# Properties

 Label 6039.2.a.c Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $1$ Dimension $11$ 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: $$1$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 2 x^{10} - 14 x^{9} + 27 x^{8} + 66 x^{7} - 125 x^{6} - 115 x^{5} + 227 x^{4} + 40 x^{3} - 129 x^{2} + 26 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2013) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + ( -1 - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + ( -1 - \beta_{9} ) q^{7} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} + ( -1 + \beta_{1} - \beta_{4} - \beta_{8} + \beta_{10} ) q^{10} - q^{11} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} ) q^{13} + ( -\beta_{1} + \beta_{7} - \beta_{9} ) q^{14} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{16} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{17} + ( -2 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{20} -\beta_{1} q^{22} + ( -\beta_{3} - \beta_{4} + \beta_{6} ) q^{23} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{25} + ( -2 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{28} + ( 1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{31} + ( 2 + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{32} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{9} ) q^{34} + ( 2 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{35} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{37} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + 3 \beta_{9} ) q^{38} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{40} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{41} + ( -3 - \beta_{2} - 2 \beta_{5} - \beta_{8} - \beta_{9} ) q^{43} + ( -1 - \beta_{2} ) q^{44} + ( -1 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} ) q^{46} + ( 3 - 3 \beta_{1} + 2 \beta_{4} + \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{47} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} ) q^{50} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{52} + ( -1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{53} + \beta_{8} q^{55} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{56} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{9} ) q^{58} + ( -1 - \beta_{3} - \beta_{5} + \beta_{8} ) q^{59} + q^{61} + ( 3 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{62} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{64} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{65} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{67} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{68} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{70} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{71} + ( -4 + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{9} + \beta_{10} ) q^{73} + ( 2 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{74} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{76} + ( 1 + \beta_{9} ) q^{77} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{79} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{80} + ( -4 + 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} ) q^{82} + ( 4 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{9} + \beta_{10} ) q^{83} + ( 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{85} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{86} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{88} + ( -3 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} - 3 \beta_{9} ) q^{89} + ( 1 + \beta_{3} - \beta_{7} + \beta_{8} ) q^{91} + ( -3 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{92} + ( -5 + 4 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{94} + ( 1 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{95} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{97} + ( 3 - 4 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q + 2q^{2} + 10q^{4} + q^{5} - 11q^{7} + 3q^{8} + O(q^{10})$$ $$11q + 2q^{2} + 10q^{4} + q^{5} - 11q^{7} + 3q^{8} - 8q^{10} - 11q^{11} - 13q^{13} - 5q^{14} + 4q^{16} + 13q^{17} - 12q^{19} + 7q^{20} - 2q^{22} + 3q^{23} + 12q^{25} - 12q^{26} - 13q^{28} - 2q^{29} + q^{31} + 23q^{32} - 14q^{34} + 4q^{35} - 14q^{37} + 8q^{38} - 34q^{40} - 3q^{41} - 21q^{43} - 10q^{44} - 12q^{46} + 