# Properties

 Label 273.2.r.b Level $273$ Weight $2$ Character orbit 273.r Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(68,273)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("273.68");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.r (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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10})$$ 64 * q - 64 * q^4 + 12 * q^6 - 10 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 64 q^{4} + 12 q^{6} - 10 q^{7} - 4 q^{9} - 6 q^{10} + 6 q^{12} + 12 q^{13} - 9 q^{15} + 32 q^{16} + 2 q^{18} - 36 q^{19} + 10 q^{21} + 10 q^{22} - 18 q^{24} - 24 q^{25} - 14 q^{28} + 8 q^{30} - 24 q^{31} - 45 q^{33} + 33 q^{39} + 90 q^{40} + 3 q^{42} - 20 q^{43} - 6 q^{48} - 6 q^{49} - 10 q^{51} - 48 q^{52} - 18 q^{55} + 4 q^{57} + 30 q^{58} + 55 q^{60} - 18 q^{61} + 31 q^{63} - 84 q^{64} + 75 q^{66} + 44 q^{67} + 33 q^{69} + 20 q^{70} + 17 q^{72} + 84 q^{73} - 18 q^{76} - 71 q^{78} + 20 q^{79} + 16 q^{81} - 18 q^{82} - 135 q^{84} - 2 q^{85} - 46 q^{88} - 10 q^{91} - 56 q^{93} + 36 q^{94} + 30 q^{96} - 24 q^{97} + 22 q^{99}+O(q^{100})$$ 64 * q - 64 * q^4 + 12 * q^6 - 10 * q^7 - 4 * q^9 - 6 * q^10 + 6 * q^12 + 12 * q^13 - 9 * q^15 + 32 * q^16 + 2 * q^18 - 36 * q^19 + 10 * q^21 + 10 * q^22 - 18 * q^24 - 24 * q^25 - 14 * q^28 + 8 * q^30 - 24 * q^31 - 45 * q^33 + 33 * q^39 + 90 * q^40 + 3 * q^42 - 20 * q^43 - 6 * q^48 - 6 * q^49 - 10 * q^51 - 48 * q^52 - 18 * q^55 + 4 * q^57 + 30 * q^58 + 55 * q^60 - 18 * q^61 + 31 * q^63 - 84 * q^64 + 75 * q^66 + 44 * q^67 + 33 * q^69 + 20 * q^70 + 17 * q^72 + 84 * q^73 - 18 * q^76 - 71 * q^78 + 20 * q^79 + 16 * q^81 - 18 * q^82 - 135 * q^84 - 2 * q^85 - 46 * q^88 - 10 * q^91 - 56 * q^93 + 36 * q^94 + 30 * q^96 - 24 * q^97 + 22 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1 2.73146i −1.52761 + 0.816332i −5.46085 0.121221 0.209960i 2.22977 + 4.17261i −2.13335 + 1.56488i 9.45317i 1.66721 2.49408i −0.573498 0.331109i
68.2 2.60090i 0.206727 1.71967i −4.76468 1.54658 2.67876i −4.47269 0.537676i 1.09877 + 2.40680i 7.19066i −2.91453 0.711004i −6.96719 4.02251i
68.3 2.39671i 1.60287 + 0.656371i −3.74420 1.10064 1.90637i 1.57313 3.84160i −1.62460 2.08822i 4.18035i 2.13835 + 2.10415i −4.56901 2.63792i
68.4 2.39579i 0.452516 + 1.67189i −3.73983 −1.50632 + 2.60901i 4.00551 1.08414i 2.64172 0.145950i 4.16828i −2.59046 + 1.51312i 6.25066 + 3.60882i
68.5 1.97104i 1.69142 0.372976i −1.88498 −0.272603 + 0.472162i −0.735148 3.33384i 2.35644 + 1.20299i 0.226699i 2.72178 1.26171i 0.930648 + 0.537310i
68.6 1.92371i −1.08390 1.35099i −1.70067 −1.94643 + 3.37132i −2.59891 + 2.08510i −0.630216 + 2.56960i 0.575834i −0.650341 + 2.92866i 6.48544 + 3.74437i
68.7 1.85355i −1.03215 + 1.39092i −1.43566 −0.852778 + 1.47706i 2.57815 + 1.91315i −1.51863 2.16651i 1.04604i −0.869331 2.87128i 2.73780 + 1.58067i
68.8 1.85049i −1.73048 + 0.0736936i −1.42433 1.21731 2.10844i 0.136370 + 3.20225i 2.52177 0.800422i 1.06527i 2.98914 0.255051i −3.90166 2.25263i
