# Properties

 Label 5290.2.a.bl Level $5290$ Weight $2$ Character orbit 5290.a Self dual yes Analytic conductor $42.241$ Analytic rank $0$ Dimension $15$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5290 = 2 \cdot 5 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5290.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.2408626693$$ Analytic rank: $$0$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ Defining polynomial: $$x^{15} - 5 x^{14} - 24 x^{13} + 142 x^{12} + 184 x^{11} - 1488 x^{10} - 426 x^{9} + 7165 x^{8} - 206 x^{7} - 16453 x^{6} + 637 x^{5} + 16290 x^{4} + 1068 x^{3} - 4992 x^{2} - 848 x - 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} + \beta_{1} q^{6} + \beta_{13} q^{7} + q^{8} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + q^{2} + \beta_{1} q^{3} + q^{4} + q^{5} + \beta_{1} q^{6} + \beta_{13} q^{7} + q^{8} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{9} + q^{10} + ( -\beta_{5} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{11} + \beta_{1} q^{12} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{13} + \beta_{13} q^{14} + \beta_{1} q^{15} + q^{16} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{17} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{18} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{19} + q^{20} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{21} + ( -\beta_{5} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{22} + \beta_{1} q^{24} + q^{25} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{26} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{11} - \beta_{12} ) q^{27} + \beta_{13} q^{28} + ( 2 - \beta_{5} - \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{29} + \beta_{1} q^{30} + ( \beta_{1} + \beta_{2} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{31} + q^{32} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{33} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{34} + \beta_{13} q^{35} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{36} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{37} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{38} + ( -1 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{39} + q^{40} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{42} + ( 2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{14} ) q^{43} + ( -\beta_{5} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{44} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{45} + ( 3 + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{47} + \beta_{1} q^{48} + ( 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + \beta_{14} ) q^{49} + q^{50} + ( -4 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{13} ) q^{51} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{52} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} - \beta_{13} ) q^{53} + ( 3 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{11} - \beta_{12} ) q^{54} + ( -\beta_{5} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{55} + \beta_{13} q^{56} + ( 2 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{57} + ( 2 - \beta_{5} - \beta_{9} + 2 \beta_{11} + \beta_{12} ) q^{58} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{59} + \beta_{1} q^{60} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{61} + ( \beta_{1} + \beta_{2} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{62} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{13} ) q^{63} + q^{64} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{65} + ( 1 - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{66} + ( 3 - 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{67} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{68} + \beta_{13} q^{70} + ( 2 + \beta_{1} + 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{71} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{72} + ( 4 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 4 \beta_{11} + \beta_{13} + \beta_{14} ) q^{73} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{74} + \beta_{1} q^{75} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{76} + ( 1 - \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{77} + ( -1 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{78} + ( -1 - \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{79} + q^{80} + ( 6 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} + \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{81} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{82} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{8} - \beta_{9} + 3 \beta_{11} - \beta_{12} - \beta_{14} ) q^{83} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{84} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{85} + ( 2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{14} ) q^{86} + ( -2 + 4 \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{14} ) q^{87} + ( -\beta_{5} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{88} + ( 3 - \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} - 3 \beta_{11} + 2 \beta_{14} ) q^{89} + ( 2 - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{90} + ( -2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 5 \beta_{11} + \beta_{12} ) q^{91} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 2 \beta_{14} ) q^{93} + ( 3 + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{94} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{95} + \beta_{1} q^{96} + ( -1 - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{14} ) q^{97} + ( 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + \beta_{14} ) q^{98} + ( 1 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 5 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15q + 15q^{2} + 5q^{3} + 15q^{4} + 15q^{5} + 5q^{6} - 4q^{7} + 15q^{8} + 28q^{9} + O(q^{10})$$ $$15q + 15q^{2} + 5q^{3} + 15q^{4} + 15q^{5} + 5q^{6} - 4q^{7} + 15q^{8} + 28q^{9} + 15q^{10} + 7q^{11} + 5q^{12} + 17q^{13} - 4q^{14} + 5q^{15} + 15q^{16} + 2q^{17} + 28q^{18} + 18q^{19} + 15q^{20} + 7q^{22} + 5q^{24} + 15q^{25} + 17q^{26} + 29q^{27} - 4q^{28} + 35q^{29} + 5q^{30} + 19q^{31} + 15q^{32} - 21q^{33} + 2q^{34} - 4q^{35} + 28q^{36} - 12q^{37} + 18q^{38} + 26q^{39} + 15q^{40} + 27q^{41} + 12q^{43} + 7q^{44} + 28q^{45} + 40q^{47} + 5q^{48} + 29q^{49} + 15q^{50} - 27q^{51} + 17q^{52} - 20q^{53} + 29q^{54} + 7q^{55} - 4q^{56} - 11q^{57} + 35q^{58} + 15q^{59} + 5q^{60} + 28q^{61} + 19q^{62} - 51q^{63} + 15q^{64} + 17q^{65} - 21q^{66} + 4q^{67} + 2q^{68} - 4q^{70} + 22q^{71} + 28q^{72} + 48q^{73} - 12q^{74} + 5q^{75} + 18q^{76} + 45q^{77} + 26q^{78} - 2q^{79} + 15q^{80} + 79q^{81} + 