# Properties

 Label 740.2.bf.a Level $740$ Weight $2$ Character orbit 740.bf Analytic conductor $5.909$ Analytic rank $0$ Dimension $76$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(97,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

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

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

chi := DirichletCharacter("740.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.bf (of order $$12$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$76$$ Relative dimension: $$19$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$76 q - 2 q^{3}+O(q^{10})$$ 76 * q - 2 * q^3 $$\operatorname{Tr}(f)(q) =$$ $$76 q - 2 q^{3} + 8 q^{13} + 2 q^{15} + 12 q^{19} - 4 q^{23} + 2 q^{25} + 28 q^{27} - 6 q^{29} + 16 q^{31} - 6 q^{33} + 20 q^{35} - 22 q^{37} + 8 q^{39} + 54 q^{41} - 16 q^{43} + 38 q^{45} + 8 q^{47} - 36 q^{49} - 24 q^{53} + 18 q^{55} - 16 q^{59} - 28 q^{61} - 6 q^{65} - 34 q^{67} + 64 q^{69} - 8 q^{71} + 4 q^{73} + 24 q^{75} - 2 q^{77} - 40 q^{79} + 2 q^{81} - 62 q^{83} + 108 q^{87} - 2 q^{89} - 20 q^{91} - 30 q^{93} - 70 q^{95}+O(q^{100})$$ 76 * q - 2 * q^3 + 8 * q^13 + 2 * q^15 + 12 * q^19 - 4 * q^23 + 2 * q^25 + 28 * q^27 - 6 * q^29 + 16 * q^31 - 6 * q^33 + 20 * q^35 - 22 * q^37 + 8 * q^39 + 54 * q^41 - 16 * q^43 + 38 * q^45 + 8 * q^47 - 36 * q^49 - 24 * q^53 + 18 * q^55 - 16 * q^59 - 28 * q^61 - 6 * q^65 - 34 * q^67 + 64 * q^69 - 8 * q^71 + 4 * q^73 + 24 * q^75 - 2 * q^77 - 40 * q^79 + 2 * q^81 - 62 * q^83 + 108 * q^87 - 2 * q^89 - 20 * q^91 - 30 * q^93 - 70 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 0 −0.804236 + 3.00145i 0 0.433448 2.19366i 0 −0.0415533 + 0.155079i 0 −5.76383 3.32775i 0
97.2 0 −0.701381 + 2.61759i 0 1.90653 + 1.16840i 0 0.951054 3.54938i 0 −3.76176 2.17185i 0
97.3 0 −0.699874 + 2.61196i 0 −2.23588 + 0.0286679i 0 −1.17325 + 4.37863i 0 −3.73445 2.15609i 0
97.4 0 −0.603899 + 2.25378i 0 −1.48644 + 1.67048i 0 1.03426 3.85990i 0 −2.11677 1.22212i 0
97.5 0 −0.564174 + 2.10553i 0 1.58691 1.57535i 0 −0.320323 + 1.19546i 0 −1.51687 0.875765i 0
97.6 0 −0.364166 + 1.35909i 0 −1.96429 1.06845i 0 0.331427 1.23690i 0 0.883578 + 0.510134i 0
97.7 0 −0.216951 + 0.809671i 0 −0.720974 + 2.11665i 0 −0.208444 + 0.777924i 0 1.98958 + 1.14868i 0
97.8 0 −0.157156 + 0.586512i 0 1.83034 + 1.28447i 0 −0.887088 + 3.31066i 0 2.27878 + 1.31565i 0
97.9 0 −0.0867145 + 0.323623i 0 1.97109 1.05584i 0 0.189910 0.708753i 0 2.50086 + 1.44387i 0
97.10 0 0.0835144 0.311680i 0 0.0788587 2.23468i 0 −0.686223 + 2.56102i 0 2.50791 + 1.44794i 0
97.11 0 0.266873 0.995984i 0 0.861533 + 2.06343i 0 0.954075 3.56066i 0 1.67731 + 0.968397i 0
97.12 0 0.358039 1.33622i 0 −2.23587 + 0.0296847i 0 −0.151686 + 0.566098i 0 0.940782 + 0.543161i 0
97.13 0 0.368630 1.37574i 0 0.199835 2.22712i 0 1.19522 4.46063i 0 0.841290 + 0.485719i 0
97.14 0 0.377301 1.40811i 0 −0.531239 2.17205i 0 −1.28827 + 4.80788i 0 0.757669 + 0.437440i 0
97.15 0 0.405966 1.51509i 0 −1.67270 + 1.48394i 0 0.553968 2.06744i 0 0.467396 + 0.269851i 0
97.16 0 0.439069 1.63863i 0 1.82834 + 1.28731i 0 0.548032 2.04528i 0 0.105753 + 0.0610568i 0
97.17 0 0.670034 2.50060i 0 −0.737701 + 2.11088i 0 −1.21971 + 4.55201i 0 −3.20599 1.85098i 0
97.18 0 0.751881 2.80606i 0 2.02352 0.951503i 0 −0.118383 + 0.441812i 0 −4.71056 2.71964i 0
97.19 0 0.843267 3.14712i 0 −2.00134 0.997312i 0 0.336984 1.25764i 0 −6.59517 3.80772i 0
273.1 0 −2.89417 0.775491i 0 −0.787145 2.09294i 0 1.23641 + 0.331294i 0 5.17676 + 2.98880i 0
See all 76 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.19 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bf.a 76
5.c odd 4 1 740.2.bi.a yes 76
37.g odd 12 1 740.2.bi.a yes 76
185.p even 12 1 inner 740.2.bf.a 76

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bf.a 76 1.a even 1 1 trivial
740.2.bf.a 76 185.p even 12 1 inner
740.2.bi.a yes 76 5.c odd 4 1
740.2.bi.a yes 76 37.g odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(740, [\chi])$$.