# Properties

 Label 6039.2.a.g Level 6039 Weight 2 Character orbit 6039.a Self dual yes Analytic conductor 48.222 Analytic rank 1 Dimension 13 CM no Inner twists 1

# Learn more about

## 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: $$13$$ Coefficient field: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} - 404 x^{4} + 98 x^{3} + 118 x^{2} - 16 x - 11$$ 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 $$1,\beta_1,\ldots,\beta_{12}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{5} -\beta_{9} q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{5} -\beta_{9} q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{10} + q^{11} + ( -\beta_{6} - \beta_{7} - \beta_{12} ) q^{13} + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} ) q^{14} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{16} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{11} ) q^{17} + ( -\beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{19} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{11} ) q^{20} -\beta_{1} q^{22} + ( -1 - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} ) q^{25} + ( -2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} ) q^{26} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{28} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{29} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{32} + ( 1 - \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{34} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{35} + ( 2 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{40} + ( -1 - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{9} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} ) q^{44} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{46} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{47} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{49} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} ) q^{50} + ( 1 - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{52} + ( -3 + 2 \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} ) q^{53} + ( -1 + \beta_{6} ) q^{55} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{56} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} ) q^{58} + ( -2 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{59} - q^{61} + ( 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{12} ) q^{62} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{64} + ( -2 + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{65} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{67} + ( 2 + \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{68} + ( -1 - \beta_{2} - \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{70} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 4 \beta_{12} ) q^{71} + ( -4 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{74} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{76} -\beta_{9} q^{77} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} - 2 \beta_{12} ) q^{79} + ( 1 - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{80} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{12} ) q^{82} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{83} + ( -2 - 3 