# Properties

 Label 8032.2.a.i Level $8032$ Weight $2$ Character orbit 8032.a Self dual yes Analytic conductor $64.136$ Analytic rank $0$ Dimension $30$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8032 = 2^{5} \cdot 251$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8032.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1358429035$$ Analytic rank: $$0$$ Dimension: $$30$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{3} + 13q^{7} + 35q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{3} + 13q^{7} + 35q^{9} + 13q^{11} - 7q^{13} + 28q^{15} - 9q^{17} - 17q^{19} - 6q^{21} + 43q^{23} + 34q^{25} + 12q^{27} - q^{29} + 39q^{31} + 17q^{35} - q^{37} + 48q^{39} - 3q^{41} - 19q^{43} + 66q^{47} + 25q^{49} - 14q^{51} + 3q^{53} + 50q^{55} - 14q^{57} + 27q^{59} + 15q^{61} + 75q^{63} - 6q^{65} + 8q^{67} + 18q^{69} + 64q^{71} - 15q^{73} + 9q^{75} + 71q^{79} + 6q^{81} + 60q^{83} + 15q^{85} + 64q^{87} - 32q^{89} - 26q^{91} - 4q^{93} + 72q^{95} - 4q^{97} + 13q^{99} + O(q^{100})$$

## 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 −3.18079 0 −0.628547 0 2.91752 0 7.11746 0
1.2 0 −2.91073 0 −4.30087 0 1.78682 0 5.47235 0
1.3 0 −2.87266 0 1.81575 0 4.67512 0 5.25220 0
1.4 0 −2.63245 0 −1.53692 0 −2.44148 0 3.92982 0
1.5 0 −2.41377 0 −0.771409 0 0.588687 0 2.82630 0
1.6 0 −2.05220 0 −2.64875 0 −3.00423 0 1.21151 0
1.7 0 −1.95667 0 −3.45496 0 1.56067 0 0.828557 0
1.8 0 −1.70694 0 1.80279 0 −0.510098 0 −0.0863705 0
1.9 0 −1.69973 0 2.39251 0 −2.42748 0 −0.110913 0
1.10 0 −1.53827 0 2.32384 0 4.76077 0 −0.633727 0
1.11 0 −1.12351 0 0.837743 0 −0.914625 0 −1.73772 0
1.12 0 −0.797542 0 −1.59101 0 1.45275 0 −2.36393 0
1.13 0 −0.513479 0 3.65851 0 −0.146814 0 −2.73634 0
1.14 0 −0.0957135 0 3.87548 0 3.14340 0 −2.99084 0
1.15 0 −0.0692070 0 0.775693 0 −4.19948 0 −2.99521 0
1.16 0 0.486172 0 −3.69982 0 0.0825276 0 −2.76364 0
1.17 0 0.679540 0 −1.36350 0 −2.99119 0 −2.53823 0
1.18 0 0.700584 0 −3.32459 0 −0.693647 0 −2.50918 0
1.19 0 1.07677 0 −3.37253 0 5.03712 0 −1.84058 0
1.20 0 1.18444 0 −0.0964158 0 −1.78364 0 −1.59711 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$251$$ $$-1$$

## 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 8032.2.a.i yes 30
4.b odd 2 1 8032.2.a.h 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8032.2.a.h 30 4.b odd 2 1
8032.2.a.i yes 30 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8032))$$:

 $$T_{3}^{30} - \cdots$$ $$T_{7}^{30} - \cdots$$ $$T_{11}^{30} - \cdots$$