# Properties

 Label 165.2.p.b Level $165$ Weight $2$ Character orbit 165.p Analytic conductor $1.318$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.p (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{3} - 4q^{4} - 20q^{6} + 10q^{7} + 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{3} - 4q^{4} - 20q^{6} + 10q^{7} + 2q^{9} + 8q^{12} + 6q^{15} + 32q^{16} - 30q^{18} - 100q^{19} - 82q^{22} + 100q^{24} + 12q^{25} + 14q^{27} + 30q^{28} + 10q^{30} + 10q^{31} - 46q^{33} - 28q^{34} + 14q^{36} + 6q^{37} - 50q^{40} - 52q^{42} + 32q^{45} + 20q^{46} - 80q^{48} - 26q^{49} - 30q^{51} + 40q^{52} + 6q^{55} - 70q^{57} + 92q^{58} + 44q^{60} + 70q^{61} - 20q^{63} + 18q^{64} + 76q^{66} + 20q^{67} + 42q^{69} - 4q^{70} - 80q^{72} + 90q^{73} - 6q^{75} - 108q^{78} - 100q^{79} + 38q^{81} - 34q^{82} + 70q^{84} + 20q^{85} + 74q^{88} + 20q^{90} - 86q^{91} + 76q^{93} + 10q^{94} - 30q^{96} + 6q^{97} - 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 −2.09233 + 1.52016i 1.72815 0.116121i 1.44890 4.45925i −0.587785 + 0.809017i −3.43934 + 2.87004i 0.326781 + 0.106178i 2.14883 + 6.61341i 2.97303 0.401348i 2.58626i
41.2 −1.54632 + 1.12347i −0.455760 1.67101i 0.510897 1.57238i −0.587785 + 0.809017i 2.58208 + 2.07189i −0.0795268 0.0258398i −0.204777 0.630238i −2.58457 + 1.52316i 1.91136i
41.3 −1.45632 + 1.05808i −1.62444 0.600999i 0.383300 1.17968i 0.587785 0.809017i 3.00160 0.843534i −0.145486 0.0472713i −0.422546 1.30046i 2.27760 + 1.95257i 1.80011i
41.4 −1.37228 + 0.997022i 1.42851 + 0.979464i 0.271073 0.834277i 0.587785 0.809017i −2.93687 + 0.0801565i 3.82974 + 1.24436i −0.588527 1.81130i 1.08130 + 2.79836i 1.69623i
41.5 −0.860692 + 0.625329i −1.27571 + 1.17157i −0.268280 + 0.825682i −0.587785 + 0.809017i 0.365379 1.80609i −2.18340 0.709430i −0.942926 2.90203i 0.254870 2.98915i 1.06387i
41.6 −0.131900 + 0.0958309i 0.642850 1.60834i −0.609820 + 1.87683i 0.587785 0.809017i 0.0693364 + 0.273744i 3.41501 + 1.10960i −0.200186 0.616109i −2.17349 2.06784i 0.163037i
41.7 0.131900 0.0958309i −1.46543 0.923314i −0.609820 + 1.87683i −0.587785 + 0.809017i −0.281772 + 0.0186487i 3.41501 + 1.10960i 0.200186 + 0.616109i 1.29498 + 2.70611i 0.163037i
41.8 0.860692 0.625329i 1.72070 + 0.197973i −0.268280 + 0.825682i 0.587785 0.809017i 1.60479 0.905610i −2.18340 0.709430i 0.942926 + 2.90203i 2.92161 + 0.681304i 1.06387i
41.9 1.37228 0.997022i −0.579977 + 1.63206i 0.271073 0.834277i −0.587785 + 0.809017i 0.831309 + 2.81790i 3.82974 + 1.24436i 0.588527 + 1.81130i −2.32725 1.89312i 1.69623i
41.10 1.45632 1.05808i 0.960940 1.44104i 0.383300 1.17968i −0.587785 + 0.809017i −0.125297 3.11536i −0.145486 0.0472713i 0.422546 + 1.30046i −1.15319 2.76950i 1.80011i
41.11 1.54632 1.12347i −0.613479 1.61977i 0.510897 1.57238i 0.587785 0.809017i −2.76839 1.81546i −0.0795268 0.0258398i 0.204777 + 0.630238i −2.24729 + 1.98739i 1.91136i
