Properties

Label 370.2.m.c
Level $370$
Weight $2$
Character orbit 370.m
Analytic conductor $2.954$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 37 x^{14} + 559 x^{12} + 4431 x^{10} + 19684 x^{8} + 48248 x^{6} + 58656 x^{4} + 25392 x^{2} + 256\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 37 x^{14} + 559 x^{12} + 4431 x^{10} + 19684 x^{8} + 48248 x^{6} + 58656 x^{4} + 25392 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{15} + 31 \nu^{13} + 381 \nu^{11} + 2361 \nu^{9} + 7734 \nu^{7} + 12700 \nu^{5} + 8872 \nu^{3} + 2272 \nu - 128 \)\()/256\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{14} - 31 \nu^{13} + 54 \nu^{12} - 381 \nu^{11} + 546 \nu^{10} - 2361 \nu^{9} + 2538 \nu^{8} - 7734 \nu^{7} + 5252 \nu^{6} - 12636 \nu^{5} + 3304 \nu^{4} - 8168 \nu^{3} - 1232 \nu^{2} - 736 \nu - 384 \)\()/256\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{15} - 2 \nu^{14} - 31 \nu^{13} - 54 \nu^{12} - 381 \nu^{11} - 546 \nu^{10} - 2361 \nu^{9} - 2538 \nu^{8} - 7734 \nu^{7} - 5252 \nu^{6} - 12636 \nu^{5} - 3304 \nu^{4} - 8168 \nu^{3} + 1232 \nu^{2} - 736 \nu + 384 \)\()/256\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 162 \nu^{12} - 273 \nu^{11} + 1670 \nu^{10} - 1237 \nu^{9} + 8286 \nu^{8} - 2018 \nu^{7} + 20652 \nu^{6} + 2188 \nu^{5} + 24280 \nu^{4} + 9576 \nu^{3} + 10640 \nu^{2} + 5632 \nu \)\()/256\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} - 6 \nu^{14} - 27 \nu^{13} - 162 \nu^{12} - 273 \nu^{11} - 1670 \nu^{10} - 1237 \nu^{9} - 8286 \nu^{8} - 2018 \nu^{7} - 20652 \nu^{6} + 2188 \nu^{5} - 24280 \nu^{4} + 9576 \nu^{3} - 10640 \nu^{2} + 5632 \nu \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{15} + 6 \nu^{14} + 27 \nu^{13} + 170 \nu^{12} + 289 \nu^{11} + 1870 \nu^{10} + 1637 \nu^{9} + 10102 \nu^{8} + 5682 \nu^{7} + 27876 \nu^{6} + 12708 \nu^{5} + 36184 \nu^{4} + 15864 \nu^{3} + 16240 \nu^{2} + 6528 \nu + 768 \)\()/256\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 170 \nu^{12} - 289 \nu^{11} + 1870 \nu^{10} - 1637 \nu^{9} + 10102 \nu^{8} - 5682 \nu^{7} + 27876 \nu^{6} - 12708 \nu^{5} + 36184 \nu^{4} - 15864 \nu^{3} + 16240 \nu^{2} - 6528 \nu + 768 \)\()/256\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 170 \nu^{12} - 381 \nu^{11} + 1870 \nu^{10} - 2361 \nu^{9} + 10102 \nu^{8} - 7766 \nu^{7} + 27876 \nu^{6} - 13084 \nu^{5} + 36120 \nu^{4} - 9800 \nu^{3} + 15536 \nu^{2} - 1952 \nu - 384 \)\()/256\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 170 \nu^{12} + 381 \nu^{11} + 1870 \nu^{10} + 2361 \nu^{9} + 10102 \nu^{8} + 7766 \nu^{7} + 27876 \nu^{6} + 13084 \nu^{5} + 36120 \nu^{4} + 9800 \nu^{3} + 15536 \nu^{2} + 1952 \nu - 384 \)\()/256\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{14} + 89 \nu^{12} + 1035 \nu^{10} + 5975 \nu^{8} + 17774 \nu^{6} + 24892 \nu^{4} + 11560 \nu^{2} + 64 \nu + 128 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{14} - 89 \nu^{12} - 1035 \nu^{10} - 5975 \nu^{8} - 17774 \nu^{6} - 24892 \nu^{4} - 11560 \nu^{2} + 64 \nu - 128 \)\()/128\)
