Properties

Label 882.3.n.c
Level $882$
Weight $3$
Character orbit 882.n
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} ) q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} ) q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{3} q^{8} + ( 4 - 4 \beta_{2} ) q^{10} -12 \beta_{1} q^{11} + ( -1 - 2 \beta_{2} ) q^{13} + 4 \beta_{2} q^{16} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{17} + ( -34 - 17 \beta_{2} ) q^{19} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{20} -24 q^{22} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( -\beta_{1} + \beta_{3} ) q^{26} -24 \beta_{3} q^{29} + ( 7 - 7 \beta_{2} ) q^{31} + 4 \beta_{1} q^{32} + ( 4 + 8 \beta_{2} ) q^{34} -47 \beta_{2} q^{37} + ( 17 \beta_{1} + 34 \beta_{3} ) q^{38} + ( -16 - 8 \beta_{2} ) q^{40} + ( -56 \beta_{1} - 28 \beta_{3} ) q^{41} + 31 q^{43} + ( 24 \beta_{1} + 24 \beta_{3} ) q^{44} + ( 12 + 12 \beta_{2} ) q^{46} + ( -34 \beta_{1} + 34 \beta_{3} ) q^{47} + \beta_{3} q^{50} + ( -2 + 2 \beta_{2} ) q^{52} -54 \beta_{1} q^{53} + ( -48 - 96 \beta_{2} ) q^{55} -48 \beta_{2} q^{58} + ( 34 \beta_{1} + 68 \beta_{3} ) q^{59} + ( 96 + 48 \beta_{2} ) q^{61} + ( -14 \beta_{1} - 7 \beta_{3} ) q^{62} + 8 q^{64} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{65} + ( 31 + 31 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{68} + 42 \beta_{3} q^{71} + ( 47 - 47 \beta_{2} ) q^{73} -47 \beta_{1} q^{74} + ( 34 + 68 \beta_{2} ) q^{76} + 41 \beta_{2} q^{79} + ( 8 \beta_{1} + 16 \beta_{3} ) q^{80} + ( -112 - 56 \beta_{2} ) q^{82} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{83} -24 q^{85} + ( -31 \beta_{1} - 31 \beta_{3} ) q^{86} + ( 48 + 48 \beta_{2} ) q^{88} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{89} -12 \beta_{3} q^{92} + ( -68 + 68 \beta_{2} ) q^{94} -102 \beta_{1} q^{95} + ( 24 + 48 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 24q^{10} - 8q^{16} - 102q^{19} - 96q^{22} - 2q^{25} + 42q^{31} + 94q^{37} - 48q^{40} + 124q^{43} + 24q^{46} - 12q^{52} + 96q^{58} + 288q^{61} + 32q^{64} + 62q^{67} + 282q^{73} - 82q^{79} - 336q^{82} - 96q^{85} + 96q^{88} - 408q^{94} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −4.24264 2.44949i 0 0 2.82843 0 6.00000 3.46410i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i 4.24264 + 2.44949i 0 0 −2.82843 0 6.00000 3.46410i
325.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.24264 + 2.44949i 0 0 2.82843 0 6.00000 + 3.46410i
325.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 4.24264 2.44949i 0 0 −2.82843 0 6.00000 + 3.46410i
\(n\): 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

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

\( T_{5}^{4} - 24 T_{5}^{2} + 576 \)
\( T_{23}^{4} + 72 T_{23}^{2} + 5184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 576 - 24 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 82944 + 288 T^{2} + T^{4} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( 576 - 24 T^{2} + T^{4} \)
$19$ \( ( 867 + 51 T + T^{2} )^{2} \)
$23$ \( 5184 + 72 T^{2} + T^{4} \)
$29$ \( ( -1152 + T^{2} )^{2} \)
$31$ \( ( 147 - 21 T + T^{2} )^{2} \)
$37$ \( ( 2209 - 47 T + T^{2} )^{2} \)
$41$ \( ( 4704 + T^{2} )^{2} \)
$43$ \( ( -31 + T )^{4} \)
$47$ \( 48108096 - 6936 T^{2} + T^{4} \)
$53$ \( 34012224 + 5832 T^{2} + T^{4} \)
$59$ \( 48108096 - 6936 T^{2} + T^{4} \)
$61$ \( ( 6912 - 144 T + T^{2} )^{2} \)
$67$ \( ( 961 - 31 T + T^{2} )^{2} \)
$71$ \( ( -3528 + T^{2} )^{2} \)
$73$ \( ( 6627 - 141 T + T^{2} )^{2} \)
$79$ \( ( 1681 + 41 T + T^{2} )^{2} \)
$83$ \( ( 24 + T^{2} )^{2} \)
$89$ \( 11943936 - 3456 T^{2} + T^{4} \)
$97$ \( ( 1728 + T^{2} )^{2} \)
show more
show less