Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (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: \(x^{16} + 22 x^{14} + 187 x^{12} + 774 x^{10} + 1619 x^{8} + 1618 x^{6} + 690 x^{4} + 96 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 22 x^{14} + 187 x^{12} + 774 x^{10} + 1619 x^{8} + 1618 x^{6} + 690 x^{4} + 96 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + 20 \nu^{12} + 147 \nu^{10} + 480 \nu^{8} + 646 \nu^{6} + 183 \nu^{4} - 105 \nu^{2} + 26 \nu - 6 \)\()/52\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} - 20 \nu^{12} - 147 \nu^{10} - 480 \nu^{8} - 646 \nu^{6} - 183 \nu^{4} + 105 \nu^{2} + 26 \nu + 6 \)\()/52\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{14} - 107 \nu^{12} - 870 \nu^{10} - 3340 \nu^{8} - 6095 \nu^{6} - 4633 \nu^{4} - 1107 \nu^{2} - 23 \)\()/104\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{14} - 107 \nu^{12} - 870 \nu^{10} - 3340 \nu^{8} - 6095 \nu^{6} - 4633 \nu^{4} - 1003 \nu^{2} + 289 \)\()/104\)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{15} + 5 \nu^{14} + 149 \nu^{13} + 107 \nu^{12} + 1210 \nu^{11} + 870 \nu^{10} + 4680 \nu^{9} + 3340 \nu^{8} + 8733 \nu^{7} + 6095 \nu^{6} + 6819 \nu^{5} + 4633 \nu^{4} + 1237 \nu^{3} + 1107 \nu^{2} - 167 \nu + 23 \)\()/208\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{15} - 13 \nu^{14} + 115 \nu^{13} - 283 \nu^{12} + 1054 \nu^{11} - 2362 \nu^{10} + 4964 \nu^{9} - 9440 \nu^{8} + 12935 \nu^{7} - 18339 \nu^{6} + 18465 \nu^{5} - 15457 \nu^{4} + 12763 \nu^{3} - 4495 \nu^{2} + 2639 \nu - 215 \)\()/208\)
\(\beta_{7}\)\(=\)\((\)\( 15 \nu^{14} + 323 \nu^{12} + 2656 \nu^{10} + 10400 \nu^{8} + 19631 \nu^{6} + 15823 \nu^{4} + 4285 \nu^{2} - 52 \nu + 203 \)\()/104\)
\(\beta_{8}\)\(=\)\((\)\( 9 \nu^{15} + 197 \nu^{13} + 2 \nu^{12} + 1662 \nu^{11} + 46 \nu^{10} + 6796 \nu^{9} + 406 \nu^{8} + 13901 \nu^{7} + 1684 \nu^{6} + 13243 \nu^{5} + 3146 \nu^{4} + 5117 \nu^{3} + 1952 \nu^{2} + 747 \nu + 160 \)\()/104\)
\(\beta_{9}\)\(=\)\((\)\( 9 \nu^{15} + 197 \nu^{13} - 2 \nu^{12} + 1662 \nu^{11} - 46 \nu^{10} + 6796 \nu^{9} - 406 \nu^{8} + 13901 \nu^{7} - 1684 \nu^{6} + 13243 \nu^{5} - 3146 \nu^{4} + 5117 \nu^{3} - 1952 \nu^{2} + 747 \nu - 160 \)\()/104\)
\(\beta_{10}\)\(=\)\((\)\( -23 \nu^{15} + 7 \nu^{14} - 501 \nu^{13} + 149 \nu^{12} - 4194 \nu^{11} + 1210 \nu^{10} - 16932 \nu^{9} + 4680 \nu^{8} - 33897 \nu^{7} + 8733 \nu^{6} - 31119 \nu^{5} + 6819 \nu^{4} - 11237 \nu^{3} + 1237 \nu^{2} - 1101 \nu - 167 \)\()/208\)
