# Properties

 Label 300.2.o.a Level $300$ Weight $2$ Character orbit 300.o Analytic conductor $2.396$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.o (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{5} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 2q^{5} + 6q^{9} - 6q^{11} + 4q^{15} + 10q^{17} + 10q^{19} - 4q^{21} + 40q^{23} - 4q^{25} + 4q^{29} + 6q^{31} + 10q^{33} - 6q^{35} - 10q^{41} + 2q^{45} - 40q^{47} - 56q^{49} + 16q^{51} - 60q^{53} - 62q^{55} - 36q^{59} - 12q^{61} - 10q^{63} + 20q^{67} + 4q^{69} + 40q^{71} + 60q^{73} + 8q^{75} - 40q^{77} + 8q^{79} - 6q^{81} - 50q^{83} + 34q^{85} - 20q^{87} - 30q^{91} - 60q^{95} - 40q^{97} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 0 −0.951057 + 0.309017i 0 −2.10592 + 0.751722i 0 0.595901i 0 0.809017 0.587785i 0
109.2 0 −0.951057 + 0.309017i 0 0.913250 + 2.04107i 0 4.62675i 0 0.809017 0.587785i 0
109.3 0 −0.951057 + 0.309017i 0 0.971442 2.01403i 0 1.04684i 0 0.809017 0.587785i 0
109.4 0 0.951057 0.309017i 0 −1.98828 1.02311i 0 3.54704i 0 0.809017 0.587785i 0
109.5 0 0.951057 0.309017i 0 −1.64247 + 1.51733i 0 3.78808i 0 0.809017 0.587785i 0
109.6 0 0.951057 0.309017i 0 2.23394 0.0974182i 0 1.31873i 0 0.809017 0.587785i 0
169.1 0 −0.587785 0.809017i 0 −1.74098 + 1.40321i 0 1.57893i 0 −0.309017 + 0.951057i 0
169.2 0 −0.587785 0.809017i 0 −0.900274 2.04683i 0 0.957526i 0 −0.309017 + 0.951057i 0
169.3 0 −0.587785 0.809017i 0 1.99921 + 1.00158i 0 3.80992i 0 −0.309017 + 0.951057i 0
169.4 0 0.587785 + 0.809017i 0 −0.921600 2.03732i 0 4.41540i 0 −0.309017 + 0.951057i 0
169.5 0 0.587785 + 0.809017i 0 0.892889 2.05006i 0 4.13266i 0 −0.309017 + 0.951057i 0
169.6 0 0.587785 + 0.809017i 0 1.28878 + 1.82730i 0 2.44380i 0 −0.309017 + 0.951057i 0
229.1 0 −0.587785 + 0.809017i 0 −1.74098 1.40321i 0 1.57893i 0 −0.309017 0.951057i 0
229.2 0 −0.587785 + 0.809017i 0 −0.900274 + 2.04683i 0 0.957526i 0 −0.309017 0.951057i 0
229.3 0 −0.587785 + 0.809017i 0 1.99921 1.00158i 0 3.80992i 0 −0.309017 0.951057i 0
229.4 0 0.587785 0.809017i 0 −0.921600 + 2.03732i 0 4.41540i 0 −0.309017 0.951057i 0
229.5 0 0.587785 0.809017i 0 0.892889 + 2.05006i 0 4.13266i 0 −0.309017 0.951057i 0
229.6 0 0.587785 0.809017i 0 1.28878 1.82730i 0 2.44380i 0 −0.309017 0.951057i 0
289.1 0 −0.951057 0.309017i 0 −2.10592 0.751722i 0 0.595901i 0 0.809017 + 0.587785i 0
289.2 0 −0.951057 0.309017i 0 0.913250 2.04107i 0 4.62675i 0 0.809017 + 0.587785i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.6 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.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.o.a 24
3.b odd 2 1 900.2.w.c 24
5.b even 2 1 1500.2.o.c 24
5.c odd 4 1 1500.2.m.c 24
5.c odd 4 1 1500.2.m.d 24
25.d even 5 1 1500.2.o.c 24
25.d even 5 1 7500.2.d.g 24
25.e even 10 1 inner 300.2.o.a 24
25.e even 10 1 7500.2.d.g 24
25.f odd 20 1 1500.2.m.c 24
25.f odd 20 1 1500.2.m.d 24
25.f odd 20 1 7500.2.a.m 12
25.f odd 20 1 7500.2.a.n 12
75.h odd 10 1 900.2.w.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.o.a 24 1.a even 1 1 trivial
300.2.o.a 24 25.e even 10 1 inner
900.2.w.c 24 3.b odd 2 1
900.2.w.c 24 75.h odd 10 1
1500.2.m.c 24 5.c odd 4 1
1500.2.m.c 24 25.f odd 20 1
1500.2.m.d 24 5.c odd 4 1
1500.2.m.d 24 25.f odd 20 1
1500.2.o.c 24 5.b even 2 1
1500.2.o.c 24 25.d even 5 1
7500.2.a.m 12 25.f odd 20 1
7500.2.a.n 12 25.f odd 20 1
7500.2.d.g 24 25.d even 5 1
7500.2.d.g 24 25.e even 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(300, [\chi])$$.