Properties

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

Related objects

Downloads

Learn more

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: \(4\)
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: 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.

\(f(q)\) \(=\) \( q + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -1 + 2 \zeta_{12}^{2} - \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} + 2 \zeta_{12}^{3} ) q^{11} + ( -1 + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} -2 \zeta_{12}^{3} q^{15} -4 \zeta_{12}^{3} q^{17} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( -1 - 4 \zeta_{12} - 2 \zeta_{12}^{2} - \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 q^{27} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{29} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{33} + ( -5 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{35} + 4 q^{37} + ( 4 - 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{39} + ( 4 - 8 \zeta_{12}^{2} + 4 \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} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( -4 + 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{51} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{53} -4 q^{55} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( -9 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( 5 - 10 \zeta_{12}^{2} - 3 \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} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{67} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{71} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + ( -7 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{75} + ( -10 - 4 \zeta_{12} + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{77} + ( -2 + 4 \zeta_{12}^{2} ) q^{79} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{81} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( -4 + 8 \zeta_{12} - 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} + ( -5 - 6 \zeta_{12} + 4 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{91} + ( 12 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( 4 - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} + 12 \zeta_{12}^{3} q^{97} + ( -6 + 12 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 8q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 8q^{7} + 4q^{9} - 12q^{19} - 8q^{21} + 4q^{25} - 16q^{27} - 12q^{35} + 16q^{37} + 4q^{49} - 16q^{55} - 36q^{59} + 8q^{63} - 28q^{75} - 24q^{77} + 4q^{81} - 36q^{83} - 16q^{85} - 48q^{87} - 12q^{91} + 48q^{93} + O(q^{100}) \)

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
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −2.73205 0 0.732051i 0 2.00000 1.73205i 0 4.46410 0
1791.2 0 −2.73205 0 0.732051i 0 2.00000 + 1.73205i 0 4.46410 0
1791.3 0 0.732051 0 2.73205i 0 2.00000 1.73205i 0 −2.46410 0
1791.4 0 0.732051 0 2.73205i 0 2.00000 + 1.73205i 0 −2.46410 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
28.d 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.f.b 4
4.b odd 2 1 1792.2.f.h 4
7.b odd 2 1 1792.2.f.h 4
8.b even 2 1 1792.2.f.i 4
8.d odd 2 1 1792.2.f.a 4
16.e even 4 1 896.2.e.a 4
16.e even 4 1 896.2.e.e yes 4
16.f odd 4 1 896.2.e.b yes 4
16.f odd 4 1 896.2.e.f yes 4
28.d even 2 1 inner 1792.2.f.b 4
56.e even 2 1 1792.2.f.i 4
56.h odd 2 1 1792.2.f.a 4
112.j even 4 1 896.2.e.a 4
112.j even 4 1 896.2.e.e yes 4
112.l odd 4 1 896.2.e.b yes 4
112.l odd 4 1 896.2.e.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.a 4 16.e even 4 1
896.2.e.a 4 112.j even 4 1
896.2.e.b yes 4 16.f odd 4 1
896.2.e.b yes 4 112.l odd 4 1
896.2.e.e yes 4 16.e even 4 1
896.2.e.e yes 4 112.j even 4 1
896.2.e.f yes 4 16.f odd 4 1
896.2.e.f yes 4 112.l odd 4 1
1792.2.f.a 4 8.d odd 2 1
1792.2.f.a 4 56.h odd 2 1
1792.2.f.b 4 1.a even 1 1 trivial
1792.2.f.b 4 28.d even 2 1 inner
1792.2.f.h 4 4.b odd 2 1
1792.2.f.h 4 7.b odd 2 1
1792.2.f.i 4 8.b even 2 1
1792.2.f.i 4 56.e even 2 1

Hecke kernels

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

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{5}^{4} + 8 T_{5}^{2} + 4 \)
\( T_{29}^{2} - 48 \)
\( T_{31}^{2} - 48 \)
\( T_{37} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -2 + 2 T + T^{2} )^{2} \)
$5$ \( 4 + 8 T^{2} + T^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( 64 + 32 T^{2} + T^{4} \)
$13$ \( 36 + 24 T^{2} + T^{4} \)
$17$ \( ( 16 + T^{2} )^{2} \)
$19$ \( ( 6 + 6 T + T^{2} )^{2} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( -48 + T^{2} )^{2} \)
$31$ \( ( -48 + T^{2} )^{2} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( 1024 + 128 T^{2} + T^{4} \)
$43$ \( 576 + 96 T^{2} + T^{4} \)
$47$ \( ( -48 + T^{2} )^{2} \)
$53$ \( ( -48 + T^{2} )^{2} \)
$59$ \( ( 54 + 18 T + T^{2} )^{2} \)
$61$ \( 4356 + 168 T^{2} + T^{4} \)
$67$ \( 576 + 96 T^{2} + T^{4} \)
$71$ \( 2704 + 152 T^{2} + T^{4} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 12 + T^{2} )^{2} \)
$83$ \( ( 78 + 18 T + T^{2} )^{2} \)
$89$ \( 256 + 224 T^{2} + T^{4} \)
$97$ \( ( 144 + T^{2} )^{2} \)
show more
show less