# Properties

 Label 6039.2.a.k Level $6039$ Weight $2$ Character orbit 6039.a Self dual yes Analytic conductor $48.222$ Analytic rank $0$ Dimension $19$ 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: $$19$$ Coefficient field: $$\mathbb{Q}[x]/(x^{19} - \cdots)$$ Defining polynomial: $$x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} - 4673 x^{11} - 15053 x^{10} + 16875 x^{9} + 20141 x^{8} - 28019 x^{7} - 11589 x^{6} + 21077 x^{5} + 1674 x^{4} - 6404 x^{3} + 241 x^{2} + 643 x - 43$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 671) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{18}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{5} -\beta_{16} q^{7} + ( -1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{5} -\beta_{16} q^{7} + ( -1 - \beta_{1} - \beta_{5} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{8} + ( 1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{13} ) q^{10} - q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} ) q^{13} + ( -1 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{14} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{16} + ( -\beta_{2} + \beta_{4} - \beta_{6} - \beta_{11} + \beta_{13} - \beta_{16} + \beta_{17} ) q^{17} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{9} + \beta_{13} + \beta_{18} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{20} + \beta_{1} q^{22} + ( 1 - \beta_{1} - \beta_{4} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{23} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{25} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{26} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{13} - \beta_{17} ) q^{28} + ( -1 + \beta_{3} + \beta_{5} - \beta_{8} + \beta_{11} + \beta_{15} - \beta_{17} ) q^{29} + ( -\beta_{3} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{31} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{32} + ( -1 + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{34} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{15} + \beta_{16} + \beta_{18} ) q^{35} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{8} - \beta_{11} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{37} + ( 3 + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} + \beta_{16} + \beta_{17} ) q^{38} + ( 2 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} ) q^{40} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} - 2 \beta_{16} ) q^{41} + ( 2 - \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{43} + ( -1 - \beta_{2} ) q^{44} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{14} - \beta_{15} + 2 \beta_{18} ) q^{46} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} + 2 \beta_{17} + \beta_{18} ) q^{47} + ( 2 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} + \beta_{13} + 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{49} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{50} + ( -3 - 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{18} ) q^{52} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{15} + \beta_{16} - \beta_{18} ) q^{53} -\beta_{3} q^{55} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{56} + ( -1 