Properties

Label 896.2.b.f
Level $896$
Weight $2$
Character orbit 896.b
Analytic conductor $7.155$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{1} q^{5} - q^{7} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{1} q^{5} - q^{7} + ( -1 - \beta_{3} ) q^{9} + \beta_{2} q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + ( 4 + \beta_{3} ) q^{15} + ( -2 - 2 \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} -\beta_{1} q^{21} + ( -4 - \beta_{3} ) q^{23} + ( 1 - \beta_{3} ) q^{25} -\beta_{2} q^{27} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{29} -2 \beta_{3} q^{31} -2 \beta_{3} q^{33} + \beta_{1} q^{35} -2 \beta_{1} q^{37} + ( -4 + \beta_{3} ) q^{39} + ( 2 - 2 \beta_{3} ) q^{41} -\beta_{2} q^{43} + ( 3 \beta_{1} + \beta_{2} ) q^{45} + 8 q^{47} + q^{49} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{51} + 4 \beta_{1} q^{53} + 2 \beta_{3} q^{55} + ( -4 - 3 \beta_{3} ) q^{57} + ( -\beta_{1} + \beta_{2} ) q^{59} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 1 + \beta_{3} ) q^{63} + ( 4 - \beta_{3} ) q^{65} + ( 2 \beta_{1} - \beta_{2} ) q^{67} + ( -6 \beta_{1} - \beta_{2} ) q^{69} + ( -8 + 2 \beta_{3} ) q^{71} + ( 6 + 2 \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{2} ) q^{75} -\beta_{2} q^{77} + ( 8 - 2 \beta_{3} ) q^{79} + ( -3 - \beta_{3} ) q^{81} -5 \beta_{1} q^{83} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -8 - 6 \beta_{3} ) q^{87} -2 q^{89} + ( -\beta_{1} + \beta_{2} ) q^{91} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 4 + 3 \beta_{3} ) q^{95} -10 q^{97} + ( -4 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{7} - 4q^{9} + 16q^{15} - 8q^{17} - 16q^{23} + 4q^{25} - 16q^{39} + 8q^{41} + 32q^{47} + 4q^{49} - 16q^{57} + 4q^{63} + 16q^{65} - 32q^{71} + 24q^{73} + 32q^{79} - 12q^{81} - 32q^{87} - 8q^{89} + 16q^{95} - 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} - \beta_{1}\)

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
0.765367i
1.84776i
1.84776i
0.765367i
0 2.61313i 0 2.61313i 0 −1.00000 0 −3.82843 0
449.2 0 1.08239i 0 1.08239i 0 −1.00000 0 1.82843 0
449.3 0 1.08239i 0 1.08239i 0 −1.00000 0 1.82843 0
449.4 0 2.61313i 0 2.61313i 0 −1.00000 0 −3.82843 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.f 4
4.b odd 2 1 896.2.b.h yes 4
8.b even 2 1 inner 896.2.b.f 4
8.d odd 2 1 896.2.b.h yes 4
16.e even 4 2 1792.2.a.w 4
16.f odd 4 2 1792.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.f 4 1.a even 1 1 trivial
896.2.b.f 4 8.b even 2 1 inner
896.2.b.h yes 4 4.b odd 2 1
896.2.b.h yes 4 8.d odd 2 1
1792.2.a.u 4 16.f odd 4 2
1792.2.a.w 4 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 \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3}^{4} + 8 T_{3}^{2} + 8 \)
\( T_{23}^{2} + 8 T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 8 + 8 T^{2} + T^{4} \)
$5$ \( 8 + 8 T^{2} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 128 + 32 T^{2} + T^{4} \)
$13$ \( 8 + 40 T^{2} + T^{4} \)
$17$ \( ( -28 + 4 T + T^{2} )^{2} \)
$19$ \( 392 + 40 T^{2} + T^{4} \)
$23$ \( ( 8 + 8 T + T^{2} )^{2} \)
$29$ \( 6272 + 160 T^{2} + T^{4} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( 128 + 32 T^{2} + T^{4} \)
$41$ \( ( -28 - 4 T + T^{2} )^{2} \)
$43$ \( 128 + 32 T^{2} + T^{4} \)
$47$ \( ( -8 + T )^{4} \)
$53$ \( 2048 + 128 T^{2} + T^{4} \)
$59$ \( 8 + 40 T^{2} + T^{4} \)
$61$ \( 7688 + 200 T^{2} + T^{4} \)
$67$ \( 512 + 64 T^{2} + T^{4} \)
$71$ \( ( 32 + 16 T + T^{2} )^{2} \)
$73$ \( ( 4 - 12 T + T^{2} )^{2} \)
$79$ \( ( 32 - 16 T + T^{2} )^{2} \)
$83$ \( 5000 + 200 T^{2} + T^{4} \)
$89$ \( ( 2 + T )^{4} \)
$97$ \( ( 10 + T )^{4} \)
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