# Properties

 Label 525.2.bs.a Level 525 Weight 2 Character orbit 525.bs Analytic conductor 4.192 Analytic rank 0 Dimension 1216 CM no Inner twists 8

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$1216$$ Relative dimension: $$76$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$1216q - 8q^{3} - 20q^{4} - 24q^{6} - 28q^{7} - 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1216q - 8q^{3} - 20q^{4} - 24q^{6} - 28q^{7} - 10q^{9} - 12q^{10} - 64q^{13} - 44q^{15} - 140q^{16} - 24q^{18} - 20q^{19} - 12q^{21} + 8q^{22} - 24q^{25} - 80q^{27} - 40q^{28} - 50q^{30} - 12q^{31} - 6q^{33} - 80q^{34} + 8q^{36} - 24q^{37} - 50q^{39} - 4q^{40} - 34q^{42} - 96q^{43} + 30q^{45} - 12q^{46} - 44q^{48} - 16q^{51} - 136q^{52} - 10q^{54} - 40q^{55} - 112q^{57} - 76q^{58} - 60q^{60} - 12q^{61} - 14q^{63} - 80q^{64} - 30q^{66} - 32q^{67} + 100q^{69} - 100q^{70} + 24q^{72} - 72q^{73} - 96q^{75} - 64q^{76} + 20q^{78} - 20q^{79} - 6q^{81} - 204q^{82} + 100q^{84} - 56q^{85} + 36q^{87} - 140q^{88} + 164q^{90} - 24q^{91} + 34q^{93} - 20q^{94} - 30q^{96} - 240q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −2.14544 1.73734i 1.64793 + 0.533222i 1.16873 + 5.49845i 2.07100 + 0.843193i −2.60915 4.00702i 0.291962 + 2.62959i 4.53862 8.90754i 2.43135 + 1.75743i −2.97829 5.40705i
2.2 −2.10462 1.70428i 0.495015 1.65981i 1.10900 + 5.21743i 0.865799 2.06165i −3.87060 + 2.64961i −2.02686 1.70054i 4.09903 8.04480i −2.50992 1.64326i −5.33581 + 2.86341i
2.3 −2.02843 1.64259i −1.68808 0.387799i 1.00060 + 4.70747i 2.22126 0.256881i 2.78716 + 3.55945i 2.15806 1.53061i 3.33287 6.54113i 2.69922 + 1.30927i −4.92763 3.12756i
2.4 −2.02020 1.63593i −1.04782 1.37916i 0.989132 + 4.65350i −1.01425 + 1.99281i −0.139395 + 4.50033i −2.56962 + 0.630106i 3.25423 6.38679i −0.804150 + 2.89022i 5.30908 2.36665i
2.5 −1.93861 1.56985i 1.72611 0.143382i 0.877938 + 4.13037i −1.96574 + 1.06578i −3.57133 2.43177i 0.878403 2.49568i 2.51712 4.94013i 2.95888 0.494984i 5.48391 + 1.01980i
2.6 −1.90594 1.54340i 0.518584 + 1.65260i 0.834698 + 3.92695i 0.362947 2.20642i 1.56222 3.95012i 2.39774 1.11841i 2.24315 4.40243i −2.46214 + 1.71402i −4.09713 + 3.64512i
2.7 −1.87622 1.51933i −1.55929 + 0.754067i 0.796003 + 3.74490i −2.20949 0.343734i 4.07124 + 0.954284i −0.488296 2.60030i 2.00418 3.93343i 1.86277 2.35162i 3.62324 + 4.00187i
2.8 −1.82030 1.47405i 1.15573 1.29008i 0.724840 + 3.41011i −1.83998 1.27061i −4.00540 + 0.644724i 0.479622 + 2.60192i 1.58048 3.10187i −0.328590 2.98195i 1.47637 + 5.02511i
2.9 −1.81817 1.47233i −1.51942 + 0.831477i 0.722180 + 3.39759i −0.209118 + 2.22627i 3.98678 + 0.725319i 0.560858 + 2.58562i 1.56505 3.07158i 1.61729 2.52673i 3.65801 3.73985i
2.10 −1.76827 1.43192i −0.780419 + 1.54627i 0.660569 + 3.10773i 2.17999 0.497639i 3.59412 1.61672i −2.64461 0.0776153i 1.21599 2.38651i −1.78189 2.41347i −4.56739 2.24160i
