3} q^{55} + ( 220 + 220 \beta_{2} ) q^{58} + ( -620 \beta_{1} - 620 \beta_{3} ) q^{59} -649 \beta_{1} q^{61} -168 \beta_{3} q^{62} + 64 q^{64} + ( -450 - 450 \beta_{2} ) q^{65} + 540 \beta_{2} q^{67} -4 \beta_{1} q^{68} -420 q^{71} + ( 205 \beta_{1} + 205 \beta_{3} ) q^{73} + 40 \beta_{2} q^{74} -32 \beta_{3} q^{76} + ( 760 + 760 \beta_{2} ) q^{79} + ( -80 \beta_{1} - 80 \beta_{3} ) q^{80} -70 \beta_{1} q^{82} -668 \beta_{3} q^{83} -10 q^{85} + ( -680 - 680 \beta_{2} ) q^{86} -320 \beta_{2} q^{88} -815 \beta_{1} q^{89} + 272 q^{92} + ( -128 \beta_{1} - 128 \beta_{3} ) q^{94} -80 \beta_{2} q^{95} + 355 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} - 80q^{11} - 32q^{16} - 320q^{22} - 136q^{23} + 150q^{25} + 440q^{29} + 64q^{32} + 40q^{37} - 1360q^{43} - 320q^{44} + 272q^{46} + 600q^{50} - 1256q^{53} + 440q^{58} + 256q^{64} - 900q^{65} - 1080q^{67} - 1680q^{71} - 80q^{74} + 1520q^{79} - 40q^{85} - 1360q^{86} + 640q^{88} + 1088q^{92} + 160q^{95} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.53553 6.12372i 0 0 −8.00000 0 7.07107 12.2474i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 3.53553 + 6.12372i 0 0 −8.00000 0 −7.07107 + 12.2474i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −3.53553 + 6.12372i 0 0 −8.00000 0 7.07107 + 12.2474i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i 3.53553 6.12372i 0 0 −8.00000 0 −7.07107 12.2474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.bg 4
3.b odd 2 1 882.4.g.bc 4
7.b odd 2 1 inner 882.4.g.bg 4
7.c even 3 1 882.4.a.y 2
7.c even 3 1 inner 882.4.g.bg 4
7.d odd 6 1 882.4.a.y 2
7.d odd 6 1 inner 882.4.g.bg 4
21.c even 2 1 882.4.g.bc 4
21.g even 6 1 882.4.a.be yes 2
21.g even 6 1 882.4.g.bc 4
21.h odd 6 1 882.4.a.be yes 2
21.h odd 6 1 882.4.g.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.y 2 7.c even 3 1
882.4.a.y 2 7.d odd 6 1
882.4.a.be yes 2 21.g even 6 1
882.4.a.be yes 2 21.h odd 6 1
882.4.g.bc 4 3.b odd 2 1
882.4.g.bc 4 21.c even 2 1
882.4.g.bc 4 21.g even 6 1
882.4.g.bc 4 21.h odd 6 1
882.4.g.bg 4 1.a even 1 1 trivial
882.4.g.bg 4 7.b odd 2 1 inner
882.4.g.bg 4 7.c even 3 1 inner
882.4.g.bg 4 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 50 T_{5}^{2} + 2500 \)
\( T_{11}^{2} + 40 T_{11} + 1600 \)
\( T_{13}^{2} - 4050 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 2500 + 50 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1600 + 40 T + T^{2} )^{2} \)
$13$ \( ( -4050 + T^{2} )^{2} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 16384 + 128 T^{2} + T^{4} \)
$23$ \( ( 4624 + 68 T + T^{2} )^{2} \)
$29$ \( ( -110 + T )^{4} \)
$31$ \( 199148544 + 14112 T^{2} + T^{4} \)
$37$ \( ( 400 - 20 T + T^{2} )^{2} \)
$41$ \( ( -2450 + T^{2} )^{2} \)
$43$ \( ( 340 + T )^{4} \)
$47$ \( 67108864 + 8192 T^{2} + T^{4} \)
$53$ \( ( 394384 + 628 T + T^{2} )^{2} \)
$59$ \( 591053440000 + 768800 T^{2} + T^{4} \)
$61$ \( 709641129604 + 842402 T^{2} + T^{4} \)
$67$ \( ( 291600 + 540 T + T^{2} )^{2} \)
$71$ \( ( 420 + T )^{4} \)
$73$ \( 7064402500 + 84050 T^{2} + T^{4} \)
$79$ \( ( 577600 - 760 T + T^{2} )^{2} \)
$83$ \( ( -892448 + T^{2} )^{2} \)
$89$ \( 1764779402500 + 1328450 T^{2} + T^{4} \)
$97$ \( ( -252050 + T^{2} )^{2} \)
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