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Properties

Label 1792.3.d.g
Level $1792$
Weight $3$
Character orbit 1792.d
Analytic conductor $48.828$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,3,Mod(1023,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} + ( - \beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_1 q^{7} + ( - \beta_{4} + 3) q^{9} + (2 \beta_{3} - 4 \beta_{2}) q^{11} + \beta_{6} q^{13} + 2 \beta_1 q^{15} + (\beta_{4} + 18) q^{17} + ( - 9 \beta_{3} + 10 \beta_{2}) q^{19} + \beta_{5} q^{21} + ( - 2 \beta_{7} - 2 \beta_1) q^{23} + ( - 7 \beta_{4} + 17) q^{25} + (6 \beta_{3} + 2 \beta_{2}) q^{27} + ( - 3 \beta_{6} + 3 \beta_{5}) q^{29} + ( - \beta_{7} - 12 \beta_1) q^{31} + ( - 2 \beta_{4} - 4) q^{33} - 7 \beta_{2} q^{35} + (3 \beta_{6} - 7 \beta_{5}) q^{37} + 2 \beta_1 q^{39} + (3 \beta_{4} + 10) q^{41} + (4 \beta_{3} + 6 \beta_{2}) q^{43} + (9 \beta_{6} - 2 \beta_{5}) q^{45} + ( - \beta_{7} - 8 \beta_1) q^{47} - 7 q^{49} + (24 \beta_{3} - 2 \beta_{2}) q^{51} + ( - \beta_{6} + 11 \beta_{5}) q^{53} + (4 \beta_{7} - 20 \beta_1) q^{55} + (9 \beta_{4} + 34) q^{57} + ( - 17 \beta_{3} - 6 \beta_{2}) q^{59} + (\beta_{6} + 4 \beta_{5}) q^{61} + (\beta_{7} - 3 \beta_1) q^{63} + ( - 7 \beta_{4} + 42) q^{65} + ( - 6 \beta_{3} - 22 \beta_{2}) q^{67} + ( - 4 \beta_{6} + 14 \beta_{5}) q^{69} + ( - 4 \beta_{7} - 16 \beta_1) q^{71} + ( - 2 \beta_{4} - 58) q^{73} + ( - 25 \beta_{3} + 14 \beta_{2}) q^{75} + ( - 4 \beta_{6} + 2 \beta_{5}) q^{77} + (4 \beta_{7} - 8 \beta_1) q^{79} + ( - 15 \beta_{4} - 13) q^{81} + ( - \beta_{3} + 12 \beta_{2}) q^{83} + (12 \beta_{6} + 2 \beta_{5}) q^{85} + (3 \beta_{7} + 12 \beta_1) q^{87} + (6 \beta_{4} - 78) q^{89} - 7 \beta_{2} q^{91} + ( - 2 \beta_{6} + 18 \beta_{5}) q^{93} + ( - 10 \beta_{7} + 42 \beta_1) q^{95} + (23 \beta_{4} - 34) q^{97} + (2 \beta_{3} - 32 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 4\nu^{6} - 4\nu^{5} - 7\nu^{3} + 20\nu^{2} + 4\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} + 4\nu^{5} + 7\nu^{3} + 20\nu^{2} - 4\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 12\nu^{6} + 4\nu^{5} - 13\nu^{3} + 36\nu^{2} + 44\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 12\nu^{6} - 4\nu^{5} + 13\nu^{3} + 36\nu^{2} - 44\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 4\nu^{5} + 13\nu^{3} + 44\nu ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 3\beta_{3} + 3\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - 2\beta_{5} + 5\beta_{4} + 10\beta_{3} - 10\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} - 11\beta_{4} + 22\beta_{3} - 22\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 5\beta_{5} + 9\beta_{3} + 9\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 14\beta_{6} - 14\beta_{5} - 13\beta_{4} - 26\beta_{3} + 26\beta_{2} ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1023.1
−0.581861 + 1.28897i
1.28897 − 0.581861i
0.581861 − 1.28897i
−1.28897 + 0.581861i
0.581861 + 1.28897i
−1.28897 − 0.581861i
−0.581861 − 1.28897i
1.28897 + 0.581861i
0 − 3.41421i 0 −1.54985 0 − 2.64575i 0 −2.65685 0
1023.2 0 − 3.41421i 0 1.54985 0 2.64575i 0 −2.65685 0
1023.3 0 − 0.585786i 0 −9.03316 0 − 2.64575i 0 8.65685 0
1023.4 0 − 0.585786i 0 9.03316 0 2.64575i 0 8.65685 0
1023.5 0 0.585786i 0 −9.03316 0 2.64575i 0 8.65685 0
1023.6 0 0.585786i 0 9.03316 0 − 2.64575i 0 8.65685 0
1023.7 0 3.41421i 0 −1.54985 0 2.64575i 0 −2.65685 0
1023.8 0 3.41421i 0 1.54985 0 − 2.64575i 0 −2.65685 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1023.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.3.d.g 8
4.b odd 2 1 inner 1792.3.d.g 8
8.b even 2 1 inner 1792.3.d.g 8
8.d odd 2 1 inner 1792.3.d.g 8
16.e even 4 1 56.3.g.a ✓ 4
16.e even 4 1 224.3.g.a 4
16.f odd 4 1 56.3.g.a ✓ 4
16.f odd 4 1 224.3.g.a 4
48.i odd 4 1 504.3.g.a 4
48.i odd 4 1 2016.3.g.a 4
48.k even 4 1 504.3.g.a 4
48.k even 4 1 2016.3.g.a 4
112.j even 4 1 392.3.g.h 4
112.j even 4 1 1568.3.g.h 4
112.l odd 4 1 392.3.g.h 4
112.l odd 4 1 1568.3.g.h 4
112.u odd 12 2 392.3.k.i 8
112.v even 12 2 392.3.k.j 8
112.w even 12 2 392.3.k.i 8
112.x odd 12 2 392.3.k.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a ✓ 4 16.e even 4 1
56.3.g.a ✓ 4 16.f odd 4 1
224.3.g.a 4 16.e even 4 1
224.3.g.a 4 16.f odd 4 1
392.3.g.h 4 112.j even 4 1
392.3.g.h 4 112.l odd 4 1
392.3.k.i 8 112.u odd 12 2
392.3.k.i 8 112.w even 12 2
392.3.k.j 8 112.v even 12 2
392.3.k.j 8 112.x odd 12 2
504.3.g.a 4 48.i odd 4 1
504.3.g.a 4 48.k even 4 1
1568.3.g.h 4 112.j even 4 1
1568.3.g.h 4 112.l odd 4 1
1792.3.d.g 8 1.a even 1 1 trivial
1792.3.d.g 8 4.b odd 2 1 inner
1792.3.d.g 8 8.b even 2 1 inner
1792.3.d.g 8 8.d odd 2 1 inner
2016.3.g.a 4 48.i odd 4 1
2016.3.g.a 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{4} + 12T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} - 84T_{5}^{2} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 12 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 84 T^{2} + 196)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 176 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 84 T^{2} + 196)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 36 T + 292)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1452 T^{2} + 515524)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1848 T^{2} + 753424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 504)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2464 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3696 T^{2} + 906304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 188)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 816 T^{2} + 153664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1344 T^{2} + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 9632 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 4716 T^{2} + 3511876)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 1652 T^{2} + 329476)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 7296 T^{2} + 6885376)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 10752 T^{2} + 3211264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 116 T + 3236)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 8064 T^{2} + 9834496)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1644 T^{2} + 21316)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 156 T + 4932)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 68 T - 15772)^{4} \) Copy content Toggle raw display
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