Properties

Label 882.4.g.bb
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: \( x^{4} + 58x^{2} + 3364 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} + 8 q^{8} + ( - 2 \beta_{3} - 2 \beta_1) q^{10} + 2 \beta_{2} q^{11} - 2 \beta_{3} q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (3 \beta_{3} + 3 \beta_1) q^{17} + 10 \beta_1 q^{19} + 4 \beta_{3} q^{20} + 4 q^{22} + ( - 30 \beta_{2} - 30) q^{23} + 107 \beta_{2} q^{25} - 4 \beta_1 q^{26} + 212 q^{29} + ( - 14 \beta_{3} - 14 \beta_1) q^{31} + 32 \beta_{2} q^{32} - 6 \beta_{3} q^{34} + ( - 246 \beta_{2} - 246) q^{37} + ( - 20 \beta_{3} - 20 \beta_1) q^{38} + 8 \beta_1 q^{40} - 21 \beta_{3} q^{41} - 284 q^{43} + ( - 8 \beta_{2} - 8) q^{44} + 60 \beta_{2} q^{46} + 4 \beta_1 q^{47} + 214 q^{50} + (8 \beta_{3} + 8 \beta_1) q^{52} + 548 \beta_{2} q^{53} + 2 \beta_{3} q^{55} + ( - 424 \beta_{2} - 424) q^{58} + (44 \beta_{3} + 44 \beta_1) q^{59} - 34 \beta_1 q^{61} + 28 \beta_{3} q^{62} + 64 q^{64} + (464 \beta_{2} + 464) q^{65} + 652 \beta_{2} q^{67} - 12 \beta_1 q^{68} + 770 q^{71} + (64 \beta_{3} + 64 \beta_1) q^{73} + 492 \beta_{2} q^{74} + 40 \beta_{3} q^{76} + ( - 472 \beta_{2} - 472) q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} - 42 \beta_1 q^{82} + 12 \beta_{3} q^{83} - 696 q^{85} + (568 \beta_{2} + 568) q^{86} + 16 \beta_{2} q^{88} + 47 \beta_1 q^{89} + 120 q^{92} + ( - 8 \beta_{3} - 8 \beta_1) q^{94} + 2320 \beta_{2} q^{95} + 20 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 4 q^{11} - 32 q^{16} + 16 q^{22} - 60 q^{23} - 214 q^{25} + 848 q^{29} - 64 q^{32} - 492 q^{37} - 1136 q^{43} - 16 q^{44} - 120 q^{46} + 856 q^{50} - 1096 q^{53} - 848 q^{58} + 256 q^{64} + 928 q^{65} - 1304 q^{67} + 3080 q^{71} - 984 q^{74} - 944 q^{79} - 2784 q^{85} + 1136 q^{86} - 32 q^{88} + 480 q^{92} - 4640 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 58\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 29\beta_{3} \) Copy content Toggle raw display

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.bb 4
3.b odd 2 1 882.4.g.bh 4
7.b odd 2 1 inner 882.4.g.bb 4
7.c even 3 1 882.4.a.bf yes 2
7.c even 3 1 inner 882.4.g.bb 4
7.d odd 6 1 882.4.a.bf yes 2
7.d odd 6 1 inner 882.4.g.bb 4
21.c even 2 1 882.4.g.bh 4
21.g even 6 1 882.4.a.x 2
21.g even 6 1 882.4.g.bh 4
21.h odd 6 1 882.4.a.x 2
21.h odd 6 1 882.4.g.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.x 2 21.g even 6 1
882.4.a.x 2 21.h odd 6 1
882.4.a.bf yes 2 7.c even 3 1
882.4.a.bf yes 2 7.d odd 6 1
882.4.g.bb 4 1.a even 1 1 trivial
882.4.g.bb 4 7.b odd 2 1 inner
882.4.g.bb 4 7.c even 3 1 inner
882.4.g.bb 4 7.d odd 6 1 inner
882.4.g.bh 4 3.b odd 2 1
882.4.g.bh 4 21.c even 2 1
882.4.g.bh 4 21.g even 6 1
882.4.g.bh 4 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 232T_{5}^{2} + 53824 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 928 \) Copy content Toggle raw display

Hecke characteristic polynomials

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