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{85} + ( 2 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{86} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} + ( -2 - \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{89} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{91} + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{12} ) q^{92} + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} ) q^{94} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{95} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} ) q^{97} + ( 2 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$13q - 4q^{2} + 12q^{4} - 7q^{5} + 5q^{7} - 15q^{8} + O(q^{10})$$ $$13q - 4q^{2} + 12q^{4} - 7q^{5} + 5q^{7} - 15q^{8} + 8q^{10} + 13q^{11} - 9q^{13} - 19q^{14} + 18q^{16} - 7q^{17} + 2q^{19} - 15q^{20} - 4q^{22} - 23q^{23} + 10q^{25} - 8q^{26} + 9q^{28} - 16q^{29} + 9q^{31} - 29q^{32} + 2q^{34} - 16q^{35} + 14q^{37} - 8q^{38} + 16q^{40} - 19q^{41} + 7q^{43} + 12q^{44} + 4q^{46} - 26q^{47} + 8q^{49} + 15q^{50} - 17q^{52} - 18q^{53} - 7q^{55} - 44q^{56} - q^{58} - 31q^{59} - 13q^{61} + 5q^{62} - 17q^{64} - 31q^{65} + 14q^{67} + 32q^{68} - 20q^{70} - 37q^{71} - 16q^{73} + 6q^{74} - 7q^{76} + 5q^{77} - 17q^{79} + 2q^{80} - 2q^{82} - 30q^{83} - 16q^{85} + 22q^{86} - 15q^{88} - 35q^{89} - q^{91} - 24q^{92} - 11q^{94} - 13q^{95} - q^{97} + q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} - 404 x^{4} + 98 x^{3} + 118 x^{2} - 16 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{12} - 5 \nu^{11} - 11 \nu^{10} + 73 \nu^{9} + 29 \nu^{8} - 374 \nu^{7} + 46 \nu^{6} + 770 \nu^{5} - 255 \nu^{4} - 495 \nu^{3} + 228 \nu^{2} + 14 \nu - 47$$$$)/17$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{12} + \nu^{11} + 20 \nu^{10} - 17 \nu^{9} - 150 \nu^{8} + 108 \nu^{7} + 506 \nu^{6} - 320 \nu^{5} - 695 \nu^{4} + 441 \nu^{3} + 201 \nu^{2} - 184 \nu + 8$$$$)/17$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{12} + 8 \nu^{11} + 18 \nu^{10} - 101 \nu^{9} - 8 \nu^{8} + 411 \nu^{7} - 292 \nu^{6} - 516 \nu^{5} + 749 \nu^{4} - 142 \nu^{3} - 386 \nu^{2} + 176 \nu + 66$$$$)/17$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{12} - 7 \nu^{11} - 33 \nu^{10} + 104 \nu^{9} + 221 \nu^{8} - 540 \nu^{7} - 764 \nu^{6} + 1126 \nu^{5} + 1350 \nu^{4} - 788 \nu^{3} - 992 \nu^{2} + 164 \nu + 186$$$$)/17$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{12} + 14 \nu^{11} + 67 \nu^{10} - 194 \nu^{9} - 312 \nu^{8} + 960 \nu^{7} + 591 \nu^{6} - 2043 \nu^{5} - 411 \nu^{4} + 1740 \nu^{3} + 129 \nu^{2} - 427 \nu - 21$$$$)/17$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{12} + 11 \nu^{11} - 11 \nu^{10} - 140 \nu^{9} + 263 \nu^{8} + 586 \nu^{7} - 1384 \nu^{6} - 833 \nu^{5} + 2700 \nu^{4} + 66 \nu^{3} - 1696 \nu^{2} + 122 \nu + 267$$$$)/17$$ $$\beta_{10}$$ $$=$$ $$($$$$-6 \nu^{12} + 26 \nu^{11} + 58 \nu^{10} - 348 \nu^{9} - 91 \nu^{8} + 1604 \nu^{7} - 489 \nu^{6} - 2912 \nu^{5} + 1362 \nu^{4} + 1675 \nu^{3} - 463 \nu^{2} - 305 \nu - 12$$$$)/17$$ $$\beta_{11}$$ $$=$$ $$($$$$6 \nu^{12} - 26 \nu^{11} - 58 \nu^{10} + 348 \nu^{9} + 91 \nu^{8} - 1604 \nu^{7} + 506 \nu^{6} + 2895 \nu^{5} - 1532 \nu^{4} - 1556 \nu^{3} + 905 \nu^{2} + 152 \nu - 141$$$$)/17$$ $$\beta_{12}$$ $$=$$ $$($$$$7 \nu^{12} - 33 \nu^{11} - 56 \nu^{10} + 432 \nu^{9} - 51 \nu^{8} - 1924 \nu^{7} + 1321 \nu^{6} + 3278 \nu^{5} - 3112 \nu^{4} - 1551 \nu^{3} + 1747 \nu^{2} + 149 \nu - 233$$$$)/17$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 8 \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{12} - \beta_{10} - \beta_{9} - \beta_{5} - \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 29 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$-\beta_{12} + \beta_{11} - 10 \beta_{10} + 9 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} - 11 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 36 \beta_{2} + 57 \beta_{1} + 84$$ $$\nu^{7}$$ $$=$$ $$-13 \beta_{12} + 2 \beta_{11} - 13 \beta_{10} - 10 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 13 \beta_{5} - 10 \beta_{4} + 57 \beta_{3} + 68 \beta_{2} + 178 \beta_{1} + 83$$ $$\nu^{8}$$ $$=$$ $$-17 \beta_{12} + 15 \beta_{11} - 79 \beta_{10} + 61 \beta_{9} + 76 \beta_{8} - 76 \beta_{7} - 89 \beta_{5} + 59 \beta_{4} + 94 \beta_{3} + 226 \beta_{2} + 392 \beta_{1} + 498$$ $$\nu^{9}$$ $$=$$ $$-123 \beta_{12} + 32 \beta_{11} - 124 \beta_{10} - 75 \beta_{9} + 31 \beta_{8} - 33 \beta_{7} - 12 \beta_{6} - 122 \beta_{5} - 75 \beta_{4} + 394 \beta_{3} + 486 \beta_{2} + 1130 \beta_{1} + 636$$ $$\nu^{10}$$ $$=$$ $$-193 \beta_{12} + 155 \beta_{11} - 581 \beta_{10} + 369 \beta_{9} + 526 \beta_{8} - 530 \beta_{7} - 2 \beta_{6} - 650 \beta_{5} + 340 \beta_{4} + 732 \beta_{3} + 1465 \beta_{2} + 2657 \beta_{1} + 3068$$ $$\nu^{11}$$ $$=$$ $$-1028 \beta_{12} + 348 \beta_{11} - 1044 \beta_{10} - 508 \beta_{9} + 328 \beta_{8} - 365 \beta_{7} - 106 \beta_{6} - 1011 \beta_{5} - 517 \beta_{4} + 2696 \beta_{3} + 3391 \beta_{2} + 7329 \beta_{1} + 4676$$ $$\nu^{12}$$ $$=$$ $$-1837 \beta_{12} + 1376 \beta_{11} - 4155 \beta_{10} + 2096 \beta_{9} + 3502 \beta_{8} - 3585 \beta_{7} - 50 \beta_{6} - 4559 \beta_{5} + 1807 \beta_{4} + 5446 \beta_{3} + 9681 \beta_{2} + 17907 \beta_{1} + 19419$$

## 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
 2.62425 2.60074 2.38255 1.88022 1.03857 0.684638 0.468970 −0.329361 −0.534142 −0.762797 −1.74673 −1.96783 −2.33909
−2.62425 0 4.88668 −0.233370 0 1.03556 −7.57538 0 0.612421
1.2 −2.60074 0 4.76385 0.540623 0 4.31275 −7.18805 0 −1.40602
1.3 −2.38255 0 3.67655 −3.76064 0 −1.09133 −3.99445 0 8.95992
1.4 −1.88022 0 1.53524 −1.27827 0 0.0298223 0.873847 0 2.40344
1.5 −1.03857 0 −0.921370 −2.59019 0 2.45800 3.03405 0 2.69009
1.6 −0.684638 0 −1.53127 1.50791 0 2.85777 2.41764 0 −1.03237
1.7 −0.468970 0 −1.78007 3.25579 0 −3.43403 1.77274 0 −1.52687
1.8 0.329361 0 −1.89152 −2.08793 0 −4.32416 −1.28171 0 −0.687681
1.9 0.534142 0 −1.71469 −3.62612 0 4.31243 −1.98417 0 −1.93686
1.10 0.762797 0 −1.41814 1.72725 0 1.91723 −2.60735 0 1.31754
1.11 1.74673 0 1.05105 2.68410 0 −0.749083 −1.65755 0 4.68839
1.12 1.96783 0 1.87234 −3.39588 0 0.824946 −0.251204 0 −6.68250
1.13 2.33909 0 3.47134 0.256721 0 −3.14990 3.44160 0 0.600493
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.13 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.g 13
3.b odd 2 1 2013.2.a.f 13

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.f 13 3.b odd 2 1
