Properties

Label 1792.2.b.p
Level $1792$
Weight $2$
Character orbit 1792.b
Analytic conductor $14.309$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} -\beta_{4} q^{5} + q^{7} + ( -2 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} -\beta_{4} q^{5} + q^{7} + ( -2 - \beta_{3} ) q^{9} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{11} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{13} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{15} + ( -1 + \beta_{1} + \beta_{3} ) q^{17} + ( -2 \beta_{2} + \beta_{5} ) q^{19} + \beta_{5} q^{21} + \beta_{1} q^{23} + ( -6 + \beta_{3} ) q^{25} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{27} + ( -\beta_{2} + 2 \beta_{5} ) q^{29} + ( -1 + \beta_{1} - \beta_{3} ) q^{31} + ( -2 - 2 \beta_{3} ) q^{33} -\beta_{4} q^{35} + ( \beta_{2} + 2 \beta_{4} ) q^{37} + ( 6 - \beta_{1} + 2 \beta_{3} ) q^{39} + ( -5 + \beta_{1} + \beta_{3} ) q^{41} + ( \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{43} + ( -4 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{45} + ( -1 - \beta_{1} - \beta_{3} ) q^{47} + q^{49} + ( -4 \beta_{2} + 2 \beta_{4} ) q^{51} -\beta_{2} q^{53} + ( 8 + 2 \beta_{1} ) q^{55} + ( -7 - 2 \beta_{1} - 3 \beta_{3} ) q^{57} + ( 4 \beta_{2} + 2 \beta_{4} + 5 \beta_{5} ) q^{59} + ( -4 \beta_{2} + \beta_{4} ) q^{61} + ( -2 - \beta_{3} ) q^{63} + ( -5 - 4 \beta_{1} - \beta_{3} ) q^{65} + ( 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{67} + ( -4 \beta_{2} - 2 \beta_{5} ) q^{69} + 2 \beta_{3} q^{73} + ( 2 \beta_{4} - 3 \beta_{5} ) q^{75} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{77} + ( -4 - 2 \beta_{1} ) q^{79} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{81} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( -4 \beta_{2} - 6 \beta_{5} ) q^{85} + ( -11 - \beta_{1} - 3 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{91} + ( -4 \beta_{2} - 2 \beta_{4} - 6 \beta_{5} ) q^{93} + ( -\beta_{1} + 4 \beta_{3} ) q^{95} + ( -3 - 3 \beta_{1} - \beta_{3} ) q^{97} + ( 3 \beta_{2} - \beta_{4} - 5 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{7} - 14q^{9} + O(q^{10}) \) \( 6q + 6q^{7} - 14q^{9} - 8q^{15} - 4q^{17} - 34q^{25} - 8q^{31} - 16q^{33} + 40q^{39} - 28q^{41} - 8q^{47} + 6q^{49} + 48q^{55} - 48q^{57} - 14q^{63} - 32q^{65} + 4q^{73} - 24q^{79} + 6q^{81} - 72q^{87} + 4q^{89} + 8q^{95} - 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{4} + 2 \nu^{3} - \nu^{2} + 2 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 2 \nu^{4} - 3 \nu^{3} + 6 \nu^{2} - 2 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{5} + 5 \nu^{3} - 4 \nu^{2} - 6 \nu - 12 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{5} - 4 \nu^{4} + 11 \nu^{3} - 16 \nu^{2} + 14 \nu - 28 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{3} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 5\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} - \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - \beta_{1} + 6\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} + 7\)\()/4\)

