Properties

Label 1792.2.e.c
Level $1792$
Weight $2$
Character orbit 1792.e
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12}^{3} q^{3} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} - q^{9} +O(q^{10})\) \( q + 2 \zeta_{12}^{3} q^{3} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} - q^{9} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( -4 + 8 \zeta_{12}^{2} ) q^{15} + 2 \zeta_{12}^{3} q^{19} + ( -4 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{21} + ( 2 - 4 \zeta_{12}^{2} ) q^{23} + 7 q^{25} + 4 \zeta_{12}^{3} q^{27} + 6 \zeta_{12}^{3} q^{29} -8 q^{31} + ( 4 - 8 \zeta_{12}^{2} ) q^{33} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( -4 + 8 \zeta_{12}^{2} ) q^{39} + ( 4 - 8 \zeta_{12}^{2} ) q^{41} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{43} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{45} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} -6 \zeta_{12}^{3} q^{53} -12 q^{55} -4 q^{57} -6 \zeta_{12}^{3} q^{59} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{61} + ( -1 - 2 \zeta_{12}^{2} ) q^{63} + 12 q^{65} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{69} + ( -2 + 4 \zeta_{12}^{2} ) q^{71} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + 14 \zeta_{12}^{3} q^{75} + ( -8 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} -11 q^{81} -6 \zeta_{12}^{3} q^{83} -12 q^{87} + ( 4 - 8 \zeta_{12}^{2} ) q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} -16 \zeta_{12}^{3} q^{93} + ( -4 + 8 \zeta_{12}^{2} ) q^{95} + ( 8 - 16 \zeta_{12}^{2} ) q^{97} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 8q^{7} - 4q^{9} + 28q^{25} - 32q^{31} + 4q^{49} - 48q^{55} - 16q^{57} - 8q^{63} + 48q^{65} - 44q^{81} - 48q^{87} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
895.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 2.00000i 0 −3.46410 0 2.00000 + 1.73205i 0 −1.00000 0
895.2 0 2.00000i 0 3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.3 0 2.00000i 0 −3.46410 0 2.00000 1.73205i 0 −1.00000 0
895.4 0 2.00000i 0 3.46410 0 2.00000 + 1.73205i 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
28.d even 2 1 inner
56.e 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.e.c 4
4.b odd 2 1 1792.2.e.a 4
7.b odd 2 1 1792.2.e.a 4
8.b even 2 1 inner 1792.2.e.c 4
8.d odd 2 1 1792.2.e.a 4
16.e even 4 1 112.2.f.b yes 2
16.e even 4 1 448.2.f.a 2
16.f odd 4 1 112.2.f.a 2
16.f odd 4 1 448.2.f.c 2
28.d even 2 1 inner 1792.2.e.c 4
48.i odd 4 1 1008.2.b.b 2
48.i odd 4 1 4032.2.b.b 2
48.k even 4 1 1008.2.b.g 2
48.k even 4 1 4032.2.b.h 2
56.e even 2 1 inner 1792.2.e.c 4
56.h odd 2 1 1792.2.e.a 4
80.i odd 4 1 2800.2.e.b 4
80.j even 4 1 2800.2.e.c 4
80.k odd 4 1 2800.2.k.e 2
80.q even 4 1 2800.2.k.b 2
80.s even 4 1 2800.2.e.c 4
80.t odd 4 1 2800.2.e.b 4
112.j even 4 1 112.2.f.b yes 2
112.j even 4 1 448.2.f.a 2
112.l odd 4 1 112.2.f.a 2
112.l odd 4 1 448.2.f.c 2
112.u odd 12 1 784.2.p.e 2
112.u odd 12 1 784.2.p.f 2
112.v even 12 1 784.2.p.a 2
112.v even 12 1 784.2.p.b 2
112.w even 12 1 784.2.p.a 2
112.w even 12 1 784.2.p.b 2
112.x odd 12 1 784.2.p.e 2
112.x odd 12 1 784.2.p.f 2
336.v odd 4 1 1008.2.b.b 2
336.v odd 4 1 4032.2.b.b 2
336.y even 4 1 1008.2.b.g 2
336.y even 4 1 4032.2.b.h 2
560.r even 4 1 2800.2.e.c 4
560.u odd 4 1 2800.2.e.b 4
560.be even 4 1 2800.2.k.b 2
560.bf odd 4 1 2800.2.k.e 2
560.bm odd 4 1 2800.2.e.b 4
560.bn even 4 1 2800.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 16.f odd 4 1
112.2.f.a 2 112.l odd 4 1
112.2.f.b yes 2 16.e even 4 1
112.2.f.b yes 2 112.j even 4 1
448.2.f.a 2 16.e even 4 1
448.2.f.a 2 112.j even 4 1
448.2.f.c 2 16.f odd 4 1
448.2.f.c 2 112.l odd 4 1
784.2.p.a 2 112.v even 12 1
784.2.p.a 2 112.w even 12 1
784.2.p.b 2 112.v even 12 1
784.2.p.b 2 112.w even 12 1
784.2.p.e 2 112.u odd 12 1
784.2.p.e 2 112.x odd 12 1
784.2.p.f 2 112.u odd 12 1
784.2.p.f 2 112.x odd 12 1
1008.2.b.b 2 48.i odd 4 1
1008.2.b.b 2 336.v odd 4 1
1008.2.b.g 2 48.k even 4 1
1008.2.b.g 2 336.y even 4 1
1792.2.e.a 4 4.b odd 2 1
1792.2.e.a 4 7.b odd 2 1
1792.2.e.a 4 8.d odd 2 1
1792.2.e.a 4 56.h odd 2 1
1792.2.e.c 4 1.a even 1 1 trivial
1792.2.e.c 4 8.b even 2 1 inner
1792.2.e.c 4 28.d even 2 1 inner
1792.2.e.c 4 56.e even 2 1 inner
2800.2.e.b 4 80.i odd 4 1
2800.2.e.b 4 80.t odd 4 1
2800.2.e.b 4 560.u odd 4 1
2800.2.e.b 4 560.bm odd 4 1
2800.2.e.c 4 80.j even 4 1
2800.2.e.c 4 80.s even 4 1
2800.2.e.c 4 560.r even 4 1
2800.2.e.c 4 560.bn even 4 1
2800.2.k.b 2 80.q even 4 1
2800.2.k.b 2 560.be even 4 1
2800.2.k.e 2 80.k odd 4 1
2800.2.k.e 2 560.bf odd 4 1
4032.2.b.b 2 48.i odd 4 1
4032.2.b.b 2 336.v odd 4 1
4032.2.b.h 2 48.k even 4 1
4032.2.b.h 2 336.y even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{11}^{2} - 12 \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 4 + T^{2} )^{2} \)
$5$ \( ( -12 + T^{2} )^{2} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( ( -12 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( ( 12 + T^{2} )^{2} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 8 + T )^{4} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 48 + T^{2} )^{2} \)
$43$ \( ( -108 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( ( -12 + T^{2} )^{2} \)
$67$ \( ( -12 + T^{2} )^{2} \)
$71$ \( ( 12 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( 12 + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 48 + T^{2} )^{2} \)
$97$ \( ( 192 + T^{2} )^{2} \)
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