27q^{82} - 29q^{83} + 2q^{85} + 12q^{86} - 7q^{87} + 7q^{88} + 20q^{89} + 28q^{90} + 6q^{91} + 63q^{93} + 40q^{94} + 18q^{95} + 5q^{96} - 22q^{97} + 29q^{98} + 23q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 5 x^{14} - 24 x^{13} + 142 x^{12} + 184 x^{11} - 1488 x^{10} - 426 x^{9} + 7165 x^{8} - 206 x^{7} - 16453 x^{6} + 637 x^{5} + 16290 x^{4} + 1068 x^{3} - 4992 x^{2} - 848 x - 32$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-22531579107 \nu^{14} + 556623134808 \nu^{13} - 513706547783 \nu^{12} - 16441966550620 \nu^{11} + 23701584536314 \nu^{10} + 183009247554540 \nu^{9} - 247267864428506 \nu^{8} - 966177662431905 \nu^{7} + 970564185268651 \nu^{6} + 2505258744545127 \nu^{5} - 1342507280183060 \nu^{4} - 2752764336504231 \nu^{3} + 437037437013280 \nu^{2} + 887978005515692 \nu + 68485306833296$$$$)/ 15304491342104$$ $$\beta_{3}$$ $$=$$ $$($$$$-76352132097 \nu^{14} + 735952770633 \nu^{13} + 927884133644 \nu^{12} - 21304517431750 \nu^{11} + 9860473593616 \nu^{10} + 230524192975872 \nu^{9} - 187765678240310 \nu^{8} - 1172857064876805 \nu^{7} + 838645457469954 \nu^{6} + 2933719207504325 \nu^{5} - 1149677262973521 \nu^{4} - 3203789103703366 \nu^{3} + 240640547258212 \nu^{2} + 1101405447280464 \nu + 112584696092304$$$$)/ 30608982684208$$ $$\beta_{4}$$ $$=$$ $$($$$$73466274029 \nu^{14} - 247894962788 \nu^{13} - 2020084509641 \nu^{12} + 6803615072642 \nu^{11} + 20354697878250 \nu^{10} - 67506509316088 \nu^{9} - 95200028441570 \nu^{8} + 296726671320599 \nu^{7} + 231505875237667 \nu^{6} - 590969101414095 \nu^{5} - 312326436631100 \nu^{4} + 495253596681307 \nu^{3} + 213552966711098 \nu^{2} - 128216623891224 \nu - 37229452420768$$$$)/ 15304491342104$$ $$\beta_{5}$$ $$=$$ $$($$$$-80463787143 \nu^{14} + 78303641679 \nu^{13} + 2688805524524 \nu^{12} - 1897780084402 \nu^{11} - 34432396948304 \nu^{10} + 14276462379264 \nu^{9} + 209865927771398 \nu^{8} - 22879078046291 \nu^{7} - 607732484056930 \nu^{6} - 110772176907541 \nu^{5} + 700942116143609 \nu^{4} + 284053368680278 \nu^{3} - 210498567301812 \nu^{2} - 126886455805560 \nu - 9518919964928$$$$)/ 15304491342104$$ $$\beta_{6}$$ $$=$$ $$($$$$119436407357 \nu^{14} - 256893932945 \nu^{13} - 3628595839476 \nu^{12} + 6836903456914 \nu^{11} + 41811306439064 \nu^{10} - 63903395705216 \nu^{9} - 229659182097186 \nu^{8} + 246639927687641 \nu^{7} + 617771505185042 \nu^{6} - 359124453187573 \nu^{5} - 701512007251103 \nu^{4} + 135090986048126 \nu^{3} + 238527016061544 \nu^{2} + 25069947955824 \nu + 2350920768928$$$$)/ 15304491342104$$ $$\beta_{7}$$ $$=$$ $$($$$$-269906479373 \nu^{14} + 210748358911 \nu^{13} + 9570441126478 \nu^{12} - 5476645894538 \nu^{11} - 131459166204524 \nu^{10} + 47306994875720 \nu^{9} + 872179404551490 \nu^{8} - 136525439756997 \nu^{7} - 2817489779489100 \nu^{6} - 17288266590951 \nu^{5} + 3878180789076449 \nu^{4} + 219262534316416 \nu^{3} - 1774377383214652 \nu^{2} - 33067015694488 \nu + 109034969279600$$$$)/ 30608982684208$$ $$\beta_{8}$$ $$=$$ $$($$$$146370896995 \nu^{14} - 1009256102743 \nu^{13} - 2990860565408 \nu^{12} + 29487593386774 \nu^{11} + 13246240820236 \nu^{10} - 322221158223872 \nu^{9} + 61792822644746 \nu^{8} + 1648907960323511 \nu^{7} - 476888446680914 \nu^{6} - 4080459887741855 \nu^{5} + 627271811980395 \nu^{4} + 4271432971923990 \nu^{3} + 21042459279344 \nu^{2} - 1310698772482756 \nu - 110034142918232$$$$)/ 15304491342104$$ $$\beta_{9}$$ $$=$$ $$($$$$88548027537 \nu^{14} - 226141759171 \nu^{13} - 2615628668494 \nu^{12} + 5977316474866 \nu^{11} + 29228055103884 \nu^{10} - 55072921628408 \nu^{9} - 156448509600450 \nu^{8} + 205729229564493 \nu^{7} + 419374394528096 \nu^{6} - 275260238706933 \nu^{5} - 490003217960809 \nu^{4} + 80546156084452 \nu^{3} + 180063900963060 \nu^{2} + 11959522018512 \nu - 610817056776$$$$)/ 7652245671052$$ $$\beta_{10}$$ $$=$$ $$($$$$-192123799646 \nu^{14} + 247377081423 \nu^{13} + 6046884060359 \nu^{12} - 5299563131908 \nu^{11} - 72910673824222 \nu^{10} + 27479291224624 \nu^{9} + 421398964388756 \nu^{8} + 78656358471656 \nu^{7} - 1171717469344535 \nu^{6} - 846329240692092 \nu^{5} + 1252893215187065 \nu^{4} + 1459079020555909 \nu^{3} - 258543886238542 \nu^{2} - 602140988646440 \nu - 99770993854232$$$$)/ 15304491342104$$ $$\beta_{11}$$ $$=$$ $$($$$$-201468395768 \nu^{14} + 602349359781 \nu^{13} + 5711115653517 \nu^{12} - 16305820196582 \nu^{11} - 59978456540802 \nu^{10} + 157681624550220 \nu^{9} + 293064920399304 \nu^{8} - 658575411873258 \nu^{7} - 700659340803051 \nu^{6} + 1190251863201360 \nu^{5} + 729922666092571 \nu^{4} - 896765145317013 \nu^{3} - 257125055451376 \nu^{2} + 258258317820040 \nu + 19650164010968$$$$)/ 15304491342104$$ $$\beta_{12}$$ $$=$$ $$($$$$443965239273 \nu^{14} - 1054464446351 \nu^{13} - 13242482317426 \nu^{12} + 27847395092002 \nu^{11} + 149482257843324 \nu^{10} - 256866317128088 \nu^{9} - 804738898130250 \nu^{8} + 965922679972609 \nu^{7} + 2134546673497656 \nu^{6} - 1328154664291901 \nu^{5} - 2385724912851201 \nu^{4} + 461101163499556 \nu^{3} + 775500362613548 \nu^{2} + 49378527750560 \nu - 721010531424$$$$)/ 15304491342104$$ $$\beta_{13}$$ $$=$$ $$($$$$951978577767 \nu^{14} - 3418982812591 \nu^{13} - 25664032257004 \nu^{12} + 94083684679650 \nu^{11} + 248569212807408 \nu^{10} - 935512825759840 \nu^{9} - 1064987748412934 \nu^{8} + 4111746246690563 \nu^{7} + 2127100543709122 \nu^{6} - 8127977690045291 \nu^{5} - 1940646544419833 \nu^{4} + 6667162207519042 \nu^{3} + 985270953740684 \nu^{2} - 1649250502282736 \nu - 248662477203472$$$$)/ 30608982684208$$ $$\beta_{14}$$ $$=$$ $$($$$$-973663711765 \nu^{14} + 5441244054933 \nu^{13} + 21689399001464 \nu^{12} - 154454079755826 \nu^{11} - 133711730294512 \nu^{10} + 1617612490072120 \nu^{9} - 24757956976814 \nu^{8} - 7787702920770361 \nu^{7} + 2032920600501898 \nu^{6} + 17908817310143229 \nu^{5} - 3870075428380421 \nu^{4} - 17810338873845690 \nu^{3} + 1140410944959288 \nu^{2} + 5570993230776200 \nu + 389574100354272$$$$)/ 30608982684208$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{4} - \beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{12} - 3 \beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 7 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{14} - 2 \beta_{13} + 9 \beta_{12} + 10 \beta_{11} + 9 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} - 2 \beta_{5} + 14 \beta_{4} + 3 \beta_{3} - 11 \beta_{2} + 2 \beta_{1} + 42$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{12} - 35 \beta_{11} + 11 \beta_{10} + \beta_{9} - 15 \beta_{8} - 2 \beta_{7} + 28 \beta_{6} + 12 \beta_{5} + 19 \beta_{4} + 32 \beta_{3} - 21 \beta_{2} + 60 \beta_{1} + 41$$ $$\nu^{6}$$ $$=$$ $$-18 \beta_{14} - 28 \beta_{13} + 77 \beta_{12} + 100 \beta_{11} + 83 \beta_{10} + 28 \beta_{9} - 106 \beta_{8} - 5 \beta_{7} + 10 \beta_{6} - 26 \beta_{5} + 171 \beta_{4} + 57 \beta_{3} - 119 \beta_{2} + 24 \beta_{1} + 397$$ $$\nu^{7}$$ $$=$$ $$-11 \beta_{14} - 10 \beta_{13} - 163 \beta_{12} - 347 \beta_{11} + 109 \beta_{10} + 25 \beta_{9} - 191 \beta_{8} - 41 \beta_{7} + 332 \beta_{6} + 125 \beta_{5} + 274 \beta_{4} + 402 \beta_{3} - 285 \beta_{2} + 551 \beta_{1} + 472$$ $$\nu^{8}$$ $$=$$ $$-251 \beta_{14} - 332 \beta_{13} + 666 \beta_{12} + 1001 \beta_{11} + 795 \beta_{10} + 317 \beta_{9} - 1151 \beta_{8} - 108 \beta_{7} + 238 \beta_{6} - 258 \beta_{5} + 1986 \beta_{4} + 841 \beta_{3} - 1302 \beta_{2} + 238 \beta_{1} + 3928$$ $$\nu^{9}$$ $$=$$ $$-276 \beta_{14} - 238 \beta_{13} - 1772 \beta_{12} - 3255 \beta_{11} + 1107 \beta_{10} + 415 \beta_{9} - 2303 \beta_{8} - 592 \beta_{7} + 3781 \beta_{6} + 1287 \beta_{5} + 3553 \beta_{4} + 4696 \beta_{3} - 3393 \beta_{2} + 5218 \beta_{1} + 5249$$ $$\nu^{10}$$ $$=$$ $$-3196 \beta_{14} - 3781 \beta_{13} + 5851 \beta_{12} + 10083 \beta_{11} + 7850 \beta_{10} + 3379 \beta_{9} - 12585 \beta_{8} - 1649 \beta_{7} + 3963 \beta_{6} - 2235 \beta_{5} + 22560 \beta_{4} + 11110 \beta_{3} - 14222 \beta_{2} + 2308 \beta_{1} + 39784$$ $$\nu^{11}$$ $$=$$ $$-4757 \beta_{14} - 3963 \beta_{13} - 18549 \beta_{12} - 29488 \beta_{11} + 11704 \beta_{10} + 5767 \beta_{9} - 27080 \beta_{8} - 7529 \beta_{7} + 42519 \beta_{6} + 13469 \beta_{5} + 43747 \beta_{4} + 53233 \beta_{3} - 38320 \beta_{2} + 50267 \beta_{1} + 58161$$ $$\nu^{12}$$ $$=$$ $$-39031 \beta_{14} - 42519 \beta_{13} + 52187 \beta_{12} + 102693 \beta_{11} + 79403 \beta_{10} + 35200 \beta_{9} - 137919 \beta_{8} - 21937 \beta_{7} + 57175 \beta_{6} - 16908 \beta_{5} + 253300 \beta_{4} + 138083 \beta_{3} - 154393 \beta_{2} + 22904 \beta_{1} + 408818$$ $$\nu^{13}$$ $$=$$ $$-70188 \beta_{14} - 57175 \beta_{13} - 189526 \beta_{12} - 258510 \beta_{11} + 127918 \beta_{10} + 72924 \beta_{9} - 314166 \beta_{8} - 90228 \beta_{7} + 476269 \beta_{6} + 144019 \beta_{5} + 523674 \beta_{4} + 594887 \beta_{3} - 422454 \beta_{2} + 489705 \beta_{1} + 646832$$ $$\nu^{14}$$ $$=$$ $$-465996 \beta_{14} - 476269 \beta_{13} + 472132 \beta_{12} + 1059540 \beta_{11} + 818817 \beta_{10} + 363138 \beta_{9} - 1513480 \beta_{8} - 272185 \beta_{7} + 765555 \beta_{6} - 100452 \beta_{5} + 2825767 \beta_{4} + 1654400 \beta_{3} - 1666519 \beta_{2} + 235288 \beta_{1} + 4244841$$

## 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
 −3.12386 −3.06629 −2.03401 −1.53931 −1.10043 −0.687865 −0.117629 −0.0566801 0.763450 1.31120 2.59599 2.75508 2.78829 3.15164 3.36041
1.00000 −3.12386 1.00000 1.00000 −3.12386 −2.47855 1.00000 6.75851 1.00000
1.2 1.00000 −3.06629 1.00000 1.00000 −3.06629 −0.998427 1.00000 6.40210 1.00000
1.3 1.00000 −2.03401 1.00000 1.00000 −2.03401 −1.56763 1.00000 1.13722 1.00000
1.4 1.00000 −1.53931 1.00000 1.00000 −1.53931 3.71769 1.00000 −0.630528 1.00000
1.5 1.00000 −1.10043 1.00000 1.00000 −1.10043 4.03086 1.00000 −1.78906 1.00000
1.6 1.00000 −0.687865 1.00000 1.00000 −0.687865 −3.00706 1.00000 −2.52684 1.00000
1.7 1.00000 −0.117629 1.00000 1.00000 −0.117629 −1.70098 1.00000 −2.98616 1.00000
1.8 1.00000 −0.0566801 1.00000 1.00000 −0.0566801 −5.00659 1.00000 −2.99679 1.00000
