# Properties

 Label 8048.2.a.t Level $8048$ Weight $2$ Character orbit 8048.a Self dual yes Analytic conductor $64.264$ Analytic rank $0$ 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: $$0$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21q + 10q^{3} - 3q^{5} + 15q^{7} + 21q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$21q + 10q^{3} - 3q^{5} + 15q^{7} + 21q^{9} + 9q^{11} - 16q^{13} + 22q^{15} + q^{17} + 12q^{19} - 2q^{21} + 22q^{23} + 18q^{25} + 43q^{27} - 13q^{29} + 28q^{31} + 3q^{33} + 12q^{35} - 39q^{37} + 11q^{39} + 4q^{41} + 50q^{43} - 6q^{45} + 27q^{47} + 16q^{49} + 37q^{51} - 24q^{53} + 49q^{55} - q^{57} + 22q^{59} - 22q^{61} + 49q^{63} - 14q^{65} + 62q^{67} - 17q^{69} + 21q^{71} - 6q^{73} + 52q^{75} - 24q^{77} + 65q^{79} + 29q^{81} + 18q^{83} - 54q^{85} + 31q^{87} + q^{89} + 45q^{91} - 26q^{93} + 53q^{95} - 2q^{97} + 58q^{99} + O(q^{100})$$

## 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 −2.92891 0 −2.56957 0 1.38532 0 5.57849 0
1.2 0 −2.20385 0 −1.17070 0 −0.239315 0 1.85694 0
1.3 0 −2.09799 0 −0.681202 0 4.42487 0 1.40156 0
1.4 0 −1.85682 0 −3.94450 0 1.81629 0 0.447775 0
1.5 0 −1.37429 0 2.26464 0 3.08988 0 −1.11131 0
1.6 0 −1.22261 0 0.941648 0 1.12930 0 −1.50523 0
1.7 0 −0.954017 0 −2.55262 0 −2.39192 0 −2.08985 0
1.8 0 −0.301406 0 3.72608 0 3.06802 0 −2.90915 0
1.9 0 −0.0551142 0 0.393007 0 −3.87485 0 −2.99696 0
1.10 0 −0.0212378 0 −0.474891 0 0.949153 0 −2.99955 0
1.11 0 0.100719 0 −0.707548 0 −0.593317 0 −2.98986 0
1.12 0 0.972125 0 0.716085 0 −2.42323 0 −2.05497 0
1.13 0 1.41685 0 2.80516 0 3.87406 0 −0.992528 0
1.14 0 1.70783 0 −2.09685 0 −2.97923 0 −0.0833023 0
1.15 0 2.02067 0 −4.25326 0 0.423427 0 1.08311 0
1.16 0 2.08528 0 2.45828 0 0.831485 0 1.34837 0
1.17 0 2.48667 0 −2.90154 0 4.76998 0 3.18353 0
1.18 0 2.49728 0 1.49435 0 1.21857 0 3.23640 0
1.19 0 3.12082 0 −2.38568 0 −0.375935 0 6.73950 0
1.20 0 3.19390 0 3.60846 0 −4.05198 0 7.20102 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.t 21
4.b odd 2 1 2012.2.a.a 21

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