# Properties

 Label 273.2.n.c Level $273$ Weight $2$ Character orbit 273.n Analytic conductor $2.180$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 8q^{3} + 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 8q^{3} + 4q^{6} + 8q^{13} - 16q^{15} - 72q^{16} - 12q^{18} + 40q^{19} + 16q^{22} + 8q^{24} - 16q^{27} + 44q^{33} - 32q^{34} - 8q^{37} - 4q^{39} - 48q^{40} - 8q^{42} + 44q^{45} - 32q^{46} + 80q^{48} - 72q^{52} + 44q^{54} - 80q^{55} - 52q^{57} + 16q^{58} + 44q^{60} - 64q^{61} + 24q^{63} - 152q^{66} + 56q^{67} + 16q^{70} + 16q^{72} + 32q^{73} + 104q^{76} - 44q^{78} + 8q^{79} + 12q^{84} - 96q^{85} - 72q^{87} - 8q^{91} - 8q^{93} + 160q^{94} + 8q^{96} - 32q^{97} - 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −1.89860 + 1.89860i −1.57000 0.731512i 5.20935i −0.367301 + 0.367301i 4.36964 1.59195i 0.707107 0.707107i 6.09327 + 6.09327i 1.92978 + 2.29694i 1.39472i
8.2 −1.76503 + 1.76503i −0.743367 + 1.56442i 4.23065i −0.269592 + 0.269592i −1.44918 4.07331i −0.707107 + 0.707107i 3.93715 + 3.93715i −1.89481 2.32587i 0.951676i
8.3 −1.72156 + 1.72156i 0.578040 1.63275i 3.92752i −1.26580 + 1.26580i 1.81574 + 3.80600i −0.707107 + 0.707107i 3.31834 + 3.31834i −2.33174 1.88759i 4.35829i
8.4 −1.65298 + 1.65298i 0.818165 + 1.52663i 3.46469i 1.96311 1.96311i −3.87590 1.17108i 0.707107 0.707107i 2.42111 + 2.42111i −1.66121 + 2.49807i 6.48995i
8.5 −1.48770 + 1.48770i 1.53135 0.809292i 2.42651i 0.854208 0.854208i −1.07421 + 3.48218i 0.707107 0.707107i 0.634513 + 0.634513i 1.69009 2.47863i 2.54161i
8.6 −1.34745 + 1.34745i −0.469658 1.66716i 1.63125i 3.06725 3.06725i 2.87926 + 1.61358i −0.707107 + 0.707107i −0.496878 0.496878i −2.55884 + 1.56599i 8.26592i
8.7 −1.08486 + 1.08486i 1.14443 + 1.30011i 0.353851i −0.879295 + 0.879295i −2.65199 0.168890i −0.707107 + 0.707107i −1.78584 1.78584i −0.380562 + 2.97576i 1.90783i
8.8 −0.946941 + 0.946941i −1.73159 0.0399728i 0.206606i 1.18824 1.18824i 1.67756 1.60186i 0.707107 0.707107i −2.08953 2.08953i 2.99680 + 0.138433i 2.25040i
8.9 −0.636874 + 0.636874i 1.71860 0.215472i 1.18878i 1.55282 1.55282i −0.957301 + 1.23176i −0.707107 + 0.707107i −2.03085 2.03085i 2.90714 0.740618i 1.97791i
8.10 −0.217303 + 0.217303i 0.368455 + 1.69241i 1.90556i −1.58813 + 1.58813i −0.447831 0.287699i 0.707107 0.707107i −0.848689 0.848689i −2.72848 + 1.24715i 0.690209i
8.11 −0.150864 + 0.150864i −1.22804 1.22144i 1.95448i 0.482343 0.482343i 0.369539 0.000996021i −0.707107 + 0.707107i −0.596590 0.596590i 0.0161717 + 2.99996i 0.145537i
8.12 −0.0762157 + 0.0762157i 1.58361 0.701549i 1.98838i −2.41817 + 2.41817i −0.0672271 + 0.174165i 0.707107 0.707107i −0.303978 0.303978i 2.01566 2.22196i 0.368606i
8.13 0.0762157 0.0762157i 1.58361 + 0.701549i 1.98838i 2.41817 2.41817i 0.174165 0.0672271i 0.707107 0.707107i 0.303978 + 0.303978i 2.01566 + 2.22196i 0.368606i
8.14 0.150864 0.150864i −1.22804 + 1.22144i 1.95448i −0.482343 + 0.482343i −0.000996021 0.369539i −0.707107 + 0.707107i 0.596590 + 0.596590i 0.0161717 2.99996i 0.145537i
8.15 0.217303 0.217303i 0.368455 1.69241i 1.90556i 1.58813 1.58813i −0.287699 0.447831i 0.707107 0.707107i 0.848689 + 0.848689i −2.72848 1.24715i 0.690209i
8.16 0.636874 0.636874i 1.71860 + 0.215472i 1.18878i −1.55282 + 1.55282i 1.23176 0.957301i −0.707107 + 0.707107i 2.03085 + 2.03085i 2.90714 + 0.740618i 1.97791i
8.17 0.946941 0.946941i −1.73159 + 0.0399728i 0.206606i −1.18824 + 1.18824i −1.60186 + 1.67756i 0.707107 0.707107i 2.08953 + 2.08953i 2.99680 0.138433i 2.25040i
8.18 1.08486 1.08486i 1.14443 1.30011i 0.353851i 0.879295 0.879295i −0.168890 2.65199i −0.707107 + 0.707107i 1.78584 + 1.78584i −0.380562 2.97576i 1.90783i
8.19 1.34745 1.34745i −0.469658 + 1.66716i 1.63125i −3.06725 + 3.06725i 1.61358 + 2.87926i −0.707107 + 0.707107i 0.496878 + 0.496878i −2.55884 1.56599i 8.26592i
8.20 1.48770 1.48770i 1.53135 + 0.809292i 2.42651i −0.854208 + 0.854208i 3.48218 1.07421i 0.707107 0.707107i −0.634513 0.634513i 1.69009 + 2.47863i 2.54161i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.24 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
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.n.c 48
3.b odd 2 1 inner 273.2.n.c 48
13.d odd 4 1 inner 273.2.n.c 48
39.f even 4 1 inner 273.2.n.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.n.c 48 1.a even 1 1 trivial
273.2.n.c 48 3.b odd 2 1 inner
273.2.n.c 48 13.d odd 4 1 inner
273.2.n.c 48 39.f even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.