6039.2.a.g 13 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{13} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 15 T^{2} + 41 T^{3} + 102 T^{4} + 222 T^{5} + 457 T^{6} + 862 T^{7} + 1569 T^{8} + 2668 T^{9} + 4388 T^{10} + 6818 T^{11} + 10284 T^{12} + 14739 T^{13} + 20568 T^{14} + 27272 T^{15} + 35104 T^{16} + 42688 T^{17} + 50208 T^{18} + 55168 T^{19} + 58496 T^{20} + 56832 T^{21} + 52224 T^{22} + 41984 T^{23} + 30720 T^{24} + 16384 T^{25} + 8192 T^{26}$$
$3$ 1
$5$ $$1 + 7 T + 52 T^{2} + 250 T^{3} + 1179 T^{4} + 4487 T^{5} + 16623 T^{6} + 53521 T^{7} + 167951 T^{8} + 473540 T^{9} + 1304029 T^{10} + 3279296 T^{11} + 8063973 T^{12} + 18231446 T^{13} + 40319865 T^{14} + 81982400 T^{15} + 163003625 T^{16} + 295962500 T^{17} + 524846875 T^{18} + 836265625 T^{19} + 1298671875 T^{20} + 1752734375 T^{21} + 2302734375 T^{22} + 2441406250 T^{23} + 2539062500 T^{24} + 1708984375 T^{25} + 1220703125 T^{26}$$
$7$ $$1 - 5 T + 54 T^{2} - 211 T^{3} + 1344 T^{4} - 4456 T^{5} + 21864 T^{6} - 64578 T^{7} + 269196 T^{8} - 723175 T^{9} + 2673431 T^{10} - 6578374 T^{11} + 22087536 T^{12} - 50044702 T^{13} + 154612752 T^{14} - 322340326 T^{15} + 916986833 T^{16} - 1736343175 T^{17} + 4524377172 T^{18} - 7597537122 T^{19} + 18005944152 T^{20} - 25687953256 T^{21} + 54235247808 T^{22} - 59602277539 T^{23} + 106775644122 T^{24} - 69206436005 T^{25} + 96889010407 T^{26}$$
$11$ $$( 1 - T )^{13}$$
$13$ $$1 + 9 T + 126 T^{2} + 897 T^{3} + 7314 T^{4} + 43127 T^{5} + 266094 T^{6} + 1341054 T^{7} + 6894558 T^{8} + 30470425 T^{9} + 136685974 T^{10} + 540624532 T^{11} + 2168911565 T^{12} + 7765534072 T^{13} + 28195850345 T^{14} + 91365545908 T^{15} + 300299084878 T^{16} + 870265808425 T^{17} + 2559901123494 T^{18} + 6473011516686 T^{19} + 16697003882598 T^{20} + 35180018804567 T^{21} + 77561308414122 T^{22} + 123659067188553 T^{23} + 225812209648662 T^{24} + 209682766102329 T^{25} + 302875106592253 T^{26}$$
$17$ $$1 + 7 T + 172 T^{2} + 1134 T^{3} + 14517 T^{4} + 87622 T^{5} + 787227 T^{6} + 4278285 T^{7} + 30327181 T^{8} + 147154624 T^{9} + 871359928 T^{10} + 3756415546 T^{11} + 19150454978 T^{12} + 72946960856 T^{13} + 325557734626 T^{14} + 1085604092794 T^{15} + 4280991326264 T^{16} + 12290501351104 T^{17} + 43060260233117 T^{18} + 103267399389165 T^{19} + 323029682529771 T^{20} + 611229818495302 T^{21} + 1721540203106949 T^{22} + 2286137083109166 T^{23} + 5894766164912876 T^{24} + 4078355660608327 T^{25} + 9904578032905937 T^{26}$$
$19$ $$1 - 2 T + 158 T^{2} - 284 T^{3} + 12041 T^{4} - 19844 T^{5} + 594592 T^{6} - 900328 T^{7} + 21505328 T^{8} - 29700114 T^{9} + 608699913 T^{10} - 762713263 T^{11} + 13995831467 T^{12} - 15936313000 T^{13} + 265920797873 T^{14} - 275339487943 T^{15} + 4175072703267 T^{16} - 3870548556594 T^{17} + 53249321155472 T^{18} - 42356723948968 T^{19} + 531488985035488 T^{20} - 337021824985604 T^{21} + 3885482568956939 T^{22} - 1741222817215484 T^{23} + 18405460905918602 T^{24} - 4426629838132322 T^{25} + 42052983462257059 T^{26}$$
