# Properties

 Label 41.2.g.a Level 41 Weight 2 Character orbit 41.g Analytic conductor 0.327 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 41.g (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.327386648287$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 10q^{2} - 6q^{3} - 10q^{5} - 2q^{6} - 8q^{7} - 10q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 10q^{2} - 6q^{3} - 10q^{5} - 2q^{6} - 8q^{7} - 10q^{8} + 6q^{10} - 16q^{11} + 2q^{12} + 14q^{14} + 8q^{15} - 20q^{16} + 8q^{17} + 16q^{19} + 20q^{20} - 10q^{21} + 6q^{22} + 12q^{23} + 68q^{24} - 8q^{25} - 28q^{26} - 6q^{27} + 18q^{28} + 40q^{29} - 36q^{30} - 12q^{31} + 10q^{33} - 16q^{34} - 36q^{35} - 40q^{36} + 46q^{38} - 50q^{39} - 44q^{40} - 4q^{41} - 40q^{42} - 48q^{44} + 16q^{45} + 70q^{46} - 12q^{47} - 50q^{48} - 30q^{49} - 24q^{51} + 20q^{52} - 26q^{53} + 68q^{54} + 20q^{55} + 106q^{56} + 10q^{57} - 20q^{58} + 6q^{59} + 76q^{60} + 30q^{61} - 10q^{62} + 92q^{63} + 70q^{64} + 68q^{65} + 34q^{66} - 22q^{67} - 20q^{68} - 38q^{69} - 20q^{70} + 4q^{71} - 74q^{72} + 10q^{74} + 4q^{75} - 128q^{76} - 20q^{77} - 10q^{78} - 2q^{79} - 70q^{80} + 28q^{81} - 90q^{82} + 80q^{83} - 30q^{84} - 56q^{85} - 46q^{86} - 10q^{87} + 10q^{88} - 72q^{89} - 70q^{90} - 6q^{93} - 18q^{94} - 40q^{95} + 66q^{96} - 22q^{97} + 6q^{98} + 14q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −2.05153 + 0.666584i 1.49921 1.49921i 2.14642 1.55947i 1.72514 + 2.37446i −2.07633 + 4.07502i −1.11329 2.18496i −0.828108 + 1.13979i 1.49525i −5.12196 3.72132i
2.2 −1.40718 + 0.457221i −1.90046 + 1.90046i 0.153075 0.111216i −0.455161 0.626475i 1.80536 3.54322i 2.12336 + 4.16732i 1.57482 2.16755i 4.22349i 0.926931 + 0.673455i
2.3 0.698642 0.227002i 0.0432913 0.0432913i −1.18146 + 0.858384i −0.422124 0.581004i 0.0204179 0.0400723i −1.42228 2.79137i −1.49413 + 2.05650i 2.99625i −0.426803 0.310091i
5.1 −1.00762 1.38687i 2.22848 + 2.22848i −0.290083 + 0.892782i −2.57687 0.837277i 0.845152 5.33608i −0.252316 1.59306i −1.73027 + 0.562198i 6.93221i 1.43532 + 4.41746i
5.2 −0.522120 0.718637i −0.983583 0.983583i 0.374204 1.15168i 1.02071 + 0.331648i −0.193291 + 1.22039i 0.625712 + 3.95059i −2.71264 + 0.881390i 1.06513i −0.294598 0.906679i
5.3 1.42655 + 1.96347i −2.02366 2.02366i −1.20216 + 3.69986i 0.110420 + 0.0358775i 1.08656 6.86025i −0.422339 2.66654i −4.36309 + 1.41766i 5.19040i 0.0870742 + 0.267987i
8.1 −1.61018 + 2.21623i 1.24382 + 1.24382i −1.70094 5.23496i −1.15434 + 0.375067i −4.75938 + 0.753811i 1.19485 + 0.189245i 9.13005 + 2.96653i 0.0941838i 1.02746 3.16221i
8.2 −0.415383 + 0.571726i −0.242613 0.242613i 0.463706 + 1.42714i 2.26179 0.734900i 0.239486 0.0379308i −4.85225 0.768522i −2.35276 0.764458i 2.88228i −0.519348 + 1.59839i
