# Properties

 Label 165.2.w.a Level $165$ Weight $2$ Character orbit 165.w Analytic conductor $1.318$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.w (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$12$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$96q - 8q^{5} - 20q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q - 8q^{5} - 20q^{7} + 8q^{11} - 16q^{12} - 12q^{15} + 8q^{16} - 20q^{17} - 60q^{20} - 32q^{22} + 32q^{23} - 32q^{25} - 60q^{28} - 40q^{30} + 16q^{31} - 16q^{33} + 24q^{36} + 8q^{37} + 56q^{38} - 120q^{41} + 12q^{42} - 200q^{46} + 60q^{47} + 48q^{48} + 80q^{50} + 40q^{51} + 40q^{52} + 36q^{53} + 80q^{55} - 80q^{56} + 40q^{57} + 44q^{58} + 48q^{60} + 40q^{61} + 80q^{62} + 20q^{63} + 56q^{66} - 48q^{67} + 80q^{68} - 92q^{70} + 32q^{71} - 60q^{73} - 24q^{77} - 96q^{78} - 80q^{80} + 24q^{81} + 32q^{82} - 200q^{83} - 80q^{85} - 80q^{86} - 144q^{88} + 56q^{91} + 20q^{92} - 72q^{93} + 60q^{95} + 32q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −2.58211 + 0.408966i 0.453990 + 0.891007i 4.59791 1.49395i −0.332092 2.21127i −1.53664 2.11501i −1.91487 0.975676i −6.60261 + 3.36420i −0.587785 + 0.809017i 1.76183 + 5.57392i
7.2 −2.36180 + 0.374072i −0.453990 0.891007i 3.53603 1.14893i −2.11284 + 0.732065i 1.40553 + 1.93455i −0.732807 0.373384i −3.66039 + 1.86506i −0.587785 + 0.809017i 4.71624 2.51934i
7.3 −1.44976 + 0.229620i −0.453990 0.891007i 0.146976 0.0477555i 2.19714 + 0.415419i 0.862772 + 1.18750i −3.00927 1.53330i 2.41359 1.22978i −0.587785 + 0.809017i −3.28072 0.0977516i
7.4 −1.27330 + 0.201671i 0.453990 + 0.891007i −0.321488 + 0.104458i −0.760317 + 2.10284i −0.757757 1.04296i −4.03315 2.05499i 2.68561 1.36839i −0.587785 + 0.809017i 0.544032 2.83088i
7.5 −1.12702 + 0.178502i 0.453990 + 0.891007i −0.663811 + 0.215685i 1.59467 1.56749i −0.670701 0.923140i 1.36271 + 0.694338i 2.74302 1.39764i −0.587785 + 0.809017i −1.51742 + 2.05123i
7.6 −0.329725 + 0.0522232i −0.453990 0.891007i −1.79612 + 0.583595i 0.0998428 + 2.23384i 0.196223 + 0.270078i 2.61195 + 1.33085i 1.15665 0.589341i −0.587785 + 0.809017i −0.149579 0.731337i
7.7 0.139396 0.0220781i −0.453990 0.891007i −1.88317 + 0.611879i −1.95846 1.07909i −0.0829560 0.114179i −1.56314 0.796462i −0.500497 + 0.255016i −0.587785 + 0.809017i −0.296825 0.107182i
7.8 0.779339 0.123435i 0.453990 + 0.891007i −1.30998 + 0.425638i 1.13825 + 1.92468i 0.463794 + 0.638358i 0.804274 + 0.409798i −2.37448 + 1.20986i −0.587785 + 0.809017i 1.12466 + 1.35948i
7.9 1.73704 0.275120i 0.453990 + 0.891007i 1.03949 0.337751i −0.351743 2.20823i 1.03373 + 1.42281i 3.16835 + 1.61435i −1.42129 + 0.724185i −0.587785 + 0.809017i −1.21852 3.73900i
7.10 1.86197 0.294907i −0.453990 0.891007i 1.47784 0.480180i 0.508223 2.17755i −1.10808 1.52514i −0.414728 0.211314i −0.749326 + 0.381801i −0.587785 + 0.809017i 0.304121 4.20440i
