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Properties

Label 546.2.ch.a
Level $546$
Weight $2$
Character orbit 546.ch
Analytic conductor $4.360$
Analytic rank $0$
Dimension $152$
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(137,546)]
 
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([6, 8, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.ch (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: \(152\)
Relative dimension: \(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q + 8 q^{7} - 152 q^{16} - 8 q^{18} + 36 q^{19} + 8 q^{21} + 24 q^{27} - 12 q^{28} - 24 q^{30} - 36 q^{31} + 12 q^{33} + 24 q^{36} - 12 q^{37} - 8 q^{39} - 36 q^{42} + 84 q^{43} - 40 q^{45} + 8 q^{46} + 4 q^{49} - 20 q^{52} - 12 q^{54} + 16 q^{55} - 8 q^{57} + 8 q^{58} + 24 q^{60} - 32 q^{61} - 28 q^{63} + 16 q^{66} + 92 q^{67} + 84 q^{69} - 8 q^{72} + 84 q^{73} - 48 q^{76} - 8 q^{78} + 32 q^{79} - 24 q^{81} - 48 q^{82} - 8 q^{84} - 152 q^{85} - 96 q^{87} - 88 q^{91} - 48 q^{93} + 32 q^{94} - 100 q^{97} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.707107 + 0.707107i −1.72681 − 0.134626i − 1.00000i −1.91810 − 0.513954i 1.31623 − 1.12584i 1.87225 − 1.86941i 0.707107 + 0.707107i 2.96375 + 0.464948i 1.71972 − 0.992884i
137.2 −0.707107 + 0.707107i −1.72614 + 0.143006i − 1.00000i −1.09645 − 0.293793i 1.11944 − 1.32168i −0.0234379 + 2.64565i 0.707107 + 0.707107i 2.95910 − 0.493697i 0.983052 − 0.567565i
137.3 −0.707107 + 0.707107i −1.47304 − 0.911131i − 1.00000i 1.94682 + 0.521648i 1.68586 − 0.397328i −2.63140 − 0.275187i 0.707107 + 0.707107i 1.33968 + 2.68426i −1.74547 + 1.00775i
137.4 −0.707107 + 0.707107i −1.45428 − 0.940775i − 1.00000i 3.45354 + 0.925372i 1.69356 − 0.363106i 2.64433 + 0.0866153i 0.707107 + 0.707107i 1.22989 + 2.73631i −3.09636 + 1.78768i
137.5 −0.707107 + 0.707107i −1.33512 + 1.10338i − 1.00000i 2.13400 + 0.571804i 0.163866 − 1.72428i −2.34532 − 1.22452i 0.707107 + 0.707107i 0.565101 − 2.94630i −1.91329 + 1.10464i
137.6 −0.707107 + 0.707107i −0.975072 + 1.43151i − 1.00000i −0.438011 − 0.117365i −0.322754 − 1.70171i 2.35961 + 1.19677i 0.707107 + 0.707107i −1.09847 − 2.79166i 0.392710 − 0.226731i
137.7 −0.707107 + 0.707107i −0.965783 − 1.43780i − 1.00000i −4.19664 − 1.12449i 1.69959 + 0.333765i −2.15090 − 1.54066i 0.707107 + 0.707107i −1.13453 + 2.77720i 3.76260 − 2.17234i
137.8 −0.707107 + 0.707107i −0.649813 + 1.60554i − 1.00000i −1.93876 − 0.519490i −0.675797 − 1.59477i −1.14094 − 2.38710i 0.707107 + 0.707107i −2.15549 − 2.08660i 1.73825 − 1.00358i
137.9 −0.707107 + 0.707107i −0.238917 − 1.71549i − 1.00000i 0.465027 + 0.124604i 1.38198 + 1.04410i −1.66275 + 2.05797i 0.707107 + 0.707107i −2.88584 + 0.819723i −0.416932 + 0.240716i
