# Properties

 Label 165.2.s.a Level $165$ Weight $2$ Character orbit 165.s 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.s (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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 12q^{4} - 4q^{5} + 4q^{6} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 12q^{4} - 4q^{5} + 4q^{6} + 12q^{9} - 12q^{10} - 4q^{14} + 10q^{15} - 44q^{16} - 16q^{19} + 46q^{20} - 32q^{21} - 12q^{24} + 14q^{25} - 76q^{26} + 4q^{30} - 20q^{31} - 24q^{34} - 40q^{35} - 12q^{36} - 8q^{39} - 72q^{40} + 60q^{41} - 48q^{44} + 4q^{45} + 108q^{46} - 28q^{49} - 38q^{50} + 28q^{51} + 16q^{54} - 20q^{55} + 24q^{56} + 60q^{59} + 48q^{60} + 40q^{61} + 64q^{64} + 20q^{65} + 12q^{66} + 20q^{69} + 86q^{70} - 32q^{71} - 32q^{74} - 40q^{75} - 136q^{76} - 52q^{79} + 42q^{80} - 12q^{81} - 70q^{85} - 104q^{86} + 40q^{89} - 8q^{90} - 40q^{91} + 72q^{94} - 2q^{95} + 28q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −1.60225 + 2.20531i −0.951057 0.309017i −1.67816 5.16484i −2.23478 + 0.0757477i 2.20531 1.60225i 0.462529 0.150285i 8.89393 + 2.88981i 0.809017 + 0.587785i 3.41365 5.04977i
4.2 −1.38939 + 1.91233i 0.951057 + 0.309017i −1.10857 3.41183i 0.389432 + 2.20190i −1.91233 + 1.38939i −1.63033 + 0.529727i 3.56862 + 1.15951i 0.809017 + 0.587785i −4.75182 2.31457i
4.3 −1.12116 + 1.54314i −0.951057 0.309017i −0.506258 1.55810i 2.14793 0.621610i 1.54314 1.12116i 4.23918 1.37739i −0.656175 0.213204i 0.809017 + 0.587785i −1.44894 + 4.01149i
4.4 −0.472206 + 0.649936i −0.951057 0.309017i 0.418596 + 1.28830i −2.11660 + 0.721122i 0.649936 0.472206i −0.483617 + 0.157137i −2.56307 0.832793i 0.809017 + 0.587785i 0.530788 1.71617i
4.5 −0.460901 + 0.634375i 0.951057 + 0.309017i 0.428031 + 1.31734i 2.23544 0.0529614i −0.634375 + 0.460901i −1.44371 + 0.469090i −2.52448 0.820252i 0.809017 + 0.587785i −0.996719 + 1.44252i
4.6 −0.169760 + 0.233654i 0.951057 + 0.309017i 0.592258 + 1.82278i −0.962959 2.01810i −0.233654 + 0.169760i 3.48791 1.13329i −1.07580 0.349548i 0.809017 + 0.587785i 0.635009 + 0.117592i
4.7 0.169760 0.233654i −0.951057 0.309017i 0.592258 + 1.82278i 1.62175 + 1.53945i −0.233654 + 0.169760i −3.48791 + 1.13329i 1.07580 + 0.349548i 0.809017 + 0.587785i 0.635009 0.117592i
4.8 0.460901 0.634375i −0.951057 0.309017i 0.428031 + 1.31734i 0.741158 2.10966i −0.634375 + 0.460901i 1.44371 0.469090i 2.52448 + 0.820252i 0.809017 + 0.587785i −0.996719 1.44252i
4.9 0.472206 0.649936i 0.951057 + 0.309017i 0.418596 + 1.28830i −1.33989 + 1.79016i 0.649936 0.472206i 0.483617 0.157137i 2.56307 + 0.832793i 0.809017 + 0.587785i 0.530788 + 1.71617i
4.10 1.12116 1.54314i 0.951057 + 0.309017i −0.506258 1.55810i 1.25493 1.85071i 1.54314 1.12116i −4.23918 + 1.37739i 0.656175 + 0.213204i 0.809017 + 0.587785i −1.44894 4.01149i
4.11 1.38939 1.91233i −0.951057 0.309017i −1.10857 3.41183i −1.97379 1.05079i −1.91233 + 1.38939i 1.63033 0.529727i −3.56862 1.15951i 0.809017 + 0.587785i −4.75182 + 2.31457i
