Properties

Label 882.4.g.ba
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: 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.

\(f(q)\) \(=\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_{1} q^{5} + 8 q^{8} +O(q^{10})\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_{1} q^{5} + 8 q^{8} + ( -28 \beta_{1} - 28 \beta_{3} ) q^{10} + 14 \beta_{2} q^{11} + 36 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} -\beta_{1} q^{19} + 56 \beta_{3} q^{20} + 28 q^{22} + ( 140 + 140 \beta_{2} ) q^{23} + 267 \beta_{2} q^{25} + 72 \beta_{1} q^{26} + 286 q^{29} + ( 66 \beta_{1} + 66 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} -2 \beta_{3} q^{34} + ( 38 + 38 \beta_{2} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{38} + 112 \beta_{1} q^{40} + 89 \beta_{3} q^{41} -34 q^{43} + ( -56 - 56 \beta_{2} ) q^{44} -280 \beta_{2} q^{46} -370 \beta_{1} q^{47} + 534 q^{50} + ( -144 \beta_{1} - 144 \beta_{3} ) q^{52} + 74 \beta_{2} q^{53} + 196 \beta_{3} q^{55} + ( -572 - 572 \beta_{2} ) q^{58} + ( 307 \beta_{1} + 307 \beta_{3} ) q^{59} + 10 \beta_{1} q^{61} -132 \beta_{3} q^{62} + 64 q^{64} + ( -1008 - 1008 \beta_{2} ) q^{65} + 684 \beta_{2} q^{67} -4 \beta_{1} q^{68} -588 q^{71} + ( 191 \beta_{1} + 191 \beta_{3} ) q^{73} -76 \beta_{2} q^{74} -4 \beta_{3} q^{76} + ( -1220 - 1220 \beta_{2} ) q^{79} + ( -224 \beta_{1} - 224 \beta_{3} ) q^{80} + 178 \beta_{1} q^{82} -299 \beta_{3} q^{83} -28 q^{85} + ( 68 + 68 \beta_{2} ) q^{86} + 112 \beta_{2} q^{88} -437 \beta_{1} q^{89} -560 q^{92} + ( 740 \beta_{1} + 740 \beta_{3} ) q^{94} -28 \beta_{2} q^{95} + 1049 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 8q^{4} + 32q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 8q^{4} + 32q^{8} - 28q^{11} - 32q^{16} + 112q^{22} + 280q^{23} - 534q^{25} + 1144q^{29} - 64q^{32} + 76q^{37} - 136q^{43} - 112q^{44} + 560q^{46} + 2136q^{50} - 148q^{53} - 1144q^{58} + 256q^{64} - 2016q^{65} - 1368q^{67} - 2352q^{71} + 152q^{74} - 2440q^{79} - 112q^{85} + 136q^{86} - 224q^{88} - 2240q^{92} + 56q^{95} + O(q^{100}) \)

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 −9.89949 17.1464i 0 0 8.00000 0 −19.7990 + 34.2929i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 9.89949 + 17.1464i 0 0 8.00000 0 19.7990 34.2929i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −9.89949 + 17.1464i 0 0 8.00000 0 −19.7990 34.2929i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 9.89949 17.1464i 0 0 8.00000 0 19.7990 + 34.2929i
\(n\): 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.ba 4
3.b odd 2 1 98.4.c.h 4
7.b odd 2 1 inner 882.4.g.ba 4
7.c even 3 1 882.4.a.bg 2
7.c even 3 1 inner 882.4.g.ba 4
7.d odd 6 1 882.4.a.bg 2
7.d odd 6 1 inner 882.4.g.ba 4
21.c even 2 1 98.4.c.h 4
21.g even 6 1 98.4.a.g 2
21.g even 6 1 98.4.c.h 4
21.h odd 6 1 98.4.a.g 2
21.h odd 6 1 98.4.c.h 4
84.j odd 6 1 784.4.a.y 2
84.n even 6 1 784.4.a.y 2
105.o odd 6 1 2450.4.a.bx 2
105.p even 6 1 2450.4.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 21.g even 6 1
98.4.a.g 2 21.h odd 6 1
98.4.c.h 4 3.b odd 2 1
98.4.c.h 4 21.c even 2 1
98.4.c.h 4 21.g even 6 1
98.4.c.h 4 21.h odd 6 1
784.4.a.y 2 84.j odd 6 1
784.4.a.y 2 84.n even 6 1
882.4.a.bg 2 7.c even 3 1
882.4.a.bg 2 7.d odd 6 1
882.4.g.ba 4 1.a even 1 1 trivial
882.4.g.ba 4 7.b odd 2 1 inner
882.4.g.ba 4 7.c even 3 1 inner
882.4.g.ba 4 7.d odd 6 1 inner
2450.4.a.bx 2 105.o odd 6 1
2450.4.a.bx 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 392 T_{5}^{2} + 153664 \)
\( T_{11}^{2} + 14 T_{11} + 196 \)
\( T_{13}^{2} - 2592 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 153664 + 392 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 196 + 14 T + T^{2} )^{2} \)
$13$ \( ( -2592 + T^{2} )^{2} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 4 + 2 T^{2} + T^{4} \)
$23$ \( ( 19600 - 140 T + T^{2} )^{2} \)
$29$ \( ( -286 + T )^{4} \)
$31$ \( 75898944 + 8712 T^{2} + T^{4} \)
$37$ \( ( 1444 - 38 T + T^{2} )^{2} \)
$41$ \( ( -15842 + T^{2} )^{2} \)
$43$ \( ( 34 + T )^{4} \)
$47$ \( 74966440000 + 273800 T^{2} + T^{4} \)
$53$ \( ( 5476 + 74 T + T^{2} )^{2} \)
$59$ \( 35531496004 + 188498 T^{2} + T^{4} \)
$61$ \( 40000 + 200 T^{2} + T^{4} \)
$67$ \( ( 467856 + 684 T + T^{2} )^{2} \)
$71$ \( ( 588 + T )^{4} \)
$73$ \( 5323453444 + 72962 T^{2} + T^{4} \)
$79$ \( ( 1488400 + 1220 T + T^{2} )^{2} \)
$83$ \( ( -178802 + T^{2} )^{2} \)
$89$ \( 145876635844 + 381938 T^{2} + T^{4} \)
$97$ \( ( -2200802 + T^{2} )^{2} \)
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