# Properties

 Label 273.2.bj.d Level $273$ Weight $2$ Character orbit 273.bj Analytic conductor $2.180$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 11 x^{14} + 85 x^{12} - 310 x^{10} + 807 x^{8} - 1196 x^{6} + 1273 x^{4} - 688 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{8} q^{2} + \beta_{3} q^{3} + ( \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{12} ) q^{4} + ( \beta_{1} + \beta_{7} + \beta_{8} - \beta_{14} ) q^{5} -\beta_{4} q^{6} + ( \beta_{8} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{7} + ( -3 \beta_{4} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{8} + ( -1 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{8} q^{2} + \beta_{3} q^{3} + ( \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{12} ) q^{4} + ( \beta_{1} + \beta_{7} + \beta_{8} - \beta_{14} ) q^{5} -\beta_{4} q^{6} + ( \beta_{8} - \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{7} + ( -3 \beta_{4} + \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{8} + ( -1 + \beta_{3} ) q^{9} + ( -\beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{9} + \beta_{12} ) q^{10} + ( \beta_{1} + \beta_{4} - \beta_{8} ) q^{11} + ( -1 + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{12} + ( \beta_{2} + \beta_{4} - \beta_{9} - \beta_{10} - \beta_{14} - \beta_{15} ) q^{13} + ( -\beta_{2} - \beta_{3} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{14} + ( \beta_{4} + \beta_{7} - \beta_{14} - \beta_{15} ) q^{15} + ( -4 + 4 \beta_{3} - \beta_{6} - 3 \beta_{10} - 3 \beta_{11} ) q^{16} + ( -\beta_{2} + \beta_{5} + \beta_{6} ) q^{17} + ( -\beta_{4} + \beta_{8} ) q^{18} + ( 3 \beta_{1} + 3 \beta_{7} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{19} + ( 3 \beta_{4} - 3 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{20} + ( \beta_{4} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{21} + ( 4 + \beta_{2} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{22} + ( 3 - 3 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{23} + ( -\beta_{1} - 3 \beta_{4} + 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + \beta_{15} ) q^{24} + ( \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{25} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{10} + 3 \beta_{11} - \beta_{14} ) q^{26} - q^{27} + ( 2 \beta_{1} + 4 \beta_{4} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{12} - \beta_{14} - 2 \beta_{15} ) q^{28} + ( 1 + 2 \beta_{2} + \beta_{13} ) q^{29} + ( 2 - 2 \beta_{3} + \beta_{6} + \beta_{10} + \beta_{11} ) q^{30} + ( -2 \beta_{1} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{31} + ( -3 \beta_{1} - 5 \beta_{4} + 5 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + 4 \beta_{15} ) q^{32} + ( \beta_{1} + \beta_{7} - \beta_{8} ) q^{33} + ( \beta_{7} - \beta_{14} - \beta_{15} ) q^{34} + ( -3 - \beta_{2} + 3 \beta_{3} + \beta_{9} - \beta_{12} - 2 \beta_{13} ) q^{35} + ( -1 - \beta_{2} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{36} + ( -\beta_{1} - \beta_{7} + 4 \beta_{8} - 