Properties

Label 1502.2.a.g
Level 1502
Weight 2
Character orbit 1502.a
Self dual yes
Analytic conductor 11.994
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1502 = 2 \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1502.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.9935303836\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} + \beta_{8} q^{5} + ( 1 - \beta_{1} ) q^{6} -\beta_{14} q^{7} + q^{8} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 - \beta_{1} ) q^{3} + q^{4} + \beta_{8} q^{5} + ( 1 - \beta_{1} ) q^{6} -\beta_{14} q^{7} + q^{8} + ( 2 - \beta_{1} + \beta_{2} ) q^{9} + \beta_{8} q^{10} + \beta_{4} q^{11} + ( 1 - \beta_{1} ) q^{12} + ( 1 - \beta_{15} ) q^{13} -\beta_{14} q^{14} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{15} + q^{16} + ( -1 - \beta_{2} - \beta_{12} + \beta_{13} ) q^{17} + ( 2 - \beta_{1} + \beta_{2} ) q^{18} + ( 3 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{12} - \beta_{13} ) q^{19} + \beta_{8} q^{20} + ( 1 + \beta_{2} + \beta_{5} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{21} + \beta_{4} q^{22} + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{23} + ( 1 - \beta_{1} ) q^{24} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{25} + ( 1 - \beta_{15} ) q^{26} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{27} -\beta_{14} q^{28} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{14} + \beta_{15} ) q^{29} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{30} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{31} + q^{32} + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{33} + ( -1 - \beta_{2} - \beta_{12} + \beta_{13} ) q^{34} + ( -1 + 3 \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{14} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( 3 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{37} + ( 3 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{12} - \beta_{13} ) q^{38} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{39} + \beta_{8} q^{40} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{41} + ( 1 + \beta_{2} + \beta_{5} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{42} + ( \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{43} + \beta_{4} q^{44} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{45} + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{46} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{47} + ( 1 - \beta_{1} ) q^{48} + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{50} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{8} + \beta_{9} - \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{51} + ( 1 - \beta_{15} ) q^{52} + ( -3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{53} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{54} + ( -\beta_{3} + \beta_{4} - \beta_{7} - \beta_{12} + \beta_{14} ) q^{55} -\beta_{14} q^{56} + ( 2 + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{57} + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{14} + \beta_{15} ) q^{58} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{59} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{60} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{61} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{62} + ( 2 + 4 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} + 3 \beta_{12} - \beta_{13} ) q^{63} + q^{64} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{65} + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{66} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{67} + ( -1 - \beta_{2} - \beta_{12} + \beta_{13} ) q^{68} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{69} + ( -1 + 3 \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{14} ) q^{70} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} ) q^{72} + ( 4 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} + 4 \beta_{12} + \beta_{15} ) q^{73} + ( 3 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{74} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - 3 \beta_{12} - \beta_{13} ) q^{75} + ( 3 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{12} - \beta_{13} ) q^{76} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{77} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{78} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} + 2 \beta_{12} ) q^{79} + \beta_{8} q^{80} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{81} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{82} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{83} + ( 1 + \beta_{2} + \beta_{5} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{84} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{85} + ( \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{86} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} - 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{87} + \beta_{4} q^{88} + ( -3 - \beta_{1} + \beta_{2} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{89} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{90} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{10} - 3 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{91} + ( -\beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} ) q^{92} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{93} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{94} + ( 5 - \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} - 2 \beta_{15} ) q^{95} + ( 1 - \beta_{1} ) q^{96} + ( -\beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{97} + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{98} + ( -5 - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{2} + 13q^{3} + 16q^{4} + 4q^{5} + 13q^{6} + 7q^{7} + 16q^{8} + 21q^{9} + O(q^{10}) \) \( 16q + 16q^{2} + 13q^{3} + 16q^{4} + 4q^{5} + 13q^{6} + 7q^{7} + 16q^{8} + 21q^{9} + 4q^{10} + 4q^{11} + 13q^{12} + 17q^{13} + 7q^{14} + 8q^{15} + 16q^{16} - q^{17} + 21q^{18} + 23q^{19} + 4q^{20} + 9q^{21} + 4q^{22} + 15q^{23} + 13q^{24} + 24q^{25} + 17q^{26} + 31q^{27} + 7q^{28} + 4q^{29} + 8q^{30} + 42q^{31} + 16q^{32} + 3q^{33} - q^{34} - 13q^{35} + 21q^{36} + 31q^{37} + 23q^{38} - 2q^{39} + 4q^{40} - 9q^{41} + 9q^{42} + 13q^{43} + 4q^{44} - 2q^{45} + 15q^{46} + 18q^{47} + 13q^{48} - 9q^{49} + 24q^{50} - 2q^{51} + 17q^{52} - 14q^{53} + 31q^{54} - 2q^{55} + 7q^{56} - 18q^{57} + 4q^{58} + 4q^{59} + 8q^{60} + q^{61} + 42q^{62} + 17q^{63} + 16q^{64} - 32q^{65} + 3q^{66} + 5q^{67} - q^{68} + 6q^{69} - 13q^{70} + 9q^{71} + 21q^{72} + 28q^{73} + 31q^{74} + 16q^{75} + 23q^{76} - 30q^{77} - 2q^{78} + 10q^{79} + 4q^{80} + 12q^{81} - 9q^{82} + 3q^{83} + 9q^{84} - 7q^{85} + 13q^{86} - 22q^{87} + 4q^{88} - 17q^{89} - 2q^{90} + 12q^{91} + 15q^{92} - q^{93} + 18q^{94} - 4q^{95} + 13q^{96} - 17q^{97} - 9q^{98} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} - 