# Properties

 Label 740.2.x.a Level $740$ Weight $2$ Character orbit 740.x Analytic conductor $5.909$ Analytic rank $0$ Dimension $28$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(101,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("740.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.x (of order $$6$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 2 q^{7} - 18 q^{9}+O(q^{10})$$ 28 * q - 2 * q^7 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 2 q^{7} - 18 q^{9} + 4 q^{11} - 6 q^{13} - 12 q^{19} + 6 q^{21} + 14 q^{25} + 2 q^{33} + 6 q^{35} + 22 q^{37} - 6 q^{39} - 14 q^{41} + 44 q^{47} - 22 q^{49} - 8 q^{53} + 12 q^{55} - 54 q^{57} + 16 q^{63} + 6 q^{65} - 12 q^{67} + 18 q^{69} - 26 q^{71} - 4 q^{73} + 14 q^{77} - 54 q^{79} - 22 q^{81} - 28 q^{83} - 4 q^{85} - 48 q^{87} + 36 q^{89} + 36 q^{91} - 18 q^{93} + 12 q^{95} - 8 q^{99}+O(q^{100})$$ 28 * q - 2 * q^7 - 18 * q^9 + 4 * q^11 - 6 * q^13 - 12 * q^19 + 6 * q^21 + 14 * q^25 + 2 * q^33 + 6 * q^35 + 22 * q^37 - 6 * q^39 - 14 * q^41 + 44 * q^47 - 22 * q^49 - 8 * q^53 + 12 * q^55 - 54 * q^57 + 16 * q^63 + 6 * q^65 - 12 * q^67 + 18 * q^69 - 26 * q^71 - 4 * q^73 + 14 * q^77 - 54 * q^79 - 22 * q^81 - 28 * q^83 - 4 * q^85 - 48 * q^87 + 36 * q^89 + 36 * q^91 - 18 * q^93 + 12 * q^95 - 8 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 0 −1.45802 2.52536i 0 −0.866025 + 0.500000i 0 −0.977514 1.69310i 0 −2.75162 + 4.76595i 0
101.2 0 −1.43328 2.48251i 0 0.866025 0.500000i 0 1.51361 + 2.62165i 0 −2.60857 + 4.51817i 0
101.3 0 −1.39306 2.41285i 0 0.866025 0.500000i 0 −1.45145 2.51399i 0 −2.38122 + 4.12440i 0
101.4 0 −0.924212 1.60078i 0 −0.866025 + 0.500000i 0 1.23057 + 2.13141i 0 −0.208337 + 0.360850i 0
101.5 0 −0.579869 1.00436i 0 −0.866025 + 0.500000i 0 −0.997854 1.72833i 0 0.827505 1.43328i 0
101.6 0 −0.376553 0.652209i 0 0.866025 0.500000i 0 0.571101 + 0.989176i 0 1.21642 2.10689i 0
101.7 0 −0.242376 0.419807i 0 −0.866025 + 0.500000i 0 2.17692 + 3.77053i 0 1.38251 2.39457i 0
101.8 0 0.0959198 + 0.166138i 0 0.866025 0.500000i 0 −1.15672 2.00350i 0 1.48160 2.56620i 0
101.9 0 0.559168 + 0.968507i 0 −0.866025 + 0.500000i 0 −1.54298 2.67252i 0 0.874663 1.51496i 0
101.10 0 0.764283 + 1.32378i 0 0.866025 0.500000i 0 2.10769 + 3.65062i 0 0.331743 0.574597i 0
101.11 0 0.926709 + 1.60511i 0 −0.866025 + 0.500000i 0 −1.34484 2.32933i 0 −0.217581 + 0.376861i 0
101.12 0 1.05756 + 1.83175i 0 0.866025 0.500000i 0 −2.39121 4.14170i 0 −0.736869 + 1.27629i 0
101.13 0 1.28513 + 2.22590i 0 0.866025 0.500000i 0 1.17302 + 2.03172i 0 −1.80310 + 3.12305i 0
101.14 0 1.71860 + 2.97669i 0 −0.866025 + 0.500000i 0 0.0896750 + 0.155322i 0 −4.40714 + 7.63339i 0
381.1 0 −1.45802 + 2.52536i 0 −0.866025 0.500000i 0 −0.977514 + 1.69310i 0 −2.75162 4.76595i 0
381.2 0 −1.43328 + 2.48251i 0 0.866025 + 0.500000i 0 1.51361 2.62165i 0 −2.60857 4.51817i 0
381.3 0 −1.39306 + 2.41285i 0 0.866025 + 0.500000i 0 −1.45145 + 2.51399i 0 −2.38122 4.12440i 0
381.4 0 −0.924212 + 1.60078i 0 −0.866025 0.500000i 0 1.23057 2.13141i 0 −0.208337 0.360850i 0
381.5 0 −0.579869 + 1.00436i 0 −0.866025 0.500000i 0 −0.997854 + 1.72833i 0 0.827505 + 1.43328i 0
381.6 0 −0.376553 + 0.652209i 0 0.866025 + 0.500000i 0 0.571101 0.989176i 0 1.21642 + 2.10689i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.x.a 28
37.e even 6 1 inner 740.2.x.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.x.a 28 1.a even 1 1 trivial
740.2.x.a 28 37.e even 6 1 inner

## Hecke kernels

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