Properties

Label 882.3.c.c
Level $882$
Weight $3$
Character orbit 882.c
Analytic conductor $24.033$
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: \( k \) \(=\) \( 3 \)
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 2 q^{4} + 2 \beta_{3} q^{5} -2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + 2 q^{4} + 2 \beta_{3} q^{5} -2 \beta_{1} q^{8} + 4 \beta_{2} q^{10} + 12 \beta_{1} q^{11} -\beta_{2} q^{13} + 4 q^{16} + 2 \beta_{3} q^{17} + 17 \beta_{2} q^{19} + 4 \beta_{3} q^{20} -24 q^{22} + 6 \beta_{1} q^{23} + q^{25} -\beta_{3} q^{26} + 24 \beta_{1} q^{29} + 7 \beta_{2} q^{31} -4 \beta_{1} q^{32} + 4 \beta_{2} q^{34} -47 q^{37} + 17 \beta_{3} q^{38} + 8 \beta_{2} q^{40} + 28 \beta_{3} q^{41} + 31 q^{43} + 24 \beta_{1} q^{44} -12 q^{46} -34 \beta_{3} q^{47} -\beta_{1} q^{50} -2 \beta_{2} q^{52} + 54 \beta_{1} q^{53} -48 \beta_{2} q^{55} -48 q^{58} + 34 \beta_{3} q^{59} -48 \beta_{2} q^{61} + 7 \beta_{3} q^{62} + 8 q^{64} -6 \beta_{1} q^{65} -31 q^{67} + 4 \beta_{3} q^{68} -42 \beta_{1} q^{71} + 47 \beta_{2} q^{73} + 47 \beta_{1} q^{74} + 34 \beta_{2} q^{76} + 41 q^{79} + 8 \beta_{3} q^{80} + 56 \beta_{2} q^{82} -2 \beta_{3} q^{83} -24 q^{85} -31 \beta_{1} q^{86} -48 q^{88} + 24 \beta_{3} q^{89} + 12 \beta_{1} q^{92} -68 \beta_{2} q^{94} + 102 \beta_{1} q^{95} + 24 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + O(q^{10}) \) \( 4 q + 8 q^{4} + 16 q^{16} - 96 q^{22} + 4 q^{25} - 188 q^{37} + 124 q^{43} - 48 q^{46} - 192 q^{58} + 32 q^{64} - 124 q^{67} + 164 q^{79} - 96 q^{85} - 192 q^{88} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

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
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.707107 + 1.22474i
−1.41421 0 2.00000 4.89898i 0 0 −2.82843 0 6.92820i
685.2 −1.41421 0 2.00000 4.89898i 0 0 −2.82843 0 6.92820i
685.3 1.41421 0 2.00000 4.89898i 0 0 2.82843 0 6.92820i
685.4 1.41421 0 2.00000 4.89898i 0 0 2.82843 0 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 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.c 4
3.b odd 2 1 inner 882.3.c.c 4
7.b odd 2 1 inner 882.3.c.c 4
7.c even 3 1 126.3.n.b 4
7.c even 3 1 882.3.n.c 4
7.d odd 6 1 126.3.n.b 4
7.d odd 6 1 882.3.n.c 4
21.c even 2 1 inner 882.3.c.c 4
21.g even 6 1 126.3.n.b 4
21.g even 6 1 882.3.n.c 4
21.h odd 6 1 126.3.n.b 4
21.h odd 6 1 882.3.n.c 4
28.f even 6 1 1008.3.cg.i 4
28.g odd 6 1 1008.3.cg.i 4
84.j odd 6 1 1008.3.cg.i 4
84.n even 6 1 1008.3.cg.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.n.b 4 7.c even 3 1
126.3.n.b 4 7.d odd 6 1
126.3.n.b 4 21.g even 6 1
126.3.n.b 4 21.h odd 6 1
882.3.c.c 4 1.a even 1 1 trivial
882.3.c.c 4 3.b odd 2 1 inner
882.3.c.c 4 7.b odd 2 1 inner
882.3.c.c 4 21.c even 2 1 inner
882.3.n.c 4 7.c even 3 1
882.3.n.c 4 7.d odd 6 1
882.3.n.c 4 21.g even 6 1
882.3.n.c 4 21.h odd 6 1
1008.3.cg.i 4 28.f even 6 1
1008.3.cg.i 4 28.g odd 6 1
1008.3.cg.i 4 84.j odd 6 1
1008.3.cg.i 4 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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 24 \)
\( T_{23}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 24 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -288 + T^{2} )^{2} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( 867 + T^{2} )^{2} \)
$23$ \( ( -72 + T^{2} )^{2} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( ( 147 + T^{2} )^{2} \)
$37$ \( ( 47 + T )^{4} \)
$41$ \( ( 4704 + T^{2} )^{2} \)
$43$ \( ( -31 + T )^{4} \)
$47$ \( ( 6936 + T^{2} )^{2} \)
$53$ \( ( -5832 + T^{2} )^{2} \)
$59$ \( ( 6936 + T^{2} )^{2} \)
$61$ \( ( 6912 + T^{2} )^{2} \)
$67$ \( ( 31 + T )^{4} \)
$71$ \( ( -3528 + T^{2} )^{2} \)
$73$ \( ( 6627 + T^{2} )^{2} \)
$79$ \( ( -41 + T )^{4} \)
$83$ \( ( 24 + T^{2} )^{2} \)
$89$ \( ( 3456 + T^{2} )^{2} \)
$97$ \( ( 1728 + T^{2} )^{2} \)
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