Properties

Label 1792.2.f.j
Level $1792$
Weight $2$
Character orbit 1792.f
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 448)
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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} - q^{9} +O(q^{10})\) \( q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} - q^{9} + ( -2 + 4 \zeta_{24}^{4} ) q^{11} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{13} -2 \zeta_{24}^{6} q^{15} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{19} + ( \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{21} + 3 q^{25} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{27} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{33} + ( -2 - \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{35} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{37} + 6 \zeta_{24}^{6} q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( -2 + 4 \zeta_{24}^{4} ) q^{43} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{45} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( -4 + 8 \zeta_{24}^{4} ) q^{51} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{55} + 6 q^{57} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{59} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{61} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} + 6 q^{65} + ( 6 - 12 \zeta_{24}^{4} ) q^{67} -6 \zeta_{24}^{6} q^{71} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{75} + ( 6 \zeta_{24} + 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{77} + 14 \zeta_{24}^{6} q^{79} -5 q^{81} + ( -7 \zeta_{24} - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} ) q^{83} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{85} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{89} + ( 6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 12 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{91} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{93} -6 \zeta_{24}^{6} q^{95} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( 2 - 4 \zeta_{24}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} + 24q^{25} + 40q^{49} + 48q^{57} + 48q^{65} - 40q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
0 −1.41421 0 1.41421i 0 −2.44949 + 1.00000i 0 −1.00000 0
1791.2 0 −1.41421 0 1.41421i 0 2.44949 + 1.00000i 0 −1.00000 0
1791.3 0 −1.41421 0 1.41421i 0 −2.44949 1.00000i 0 −1.00000 0
1791.4 0 −1.41421 0 1.41421i 0 2.44949 1.00000i 0 −1.00000 0
1791.5 0 1.41421 0 1.41421i 0 −2.44949 1.00000i 0 −1.00000 0
1791.6 0 1.41421 0 1.41421i 0 2.44949 1.00000i 0 −1.00000 0
1791.7 0 1.41421 0 1.41421i 0 −2.44949 + 1.00000i 0 −1.00000 0
1791.8 0 1.41421 0 1.41421i 0 2.44949 + 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1791.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.f.j 8
4.b odd 2 1 inner 1792.2.f.j 8
7.b odd 2 1 inner 1792.2.f.j 8
8.b even 2 1 inner 1792.2.f.j 8
8.d odd 2 1 inner 1792.2.f.j 8
16.e even 4 2 448.2.e.b 8
16.f odd 4 2 448.2.e.b 8
28.d even 2 1 inner 1792.2.f.j 8
48.i odd 4 2 4032.2.p.f 8
48.k even 4 2 4032.2.p.f 8
56.e even 2 1 inner 1792.2.f.j 8
56.h odd 2 1 inner 1792.2.f.j 8
112.j even 4 2 448.2.e.b 8
112.l odd 4 2 448.2.e.b 8
336.v odd 4 2 4032.2.p.f 8
336.y even 4 2 4032.2.p.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.b 8 16.e even 4 2
448.2.e.b 8 16.f odd 4 2
448.2.e.b 8 112.j even 4 2
448.2.e.b 8 112.l odd 4 2
1792.2.f.j 8 1.a even 1 1 trivial
1792.2.f.j 8 4.b odd 2 1 inner
1792.2.f.j 8 7.b odd 2 1 inner
1792.2.f.j 8 8.b even 2 1 inner
1792.2.f.j 8 8.d odd 2 1 inner
1792.2.f.j 8 28.d even 2 1 inner
1792.2.f.j 8 56.e even 2 1 inner
1792.2.f.j 8 56.h odd 2 1 inner
4032.2.p.f 8 48.i odd 4 2
4032.2.p.f 8 48.k even 4 2
4032.2.p.f 8 336.v odd 4 2
4032.2.p.f 8 336.y even 4 2

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} + 2 \)
\( T_{29} \)
\( T_{31}^{2} - 24 \)
\( T_{37}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -2 + T^{2} )^{4} \)
$5$ \( ( 2 + T^{2} )^{4} \)
$7$ \( ( 49 - 10 T^{2} + T^{4} )^{2} \)
$11$ \( ( 12 + T^{2} )^{4} \)
$13$ \( ( 18 + T^{2} )^{4} \)
$17$ \( ( 24 + T^{2} )^{4} \)
$19$ \( ( -18 + T^{2} )^{4} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( ( -24 + T^{2} )^{4} \)
$37$ \( ( -48 + T^{2} )^{4} \)
$41$ \( ( 96 + T^{2} )^{4} \)
$43$ \( ( 12 + T^{2} )^{4} \)
$47$ \( ( -24 + T^{2} )^{4} \)
$53$ \( ( -192 + T^{2} )^{4} \)
$59$ \( ( -50 + T^{2} )^{4} \)
$61$ \( ( 18 + T^{2} )^{4} \)
$67$ \( ( 108 + T^{2} )^{4} \)
$71$ \( ( 36 + T^{2} )^{4} \)
$73$ \( ( 24 + T^{2} )^{4} \)
$79$ \( ( 196 + T^{2} )^{4} \)
$83$ \( ( -98 + T^{2} )^{4} \)
$89$ \( ( 216 + T^{2} )^{4} \)
$97$ \( ( 24 + T^{2} )^{4} \)
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