8.3 0.746800 1.02788i −0.604406 0.604406i 0.119203 + 0.366868i −3.27974 + 1.06565i −1.07263 + 0.169888i 1.70635 + 0.270260i 2.88281 + 0.936683i 2.26939i −1.35394 + 4.16701i
20.1 −2.29671 0.746246i −1.67347 + 1.67347i 3.09996 + 2.25225i −1.68856 + 2.32411i 5.09230 2.59466i −1.86997 0.952797i −2.60008 3.57870i 2.60102i 5.61249 4.07771i
20.2 0.290902 + 0.0945196i 0.964912 0.964912i −1.54234 1.12058i −2.03721 + 2.80398i 0.371897 0.189491i 1.29765 + 0.661185i −0.702328 0.966671i 1.13789i −0.857659 + 0.623126i
20.3 1.14785 + 0.372958i −1.55151 + 1.55151i −0.439578 0.319372i 1.49595 2.05900i −2.35955 + 1.20225i −1.01547 0.517405i −1.80427 2.48337i 1.81438i 2.48504 1.80549i
21.1 −2.05153 0.666584i 1.49921 + 1.49921i 2.14642 + 1.55947i 1.72514 2.37446i −2.07633 4.07502i −1.11329 + 2.18496i −0.828108 1.13979i 1.49525i −5.12196 + 3.72132i
21.2 −1.40718 0.457221i −1.90046 1.90046i 0.153075 + 0.111216i −0.455161 + 0.626475i 1.80536 + 3.54322i 2.12336 4.16732i 1.57482 + 2.16755i 4.22349i 0.926931 0.673455i
21.3 0.698642 + 0.227002i 0.0432913 + 0.0432913i −1.18146 0.858384i −0.422124 + 0.581004i 0.0204179 + 0.0400723i −1.42228 + 2.79137i −1.49413 2.05650i 2.99625i −0.426803 + 0.310091i
33.1 −1.00762 + 1.38687i 2.22848 2.22848i −0.290083 0.892782i −2.57687 + 0.837277i 0.845152 + 5.33608i −0.252316 + 1.59306i −1.73027 0.562198i 6.93221i 1.43532 4.41746i
33.2 −0.522120 + 0.718637i −0.983583 + 0.983583i 0.374204 + 1.15168i 1.02071 0.331648i −0.193291 1.22039i 0.625712 3.95059i −2.71264 0.881390i 1.06513i −0.294598 + 0.906679i
33.3 1.42655 1.96347i −2.02366 + 2.02366i −1.20216 3.69986i 0.110420 0.0358775i 1.08656 + 6.86025i −0.422339 + 2.66654i −4.36309 1.41766i 5.19040i 0.0870742 0.267987i
36.1 −1.61018 2.21623i 1.24382 1.24382i −1.70094 + 5.23496i −1.15434 0.375067i −4.75938 0.753811i 1.19485 0.189245i 9.13005 2.96653i 0.0941838i 1.02746 + 3.16221i
36.2 −0.415383 0.571726i −0.242613 + 0.242613i 0.463706 1.42714i 2.26179 + 0.734900i 0.239486 + 0.0379308i −4.85225 + 0.768522i −2.35276 + 0.764458i 2.88228i −0.519348 1.59839i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.g even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.2.g.a 24
3.b odd 2 1 369.2.u.a 24
4.b odd 2 1 656.2.bs.d 24
41.g even 20 1 inner 41.2.g.a 24
41.h odd 40 2 1681.2.a.m 24
123.m odd 20 1 369.2.u.a 24
164.n odd 20 1 656.2.bs.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a 24 1.a even 1 1 trivial
41.2.g.a 24 41.g even 20 1 inner
369.2.u.a 24 3.b odd 2 1
369.2.u.a 24 123.m odd 20 1
656.2.bs.d 24 4.b odd 2 1
656.2.bs.d 24 164.n odd 20 1
1681.2.a.m 24 41.h odd 40 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(41, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database