41.12 2.09233 1.52016i −1.46636 + 0.921840i 1.44890 4.45925i 0.587785 0.809017i −1.66676 + 4.15790i 0.326781 + 0.106178i −2.14883 6.61341i 1.30042 2.70350i 2.58626i
101.1 −0.848632 + 2.61182i −0.139896 + 1.72639i −4.48340 3.25738i −0.951057 + 0.309017i −4.39031 1.83046i −0.0372155 + 0.0512227i 7.86897 5.71714i −2.96086 0.483031i 2.74623i
101.2 −0.664086 + 2.04385i 1.72041 + 0.200478i −2.11826 1.53901i 0.951057 0.309017i −1.55225 + 3.38312i −1.16392 + 1.60200i 1.07501 0.781038i 2.91962 + 0.689809i 2.14903i
101.3 −0.491121 + 1.51151i −1.07720 1.35634i −0.425442 0.309102i −0.951057 + 0.309017i 2.57916 0.962071i −1.78891 + 2.46222i −1.89539 + 1.37708i −0.679300 + 2.92208i 1.58930i
101.4 −0.212587 + 0.654275i 1.00646 1.40962i 1.23515 + 0.897390i 0.951057 0.309017i 0.708322 + 0.958169i −0.891460 + 1.22699i −1.96284 + 1.42608i −0.974079 2.83746i 0.687946i
101.5 −0.140144 + 0.431318i 0.180419 + 1.72263i 1.45164 + 1.05468i 0.951057 0.309017i −0.768285 0.163597i 2.72178 3.74621i −1.39214 + 1.01145i −2.93490 + 0.621591i 0.453514i
101.6 −0.0403345 + 0.124137i −0.644015 + 1.60787i 1.60425 + 1.16556i −0.951057 + 0.309017i −0.173620 0.144799i −1.50339 + 2.06924i −0.420589 + 0.305576i −2.17049 2.07099i 0.130525i
101.7 0.0403345 0.124137i −1.72819 + 0.115636i 1.60425 + 1.16556i 0.951057 0.309017i −0.0553509 + 0.219196i −1.50339 + 2.06924i 0.420589 0.305576i 2.97326 0.399681i 0.130525i
101.8 0.140144 0.431318i −1.58256 0.703911i 1.45164 + 1.05468i −0.951057 + 0.309017i −0.525396 + 0.583940i 2.72178 3.74621i 1.39214 1.01145i 2.00902 + 2.22797i 0.453514i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.12 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
11.d odd 10 1 inner
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.p.b 48
3.b odd 2 1 inner 165.2.p.b 48
5.b even 2 1 825.2.bi.e 48
5.c odd 4 1 825.2.bs.g 48
5.c odd 4 1 825.2.bs.h 48
11.d odd 10 1 inner 165.2.p.b 48
15.d odd 2 1 825.2.bi.e 48
15.e even 4 1 825.2.bs.g 48
15.e even 4 1 825.2.bs.h 48
33.f even 10 1 inner 165.2.p.b 48
55.h odd 10 1 825.2.bi.e 48
55.l even 20 1 825.2.bs.g 48
55.l even 20 1 825.2.bs.h 48
165.r even 10 1 825.2.bi.e 48
165.u odd 20 1 825.2.bs.g 48
165.u odd 20 1 825.2.bs.h 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.p.b 48 1.a even 1 1 trivial
165.2.p.b 48 3.b odd 2 1 inner
165.2.p.b 48 11.d odd 10 1 inner
165.2.p.b 48 33.f even 10 1 inner
825.2.bi.e 48 5.b even 2 1
825.2.bi.e 48 15.d odd 2 1
825.2.bi.e 48 55.h odd 10 1
825.2.bi.e 48 165.r even 10 1
825.2.bs.g 48 5.c odd 4 1
825.2.bs.g 48 15.e even 4 1
825.2.bs.g 48 55.l even 20 1
825.2.bs.g 48 165.u odd 20 1
825.2.bs.h 48 5.c odd 4 1
825.2.bs.h 48 15.e even 4 1
825.2.bs.h 48 55.l even 20 1
825.2.bs.h 48 165.u odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(165, [\chi])$$.