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: \(x^{2} + 1\)
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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3} \)
\( T_{5}^{2} + 4 \)
\( T_{11}^{2} + 16 \)
\( T_{23} \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

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