Properties

Label 896.2.b.b
Level $896$
Weight $2$
Character orbit 896.b
Analytic conductor $7.155$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta q^{5} - q^{7} + 3 q^{9} -\beta q^{11} + \beta q^{13} -2 q^{17} + 2 \beta q^{19} + 8 q^{23} -3 q^{25} + 2 \beta q^{29} -8 q^{31} -\beta q^{35} + 2 \beta q^{37} -6 q^{41} -\beta q^{43} + 3 \beta q^{45} -8 q^{47} + q^{49} + 4 \beta q^{53} + 8 q^{55} + 4 \beta q^{59} -\beta q^{61} -3 q^{63} -8 q^{65} -3 \beta q^{67} + 8 q^{71} + 6 q^{73} + \beta q^{77} + 8 q^{79} + 9 q^{81} -2 \beta q^{83} -2 \beta q^{85} + 6 q^{89} -\beta q^{91} -16 q^{95} + 14 q^{97} -3 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{7} + 6q^{9} - 4q^{17} + 16q^{23} - 6q^{25} - 16q^{31} - 12q^{41} - 16q^{47} + 2q^{49} + 16q^{55} - 6q^{63} - 16q^{65} + 16q^{71} + 12q^{73} + 16q^{79} + 18q^{81} + 12q^{89} - 32q^{95} + 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.41421i
1.41421i
0 0 0 2.82843i 0 −1.00000 0 3.00000 0
449.2 0 0 0 2.82843i 0 −1.00000 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.b.b 2
4.b odd 2 1 896.2.b.d yes 2
8.b even 2 1 inner 896.2.b.b 2
8.d odd 2 1 896.2.b.d yes 2
16.e even 4 2 1792.2.a.o 2
16.f odd 4 2 1792.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.b 2 1.a even 1 1 trivial
896.2.b.b 2 8.b even 2 1 inner
896.2.b.d yes 2 4.b odd 2 1
896.2.b.d yes 2 8.d odd 2 1
1792.2.a.m 2 16.f odd 4 2
1792.2.a.o 2 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3} \)
\( T_{23} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 8 + T^{2} \)
$13$ \( 8 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 32 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( 32 + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 32 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 8 + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 128 + T^{2} \)
$59$ \( 128 + T^{2} \)
$61$ \( 8 + T^{2} \)
$67$ \( 72 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 32 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( -14 + T )^{2} \)
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