# Properties

 Label 525.2.bo.a Level 525 Weight 2 Character orbit 525.bo Analytic conductor 4.192 Analytic rank 0 Dimension 320 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.bo (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$320q - 40q^{4} - 2q^{5} + 8q^{6} + 60q^{8} - 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$320q - 40q^{4} - 2q^{5} + 8q^{6} + 60q^{8} - 40q^{9} + 4q^{10} - 6q^{11} + 12q^{14} + 4q^{15} + 40q^{16} - 20q^{17} + 16q^{19} + 8q^{20} + 40q^{22} - 60q^{23} - 48q^{24} + 4q^{25} - 30q^{28} + 24q^{29} - 48q^{30} - 30q^{31} - 80q^{36} + 30q^{38} - 32q^{40} + 36q^{41} + 10q^{42} - 16q^{44} + 2q^{45} + 32q^{46} + 16q^{49} - 140q^{50} + 190q^{52} - 60q^{53} + 4q^{54} - 8q^{55} + 60q^{58} + 24q^{59} - 46q^{60} - 20q^{61} + 120q^{62} + 10q^{63} - 4q^{64} - 30q^{65} - 16q^{66} - 60q^{67} - 32q^{69} - 76q^{70} + 32q^{71} - 30q^{72} - 40q^{73} - 12q^{74} - 8q^{75} - 344q^{76} + 4q^{79} - 52q^{80} + 40q^{81} - 80q^{83} + 48q^{84} - 76q^{85} + 24q^{86} - 200q^{88} - 52q^{90} + 26q^{91} + 180q^{92} + 16q^{94} - 38q^{95} - 58q^{96} - 140q^{97} + 360q^{98} + 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.576680 2.71307i 0.406737 + 0.913545i −5.20107 + 2.31567i 0.442104 2.19193i 2.24395 1.63033i 1.97794 + 1.75720i 6.02126 + 8.28755i −0.669131 + 0.743145i −6.20179 + 0.0645817i
4.2 −0.562446 2.64610i −0.406737 0.913545i −4.85841 + 2.16310i −2.12768 + 0.687738i −2.18856 + 1.59009i 0.530213 + 2.59208i 5.27620 + 7.26207i −0.669131 + 0.743145i 3.01653 + 5.24323i
4.3 −0.550698 2.59083i 0.406737 + 0.913545i −4.58205 + 2.04006i 0.479670 + 2.18401i 2.14285 1.55687i 0.0455325 2.64536i 4.69504 + 6.46216i −0.669131 + 0.743145i 5.39426 2.44548i
4.4 −0.489241 2.30170i −0.406737 0.913545i −3.23137 + 1.43870i −1.95437 1.08648i −1.90371 + 1.38313i −0.927957 2.47768i 2.12611 + 2.92634i −0.669131 + 0.743145i −1.54459 + 5.02992i
4.5 −0.485930 2.28612i −0.406737 0.913545i −3.16312 + 1.40831i 1.52274 + 1.63746i −1.89083 + 1.37377i 2.64502 0.0621505i 2.00910 + 2.76528i −0.669131 + 0.743145i 3.00348 4.27685i
4.6 −0.459756 2.16298i 0.406737 + 0.913545i −2.64003 + 1.17542i −2.14884 + 0.618467i 1.78898 1.29977i −2.46658 0.957073i 1.15663 + 1.59196i −0.669131 + 0.743145i 2.32567 + 4.36355i
4.7 −0.408094 1.91993i −0.406737 0.913545i −1.69250 + 0.753548i 1.10575 + 1.94353i −1.58796 + 1.15372i −2.49128 + 0.890806i −0.169980 0.233957i −0.669131 + 0.743145i 3.28020 2.91610i
4.8 −0.397148 1.86843i 0.406737 + 0.913545i −1.50623 + 0.670617i 0.571665 2.16176i 1.54537 1.12277i −2.41699 + 1.07617i −0.394346 0.542771i −0.669131 + 0.743145i −4.26614 0.209581i
