# 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

# Learn more about

## Newspace parameters

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

## Newform invariants

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

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

## 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

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

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