1.9 1.00000 0.763450 1.00000 1.00000 0.763450 4.04396 1.00000 −2.41714 1.00000
1.10 1.00000 1.31120 1.00000 1.00000 1.31120 3.61614 1.00000 −1.28075 1.00000
1.11 1.00000 2.59599 1.00000 1.00000 2.59599 1.59863 1.00000 3.73916 1.00000
1.12 1.00000 2.75508 1.00000 1.00000 2.75508 −4.44855 1.00000 4.59049 1.00000
1.13 1.00000 2.78829 1.00000 1.00000 2.78829 −0.104808 1.00000 4.77457 1.00000
1.14 1.00000 3.15164 1.00000 1.00000 3.15164 0.586101 1.00000 6.93286 1.00000
1.15 1.00000 3.36041 1.00000 1.00000 3.36041 −2.28078 1.00000 8.29236 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 5290.2.a.bl 15
23.b odd 2 1 5290.2.a.bk 15
23.d odd 22 2 230.2.g.d 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.d 30 23.d odd 22 2
5290.2.a.bk 15 23.b odd 2 1
5290.2.a.bl 15 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5290))$$:

 $$T_{3}^{15} - \cdots$$ $$T_{7}^{15} + \cdots$$ $$T_{11}^{15} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{15}$$
$3$ $$-32 - 848 T - 4992 T^{2} + 1068 T^{3} + 16290 T^{4} + 637 T^{5} - 16453 T^{6} - 206 T^{7} + 7165 T^{8} - 426 T^{9} - 1488 T^{10} + 184 T^{11} + 142 T^{12} - 24 T^{13} - 5 T^{14} + T^{15}$$
$5$ $$( -1 + T )^{15}$$
$7$ $$21691 + 216802 T + 32308 T^{2} - 642460 T^{3} - 500878 T^{4} + 235749 T^{5} + 318791 T^{6} + 13119 T^{7} - 63569 T^{8} - 10661 T^{9} + 5710 T^{10} + 1256 T^{11} - 243 T^{12} - 59 T^{13} + 4 T^{14} + T^{15}$$
$11$ $$-563872 - 2940528 T - 2525384 T^{2} + 2686880 T^{3} + 2705958 T^{4} - 974757 T^{5} - 981653 T^{6} + 200282 T^{7} + 164925 T^{8} - 25846 T^{9} - 13514 T^{10} + 1934 T^{11} + 510 T^{12} - 72 T^{13} - 7 T^{14} + T^{15}$$
$13$ $$-34700192 - 28066912 T + 50452152 T^{2} + 15929496 T^{3} - 31620508 T^{4} + 3754085 T^{5} + 6037813 T^{6} - 1930387 T^{7} - 289521 T^{8} + 205900 T^{9} - 14979 T^{10} - 6431 T^{11} + 1138 T^{12} + 23 T^{13} - 17 T^{14} + T^{15}$$
$17$ $$13087744 - 81488896 T - 15870464 T^{2} + 95214848 T^{3} + 6837312 T^{4} - 30907264 T^{5} - 1349760 T^{6} + 4238592 T^{7} + 151144 T^{8} - 282120 T^{9} - 8693 T^{10} + 9529 T^{11} + 223 T^{12} - 157 T^{13} - 2 T^{14} + T^{15}$$
$19$ $$-82821344 + 61043216 T + 97058952 T^{2} - 83741740 T^{3} - 18390200 T^{4} + 25512847 T^{5} - 268499 T^{6} - 3279940 T^{7} + 345972 T^{8} + 200836 T^{9} - 33648 T^{10} - 5289 T^{11} + 1297 T^{12} + 18 T^{13} - 18 T^{14} + T^{15}$$
$23$ $$T^{15}$$
$29$ $$-28157152 + 53754704 T + 279770088 T^{2} - 831715620 T^{3} + 773828236 T^{4} - 288472049 T^{5} + 8105911 T^{6} + 26215473 T^{7} - 7050837 T^{8} + 131262 T^{9} + 222049 T^{10} - 33809 T^{11} + 136 T^{12} + 385 T^{13} - 35 T^{14} + T^{15}$$
$31$ $$1768741888 + 547782656 T - 1900841472 T^{2} - 781597440 T^{3} + 551926592 T^{4} + 244608928 T^{5} - 62417424 T^{6} - 24538480 T^{7} + 4779448 T^{8} + 979580 T^{9} - 220971 T^{10} - 7836 T^{11} + 3645 T^{12} - 93 T^{13} - 19 T^{14} + T^{15}$$
$37$ $$-180896 - 4091936 T - 21080856 T^{2} - 30312236 T^{3} + 19357618 T^{4} + 55690797 T^{5} + 3046456 T^{6} - 14735054 T^{7} - 4050912 T^{8} + 266111 T^{9} + 181329 T^{10} + 6909 T^{11} - 2610 T^{12} - 182 T^{13} + 12 T^{14} + T^{15}$$
$41$ $$5052544199 - 5395538597 T - 283725266 T^{2} + 2251807217 T^{3} - 653644185 T^{4} - 232319130 T^{5} + 139730100 T^{6} - 7095715 T^{7} - 8269868 T^{8} + 1738952 T^{9} + 21220 T^{10} - 41273 T^{11} + 3772 T^{12} + 89 T^{13} - 27 T^{14} + T^{15}$$
