Properties

Label 1792.2.b.j
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{7} + 3 q^{9} +O(q^{10})\) \( q + q^{7} + 3 q^{9} + 2 i q^{11} + 4 i q^{13} -2 q^{17} -4 i q^{19} + 4 q^{23} + 5 q^{25} + 6 i q^{29} -8 q^{31} + 2 i q^{37} + 2 q^{41} + 10 i q^{43} + q^{49} -2 i q^{53} -8 i q^{59} + 8 i q^{61} + 3 q^{63} -2 i q^{67} + 14 q^{73} + 2 i q^{77} -4 q^{79} + 9 q^{81} -12 i q^{83} + 6 q^{89} + 4 i q^{91} + 6 q^{97} + 6 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{7} + 6q^{9} - 4q^{17} + 8q^{23} + 10q^{25} - 16q^{31} + 4q^{41} + 2q^{49} + 6q^{63} + 28q^{73} - 8q^{79} + 18q^{81} + 12q^{89} + 12q^{97} + O(q^{100}) \)

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 0 0 1.00000 0 3.00000 0
897.2 0 0 0 0 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.j 2
4.b odd 2 1 1792.2.b.e 2
8.b even 2 1 inner 1792.2.b.j 2
8.d odd 2 1 1792.2.b.e 2
16.e even 4 1 896.2.a.a 1
16.e even 4 1 896.2.a.b yes 1
16.f odd 4 1 896.2.a.c yes 1
16.f odd 4 1 896.2.a.d yes 1
48.i odd 4 1 8064.2.a.h 1
48.i odd 4 1 8064.2.a.m 1
48.k even 4 1 8064.2.a.q 1
48.k even 4 1 8064.2.a.t 1
112.j even 4 1 6272.2.a.d 1
112.j even 4 1 6272.2.a.e 1
112.l odd 4 1 6272.2.a.c 1
112.l odd 4 1 6272.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.a.a 1 16.e even 4 1
896.2.a.b yes 1 16.e even 4 1
896.2.a.c yes 1 16.f odd 4 1
896.2.a.d yes 1 16.f odd 4 1
1792.2.b.e 2 4.b odd 2 1
1792.2.b.e 2 8.d odd 2 1
1792.2.b.j 2 1.a even 1 1 trivial
1792.2.b.j 2 8.b even 2 1 inner
6272.2.a.c 1 112.l odd 4 1
6272.2.a.d 1 112.j even 4 1
6272.2.a.e 1 112.j even 4 1
6272.2.a.f 1 112.l odd 4 1
8064.2.a.h 1 48.i odd 4 1
8064.2.a.m 1 48.i odd 4 1
8064.2.a.q 1 48.k even 4 1
8064.2.a.t 1 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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 36 + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( 64 + T^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -14 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( -6 + T )^{2} \)
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