\(\beta_{12}\)\(=\)\((\)\( 3 \nu^{15} + 6 \nu^{14} + 81 \nu^{13} + 170 \nu^{12} + 835 \nu^{11} + 1870 \nu^{10} + 4143 \nu^{9} + 10102 \nu^{8} + 10358 \nu^{7} + 27876 \nu^{6} + 12588 \nu^{5} + 36184 \nu^{4} + 7016 \nu^{3} + 16112 \nu^{2} + 1664 \nu + 256 \)\()/256\)
\(\beta_{13}\)\(=\)\((\)\( -3 \nu^{14} - 85 \nu^{12} - 935 \nu^{10} - 5051 \nu^{8} - 13938 \nu^{6} - 18092 \nu^{4} - 8056 \nu^{2} - 128 \)\()/64\)
\(\beta_{14}\)\(=\)\((\)\( 3 \nu^{14} + 87 \nu^{12} + 985 \nu^{10} + 5513 \nu^{8} + 8 \nu^{7} + 15872 \nu^{6} + 112 \nu^{5} + 21684 \nu^{4} + 440 \nu^{3} + 10336 \nu^{2} + 496 \nu + 320 \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( -3 \nu^{14} - 87 \nu^{12} - 985 \nu^{10} - 5513 \nu^{8} + 8 \nu^{7} - 15872 \nu^{6} + 112 \nu^{5} - 21684 \nu^{4} + 440 \nu^{3} - 10336 \nu^{2} + 496 \nu - 320 \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{11} + \beta_{10}\)
\(\nu^{2}\)\(=\)\(\beta_{13} + \beta_{7} + \beta_{6} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} - 6 \beta_{11} - 6 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2}\)
\(\nu^{4}\)\(=\)\(-11 \beta_{13} - 2 \beta_{9} - 2 \beta_{8} - 9 \beta_{7} - 9 \beta_{6} + 26\)
\(\nu^{5}\)\(=\)\(-11 \beta_{15} - 11 \beta_{14} + 42 \beta_{11} + 42 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + 13 \beta_{3} + 13 \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-2 \beta_{15} + 2 \beta_{14} + 101 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 24 \beta_{9} + 24 \beta_{8} + 75 \beta_{7} + 75 \beta_{6} - 192\)
\(\nu^{7}\)\(=\)\(103 \beta_{15} + 103 \beta_{14} - 320 \beta_{11} - 320 \beta_{10} - 99 \beta_{9} + 99 \beta_{8} - 127 \beta_{3} - 127 \beta_{2} - 56 \beta_{1} - 28\)
\(\nu^{8}\)\(=\)\(28 \beta_{15} - 28 \beta_{14} - 871 \beta_{13} - 32 \beta_{11} + 32 \beta_{10} - 230 \beta_{9} - 230 \beta_{8} - 621 \beta_{7} - 621 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 1510\)
\(\nu^{9}\)\(=\)\(-919 \beta_{15} - 919 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + 2546 \beta_{11} + 2546 \beta_{10} + 839 \beta_{9} - 839 \beta_{8} + 4 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 1133 \beta_{3} + 1133 \beta_{2} + 588 \beta_{1} + 294\)
