# Properties

 Label 525.2.n.d Level 525 Weight 2 Character orbit 525.n Analytic conductor 4.192 Analytic rank 0 Dimension 32 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{5})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + q^{2} - 8q^{3} - 15q^{4} - 3q^{5} + q^{6} - 32q^{7} - 3q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + q^{2} - 8q^{3} - 15q^{4} - 3q^{5} + q^{6} - 32q^{7} - 3q^{8} - 8q^{9} + 8q^{10} + 2q^{11} - 5q^{12} + 12q^{13} - q^{14} - 8q^{15} - 17q^{16} + 12q^{17} - 4q^{18} - 13q^{19} - 27q^{20} + 8q^{21} - 21q^{22} - 12q^{23} - 18q^{24} + 11q^{25} - 2q^{26} - 8q^{27} + 15q^{28} + 21q^{29} - 12q^{30} + 3q^{31} - 50q^{32} - 13q^{33} - 41q^{34} + 3q^{35} - 5q^{36} - 22q^{37} + 44q^{38} + 12q^{39} - 39q^{40} - 3q^{41} - q^{42} + 24q^{43} - 43q^{44} + 2q^{45} + 10q^{46} + 8q^{47} - 17q^{48} + 32q^{49} + 19q^{50} - 8q^{51} + 53q^{52} + 18q^{53} + q^{54} + 23q^{55} + 3q^{56} + 42q^{57} - 32q^{58} + 28q^{59} + 73q^{60} + 36q^{61} + 10q^{62} + 8q^{63} + 9q^{64} - 34q^{65} + 4q^{66} - 22q^{67} - 78q^{68} - 2q^{69} - 8q^{70} - 40q^{71} - 3q^{72} - 10q^{73} - 34q^{74} + 6q^{75} + 132q^{76} - 2q^{77} + 28q^{78} + 18q^{79} + 148q^{80} - 8q^{81} + 102q^{82} + 16q^{83} + 5q^{84} + 18q^{85} + 16q^{86} - 34q^{87} + 13q^{88} - 17q^{89} - 2q^{90} - 12q^{91} - 106q^{92} + 18q^{93} - 20q^{94} - 92q^{95} - 15q^{96} + 30q^{97} + q^{98} + 22q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1 −0.864713 2.66131i −0.809017 + 0.587785i −4.71683 + 3.42698i −1.60184 + 1.56017i 2.26385 + 1.64478i −1.00000 8.67128 + 6.30005i 0.309017 0.951057i 5.53723 + 2.91389i
106.2 −0.590098 1.81614i −0.809017 + 0.587785i −1.33210 + 0.967825i 1.37624 + 1.76238i 1.54490 + 1.12243i −1.00000 −0.546024 0.396709i 0.309017 0.951057i 2.38860 3.53941i
106.3 −0.573156 1.76399i −0.809017 + 0.587785i −1.16513 + 0.846516i 1.88413 1.20418i 1.50054 + 1.09021i −1.00000 −0.840036 0.610322i 0.309017 0.951057i −3.20407 2.63341i
106.4 −0.198286 0.610262i −0.809017 + 0.587785i 1.28493 0.933558i −1.63030 + 1.53040i 0.519120 + 0.377162i −1.00000 −1.86274 1.35336i 0.309017 0.951057i 1.25721 + 0.691449i
106.5 0.143867 + 0.442776i −0.809017 + 0.587785i 1.44268 1.04817i 0.251880 2.22184i −0.376647 0.273650i −1.00000 1.42495 + 1.03529i 0.309017 0.951057i 1.02001 0.208121i
106.6 0.276260 + 0.850242i −0.809017 + 0.587785i 0.971442 0.705794i −1.96138 1.07378i −0.723259 0.525478i −1.00000 2.31498 + 1.68193i 0.309017 0.951057i 0.371124 1.96429i
106.7 0.662392 + 2.03863i −0.809017 + 0.587785i −2.09923 + 1.52518i 2.11075 0.738048i −1.73416 1.25994i −1.00000 −1.03146 0.749399i 0.309017 0.951057i 2.90276 + 3.81418i
106.8 0.834718 + 2.56900i −0.809017 + 0.587785i −4.28496 + 3.11320i −0.620482 + 2.14826i −2.18532 1.58773i −1.00000 −7.20390 5.23394i 0.309017 0.951057i −6.03679 + 0.199170i
211.1 −2.01265 + 1.46227i 0.309017 0.951057i 1.29447 3.98396i −1.44782 + 1.70406i 0.768762 + 2.36601i −1.00000 1.68281 + 5.17916i −0.809017 0.587785i 0.422145 5.54678i
211.2 −1.24444 + 0.904140i 0.309017 0.951057i 0.113133 0.348187i −1.50439 1.65433i 0.475335 + 1.46293i −1.00000 −0.776647 2.39027i −0.809017 0.587785i 3.36787 + 0.698540i
211.3 −1.04196 + 0.757031i 0.309017 0.951057i −0.105441 + 0.324514i 1.64809 1.51123i 0.397995 + 1.22490i −1.00000 −0.931791 2.86776i −0.809017 0.587785i −0.573204 + 2.82230i
211.4 0.0219060 0.0159156i 0.309017 0.951057i −0.617807 + 1.90142i 1.28481 + 1.83010i −0.00836734 0.0257520i −1.00000 0.0334632 + 0.102989i −0.809017 0.587785i 0.0572722 + 0.0196417i
211.5 0.464153 0.337227i 0.309017 0.951057i −0.516318 + 1.58906i 1.08776 1.95366i −0.177291 0.545645i −1.00000 0.650806 + 2.00297i −0.809017 0.587785i −0.153940 1.27362i
211.6 0.546644 0.397160i 0.309017 0.951057i −0.476951 + 1.46790i −2.15335 0.602582i −0.208799 0.642618i −1.00000 0.739869 + 2.27708i −0.809017 0.587785i −1.41643 + 0.525825i
211.7 1.88023 1.36607i 0.309017 0.951057i 1.05109 3.23494i 1.98079 1.03753i −0.718185 2.21034i −1.00000 −1.00647 3.09761i −0.809017 0.587785i 2.30701 4.65669i
211.8 2.19514 1.59486i 0.309017 0.951057i 1.65701 5.09975i −2.20491 + 0.371995i −0.838467 2.58054i −1.00000 −2.81909 8.67626i −0.809017 0.587785i −4.24679 + 4.33310i
316.1 −2.01265 1.46227i 0.309017 + 0.951057i 1.29447 + 3.98396i −1.44782 1.70406i 0.768762 2.36601i −1.00000 1.68281 5.17916i −0.809017 + 0.587785i 0.422145 + 5.54678i
316.2 −1.24444 0.904140i 0.309017 + 0.951057i 0.113133 + 0.348187i −1.50439 + 1.65433i 0.475335 1.46293i −1.00000 −0.776647 + 2.39027i −0.809017 + 0.587785i 3.36787 0.698540i
316.3 −1.04196 0.757031i 0.309017 + 0.951057i −0.105441 0.324514i 1.64809 + 1.51123i 0.397995 1.22490i −1.00000 −0.931791 + 2.86776i −0.809017 + 0.587785i −0.573204 2.82230i
316.4 0.0219060 + 0.0159156i 0.309017 + 0.951057i −0.617807 1.90142i 1.28481 1.83010i −0.00836734 + 0.0257520i −1.00000 0.0334632 0.102989i −0.809017 + 0.587785i 0.0572722 0.0196417i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.n.d 32
25.d even 5 1 inner 525.2.n.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.d 32 1.a even 1 1 trivial
525.2.n.d 32 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database