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{58} + ( 4 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} - \beta_{17} - 2 \beta_{18} ) q^{59} - q^{61} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{18} ) q^{62} + ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{64} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{16} + \beta_{17} + 3 \beta_{18} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - \beta_{17} + \beta_{18} ) q^{67} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{9} - 2 \beta_{11} + \beta_{12} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{68} + ( 3 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} + \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{70} + ( 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{15} - \beta_{16} - 2 \beta_{18} ) q^{71} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{73} + ( 6 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 4 \beta_{9} + 3 \beta_{10} + \beta_{11} + 4 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{74} + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{76} + \beta_{16} q^{77} + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{11} + \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{80} + ( -3 + 3 \beta_{1} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{82} + ( 1 - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{83} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - 3 \beta_{14} + 2 \beta_{17} + \beta_{18} ) q^{85} + ( 5 - 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + 4 \beta_{11} + 2 \beta_{12} - \beta_{14} + 3 \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{86} + ( 1 + \beta_{1} + \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} ) q^{88} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{10} - 2 \beta_{12} + \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{89} + ( 3 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{15} + 2 \beta_{16} - \beta_{17} ) q^{91} + ( 2 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{92} + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{94} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} ) q^{95} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{15} + \beta_{16} ) q^{97} + ( 3 - \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} + 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$19q - 5q^{2} + 23q^{4} + 9q^{7} - 9q^{8} + O(q^{10})$$ $$19q - 5q^{2} + 23q^{4} + 9q^{7} - 9q^{8} + 7q^{10} - 19q^{11} + 8q^{13} + 11q^{14} + 31q^{16} - 9q^{17} + 17q^{19} + 6q^{20} + 5q^{22} + 10q^{23} + 45q^{25} - 5q^{26} + 36q^{28} - 27q^{29} + 7q^{31} - 8q^{32} - 5q^{34} - 17q^{35} + 20q^{37} + 37q^{38} + 10q^{40} - 19q^{41} + 20q^{43} - 23q^{44} + 41q^{46} + 19q^{47} + 42q^{49} - 36q^{50} - 28q^{52} - 3q^{53} + 44q^{56} + 23q^{58} + 28q^{59} - 19q^{61} + 11q^{62} + 47q^{64} - 25q^{65} + 3q^{67} - 38q^{68} + 3q^{70} + 19q^{71} + 20q^{73} + 22q^{74} - 25q^{76} - 9q^{77} + 69q^{79} + 36q^{80} - 61q^{82} - q^{83} + 24q^{85} + 27q^{86} + 9q^{88} + 