# Properties

 Label 6008.2.a.c Level 6008 Weight 2 Character orbit 6008.a Self dual Yes Analytic conductor 47.974 Analytic rank 1 Dimension 44 CM No

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$6008 = 2^{3} \cdot 751$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6008.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$47.9741215344$$ Analytic rank: $$1$$ Dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - 4q^{3} - 21q^{5} - 10q^{7} + 38q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - 4q^{3} - 21q^{5} - 10q^{7} + 38q^{9} + 11q^{11} - 36q^{13} - 5q^{15} - 10q^{17} - 7q^{19} - 42q^{21} - 5q^{23} + 29q^{25} - 16q^{27} - 57q^{29} - 21q^{31} - 32q^{33} + 17q^{35} - 52q^{37} + 8q^{39} - 16q^{41} - 9q^{43} - 84q^{45} - q^{47} + 28q^{49} - q^{51} - 52q^{53} - 39q^{55} - 15q^{57} + 7q^{59} - 85q^{61} - 25q^{63} - 9q^{65} - 36q^{67} - 72q^{69} + 12q^{71} - 60q^{73} - 5q^{75} - 81q^{77} - 13q^{79} + 20q^{81} + 5q^{83} - 72q^{85} + 9q^{87} - 37q^{89} - 23q^{91} - 60q^{93} + 24q^{95} - 79q^{97} + 42q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 0 −3.39974 0 −1.61655 0 0.842666 0 8.55822 0
1.2 0 −3.30310 0 −2.23921 0 4.70410 0 7.91049 0
1.3 0 −2.87868 0 −2.95953 0 −5.20423 0 5.28682 0
1.4 0 −2.71670 0 1.94897 0 −1.05398 0 4.38045 0
1.5 0 −2.66760 0 2.00995 0 0.318046 0 4.11611 0
1.6 0 −2.66129 0 −2.69520 0 −1.29192 0 4.08246 0
1.7 0 −2.63254 0 2.34289 0 3.66913 0 3.93027 0
1.8 0 −2.37283 0 −3.87547 0 0.916538 0 2.63034 0
1.9 0 −2.15961 0 0.185718 0 3.00693 0 1.66392 0
1.10 0 −2.08546 0 −0.642029 0 1.54125 0 1.34913 0
1.11 0 −1.92521 0 1.08205 0 −3.74499 0 0.706428 0
1.12 0 −1.66100 0 −3.23151 0 4.31694 0 −0.241072 0
1.13 0 −1.63270 0 2.45098 0 −0.216771 0 −0.334292 0
1.14 0 −1.49682 0 1.87951 0 2.30247 0 −0.759523 0
1.15 0 −1.30016 0 −0.475831 0 −2.70311 0 −1.30958 0
1.16 0 −1.27703 0 −2.93535 0 −5.00244 0 −1.36920 0
1.17 0 −1.16746 0 2.71362 0 −1.33257 0 −1.63703 0
1.18 0 −1.08162 0 −1.48750 0 −1.61434 0 −1.83009 0
1.19 0 −0.644887 0 −2.11044 0 0.769206 0 −2.58412 0
1.20 0 −0.214146 0 −4.30470 0 −2.13393 0 −2.95414 0
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.44 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
$$2$$ $$-1$$
$$751$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{44} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6008))$$.