# Properties

 Label 740.2.bb.a Level $740$ Weight $2$ Character orbit 740.bb Analytic conductor $5.909$ Analytic rank $0$ Dimension $36$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(269,740)]

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, 3, 4]))

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

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

chi := DirichletCharacter("740.269");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.bb (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: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 2 q^{5} + 12 q^{9}+O(q^{10})$$ 36 * q - 2 * q^5 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 2 q^{5} + 12 q^{9} - 12 q^{11} + 3 q^{15} - 10 q^{19} + 4 q^{21} + 8 q^{25} + 4 q^{29} + 40 q^{31} - q^{35} + 4 q^{39} + 22 q^{41} + 22 q^{45} + 24 q^{49} + 40 q^{51} - 6 q^{55} - 14 q^{59} + 12 q^{61} + 9 q^{65} + 2 q^{69} - 24 q^{71} - 62 q^{75} - 12 q^{79} + 26 q^{81} - 24 q^{85} + 30 q^{91} + 9 q^{95} - 16 q^{99}+O(q^{100})$$ 36 * q - 2 * q^5 + 12 * q^9 - 12 * q^11 + 3 * q^15 - 10 * q^19 + 4 * q^21 + 8 * q^25 + 4 * q^29 + 40 * q^31 - q^35 + 4 * q^39 + 22 * q^41 + 22 * q^45 + 24 * q^49 + 40 * q^51 - 6 * q^55 - 14 * q^59 + 12 * q^61 + 9 * q^65 + 2 * q^69 - 24 * q^71 - 62 * q^75 - 12 * q^79 + 26 * q^81 - 24 * q^85 + 30 * q^91 + 9 * q^95 - 16 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1 0 −2.52616 + 1.45848i 0 −1.70568 + 1.44591i 0 1.93278 1.11589i 0 2.75432 4.77062i 0
269.2 0 −2.34777 + 1.35549i 0 2.18160 + 0.490514i 0 −3.51625 + 2.03011i 0 2.17468 3.76666i 0
269.3 0 −2.31618 + 1.33725i 0 −1.42198 1.72568i 0 −0.105556 + 0.0609429i 0 2.07646 3.59654i 0
269.4 0 −1.69975 + 0.981350i 0 2.05545 + 0.880424i 0 4.10335 2.36907i 0 0.426096 0.738020i 0
269.5 0 −1.39171 + 0.803502i 0 −2.13362 + 0.669088i 0 −4.16585 + 2.40515i 0 −0.208768 + 0.361596i 0
269.6 0 −1.00394 + 0.579623i 0 1.49599 1.66193i 0 −0.211890 + 0.122335i 0 −0.828074 + 1.43427i 0
269.7 0 −0.954843 + 0.551279i 0 0.369612 2.20531i 0 −0.195612 + 0.112936i 0 −0.892183 + 1.54531i 0
269.8 0 −0.755229 + 0.436031i 0 0.0540287 + 2.23542i 0 0.453259 0.261689i 0 −1.11975 + 1.93947i 0
269.9 0 −0.419315 + 0.242092i 0 −2.16774 0.548549i 0 2.37885 1.37343i 0 −1.38278 + 2.39505i 0
269.10 0 0.419315 0.242092i 0 0.608812 2.15159i 0 −2.37885 + 1.37343i 0 −1.38278 + 2.39505i 0
269.11 0 0.755229 0.436031i 0 1.90891 + 1.16450i 0 −0.453259 + 0.261689i 0 −1.11975 + 1.93947i 0
269.12 0 0.954843 0.551279i 0 −2.09466 0.782561i 0 0.195612 0.112936i 0 −0.892183 + 1.54531i 0
269.13 0 1.00394 0.579623i 0 −2.18727 + 0.464599i 0 0.211890 0.122335i 0 −0.828074 + 1.43427i 0
269.14 0 1.39171 0.803502i 0 1.64626 1.51322i 0 4.16585 2.40515i 0 −0.208768 + 0.361596i 0
269.15 0 1.69975 0.981350i 0 −0.265253 + 2.22028i 0 −4.10335 + 2.36907i 0 0.426096 0.738020i 0
269.16 0 2.31618 1.33725i 0 −0.783499 2.09431i 0 0.105556 0.0609429i 0 2.07646 3.59654i 0
269.17 0 2.34777 1.35549i 0 −0.666005 + 2.13458i 0 3.51625 2.03011i 0 2.17468 3.76666i 0
269.18 0 2.52616 1.45848i 0 2.10504 0.754203i 0 −1.93278 + 1.11589i 0 2.75432 4.77062i 0
729.1 0 −2.52616 1.45848i 0 −1.70568 1.44591i 0 1.93278 + 1.11589i 0 2.75432 + 4.77062i 0
729.2 0 −2.34777 1.35549i 0 2.18160 0.490514i 0 −3.51625 2.03011i 0 2.17468 + 3.76666i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n 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.bb.a 36
5.b even 2 1 inner 740.2.bb.a 36
37.c even 3 1 inner 740.2.bb.a 36
185.n even 6 1 inner 740.2.bb.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.bb.a 36 1.a even 1 1 trivial
740.2.bb.a 36 5.b even 2 1 inner
740.2.bb.a 36 37.c even 3 1 inner
740.2.bb.a 36 185.n even 6 1 inner

## Hecke kernels

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