24q^{91} + 67q^{92} + 64q^{94} + 3q^{95} + 21q^{97} + 87q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} - 4673 x^{11} - 15053 x^{10} + 16875 x^{9} + 20141 x^{8} - 28019 x^{7} - 11589 x^{6} + 21077 x^{5} + 1674 x^{4} - 6404 x^{3} + 241 x^{2} + 643 x - 43$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$8867201804 \nu^{18} - 28397403169 \nu^{17} - 153963852608 \nu^{16} + 585160018840 \nu^{15} + 420015084215 \nu^{14} - 3896315485489 \nu^{13} + 7816695786848 \nu^{12} + 4170318228153 \nu^{11} - 74752830871939 \nu^{10} + 61181608853165 \nu^{9} + 280962923414937 \nu^{8} - 283316198675975 \nu^{7} - 514007340828783 \nu^{6} + 474088918297633 \nu^{5} + 431277028203763 \nu^{4} - 300469578965604 \nu^{3} - 132758950019250 \nu^{2} + 50083912844543 \nu + 7406641729547$$$$)/ 2584793613763$$ $$\beta_{4}$$ $$=$$ $$($$$$-9925308116 \nu^{18} + 211585433401 \nu^{17} - 157290256641 \nu^{16} - 5438380518759 \nu^{15} + 7280102119263 \nu^{14} + 56715822193005 \nu^{13} - 79490940840104 \nu^{12} - 311772296964757 \nu^{11} + 401624582318481 \nu^{10} + 976030566478054 \nu^{9} - 1042517331908497 \nu^{8} - 1744313483302286 \nu^{7} + 1352625850873237 \nu^{6} + 1686914951359080 \nu^{5} - 761921796525769 \nu^{4} - 794326099908997 \nu^{3} + 121134977398858 \nu^{2} + 133177244880778 \nu + 4643140412416$$$$)/ 2584793613763$$ $$\beta_{5}$$ $$=$$ $$($$$$16270525372 \nu^{18} - 35753207242 \nu^{17} - 519467802678 \nu^{16} + 1065729788540 \nu^{15} + 6910095072255 \nu^{14} - 13113396066991 \nu^{13} - 49478643130019 \nu^{12} + 86061802478689 \nu^{11} + 204922845192456 \nu^{10} - 323580558431117 \nu^{9} - 488697894791415 \nu^{8} + 692134937669964 \nu^{7} + 625158766804376 \nu^{6} - 777549565519077 \nu^{5} - 360901712332219 \nu^{4} + 372817093364514 \nu^{3} + 64006649549778 \nu^{2} - 47424793625471 \nu + 1737797285981$$$$)/ 2584793613763$$ $$\beta_{6}$$ $$=$$ $$($$$$37762226457 \nu^{18} - 165903350608 \nu^{17} - 747153495518 \nu^{16} + 3973232345087 \nu^{15} + 4711323004499 \nu^{14} - 37022753494585 \nu^{13} - 4098672552276 \nu^{12} + 168673178705194 \nu^{11} - 72597790484259 \nu^{10} - 372792814215164 \nu^{9} + 311955896020022 \nu^{8} + 273746173403669 \nu^{7} - 483092267921848 \nu^{6} + 262742744285573 \nu^{5} + 256243772589666 \nu^{4} - 424688104742192 \nu^{3} - 12717902390185 \nu^{2} + 114678199755666 \nu - 2002160866471$$$$)/ 2584793613763$$ $$\beta_{7}$$ $$=$$ $$($$$$41025764486 \nu^{18} + 68688107954 \nu^{17} - 1546829355593 \nu^{16} - 1494342049450 \nu^{15} + 22839785265251 \nu^{14} + 12033133930515 \nu^{13} - 172638105605592 \nu^{12} - 42177686787597 \nu^{11} + 725805637025464 \nu^{10} + 41070927048001 \nu^{9} - 1708211052099464 \nu^{8} + 119332311754315 \nu^{7} + 2128704632011736 \nu^{6} - 311958245977134 \nu^{5} - 1222257229443266 \nu^{4} + 179493755356720 \nu^{3} + 237228681118952 \nu^{2} - 19813780702 \nu - 165641941058$$$$)/ 2584793613763$$ $$\beta_{8}$$ $$=$$ $$($$$$-44058942396 \nu^{18} + 100366607650 \nu^{17} + 1163734380318 \nu^{16} - 2544906190927 \nu^{15} - 12599484691005 \nu^{14} + 26056387605249 \nu^{13} + 72759363883607 \nu^{12} - 139981328381100 \nu^{11} - 242182034413731 \nu^{10} + 427053290996220 \nu^{9} + 464054899866586 \nu^{8} - 743045495285101 \nu^{7} - 477531549805097 \nu^{6} + 689021693044756 \nu^{5} + 222408147075718 \nu^{4} - 272000230119563 \nu^{3} - 27572301495516 \nu^{2} + 23059854361824 \nu - 246530814813$$$$)/ 2584793613763$$ $$\beta_{9}$$ $$=$$ $$($$$$-57136613064 \nu^{18} + 46120188886 \nu^{17} + 1573939504037 \nu^{16} - 711542003465 \nu^{15} - 18089060782894 \nu^{14} + 888172948827 \nu^{13} + 113473718291916 \nu^{12} + 41171189048558 \nu^{11} - 423076214974525 \nu^{10} - 308367411917029 \nu^{9} + 949735366352185 \nu^{8} + 935542592921969 \nu^{7} - 1236732975882466 \nu^{6} - 1317806710310478 \nu^{5} + 850923535863497 \nu^{4} + 778848775823786 \nu^{3} - 236302917332312 \nu^{2} - 127878466080227 \nu + 8443491451865$$$$)/ 2584793613763$$ $$\beta_{10}$$ $$=$$ $$($$$$-81254538506 \nu^{18} + 654428683551 \nu^{17} + 970372146851 \nu^{16} - 16384821425649 \nu^{15} + 5490595341552 \nu^{14} + 164287162414465 \nu^{13} - 143000610338671 \nu^{12} - 851967355105863 \nu^{11} + 907306072440698 \nu^{10} + 2447482008098315 \nu^{9} - 2686226489462764 \nu^{8} - 3851875830463548 \nu^{7} + 3918184841562988 \nu^{6} + 3081313386395895 \nu^{5} - 2559452836437059 \nu^{4} - 1114097846357755 \nu^{3} + 587551380088491 \nu^{2} + 144837697384176 \nu - 25082161671395$$$$)/ 2584793613763$$ $$\beta_{11}$$ $$=$$ $$($$$$-91424960248 \nu^{18} + 290362672773 \nu^{17} + 2163987038842 \nu^{16} - 7229272850781 \nu^{15} - 19906864553003 \nu^{14} + 71928744979165 \nu^{13} + 89628292185599 \nu^{12} - 369628093368947 \nu^{11} - 196989043393835 \nu^{10} + 1054469865039305 \nu^{9} + 148090368081498 \nu^{8} - 1669129966062270 \nu^{7} + 134975892121205 \nu^{6} + 1394627423083597 \nu^{5} - 249311959857695 \nu^{4} - 557432190432943 \nu^{3} + 94000023879346 \nu^{2} + 76490070141423 \nu - 5331850884506$$$$)/ 2584793613763$$ $$\beta_{12}$$ $$=$$ $$($$$$110220246484 \nu^{18} - 467343546422 \nu^{17} - 2253974970877 \nu^{16} + 11406225381357 \nu^{15} + 15508655859118 \nu^{14} - 110076102275048 \nu^{13} - 28689902332059 \nu^{12} + 538405030181766 \nu^{11} - 130776269174957 \nu^{10} - 1408591428559113 \nu^{9} + 700069942254179 \nu^{8} + 1883801315016804 \nu^{7} - 1141430209449476 \nu^{6} - 1082982581796351 \nu^{5} + 635720108720707 \nu^{4} + 170395081372216 \nu^{3} - 54746227366930 \nu^{2} + 3421690058101 \nu - 8426703234447$$$$)/ 2584793613763$$ $$\beta_{13}$$ $$=$$ $$($$$$112916661333 \nu^{18} - 420512816139 \nu^{17} - 2391672731682 \nu^{16} + 10136775407328 \nu^{15} + 17708092485247 \nu^{14} - 95838102406759 \nu^{13} - 44248321607856 \nu^{12} + 452239469595452 \nu^{11} - 80531592282880 \nu^{10} - 1103682120823352 \nu^{9} + 652563422729939 \nu^{8} + 1250741201795975 \nu^{7} - 1243226926847429 \nu^{6} - 354335113278947 \nu^{5} + 866457444779580 \nu^{4} - 237488214060000 \nu^{3} - 170724575819309 \nu^{2} + 72688955170899 \nu - 992900881709$$$$)/ 2584793613763$$ $$\beta_{14}$$ $$=$$ $$($$$$134524452259 \nu^{18} - 595936463557 \nu^{17} - 2794615480726 \nu^{16} + 14870979403509 \nu^{15} + 19764923687112 \nu^{14} - 148126283460976 \nu^{13} - 39968403213030 \nu^{12} + 759359884142781 \nu^{11} - 157363017924653 \nu^{10} - 2143032129859066 \nu^{9} + 934401836140697 \nu^{8} + 3296195916484018 \nu^{7} - 1683719333735166 \nu^{6} - 2591704634942325 \nu^{5} + 1136718336996770 \nu^{4} + 957988530198262 \nu^{3} - 195065901784280 \nu^{2} - 131669286419725 \nu - 6942619491638$$$$)/ 2584793613763$$ $$\beta_{15}$$ $$=$$ $$($$$$268063867548 \nu^{18} - 975838766429 \nu^{17} - 5844382212784 \nu^{16} + 23718172730414 \nu^{15} + 46258916319239 \nu^{14} - 227625432361619 \nu^{13} - 147385979907443 \nu^{12} + 1105726169117787 \nu^{11} + 23829240400472 \nu^{10} - 2873186468193146 \nu^{9} + 965965174706451 \nu^{8} + 3844359687237313 \nu^{7} - 2136542032244486 \nu^{6} - 2308864017778112 \nu^{5} + 1541694664772399 \nu^{4} + 490128717085397 \nu^{3} - 296741969124174 \nu^{2} - 29439287899993 \nu + 2625408551097$$$$)/ 2584793613763$$ $$\beta_{16}$$ $$=$$ $$($$$$-386309035457 \nu^{18} + 1365268186173 \nu^{17} + 8692512067202 \nu^{16} - 33566135421448 \nu^{15} - 73247230474246 \nu^{14} + 327363575480166 \nu^{13} + 276164257045428 \nu^{12} - 1627427645433792 \nu^{11} - 350590014826373 \nu^{10} + 4379858371730078 \nu^{9} - 537601793521600 \nu^{8} - 6212724005068246 \nu^{7} + 1859899256299250 \nu^{6} + 4173706077317173 \nu^{5} - 1430231827654368 \nu^{4} - 1122823610741571 \nu^{3} + 279954024772659 \nu^{2} + 99943970595050 \nu - 8519539396534$$$$)/ 2584793613763$$ $$\beta_{17}$$ $$=$$ $$($$$$-407596389827 \nu^{18} + 996406897672 \nu^{17} + 10303884149056 \nu^{16} - 24506168703039 \nu^{15} - 104725686151822 \nu^{14} + 239031289075534 \nu^{13} + 553369806506455 \nu^{12} - 1188615976491545 \nu^{11} - 1630922731137518 \nu^{10} + 3205416930871435 \nu^{9} + 2656196462753142 \nu^{8} - 4586218801226622 \nu^{7} - 2216000497099625 \nu^{6} + 3162740167704228 \nu^{5} + 818360171228219 \nu^{4} - 877910452999009 \nu^{3} - 76416261707646 \nu^{2} + 75002246004675 \nu - 7512666382563$$$$)/ 2584793613763$$ $$\beta_{18}$$ $$=$$ $$($$$$433257329190 \nu^{18} - 1312562946260 \nu^{17} - 10331601737881 \nu^{16} + 32317494993362 \nu^{15} + 96395574289456 \nu^{14} - 315991261600359 \nu^{13} - 447303281907149 \nu^{12} + 1578531396957927 \nu^{11} + 1065197051025780 \nu^{10} - 4290881606498135 \nu^{9} - 1147835903097151 \nu^{8} + 6222910401208218 \nu^{7} + 239835296100884 \nu^{6} - 4402846506872819 \nu^{5} + 270534203277648 \nu^{4} + 1308086020776475 \nu^{3} - 97910802478304 \nu^{2} - 115787993029379 \nu + 7601132183693$$$$)/ 2584793613763$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{11} - \beta_{6} + \beta_{5} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{3} + 7 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{18} + 2 \beta_{17} - \beta_{16} - \beta_{14} + 11 \beta_{13} - \beta_{12} + 7 \beta_{11} + 2 \beta_{10} + \beta_{9} - 9 \beta_{6} + 11 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 30 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$-\beta_{17} + 2 \beta_{16} + \beta_{15} + \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} - 