# Properties

 Label 560.2.r.a Level 560 Weight 2 Character orbit 560.r Analytic conductor 4.472 Analytic rank 0 Dimension 184 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.r (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$92$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 4q^{2} + 8q^{8} - 168q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 4q^{2} + 8q^{8} - 168q^{9} - 8q^{11} - 8q^{15} - 8q^{16} - 4q^{21} - 12q^{22} - 32q^{28} - 44q^{30} - 24q^{32} + 20q^{35} - 40q^{36} - 8q^{37} + 48q^{39} - 32q^{42} + 24q^{44} - 8q^{46} - 4q^{50} - 8q^{51} - 4q^{56} + 24q^{57} - 4q^{58} + 64q^{60} - 32q^{63} - 8q^{65} + 12q^{70} - 64q^{72} + 40q^{74} - 28q^{78} + 104q^{81} - 48q^{84} + 16q^{85} + 8q^{86} + 96q^{88} - 20q^{91} - 56q^{92} + 16q^{93} - 56q^{95} - 80q^{98} + 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
237.1 −1.41385 + 0.0320021i 2.37744i 1.99795 0.0904924i −1.88118 1.20879i 0.0760829 + 3.36134i 2.53631 + 0.753095i −2.82191 + 0.191881i −2.65221 2.69839 + 1.64884i
237.2 −1.41385 + 0.0320021i 2.37744i 1.99795 0.0904924i 1.88118 + 1.20879i −0.0760829 3.36134i −0.753095 2.53631i −2.82191 + 0.191881i −2.65221 −2.69839 1.64884i
237.3 −1.40124 0.191148i 0.303303i 1.92692 + 0.535688i −0.412090 + 2.19777i −0.0579760 + 0.425000i −2.04400 + 1.67990i −2.59768 1.11895i 2.90801 0.997536 3.00082i
237.4 −1.40124 0.191148i 0.303303i 1.92692 + 0.535688i 0.412090 2.19777i 0.0579760 0.425000i −1.67990 + 2.04400i −2.59768 1.11895i 2.90801 −0.997536 + 3.00082i
237.5 −1.39265 0.246023i 2.36202i 1.87894 + 0.685249i 2.19592 + 0.421835i −0.581111 + 3.28946i 1.13552 + 2.38968i −2.44812 1.41658i −2.57912 −2.95436 1.12772i
237.6 −1.39265 0.246023i 2.36202i 1.87894 + 0.685249i −2.19592 0.421835i 0.581111 3.28946i −2.38968 1.13552i −2.44812 1.41658i −2.57912 2.95436 + 1.12772i
237.7 −1.34754 0.429109i 2.77875i 1.63173 + 1.15648i 1.52419 1.63611i −1.19239 + 3.74447i −0.904951 2.48617i −1.70257 2.25860i −4.72144 −2.75598 + 1.55067i
237.8 −1.34754 0.429109i 2.77875i 1.63173 + 1.15648i −1.52419 + 1.63611i 1.19239 3.74447i 2.48617 + 0.904951i −1.70257 2.25860i −4.72144 2.75598 1.55067i
237.9 −1.33659 + 0.462100i 0.445450i 1.57293 1.23527i −2.22590 0.213020i 0.205843 + 0.595383i −1.93702 1.80221i −1.53153 + 2.37790i 2.80157 3.07354 0.743868i
237.10 −1.33659 + 0.462100i 0.445450i 1.57293 1.23527i 2.22590 + 0.213020i −0.205843 0.595383i 1.80221 + 1.93702i −1.53153 + 2.37790i 2.80157 −3.07354 + 0.743868i
237.11 −1.33522 + 0.466046i 0.958223i 1.56560 1.24454i 0.666131 2.13454i 0.446576 + 1.27943i 0.729292 2.54325i −1.51040 + 2.39138i 2.08181 0.105366 + 3.16052i
237.12 −1.33522 + 0.466046i 0.958223i 1.56560 1.24454i −0.666131 + 2.13454i −0.446576 1.27943i 2.54325 0.729292i −1.51040 + 2.39138i 2.08181 −0.105366 3.16052i
237.13 −1.31458 + 0.521428i 2.31341i 1.45623 1.37091i 1.44475 + 1.70666i 1.20628 + 3.04115i −2.55188 0.698510i −1.19949 + 2.56149i −2.35185 −2.78914 1.49020i
237.14 −1.31458 + 0.521428i 2.31341i 1.45623 1.37091i −1.44475 1.70666i −1.20628 3.04115i 0.698510 + 2.55188i −1.19949 + 2.56149i −2.35185 2.78914 + 1.49020i
237.15 −1.24845 0.664358i 1.05190i 1.11726 + 1.65884i −2.04569 + 0.902849i −0.698839 + 1.31325i 0.442290 2.60852i −0.292782 2.81323i 1.89350 3.15376 + 0.231910i
237.16 −1.24845 0.664358i 1.05190i 1.11726 + 1.65884i 2.04569 0.902849i 0.698839 1.31325i 2.60852 0.442290i −0.292782 2.81323i 1.89350 −3.15376 0.231910i
237.17 −1.18156 + 0.777113i 2.96802i 0.792190 1.83642i −0.835108 + 2.07427i 2.30649 + 3.50691i 2.64524 0.0522070i 0.491081 + 2.78547i −5.80916 −0.625209 3.09986i
237.18 −1.18156 + 0.777113i 2.96802i 0.792190 1.83642i 0.835108 2.07427i −2.30649 3.50691i 0.0522070 2.64524i 0.491081 + 2.78547i −5.80916 0.625209 + 3.09986i
237.19 −1.13458 0.844232i 3.14518i 0.574546 + 1.91570i −1.64124 + 1.51866i −2.65526 + 3.56847i −1.68783 + 2.03745i 0.965423 2.65856i −6.89218 3.14422 0.337448i
237.20 −1.13458 0.844232i 3.14518i 0.574546 + 1.91570i 1.64124 1.51866i 2.65526 3.56847i −2.03745 + 1.68783i 0.965423 2.65856i −6.89218 −3.14422 + 0.337448i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.92 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
80.t odd 4 1 inner
560.r even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.r.a 184
5.c odd 4 1 560.2.bn.a yes 184
7.b odd 2 1 inner 560.2.r.a 184
16.e even 4 1 560.2.bn.a yes 184
35.f even 4 1 560.2.bn.a yes 184
80.t odd 4 1 inner 560.2.r.a 184
112.l odd 4 1 560.2.bn.a yes 184
560.r even 4 1 inner 560.2.r.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.r.a 184 1.a even 1 1 trivial
560.2.r.a 184 7.b odd 2 1 inner
560.2.r.a 184 80.t odd 4 1 inner
560.2.r.a 184 560.r even 4 1 inner
560.2.bn.a yes 184 5.c odd 4 1
560.2.bn.a yes 184 16.e even 4 1
560.2.bn.a yes 184 35.f even 4 1
560.2.bn.a yes 184 112.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database