Properties

Label 882.3.b.i
Level $882$
Weight $3$
Character orbit 882.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 2 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 2 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{11} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{13} + 4 q^{16} -22 \zeta_{8}^{2} q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} -4 \zeta_{8}^{2} q^{20} + 4 q^{22} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + 21 q^{25} + 18 \zeta_{8}^{2} q^{26} + ( 25 \zeta_{8} + 25 \zeta_{8}^{3} ) q^{29} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{31} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( -22 \zeta_{8} + 22 \zeta_{8}^{3} ) q^{34} + 64 q^{37} -8 \zeta_{8}^{2} q^{38} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{40} + 20 \zeta_{8}^{2} q^{41} + 44 q^{43} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{44} -4 q^{46} + 68 \zeta_{8}^{2} q^{47} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{50} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{52} + ( -13 \zeta_{8} - 13 \zeta_{8}^{3} ) q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{55} + 50 q^{58} -100 \zeta_{8}^{2} q^{59} + ( -37 \zeta_{8} + 37 \zeta_{8}^{3} ) q^{61} -48 \zeta_{8}^{2} q^{62} -8 q^{64} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{65} + 120 q^{67} + 44 \zeta_{8}^{2} q^{68} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} + ( 53 \zeta_{8} - 53 \zeta_{8}^{3} ) q^{73} + ( -64 \zeta_{8} - 64 \zeta_{8}^{3} ) q^{74} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{76} + 92 q^{79} + 8 \zeta_{8}^{2} q^{80} + ( 20 \zeta_{8} - 20 \zeta_{8}^{3} ) q^{82} + 112 \zeta_{8}^{2} q^{83} + 44 q^{85} + ( -44 \zeta_{8} - 44 \zeta_{8}^{3} ) q^{86} -8 q^{88} -20 \zeta_{8}^{2} q^{89} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{92} + ( 68 \zeta_{8} - 68 \zeta_{8}^{3} ) q^{94} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + ( 19 \zeta_{8} - 19 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} + 16q^{22} + 84q^{25} + 256q^{37} + 176q^{43} - 16q^{46} + 200q^{58} - 32q^{64} + 480q^{67} + 368q^{79} + 176q^{85} - 32q^{88} + O(q^{100}) \)

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
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 −2.82843
197.2 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
197.3 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
197.4 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 −2.82843
\(n\): 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.b.i 4
3.b odd 2 1 inner 882.3.b.i 4
7.b odd 2 1 inner 882.3.b.i 4
7.c even 3 2 882.3.s.f 8
7.d odd 6 2 882.3.s.f 8
21.c even 2 1 inner 882.3.b.i 4
21.g even 6 2 882.3.s.f 8
21.h odd 6 2 882.3.s.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.i 4 1.a even 1 1 trivial
882.3.b.i 4 3.b odd 2 1 inner
882.3.b.i 4 7.b odd 2 1 inner
882.3.b.i 4 21.c even 2 1 inner
882.3.s.f 8 7.c even 3 2
882.3.s.f 8 7.d odd 6 2
882.3.s.f 8 21.g even 6 2
882.3.s.f 8 21.h odd 6 2

Hecke kernels

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

\( T_{5}^{2} + 4 \)
\( T_{11}^{2} + 8 \)
\( T_{13}^{2} - 162 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 4 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 8 + T^{2} )^{2} \)
$13$ \( ( -162 + T^{2} )^{2} \)
$17$ \( ( 484 + T^{2} )^{2} \)
$19$ \( ( -32 + T^{2} )^{2} \)
$23$ \( ( 8 + T^{2} )^{2} \)
$29$ \( ( 1250 + T^{2} )^{2} \)
$31$ \( ( -1152 + T^{2} )^{2} \)
$37$ \( ( -64 + T )^{4} \)
$41$ \( ( 400 + T^{2} )^{2} \)
$43$ \( ( -44 + T )^{4} \)
$47$ \( ( 4624 + T^{2} )^{2} \)
$53$ \( ( 338 + T^{2} )^{2} \)
$59$ \( ( 10000 + T^{2} )^{2} \)
$61$ \( ( -2738 + T^{2} )^{2} \)
$67$ \( ( -120 + T )^{4} \)
$71$ \( ( 72 + T^{2} )^{2} \)
$73$ \( ( -5618 + T^{2} )^{2} \)
$79$ \( ( -92 + T )^{4} \)
$83$ \( ( 12544 + T^{2} )^{2} \)
$89$ \( ( 400 + T^{2} )^{2} \)
$97$ \( ( -722 + T^{2} )^{2} \)
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