# Properties

 Label 555.2.g.a Level $555$ Weight $2$ Character orbit 555.g Analytic conductor $4.432$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$555 = 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 555.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.43169731218$$ Analytic rank: $$0$$ Dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 44q^{4} - 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 44q^{4} - 40q^{9} + 4q^{10} + 8q^{11} + 52q^{16} - 16q^{21} + 8q^{25} - 16q^{26} + 16q^{30} - 32q^{34} - 44q^{36} - 28q^{40} + 8q^{41} + 16q^{44} - 8q^{46} - 24q^{49} + 92q^{64} - 48q^{65} - 56q^{70} + 24q^{71} - 68q^{74} + 8q^{75} + 40q^{81} - 16q^{84} - 64q^{85} + 80q^{86} - 4q^{90} + 32q^{95} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
184.1 −2.71549 1.00000i 5.37390 0.734693 2.11192i 2.71549i 1.35808i −9.16180 −1.00000 −1.99505 + 5.73492i
184.2 −2.71549 1.00000i 5.37390 0.734693 + 2.11192i 2.71549i 1.35808i −9.16180 −1.00000 −1.99505 5.73492i
184.3 −2.49663 1.00000i 4.23317 2.05572 + 0.879776i 2.49663i 3.27811i −5.57542 −1.00000 −5.13238 2.19648i
184.4 −2.49663 1.00000i 4.23317 2.05572 0.879776i 2.49663i 3.27811i −5.57542 −1.00000 −5.13238 + 2.19648i
184.5 −2.46809 1.00000i 4.09146 −1.94703 + 1.09958i 2.46809i 1.91769i −5.16192 −1.00000 4.80544 2.71386i
184.6 −2.46809 1.00000i 4.09146 −1.94703 1.09958i 2.46809i 1.91769i −5.16192 −1.00000 4.80544 + 2.71386i
184.7 −1.81177 1.00000i 1.28250 −2.16949 0.541568i 1.81177i 4.65490i 1.29994 −1.00000 3.93062 + 0.981196i
184.8 −1.81177 1.00000i 1.28250 −2.16949 + 0.541568i 1.81177i 4.65490i 1.29994 −1.00000 3.93062 0.981196i
184.9 −1.72645 1.00000i 0.980624 −1.26006 1.84723i 1.72645i 2.83159i 1.75990 −1.00000 2.17542 + 3.18915i
184.10 −1.72645 1.00000i 0.980624 −1.26006 + 1.84723i 1.72645i 2.83159i 1.75990 −1.00000 2.17542 3.18915i
184.11 −1.47721 1.00000i 0.182137 −0.357763 + 2.20726i 1.47721i 1.62051i 2.68536 −1.00000 0.528490 3.26058i
184.12 −1.47721 1.00000i 0.182137 −0.357763 2.20726i 1.47721i 1.62051i 2.68536 −1.00000 0.528490 + 3.26058i
184.13 −1.39280 1.00000i −0.0601015 2.20287 + 0.383884i 1.39280i 0.344118i 2.86931 −1.00000 −3.06816 0.534675i
184.14 −1.39280 1.00000i −0.0601015 2.20287 0.383884i 1.39280i 0.344118i 2.86931 −1.00000 −3.06816 + 0.534675i
184.15 −0.818204 1.00000i −1.33054 0.157839 2.23049i 0.818204i 3.66373i 2.72506 −1.00000 −0.129144 + 1.82500i
184.16 −0.818204 1.00000i −1.33054 0.157839 + 2.23049i 0.818204i 3.66373i 2.72506 −1.00000 −0.129144 1.82500i
184.17 −0.414618 1.00000i −1.82809 1.55319 1.60860i 0.414618i 0.262182i 1.58720 −1.00000 −0.643980 + 0.666956i
184.18 −0.414618 1.00000i −1.82809 1.55319 + 1.60860i 0.414618i 0.262182i 1.58720 −1.00000 −0.643980 0.666956i
184.19 −0.273741 1.00000i −1.92507 −1.93156 1.12653i 0.273741i 3.71619i 1.07445 −1.00000 0.528747 + 0.308378i
184.20 −0.273741 1.00000i −1.92507 −1.93156 + 1.12653i 0.273741i 3.71619i 1.07445 −1.00000 0.528747 0.308378i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 184.40 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.b even 2 1 inner
37.b even 2 1 inner
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.g.a 40
3.b odd 2 1 1665.2.g.e 40
5.b even 2 1 inner 555.2.g.a 40
15.d odd 2 1 1665.2.g.e 40
37.b even 2 1 inner 555.2.g.a 40
111.d odd 2 1 1665.2.g.e 40
185.d even 2 1 inner 555.2.g.a 40
555.b odd 2 1 1665.2.g.e 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.g.a 40 1.a even 1 1 trivial
555.2.g.a 40 5.b even 2 1 inner
555.2.g.a 40 37.b even 2 1 inner
555.2.g.a 40 185.d even 2 1 inner
1665.2.g.e 40 3.b odd 2 1
1665.2.g.e 40 15.d odd 2 1
1665.2.g.e 40 111.d odd 2 1
1665.2.g.e 40 555.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(555, [\chi])$$.