# Properties

 Label 4004.2.m.c Level 4004 Weight 2 Character orbit 4004.m Analytic conductor 31.972 Analytic rank 0 Dimension 36 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.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.9721009693$$ Analytic rank: $$0$$ Dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 4q^{3} + 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 4q^{3} + 40q^{9} - 4q^{17} + 8q^{23} - 80q^{25} + 8q^{27} + 8q^{29} - 24q^{39} + 32q^{43} - 36q^{49} - 20q^{51} + 12q^{53} + 32q^{61} - 24q^{65} + 80q^{69} - 36q^{75} + 36q^{77} + 16q^{79} + 132q^{81} + 8q^{91} + 56q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2157.1 0 −3.27485 0 2.17871i 0 1.00000i 0 7.72466 0
2157.2 0 −3.27485 0 2.17871i 0 1.00000i 0 7.72466 0
2157.3 0 −3.10315 0 1.71558i 0 1.00000i 0 6.62954 0
2157.4 0 −3.10315 0 1.71558i 0 1.00000i 0 6.62954 0
2157.5 0 −2.75761 0 1.98236i 0 1.00000i 0 4.60441 0
2157.6 0 −2.75761 0 1.98236i 0 1.00000i 0 4.60441 0
2157.7 0 −2.10949 0 3.39046i 0 1.00000i 0 1.44995 0
2157.8 0 −2.10949 0 3.39046i 0 1.00000i 0 1.44995 0
2157.9 0 −1.64159 0 4.02832i 0 1.00000i 0 −0.305195 0
2157.10 0 −1.64159 0 4.02832i 0 1.00000i 0 −0.305195 0
2157.11 0 −1.08723 0 3.34996i 0 1.00000i 0 −1.81792 0
2157.12 0 −1.08723 0 3.34996i 0 1.00000i 0 −1.81792 0
2157.13 0 −1.07570 0 3.26058i 0 1.00000i 0 −1.84286 0
2157.14 0 −1.07570 0 3.26058i 0 1.00000i 0 −1.84286 0
2157.15 0 −0.819032 0 1.69723i 0 1.00000i 0 −2.32919 0
2157.16 0 −0.819032 0 1.69723i 0 1.00000i 0 −2.32919 0
2157.17 0 −0.722868 0 0.383791i 0 1.00000i 0 −2.47746 0
2157.18 0 −0.722868 0 0.383791i 0 1.00000i 0 −2.47746 0
2157.19 0 −0.189991 0 1.01340i 0 1.00000i 0 −2.96390 0
2157.20 0 −0.189991 0 1.01340i 0 1.00000i 0 −2.96390 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2157.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.m.c 36
13.b even 2 1 inner 4004.2.m.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.c 36 1.a even 1 1 trivial
4004.2.m.c 36 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(4004, [\chi])$$.