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Properties

Label 261.2.k.d
Level $261$
Weight $2$
Character orbit 261.k
Analytic conductor $2.084$
Analytic rank $0$
Dimension $24$
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] = [261,2,Mod(82,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.k (of order \(7\), degree \(6\), 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} - 4 q^{10} + 18 q^{13} - 22 q^{16} - 40 q^{22} + 18 q^{25} - 24 q^{28} - 28 q^{31} - 50 q^{34} - 40 q^{37} + 30 q^{43} + 40 q^{46} + 36 q^{49} - 52 q^{52} + 10 q^{55} + 54 q^{58} + 64 q^{61} + 20 q^{64} - 24 q^{67} + 120 q^{70} - 76 q^{73} + 126 q^{76} + 10 q^{79} - 10 q^{82} + 62 q^{85} - 20 q^{88} + 82 q^{91} - 36 q^{94} - 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −1.52747 − 1.91539i 0 −0.890498 + 3.90153i 0.135907 + 0.170422i 0 0.341351 + 1.49556i 4.41863 − 2.12790i 0 0.118830 − 0.520628i
82.2 −0.557692 − 0.699324i 0 0.267009 − 1.16984i −1.24144 − 1.55672i 0 −0.173788 − 0.761415i −2.57878 + 1.24188i 0 −0.396309 + 1.73634i
82.3 0.557692 + 0.699324i 0 0.267009 − 1.16984i 1.24144 + 1.55672i 0 −0.173788 − 0.761415i 2.57878 − 1.24188i 0 −0.396309 + 1.73634i
82.4 1.52747 + 1.91539i 0 −0.890498 + 3.90153i −0.135907 − 0.170422i 0 0.341351 + 1.49556i −4.41863 + 2.12790i 0 0.118830 − 0.520628i
136.1 −1.95622 + 0.942065i 0 1.69232 − 2.12210i −0.656754 + 0.316276i 0 0.363311 + 0.455578i −0.345098 + 1.51197i 0 0.986801 − 1.23741i
136.2 −0.770193 + 0.370905i 0 −0.791353 + 0.992325i 3.56724 − 1.71789i 0 −2.73378 − 3.42805i 0.621880 − 2.72463i 0 −2.11029 + 2.64622i
136.3 0.770193 − 0.370905i 0 −0.791353 + 0.992325i −3.56724 + 1.71789i 0 −2.73378 − 3.42805i −0.621880 + 2.72463i 0 −2.11029 + 2.64622i
136.4 1.95622 − 0.942065i 0 1.69232 − 2.12210i 0.656754 − 0.316276i 0 0.363311 + 0.455578i 0.345098 − 1.51197i 0 0.986801 − 1.23741i
181.1 −0.398448 − 1.74571i 0 −1.08681 + 0.523382i 0.325426 + 1.42578i 0 3.26031 + 1.57008i −0.886136 − 1.11118i 0 2.35934 − 1.13620i
181.2 −0.164537 − 0.720885i 0 1.30934 − 0.630543i −0.654129 − 2.86592i 0 −1.05740 − 0.509219i −1.59203 − 1.99634i 0 −1.95837 + 0.943103i
181.3 0.164537 + 0.720885i 0 1.30934 − 0.630543i 0.654129 + 2.86592i 0 −1.05740 − 0.509219i 1.59203 + 1.99634i 0 −1.95837 + 0.943103i
181.4 0.398448 + 1.74571i 0 −1.08681 + 0.523382i −0.325426 − 1.42578i 0 3.26031 + 1.57008i 0.886136 + 1.11118i 0 2.35934 − 1.13620i
190.1 −1.95622 − 0.942065i 0 1.69232 + 2.12210i −0.656754 − 0.316276i 0 0.363311 − 0.455578i −0.345098 − 1.51197i 0 0.986801 + 1.23741i
190.2 −0.770193 − 0.370905i 0 −0.791353 − 0.992325i 3.56724 + 1.71789i 0 −2.73378 + 3.42805i 0.621880 + 2.72463i 0 −2.11029 − 2.64622i
190.3 0.770193 + 0.370905i 0 −0.791353 − 0.992325i −3.56724 − 1.71789i 0 −2.73378 + 3.42805i −0.621880 − 2.72463i 0 −2.11029 − 2.64622i
190.4 1.95622 + 0.942065i 0 1.69232 + 2.12210i 0.656754 + 0.316276i 0 0.363311 − 0.455578i 0.345098 + 1.51197i 0 0.986801 + 1.23741i
199.1 −0.398448 + 1.74571i 0 −1.08681 − 0.523382i 0.325426 − 1.42578i 0 3.26031 − 1.57008i −0.886136 + 1.11118i 0 2.35934 + 1.13620i
199.2 −0.164537 + 0.720885i 0 1.30934 + 0.630543i −0.654129 + 2.86592i 0 −1.05740 + 0.509219i −1.59203 + 1.99634i 0 −1.95837 − 0.943103i
199.3 0.164537 − 0.720885i 0 1.30934 + 0.630543i 0.654129 − 2.86592i 0 −1.05740 + 0.509219i 1.59203 − 1.99634i 0 −1.95837 − 0.943103i
199.4 0.398448 − 1.74571i 0 −1.08681 − 0.523382i −0.325426 + 1.42578i 0 3.26031 − 1.57008i 0.886136 − 1.11118i 0 2.35934 + 1.13620i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.d even 7 1 inner
87.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.k.d ✓ 24
3.b odd 2 1 inner 261.2.k.d ✓ 24
29.d even 7 1 inner 261.2.k.d ✓ 24
29.d even 7 1 7569.2.a.bq 12
29.e even 14 1 7569.2.a.br 12
87.h odd 14 1 7569.2.a.br 12
87.j odd 14 1 inner 261.2.k.d ✓ 24
87.j odd 14 1 7569.2.a.bq 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.k.d ✓ 24 1.a even 1 1 trivial
261.2.k.d ✓ 24 3.b odd 2 1 inner
261.2.k.d ✓ 24 29.d even 7 1 inner
261.2.k.d ✓ 24 87.j odd 14 1 inner
7569.2.a.bq 12 29.d even 7 1
7569.2.a.bq 12 87.j odd 14 1
7569.2.a.br 12 29.e even 14 1
7569.2.a.br 12 87.h odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 3 T_{2}^{22} + 36 T_{2}^{20} + 77 T_{2}^{18} + 405 T_{2}^{16} + 3259 T_{2}^{14} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display