Properties

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

Related objects

Downloads

Learn more about

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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 224)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( -3 + \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} -4 q^{15} + \beta_{3} q^{17} -\beta_{1} q^{19} + \beta_{1} q^{21} + 4 q^{23} + ( -1 - \beta_{3} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{27} + ( 2 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + \beta_{3} ) q^{31} -8 q^{33} + ( \beta_{1} + \beta_{2} ) q^{35} + ( -2 \beta_{1} - \beta_{2} ) q^{37} + ( -8 + 2 \beta_{3} ) q^{39} + ( 4 + \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -\beta_{1} + 3 \beta_{2} ) q^{45} + ( -6 - \beta_{3} ) q^{47} + q^{49} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{51} -5 \beta_{2} q^{53} + ( -12 - 2 \beta_{3} ) q^{55} + ( 6 - \beta_{3} ) q^{57} + ( \beta_{1} + 4 \beta_{2} ) q^{59} + ( -\beta_{1} - 5 \beta_{2} ) q^{61} + ( -3 + \beta_{3} ) q^{63} + ( -2 + \beta_{3} ) q^{65} + 2 \beta_{2} q^{67} + 4 \beta_{1} q^{69} + ( 4 - 2 \beta_{3} ) q^{71} + ( -6 + 2 \beta_{3} ) q^{73} + ( \beta_{1} - 4 \beta_{2} ) q^{75} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -4 - 2 \beta_{3} ) q^{79} + ( 11 - 3 \beta_{3} ) q^{81} + ( -\beta_{1} - 4 \beta_{2} ) q^{83} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{85} + ( -10 + \beta_{3} ) q^{87} + 6 q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{93} + 4 q^{95} + ( 8 + \beta_{3} ) q^{97} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} - 12q^{9} + O(q^{10}) \) \( 4q + 4q^{7} - 12q^{9} - 16q^{15} + 16q^{23} - 4q^{25} - 8q^{31} - 32q^{33} - 32q^{39} + 16q^{41} - 24q^{47} + 4q^{49} - 48q^{55} + 24q^{57} - 12q^{63} - 8q^{65} + 16q^{71} - 24q^{73} - 16q^{79} + 44q^{81} - 40q^{87} + 24q^{89} + 16q^{95} + 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/2\)

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
1.61803i
0.618034i
0.618034i
1.61803i
0 3.23607i 0 1.23607i 0 1.00000 0 −7.47214 0
897.2 0 1.23607i 0 3.23607i 0 1.00000 0 1.47214 0
897.3 0 1.23607i 0 3.23607i 0 1.00000 0 1.47214 0
897.4 0 3.23607i 0 1.23607i 0 1.00000 0 −7.47214 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.m 4
4.b odd 2 1 1792.2.b.k 4
8.b even 2 1 inner 1792.2.b.m 4
8.d odd 2 1 1792.2.b.k 4
16.e even 4 1 224.2.a.d yes 2
16.e even 4 1 448.2.a.i 2
16.f odd 4 1 224.2.a.c 2
16.f odd 4 1 448.2.a.j 2
48.i odd 4 1 2016.2.a.o 2
48.i odd 4 1 4032.2.a.bv 2
48.k even 4 1 2016.2.a.r 2
48.k even 4 1 4032.2.a.bw 2
80.k odd 4 1 5600.2.a.bk 2
80.q even 4 1 5600.2.a.z 2
112.j even 4 1 1568.2.a.v 2
112.j even 4 1 3136.2.a.bf 2
112.l odd 4 1 1568.2.a.k 2
112.l odd 4 1 3136.2.a.by 2
112.u odd 12 2 1568.2.i.v 4
112.v even 12 2 1568.2.i.n 4
112.w even 12 2 1568.2.i.m 4
112.x odd 12 2 1568.2.i.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 16.f odd 4 1
224.2.a.d yes 2 16.e even 4 1
448.2.a.i 2 16.e even 4 1
448.2.a.j 2 16.f odd 4 1
1568.2.a.k 2 112.l odd 4 1
1568.2.a.v 2 112.j even 4 1
1568.2.i.m 4 112.w even 12 2
1568.2.i.n 4 112.v even 12 2
1568.2.i.v 4 112.u odd 12 2
1568.2.i.w 4 112.x odd 12 2
1792.2.b.k 4 4.b odd 2 1
1792.2.b.k 4 8.d odd 2 1
1792.2.b.m 4 1.a even 1 1 trivial
1792.2.b.m 4 8.b even 2 1 inner
2016.2.a.o 2 48.i odd 4 1
2016.2.a.r 2 48.k even 4 1
3136.2.a.bf 2 112.j even 4 1
3136.2.a.by 2 112.l odd 4 1
4032.2.a.bv 2 48.i odd 4 1
4032.2.a.bw 2 48.k even 4 1
5600.2.a.z 2 80.q even 4 1
5600.2.a.bk 2 80.k odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 12 T_{3}^{2} + 16 \)
\( T_{5}^{4} + 12 T_{5}^{2} + 16 \)
\( T_{11}^{4} + 48 T_{11}^{2} + 256 \)
\( T_{23} - 4 \)
\( T_{31}^{2} + 4 T_{31} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 12 T^{2} + T^{4} \)
$5$ \( 16 + 12 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 256 + 48 T^{2} + T^{4} \)
$13$ \( 16 + 28 T^{2} + T^{4} \)
$17$ \( ( -20 + T^{2} )^{2} \)
$19$ \( 16 + 12 T^{2} + T^{4} \)
$23$ \( ( -4 + T )^{4} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( -16 + 4 T + T^{2} )^{2} \)
$37$ \( ( 20 + T^{2} )^{2} \)
$41$ \( ( -4 - 8 T + T^{2} )^{2} \)
$43$ \( 256 + 48 T^{2} + T^{4} \)
$47$ \( ( 16 + 12 T + T^{2} )^{2} \)
$53$ \( ( 100 + T^{2} )^{2} \)
$59$ \( 1936 + 108 T^{2} + T^{4} \)
$61$ \( 5776 + 172 T^{2} + T^{4} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -64 - 8 T + T^{2} )^{2} \)
$73$ \( ( -44 + 12 T + T^{2} )^{2} \)
$79$ \( ( -64 + 8 T + T^{2} )^{2} \)
$83$ \( 1936 + 108 T^{2} + T^{4} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( 44 - 16 T + T^{2} )^{2} \)
show more
show less