Properties

Label 1792.2.e.f
Level $1792$
Weight $2$
Character orbit 1792.e
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
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 a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{3} + ( \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{5} + ( -\zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( -1 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{3} + ( \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{5} + ( -\zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( -1 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{9} -2 q^{11} + ( \zeta_{16} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{13} + ( 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{15} + ( 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{17} + ( \zeta_{16} - 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{19} + ( 3 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} - 2 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{21} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{23} + ( -1 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{25} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{27} + ( 4 \zeta_{16}^{2} + 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{31} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{33} + ( -4 - \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{35} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{37} + ( 2 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{39} + ( 6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{41} + ( -2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{43} + ( -\zeta_{16} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{45} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{47} + ( 1 + 2 \zeta_{16} - 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{49} + ( -8 - 8 \zeta_{16}^{2} + 8 \zeta_{16}^{6} ) q^{51} -2 \zeta_{16}^{4} q^{53} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{55} + ( 4 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{57} + ( 5 \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{59} + ( \zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{61} + ( \zeta_{16} + 3 \zeta_{16}^{2} + 3 \zeta_{16}^{3} + 5 \zeta_{16}^{4} - 3 \zeta_{16}^{5} + 3 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{63} + ( -4 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{65} + ( 10 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{67} + ( -6 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{69} + ( -2 \zeta_{16}^{2} + 6 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{71} + ( -6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{73} + ( -\zeta_{16} - 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{75} + ( 2 \zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( -2 \zeta_{16}^{2} + 10 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{79} + ( -3 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{81} + ( 5 \zeta_{16} - 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{83} + 8 \zeta_{16}^{4} q^{85} + ( -10 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 10 \zeta_{16}^{7} ) q^{87} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{89} + ( 4 - \zeta_{16} - 6 \zeta_{16}^{2} - 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} + 6 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{91} + ( -8 \zeta_{16}^{2} - 8 \zeta_{16}^{4} - 8 \zeta_{16}^{6} ) q^{93} + ( 2 \zeta_{16}^{2} - 8 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{95} + ( 4 \zeta_{16} - 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{97} + ( 2 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{9} - 16q^{11} - 8q^{25} - 32q^{35} - 16q^{43} + 8q^{49} - 64q^{51} + 32q^{57} - 32q^{65} + 80q^{67} - 24q^{81} + 32q^{91} + 16q^{99} + O(q^{100}) \)