12 \beta_{10} - 12 \beta_{9} + 12 \beta_{8} + 12 \beta_{7} + \beta_{6} - 12 \beta_{5} - \beta_{4} - 11 \beta_{3} + 50 \beta_{2} - \beta_{1} + 88$$ $$\nu^{7}$$ $$=$$ $$10 \beta_{18} + 24 \beta_{17} - 15 \beta_{16} - 14 \beta_{14} + 94 \beta_{13} - 12 \beta_{12} + 46 \beta_{11} + 26 \beta_{10} + 12 \beta_{9} - \beta_{8} - 71 \beta_{6} + 97 \beta_{5} - 16 \beta_{4} + 17 \beta_{3} - 14 \beta_{2} + 200 \beta_{1} + 76$$ $$\nu^{8}$$ $$=$$ $$3 \beta_{18} - 15 \beta_{17} + 31 \beta_{16} + 14 \beta_{15} + 14 \beta_{14} - 18 \beta_{13} - 93 \beta_{12} + 19 \beta_{11} - 112 \beta_{10} - 109 \beta_{9} + 110 \beta_{8} + 109 \beta_{7} + 14 \beta_{6} - 112 \beta_{5} - 12 \beta_{4} - 94 \beta_{3} + 367 \beta_{2} - 23 \beta_{1} + 567$$ $$\nu^{9}$$ $$=$$ $$78 \beta_{18} + 218 \beta_{17} - 153 \beta_{16} + 2 \beta_{15} - 142 \beta_{14} + 746 \beta_{13} - 99 \beta_{12} + 317 \beta_{11} + 249 \beta_{10} + 111 \beta_{9} - 22 \beta_{8} + 2 \beta_{7} - 549 \beta_{6} + 799 \beta_{5} - 171 \beta_{4} + 196 \beta_{3} - 145 \beta_{2} + 1410 \beta_{1} + 534$$ $$\nu^{10}$$ $$=$$ $$47 \beta_{18} - 163 \beta_{17} + 334 \beta_{16} + 143 \beta_{15} + 143 \beta_{14} - 224 \beta_{13} - 724 \beta_{12} + 229 \beta_{11} - 958 \beta_{10} - 905 \beta_{9} + 915 \beta_{8} + 905 \beta_{7} + 148 \beta_{6} - 967 \beta_{5} - 107 \beta_{4} - 744 \beta_{3} + 2742 \beta_{2} - 313 \beta_{1} + 3874$$ $$\nu^{11}$$ $$=$$ $$570 \beta_{18} + 1806 \beta_{17} - 1356 \beta_{16} + 33 \beta_{15} - 1270 \beta_{14} + 5763 \beta_{13} - 699 \beta_{12} + 2290 \beta_{11} + 2145 \beta_{10} + 945 \beta_{9} - 306 \beta_{8} + 29 \beta_{7} - 4239 \beta_{6} + 6393 \beta_{5} - 1566 \beta_{4} + 1931 \beta_{3} - 1357 \beta_{2} + 10260 \beta_{1} + 3645$$ $$\nu^{12}$$ $$=$$ $$492 \beta_{18} - 1566 \beta_{17} + 3118 \beta_{16} + 1289 \beta_{15} + 1305 \beta_{14} - 2393 \beta_{13} - 5465 \beta_{12} + 2259 \beta_{11} - 7879 \beta_{10} - 7253 \beta_{9} + 7277 \beta_{8} + 7237 \beta_{7} + 1434 \beta_{6} - 8102 \beta_{5} - 862 \beta_{4} - 5721 \beta_{3} + 20716 \beta_{2} - 3451 \beta_{1} + 27441$$ $$\nu^{13}$$ $$=$$ $$4103 \beta_{18} + 14404 \beta_{17} - 11290 \beta_{16} + 341 \beta_{15} - 10672 \beta_{14} + 44078 \beta_{13} - 4515 \beta_{12} + 17038 \beta_{11} + 17659 \beta_{10} + 7783 \beta_{9} - 3478 \beta_{8} + 235 \beta_{7} - 32754 \beta_{6} + 50414 \beta_{5} - 13319 \beta_{4} + 17533 \beta_{3} - 12149 \beta_{2} + 76103 \beta_{1} + 24525$$ $$\nu^{14}$$ $$=$$ $$4351 \beta_{18} - 14133 \beta_{17} + 27117 \beta_{16} + 10875 \beta_{15} + 11326 \beta_{14} - 23509 \beta_{13} - 40789 \beta_{12} + 19984 \beta_{11} - 63529 \beta_{10} - 57279 \beta_{9} + 56578 \beta_{8} + 56857 \beta_{7} + 13359 \beta_{6} - 67001 \beta_{5} - 6621 \beta_{4} - 43553 \beta_{3} + 157618 \beta_{2} - 34216 \beta_{1} + 198716$$ $$\nu^{15}$$ $$=$$ $$29563 \beta_{18} + 112951 \beta_{17} - 91244 \beta_{16} + 2777 \beta_{15} - 86679 \beta_{14} + 336132 \beta_{13} - 27266 \beta_{12} + 128648 \beta_{11} + 142361 \beta_{10} + 63190 \beta_{9} - 35398 \beta_{8} + 1034 \beta_{7} - 253130 \beta_{6} + 394442 \beta_{5} - 