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Properties

Label 1000.2.f.e
Level $1000$
Weight $2$
Character orbit 1000.f
Analytic conductor $7.985$
Analytic rank $0$
Dimension $48$
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(749,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.f (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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{4} - 4 q^{6} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{4} - 4 q^{6} + 48 q^{9} + 12 q^{14} + 16 q^{16} + 20 q^{24} - 12 q^{26} - 16 q^{31} + 40 q^{36} + 12 q^{44} - 12 q^{46} - 48 q^{49} - 68 q^{54} + 8 q^{56} + 28 q^{64} - 88 q^{66} - 48 q^{71} - 20 q^{74} + 24 q^{76} - 16 q^{79} + 72 q^{81} - 28 q^{84} - 96 q^{86} + 24 q^{89} - 32 q^{94} - 40 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1 −1.41364 − 0.0403071i 1.98793 1.99675 + 0.113959i 0 −2.81021 − 0.0801275i − 3.65314i −2.81809 − 0.241581i 0.951849 0
749.2 −1.41364 + 0.0403071i 1.98793 1.99675 − 0.113959i 0 −2.81021 + 0.0801275i 3.65314i −2.81809 + 0.241581i 0.951849 0
749.3 −1.39764 − 0.215859i 0.671626 1.90681 + 0.603388i 0 −0.938694 − 0.144977i 0.0460000i −2.53479 − 1.25492i −2.54892 0
749.4 −1.39764 + 0.215859i 0.671626 1.90681 − 0.603388i 0 −0.938694 + 0.144977i − 0.0460000i −2.53479 + 1.25492i −2.54892 0
749.5 −1.36516 − 0.369256i −2.68433 1.72730 + 1.00818i 0 3.66453 + 0.991206i − 3.97895i −1.98576 − 2.01414i 4.20565 0
749.6 −1.36516 + 0.369256i −2.68433 1.72730 − 1.00818i 0 3.66453 − 0.991206i 3.97895i −1.98576 + 2.01414i 4.20565 0
749.7 −1.33620 − 0.463220i −2.27266 1.57086 + 1.23791i 0 3.03672 + 1.05274i 0.712051i −1.52555 − 2.38174i 2.16497 0
749.8 −1.33620 + 0.463220i −2.27266 1.57086 − 1.23791i 0 3.03672 − 1.05274i − 0.712051i −1.52555 + 2.38174i 2.16497 0
749.9 −1.21979 − 0.715619i 3.15903 0.975778 + 1.74581i 0 −3.85335 − 2.26066i 2.63004i 0.0590907 − 2.82781i 6.97946 0
749.10 −1.21979 + 0.715619i 3.15903 0.975778 − 1.74581i 0 −3.85335 + 2.26066i − 2.63004i 0.0590907 + 2.82781i 6.97946 0
749.11 −1.10251 − 0.885699i −0.211027 0.431074 + 1.95299i 0 0.232660 + 0.186907i − 0.0523737i 1.25450 − 2.53500i −2.95547 0
749.12 −1.10251 + 0.885699i −0.211027 0.431074 − 1.95299i 0 0.232660 − 0.186907i 0.0523737i 1.25450 + 2.53500i −2.95547 0
749.13 −0.865034 − 1.11880i −0.323264 −0.503434 + 1.93560i 0 0.279634 + 0.361668i 4.71845i 2.60104 − 1.11112i −2.89550 0
749.14 −0.865034 + 1.11880i −0.323264 −0.503434 − 1.93560i 0 0.279634 − 0.361668i − 4.71845i 2.60104 + 1.11112i −2.89550 0
749.15 −0.851143 − 1.12941i −2.02726 −0.551112 + 1.92257i 0 1.72549 + 2.28960i − 0.893028i 2.64044 − 1.01395i 1.10978 0
749.16 −0.851143 + 1.12941i −2.02726 −0.551112 − 1.92257i 0 1.72549 − 2.28960i 0.893028i 2.64044 + 1.01395i 1.10978 0
749.17 −0.679576 − 1.24023i 3.23939 −1.07635 + 1.68566i 0 −2.20141 − 4.01760i − 1.45476i 2.82208 + 0.189390i 7.49366 0
749.18 −0.679576 + 1.24023i 3.23939 −1.07635 − 1.68566i 0 −2.20141 + 4.01760i 1.45476i 2.82208 − 0.189390i 7.49366 0
749.19 −0.442189 − 1.34331i 0.559923 −1.60894 + 1.18799i 0 −0.247592 − 0.752148i − 3.28431i 2.30729 + 1.63598i −2.68649 0
749.20 −0.442189 + 1.34331i 0.559923 −1.60894 − 1.18799i 0 −0.247592 + 0.752148i 3.28431i 2.30729 − 1.63598i −2.68649 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.f.e 48
4.b odd 2 1 4000.2.f.e 48
5.b even 2 1 inner 1000.2.f.e 48
5.c odd 4 2 1000.2.d.d ✓ 48
8.b even 2 1 inner 1000.2.f.e 48
8.d odd 2 1 4000.2.f.e 48
20.d odd 2 1 4000.2.f.e 48
20.e even 4 2 4000.2.d.d 48
40.e odd 2 1 4000.2.f.e 48
40.f even 2 1 inner 1000.2.f.e 48
40.i odd 4 2 1000.2.d.d ✓ 48
40.k even 4 2 4000.2.d.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.d.d ✓ 48 5.c odd 4 2
1000.2.d.d ✓ 48 40.i odd 4 2
1000.2.f.e 48 1.a even 1 1 trivial
1000.2.f.e 48 5.b even 2 1 inner
1000.2.f.e 48 8.b even 2 1 inner
1000.2.f.e 48 40.f even 2 1 inner
4000.2.d.d 48 20.e even 4 2
4000.2.d.d 48 40.k even 4 2
4000.2.f.e 48 4.b odd 2 1
4000.2.f.e 48 8.d odd 2 1
4000.2.f.e 48 20.d odd 2 1
4000.2.f.e 48 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 48 T_{3}^{22} + 981 T_{3}^{20} - 11180 T_{3}^{18} + 78134 T_{3}^{16} - 346436 T_{3}^{14} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display