25 x^{14} + 59 x^{13} + 273 x^{12} - 443 x^{11} - 1620 x^{10} + 1595 x^{9} + 5490 x^{8} - 2787 x^{7} - 10540 x^{6} + 1919 x^{5} + 10822 x^{4} + 132 x^{3} - 5202 x^{2} - 408 x + 864\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-251329 \nu^{15} - 1584507 \nu^{14} + 18369211 \nu^{13} + 16293859 \nu^{12} - 277881195 \nu^{11} - 19041811 \nu^{10} + 1798435782 \nu^{9} - 367991447 \nu^{8} - 5839296552 \nu^{7} + 1916668527 \nu^{6} + 9731173486 \nu^{5} - 3860286863 \nu^{4} - 7712219392 \nu^{3} + 3338170596 \nu^{2} + 2228921478 \nu - 995600016\)\()/15170388\)
\(\beta_{4}\)\(=\)\((\)\(2152961 \nu^{15} - 2992767 \nu^{14} - 75505901 \nu^{13} + 101357887 \nu^{12} + 942358629 \nu^{11} - 1106869447 \nu^{10} - 5673325680 \nu^{9} + 5644850179 \nu^{8} + 17996454078 \nu^{7} - 14996722779 \nu^{6} - 30039066440 \nu^{5} + 20895440935 \nu^{4} + 24175232882 \nu^{3} - 14156562972 \nu^{2} - 7154663178 \nu + 3722926488\)\()/30340776\)
\(\beta_{5}\)\(=\)\((\)\(1163869 \nu^{15} + 566367 \nu^{14} - 54227851 \nu^{13} + 33414857 \nu^{12} + 747190683 \nu^{11} - 608856389 \nu^{10} - 4740141690 \nu^{9} + 3887672003 \nu^{8} + 15625211376 \nu^{7} - 11924688423 \nu^{6} - 26979358726 \nu^{5} + 18408067175 \nu^{4} + 22359231148 \nu^{3} - 13284257688 \nu^{2} - 6739432566 \nu + 3546141852\)\()/15170388\)
\(\beta_{6}\)\(=\)\((\)\(703019 \nu^{15} - 1312695 \nu^{14} - 24512099 \nu^{13} + 45337273 \nu^{12} + 299715009 \nu^{11} - 505524805 \nu^{10} - 1785824604 \nu^{9} + 2637272827 \nu^{8} + 5708655198 \nu^{7} - 7151499639 \nu^{6} - 9793834760 \nu^{5} + 10073866207 \nu^{4} + 8254709780 \nu^{3} - 6716775792 \nu^{2} - 2590394766 \nu + 1619413560\)\()/7585194\)
\(\beta_{7}\)\(=\)\((\)\(3933517 \nu^{15} - 12690411 \nu^{14} - 100730473 \nu^{13} + 283037219 \nu^{12} + 1078589433 \nu^{11} - 2490320003 \nu^{10} - 6051990768 \nu^{9} + 11100949679 \nu^{8} + 18727709070 \nu^{7} - 26695453431 \nu^{6} - 31298943400 \nu^{5} + 33854742995 \nu^{4} + 25510417498 \nu^{3} - 20331685164 \nu^{2} - 7541723994 \nu + 4476605448\)\()/30340776\)
\(\beta_{8}\)\(=\)\((\)\(-5141315 \nu^{15} + 19777701 \nu^{14} + 114313127 \nu^{13} - 407970373 \nu^{12} - 1116678999 \nu^{11} + 3377150125 \nu^{10} + 6001094040 \nu^{9} - 14360727577 \nu^{8} - 18440525034 \nu^{7} + 33370762041 \nu^{6} + 31302228416 \nu^{5} - 41553289765 \nu^{4} - 26280106694 \nu^{3} + 25105053516 \nu^{2} + 8148347334 \nu - 5633312592\)\()/30340776\)
\(\beta_{9}\)\(=\)\((\)\(-7097413 \nu^{15} + 30581883 \nu^{14} + 138236065 \nu^{13} - 603673235 \nu^{12} - 1157390361 \nu^{11} + 4710760955 \nu^{10} + 5402817528 \nu^{9} - 18628890671 \nu^{8} - 14997319518 \nu^{7} + 39797622327 \nu^{6} + 24184584640 \nu^{5} - 45097771715 \nu^{4} - 20209515514 \nu^{3} + 24564373956 \nu^{2} + 6263525418 \nu - 4970534088\)\()/30340776\)
\(\beta_{10}\)\(=\)\((\)\(4543691 \nu^{15} - 20381685 \nu^{14} - 83574707 \nu^{13} + 392059489 \nu^{12} + 667688319 \nu^{11} - 3000681097 \nu^{10} - 3022863048 \nu^{9} + 11678367937 \nu^{8} + 8270149962 \nu^{7} - 24554922765 \nu^{6} - 13281073880 \nu^{5} + 27330854257 \nu^{4} + 11105810834 \nu^{3} - 14605002864 \nu^{2} - 3519302850 \nu + 2886895020\)\()/15170388\)
\(\beta_{11}\)\(=\)\((\)\(14428739 \nu^{15} - 53907381 \nu^{14} - 327245255 \nu^{13} + 1125212773 \nu^{12} + 3188169327 \nu^{11} - 9294142285 \nu^{10} - 16853828448 \nu^{9} + 39069502297 \nu^{8} + 50763253386 \nu^{7} - 89372958825 \nu^{6} - 84924755768 \nu^{5} + 109577189557 \nu^{4} + 71154890678 \nu^{3} - 65483652324 \nu^{2} - 22295790006 \nu + 14760195528\)\()/30340776\)
