# Properties

 Label 169.3.d.e Level $169$ Weight $3$ Character orbit 169.d Analytic conductor $4.605$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,3,Mod(70,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("169.70");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 169.d (of order $$4$$, degree $$2$$, 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: $$4.60491646769$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{3} + 84 q^{9}+O(q^{10})$$ 24 * q + 12 * q^3 + 84 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{3} + 84 q^{9} + 188 q^{14} + 188 q^{16} - 136 q^{22} + 60 q^{27} + 84 q^{29} + 176 q^{35} - 524 q^{40} - 368 q^{42} - 368 q^{48} - 44 q^{53} - 704 q^{55} - 8 q^{61} - 740 q^{66} - 168 q^{68} + 744 q^{74} - 300 q^{79} + 240 q^{81} - 952 q^{87} + 1736 q^{92} + 1132 q^{94}+O(q^{100})$$ 24 * q + 12 * q^3 + 84 * q^9 + 188 * q^14 + 188 * q^16 - 136 * q^22 + 60 * q^27 + 84 * q^29 + 176 * q^35 - 524 * q^40 - 368 * q^42 - 368 * q^48 - 44 * q^53 - 704 * q^55 - 8 * q^61 - 740 * q^66 - 168 * q^68 + 744 * q^74 - 300 * q^79 + 240 * q^81 - 952 * q^87 + 1736 * q^92 + 1132 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
70.1 −2.28029 + 2.28029i −2.04607 6.39942i −5.50472 + 5.50472i 4.66563 4.66563i −2.31365 2.31365i 5.47136 + 5.47136i −4.81360 25.1047i
70.2 −2.02149 + 2.02149i 4.64845 4.17281i 2.43155 2.43155i −9.39676 + 9.39676i 0.759383 + 0.759383i 0.349324 + 0.349324i 12.6080 9.83067i
70.3 −1.66446 + 1.66446i −5.02350 1.54085i 1.10964 1.10964i 8.36141 8.36141i −4.64797 4.64797i −4.09315 4.09315i 16.2355 3.69391i
70.4 −1.40788 + 1.40788i −0.599528 0.0357421i 5.65111 5.65111i 0.844064 0.844064i −1.43861 1.43861i −5.68184 5.68184i −8.64057 15.9122i
70.5 −1.17461 + 1.17461i 1.35405 1.24058i −1.57408 + 1.57408i −1.59048 + 1.59048i −8.90433 8.90433i −6.15564 6.15564i −7.16655 3.69787i
70.6 −0.285703 + 0.285703i 4.66660 3.83675i 4.14024 4.14024i −1.33326 + 1.33326i 1.61555 + 1.61555i −2.23898 2.23898i 12.7772 2.36576i
70.7 0.285703 0.285703i 4.66660 3.83675i −4.14024 + 4.14024i 1.33326 1.33326i −1.61555 1.61555i 2.23898 + 2.23898i 12.7772 2.36576i
70.8 1.17461 1.17461i 1.35405 1.24058i 1.57408 1.57408i 1.59048 1.59048i 8.90433 + 8.90433i 6.15564 + 6.15564i −7.16655 3.69787i
70.9 1.40788 1.40788i −0.599528 0.0357421i −5.65111 + 5.65111i −0.844064 + 0.844064i 1.43861 + 1.43861i 5.68184 + 5.68184i −8.64057 15.9122i
70.10 1.66446 1.66446i −5.02350 1.54085i −1.10964 + 1.10964i −8.36141 + 8.36141i 4.64797 + 4.64797i 4.09315 + 4.09315i 16.2355 3.69391i
70.11 2.02149 2.02149i 4.64845 4.17281i −2.43155 + 2.43155i 9.39676 9.39676i −0.759383 0.759383i −0.349324 0.349324i 12.6080 9.83067i
70.12 2.28029 2.28029i −2.04607 6.39942i 5.50472 5.50472i −4.66563 + 4.66563i 2.31365 + 2.31365i −5.47136 5.47136i −4.81360 25.1047i
99.1 −2.28029 2.28029i −2.04607 6.39942i −5.50472 5.50472i 4.66563 + 4.66563i −2.31365 + 2.31365i 5.47136 5.47136i −4.81360 25.1047i
99.2 −2.02149 2.02149i 4.64845 4.17281i 2.43155 + 2.43155i −9.39676 9.39676i 0.759383 0.759383i 0.349324 0.349324i 12.6080 9.83067i
99.3 −1.66446 1.66446i −5.02350 1.54085i 1.10964 + 1.10964i 8.36141 + 8.36141i −4.64797 + 4.64797i −4.09315 + 4.09315i 16.2355 3.69391i
99.4 −1.40788 1.40788i −0.599528 0.0357421i 5.65111 + 5.65111i 0.844064 + 0.844064i −1.43861 + 1.43861i −5.68184 + 5.68184i −8.64057 15.9122i
99.5 −1.17461 1.17461i 1.35405 1.24058i −1.57408 1.57408i −1.59048 1.59048i −8.90433 + 8.90433i −6.15564 + 6.15564i −7.16655 3.69787i
99.6 −0.285703 0.285703i 4.66660 3.83675i 4.14024 + 4.14024i −1.33326 1.33326i 1.61555 1.61555i −2.23898 + 2.23898i 12.7772 2.36576i
99.7 0.285703 + 0.285703i 4.66660 3.83675i −4.14024 4.14024i 1.33326 + 1.33326i −1.61555 + 1.61555i 2.23898 2.23898i 12.7772 2.36576i
99.8 1.17461 + 1.17461i 1.35405 1.24058i 1.57408 + 1.57408i 1.59048 + 1.59048i 8.90433 8.90433i 6.15564 6.15564i −7.16655 3.69787i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 70.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.d odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.d.e 24
13.b even 2 1 inner 169.3.d.e 24
13.c even 3 2 169.3.f.g 48
13.d odd 4 2 inner 169.3.d.e 24
13.e even 6 2 169.3.f.g 48
13.f odd 12 4 169.3.f.g 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.d.e 24 1.a even 1 1 trivial
169.3.d.e 24 13.b even 2 1 inner
169.3.d.e 24 13.d odd 4 2 inner
169.3.f.g 48 13.c even 3 2
169.3.f.g 48 13.e even 6 2
169.3.f.g 48 13.f odd 12 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + 229T_{2}^{20} + 17518T_{2}^{16} + 540679T_{2}^{12} + 6695460T_{2}^{8} + 26716280T_{2}^{4} + 707281$$ acting on $$S_{3}^{\mathrm{new}}(169, [\chi])$$.