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Properties

Label 1024.2.b.h
Level $1024$
Weight $2$
Character orbit 1024.b
Analytic conductor $8.177$
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] = [1024,2,Mod(513,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.513");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.b (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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 256)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{4} q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{4} q^{7} + ( - \beta_{2} - 1) q^{9} - \beta_{6} q^{11} + (\beta_{5} + \beta_{3}) q^{13} + \beta_{7} q^{15} + \beta_{2} q^{17} + \beta_{6} q^{19} + \beta_{3} q^{21} + \beta_{7} q^{23} - q^{25} + ( - \beta_{6} - \beta_1) q^{27} + \beta_{5} q^{29} + (\beta_{7} + \beta_{4}) q^{31} + \beta_{2} q^{33} + 2 \beta_{6} q^{35} + ( - \beta_{5} + \beta_{3}) q^{37} + (2 \beta_{7} - \beta_{4}) q^{39} + 2 \beta_{2} q^{41} - 3 \beta_1 q^{43} + ( - \beta_{5} - 3 \beta_{3}) q^{45} + ( - \beta_{7} + 3 \beta_{4}) q^{47} + ( - 2 \beta_{2} + 1) q^{49} + (\beta_{6} + 3 \beta_1) q^{51} + (\beta_{5} - 3 \beta_{3}) q^{53} + 3 \beta_{4} q^{55} - \beta_{2} q^{57} + ( - 2 \beta_{6} - 3 \beta_1) q^{59} + ( - \beta_{5} + \beta_{3}) q^{61} + (\beta_{7} + 2 \beta_{4}) q^{63} + ( - 2 \beta_{2} - 6) q^{65} + (\beta_{6} - 2 \beta_1) q^{67} + ( - 4 \beta_{5} - 3 \beta_{3}) q^{69} - \beta_{7} q^{71} + (\beta_{2} + 6) q^{73} - \beta_1 q^{75} + ( - 4 \beta_{5} + 3 \beta_{3}) q^{77} + (2 \beta_{7} - 2 \beta_{4}) q^{79} + ( - \beta_{2} + 1) q^{81} + 3 \beta_1 q^{83} + 3 \beta_{3} q^{85} + \beta_{7} q^{87} + (\beta_{2} + 6) q^{89} + 2 \beta_1 q^{91} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{93} - 3 \beta_{4} q^{95} - \beta_{2} q^{97} + ( - 2 \beta_{6} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} - 48 q^{65} + 48 q^{73} + 8 q^{81} + 48 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} + 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\zeta_{24}^{6} - 2\zeta_{24}^{4} + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{4} + \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{4} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{6} + 3\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{6} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} - \beta_{3} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
513.1
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0 − 2.73205i 0 − 2.44949i 0 1.03528 0 −4.46410 0
513.2 0 − 2.73205i 0 2.44949i 0 −1.03528 0 −4.46410 0
513.3 0 − 0.732051i 0 − 2.44949i 0 −3.86370 0 2.46410 0
513.4 0 − 0.732051i 0 2.44949i 0 3.86370 0 2.46410 0
513.5 0 0.732051i 0 − 2.44949i 0 3.86370 0 2.46410 0
513.6 0 0.732051i 0 2.44949i 0 −3.86370 0 2.46410 0
513.7 0 2.73205i 0 − 2.44949i 0 −1.03528 0 −4.46410 0
513.8 0 2.73205i 0 2.44949i 0 1.03528 0 −4.46410 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 513.8
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 1024.2.b.h 8
4.b odd 2 1 inner 1024.2.b.h 8
8.b even 2 1 inner 1024.2.b.h 8
8.d odd 2 1 inner 1024.2.b.h 8
16.e even 4 1 1024.2.a.g 4
16.e even 4 1 1024.2.a.j 4
16.f odd 4 1 1024.2.a.g 4
16.f odd 4 1 1024.2.a.j 4
32.g even 8 2 256.2.e.a ✓ 8
32.g even 8 2 256.2.e.b yes 8
32.h odd 8 2 256.2.e.a ✓ 8
32.h odd 8 2 256.2.e.b yes 8
48.i odd 4 1 9216.2.a.bb 4
48.i odd 4 1 9216.2.a.bk 4
48.k even 4 1 9216.2.a.bb 4
48.k even 4 1 9216.2.a.bk 4
96.o even 8 2 2304.2.k.f 8
96.o even 8 2 2304.2.k.k 8
96.p odd 8 2 2304.2.k.f 8
96.p odd 8 2 2304.2.k.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a ✓ 8 32.g even 8 2
256.2.e.a ✓ 8 32.h odd 8 2
256.2.e.b yes 8 32.g even 8 2
256.2.e.b yes 8 32.h odd 8 2
1024.2.a.g 4 16.e even 4 1
1024.2.a.g 4 16.f odd 4 1
1024.2.a.j 4 16.e even 4 1
1024.2.a.j 4 16.f odd 4 1
1024.2.b.h 8 1.a even 1 1 trivial
1024.2.b.h 8 4.b odd 2 1 inner
1024.2.b.h 8 8.b even 2 1 inner
1024.2.b.h 8 8.d odd 2 1 inner
2304.2.k.f 8 96.o even 8 2
2304.2.k.f 8 96.p odd 8 2
2304.2.k.k 8 96.o even 8 2
2304.2.k.k 8 96.p odd 8 2
9216.2.a.bb 4 48.i odd 4 1
9216.2.a.bb 4 48.k even 4 1
9216.2.a.bk 4 48.i odd 4 1
9216.2.a.bk 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 \(S_{2}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 24 T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 72 T^{2} + 324)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 156 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 168 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 256 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 72 T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
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