Properties

Label 896.2.b.c
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 2 i q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} + 2 i q^{5} + q^{7} - q^{9} + 2 i q^{13} -4 q^{15} + 6 q^{17} + 6 i q^{19} + 2 i q^{21} -4 q^{23} + q^{25} + 4 i q^{27} -8 q^{31} + 2 i q^{35} -8 i q^{37} -4 q^{39} -6 q^{41} -8 i q^{43} -2 i q^{45} + q^{49} + 12 i q^{51} + 4 i q^{53} -12 q^{57} -6 i q^{59} -2 i q^{61} - q^{63} -4 q^{65} + 4 i q^{67} -8 i q^{69} -16 q^{71} + 6 q^{73} + 2 i q^{75} + 16 q^{79} -11 q^{81} + 14 i q^{83} + 12 i q^{85} + 14 q^{89} + 2 i q^{91} -16 i q^{93} -12 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{9} - 8q^{15} + 12q^{17} - 8q^{23} + 2q^{25} - 16q^{31} - 8q^{39} - 12q^{41} + 2q^{49} - 24q^{57} - 2q^{63} - 8q^{65} - 32q^{71} + 12q^{73} + 32q^{79} - 22q^{81} + 28q^{89} - 24q^{95} - 4q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
0 2.00000i 0 2.00000i 0 1.00000 0 −1.00000 0
449.2 0 2.00000i 0 2.00000i 0 1.00000 0 −1.00000 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.c yes 2
4.b odd 2 1 896.2.b.a 2
8.b even 2 1 inner 896.2.b.c yes 2
8.d odd 2 1 896.2.b.a 2
16.e even 4 1 1792.2.a.c 1
16.e even 4 1 1792.2.a.f 1
16.f odd 4 1 1792.2.a.b 1
16.f odd 4 1 1792.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.a 2 4.b odd 2 1
896.2.b.a 2 8.d odd 2 1
896.2.b.c yes 2 1.a even 1 1 trivial
896.2.b.c yes 2 8.b even 2 1 inner
1792.2.a.b 1 16.f odd 4 1
1792.2.a.c 1 16.e even 4 1
1792.2.a.f 1 16.e even 4 1
1792.2.a.g 1 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 16 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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