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Properties

Label 132.2.j.a
Level $132$
Weight $2$
Character orbit 132.j
Analytic conductor $1.054$
Analytic rank $0$
Dimension $48$
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] = [132,2,Mod(7,132)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(132, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 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("132.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 132 = 2^{2} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 132.j (of order \(10\), degree \(4\), 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: \(1.05402530668\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} + 12 q^{9} + 8 q^{12} - 14 q^{14} - 24 q^{16} - 10 q^{18} - 22 q^{20} - 26 q^{22} - 30 q^{24} - 20 q^{25} - 38 q^{26} - 10 q^{28} - 20 q^{30} - 4 q^{33} + 4 q^{36} - 48 q^{37} + 58 q^{38} + 70 q^{40} - 40 q^{41} + 12 q^{42} + 34 q^{44} + 70 q^{46} - 16 q^{48} - 28 q^{49} + 70 q^{50} + 30 q^{52} - 64 q^{53} + 60 q^{56} - 54 q^{58} - 12 q^{60} - 40 q^{64} - 8 q^{66} + 4 q^{70} + 10 q^{72} + 20 q^{73} + 50 q^{74} - 8 q^{77} + 40 q^{78} + 58 q^{80} - 12 q^{81} + 62 q^{82} + 60 q^{84} + 40 q^{85} + 8 q^{86} + 8 q^{88} + 48 q^{89} + 20 q^{90} + 42 q^{92} + 24 q^{93} - 10 q^{94} + 50 q^{96} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.38167 + 0.301665i −0.951057 − 0.309017i 1.81800 − 0.833600i −0.213696 + 0.155259i 1.40726 + 0.140058i −1.03321 − 3.17989i −2.26039 + 1.70018i 0.809017 + 0.587785i 0.248420 − 0.278981i
7.2 −1.24828 + 0.664670i 0.951057 + 0.309017i 1.11643 − 1.65939i −3.06643 + 2.22789i −1.39258 + 0.246398i 0.793217 + 2.44127i −0.290669 + 2.81345i 0.809017 + 0.587785i 2.34696 − 4.81920i
7.3 −1.12117 − 0.861957i −0.951057 − 0.309017i 0.514062 + 1.93281i −1.53251 + 1.11343i 0.799941 + 1.16623i 1.07196 + 3.29914i 1.08964 − 2.61011i 0.809017 + 0.587785i 2.67794 + 0.0726042i
7.4 −0.989218 + 1.01067i 0.951057 + 0.309017i −0.0428958 − 1.99954i 2.66345 − 1.93511i −1.25312 + 0.655516i −1.40437 − 4.32221i 2.06330 + 1.93463i 0.809017 + 0.587785i −0.678982 + 4.60611i
7.5 −0.877083 − 1.10938i 0.951057 + 0.309017i −0.461450 + 1.94604i 1.42356 − 1.03428i −0.491338 − 1.32612i 0.160112 + 0.492774i 2.56363 − 1.19491i 0.809017 + 0.587785i −2.39599 − 0.672124i
7.6 −0.655516 + 1.25312i −0.951057 − 0.309017i −1.14060 − 1.64288i 2.66345 − 1.93511i 1.01067 − 0.989218i 1.40437 + 4.32221i 2.80639 − 0.352368i 0.809017 + 0.587785i 0.678982 + 4.60611i
7.7 −0.246398 + 1.39258i −0.951057 − 0.309017i −1.87858 − 0.686259i −3.06643 + 2.22789i 0.664670 − 1.24828i −0.793217 − 2.44127i 1.41855 − 2.44698i 0.809017 + 0.587785i −2.34696 − 4.81920i
7.8 0.140058 + 1.40726i 0.951057 + 0.309017i −1.96077 + 0.394195i −0.213696 + 0.155259i −0.301665 + 1.38167i 1.03321 + 3.17989i −0.829356 − 2.70410i 0.809017 + 0.587785i −0.248420 − 0.278981i
