# Properties

 Label 525.3.l.e Level 525 Weight 3 Character orbit 525.l Analytic conductor 14.305 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3052138789$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} + 24q^{6} + 48q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} + 24q^{6} + 48q^{8} + 48q^{12} - 64q^{13} - 184q^{16} - 24q^{17} - 24q^{18} - 8q^{22} - 8q^{23} - 80q^{26} + 96q^{31} - 56q^{32} + 72q^{33} + 168q^{36} - 8q^{37} - 56q^{38} + 320q^{41} + 112q^{43} + 320q^{46} - 64q^{47} - 192q^{48} - 192q^{51} - 96q^{52} + 72q^{53} - 336q^{56} - 48q^{57} + 512q^{58} - 496q^{61} + 776q^{62} - 192q^{66} + 192q^{67} - 568q^{68} - 144q^{71} - 144q^{72} - 224q^{73} + 416q^{76} - 112q^{77} + 216q^{78} - 216q^{81} - 352q^{82} + 32q^{83} + 240q^{86} - 384q^{87} - 216q^{88} - 1304q^{92} + 168q^{96} + 816q^{97} + 56q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −2.74240 + 2.74240i 1.22474 + 1.22474i 11.0415i 0 −6.71747 −1.87083 + 1.87083i 19.3105 + 19.3105i 3.00000i 0
43.2 −2.41688 + 2.41688i −1.22474 1.22474i 7.68258i 0 5.92011 1.87083 1.87083i 8.90034 + 8.90034i 3.00000i 0
43.3 −2.24469 + 2.24469i −1.22474 1.22474i 6.07726i 0 5.49834 −1.87083 + 1.87083i 4.66280 + 4.66280i 3.00000i 0
43.4 −2.08980 + 2.08980i 1.22474 + 1.22474i 4.73454i 0 −5.11895 1.87083 1.87083i 1.53505 + 1.53505i 3.00000i 0
43.5 −0.992944 + 0.992944i −1.22474 1.22474i 2.02813i 0 2.43221 −1.87083 + 1.87083i −5.98559 5.98559i 3.00000i 0
43.6 −0.675544 + 0.675544i −1.22474 1.22474i 3.08728i 0 1.65474 1.87083 1.87083i −4.78777 4.78777i 3.00000i 0
43.7 −0.408558 + 0.408558i 1.22474 + 1.22474i 3.66616i 0 −1.00076 1.87083 1.87083i −3.13207 3.13207i 3.00000i 0
43.8 0.867675 0.867675i −1.22474 1.22474i 2.49428i 0 −2.12536 1.87083 1.87083i 5.63493 + 5.63493i 3.00000i 0
43.9 1.01289 1.01289i −1.22474 1.22474i 1.94811i 0 −2.48106 −1.87083 + 1.87083i 6.02478 + 6.02478i 3.00000i 0
43.10 1.36784 1.36784i 1.22474 + 1.22474i 0.258033i 0 3.35051 −1.87083 + 1.87083i 5.82430 + 5.82430i 3.00000i 0
43.11 1.59930 1.59930i 1.22474 + 1.22474i 1.11554i 0 3.91747 −1.87083 + 1.87083i 4.61313 + 4.61313i 3.00000i 0
43.12 2.72310 2.72310i 1.22474 + 1.22474i 10.8306i 0 6.67022 1.87083 1.87083i −18.6004 18.6004i 3.00000i 0
232.1 −2.74240 2.74240i 1.22474 1.22474i 11.0415i 0 −6.71747 −1.87083 1.87083i 19.3105 19.3105i 3.00000i 0
232.2 −2.41688 2.41688i −1.22474 + 1.22474i 7.68258i 0 5.92011 1.87083 + 1.87083i 8.90034 8.90034i 3.00000i 0
232.3 −2.24469 2.24469i −1.22474 + 1.22474i 6.07726i 0 5.49834 −1.87083 1.87083i 4.66280 4.66280i 3.00000i 0
232.4 −2.08980 2.08980i 1.22474 1.22474i 4.73454i 0 −5.11895 1.87083 + 1.87083i 1.53505 1.53505i 3.00000i 0
232.5 −0.992944 0.992944i −1.22474 + 1.22474i 2.02813i 0 2.43221 −1.87083 1.87083i −5.98559 + 5.98559i 3.00000i 0
232.6 −0.675544 0.675544i −1.22474 + 1.22474i 3.08728i 0 1.65474 1.87083 + 1.87083i −4.78777 + 4.78777i 3.00000i 0
232.7 −0.408558 0.408558i 1.22474 1.22474i 3.66616i 0 −1.00076 1.87083 + 1.87083i −3.13207 + 3.13207i 3.00000i 0
232.8 0.867675 + 0.867675i −1.22474 + 1.22474i 2.49428i 0 −2.12536 1.87083 + 1.87083i 5.63493 5.63493i 3.00000i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 232.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.l.e 24
5.b even 2 1 105.3.l.a 24
5.c odd 4 1 105.3.l.a 24
5.c odd 4 1 inner 525.3.l.e 24
15.d odd 2 1 315.3.o.b 24
15.e even 4 1 315.3.o.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.l.a 24 5.b even 2 1
105.3.l.a 24 5.c odd 4 1
315.3.o.b 24 15.d odd 2 1
315.3.o.b 24 15.e even 4 1
525.3.l.e 24 1.a even 1 1 trivial
525.3.l.e 24 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(525, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database