7.11 2.13992 0.338930i −0.453990 0.891007i 2.56227 0.832532i 1.12340 + 1.93339i −1.27349 1.75281i −0.770101 0.392386i 1.33999 0.682757i −0.587785 + 0.809017i 3.05926 + 3.75654i
7.12 2.46605 0.390584i 0.453990 + 0.891007i 4.02673 1.30836i −2.14608 + 0.627950i 1.46758 + 2.01994i −3.26542 1.66382i 4.96977 2.53222i −0.587785 + 0.809017i −5.04708 + 2.38678i
13.1 −2.21461 + 1.12840i −0.987688 0.156434i 2.45566 3.37992i −0.704990 2.12202i 2.36387 0.768068i 0.619606 + 3.91204i −0.846779 + 5.34635i 0.951057 + 0.309017i 3.95578 + 3.90395i
13.2 −1.93567 + 0.986271i −0.987688 0.156434i 1.59850 2.20015i 1.61795 + 1.54345i 2.06612 0.671323i −0.626599 3.95619i −0.244529 + 1.54390i 0.951057 + 0.309017i −4.65407 1.39188i
13.3 −1.78343 + 0.908700i 0.987688 + 0.156434i 1.17930 1.62316i −1.76846 1.36841i −1.90362 + 0.618524i −0.663220 4.18741i −0.00198567 + 0.0125370i 0.951057 + 0.309017i 4.39739 + 0.833463i
13.4 −1.48575 + 0.757027i 0.987688 + 0.156434i 0.458789 0.631469i 1.81791 1.30201i −1.58588 + 0.515284i 0.372318 + 2.35072i 0.318101 2.00841i 0.951057 + 0.309017i −1.71530 + 3.31066i
13.5 −0.584989 + 0.298067i −0.987688 0.156434i −0.922202 + 1.26930i −0.0319712 2.23584i 0.624415 0.202885i −0.357848 2.25936i 0.366555 2.31434i 0.951057 + 0.309017i 0.685133 + 1.29841i
13.6 −0.169439 + 0.0863333i 0.987688 + 0.156434i −1.15431 + 1.58878i −2.23301 0.116851i −0.180858 + 0.0587644i 0.692240 + 4.37063i 0.117918 0.744505i 0.951057 + 0.309017i 0.388447 0.172984i
13.7 0.333685 0.170021i 0.987688 + 0.156434i −1.09313 + 1.50457i 1.24112 + 1.86001i 0.356174 0.115728i −0.158351 0.999786i −0.226124 + 1.42769i 0.951057 + 0.309017i 0.730383 + 0.409641i
13.8 0.649706 0.331042i −0.987688 0.156434i −0.863041 + 1.18787i 0.229865 + 2.22422i −0.693493 + 0.225330i 0.281760 + 1.77896i −0.395626 + 2.49788i 0.951057 + 0.309017i 0.885655 + 1.36900i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.12 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.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.w.a 96
3.b odd 2 1 495.2.bj.c 96
5.b even 2 1 825.2.cw.b 96
5.c odd 4 1 inner 165.2.w.a 96
5.c odd 4 1 825.2.cw.b 96
11.d odd 10 1 inner 165.2.w.a 96
15.e even 4 1 495.2.bj.c 96
33.f even 10 1 495.2.bj.c 96
55.h odd 10 1 825.2.cw.b 96
55.l even 20 1 inner 165.2.w.a 96
55.l even 20 1 825.2.cw.b 96
165.u odd 20 1 495.2.bj.c 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.w.a 96 1.a even 1 1 trivial
165.2.w.a 96 5.c odd 4 1 inner
165.2.w.a 96 11.d odd 10 1 inner
165.2.w.a 96 55.l even 20 1 inner
495.2.bj.c 96 3.b odd 2 1
495.2.bj.c 96 15.e even 4 1
495.2.bj.c 96 33.f even 10 1
495.2.bj.c 96 165.u odd 20 1
825.2.cw.b 96 5.b even 2 1
825.2.cw.b 96 5.c odd 4 1
825.2.cw.b 96 55.h odd 10 1
825.2.cw.b 96 55.l even 20 1

## Hecke kernels

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