2.11 −1.63394 1.32314i 0.407314 + 1.68348i 0.503241 + 2.36756i −1.47395 + 1.68151i 1.56195 3.28963i 2.44053 + 1.02168i 0.401327 0.787649i −2.66819 + 1.37141i 4.63321 0.797245i
2.12 −1.61952 1.31146i 0.852586 + 1.50768i 0.487094 + 2.29160i 0.657350 + 2.13726i 0.596485 3.55986i −2.28662 1.33093i 0.324313 0.636500i −1.54619 + 2.57085i 1.73835 4.32344i
2.13 −1.49559 1.21111i −0.875641 1.49441i 0.354192 + 1.66634i 2.19499 0.426619i −0.500284 + 3.29552i 0.855774 + 2.50353i −0.258988 + 0.508293i −1.46651 + 2.61713i −3.79949 2.02032i
2.14 −1.49464 1.21034i 1.52200 0.826756i 0.353218 + 1.66176i 2.03858 + 0.918808i −3.27550 0.606427i −0.413910 2.61317i −0.262916 + 0.516001i 1.63295 2.51664i −1.93488 3.84066i
2.15 −1.47969 1.19823i 1.49107 + 0.881317i 0.337908 + 1.58973i −1.81017 1.31275i −1.15030 3.09072i −1.79797 + 1.94095i −0.323938 + 0.635763i 1.44656 + 2.62820i 1.10552 + 4.11146i
2.16 −1.40992 1.14173i −1.65433 0.513008i 0.268505 + 1.26322i −0.187678 2.22818i 1.74677 + 2.61211i −2.52135 + 0.801735i −0.583603 + 1.14539i 2.47365 + 1.69737i −2.27937 + 3.35584i
2.17 −1.40053 1.13413i 0.668427 1.59788i 0.259417 + 1.22046i 1.53018 + 1.63051i −2.74835 + 1.47979i −1.57002 + 2.12957i −0.615476 + 1.20794i −2.10641 2.13612i −0.293854 4.01899i
2.18 −1.29573 1.04926i 1.71222 0.261325i 0.162144 + 0.762826i 1.09567 1.94923i −2.49278 1.45796i 2.61928 + 0.373301i −0.923561 + 1.81259i 2.86342 0.894893i −3.46495 + 1.37604i
2.19 −1.27995 1.03648i −0.162421 1.72442i 0.148148 + 0.696980i 0.135304 2.23197i −1.57943 + 2.37551i 1.74728 1.98671i −0.962644 + 1.88930i −2.94724 + 0.560165i −2.48657 + 2.71656i
2.20 −1.20229 0.973599i −1.55126 0.770447i 0.0817938 + 0.384809i 0.300868 + 2.21573i 1.11497 + 2.43661i 0.513705 2.59540i −1.12839 + 2.21460i 1.81282 + 2.39033i 1.79550 2.95689i
See next 80 embeddings (of 1216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 473.76 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
7.c even 3 1 inner
21.h odd 6 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
175.w odd 60 1 inner
525.bs 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.bs.a 1216
3.b odd 2 1 inner 525.2.bs.a 1216
7.c even 3 1 inner 525.2.bs.a 1216
21.h odd 6 1 inner 525.2.bs.a 1216
25.f odd 20 1 inner 525.2.bs.a 1216
75.l even 20 1 inner 525.2.bs.a 1216
175.w odd 60 1 inner 525.2.bs.a 1216
525.bs even 60 1 inner 525.2.bs.a 1216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bs.a 1216 1.a even 1 1 trivial
525.2.bs.a 1216 3.b odd 2 1 inner
525.2.bs.a 1216 7.c even 3 1 inner
525.2.bs.a 1216 21.h odd 6 1 inner
525.2.bs.a 1216 25.f odd 20 1 inner
525.2.bs.a 1216 75.l even 20 1 inner
525.2.bs.a 1216 175.w odd 60 1 inner
525.2.bs.a 1216 525.bs 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