Properties

Label 546.2.cg.b
Level $546$
Weight $2$
Character orbit 546.cg
Analytic conductor $4.360$
Analytic rank $0$
Dimension $40$
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] = [546,2,Mod(145,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 10, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.cg (of order \(12\), 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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{7} + 20 q^{9} + 8 q^{11} - 20 q^{12} + 4 q^{14} - 40 q^{16} + 16 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{22} - 24 q^{25} + 8 q^{26} - 4 q^{28} - 12 q^{29} - 8 q^{33} - 8 q^{34} - 32 q^{35} + 40 q^{37} + 8 q^{38} + 20 q^{39} + 20 q^{41} + 12 q^{42} - 24 q^{43} - 4 q^{44} + 32 q^{46} + 4 q^{47} + 32 q^{49} - 16 q^{50} + 24 q^{51} + 16 q^{52} - 4 q^{53} + 24 q^{55} + 12 q^{56} + 8 q^{57} + 12 q^{58} + 24 q^{59} + 60 q^{61} + 32 q^{62} - 4 q^{63} + 44 q^{65} - 12 q^{67} - 8 q^{69} - 24 q^{70} - 28 q^{71} + 60 q^{73} - 40 q^{74} - 72 q^{75} + 4 q^{76} - 12 q^{77} + 4 q^{78} - 20 q^{81} - 24 q^{82} + 60 q^{83} - 8 q^{84} - 4 q^{85} - 20 q^{86} - 36 q^{89} - 16 q^{92} - 24 q^{93} - 48 q^{97} - 88 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −4.23240 + 1.13407i 0.965926 0.258819i 1.34853 + 2.27629i 0.707107 + 0.707107i 0.500000 + 0.866025i 2.19085 3.79467i
145.2 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −1.30432 + 0.349490i 0.965926 0.258819i −1.83449 1.90648i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.675164 1.16942i
145.3 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −0.981662 + 0.263036i 0.965926 0.258819i 2.09303 1.61840i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.508146 0.880134i
145.4 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i 3.01304 0.807342i 0.965926 0.258819i 2.56888 + 0.633142i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.55967 + 2.70142i
145.5 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i 3.50534 0.939253i 0.965926 0.258819i −2.61012 0.432735i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.81450 + 3.14280i
145.6 0.707107 0.707107i −0.866025 0.500000i 1.00000i −2.31481 + 0.620250i −0.965926 + 0.258819i −2.57277 + 0.617120i −0.707107 0.707107i 0.500000 + 0.866025i −1.19823 + 2.07540i
145.7 0.707107 0.707107i −0.866025 0.500000i 1.00000i −1.42776 + 0.382567i −0.965926 + 0.258819i 0.349274 + 2.62260i −0.707107 0.707107i 0.500000 + 0.866025i −0.739062 + 1.28009i
145.8 0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.499350 0.133800i −0.965926 + 0.258819i 0.430450 + 2.61050i −0.707107 0.707107i 0.500000 + 0.866025i 0.258483 0.447705i
145.9 0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.589284 0.157898i −0.965926 + 0.258819i −1.91156 1.82919i −0.707107 0.707107i 0.500000 + 0.866025i 0.305036 0.528338i
145.10 0.707107 0.707107i −0.866025 0.500000i 1.00000i 2.65393 0.711118i −0.965926 + 0.258819i 1.40673 2.24078i −0.707107 0.707107i 0.500000 + 0.866025i 1.37378 2.37945i
241.1 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −4.23240 1.13407i 0.965926 + 0.258819i 1.34853 2.27629i 0.707107 0.707107i 0.500000 0.866025i 2.19085 + 3.79467i
241.2 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −1.30432 0.349490i 0.965926 + 0.258819i −1.83449 + 1.90648i 0.707107 0.707107i 0.500000 0.866025i 0.675164 + 1.16942i
241.3 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −0.981662 0.263036i 0.965926 + 0.258819i 2.09303 + 1.61840i 0.707107 0.707107i 0.500000 0.866025i 0.508146 + 0.880134i
241.4 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i 3.01304 + 0.807342i 0.965926 + 0.258819i 2.56888 0.633142i 0.707107 0.707107i 0.500000 0.866025i −1.55967 2.70142i
241.5 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i 3.50534 + 0.939253i 0.965926 + 0.258819i −2.61012 + 0.432735i 0.707107 0.707107i 0.500000 0.866025i −1.81450 3.14280i
241.6 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i −2.31481 0.620250i −0.965926 0.258819i −2.57277 0.617120i −0.707107 + 0.707107i 0.500000 0.866025i −1.19823 2.07540i
241.7 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i −1.42776 0.382567i −0.965926 0.258819i 0.349274 2.62260i −0.707107 + 0.707107i 0.500000 0.866025i −0.739062 1.28009i
241.8 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 0.499350 + 0.133800i −0.965926 0.258819i 0.430450 2.61050i −0.707107 + 0.707107i 0.500000 0.866025i 0.258483 + 0.447705i
241.9 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 0.589284 + 0.157898i −0.965926 0.258819i −1.91156 + 1.82919i −0.707107 + 0.707107i 0.500000 0.866025i 0.305036 + 0.528338i
241.10 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 2.65393 + 0.711118i −0.965926 0.258819i 1.40673 + 2.24078i −0.707107 + 0.707107i 0.500000 0.866025i 1.37378 + 2.37945i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.cg.b yes 40
7.d odd 6 1 546.2.by.b 40
13.f odd 12 1 546.2.by.b 40
91.ba even 12 1 inner 546.2.cg.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.by.b 40 7.d odd 6 1
546.2.by.b 40 13.f odd 12 1
546.2.cg.b yes 40 1.a even 1 1 trivial
546.2.cg.b yes 40 91.ba even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 12 T_{5}^{38} + 12 T_{5}^{37} - 348 T_{5}^{36} + 128 T_{5}^{35} - 4680 T_{5}^{34} - 5592 T_{5}^{33} + 108147 T_{5}^{32} - 132044 T_{5}^{31} + 1446062 T_{5}^{30} - 752596 T_{5}^{29} - 6554768 T_{5}^{28} + \cdots + 14841086976 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display