↑

Properties

Label 546.2.by.b
Level $546$
Weight $2$
Character orbit 546.by
Analytic conductor $4.360$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{7} - 40 q^{9} - 4 q^{11} + 20 q^{12} + 4 q^{14} + 20 q^{16} + 8 q^{17} + 8 q^{19} - 8 q^{21} - 4 q^{22} + 24 q^{23} + 24 q^{25} - 8 q^{26} + 4 q^{28} - 12 q^{29} + 24 q^{31} - 4 q^{33} + 8 q^{34} + 28 q^{35} - 8 q^{37} - 8 q^{38} - 16 q^{39} - 20 q^{41} - 12 q^{42} - 24 q^{43} + 8 q^{44} - 4 q^{46} - 16 q^{47} + 4 q^{49} - 16 q^{50} - 24 q^{51} - 4 q^{52} - 4 q^{53} - 24 q^{55} + 12 q^{56} + 8 q^{57} + 24 q^{58} - 12 q^{59} - 32 q^{62} + 4 q^{63} - 4 q^{65} - 24 q^{67} + 24 q^{68} + 8 q^{69} + 52 q^{70} - 28 q^{71} + 108 q^{73} + 20 q^{74} - 36 q^{75} - 4 q^{76} + 12 q^{77} + 4 q^{78} + 40 q^{81} - 48 q^{82} - 60 q^{83} - 8 q^{84} - 4 q^{85} - 20 q^{86} - 36 q^{87} - 60 q^{89} - 40 q^{91} - 16 q^{92} - 48 q^{95} + 48 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 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 −4.03777 + 1.08192i −0.258819 + 0.965926i −0.0127227 + 2.64572i −0.707107 − 0.707107i −1.00000 4.18021
19.2 −0.965926 − 0.258819i − 1.00000i 0.866025 + 0.500000i −1.37297 + 0.367885i −0.258819 + 0.965926i −0.220772 − 2.63652i −0.707107 − 0.707107i −1.00000 1.42140
19.3 −0.965926 − 0.258819i − 1.00000i 0.866025 + 0.500000i 0.195603 − 0.0524115i −0.258819 + 0.965926i −2.35070 + 1.21417i −0.707107 − 0.707107i −1.00000 −0.202503
19.4 −0.965926 − 0.258819i − 1.00000i 0.866025 + 0.500000i 1.30611 − 0.349971i −0.258819 + 0.965926i 2.64352 − 0.108591i −0.707107 − 0.707107i −1.00000 −1.35218
19.5 −0.965926 − 0.258819i − 1.00000i 0.866025 + 0.500000i 3.90903 − 1.04742i −0.258819 + 0.965926i −1.71472 − 2.01488i −0.707107 − 0.707107i −1.00000 −4.04692
19.6 0.965926 + 0.258819i − 1.00000i 0.866025 + 0.500000i −3.15172 + 0.844500i 0.258819 − 0.965926i −2.64535 − 0.0462630i 0.707107 + 0.707107i −1.00000 −3.26290
19.7 0.965926 + 0.258819i − 1.00000i 0.866025 + 0.500000i −1.66136 + 0.445159i 0.258819 − 0.965926i 2.48285 + 0.914026i 0.707107 + 0.707107i −1.00000 −1.71996
19.8 0.965926 + 0.258819i − 1.00000i 0.866025 + 0.500000i −1.45998 + 0.391201i 0.258819 − 0.965926i 0.541277 − 2.58979i 0.707107 + 0.707107i −1.00000 −1.51148
19.9 0.965926 + 0.258819i − 1.00000i 0.866025 + 0.500000i 2.82421 − 0.756745i 0.258819 − 0.965926i −2.36822 − 1.17963i 0.707107 + 0.707107i −1.00000 2.92384
19.10 0.965926 + 0.258819i − 1.00000i 0.866025 + 0.500000i 3.44884 − 0.924115i 0.258819 − 0.965926i 2.64483 + 0.0697092i 0.707107 + 0.707107i −1.00000 3.57050
115.1 −0.965926 + 0.258819i 1.00000i 0.866025 − 0.500000i −4.03777 − 1.08192i −0.258819 − 0.965926i −0.0127227 − 2.64572i −0.707107 + 0.707107i −1.00000 4.18021
115.2 −0.965926 + 0.258819i 1.00000i 0.866025 − 0.500000i −1.37297 − 0.367885i −0.258819 − 0.965926i −0.220772 + 2.63652i −0.707107 + 0.707107i −1.00000 1.42140
115.3 −0.965926 + 0.258819i 1.00000i 0.866025 − 0.500000i 0.195603 + 0.0524115i −0.258819 − 0.965926i −2.35070 − 1.21417i −0.707107 + 0.707107i −1.00000 −0.202503
115.4 −0.965926 + 0.258819i 1.00000i 0.866025 − 0.500000i 1.30611 + 0.349971i −0.258819 − 0.965926i 2.64352 + 0.108591i −0.707107 + 0.707107i −1.00000 −1.35218
115.5 −0.965926 + 0.258819i 1.00000i 0.866025 − 0.500000i 3.90903 + 1.04742i −0.258819 − 0.965926i −1.71472 + 2.01488i −0.707107 + 0.707107i −1.00000 −4.04692
115.6 0.965926 − 0.258819i 1.00000i 0.866025 − 0.500000i −3.15172 − 0.844500i 0.258819 + 0.965926i −2.64535 + 0.0462630i 0.707107 − 0.707107i −1.00000 −3.26290
115.7 0.965926 − 0.258819i 1.00000i 0.866025 − 0.500000i −1.66136 − 0.445159i 0.258819 + 0.965926i 2.48285 − 0.914026i 0.707107 − 0.707107i −1.00000 −1.71996
115.8 0.965926 − 0.258819i 1.00000i 0.866025 − 0.500000i −1.45998 − 0.391201i 0.258819 + 0.965926i 0.541277 + 2.58979i 0.707107 − 0.707107i −1.00000 −1.51148
115.9 0.965926 − 0.258819i 1.00000i 0.866025 − 0.500000i 2.82421 + 0.756745i 0.258819 + 0.965926i −2.36822 + 1.17963i 0.707107 − 0.707107i −1.00000 2.92384
115.10 0.965926 − 0.258819i 1.00000i 0.866025 − 0.500000i 3.44884 + 0.924115i 0.258819 + 0.965926i 2.64483 − 0.0697092i 0.707107 − 0.707107i −1.00000 3.57050
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - 12 T_{5}^{38} - 12 T_{5}^{37} - 348 T_{5}^{36} + 364 T_{5}^{35} + 4824 T_{5}^{34} + \cdots + 14841086976 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display