# Properties

 Label 6039.2.a.l Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $0$ Dimension $21$ 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: $$21$$ Twist minimal: no (minimal twist has level 671) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21q + 32q^{4} - 7q^{5} + 5q^{7} + 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$21q + 32q^{4} - 7q^{5} + 5q^{7} + 6q^{8} + q^{10} + 21q^{11} + 20q^{13} - 17q^{14} + 50q^{16} - q^{17} + 15q^{19} + 2q^{20} - 11q^{23} + 48q^{25} + 5q^{26} - 16q^{28} + 9q^{29} + 22q^{31} - 3q^{32} + 33q^{34} + 39q^{35} + 21q^{37} - 11q^{38} - 16q^{40} - 7q^{41} + 16q^{43} + 32q^{44} - 3q^{46} - 5q^{47} + 80q^{49} + 33q^{50} + 60q^{52} - 9q^{53} - 7q^{55} - 44q^{56} - 27q^{58} - 13q^{59} + 21q^{61} + 23q^{62} + 66q^{64} - 25q^{65} + 38q^{67} + 74q^{68} - 33q^{70} - 12q^{71} + 20q^{73} + 12q^{74} + 59q^{76} + 5q^{77} + q^{79} + 38q^{80} + 7q^{82} + 19q^{83} + 38q^{85} + 3q^{86} + 6q^{88} - 37q^{89} + 24q^{91} - 31q^{92} - 64q^{94} + 43q^{95} + 68q^{97} + 2q^{98} + O(q^{100})$$

## 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 −2.68161 0 5.19104 3.56342 0 3.18744 −8.55713 0 −9.55570
1.2 −2.67686 0 5.16560 −1.61394 0 −4.48909 −8.47390 0 4.32029
1.3 −2.54333 0 4.46852 0.268217 0 2.27426 −6.27827 0 −0.682163
1.4 −2.16302 0 2.67865 −3.42546 0 1.52570 −1.46794 0 7.40934
1.5 −2.08860 0 2.36225 0.0546248 0 −2.53368 −0.756597 0 −0.114089
1.6 −1.61525 0 0.609041 −2.50050 0 4.06392 2.24675 0 4.03894
1.7 −1.29392 0 −0.325780 3.15298 0 5.03078 3.00937 0 −4.07969
1.8 −0.942064 0 −1.11252 −4.16220 0 −0.914635 2.93219 0 3.92106
1.9 −0.731918 0 −1.46430 −2.76447 0 −1.34613 2.53558 0 2.02337
1.10 −0.543169 0 −1.70497 0.382346 0 −4.84691 2.01242 0 −0.207678
1.11 −0.472572 0 −1.77668 3.20957 0 4.06194 1.78475 0 −1.51675
1.12 0.404310 0 −1.83653 −0.0975766 0 1.38056 −1.55115 0 −0.0394512
1.13 0.634118 0 −1.59789 1.64555 0 −3.41198 −2.28149 0 1.04347
1.14 1.29453 0 −0.324186 −2.94198 0 −0.962580 −3.00873 0 −3.80849
1.15 1.45442 0 0.115325 −3.82594 0 3.26042 −2.74110 0 −5.56450
1.16 1.47389 0 0.172365 −0.593969 0 3.31761 −2.69374 0 −0.875448
1.17 2.19302 0 2.80936 3.87665 0 3.34814 1.77494 0 8.50160
1.18 2.40042 0 3.76203 3.90458 0 −3.05635 4.22962 0 9.37264
1.19 2.55546 0 4.53040 −4.10969 0 −4.64385 6.46634 0 −10.5022
1.20 2.60374 0 4.77947 −0.851432 0 3.04314 7.23702 0 −2.21691
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
$$3$$ $$-1$$
$$11$$ $$-1$$
$$61$$ $$-1$$

## 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.l 21
3.b odd 2 1 671.2.a.d 21
33.d even 2 1 7381.2.a.j 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.d 21 3.b odd 2 1
6039.2.a.l 21 1.a even 1 1 trivial
7381.2.a.j 21 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{21} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.