\(\beta_{11}\)\(=\)\((\)\( -23 \nu^{15} - 7 \nu^{14} - 501 \nu^{13} - 149 \nu^{12} - 4194 \nu^{11} - 1210 \nu^{10} - 16932 \nu^{9} - 4680 \nu^{8} - 33897 \nu^{7} - 8733 \nu^{6} - 31119 \nu^{5} - 6819 \nu^{4} - 11237 \nu^{3} - 1237 \nu^{2} - 1101 \nu + 167 \)\()/208\)
\(\beta_{12}\)\(=\)\((\)\( 6 \nu^{15} + 133 \nu^{13} + 1142 \nu^{11} + 4791 \nu^{9} + 10194 \nu^{7} + 10354 \nu^{5} + 4323 \nu^{3} + 471 \nu + 26 \)\()/52\)
\(\beta_{13}\)\(=\)\((\)\( -41 \nu^{15} + 5 \nu^{14} - 895 \nu^{13} + 119 \nu^{12} - 7518 \nu^{11} + 1094 \nu^{10} - 30524 \nu^{9} + 4840 \nu^{8} - 61699 \nu^{7} + 10375 \nu^{6} - 57605 \nu^{5} + 9313 \nu^{4} - 21367 \nu^{3} + 2055 \nu^{2} - 2075 \nu - 317 \)\()/208\)
\(\beta_{14}\)\(=\)\((\)\( -41 \nu^{15} - 5 \nu^{14} - 895 \nu^{13} - 119 \nu^{12} - 7518 \nu^{11} - 1094 \nu^{10} - 30524 \nu^{9} - 4840 \nu^{8} - 61699 \nu^{7} - 10375 \nu^{6} - 57605 \nu^{5} - 9313 \nu^{4} - 21367 \nu^{3} - 2055 \nu^{2} - 2075 \nu + 317 \)\()/208\)
\(\beta_{15}\)\(=\)\((\)\( 59 \nu^{15} + 5 \nu^{14} + 1293 \nu^{13} + 107 \nu^{12} + 10934 \nu^{11} + 870 \nu^{10} + 44928 \nu^{9} + 3340 \nu^{8} + 92869 \nu^{7} + 6095 \nu^{6} + 90383 \nu^{5} + 4633 \nu^{4} + 35401 \nu^{3} + 1003 \nu^{2} + 3577 \nu - 185 \)\()/208\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 5 \beta_{2} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{10} + \beta_{7} - 6 \beta_{4} + 8 \beta_{3} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(-9 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} - 2 \beta_{5} - \beta_{3} + 29 \beta_{2} + 29 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(\beta_{14} - \beta_{13} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{7} + 35 \beta_{4} - 57 \beta_{3} + 2 \beta_{2} - 11 \beta_{1} - 84\)
\(\nu^{7}\)\(=\)\(-4 \beta_{15} + 66 \beta_{14} + 66 \beta_{13} + 24 \beta_{12} - 79 \beta_{11} - 79 \beta_{10} + 35 \beta_{9} + 35 \beta_{8} + \beta_{7} + 2 \beta_{6} + 24 \beta_{5} - 2 \beta_{4} + 12 \beta_{3} - 176 \beta_{2} - 177 \beta_{1} - 10\)
\(\nu^{8}\)\(=\)\(-13 \beta_{14} + 13 \beta_{13} + 57 \beta_{11} - 57 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 66 \beta_{7} - 211 \beta_{4} + 391 \beta_{3} - 28 \beta_{2} + 94 \beta_{1} + 500\)
\(\nu^{9}\)\(=\)\(60 \beta_{15} - 453 \beta_{14} - 453 \beta_{13} - 218 \beta_{12} + 569 \beta_{11} + 569 \beta_{10} - 211 \beta_{9} - 211 \beta_{8} - 13 \beta_{7} - 26 \beta_{6} - 222 \beta_{5} + 30 \beta_{4} - 111 \beta_{3} + 1098 \beta_{2} + 1111 \beta_{1} + 79\)
\(\nu^{10}\)\(=\)\(124 \beta_{14} - 124 \beta_{13} - 403 \beta_{11} + 403 \beta_{10} + 30 \beta_{9} - 30 \beta_{8} - 453 \beta_{7} + 1309 \beta_{4} - 2633 \beta_{3} + 280 \beta_{2} - 733 \beta_{1} - 3091\)