7.9 0.690263 − 1.23432i 0.951057 + 0.309017i −1.04707 − 1.70401i 0.725620 − 0.527194i 1.03790 − 0.960602i 0.489790 + 1.50742i −2.82604 + 0.116210i 0.809017 + 0.587785i −0.149856 − 1.25955i
7.10 0.960602 − 1.03790i −0.951057 − 0.309017i −0.154488 − 1.99402i 0.725620 − 0.527194i −1.23432 + 0.690263i −0.489790 − 1.50742i −2.21801 − 1.75512i 0.809017 + 0.587785i 0.149856 − 1.25955i
7.11 1.16623 + 0.799941i 0.951057 + 0.309017i 0.720190 + 1.86583i −1.53251 + 1.11343i 0.861957 + 1.12117i −1.07196 − 3.29914i −0.652646 + 2.75210i 0.809017 + 0.587785i −2.67794 + 0.0726042i
7.12 1.32612 + 0.491338i −0.951057 − 0.309017i 1.51717 + 1.30314i 1.42356 − 1.03428i −1.10938 − 0.877083i −0.160112 − 0.492774i 1.37166 + 2.47357i 0.809017 + 0.587785i 2.39599 − 0.672124i
19.1 −1.38167 − 0.301665i −0.951057 + 0.309017i 1.81800 + 0.833600i −0.213696 − 0.155259i 1.40726 − 0.140058i −1.03321 + 3.17989i −2.26039 − 1.70018i 0.809017 − 0.587785i 0.248420 + 0.278981i
19.2 −1.24828 − 0.664670i 0.951057 − 0.309017i 1.11643 + 1.65939i −3.06643 − 2.22789i −1.39258 − 0.246398i 0.793217 − 2.44127i −0.290669 − 2.81345i 0.809017 − 0.587785i 2.34696 + 4.81920i
19.3 −1.12117 + 0.861957i −0.951057 + 0.309017i 0.514062 − 1.93281i −1.53251 − 1.11343i 0.799941 − 1.16623i 1.07196 − 3.29914i 1.08964 + 2.61011i 0.809017 − 0.587785i 2.67794 − 0.0726042i
19.4 −0.989218 − 1.01067i 0.951057 − 0.309017i −0.0428958 + 1.99954i 2.66345 + 1.93511i −1.25312 − 0.655516i −1.40437 + 4.32221i 2.06330 − 1.93463i 0.809017 − 0.587785i −0.678982 − 4.60611i
19.5 −0.877083 + 1.10938i 0.951057 − 0.309017i −0.461450 − 1.94604i 1.42356 + 1.03428i −0.491338 + 1.32612i 0.160112 − 0.492774i 2.56363 + 1.19491i 0.809017 − 0.587785i −2.39599 + 0.672124i
19.6 −0.655516 − 1.25312i −0.951057 + 0.309017i −1.14060 + 1.64288i 2.66345 + 1.93511i 1.01067 + 0.989218i 1.40437 − 4.32221i 2.80639 + 0.352368i 0.809017 − 0.587785i 0.678982 − 4.60611i
19.7 −0.246398 − 1.39258i −0.951057 + 0.309017i −1.87858 + 0.686259i −3.06643 − 2.22789i 0.664670 + 1.24828i −0.793217 + 2.44127i 1.41855 + 2.44698i 0.809017 − 0.587785i −2.34696 + 4.81920i
19.8 0.140058 − 1.40726i 0.951057 − 0.309017i −1.96077 − 0.394195i −0.213696 − 0.155259i −0.301665 − 1.38167i 1.03321 − 3.17989i −0.829356 + 2.70410i 0.809017 − 0.587785i −0.248420 + 0.278981i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 132.2.j.a ✓ 48
3.b odd 2 1 396.2.r.b 48
4.b odd 2 1 inner 132.2.j.a ✓ 48
11.d odd 10 1 inner 132.2.j.a ✓ 48
12.b even 2 1 396.2.r.b 48
33.f even 10 1 396.2.r.b 48
44.g even 10 1 inner 132.2.j.a ✓ 48
132.n odd 10 1 396.2.r.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.j.a ✓ 48 1.a even 1 1 trivial
132.2.j.a ✓ 48 4.b odd 2 1 inner
132.2.j.a ✓ 48 11.d odd 10 1 inner
132.2.j.a ✓ 48 44.g even 10 1 inner
396.2.r.b 48 3.b odd 2 1
396.2.r.b 48 12.b even 2 1
396.2.r.b 48 33.f even 10 1
396.2.r.b 48 132.n odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(132, [\chi])\).