Properties

Label 882.3.c.a
Level $882$
Weight $3$
Character orbit 882.c
Analytic conductor $24.033$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 4 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 4 \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} + ( 3 \beta_{1} + 5 \beta_{3} ) q^{10} + ( -12 - 4 \beta_{2} ) q^{11} + ( -4 \beta_{1} + \beta_{3} ) q^{13} + 4 q^{16} + 9 \beta_{3} q^{17} + ( 14 \beta_{1} - 6 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 8 \beta_{3} ) q^{20} + ( -8 - 12 \beta_{2} ) q^{22} + ( -20 + 12 \beta_{2} ) q^{23} + ( -9 - 23 \beta_{2} ) q^{25} + ( 5 \beta_{1} - 3 \beta_{3} ) q^{26} + ( 16 - 13 \beta_{2} ) q^{29} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{31} + 4 \beta_{2} q^{32} + ( 9 \beta_{1} + 9 \beta_{3} ) q^{34} + ( -32 + 3 \beta_{2} ) q^{37} + ( -20 \beta_{1} + 8 \beta_{3} ) q^{38} + ( 6 \beta_{1} + 10 \beta_{3} ) q^{40} + ( \beta_{1} - 16 \beta_{3} ) q^{41} + ( -44 - 8 \beta_{2} ) q^{43} + ( -24 - 8 \beta_{2} ) q^{44} + ( 24 - 20 \beta_{2} ) q^{46} + ( 28 \beta_{1} + 20 \beta_{3} ) q^{47} + ( -46 - 9 \beta_{2} ) q^{50} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{52} + ( 24 - 16 \beta_{2} ) q^{53} + ( -24 \beta_{1} - 68 \beta_{3} ) q^{55} + ( -26 + 16 \beta_{2} ) q^{58} + ( -18 \beta_{1} - 46 \beta_{3} ) q^{59} + ( 52 \beta_{1} + 23 \beta_{3} ) q^{61} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{62} + 8 q^{64} + 7 \beta_{2} q^{65} + ( 64 + 24 \beta_{2} ) q^{67} + 18 \beta_{3} q^{68} + ( -8 - 40 \beta_{2} ) q^{71} + ( -15 \beta_{1} + 56 \beta_{3} ) q^{73} + ( 6 - 32 \beta_{2} ) q^{74} + ( 28 \beta_{1} - 12 \beta_{3} ) q^{76} + ( -48 + 20 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 16 \beta_{3} ) q^{80} + ( -17 \beta_{1} - 15 \beta_{3} ) q^{82} + ( 6 \beta_{1} - 26 \beta_{3} ) q^{83} + ( -72 - 45 \beta_{2} ) q^{85} + ( -16 - 44 \beta_{2} ) q^{86} + ( -16 - 24 \beta_{2} ) q^{88} + ( 32 \beta_{1} - 9 \beta_{3} ) q^{89} + ( -40 + 24 \beta_{2} ) q^{92} + ( -8 \beta_{1} + 48 \beta_{3} ) q^{94} + ( 20 - 12 \beta_{2} ) q^{95} + ( 81 \beta_{1} - 8 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + O(q^{10}) \) \( 4 q + 8 q^{4} - 48 q^{11} + 16 q^{16} - 32 q^{22} - 80 q^{23} - 36 q^{25} + 64 q^{29} - 128 q^{37} - 176 q^{43} - 96 q^{44} + 96 q^{46} - 184 q^{50} + 96 q^{53} - 104 q^{58} + 32 q^{64} + 256 q^{67} - 32 q^{71} + 24 q^{74} - 192 q^{79} - 288 q^{85} - 64 q^{86} - 64 q^{88} - 160 q^{92} + 80 q^{95} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.84776i
1.84776i
0.765367i
0.765367i
−1.41421 0 2.00000 1.21371i 0 0 −2.82843 0 1.71644i
685.2 −1.41421 0 2.00000 1.21371i 0 0 −2.82843 0 1.71644i
685.3 1.41421 0 2.00000 8.15640i 0 0 2.82843 0 11.5349i
685.4 1.41421 0 2.00000 8.15640i 0 0 2.82843 0 11.5349i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.c.a 4
3.b odd 2 1 98.3.b.a 4
7.b odd 2 1 inner 882.3.c.a 4
7.c even 3 2 882.3.n.j 8
7.d odd 6 2 882.3.n.j 8
12.b even 2 1 784.3.c.b 4
21.c even 2 1 98.3.b.a 4
21.g even 6 2 98.3.d.b 8
21.h odd 6 2 98.3.d.b 8
84.h odd 2 1 784.3.c.b 4
84.j odd 6 2 784.3.s.j 8
84.n even 6 2 784.3.s.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.3.b.a 4 3.b odd 2 1
98.3.b.a 4 21.c even 2 1
98.3.d.b 8 21.g even 6 2
98.3.d.b 8 21.h odd 6 2
784.3.c.b 4 12.b even 2 1
784.3.c.b 4 84.h odd 2 1
784.3.s.j 8 84.j odd 6 2
784.3.s.j 8 84.n even 6 2
882.3.c.a 4 1.a even 1 1 trivial
882.3.c.a 4 7.b odd 2 1 inner
882.3.n.j 8 7.c even 3 2
882.3.n.j 8 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 68 T_{5}^{2} + 98 \)
\( T_{23}^{2} + 40 T_{23} + 112 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 98 + 68 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 112 + 24 T + T^{2} )^{2} \)
$13$ \( 98 + 68 T^{2} + T^{4} \)
$17$ \( 13122 + 324 T^{2} + T^{4} \)
$19$ \( 128 + 928 T^{2} + T^{4} \)
$23$ \( ( 112 + 40 T + T^{2} )^{2} \)
$29$ \( ( -82 - 32 T + T^{2} )^{2} \)
$31$ \( 512 + 320 T^{2} + T^{4} \)
$37$ \( ( 1006 + 64 T + T^{2} )^{2} \)
$41$ \( 164738 + 1028 T^{2} + T^{4} \)
$43$ \( ( 1808 + 88 T + T^{2} )^{2} \)
$47$ \( 4524032 + 4736 T^{2} + T^{4} \)
$53$ \( ( 64 - 48 T + T^{2} )^{2} \)
$59$ \( 36992 + 9760 T^{2} + T^{4} \)
$61$ \( 41714978 + 12932 T^{2} + T^{4} \)
$67$ \( ( 2944 - 128 T + T^{2} )^{2} \)
$71$ \( ( -3136 + 16 T + T^{2} )^{2} \)
$73$ \( 42154562 + 13444 T^{2} + T^{4} \)
$79$ \( ( 1504 + 96 T + T^{2} )^{2} \)
$83$ \( 1812608 + 2848 T^{2} + T^{4} \)
$89$ \( 269378 + 4420 T^{2} + T^{4} \)
$97$ \( 54100802 + 26500 T^{2} + T^{4} \)
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