# Properties

 Label 4004.2.e.a Level 4004 Weight 2 Character orbit 4004.e Analytic conductor 31.972 Analytic rank 0 Dimension 48 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4004.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.9721009693$$ Analytic rank: $$0$$ Dimension: $$48$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{7} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 4q^{7} - 48q^{9} + 2q^{11} + 48q^{13} + 8q^{15} + 4q^{17} + 10q^{21} + 4q^{23} - 44q^{25} + 10q^{33} - 14q^{35} - 16q^{37} + 10q^{49} - 8q^{53} - 12q^{55} + 16q^{61} + 16q^{63} + 4q^{67} - 16q^{73} + 2q^{77} + 64q^{81} - 4q^{83} - 12q^{87} - 4q^{91} + 36q^{93} - 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3849.1 0 3.23317i 0 0.717019i 0 −0.284426 + 2.63042i 0 −7.45341 0
3849.2 0 3.15005i 0 2.09918i 0 1.59876 2.10807i 0 −6.92283 0
3849.3 0 3.04894i 0 4.30537i 0 −2.63608 0.225969i 0 −6.29604 0
3849.4 0 2.99006i 0 1.25381i 0 −2.36377 1.18853i 0 −5.94047 0
3849.5 0 2.95250i 0 1.47372i 0 0.700125 + 2.55144i 0 −5.71727 0
3849.6 0 2.73693i 0 3.14756i 0 −1.11363 2.39996i 0 −4.49077 0
3849.7 0 2.42199i 0 0.343428i 0 2.23070 1.42266i 0 −2.86603 0
3849.8 0 2.41288i 0 2.25637i 0 2.16066 1.52694i 0 −2.82198 0
3849.9 0 2.39724i 0 2.83291i 0 2.55736 + 0.678178i 0 −2.74675 0
3849.10 0 2.30818i 0 0.586386i 0 −2.63094 0.279585i 0 −2.32771 0
3849.11 0 1.96820i 0 3.88595i 0 −0.0756292 + 2.64467i 0 −0.873803 0
3849.12 0 1.77770i 0 3.16364i 0 0.435138 + 2.60972i 0 −0.160220 0
3849.13 0 1.76248i 0 1.09765i 0 −1.82169 + 1.91871i 0 −0.106338 0
3849.14 0 1.33626i 0 2.57665i 0 2.48110 + 0.918779i 0 1.21441 0
3849.15 0 1.12498i 0 0.542613i 0 1.59076 2.11411i 0 1.73443 0
3849.16 0 1.04079i 0 4.22530i 0 −2.02914 + 1.69782i 0 1.91676 0
3849.17 0 1.00653i 0 1.31275i 0 −1.40205 2.24371i 0 1.98690 0
3849.18 0 0.968020i 0 0.815183i 0 −2.53164 + 0.768635i 0 2.06294 0
3849.19 0 0.950260i 0 3.52969i 0 −2.60374 + 0.469588i 0 2.09701 0
3849.20 0 0.867297i 0 0.687824i 0 2.28648 + 1.33116i 0 2.24780 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3849.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.e.a 48
7.b odd 2 1 4004.2.e.b yes 48
11.b odd 2 1 4004.2.e.b yes 48
77.b even 2 1 inner 4004.2.e.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.e.a 48 1.a even 1 1 trivial
4004.2.e.a 48 77.b even 2 1 inner
4004.2.e.b yes 48 7.b odd 2 1
4004.2.e.b yes 48 11.b odd 2 1

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database