$43$ $$-40533032992 + 49570984336 T + 25710884928 T^{2} - 44203534196 T^{3} + 11468455182 T^{4} + 2141962515 T^{5} - 1020763084 T^{6} - 7044191 T^{7} + 33331738 T^{8} - 1432123 T^{9} - 525514 T^{10} + 34192 T^{11} + 4028 T^{12} - 309 T^{13} - 12 T^{14} + T^{15}$$
$47$ $$926946641567 - 1687208400459 T + 526355841918 T^{2} + 125685710760 T^{3} - 73918958286 T^{4} + 2064985357 T^{5} + 3220373273 T^{6} - 380484880 T^{7} - 50614675 T^{8} + 11072982 T^{9} - 6812 T^{10} - 118496 T^{11} + 6561 T^{12} + 330 T^{13} - 40 T^{14} + T^{15}$$
$53$ $$-709550368 - 14045473120 T - 25049420680 T^{2} - 8069540412 T^{3} + 6776756926 T^{4} + 4193703525 T^{5} + 88490479 T^{6} - 333893604 T^{7} - 52555074 T^{8} + 3748074 T^{9} + 1138104 T^{10} + 12681 T^{11} - 8311 T^{12} - 312 T^{13} + 20 T^{14} + T^{15}$$
$59$ $$-6952492832 - 6243244384 T + 3475788672 T^{2} + 4150964412 T^{3} - 303531496 T^{4} - 951776191 T^{5} - 77072025 T^{6} + 85739023 T^{7} + 12559663 T^{8} - 2570452 T^{9} - 385019 T^{10} + 38185 T^{11} + 4226 T^{12} - 309 T^{13} - 15 T^{14} + T^{15}$$
$61$ $$-2359770656 - 4554444464 T + 11700361560 T^{2} + 11586520236 T^{3} - 22224398444 T^{4} + 5975492421 T^{5} + 831238399 T^{6} - 461279734 T^{7} + 23916456 T^{8} + 8431700 T^{9} - 928458 T^{10} - 43845 T^{11} + 9079 T^{12} - 104 T^{13} - 28 T^{14} + T^{15}$$
$67$ $$-41308152544 + 62623926096 T - 18117350168 T^{2} - 10701883148 T^{3} + 5801389876 T^{4} + 222972069 T^{5} - 516206338 T^{6} + 42620884 T^{7} + 18542669 T^{8} - 2477948 T^{9} - 299568 T^{10} + 49805 T^{11} + 2061 T^{12} - 399 T^{13} - 4 T^{14} + T^{15}$$
$71$ $$105724928 + 414075392 T - 169794560 T^{2} - 545899264 T^{3} + 63327424 T^{4} + 214071712 T^{5} - 14468016 T^{6} - 31242800 T^{7} + 3105500 T^{8} + 1549596 T^{9} - 235529 T^{10} - 17285 T^{11} + 4379 T^{12} - 65 T^{13} - 22 T^{14} + T^{15}$$
$73$ $$396062989312 - 482943027200 T + 28870180096 T^{2} + 141787485184 T^{3} - 36817293888 T^{4} - 9570342240 T^{5} + 4080109968 T^{6} - 75588816 T^{7} - 119575888 T^{8} + 12826838 T^{9} + 759921 T^{10} - 189285 T^{11} + 6291 T^{12} + 583 T^{13} - 48 T^{14} + T^{15}$$
$79$ $$4511302872064 - 5510733801984 T + 655505695488 T^{2} + 908182974336 T^{3} - 94157044096 T^{4} - 54157104480 T^{5} + 3664956336 T^{6} + 1558653376 T^{7} - 60665644 T^{8} - 23318166 T^{9} + 461513 T^{10} + 181226 T^{11} - 1585 T^{12} - 688 T^{13} + 2 T^{14} + T^{15}$$
$83$ $$-1226113860256 - 4120596646016 T - 444428656400 T^{2} + 1666588242756 T^{3} + 819936744884 T^{4} + 118236372977 T^{5} - 8655272027 T^{6} - 3870169844 T^{7} - 204557361 T^{8} + 32754676 T^{9} + 3606576 T^{10} - 49696 T^{11} - 18096 T^{12} - 362 T^{13} + 29 T^{14} + T^{15}$$
$89$ $$44903107363643 + 22571062890032 T - 8949705884037 T^{2} - 2482145937407 T^{3} + 739352250203 T^{4} + 79114698462 T^{5} - 26322714480 T^{6} - 780539748 T^{7} + 452242803 T^{8} - 4284957 T^{9} - 3824349 T^{10} + 118927 T^{11} + 14644 T^{12} - 630 T^{13} - 20 T^{14} + T^{15}$$
$97$ $$19390524416 - 142109335040 T - 24677895424 T^{2} + 99318398208 T^{3} + 52965016768 T^{4} + 3586446432 T^{5} - 3275200448 T^{6} - 745782232 T^{7} + 9032764 T^{8} + 19153938 T^{9} + 1854905 T^{10} - 34512 T^{11} - 12963 T^{12} - 376 T^{13} + 22 T^{14} + T^{15}$$