Properties

Label 896.2.e.f
Level $896$
Weight $2$
Character orbit 896.e
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{13} -2 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{3} q^{17} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{19} + ( 5 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{21} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} -4 \zeta_{12}^{3} q^{27} + ( 4 - 8 \zeta_{12}^{2} ) q^{29} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{33} + ( 1 + 4 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} -4 \zeta_{12}^{3} q^{37} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{39} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{41} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{43} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{45} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( -4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{51} + ( 4 - 8 \zeta_{12}^{2} ) q^{53} -4 q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 3 - 6 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{59} + ( -3 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{61} + ( -1 + 8 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{63} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( 6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{71} + ( 4 - 8 \zeta_{12}^{2} ) q^{73} + ( -3 + 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{75} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( 1 - 2 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{83} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} + ( -12 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( 4 - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{89} + ( 3 - 4 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( 4 - 8 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{93} + ( -4 + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} -12 \zeta_{12}^{3} q^{97} + ( -14 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} + 8q^{7} - 4q^{9} + 8q^{11} + 12q^{13} + 12q^{21} - 4q^{25} + 8q^{35} - 24q^{43} + 20q^{45} + 4q^{49} - 16q^{51} - 16q^{55} - 12q^{61} - 8q^{63} + 24q^{67} + 32q^{69} + 16q^{77} + 4q^{81} - 48q^{87} + 24q^{91} - 56q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 2.73205i 0 −0.732051 0 2.00000 + 1.73205i 0 −4.46410 0
447.2 0 0.732051i 0 2.73205 0 2.00000 + 1.73205i 0 2.46410 0
447.3 0 0.732051i 0 2.73205 0 2.00000 1.73205i 0 2.46410 0
447.4 0 2.73205i 0 −0.732051 0 2.00000 1.73205i 0 −4.46410 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
56.e 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.e.f yes 4
4.b odd 2 1 896.2.e.e yes 4
7.b odd 2 1 896.2.e.a 4
8.b even 2 1 896.2.e.b yes 4
8.d odd 2 1 896.2.e.a 4
16.e even 4 1 1792.2.f.a 4
16.e even 4 1 1792.2.f.h 4
16.f odd 4 1 1792.2.f.b 4
16.f odd 4 1 1792.2.f.i 4
28.d even 2 1 896.2.e.b yes 4
56.e even 2 1 inner 896.2.e.f yes 4
56.h odd 2 1 896.2.e.e yes 4
112.j even 4 1 1792.2.f.a 4
112.j even 4 1 1792.2.f.h 4
112.l odd 4 1 1792.2.f.b 4
112.l odd 4 1 1792.2.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.a 4 7.b odd 2 1
896.2.e.a 4 8.d odd 2 1
896.2.e.b yes 4 8.b even 2 1
896.2.e.b yes 4 28.d even 2 1
896.2.e.e yes 4 4.b odd 2 1
896.2.e.e yes 4 56.h odd 2 1
896.2.e.f yes 4 1.a even 1 1 trivial
896.2.e.f yes 4 56.e even 2 1 inner
1792.2.f.a 4 16.e even 4 1
1792.2.f.a 4 112.j even 4 1
1792.2.f.b 4 16.f odd 4 1
1792.2.f.b 4 112.l odd 4 1
1792.2.f.h 4 16.e even 4 1
1792.2.f.h 4 112.j even 4 1
1792.2.f.i 4 16.f odd 4 1
1792.2.f.i 4 112.l odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 8 T_{3}^{2} + 4 \)
\( T_{5}^{2} - 2 T_{5} - 2 \)
\( T_{11}^{2} - 4 T_{11} - 8 \)
\( T_{31}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 8 T^{2} + T^{4} \)
$5$ \( ( -2 - 2 T + T^{2} )^{2} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( ( -8 - 4 T + T^{2} )^{2} \)
$13$ \( ( 6 - 6 T + T^{2} )^{2} \)
$17$ \( ( 16 + T^{2} )^{2} \)
$19$ \( 36 + 24 T^{2} + T^{4} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( 48 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( 1024 + 128 T^{2} + T^{4} \)
$43$ \( ( 24 + 12 T + T^{2} )^{2} \)
$47$ \( ( -48 + T^{2} )^{2} \)
$53$ \( ( 48 + T^{2} )^{2} \)
$59$ \( 2916 + 216 T^{2} + T^{4} \)
$61$ \( ( -66 + 6 T + T^{2} )^{2} \)
$67$ \( ( 24 - 12 T + T^{2} )^{2} \)
$71$ \( 2704 + 152 T^{2} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 12 + T^{2} )^{2} \)
$83$ \( 6084 + 168 T^{2} + T^{4} \)
$89$ \( 256 + 224 T^{2} + T^{4} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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