# Properties

 Label 6036.2.a.i Level 6036 Weight 2 Character orbit 6036.a Self dual yes Analytic conductor 48.198 Analytic rank 0 Dimension 26 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6036 = 2^{2} \cdot 3 \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6036.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1977026600$$ Analytic rank: $$0$$ Dimension: $$26$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$26q - 26q^{3} + 6q^{5} + 5q^{7} + 26q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$26q - 26q^{3} + 6q^{5} + 5q^{7} + 26q^{9} - 11q^{11} + 13q^{13} - 6q^{15} + 12q^{17} - q^{19} - 5q^{21} - 22q^{23} + 48q^{25} - 26q^{27} + 6q^{29} + 19q^{31} + 11q^{33} - 21q^{35} + 20q^{37} - 13q^{39} + 25q^{41} + 4q^{43} + 6q^{45} + 8q^{47} + 67q^{49} - 12q^{51} - 5q^{53} + 20q^{55} + q^{57} - 18q^{59} + 43q^{61} + 5q^{63} + 41q^{65} + 5q^{67} + 22q^{69} - q^{71} + 22q^{73} - 48q^{75} + 23q^{77} + 16q^{79} + 26q^{81} - 19q^{83} + 29q^{85} - 6q^{87} + 49q^{89} - 13q^{91} - 19q^{93} - 26q^{95} + 25q^{97} - 11q^{99} + O(q^{100})$$

## 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 −1.00000 0 −4.35282 0 4.52941 0 1.00000 0
1.2 0 −1.00000 0 −3.83715 0 0.0390685 0 1.00000 0
1.3 0 −1.00000 0 −3.00702 0 4.42560 0 1.00000 0
1.4 0 −1.00000 0 −2.74974 0 1.46678 0 1.00000 0
1.5 0 −1.00000 0 −2.60444 0 −1.53102 0 1.00000 0
1.6 0 −1.00000 0 −2.57233 0 −0.722776 0 1.00000 0
1.7 0 −1.00000 0 −2.38993 0 0.588107 0 1.00000 0
1.8 0 −1.00000 0 −1.79097 0 −4.44185 0 1.00000 0
1.9 0 −1.00000 0 −1.78585 0 −2.07926 0 1.00000 0
1.10 0 −1.00000 0 −0.970233 0 3.74274 0 1.00000 0
1.11 0 −1.00000 0 −0.461317 0 −3.07386 0 1.00000 0
1.12 0 −1.00000 0 −0.244833 0 2.69533 0 1.00000 0
1.13 0 −1.00000 0 0.0581310 0 −2.78315 0 1.00000 0
1.14 0 −1.00000 0 0.344611 0 2.72948 0 1.00000 0
1.15 0 −1.00000 0 0.754498 0 −4.03319 0 1.00000 0
1.16 0 −1.00000 0 1.62125 0 5.08723 0 1.00000 0
1.17 0 −1.00000 0 1.66086 0 −2.19144 0 1.00000 0
1.18 0 −1.00000 0 1.90366 0 −3.61918 0 1.00000 0
1.19 0 −1.00000 0 2.09204 0 −1.11995 0 1.00000 0
1.20 0 −1.00000 0 2.53458 0 3.60576 0 1.00000 0
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$503$$ $$1$$

## 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 6036.2.a.i 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6036.2.a.i 26 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6036))$$:

 $$T_{5}^{26} - \cdots$$ $$T_{7}^{26} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database