Properties

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

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + ( - 6 \beta_{2} - \beta_1 - 6) q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + ( - 6 \beta_{2} - \beta_1 - 6) q^{5} + 8 q^{8} + (2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{10} + ( - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{11} + ( - 3 \beta_{3} - 24) q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (\beta_{3} + 42 \beta_{2} + \beta_1) q^{17} + ( - 36 \beta_{2} + 2 \beta_1 - 36) q^{19} + ( - 4 \beta_{3} + 24) q^{20} + (12 \beta_{3} + 4) q^{22} + ( - 154 \beta_{2} - 6 \beta_1 - 154) q^{23} + (12 \beta_{3} + 9 \beta_{2} + 12 \beta_1) q^{25} + (48 \beta_{2} - 6 \beta_1 + 48) q^{26} + ( - 18 \beta_{3} + 40) q^{29} + ( - 6 \beta_{3} - 192 \beta_{2} - 6 \beta_1) q^{31} + 32 \beta_{2} q^{32} + ( - 2 \beta_{3} + 84) q^{34} + ( - 268 \beta_{2} - 12 \beta_1 - 268) q^{37} + ( - 4 \beta_{3} + 72 \beta_{2} - 4 \beta_1) q^{38} + ( - 48 \beta_{2} - 8 \beta_1 - 48) q^{40} + (5 \beta_{3} + 378) q^{41} + (24 \beta_{3} + 200) q^{43} + ( - 8 \beta_{2} + 24 \beta_1 - 8) q^{44} + (12 \beta_{3} + 308 \beta_{2} + 12 \beta_1) q^{46} + ( - 156 \beta_{2} - 10 \beta_1 - 156) q^{47} + ( - 24 \beta_{3} + 18) q^{50} + (12 \beta_{3} - 96 \beta_{2} + 12 \beta_1) q^{52} + ( - 24 \beta_{3} + 26 \beta_{2} - 24 \beta_1) q^{53} + (34 \beta_{3} - 576) q^{55} + ( - 80 \beta_{2} - 36 \beta_1 - 80) q^{58} + ( - 2 \beta_{3} + 432 \beta_{2} - 2 \beta_1) q^{59} + (708 \beta_{2} + 13 \beta_1 + 708) q^{61} + (12 \beta_{3} - 384) q^{62} + 64 q^{64} + ( - 150 \beta_{2} + 6 \beta_1 - 150) q^{65} + ( - 24 \beta_{3} + 72 \beta_{2} - 24 \beta_1) q^{67} + ( - 168 \beta_{2} - 4 \beta_1 - 168) q^{68} + (30 \beta_{3} + 762) q^{71} + (83 \beta_{3} - 372 \beta_{2} + 83 \beta_1) q^{73} + (24 \beta_{3} + 536 \beta_{2} + 24 \beta_1) q^{74} + (8 \beta_{3} + 144) q^{76} + ( - 488 \beta_{2} + 84 \beta_1 - 488) q^{79} + (16 \beta_{3} + 96 \beta_{2} + 16 \beta_1) q^{80} + ( - 756 \beta_{2} + 10 \beta_1 - 756) q^{82} + (136 \beta_{3} - 156) q^{83} + ( - 48 \beta_{3} + 350) q^{85} + ( - 400 \beta_{2} + 48 \beta_1 - 400) q^{86} + ( - 48 \beta_{3} + 16 \beta_{2} - 48 \beta_1) q^{88} + ( - 54 \beta_{2} - 29 \beta_1 - 54) q^{89} + ( - 24 \beta_{3} + 616) q^{92} + (20 \beta_{3} + 312 \beta_{2} + 20 \beta_1) q^{94} + (24 \beta_{3} + 20 \beta_{2} + 24 \beta_1) q^{95} + (125 \beta_{3} + 372) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8} - 24 q^{10} - 4 q^{11} - 96 q^{13} - 32 q^{16} - 84 q^{17} - 72 q^{19} + 96 q^{20} + 16 q^{22} - 308 q^{23} - 18 q^{25} + 96 q^{26} + 160 q^{29} + 384 q^{31} - 64 q^{32} + 336 q^{34} - 536 q^{37} - 144 q^{38} - 96 q^{40} + 1512 q^{41} + 800 q^{43} - 16 q^{44} - 616 q^{46} - 312 q^{47} + 72 q^{50} + 192 q^{52} - 52 q^{53} - 2304 q^{55} - 160 q^{58} - 864 q^{59} + 1416 q^{61} - 1536 q^{62} + 256 q^{64} - 300 q^{65} - 144 q^{67} - 336 q^{68} + 3048 q^{71} + 744 q^{73} - 1072 q^{74} + 576 q^{76} - 976 q^{79} - 192 q^{80} - 1512 q^{82} - 624 q^{83} + 1400 q^{85} - 800 q^{86} - 32 q^{88} - 108 q^{89} + 2464 q^{92} - 624 q^{94} - 40 q^{95} + 1488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 7\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} ) / 7 \) 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
