# Properties

 Label 525.2.bv.a Level 525 Weight 2 Character orbit 525.bv Analytic conductor 4.192 Analytic rank 0 Dimension 640 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bv (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$40$$ over $$\Q(\zeta_{60})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$640q + 12q^{5} - 8q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q + 12q^{5} - 8q^{7} + 24q^{8} + 12q^{10} + 8q^{15} - 80q^{16} - 72q^{22} + 48q^{23} - 12q^{25} - 36q^{28} - 80q^{29} + 32q^{30} - 24q^{32} + 36q^{33} - 64q^{35} + 160q^{36} - 4q^{37} - 192q^{38} - 12q^{40} - 16q^{42} + 120q^{43} + 60q^{47} + 288q^{50} - 432q^{52} - 136q^{53} - 16q^{57} - 4q^{58} - 240q^{59} - 20q^{60} - 4q^{63} + 120q^{64} + 4q^{65} - 8q^{67} - 132q^{68} + 76q^{70} - 12q^{72} - 36q^{73} - 48q^{75} - 60q^{77} - 80q^{78} + 12q^{80} - 80q^{81} - 252q^{82} - 160q^{84} + 72q^{85} + 24q^{87} + 152q^{88} + 56q^{92} - 96q^{93} + 172q^{95} - 488q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
52.1 −1.00045 2.60626i 0.0523360 0.998630i −4.30539 + 3.87659i 1.46116 + 1.69263i −2.65504 + 0.862676i 1.56547 2.13291i 9.43590 + 4.80783i −0.994522 0.104528i 2.94961 5.50156i
52.2 −0.952068 2.48022i −0.0523360 + 0.998630i −3.75878 + 3.38442i 2.10803 0.745796i 2.52665 0.820958i −2.63239 + 0.265557i 7.23848 + 3.68819i −0.994522 0.104528i −3.85673 4.51833i
52.3 −0.944724 2.46109i 0.0523360 0.998630i −3.67817 + 3.31184i −0.481208 2.18368i −2.50716 + 0.814626i 0.854940 + 2.50381i 6.92789 + 3.52994i −0.994522 0.104528i −4.91961 + 3.24727i
52.4 −0.872278 2.27236i −0.0523360 + 0.998630i −2.91647 + 2.62600i −2.11358 0.729921i 2.31490 0.752156i 0.892264 2.49076i 4.17372 + 2.12662i −0.994522 0.104528i 0.184983 + 5.43951i
52.5 −0.802228 2.08988i −0.0523360 + 0.998630i −2.23772 + 2.01485i 0.318271 + 2.21330i 2.12900 0.691753i −0.914813 + 2.48256i 2.01681 + 1.02761i −0.994522 0.104528i 4.37020 2.44072i
52.6 −0.755992 1.96943i 0.0523360 0.998630i −1.82083 + 1.63948i 0.924367 + 2.03606i −2.00629 + 0.651884i −1.10577 + 2.40360i 0.846132 + 0.431126i −0.994522 0.104528i 3.31106 3.35972i
52.7 −0.754243 1.96487i −0.0523360 + 0.998630i −1.80554 + 1.62572i 2.18621 + 0.469548i 2.00165 0.650376i 2.57259 + 0.617866i 0.805604 + 0.410476i −0.994522 0.104528i −0.726334 4.64978i
52.8 −0.753962 1.96414i 0.0523360 0.998630i −1.80309 + 1.62351i 0.485187 2.18279i −2.00090 + 0.650133i −0.829509 2.51235i 0.799114 + 0.407169i −0.994522 0.104528i −4.65312 + 0.692770i
52.9 −0.641755 1.67183i −0.0523360 + 0.998630i −0.896875 + 0.807550i −0.674040 + 2.13206i 1.70313 0.553379i 0.561772 2.58542i −1.26552 0.644814i −0.994522 0.104528i 3.99701 0.241379i
52.10 −0.611789 1.59376i −0.0523360 + 0.998630i −0.679511 + 0.611834i −1.29445 1.82330i 1.62360 0.527539i −2.56289 0.656955i −1.65133 0.841396i −0.994522 0.104528i −2.11398 + 3.17852i
52.11 −0.498488 1.29861i 0.0523360 0.998630i 0.0484028 0.0435821i −0.935202 + 2.03111i −1.32292 + 0.429841i −2.07682 1.63916i −2.55950 1.30413i −0.994522 0.104528i 3.10380 + 0.201975i
52.12 −0.496455 1.29331i 0.0523360 0.998630i 0.0601081 0.0541216i 2.05147 0.889639i −1.31752 + 0.428088i 2.12828 + 1.57176i −2.56850 1.30872i −0.994522 0.104528i −2.16904 2.21152i
52.13 −0.423361 1.10289i 0.0523360 0.998630i 0.449149 0.404416i −1.56576 1.59637i −1.12354 + 0.365060i 1.78771 1.95041i −2.74138 1.39680i −0.994522 0.104528i −1.09774 + 2.40271i
52.14 −0.385788 1.00501i 0.0523360 0.998630i 0.625072 0.562818i −2.23091 + 0.151798i −1.02383 + 0.332661i −1.22058 + 2.34738i −2.72515 1.38853i −0.994522 0.104528i 1.01322 + 2.18353i
52.15 −0.377442 0.983270i −0.0523360 + 0.998630i 0.661931 0.596006i 2.00799 0.983866i 1.00168 0.325465i −2.28125 1.34011i −2.71274 1.38221i −0.994522 0.104528i −1.72530 1.60304i
52.16 −0.221735 0.577641i −0.0523360 + 0.998630i 1.20179 1.08209i 1.83440 + 1.27866i 0.588454 0.191200i 2.58948 0.542755i −1.99414 1.01606i −0.994522 0.104528i 0.331851 1.34315i
52.17 −0.206383 0.537647i −0.0523360 + 0.998630i 1.23982 1.11634i −1.08016 1.95787i 0.547712 0.177962i 2.64165 0.147266i −1.88233 0.959096i −0.994522 0.104528i −0.829717 + 0.984816i
52.18 −0.107591 0.280285i 0.0523360 0.998630i 1.41931 1.27795i 1.68997 + 1.46424i −0.285532 + 0.0927749i −1.90734 1.83359i −1.04590 0.532913i −0.994522 0.104528i 0.228576 0.631214i
52.19 −0.0754782 0.196628i −0.0523360 + 0.998630i 1.45332 1.30858i −2.18626 + 0.469349i 0.200308 0.0650841i −2.39330 + 1.12789i −0.742319 0.378230i −0.994522 0.104528i 0.257302 + 0.394452i
52.20 −0.0670074 0.174560i 0.0523360 0.998630i 1.46031 1.31487i 0.297475 2.21619i −0.177828 + 0.0577798i −2.22491 + 1.43170i −0.660575 0.336580i −0.994522 0.104528i −0.406792 + 0.0965741i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 523.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bv.a 640
7.d odd 6 1 inner 525.2.bv.a 640
25.f odd 20 1 inner 525.2.bv.a 640
175.x even 60 1 inner 525.2.bv.a 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bv.a 640 1.a even 1 1 trivial
525.2.bv.a 640 7.d odd 6 1 inner
525.2.bv.a 640 25.f odd 20 1 inner
525.2.bv.a 640 175.x even 60 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database