# Properties

 Label 8048.2.a.s Level 8048 Weight 2 Character orbit 8048.a Self dual yes Analytic conductor 64.264 Analytic rank 1 Dimension 21 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8048 = 2^{4} \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8048.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2636035467$$ Analytic rank: $$1$$ Dimension: $$21$$ Twist minimal: no (minimal twist has level 2012) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21q - 10q^{3} + 3q^{5} - 13q^{7} + 21q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$21q - 10q^{3} + 3q^{5} - 13q^{7} + 21q^{9} - 7q^{11} + 12q^{13} - 14q^{15} + q^{17} - 14q^{19} + 14q^{21} - 26q^{23} + 18q^{25} - 37q^{27} + 9q^{29} - 28q^{31} + 3q^{33} - 20q^{35} + 31q^{37} - 29q^{39} + 4q^{41} - 38q^{43} + 24q^{45} - 9q^{47} + 16q^{49} - 15q^{51} + 22q^{53} - 35q^{55} - q^{57} - 10q^{59} + 22q^{61} - 35q^{63} - 14q^{65} - 58q^{67} + 15q^{69} - 27q^{71} - 6q^{73} - 48q^{75} + 16q^{77} - 47q^{79} + 29q^{81} - 22q^{83} + 14q^{85} - 29q^{87} + q^{89} - 51q^{91} + 34q^{93} - 23q^{95} - 2q^{97} - 22q^{99} + O(q^{100})$$

## 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 −3.31593 0 −2.21364 0 1.66896 0 7.99538 0
1.2 0 −3.16551 0 −0.445451 0 −3.10818 0 7.02047 0
1.3 0 −3.07762 0 3.88947 0 0.0828849 0 6.47174 0
1.4 0 −2.69810 0 1.28348 0 −4.83591 0 4.27974 0
1.5 0 −2.44429 0 3.11768 0 1.05875 0 2.97454 0
1.6 0 −2.30121 0 1.38461 0 2.51996 0 2.29558 0
1.7 0 −1.93158 0 −3.25684 0 −2.49607 0 0.730999 0
1.8 0 −1.38206 0 −2.65136 0 −3.15617 0 −1.08990 0
1.9 0 −1.32334 0 2.33991 0 −2.12211 0 −1.24878 0
1.10 0 −1.10653 0 2.64866 0 −3.45308 0 −1.77559 0
1.11 0 −0.207528 0 −1.91018 0 4.59256 0 −2.95693 0
1.12 0 0.112957 0 0.539075 0 1.01688 0 −2.98724 0
1.13 0 0.175511 0 3.29961 0 −3.77232 0 −2.96920 0
1.14 0 0.273229 0 −1.56489 0 3.17063 0 −2.92535 0
1.15 0 0.388717 0 −4.11083 0 1.27214 0 −2.84890 0
1.16 0 1.27408 0 3.64848 0 0.903884 0 −1.37671 0
1.17 0 1.30141 0 −2.99143 0 −4.27894 0 −1.30633 0
1.18 0 1.91007 0 1.16052 0 −1.88100 0 0.648366 0
1.19 0 2.22301 0 −1.09600 0 1.95566 0 1.94176 0
1.20 0 2.41348 0 −0.469286 0 1.19654 0 2.82487 0
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$503$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8048.2.a.s 21
4.b odd 2 1 2012.2.a.b 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2012.2.a.b 21 4.b odd 2 1
8048.2.a.s 21 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8048))$$:

 $$T_{3}^{21} + \cdots$$ $$T_{5}^{21} - \cdots$$ $$T_{7}^{21} + \cdots$$ $$T_{13}^{21} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database