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Properties

Label 1000.2.be.a
Level $1000$
Weight $2$
Character orbit 1000.be
Analytic conductor $7.985$
Analytic rank $0$
Dimension $760$
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(9,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(50))
 
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("1000.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.be (of order \(50\), degree \(20\), 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: \(760\)
Relative dimension: \(38\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 760 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 760 q - 20 q^{15} - 120 q^{23} - 10 q^{31} + 40 q^{37} + 20 q^{39} + 160 q^{45} + 20 q^{47} + 210 q^{49} + 60 q^{51} + 30 q^{53} + 20 q^{55} + 20 q^{59} + 20 q^{63} + 10 q^{65} + 80 q^{71} - 40 q^{73} + 60 q^{81} - 30 q^{83} - 60 q^{85} - 110 q^{87} - 20 q^{89} - 300 q^{93} - 60 q^{95} - 30 q^{97} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −3.30633 + 0.208016i 0 2.20842 − 0.350535i 0 1.00963 − 0.328050i 0 7.91218 − 0.999541i 0
9.2 0 −3.08941 + 0.194369i 0 −0.326222 + 2.21214i 0 −1.84070 + 0.598081i 0 6.53033 − 0.824972i 0
9.3 0 −3.05187 + 0.192008i 0 −0.717620 − 2.11779i 0 −4.07645 + 1.32452i 0 6.30072 − 0.795966i 0
9.4 0 −2.71941 + 0.171090i 0 0.0311479 + 2.23585i 0 3.14886 − 1.02313i 0 4.38955 − 0.554529i 0
9.5 0 −2.49602 + 0.157036i 0 −2.22694 − 0.201806i 0 1.41211 − 0.458821i 0 3.22910 − 0.407930i 0
9.6 0 −2.40609 + 0.151378i 0 −2.08086 − 0.818541i 0 2.96184 − 0.962360i 0 2.78999 − 0.352458i 0
9.7 0 −2.28863 + 0.143988i 0 2.23440 + 0.0862474i 0 0.0280099 − 0.00910096i 0 2.24073 − 0.283070i 0
9.8 0 −2.19258 + 0.137946i 0 −1.34869 + 1.78355i 0 −3.32818 + 1.08139i 0 1.81205 − 0.228915i 0
9.9 0 −2.16765 + 0.136377i 0 0.0556529 − 2.23538i 0 1.90109 − 0.617701i 0 1.70376 − 0.215234i 0
9.10 0 −1.92586 + 0.121165i 0 1.29232 + 1.82480i 0 4.63237 − 1.50515i 0 0.717912 − 0.0906934i 0
9.11 0 −1.89013 + 0.118917i 0 2.23485 − 0.0737010i 0 −4.52314 + 1.46966i 0 0.582114 − 0.0735380i 0
9.12 0 −1.44519 + 0.0909235i 0 0.0207097 − 2.23597i 0 0.0982792 − 0.0319328i 0 −0.896044 + 0.113197i 0
9.13 0 −1.29861 + 0.0817016i 0 −1.91537 + 1.15385i 0 −0.137994 + 0.0448369i 0 −1.29663 + 0.163803i 0
9.14 0 −0.921188 + 0.0579562i 0 −1.71599 − 1.43366i 0 −2.63878 + 0.857391i 0 −2.13112 + 0.269223i 0
9.15 0 −0.736930 + 0.0463637i 0 0.897352 + 2.04811i 0 −0.374615 + 0.121720i 0 −2.43543 + 0.307666i 0
9.16 0 −0.706933 + 0.0444765i 0 1.67709 − 1.47898i 0 −0.611884 + 0.198813i 0 −2.47857 + 0.313116i 0
9.17 0 −0.561514 + 0.0353275i 0 2.00267 − 0.994651i 0 4.06306 − 1.32017i 0 −2.66229 + 0.336326i 0
9.18 0 −0.305057 + 0.0191926i 0 0.267377 + 2.22002i 0 −2.88902 + 0.938699i 0 −2.88365 + 0.364290i 0
9.19 0 −0.174234 + 0.0109619i 0 −1.87076 + 1.22485i 0 1.22110 − 0.396761i 0 −2.94611 + 0.372180i 0
9.20 0 0.354208 − 0.0222849i 0 1.40549 − 1.73914i 0 2.33764 − 0.759546i 0 −2.85138 + 0.360213i 0
See next 80 embeddings (of 760 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.h even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.be.a ✓ 760
125.h even 50 1 inner 1000.2.be.a ✓ 760
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.be.a ✓ 760 1.a even 1 1 trivial
1000.2.be.a ✓ 760 125.h even 50 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1000, [\chi])\).