16q^{47} - 18q^{49} + 13q^{50} - 33q^{52} - q^{55} - 16q^{56} - 17q^{58} - 3q^{59} + 11q^{61} + 21q^{62} - 7q^{64} + q^{65} - 24q^{67} - 2q^{68} + 4q^{70} - 7q^{71} - 42q^{73} + 16q^{74} - 13q^{76} + 11q^{77} - 11q^{79} - 42q^{80} - 38q^{82} + 34q^{83} - 14q^{85} - 42q^{86} - 3q^{88} - 29q^{89} + 9q^{91} - 42q^{92} - 33q^{94} + 31q^{95} - 45q^{97} + 33q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 2 x^{10} - 14 x^{9} + 27 x^{8} + 66 x^{7} - 125 x^{6} - 115 x^{5} + 227 x^{4} + 40 x^{3} - 129 x^{2} + 26 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{10} - 3 \nu^{9} - 80 \nu^{8} + 42 \nu^{7} + 340 \nu^{6} - 257 \nu^{5} - 506 \nu^{4} + 688 \nu^{3} + 167 \nu^{2} - 532 \nu + 72$$$$)/17$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{10} + 5 \nu^{9} - 99 \nu^{8} - 70 \nu^{7} + 459 \nu^{6} + 264 \nu^{5} - 834 \nu^{4} - 257 \nu^{3} + 583 \nu^{2} + 48 \nu - 86$$$$)/17$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{10} - 9 \nu^{9} - 19 \nu^{8} + 126 \nu^{7} + 136 \nu^{6} - 584 \nu^{5} - 379 \nu^{4} + 1061 \nu^{3} + 280 \nu^{2} - 678 \nu + 46$$$$)/17$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{10} - 2 \nu^{9} + 179 \nu^{8} + 45 \nu^{7} - 799 \nu^{6} - 211 \nu^{5} + 1323 \nu^{4} + 266 \nu^{3} - 699 \nu^{2} - 111 \nu + 31$$$$)/17$$ $$\beta_{7}$$ $$=$$ $$($$$$-20 \nu^{10} - 7 \nu^{9} + 278 \nu^{8} + 115 \nu^{7} - 1258 \nu^{6} - 475 \nu^{5} + 2157 \nu^{4} + 506 \nu^{3} - 1265 \nu^{2} - 74 \nu + 66$$$$)/17$$ $$\beta_{8}$$ $$=$$ $$($$$$8 \nu^{10} - 21 \nu^{9} - 118 \nu^{8} + 277 \nu^{7} + 612 \nu^{6} - 1204 \nu^{5} - 1247 \nu^{4} + 1960 \nu^{3} + 676 \nu^{2} - 1004 \nu + 113$$$$)/17$$ $$\beta_{9}$$ $$=$$ $$($$$$23 \nu^{10} - 3 \nu^{9} - 318 \nu^{8} + 25 \nu^{7} + 1428 \nu^{6} - 189 \nu^{5} - 2359 \nu^{4} + 705 \nu^{3} + 1119 \nu^{2} - 583 \nu + 89$$$$)/17$$ $$\beta_{10}$$ $$=$$ $$($$$$20 \nu^{10} - 10 \nu^{9} - 278 \nu^{8} + 123 \nu^{7} + 1275 \nu^{6} - 613 \nu^{5} - 2225 \nu^{4} + 1347 \nu^{3} + 1231 \nu^{2} - 878 \nu + 36$$$$)/17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 6 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{8} - 9 \beta_{7} + 10 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + 9 \beta_{2} + 28 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - \beta_{4} + 11 \beta_{3} + 37 \beta_{2} + 13 \beta_{1} + 75$$ $$\nu^{7}$$ $$=$$ $$-\beta_{10} + 13 \beta_{8} - 68 \beta_{7} + 81 \beta_{6} - 25 \beta_{5} - 54 \beta_{4} + 2 \beta_{3} + 70 \beta_{2} + 167 \beta_{1} + 28$$ $$\nu^{8}$$ $$=$$ $$-65 \beta_{10} - 25 \beta_{9} + 92 \beta_{8} - 123 \beta_{7} + 125 \beta_{6} - 108 \beta_{5} - 20 \beta_{4} + 92 \beta_{3} + 242 \beta_{2} + 128 \beta_{1} + 434$$ $$\nu^{9}$$ $$=$$ $$-20 \beta_{10} - 2 \beta_{9} + 125 \beta_{8} - 495 \beta_{7} + 612 \beta_{6} - 230 \beta_{5} - 354 \beta_{4} + 35 \beta_{3} + 524 \beta_{2} + 1044 \beta_{1} + 283$$ $$\nu^{10}$$ $$=$$ $$-444 \beta_{10} - 221 \beta_{9} + 702 \beta_{8} - 1030 \beta_{7} + 1067 \beta_{6} - 870 \beta_{5} - 237 \beta_{4} + 694 \beta_{3} + 1651 \beta_{2} + 1122 \beta_{1} + 2653$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.39095 −2.09663 −1.65258 −1.03852 0.0512060 0.175402 0.942842 1.21038 2.04468 2.05394 2.70023
−2.39095 0 3.71663 2.34211 0 −0.680854 −4.10437 0 −5.59986
1.2 −2.09663 0 2.39586 1.65820 0 −1.74412 −0.829977 0 −3.47663
1.3 −1.65258 0 0.731020 −2.07534 0 −0.262572 2.09709 0 3.42966
1.4 −1.03852 0 −0.921477 1.07279 0 1.21131 3.03401 0 −1.11412
1.5 0.0512060 0 −1.99738 −3.74211 0 −4.65643 −0.204690 0 −0.191618
1.6 0.175402 0 −1.96923 1.94648 0 −2.33819 −0.696212 0 0.341417
1.7 0.942842 0 −1.11105 −3.49194 0 1.81837 −2.93323 0 −3.29234
1.8 1.21038 0 −0.534974 4.03730 0 0.314382 −3.06829 0 4.88668
1.9 2.04468 0 2.18073 1.44717 0 −4.07149 0.369530 0 2.95900
1.10 2.05394 0 2.21866 0.0248942 0 1.52855 0.449108 0 0.0511312
1.11 2.70023 0 5.29122 −2.21957 0 −2.11895 8.88702 0 −5.99333
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 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.c 11
3.b odd 2 1 2013.2.a.b 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.b 11 3.b odd 2 1
6039.2.a.c 11 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{11} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.