4.9 −0.390171 1.83561i 0.406737 + 0.913545i −1.39014 + 0.618929i −2.23534 0.0570506i 1.51822 1.10305i 1.21775 + 2.34885i −0.527594 0.726171i −0.669131 + 0.743145i 0.767442 + 4.12547i
4.10 −0.383892 1.80607i 0.406737 + 0.913545i −1.28743 + 0.573200i 2.20206 0.388478i 1.49378 1.08530i 0.977325 2.45862i −0.641123 0.882430i −0.669131 + 0.743145i −1.54697 3.82795i
4.11 −0.323952 1.52407i −0.406737 0.913545i −0.390761 + 0.173978i −1.32238 + 1.80314i −1.26055 + 0.915841i 1.46476 2.20329i −1.43994 1.98190i −0.669131 + 0.743145i 3.17650 + 1.43128i
4.12 −0.268486 1.26313i 0.406737 + 0.913545i 0.303680 0.135207i 0.134729 + 2.23201i 1.04472 0.759036i 2.58880 + 0.545993i −1.77039 2.43673i −0.669131 + 0.743145i 2.78314 0.769443i
4.13 −0.260051 1.22344i −0.406737 0.913545i 0.397901 0.177157i 1.57700 1.58527i −1.01190 + 0.735188i −1.04710 2.42973i −1.79059 2.46454i −0.669131 + 0.743145i −2.34959 1.51712i
4.14 −0.208473 0.980788i −0.406737 0.913545i 0.908606 0.404537i 2.23584 + 0.0318845i −0.811201 + 0.589372i −0.130934 + 2.64251i −1.76493 2.42922i −0.669131 + 0.743145i −0.434841 2.19953i
4.15 −0.183117 0.861499i 0.406737 + 0.913545i 1.11844 0.497963i 2.08943 + 0.796430i 0.712538 0.517689i −1.18383 + 2.36612i −1.66918 2.29743i −0.669131 + 0.743145i 0.303514 1.94588i
4.16 −0.156432 0.735953i −0.406737 0.913545i 1.30993 0.583221i −0.658939 2.13677i −0.608700 + 0.442247i 2.62185 0.354813i −1.51863 2.09022i −0.669131 + 0.743145i −1.46949 + 0.819207i
4.17 −0.0711168 0.334578i −0.406737 0.913545i 1.72021 0.765885i −2.11153 0.735838i −0.276727 + 0.201054i −2.20361 + 1.46427i −0.780691 1.07453i −0.669131 + 0.743145i −0.0960305 + 0.758801i
4.18 −0.0676435 0.318238i 0.406737 + 0.913545i 1.73039 0.770420i −2.23499 + 0.0693262i 0.263211 0.191234i −0.802430 2.52113i −0.744695 1.02498i −0.669131 + 0.743145i 0.173245 + 0.706569i
4.19 −0.0648139 0.304925i 0.406737 + 0.913545i 1.73831 0.773947i −1.65210 1.50684i 0.252201 0.183235i −0.00949525 + 2.64573i −0.715132 0.984294i −0.669131 + 0.743145i −0.352395 + 0.601431i
4.20 −0.0604608 0.284445i 0.406737 + 0.913545i 1.74984 0.779078i 0.835291 2.07420i 0.235262 0.170928i 2.56191 0.660769i −0.669258 0.921154i −0.669131 + 0.743145i −0.640498 0.112187i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 394.40 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.c even 3 1 inner
25.e even 10 1 inner
175.t 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.bo.a 320
7.c even 3 1 inner 525.2.bo.a 320
25.e even 10 1 inner 525.2.bo.a 320
175.t even 30 1 inner 525.2.bo.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bo.a 320 1.a even 1 1 trivial
525.2.bo.a 320 7.c even 3 1 inner
525.2.bo.a 320 25.e even 10 1 inner
525.2.bo.a 320 175.t 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