# Properties

 Label 525.2.bm.a Level 525 Weight 2 Character orbit 525.bm Analytic conductor 4.192 Analytic rank 0 Dimension 608 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.bm (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$608$$ Relative dimension: $$76$$ over $$\Q(\zeta_{30})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$608q - 9q^{3} - 78q^{4} - 36q^{7} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$608q - 9q^{3} - 78q^{4} - 36q^{7} - 3q^{9} - 30q^{10} + 3q^{12} + 16q^{15} + 50q^{16} - 8q^{18} - 18q^{19} - 21q^{21} - 24q^{22} - 66q^{24} - 10q^{25} - 30q^{28} - 7q^{30} - 36q^{31} + 36q^{33} - 100q^{36} - 14q^{37} - 31q^{39} - 30q^{40} + 20q^{42} - 96q^{43} - 117q^{45} + 42q^{46} - 28q^{49} - 8q^{51} - 66q^{52} + 3q^{54} + 48q^{57} - 38q^{58} - 49q^{60} - 18q^{61} + 4q^{63} - 32q^{64} - 3q^{66} - 22q^{67} - 270q^{70} + 45q^{72} + 102q^{73} + 135q^{75} + 58q^{78} - 34q^{79} - 55q^{81} + 108q^{82} - 75q^{84} - 96q^{85} - 9q^{87} + 36q^{88} + 38q^{91} - 22q^{93} + 30q^{94} - 81q^{96} + 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −0.578671 + 2.72243i 1.50155 + 0.863338i −5.24969 2.33731i 2.12367 + 0.700010i −3.21928 + 3.58828i 2.47291 + 0.940586i 6.12911 8.43599i 1.50929 + 2.59269i −3.13464 + 5.37648i
131.2 −0.572271 + 2.69232i −1.59928 + 0.665065i −5.09402 2.26800i −1.14863 + 1.91850i −0.875350 4.68637i −2.63822 0.199474i 5.78563 7.96324i 2.11538 2.12725i −4.50790 4.19039i
131.3 −0.548566 + 2.58080i 1.24592 1.20320i −4.53252 2.01801i −1.83369 + 1.27969i 2.42176 + 3.87550i 1.29192 2.30888i 4.59278 6.32141i 0.104613 2.99818i −2.29672 5.43437i
131.4 −0.540793 + 2.54423i −1.70841 0.285183i −4.35356 1.93833i 0.488675 2.18202i 1.64947 4.19237i 1.22413 + 2.34553i 4.22820 5.81962i 2.83734 + 0.974421i 5.28728 + 2.42332i
131.5 −0.530626 + 2.49640i −0.257292 + 1.71283i −4.12336 1.83584i 1.58898 1.57326i −4.13939 1.55118i −1.47493 2.19649i 3.77069 5.18991i −2.86760 0.881397i 3.08432 + 4.80154i
131.6 −0.528347 + 2.48568i 1.52678 0.817885i −4.07236 1.81313i −0.273491 2.21928i 1.22633 + 4.22722i −2.41288 + 1.08537i 3.67110 5.05284i 1.66213 2.49746i 5.66091 + 0.492741i
131.7 −0.505077 + 2.37620i 0.0606037 1.73099i −3.56413 1.58685i 2.07975 + 0.821374i 4.08257 + 1.01829i −2.37564 + 1.16461i 2.71505 3.73695i −2.99265 0.209809i −3.00218 + 4.52704i
131.8 −0.489122 + 2.30114i −1.23199 1.21745i −3.22892 1.43761i 1.02927 + 1.98509i 3.40412 2.23951i 1.46353 2.20410i 2.12188 2.92052i 0.0356207 + 2.99979i −5.07142 + 1.39755i
131.9 −0.478695 + 2.25208i −0.600734 1.62454i −3.01563 1.34265i −2.22631 + 0.208681i 3.94616 0.575244i 0.695446 + 2.55272i 1.76069 2.42338i −2.27824 + 1.95183i 0.595755 5.11372i
131.10 −0.462774 + 2.17718i 1.15009 + 1.29510i −2.69886 1.20161i −2.18803 0.460993i −3.35190 + 1.90461i −0.462395 2.60503i 1.24847 1.71838i −0.354583 + 2.97897i 2.01623 4.55040i
