# Properties

 Label 525.2.bp.a Level 525 Weight 2 Character orbit 525.bp 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.bp (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.

 $$\operatorname{Tr}(f)(q) =$$ $$608q - 15q^{3} + 66q^{4} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$608q - 15q^{3} + 66q^{4} - 3q^{9} - 30q^{10} - 15q^{12} - 36q^{15} + 66q^{16} - 18q^{19} + 9q^{21} - 80q^{22} - 30q^{24} + 2q^{25} - 90q^{28} - 23q^{30} - 90q^{33} + 44q^{36} - 10q^{37} - 19q^{39} + 42q^{40} - 70q^{42} - 117q^{45} - 54q^{46} - 28q^{49} - 8q^{51} - 30q^{52} - 21q^{54} + 50q^{58} - 67q^{60} - 18q^{61} - 70q^{63} - 176q^{64} + 57q^{66} - 10q^{67} + 42q^{70} - 45q^{72} - 150q^{73} + 33q^{75} + 10q^{78} - 34q^{79} + 49q^{81} - 53q^{84} - 8q^{85} - 15q^{87} + 80q^{88} - 62q^{91} + 30q^{94} - 9q^{96} + 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −0.289436 + 2.75380i 1.37833 1.04891i −5.54333 1.17827i 2.23018 0.162109i 2.48954 + 4.09923i 2.22240 + 1.43560i 3.13784 9.65728i 0.799592 2.89148i −0.199078 + 6.18839i
59.2 −0.285976 + 2.72088i 0.119854 + 1.72790i −5.36510 1.14039i −1.52908 + 1.63154i −4.73568 0.168030i −1.45661 + 2.20869i 2.94628 9.06773i −2.97127 + 0.414190i −4.00193 4.62702i
59.3 −0.276821 + 2.63377i −1.55725 + 0.758264i −4.90384 1.04234i 1.91558 1.15349i −1.56602 4.31135i −1.04939 2.42874i 2.46605 7.58974i 1.85007 2.36162i 2.50776 + 5.36452i
59.4 −0.275952 + 2.62550i −1.15521 1.29053i −4.86083 1.03320i −1.85485 1.24881i 3.70709 2.67690i 2.62092 0.361651i 2.42244 7.45550i −0.330959 + 2.98169i 3.79060 4.52531i
59.5 −0.265957 + 2.53041i −1.34519 1.09109i −4.37597 0.930141i 1.06957 + 1.96368i 3.11866 3.11370i −2.10244 + 1.60616i 1.94497 5.98600i 0.619064 + 2.93543i −5.25338 + 2.18420i
59.6 −0.253722 + 2.41400i 0.845155 1.51186i −3.80673 0.809146i −1.62101 + 1.54023i 3.43519 + 2.42380i −1.20580 2.35501i 1.41898 4.36716i −1.57143 2.55551i −3.30684 4.30391i
59.7 −0.246790 + 2.34805i 0.854666 1.50650i −3.49616 0.743131i −0.832478 2.07533i 3.32642 + 2.37859i −2.19679 + 1.47449i 1.14856 3.53490i −1.53909 2.57511i 5.07843 1.44253i
59.8 −0.243441 + 2.31618i 1.63205 + 0.580002i −3.34915 0.711884i −1.08008 + 1.95791i −1.74070 + 3.63894i 2.37727 1.16128i 1.02481 3.15403i 2.32719 + 1.89319i −4.27196 2.97830i
59.9 −0.235548 + 2.24109i −1.62356 + 0.603364i −3.01072 0.639948i −0.938365 2.02965i −0.969767 3.78067i −0.601362 + 2.57650i 0.750653 2.31027i 2.27190 1.95920i 4.76966 1.62488i
59.10 −0.229678 + 2.18524i −0.865425 + 1.50035i −2.76624 0.587982i 1.99819 + 1.00362i −3.07985 2.23576i 2.32529 + 1.26215i 0.562233 1.73037i −1.50208 2.59688i −2.65209 + 4.13601i
59.11 −0.229384 + 2.18244i 1.68679 + 0.393368i −2.75415 0.585412i 1.91043 + 1.16200i −1.24543 + 3.59109i −2.57372 0.613158i 0.553131 1.70236i 2.69052 + 1.32706i −2.97422 + 3.90287i
59.12 −0.227988 + 2.16916i 1.13976 + 1.30421i −2.69700 0.573265i 0.504609 2.17839i −3.08889 + 2.17498i 2.46786 + 0.953758i 0.510390 1.57082i −0.401902 + 2.97296i 4.61024 + 1.59123i
59.13 −0.227792 + 2.16729i −0.382724 + 1.68924i −2.68897 0.571559i −2.01231 0.974992i −3.57389 1.21427i 0.849883 2.50553i 0.504421 1.55245i −2.70705 1.29302i 2.57148 4.13917i
59.14 −0.211264 + 2.01005i −0.544642 1.64419i −2.03936 0.433479i 1.73522 1.41032i 3.41996 0.747397i 0.844505 2.50735i 0.0530372 0.163232i −2.40673 + 1.79099i 2.46822 + 3.78582i
59.15 −0.191732 + 1.82421i −1.65023 0.526055i −1.33469 0.283698i 0.748065 + 2.10723i 1.27604 2.90951i 1.76046 1.97503i −0.360206 + 1.10860i 2.44653 + 1.73623i −3.98745 + 0.960606i
59.16 −0.182544 + 1.73679i 1.67865 0.426758i −1.02683 0.218259i 0.0202789 2.23598i 0.434762 + 2.99337i −0.809322 2.51893i −0.512796 + 1.57822i 2.63575 1.43276i 3.87972 + 0.443385i
59.17 −0.177654 + 1.69027i −0.149616 1.72558i −0.869147 0.184743i 2.18663 0.467578i 2.94327 + 0.0536651i −0.898656 + 2.48846i −0.583724 + 1.79652i −2.95523 + 0.516348i 0.401867 + 3.77906i
59.18 −0.176872 + 1.68282i −1.68990 + 0.379800i −0.844320 0.179466i −2.01435 + 0.970779i −0.340242 2.91098i 0.713523 + 2.54772i −0.594426 + 1.82945i 2.71150 1.28365i −1.27737 3.56149i
59.19 −0.169271 + 1.61051i −0.223032 1.71763i −0.608788 0.129402i −1.76715 + 1.37010i 2.80401 0.0684501i 1.20608 + 2.35486i −0.689378 + 2.12169i −2.90051 + 0.766175i −1.90744 3.07793i
59.20 −0.166978 + 1.58869i −1.07806 + 1.35565i −0.539759 0.114729i −0.154769 + 2.23071i −1.97370 1.93906i −2.37636 1.16314i −0.714876 + 2.20016i −0.675580 2.92294i −3.51806 0.618358i
See next 80 embeddings (of 608 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 509.76 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.e even 10 1 inner
75.h odd 10 1 inner
175.u odd 30 1 inner
525.bp 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.bp.a 608
3.b odd 2 1 inner 525.2.bp.a 608
7.d odd 6 1 inner 525.2.bp.a 608
21.g even 6 1 inner 525.2.bp.a 608
25.e even 10 1 inner 525.2.bp.a 608
75.h odd 10 1 inner 525.2.bp.a 608
175.u odd 30 1 inner 525.2.bp.a 608
525.bp even 30 1 inner 525.2.bp.a 608

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