137.10 −0.707107 + 0.707107i 0.116244 − 1.72815i − 1.00000i −1.15194 − 0.308662i 1.13979 + 1.30418i 1.79371 + 1.94489i 0.707107 + 0.707107i −2.97297 − 0.401773i 1.03280 − 0.596290i
137.11 −0.707107 + 0.707107i 0.396140 − 1.68614i − 1.00000i 0.940755 + 0.252074i 0.912169 + 1.47240i 1.13831 − 2.38836i 0.707107 + 0.707107i −2.68615 − 1.33590i −0.843457 + 0.486970i
137.12 −0.707107 + 0.707107i 0.480095 + 1.66418i − 1.00000i 3.07620 + 0.824266i −1.51623 − 0.837278i 1.73277 − 1.99938i 0.707107 + 0.707107i −2.53902 + 1.59793i −2.75805 + 1.59236i
137.13 −0.707107 + 0.707107i 0.642070 + 1.60865i − 1.00000i −0.758733 − 0.203302i −1.59150 − 0.683474i −1.58539 + 2.11814i 0.707107 + 0.707107i −2.17549 + 2.06573i 0.680261 − 0.392749i
137.14 −0.707107 + 0.707107i 0.922039 + 1.46623i − 1.00000i −3.25307 − 0.871657i −1.68876 − 0.384805i 2.64559 − 0.0294574i 0.707107 + 0.707107i −1.29969 + 2.70385i 2.91662 − 1.68391i
137.15 −0.707107 + 0.707107i 1.47583 − 0.906598i − 1.00000i 3.71236 + 0.994723i −0.402510 + 1.68463i 1.97669 + 1.75860i 0.707107 + 0.707107i 1.35616 − 2.67597i −3.32841 + 1.92166i
137.16 −0.707107 + 0.707107i 1.52325 + 0.824446i − 1.00000i 2.39134 + 0.640759i −1.66007 + 0.494128i −0.703385 − 2.55054i 0.707107 + 0.707107i 1.64058 + 2.51167i −2.14402 + 1.23785i
137.17 −0.707107 + 0.707107i 1.59861 − 0.666660i − 1.00000i −1.41769 − 0.379869i −0.658990 + 1.60179i 1.80686 − 1.93268i 0.707107 + 0.707107i 2.11113 − 2.13146i 1.27107 − 0.733851i
137.18 −0.707107 + 0.707107i 1.66850 + 0.464889i − 1.00000i 1.76812 + 0.473766i −1.50853 + 0.851078i −1.72527 + 2.00586i 0.707107 + 0.707107i 2.56776 + 1.55133i −1.58525 + 0.915246i
137.19 −0.707107 + 0.707107i 1.72220 − 0.184470i − 1.00000i −3.71876 − 0.996438i −1.08734 + 1.34822i −0.403232 + 2.61484i 0.707107 + 0.707107i 2.93194 − 0.635390i 3.33415 − 1.92497i
137.20 0.707107 − 0.707107i −1.65831 − 0.500003i − 1.00000i −0.940755 − 0.252074i −1.52616 + 0.819048i 1.13831 − 2.38836i −0.707107 − 0.707107i 2.49999 + 1.65832i −0.843457 + 0.486970i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.x odd 12 1 inner
273.bv 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.ch.a yes 152
3.b odd 2 1 inner 546.2.ch.a yes 152
7.c even 3 1 546.2.bw.a ✓ 152
13.f odd 12 1 546.2.bw.a ✓ 152
21.h odd 6 1 546.2.bw.a ✓ 152
39.k even 12 1 546.2.bw.a ✓ 152
91.x odd 12 1 inner 546.2.ch.a yes 152
273.bv even 12 1 inner 546.2.ch.a yes 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bw.a ✓ 152 7.c even 3 1
546.2.bw.a ✓ 152 13.f odd 12 1
546.2.bw.a ✓ 152 21.h odd 6 1
546.2.bw.a ✓ 152 39.k even 12 1
546.2.ch.a yes 152 1.a even 1 1 trivial
546.2.ch.a yes 152 3.b odd 2 1 inner
546.2.ch.a yes 152 91.x odd 12 1 inner
546.2.ch.a yes 152 273.bv even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(546, [\chi])\).