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Properties

Label 546.2.z.a
Level $546$
Weight $2$
Character orbit 546.z
Analytic conductor $4.360$
Analytic rank $0$
Dimension $32$
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(131,546)]
 
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, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.z (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: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{10} - 24 q^{11} + 4 q^{14} - 12 q^{15} - 16 q^{16} + 4 q^{17} + 24 q^{18} - 12 q^{21} - 12 q^{22} - 6 q^{24} - 18 q^{25} + 16 q^{26} - 6 q^{27} - 2 q^{28} + 10 q^{30}+ \cdots + 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −0.866025 − 0.500000i −1.70193 + 0.321635i 0.500000 + 0.866025i 1.63964 − 2.83994i 1.63473 + 0.572419i 1.76715 + 1.96906i − 1.00000i 2.79310 − 1.09480i −2.83994 + 1.63964i
131.2 −0.866025 − 0.500000i −1.54393 − 0.785030i 0.500000 + 0.866025i −0.890016 + 1.54155i 0.944570 + 1.45182i −1.51727 + 2.16746i − 1.00000i 1.76746 + 2.42407i 1.54155 − 0.890016i
131.3 −0.866025 − 0.500000i −0.631675 − 1.61276i 0.500000 + 0.866025i −1.49759 + 2.59391i −0.259332 + 1.71253i −0.0366585 − 2.64550i − 1.00000i −2.20197 + 2.03748i 2.59391 − 1.49759i
131.4 −0.866025 − 0.500000i −0.526573 − 1.65007i 0.500000 + 0.866025i 1.35360 − 2.34450i −0.369008 + 1.69229i −2.61407 + 0.408207i − 1.00000i −2.44544 + 1.73776i −2.34450 + 1.35360i
131.5 −0.866025 − 0.500000i 0.115738 + 1.72818i 0.500000 + 0.866025i −1.85995 + 3.22152i 0.763858 − 1.55452i −1.49982 − 2.17958i − 1.00000i −2.97321 + 0.400031i 3.22152 − 1.85995i
131.6 −0.866025 − 0.500000i 0.587248 + 1.62946i 0.500000 + 0.866025i 0.386271 − 0.669041i 0.306158 − 1.70478i 1.82832 + 1.91239i − 1.00000i −2.31028 + 1.91379i −0.669041 + 0.386271i
131.7 −0.866025 − 0.500000i 1.11404 − 1.32624i 0.500000 + 0.866025i 0.737837 − 1.27797i −1.62791 + 0.591542i 2.24993 − 1.39206i − 1.00000i −0.517844 − 2.95497i −1.27797 + 0.737837i
131.8 −0.866025 − 0.500000i 1.72106 + 0.194824i 0.500000 + 0.866025i −0.735814 + 1.27447i −1.39307 − 1.02925i 0.322414 + 2.62603i − 1.00000i 2.92409 + 0.670607i 1.27447 − 0.735814i
131.9 0.866025 + 0.500000i −1.73120 − 0.0542481i 0.500000 + 0.866025i −0.401573 + 0.695544i −1.47214 − 0.912581i 1.69379 − 2.03250i 1.00000i 2.99411 + 0.187829i −0.695544 + 0.401573i
131.10 0.866025 + 0.500000i −1.61832 + 0.617297i 0.500000 + 0.866025i −0.166488 + 0.288366i −1.71015 − 0.274563i −2.56503 + 0.648546i 1.00000i 2.23789 − 1.99796i −0.288366 + 0.166488i
131.11 0.866025 + 0.500000i −0.913445 + 1.47160i 0.500000 + 0.866025i 1.72310 − 2.98449i −1.52687 + 0.817724i 2.50296 + 0.857422i 1.00000i −1.33124 − 2.68846i 2.98449 − 1.72310i
131.12 0.866025 + 0.500000i 0.110315 + 1.72853i 0.500000 + 0.866025i −1.50901 + 2.61368i −0.768732 + 1.55211i 0.237102 − 2.63511i 1.00000i −2.97566 + 0.381366i −2.61368 + 1.50901i
131.13 0.866025 + 0.500000i 0.305007 − 1.70498i 0.500000 + 0.866025i 1.04730 − 1.81398i 1.11664 − 1.32406i 1.84957 − 1.89185i 1.00000i −2.81394 − 1.04007i 1.81398 − 1.04730i
131.14 0.866025 + 0.500000i 1.48032 − 0.899243i 0.500000 + 0.866025i −1.85844 + 3.21892i 1.73162 − 0.0386050i −1.15681 + 2.37945i 1.00000i 1.38272 − 2.66234i −3.21892 + 1.85844i
131.15 0.866025 + 0.500000i 1.55397 + 0.764969i 0.500000 + 0.866025i 0.698212 − 1.20934i 0.963293 + 1.43947i 0.352771 + 2.62213i 1.00000i 1.82964 + 2.37748i 1.20934 − 0.698212i
131.16 0.866025 + 0.500000i 1.67937 − 0.423929i 0.500000 + 0.866025i 1.33293 − 2.30870i 1.66634 + 0.472552i −2.41436 − 1.08207i 1.00000i 2.64057 − 1.42387i 2.30870 − 1.33293i
521.1 −0.866025 + 0.500000i −1.70193 − 0.321635i 0.500000 − 0.866025i 1.63964 + 2.83994i 1.63473 − 0.572419i 1.76715 − 1.96906i 1.00000i 2.79310 + 1.09480i −2.83994 − 1.63964i
521.2 −0.866025 + 0.500000i −1.54393 + 0.785030i 0.500000 − 0.866025i −0.890016 − 1.54155i 0.944570 − 1.45182i −1.51727 − 2.16746i 1.00000i 1.76746 − 2.42407i 1.54155 + 0.890016i
521.3 −0.866025 + 0.500000i −0.631675 + 1.61276i 0.500000 − 0.866025i −1.49759 − 2.59391i −0.259332 − 1.71253i −0.0366585 + 2.64550i 1.00000i −2.20197 − 2.03748i 2.59391 + 1.49759i
521.4 −0.866025 + 0.500000i −0.526573 + 1.65007i 0.500000 − 0.866025i 1.35360 + 2.34450i −0.369008 − 1.69229i −2.61407 − 0.408207i 1.00000i −2.44544 − 1.73776i −2.34450 − 1.35360i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g 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.z.a ✓ 32
3.b odd 2 1 546.2.z.b yes 32
7.d odd 6 1 546.2.z.b yes 32
21.g even 6 1 inner 546.2.z.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.z.a ✓ 32 1.a even 1 1 trivial
546.2.z.a ✓ 32 21.g even 6 1 inner
546.2.z.b yes 32 3.b odd 2 1
546.2.z.b yes 32 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{32} + 49 T_{5}^{30} - 24 T_{5}^{29} + 1438 T_{5}^{28} - 1028 T_{5}^{27} + 28007 T_{5}^{26} + \cdots + 567011344 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display