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Properties

Label 1681.2.a.m
Level $1681$
Weight $2$
Character orbit 1681.a
Self dual yes
Analytic conductor $13.423$
Analytic rank $0$
Dimension $24$
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] = [1681,2,Mod(1,1681)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1681, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1681.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1681 = 41^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1681.a (trivial)

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: yes
Analytic conductor: \(13.4228525798\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 41)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9} - 4 q^{10} + 20 q^{16} + 20 q^{18} + 40 q^{20} + 56 q^{21} + 48 q^{23} + 8 q^{25} + 32 q^{31} + 36 q^{32} + 24 q^{33} + 108 q^{36} + 60 q^{39} + 44 q^{40} - 40 q^{42} + 36 q^{43} - 36 q^{45} + 36 q^{46} + 32 q^{49} - 60 q^{50} + 36 q^{51} - 60 q^{57} + 116 q^{59} + 40 q^{61} + 48 q^{62} + 60 q^{64} + 16 q^{66} + 4 q^{72} + 16 q^{73} + 24 q^{74} - 16 q^{77} - 60 q^{78} + 84 q^{80} - 28 q^{81} + 80 q^{83} - 24 q^{84} - 24 q^{86} + 76 q^{87} + 56 q^{90} + 40 q^{91} - 40 q^{92} + 36 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.41490 −2.36665 3.83176 2.87275 5.71522 −2.09872 −4.42352 2.60102 −6.93742
1.2 −2.41490 2.36665 3.83176 2.87275 −5.71522 2.09872 −4.42352 2.60102 −6.93742
1.3 −1.71427 −3.15154 0.938727 −2.70948 5.40260 −1.61292 1.81931 6.93221 4.64479
1.4 −1.71427 3.15154 0.938727 −2.70948 −5.40260 1.61292 1.81931 6.93221 4.64479
1.5 −1.27053 −0.854760 −0.385748 3.44852 1.08600 −1.72762 3.03117 −2.26939 −4.38146
1.6 −1.27053 0.854760 −0.385748 3.44852 −1.08600 1.72762 3.03117 −2.26939 −4.38146
1.7 −0.888284 −1.39100 −1.21095 1.07324 1.23560 −3.99983 2.85224 −1.06513 −0.953338
1.8 −0.888284 1.39100 −1.21095 1.07324 −1.23560 3.99983 2.85224 −1.06513 −0.953338
1.9 −0.734595 −0.0612231 −1.46037 −0.718161 0.0449742 −3.13283 2.54197 −2.99625 0.527558
1.10 −0.734595 0.0612231 −1.46037 −0.718161 −0.0449742 3.13283 2.54197 −2.99625 0.527558
1.11 0.305872 −1.36459 −1.90644 3.46591 −0.417390 −1.45639 −1.19487 −1.13789 1.06013
1.12 0.305872 1.36459 −1.90644 3.46591 0.417390 1.45639 −1.19487 −1.13789 1.06013
1.13 0.706693 −0.343106 −1.50059 −2.37819 −0.242471 4.91274 −2.47384 −2.88228 −1.68065
1.14 0.706693 0.343106 −1.50059 −2.37819 0.242471 −4.91274 −2.47384 −2.88228 −1.68065
1.15 1.20692 −2.19417 −0.543349 −2.54506 −2.64818 −1.13968 −3.06961 1.81438 −3.07168
1.16 1.20692 2.19417 −0.543349 −2.54506 2.64818 1.13968 −3.06961 1.81438 −3.07168
1.17 1.47960 −2.68766 0.189211 −0.774365 −3.97665 −4.67709 −2.67924 4.22349 −1.14575
1.18 1.47960 2.68766 0.189211 −0.774365 3.97665 4.67709 −2.67924 4.22349 −1.14575
1.19 2.15711 −2.12020 2.65312 2.93499 −4.57350 −2.45224 1.40886 1.49525 6.33110
1.20 2.15711 2.12020 2.65312 2.93499 4.57350 2.45224 1.40886 1.49525 6.33110
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(41\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1681.2.a.m 24
41.b even 2 1 inner 1681.2.a.m 24
41.h odd 40 2 41.2.g.a ✓ 24
123.o even 40 2 369.2.u.a 24
164.o even 40 2 656.2.bs.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a ✓ 24 41.h odd 40 2
369.2.u.a 24 123.o even 40 2
656.2.bs.d 24 164.o even 40 2
1681.2.a.m 24 1.a even 1 1 trivial
1681.2.a.m 24 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1681))\):

\( T_{2}^{12} - 4 T_{2}^{11} - 9 T_{2}^{10} + 48 T_{2}^{9} + 15 T_{2}^{8} - 194 T_{2}^{7} + 37 T_{2}^{6} + \cdots - 19 \) Copy content Toggle raw display
\( T_{3}^{24} - 48 T_{3}^{22} + 997 T_{3}^{20} - 11768 T_{3}^{18} + 87134 T_{3}^{16} - 421344 T_{3}^{14} + \cdots + 256 \) Copy content Toggle raw display