Properties

Label 1792.2.b.d
Level $1792$
Weight $2$
Character orbit 1792.b
Analytic conductor $14.309$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} - q^{7} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} - q^{7} + 3 q^{9} - 2 \beta q^{11} + \beta q^{13} - 6 q^{17} - 4 \beta q^{19} + q^{25} + 3 \beta q^{29} - 8 q^{31} + \beta q^{35} + \beta q^{37} - 2 q^{41} - 2 \beta q^{43} - 3 \beta q^{45} + 8 q^{47} + q^{49} - 3 \beta q^{53} - 8 q^{55} - 3 \beta q^{61} - 3 q^{63} + 4 q^{65} + 2 \beta q^{67} - 8 q^{71} - 10 q^{73} + 2 \beta q^{77} - 16 q^{79} + 9 q^{81} - 4 \beta q^{83} + 6 \beta q^{85} + 6 q^{89} - \beta q^{91} - 16 q^{95} - 6 q^{97} - 6 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 6 q^{9} - 12 q^{17} + 2 q^{25} - 16 q^{31} - 4 q^{41} + 16 q^{47} + 2 q^{49} - 16 q^{55} - 6 q^{63} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 32 q^{79} + 18 q^{81} + 12 q^{89} - 32 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
0 0 0 2.00000i 0 −1.00000 0 3.00000 0
897.2 0 0 0 2.00000i 0 −1.00000 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
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.d 2
4.b odd 2 1 1792.2.b.i 2
8.b even 2 1 inner 1792.2.b.d 2
8.d odd 2 1 1792.2.b.i 2
16.e even 4 1 112.2.a.b 1
16.e even 4 1 448.2.a.e 1
16.f odd 4 1 56.2.a.a 1
16.f odd 4 1 448.2.a.d 1
48.i odd 4 1 1008.2.a.d 1
48.i odd 4 1 4032.2.a.bk 1
48.k even 4 1 504.2.a.c 1
48.k even 4 1 4032.2.a.bb 1
80.i odd 4 1 2800.2.g.p 2
80.j even 4 1 1400.2.g.g 2
80.k odd 4 1 1400.2.a.g 1
80.q even 4 1 2800.2.a.p 1
80.s even 4 1 1400.2.g.g 2
80.t odd 4 1 2800.2.g.p 2
112.j even 4 1 392.2.a.d 1
112.j even 4 1 3136.2.a.q 1
112.l odd 4 1 784.2.a.e 1
112.l odd 4 1 3136.2.a.p 1
112.u odd 12 2 392.2.i.c 2
112.v even 12 2 392.2.i.d 2
112.w even 12 2 784.2.i.e 2
112.x odd 12 2 784.2.i.g 2
176.i even 4 1 6776.2.a.g 1
208.o odd 4 1 9464.2.a.c 1
336.v odd 4 1 3528.2.a.x 1
336.y even 4 1 7056.2.a.bo 1
336.br odd 12 2 3528.2.s.e 2
336.bu even 12 2 3528.2.s.t 2
560.be even 4 1 9800.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 16.f odd 4 1
112.2.a.b 1 16.e even 4 1
392.2.a.d 1 112.j even 4 1
392.2.i.c 2 112.u odd 12 2
392.2.i.d 2 112.v even 12 2
448.2.a.d 1 16.f odd 4 1
448.2.a.e 1 16.e even 4 1
504.2.a.c 1 48.k even 4 1
784.2.a.e 1 112.l odd 4 1
784.2.i.e 2 112.w even 12 2
784.2.i.g 2 112.x odd 12 2
1008.2.a.d 1 48.i odd 4 1
1400.2.a.g 1 80.k odd 4 1
1400.2.g.g 2 80.j even 4 1
1400.2.g.g 2 80.s even 4 1
1792.2.b.d 2 1.a even 1 1 trivial
1792.2.b.d 2 8.b even 2 1 inner
1792.2.b.i 2 4.b odd 2 1
1792.2.b.i 2 8.d odd 2 1
2800.2.a.p 1 80.q even 4 1
2800.2.g.p 2 80.i odd 4 1
2800.2.g.p 2 80.t odd 4 1
3136.2.a.p 1 112.l odd 4 1
3136.2.a.q 1 112.j even 4 1
3528.2.a.x 1 336.v odd 4 1
3528.2.s.e 2 336.br odd 12 2
3528.2.s.t 2 336.bu even 12 2
4032.2.a.bb 1 48.k even 4 1
4032.2.a.bk 1 48.i odd 4 1
6776.2.a.g 1 176.i even 4 1
7056.2.a.bo 1 336.y even 4 1
9464.2.a.c 1 208.o odd 4 1
9800.2.a.u 1 560.be even 4 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{31} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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