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Properties

Label 663.2.ba.a
Level $663$
Weight $2$
Character orbit 663.ba
Analytic conductor $5.294$
Analytic rank $0$
Dimension $80$
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] = [663,2,Mod(322,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.322");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.ba (of order \(6\), degree \(2\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 36 q^{4} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 36 q^{4} + 40 q^{9} - 4 q^{13} - 6 q^{15} - 12 q^{16} - 4 q^{17} + 44 q^{25} - 16 q^{26} - 8 q^{30} - 60 q^{32} - 6 q^{33} - 36 q^{36} + 96 q^{38} + 4 q^{42} + 10 q^{43} - 8 q^{49} + 48 q^{50} + 16 q^{51} + 72 q^{52} - 40 q^{53} + 44 q^{55} + 144 q^{59} - 56 q^{64} - 16 q^{66} - 48 q^{67} + 40 q^{68} - 26 q^{69} - 48 q^{76} - 64 q^{77} - 40 q^{81} + 6 q^{85} - 4 q^{87} - 48 q^{89} - 44 q^{94} + 36 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
322.1 −2.27119 + 1.31127i −0.866025 + 0.500000i 2.43887 − 4.22425i 0.0141426 1.31127 − 2.27119i 0.545766 − 0.945294i 7.54702i 0.500000 − 0.866025i −0.0321205 + 0.0185448i
322.2 −2.27119 + 1.31127i 0.866025 − 0.500000i 2.43887 − 4.22425i −0.0141426 −1.31127 + 2.27119i −0.545766 + 0.945294i 7.54702i 0.500000 − 0.866025i 0.0321205 − 0.0185448i
322.3 −2.03864 + 1.17701i −0.866025 + 0.500000i 1.77070 − 3.06694i −2.72761 1.17701 − 2.03864i −2.04815 + 3.54750i 3.62848i 0.500000 − 0.866025i 5.56062 − 3.21043i
322.4 −2.03864 + 1.17701i 0.866025 − 0.500000i 1.77070 − 3.06694i 2.72761 −1.17701 + 2.03864i 2.04815 − 3.54750i 3.62848i 0.500000 − 0.866025i −5.56062 + 3.21043i
322.5 −1.91302 + 1.10448i −0.866025 + 0.500000i 1.43977 − 2.49376i 3.77156 1.10448 − 1.91302i −0.302258 + 0.523526i 1.94288i 0.500000 − 0.866025i −7.21508 + 4.16563i
322.6 −1.91302 + 1.10448i 0.866025 − 0.500000i 1.43977 − 2.49376i −3.77156 −1.10448 + 1.91302i 0.302258 − 0.523526i 1.94288i 0.500000 − 0.866025i 7.21508 − 4.16563i
322.7 −1.74752 + 1.00893i −0.866025 + 0.500000i 1.03589 − 1.79422i 0.473184 1.00893 − 1.74752i 2.37085 − 4.10644i 0.144854i 0.500000 − 0.866025i −0.826900 + 0.477411i
322.8 −1.74752 + 1.00893i 0.866025 − 0.500000i 1.03589 − 1.79422i −0.473184 −1.00893 + 1.74752i −2.37085 + 4.10644i 0.144854i 0.500000 − 0.866025i 0.826900 − 0.477411i
322.9 −1.55800 + 0.899513i −0.866025 + 0.500000i 0.618249 − 1.07084i 1.31246 0.899513 − 1.55800i −0.177369 + 0.307212i − 1.37356i 0.500000 − 0.866025i −2.04481 + 1.18057i
322.10 −1.55800 + 0.899513i 0.866025 − 0.500000i 0.618249 − 1.07084i −1.31246 −0.899513 + 1.55800i 0.177369 − 0.307212i − 1.37356i 0.500000 − 0.866025i 2.04481 − 1.18057i
322.11 −1.16685 + 0.673680i −0.866025 + 0.500000i −0.0923095 + 0.159885i −2.43458 0.673680 − 1.16685i −0.507592 + 0.879176i − 2.94347i 0.500000 − 0.866025i 2.84078 − 1.64013i
322.12 −1.16685 + 0.673680i 0.866025 − 0.500000i −0.0923095 + 0.159885i 2.43458 −0.673680 + 1.16685i 0.507592 − 0.879176i − 2.94347i 0.500000 − 0.866025i −2.84078 + 1.64013i
322.13 −1.11383 + 0.643070i −0.866025 + 0.500000i −0.172922 + 0.299509i −0.174829 0.643070 − 1.11383i −1.19914 + 2.07697i − 3.01708i 0.500000 − 0.866025i 0.194729 − 0.112427i
322.14 −1.11383 + 0.643070i 0.866025 − 0.500000i −0.172922 + 0.299509i 0.174829 −0.643070 + 1.11383i 1.19914 − 2.07697i − 3.01708i 0.500000 − 0.866025i −0.194729 + 0.112427i
322.15 −0.788209 + 0.455073i −0.866025 + 0.500000i −0.585818 + 1.01467i −3.08206 0.455073 − 0.788209i 1.42728 − 2.47212i − 2.88665i 0.500000 − 0.866025i 2.42930 − 1.40256i
322.16 −0.788209 + 0.455073i 0.866025 − 0.500000i −0.585818 + 1.01467i 3.08206 −0.455073 + 0.788209i −1.42728 + 2.47212i − 2.88665i 0.500000 − 0.866025i −2.42930 + 1.40256i
322.17 −0.225430 + 0.130152i −0.866025 + 0.500000i −0.966121 + 1.67337i 1.15644 0.130152 − 0.225430i 1.23225 − 2.13432i − 1.02358i 0.500000 − 0.866025i −0.260696 + 0.150513i
322.18 −0.225430 + 0.130152i 0.866025 − 0.500000i −0.966121 + 1.67337i −1.15644 −0.130152 + 0.225430i −1.23225 + 2.13432i − 1.02358i 0.500000 − 0.866025i 0.260696 − 0.150513i
322.19 −0.211314 + 0.122002i −0.866025 + 0.500000i −0.970231 + 1.68049i 2.78703 0.122002 − 0.211314i 0.358946 − 0.621712i − 0.961491i 0.500000 − 0.866025i −0.588939 + 0.340024i
322.20 −0.211314 + 0.122002i 0.866025 − 0.500000i −0.970231 + 1.68049i −2.78703 −0.122002 + 0.211314i −0.358946 + 0.621712i − 0.961491i 0.500000 − 0.866025i 0.588939 − 0.340024i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 322.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner
17.b even 2 1 inner
221.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.ba.a ✓ 80
13.e even 6 1 inner 663.2.ba.a ✓ 80
17.b even 2 1 inner 663.2.ba.a ✓ 80
221.n even 6 1 inner 663.2.ba.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.ba.a ✓ 80 1.a even 1 1 trivial
663.2.ba.a ✓ 80 13.e even 6 1 inner
663.2.ba.a ✓ 80 17.b even 2 1 inner
663.2.ba.a ✓ 80 221.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(663, [\chi])\).