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Properties

Label 546.2.bv.a
Level $546$
Weight $2$
Character orbit 546.bv
Analytic conductor $4.360$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(317,546)]
 
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, 4, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.317");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bv (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: \(144\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 16 q^{7} + 72 q^{16} + 16 q^{18} + 8 q^{19} + 8 q^{21} + 24 q^{27} - 8 q^{28} - 8 q^{31} + 12 q^{33} - 8 q^{37} + 16 q^{39} - 40 q^{45} + 8 q^{46} + 8 q^{52} - 12 q^{54} - 32 q^{55} + 16 q^{57} + 8 q^{58} - 32 q^{61} - 40 q^{63} - 32 q^{66} - 8 q^{67} + 16 q^{72} + 80 q^{73} - 16 q^{76} + 16 q^{78} - 112 q^{79} - 48 q^{81} + 16 q^{84} - 80 q^{85} + 96 q^{87} - 32 q^{91} + 36 q^{93} - 64 q^{94} - 144 q^{97} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
317.1 −0.965926 + 0.258819i −1.70001 − 0.331586i 0.866025 − 0.500000i −1.16729 + 0.312774i 1.72791 − 0.119709i 0.138253 + 2.64214i −0.707107 + 0.707107i 2.78010 + 1.12740i 1.04656 − 0.604234i
317.2 −0.965926 + 0.258819i −1.65689 + 0.504710i 0.866025 − 0.500000i 1.11432 − 0.298580i 1.46980 − 0.916345i 0.578898 − 2.58164i −0.707107 + 0.707107i 2.49054 − 1.67249i −0.999069 + 0.576813i
317.3 −0.965926 + 0.258819i −1.40623 − 1.01120i 0.866025 − 0.500000i −1.26981 + 0.340244i 1.62003 + 0.612783i −1.12113 − 2.39647i −0.707107 + 0.707107i 0.954959 + 2.84395i 1.13848 − 0.657301i
317.4 −0.965926 + 0.258819i −1.35728 + 1.07601i 0.866025 − 0.500000i −2.88157 + 0.772115i 1.03254 − 1.39063i −2.53580 + 0.754783i −0.707107 + 0.707107i 0.684406 − 2.92089i 2.58355 − 1.49161i
317.5 −0.965926 + 0.258819i −1.11793 + 1.32296i 0.866025 − 0.500000i 1.82490 − 0.488980i 0.737434 − 1.56722i 2.57215 + 0.619719i −0.707107 + 0.707107i −0.500449 − 2.95796i −1.63616 + 0.944637i
317.6 −0.965926 + 0.258819i −0.977016 − 1.43019i 0.866025 − 0.500000i 1.28039 − 0.343079i 1.31389 + 1.12859i −2.62526 + 0.328643i −0.707107 + 0.707107i −1.09088 + 2.79463i −1.14797 + 0.662779i
317.7 −0.965926 + 0.258819i −0.540391 − 1.64559i 0.866025 − 0.500000i −1.97621 + 0.529524i 0.947888 + 1.44966i 2.60708 − 0.450728i −0.707107 + 0.707107i −2.41596 + 1.77853i 1.77182 − 1.02296i
317.8 −0.965926 + 0.258819i −0.357658 + 1.69472i 0.866025 − 0.500000i 3.24035 − 0.868248i −0.0931546 − 1.72954i −1.25441 + 2.32948i −0.707107 + 0.707107i −2.74416 − 1.21226i −2.90521 + 1.67733i
317.9 −0.965926 + 0.258819i −0.214218 + 1.71875i 0.866025 − 0.500000i −0.579405 + 0.155251i −0.237927 − 1.71563i 0.0183841 − 2.64569i −0.707107 + 0.707107i −2.90822 − 0.736375i 0.519481 − 0.299922i
317.10 −0.965926 + 0.258819i 0.0392246 − 1.73161i 0.866025 − 0.500000i 1.98929 − 0.533030i 0.410285 + 1.68276i −2.22758 + 1.42754i −0.707107 + 0.707107i −2.99692 − 0.135843i −1.78355 + 1.02973i
