Properties

Label 882.3.b.e
Level $882$
Weight $3$
Character orbit 882.b
Analytic conductor $24.033$
Analytic rank $0$
Dimension $2$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 2 \beta q^{8} +O(q^{10})\) \( q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 2 \beta q^{8} + 6 q^{10} + 9 \beta q^{11} + q^{13} + 4 q^{16} -12 \beta q^{17} -23 q^{19} -6 \beta q^{20} + 18 q^{22} + 12 \beta q^{23} + 7 q^{25} -\beta q^{26} -24 \beta q^{29} -47 q^{31} -4 \beta q^{32} -24 q^{34} -55 q^{37} + 23 \beta q^{38} -12 q^{40} -33 \beta q^{41} + 23 q^{43} -18 \beta q^{44} + 24 q^{46} -3 \beta q^{47} -7 \beta q^{50} -2 q^{52} + 36 \beta q^{53} -54 q^{55} -48 q^{58} + 60 \beta q^{59} -104 q^{61} + 47 \beta q^{62} -8 q^{64} + 3 \beta q^{65} -97 q^{67} + 24 \beta q^{68} -69 \beta q^{71} -65 q^{73} + 55 \beta q^{74} + 46 q^{76} + 113 q^{79} + 12 \beta q^{80} -66 q^{82} -21 \beta q^{83} + 72 q^{85} -23 \beta q^{86} -36 q^{88} + 96 \beta q^{89} -24 \beta q^{92} -6 q^{94} -69 \beta q^{95} -104 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} + 12q^{10} + 2q^{13} + 8q^{16} - 46q^{19} + 36q^{22} + 14q^{25} - 94q^{31} - 48q^{34} - 110q^{37} - 24q^{40} + 46q^{43} + 48q^{46} - 4q^{52} - 108q^{55} - 96q^{58} - 208q^{61} - 16q^{64} - 194q^{67} - 130q^{73} + 92q^{76} + 226q^{79} - 132q^{82} + 144q^{85} - 72q^{88} - 12q^{94} - 208q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.41421i
1.41421i
1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 6.00000
197.2 1.41421i 0 −2.00000 4.24264i 0 0 2.82843i 0 6.00000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.b.e 2
3.b odd 2 1 inner 882.3.b.e 2
7.b odd 2 1 882.3.b.b 2
7.c even 3 2 882.3.s.a 4
7.d odd 6 2 126.3.s.a 4
21.c even 2 1 882.3.b.b 2
21.g even 6 2 126.3.s.a 4
21.h odd 6 2 882.3.s.a 4
28.f even 6 2 1008.3.dc.b 4
84.j odd 6 2 1008.3.dc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.a 4 7.d odd 6 2
126.3.s.a 4 21.g even 6 2
882.3.b.b 2 7.b odd 2 1
882.3.b.b 2 21.c even 2 1
882.3.b.e 2 1.a even 1 1 trivial
882.3.b.e 2 3.b odd 2 1 inner
882.3.s.a 4 7.c even 3 2
882.3.s.a 4 21.h odd 6 2
1008.3.dc.b 4 28.f even 6 2
1008.3.dc.b 4 84.j odd 6 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 18 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 162 + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 288 + T^{2} \)
$19$ \( ( 23 + T )^{2} \)
$23$ \( 288 + T^{2} \)
$29$ \( 1152 + T^{2} \)
$31$ \( ( 47 + T )^{2} \)
$37$ \( ( 55 + T )^{2} \)
$41$ \( 2178 + T^{2} \)
$43$ \( ( -23 + T )^{2} \)
$47$ \( 18 + T^{2} \)
$53$ \( 2592 + T^{2} \)
$59$ \( 7200 + T^{2} \)
$61$ \( ( 104 + T )^{2} \)
$67$ \( ( 97 + T )^{2} \)
$71$ \( 9522 + T^{2} \)
$73$ \( ( 65 + T )^{2} \)
$79$ \( ( -113 + T )^{2} \)
$83$ \( 882 + T^{2} \)
$89$ \( 18432 + T^{2} \)
$97$ \( ( 104 + T )^{2} \)
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