Properties

Label 1792.1.o.b
Level 1792
Weight 1
Character orbit 1792.o
Analytic conductor 0.894
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1792.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.3136.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{4} q^{3} + \zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q -\zeta_{12}^{4} q^{3} + \zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{7} -\zeta_{12}^{4} q^{11} + \zeta_{12}^{3} q^{15} + \zeta_{12}^{4} q^{17} + \zeta_{12}^{2} q^{19} + \zeta_{12} q^{21} -\zeta_{12}^{5} q^{23} + q^{27} + \zeta_{12} q^{31} -\zeta_{12}^{2} q^{33} -\zeta_{12}^{2} q^{35} -\zeta_{12}^{5} q^{37} -\zeta_{12}^{5} q^{47} - q^{49} + \zeta_{12}^{2} q^{51} -\zeta_{12} q^{53} + \zeta_{12}^{3} q^{55} + q^{57} + \zeta_{12}^{4} q^{59} + \zeta_{12}^{5} q^{61} -\zeta_{12}^{4} q^{67} -\zeta_{12}^{3} q^{69} + \zeta_{12}^{4} q^{73} + \zeta_{12} q^{77} + \zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{81} -\zeta_{12}^{3} q^{85} -\zeta_{12}^{2} q^{89} -\zeta_{12}^{5} q^{93} -\zeta_{12} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + O(q^{10}) \) \( 4q + 2q^{3} + 2q^{11} - 2q^{17} + 2q^{19} + 4q^{27} - 2q^{33} - 2q^{35} - 4q^{49} + 2q^{51} + 4q^{57} - 2q^{59} + 2q^{67} - 2q^{73} + 2q^{81} - 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0.500000 + 0.866025i 0 −0.866025 0.500000i 0 1.00000i 0 0 0
639.2 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 1.00000i 0 0 0
1663.1 0 0.500000 0.866025i 0 −0.866025 + 0.500000i 0 1.00000i 0 0 0
1663.2 0 0.500000 0.866025i 0 0.866025 0.500000i 0 1.00000i 0 0 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.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.o.b 4
4.b odd 2 1 1792.1.o.a 4
7.c even 3 1 inner 1792.1.o.b 4
8.b even 2 1 1792.1.o.a 4
8.d odd 2 1 inner 1792.1.o.b 4
16.e even 4 1 224.1.r.a 4
16.e even 4 1 448.1.r.a 4
16.f odd 4 1 224.1.r.a 4
16.f odd 4 1 448.1.r.a 4
28.g odd 6 1 1792.1.o.a 4
48.i odd 4 1 2016.1.cd.a 4
48.k even 4 1 2016.1.cd.a 4
56.k odd 6 1 inner 1792.1.o.b 4
56.p even 6 1 1792.1.o.a 4
112.j even 4 1 1568.1.r.a 4
112.j even 4 1 3136.1.r.b 4
112.l odd 4 1 1568.1.r.a 4
112.l odd 4 1 3136.1.r.b 4
112.u odd 12 1 224.1.r.a 4
112.u odd 12 1 448.1.r.a 4
112.u odd 12 1 1568.1.d.b 2
112.u odd 12 1 3136.1.d.b 2
112.v even 12 1 1568.1.d.a 2
112.v even 12 1 1568.1.r.a 4
112.v even 12 1 3136.1.d.d 2
112.v even 12 1 3136.1.r.b 4
112.w even 12 1 224.1.r.a 4
112.w even 12 1 448.1.r.a 4
112.w even 12 1 1568.1.d.b 2
112.w even 12 1 3136.1.d.b 2
112.x odd 12 1 1568.1.d.a 2
112.x odd 12 1 1568.1.r.a 4
112.x odd 12 1 3136.1.d.d 2
112.x odd 12 1 3136.1.r.b 4
336.bt odd 12 1 2016.1.cd.a 4
336.bu even 12 1 2016.1.cd.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 16.e even 4 1
224.1.r.a 4 16.f odd 4 1
224.1.r.a 4 112.u odd 12 1
224.1.r.a 4 112.w even 12 1
448.1.r.a 4 16.e even 4 1
448.1.r.a 4 16.f odd 4 1
448.1.r.a 4 112.u odd 12 1
448.1.r.a 4 112.w even 12 1
1568.1.d.a 2 112.v even 12 1
1568.1.d.a 2 112.x odd 12 1
1568.1.d.b 2 112.u odd 12 1
1568.1.d.b 2 112.w even 12 1
1568.1.r.a 4 112.j even 4 1
1568.1.r.a 4 112.l odd 4 1
1568.1.r.a 4 112.v even 12 1
1568.1.r.a 4 112.x odd 12 1
1792.1.o.a 4 4.b odd 2 1
1792.1.o.a 4 8.b even 2 1
1792.1.o.a 4 28.g odd 6 1
1792.1.o.a 4 56.p even 6 1
1792.1.o.b 4 1.a even 1 1 trivial
1792.1.o.b 4 7.c even 3 1 inner
1792.1.o.b 4 8.d odd 2 1 inner
1792.1.o.b 4 56.k odd 6 1 inner
2016.1.cd.a 4 48.i odd 4 1
2016.1.cd.a 4 48.k even 4 1
2016.1.cd.a 4 336.bt odd 12 1
2016.1.cd.a 4 336.bu even 12 1
3136.1.d.b 2 112.u odd 12 1
3136.1.d.b 2 112.w even 12 1
3136.1.d.d 2 112.v even 12 1
3136.1.d.d 2 112.x odd 12 1
3136.1.r.b 4 112.j even 4 1
3136.1.r.b 4 112.l odd 4 1
3136.1.r.b 4 112.v even 12 1
3136.1.r.b 4 112.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1792, [\chi])\).

Hecke characteristic polynomials

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