# Properties

 Label 1045.2.f.b Level $1045$ Weight $2$ Character orbit 1045.f Analytic conductor $8.344$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(626,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.626");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.f (of order $$2$$, degree $$1$$, 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: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 40 q^{4} + 40 q^{5} - 28 q^{9}+O(q^{10})$$ 40 * q + 40 * q^4 + 40 * q^5 - 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 40 q^{4} + 40 q^{5} - 28 q^{9} - 4 q^{11} + 48 q^{16} + 40 q^{20} + 24 q^{23} + 40 q^{25} - 24 q^{26} + 24 q^{36} - 28 q^{38} - 60 q^{42} - 48 q^{44} - 28 q^{45} - 36 q^{49} - 4 q^{55} - 36 q^{58} - 40 q^{64} - 28 q^{66} + 8 q^{77} + 48 q^{80} + 80 q^{81} + 8 q^{82} + 4 q^{92} + 56 q^{93} - 16 q^{99}+O(q^{100})$$ 40 * q + 40 * q^4 + 40 * q^5 - 28 * q^9 - 4 * q^11 + 48 * q^16 + 40 * q^20 + 24 * q^23 + 40 * q^25 - 24 * q^26 + 24 * q^36 - 28 * q^38 - 60 * q^42 - 48 * q^44 - 28 * q^45 - 36 * q^49 - 4 * q^55 - 36 * q^58 - 40 * q^64 - 28 * q^66 + 8 * q^77 + 48 * q^80 + 80 * q^81 + 8 * q^82 + 4 * q^92 + 56 * q^93 - 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
626.1 −2.69018 1.09158i 5.23709 1.00000 2.93656i 4.34570i −8.70838 1.80845 −2.69018
626.2 −2.69018 1.09158i 5.23709 1.00000 2.93656i 4.34570i −8.70838 1.80845 −2.69018
626.3 −2.36088 2.84862i 3.57375 1.00000 6.72526i 0.816397i −3.71543 −5.11466 −2.36088
626.4 −2.36088 2.84862i 3.57375 1.00000 6.72526i 0.816397i −3.71543 −5.11466 −2.36088
626.5 −2.34872 1.68631i 3.51646 1.00000 3.96066i 0.129031i −3.56174 0.156361 −2.34872
626.6 −2.34872 1.68631i 3.51646 1.00000 3.96066i 0.129031i −3.56174 0.156361 −2.34872
626.7 −2.17500 1.39639i 2.73061 1.00000 3.03714i 0.972912i −1.58909 1.05010 −2.17500
626.8 −2.17500 1.39639i 2.73061 1.00000 3.03714i 0.972912i −1.58909 1.05010 −2.17500
626.9 −1.83534 0.556027i 1.36849 1.00000 1.02050i 4.08954i 1.15904 2.69083 −1.83534
626.10 −1.83534 0.556027i 1.36849 1.00000 1.02050i 4.08954i 1.15904 2.69083 −1.83534
626.11 −1.23717 3.34487i −0.469419 1.00000 4.13816i 3.14849i 3.05508 −8.18815 −1.23717
626.12 −1.23717 3.34487i −0.469419 1.00000 4.13816i 3.14849i 3.05508 −8.18815 −1.23717
626.13 −1.10652 0.279240i −0.775612 1.00000 0.308985i 2.99424i 3.07127 2.92202 −1.10652
626.14 −1.10652 0.279240i −0.775612 1.00000 0.308985i 2.99424i 3.07127 2.92202 −1.10652
626.15 −0.863466 0.924425i −1.25443 1.00000 0.798209i 0.285490i 2.81009 2.14544 −0.863466
626.16 −0.863466 0.924425i −1.25443 1.00000 0.798209i 0.285490i 2.81009 2.14544 −0.863466
626.17 −0.233551 2.00145i −1.94545 1.00000 0.467439i 1.43314i 0.921463 −1.00579 −0.233551
626.18 −0.233551 2.00145i −1.94545 1.00000 0.467439i 1.43314i 0.921463 −1.00579 −0.233551
626.19 −0.136022 2.54256i −1.98150 1.00000 0.345844i 4.55489i 0.541571 −3.46461 −0.136022
626.20 −0.136022 2.54256i −1.98150 1.00000 0.345844i 4.55489i 0.541571 −3.46461 −0.136022
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 626.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.b odd 2 1 inner
209.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.f.b 40
11.b odd 2 1 inner 1045.2.f.b 40
19.b odd 2 1 inner 1045.2.f.b 40
209.d even 2 1 inner 1045.2.f.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.f.b 40 1.a even 1 1 trivial
1045.2.f.b 40 11.b odd 2 1 inner
1045.2.f.b 40 19.b odd 2 1 inner
1045.2.f.b 40 209.d even 2 1 inner