$23$ $$1 + 23 T + 405 T^{2} + 5212 T^{3} + 57841 T^{4} + 550153 T^{5} + 4713705 T^{6} + 36379173 T^{7} + 258356891 T^{8} + 1689249269 T^{9} + 10279724648 T^{10} + 58203767172 T^{11} + 308458660933 T^{12} + 1527720335132 T^{13} + 7094549201459 T^{14} + 30789792833988 T^{15} + 125073409792216 T^{16} + 472721204686229 T^{17} + 1662873566889613 T^{18} + 5385423216139797 T^{19} + 16049342733651135 T^{20} + 43083023485297993 T^{21} + 104180471091681383 T^{22} + 215914976445538588 T^{23} + 385887951955140435 T^{24} + 504036361936467383 T^{25} + 504036361936467383 T^{26}$$
$29$ $$1 + 16 T + 339 T^{2} + 3906 T^{3} + 49572 T^{4} + 460161 T^{5} + 4444691 T^{6} + 35015453 T^{7} + 280272252 T^{8} + 1924075768 T^{9} + 13277547428 T^{10} + 80523088202 T^{11} + 488644075869 T^{12} + 2632299221416 T^{13} + 14170678200201 T^{14} + 67719917177882 T^{15} + 323826104221492 T^{16} + 1360862233266808 T^{17} + 5748705921337548 T^{18} + 20828008039779413 T^{19} + 76670369981725519 T^{20} + 230193889634546721 T^{21} + 719148240315778068 T^{22} + 1643282453270585106 T^{23} + 4135972810574276031 T^{24} + 5661036531287504656 T^{25} + 10260628712958602189 T^{26}$$
$31$ $$1 - 9 T + 244 T^{2} - 1906 T^{3} + 27879 T^{4} - 201154 T^{5} + 2080402 T^{6} - 14398334 T^{7} + 116789053 T^{8} - 780130930 T^{9} + 5250384513 T^{10} - 33363036399 T^{11} + 194535543362 T^{12} - 1148596132950 T^{13} + 6030601844222 T^{14} - 32061877979439 T^{15} + 156414205026783 T^{16} - 720467296604530 T^{17} + 3343571433484003 T^{18} - 12778574425267454 T^{19} + 57237297421752622 T^{20} - 171562443745406914 T^{21} + 737110226217346809 T^{22} - 1562211514985406706 T^{23} + 6199668362722778764 T^{24} - 7088965054096947849 T^{25} + 24417546297445042591 T^{26}$$
$37$ $$1 - 14 T + 415 T^{2} - 4761 T^{3} + 79829 T^{4} - 777369 T^{5} + 9566800 T^{6} - 80759411 T^{7} + 803041239 T^{8} - 5948863461 T^{9} + 50052301916 T^{10} - 327293775093 T^{11} + 2387786137488 T^{12} - 13787787297494 T^{13} + 88348087087056 T^{14} - 448065178102317 T^{15} + 2535299248951148 T^{16} - 11149127892931221 T^{17} + 55686057146442723 T^{18} - 207206553577985099 T^{19} + 908194282155984400 T^{20} - 2730492640615113849 T^{21} + 10374715726101201833 T^{22} - 22893670197081379089 T^{23} + 73835813038476071395 T^{24} - 92161328081760493934 T^{25} +$$$$24\!\cdots\!97$$$$T^{26}$$
$41$ $$1 + 19 T + 570 T^{2} + 8078 T^{3} + 140623 T^{4} + 1604457 T^{5} + 20664839 T^{6} + 197699421 T^{7} + 2057334849 T^{8} + 16890207028 T^{9} + 148144053509 T^{10} + 1056883015886 T^{11} + 7987872705573 T^{12} + 49754457667820 T^{13} + 327502780928493 T^{14} + 1776620349704366 T^{15} + 10210236311893789 T^{16} + 47727688301648308 T^{17} + 238354999790048649 T^{18} + 939092858135344461 T^{19} + 4024565714312770159 T^{20} + 12811469178339792297 T^{21} + 46037429760281977703 T^{22} +$$$$10\!\cdots\!78$$$$T^{23} +$$$$31\!\cdots\!70$$$$T^{24} +$$$$42\!\cdots\!39$$$$T^{25} +$$$$92\!\cdots\!21$$$$T^{26}$$
$43$ $$1 - 7 T + 411 T^{2} - 3130 T^{3} + 81725 T^{4} - 651656 T^{5} + 10451069 T^{6} - 84087081 T^{7} + 960506150 T^{8} - 7525014680 T^{9} + 66998871224 T^{10} - 493515264974 T^{11} + 3647340651788 T^{12} - 24361840908780 T^{13} + 156835648026884 T^{14} - 912509724936926 T^{15} + 5326879254406568 T^{16} - 25726527712998680 T^{17} + 141202513603424450 T^{18} - 531544966731669969 T^{19} + 2840795060163423383 T^{20} - 7616685840100357256 T^{21} + 41074381210538494175 T^{22} - 67643939640579699370 T^{23} +$$$$38\!\cdots\!77$$$$T^{24} -$$$$27\!\cdots\!07$$$$T^{25} +$$$$17\!\cdots\!43$$$$T^{26}$$
