# Properties

 Label 525.2.bg.b Level 525 Weight 2 Character orbit 525.bg Analytic conductor 4.192 Analytic rank 0 Dimension 160 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.bg (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$20$$ over $$\Q(\zeta_{15})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 2q^{2} + 20q^{3} + 20q^{4} - 2q^{5} - 4q^{6} + 4q^{7} - 30q^{8} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 2q^{2} + 20q^{3} + 20q^{4} - 2q^{5} - 4q^{6} + 4q^{7} - 30q^{8} + 20q^{9} - 6q^{10} + 12q^{11} + 20q^{12} + 12q^{14} + 4q^{15} + 12q^{16} - 6q^{17} - 8q^{18} + 12q^{19} - 7q^{21} + 14q^{23} - 30q^{24} - 4q^{25} - 56q^{26} - 40q^{27} - 27q^{28} - 2q^{29} + 4q^{30} + 27q^{31} - 78q^{32} - 8q^{33} + 76q^{34} + 21q^{35} - 40q^{36} + 14q^{37} - 17q^{38} + 10q^{40} - 6q^{41} - 2q^{42} + 28q^{43} - 22q^{44} + 3q^{45} - 22q^{46} + 7q^{47} - 24q^{48} + 152q^{49} - 62q^{50} + 4q^{51} + 13q^{52} - 26q^{53} + 2q^{54} + 64q^{55} - 33q^{56} + 56q^{57} + 48q^{59} + 15q^{60} + 36q^{61} - 88q^{62} - 7q^{63} - 2q^{64} + 46q^{65} + 15q^{66} + 42q^{67} - 28q^{68} - 8q^{69} - 32q^{70} + 36q^{71} + 15q^{72} + 10q^{73} - 16q^{74} + q^{75} - 76q^{76} + 12q^{77} - 8q^{78} - 10q^{79} + 100q^{80} + 20q^{81} + 14q^{82} - 74q^{83} - 6q^{84} + 90q^{85} + 6q^{86} - 4q^{87} + 102q^{88} + 33q^{89} - 18q^{90} - 31q^{91} - 94q^{92} - 98q^{93} - 30q^{94} - 106q^{95} + 47q^{96} - 26q^{97} + 26q^{98} + 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −2.43210 + 1.08284i 0.669131 0.743145i 3.40432 3.78088i −1.23217 + 1.86595i −0.822687 + 2.53197i 1.91069 1.83010i −2.54019 + 7.81792i −0.104528 0.994522i 0.976241 5.87242i
16.2 −2.19023 + 0.975155i 0.669131 0.743145i 2.50794 2.78535i −0.992732 2.00362i −0.740871 + 2.28017i 2.55323 + 0.693547i −1.29508 + 3.98585i −0.104528 0.994522i 4.12815 + 3.42032i
16.3 −2.04004 + 0.908286i 0.669131 0.743145i 1.99854 2.21960i 1.97582 1.04697i −0.690068 + 2.12381i −0.0626701 2.64501i −0.680936 + 2.09571i −0.104528 0.994522i −3.07982 + 3.93047i
16.4 −1.72477 + 0.767919i 0.669131 0.743145i 1.04689 1.16268i −2.14015 + 0.647903i −0.583424 + 1.79560i −2.28147 + 1.33974i 0.254054 0.781898i −0.104528 0.994522i 3.19373 2.76095i
16.5 −1.53240 + 0.682267i 0.669131 0.743145i 0.544489 0.604717i 1.08380 + 1.95586i −0.518350 + 1.59532i −2.12083 1.58180i 0.614905 1.89248i −0.104528 0.994522i −2.99523 2.25771i
16.6 −1.19605 + 0.532515i 0.669131 0.743145i −0.191302 + 0.212462i 1.53481 + 1.62614i −0.404577 + 1.24516i 2.55996 + 0.668275i 0.924820 2.84630i −0.104528 0.994522i −2.70166 1.12764i
16.7 −0.989117 + 0.440383i 0.669131 0.743145i −0.553846 + 0.615109i −0.228741 2.22434i −0.334580 + 1.02973i −1.49334 + 2.18402i 0.946095 2.91178i −0.104528 0.994522i 1.20581 + 2.09940i
16.8 −0.504477 + 0.224608i 0.669131 0.743145i −1.13421 + 1.25967i −1.89312 + 1.19000i −0.170645 + 0.525192i 0.160877 2.64086i 0.630543 1.94061i −0.104528 0.994522i 0.687754 1.02554i
