Properties

Label 1792.2.b.n
Level $1792$
Weight $2$
Character orbit 1792.b
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 896)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{13} + 2 q^{15} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -3 + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{19} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{21} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{23} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{25} -4 \zeta_{12}^{3} q^{27} + ( 2 - 4 \zeta_{12}^{2} ) q^{29} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{33} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( 2 - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{37} -2 q^{39} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{45} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} + q^{49} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} + 6 \zeta_{12}^{3} q^{53} -4 q^{55} + ( 8 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 3 - 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{59} + ( -7 + 14 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{61} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( -4 + 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{67} + ( 2 - 4 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{69} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{71} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + ( 3 - 6 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{75} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{77} + 8 q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( -1 + 2 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{83} + ( -6 + 12 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{85} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{87} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( 8 - 16 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{93} + ( -10 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{95} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{97} + ( 6 - 12 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{7} - 4q^{9} + 8q^{15} + 16q^{17} - 8q^{23} + 4q^{25} + 24q^{31} + 32q^{33} - 8q^{39} + 8q^{47} + 4q^{49} - 16q^{55} + 32q^{57} - 4q^{63} + 16q^{65} - 8q^{73} + 32q^{79} + 4q^{81} - 24q^{87} + 8q^{89} - 40q^{95} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 2.73205i 0 0.732051i 0 1.00000 0 −4.46410 0
897.2 0 0.732051i 0 2.73205i 0 1.00000 0 2.46410 0
897.3 0 0.732051i 0 2.73205i 0 1.00000 0 2.46410 0
897.4 0 2.73205i 0 0.732051i 0 1.00000 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
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 1792.2.b.n 4
4.b odd 2 1 1792.2.b.l 4
8.b even 2 1 inner 1792.2.b.n 4
8.d odd 2 1 1792.2.b.l 4
16.e even 4 1 896.2.a.f yes 2
16.e even 4 1 896.2.a.g yes 2
16.f odd 4 1 896.2.a.e 2
16.f odd 4 1 896.2.a.h yes 2
48.i odd 4 1 8064.2.a.be 2
48.i odd 4 1 8064.2.a.bm 2
48.k even 4 1 8064.2.a.bf 2
48.k even 4 1 8064.2.a.br 2
112.j even 4 1 6272.2.a.i 2
112.j even 4 1 6272.2.a.t 2
112.l odd 4 1 6272.2.a.j 2
112.l odd 4 1 6272.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.a.e 2 16.f odd 4 1
896.2.a.f yes 2 16.e even 4 1
896.2.a.g yes 2 16.e even 4 1
896.2.a.h yes 2 16.f odd 4 1
1792.2.b.l 4 4.b odd 2 1
1792.2.b.l 4 8.d odd 2 1
1792.2.b.n 4 1.a even 1 1 trivial
1792.2.b.n 4 8.b even 2 1 inner
6272.2.a.i 2 112.j even 4 1
6272.2.a.j 2 112.l odd 4 1
6272.2.a.s 2 112.l odd 4 1
6272.2.a.t 2 112.j even 4 1
8064.2.a.be 2 48.i odd 4 1
8064.2.a.bf 2 48.k even 4 1
8064.2.a.bm 2 48.i odd 4 1
8064.2.a.br 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{4} + 8 T_{3}^{2} + 4 \)
\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{11}^{4} + 32 T_{11}^{2} + 64 \)
\( T_{23}^{2} + 4 T_{23} - 44 \)
\( T_{31}^{2} - 12 T_{31} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 8 T^{2} + T^{4} \)
$5$ \( 4 + 8 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 64 + 32 T^{2} + T^{4} \)
$13$ \( 4 + 8 T^{2} + T^{4} \)
$17$ \( ( 4 - 8 T + T^{2} )^{2} \)
$19$ \( 676 + 56 T^{2} + T^{4} \)
$23$ \( ( -44 + 4 T + T^{2} )^{2} \)
$29$ \( ( 12 + T^{2} )^{2} \)
$31$ \( ( 24 - 12 T + T^{2} )^{2} \)
$37$ \( 2704 + 152 T^{2} + T^{4} \)
$41$ \( ( -12 + T^{2} )^{2} \)
$43$ \( 576 + 96 T^{2} + T^{4} \)
$47$ \( ( -8 - 4 T + T^{2} )^{2} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( 676 + 56 T^{2} + T^{4} \)
$61$ \( 21316 + 296 T^{2} + T^{4} \)
$67$ \( 256 + 224 T^{2} + T^{4} \)
$71$ \( ( -192 + T^{2} )^{2} \)
$73$ \( ( -44 + 4 T + T^{2} )^{2} \)
$79$ \( ( -8 + T )^{4} \)
$83$ \( 6084 + 168 T^{2} + T^{4} \)
$89$ \( ( -44 - 4 T + T^{2} )^{2} \)
$97$ \( ( 4 - 8 T + T^{2} )^{2} \)
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