# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bh (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{4} - 4q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 32q^{4} - 4q^{7} - 4q^{9} - 12q^{10} - 30q^{12} + 12q^{15} - 16q^{16} - 10q^{18} + 10q^{21} - 8q^{22} + 36q^{24} - 36q^{25} - 20q^{28} - 22q^{30} + 12q^{31} + 36q^{36} - 36q^{40} + 48q^{42} - 32q^{43} - 6q^{45} - 48q^{46} + 36q^{49} - 16q^{51} - 54q^{54} - 8q^{57} - 12q^{58} + 16q^{60} - 72q^{61} - 86q^{63} - 48q^{64} - 78q^{66} + 32q^{67} - 4q^{70} + 62q^{72} + 48q^{73} + 48q^{75} - 20q^{78} - 64q^{79} + 28q^{81} + 72q^{82} - 18q^{84} + 64q^{85} + 60q^{87} + 44q^{88} + 8q^{91} - 38q^{93} + 72q^{94} + 66q^{96} - 68q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −2.34239 1.35238i −1.69032 + 0.377930i 2.65786 + 4.60354i 0.200745 0.347701i 4.47048 + 1.40069i −0.906582 2.48558i 8.96820i 2.71434 1.27764i −0.940448 + 0.542968i
131.2 −2.16068 1.24747i 1.72610 0.143491i 2.11235 + 3.65869i −0.688862 + 1.19314i −3.90853 1.84321i −2.58242 + 0.575427i 5.55045i 2.95882 0.495358i 2.97682 1.71866i
131.3 −2.13803 1.23439i 1.47880 + 0.901745i 2.04745 + 3.54630i 1.98059 3.43049i −2.04862 3.75338i 2.62523 0.328920i 5.17189i 1.37371 + 2.66701i −8.46914 + 4.88966i
131.4 −2.13534 1.23284i −0.105557 + 1.72883i 2.03979 + 3.53302i −1.46641 + 2.53990i 2.35677 3.56151i 0.805632 + 2.52011i 5.12758i −2.97772 0.364982i 6.26259 3.61571i
131.5 −1.75475 1.01310i 0.351400 1.69603i 1.05276 + 1.82344i 0.867268 1.50215i −2.33487 + 2.62010i −1.44131 2.21870i 0.213809i −2.75304 1.19197i −3.04367 + 1.75727i
131.6 −1.71340 0.989233i −1.24374 1.20545i 0.957162 + 1.65785i −0.830302 + 1.43813i 0.938561 + 3.29577i 2.61452 0.405294i 0.169506i 0.0937900 + 2.99853i 2.84528 1.64272i
131.7 −1.61355 0.931585i −1.54166 0.789478i 0.735702 + 1.27427i 0.650945 1.12747i 1.75209 + 2.71005i −0.953821 + 2.46784i 0.984865i 1.75345 + 2.43422i −2.10067 + 1.21282i
131.8 −1.59036 0.918192i −0.540046 + 1.64571i 0.686154 + 1.18845i 1.90624 3.30170i 2.36994 2.12139i −2.46739 + 0.954988i 1.15268i −2.41670 1.77751i −6.06319 + 3.50058i
131.9 −1.15260 0.665454i −0.732991 + 1.56931i −0.114341 0.198044i −0.0955042 + 0.165418i 1.88915 1.32101i 2.46727 0.955290i 2.96617i −1.92545 2.30058i 0.220156 0.127107i
131.10 −1.09109 0.629943i −1.60797 + 0.643773i −0.206343 0.357397i −2.03837 + 3.53057i 2.15998 + 0.310511i −1.74546 1.98831i 3.03971i 2.17111 2.07033i 4.44811 2.56812i
131.11 −1.00971 0.582955i 1.67611 0.436646i −0.320327 0.554823i −0.156706 + 0.271422i −1.94692 0.536211i 2.13875 + 1.55748i 3.07876i 2.61868 1.46373i 0.316454 0.182705i
131.12 −0.987309 0.570023i 1.12414 + 1.31769i −0.350147 0.606472i −0.578962 + 1.00279i −0.358764 1.94175i −2.55265 + 0.695677i 3.07846i −0.472604 + 2.96254i 1.14323 0.660043i
131.13 −0.734739 0.424202i 0.243844 1.71480i −0.640106 1.10870i −1.44264 + 2.49872i −0.906583 + 1.15649i −1.51284 + 2.17056i 2.78294i −2.88108 0.836289i 2.11992 1.22394i
131.14 −0.636389 0.367419i 1.70069 0.328105i −0.730006 1.26441i 1.03100 1.78574i −1.20285 0.416064i −0.617429 2.57270i 2.54255i 2.78469 1.11601i −1.31223 + 0.757615i
131.15 −0.326193 0.188328i −0.329203 1.70048i −0.929065 1.60919i 2.06368 3.57439i −0.212864 + 0.616683i 2.50486 + 0.851866i 1.45319i −2.78325 + 1.11960i −1.34632 + 0.777295i
131.16 −0.0408006 0.0235563i 0.539158 + 1.64600i −0.998890 1.73013i 0.697336 1.20782i 0.0167755 0.0798583i 0.623634 2.57120i 0.188345i −2.41862 + 1.77491i −0.0569035 + 0.0328532i
131.17 0.0408006 + 0.0235563i −1.15590 1.28992i −0.998890 1.73013i −0.697336 + 1.20782i −0.0167755 0.0798583i 0.623634 2.57120i 0.188345i −0.327806 + 2.98204i −0.0569035 + 0.0328532i
131.18 0.326193 + 0.188328i 1.30806 + 1.13534i −0.929065 1.60919i −2.06368 + 3.57439i 0.212864 + 0.616683i 2.50486 + 0.851866i 1.45319i 0.422020 + 2.97017i −1.34632 + 0.777295i
131.19 0.636389 + 0.367419i 1.13449 1.30879i −0.730006 1.26441i −1.03100 + 1.78574i 1.20285 0.416064i −0.617429 2.57270i 2.54255i −0.425855 2.96962i −1.31223 + 0.757615i
131.20 0.734739 + 0.424202i 1.60698 + 0.646225i −0.640106 1.10870i 1.44264 2.49872i 0.906583 + 1.15649i −1.51284 + 2.17056i 2.78294i 2.16479 + 2.07694i 2.11992 1.22394i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 248.32 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
21.g 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.bh.a 64
3.b odd 2 1 inner 273.2.bh.a 64
7.d odd 6 1 inner 273.2.bh.a 64
21.g even 6 1 inner 273.2.bh.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bh.a 64 1.a even 1 1 trivial
273.2.bh.a 64 3.b odd 2 1 inner
273.2.bh.a 64 7.d odd 6 1 inner
273.2.bh.a 64 21.g even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(273, [\chi])$$.