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Properties

Label 668.3.d.a
Level $668$
Weight $3$
Character orbit 668.d
Analytic conductor $18.202$
Analytic rank $0$
Dimension $28$
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] = [668,3,Mod(333,668)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(668, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("668.333");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 668.d (of order \(2\), degree \(1\), 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: \(18.2016816593\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 64 q^{9} - 2 q^{11} + 10 q^{19} + 64 q^{21} - 100 q^{25} + 48 q^{27} + 38 q^{29} + 38 q^{31} - 92 q^{33} + 46 q^{47} + 184 q^{49} + 12 q^{57} + 38 q^{61} + 190 q^{63} - 10 q^{65} + 102 q^{75} + 8 q^{77} - 108 q^{81} + 10 q^{85} - 118 q^{87} + 62 q^{89} - 44 q^{93} - 204 q^{97} + 254 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
333.1 0 −5.13795 0 2.98582i 0 −2.74320 0 17.3985 0
333.2 0 −5.13795 0 − 2.98582i 0 −2.74320 0 17.3985 0
333.3 0 −4.70135 0 7.20552i 0 10.9529 0 13.1027 0
333.4 0 −4.70135 0 − 7.20552i 0 10.9529 0 13.1027 0
333.5 0 −3.87260 0 3.72589i 0 −6.41982 0 5.99703 0
333.6 0 −3.87260 0 − 3.72589i 0 −6.41982 0 5.99703 0
333.7 0 −2.32033 0 − 8.29023i 0 −12.0965 0 −3.61607 0
333.8 0 −2.32033 0 8.29023i 0 −12.0965 0 −3.61607 0
333.9 0 −2.05814 0 − 4.42896i 0 4.08912 0 −4.76405 0
333.10 0 −2.05814 0 4.42896i 0 4.08912 0 −4.76405 0
333.11 0 −1.68366 0 − 4.61482i 0 −7.58466 0 −6.16528 0
333.12 0 −1.68366 0 4.61482i 0 −7.58466 0 −6.16528 0
333.13 0 −1.09624 0 − 0.673039i 0 7.86393 0 −7.79825 0
333.14 0 −1.09624 0 0.673039i 0 7.86393 0 −7.79825 0
333.15 0 0.242608 0 8.99503i 0 2.89606 0 −8.94114 0
333.16 0 0.242608 0 − 8.99503i 0 2.89606 0 −8.94114 0
333.17 0 1.67933 0 0.162378i 0 −2.80548 0 −6.17986 0
333.18 0 1.67933 0 − 0.162378i 0 −2.80548 0 −6.17986 0
333.19 0 2.62668 0 − 7.30740i 0 1.58060 0 −2.10055 0
333.20 0 2.62668 0 7.30740i 0 1.58060 0 −2.10055 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 333.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.3.d.a ✓ 28
167.b odd 2 1 inner 668.3.d.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.3.d.a ✓ 28 1.a even 1 1 trivial
668.3.d.a ✓ 28 167.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(668, [\chi])\).