# Properties

 Label 547.3.b.b Level 547 Weight 3 Character orbit 547.b Analytic conductor 14.905 Analytic rank 0 Dimension 88 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 547.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.9046704605$$ Analytic rank: $$0$$ Dimension: $$88$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 192q^{4} - 306q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 192q^{4} - 306q^{9} - 4q^{10} - 32q^{11} + 26q^{13} - 26q^{14} + 22q^{15} + 236q^{16} - 12q^{19} - 16q^{21} - 2q^{24} - 544q^{25} - 96q^{29} + 26q^{34} + 10q^{35} + 364q^{36} + 44q^{40} + 124q^{44} - 288q^{46} - 310q^{47} - 694q^{49} + 86q^{51} - 316q^{52} + 24q^{53} - 266q^{54} + 158q^{56} - 80q^{60} + 40q^{62} - 652q^{64} + 528q^{66} + 28q^{67} + 16q^{69} + 94q^{73} - 614q^{74} - 28q^{76} - 98q^{78} + 928q^{81} - 772q^{82} + 358q^{84} + 74q^{85} - 410q^{86} - 214q^{90} + 656q^{93} - 724q^{96} + 346q^{97} + 50q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
546.1 3.90365i 5.00487i −11.2385 3.32024i 19.5373 0.969581i 28.2567i −16.0487 −12.9611
546.2 3.88566i 2.17695i −11.0983 9.52039i −8.45889 6.36562i 27.5816i 4.26087 −36.9930
546.3 3.79607i 0.894948i −10.4101 3.75049i −3.39728 0.715898i 24.3333i 8.19907 14.2371
546.4 3.76304i 4.52828i −10.1605 3.85159i −17.0401 7.12637i 23.1820i −11.5053 14.4937
546.5 3.58531i 0.541221i −8.85444 2.75520i −1.94044 8.99411i 17.4047i 8.70708 −9.87823
546.6 3.54864i 2.90313i −8.59283 1.02412i 10.3021 3.84607i 16.2983i 0.571853 3.63422
546.7 3.54186i 4.64741i −8.54479 0.255244i −16.4605 11.6148i 16.0970i −12.5984 0.904038
546.8 3.52162i 0.549187i −8.40183 7.97338i 1.93403 12.0038i 15.5016i 8.69839 28.0792
546.9 3.42438i 4.48969i −7.72636 7.63002i 15.3744 10.8574i 12.7605i −11.1573 26.1281
546.10 3.26612i 1.63931i −6.66753 7.05581i 5.35418 7.78718i 8.71245i 6.31266 −23.0451
546.11 3.08562i 1.35400i −5.52104 5.29090i 4.17794 11.3849i 4.69333i 7.16667 −16.3257
546.12 3.05754i 3.93936i −5.34857 2.27125i −12.0448 6.60828i 4.12330i −6.51856 −6.94444
546.13 3.03308i 3.99539i −5.19958 9.10946i −12.1184 2.89551i 3.63844i −6.96316 27.6297
546.14 2.90201i 2.26440i −4.42164 8.06772i 6.57132 4.29626i 1.22362i 3.87247 23.4126
546.15 2.87053i 4.81682i −4.23994 8.18119i −13.8268 5.20188i 0.688763i −14.2018 −23.4844
546.16 2.83323i 2.06299i −4.02719 2.31552i −5.84492 4.45862i 0.0770388i 4.74408 −6.56039
546.17 2.78422i 5.61272i −3.75190 5.19773i 15.6271 12.7126i 0.690758i −22.5026 14.4716
546.18 2.76044i 1.30646i −3.62000 3.69969i −3.60639 4.19330i 1.04896i 7.29317 10.2127
546.19 2.71290i 2.08874i −3.35982 3.69381i 5.66654 8.32703i 1.73675i 4.63716 −10.0209
546.20 2.64774i 5.22856i −3.01055 9.23817i 13.8439 6.29191i 2.61981i −18.3379 −24.4603
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 546.88 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.3.b.b 88
547.b odd 2 1 inner 547.3.b.b 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.3.b.b 88 1.a even 1 1 trivial
547.3.b.b 88 547.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{88} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(547, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database