# Properties

 Label 8048.2.a.v Level 8048 Weight 2 Character orbit 8048.a Self dual yes Analytic conductor 64.264 Analytic rank 1 Dimension 28 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: $$28$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 4024) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{3} - 12q^{5} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{3} - 12q^{5} + 18q^{9} + 14q^{11} - 31q^{13} + 2q^{15} - 9q^{17} + 8q^{19} - 28q^{21} + 4q^{23} + 22q^{25} + 4q^{27} - 47q^{29} + 5q^{31} - 26q^{33} + 13q^{35} - 67q^{37} + 9q^{39} - 28q^{41} - 15q^{43} - 57q^{45} + 10q^{47} + 20q^{49} + 11q^{51} - 58q^{53} - 15q^{55} - 31q^{57} + 32q^{59} - 55q^{61} + 16q^{63} - 44q^{65} - 22q^{67} - 44q^{69} + 47q^{71} - 5q^{73} + 25q^{75} - 50q^{77} + 14q^{79} - 28q^{81} + 16q^{83} - 78q^{85} + 11q^{87} - 20q^{89} + 15q^{91} - 83q^{93} + 27q^{95} - 8q^{97} + 70q^{99} + O(q^{100})$$

## 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 −3.18633 0 −1.57145 0 3.06067 0 7.15268 0
1.2 0 −2.89723 0 −1.96349 0 −2.82850 0 5.39392 0
1.3 0 −2.84528 0 −2.72112 0 3.71681 0 5.09562 0
1.4 0 −2.31121 0 0.927476 0 1.63809 0 2.34167 0
1.5 0 −2.11208 0 2.26631 0 4.06526 0 1.46088 0
1.6 0 −1.95835 0 0.337153 0 −0.830729 0 0.835124 0
1.7 0 −1.67525 0 −2.92493 0 1.65914 0 −0.193522 0
1.8 0 −1.66428 0 −3.44274 0 −4.43077 0 −0.230172 0
1.9 0 −1.59308 0 2.75433 0 −1.46360 0 −0.462108 0
1.10 0 −1.49100 0 −1.09730 0 0.799854 0 −0.776913 0
1.11 0 −0.823496 0 −2.12271 0 −4.15456 0 −2.32185 0
1.12 0 −0.799349 0 0.626001 0 0.555593 0 −2.36104 0
1.13 0 −0.682343 0 3.69511 0 3.46637 0 −2.53441 0
1.14 0 −0.659121 0 0.997881 0 −0.366743 0 −2.56556 0
1.15 0 0.0730681 0 −4.10981 0 1.52145 0 −2.99466 0
1.16 0 0.447624 0 0.901345 0 1.57888 0 −2.79963 0
1.17 0 0.596863 0 3.74289 0 −1.10831 0 −2.64375 0
1.18 0 0.667261 0 0.307121 0 −4.25072 0 −2.55476 0
1.19 0 0.893789 0 −3.85908 0 −3.77625 0 −2.20114 0
1.20 0 1.32661 0 −1.90684 0 4.79236 0 −1.24010 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.v 28
4.b odd 2 1 4024.2.a.d 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.d 28 4.b odd 2 1
8048.2.a.v 28 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{3}^{28} + \cdots$$ $$T_{5}^{28} + \cdots$$ $$T_{7}^{28} - \cdots$$ $$T_{13}^{28} + \cdots$$