# Properties

 Label 273.2.ba.c Level $273$ Weight $2$ Character orbit 273.ba Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(38,273)]

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, 1, 3]))

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

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

chi := DirichletCharacter("273.38");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.ba (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.

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 12 q^{3} - 40 q^{4} - 12 q^{9}+O(q^{10})$$ 64 * q - 12 * q^3 - 40 * q^4 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 12 q^{3} - 40 q^{4} - 12 q^{9} - 12 q^{10} - 6 q^{12} - 32 q^{16} - 24 q^{22} + 28 q^{25} - 22 q^{30} + 44 q^{36} + 6 q^{39} - 108 q^{40} + 24 q^{42} + 20 q^{49} + 28 q^{51} + 36 q^{52} + 48 q^{61} + 32 q^{64} + 90 q^{66} - 60 q^{75} - 68 q^{78} + 16 q^{79} + 60 q^{81} + 120 q^{82} + 60 q^{87} + 12 q^{88} - 32 q^{91} + 168 q^{94}+O(q^{100})$$ 64 * q - 12 * q^3 - 40 * q^4 - 12 * q^9 - 12 * q^10 - 6 * q^12 - 32 * q^16 - 24 * q^22 + 28 * q^25 - 22 * q^30 + 44 * q^36 + 6 * q^39 - 108 * q^40 + 24 * q^42 + 20 * q^49 + 28 * q^51 + 36 * q^52 + 48 * q^61 + 32 * q^64 + 90 * q^66 - 60 * q^75 - 68 * q^78 + 16 * q^79 + 60 * q^81 + 120 * q^82 + 60 * q^87 + 12 * q^88 - 32 * q^91 + 168 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1 −1.32290 2.29133i −1.08139 + 1.35300i −2.50014 + 4.33037i −2.63347 + 1.52043i 4.53074 + 0.687935i 2.25766 1.37948i 7.93816 −0.661205 2.92623i 6.96765 + 4.02277i
38.2 −1.32290 2.29133i 0.631037 1.61301i −2.50014 + 4.33037i −2.63347 + 1.52043i −4.53074 + 0.687935i −2.25766 + 1.37948i 7.93816 −2.20359 2.03573i 6.96765 + 4.02277i
38.3 −1.22551 2.12265i 0.680272 + 1.59287i −2.00377 + 3.47064i 0.770070 0.444600i 2.54743 3.39606i −1.88453 + 1.85703i 4.92055 −2.07446 + 2.16717i −1.88746 1.08973i
38.4 −1.22551 2.12265i 1.71960 0.207302i −2.00377 + 3.47064i 0.770070 0.444600i −2.54743 3.39606i 1.88453 1.85703i 4.92055 2.91405 0.712952i −1.88746 1.08973i
38.5 −1.16062 2.01026i −1.69161 + 0.372094i −1.69410 + 2.93426i 2.66615 1.53930i 2.71133 + 2.96871i 0.235284 + 2.63527i 3.22233 2.72309 1.25888i −6.18879 3.57310i
38.6 −1.16062 2.01026i −0.523562 1.65102i −1.69410 + 2.93426i 2.66615 1.53930i −2.71133 + 2.96871i −0.235284 2.63527i 3.22233 −2.45177 + 1.72883i −6.18879 3.57310i
38.7 −0.903010 1.56406i −1.69189 0.370830i −0.630855 + 1.09267i −1.48165 + 0.855432i 0.947792 + 2.98108i −1.44874 2.21386i −1.33337 2.72497 + 1.25481i 2.67589 + 1.54493i
38.8 −0.903010 1.56406i −1.16709 1.27980i −0.630855 + 1.09267i −1.48165 + 0.855432i −0.947792 + 2.98108i 1.44874 + 2.21386i −1.33337 −0.275792 + 2.98730i 2.67589 + 1.54493i
38.9 −0.697858 1.20873i −0.179489 + 1.72273i 0.0259885 0.0450133i −0.529415 + 0.305658i 2.20756 0.985265i −2.05092 1.67145i −2.86398 −2.93557 0.618419i 0.738913 + 0.426611i
38.10 −0.697858 1.20873i 1.40218 1.01680i 0.0259885 0.0450133i −0.529415 + 0.305658i −2.20756 0.985265i 2.05092 + 1.67145i −2.86398 0.932217 2.85149i 0.738913 + 0.426611i
38.11 −0.662291 1.14712i 0.379710 + 1.68992i 0.122740 0.212592i 3.12151 1.80220i 1.68706 1.55479i 2.63492 0.239193i −2.97432 −2.71164 + 1.28336i −4.13469 2.38717i
38.12 −0.662291 1.14712i 1.65337 0.516120i 0.122740 0.212592i 3.12151 1.80220i −1.68706 1.55479i −2.63492 + 0.239193i −2.97432 2.46724 1.70667i −4.13469 2.38717i
38.13 −0.387119 0.670510i −1.14149 + 1.30269i 0.700278 1.21292i −1.11726 + 0.645050i 1.31536 + 0.261085i −0.867295 + 2.49956i −2.63284 −0.393998 2.97402i 0.865025 + 0.499422i
38.14 −0.387119 0.670510i 0.557417 1.63990i 0.700278 1.21292i −1.11726 + 0.645050i −1.31536 + 0.261085i 0.867295 2.49956i −2.63284 −2.37857 1.82822i 0.865025 + 0.499422i
38.15 −0.100354 0.173818i −1.73110 + 0.0572655i 0.979858 1.69716i 2.67317 1.54335i 0.183677 + 0.295150i −2.55853 0.673741i −0.794745 2.99344 0.198265i −0.536525 0.309763i
38.16 −0.100354 0.173818i −0.815959 1.52781i 0.979858 1.69716i 2.67317 1.54335i −0.183677 + 0.295150i 2.55853 + 0.673741i −0.794745 −1.66842 + 2.49326i −0.536525 0.309763i
38.17 0.100354 + 0.173818i −1.73110 + 0.0572655i 0.979858 1.69716i −2.67317 + 1.54335i −0.183677 0.295150i 2.55853 + 0.673741i 0.794745 2.99344 0.198265i −0.536525 0.309763i
38.18 0.100354 + 0.173818i −0.815959 1.52781i 0.979858 1.69716i −2.67317 + 1.54335i 0.183677 0.295150i −2.55853 0.673741i 0.794745 −1.66842 + 2.49326i −0.536525 0.309763i
38.19 0.387119 + 0.670510i −1.14149 + 1.30269i 0.700278 1.21292i 1.11726 0.645050i −1.31536 0.261085i 0.867295 2.49956i 2.63284 −0.393998 2.97402i 0.865025 + 0.499422i
38.20 0.387119 + 0.670510i 0.557417 1.63990i 0.700278 1.21292i 1.11726 0.645050i 1.31536 0.261085i −0.867295 + 2.49956i 2.63284 −2.37857 1.82822i 0.865025 + 0.499422i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 38.32 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
7.d odd 6 1 inner
13.b even 2 1 inner
21.g even 6 1 inner
39.d odd 2 1 inner
91.s odd 6 1 inner
273.ba 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.ba.c 64
3.b odd 2 1 inner 273.2.ba.c 64
7.d odd 6 1 inner 273.2.ba.c 64
13.b even 2 1 inner 273.2.ba.c 64
21.g even 6 1 inner 273.2.ba.c 64
39.d odd 2 1 inner 273.2.ba.c 64
91.s odd 6 1 inner 273.2.ba.c 64
273.ba even 6 1 inner 273.2.ba.c 64