3 \beta_{10} + 3 \beta_{11} ) q^{37} + ( \beta_{3} - \beta_{5} + 4 \beta_{9} - 4 \beta_{12} ) q^{38} + ( \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{15} ) q^{39} + ( 4 - 4 \beta_{3} - 3 \beta_{5} + \beta_{6} + 4 \beta_{10} + 4 \beta_{11} - 3 \beta_{13} ) q^{40} + ( -3 \beta_{4} - 4 \beta_{7} ) q^{41} + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{42} + ( -5 + \beta_{2} - \beta_{13} ) q^{43} + ( \beta_{1} + \beta_{7} - 8 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} + 3 \beta_{14} ) q^{44} + ( -\beta_{1} + \beta_{4} - \beta_{8} - \beta_{15} ) q^{45} + ( -\beta_{1} + \beta_{4} - \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + 3 \beta_{15} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{7} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{14} ) q^{47} + ( -4 - \beta_{2} - 3 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} ) q^{48} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{49} + ( \beta_{4} - \beta_{7} + 3 \beta_{14} + 3 \beta_{15} ) q^{50} + ( \beta_{5} + \beta_{6} + \beta_{13} ) q^{51} + ( 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{4} + \beta_{5} + 2 \beta_{6} - 5 \beta_{8} - 5 \beta_{9} - \beta_{12} - 2 \beta_{15} ) q^{52} + ( 3 \beta_{2} + \beta_{3} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{9} - 2 \beta_{12} ) q^{53} + \beta_{8} q^{54} + ( -1 - 2 \beta_{2} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{55} + ( 9 + \beta_{2} - 8 \beta_{3} + \beta_{6} + 2 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{56} + ( 3 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{7} - 2 \beta_{8} + \beta_{14} ) q^{58} + ( -\beta_{1} + \beta_{4} - \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + \beta_{15} ) q^{59} + ( 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} ) q^{60} + ( 4 - 4 \beta_{3} + 3 \beta_{5} - \beta_{6} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{13} ) q^{61} + ( -7 - \beta_{2} - 3 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{62} + ( \beta_{4} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{14} ) q^{63} + ( -6 - 3 \beta_{2} - 6 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} + 6 \beta_{12} + \beta_{13} ) q^{64} + ( -4 + 2 \beta_{1} + 4 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} - 4 \beta_{11} - \beta_{13} - \beta_{14} ) q^{65} + ( \beta_{2} + 4 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{9} - 2 \beta_{12} ) q^{66} + ( -7 \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{12} - 2 \beta_{15} ) q^{67} + ( -1 + \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{13} ) q^{68} + ( 3 - \beta_{2} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{69} + ( \beta_{1} - 5 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{70} + ( \beta_{4} - \beta_{7} + 3 \beta_{14} + 3 \beta_{15} ) q^{71} + ( -\beta_{1} - \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{14} ) q^{72} + ( -3 \beta_{4} + 3 \beta_{8} + \beta_{9} + \beta_{12} ) q^{73} + ( -4 \beta_{2} - 7 