Properties

Label 882.4.g.bi
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-3}, \sqrt{22})\)
Defining polynomial: \(x^{4} + 22 x^{2} + 484\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 98)
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.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -8 q^{8} +O(q^{10})\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -8 q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{10} -20 \beta_{2} q^{11} -7 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{17} -\beta_{1} q^{19} + 4 \beta_{3} q^{20} + 40 q^{22} + ( 48 + 48 \beta_{2} ) q^{23} -37 \beta_{2} q^{25} + 14 \beta_{1} q^{26} + 166 q^{29} + ( -22 \beta_{1} - 22 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} -12 \beta_{3} q^{34} + ( 78 + 78 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} -8 \beta_{1} q^{40} + 42 \beta_{3} q^{41} + 436 q^{43} + ( 80 + 80 \beta_{2} ) q^{44} + 96 \beta_{2} q^{46} + 22 \beta_{1} q^{47} + 74 q^{50} + ( 28 \beta_{1} + 28 \beta_{3} ) q^{52} -62 \beta_{2} q^{53} -20 \beta_{3} q^{55} + ( 332 + 332 \beta_{2} ) q^{58} + ( 71 \beta_{1} + 71 \beta_{3} ) q^{59} -29 \beta_{1} q^{61} -44 \beta_{3} q^{62} + 64 q^{64} + ( 616 + 616 \beta_{2} ) q^{65} + 580 \beta_{2} q^{67} + 24 \beta_{1} q^{68} + 544 q^{71} + ( -64 \beta_{1} - 64 \beta_{3} ) q^{73} + 156 \beta_{2} q^{74} -4 \beta_{3} q^{76} + ( 680 + 680 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{80} -84 \beta_{1} q^{82} + 21 \beta_{3} q^{83} + 528 q^{85} + ( 872 + 872 \beta_{2} ) q^{86} + 160 \beta_{2} q^{88} -160 \beta_{1} q^{89} -192 q^{92} + ( 44 \beta_{1} + 44 \beta_{3} ) q^{94} -88 \beta_{2} q^{95} + 70 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} + 40q^{11} - 32q^{16} + 160q^{22} + 96q^{23} + 74q^{25} + 664q^{29} + 64q^{32} + 156q^{37} + 1744q^{43} + 160q^{44} - 192q^{46} + 296q^{50} + 124q^{53} + 664q^{58} + 256q^{64} + 1232q^{65} - 1160q^{67} + 2176q^{71} - 312q^{74} + 1360q^{79} + 2112q^{85} + 1744q^{86} - 320q^{88} - 768q^{92} + 176q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 22 x^{2} + 484\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/22\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/11\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(22 \beta_{2}\)
\(\nu^{3}\)\(=\)\(11 \beta_{3}\)

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
−2.34521 4.06202i
2.34521 + 4.06202i
−2.34521 + 4.06202i
2.34521 4.06202i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.69042 8.12404i 0 0 −8.00000 0 9.38083 16.2481i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.69042 + 8.12404i 0 0 −8.00000 0 −9.38083 + 16.2481i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −4.69042 + 8.12404i 0 0 −8.00000 0 9.38083 + 16.2481i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i 4.69042 8.12404i 0 0 −8.00000 0 −9.38083 16.2481i
\(n\): 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.bi 4
3.b odd 2 1 98.4.c.g 4
7.b odd 2 1 inner 882.4.g.bi 4
7.c even 3 1 882.4.a.w 2
7.c even 3 1 inner 882.4.g.bi 4
7.d odd 6 1 882.4.a.w 2
7.d odd 6 1 inner 882.4.g.bi 4
21.c even 2 1 98.4.c.g 4
21.g even 6 1 98.4.a.h 2
21.g even 6 1 98.4.c.g 4
21.h odd 6 1 98.4.a.h 2
21.h odd 6 1 98.4.c.g 4
84.j odd 6 1 784.4.a.z 2
84.n even 6 1 784.4.a.z 2
105.o odd 6 1 2450.4.a.bs 2
105.p even 6 1 2450.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 21.g even 6 1
98.4.a.h 2 21.h odd 6 1
98.4.c.g 4 3.b odd 2 1
98.4.c.g 4 21.c even 2 1
98.4.c.g 4 21.g even 6 1
98.4.c.g 4 21.h odd 6 1
784.4.a.z 2 84.j odd 6 1
784.4.a.z 2 84.n even 6 1
882.4.a.w 2 7.c even 3 1
882.4.a.w 2 7.d odd 6 1
882.4.g.bi 4 1.a even 1 1 trivial
882.4.g.bi 4 7.b odd 2 1 inner
882.4.g.bi 4 7.c even 3 1 inner
882.4.g.bi 4 7.d odd 6 1 inner
2450.4.a.bs 2 105.o odd 6 1
2450.4.a.bs 2 105.p even 6 1

Hecke kernels

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

\( T_{5}^{4} + 88 T_{5}^{2} + 7744 \)
\( T_{11}^{2} - 20 T_{11} + 400 \)
\( T_{13}^{2} - 4312 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 7744 + 88 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 400 - 20 T + T^{2} )^{2} \)
$13$ \( ( -4312 + T^{2} )^{2} \)
$17$ \( 10036224 + 3168 T^{2} + T^{4} \)
$19$ \( 7744 + 88 T^{2} + T^{4} \)
$23$ \( ( 2304 - 48 T + T^{2} )^{2} \)
$29$ \( ( -166 + T )^{4} \)
$31$ \( 1814078464 + 42592 T^{2} + T^{4} \)
$37$ \( ( 6084 - 78 T + T^{2} )^{2} \)
$41$ \( ( -155232 + T^{2} )^{2} \)
$43$ \( ( -436 + T )^{4} \)
$47$ \( 1814078464 + 42592 T^{2} + T^{4} \)
$53$ \( ( 3844 - 62 T + T^{2} )^{2} \)
$59$ \( 196788057664 + 443608 T^{2} + T^{4} \)
$61$ \( 5477184064 + 74008 T^{2} + T^{4} \)
$67$ \( ( 336400 + 580 T + T^{2} )^{2} \)
$71$ \( ( -544 + T )^{4} \)
$73$ \( 129922760704 + 360448 T^{2} + T^{4} \)
$79$ \( ( 462400 - 680 T + T^{2} )^{2} \)
$83$ \( ( -38808 + T^{2} )^{2} \)
$89$ \( 5075107840000 + 2252800 T^{2} + T^{4} \)
$97$ \( ( -431200 + T^{2} )^{2} \)
show more
show less