Properties

Label 882.4.g.bh
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: \(\Q(\sqrt{-3}, \sqrt{58})\)
Defining polynomial: \(x^{4} + 58 x^{2} + 3364\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -8 q^{8} +O(q^{10})\) \( q + ( 2 + 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + \beta_{1} q^{5} -8 q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{10} -2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{17} -10 \beta_{1} q^{19} + 4 \beta_{3} q^{20} + 4 q^{22} + ( 30 + 30 \beta_{2} ) q^{23} + 107 \beta_{2} q^{25} -4 \beta_{1} q^{26} -212 q^{29} + ( 14 \beta_{1} + 14 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + 6 \beta_{3} q^{34} + ( -246 - 246 \beta_{2} ) q^{37} + ( -20 \beta_{1} - 20 \beta_{3} ) q^{38} -8 \beta_{1} q^{40} -21 \beta_{3} q^{41} -284 q^{43} + ( 8 + 8 \beta_{2} ) q^{44} + 60 \beta_{2} q^{46} + 4 \beta_{1} q^{47} -214 q^{50} + ( -8 \beta_{1} - 8 \beta_{3} ) q^{52} -548 \beta_{2} q^{53} -2 \beta_{3} q^{55} + ( -424 - 424 \beta_{2} ) q^{58} + ( 44 \beta_{1} + 44 \beta_{3} ) q^{59} + 34 \beta_{1} q^{61} + 28 \beta_{3} q^{62} + 64 q^{64} + ( -464 - 464 \beta_{2} ) q^{65} + 652 \beta_{2} q^{67} -12 \beta_{1} q^{68} -770 q^{71} + ( -64 \beta_{1} - 64 \beta_{3} ) q^{73} -492 \beta_{2} q^{74} -40 \beta_{3} q^{76} + ( -472 - 472 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{80} + 42 \beta_{1} q^{82} + 12 \beta_{3} q^{83} -696 q^{85} + ( -568 - 568 \beta_{2} ) q^{86} + 16 \beta_{2} q^{88} + 47 \beta_{1} q^{89} -120 q^{92} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{94} -2320 \beta_{2} q^{95} -20 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{4} - 32q^{8} + 4q^{11} - 32q^{16} + 16q^{22} + 60q^{23} - 214q^{25} - 848q^{29} + 64q^{32} - 492q^{37} - 1136q^{43} + 16q^{44} - 120q^{46} - 856q^{50} + 1096q^{53} - 848q^{58} + 256q^{64} - 928q^{65} - 1304q^{67} - 3080q^{71} + 984q^{74} - 944q^{79} - 2784q^{85} - 1136q^{86} - 32q^{88} - 480q^{92} + 4640q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 58 x^{2} + 3364\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/58\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/29\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(58 \beta_{2}\)
\(\nu^{3}\)\(=\)\(29 \beta_{3}\)

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
−3.80789 6.59545i
3.80789 + 6.59545i
−3.80789 + 6.59545i
3.80789 6.59545i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −7.61577 13.1909i 0 0 −8.00000 0 15.2315 26.3818i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 7.61577 + 13.1909i 0 0 −8.00000 0 −15.2315 + 26.3818i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −7.61577 + 13.1909i 0 0 −8.00000 0 15.2315 + 26.3818i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i 7.61577 13.1909i 0 0 −8.00000 0 −15.2315 26.3818i
\(n\): 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.bh 4
3.b odd 2 1 882.4.g.bb 4
7.b odd 2 1 inner 882.4.g.bh 4
7.c even 3 1 882.4.a.x 2
7.c even 3 1 inner 882.4.g.bh 4
7.d odd 6 1 882.4.a.x 2
7.d odd 6 1 inner 882.4.g.bh 4
21.c even 2 1 882.4.g.bb 4
21.g even 6 1 882.4.a.bf yes 2
21.g even 6 1 882.4.g.bb 4
21.h odd 6 1 882.4.a.bf yes 2
21.h odd 6 1 882.4.g.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x 2 7.c even 3 1
882.4.a.x 2 7.d odd 6 1
882.4.a.bf yes 2 21.g even 6 1
882.4.a.bf yes 2 21.h odd 6 1
882.4.g.bb 4 3.b odd 2 1
882.4.g.bb 4 21.c even 2 1
882.4.g.bb 4 21.g even 6 1
882.4.g.bb 4 21.h odd 6 1
882.4.g.bh 4 1.a even 1 1 trivial
882.4.g.bh 4 7.b odd 2 1 inner
882.4.g.bh 4 7.c even 3 1 inner
882.4.g.bh 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 232 T_{5}^{2} + 53824 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13}^{2} - 928 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 53824 + 232 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 4 - 2 T + T^{2} )^{2} \)
$13$ \( ( -928 + T^{2} )^{2} \)
$17$ \( 4359744 + 2088 T^{2} + T^{4} \)
$19$ \( 538240000 + 23200 T^{2} + T^{4} \)
$23$ \( ( 900 - 30 T + T^{2} )^{2} \)
$29$ \( ( 212 + T )^{4} \)
$31$ \( 2067702784 + 45472 T^{2} + T^{4} \)
$37$ \( ( 60516 + 246 T + T^{2} )^{2} \)
$41$ \( ( -102312 + T^{2} )^{2} \)
$43$ \( ( 284 + T )^{4} \)
$47$ \( 13778944 + 3712 T^{2} + T^{4} \)
$53$ \( ( 300304 - 548 T + T^{2} )^{2} \)
$59$ \( 201737519104 + 449152 T^{2} + T^{4} \)
$61$ \( 71926948864 + 268192 T^{2} + T^{4} \)
$67$ \( ( 425104 + 652 T + T^{2} )^{2} \)
$71$ \( ( 770 + T )^{4} \)
$73$ \( 903016873984 + 950272 T^{2} + T^{4} \)
$79$ \( ( 222784 + 472 T + T^{2} )^{2} \)
$83$ \( ( -33408 + T^{2} )^{2} \)
$89$ \( 262643950144 + 512488 T^{2} + T^{4} \)
$97$ \( ( -92800 + T^{2} )^{2} \)
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