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} - 5 \beta_{9} + 5 \beta_{12} ) q^{74} + ( -1 + \beta_{3} - 3 \beta_{5} - \beta_{6} - 3 \beta_{13} ) q^{75} + ( -8 \beta_{4} + 2 \beta_{7} + 6 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} + 6 \beta_{12} + 7 \beta_{14} + 7 \beta_{15} ) q^{76} + ( -3 \beta_{2} - \beta_{3} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{77} + ( 2 + \beta_{2} - 2 \beta_{7} + \beta_{9} + \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - \beta_{14} - \beta_{15} ) q^{78} + ( -1 + \beta_{3} + \beta_{5} + 4 \beta_{6} + 5 \beta_{10} + 5 \beta_{11} + \beta_{13} ) q^{79} + ( \beta_{1} + 10 \beta_{4} - 10 \beta_{8} - 4 \beta_{9} - 4 \beta_{12} - 5 \beta_{15} ) q^{80} -\beta_{3} q^{81} + ( -5 + 5 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} + \beta_{10} + \beta_{11} + 3 \beta_{13} ) q^{82} + ( 3 \beta_{4} - 2 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - 4 \beta_{12} - \beta_{14} - \beta_{15} ) q^{83} + ( 3 \beta_{1} + 3 \beta_{4} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{84} + ( -4 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - 4 \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{85} + ( -\beta_{1} - \beta_{7} + 3 \beta_{8} - \beta_{14} ) q^{86} + ( 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{87} + ( 6 \beta_{2} + 9 \beta_{3} + 3 \beta_{5} - 6 \beta_{6} + 8 \beta_{9} - 8 \beta_{12} ) q^{88} + ( 3 \beta_{1} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{89} + ( 2 + \beta_{2} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{90} + ( -4 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{11} + \beta_{13} - 3 \beta_{15} ) q^{91} + ( 3 \beta_{2} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{93} + ( -6 \beta_{3} - 2 \beta_{5} ) q^{94} + ( -2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{9} + 3 \beta_{12} ) q^{95} + ( -3 \beta_{1} - 3 \beta_{7} + 5 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - 4 \beta_{14} ) q^{96} + ( \beta_{7} - 2 \beta_{14} - 2 \beta_{15} ) q^{97} + ( -\beta_{1} - 5 \beta_{4} + 4 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{98} + ( -\beta_{4} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{3} + 8q^{4} - 8q^{9} + O(q^{10})$$ $$16q + 8q^{3} + 8q^{4} - 8q^{9} - 16q^{10} - 8q^{12} + 6q^{13} - 10q^{14} - 28q^{16} - 2q^{17} + 60q^{22} + 24q^{23} + 10q^{25} + 14q^{26} - 16q^{27} + 24q^{29} + 16q^{30} - 30q^{35} - 16q^{36} + 3q^{39} + 26q^{40} + 4q^{42} - 76q^{43} - 56q^{48} + 2q^{49} + 2q^{51} + 10q^{53} - 16q^{55} + 72q^{56} + 26q^{61} - 104q^{62} - 84q^{64} - 32q^{65} + 30q^{66} - 12q^{68} + 48q^{69} - 54q^{74} - 10q^{75} - 10q^{77} + 28q^{78} - 10q^{79} - 8q^{81} - 48q^{82} + 12q^{87} + 68q^{88} + 32q^{90} - 57q^{91} + 16q^{92} - 48q^{94} + 18q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11 x^{14} + 85 x^{12} - 310 x^{10} + 807 x^{8} - 1196 x^{6} + 1273 x^{4} - 688 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$93708 \nu^{14} - 905544 \nu^{12} + 6775609 \nu^{10} - 19993643 \nu^{8} + 48900157 \nu^{6} - 34658945 \nu^{4} + 19033136 \nu^{2} + 122770133$$$$)/53934263$$ $$\beta_{3}$$ $$=$$ $$($$$$-2439541 \nu^{14} + 25335623 \nu^{12} - 192872281 \nu^{10} + 647847966 \nu^{8} - 1648811299 \nu^{6} + 2135288524 \nu^{4} - 2550992573 \nu^{2} + 1373874032$$$$)/ 862948208$$ $$\beta_{4}$$ $$=$$ $$($$$$2051459 \nu^{15} - 31489585 \nu^{13} + 239720495 \nu^{11} - 1065436466 \nu^{9} + 2049303605 \nu^{7} - 2653944980 \nu^{5} - 344398261 \nu^{3} - 635028480 \nu$$$$)/ 1725896416$$ $$\beta_{5}$$ $$=$$ $$($$$$5499477 \nu^{14} - 79437735 \nu^{12} + 653830489 \nu^{10} - 3074557294 \nu^{8} + 8479881587 \nu^{6} - 15396256332 \nu^{4} + 14007293741 \nu^{2} - 7631273072$$$$)/ 862948208$$ $$\beta_{6}$$ $$=$$ $$($$$$7318623 \nu^{14} - 76006869 \nu^{12} + 578616843 \nu^{10} - 1943543898 \nu^{8} + 4946433897 \nu^{6} - 6405865572 \nu^{4} + 6790029511 \nu^{2} - 1532777472$$$$)/ 862948208$$ $$\beta_{7}$$ $$=$$ $$($$$$-2439541 \nu^{15} + 25335623 \nu^{13} - 192872281 \nu^{11} + 647847966 \nu^{9} - 1648811299 \nu^{7} + 2135288524 \nu^{5} - 2550992573 \nu^{3} + 510925824 \nu$$$$)/ 862948208$$ $$\beta_{8}$$ $$=$$ $$($$$$-6901309 \nu^{15} + 58008863 \nu^{13} - 407614369 \nu^{11} + 844736062 \nu^{9} - 1749011307 \nu^{7} + 657335740 \nu^{5} - 2162804917 \nu^{3} + 1111287280 \nu$$$$)/ 1725896416$$ $$\beta_{9}$$ $$=$$ $$($$$$-7983216 \nu^{15} + 6901309 \nu^{14} + 78057212 \nu^{13} - 58008863 \nu^{12} - 577230868 \nu^{11} + 407614369 \nu^{10} + 1703307836 \nu^{9} - 844736062 \nu^{8} - 3851063448 \nu^{7} + 1749011307 \nu^{6} + 2952681140 \nu^{5} - 657335740 \nu^{4} - 1621479872 \nu^{3} + 2162804917 \nu^{2} - 2985621268 \nu - 1111287280$$$$)/ 1725896416$$ $$\beta_{10}$$ $$=$$ $$($$$$-34417935 \nu^{15} + 8205836 \nu^{14} + 390591909 \nu^{13} - 125958340 \nu^{12} - 3041434107 \nu^{11} + 958881980 \nu^{10} + 11536837802 \nu^{9} - 4261745864 \nu^{8} - 30334459849 \nu^{7} + 8197214420 \nu^{6} + 47423070356 \nu^{5} - 10615779920 \nu^{4} - 48250376215 \nu^{3} - 1377593044 \nu^{2} + 26115780688 \nu - 2540113920$$$$)/ 6903585664$$ $$\beta_{11}$$ $$=$$ $$($$$$34417935 \nu^{15} + 8205836 \nu^{14} - 390591909 \nu^{13} - 125958340 \nu^{12} + 3041434107 \nu^{11} + 958881980 \nu^{10} - 11536837802 \nu^{9} - 4261745864 \nu^{8} + 30334459849 \nu^{7} + 8197214420 \nu^{6} - 47423070356 \nu^{5} - 10615779920 \nu^{4} + 48250376215 \nu^{3} - 1377593044 \nu^{2} - 26115780688 \nu - 2540113920$$$$)/ 6903585664$$ $$\beta_{12}$$ $$=$$ $$($$$$-7983216 \nu^{15} - 6901309 \nu^{14} + 78057212 \nu^{13} + 58008863 \nu^{12} - 577230868 \nu^{11} - 407614369 \nu^{10} + 1703307836 \nu^{9} + 844736062 \nu^{8} - 3851063448 \nu^{7} - 1749011307 \nu^{6} + 2952681140 \nu^{5} + 657335740 \nu^{4} - 1621479872 \nu^{3} - 2162804917 \nu^{2} - 2985621268 \nu + 1111287280$$$$)/ 1725896416$$ $$\beta_{13}$$ $$=$$ $$($$$$1183968 \nu^{14} - 11648434 \nu^{12} + 85607464 \nu^{10} - 252612728 \nu^{8} + 551180115 \nu^{6} - 437903720 \nu^{4} + 240477056 \nu^{2} + 