# 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 $8$

# Related objects

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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 16 q^{7}+O(q^{10})$$ 144 * q - 16 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 16 q^{7} + 72 q^{16} + 16 q^{18} + 8 q^{19} + 8 q^{21} + 24 q^{27} - 8 q^{28} - 8 q^{31} + 12 q^{33} - 8 q^{37} + 16 q^{39} - 40 q^{45} + 8 q^{46} + 8 q^{52} - 12 q^{54} - 32 q^{55} + 16 q^{57} + 8 q^{58} - 32 q^{61} - 40 q^{63} - 32 q^{66} - 8 q^{67} + 16 q^{72} + 80 q^{73} - 16 q^{76} + 16 q^{78} - 112 q^{79} - 48 q^{81} + 16 q^{84} - 80 q^{85} + 96 q^{87} - 32 q^{91} + 36 q^{93} - 64 q^{94} - 144 q^{97} - 32 q^{99}+O(q^{100})$$ 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

## 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 515.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(546, [\chi])$$.