\(\nu^{11}\)\(=\)\(-620 \beta_{15} + 3026 \beta_{14} + 3026 \beta_{13} + 1778 \beta_{12} - 3927 \beta_{11} - 3927 \beta_{10} + 1309 \beta_{9} + 1309 \beta_{8} + 124 \beta_{7} + 248 \beta_{6} + 1862 \beta_{5} - 310 \beta_{4} + 931 \beta_{3} - 6973 \beta_{2} - 7097 \beta_{1} - 579\)
\(\nu^{12}\)\(=\)\(-1055 \beta_{14} + 1055 \beta_{13} + 2861 \beta_{11} - 2861 \beta_{10} - 310 \beta_{9} + 310 \beta_{8} + 3026 \beta_{7} - 8282 \beta_{4} + 17572 \beta_{3} - 2440 \beta_{2} + 5466 \beta_{1} + 19574\)
\(\nu^{13}\)\(=\)\(5500 \beta_{15} - 19978 \beta_{14} - 19978 \beta_{13} - 13702 \beta_{12} + 26552 \beta_{11} + 26552 \beta_{10} - 8282 \beta_{9} - 8282 \beta_{8} - 1055 \beta_{7} - 2110 \beta_{6} - 14822 \beta_{5} + 2750 \beta_{4} - 7411 \beta_{3} + 44808 \beta_{2} + 45863 \beta_{1} + 4101\)
\(\nu^{14}\)\(=\)\(8466 \beta_{14} - 8466 \beta_{13} - 20354 \beta_{11} + 20354 \beta_{10} + 2750 \beta_{9} - 2750 \beta_{8} - 19978 \beta_{7} + 53090 \beta_{4} - 116816 \beta_{3} + 19762 \beta_{2} - 39740 \beta_{1} - 125893\)
\(\nu^{15}\)\(=\)\(-45024 \beta_{15} + 131294 \beta_{14} + 131294 \beta_{13} + 102072 \beta_{12} - 177792 \beta_{11} - 177792 \beta_{10} + 53090 \beta_{9} + 53090 \beta_{8} + 8466 \beta_{7} + 16932 \beta_{6} + 114096 \beta_{5} - 22512 \beta_{4} + 57048 \beta_{3} - 290299 \beta_{2} - 298765 \beta_{1} - 28524\)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(\beta_{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
2.60802i
1.77930i
0.775848i
0.485989i
0.106359i
1.10207i
1.98765i
2.45308i
2.60802i
1.77930i
0.775848i
0.485989i
0.106359i
1.10207i
1.98765i
2.45308i
−2.25861 1.30401i −0.500000 + 0.866025i 2.40088 + 4.15844i 1.50528i 2.25861 1.30401i −0.866025 + 0.500000i 7.30704i −0.500000 0.866025i −1.96290 + 3.39983i
43.2 −1.54092 0.889651i −0.500000 + 0.866025i 0.582956 + 1.00971i 0.681820i 1.54092 0.889651i 0.866025 0.500000i 1.48409i −0.500000 0.866025i −0.606581 + 1.05063i
43.3 −0.671904 0.387924i −0.500000 + 0.866025i −0.699030 1.21076i 3.03444i 0.671904 0.387924i 0.866025 0.500000i 2.63638i −0.500000 0.866025i 1.17713 2.03885i
43.4 −0.420879 0.242995i −0.500000 + 0.866025i −0.881907 1.52751i 1.06536i 0.420879 0.242995i −0.866025 + 0.500000i 1.82917i −0.500000 0.866025i 0.258876 0.448387i
43.5 0.0921099 + 0.0531797i −0.500000 + 0.866025i −0.994344 1.72225i 1.41292i −0.0921099 + 0.0531797i −0.866025 + 0.500000i 0.424234i −0.500000 0.866025i 0.0751388 0.130144i
43.6 0.954423 + 0.551037i −0.500000 + 0.866025i −0.392717 0.680206i 3.28432i −0.954423 + 0.551037i 0.866025 0.500000i 3.06975i −0.500000 0.866025i 1.80978 3.13463i