141925895$$$$)/53934263$$ $$\beta_{14}$$ $$=$$ $$($$$$5499477 \nu^{15} - 79437735 \nu^{13} + 653830489 \nu^{11} - 3074557294 \nu^{9} + 8479881587 \nu^{7} - 15396256332 \nu^{5} + 14007293741 \nu^{3} - 7631273072 \nu$$$$)/ 862948208$$ $$\beta_{15}$$ $$=$$ $$($$$$1183968 \nu^{15} - 11648434 \nu^{13} + 85607464 \nu^{11} - 252612728 \nu^{9} + 551180115 \nu^{7} - 437903720 \nu^{5} + 240477056 \nu^{3} + 141925895 \nu$$$$)/53934263$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} - 3 \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 4 \beta_{7} - \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{12} + \beta_{9} - 8 \beta_{6} + \beta_{5} - 14 \beta_{3} + 8 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{14} - 16 \beta_{11} + 16 \beta_{10} - 10 \beta_{8} - 22 \beta_{7} - 22 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{13} - 10 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} + 10 \beta_{9} + 54 \beta_{2} - 79$$ $$\nu^{7}$$ $$=$$ $$-63 \beta_{15} - 108 \beta_{12} - 108 \beta_{9} - 74 \beta_{8} + 74 \beta_{4} - 133 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-63 \beta_{13} + 74 \beta_{11} + 74 \beta_{10} + 349 \beta_{6} - 63 \beta_{5} + 478 \beta_{3} - 478$$ $$\nu^{9}$$ $$=$$ $$-412 \beta_{15} - 412 \beta_{14} - 698 \beta_{12} + 698 \beta_{11} - 698 \beta_{10} - 698 \beta_{9} + 827 \beta_{7} + 497 \beta_{4}$$ $$\nu^{10}$$ $$=$$ $$497 \beta_{12} - 497 \beta_{9} + 2223 \beta_{6} - 412 \beta_{5} + 2968 \beta_{3} - 2223 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$-2635 \beta_{14} + 4446 \beta_{11} - 4446 \beta_{10} + 3217 \beta_{8} + 5191 \beta_{7} + 5191 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$2635 \beta_{13} + 3217 \beta_{12} - 3217 \beta_{11} - 3217 \beta_{10} - 3217 \beta_{9} - 14083 \beta_{2} + 18613$$ $$\nu^{13}$$ $$=$$ $$16718 \beta_{15} + 28166 \beta_{12} + 28166 \beta_{9} + 20517 \beta_{8} - 20517 \beta_{4} + 32696 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$16718 \beta_{13} - 20517 \beta_{11} - 20517 \beta_{10} - 89028 \beta_{6} + 16718 \beta_{5} - 117185 \beta_{3} + 117185$$ $$\nu^{15}$$ $$=$$ $$105746 \beta_{15} + 105746 \beta_{14} + 178056 \beta_{12} - 178056 \beta_{11} + 178056 \beta_{10} + 178056 \beta_{9} - 206213 \beta_{7} - 130062 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −1.39394 − 0.804793i 1.02312 + 0.590698i 0.725066 + 0.418617i −2.17587 − 1.25624i 2.17587 + 1.25624i −0.725066 − 0.418617i −1.02312 − 0.590698i 1.39394 + 0.804793i −1.39394 + 0.804793i 1.02312 − 0.590698i 0.725066 − 0.418617i −2.17587 + 1.25624i 2.17587 − 1.25624i −0.725066 + 0.418617i −1.02312 + 0.590698i 1.39394 − 0.804793i
−2.35955 + 1.36229i 0.500000 0.866025i 2.71165 4.69671i 1.84188 1.06341i 2.72457i 1.72383 2.00709i 9.32701i −0.500000 0.866025i −2.89733 + 5.01833i
25.2 −1.69305 + 0.977485i 0.500000 0.866025i 0.910952 1.57782i 0.130553 0.0753750i 1.95497i 0.807080 + 2.51965i 0.348171i −0.500000 0.866025i −0.147356 + 0.255228i