\(\beta_{12}\)\(=\)\((\)\(15215297 \nu^{15} - 64081299 \nu^{14} - 304999169 \nu^{13} + 1273696411 \nu^{12} + 2660739345 \nu^{11} - 10063616635 \nu^{10} - 12955139340 \nu^{9} + 40492006795 \nu^{8} + 37081420506 \nu^{7} - 88374269139 \nu^{6} - 60491827340 \nu^{5} + 102689380255 \nu^{4} + 50315415662 \nu^{3} - 57565192932 \nu^{2} - 15794686386 \nu + 11973523776\)\()/30340776\)
\(\beta_{13}\)\(=\)\((\)\(-2852135 \nu^{15} + 14248853 \nu^{14} + 43424983 \nu^{13} - 259121861 \nu^{12} - 264026759 \nu^{11} + 1864655217 \nu^{10} + 846344956 \nu^{9} - 6764844181 \nu^{8} - 1632100274 \nu^{7} + 13082455793 \nu^{6} + 2094764512 \nu^{5} - 13094073201 \nu^{4} - 1733959858 \nu^{3} + 6049518780 \nu^{2} + 604543938 \nu - 961474808\)\()/5056796\)
\(\beta_{14}\)\(=\)\((\)\(-18237533 \nu^{15} + 84027087 \nu^{14} + 320640653 \nu^{13} - 1589923375 \nu^{12} - 2424683013 \nu^{11} + 11965715695 \nu^{10} + 10368322836 \nu^{9} - 45794255239 \nu^{8} - 26980406706 \nu^{7} + 94729495911 \nu^{6} + 41768330372 \nu^{5} - 103654349683 \nu^{4} - 34212109358 \nu^{3} + 54087013740 \nu^{2} + 10744813050 \nu - 10233058152\)\()/30340776\)
\(\beta_{15}\)\(=\)\((\)\(3203661 \nu^{15} - 14259529 \nu^{14} - 58957967 \nu^{13} + 273586729 \nu^{12} + 465968347 \nu^{11} - 2077186025 \nu^{10} - 2054153162 \nu^{9} + 7978844087 \nu^{8} + 5381233920 \nu^{7} - 16479785559 \nu^{6} - 8176382814 \nu^{5} + 17909453195 \nu^{4} + 6455689520 \nu^{3} - 9239471536 \nu^{2} - 1951767018 \nu + 1721500680\)\()/5056796\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-3 \beta_{14} + 3 \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 3 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 11 \beta_{2} + 18 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(-16 \beta_{14} + 18 \beta_{13} - \beta_{12} + 17 \beta_{11} - 3 \beta_{10} + 10 \beta_{9} + 10 \beta_{8} + \beta_{7} - 19 \beta_{6} - 4 \beta_{5} + \beta_{4} + 15 \beta_{3} + 32 \beta_{2} + 87 \beta_{1} + 50\)
\(\nu^{6}\)\(=\)\(2 \beta_{15} - 51 \beta_{14} + 64 \beta_{13} - 7 \beta_{12} + 64 \beta_{11} - 21 \beta_{10} + 23 \beta_{9} + 17 \beta_{8} + 6 \beta_{7} - 71 \beta_{6} - 26 \beta_{5} + 6 \beta_{4} + 50 \beta_{3} + 128 \beta_{2} + 273 \beta_{1} + 243\)
\(\nu^{7}\)\(=\)\(11 \beta_{15} - 207 \beta_{14} + 284 \beta_{13} - \beta_{12} + 278 \beta_{11} - 77 \beta_{10} + 118 \beta_{9} + 85 \beta_{8} + 33 \beta_{7} - 323 \beta_{6} - 111 \beta_{5} - 6 \beta_{4} + 196 \beta_{3} + 425 \beta_{2} + 1113 \beta_{1} + 694\)
\(\nu^{8}\)\(=\)\(73 \beta_{15} - 697 \beta_{14} + 1074 \beta_{13} + 2 \beta_{12} + 1092 \beta_{11} - 368 \beta_{10} + 379 \beta_{9} + 168 \beta_{8} + 155 \beta_{7} - 1266 \beta_{6} - 522 \beta_{5} - 53 \beta_{4} + 686 \beta_{3} + 1566 \beta_{2} + 3935 \beta_{1} + 2786\)
\(\nu^{9}\)\(=\)\(348 \beta_{15} - 2603 \beta_{14} + 4351 \beta_{13} + 206 \beta_{12} + 4422 \beta_{11} - 1431 \beta_{10} + 1588 \beta_{9} + 592 \beta_{8} + 693 \beta_{7} - 5218 \beta_{6} - 2181 \beta_{5} - 474 \beta_{4} + 2544 \beta_{3} + 5456 \beta_{2} + 15185 \beta_{1} + 9302\)
\(\nu^{10}\)\(=\)\(1685 \beta_{15} - 9070 \beta_{14} + 16713 \beta_{13} + 1015 \beta_{12} + 17357 \beta_{11} - 6027 \beta_{10} + 5727 \beta_{9} + 990 \beta_{8} + 2960 \beta_{7} - 20487 \beta_{6} - 9233 \beta_{5} - 2381 \beta_{4} + 9127 \beta_{3} + 19735 \beta_{2} + 55948 \beta_{1} + 35123\)