\(\nu^{10}\)\(=\)\(-282 \beta_{15} + 282 \beta_{14} + 7329 \beta_{13} + 366 \beta_{11} - 366 \beta_{10} + 2056 \beta_{9} + 2056 \beta_{8} + 5159 \beta_{7} + 5159 \beta_{6} + 80 \beta_{5} - 80 \beta_{4} + 12 \beta_{3} - 12 \beta_{2} - 12252\)
\(\nu^{11}\)\(=\)\(8039 \beta_{15} + 8039 \beta_{14} - 100 \beta_{13} - 200 \beta_{12} - 20692 \beta_{11} - 20692 \beta_{10} - 6955 \beta_{9} + 6955 \beta_{8} - 108 \beta_{7} + 108 \beta_{6} - 192 \beta_{5} - 192 \beta_{4} - 9755 \beta_{3} - 9755 \beta_{2} - 5600 \beta_{1} - 2800\)
\(\nu^{12}\)\(=\)\(2500 \beta_{15} - 2500 \beta_{14} - 61043 \beta_{13} - 3692 \beta_{11} + 3692 \beta_{10} - 17886 \beta_{9} - 17886 \beta_{8} - 43025 \beta_{7} - 43025 \beta_{6} - 1076 \beta_{5} + 1076 \beta_{4} - 300 \beta_{3} + 300 \beta_{2} + 100922\)
\(\nu^{13}\)\(=\)\(-69671 \beta_{15} - 69671 \beta_{14} + 1576 \beta_{13} + 3152 \beta_{12} + 169910 \beta_{11} + 169910 \beta_{10} + 57167 \beta_{9} - 57167 \beta_{8} + 1792 \beta_{7} - 1792 \beta_{6} + 2968 \beta_{5} + 2968 \beta_{4} + 82721 \beta_{3} + 82721 \beta_{2} + 51172 \beta_{1} + 25586\)
\(\nu^{14}\)\(=\)\(-20794 \beta_{15} + 20794 \beta_{14} + 506205 \beta_{13} + 35122 \beta_{11} - 35122 \beta_{10} + 153784 \beta_{9} + 153784 \beta_{8} + 359851 \beta_{7} + 359851 \beta_{6} + 12288 \beta_{5} - 12288 \beta_{4} + 4760 \beta_{3} - 4760 \beta_{2} - 837320\)
\(\nu^{15}\)\(=\)\(600927 \beta_{15} + 600927 \beta_{14} - 20200 \beta_{13} - 40400 \beta_{12} - 1402224 \beta_{11} - 1402224 \beta_{10} - 468363 \beta_{9} + 468363 \beta_{8} - 23848 \beta_{7} + 23848 \beta_{6} - 37744 \beta_{5} - 37744 \beta_{4} - 696719 \beta_{3} - 696719 \beta_{2} - 458440 \beta_{1} - 229220\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
2.85998i
2.06794i
1.93403i
0.101618i
0.926756i
1.76701i
2.88937i
2.90925i
2.85998i
2.06794i
1.93403i
0.101618i
0.926756i
1.76701i
2.88937i
2.90925i
−0.500000 0.866025i −2.47681 1.42999i −0.500000 + 0.866025i 1.52446 1.63586i 2.85998i 4.15112 + 2.39665i 1.00000 2.58973 + 4.48555i −2.17892 0.502291i
159.2 −0.500000 0.866025i −1.79089 1.03397i −0.500000 + 0.866025i 1.86778 + 1.22939i 2.06794i −1.25580 0.725037i 1.00000 0.638185 + 1.10537i 0.130790 2.23224i
159.3 −0.500000 0.866025i −1.67492 0.967016i −0.500000 + 0.866025i −0.564048 + 2.16376i 1.93403i 0.358580 + 0.207027i 1.00000 0.370239 + 0.641273i 2.15589 0.593399i
159.4 −0.500000 0.866025i 0.0880035 + 0.0508088i −0.500000 + 0.866025i −1.69259 1.46122i 0.101618i 1.94895 + 1.12523i 1.00000 −1.49484 2.58913i −0.419156 + 2.19643i
159.5 −0.500000 0.866025i 0.802594 + 0.463378i −0.500000 + 0.866025i 1.82024 1.29873i 0.926756i −3.13584 1.81048i 1.00000 −1.07056 1.85427i −2.03486 0.927011i
