# Properties

 Label 369.2.u.a Level 369 Weight 2 Character orbit 369.u Analytic conductor 2.946 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$369 = 3^{2} \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 369.u (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94647983459$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 41) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 10q^{2} + 10q^{5} - 8q^{7} + 10q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 10q^{2} + 10q^{5} - 8q^{7} + 10q^{8} + 6q^{10} + 16q^{11} - 14q^{14} - 20q^{16} - 8q^{17} + 16q^{19} - 20q^{20} + 6q^{22} - 12q^{23} - 8q^{25} + 28q^{26} + 18q^{28} - 40q^{29} - 12q^{31} - 16q^{34} + 36q^{35} - 46q^{38} - 44q^{40} + 4q^{41} + 48q^{44} + 70q^{46} + 12q^{47} - 30q^{49} + 20q^{52} + 26q^{53} + 20q^{55} - 106q^{56} - 20q^{58} - 6q^{59} + 30q^{61} + 10q^{62} + 70q^{64} - 68q^{65} - 22q^{67} + 20q^{68} - 20q^{70} - 4q^{71} - 10q^{74} - 128q^{76} + 20q^{77} - 2q^{79} + 70q^{80} - 90q^{82} - 80q^{83} - 56q^{85} + 46q^{86} + 10q^{88} + 72q^{89} - 18q^{94} + 40q^{95} - 22q^{97} - 6q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1 −1.42655 1.96347i 0 −1.20216 + 3.69986i −0.110420 0.0358775i 0 −0.422339 2.66654i 4.36309 1.41766i 0 0.0870742 + 0.267987i
46.2 0.522120 + 0.718637i 0 0.374204 1.15168i −1.02071 0.331648i 0 0.625712 + 3.95059i 2.71264 0.881390i 0 −0.294598 0.906679i
46.3 1.00762 + 1.38687i 0 −0.290083 + 0.892782i 2.57687 + 0.837277i 0 −0.252316 1.59306i 1.73027 0.562198i 0 1.43532 + 4.41746i
118.1 −0.746800 1.02788i 0 0.119203 0.366868i 3.27974 + 1.06565i 0 1.70635 0.270260i −2.88281 + 0.936683i 0 −1.35394 4.16701i
118.2 0.415383 + 0.571726i 0 0.463706 1.42714i −2.26179 0.734900i 0 −4.85225 + 0.768522i 2.35276 0.764458i 0 −0.519348 1.59839i
118.3 1.61018 + 2.21623i 0 −1.70094 + 5.23496i 1.15434 + 0.375067i 0 1.19485 0.189245i −9.13005 + 2.96653i 0 1.02746 + 3.16221i
172.1 −0.746800 + 1.02788i 0 0.119203 + 0.366868i 3.27974 1.06565i 0 1.70635 + 0.270260i −2.88281 0.936683i 0 −1.35394 + 4.16701i
172.2 0.415383 0.571726i 0 0.463706 + 1.42714i −2.26179 + 0.734900i 0 −4.85225 0.768522i 2.35276 + 0.764458i 0 −0.519348 + 1.59839i
172.3 1.61018 2.21623i 0 −1.70094 5.23496i 1.15434 0.375067i 0 1.19485 + 0.189245i −9.13005 2.96653i 0 1.02746 3.16221i
226.1 −0.698642 0.227002i 0 −1.18146 0.858384i 0.422124 0.581004i 0 −1.42228 + 2.79137i 1.49413 + 2.05650i 0 −0.426803 + 0.310091i
226.2 1.40718 + 0.457221i 0 0.153075 + 0.111216i 0.455161 0.626475i 0 2.12336 4.16732i −1.57482 2.16755i 0 0.926931 0.673455i
226.3 2.05153 + 0.666584i 0 2.14642 + 1.55947i −1.72514 + 2.37446i 0 −1.11329 + 2.18496i 0.828108 + 1.13979i 0 −5.12196 + 3.72132i
244.1 −1.14785 + 0.372958i 0 −0.439578 + 0.319372i −1.49595 2.05900i 0 −1.01547 + 0.517405i 1.80427 2.48337i 0 2.48504 + 1.80549i
244.2 −0.290902 + 0.0945196i 0 −1.54234 + 1.12058i 2.03721 + 2.80398i 0 1.29765 0.661185i 0.702328 0.966671i 0 −0.857659 0.623126i
244.3 2.29671 0.746246i 0 3.09996 2.25225i 1.68856 + 2.32411i 0 −1.86997 + 0.952797i 2.60008 3.57870i 0 5.61249 + 4.07771i
289.1 −0.698642 + 0.227002i 0 −1.18146 + 0.858384i 0.422124 + 0.581004i 0 −1.42228 2.79137i 1.49413 2.05650i 0 −0.426803 0.310091i
289.2 1.40718 0.457221i 0 0.153075 0.111216i 0.455161 + 0.626475i 0 2.12336 + 4.16732i −1.57482 + 2.16755i 0 0.926931 + 0.673455i
289.3 2.05153 0.666584i 0 2.14642 1.55947i −1.72514 2.37446i 0 −1.11329 2.18496i 0.828108 1.13979i 0 −5.12196 3.72132i
307.1 −1.14785 0.372958i 0 −0.439578 0.319372i −1.49595 + 2.05900i 0 −1.01547 0.517405i 1.80427 + 2.48337i 0 2.48504 1.80549i
307.2 −0.290902 0.0945196i 0 −1.54234 1.12058i 2.03721 2.80398i 0 1.29765 + 0.661185i 0.702328 + 0.966671i 0 −0.857659 + 0.623126i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 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 369.2.u.a 24
3.b odd 2 1 41.2.g.a 24
12.b even 2 1 656.2.bs.d 24
41.g even 20 1 inner 369.2.u.a 24
123.m odd 20 1 41.2.g.a 24
123.o even 40 2 1681.2.a.m 24
492.y even 20 1 656.2.bs.d 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.g.a 24 3.b odd 2 1
41.2.g.a 24 123.m odd 20 1
369.2.u.a 24 1.a even 1 1 trivial
369.2.u.a 24 41.g even 20 1 inner
656.2.bs.d 24 12.b even 2 1
656.2.bs.d 24 492.y even 20 1
1681.2.a.m 24 123.o even 40 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(369, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database