Properties

Label 882.3.b.d
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: \( x^{2} + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 2 q^{4} + \beta q^{5} + 2 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 2 q^{4} + \beta q^{5} + 2 \beta q^{8} + 2 q^{10} - 5 \beta q^{11} - 15 q^{13} + 4 q^{16} + 8 \beta q^{17} + 13 q^{19} - 2 \beta q^{20} - 10 q^{22} + 16 \beta q^{23} + 23 q^{25} + 15 \beta q^{26} - 16 \beta q^{29} - 3 q^{31} - 4 \beta q^{32} + 16 q^{34} + 17 q^{37} - 13 \beta q^{38} - 4 q^{40} + 57 \beta q^{41} - 85 q^{43} + 10 \beta q^{44} + 32 q^{46} + 51 \beta q^{47} - 23 \beta q^{50} + 30 q^{52} + 24 \beta q^{53} + 10 q^{55} - 32 q^{58} + 64 \beta q^{59} + 72 q^{61} + 3 \beta q^{62} - 8 q^{64} - 15 \beta q^{65} + 43 q^{67} - 16 \beta q^{68} + 37 \beta q^{71} + 95 q^{73} - 17 \beta q^{74} - 26 q^{76} + 69 q^{79} + 4 \beta q^{80} + 114 q^{82} - 43 \beta q^{83} - 16 q^{85} + 85 \beta q^{86} + 20 q^{88} + 96 \beta q^{89} - 32 \beta q^{92} + 102 q^{94} + 13 \beta q^{95} - 16 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{10} - 30 q^{13} + 8 q^{16} + 26 q^{19} - 20 q^{22} + 46 q^{25} - 6 q^{31} + 32 q^{34} + 34 q^{37} - 8 q^{40} - 170 q^{43} + 64 q^{46} + 60 q^{52} + 20 q^{55} - 64 q^{58} + 144 q^{61} - 16 q^{64} + 86 q^{67} + 190 q^{73} - 52 q^{76} + 138 q^{79} + 228 q^{82} - 32 q^{85} + 40 q^{88} + 204 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 1.41421i 0 0 2.82843i 0 2.00000
197.2 1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
\(n\): 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.d 2
3.b odd 2 1 inner 882.3.b.d 2
7.b odd 2 1 882.3.b.c 2
7.c even 3 2 882.3.s.c 4
7.d odd 6 2 126.3.s.b 4
21.c even 2 1 882.3.b.c 2
21.g even 6 2 126.3.s.b 4
21.h odd 6 2 882.3.s.c 4
28.f even 6 2 1008.3.dc.a 4
84.j odd 6 2 1008.3.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 7.d odd 6 2
126.3.s.b 4 21.g even 6 2
882.3.b.c 2 7.b odd 2 1
882.3.b.c 2 21.c even 2 1
882.3.b.d 2 1.a even 1 1 trivial
882.3.b.d 2 3.b odd 2 1 inner
882.3.s.c 4 7.c even 3 2
882.3.s.c 4 21.h odd 6 2
1008.3.dc.a 4 28.f even 6 2
1008.3.dc.a 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 \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 50 \) Copy content Toggle raw display
\( T_{13} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 50 \) Copy content Toggle raw display
$13$ \( (T + 15)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 128 \) Copy content Toggle raw display
$19$ \( (T - 13)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 512 \) Copy content Toggle raw display
$29$ \( T^{2} + 512 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6498 \) Copy content Toggle raw display
$43$ \( (T + 85)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5202 \) Copy content Toggle raw display
$53$ \( T^{2} + 1152 \) Copy content Toggle raw display
$59$ \( T^{2} + 8192 \) Copy content Toggle raw display
$61$ \( (T - 72)^{2} \) Copy content Toggle raw display
$67$ \( (T - 43)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2738 \) Copy content Toggle raw display
$73$ \( (T - 95)^{2} \) Copy content Toggle raw display
$79$ \( (T - 69)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3698 \) Copy content Toggle raw display
$89$ \( T^{2} + 18432 \) Copy content Toggle raw display
$97$ \( (T + 16)^{2} \) Copy content Toggle raw display
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