# Properties

 Label 6039.2.a.j Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $0$ Dimension $14$ 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: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{10} q^{5} + ( 1 - \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{10} q^{5} + ( 1 - \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -\beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{10} + q^{11} + ( 1 - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{13} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{9} - \beta_{10} ) q^{14} + ( 1 + \beta_{2} - \beta_{5} + \beta_{10} + \beta_{11} ) q^{16} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{17} + ( 1 - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{19} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} - 2 \beta_{13} ) q^{20} + \beta_{1} q^{22} + ( -1 + \beta_{2} + \beta_{6} - \beta_{8} + \beta_{12} + \beta_{13} ) q^{23} + ( 1 + \beta_{1} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{25} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{26} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{28} + ( -1 + \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} ) q^{29} + ( 2 + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} ) q^{31} + ( \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{32} + ( 2 - \beta_{1} - \beta_{3} - \beta_{7} + \beta_{11} - 2 \beta_{12} ) q^{34} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{35} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{13} ) q^{37} + ( -1 + 2 \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{38} + ( -2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} + \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{40} + ( 3 - \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{41} + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{8} - \beta_{12} + \beta_{13} ) q^{43} + ( 1 + \beta_{2} ) q^{44} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{46} + ( -4 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} ) q^{47} + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{49} + ( 3 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - 4 \beta_{12} ) q^{50} + ( 2 \beta_{1} - \beta_{2} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - 3 \beta_{13} ) q^{52} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{9} - 2 \beta_{10} - \beta_{12} - \beta_{13} ) q^{53} -\beta_{10} q^{55} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{56} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{9} + 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} ) q^{58} + ( -\beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{59} + q^{61} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{62} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{64} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{65} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{67} + ( -6 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{68} + ( 4 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{70} + ( 2 + 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} ) q^{71} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{73} + ( 3 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{74} + ( 