$47$ $$1 + 26 T + 650 T^{2} + 11372 T^{3} + 180271 T^{4} + 2414607 T^{5} + 29737747 T^{6} + 326628724 T^{7} + 3338580611 T^{8} + 31177483834 T^{9} + 272772989422 T^{10} + 2206112519221 T^{11} + 16781305763628 T^{12} + 118546280860676 T^{13} + 788721370890516 T^{14} + 4873302554959189 T^{15} + 28320110080760306 T^{16} + 152136175492576954 T^{17} + 765686793599859277 T^{18} + 3520801348632510196 T^{19} + 15065830180679216861 T^{20} + 57494899452494742927 T^{21} +$$$$20\!\cdots\!57$$$$T^{22} +$$$$59\!\cdots\!28$$$$T^{23} +$$$$16\!\cdots\!50$$$$T^{24} +$$$$30\!\cdots\!66$$$$T^{25} +$$$$54\!\cdots\!27$$$$T^{26}$$
$53$ $$1 + 18 T + 397 T^{2} + 5159 T^{3} + 69779 T^{4} + 685382 T^{5} + 6697901 T^{6} + 48698795 T^{7} + 339648838 T^{8} + 1360288879 T^{9} + 2739046927 T^{10} - 66090461863 T^{11} - 752023040015 T^{12} - 7444225952930 T^{13} - 39857221120795 T^{14} - 185648107373167 T^{15} + 407781089350979 T^{16} + 10733333554260799 T^{17} + 142039613254287134 T^{18} + 1079377678927139555 T^{19} + 7868098918225382137 T^{20} + 42671671133519424902 T^{21} +$$$$23\!\cdots\!07$$$$T^{22} +$$$$90\!\cdots\!91$$$$T^{23} +$$$$36\!\cdots\!09$$$$T^{24} +$$$$88\!\cdots\!38$$$$T^{25} +$$$$26\!\cdots\!73$$$$T^{26}$$
$59$ $$1 + 31 T + 758 T^{2} + 11717 T^{3} + 161239 T^{4} + 1691848 T^{5} + 17364310 T^{6} + 146609274 T^{7} + 1343833743 T^{8} + 10554092416 T^{9} + 97314500021 T^{10} + 763530615380 T^{11} + 6882032222794 T^{12} + 50283233012916 T^{13} + 406039901144846 T^{14} + 2657850072137780 T^{15} + 19986354699812959 T^{16} + 127887747832034176 T^{17} + 960739396686821157 T^{18} + 6184057414039586634 T^{19} + 43213715864357409890 T^{20} +$$$$24\!\cdots\!08$$$$T^{21} +$$$$13\!\cdots\!21$$$$T^{22} +$$$$59\!\cdots\!17$$$$T^{23} +$$$$22\!\cdots\!22$$$$T^{24} +$$$$55\!\cdots\!11$$$$T^{25} +$$$$10\!\cdots\!79$$$$T^{26}$$
$61$ $$( 1 + T )^{13}$$
$67$ $$1 - 14 T + 462 T^{2} - 5387 T^{3} + 105068 T^{4} - 1066907 T^{5} + 15781773 T^{6} - 145190299 T^{7} + 1792731622 T^{8} - 15258806275 T^{9} + 165152548835 T^{10} - 1313267597677 T^{11} + 12847016504773 T^{12} - 95311824122652 T^{13} + 860750105819791 T^{14} - 5895258245972053 T^{15} + 49671776045261105 T^{16} - 307482051563084275 T^{17} + 2420411972975033554 T^{18} - 13133679554173378531 T^{19} + 95648774773673177679 T^{20} -$$$$43\!\cdots\!87$$$$T^{21} +$$$$28\!\cdots\!96$$$$T^{22} -$$$$98\!\cdots\!63$$$$T^{23} +$$$$56\!\cdots\!46$$$$T^{24} -$$$$11\!\cdots\!54$$$$T^{25} +$$$$54\!\cdots\!87$$$$T^{26}$$
$71$ $$1 + 37 T + 901 T^{2} + 14746 T^{3} + 206290 T^{4} + 2506621 T^{5} + 30453105 T^{6} + 345876887 T^{7} + 3803389784 T^{8} + 37856060224 T^{9} + 366954231344 T^{10} + 3367335479732 T^{11} + 30842666414903 T^{12} + 263798235920490 T^{13} + 2189829315458113 T^{14} + 16974738153329012 T^{15} + 131336955894562384 T^{16} + 961986126329076544 T^{17} + 6862187481586350184 T^{18} + 44306927426411633927 T^{19} +$$$$27\!\cdots\!55$$$$T^{20} +$$$$16\!\cdots\!81$$$$T^{21} +$$$$94\!\cdots\!90$$$$T^{22} +$$$$48\!\cdots\!46$$$$T^{23} +$$$$20\!\cdots\!71$$$$T^{24} +$$$$60\!\cdots\!17$$$$T^{25} +$$$$11\!\cdots\!11$$$$T^{26}$$
