# Properties

 Label 1008.2.v.e Level 1008 Weight 2 Character orbit 1008.v Analytic conductor 8.049 Analytic rank 0 Dimension 40 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1008.v (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 40q^{7} + 48q^{10} - 24q^{13} + 12q^{16} - 32q^{19} - 8q^{22} - 56q^{34} - 8q^{37} + 32q^{43} - 52q^{46} + 40q^{49} - 8q^{52} + 48q^{55} + 56q^{58} - 24q^{61} + 48q^{64} + 48q^{70} - 24q^{76} - 64q^{82} + 64q^{85} - 120q^{88} - 24q^{91} - 128q^{94} + 64q^{97} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
323.1 −1.40729 0.139782i 0 1.96092 + 0.393427i −1.17902 1.17902i 0 1.00000 −2.70459 0.827766i 0 1.49442 + 1.82403i
323.2 −1.39680 + 0.221233i 0 1.90211 0.618037i −2.51504 2.51504i 0 1.00000 −2.52014 + 1.28408i 0 4.06942 + 2.95660i
323.3 −1.37088 0.347400i 0 1.75863 + 0.952488i 0.111394 + 0.111394i 0 1.00000 −2.07997 1.91669i 0 −0.114009 0.191406i
323.4 −1.13420 0.844744i 0 0.572816 + 1.91622i 2.12043 + 2.12043i 0 1.00000 0.969023 2.65725i 0 −0.613769 4.19620i
323.5 −1.11731 + 0.866958i 0 0.496766 1.93732i 0.925496 + 0.925496i 0 1.00000 1.12454 + 2.59527i 0 −1.83643 0.231700i
323.6 −0.890883 1.09833i 0 −0.412656 + 1.95697i −1.65702 1.65702i 0 1.00000 2.51702 1.29019i 0 −0.343745 + 3.29617i
323.7 −0.614527 + 1.27372i 0 −1.24471 1.56547i −2.62814 2.62814i 0 1.00000 2.75887 0.623393i 0 4.96257 1.73245i
323.8 −0.579268 + 1.29014i 0 −1.32890 1.49467i −0.667815 0.667815i 0 1.00000 2.69811 0.848644i 0 1.24842 0.474728i
323.9 −0.351970 1.36971i 0 −1.75223 + 0.964197i 2.96859 + 2.96859i 0 1.00000 1.93741 + 2.06069i 0 3.02127 5.11098i
323.10 −0.153718 1.40583i 0 −1.95274 + 0.432203i 0.0893433 + 0.0893433i 0 1.00000 0.907777 + 2.67879i 0 0.111868 0.139335i
323.11 0.153718 + 1.40583i 0 −1.95274 + 0.432203i −0.0893433 0.0893433i 0 1.00000 −0.907777 2.67879i 0 0.111868 0.139335i
323.12 0.351970 + 1.36971i 0 −1.75223 + 0.964197i −2.96859 2.96859i 0 1.00000 −1.93741 2.06069i 0 3.02127 5.11098i
323.13 0.579268 1.29014i 0 −1.32890 1.49467i 0.667815 + 0.667815i 0 1.00000 −2.69811 + 0.848644i 0 1.24842 0.474728i
323.14 0.614527 1.27372i 0 −1.24471 1.56547i 2.62814 + 2.62814i 0 1.00000 −2.75887 + 0.623393i 0 4.96257 1.73245i
323.15 0.890883 + 1.09833i 0 −0.412656 + 1.95697i 1.65702 + 1.65702i 0 1.00000 −2.51702 + 1.29019i 0 −0.343745 + 3.29617i
323.16 1.11731 0.866958i 0 0.496766 1.93732i −0.925496 0.925496i 0 1.00000 −1.12454 2.59527i 0 −1.83643 0.231700i
323.17 1.13420 + 0.844744i 0 0.572816 + 1.91622i −2.12043 2.12043i 0 1.00000 −0.969023 + 2.65725i 0 −0.613769 4.19620i
323.18 1.37088 + 0.347400i 0 1.75863 + 0.952488i −0.111394 0.111394i 0 1.00000 2.07997 + 1.91669i 0 −0.114009 0.191406i
323.19 1.39680 0.221233i 0 1.90211 0.618037i 2.51504 + 2.51504i 0 1.00000 2.52014 1.28408i 0 4.06942 + 2.95660i
323.20 1.40729 + 0.139782i 0 1.96092 + 0.393427i 1.17902 + 1.17902i 0 1.00000 2.70459 + 0.827766i 0 1.49442 + 1.82403i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 827.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.v.e 40
3.b odd 2 1 inner 1008.2.v.e 40
4.b odd 2 1 4032.2.v.e 40
12.b even 2 1 4032.2.v.e 40
16.e even 4 1 4032.2.v.e 40
16.f odd 4 1 inner 1008.2.v.e 40
48.i odd 4 1 4032.2.v.e 40
48.k even 4 1 inner 1008.2.v.e 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.e 40 1.a even 1 1 trivial
1008.2.v.e 40 3.b odd 2 1 inner
1008.2.v.e 40 16.f odd 4 1 inner
1008.2.v.e 40 48.k even 4 1 inner
4032.2.v.e 40 4.b odd 2 1
4032.2.v.e 40 12.b even 2 1
4032.2.v.e 40 16.e even 4 1
4032.2.v.e 40 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}^{40} + \cdots$$ $$T_{11}^{40} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database