# Properties

 Label 740.2.l.b Level $740$ Weight $2$ Character orbit 740.l Analytic conductor $5.909$ Analytic rank $0$ Dimension $32$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(413,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("740.413");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.l (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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 2 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10})$$ 32 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 2 q^{3} + 4 q^{5} + 4 q^{7} - 8 q^{15} + 10 q^{19} + 8 q^{25} + 26 q^{27} + 12 q^{29} + 40 q^{31} - 28 q^{33} + 30 q^{35} + 6 q^{37} + 38 q^{39} - 14 q^{45} + 4 q^{47} - 26 q^{51} + 8 q^{53} + 2 q^{55} - 16 q^{59} - 8 q^{61} + 18 q^{63} + 4 q^{65} + 18 q^{67} + 12 q^{69} - 4 q^{71} + 86 q^{75} + 4 q^{77} + 32 q^{79} - 76 q^{81} + 6 q^{83} + 4 q^{85} - 60 q^{87} - 14 q^{91} + 18 q^{95} - 20 q^{97} - 56 q^{99}+O(q^{100})$$ 32 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 - 8 * q^15 + 10 * q^19 + 8 * q^25 + 26 * q^27 + 12 * q^29 + 40 * q^31 - 28 * q^33 + 30 * q^35 + 6 * q^37 + 38 * q^39 - 14 * q^45 + 4 * q^47 - 26 * q^51 + 8 * q^53 + 2 * q^55 - 16 * q^59 - 8 * q^61 + 18 * q^63 + 4 * q^65 + 18 * q^67 + 12 * q^69 - 4 * q^71 + 86 * q^75 + 4 * q^77 + 32 * q^79 - 76 * q^81 + 6 * q^83 + 4 * q^85 - 60 * q^87 - 14 * q^91 + 18 * q^95 - 20 * q^97 - 56 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
413.1 0 −2.33096 + 2.33096i 0 0.800627 2.08782i 0 −1.35614 + 1.35614i 0 7.86678i 0
413.2 0 −2.24402 + 2.24402i 0 −0.742314 + 2.10926i 0 2.16004 2.16004i 0 7.07129i 0
413.3 0 −1.43758 + 1.43758i 0 −2.07292 0.838464i 0 2.28736 2.28736i 0 1.13328i 0
413.4 0 −1.39300 + 1.39300i 0 1.82670 + 1.28963i 0 −0.266034 + 0.266034i 0 0.880892i 0
413.5 0 −0.773163 + 0.773163i 0 −0.424269 2.19545i 0 0.328311 0.328311i 0 1.80444i 0
413.6 0 −0.527387 + 0.527387i 0 1.81742 1.30268i 0 −2.46461 + 2.46461i 0 2.44373i 0
413.7 0 −0.387665 + 0.387665i 0 1.52697 1.63352i 0 3.12801 3.12801i 0 2.69943i 0
413.8 0 0.105893 0.105893i 0 1.04190 + 1.97849i 0 3.04554 3.04554i 0 2.97757i 0
413.9 0 0.204117 0.204117i 0 −2.23286 + 0.119792i 0 −0.599893 + 0.599893i 0 2.91667i 0
413.10 0 0.808285 0.808285i 0 −0.906222 2.04420i 0 −3.10505 + 3.10505i 0 1.69335i 0
413.11 0 0.872562 0.872562i 0 −2.11806 + 0.716816i 0 0.815420 0.815420i 0 1.47727i 0
413.12 0 1.01791 1.01791i 0 1.70209 + 1.45014i 0 −0.533882 + 0.533882i 0 0.927718i 0
413.13 0 1.27465 1.27465i 0 1.90893 1.16447i 0 −1.13749 + 1.13749i 0 0.249450i 0
413.14 0 1.73414 1.73414i 0 2.18987 + 0.452188i 0 0.485073 0.485073i 0 3.01448i 0
413.15 0 1.87353 1.87353i 0 −0.169498 2.22963i 0 2.26756 2.26756i 0 4.02023i 0
413.16 0 2.20270 2.20270i 0 −2.14837 0.620077i 0 −3.05423 + 3.05423i 0 6.70378i 0
697.1 0 −2.33096 2.33096i 0 0.800627 + 2.08782i 0 −1.35614 1.35614i 0 7.86678i 0
697.2 0 −2.24402 2.24402i 0 −0.742314 2.10926i 0 2.16004 + 2.16004i 0 7.07129i 0
697.3 0 −1.43758 1.43758i 0 −2.07292 + 0.838464i 0 2.28736 + 2.28736i 0 1.13328i 0
697.4 0 −1.39300 1.39300i 0 1.82670 1.28963i 0 −0.266034 0.266034i 0 0.880892i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.16 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.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.l.b 32
5.c odd 4 1 740.2.o.b yes 32
37.d odd 4 1 740.2.o.b yes 32
185.f even 4 1 inner 740.2.l.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.l.b 32 1.a even 1 1 trivial
740.2.l.b 32 185.f even 4 1 inner
740.2.o.b yes 32 5.c odd 4 1
740.2.o.b yes 32 37.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{32} - 2 T_{3}^{31} + 2 T_{3}^{30} - 6 T_{3}^{29} + 236 T_{3}^{28} - 528 T_{3}^{27} + 602 T_{3}^{26} + \cdots + 3600$$ acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$.