Properties

Label 882.4.g.bd
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \( 7^{2} \)
Twist minimal: no (minimal twist has level 294)
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.

\(f(q)\) \(=\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} + 8 q^{8} +O(q^{10})\) \( q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} + 8 q^{8} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{10} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{11} + ( 24 + 3 \beta_{3} ) q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( -\beta_{1} - 42 \beta_{2} - \beta_{3} ) q^{17} + ( 36 - 2 \beta_{1} + 36 \beta_{2} ) q^{19} + ( -24 + 4 \beta_{3} ) q^{20} + ( 4 + 12 \beta_{3} ) q^{22} + ( -154 - 6 \beta_{1} - 154 \beta_{2} ) q^{23} + ( 12 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} ) q^{25} + ( -48 + 6 \beta_{1} - 48 \beta_{2} ) q^{26} + ( 40 - 18 \beta_{3} ) q^{29} + ( 6 \beta_{1} + 192 \beta_{2} + 6 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} + ( -84 + 2 \beta_{3} ) q^{34} + ( -268 - 12 \beta_{1} - 268 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 72 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 48 + 8 \beta_{1} + 48 \beta_{2} ) q^{40} + ( -378 - 5 \beta_{3} ) q^{41} + ( 200 + 24 \beta_{3} ) q^{43} + ( -8 + 24 \beta_{1} - 8 \beta_{2} ) q^{44} + ( 12 \beta_{1} + 308 \beta_{2} + 12 \beta_{3} ) q^{46} + ( 156 + 10 \beta_{1} + 156 \beta_{2} ) q^{47} + ( 18 - 24 \beta_{3} ) q^{50} + ( -12 \beta_{1} + 96 \beta_{2} - 12 \beta_{3} ) q^{52} + ( -24 \beta_{1} + 26 \beta_{2} - 24 \beta_{3} ) q^{53} + ( 576 - 34 \beta_{3} ) q^{55} + ( -80 - 36 \beta_{1} - 80 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 432 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -708 - 13 \beta_{1} - 708 \beta_{2} ) q^{61} + ( 384 - 12 \beta_{3} ) q^{62} + 64 q^{64} + ( -150 + 6 \beta_{1} - 150 \beta_{2} ) q^{65} + ( -24 \beta_{1} + 72 \beta_{2} - 24 \beta_{3} ) q^{67} + ( 168 + 4 \beta_{1} + 168 \beta_{2} ) q^{68} + ( 762 + 30 \beta_{3} ) q^{71} + ( -83 \beta_{1} + 372 \beta_{2} - 83 \beta_{3} ) q^{73} + ( 24 \beta_{1} + 536 \beta_{2} + 24 \beta_{3} ) q^{74} + ( -144 - 8 \beta_{3} ) q^{76} + ( -488 + 84 \beta_{1} - 488 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 96 \beta_{2} - 16 \beta_{3} ) q^{80} + ( 756 - 10 \beta_{1} + 756 \beta_{2} ) q^{82} + ( 156 - 136 \beta_{3} ) q^{83} + ( 350 - 48 \beta_{3} ) q^{85} + ( -400 + 48 \beta_{1} - 400 \beta_{2} ) q^{86} + ( -48 \beta_{1} + 16 \beta_{2} - 48 \beta_{3} ) q^{88} + ( 54 + 29 \beta_{1} + 54 \beta_{2} ) q^{89} + ( 616 - 24 \beta_{3} ) q^{92} + ( -20 \beta_{1} - 312 \beta_{2} - 20 \beta_{3} ) q^{94} + ( 24 \beta_{1} + 20 \beta_{2} + 24 \beta_{3} ) q^{95} + ( -372 - 125 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 8q^{4} + 12q^{5} + 32q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 8q^{4} + 12q^{5} + 32q^{8} + 24q^{10} - 4q^{11} + 96q^{13} - 32q^{16} + 84q^{17} + 72q^{19} - 96q^{20} + 16q^{22} - 308q^{23} - 18q^{25} - 96q^{26} + 160q^{29} - 384q^{31} - 64q^{32} - 336q^{34} - 536q^{37} + 144q^{38} + 96q^{40} - 1512q^{41} + 800q^{43} - 16q^{44} - 616q^{46} + 312q^{47} + 72q^{50} - 192q^{52} - 52q^{53} + 2304q^{55} - 160q^{58} + 864q^{59} - 1416q^{61} + 1536q^{62} + 256q^{64} - 300q^{65} - 144q^{67} + 336q^{68} + 3048q^{71} - 744q^{73} - 1072q^{74} - 576q^{76} - 976q^{79} + 192q^{80} + 1512q^{82} + 624q^{83} + 1400q^{85} - 800q^{86} - 32q^{88} + 108q^{89} + 2464q^{92} + 624q^{94} - 40q^{95} - 1488q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 7 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 7 \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/7\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)\(/7\)

