Properties

Label 882.3.s.a
Level $882$
Weight $3$
Character orbit 882.s
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.s (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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} -2 \beta_{3} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + 3 \beta_{1} q^{5} -2 \beta_{3} q^{8} -6 \beta_{2} q^{10} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{11} + q^{13} + ( -4 + 4 \beta_{2} ) q^{16} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{17} + ( 23 - 23 \beta_{2} ) q^{19} + 6 \beta_{3} q^{20} + 18 q^{22} + 12 \beta_{1} q^{23} -7 \beta_{2} q^{25} -\beta_{1} q^{26} + 24 \beta_{3} q^{29} + 47 \beta_{2} q^{31} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{32} -24 q^{34} + ( 55 - 55 \beta_{2} ) q^{37} + ( -23 \beta_{1} + 23 \beta_{3} ) q^{38} + ( 12 - 12 \beta_{2} ) q^{40} + 33 \beta_{3} q^{41} + 23 q^{43} -18 \beta_{1} q^{44} -24 \beta_{2} q^{46} -3 \beta_{1} q^{47} + 7 \beta_{3} q^{50} + 2 \beta_{2} q^{52} + ( -36 \beta_{1} + 36 \beta_{3} ) q^{53} -54 q^{55} + ( 48 - 48 \beta_{2} ) q^{58} + ( -60 \beta_{1} + 60 \beta_{3} ) q^{59} + ( 104 - 104 \beta_{2} ) q^{61} -47 \beta_{3} q^{62} -8 q^{64} + 3 \beta_{1} q^{65} + 97 \beta_{2} q^{67} + 24 \beta_{1} q^{68} + 69 \beta_{3} q^{71} + 65 \beta_{2} q^{73} + ( -55 \beta_{1} + 55 \beta_{3} ) q^{74} + 46 q^{76} + ( -113 + 113 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{80} + ( 66 - 66 \beta_{2} ) q^{82} + 21 \beta_{3} q^{83} + 72 q^{85} -23 \beta_{1} q^{86} + 36 \beta_{2} q^{88} + 96 \beta_{1} q^{89} + 24 \beta_{3} q^{92} + 6 \beta_{2} q^{94} + ( 69 \beta_{1} - 69 \beta_{3} ) q^{95} -104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 12q^{10} + 4q^{13} - 8q^{16} + 46q^{19} + 72q^{22} - 14q^{25} + 94q^{31} - 96q^{34} + 110q^{37} + 24q^{40} + 92q^{43} - 48q^{46} + 4q^{52} - 216q^{55} + 96q^{58} + 208q^{61} - 32q^{64} + 194q^{67} + 130q^{73} + 184q^{76} - 226q^{79} + 132q^{82} + 288q^{85} + 72q^{88} + 12q^{94} - 416q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 3.67423 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
557.2 1.22474 0.707107i 0 1.00000 1.73205i −3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 + 5.19615i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.67423 + 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
863.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.67423 2.12132i 0 0 2.82843i 0 −3.00000 5.19615i
\(n\): 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

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

\( T_{5}^{4} - 18 T_{5}^{2} + 324 \)
\( T_{11}^{4} - 162 T_{11}^{2} + 26244 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 - 18 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 26244 - 162 T^{2} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 82944 - 288 T^{2} + T^{4} \)
$19$ \( ( 529 - 23 T + T^{2} )^{2} \)
$23$ \( 82944 - 288 T^{2} + T^{4} \)
$29$ \( ( 1152 + T^{2} )^{2} \)
$31$ \( ( 2209 - 47 T + T^{2} )^{2} \)
$37$ \( ( 3025 - 55 T + T^{2} )^{2} \)
$41$ \( ( 2178 + T^{2} )^{2} \)
$43$ \( ( -23 + T )^{4} \)
$47$ \( 324 - 18 T^{2} + T^{4} \)
$53$ \( 6718464 - 2592 T^{2} + T^{4} \)
$59$ \( 51840000 - 7200 T^{2} + T^{4} \)
$61$ \( ( 10816 - 104 T + T^{2} )^{2} \)
$67$ \( ( 9409 - 97 T + T^{2} )^{2} \)
$71$ \( ( 9522 + T^{2} )^{2} \)
$73$ \( ( 4225 - 65 T + T^{2} )^{2} \)
$79$ \( ( 12769 + 113 T + T^{2} )^{2} \)
$83$ \( ( 882 + T^{2} )^{2} \)
$89$ \( 339738624 - 18432 T^{2} + T^{4} \)
$97$ \( ( 104 + T )^{4} \)
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