$73$ $$1 + 16 T + 630 T^{2} + 8528 T^{3} + 185642 T^{4} + 2185728 T^{5} + 34781245 T^{6} + 364567989 T^{7} + 4741032295 T^{8} + 45021431638 T^{9} + 506918130739 T^{10} + 4405175370490 T^{11} + 44355380310892 T^{12} + 353337388110734 T^{13} + 3237942762695116 T^{14} + 23475179549341210 T^{15} + 197199770465693563 T^{16} + 1278529465820948758 T^{17} + 9828499372260095935 T^{18} + 55171614545051662821 T^{19} +$$$$38\!\cdots\!65$$$$T^{20} +$$$$17\!\cdots\!68$$$$T^{21} +$$$$10\!\cdots\!46$$$$T^{22} +$$$$36\!\cdots\!72$$$$T^{23} +$$$$19\!\cdots\!10$$$$T^{24} +$$$$36\!\cdots\!36$$$$T^{25} +$$$$16\!\cdots\!33$$$$T^{26}$$
$79$ $$1 + 17 T + 622 T^{2} + 8010 T^{3} + 182593 T^{4} + 1960260 T^{5} + 35114164 T^{6} + 328234756 T^{7} + 5053274170 T^{8} + 42318143615 T^{9} + 581709871210 T^{10} + 4424991480327 T^{11} + 55189321376644 T^{12} + 383304844703314 T^{13} + 4359956388754876 T^{14} + 27616371828720807 T^{15} + 286805653191507190 T^{16} + 1648295121573882815 T^{17} + 15549209620699913830 T^{18} + 79789751649596287876 T^{19} +$$$$67\!\cdots\!76$$$$T^{20} +$$$$29\!\cdots\!60$$$$T^{21} +$$$$21\!\cdots\!67$$$$T^{22} +$$$$75\!\cdots\!10$$$$T^{23} +$$$$46\!\cdots\!38$$$$T^{24} +$$$$10\!\cdots\!97$$$$T^{25} +$$$$46\!\cdots\!39$$$$T^{26}$$
$83$ $$1 + 30 T + 974 T^{2} + 18458 T^{3} + 366294 T^{4} + 5398142 T^{5} + 83760451 T^{6} + 1050757230 T^{7} + 13881448384 T^{8} + 153604165536 T^{9} + 1782849294068 T^{10} + 17626677690403 T^{11} + 182574569560402 T^{12} + 1624662755159746 T^{13} + 15153689273513366 T^{14} + 121430182609186267 T^{15} + 1019410049307259516 T^{16} + 7289795794944625056 T^{17} + 54679589368282670912 T^{18} +$$$$34\!\cdots\!70$$$$T^{19} +$$$$22\!\cdots\!77$$$$T^{20} +$$$$12\!\cdots\!22$$$$T^{21} +$$$$68\!\cdots\!82$$$$T^{22} +$$$$28\!\cdots\!42$$$$T^{23} +$$$$12\!\cdots\!58$$$$T^{24} +$$$$32\!\cdots\!30$$$$T^{25} +$$$$88\!\cdots\!63$$$$T^{26}$$
$89$ $$1 + 35 T + 1184 T^{2} + 25669 T^{3} + 516942 T^{4} + 8370310 T^{5} + 127074742 T^{6} + 1696233422 T^{7} + 21565020894 T^{8} + 253402493441 T^{9} + 2858507560085 T^{10} + 30474939017346 T^{11} + 310586541225446 T^{12} + 3003263361737920 T^{13} + 27642202169064694 T^{14} + 241391991956397666 T^{15} + 2015159216125562365 T^{16} + 15899040313476141281 T^{17} +$$$$12\!\cdots\!06$$$$T^{18} +$$$$84\!\cdots\!42$$$$T^{19} +$$$$56\!\cdots\!18$$$$T^{20} +$$$$32\!\cdots\!10$$$$T^{21} +$$$$18\!\cdots\!78$$$$T^{22} +$$$$80\!\cdots\!69$$$$T^{23} +$$$$32\!\cdots\!76$$$$T^{24} +$$$$86\!\cdots\!35$$$$T^{25} +$$$$21\!\cdots\!69$$$$T^{26}$$
$97$ $$1 + T + 606 T^{2} - 479 T^{3} + 180587 T^{4} - 591622 T^{5} + 35070746 T^{6} - 224773222 T^{7} + 5007271033 T^{8} - 50444281573 T^{9} + 575603368072 T^{10} - 7793122388836 T^{11} + 58515039743159 T^{12} - 878038609738246 T^{13} + 5675958855086423 T^{14} - 73325488556557924 T^{15} + 525337652748376456 T^{16} - 4465795978219239013 T^{17} + 42999140119390875481 T^{18} -$$$$18\!\cdots\!38$$$$T^{19} +$$$$28\!\cdots\!98$$$$T^{20} -$$$$46\!\cdots\!42$$$$T^{21} +$$$$13\!\cdots\!79$$$$T^{22} -$$$$35\!\cdots\!71$$$$T^{23} +$$$$43\!\cdots\!18$$$$T^{24} +$$$$69\!\cdots\!41$$$$T^{25} +$$$$67\!\cdots\!77$$$$T^{26}$$
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