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Properties

Label 1024.2.g.g
Level $1024$
Weight $2$
Character orbit 1024.g
Analytic conductor $8.177$
Analytic rank $0$
Dimension $16$
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] = [1024,2,Mod(129,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.g (of order \(8\), degree \(4\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{14} + \beta_{11}) q^{3} + (\beta_{13} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{10} + \beta_{8} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} + \beta_{11}) q^{3} + (\beta_{13} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{14} + \beta_{12} + \cdots - 2 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} + 16 q^{9} - 24 q^{13} + 48 q^{21} + 32 q^{25} + 8 q^{29} - 80 q^{33} - 8 q^{37} + 16 q^{41} - 8 q^{45} - 40 q^{53} + 16 q^{57} - 8 q^{61} - 32 q^{65} - 32 q^{73} - 32 q^{77} + 32 q^{85} - 32 q^{89} - 48 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{10} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{48}^{14} + \zeta_{48}^{8} + \zeta_{48}^{4} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{48}^{15} + \zeta_{48}^{5} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{48}^{13} + \zeta_{48}^{9} + \zeta_{48}^{3} - \zeta_{48} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{48}^{15} - \zeta_{48}^{5} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{5} - \zeta_{48} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{8} - \zeta_{48}^{4} - \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{9} - \zeta_{48}^{3} - \zeta_{48} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{9} + \zeta_{48}^{3} - \zeta_{48} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( -\zeta_{48}^{14} - \zeta_{48}^{12} - \zeta_{48}^{8} + \zeta_{48}^{4} - \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{13} + \zeta_{48}^{11} + \zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{7} + \zeta_{48}^{5} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{12} - \beta_{11} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( ( \beta_{13} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{9} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{9} - \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( ( \beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{11} + \beta_{7} + \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( -\beta_{15} + \beta_{14} + \beta_{12} - \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( -\beta_{12} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( -\beta_{13} - \beta_{10} - \beta_{8} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{9} - \beta_{7} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.991445 + 0.130526i
−0.608761 − 0.793353i
0.608761 + 0.793353i
0.991445 − 0.130526i
−0.793353 − 0.608761i
−0.130526 + 0.991445i
0.130526 − 0.991445i
0.793353 + 0.608761i
−0.793353 + 0.608761i
−0.130526 − 0.991445i
0.130526 + 0.991445i
0.793353 − 0.608761i
−0.991445 − 0.130526i
−0.608761 + 0.793353i
0.608761 − 0.793353i
0.991445 + 0.130526i
0 −0.580775 + 1.40211i 0 3.29788 − 1.36603i 0 −1.02642 + 1.02642i 0 0.492694 + 0.492694i 0
129.2 0 −0.356604 + 0.860919i 0 −0.883663 + 0.366025i 0 2.35207 − 2.35207i 0 1.50731 + 1.50731i 0
129.3 0 0.356604 − 0.860919i 0 −0.883663 + 0.366025i 0 −2.35207 + 2.35207i 0 1.50731 + 1.50731i 0
129.4 0 0.580775 − 1.40211i 0 3.29788 − 1.36603i 0 1.02642 − 1.02642i 0 0.492694 + 0.492694i 0
385.1 0 −2.70868 + 1.12197i 0 0.151613 − 0.366025i 0 −3.06528 − 3.06528i 0 3.95680 − 3.95680i 0
385.2 0 −0.445644 + 0.184592i 0 −0.565826 + 1.36603i 0 0.135131 + 0.135131i 0 −1.95680 + 1.95680i 0
385.3 0 0.445644 − 0.184592i 0 −0.565826 + 1.36603i 0 −0.135131 − 0.135131i 0 −1.95680 + 1.95680i 0
385.4 0 2.70868 − 1.12197i 0 0.151613 − 0.366025i 0 3.06528 + 3.06528i 0 3.95680 − 3.95680i 0
