# Properties

 Label 6039.2.a.p Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $0$ Dimension $25$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6039 = 3^{2} \cdot 11 \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6039.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2216577807$$ Analytic rank: $$0$$ Dimension: $$25$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$25q + 5q^{2} + 25q^{4} + 12q^{5} - 4q^{7} + 15q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$25q + 5q^{2} + 25q^{4} + 12q^{5} - 4q^{7} + 15q^{8} - 12q^{10} - 25q^{11} - 4q^{13} + 14q^{14} + 21q^{16} + 16q^{17} - 18q^{19} + 28q^{20} - 5q^{22} + 8q^{23} + 29q^{25} + 16q^{26} + 18q^{28} + 28q^{29} - 8q^{31} + 35q^{32} + 6q^{34} + 22q^{35} + 4q^{37} - 4q^{38} - 12q^{40} + 58q^{41} - 26q^{43} - 25q^{44} + 8q^{46} + 20q^{47} + 23q^{49} + 27q^{50} - 2q^{52} + 36q^{53} - 12q^{55} + 70q^{56} + 12q^{58} + 18q^{59} + 25q^{61} + 42q^{62} + 35q^{64} + 76q^{65} - 8q^{67} + 28q^{68} + 76q^{70} + 24q^{71} + 2q^{73} + 40q^{74} - 64q^{76} + 4q^{77} - 22q^{79} + 36q^{80} + 30q^{82} + 14q^{83} + 70q^{86} - 15q^{88} + 82q^{89} - 6q^{91} + 48q^{92} - 16q^{94} + 34q^{95} + 16q^{97} + 35q^{98} + O(q^{100})$$

## 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 −2.57687 0 4.64028 2.12696 0 −3.45468 −6.80367 0 −5.48091
1.2 −2.42221 0 3.86709 1.62300 0 2.13922 −4.52249 0 −3.93125
1.3 −2.21594 0 2.91037 4.37397 0 0.199638 −2.01732 0 −9.69243
1.4 −1.92069 0 1.68905 −2.68560 0 −3.03402 0.597238 0 5.15821
1.5 −1.87622 0 1.52020 −0.769125 0 −0.850176 0.900213 0 1.44305
1.6 −1.53628 0 0.360147 2.16725 0 2.96367 2.51927 0 −3.32950
1.7 −1.23685 0 −0.470198 0.403292 0 −0.303505 3.05527 0 −0.498813
1.8 −1.04320 0 −0.911728 −1.64856 0 5.01087 3.03752 0 1.71978
1.9 −0.870092 0 −1.24294 3.50217 0 −4.57282 2.82166 0 −3.04721
1.10 −0.608354 0 −1.62991 −1.59846 0 −1.08777 2.20827 0 0.972431
1.11 −0.325798 0 −1.89386 3.19527 0 −2.43974 1.26861 0 −1.04101
1.12 −0.228656 0 −1.94772 −0.872276 0 −1.27506 0.902667 0 0.199451
1.13 0.536820 0 −1.71182 1.82707 0 2.96010 −1.99258 0 0.980806
1.14 0.649968 0 −1.57754 −3.31582 0 1.03899 −2.32529 0 −2.15517
1.15 0.800654 0 −1.35895 0.757549 0 0.484724 −2.68936 0 0.606535
1.16 1.12582 0 −0.732534 −0.691275 0 −1.12590 −3.07634 0 −0.778249
1.17 1.18061 0 −0.606158 −0.892999 0 −4.86985 −3.07686 0 −1.05428
1.18 1.38389 0 −0.0848502 4.17750 0 3.01863 −2.88520 0 5.78120
1.19 1.84847 0 1.41684 −3.30338 0 −3.95713 −1.07795 0 −6.10619
1.20 1.98740 0 1.94976 −1.69625 0 1.47319 −0.0998540 0 −3.37113
See all 25 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.25 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$11$$ $$1$$
$$61$$ $$-1$$

## 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 6039.2.a.p yes 25
3.b odd 2 1 6039.2.a.m 25

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.m 25 3.b odd 2 1
6039.2.a.p yes 25 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{25} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.