# Properties

 Label 6040.2.a.s Level 6040 Weight 2 Character orbit 6040.a Self dual Yes Analytic conductor 48.230 Analytic rank 0 Dimension 24 CM No

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$6040 = 2^{3} \cdot 5 \cdot 151$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6040.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.2296428209$$ Analytic rank: $$0$$ Dimension: $$24$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q$$ $$\mathstrut +\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 24q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut +\mathstrut 40q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q$$ $$\mathstrut +\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 24q^{5}$$ $$\mathstrut +\mathstrut 3q^{7}$$ $$\mathstrut +\mathstrut 40q^{9}$$ $$\mathstrut +\mathstrut 17q^{11}$$ $$\mathstrut +\mathstrut 16q^{13}$$ $$\mathstrut +\mathstrut 2q^{15}$$ $$\mathstrut +\mathstrut 22q^{17}$$ $$\mathstrut +\mathstrut 16q^{19}$$ $$\mathstrut -\mathstrut q^{21}$$ $$\mathstrut +\mathstrut 7q^{23}$$ $$\mathstrut +\mathstrut 24q^{25}$$ $$\mathstrut -\mathstrut 4q^{27}$$ $$\mathstrut +\mathstrut 25q^{29}$$ $$\mathstrut +\mathstrut 28q^{31}$$ $$\mathstrut +\mathstrut 11q^{33}$$ $$\mathstrut +\mathstrut 3q^{35}$$ $$\mathstrut +\mathstrut 26q^{37}$$ $$\mathstrut +\mathstrut 13q^{39}$$ $$\mathstrut +\mathstrut 38q^{41}$$ $$\mathstrut -\mathstrut 13q^{43}$$ $$\mathstrut +\mathstrut 40q^{45}$$ $$\mathstrut +\mathstrut 12q^{47}$$ $$\mathstrut +\mathstrut 61q^{49}$$ $$\mathstrut +\mathstrut 53q^{53}$$ $$\mathstrut +\mathstrut 17q^{55}$$ $$\mathstrut +\mathstrut 30q^{57}$$ $$\mathstrut +\mathstrut 35q^{59}$$ $$\mathstrut +\mathstrut 44q^{61}$$ $$\mathstrut -\mathstrut 9q^{63}$$ $$\mathstrut +\mathstrut 16q^{65}$$ $$\mathstrut -\mathstrut 15q^{67}$$ $$\mathstrut +\mathstrut 9q^{69}$$ $$\mathstrut +\mathstrut 22q^{71}$$ $$\mathstrut +\mathstrut 31q^{73}$$ $$\mathstrut +\mathstrut 2q^{75}$$ $$\mathstrut +\mathstrut 26q^{77}$$ $$\mathstrut +\mathstrut 20q^{79}$$ $$\mathstrut +\mathstrut 88q^{81}$$ $$\mathstrut -\mathstrut 14q^{83}$$ $$\mathstrut +\mathstrut 22q^{85}$$ $$\mathstrut -\mathstrut 18q^{87}$$ $$\mathstrut +\mathstrut 37q^{89}$$ $$\mathstrut -\mathstrut 26q^{91}$$ $$\mathstrut +\mathstrut 13q^{93}$$ $$\mathstrut +\mathstrut 16q^{95}$$ $$\mathstrut +\mathstrut 21q^{97}$$ $$\mathstrut -\mathstrut 12q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 0 −3.28894 0 1.00000 0 4.04677 0 7.81711 0
1.2 0 −3.20286 0 1.00000 0 −3.67370 0 7.25831 0
1.3 0 −3.03909 0 1.00000 0 −1.69120 0 6.23604 0
1.4 0 −2.50341 0 1.00000 0 −2.70762 0 3.26705 0
1.5 0 −2.45370 0 1.00000 0 −1.79083 0 3.02066 0
1.6 0 −2.29572 0 1.00000 0 4.99611 0 2.27034 0
1.7 0 −2.03886 0 1.00000 0 1.67649 0 1.15693 0
1.8 0 −1.08470 0 1.00000 0 4.94762 0 −1.82343 0
1.9 0 −0.875996 0 1.00000 0 −1.42220 0 −2.23263 0
1.10 0 −0.559764 0 1.00000 0 −0.837547 0 −2.68666 0
1.11 0 0.0496662 0 1.00000 0 1.60144 0 −2.99753 0
1.12 0 0.161432 0 1.00000 0 −4.19061 0 −2.97394 0
1.13 0 0.225898 0 1.00000 0 2.54851 0 −2.94897 0
1.14 0 0.549309 0 1.00000 0 −0.285162 0 −2.69826 0
1.15 0 0.568728 0 1.00000 0 −5.03281 0 −2.67655 0
1.16 0 1.21987 0 1.00000 0 0.325030 0 −1.51192 0
1.17 0 1.49843 0 1.00000 0 1.73584 0 −0.754697 0
1.18 0 1.94109 0 1.00000 0 3.18961 0 0.767827 0
1.19 0 2.40154 0 1.00000 0 4.37326 0 2.76742 0
1.20 0 2.57003 0 1.00000 0 −3.15345 0 3.60506 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 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.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$151$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6040))$$:

 $$T_{3}^{24} - \cdots$$ $$T_{7}^{24} - \cdots$$ $$T_{11}^{24} - \cdots$$