# Properties

 Label 1681.2.a.m Level 1681 Weight 2 Character orbit 1681.a Self dual yes Analytic conductor 13.423 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1681 = 41^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1681.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.4228525798$$ Analytic rank: $$0$$ Dimension: $$24$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 41) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 8q^{2} + 20q^{4} + 12q^{5} + 24q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 8q^{2} + 20q^{4} + 12q^{5} + 24q^{8} + 24q^{9} - 4q^{10} + 20q^{16} + 20q^{18} + 40q^{20} + 56q^{21} + 48q^{23} + 8q^{25} + 32q^{31} + 36q^{32} + 24q^{33} + 108q^{36} + 60q^{39} + 44q^{40} - 40q^{42} + 36q^{43} - 36q^{45} + 36q^{46} + 32q^{49} - 60q^{50} + 36q^{51} - 60q^{57} + 116q^{59} + 40q^{61} + 48q^{62} + 60q^{64} + 16q^{66} + 4q^{72} + 16q^{73} + 24q^{74} - 16q^{77} - 60q^{78} + 84q^{80} - 28q^{81} + 80q^{83} - 24q^{84} - 24q^{86} + 76q^{87} + 56q^{90} + 40q^{91} - 40q^{92} + 36q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.41490 −2.36665 3.83176 2.87275 5.71522 −2.09872 −4.42352 2.60102 −6.93742
1.2 −2.41490 2.36665 3.83176 2.87275 −5.71522 2.09872 −4.42352 2.60102 −6.93742
1.3 −1.71427 −3.15154 0.938727 −2.70948 5.40260 −1.61292 1.81931 6.93221 4.64479
1.4 −1.71427 3.15154 0.938727 −2.70948 −5.40260 1.61292 1.81931 6.93221 4.64479
1.5 −1.27053 −0.854760 −0.385748 3.44852 1.08600 −1.72762 3.03117 −2.26939 −4.38146
1.6 −1.27053 0.854760 −0.385748 3.44852 −1.08600 1.72762 3.03117 −2.26939 −4.38146
1.7 −0.888284 −1.39100 −1.21095 1.07324 1.23560 −3.99983 2.85224 −1.06513 −0.953338
1.8 −0.888284 1.39100 −1.21095 1.07324 −1.23560 3.99983 2.85224 −1.06513 −0.953338
1.9 −0.734595 −0.0612231 −1.46037 −0.718161 0.0449742 −3.13283 2.54197 −2.99625 0.527558
1.10 −0.734595 0.0612231 −1.46037 −0.718161 −0.0449742 3.13283 2.54197 −2.99625 0.527558
1.11 0.305872 −1.36459 −1.90644 3.46591 −0.417390 −1.45639 −1.19487 −1.13789 1.06013
1.12 0.305872 1.36459 −1.90644 3.46591 0.417390 1.45639 −1.19487 −1.13789 1.06013
1.13 0.706693 −0.343106 −1.50059 −2.37819 −0.242471 4.91274 −2.47384 −2.88228 −1.68065
1.14 0.706693 0.343106 −1.50059 −2.37819 0.242471 −4.91274 −2.47384 −2.88228 −1.68065
1.15 1.20692 −2.19417 −0.543349 −2.54506 −2.64818 −1.13968 −3.06961 1.81438 −3.07168
1.16 1.20692 2.19417 −0.543349 −2.54506 2.64818 1.13968 −3.06961 1.81438 −3.07168
1.17 1.47960 −2.68766 0.189211 −0.774365 −3.97665 −4.67709 −2.67924 4.22349 −1.14575
1.18 1.47960 2.68766 0.189211 −0.774365 3.97665 4.67709 −2.67924 4.22349 −1.14575
1.19 2.15711 −2.12020 2.65312 2.93499 −4.57350 −2.45224 1.40886 1.49525 6.33110
1.20 2.15711 2.12020 2.65312 2.93499 4.57350 2.45224 1.40886 1.49525 6.33110
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 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.b even 2 1 inner

## Twists

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

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

## Atkin-Lehner signs

$$p$$ Sign
$$41$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1681))$$:

 $$T_{2}^{12} - \cdots$$ $$T_{3}^{24} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database