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Properties

Label 1000.2.q.c
Level $1000$
Weight $2$
Character orbit 1000.q
Analytic conductor $7.985$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 10 q^{9} + 6 q^{11} - 6 q^{19} - 4 q^{21} + 30 q^{23} - 2 q^{29} + 6 q^{31} + 40 q^{37} - 12 q^{39} + 20 q^{47} - 60 q^{49} - 60 q^{51} + 30 q^{53} - 30 q^{59} + 14 q^{61} + 20 q^{63} - 4 q^{69} + 12 q^{71} - 40 q^{73} + 16 q^{79} - 52 q^{81} - 30 q^{83} - 110 q^{87} + 24 q^{89} - 4 q^{91} - 30 q^{97} + 92 q^{99}+O(q^{100}) \) 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 −2.79539 + 0.908276i 0 0 0 3.13612i 0 4.56217 − 3.31461i 0
49.2 0 −1.92858 + 0.626633i 0 0 0 − 0.498275i 0 0.899692 − 0.653665i 0
49.3 0 −1.54478 + 0.501931i 0 0 0 − 2.51754i 0 −0.292627 + 0.212606i 0
49.4 0 −0.726954 + 0.236202i 0 0 0 4.32704i 0 −1.95438 + 1.41994i 0
49.5 0 0.685617 − 0.222770i 0 0 0 − 4.42421i 0 −2.00661 + 1.45789i 0
49.6 0 0.811472 − 0.263663i 0 0 0 − 1.47738i 0 −1.83808 + 1.33545i 0
49.7 0 2.57764 − 0.837527i 0 0 0 0.760910i 0 3.51574 − 2.55433i 0
49.8 0 2.92097 − 0.949081i 0 0 0 4.49756i 0 5.20427 − 3.78112i 0
449.1 0 −2.79539 − 0.908276i 0 0 0 − 3.13612i 0 4.56217 + 3.31461i 0
449.2 0 −1.92858 − 0.626633i 0 0 0 0.498275i 0 0.899692 + 0.653665i 0
449.3 0 −1.54478 − 0.501931i 0 0 0 2.51754i 0 −0.292627 − 0.212606i 0
449.4 0 −0.726954 − 0.236202i 0 0 0 − 4.32704i 0 −1.95438 − 1.41994i 0
449.5 0 0.685617 + 0.222770i 0 0 0 4.42421i 0 −2.00661 − 1.45789i 0
449.6 0 0.811472 + 0.263663i 0 0 0 1.47738i 0 −1.83808 − 1.33545i 0
449.7 0 2.57764 + 0.837527i 0 0 0 − 0.760910i 0 3.51574 + 2.55433i 0
449.8 0 2.92097 + 0.949081i 0 0 0 − 4.49756i 0 5.20427 + 3.78112i 0
649.1 0 −1.76114 + 2.42400i 0 0 0 − 4.21442i 0 −1.84712 − 5.68486i 0
649.2 0 −1.35643 + 1.86696i 0 0 0 4.74404i 0 −0.718596 − 2.21161i 0
649.3 0 −0.919962 + 1.26622i 0 0 0 0.0338937i 0 0.170070 + 0.523422i 0
649.4 0 −0.0350089 + 0.0481856i 0 0 0 − 1.71675i 0 0.925955 + 2.84980i 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
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.c 32
5.b even 2 1 200.2.q.a ✓ 32
5.c odd 4 1 1000.2.m.d 32
5.c odd 4 1 1000.2.m.e 32
20.d odd 2 1 400.2.y.d 32
25.d even 5 1 200.2.q.a ✓ 32
25.e even 10 1 inner 1000.2.q.c 32
25.f odd 20 1 1000.2.m.d 32
25.f odd 20 1 1000.2.m.e 32
25.f odd 20 1 5000.2.a.q 16
25.f odd 20 1 5000.2.a.r 16
100.j odd 10 1 400.2.y.d 32
100.l even 20 1 10000.2.a.bq 16
100.l even 20 1 10000.2.a.br 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.q.a ✓ 32 5.b even 2 1
200.2.q.a ✓ 32 25.d even 5 1
400.2.y.d 32 20.d odd 2 1
400.2.y.d 32 100.j odd 10 1
1000.2.m.d 32 5.c odd 4 1
1000.2.m.d 32 25.f odd 20 1
1000.2.m.e 32 5.c odd 4 1
1000.2.m.e 32 25.f odd 20 1
1000.2.q.c 32 1.a even 1 1 trivial
1000.2.q.c 32 25.e even 10 1 inner
5000.2.a.q 16 25.f odd 20 1
5000.2.a.r 16 25.f odd 20 1
10000.2.a.bq 16 100.l even 20 1
10000.2.a.br 16 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 17 T_{3}^{30} + 216 T_{3}^{28} - 2500 T_{3}^{26} - 460 T_{3}^{25} + 25770 T_{3}^{24} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display