# Properties

 Label 6039.2.a.d Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $0$ 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: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 4 x^{10} - 6 x^{9} + 37 x^{8} - 2 x^{7} - 109 x^{6} + 55 x^{5} + 115 x^{4} - 76 x^{3} - 29 x^{2} + 14 x + 3$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( 1 - \beta_{10} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{7} + ( 2 - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( 1 - \beta_{10} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{7} + ( 2 - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{8} + ( -1 + 3 \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{10} + q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{14} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{16} + ( 2 - 2 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{17} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{19} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{9} ) q^{20} + \beta_{1} q^{22} + ( 2 - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{26} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{28} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{29} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( 3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{32} + ( -3 + \beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{10} ) q^{34} + ( 3 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{35} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{37} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{38} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{40} + ( -3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{8} - 2 \beta_{10} ) q^{41} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{43} + ( 1 + \beta_{5} + \beta_{6} ) q^{44} + ( 5 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{46} + ( 5 + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{47} + ( -4 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{49} + ( -6 + 7 \beta_{1} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} ) q^{50} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{52} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} + 4 \beta_{9} + 2 \beta_{10} ) q^{53} + ( 1 - \beta_{10} ) q^{55} + ( 1 + 5 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{56} + ( -3 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{58} + ( 2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{59} + q^{61} + ( -1 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{62} + ( 5 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{64} + ( -3 + 5 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} ) q^{65} + ( -4 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} - 2 \beta_{10} ) q^{67} + ( -1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{68} + ( 4 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} ) q^{70} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 5 \beta_{9} - \beta_{10} ) q^{71} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{6} + 5 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{73} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{74} + ( 2 \beta_{2} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{77} + ( 4 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{79} + ( 4 + 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} + 4 \beta_{9} + 4 \beta_{10} ) q^{80} + ( -4 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{82} + ( -2 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 4 - 7 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{86} + ( 2 - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{88} + ( 7 - \beta_{1} + 5 \beta_{2} + \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{89} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} ) q^{91} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{92} + ( 7 - \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{94} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{95} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{97} + ( 4 - 2 \beta_{1} + 5 \beta_{2} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} + 6 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 5 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q + 4q^{2} + 6q^{4} + 13q^{5} - 5q^{7} + 9q^{8} + O(q^{10})$$ $$11q + 4q^{2} + 6q^{4} + 13q^{5} - 5q^{7} + 9q^{8} + 6q^{10} + 11q^{11} - 3q^{13} + 9q^{14} + 4q^{16} + 7q^{17} - 8q^{19} + 25q^{20} + 4q^{22} + 15q^{23} + 4q^{25} + 2q^{26} + 13q^{28} + 8q^{29} - 17q^{31} + 