# Properties

 Label 740.2.bc.d Level $740$ Weight $2$ Character orbit 740.bc Analytic conductor $5.909$ Analytic rank $0$ Dimension $36$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(81,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 16]))

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

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

chi := DirichletCharacter("740.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.bc (of order $$9$$, degree $$6$$, 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: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 12 q^{7}+O(q^{10})$$ 36 * q + 12 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 12 q^{7} - 6 q^{11} - 12 q^{13} - 6 q^{17} + 15 q^{19} - 24 q^{23} + 12 q^{27} - 3 q^{29} + 30 q^{31} + 18 q^{33} - 3 q^{35} + 36 q^{37} - 6 q^{39} - 9 q^{41} + 12 q^{43} + 21 q^{45} - 21 q^{47} - 30 q^{49} - 24 q^{51} - 3 q^{53} + 3 q^{55} + 3 q^{57} - 27 q^{59} + 30 q^{61} - 27 q^{63} + 12 q^{65} - 21 q^{67} - 45 q^{69} + 24 q^{71} - 48 q^{73} - 6 q^{75} - 42 q^{77} + 108 q^{81} - 9 q^{83} + 3 q^{85} - 81 q^{87} - 60 q^{89} - 66 q^{91} + 75 q^{93} + 12 q^{95} - 36 q^{97} + 108 q^{99}+O(q^{100})$$ 36 * q + 12 * q^7 - 6 * q^11 - 12 * q^13 - 6 * q^17 + 15 * q^19 - 24 * q^23 + 12 * q^27 - 3 * q^29 + 30 * q^31 + 18 * q^33 - 3 * q^35 + 36 * q^37 - 6 * q^39 - 9 * q^41 + 12 * q^43 + 21 * q^45 - 21 * q^47 - 30 * q^49 - 24 * q^51 - 3 * q^53 + 3 * q^55 + 3 * q^57 - 27 * q^59 + 30 * q^61 - 27 * q^63 + 12 * q^65 - 21 * q^67 - 45 * q^69 + 24 * q^71 - 48 * q^73 - 6 * q^75 - 42 * q^77 + 108 * q^81 - 9 * q^83 + 3 * q^85 - 81 * q^87 - 60 * q^89 - 66 * q^91 + 75 * q^93 + 12 * q^95 - 36 * q^97 + 108 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1 0 −2.51079 0.913852i 0 −0.173648 0.984808i 0 −0.774141 4.39037i 0 3.17079 + 2.66061i 0
81.2 0 −2.05044 0.746301i 0 −0.173648 0.984808i 0 0.299868 + 1.70063i 0 1.34922 + 1.13213i 0
81.3 0 −0.210487 0.0766109i 0 −0.173648 0.984808i 0 −0.202628 1.14916i 0 −2.25970 1.89611i 0
81.4 0 1.26990 + 0.462206i 0 −0.173648 0.984808i 0 −0.591039 3.35195i 0 −0.899118 0.754450i 0
81.5 0 2.06789 + 0.752652i 0 −0.173648 0.984808i 0 0.773432 + 4.38635i 0 1.41157 + 1.18444i 0
81.6 0 2.37361 + 0.863925i 0 −0.173648 0.984808i 0 −0.150921 0.855916i 0 2.58955 + 2.17289i 0
181.1 0 −0.593287 3.36470i 0 −0.766044 + 0.642788i 0 2.77336 2.32713i 0 −8.15013 + 2.96641i 0
181.2 0 −0.271716 1.54098i 0 −0.766044 + 0.642788i 0 0.504138 0.423022i 0 0.518293 0.188643i 0
181.3 0 −0.116920 0.663085i 0 −0.766044 + 0.642788i 0 −3.07084 + 2.57674i 0 2.39307 0.871005i 0
181.4 0 0.0765251 + 0.433995i 0 −0.766044 + 0.642788i 0 0.108070 0.0906818i 0 2.63658 0.959637i 0
181.5 0 0.223103 + 1.26528i 0 −0.766044 + 0.642788i 0 3.02839 2.54112i 0 1.26792 0.461486i 0
181.6 0 0.508647 + 2.88468i 0 −0.766044 + 0.642788i 0 −0.0561339 + 0.0471019i 0 −5.24358 + 1.90851i 0
201.1 0 −2.51079 + 0.913852i 0 −0.173648 + 0.984808i 0 −0.774141 + 4.39037i 0 3.17079 2.66061i 0
201.2 0 −2.05044 + 0.746301i 0 −0.173648 + 0.984808i 0 0.299868 1.70063i 0 1.34922 1.13213i 0
201.3 0 −0.210487 + 0.0766109i 0 −0.173648 + 0.984808i 0 −0.202628 + 1.14916i 0 −2.25970 + 1.89611i 0
201.4 0 1.26990 0.462206i 0 −0.173648 + 0.984808i 0 −0.591039 + 3.35195i 0 −0.899118 + 0.754450i 0
201.5 0 2.06789 0.752652i 0 −0.173648 + 0.984808i 0 0.773432 4.38635i 0 1.41157 1.18444i 0
201.6 0 2.37361 0.863925i 0 −0.173648 + 0.984808i 0 −0.150921 + 0.855916i 0 2.58955 2.17289i 0
441.1 0 −2.31219 + 1.94016i 0 0.939693 + 0.342020i 0 0.459860 + 0.167375i 0 1.06107 6.01760i 0
441.2 0 −1.32324 + 1.11033i 0 0.939693 + 0.342020i 0 −3.84213 1.39842i 0 −0.00281124 + 0.0159433i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.bc.d 36
37.f even 9 1 inner 740.2.bc.d 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bc.d 36 1.a even 1 1 trivial
740.2.bc.d 36 37.f even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{36} - 10 T_{3}^{33} - 27 T_{3}^{32} + 15 T_{3}^{31} + 739 T_{3}^{30} - 672 T_{3}^{29} + \cdots + 3560769$$ acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$.