131.11 −0.457158 + 2.15076i −0.177450 + 1.72294i −2.58969 1.15300i −0.910547 + 2.04228i −3.62450 1.16931i 2.32202 + 1.26816i 1.07888 1.48495i −2.93702 0.611470i −3.97619 2.89201i
131.12 −0.427547 + 2.01145i −1.51386 + 0.841555i −2.03605 0.906508i 2.19435 + 0.429892i −1.04550 3.40487i 1.64883 2.06914i 0.276476 0.380537i 1.58357 2.54800i −1.80290 + 4.23004i
131.13 −0.421130 + 1.98126i 0.963172 + 1.43955i −1.92096 0.855264i 0.372730 + 2.20478i −3.25774 + 1.30206i −2.59804 0.500191i 0.122328 0.168370i −1.14460 + 2.77306i −4.52522 0.190026i
131.14 −0.414854 + 1.95173i −1.61597 0.623422i −1.81007 0.805895i −1.81656 1.30389i 1.88714 2.89531i −0.995488 2.45133i −0.0218510 + 0.0300753i 2.22269 + 2.01486i 3.29845 3.00451i
131.15 −0.406505 + 1.91245i 1.66519 + 0.476610i −1.66514 0.741369i −0.898652 2.04754i −1.58840 + 2.99085i 2.59781 + 0.501390i −0.203728 + 0.280407i 2.54569 + 1.58729i 4.28113 0.886296i
131.16 −0.392327 + 1.84575i 0.000111749 1.73205i −1.42579 0.634803i 0.827237 2.07742i −3.19698 0.679324i −0.468357 + 2.60397i −0.487221 + 0.670602i −3.00000 0.000387111i 3.50986 + 2.34190i
131.17 −0.352641 + 1.65904i −1.23560 + 1.21379i −0.800979 0.356619i −2.12372 0.699871i −1.57801 2.47795i −1.86600 + 1.87565i −1.11979 + 1.54126i 0.0534131 2.99952i 1.91003 3.27654i
131.18 −0.350482 + 1.64889i 1.30612 1.13757i −0.768903 0.342338i 0.782565 + 2.09466i 1.41795 + 2.55234i 1.46624 + 2.20230i −1.14773 + 1.57971i 0.411886 2.97159i −3.72813 + 0.556222i
131.19 −0.343956 + 1.61819i 1.58851 0.690379i −0.673133 0.299698i 2.12496 0.696100i 0.570782 + 2.80797i 0.539750 2.59011i −1.22830 + 1.69060i 2.04676 2.19335i 0.395527 + 3.67801i
131.20 −0.318971 + 1.50064i 0.109335 1.72860i −0.323084 0.143846i 0.267996 2.21995i 2.55912 + 0.715443i −1.46253 2.20477i −1.48460 + 2.04338i −2.97609 0.377991i 3.24586 + 1.11026i
See next 80 embeddings (of 608 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 521.76 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.d odd 6 1 inner
21.g even 6 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
175.v odd 30 1 inner
525.bm even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bm.a 608
3.b odd 2 1 inner 525.2.bm.a 608
7.d odd 6 1 inner 525.2.bm.a 608
21.g even 6 1 inner 525.2.bm.a 608
25.d even 5 1 inner 525.2.bm.a 608
75.j odd 10 1 inner 525.2.bm.a 608
175.v odd 30 1 inner 525.2.bm.a 608
525.bm even 30 1 inner 525.2.bm.a 608

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bm.a 608 1.a even 1 1 trivial
525.2.bm.a 608 3.b odd 2 1 inner
525.2.bm.a 608 7.d odd 6 1 inner
525.2.bm.a 608 21.g even 6 1 inner
525.2.bm.a 608 25.d even 5 1 inner
525.2.bm.a 608 75.j odd 10 1 inner
525.2.bm.a 608 175.v odd 30 1 inner
525.2.bm.a 608 525.bm even 30 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