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
−0.923880 0.382683i
0.923880 0.382683i
−0.382683 0.923880i
0.382683 0.923880i
−0.382683 + 0.923880i
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 + 0.382683i
0 2.61313i 0 −1.08239 0 1.08239 2.41421i 0 −3.82843 0
895.2 0 2.61313i 0 1.08239 0 −1.08239 + 2.41421i 0 −3.82843 0
895.3 0 1.08239i 0 −2.61313 0 2.61313 0.414214i 0 1.82843 0
895.4 0 1.08239i 0 2.61313 0 −2.61313 + 0.414214i 0 1.82843 0
895.5 0 1.08239i 0 −2.61313 0 2.61313 + 0.414214i 0 1.82843 0
895.6 0 1.08239i 0 2.61313 0 −2.61313 0.414214i 0 1.82843 0
895.7 0 2.61313i 0 −1.08239 0 1.08239 + 2.41421i 0 −3.82843 0
895.8 0 2.61313i 0 1.08239 0 −1.08239 2.41421i 0 −3.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 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.f 8
4.b odd 2 1 1792.2.e.g 8
7.b odd 2 1 inner 1792.2.e.f 8
8.b even 2 1 1792.2.e.g 8
8.d odd 2 1 inner 1792.2.e.f 8
16.e even 4 1 224.2.f.a 8
16.e even 4 1 448.2.f.d 8
16.f odd 4 1 224.2.f.a 8
16.f odd 4 1 448.2.f.d 8
28.d even 2 1 1792.2.e.g 8
48.i odd 4 1 2016.2.b.b 8
48.i odd 4 1 4032.2.b.p 8
48.k even 4 1 2016.2.b.b 8
48.k even 4 1 4032.2.b.p 8
56.e even 2 1 inner 1792.2.e.f 8
56.h odd 2 1 1792.2.e.g 8
112.j even 4 1 224.2.f.a 8
112.j even 4 1 448.2.f.d 8
112.l odd 4 1 224.2.f.a 8
112.l odd 4 1 448.2.f.d 8
112.u odd 12 2 1568.2.p.a 16
112.v even 12 2 1568.2.p.a 16
112.w even 12 2 1568.2.p.a 16
112.x odd 12 2 1568.2.p.a 16
336.v odd 4 1 2016.2.b.b 8
336.v odd 4 1 4032.2.b.p 8
336.y even 4 1 2016.2.b.b 8
336.y even 4 1 4032.2.b.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.f.a 8 16.e even 4 1
224.2.f.a 8 16.f odd 4 1
224.2.f.a 8 112.j even 4 1
224.2.f.a 8 112.l odd 4 1
448.2.f.d 8 16.e even 4 1
448.2.f.d 8 16.f odd 4 1
448.2.f.d 8 112.j even 4 1
448.2.f.d 8 112.l odd 4 1
1568.2.p.a 16 112.u odd 12 2
1568.2.p.a 16 112.v even 12 2
1568.2.p.a 16 112.w even 12 2
1568.2.p.a 16 112.x odd 12 2
1792.2.e.f 8 1.a even 1 1 trivial
1792.2.e.f 8 7.b odd 2 1 inner
1792.2.e.f 8 8.d odd 2 1 inner
1792.2.e.f 8 56.e even 2 1 inner
1792.2.e.g 8 4.b odd 2 1
1792.2.e.g 8 8.b even 2 1
1792.2.e.g 8 28.d even 2 1
1792.2.e.g 8 56.h odd 2 1
2016.2.b.b 8 48.i odd 4 1
2016.2.b.b 8 48.k even 4 1
2016.2.b.b 8 336.v odd 4 1
2016.2.b.b 8 336.y even 4 1
4032.2.b.p 8 48.i odd 4 1
4032.2.b.p 8 48.k even 4 1
4032.2.b.p 8 336.v odd 4 1
4032.2.b.p 8 336.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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{4} + 8 T_{3}^{2} + 8 \)
\( T_{11} + 2 \)
\( T_{31}^{4} - 64 T_{31}^{2} + 512 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$5$ \( ( 8 - 8 T^{2} + T^{4} )^{2} \)
$7$ \( 2401 - 196 T^{2} - 26 T^{4} - 4 T^{6} + T^{8} \)
$11$ \( ( 2 + T )^{8} \)
$13$ \( ( 392 - 40 T^{2} + T^{4} )^{2} \)
$17$ \( ( 512 + 64 T^{2} + T^{4} )^{2} \)
$19$ \( ( 392 + 40 T^{2} + T^{4} )^{2} \)
$23$ \( ( 784 + 72 T^{2} + T^{4} )^{2} \)
$29$ \( ( 784 + 72 T^{2} + T^{4} )^{2} \)
$31$ \( ( 512 - 64 T^{2} + T^{4} )^{2} \)
$37$ \( ( 784 + 72 T^{2} + T^{4} )^{2} \)
$41$ \( ( 6272 + 160 T^{2} + T^{4} )^{2} \)
$43$ \( ( -28 + 4 T + T^{2} )^{4} \)
$47$ \( ( 512 - 64 T^{2} + T^{4} )^{2} \)
$53$ \( ( 4 + T^{2} )^{4} \)
$59$ \( ( 2312 + 104 T^{2} + T^{4} )^{2} \)
$61$ \( ( 8 - 8 T^{2} + T^{4} )^{2} \)
$67$ \( ( 68 - 20 T + T^{2} )^{4} \)
$71$ \( ( 784 + 88 T^{2} + T^{4} )^{2} \)
$73$ \( ( 128 + 160 T^{2} + T^{4} )^{2} \)
$79$ \( ( 8464 + 216 T^{2} + T^{4} )^{2} \)
$83$ \( ( 392 + 136 T^{2} + T^{4} )^{2} \)
$89$ \( ( 128 + 32 T^{2} + T^{4} )^{2} \)
$97$ \( ( 25088 + 320 T^{2} + T^{4} )^{2} \)
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