4.12 1.60225 2.20531i 0.951057 + 0.309017i −1.67816 5.16484i −0.762627 + 2.10200i 2.20531 1.60225i −0.462529 + 0.150285i −8.89393 2.88981i 0.809017 + 0.587785i 3.41365 + 5.04977i
49.1 −2.46515 + 0.800975i 0.587785 0.809017i 3.81735 2.77347i 2.09522 + 0.781069i −0.800975 + 2.46515i −2.00926 2.76550i −4.14176 + 5.70065i −0.309017 0.951057i −5.79063 0.247234i
49.2 −2.10559 + 0.684148i −0.587785 + 0.809017i 2.34742 1.70550i −0.0486578 + 2.23554i 0.684148 2.10559i 0.936570 + 1.28908i −1.17323 + 1.61482i −0.309017 0.951057i −1.42699 4.74042i
49.3 −2.00341 + 0.650947i 0.587785 0.809017i 1.97188 1.43266i −2.23165 + 0.140493i −0.650947 + 2.00341i 1.94929 + 2.68296i −0.541555 + 0.745386i −0.309017 0.951057i 4.37946 1.73415i
49.4 −1.20873 + 0.392740i −0.587785 + 0.809017i −0.311255 + 0.226140i −1.58253 1.57975i 0.392740 1.20873i −0.284071 0.390991i 1.78148 2.45199i −0.309017 0.951057i 2.53328 + 1.28796i
49.5 −0.464400 + 0.150893i 0.587785 0.809017i −1.42514 + 1.03542i −0.0541280 2.23541i −0.150893 + 0.464400i −3.00296 4.13322i 1.07962 1.48598i −0.309017 0.951057i 0.362444 + 1.02996i
49.6 −0.283476 + 0.0921069i 0.587785 0.809017i −1.54616 + 1.12335i 0.882995 + 2.05434i −0.0921069 + 0.283476i 1.36429 + 1.87778i 0.685226 0.943133i −0.309017 0.951057i −0.439527 0.501026i
49.7 0.283476 0.0921069i −0.587785 + 0.809017i −1.54616 + 1.12335i −1.92187 + 1.14299i −0.0921069 + 0.283476i −1.36429 1.87778i −0.685226 + 0.943133i −0.309017 0.951057i −0.439527 + 0.501026i
49.8 0.464400 0.150893i −0.587785 + 0.809017i −1.42514 + 1.03542i 1.35773 1.77667i −0.150893 + 0.464400i 3.00296 + 4.13322i −1.07962 + 1.48598i −0.309017 0.951057i 0.362444 1.02996i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j 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.s.a 48
3.b odd 2 1 495.2.ba.c 48
5.b even 2 1 inner 165.2.s.a 48
5.c odd 4 1 825.2.n.o 24
5.c odd 4 1 825.2.n.p 24
11.c even 5 1 inner 165.2.s.a 48
11.c even 5 1 1815.2.c.j 24
11.d odd 10 1 1815.2.c.k 24
15.d odd 2 1 495.2.ba.c 48
33.h odd 10 1 495.2.ba.c 48
55.h odd 10 1 1815.2.c.k 24
55.j even 10 1 inner 165.2.s.a 48
55.j even 10 1 1815.2.c.j 24
55.k odd 20 1 825.2.n.o 24
55.k odd 20 1 825.2.n.p 24
55.k odd 20 1 9075.2.a.dy 12
55.k odd 20 1 9075.2.a.dz 12
55.l even 20 1 9075.2.a.dx 12
55.l even 20 1 9075.2.a.ea 12
165.o odd 10 1 495.2.ba.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a 48 1.a even 1 1 trivial
165.2.s.a 48 5.b even 2 1 inner
165.2.s.a 48 11.c even 5 1 inner
165.2.s.a 48 55.j even 10 1 inner
495.2.ba.c 48 3.b odd 2 1
495.2.ba.c 48 15.d odd 2 1
495.2.ba.c 48 33.h odd 10 1
495.2.ba.c 48 165.o odd 10 1
825.2.n.o 24 5.c odd 4 1
825.2.n.o 24 55.k odd 20 1
825.2.n.p 24 5.c odd 4 1
825.2.n.p 24 55.k odd 20 1
1815.2.c.j 24 11.c even 5 1
1815.2.c.j 24 55.j even 10 1
1815.2.c.k 24 11.d odd 10 1
1815.2.c.k 24 55.h odd 10 1
9075.2.a.dx 12 55.l even 20 1
9075.2.a.dy 12 55.k odd 20 1
9075.2.a.dz 12 55.k odd 20 1
9075.2.a.ea 12 55.l even 20 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(165, [\chi])$$.