68.9 1.53974i −1.04458 1.38161i −0.370785 0.956175 1.65614i −2.12732 + 1.60838i −2.63109 0.278191i 2.50856i −0.817709 + 2.88641i −2.55002 1.47226i
68.10 1.40538i −0.386542 + 1.68837i 0.0248937 1.48377 2.56997i 2.37281 + 0.543240i 0.0176221 + 2.64569i 2.84575i −2.70117 1.30525i −3.61180 2.08527i
68.11 1.05456i 1.46610 0.922247i 0.887893 −0.000964697 0.00167090i −0.972570 1.54610i −2.64511 + 0.0584461i 3.04547i 1.29892 2.70422i 0.00176208 + 0.00101734i
68.12 0.883413i 1.41202 + 1.00309i 1.21958 −1.72131 + 2.98140i 0.886146 1.24740i −2.26618 + 1.36544i 2.84422i 0.987606 + 2.83278i 2.63381 + 1.52063i
68.13 0.843135i 0.346738 1.69699i 1.28912 −1.06280 + 1.84082i −1.43079 0.292347i 2.34470 1.22572i 2.77318i −2.75955 1.17682i 1.55206 + 0.896082i
68.14 0.771110i 0.714765 + 1.57769i 1.40539 0.907647 1.57209i 1.21657 0.551162i 0.574285 2.58267i 2.62593i −1.97822 + 2.25536i −1.21225 0.699896i
68.15 0.546208i −1.72803 + 0.117898i 1.70166 −1.25657 + 2.17644i 0.0643970 + 0.943866i −0.612705 2.57383i 2.02188i 2.97220 0.407464i 1.18879 + 0.686349i
68.16 0.0504397i −1.05075 + 1.37693i 1.99746 −1.19579 + 2.07117i 0.0694519 + 0.0529993i 0.00656452 + 2.64574i 0.201631i −0.791866 2.89360i 0.104469 + 0.0603154i
68.17 0.0504397i 0.667082 1.59844i 1.99746 1.19579 2.07117i 0.0806247 + 0.0336474i 0.00656452 + 2.64574i 0.201631i −2.11000 2.13258i 0.104469 + 0.0603154i
68.18 0.546208i −0.761914 1.55547i 1.70166 1.25657 2.17644i 0.849611 0.416164i −0.612705 2.57383i 2.02188i −1.83897 + 2.37027i 1.18879 + 0.686349i
68.19 0.771110i 1.72370 0.169842i 1.40539 −0.907647 + 1.57209i 0.130967 + 1.32917i 0.574285 2.58267i 2.62593i 2.94231 0.585513i −1.21225 0.699896i
68.20 0.843135i −1.29627 + 1.14878i 1.28912 1.06280 1.84082i −0.968576 1.09293i 2.34470 1.22572i 2.77318i 0.360617 2.97825i 1.55206 + 0.896082i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 68.32 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
91.v odd 6 1 inner
273.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.r.b 64
3.b odd 2 1 inner 273.2.r.b 64
7.d odd 6 1 273.2.bf.b yes 64
13.c even 3 1 273.2.bf.b yes 64
21.g even 6 1 273.2.bf.b yes 64
39.i odd 6 1 273.2.bf.b yes 64
91.v odd 6 1 inner 273.2.r.b 64
273.r even 6 1 inner 273.2.r.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.r.b 64 1.a even 1 1 trivial
273.2.r.b 64 3.b odd 2 1 inner
273.2.r.b 64 91.v odd 6 1 inner
273.2.r.b 64 273.r even 6 1 inner
273.2.bf.b yes 64 7.d odd 6 1
273.2.bf.b yes 64 13.c even 3 1
273.2.bf.b yes 64 21.g even 6 1
273.2.bf.b yes 64 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} + 48 T_{2}^{30} + 1036 T_{2}^{28} + 13303 T_{2}^{26} + 113325 T_{2}^{24} + 676187 T_{2}^{22} + 2908599 T_{2}^{20} + 9145849 T_{2}^{18} + 21096866 T_{2}^{16} + 35519813 T_{2}^{14} + 43068692 T_{2}^{12} + \cdots + 367$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.