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Properties

Label 546.2.by.a
Level $546$
Weight $2$
Character orbit 546.by
Analytic conductor $4.360$
Analytic rank $0$
Dimension $32$
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(19,546)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.by (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: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{7} - 32 q^{9} - 4 q^{11} - 16 q^{12} + 4 q^{14} + 16 q^{16} - 8 q^{17} - 12 q^{19} - 8 q^{21} - 4 q^{22} - 24 q^{23} - 24 q^{25} + 8 q^{26} - 4 q^{28} - 12 q^{29} - 28 q^{31} + 4 q^{33} - 8 q^{34} - 44 q^{35} - 36 q^{37} + 8 q^{38} - 20 q^{39} - 28 q^{41} + 12 q^{42} + 84 q^{43} + 8 q^{44} - 4 q^{46} + 16 q^{47} + 24 q^{49} - 16 q^{50} + 8 q^{52} - 4 q^{53} + 48 q^{55} + 12 q^{56} + 12 q^{57} + 24 q^{58} + 12 q^{59} - 40 q^{62} - 8 q^{63} - 4 q^{65} + 8 q^{67} - 8 q^{69} + 60 q^{70} + 20 q^{71} + 4 q^{73} + 20 q^{74} + 8 q^{75} + 36 q^{76} - 12 q^{77} - 20 q^{78} + 32 q^{81} - 48 q^{82} + 12 q^{83} - 16 q^{84} - 4 q^{85} + 52 q^{86} - 36 q^{87} - 36 q^{89} - 16 q^{92} + 4 q^{93} - 48 q^{95} + 76 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.965926 − 0.258819i 1.00000i 0.866025 + 0.500000i −2.19856 + 0.589103i 0.258819 − 0.965926i 1.83682 − 1.90423i −0.707107 − 0.707107i −1.00000 2.27612
19.2 −0.965926 − 0.258819i 1.00000i 0.866025 + 0.500000i −1.43558 + 0.384662i 0.258819 − 0.965926i 1.74877 + 1.98539i −0.707107 − 0.707107i −1.00000 1.48622
19.3 −0.965926 − 0.258819i 1.00000i 0.866025 + 0.500000i −0.106603 + 0.0285643i 0.258819 − 0.965926i −1.53649 + 2.15388i −0.707107 − 0.707107i −1.00000 0.110364
19.4 −0.965926 − 0.258819i 1.00000i 0.866025 + 0.500000i 3.74075 − 1.00233i 0.258819 − 0.965926i −2.20451 + 1.46293i −0.707107 − 0.707107i −1.00000 −3.87271
19.5 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i −2.85793 + 0.765779i −0.258819 + 0.965926i 0.565013 + 2.58472i 0.707107 + 0.707107i −1.00000 −2.95874
19.6 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 0.128673 − 0.0344778i −0.258819 + 0.965926i −2.64307 + 0.119073i 0.707107 + 0.707107i −1.00000 0.133212
19.7 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 0.421063 − 0.112824i −0.258819 + 0.965926i 2.44466 + 1.01175i 0.707107 + 0.707107i −1.00000 0.435917
19.8 0.965926 + 0.258819i 1.00000i 0.866025 + 0.500000i 2.30819 − 0.618478i −0.258819 + 0.965926i 1.78879 − 1.94942i 0.707107 + 0.707107i −1.00000 2.38962
115.1 −0.965926 + 0.258819i − 1.00000i 0.866025 − 0.500000i −2.19856 − 0.589103i 0.258819 + 0.965926i 1.83682 + 1.90423i −0.707107 + 0.707107i −1.00000 2.27612
115.2 −0.965926 + 0.258819i − 1.00000i 0.866025 − 0.500000i −1.43558 − 0.384662i 0.258819 + 0.965926i 1.74877 − 1.98539i −0.707107 + 0.707107i −1.00000 1.48622
115.3 −0.965926 + 0.258819i − 1.00000i 0.866025 − 0.500000i −0.106603 − 0.0285643i 0.258819 + 0.965926i −1.53649 − 2.15388i −0.707107 + 0.707107i −1.00000 0.110364
115.4 −0.965926 + 0.258819i − 1.00000i 0.866025 − 0.500000i 3.74075 + 1.00233i 0.258819 + 0.965926i −2.20451 − 1.46293i −0.707107 + 0.707107i −1.00000 −3.87271
115.5 0.965926 − 0.258819i − 1.00000i 0.866025 − 0.500000i −2.85793 − 0.765779i −0.258819 − 0.965926i 0.565013 − 2.58472i 0.707107 − 0.707107i −1.00000 −2.95874
115.6 0.965926 − 0.258819i − 1.00000i 0.866025 − 0.500000i 0.128673 + 0.0344778i −0.258819 − 0.965926i −2.64307 − 0.119073i 0.707107 − 0.707107i −1.00000 0.133212
115.7 0.965926 − 0.258819i − 1.00000i 0.866025 − 0.500000i 0.421063 + 0.112824i −0.258819 − 0.965926i 2.44466 − 1.01175i 0.707107 − 0.707107i −1.00000 0.435917
115.8 0.965926 − 0.258819i − 1.00000i 0.866025 − 0.500000i 2.30819 + 0.618478i −0.258819 − 0.965926i 1.78879 + 1.94942i 0.707107 − 0.707107i −1.00000 2.38962
397.1 −0.258819 + 0.965926i − 1.00000i −0.866025 − 0.500000i −0.995376 − 3.71479i 0.965926 + 0.258819i 2.54668 − 0.717240i 0.707107 − 0.707107i −1.00000 3.84584
397.2 −0.258819 + 0.965926i − 1.00000i −0.866025 − 0.500000i −0.166844 − 0.622671i 0.965926 + 0.258819i −2.38407 + 1.14726i 0.707107 − 0.707107i −1.00000 0.644637
397.3 −0.258819 + 0.965926i − 1.00000i −0.866025 − 0.500000i 0.379698 + 1.41705i 0.965926 + 0.258819i 1.63697 − 2.07854i 0.707107 − 0.707107i −1.00000 −1.46704
397.4 −0.258819 + 0.965926i − 1.00000i −0.866025 − 0.500000i 0.782522 + 2.92041i 0.965926 + 0.258819i 1.58056 + 2.12175i 0.707107 − 0.707107i −1.00000 −3.02343
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w 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.by.a ✓ 32
7.d odd 6 1 546.2.cg.a yes 32
13.f odd 12 1 546.2.cg.a yes 32
91.w even 12 1 inner 546.2.by.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.by.a ✓ 32 1.a even 1 1 trivial
546.2.by.a ✓ 32 91.w even 12 1 inner
546.2.cg.a yes 32 7.d odd 6 1
546.2.cg.a yes 32 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 12 T_{5}^{30} - 36 T_{5}^{29} - 156 T_{5}^{28} - 124 T_{5}^{27} - 1800 T_{5}^{26} + \cdots + 9409 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display