# Properties

 Label 165.6.f.a Level $165$ Weight $6$ Character orbit 165.f Analytic conductor $26.463$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,6,Mod(131,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("165.131");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 165.f (of order $$2$$, degree $$1$$, 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: $$26.4633302691$$ Analytic rank: $$0$$ Dimension: $$80$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 44 q^{3} + 1280 q^{4} - 352 q^{9}+O(q^{10})$$ 80 * q + 44 * q^3 + 1280 * q^4 - 352 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 44 q^{3} + 1280 q^{4} - 352 q^{9} + 2112 q^{12} + 1100 q^{15} + 13792 q^{16} - 9892 q^{22} - 50000 q^{25} - 10780 q^{27} + 2112 q^{31} + 6316 q^{33} + 69560 q^{34} - 17268 q^{36} - 7456 q^{37} - 100712 q^{42} + 61352 q^{48} - 233408 q^{49} - 15800 q^{55} - 93728 q^{58} + 62700 q^{60} + 212400 q^{64} + 203724 q^{66} + 182072 q^{67} - 122584 q^{69} + 6600 q^{70} - 27500 q^{75} - 489128 q^{78} + 194872 q^{81} - 237544 q^{82} - 641716 q^{88} + 168272 q^{91} + 433336 q^{93} + 949008 q^{97} + 328952 q^{99}+O(q^{100})$$ 80 * q + 44 * q^3 + 1280 * q^4 - 352 * q^9 + 2112 * q^12 + 1100 * q^15 + 13792 * q^16 - 9892 * q^22 - 50000 * q^25 - 10780 * q^27 + 2112 * q^31 + 6316 * q^33 + 69560 * q^34 - 17268 * q^36 - 7456 * q^37 - 100712 * q^42 + 61352 * q^48 - 233408 * q^49 - 15800 * q^55 - 93728 * q^58 + 62700 * q^60 + 212400 * q^64 + 203724 * q^66 + 182072 * q^67 - 122584 * q^69 + 6600 * q^70 - 27500 * q^75 - 489128 * q^78 + 194872 * q^81 - 237544 * q^82 - 641716 * q^88 + 168272 * q^91 + 433336 * q^93 + 949008 * q^97 + 328952 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −10.9622 −2.76318 15.3416i 88.1691 25.0000i 30.2904 + 168.177i 90.0333i −615.734 −227.730 + 84.7831i 274.054i
131.2 −10.9622 −2.76318 + 15.3416i 88.1691 25.0000i 30.2904 168.177i 90.0333i −615.734 −227.730 84.7831i 274.054i
131.3 −10.5656 14.6766 + 5.25334i 79.6325 25.0000i −155.067 55.5049i 67.9017i −503.268 187.805 + 154.202i 264.141i
131.4 −10.5656 14.6766 5.25334i 79.6325 25.0000i −155.067 + 55.5049i 67.9017i −503.268 187.805 154.202i 264.141i
131.5 −9.99646 −15.1349 + 3.73284i 67.9292 25.0000i 151.296 37.3152i 75.9052i −359.165 215.132 112.993i 249.911i
131.6 −9.99646 −15.1349 3.73284i 67.9292 25.0000i 151.296 + 37.3152i 75.9052i −359.165 215.132 + 112.993i 249.911i
131.7 −9.78187 5.46056 14.6008i 63.6850 25.0000i −53.4145 + 142.823i 214.687i −309.938 −183.364 159.457i 244.547i
131.8 −9.78187 5.46056 + 14.6008i 63.6850 25.0000i −53.4145 142.823i 214.687i −309.938 −183.364 + 159.457i 244.547i
131.9 −9.43613 12.0254 + 9.91910i 57.0405 25.0000i −113.474 93.5979i 156.233i −236.286 46.2229 + 238.563i 235.903i
131.10 −9.43613 12.0254 9.91910i 57.0405 25.0000i −113.474 + 93.5979i 156.233i −236.286 46.2229 238.563i 235.903i
131.11 −8.20151 −3.15142 15.2666i 35.2648 25.0000i 25.8464 + 125.209i 179.318i −26.7763 −223.137 + 96.2227i 205.038i
131.12 −8.20151 −3.15142 + 15.2666i 35.2648 25.0000i 25.8464 125.209i 179.318i −26.7763 −223.137 96.2227i 205.038i
131.13 −7.77066 −12.8111 8.88121i 28.3832 25.0000i 99.5507 + 69.0129i 223.165i 28.1048 85.2482 + 227.556i 194.267i
131.14 −7.77066 −12.8111 + 8.88121i 28.3832 25.0000i 99.5507 69.0129i 223.165i 28.1048 85.2482 227.556i 194.267i
131.15 −7.54491 −0.606932 + 15.5766i 24.9257 25.0000i 4.57925 117.524i 74.1145i 53.3749 −242.263 18.9079i 188.623i
131.16 −7.54491 −0.606932 15.5766i 24.9257 25.0000i 4.57925 + 117.524i 74.1145i 53.3749 −242.263 + 18.9079i 188.623i
131.17 −7.04255 −14.9021 + 4.57456i 17.5974 25.0000i 104.949 32.2165i 17.4527i 101.431 201.147 136.341i 176.064i
131.18 −7.04255 −14.9021 4.57456i 17.5974 25.0000i 104.949 + 32.2165i 17.4527i 101.431 201.147 + 136.341i 176.064i
131.19 −6.89239 14.9703 + 4.34640i 15.5051 25.0000i −103.181 29.9571i 243.497i 113.689 205.218 + 130.134i 172.310i
131.20 −6.89239 14.9703 4.34640i 15.5051 25.0000i −103.181 + 29.9571i 243.497i 113.689 205.218 130.134i 172.310i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.80 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.6.f.a 80
3.b odd 2 1 inner 165.6.f.a 80
11.b odd 2 1 inner 165.6.f.a 80
33.d even 2 1 inner 165.6.f.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.f.a 80 1.a even 1 1 trivial
165.6.f.a 80 3.b odd 2 1 inner
165.6.f.a 80 11.b odd 2 1 inner
165.6.f.a 80 33.d even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(165, [\chi])$$.