3 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{76} + ( 1 - \beta_{4} ) q^{77} + ( 2 - 3 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{79} + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - 2 \beta_{12} ) q^{80} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{82} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{83} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{85} + ( -1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} + 3 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{86} + ( \beta_{1} + \beta_{3} ) q^{88} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{89} + ( 5 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{91} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{92} + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} + 4 \beta_{9} - 5 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{94} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{95} + ( 2 + 4 \beta_{1} + \beta_{2} - \beta_{4} - 4 \beta_{6} + 2 \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{97} + ( 1 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + q^{2} + 15q^{4} - q^{5} + 9q^{7} + O(q^{10})$$ $$14q + q^{2} + 15q^{4} - q^{5} + 9q^{7} + 6q^{10} + 14q^{11} + q^{13} + 7q^{14} + 17q^{16} + 9q^{17} + 22q^{19} - 23q^{20} + q^{22} - q^{23} + 25q^{25} - 4q^{26} + 37q^{28} + 6q^{29} + 9q^{31} - 4q^{32} + 8q^{34} - 18q^{35} + 18q^{37} - 8q^{38} + 16q^{40} + 25q^{41} + 25q^{43} + 15q^{44} + 20q^{46} - 36q^{47} + 25q^{49} - 2q^{50} - 13q^{52} - q^{55} + 40q^{56} + 33q^{58} - 17q^{59} + 14q^{61} + 13q^{62} - 6q^{64} + 61q^{65} + 22q^{67} - 66q^{68} + 44q^{70} + 13q^{71} + 20q^{73} + 12q^{74} + 49q^{76} + 9q^{77} + 31q^{79} - 88q^{80} + 2q^{82} - 32q^{83} + 2q^{85} + 14q^{86} + 21q^{89} + 45q^{91} + 14q^{92} - 31q^{94} - 23q^{95} + 37q^{97} + 38q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{13} - 71 \nu^{12} - 586 \nu^{11} + 2838 \nu^{10} + 10454 \nu^{9} - 35539 \nu^{8} - 62411 \nu^{7} + 186606 \nu^{6} + 140254 \nu^{5} - 411132 \nu^{4} - 83942 \nu^{3} + 307840 \nu^{2} - 27224 \nu - 13953$$$$)/4505$$ $$\beta_{5}$$ $$=$$ $$($$$$-82 \nu^{13} - 1317 \nu^{12} + 1503 \nu^{11} + 25486 \nu^{10} - 7732 \nu^{9} - 179663 \nu^{8} - 4527 \nu^{7} + 556827 \nu^{6} + 125703 \nu^{5} - 713699 \nu^{4} - 233864 \nu^{3} + 253645 \nu^{2} + 24637 \nu - 22401$$$$)/4505$$ $$\beta_{6}$$ $$=$$ $$($$$$-311 \nu^{13} + 444 \nu^{12} + 6964 \nu^{11} - 9372 \nu^{10} - 59981 \nu^{9} + 74721 \nu^{8} + 245494 \nu^{7} - 273749 \nu^{6} - 466936 \nu^{5} + 435843 \nu^{4} + 306853 \nu^{3} - 209250 \nu^{2} + 43281 \nu - 1068$$$$)/4505$$ $$\beta_{7}$$ $$=$$ $$($$$$-349 \nu^{13} - 2254 \nu^{12} + 7221 \nu^{11} + 44412 \nu^{10} - 54664 \nu^{9} - 320701 \nu^{8} + 180436 \nu^{7} + 1032859 \nu^{6} - 232439 \nu^{5} - 1428903 \nu^{4} + 76842 \nu^{3} + 618105 \nu^{2} - 67706 \nu - 17712$$$$)/4505$$ $$\beta_{8}$$ $$=$$ $$($$$$-251 \nu^{13} + 199 \nu^{12} + 5183 \nu^{11} - 3957 \nu^{10} - 40303 \nu^{9} + 29344 \nu^{8} + 146488 \nu^{7} - 100390 \nu^{6} - 248794 \nu^{5} + 158677 \nu^{4} + 158118 \nu^{3} - 92020 \nu^{2} - 2743 \nu + 885$$$$)/901$$ $$\beta_{9}$$ $$=$$ $$($$$$1423 \nu^{13} - 2582 \nu^{12} - 31082 \nu^{11} + 52066 \nu^{10} + 260758 \nu^{9} - 