# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(109,300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 7]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("300.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 109.6 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 $$S_{2}^{\mathrm{new}}(300, [\chi])$$.