159.6 −0.500000 0.866025i 1.53027 + 0.883503i −0.500000 + 0.866025i 2.03272 0.931689i 1.76701i 3.45647 + 1.99560i 1.00000 0.0611550 + 0.105924i −1.82323 1.29454i
159.7 −0.500000 0.866025i 2.50227 + 1.44468i −0.500000 + 0.866025i −0.791839 + 2.09117i 2.88937i 0.668346 + 0.385870i 1.00000 2.67422 + 4.63189i 2.20693 0.359832i
159.8 −0.500000 0.866025i 2.51948 + 1.45462i −0.500000 + 0.866025i −1.19673 1.88887i 2.90925i −0.191824 0.110750i 1.00000 2.73187 + 4.73173i −1.03745 + 1.98083i
249.1 −0.500000 + 0.866025i −2.47681 + 1.42999i −0.500000 0.866025i 1.52446 + 1.63586i 2.85998i 4.15112 2.39665i 1.00000 2.58973 4.48555i −2.17892 + 0.502291i
249.2 −0.500000 + 0.866025i −1.79089 + 1.03397i −0.500000 0.866025i 1.86778 1.22939i 2.06794i −1.25580 + 0.725037i 1.00000 0.638185 1.10537i 0.130790 + 2.23224i
249.3 −0.500000 + 0.866025i −1.67492 + 0.967016i −0.500000 0.866025i −0.564048 2.16376i 1.93403i 0.358580 0.207027i 1.00000 0.370239 0.641273i 2.15589 + 0.593399i
249.4 −0.500000 + 0.866025i 0.0880035 0.0508088i −0.500000 0.866025i −1.69259 + 1.46122i 0.101618i 1.94895 1.12523i 1.00000 −1.49484 + 2.58913i −0.419156 2.19643i
249.5 −0.500000 + 0.866025i 0.802594 0.463378i −0.500000 0.866025i 1.82024 + 1.29873i 0.926756i −3.13584 + 1.81048i 1.00000 −1.07056 + 1.85427i −2.03486 + 0.927011i
249.6 −0.500000 + 0.866025i 1.53027 0.883503i −0.500000 0.866025i 2.03272 + 0.931689i 1.76701i 3.45647 1.99560i 1.00000 0.0611550 0.105924i −1.82323 + 1.29454i
249.7 −0.500000 + 0.866025i 2.50227 1.44468i −0.500000 0.866025i −0.791839 2.09117i 2.88937i 0.668346 0.385870i 1.00000 2.67422 4.63189i 2.20693 + 0.359832i
249.8 −0.500000 + 0.866025i 2.51948 1.45462i −0.500000 0.866025i −1.19673 + 1.88887i 2.90925i −0.191824 + 0.110750i 1.00000 2.73187 4.73173i −1.03745 1.98083i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.m.c 16
5.b even 2 1 370.2.m.d yes 16
37.e even 6 1 370.2.m.d yes 16
185.l even 6 1 inner 370.2.m.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.m.c 16 1.a even 1 1 trivial
370.2.m.c 16 185.l even 6 1 inner
370.2.m.d yes 16 5.b even 2 1
370.2.m.d yes 16 37.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{8} \)
$3$ \( 256 - 4800 T + 32304 T^{2} - 43200 T^{3} + 1776 T^{4} + 30096 T^{5} - 4132 T^{6} - 13308 T^{7} + 3208 T^{8} + 3318 T^{9} - 771 T^{10} - 561 T^{11} + 154 T^{12} + 51 T^{13} - 14 T^{14} - 3 T^{15} + T^{16} \)
$5$ \( 390625 - 468750 T + 328125 T^{2} - 218750 T^{3} + 158125 T^{4} - 88000 T^{5} + 42550 T^{6} - 21700 T^{7} + 10810 T^{8} - 4340 T^{9} + 1702 T^{10} - 704 T^{11} + 253 T^{12} - 70 T^{13} + 21 T^{14} - 6 T^{15} + T^{16} \)
