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Properties

Label 546.2.bg.a
Level $546$
Weight $2$
Character orbit 546.bg
Analytic conductor $4.360$
Analytic rank $0$
Dimension $36$
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] = [546,2,Mod(311,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.311");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bg (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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{2} - 18 q^{4} + 36 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 18 q^{2} - 18 q^{4} + 36 q^{8} + 4 q^{9} - 14 q^{15} - 18 q^{16} + 4 q^{18} + 23 q^{21} + 14 q^{25} - 6 q^{26} + 7 q^{30} - 18 q^{32} + 24 q^{33} - 8 q^{36} - 10 q^{39} - 16 q^{42} - 16 q^{43} - 9 q^{45} + 72 q^{47} + 12 q^{49} - 28 q^{50} - 3 q^{51} + 6 q^{52} + 9 q^{54} - 8 q^{57} + 24 q^{59} + 7 q^{60} - 36 q^{61} - 39 q^{63} + 36 q^{64} + 18 q^{65} - 24 q^{66} - 72 q^{71} + 4 q^{72} + 54 q^{75} + 20 q^{78} + 20 q^{79} - 20 q^{81} - 24 q^{82} - 7 q^{84} + 8 q^{86} - 24 q^{87} - 72 q^{89} - 2 q^{91} - 14 q^{93} - 72 q^{94} - 12 q^{98} + 72 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1 −0.500000 − 0.866025i −1.71076 − 0.270722i −0.500000 + 0.866025i 2.20332 − 1.27209i 0.620929 + 1.61693i 2.32714 − 1.25873i 1.00000 2.85342 + 0.926282i −2.20332 − 1.27209i
311.2 −0.500000 − 0.866025i −1.63622 − 0.568150i −0.500000 + 0.866025i −1.08726 + 0.627730i 0.326076 + 1.70108i 1.14208 − 2.38656i 1.00000 2.35441 + 1.85923i 1.08726 + 0.627730i
311.3 −0.500000 − 0.866025i −1.62062 + 0.611211i −0.500000 + 0.866025i 0.0759308 − 0.0438386i 1.33964 + 1.09790i −2.62099 − 0.361112i 1.00000 2.25284 − 1.98109i −0.0759308 − 0.0438386i
311.4 −0.500000 − 0.866025i −1.34147 + 1.09565i −0.500000 + 0.866025i −2.44950 + 1.41422i 1.61960 + 0.613923i −2.60728 + 0.449563i 1.00000 0.599095 − 2.93957i 2.44950 + 1.41422i
311.5 −0.500000 − 0.866025i −1.31014 − 1.13293i −0.500000 + 0.866025i −1.08726 + 0.627730i −0.326076 + 1.70108i −1.14208 + 2.38656i 1.00000 0.432936 + 2.96860i 1.08726 + 0.627730i
311.6 −0.500000 − 0.866025i −1.08983 − 1.34620i −0.500000 + 0.866025i 2.20332 − 1.27209i −0.620929 + 1.61693i −2.32714 + 1.25873i 1.00000 −0.624526 + 2.93427i −2.20332 − 1.27209i
311.7 −0.500000 − 0.866025i −0.924091 + 1.46494i −0.500000 + 0.866025i 3.23425 − 1.86729i 1.73072 + 0.0678152i −0.481958 − 2.60148i 1.00000 −1.29211 − 2.70748i −3.23425 − 1.86729i
311.8 −0.500000 − 0.866025i −0.418670 + 1.68069i −0.500000 + 0.866025i 0.0916492 − 0.0529137i 1.66485 − 0.477766i 2.29863 + 1.31008i 1.00000 −2.64943 − 1.40731i −0.0916492 − 0.0529137i
311.9 −0.500000 − 0.866025i −0.280988 − 1.70911i −0.500000 + 0.866025i 0.0759308 − 0.0438386i −1.33964 + 1.09790i 2.62099 + 0.361112i 1.00000 −2.84209 + 0.960476i −0.0759308 − 0.0438386i
311.10 −0.500000 − 0.866025i 0.278126 − 1.70957i −0.500000 + 0.866025i −2.44950 + 1.41422i −1.61960 + 0.613923i 2.60728 − 0.449563i 1.00000 −2.84529 − 0.950954i 2.44950 + 1.41422i
