# Properties

 Label 185.2.n.a Level $185$ Weight $2$ Character orbit 185.n Analytic conductor $1.477$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$185 = 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 185.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 16q^{4} - 2q^{5} - 16q^{6} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 16q^{4} - 2q^{5} - 16q^{6} + 14q^{9} - 12q^{10} + 12q^{11} - 8q^{14} - 10q^{15} - 16q^{16} - 8q^{19} + 22q^{20} - 26q^{21} - 42q^{24} + 12q^{26} - 16q^{29} + 18q^{34} - 16q^{35} + 32q^{36} - 2q^{39} - 42q^{40} + 2q^{41} - 10q^{44} - 56q^{45} + 52q^{46} + 10q^{49} + 34q^{50} - 28q^{51} - 42q^{54} + 4q^{55} + 18q^{56} - 28q^{59} + 44q^{60} + 20q^{61} + 36q^{64} + 10q^{65} - 148q^{66} + 70q^{69} - 10q^{70} - 46q^{71} + 56q^{74} + 32q^{75} + 24q^{76} + 2q^{79} + 132q^{80} - 2q^{81} - 168q^{84} - 28q^{85} - 22q^{86} + 8q^{89} - 28q^{90} - 48q^{91} + 32q^{94} - 10q^{95} + 106q^{96} + 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
84.1 −2.33663 1.34906i 2.35586 1.36016i 2.63990 + 4.57244i 0.261363 2.22074i −7.33972 −1.64348 + 0.948866i 8.84926i 2.20006 3.81062i −3.60661 + 4.83646i
84.2 −1.99875 1.15398i −1.81123 + 1.04571i 1.66334 + 2.88098i −0.0902553 2.23425i 4.82692 3.28655 1.89749i 3.06190i 0.687027 1.18997i −2.39787 + 4.56985i
84.3 −1.85494 1.07095i 0.858786 0.495820i 1.29387 + 2.24104i −1.59352 + 1.56866i −2.12399 −1.20076 + 0.693262i 1.25887i −1.00832 + 1.74647i 4.63584 1.20318i
84.4 −1.64422 0.949293i 1.16436 0.672243i 0.802314 + 1.38965i 1.62372 + 1.53738i −2.55262 2.34261 1.35251i 0.750648i −0.596178 + 1.03261i −1.21033 4.06918i
84.5 −1.58456 0.914845i −1.14999 + 0.663947i 0.673882 + 1.16720i 2.12799 0.686767i 2.42963 −2.49908 + 1.44285i 1.19339i −0.618348 + 1.07101i −4.00021 0.858562i
84.6 −0.843193 0.486818i −1.64972 + 0.952467i −0.526017 0.911089i −1.79344 + 1.33551i 1.85471 3.89368 2.24802i 2.97157i 0.314386 0.544533i 2.16236 0.253015i
84.7 −0.704704 0.406861i −0.142682 + 0.0823772i −0.668928 1.15862i −1.50396 1.65472i 0.134064 −2.76887 + 1.59861i 2.71609i −1.48643 + 2.57457i 0.386605 + 1.77799i
84.8 −0.398514 0.230082i 2.88437 1.66529i −0.894124 1.54867i −2.18772 + 0.462462i −1.53262 −0.0944407 + 0.0545254i 1.74322i 4.04641 7.00858i 0.978243 + 0.319058i
84.9 −0.153805 0.0887996i −1.48057 + 0.854808i −0.984229 1.70474i 0.854987 + 2.06616i 0.303627 −1.21910 + 0.703845i 0.704795i −0.0386059 + 0.0668674i 0.0519722 0.393709i
84.10 0.153805 + 0.0887996i 1.48057 0.854808i −0.984229 1.70474i 1.36185 + 1.77352i 0.303627 1.21910 0.703845i 0.704795i −0.0386059 + 0.0668674i 0.0519722 + 0.393709i
84.11 0.398514 + 0.230082i −2.88437 + 1.66529i −0.894124 1.54867i 1.49437 1.66339i −1.53262 0.0944407 0.0545254i 1.74322i 4.04641 7.00858i 0.978243 0.319058i
84.12 0.704704 + 0.406861i 0.142682 0.0823772i −0.668928 1.15862i −0.681051 2.12983i 0.134064 2.76887 1.59861i 2.71609i −1.48643 + 2.57457i 0.386605 1.77799i
84.13 0.843193 + 0.486818i 1.64972 0.952467i −0.526017 0.911089i 2.05330 0.885408i 1.85471 −3.89368 + 2.24802i 2.97157i 0.314386 0.544533i 2.16236 + 0.253015i
84.14 1.58456 + 0.914845i 1.14999 0.663947i 0.673882 + 1.16720i −1.65875 + 1.49951i 2.42963 2.49908 1.44285i 1.19339i −0.618348 + 1.07101i −4.00021 + 0.858562i
84.15 1.64422 + 0.949293i −1.16436 + 0.672243i 0.802314 + 1.38965i 0.519553 + 2.17487i −2.55262 −2.34261 + 1.35251i 0.750648i −0.596178 + 1.03261i −1.21033 + 4.06918i
84.16 1.85494 + 1.07095i −0.858786 + 0.495820i 1.29387 + 2.24104i 2.15526 0.595704i −2.12399 1.20076 0.693262i 1.25887i −1.00832 + 1.74647i 4.63584 + 1.20318i
84.17 1.99875 + 1.15398i 1.81123 1.04571i 1.66334 + 2.88098i −1.88979 1.19529i 4.82692 −3.28655 + 1.89749i 3.06190i 0.687027 1.18997i −2.39787 4.56985i
84.18 2.33663 + 1.34906i −2.35586 + 1.36016i 2.63990 + 4.57244i −2.05390 0.884024i −7.33972 1.64348 0.948866i 8.84926i 2.20006 3.81062i −3.60661 4.83646i
174.1 −2.33663 + 1.34906i 2.35586 + 1.36016i 2.63990 4.57244i 0.261363 + 2.22074i −7.33972 −1.64348 0.948866i 8.84926i 2.20006 + 3.81062i −3.60661 4.83646i
174.2 −1.99875 + 1.15398i −1.81123 1.04571i 1.66334 2.88098i −0.0902553 + 2.23425i 4.82692 3.28655 + 1.89749i 3.06190i 0.687027 + 1.18997i −2.39787 4.56985i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 174.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.n.a 36
5.b even 2 1 inner 185.2.n.a 36
5.c odd 4 2 925.2.e.f 36
37.c even 3 1 inner 185.2.n.a 36
185.n even 6 1 inner 185.2.n.a 36
185.s odd 12 2 925.2.e.f 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.n.a 36 1.a even 1 1 trivial
185.2.n.a 36 5.b even 2 1 inner
185.2.n.a 36 37.c even 3 1 inner
185.2.n.a 36 185.n even 6 1 inner
925.2.e.f 36 5.c odd 4 2
925.2.e.f 36 185.s odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(185, [\chi])$$.