641.1 0 −2.70868 − 1.12197i 0 0.151613 + 0.366025i 0 −3.06528 + 3.06528i 0 3.95680 + 3.95680i 0
641.2 0 −0.445644 − 0.184592i 0 −0.565826 − 1.36603i 0 0.135131 − 0.135131i 0 −1.95680 − 1.95680i 0
641.3 0 0.445644 + 0.184592i 0 −0.565826 − 1.36603i 0 −0.135131 + 0.135131i 0 −1.95680 − 1.95680i 0
641.4 0 2.70868 + 1.12197i 0 0.151613 + 0.366025i 0 3.06528 − 3.06528i 0 3.95680 + 3.95680i 0
897.1 0 −0.580775 − 1.40211i 0 3.29788 + 1.36603i 0 −1.02642 − 1.02642i 0 0.492694 − 0.492694i 0
897.2 0 −0.356604 − 0.860919i 0 −0.883663 − 0.366025i 0 2.35207 + 2.35207i 0 1.50731 − 1.50731i 0
897.3 0 0.356604 + 0.860919i 0 −0.883663 − 0.366025i 0 −2.35207 − 2.35207i 0 1.50731 − 1.50731i 0
897.4 0 0.580775 + 1.40211i 0 3.29788 + 1.36603i 0 1.02642 + 1.02642i 0 0.492694 − 0.492694i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.g.g yes 16
4.b odd 2 1 inner 1024.2.g.g yes 16
8.b even 2 1 1024.2.g.d yes 16
8.d odd 2 1 1024.2.g.d yes 16
16.e even 4 1 1024.2.g.a ✓ 16
16.e even 4 1 1024.2.g.f yes 16
16.f odd 4 1 1024.2.g.a ✓ 16
16.f odd 4 1 1024.2.g.f yes 16
32.g even 8 1 1024.2.g.a ✓ 16
32.g even 8 1 1024.2.g.d yes 16
32.g even 8 1 1024.2.g.f yes 16
32.g even 8 1 inner 1024.2.g.g yes 16
32.h odd 8 1 1024.2.g.a ✓ 16
32.h odd 8 1 1024.2.g.d yes 16
32.h odd 8 1 1024.2.g.f yes 16
32.h odd 8 1 inner 1024.2.g.g yes 16
64.i even 16 1 4096.2.a.i 8
64.i even 16 1 4096.2.a.s 8
64.j odd 16 1 4096.2.a.i 8
64.j odd 16 1 4096.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1024.2.g.a ✓ 16 16.e even 4 1
1024.2.g.a ✓ 16 16.f odd 4 1
1024.2.g.a ✓ 16 32.g even 8 1
1024.2.g.a ✓ 16 32.h odd 8 1
1024.2.g.d yes 16 8.b even 2 1
1024.2.g.d yes 16 8.d odd 2 1
1024.2.g.d yes 16 32.g even 8 1
1024.2.g.d yes 16 32.h odd 8 1
1024.2.g.f yes 16 16.e even 4 1
1024.2.g.f yes 16 16.f odd 4 1
1024.2.g.f yes 16 32.g even 8 1
1024.2.g.f yes 16 32.h odd 8 1
1024.2.g.g yes 16 1.a even 1 1 trivial
1024.2.g.g yes 16 4.b odd 2 1 inner
1024.2.g.g yes 16 32.g even 8 1 inner
1024.2.g.g yes 16 32.h odd 8 1 inner
4096.2.a.i 8 64.i even 16 1
4096.2.a.i 8 64.j odd 16 1
4096.2.a.s 8 64.i even 16 1
4096.2.a.s 8 64.j odd 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{16} - 8T_{3}^{14} + 32T_{3}^{12} + 208T_{3}^{10} + 568T_{3}^{8} + 416T_{3}^{6} + 128T_{3}^{4} - 64T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} + 8T_{5}^{5} + 32T_{5}^{4} + 40T_{5}^{3} + 16T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + 8 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 480 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 406586896 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 72 T^{6} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 24 T^{14} + \cdots + 4477456 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 6505390336 \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 1119364)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 224 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 8836)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1749006250000 \) Copy content Toggle raw display
$47$ \( (T^{8} + 256 T^{6} + \cdots + 1364224)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 20 T^{7} + \cdots + 21316)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 36030006250000 \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 78650008458256 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 355196928256 \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 2062096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 128 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21848556971536 \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} + \cdots + 85264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 376)^{4} \) Copy content Toggle raw display
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