# Properties

 Label 740.2.ba.a Level $740$ Weight $2$ Character orbit 740.ba Analytic conductor $5.909$ Analytic rank $0$ Dimension $40$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(249,740)]

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, 1]))

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

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

chi := DirichletCharacter("740.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.ba (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: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 3 q^{5} + 22 q^{9}+O(q^{10})$$ 40 * q - 3 * q^5 + 22 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 3 q^{5} + 22 q^{9} - 12 q^{11} - 3 q^{15} - 6 q^{19} + 12 q^{21} + 5 q^{25} + 21 q^{35} + 12 q^{39} + 32 q^{41} + 22 q^{49} + 30 q^{55} + 42 q^{59} + 42 q^{61} - 7 q^{65} - 54 q^{69} + 16 q^{71} - 50 q^{75} + 12 q^{79} - 56 q^{81} - 10 q^{85} + 30 q^{89} - 6 q^{91} - 7 q^{95} - 40 q^{99}+O(q^{100})$$ 40 * q - 3 * q^5 + 22 * q^9 - 12 * q^11 - 3 * q^15 - 6 * q^19 + 12 * q^21 + 5 * q^25 + 21 * q^35 + 12 * q^39 + 32 * q^41 + 22 * q^49 + 30 * q^55 + 42 * q^59 + 42 * q^61 - 7 * q^65 - 54 * q^69 + 16 * q^71 - 50 * q^75 + 12 * q^79 - 56 * q^81 - 10 * q^85 + 30 * q^89 - 6 * q^91 - 7 * q^95 - 40 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1 0 −2.92356 + 1.68792i 0 −1.87111 1.22432i 0 −3.35180 + 1.93516i 0 4.19812 7.27135i 0
249.2 0 −2.49411 + 1.43997i 0 2.16123 + 0.573672i 0 −0.134772 + 0.0778107i 0 2.64704 4.58481i 0
249.3 0 −2.10840 + 1.21728i 0 −0.272013 + 2.21946i 0 −0.549920 + 0.317496i 0 1.46355 2.53495i 0
249.4 0 −2.10722 + 1.21660i 0 −1.53139 + 1.62937i 0 4.29249 2.47827i 0 1.46025 2.52922i 0
249.5 0 −2.06209 + 1.19055i 0 2.05385 0.884133i 0 −0.155067 + 0.0895280i 0 1.33480 2.31194i 0
249.6 0 −1.27685 + 0.737190i 0 −2.02720 0.943639i 0 1.28215 0.740252i 0 −0.413101 + 0.715511i 0
249.7 0 −0.844980 + 0.487850i 0 −0.187442 2.22820i 0 −2.52014 + 1.45501i 0 −1.02401 + 1.77363i 0
249.8 0 −0.676419 + 0.390531i 0 −1.31592 + 1.80786i 0 −2.25770 + 1.30348i 0 −1.19497 + 2.06975i 0
249.9 0 −0.200277 + 0.115630i 0 1.15125 + 1.91693i 0 −0.567064 + 0.327394i 0 −1.47326 + 2.55176i 0
249.10 0 −0.0486079 + 0.0280638i 0 1.58113 1.58115i 0 4.16312 2.40358i 0 −1.49842 + 2.59535i 0
249.11 0 0.0486079 0.0280638i 0 2.15988 0.578723i 0 −4.16312 + 2.40358i 0 −1.49842 + 2.59535i 0
249.12 0 0.200277 0.115630i 0 −1.08448 1.95548i 0 0.567064 0.327394i 0 −1.47326 + 2.55176i 0
249.13 0 0.676419 0.390531i 0 −2.22361 + 0.235690i 0 2.25770 1.30348i 0 −1.19497 + 2.06975i 0
249.14 0 0.844980 0.487850i 0 1.83595 + 1.27643i 0 2.52014 1.45501i 0 −1.02401 + 1.77363i 0
249.15 0 1.27685 0.737190i 0 −0.196386 + 2.22743i 0 −1.28215 + 0.740252i 0 −0.413101 + 0.715511i 0
249.16 0 2.06209 1.19055i 0 1.79261 1.33662i 0 0.155067 0.0895280i 0 1.33480 2.31194i 0
249.17 0 2.10722 1.21660i 0 −2.17677 + 0.511535i 0 −4.29249 + 2.47827i 0 1.46025 2.52922i 0
249.18 0 2.10840 1.21728i 0 −2.05812 0.874161i 0 0.549920 0.317496i 0 1.46355 2.53495i 0
249.19 0 2.49411 1.43997i 0 0.583799 2.15851i 0 0.134772 0.0778107i 0 2.64704 4.58481i 0
249.20 0 2.92356 1.68792i 0 0.124735 + 2.23259i 0 3.35180 1.93516i 0 4.19812 7.27135i 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 249.20 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.e even 6 1 inner
185.l 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.ba.a 40
5.b even 2 1 inner 740.2.ba.a 40
37.e even 6 1 inner 740.2.ba.a 40
185.l even 6 1 inner 740.2.ba.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.ba.a 40 1.a even 1 1 trivial
740.2.ba.a 40 5.b even 2 1 inner
740.2.ba.a 40 37.e even 6 1 inner
740.2.ba.a 40 185.l even 6 1 inner

## Hecke kernels

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