388563 \nu^{8} - 1050382 \nu^{7} + 1308832 \nu^{6} + 2048843 \nu^{5} - 1915614 \nu^{4} - 1595379 \nu^{3} + 883850 \nu^{2} + 113882 \nu - 20941$$$$)/4505$$ $$\beta_{10}$$ $$=$$ $$($$$$1433 \nu^{13} - 1872 \nu^{12} - 29727 \nu^{11} + 37201 \nu^{10} + 232803 \nu^{9} - 271938 \nu^{8} - 858752 \nu^{7} + 893382 \nu^{6} + 1511263 \nu^{5} - 1286659 \nu^{4} - 1048784 \nu^{3} + 607560 \nu^{2} + 52752 \nu - 12056$$$$)/4505$$ $$\beta_{11}$$ $$=$$ $$($$$$-303 \nu^{13} + 111 \nu^{12} + 6246 \nu^{11} - 2343 \nu^{10} - 48107 \nu^{9} + 18455 \nu^{8} + 170845 \nu^{7} - 67311 \nu^{6} - 277112 \nu^{5} + 115493 \nu^{4} + 162984 \nu^{3} - 77090 \nu^{2} - 5623 \nu + 3337$$$$)/901$$ $$\beta_{12}$$ $$=$$ $$($$$$-1576 \nu^{13} + 729 \nu^{12} + 31524 \nu^{11} - 14292 \nu^{10} - 233541 \nu^{9} + 104816 \nu^{8} + 786304 \nu^{7} - 360249 \nu^{6} - 1177831 \nu^{5} + 604778 \nu^{4} + 591393 \nu^{3} - 424595 \nu^{2} + 14111 \nu + 21502$$$$)/4505$$ $$\beta_{13}$$ $$=$$ $$($$$$-3393 \nu^{13} + 2367 \nu^{12} + 70487 \nu^{11} - 47406 \nu^{10} - 551558 \nu^{9} + 352898 \nu^{8} + 2019747 \nu^{7} - 1203902 \nu^{6} - 3477208 \nu^{5} + 1876454 \nu^{4} + 2324484 \nu^{3} - 1064330 \nu^{2} - 176207 \nu + 36556$$$$)/4505$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{5} + 7 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$-\beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + 9 \beta_{3} + 28 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{13} + 9 \beta_{11} + 9 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - 10 \beta_{5} + \beta_{3} + 48 \beta_{2} - \beta_{1} + 84$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{13} - 12 \beta_{12} + 10 \beta_{11} - \beta_{10} + 13 \beta_{9} + 9 \beta_{8} + 12 \beta_{6} - 12 \beta_{5} - \beta_{4} + 67 \beta_{3} + 2 \beta_{2} + 168 \beta_{1} - 1$$ $$\nu^{8}$$ $$=$$ $$16 \beta_{13} - 2 \beta_{12} + 64 \beta_{11} + 66 \beta_{10} + 16 \beta_{9} - 26 \beta_{8} - \beta_{7} + 28 \beta_{6} - 79 \beta_{5} - \beta_{4} + 14 \beta_{3} + 326 \beta_{2} - 13 \beta_{1} + 504$$ $$\nu^{9}$$ $$=$$ $$35 \beta_{13} - 109 \beta_{12} + 79 \beta_{11} - 10 \beta_{10} + 125 \beta_{9} + 57 \beta_{8} - 4 \beta_{7} + 114 \beta_{6} - 106 \beta_{5} - 12 \beta_{4} + 472 \beta_{3} + 33 \beta_{2} + 1046 \beta_{1} - 12$$ $$\nu^{10}$$ $$=$$ $$176 \beta_{13} - 37 \beta_{12} + 426 \beta_{11} + 464 \beta_{10} + 174 \beta_{9} - 243 \beta_{8} - 20 \beta_{7} + 282 \beta_{6} - 579 \beta_{5} - 11 \beta_{4} + 141 \beta_{3} + 2204 \beta_{2} - 118 \beta_{1} + 3147$$ $$\nu^{11}$$ $$=$$ $$405 \beta_{13} - 898 \beta_{12} + 586 \beta_{11} - 56 \beta_{10} + 1066 \beta_{9} + 302 \beta_{8} - 75 \beta_{7} + 984 \beta_{6} - 839 \beta_{5} - 98 \beta_{4} + 3258 \beta_{3} + 370 \beta_{2} + 6644 \beta_{1} - 89$$ $$\nu^{12}$$ $$=$$ $$1652 \beta_{13} - 443 \beta_{12} + 2777 \beta_{11} + 3245 \beta_{10} + 1609 \beta_{9} - 2010 \beta_{8} - 256 \beta_{7} + 2491 \beta_{6} - 4113 \beta_{5} - 72 \beta_{4} + 1251 \beta_{3} + 14874 \beta_{2} - 923 \beta_{1} + 20093$$ $$\nu^{13}$$ $$=$$ $$3916 \beta_{13} - 7055 \beta_{12} + 4261 \beta_{11} - 128 \beta_{10} + 8540 \beta_{9} + 1339 \beta_{8} - 911 \beta_{7} + 8029 \beta_{6} - 6318 \beta_{5} - 681 \beta_{4} + 22304 \beta_{3} + 3528 \beta_{2} + 42680 \beta_{1} - 465$$