25.3 −0.558156 + 0.322252i 0.500000 0.866025i −0.792308 + 1.37232i 3.70788 2.14075i 0.644503i −2.12926 1.57043i 2.31030i −0.500000 0.866025i −1.37972 + 2.38974i
25.4 −0.504542 + 0.291297i 0.500000 0.866025i −0.830292 + 1.43811i −1.26176 + 0.728479i 0.582595i −2.46845 + 0.952230i 2.13264i −0.500000 0.866025i 0.424408 0.735096i
25.5 0.504542 0.291297i 0.500000 0.866025i −0.830292 + 1.43811i 1.26176 0.728479i 0.582595i 2.46845 0.952230i 2.13264i −0.500000 0.866025i 0.424408 0.735096i
25.6 0.558156 0.322252i 0.500000 0.866025i −0.792308 + 1.37232i −3.70788 + 2.14075i 0.644503i 2.12926 + 1.57043i 2.31030i −0.500000 0.866025i −1.37972 + 2.38974i
25.7 1.69305 0.977485i 0.500000 0.866025i 0.910952 1.57782i −0.130553 + 0.0753750i 1.95497i −0.807080 2.51965i 0.348171i −0.500000 0.866025i −0.147356 + 0.255228i
25.8 2.35955 1.36229i 0.500000 0.866025i 2.71165 4.69671i −1.84188 + 1.06341i 2.72457i −1.72383 + 2.00709i 9.32701i −0.500000 0.866025i −2.89733 + 5.01833i
142.1 −2.35955 1.36229i 0.500000 + 0.866025i 2.71165 + 4.69671i 1.84188 + 1.06341i 2.72457i 1.72383 + 2.00709i 9.32701i −0.500000 + 0.866025i −2.89733 5.01833i
142.2 −1.69305 0.977485i 0.500000 + 0.866025i 0.910952 + 1.57782i 0.130553 + 0.0753750i 1.95497i 0.807080 2.51965i 0.348171i −0.500000 + 0.866025i −0.147356 0.255228i
142.3 −0.558156 0.322252i 0.500000 + 0.866025i −0.792308 1.37232i 3.70788 + 2.14075i 0.644503i −2.12926 + 1.57043i 2.31030i −0.500000 + 0.866025i −1.37972 2.38974i
142.4 −0.504542 0.291297i 0.500000 + 0.866025i −0.830292 1.43811i −1.26176 0.728479i 0.582595i −2.46845 0.952230i 2.13264i −0.500000 + 0.866025i 0.424408 + 0.735096i
142.5 0.504542 + 0.291297i 0.500000 + 0.866025i −0.830292 1.43811i 1.26176 + 0.728479i 0.582595i 2.46845 + 0.952230i 2.13264i −0.500000 + 0.866025i 0.424408 + 0.735096i
142.6 0.558156 + 0.322252i 0.500000 + 0.866025i −0.792308 1.37232i −3.70788 2.14075i 0.644503i 2.12926 1.57043i 2.31030i −0.500000 + 0.866025i −1.37972 2.38974i
142.7 1.69305 + 0.977485i 0.500000 + 0.866025i 0.910952 + 1.57782i −0.130553 0.0753750i 1.95497i −0.807080 + 2.51965i 0.348171i −0.500000 + 0.866025i −0.147356 0.255228i
142.8 2.35955 + 1.36229i 0.500000 + 0.866025i 2.71165 + 4.69671i −1.84188 1.06341i 2.72457i −1.72383 2.00709i 9.32701i −0.500000 + 0.866025i −2.89733 5.01833i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 142.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bj.d 16
3.b odd 2 1 819.2.dl.g 16
7.c even 3 1 inner 273.2.bj.d 16
7.c even 3 1 1911.2.c.j 8
7.d odd 6 1 1911.2.c.m 8
13.b even 2 1 inner 273.2.bj.d 16
21.h odd 6 1 819.2.dl.g 16
39.d odd 2 1 819.2.dl.g 16
91.r even 6 1 inner 273.2.bj.d 16
91.r even 6 1 1911.2.c.j 8
91.s odd 6 1 1911.2.c.m 8
273.w odd 6 1 819.2.dl.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.d 16 1.a even 1 1 trivial
273.2.bj.d 16 7.c even 3 1 inner
273.2.bj.d 16 13.b even 2 1 inner
273.2.bj.d 16 91.r even 6 1 inner
819.2.dl.g 16 3.b odd 2 1
819.2.dl.g 16 21.h odd 6 1
819.2.dl.g 16 39.d odd 2 1
819.2.dl.g 16 273.w odd 6 1