27q^{32} - 18q^{34} + 2q^{35} - 10q^{37} + 30q^{38} + 10q^{40} + 25q^{41} - 7q^{43} + 6q^{44} + 32q^{46} + 30q^{47} - 2q^{49} - 11q^{50} - 7q^{52} + 18q^{53} + 13q^{55} + 20q^{56} - 13q^{58} + 43q^{59} + 11q^{61} - 7q^{62} + 25q^{64} + 27q^{65} - 30q^{67} - 10q^{68} - 4q^{70} + 7q^{71} + 6q^{73} + 44q^{74} - 19q^{76} - 5q^{77} + 17q^{79} + 22q^{80} + 8q^{82} + 34q^{83} + 10q^{85} - 2q^{86} + 9q^{88} + 41q^{89} - 39q^{91} + 32q^{92} + 55q^{94} + 9q^{95} - 41q^{97} + 29q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 4 x^{10} - 6 x^{9} + 37 x^{8} - 2 x^{7} - 109 x^{6} + 55 x^{5} + 115 x^{4} - 76 x^{3} - 29 x^{2} + 14 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} - 2 \nu^{9} - 10 \nu^{8} + 17 \nu^{7} + 32 \nu^{6} - 45 \nu^{5} - 35 \nu^{4} + 54 \nu^{3} + 14 \nu^{2} - 37 \nu - 6$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 55 \nu^{4} + 378 \nu^{3} - 38 \nu^{2} - 35 \nu + 6$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{10} - 16 \nu^{9} - 41 \nu^{8} + 145 \nu^{7} + 97 \nu^{6} - 408 \nu^{5} - 64 \nu^{4} + 387 \nu^{3} + 16 \nu^{2} - 62 \nu - 30$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-4 \nu^{10} + 11 \nu^{9} + 40 \nu^{8} - 107 \nu^{7} - 137 \nu^{6} + 339 \nu^{5} + 188 \nu^{4} - 396 \nu^{3} - 83 \nu^{2} + 109 \nu + 6$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$($$$$4 \nu^{10} - 11 \nu^{9} - 40 \nu^{8} + 107 \nu^{7} + 137 \nu^{6} - 339 \nu^{5} - 188 \nu^{4} + 396 \nu^{3} + 92 \nu^{2} - 109 \nu - 33$$$$)/9$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{10} - 20 \nu^{9} - 61 \nu^{8} + 179 \nu^{7} + 152 \nu^{6} - 489 \nu^{5} - 53 \nu^{4} + 432 \nu^{3} - 136 \nu^{2} - 19 \nu + 21$$$$)/9$$ $$\beta_{8}$$ $$=$$ $$($$$$-2 \nu^{10} + 5 \nu^{9} + 20 \nu^{8} - 44 \nu^{7} - 73 \nu^{6} + 119 \nu^{5} + 125 \nu^{4} - 114 \nu^{3} - 94 \nu^{2} + 28 \nu + 15$$$$)/3$$ $$\beta_{9}$$ $$=$$ $$($$$$7 \nu^{10} - 14 \nu^{9} - 79 \nu^{8} + 128 \nu^{7} + 314 \nu^{6} - 360 \nu^{5} - 506 \nu^{4} + 342 \nu^{3} + 296 \nu^{2} - 43 \nu - 42$$$$)/9$$ $$\beta_{10}$$ $$=$$ $$($$$$16 \nu^{10} - 47 \nu^{9} - 142 \nu^{8} + 431 \nu^{7} + 386 \nu^{6} - 1236 \nu^{5} - 305 \nu^{4} + 1224 \nu^{3} - 46 \nu^{2} - 226 \nu - 6$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 4 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} - \beta_{7} + 7 \beta_{6} + 8 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{10} + \beta_{9} + \beta_{8} - 9 \beta_{7} + 9 \beta_{6} + 18 \beta_{5} - 8 \beta_{4} + 2 \beta_{3} + 21 \beta_{1} + 19$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{10} + \beta_{9} + \beta_{8} - 12 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} - 18 \beta_{4} + 11 \beta_{3} + 2 \beta_{2} + 15 \beta_{1} + 96$$ $$\nu^{7}$$ $$=$$ $$53 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} - 65 \beta_{7} + 72 \beta_{6} + 134 \beta_{5} - 56 \beta_{4} + 25 \beta_{3} + 6 \beta_{2} + 124 \beta_{1} + 154$$ $$\nu^{8}$$ $$=$$ $$80 \beta_{10} + 14 \beta_{9} + 16 \beta_{8} - 107 \beta_{7} + 292 \beta_{6} + 386 \beta_{5} - 134 \beta_{4} + 96 \beta_{3} + 33 \beta_{2} + 148 \beta_{1} + 613$$ $$\nu^{9}$$ $$=$$ $$339 \beta_{10} + 80 \beta_{9} + 93 \beta_{8} - 448 \beta_{7} + 551 \beta_{6} + 960 \beta_{5} - 386 \beta_{4} + 236 \beta_{3} + 96 \beta_{2} + 781 \beta_{1} + 1180$$ $$\nu^{10}$$ $$=$$ $$598 \beta_{10} + 143 \beta_{9} + 172 \beta_{8} - 863 \beta_{7} + 1940 \beta_{6} + 2678 \beta_{5} - 960 \beta_{4} + 780 \beta_{3} + 365 \beta_{2} + 1255 \beta_{1} + 4071$$

## 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
 −2.14662 −1.57504 −1.35090 −0.423080 −0.186189 0.536504 1.14470 1.36137 1.53948 2.39607 2.70371
−2.14662 0 2.60797 4.24506 0 0.408872 −1.30509 0 −9.11254
1.2 −1.57504 0 0.480756 −0.664958 0 −4.15209 2.39287 0 1.04734
1.3 −1.35090 0 −0.175068 0.638467 0 −0.931245 2.93830 0 −0.862505
1.4 −0.423080 0 −1.82100 1.36483 0 0.865577 1.61659 0 −0.577432
1.5 −0.186189 0 −1.96533 −1.94515 0 0.0328573 0.738300 0 0.362164
1.6 0.536504 0 −1.71216 3.00877 0 0.211107 −1.99159 0 1.61422
1.7 1.14470 0 −0.689651 −1.33318 0 −3.52170 −3.07886 0 −1.52610
1.8 1.36137 0 −0.146675 3.87776 0 −4.18708 −2.92242 0 5.27907
1.9 1.53948 0 0.369988 0.258725 0 4.37144 −2.50937 0 0.398300
1.10 2.39607 0 3.74115 0.714370 0 −0.642303 4.17191 0 1.71168
1.11 2.70371 0 5.31003 2.83529 0 2.54457 8.94935 0 7.66580
 $$n$$: 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.d 11
3.b odd 2 1 2013.2.a.a 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.a 11 3.b odd 2 1
6039.2.a.d 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.