Properties

Label 1792.2.f.f
Level $1792$
Weight $2$
Character orbit 1792.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 3 q^{9} -2 \beta_{1} q^{11} -\beta_{2} q^{13} -6 \beta_{1} q^{15} + 2 \beta_{2} q^{17} -\beta_{3} q^{19} + ( 6 - \beta_{2} ) q^{21} -4 \beta_{1} q^{23} - q^{25} + 4 q^{29} + 2 \beta_{3} q^{31} -2 \beta_{2} q^{33} + ( -6 \beta_{1} - \beta_{3} ) q^{35} -8 q^{37} -6 \beta_{1} q^{39} + 6 \beta_{1} q^{43} -3 \beta_{2} q^{45} -2 \beta_{3} q^{47} + ( 5 - 2 \beta_{2} ) q^{49} + 12 \beta_{1} q^{51} + 4 q^{53} -2 \beta_{3} q^{55} -6 q^{57} + \beta_{3} q^{59} -3 \beta_{2} q^{61} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{63} -6 q^{65} -2 \beta_{1} q^{67} -4 \beta_{2} q^{69} + 10 \beta_{1} q^{71} + 6 \beta_{2} q^{73} -\beta_{3} q^{75} + ( -2 - 2 \beta_{2} ) q^{77} + 6 \beta_{1} q^{79} -9 q^{81} + \beta_{3} q^{83} + 12 q^{85} + 4 \beta_{3} q^{87} -6 \beta_{2} q^{89} + ( -6 \beta_{1} - \beta_{3} ) q^{91} + 12 q^{93} + 6 \beta_{1} q^{95} + 2 \beta_{2} q^{97} -6 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 24q^{21} - 4q^{25} + 16q^{29} - 32q^{37} + 20q^{49} + 16q^{53} - 24q^{57} - 24q^{65} - 8q^{77} - 36q^{81} + 48q^{85} + 48q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.22474 1.22474i
0 −2.44949 0 2.44949i 0 −2.44949 + 1.00000i 0 3.00000 0
1791.2 0 −2.44949 0 2.44949i 0 −2.44949 1.00000i 0 3.00000 0
1791.3 0 2.44949 0 2.44949i 0 2.44949 1.00000i 0 3.00000 0
1791.4 0 2.44949 0 2.44949i 0 2.44949 + 1.00000i 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
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.f.f 4
4.b odd 2 1 inner 1792.2.f.f 4
7.b odd 2 1 inner 1792.2.f.f 4
8.b even 2 1 1792.2.f.e 4
8.d odd 2 1 1792.2.f.e 4
16.e even 4 1 56.2.e.b 4
16.e even 4 1 224.2.e.b 4
16.f odd 4 1 56.2.e.b 4
16.f odd 4 1 224.2.e.b 4
28.d even 2 1 inner 1792.2.f.f 4
48.i odd 4 1 504.2.p.f 4
48.i odd 4 1 2016.2.p.e 4
48.k even 4 1 504.2.p.f 4
48.k even 4 1 2016.2.p.e 4
56.e even 2 1 1792.2.f.e 4
56.h odd 2 1 1792.2.f.e 4
112.j even 4 1 56.2.e.b 4
112.j even 4 1 224.2.e.b 4
112.l odd 4 1 56.2.e.b 4
112.l odd 4 1 224.2.e.b 4
112.u odd 12 2 392.2.m.f 8
112.u odd 12 2 1568.2.q.e 8
112.v even 12 2 392.2.m.f 8
112.v even 12 2 1568.2.q.e 8
112.w even 12 2 392.2.m.f 8
112.w even 12 2 1568.2.q.e 8
112.x odd 12 2 392.2.m.f 8
112.x odd 12 2 1568.2.q.e 8
336.v odd 4 1 504.2.p.f 4
336.v odd 4 1 2016.2.p.e 4
336.y even 4 1 504.2.p.f 4
336.y even 4 1 2016.2.p.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 16.e even 4 1
56.2.e.b 4 16.f odd 4 1
56.2.e.b 4 112.j even 4 1
56.2.e.b 4 112.l odd 4 1
224.2.e.b 4 16.e even 4 1
224.2.e.b 4 16.f odd 4 1
224.2.e.b 4 112.j even 4 1
224.2.e.b 4 112.l odd 4 1
392.2.m.f 8 112.u odd 12 2
392.2.m.f 8 112.v even 12 2
392.2.m.f 8 112.w even 12 2
392.2.m.f 8 112.x odd 12 2
504.2.p.f 4 48.i odd 4 1
504.2.p.f 4 48.k even 4 1
504.2.p.f 4 336.v odd 4 1
504.2.p.f 4 336.y even 4 1
1568.2.q.e 8 112.u odd 12 2
1568.2.q.e 8 112.v even 12 2
1568.2.q.e 8 112.w even 12 2
1568.2.q.e 8 112.x odd 12 2
1792.2.f.e 4 8.b even 2 1
1792.2.f.e 4 8.d odd 2 1
1792.2.f.e 4 56.e even 2 1
1792.2.f.e 4 56.h odd 2 1
1792.2.f.f 4 1.a even 1 1 trivial
1792.2.f.f 4 4.b odd 2 1 inner
1792.2.f.f 4 7.b odd 2 1 inner
1792.2.f.f 4 28.d even 2 1 inner
2016.2.p.e 4 48.i odd 4 1
2016.2.p.e 4 48.k even 4 1
2016.2.p.e 4 336.v odd 4 1
2016.2.p.e 4 336.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 \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{2} - 6 \)
\( T_{5}^{2} + 6 \)
\( T_{29} - 4 \)
\( T_{31}^{2} - 24 \)
\( T_{37} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -6 + T^{2} )^{2} \)
$5$ \( ( 6 + T^{2} )^{2} \)
$7$ \( 49 - 10 T^{2} + T^{4} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( 6 + T^{2} )^{2} \)
$17$ \( ( 24 + T^{2} )^{2} \)
$19$ \( ( -6 + T^{2} )^{2} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( ( -24 + T^{2} )^{2} \)
$37$ \( ( 8 + T )^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( ( -24 + T^{2} )^{2} \)
$53$ \( ( -4 + T )^{4} \)
$59$ \( ( -6 + T^{2} )^{2} \)
$61$ \( ( 54 + T^{2} )^{2} \)
$67$ \( ( 4 + T^{2} )^{2} \)
$71$ \( ( 100 + T^{2} )^{2} \)
$73$ \( ( 216 + T^{2} )^{2} \)
$79$ \( ( 36 + T^{2} )^{2} \)
$83$ \( ( -6 + T^{2} )^{2} \)
$89$ \( ( 216 + T^{2} )^{2} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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