Properties

Label 882.3.c.e
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{2} q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{10} -2 \beta_{2} q^{11} + ( -5 \beta_{1} - 8 \beta_{3} ) q^{13} + 4 q^{16} + ( -6 \beta_{1} + 9 \beta_{3} ) q^{17} + ( -6 \beta_{1} + 10 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{20} + 4 q^{22} + ( 28 + 6 \beta_{2} ) q^{23} + ( 15 - 7 \beta_{2} ) q^{25} + ( 3 \beta_{1} + 13 \beta_{3} ) q^{26} + ( 28 - 11 \beta_{2} ) q^{29} + ( 14 \beta_{1} - 14 \beta_{3} ) q^{31} -4 \beta_{2} q^{32} + ( -15 \beta_{1} - 3 \beta_{3} ) q^{34} + ( 16 + 21 \beta_{2} ) q^{37} + ( -16 \beta_{1} - 4 \beta_{3} ) q^{38} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{40} + ( 35 \beta_{1} + 14 \beta_{3} ) q^{41} + ( -32 - 28 \beta_{2} ) q^{43} -4 \beta_{2} q^{44} + ( -12 - 28 \beta_{2} ) q^{46} + ( -10 \beta_{1} + 50 \beta_{3} ) q^{47} + ( 14 - 15 \beta_{2} ) q^{50} + ( -10 \beta_{1} - 16 \beta_{3} ) q^{52} + ( -42 - 44 \beta_{2} ) q^{53} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{55} + ( 22 - 28 \beta_{2} ) q^{58} + ( -46 \beta_{1} + 6 \beta_{3} ) q^{59} + ( -37 \beta_{1} + 8 \beta_{3} ) q^{61} + 28 \beta_{1} q^{62} + 8 q^{64} + ( 42 + 29 \beta_{2} ) q^{65} + 4 q^{67} + ( -12 \beta_{1} + 18 \beta_{3} ) q^{68} + ( 112 - 14 \beta_{2} ) q^{71} + ( 32 \beta_{1} + 33 \beta_{3} ) q^{73} + ( -42 - 16 \beta_{2} ) q^{74} + ( -12 \beta_{1} + 20 \beta_{3} ) q^{76} + ( -36 + 28 \beta_{2} ) q^{79} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{80} + ( 21 \beta_{1} - 49 \beta_{3} ) q^{82} + ( 76 \beta_{1} + 12 \beta_{3} ) q^{83} + ( -24 - 21 \beta_{2} ) q^{85} + ( 56 + 32 \beta_{2} ) q^{86} + 8 q^{88} + ( -30 \beta_{1} + 73 \beta_{3} ) q^{89} + ( 56 + 12 \beta_{2} ) q^{92} + ( -60 \beta_{1} - 40 \beta_{3} ) q^{94} + ( -28 - 24 \beta_{2} ) q^{95} + ( 64 \beta_{1} - 39 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} + 16q^{16} + 16q^{22} + 112q^{23} + 60q^{25} + 112q^{29} + 64q^{37} - 128q^{43} - 48q^{46} + 56q^{50} - 168q^{53} + 88q^{58} + 32q^{64} + 168q^{65} + 16q^{67} + 448q^{71} - 168q^{74} - 144q^{79} - 96q^{85} + 224q^{86} + 32q^{88} + 224q^{92} - 112q^{95} + O(q^{100}) \)

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
0.765367i
0.765367i
1.84776i
1.84776i
−1.41421 0 2.00000 4.46088i 0 0 −2.82843 0 6.30864i
685.2 −1.41421 0 2.00000 4.46088i 0 0 −2.82843 0 6.30864i
685.3 1.41421 0 2.00000 0.317025i 0 0 2.82843 0 0.448342i
685.4 1.41421 0 2.00000 0.317025i 0 0 2.82843 0 0.448342i
\(n\): 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.e yes 4
3.b odd 2 1 882.3.c.d 4
7.b odd 2 1 inner 882.3.c.e yes 4
7.c even 3 2 882.3.n.g 8
7.d odd 6 2 882.3.n.g 8
21.c even 2 1 882.3.c.d 4
21.g even 6 2 882.3.n.h 8
21.h odd 6 2 882.3.n.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.c.d 4 3.b odd 2 1
882.3.c.d 4 21.c even 2 1
882.3.c.e yes 4 1.a even 1 1 trivial
882.3.c.e yes 4 7.b odd 2 1 inner
882.3.n.g 8 7.c even 3 2
882.3.n.g 8 7.d odd 6 2
882.3.n.h 8 21.g even 6 2
882.3.n.h 8 21.h odd 6 2

Hecke kernels

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

\( T_{5}^{4} + 20 T_{5}^{2} + 2 \)
\( T_{23}^{2} - 56 T_{23} + 712 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 2 + 20 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -8 + T^{2} )^{2} \)
$13$ \( 3362 + 356 T^{2} + T^{4} \)
$17$ \( 46818 + 468 T^{2} + T^{4} \)
$19$ \( 67712 + 544 T^{2} + T^{4} \)
$23$ \( ( 712 - 56 T + T^{2} )^{2} \)
$29$ \( ( 542 - 56 T + T^{2} )^{2} \)
$31$ \( 307328 + 1568 T^{2} + T^{4} \)
$37$ \( ( -626 - 32 T + T^{2} )^{2} \)
$41$ \( 8072162 + 5684 T^{2} + T^{4} \)
$43$ \( ( -544 + 64 T + T^{2} )^{2} \)
$47$ \( 23120000 + 10400 T^{2} + T^{4} \)
$53$ \( ( -2108 + 84 T + T^{2} )^{2} \)
$59$ \( 4669568 + 8608 T^{2} + T^{4} \)
$61$ \( 1016738 + 5732 T^{2} + T^{4} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( ( 12152 - 224 T + T^{2} )^{2} \)
$73$ \( 8380418 + 8452 T^{2} + T^{4} \)
$79$ \( ( -272 + 72 T + T^{2} )^{2} \)
$83$ \( 111183872 + 23680 T^{2} + T^{4} \)
$89$ \( 155196962 + 24916 T^{2} + T^{4} \)
$97$ \( 11683778 + 22468 T^{2} + T^{4} \)
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