43.7 1.72135 + 0.993824i −0.500000 + 0.866025i 0.975372 + 1.68939i 2.85284i −1.72135 + 0.993824i −0.866025 + 0.500000i 0.0979034i −0.500000 0.866025i −2.83522 + 4.91075i
43.8 2.12443 + 1.22654i −0.500000 + 0.866025i 2.00879 + 3.47933i 0.0682999i −2.12443 + 1.22654i 0.866025 0.500000i 4.94928i −0.500000 0.866025i 0.0837724 0.145098i
127.1 −2.25861 + 1.30401i −0.500000 0.866025i 2.40088 4.15844i 1.50528i 2.25861 + 1.30401i −0.866025 0.500000i 7.30704i −0.500000 + 0.866025i −1.96290 3.39983i
127.2 −1.54092 + 0.889651i −0.500000 0.866025i 0.582956 1.00971i 0.681820i 1.54092 + 0.889651i 0.866025 + 0.500000i 1.48409i −0.500000 + 0.866025i −0.606581 1.05063i
127.3 −0.671904 + 0.387924i −0.500000 0.866025i −0.699030 + 1.21076i 3.03444i 0.671904 + 0.387924i 0.866025 + 0.500000i 2.63638i −0.500000 + 0.866025i 1.17713 + 2.03885i
127.4 −0.420879 + 0.242995i −0.500000 0.866025i −0.881907 + 1.52751i 1.06536i 0.420879 + 0.242995i −0.866025 0.500000i 1.82917i −0.500000 + 0.866025i 0.258876 + 0.448387i
127.5 0.0921099 0.0531797i −0.500000 0.866025i −0.994344 + 1.72225i 1.41292i −0.0921099 0.0531797i −0.866025 0.500000i 0.424234i −0.500000 + 0.866025i 0.0751388 + 0.130144i
127.6 0.954423 0.551037i −0.500000 0.866025i −0.392717 + 0.680206i 3.28432i −0.954423 0.551037i 0.866025 + 0.500000i 3.06975i −0.500000 + 0.866025i 1.80978 + 3.13463i
127.7 1.72135 0.993824i −0.500000 0.866025i 0.975372 1.68939i 2.85284i −1.72135 0.993824i −0.866025 0.500000i 0.0979034i −0.500000 + 0.866025i −2.83522 4.91075i
127.8 2.12443 1.22654i −0.500000 0.866025i 2.00879 3.47933i 0.0682999i −2.12443 1.22654i 0.866025 + 0.500000i 4.94928i −0.500000 + 0.866025i 0.0837724 + 0.145098i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.bd.a 16
3.b odd 2 1 819.2.ct.b 16
13.e even 6 1 inner 273.2.bd.a 16
13.f odd 12 1 3549.2.a.bb 8
13.f odd 12 1 3549.2.a.bd 8
39.h odd 6 1 819.2.ct.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.a 16 1.a even 1 1 trivial
273.2.bd.a 16 13.e even 6 1 inner
819.2.ct.b 16 3.b odd 2 1
819.2.ct.b 16 39.h odd 6 1
3549.2.a.bb 8 13.f odd 12 1
3549.2.a.bd 8 13.f odd 12 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

$p$ $F_p(T)$
$2$ \( 1 - 12 T + 24 T^{2} + 288 T^{3} + 519 T^{4} - 144 T^{5} - 758 T^{6} + 66 T^{7} + 824 T^{8} + 12 T^{9} - 315 T^{10} - 6 T^{11} + 88 T^{12} - 11 T^{14} + T^{16} \)
$3$ \( ( 1 + T + T^{2} )^{8} \)
