# Properties

 Label 572.2.n.b Level $572$ Weight $2$ Character orbit 572.n Analytic conductor $4.567$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + q^{3} - 7q^{5} + 11q^{7} - 22q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + q^{3} - 7q^{5} + 11q^{7} - 22q^{9} + q^{11} + 7q^{13} - 24q^{15} + 7q^{17} - 7q^{19} - 12q^{21} + 34q^{23} - 26q^{25} - 11q^{27} + 8q^{29} - 17q^{31} - 2q^{33} - 6q^{35} - 15q^{37} + 4q^{39} + 29q^{41} - 8q^{43} + 62q^{45} - 27q^{47} - 4q^{49} + 69q^{51} - 32q^{53} + 19q^{55} + 9q^{57} - 37q^{59} - 19q^{61} + 46q^{63} - 18q^{65} + 10q^{67} - 32q^{69} - 27q^{71} - 79q^{75} + 13q^{77} + 21q^{79} - 18q^{81} + 27q^{83} + 7q^{85} + 56q^{87} + 82q^{89} - 11q^{91} - 53q^{93} + 51q^{95} - 88q^{97} + 86q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 0 −2.46647 + 1.79199i 0 0.518170 + 1.59476i 0 −2.24130 1.62840i 0 1.94517 5.98661i 0
53.2 0 −1.94312 + 1.41176i 0 0.879947 + 2.70820i 0 3.92186 + 2.84939i 0 0.855592 2.63324i 0
53.3 0 −0.634331 + 0.460868i 0 −0.799372 2.46021i 0 0.289986 + 0.210687i 0 −0.737075 + 2.26848i 0
53.4 0 −0.0637055 + 0.0462848i 0 −0.454508 1.39883i 0 1.50470 + 1.09323i 0 −0.925135 + 2.84727i 0
53.5 0 0.178116 0.129409i 0 1.35047 + 4.15632i 0 −3.13912 2.28071i 0 −0.912072 + 2.80707i 0
53.6 0 2.12584 1.54451i 0 0.674037 + 2.07447i 0 1.61142 + 1.17077i 0 1.20663 3.71362i 0
53.7 0 2.49464 1.81246i 0 −1.12366 3.45827i 0 1.36148 + 0.989171i 0 2.01117 6.18974i 0
157.1 0 −1.05349 3.24229i 0 −2.42026 1.75842i 0 1.07034 3.29416i 0 −6.97559 + 5.06806i 0
157.2 0 −0.527342 1.62299i 0 1.55733 + 1.13147i 0 −0.0663952 + 0.204343i 0 0.0710371 0.0516114i 0
157.3 0 −0.344293 1.05962i 0 −0.665019 0.483164i 0 0.816463 2.51281i 0 1.42278 1.03371i 0
157.4 0 0.288033 + 0.886474i 0 −2.56853 1.86615i 0 −0.780072 + 2.40081i 0 1.72418 1.25269i 0
157.5 0 0.525215 + 1.61645i 0 1.11280 + 0.808497i 0 1.47468 4.53859i 0 0.0900014 0.0653899i 0
157.6 0 0.955130 + 2.93959i 0 −3.52306 2.55966i 0 0.357931 1.10160i 0 −5.30186 + 3.85203i 0
157.7 0 0.965759 + 2.97230i 0 1.96166 + 1.42523i 0 −0.681960 + 2.09886i 0 −5.47482 + 3.97769i 0
313.1 0 −2.46647 1.79199i 0 0.518170 1.59476i 0 −2.24130 + 1.62840i 0 1.94517 + 5.98661i 0
313.2 0 −1.94312 1.41176i 0 0.879947 2.70820i 0 3.92186 2.84939i 0 0.855592 + 2.63324i 0
313.3 0 −0.634331 0.460868i 0 −0.799372 + 2.46021i 0 0.289986 0.210687i 0 −0.737075 2.26848i 0
313.4 0 −0.0637055 0.0462848i 0 −0.454508 + 1.39883i 0 1.50470 1.09323i 0 −0.925135 2.84727i 0
313.5 0 0.178116 + 0.129409i 0 1.35047 4.15632i 0 −3.13912 + 2.28071i 0 −0.912072 2.80707i 0
313.6 0 2.12584 + 1.54451i 0 0.674037 2.07447i 0 1.61142 1.17077i 0 1.20663 + 3.71362i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 521.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.n.b 28
11.c even 5 1 inner 572.2.n.b 28
11.c even 5 1 6292.2.a.z 14
11.d odd 10 1 6292.2.a.y 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.n.b 28 1.a even 1 1 trivial
572.2.n.b 28 11.c even 5 1 inner
6292.2.a.y 14 11.d odd 10 1
6292.2.a.z 14 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.