317.11 −0.965926 + 0.258819i 0.398691 + 1.68554i 0.866025 − 0.500000i −1.89818 + 0.508616i −0.821356 − 1.52492i 1.71379 + 2.01567i −0.707107 + 0.707107i −2.68209 + 1.34402i 1.70186 − 0.982571i
317.12 −0.965926 + 0.258819i 0.664489 − 1.59952i 0.866025 − 0.500000i 2.30125 − 0.616617i −0.227862 + 1.71700i 2.49307 − 0.885767i −0.707107 + 0.707107i −2.11691 − 2.12572i −2.06324 + 1.19121i
317.13 −0.965926 + 0.258819i 0.910127 − 1.47366i 0.866025 − 0.500000i −2.92144 + 0.782798i −0.497704 + 1.65900i −0.584700 + 2.58033i −0.707107 + 0.707107i −1.34334 − 2.68243i 2.61929 − 1.51225i
317.14 −0.965926 + 0.258819i 1.15401 + 1.29162i 0.866025 − 0.500000i −2.59930 + 0.696480i −1.44898 − 0.948926i −2.64286 + 0.123744i −0.707107 + 0.707107i −0.336539 + 2.98106i 2.33047 − 1.34550i
317.15 −0.965926 + 0.258819i 1.19913 + 1.24984i 0.866025 − 0.500000i 4.04433 − 1.08367i −1.48175 − 0.896896i −1.05878 − 2.42466i −0.707107 + 0.707107i −0.124198 + 2.99743i −3.62604 + 2.09350i
317.16 −0.965926 + 0.258819i 1.54470 − 0.783515i 0.866025 − 0.500000i 2.31362 − 0.619932i −1.28928 + 1.15662i −1.72913 − 2.00252i −0.707107 + 0.707107i 1.77221 − 2.42060i −2.07433 + 1.19762i
317.17 −0.965926 + 0.258819i 1.69438 − 0.359266i 0.866025 − 0.500000i −3.75102 + 1.00508i −1.54366 + 0.785563i 1.18204 − 2.36702i −0.707107 + 0.707107i 2.74186 − 1.21747i 3.36308 − 1.94167i
317.18 −0.965926 + 0.258819i 1.72288 − 0.178019i 0.866025 − 0.500000i 0.935798 − 0.250746i −1.61810 + 0.617867i 2.47599 + 0.932449i −0.707107 + 0.707107i 2.93662 − 0.613411i −0.839013 + 0.484404i
317.19 0.965926 − 0.258819i −1.73129 + 0.0513637i 0.866025 − 0.500000i 2.92144 − 0.782798i −1.65900 + 0.497704i −0.584700 + 2.58033i 0.707107 − 0.707107i 2.99472 − 0.177851i 2.61929 − 1.51225i
317.20 0.965926 − 0.258819i −1.71747 − 0.224294i 0.866025 − 0.500000i −2.30125 + 0.616617i −1.71700 + 0.227862i 2.49307 − 0.885767i 0.707107 − 0.707107i 2.89938 + 0.770435i −2.06324 + 1.19121i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 317.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
13.d odd 4 1 inner
21.h odd 6 1 inner
39.f even 4 1 inner
91.z odd 12 1 inner
273.cd 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.bv.a ✓ 144
3.b odd 2 1 inner 546.2.bv.a ✓ 144
7.c even 3 1 inner 546.2.bv.a ✓ 144
13.d odd 4 1 inner 546.2.bv.a ✓ 144
21.h odd 6 1 inner 546.2.bv.a ✓ 144
39.f even 4 1 inner 546.2.bv.a ✓ 144
91.z odd 12 1 inner 546.2.bv.a ✓ 144
273.cd even 12 1 inner 546.2.bv.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bv.a ✓ 144 1.a even 1 1 trivial
546.2.bv.a ✓ 144 3.b odd 2 1 inner
546.2.bv.a ✓ 144 7.c even 3 1 inner
546.2.bv.a ✓ 144 13.d odd 4 1 inner
546.2.bv.a ✓ 144 21.h odd 6 1 inner
546.2.bv.a ✓ 144 39.f even 4 1 inner
546.2.bv.a ✓ 144 91.z odd 12 1 inner
546.2.bv.a ✓ 144 273.cd even 12 1 inner

Hecke kernels

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