108916 \beta_{4} + 151745 \beta_{3} - 106241 \beta_{2} + 571467 \beta_{1} + 163128$$ $$\nu^{16}$$ $$=$$ $$35212 \beta_{18} - 122951 \beta_{17} + 226841 \beta_{16} + 88185 \beta_{15} + 95848 \beta_{14} - 219042 \beta_{13} - 303570 \beta_{12} + 165507 \beta_{11} - 506776 \beta_{10} - 449554 \beta_{9} + 434837 \beta_{8} + 442742 \beta_{7} + 121414 \beta_{6} - 550426 \beta_{5} - 49387 \beta_{4} - 331180 \beta_{3} + 1204731 \beta_{2} - 319245 \beta_{1} + 1458972$$ $$\nu^{17}$$ $$=$$ $$214159 \beta_{18} + 879024 \beta_{17} - 726921 \beta_{16} + 18877 \beta_{15} - 690523 \beta_{14} + 2563542 \beta_{13} - 153522 \beta_{12} + 977051 \beta_{11} + 1136319 \beta_{10} + 509856 \beta_{9} - 336785 \beta_{8} - 4112 \beta_{7} - 1955646 \beta_{6} + 3072741 \beta_{5} - 871174 \beta_{4} + 1274242 \beta_{3} - 915300 \beta_{2} + 4326695 \beta_{1} + 1070635$$ $$\nu^{18}$$ $$=$$ $$270373 \beta_{18} - 1044853 \beta_{17} + 1855069 \beta_{16} + 697382 \beta_{15} + 799771 \beta_{14} - 1968542 \beta_{13} - 2261487 \beta_{12} + 1313376 \beta_{11} - 4017999 \beta_{10} - 3519120 \beta_{9} + 3323516 \beta_{8} + 3431715 \beta_{7} + 1082768 \beta_{6} - 4503238 \beta_{5} - 360082 \beta_{4} - 2526317 \beta_{3} + 9236634 \beta_{2} - 2867553 \beta_{1} + 10807509$$

## 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
 2.72847 2.69542 2.34443 2.13634 2.04293 1.98166 1.31423 1.04858 0.686089 0.488177 0.0681863 −0.436741 −0.556474 −0.976920 −1.62941 −1.82302 −2.08441 −2.21976 −2.80778
−2.72847 0 5.44452 3.87909 0 0.707409 −9.39826 0 −10.5840
1.2 −2.69542 0 5.26531 −1.92020 0 2.04756 −8.80138 0 5.17575
1.3 −2.34443 0 3.49635 −3.40234 0 −2.95885 −3.50809 0 7.97654
1.4 −2.13634 0 2.56396 0.997990 0 −0.526710 −1.20482 0 −2.13205
1.5 −2.04293 0 2.17358 −0.258135 0 3.70333 −0.354615 0 0.527352
1.6 −1.98166 0 1.92698 −2.60057 0 −2.18138 0.144707 0 5.15344
1.7 −1.31423 0 −0.272811 1.38757 0 2.03012 2.98699 0 −1.82358
1.8 −1.04858 0 −0.900472 4.37157 0 −3.70774 3.04139 0 −4.58396
1.9 −0.686089 0 −1.52928 −3.77631 0 4.48234 2.42140 0 2.59089
1.10 −0.488177 0 −1.76168 −1.92726 0 0.259128 1.83637 0 0.940847
1.11 −0.0681863 0 −1.99535 3.91488 0 0.303195 0.272428 0 −0.266941
1.12 0.436741 0 −1.80926 −3.47884 0 −1.93516 −1.66366 0 −1.51935
1.13 0.556474 0 −1.69034 −0.874867 0 4.50273 −2.05358 0 −0.486841
1.14 0.976920 0 −1.04563 2.45987 0 −4.72156 −2.97533 0 2.40309
1.15 1.62941 0 0.654988 −2.04510 0 −3.70972 −2.19158 0 −3.33231
1.16 1.82302 0 1.32341 3.15385 0 −0.287436 −1.23344 0 5.74953
1.17 2.08441 0 2.34475 −2.98183 0 3.78550 0.718591 0 −6.21535
1.18 2.21976 0 2.92734 2.17512 0 5.05675 2.05847 0 4.82825
1.19 2.80778 0 5.88364 0.925514 0 2.15050 10.9044 0 2.59864
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.19 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.k 19
3.b odd 2 1 671.2.a.c 19
33.d even 2 1 7381.2.a.i 19

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.c 19 3.b odd 2 1
6039.2.a.k 19 1.a even 1 1 trivial
7381.2.a.i 19 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{19} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6039))$$.