Properties

Label 1792.2.b.b
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 224)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} - q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} - q^{7} - q^{9} -4 i q^{11} -4 i q^{13} -2 q^{17} + 6 i q^{19} -2 i q^{21} + 8 q^{23} + 5 q^{25} + 4 i q^{27} + 2 i q^{29} + 4 q^{31} + 8 q^{33} -10 i q^{37} + 8 q^{39} + 10 q^{41} + 4 i q^{43} -4 q^{47} + q^{49} -4 i q^{51} + 2 i q^{53} -12 q^{57} + 10 i q^{59} -8 i q^{61} + q^{63} + 8 i q^{67} + 16 i q^{69} + 6 q^{73} + 10 i q^{75} + 4 i q^{77} + 16 q^{79} -11 q^{81} -2 i q^{83} -4 q^{87} -18 q^{89} + 4 i q^{91} + 8 i q^{93} -2 q^{97} + 4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{9} - 4q^{17} + 16q^{23} + 10q^{25} + 8q^{31} + 16q^{33} + 16q^{39} + 20q^{41} - 8q^{47} + 2q^{49} - 24q^{57} + 2q^{63} + 12q^{73} + 32q^{79} - 22q^{81} - 8q^{87} - 36q^{89} - 4q^{97} + O(q^{100}) \)

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 2.00000i 0 0 0 −1.00000 0 −1.00000 0
897.2 0 2.00000i 0 0 0 −1.00000 0 −1.00000 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.b 2
4.b odd 2 1 1792.2.b.f 2
8.b even 2 1 inner 1792.2.b.b 2
8.d odd 2 1 1792.2.b.f 2
16.e even 4 1 224.2.a.b yes 1
16.e even 4 1 448.2.a.b 1
16.f odd 4 1 224.2.a.a 1
16.f odd 4 1 448.2.a.f 1
48.i odd 4 1 2016.2.a.g 1
48.i odd 4 1 4032.2.a.z 1
48.k even 4 1 2016.2.a.e 1
48.k even 4 1 4032.2.a.p 1
80.k odd 4 1 5600.2.a.t 1
80.q even 4 1 5600.2.a.c 1
112.j even 4 1 1568.2.a.h 1
112.j even 4 1 3136.2.a.f 1
112.l odd 4 1 1568.2.a.b 1
112.l odd 4 1 3136.2.a.y 1
112.u odd 12 2 1568.2.i.k 2
112.v even 12 2 1568.2.i.c 2
112.w even 12 2 1568.2.i.b 2
112.x odd 12 2 1568.2.i.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 16.f odd 4 1
224.2.a.b yes 1 16.e even 4 1
448.2.a.b 1 16.e even 4 1
448.2.a.f 1 16.f odd 4 1
1568.2.a.b 1 112.l odd 4 1
1568.2.a.h 1 112.j even 4 1
1568.2.i.b 2 112.w even 12 2
1568.2.i.c 2 112.v even 12 2
1568.2.i.j 2 112.x odd 12 2
1568.2.i.k 2 112.u odd 12 2
1792.2.b.b 2 1.a even 1 1 trivial
1792.2.b.b 2 8.b even 2 1 inner
1792.2.b.f 2 4.b odd 2 1
1792.2.b.f 2 8.d odd 2 1
2016.2.a.e 1 48.k even 4 1
2016.2.a.g 1 48.i odd 4 1
3136.2.a.f 1 112.j even 4 1
3136.2.a.y 1 112.l odd 4 1
4032.2.a.p 1 48.k even 4 1
4032.2.a.z 1 48.i odd 4 1
5600.2.a.c 1 80.q even 4 1
5600.2.a.t 1 80.k odd 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -8 + T )^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( 4 + T )^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( 100 + T^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 4 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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