1911.2.c.j 8 7.c even 3 1
1911.2.c.j 8 91.r even 6 1
1911.2.c.m 8 7.d odd 6 1
1911.2.c.m 8 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 92 T^{2} + 381 T^{4} - 755 T^{6} + 1089 T^{8} - 398 T^{10} + 107 T^{12} - 12 T^{14} + T^{16}$$
$3$ $$( 1 - T + T^{2} )^{8}$$
$5$ $$16 - 716 T^{2} + 31513 T^{4} - 23428 T^{6} + 12945 T^{8} - 2942 T^{10} + 493 T^{12} - 25 T^{14} + T^{16}$$
$7$ $$5764801 - 117649 T^{2} + 160867 T^{4} + 4067 T^{6} + 2797 T^{8} + 83 T^{10} + 67 T^{12} - T^{14} + T^{16}$$
$11$ $$16 - 460 T^{2} + 12433 T^{4} - 22538 T^{6} + 35865 T^{8} - 5512 T^{10} + 643 T^{12} - 29 T^{14} + T^{16}$$
$13$ $$( 28561 - 6591 T - 2028 T^{2} + 507 T^{3} + 78 T^{4} + 39 T^{5} - 12 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$17$ $$( 16 - 92 T + 441 T^{2} - 498 T^{3} + 465 T^{4} - 68 T^{5} + 23 T^{6} + T^{7} + T^{8} )^{2}$$
$19$ $$61013446081 - 14020230840 T^{2} + 2220323114 T^{4} - 174775024 T^{6} + 9830787 T^{8} - 340528 T^{10} + 8490 T^{12} - 112 T^{14} + T^{16}$$
$23$ $$( 64 + 424 T + 2593 T^{2} + 1623 T^{3} + 1357 T^{4} - 430 T^{5} + 117 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$29$ $$( -104 + 143 T - 41 T^{2} - 6 T^{3} + T^{4} )^{4}$$
$31$ $$35153041 - 102275250 T^{2} + 272441327 T^{4} - 71534852 T^{6} + 15686490 T^{8} - 520547 T^{10} + 12924 T^{12} - 131 T^{14} + T^{16}$$
$37$ $$5323914784321 - 739793005903 T^{2} + 71850475036 T^{4} - 3349883567 T^{6} + 111552870 T^{8} - 2121832 T^{10} + 29023 T^{12} - 206 T^{14} + T^{16}$$
$41$ $$( 649636 + 254215 T^{2} + 13413 T^{4} + 212 T^{6} + T^{8} )^{2}$$
$43$ $$( 169 + 286 T + 121 T^{2} + 19 T^{3} + T^{4} )^{4}$$
$47$ $$959512576 - 1015021568 T^{2} + 916135936 T^{4} - 156811264 T^{6} + 20613888 T^{8} - 748544 T^{10} + 20512 T^{12} - 160 T^{14} + T^{16}$$
$53$ $$( 44249104 + 2667452 T + 1225121 T^{2} + 2360 T^{3} + 20953 T^{4} - 2 T^{5} + 185 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$59$ $$6146560000 - 3147524800 T^{2} + 1323896809 T^{4} - 129701384 T^{6} + 8868573 T^{8} - 334642 T^{10} + 9097 T^{12} - 113 T^{14} + T^{16}$$
$61$ $$( 44169316 - 12328330 T + 2856177 T^{2} - 336036 T^{3} + 38505 T^{4} - 2566 T^{5} + 257 T^{6} - 13 T^{7} + T^{8} )^{2}$$
$67$ $$1180013254445041 - 90149030483325 T^{2} + 4765784579912 T^{4} - 131006347541 T^{6} + 2592886788 T^{8} - 22663706 T^{10} + 142551 T^{12} - 452 T^{14} + T^{16}$$
$71$ $$( 33124 + 32227 T^{2} + 7752 T^{4} + 173 T^{6} + T^{8} )^{2}$$
$73$ $$25411681 - 29066406 T^{2} + 25014803 T^{4} - 8579072 T^{6} + 2183070 T^{8} - 124007 T^{10} + 5256 T^{12} - 83 T^{14} + T^{16}$$
$79$ $$( 116014441 - 18978502 T + 6389799 T^{2} + 429700 T^{3} + 91064 T^{4} + 1999 T^{5} + 330 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$83$ $$( 4218916 + 525551 T^{2} + 18099 T^{4} + 232 T^{6} + T^{8} )^{2}$$
$89$ $$9834496 - 12005144256 T^{2} + 14654651463545 T^{4} - 295095297610 T^{6} + 4054959957 T^{8} - 30347056 T^{10} + 165963 T^{12} - 493 T^{14} + T^{16}$$
$97$ $$( 400 + 2523 T^{2} + 1120 T^{4} + 67 T^{6} + T^{8} )^{2}$$