$5$ \( 9 + 1968 T^{2} + 8350 T^{4} + 12186 T^{6} + 8131 T^{8} + 2690 T^{10} + 439 T^{12} + 34 T^{14} + T^{16} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$11$ \( 144 - 2448 T + 10860 T^{2} + 51204 T^{3} + 37501 T^{4} - 78402 T^{5} - 60765 T^{6} + 139170 T^{7} + 251713 T^{8} + 173040 T^{9} + 59006 T^{10} + 7974 T^{11} - 677 T^{12} - 240 T^{13} + 28 T^{14} + 12 T^{15} + T^{16} \)
$13$ \( 815730721 - 250994068 T - 120670225 T^{2} + 22277580 T^{3} + 14023451 T^{4} - 628342 T^{5} - 1155115 T^{6} - 28002 T^{7} + 98049 T^{8} - 2154 T^{9} - 6835 T^{10} - 286 T^{11} + 491 T^{12} + 60 T^{13} - 25 T^{14} - 4 T^{15} + T^{16} \)
$17$ \( 2883582601 + 234664630 T + 2330838850 T^{2} - 1885876084 T^{3} + 1760432883 T^{4} - 727258022 T^{5} + 278358164 T^{6} - 60927516 T^{7} + 15319882 T^{8} - 2527264 T^{9} + 540459 T^{10} - 68558 T^{11} + 11448 T^{12} - 1046 T^{13} + 155 T^{14} - 10 T^{15} + T^{16} \)
$19$ \( 25335725584 + 7202851344 T - 5761497252 T^{2} - 1832027220 T^{3} + 1083013485 T^{4} + 141572916 T^{5} - 93001043 T^{6} - 7551360 T^{7} + 5732258 T^{8} + 180468 T^{9} - 215127 T^{10} - 3060 T^{11} + 5890 T^{12} - 95 T^{14} + T^{16} \)
$23$ \( 1446433024 + 1528886400 T + 2667358576 T^{2} - 889293848 T^{3} + 772743593 T^{4} - 149196990 T^{5} + 80220610 T^{6} - 12273552 T^{7} + 5603725 T^{8} - 586894 T^{9} + 227780 T^{10} - 14156 T^{11} + 6555 T^{12} - 190 T^{13} + 101 T^{14} + 2 T^{15} + T^{16} \)
$29$ \( 2538849769 + 2586566258 T + 6606128639 T^{2} - 6235198678 T^{3} + 5013493680 T^{4} - 1935836228 T^{5} + 663667199 T^{6} - 137598688 T^{7} + 32835814 T^{8} - 5339730 T^{9} + 1056916 T^{10} - 125736 T^{11} + 18431 T^{12} - 1544 T^{13} + 202 T^{14} - 12 T^{15} + T^{16} \)
$31$ \( 7929856 + 112898560 T^{2} + 170903929 T^{4} + 64267032 T^{6} + 9194570 T^{8} + 582754 T^{10} + 17145 T^{12} + 222 T^{14} + T^{16} \)
$37$ \( 16292735449 - 50156040420 T + 45278382702 T^{2} + 19052088840 T^{3} - 6265438547 T^{4} - 2867954790 T^{5} + 1090759412 T^{6} + 31084044 T^{7} - 46404942 T^{8} + 942702 T^{9} + 1715827 T^{10} - 231432 T^{11} - 2470 T^{12} + 2070 T^{13} - 7 T^{14} - 18 T^{15} + T^{16} \)
$41$ \( 692224 - 5271552 T + 5394432 T^{2} + 60825600 T^{3} + 108961024 T^{4} + 78628608 T^{5} + 17044352 T^{6} - 6527040 T^{7} - 1928880 T^{8} + 411120 T^{9} + 131908 T^{10} - 19716 T^{11} - 3931 T^{12} + 666 T^{13} + 71 T^{14} - 18 T^{15} + T^{16} \)
$43$ \( 15764309136 - 13166806608 T + 20996953932 T^{2} + 2947067436 T^{3} + 7959097345 T^{4} - 758584324 T^{5} + 744793030 T^{6} - 23007624 T^{7} + 34448407 T^{8} - 638086 T^{9} + 1055388 T^{10} + 24974 T^{11} + 17299 T^{12} + 686 T^{13} + 215 T^{14} + 10 T^{15} + T^{16} \)
