# Properties

 Label 525.2.bk.a Level 525 Weight 2 Character orbit 525.bk Analytic conductor 4.192 Analytic rank 0 Dimension 480 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.bk (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$480$$ Relative dimension: $$60$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$480q + 4q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$480q + 4q^{3} + 16q^{10} - 16q^{12} + 8q^{13} + 16q^{15} + 120q^{16} + 20q^{18} - 40q^{19} - 48q^{22} - 104q^{25} + 16q^{27} - 20q^{30} - 28q^{33} - 80q^{34} + 16q^{37} - 40q^{39} - 64q^{40} - 80q^{42} + 40q^{43} - 20q^{45} - 276q^{48} - 260q^{54} - 40q^{55} - 4q^{57} - 40q^{58} - 52q^{60} - 32q^{63} + 160q^{64} + 56q^{67} + 8q^{70} + 288q^{72} + 48q^{73} + 60q^{75} + 140q^{78} + 80q^{79} - 40q^{81} - 184q^{85} + 96q^{87} - 56q^{88} + 24q^{90} + 96q^{93} - 560q^{94} - 24q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −1.27716 + 2.50657i 1.72730 0.128199i −3.47619 4.78457i −0.295036 + 2.21652i −1.88470 + 4.49333i −0.707107 + 0.707107i 10.8754 1.72250i 2.96713 0.442878i −5.17905 3.57038i
8.2 −1.24682 + 2.44703i 0.890522 + 1.48559i −3.25781 4.48399i −1.63146 1.52917i −4.74560 + 0.326866i 0.707107 0.707107i 9.60928 1.52196i −1.41394 + 2.64590i 5.77605 2.08562i
8.3 −1.21549 + 2.38554i −0.0368220 1.73166i −3.03780 4.18118i 0.817695 2.08120i 4.17570 + 2.01698i −0.707107 + 0.707107i 8.37801 1.32695i −2.99729 + 0.127526i 3.97087 + 4.48032i
8.4 −1.14547 + 2.24812i −1.38991 + 1.03351i −2.56637 3.53230i 1.60960 1.55216i −0.731341 4.30855i 0.707107 0.707107i 5.89662 0.933933i 0.863721 2.87298i 1.64569 + 5.39653i
8.5 −1.13379 + 2.22520i −1.71426 + 0.247589i −2.49044 3.42779i 1.16615 + 1.90790i 1.39269 4.09529i −0.707107 + 0.707107i 5.51785 0.873941i 2.87740 0.848865i −5.56763 + 0.431745i
8.6 −1.09213 + 2.14343i −0.0951717 1.72943i −2.22596 3.06377i −1.39499 + 1.74757i 3.81085 + 1.68477i 0.707107 0.707107i 4.24598 0.672496i −2.98188 + 0.329187i −2.22229 4.89863i
8.7 −1.04373 + 2.04843i 1.54603 0.780894i −1.93114 2.65798i 1.20254 1.88518i −0.0140247 + 3.98198i 0.707107 0.707107i 2.91887 0.462304i 1.78041 2.41457i 2.60653 + 4.43094i
8.8 −1.00540 + 1.97320i −0.446385 + 1.67354i −1.70714 2.34967i −2.11390 + 0.728982i −2.85344 2.56338i −0.707107 + 0.707107i 1.97810 0.313301i −2.60148 1.49409i 0.686881 4.90408i
8.9 −0.996070 + 1.95490i 1.41904 + 0.993139i −1.65390 2.27640i 2.07107 + 0.843023i −3.35495 + 1.78484i 0.707107 0.707107i 1.76348 0.279307i 1.02735 + 2.81861i −3.71095 + 3.20901i
8.10 −0.995192 + 1.95317i −0.685670 + 1.59055i −1.64891 2.26954i 0.282336 + 2.21817i −2.42425 2.92214i 0.707107 0.707107i 1.74356 0.276153i −2.05971 2.18119i −4.61346 1.65606i
8.11 −0.916279 + 1.79830i 1.53212 0.807837i −1.21874 1.67745i −2.08851 0.798830i 0.0488799 + 3.49542i −0.707107 + 0.707107i 0.146404 0.0231881i 1.69480 2.47541i 3.35019 3.02381i
8.12 −0.878575 + 1.72430i 1.34260 + 1.09426i −1.02575 1.41182i −0.562535 2.16415i −3.06642 + 1.35366i −0.707107 + 0.707107i −0.487197 + 0.0771645i 0.605175 + 2.93833i 4.22588 + 0.931390i
8.13 −0.851523 + 1.67121i −0.524400 1.65076i −0.892275 1.22811i 2.19833 + 0.409071i 3.20530 + 0.529279i −0.707107 + 0.707107i −0.892875 + 0.141417i −2.45001 + 1.73131i −2.55557 + 3.32554i
8.14 −0.764791 + 1.50099i −1.73141 + 0.0470098i −0.492485 0.677848i 1.36230 1.77317i 1.25361 2.63478i −0.707107 + 0.707107i −1.93362 + 0.306256i 2.99558 0.162787i 1.61963 + 3.40090i
8.15 −0.672489 + 1.31983i −1.30188 + 1.14242i −0.114150 0.157113i −1.72356 1.42455i −0.632301 2.48652i 0.707107 0.707107i −2.64197 + 0.418446i 0.389772 2.97457i 3.03924 1.31682i
8.16 −0.649632 + 1.27497i 0.191343 1.72145i −0.0279682 0.0384950i −1.41872 1.72836i 2.07050 + 1.36227i 0.707107 0.707107i −2.75939 + 0.437045i −2.92678 0.658773i 3.12526 0.686035i
8.17 −0.641887 + 1.25977i 1.23644 + 1.21293i 0.000559265 0 0.000769763i 0.0490481 + 2.23553i −2.32168 + 0.779075i −0.707107 + 0.707107i −2.79427 + 0.442569i 0.0575908 + 2.99945i −2.84775 1.37317i
8.18 −0.640516 + 1.25708i −1.19705 1.25183i 0.00557249 + 0.00766987i 2.19052 0.449020i 2.34038 0.702977i 0.707107 0.707107i −2.80019 + 0.443506i −0.134142 + 2.99700i −0.838608 + 3.04127i
8.19 −0.634690 + 1.24565i −1.69875 0.338004i 0.0267608 + 0.0368331i 0.131414 + 2.23220i 1.49921 1.90152i 0.707107 0.707107i −2.82449 + 0.447355i 2.77151 + 1.14837i −2.86395 1.25306i
8.20 −0.537657 + 1.05521i −1.08000 1.35411i 0.351174 + 0.483349i −1.77494 + 1.35999i 2.00954 0.411579i −0.707107 + 0.707107i −3.03827 + 0.481215i −0.667215 + 2.92486i −0.480765 2.60415i
See next 80 embeddings (of 480 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 512.60 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bk.a 480
3.b odd 2 1 inner 525.2.bk.a 480
25.f odd 20 1 inner 525.2.bk.a 480
75.l even 20 1 inner 525.2.bk.a 480

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bk.a 480 1.a even 1 1 trivial
525.2.bk.a 480 3.b odd 2 1 inner
525.2.bk.a 480 25.f odd 20 1 inner
525.2.bk.a 480 75.l even 20 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