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{32} + 26 T_{2}^{30} + 404 T_{2}^{28} + 4136 T_{2}^{26} + 31455 T_{2}^{24} + 179569 T_{2}^{22} + 796976 T_{2}^{20} + 2718281 T_{2}^{18} + 7227508 T_{2}^{16} + 14586711 T_{2}^{14} + 22453924 T_{2}^{12} + \cdots + 3721$$ T2^32 + 26*T2^30 + 404*T2^28 + 4136*T2^26 + 31455*T2^24 + 179569*T2^22 + 796976*T2^20 + 2718281*T2^18 + 7227508*T2^16 + 14586711*T2^14 + 22453924*T2^12 + 24845331*T2^10 + 19540733*T2^8 + 8932777*T2^6 + 2651284*T2^4 + 105591*T2^2 + 3721 $$T_{19}^{32} + 100 T_{19}^{30} + 6474 T_{19}^{28} + 248412 T_{19}^{26} + 6930322 T_{19}^{24} + 125670468 T_{19}^{22} + 1623897188 T_{19}^{20} + 11228945442 T_{19}^{18} + 54238711263 T_{19}^{16} + \cdots + 15752961$$ T19^32 + 100*T19^30 + 6474*T19^28 + 248412*T19^26 + 6930322*T19^24 + 125670468*T19^22 + 1623897188*T19^20 + 11228945442*T19^18 + 54238711263*T19^16 + 125273490896*T19^14 + 204799184636*T19^12 + 195077579046*T19^10 + 129535430122*T19^8 + 32333317380*T19^6 + 5848742214*T19^4 + 344564766*T19^2 + 15752961