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 −7.94975 13.7694i 0 0 8.00000 0 −15.8995 + 27.5387i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 1.94975 + 3.37706i 0 0 8.00000 0 3.89949 6.75412i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −7.94975 + 13.7694i 0 0 8.00000 0 −15.8995 27.5387i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.94975 3.37706i 0 0 8.00000 0 3.89949 + 6.75412i
\(n\): 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.y 4
3.b odd 2 1 294.4.e.n 4
7.b odd 2 1 882.4.g.bd 4
7.c even 3 1 882.4.a.bi 2
7.c even 3 1 inner 882.4.g.y 4
7.d odd 6 1 882.4.a.bc 2
7.d odd 6 1 882.4.g.bd 4
21.c even 2 1 294.4.e.o 4
21.g even 6 1 294.4.a.j 2
21.g even 6 1 294.4.e.o 4
21.h odd 6 1 294.4.a.k yes 2
21.h odd 6 1 294.4.e.n 4
84.j odd 6 1 2352.4.a.cd 2
84.n even 6 1 2352.4.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 21.g even 6 1
294.4.a.k yes 2 21.h odd 6 1
294.4.e.n 4 3.b odd 2 1
294.4.e.n 4 21.h odd 6 1
294.4.e.o 4 21.c even 2 1
294.4.e.o 4 21.g even 6 1
882.4.a.bc 2 7.d odd 6 1
882.4.a.bi 2 7.c even 3 1
882.4.g.y 4 1.a even 1 1 trivial
882.4.g.y 4 7.c even 3 1 inner
882.4.g.bd 4 7.b odd 2 1
882.4.g.bd 4 7.d odd 6 1
2352.4.a.bn 2 84.n even 6 1
2352.4.a.cd 2 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 12T_{5}^{3} + 206T_{5}^{2} - 744T_{5} + 3844 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} + 3540T_{11}^{2} - 14096T_{11} + 12418576 \) Copy content Toggle raw display
\( T_{13}^{2} + 48T_{13} - 306 \) 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
$5$ \( T^{4} + 12 T^{3} + 206 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + 3540 T^{2} + \cdots + 12418576 \) Copy content Toggle raw display
$13$ \( (T^{2} + 48 T - 306)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 84 T^{3} + 5390 T^{2} + \cdots + 2775556 \) Copy content Toggle raw display
$19$ \( T^{4} + 72 T^{3} + 4280 T^{2} + \cdots + 817216 \) Copy content Toggle raw display
$23$ \( T^{4} + 308 T^{3} + \cdots + 407555344 \) Copy content Toggle raw display
$29$ \( (T^{2} - 80 T - 30152)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 384 T^{3} + \cdots + 1111288896 \) Copy content Toggle raw display
$37$ \( T^{4} + 536 T^{3} + \cdots + 3330674944 \) Copy content Toggle raw display
$41$ \( (T^{2} - 756 T + 140434)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 400 T - 16448)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 312 T^{3} + \cdots + 211295296 \) Copy content Toggle raw display
$53$ \( T^{4} + 52 T^{3} + \cdots + 3110515984 \) Copy content Toggle raw display
$59$ \( T^{4} + 864 T^{3} + \cdots + 34682357824 \) Copy content Toggle raw display
$61$ \( T^{4} - 1416 T^{3} + \cdots + 234936028804 \) Copy content Toggle raw display
$67$ \( T^{4} + 144 T^{3} + \cdots + 2627997696 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1524 T + 492444)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 744 T^{3} + \cdots + 288087680644 \) Copy content Toggle raw display
$79$ \( T^{4} + 976 T^{3} + \cdots + 205520782336 \) Copy content Toggle raw display
$83$ \( (T^{2} + 312 T - 1788272)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108 T^{3} + \cdots + 6320568004 \) Copy content Toggle raw display
$97$ \( (T^{2} - 744 T - 1392866)^{2} \) Copy content Toggle raw display
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