16.9 −0.308545 + 0.137373i 0.669131 0.743145i −1.26193 + 1.40152i −2.19353 0.434088i −0.104369 + 0.321214i 2.54788 + 0.712971i 0.405570 1.24822i −0.104528 0.994522i 0.736435 0.167396i
16.10 −0.195464 + 0.0870263i 0.669131 0.743145i −1.30763 + 1.45227i 2.23437 0.0871077i −0.0661180 + 0.203490i −2.62302 0.346096i 0.261445 0.804645i −0.104528 0.994522i −0.429159 + 0.211475i
16.11 0.340400 0.151556i 0.669131 0.743145i −1.24536 + 1.38311i 2.23604 0.0110432i 0.115144 0.354377i 0.0639980 + 2.64498i −0.444590 + 1.36831i −0.104528 0.994522i 0.759475 0.342644i
16.12 0.811470 0.361290i 0.669131 0.743145i −0.810308 + 0.899939i −1.71756 1.43178i 0.274489 0.844789i −1.96006 1.77712i −0.881380 + 2.71261i −0.104528 0.994522i −1.91103 0.541305i
16.13 0.845870 0.376606i 0.669131 0.743145i −0.764597 + 0.849171i 0.707816 2.12108i 0.286125 0.880602i 2.58582 0.559924i −0.899197 + 2.76744i −0.104528 0.994522i −0.200092 2.06073i
16.14 1.05172 0.468254i 0.669131 0.743145i −0.451416 + 0.501348i −1.26593 + 1.84321i 0.355755 1.09490i −2.58874 + 0.546295i −0.951513 + 2.92846i −0.104528 0.994522i −0.468306 + 2.53131i
16.15 1.23117 0.548154i 0.669131 0.743145i −0.122944 + 0.136543i −0.120990 + 2.23279i 0.416458 1.28173i 2.20566 1.46119i −0.909436 + 2.79896i −0.104528 0.994522i 1.07495 + 2.81528i
16.16 1.88139 0.837651i 0.669131 0.743145i 1.49973 1.66561i 2.15442 0.598733i 0.636403 1.95865i −0.705516 2.54995i 0.153568 0.472633i −0.104528 0.994522i 3.55178 2.93110i
16.17 1.90891 0.849901i 0.669131 0.743145i 1.58334 1.75848i 1.75701 + 1.38308i 0.645710 1.98729i −1.00807 + 2.44618i 0.236500 0.727873i −0.104528 0.994522i 4.52946 + 1.14688i
16.18 2.10352 0.936548i 0.669131 0.743145i 2.20942 2.45381i −0.282304 2.21818i 0.711540 2.18990i 0.729324 + 2.54324i 0.926375 2.85109i −0.104528 0.994522i −2.67126 4.40159i
16.19 2.34261 1.04299i 0.669131 0.743145i 3.06170 3.40036i −1.80799 1.31574i 0.792412 2.43879i −2.61898 0.375455i 2.04097 6.28145i −0.104528 0.994522i −5.60772 1.19654i
16.20 2.42324 1.07889i 0.669131 0.743145i 3.36980 3.74254i −0.617899 + 2.14900i 0.819687 2.52274i 2.64524 + 0.0517719i 2.48864 7.65924i −0.104528 0.994522i 0.821228 + 5.87418i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 466.20 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.d even 5 1 inner
175.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bg.b 160
7.c even 3 1 inner 525.2.bg.b 160
25.d even 5 1 inner 525.2.bg.b 160
175.q even 15 1 inner 525.2.bg.b 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bg.b 160 1.a even 1 1 trivial
525.2.bg.b 160 7.c even 3 1 inner
525.2.bg.b 160 25.d even 5 1 inner
525.2.bg.b 160 175.q even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{160} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database