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 −1.94975 3.37706i 0 0 8.00000 0 −3.89949 + 6.75412i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 7.94975 + 13.7694i 0 0 8.00000 0 15.8995 27.5387i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −1.94975 + 3.37706i 0 0 8.00000 0 −3.89949 6.75412i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.94975 13.7694i 0 0 8.00000 0 15.8995 + 27.5387i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.bd 4
3.b odd 2 1 294.4.e.o 4
7.b odd 2 1 882.4.g.y 4
7.c even 3 1 882.4.a.bc 2
7.c even 3 1 inner 882.4.g.bd 4
7.d odd 6 1 882.4.a.bi 2
7.d odd 6 1 882.4.g.y 4
21.c even 2 1 294.4.e.n 4
21.g even 6 1 294.4.a.k yes 2
21.g even 6 1 294.4.e.n 4
21.h odd 6 1 294.4.a.j 2
21.h odd 6 1 294.4.e.o 4
84.j odd 6 1 2352.4.a.bn 2
84.n even 6 1 2352.4.a.cd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 21.h odd 6 1
294.4.a.k yes 2 21.g even 6 1
294.4.e.n 4 21.c even 2 1
294.4.e.n 4 21.g even 6 1
294.4.e.o 4 3.b odd 2 1
294.4.e.o 4 21.h odd 6 1
882.4.a.bc 2 7.c even 3 1
882.4.a.bi 2 7.d odd 6 1
882.4.g.y 4 7.b odd 2 1
882.4.g.y 4 7.d odd 6 1
882.4.g.bd 4 1.a even 1 1 trivial
882.4.g.bd 4 7.c even 3 1 inner
2352.4.a.bn 2 84.j odd 6 1
2352.4.a.cd 2 84.n even 6 1

Hecke kernels

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

\( T_{5}^{4} - 12 T_{5}^{3} + 206 T_{5}^{2} + 744 T_{5} + 3844 \)
\( T_{11}^{4} + 4 T_{11}^{3} + 3540 T_{11}^{2} - 14096 T_{11} + 12418576 \)
\( T_{13}^{2} - 48 T_{13} - 306 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 3844 + 744 T + 206 T^{2} - 12 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 12418576 - 14096 T + 3540 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( ( -306 - 48 T + T^{2} )^{2} \)
$17$ \( 2775556 - 139944 T + 5390 T^{2} - 84 T^{3} + T^{4} \)
$19$ \( 817216 - 65088 T + 4280 T^{2} - 72 T^{3} + T^{4} \)
$23$ \( 407555344 + 6217904 T + 74676 T^{2} + 308 T^{3} + T^{4} \)
$29$ \( ( -30152 - 80 T + T^{2} )^{2} \)
$31$ \( 1111288896 + 12801024 T + 114120 T^{2} + 384 T^{3} + T^{4} \)
$37$ \( 3330674944 + 30933632 T + 229584 T^{2} + 536 T^{3} + T^{4} \)
$41$ \( ( 140434 + 756 T + T^{2} )^{2} \)
$43$ \( ( -16448 - 400 T + T^{2} )^{2} \)
$47$ \( 211295296 - 4535232 T + 82808 T^{2} - 312 T^{3} + T^{4} \)
$53$ \( 3110515984 - 2900144 T + 58476 T^{2} + 52 T^{3} + T^{4} \)
$59$ \( 34682357824 - 160904448 T + 560264 T^{2} - 864 T^{3} + T^{4} \)
$61$ \( 234936028804 + 686338032 T + 1520354 T^{2} + 1416 T^{3} + T^{4} \)
$67$ \( 2627997696 - 7382016 T + 72000 T^{2} + 144 T^{3} + T^{4} \)
$71$ \( ( 492444 - 1524 T + T^{2} )^{2} \)
$73$ \( 288087680644 - 399333072 T + 1090274 T^{2} + 744 T^{3} + T^{4} \)
$79$ \( 205520782336 - 442463744 T + 1405920 T^{2} + 976 T^{3} + T^{4} \)
$83$ \( ( -1788272 - 312 T + T^{2} )^{2} \)
$89$ \( 6320568004 + 8586216 T + 91166 T^{2} - 108 T^{3} + T^{4} \)
$97$ \( ( -1392866 + 744 T + T^{2} )^{2} \)
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