# Properties

 Label 900.3.u.c Level $900$ Weight $3$ Character orbit 900.u Analytic conductor $24.523$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.u (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 52q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 52q^{9} + 96q^{11} - 144q^{19} - 256q^{21} - 300q^{29} - 24q^{31} - 80q^{39} + 180q^{41} - 96q^{49} - 288q^{51} - 96q^{59} - 156q^{61} + 300q^{69} - 240q^{79} + 868q^{81} + 240q^{91} + 240q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1 0 −2.96464 + 0.459278i 0 0 0 7.16186 4.13490i 0 8.57813 2.72318i 0
149.2 0 −2.87949 0.841761i 0 0 0 10.3705 5.98742i 0 7.58288 + 4.84768i 0
149.3 0 −1.39478 + 2.65605i 0 0 0 2.45229 1.41583i 0 −5.10917 7.40921i 0
149.4 0 −0.920635 2.85525i 0 0 0 1.02985 0.594587i 0 −7.30486 + 5.25728i 0
149.5 0 −0.783177 2.89597i 0 0 0 4.08158 2.35650i 0 −7.77327 + 4.53611i 0
149.6 0 −0.114662 2.99781i 0 0 0 −1.38806 + 0.801399i 0 −8.97371 + 0.687471i 0
149.7 0 0.114662 + 2.99781i 0 0 0 1.38806 0.801399i 0 −8.97371 + 0.687471i 0
149.8 0 0.783177 + 2.89597i 0 0 0 −4.08158 + 2.35650i 0 −7.77327 + 4.53611i 0
149.9 0 0.920635 + 2.85525i 0 0 0 −1.02985 + 0.594587i 0 −7.30486 + 5.25728i 0
149.10 0 1.39478 2.65605i 0 0 0 −2.45229 + 1.41583i 0 −5.10917 7.40921i 0
149.11 0 2.87949 + 0.841761i 0 0 0 −10.3705 + 5.98742i 0 7.58288 + 4.84768i 0
149.12 0 2.96464 0.459278i 0 0 0 −7.16186 + 4.13490i 0 8.57813 2.72318i 0
749.1 0 −2.96464 0.459278i 0 0 0 7.16186 + 4.13490i 0 8.57813 + 2.72318i 0
749.2 0 −2.87949 + 0.841761i 0 0 0 10.3705 + 5.98742i 0 7.58288 4.84768i 0
749.3 0 −1.39478 2.65605i 0 0 0 2.45229 + 1.41583i 0 −5.10917 + 7.40921i 0
749.4 0 −0.920635 + 2.85525i 0 0 0 1.02985 + 0.594587i 0 −7.30486 5.25728i 0
749.5 0 −0.783177 + 2.89597i 0 0 0 4.08158 + 2.35650i 0 −7.77327 4.53611i 0
749.6 0 −0.114662 + 2.99781i 0 0 0 −1.38806 0.801399i 0 −8.97371 0.687471i 0
749.7 0 0.114662 2.99781i 0 0 0 1.38806 + 0.801399i 0 −8.97371 0.687471i 0
749.8 0 0.783177 2.89597i 0 0 0 −4.08158 2.35650i 0 −7.77327 4.53611i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 749.12 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.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.c 24
3.b odd 2 1 2700.3.u.c 24
5.b even 2 1 inner 900.3.u.c 24
5.c odd 4 1 180.3.o.b 12
5.c odd 4 1 900.3.p.c 12
9.c even 3 1 2700.3.u.c 24
9.d odd 6 1 inner 900.3.u.c 24
15.d odd 2 1 2700.3.u.c 24
15.e even 4 1 540.3.o.b 12
15.e even 4 1 2700.3.p.c 12
20.e even 4 1 720.3.bs.b 12
45.h odd 6 1 inner 900.3.u.c 24
45.j even 6 1 2700.3.u.c 24
45.k odd 12 1 540.3.o.b 12
45.k odd 12 1 1620.3.g.b 12
45.k odd 12 1 2700.3.p.c 12
45.l even 12 1 180.3.o.b 12
45.l even 12 1 900.3.p.c 12
45.l even 12 1 1620.3.g.b 12
60.l odd 4 1 2160.3.bs.b 12
180.v odd 12 1 720.3.bs.b 12
180.x even 12 1 2160.3.bs.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b 12 5.c odd 4 1
180.3.o.b 12 45.l even 12 1
540.3.o.b 12 15.e even 4 1
540.3.o.b 12 45.k odd 12 1
720.3.bs.b 12 20.e even 4 1
720.3.bs.b 12 180.v odd 12 1
900.3.p.c 12 5.c odd 4 1
900.3.p.c 12 45.l even 12 1
900.3.u.c 24 1.a even 1 1 trivial
900.3.u.c 24 5.b even 2 1 inner
900.3.u.c 24 9.d odd 6 1 inner
900.3.u.c 24 45.h odd 6 1 inner
1620.3.g.b 12 45.k odd 12 1
1620.3.g.b 12 45.l even 12 1
2160.3.bs.b 12 60.l odd 4 1
2160.3.bs.b 12 180.x even 12 1
2700.3.p.c 12 15.e even 4 1
2700.3.p.c 12 45.k odd 12 1
2700.3.u.c 24 3.b odd 2 1
2700.3.u.c 24 9.c even 3 1
2700.3.u.c 24 15.d odd 2 1
2700.3.u.c 24 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$64\!\cdots\!30$$$$T_{7}^{8} -$$$$20\!\cdots\!90$$$$T_{7}^{6} +$$$$46\!\cdots\!61$$$$T_{7}^{4} -$$$$51\!\cdots\!26$$$$T_{7}^{2} +$$$$40\!\cdots\!21$$">$$T_{7}^{24} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.