## 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
 −2.57087 −2.54331 −1.76637 −1.69494 −1.18140 −0.231279 −0.0561655 0.179763 0.546298 1.61392 1.67203 1.93923 2.45909 2.63401
−2.57087 0 4.60938 −4.00721 0 2.04273 −6.70837 0 10.3020
1.2 −2.54331 0 4.46845 −0.329781 0 0.595463 −6.27804 0 0.838736
1.3 −1.76637 0 1.12006 2.50284 0 −4.08919 1.55430 0 −4.42094
1.4 −1.69494 0 0.872835 −1.13650 0 4.14659 1.91048 0 1.92630
1.5 −1.18140 0 −0.604292 2.90081 0 −0.528840 3.07671 0 −3.42702
1.6 −0.231279 0 −1.94651 −3.70694 0 −0.911108 0.912746 0 0.857338
1.7 −0.0561655 0 −1.99685 2.87016 0 3.53988 0.224485 0 −0.161204
1.8 0.179763 0 −1.96769 −2.20477 0 3.17177 −0.713244 0 −0.396336
1.9 0.546298 0 −1.70156 0.842631 0 −4.19208 −2.02216 0 0.460328
1.10 1.61392 0 0.604750 3.65776 0 0.985739 −2.25183 0 5.90335
1.11 1.67203 0 0.795679 1.45953 0 −0.113904 −2.01366 0 2.44037
1.12 1.93923 0 1.76061 −3.07794 0 −3.15900 −0.464237 0 −5.96883
1.13 2.45909 0 4.04715 1.85442 0 2.32343 5.03413 0 4.56020
1.14 2.63401 0 4.93799 −2.62502 0 5.18852 7.73867 0 −6.91432
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 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.j 14
3.b odd 2 1 2013.2.a.h 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.2.a.h 14 3.b odd 2 1
6039.2.a.j 14 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{14} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 15 T + 74 T^{2} + 386 T^{3} - 802 T^{4} - 745 T^{5} + 1123 T^{6} + 497 T^{7} - 627 T^{8} - 148 T^{9} + 167 T^{10} + 20 T^{11} - 21 T^{12} - T^{13} + T^{14}$$
$3$ $$T^{14}$$
$5$ $$-17240 - 31268 T + 85462 T^{2} + 37207 T^{3} - 94641 T^{4} - 9611 T^{5} + 40005 T^{6} + 5 T^{7} - 8198 T^{8} + 261 T^{9} + 876 T^{10} - 30 T^{11} - 47 T^{12} + T^{13} + T^{14}$$
$7$ $$2000 + 15844 T - 23864 T^{2} - 67987 T^{3} + 88676 T^{4} + 46348 T^{5} - 87098 T^{6} + 18359 T^{7} + 13200 T^{8} - 5204 T^{9} - 425 T^{10} + 387 T^{11} - 21 T^{12} - 9 T^{13} + T^{14}$$
$11$ $$( -1 + T )^{14}$$
$13$ $$-30904 + 161844 T + 2526238 T^{2} + 311705 T^{3} - 2104113 T^{4} - 142588 T^{5} + 565498 T^{6} + 22553 T^{7} - 66920 T^{8} - 1837 T^{9} + 3799 T^{10} + 71 T^{11} - 101 T^{12} - T^{13} + T^{14}$$
$17$ $$-2464 - 34560 T + 52466 T^{2} + 321879 T^{3} - 194290 T^{4} - 469545 T^{5} + 103619 T^{6} + 170783 T^{7} - 16478 T^{8} - 19030 T^{9} + 1680 T^{10} + 760 T^{11} - 77 T^{12} - 9 T^{13} + T^{14}$$
$19$ $$-372904 - 907542 T + 1669088 T^{2} + 1826469 T^{3} - 3898270 T^{4} + 1662652 T^{5} + 296423 T^{6} - 371220 T^{7} + 59143 T^{8} + 17420 T^{9} - 6374 T^{10} + 272 T^{11} + 139 T^{12} - 22 T^{13} + T^{14}$$
$23$ $$-101900800 + 57121760 T + 99520576 T^{2} - 27702169 T^{3} - 32706597 T^{4} + 3961472 T^{5} + 4413930 T^{6} - 270777 T^{7} - 290230 T^{8} + 9821 T^{9} + 9757 T^{10} - 170 T^{11} - 160 T^{12} + T^{13} + T^{14}$$
$29$ $$4669602512 - 1405884280 T - 1852620762 T^{2} + 697279679 T^{3} + 219514428 T^{4} - 113502316 T^{5} - 4863380 T^{6} + 7344831 T^{7} - 488125 T^{8} - 174531 T^{9} + 19390 T^{10} + 1714 T^{11} - 242 T^{12} - 6 T^{13} + T^{14}$$
$31$ $$302065736 - 17480178 T - 325278958 T^{2} + 103540831 T^{3} + 73299057 T^{4} - 31318191 T^{5} - 5283630 T^{6} + 3335674 T^{7} + 11491 T^{8} - 141160 T^{9} + 9594 T^{10} + 2032 T^{11} - 195 T^{12} - 9 T^{13} + T^{14}$$