311.11 −0.500000 − 0.866025i 0.658943 + 1.60181i −0.500000 + 0.866025i −1.00187 + 0.578432i 1.05774 − 1.37157i −0.296298 − 2.62911i 1.00000 −2.13159 + 2.11100i 1.00187 + 0.578432i
311.12 −0.500000 − 0.866025i 0.806632 − 1.53276i −0.500000 + 0.866025i 3.23425 − 1.86729i −1.73072 + 0.0678152i 0.481958 + 2.60148i 1.00000 −1.69869 − 2.47274i −3.23425 − 1.86729i
311.13 −0.500000 − 0.866025i 0.909476 + 1.47406i −0.500000 + 0.866025i −3.26421 + 1.88459i 0.821836 − 1.52466i −0.385108 + 2.61757i 1.00000 −1.34571 + 2.68124i 3.26421 + 1.88459i
311.14 −0.500000 − 0.866025i 1.24618 − 1.20292i −0.500000 + 0.866025i 0.0916492 − 0.0529137i −1.66485 − 0.477766i −2.29863 − 1.31008i 1.00000 0.105953 − 2.99813i −0.0916492 − 0.0529137i
311.15 −0.500000 − 0.866025i 1.40752 + 1.00940i −0.500000 + 0.866025i 2.19770 − 1.26884i 0.170404 − 1.72365i −2.61925 − 0.373536i 1.00000 0.962231 + 2.84150i −2.19770 − 1.26884i
311.16 −0.500000 − 0.866025i 1.57792 + 0.714250i −0.500000 + 0.866025i 2.19770 − 1.26884i −0.170404 − 1.72365i 2.61925 + 0.373536i 1.00000 1.97969 + 2.25407i −2.19770 − 1.26884i
311.17 −0.500000 − 0.866025i 1.71668 − 0.230243i −0.500000 + 0.866025i −1.00187 + 0.578432i −1.05774 − 1.37157i 0.296298 + 2.62911i 1.00000 2.89398 − 0.790508i 1.00187 + 0.578432i
311.18 −0.500000 − 0.866025i 1.73131 + 0.0505991i −0.500000 + 0.866025i −3.26421 + 1.88459i −0.821836 − 1.52466i 0.385108 − 2.61757i 1.00000 2.99488 + 0.175206i 3.26421 + 1.88459i
467.1 −0.500000 + 0.866025i −1.71076 + 0.270722i −0.500000 − 0.866025i 2.20332 + 1.27209i 0.620929 − 1.61693i 2.32714 + 1.25873i 1.00000 2.85342 − 0.926282i −2.20332 + 1.27209i
467.2 −0.500000 + 0.866025i −1.63622 + 0.568150i −0.500000 − 0.866025i −1.08726 − 0.627730i 0.326076 − 1.70108i 1.14208 + 2.38656i 1.00000 2.35441 − 1.85923i 1.08726 − 0.627730i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 467.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
39.d odd 2 1 inner
273.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bg.a ✓ 36
3.b odd 2 1 546.2.bg.b yes 36
7.d odd 6 1 inner 546.2.bg.a ✓ 36
13.b even 2 1 546.2.bg.b yes 36
21.g even 6 1 546.2.bg.b yes 36
39.d odd 2 1 inner 546.2.bg.a ✓ 36
91.s odd 6 1 546.2.bg.b yes 36
273.ba even 6 1 inner 546.2.bg.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bg.a ✓ 36 1.a even 1 1 trivial
546.2.bg.a ✓ 36 7.d odd 6 1 inner
546.2.bg.a ✓ 36 39.d odd 2 1 inner
546.2.bg.a ✓ 36 273.ba even 6 1 inner
546.2.bg.b yes 36 3.b odd 2 1
546.2.bg.b yes 36 13.b even 2 1
546.2.bg.b yes 36 21.g even 6 1
546.2.bg.b yes 36 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 26 T_{5}^{16} + 475 T_{5}^{14} - 3 T_{5}^{13} - 4279 T_{5}^{12} - 111 T_{5}^{11} + 27980 T_{5}^{10} + 3489 T_{5}^{9} - 89641 T_{5}^{8} - 38184 T_{5}^{7} + 195820 T_{5}^{6} + 233430 T_{5}^{5} + 48754 T_{5}^{4} + \cdots + 12 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display