\(\nu^{11}\)\(=\)\(7370 \beta_{15} - 33049 \beta_{14} + 65481 \beta_{13} + 5455 \beta_{12} + 68406 \beta_{11} - 23636 \beta_{10} + 22492 \beta_{9} + 1939 \beta_{8} + 12326 \beta_{7} - 81289 \beta_{6} - 37493 \beta_{5} - 11721 \beta_{4} + 33497 \beta_{3} + 70303 \beta_{2} + 212281 \beta_{1} + 124689\)
\(\nu^{12}\)\(=\)\(31884 \beta_{15} - 117636 \beta_{14} + 251712 \beta_{13} + 23845 \beta_{12} + 266326 \beta_{11} - 94698 \beta_{10} + 84142 \beta_{9} - 3925 \beta_{8} + 50265 \beta_{7} - 316733 \beta_{6} - 151659 \beta_{5} - 51783 \beta_{4} + 121922 \beta_{3} + 254449 \beta_{2} + 793659 \beta_{1} + 463117\)
\(\nu^{13}\)\(=\)\(132451 \beta_{15} - 427293 \beta_{14} + 972095 \beta_{13} + 104550 \beta_{12} + 1035125 \beta_{11} - 369408 \beta_{10} + 323342 \beta_{9} - 40480 \beta_{8} + 202083 \beta_{7} - 1234443 \beta_{6} - 602597 \beta_{5} - 223671 \beta_{4} + 448323 \beta_{3} + 918978 \beta_{2} + 2997310 \beta_{1} + 1687445\)
\(\nu^{14}\)\(=\)\(543533 \beta_{15} - 1542977 \beta_{14} + 3725861 \beta_{13} + 432193 \beta_{12} + 3999346 \beta_{11} - 1446602 \beta_{10} + 1222586 \beta_{9} - 262160 \beta_{8} + 803003 \beta_{7} - 4772847 \beta_{6} - 2380238 \beta_{5} - 930103 \beta_{4} + 1647593 \beta_{3} + 3346632 \beta_{2} + 11268795 \beta_{1} + 6257237\)
\(\nu^{15}\)\(=\)\(2187672 \beta_{15} - 5629182 \beta_{14} + 14284411 \beta_{13} + 1771938 \beta_{12} + 15413987 \beta_{11} - 5601867 \beta_{10} + 4660886 \beta_{9} - 1293022 \beta_{8} + 3163330 \beta_{7} - 18418463 \beta_{6} - 9316307 \beta_{5} - 3807019 \beta_{4} + 6087977 \beta_{3} + 12210668 \beta_{2} + 42520221 \beta_{1} + 23090620\)

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
3.78209
3.23864
2.25035
1.75595
1.64226
1.30501
0.861451
0.467096
−0.832294
−1.04534
−1.11488
−1.57335
−1.62117
−1.78129
−2.15895
−2.17558
1.00000 −2.78209 1.00000 −0.612854 −2.78209 3.29799 1.00000 4.74004 −0.612854
1.2 1.00000 −2.23864 1.00000 1.50415 −2.23864 −1.71641 1.00000 2.01150 1.50415
1.3 1.00000 −1.25035 1.00000 −3.07514 −1.25035 −0.487959 1.00000 −1.43661 −3.07514
1.4 1.00000 −0.755947 1.00000 −3.67615 −0.755947 −3.29606 1.00000 −2.42854 −3.67615
1.5 1.00000 −0.642262 1.00000 4.25875 −0.642262 0.198430 1.00000 −2.58750 4.25875
1.6 1.00000 −0.305006 1.00000 2.67935 −0.305006 2.41108 1.00000 −2.90697 2.67935
1.7 1.00000 0.138549 1.00000 1.13854 0.138549 −2.28657 1.00000 −2.98080 1.13854
1.8 1.00000 0.532904 1.00000 −2.56419 0.532904 4.62175 1.00000 −2.71601 −2.56419
1.9 1.00000 1.83229 1.00000 1.86915 1.83229 −1.62212 1.00000 0.357300 1.86915
1.10 1.00000 2.04534 1.00000 3.81409 2.04534 0.856873 1.00000 1.18340 3.81409
1.11 1.00000 2.11488 1.00000 −1.27093 2.11488 2.13923 1.00000 1.47271 −1.27093
1.12 1.00000 2.57335 1.00000 −0.396646 2.57335 −0.271369 1.00000 3.62214 −0.396646
1.13 1.00000 2.62117 1.00000 1.35478 2.62117 4.32530 1.00000 3.87051 1.35478
1.14 1.00000 2.78129 1.00000 0.221151 2.78129 0.625043 1.00000 4.73559 0.221151
1.15 1.00000 3.15895 1.00000 −3.99340 3.15895 2.00996 1.00000 6.97896 −3.99340
1.16 1.00000 3.17558 1.00000 2.74937 3.17558 −3.80515 1.00000 7.08430 2.74937
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

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 1502.2.a.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.g 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).