$37$ $$-207834696 - 110411748 T + 285888846 T^{2} + 229415769 T^{3} - 18042521 T^{4} - 45409895 T^{5} - 2052421 T^{6} + 3619646 T^{7} + 158967 T^{8} - 145265 T^{9} - 957 T^{10} + 2753 T^{11} - 86 T^{12} - 18 T^{13} + T^{14}$$
$41$ $$7101316 - 124448906 T + 373431448 T^{2} - 291274201 T^{3} - 37905795 T^{4} + 122990661 T^{5} - 40604543 T^{6} - 665011 T^{7} + 2146054 T^{8} - 208935 T^{9} - 28550 T^{10} + 4844 T^{11} - 19 T^{12} - 25 T^{13} + T^{14}$$
$43$ $$6851179168 - 19783965700 T - 10187580836 T^{2} + 4352781779 T^{3} + 1632097208 T^{4} - 403996347 T^{5} - 93038263 T^{6} + 19322373 T^{7} + 2205340 T^{8} - 481578 T^{9} - 17181 T^{10} + 5736 T^{11} - 70 T^{12} - 25 T^{13} + T^{14}$$
$47$ $$-2272971239168 - 1005157933468 T + 143891921668 T^{2} + 126973424103 T^{3} + 8498161938 T^{4} - 4807400482 T^{5} - 768562155 T^{6} + 46325626 T^{7} + 17027636 T^{8} + 571721 T^{9} - 123454 T^{10} - 10114 T^{11} + 123 T^{12} + 36 T^{13} + T^{14}$$
$53$ $$-50664924 + 338642262 T + 802110732 T^{2} - 1644210867 T^{3} - 872460391 T^{4} + 142381689 T^{5} + 79199589 T^{6} - 4285658 T^{7} - 2730160 T^{8} + 52884 T^{9} + 44255 T^{10} - 227 T^{11} - 340 T^{12} + T^{14}$$
$59$ $$10127082816 - 1059850512 T - 18546914676 T^{2} - 7042253019 T^{3} + 4361986283 T^{4} + 2106539067 T^{5} - 117743352 T^{6} - 92229603 T^{7} - 1565679 T^{8} + 1378822 T^{9} + 56912 T^{10} - 8299 T^{11} - 437 T^{12} + 17 T^{13} + T^{14}$$
$61$ $$( -1 + T )^{14}$$
$67$ $$-11152814800 + 85007231770 T - 40076549556 T^{2} - 10386722411 T^{3} + 7619098256 T^{4} - 236110449 T^{5} - 370205235 T^{6} + 39840548 T^{7} + 6092263 T^{8} - 1048201 T^{9} - 15175 T^{10} + 9005 T^{11} - 257 T^{12} - 22 T^{13} + T^{14}$$
$71$ $$6346448960 - 40987926296 T + 86126251160 T^{2} - 71654183149 T^{3} + 26599299002 T^{4} - 3518799783 T^{5} - 370757592 T^{6} + 134878051 T^{7} - 3758419 T^{8} - 1533539 T^{9} + 90246 T^{10} + 7366 T^{11} - 528 T^{12} - 13 T^{13} + T^{14}$$
$73$ $$457179272 + 1312877468 T + 1231956290 T^{2} + 229615621 T^{3} - 269083547 T^{4} - 124095806 T^{5} + 9128612 T^{6} + 11054479 T^{7} + 251085 T^{8} - 377088 T^{9} - 3239 T^{10} + 5858 T^{11} - 207 T^{12} - 20 T^{13} + T^{14}$$
$79$ $$-33860837584 + 291930714652 T - 124989959652 T^{2} - 18249345063 T^{3} + 14077876019 T^{4} - 292880422 T^{5} - 563778923 T^{6} + 44156274 T^{7} + 9527777 T^{8} - 1116838 T^{9} - 55576 T^{10} + 10514 T^{11} - 83 T^{12} - 31 T^{13} + T^{14}$$
$83$ $$1517677024 + 1293192456 T - 3066820452 T^{2} - 506512067 T^{3} + 997326301 T^{4} + 90999358 T^{5} - 101051497 T^{6} - 10457714 T^{7} + 3703689 T^{8} + 454634 T^{9} - 45273 T^{10} - 7162 T^{11} + 35 T^{12} + 32 T^{13} + T^{14}$$
$89$ $$12934626280804 - 6966045100834 T - 2347220343922 T^{2} + 866869845971 T^{3} + 76019389600 T^{4} - 32454957798 T^{5} - 546384186 T^{6} + 532025413 T^{7} - 8001968 T^{8} - 4181220 T^{9} + 132903 T^{10} + 15377 T^{11} - 639 T^{12} - 21 T^{13} + T^{14}$$
$97$ $$39289496832 - 154414314432 T + 25496215194 T^{2} + 120933892323 T^{3} + 15836415500 T^{4} - 7323227966 T^{5} - 828850102 T^{6} + 185702483 T^{7} + 13312351 T^{8} - 2433910 T^{9} - 68176 T^{10} + 15695 T^{11} - 93 T^{12} - 37 T^{13} + T^{14}$$