Properties

Label 1792.2.e.b
Level $1792$
Weight $2$
Character orbit 1792.e
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \(x^{4} - 3 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{7} + 3 q^{9} -\beta_{2} q^{11} -2 \beta_{3} q^{23} -5 q^{25} -\beta_{1} q^{29} -3 \beta_{1} q^{37} -\beta_{2} q^{43} -7 q^{49} + 5 \beta_{1} q^{53} -3 \beta_{3} q^{63} -3 \beta_{2} q^{67} -2 \beta_{3} q^{71} + 7 \beta_{1} q^{77} -6 \beta_{3} q^{79} + 9 q^{81} -3 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 20q^{25} - 28q^{49} + 36q^{81} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 5 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
1.32288 + 0.500000i
−1.32288 0.500000i
1.32288 0.500000i
−1.32288 + 0.500000i
0 0 0 0 0 2.64575i 0 3.00000 0
895.2 0 0 0 0 0 2.64575i 0 3.00000 0
895.3 0 0 0 0 0 2.64575i 0 3.00000 0
895.4 0 0 0 0 0 2.64575i 0 3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.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.e.b 4
4.b odd 2 1 inner 1792.2.e.b 4
7.b odd 2 1 CM 1792.2.e.b 4
8.b even 2 1 inner 1792.2.e.b 4
8.d odd 2 1 inner 1792.2.e.b 4
16.e even 4 1 28.2.d.a 2
16.e even 4 1 448.2.f.b 2
16.f odd 4 1 28.2.d.a 2
16.f odd 4 1 448.2.f.b 2
28.d even 2 1 inner 1792.2.e.b 4
48.i odd 4 1 252.2.b.a 2
48.i odd 4 1 4032.2.b.e 2
48.k even 4 1 252.2.b.a 2
48.k even 4 1 4032.2.b.e 2
56.e even 2 1 inner 1792.2.e.b 4
56.h odd 2 1 inner 1792.2.e.b 4
80.i odd 4 1 700.2.c.d 4
80.j even 4 1 700.2.c.d 4
80.k odd 4 1 700.2.g.a 2
80.q even 4 1 700.2.g.a 2
80.s even 4 1 700.2.c.d 4
80.t odd 4 1 700.2.c.d 4
112.j even 4 1 28.2.d.a 2
112.j even 4 1 448.2.f.b 2
112.l odd 4 1 28.2.d.a 2
112.l odd 4 1 448.2.f.b 2
112.u odd 12 2 196.2.f.b 4
112.v even 12 2 196.2.f.b 4
112.w even 12 2 196.2.f.b 4
112.x odd 12 2 196.2.f.b 4
336.v odd 4 1 252.2.b.a 2
336.v odd 4 1 4032.2.b.e 2
336.y even 4 1 252.2.b.a 2
336.y even 4 1 4032.2.b.e 2
560.r even 4 1 700.2.c.d 4
560.u odd 4 1 700.2.c.d 4
560.be even 4 1 700.2.g.a 2
560.bf odd 4 1 700.2.g.a 2
560.bm odd 4 1 700.2.c.d 4
560.bn even 4 1 700.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 16.e even 4 1
28.2.d.a 2 16.f odd 4 1
28.2.d.a 2 112.j even 4 1
28.2.d.a 2 112.l odd 4 1
196.2.f.b 4 112.u odd 12 2
196.2.f.b 4 112.v even 12 2
196.2.f.b 4 112.w even 12 2
196.2.f.b 4 112.x odd 12 2
252.2.b.a 2 48.i odd 4 1
252.2.b.a 2 48.k even 4 1
252.2.b.a 2 336.v odd 4 1
252.2.b.a 2 336.y even 4 1
448.2.f.b 2 16.e even 4 1
448.2.f.b 2 16.f odd 4 1
448.2.f.b 2 112.j even 4 1
448.2.f.b 2 112.l odd 4 1
700.2.c.d 4 80.i odd 4 1
700.2.c.d 4 80.j even 4 1
700.2.c.d 4 80.s even 4 1
700.2.c.d 4 80.t odd 4 1
700.2.c.d 4 560.r even 4 1
700.2.c.d 4 560.u odd 4 1
700.2.c.d 4 560.bm odd 4 1
700.2.c.d 4 560.bn even 4 1
700.2.g.a 2 80.k odd 4 1
700.2.g.a 2 80.q even 4 1
700.2.g.a 2 560.be even 4 1
700.2.g.a 2 560.bf odd 4 1
1792.2.e.b 4 1.a even 1 1 trivial
1792.2.e.b 4 4.b odd 2 1 inner
1792.2.e.b 4 7.b odd 2 1 CM
1792.2.e.b 4 8.b even 2 1 inner
1792.2.e.b 4 8.d odd 2 1 inner
1792.2.e.b 4 28.d even 2 1 inner
1792.2.e.b 4 56.e even 2 1 inner
1792.2.e.b 4 56.h odd 2 1 inner
4032.2.b.e 2 48.i odd 4 1
4032.2.b.e 2 48.k even 4 1
4032.2.b.e 2 336.v odd 4 1
4032.2.b.e 2 336.y even 4 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} - 28 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( -28 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 28 + T^{2} )^{2} \)
$29$ \( ( 4 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -28 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 100 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -252 + T^{2} )^{2} \)
$71$ \( ( 28 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 252 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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