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
1.40680 0.144584i
0.264658 1.38923i
−0.671462 1.24464i
−0.671462 + 1.24464i
0.264658 + 1.38923i
1.40680 + 0.144584i
0 3.10278i 0 2.52444i 0 1.00000 0 −6.62721 0
897.2 0 2.24914i 0 3.30777i 0 1.00000 0 −2.05863 0
897.3 0 1.14637i 0 3.83221i 0 1.00000 0 1.68585 0
897.4 0 1.14637i 0 3.83221i 0 1.00000 0 1.68585 0
897.5 0 2.24914i 0 3.30777i 0 1.00000 0 −2.05863 0
897.6 0 3.10278i 0 2.52444i 0 1.00000 0 −6.62721 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 897.6
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.p 6
4.b odd 2 1 1792.2.b.o 6
8.b even 2 1 inner 1792.2.b.p 6
8.d odd 2 1 1792.2.b.o 6
16.e even 4 1 896.2.a.i 3
16.e even 4 1 896.2.a.l yes 3
16.f odd 4 1 896.2.a.j yes 3
16.f odd 4 1 896.2.a.k yes 3
48.i odd 4 1 8064.2.a.bu 3
48.i odd 4 1 8064.2.a.ce 3
48.k even 4 1 8064.2.a.cb 3
48.k even 4 1 8064.2.a.ch 3
112.j even 4 1 6272.2.a.v 3
112.j even 4 1 6272.2.a.w 3
112.l odd 4 1 6272.2.a.u 3
112.l odd 4 1 6272.2.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.a.i 3 16.e even 4 1
896.2.a.j yes 3 16.f odd 4 1
896.2.a.k yes 3 16.f odd 4 1
896.2.a.l yes 3 16.e even 4 1
1792.2.b.o 6 4.b odd 2 1
1792.2.b.o 6 8.d odd 2 1
1792.2.b.p 6 1.a even 1 1 trivial
1792.2.b.p 6 8.b even 2 1 inner
6272.2.a.u 3 112.l odd 4 1
6272.2.a.v 3 112.j even 4 1
6272.2.a.w 3 112.j even 4 1
6272.2.a.x 3 112.l odd 4 1
8064.2.a.bu 3 48.i odd 4 1
8064.2.a.cb 3 48.k even 4 1
8064.2.a.ce 3 48.i odd 4 1
8064.2.a.ch 3 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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{6} + 16 T_{3}^{4} + 68 T_{3}^{2} + 64 \)
\( T_{5}^{6} + 32 T_{5}^{4} + 324 T_{5}^{2} + 1024 \)
\( T_{11}^{6} + 36 T_{11}^{4} + 320 T_{11}^{2} + 256 \)
\( T_{23}^{3} - 28 T_{23} + 16 \)
\( T_{31}^{3} + 4 T_{31}^{2} - 56 T_{31} - 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 64 + 68 T^{2} + 16 T^{4} + T^{6} \)
$5$ \( 1024 + 324 T^{2} + 32 T^{4} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 256 + 320 T^{2} + 36 T^{4} + T^{6} \)
$13$ \( 18496 + 2116 T^{2} + 80 T^{4} + T^{6} \)
$17$ \( ( 8 - 28 T + 2 T^{2} + T^{3} )^{2} \)
$19$ \( 4096 + 1572 T^{2} + 80 T^{4} + T^{6} \)
$23$ \( ( 16 - 28 T + T^{3} )^{2} \)
$29$ \( 7744 + 1776 T^{2} + 92 T^{4} + T^{6} \)
$31$ \( ( -256 - 56 T + 4 T^{2} + T^{3} )^{2} \)
$37$ \( 18496 + 3056 T^{2} + 124 T^{4} + T^{6} \)
$41$ \( ( -8 + 36 T + 14 T^{2} + T^{3} )^{2} \)
$43$ \( 92416 + 8000 T^{2} + 196 T^{4} + T^{6} \)
$47$ \( ( -64 - 24 T + 4 T^{2} + T^{3} )^{2} \)
$53$ \( ( 4 + T^{2} )^{3} \)
$59$ \( 2483776 + 56484 T^{2} + 416 T^{4} + T^{6} \)
$61$ \( 246016 + 18308 T^{2} + 256 T^{4} + T^{6} \)
$67$ \( 1024 + 5248 T^{2} + 164 T^{4} + T^{6} \)
$71$ \( T^{6} \)
$73$ \( ( 8 - 68 T - 2 T^{2} + T^{3} )^{2} \)
$79$ \( ( -512 - 64 T + 12 T^{2} + T^{3} )^{2} \)
$83$ \( 135424 + 10500 T^{2} + 224 T^{4} + T^{6} \)
$89$ \( ( 296 - 116 T - 2 T^{2} + T^{3} )^{2} \)
$97$ \( ( -1816 - 188 T + 10 T^{2} + T^{3} )^{2} \)
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