$47$ \( 1336336 + 33845368 T^{2} + 98680561 T^{4} + 44803608 T^{6} + 6887354 T^{8} + 471742 T^{10} + 15381 T^{12} + 222 T^{14} + T^{16} \)
$53$ \( ( -76707 + 89910 T - 5832 T^{2} - 19170 T^{3} + 2754 T^{4} + 1026 T^{5} - 168 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$59$ \( 3887273104 + 8125939536 T + 6748307916 T^{2} + 2270513772 T^{3} - 145310295 T^{4} - 113420976 T^{5} + 344300344 T^{6} + 371997060 T^{7} + 195076667 T^{8} + 65948376 T^{9} + 15608430 T^{10} + 2660790 T^{11} + 327337 T^{12} + 28500 T^{13} + 1675 T^{14} + 60 T^{15} + T^{16} \)
$61$ \( 103866332089 + 127056205354 T + 146407411436 T^{2} + 47476831324 T^{3} + 26038308426 T^{4} + 6119158214 T^{5} + 3051479096 T^{6} + 530968738 T^{7} + 152908123 T^{8} + 12785262 T^{9} + 3703384 T^{10} + 271482 T^{11} + 53042 T^{12} + 1412 T^{13} + 268 T^{14} + 6 T^{15} + T^{16} \)
$67$ \( 5866334464 - 45343076736 T + 133462187888 T^{2} - 128598937800 T^{3} + 54486257009 T^{4} - 7326684648 T^{5} - 1485775154 T^{6} + 431145240 T^{7} + 38020119 T^{8} - 14682954 T^{9} - 346924 T^{10} + 262914 T^{11} + 6427 T^{12} - 2970 T^{13} - 57 T^{14} + 18 T^{15} + T^{16} \)
$71$ \( 8447342955287184 + 289151839778256 T - 387082496566740 T^{2} - 13362733173972 T^{3} + 11689183350061 T^{4} + 407758474068 T^{5} - 202198889719 T^{6} - 6985442844 T^{7} + 2548849493 T^{8} + 83725368 T^{9} - 20593578 T^{10} - 599316 T^{11} + 122673 T^{12} + 2640 T^{13} - 428 T^{14} - 6 T^{15} + T^{16} \)
$73$ \( 17309352523401 + 8472674581404 T^{2} + 1089036344158 T^{4} + 56758848702 T^{6} + 1487243191 T^{8} + 21177398 T^{10} + 164251 T^{12} + 646 T^{14} + T^{16} \)
$79$ \( ( 75240852 + 5078148 T - 6074987 T^{2} + 70920 T^{3} + 99754 T^{4} - 974 T^{5} - 563 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$83$ \( 2140872301584 + 1825221854616 T^{2} + 369726290449 T^{4} + 27837898786 T^{6} + 990922729 T^{8} + 18015746 T^{10} + 164878 T^{12} + 684 T^{14} + T^{16} \)
$89$ \( 72734154590831616 + 12225086128196352 T - 3334789752264864 T^{2} - 675630069691464 T^{3} + 188710877512633 T^{4} - 5245161354102 T^{5} - 1798621372831 T^{6} + 105489558138 T^{7} + 14875953002 T^{8} - 1755119190 T^{9} + 7020273 T^{10} + 7056522 T^{11} - 190830 T^{12} - 27222 T^{13} + 2377 T^{14} - 78 T^{15} + T^{16} \)
$97$ \( 941955655936 - 3208618464000 T + 1735107937840 T^{2} + 6499645590000 T^{3} + 4070133756097 T^{4} + 471017902812 T^{5} - 99402588504 T^{6} - 17064046512 T^{7} + 3553986407 T^{8} + 1015553094 T^{9} + 81915010 T^{10} + 669720 T^{11} - 216495 T^{12} - 3078 T^{13} + 915 T^{14} + 54 T^{15} + T^{16} \)
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