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Properties

Label 1000.2.q.d
Level $1000$
Weight $2$
Character orbit 1000.q
Analytic conductor $7.985$
Analytic rank $0$
Dimension $32$
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] = [1000,2,Mod(49,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 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("1000.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.q (of order \(10\), degree \(4\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 22 q^{9} - 20 q^{11} - 6 q^{21} - 10 q^{29} - 18 q^{31} - 28 q^{39} - 4 q^{41} - 36 q^{49} + 200 q^{51} + 106 q^{59} + 8 q^{61} + 14 q^{69} - 12 q^{71} + 44 q^{79} - 12 q^{81} - 40 q^{89} - 54 q^{91} - 244 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −3.05115 + 0.991378i 0 0 0 2.69216i 0 5.89962 − 4.28633i 0
49.2 0 −2.37758 + 0.772523i 0 0 0 1.96923i 0 2.62905 − 1.91012i 0
49.3 0 −1.42381 + 0.462625i 0 0 0 − 4.94031i 0 −0.613832 + 0.445975i 0
49.4 0 −1.14632 + 0.372462i 0 0 0 − 1.59935i 0 −1.25173 + 0.909432i 0
49.5 0 1.14632 − 0.372462i 0 0 0 1.59935i 0 −1.25173 + 0.909432i 0
49.6 0 1.42381 − 0.462625i 0 0 0 4.94031i 0 −0.613832 + 0.445975i 0
49.7 0 2.37758 − 0.772523i 0 0 0 − 1.96923i 0 2.62905 − 1.91012i 0
49.8 0 3.05115 − 0.991378i 0 0 0 − 2.69216i 0 5.89962 − 4.28633i 0
449.1 0 −3.05115 − 0.991378i 0 0 0 − 2.69216i 0 5.89962 + 4.28633i 0
449.2 0 −2.37758 − 0.772523i 0 0 0 − 1.96923i 0 2.62905 + 1.91012i 0
449.3 0 −1.42381 − 0.462625i 0 0 0 4.94031i 0 −0.613832 − 0.445975i 0
449.4 0 −1.14632 − 0.372462i 0 0 0 1.59935i 0 −1.25173 − 0.909432i 0
449.5 0 1.14632 + 0.372462i 0 0 0 − 1.59935i 0 −1.25173 − 0.909432i 0
449.6 0 1.42381 + 0.462625i 0 0 0 − 4.94031i 0 −0.613832 − 0.445975i 0
449.7 0 2.37758 + 0.772523i 0 0 0 1.96923i 0 2.62905 + 1.91012i 0
449.8 0 3.05115 + 0.991378i 0 0 0 2.69216i 0 5.89962 + 4.28633i 0
649.1 0 −1.80328 + 2.48200i 0 0 0 1.36851i 0 −1.98145 − 6.09828i 0
649.2 0 −1.27109 + 1.74951i 0 0 0 − 3.20389i 0 −0.518051 − 1.59440i 0
649.3 0 −0.565053 + 0.777729i 0 0 0 − 0.364298i 0 0.641474 + 1.97425i 0
649.4 0 −0.509453 + 0.701201i 0 0 0 3.82614i 0 0.694910 + 2.13871i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.q.d 32
5.b even 2 1 inner 1000.2.q.d 32
5.c odd 4 1 200.2.m.c ✓ 16
5.c odd 4 1 1000.2.m.c 16
20.e even 4 1 400.2.u.g 16
25.d even 5 1 inner 1000.2.q.d 32
25.e even 10 1 inner 1000.2.q.d 32
25.f odd 20 1 200.2.m.c ✓ 16
25.f odd 20 1 1000.2.m.c 16
25.f odd 20 1 5000.2.a.l 8
25.f odd 20 1 5000.2.a.m 8
100.l even 20 1 400.2.u.g 16
100.l even 20 1 10000.2.a.bh 8
100.l even 20 1 10000.2.a.bk 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.c ✓ 16 5.c odd 4 1
200.2.m.c ✓ 16 25.f odd 20 1
400.2.u.g 16 20.e even 4 1
400.2.u.g 16 100.l even 20 1
1000.2.m.c 16 5.c odd 4 1
1000.2.m.c 16 25.f odd 20 1
1000.2.q.d 32 1.a even 1 1 trivial
1000.2.q.d 32 5.b even 2 1 inner
1000.2.q.d 32 25.d even 5 1 inner
1000.2.q.d 32 25.e even 10 1 inner
5000.2.a.l 8 25.f odd 20 1
5000.2.a.m 8 25.f odd 20 1
10000.2.a.bh 8 100.l even 20 1
10000.2.a.bk 8 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 23 T_{3}^{30} + 308 T_{3}^{28} - 3256 T_{3}^{26} + 33230 T_{3}^{24} - 242531 T_{3}^{22} + \cdots + 40960000 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display