# Properties

 Label 900.3.p.f Level $900$ Weight $3$ Character orbit 900.p 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.p (of order $$6$$, degree $$2$$, 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{9} - 18q^{11} - 26q^{21} - 36q^{29} + 30q^{31} - 6q^{39} - 36q^{41} - 108q^{49} + 124q^{51} + 306q^{59} + 48q^{61} + 268q^{69} - 114q^{79} - 14q^{81} - 84q^{91} - 418q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 0 −2.88925 + 0.807601i 0 0 0 −6.19003 + 10.7214i 0 7.69556 4.66673i 0
101.2 0 −2.69243 + 1.32320i 0 0 0 1.36039 2.35627i 0 5.49831 7.12521i 0
101.3 0 −2.55773 1.56781i 0 0 0 5.75148 9.96186i 0 4.08395 + 8.02006i 0
101.4 0 −1.92469 2.30121i 0 0 0 0.382345 0.662241i 0 −1.59110 + 8.85824i 0
101.5 0 −0.947625 + 2.84640i 0 0 0 −0.333962 + 0.578439i 0 −7.20401 5.39464i 0
101.6 0 −0.0930019 + 2.99856i 0 0 0 3.67363 6.36292i 0 −8.98270 0.557743i 0
101.7 0 0.0930019 2.99856i 0 0 0 −3.67363 + 6.36292i 0 −8.98270 0.557743i 0
101.8 0 0.947625 2.84640i 0 0 0 0.333962 0.578439i 0 −7.20401 5.39464i 0
101.9 0 1.92469 + 2.30121i 0 0 0 −0.382345 + 0.662241i 0 −1.59110 + 8.85824i 0
101.10 0 2.55773 + 1.56781i 0 0 0 −5.75148 + 9.96186i 0 4.08395 + 8.02006i 0
101.11 0 2.69243 1.32320i 0 0 0 −1.36039 + 2.35627i 0 5.49831 7.12521i 0
101.12 0 2.88925 0.807601i 0 0 0 6.19003 10.7214i 0 7.69556 4.66673i 0
401.1 0 −2.88925 0.807601i 0 0 0 −6.19003 10.7214i 0 7.69556 + 4.66673i 0
401.2 0 −2.69243 1.32320i 0 0 0 1.36039 + 2.35627i 0 5.49831 + 7.12521i 0
401.3 0 −2.55773 + 1.56781i 0 0 0 5.75148 + 9.96186i 0 4.08395 8.02006i 0
401.4 0 −1.92469 + 2.30121i 0 0 0 0.382345 + 0.662241i 0 −1.59110 8.85824i 0
401.5 0 −0.947625 2.84640i 0 0 0 −0.333962 0.578439i 0 −7.20401 + 5.39464i 0
401.6 0 −0.0930019 2.99856i 0 0 0 3.67363 + 6.36292i 0 −8.98270 + 0.557743i 0
401.7 0 0.0930019 + 2.99856i 0 0 0 −3.67363 6.36292i 0 −8.98270 + 0.557743i 0
401.8 0 0.947625 + 2.84640i 0 0 0 0.333962 + 0.578439i 0 −7.20401 + 5.39464i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 401.12 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.p.f 24
3.b odd 2 1 2700.3.p.f 24
5.b even 2 1 inner 900.3.p.f 24
5.c odd 4 2 180.3.t.a 24
9.c even 3 1 2700.3.p.f 24
9.d odd 6 1 inner 900.3.p.f 24
15.d odd 2 1 2700.3.p.f 24
15.e even 4 2 540.3.t.a 24
45.h odd 6 1 inner 900.3.p.f 24
45.j even 6 1 2700.3.p.f 24
45.k odd 12 2 540.3.t.a 24
45.k odd 12 2 1620.3.b.b 24
45.l even 12 2 180.3.t.a 24
45.l even 12 2 1620.3.b.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.t.a 24 5.c odd 4 2
180.3.t.a 24 45.l even 12 2
540.3.t.a 24 15.e even 4 2
540.3.t.a 24 45.k odd 12 2
900.3.p.f 24 1.a even 1 1 trivial
900.3.p.f 24 5.b even 2 1 inner
900.3.p.f 24 9.d odd 6 1 inner
900.3.p.f 24 45.h odd 6 1 inner
1620.3.b.b 24 45.k odd 12 2
1620.3.b.b 24 45.l even 12 2
2700.3.p.f 24 3.b odd 2 1
2700.3.p.f 24 9.c even 3 1
2700.3.p.f 24 15.d odd 2 1
2700.3.p.f 24 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$15\!\cdots\!88$$$$T_{7}^{12} +$$$$12\!\cdots\!46$$$$T_{7}^{10} +$$$$78\!\cdots\!25$$$$T_{7}^{8} +$$$$76\!\cdots\!80$$$$T_{7}^{6} +$$$$55\!\cdots\!56$$$$T_{7}^{4} +$$$$18\!\cdots\!28$$$$T_{7}^{2} +$$$$44\!\cdots\!56$$">$$T_{7}^{24} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.