$7$ \( 256 + 384 T - 3392 T^{2} - 5376 T^{3} + 51328 T^{4} - 77664 T^{5} + 20304 T^{6} + 38544 T^{7} - 9488 T^{8} - 17376 T^{9} + 10292 T^{10} - 756 T^{11} - 600 T^{12} + 72 T^{13} + 42 T^{14} - 12 T^{15} + T^{16} \)
$11$ \( ( -11840 - 15360 T - 1688 T^{2} + 3384 T^{3} + 798 T^{4} - 198 T^{5} - 56 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$13$ \( 1849600 - 15558400 T + 110517120 T^{2} - 155272960 T^{3} + 156844464 T^{4} - 86183424 T^{5} + 38008152 T^{6} - 11361740 T^{7} + 3133569 T^{8} - 645564 T^{9} + 147624 T^{10} - 23904 T^{11} + 4583 T^{12} - 488 T^{13} + 84 T^{14} - 6 T^{15} + T^{16} \)
$17$ \( 226576 + 105672 T + 879428 T^{2} + 1612032 T^{3} + 3807140 T^{4} + 3990228 T^{5} + 3345596 T^{6} + 1689552 T^{7} + 711469 T^{8} + 179220 T^{9} + 50374 T^{10} + 8016 T^{11} + 2735 T^{12} + 204 T^{13} + 58 T^{14} + T^{16} \)
$19$ \( 1435500544 + 4073717760 T + 4482760704 T^{2} + 1785692160 T^{3} - 57965568 T^{4} - 212138496 T^{5} - 13862144 T^{6} + 19838976 T^{7} + 3487552 T^{8} - 741216 T^{9} - 170160 T^{10} + 21888 T^{11} + 5980 T^{12} - 282 T^{13} - 91 T^{14} + 3 T^{15} + T^{16} \)
$23$ \( ( -9224 - 19864 T + 43960 T^{2} - 15584 T^{3} - 606 T^{4} + 860 T^{5} - 56 T^{6} - 11 T^{7} + T^{8} )^{2} \)
$29$ \( 14500864 + 256263936 T^{2} + 198324352 T^{4} + 54563244 T^{6} + 7007273 T^{8} + 446220 T^{10} + 13646 T^{12} + 192 T^{14} + T^{16} \)
$31$ \( 16384 + 474112 T^{2} + 1391616 T^{4} + 1378800 T^{6} + 548664 T^{8} + 86971 T^{10} + 5087 T^{12} + 121 T^{14} + T^{16} \)
$37$ \( 3512479453921 + 1518910034128 T + 159075037358 T^{2} - 19000244218 T^{3} - 1690493222 T^{4} + 640355226 T^{5} + 1730416 T^{6} - 34110522 T^{7} - 7424265 T^{8} - 921906 T^{9} + 1264 T^{10} + 12642 T^{11} - 902 T^{12} - 274 T^{13} + 62 T^{14} + 16 T^{15} + T^{16} \)
$41$ \( 23754764740996 - 11245921700338 T + 5235862332607 T^{2} - 1257368534669 T^{3} + 346531080914 T^{4} - 61457739217 T^{5} + 13883750897 T^{6} - 1890879656 T^{7} + 326675395 T^{8} - 30100061 T^{9} + 4593461 T^{10} - 322528 T^{11} + 45395 T^{12} - 1943 T^{13} + 262 T^{14} - 7 T^{15} + T^{16} \)
$43$ \( ( 177208 - 262724 T + 137352 T^{2} - 24832 T^{3} - 2200 T^{4} + 1199 T^{5} - 69 T^{6} - 11 T^{7} + T^{8} )^{2} \)
$47$ \( 35452370944 + 45429778432 T^{2} + 12943589952 T^{4} + 1582370592 T^{6} + 97315044 T^{8} + 3104440 T^{10} + 49616 T^{12} + 367 T^{14} + T^{16} \)
$53$ \( 802816 + 21547008 T + 167222016 T^{2} - 685656576 T^{3} + 402965120 T^{4} + 1418504832 T^{5} + 470242512 T^{6} - 614103792 T^{7} + 148140344 T^{8} + 397692 T^{9} - 2694681 T^{10} + 43677 T^{11} + 38200 T^{12} - 675 T^{13} - 222 T^{14} + 3 T^{15} + T^{16} \)
$59$ \( 159386189824 - 54499958784 T - 70785638400 T^{2} + 26328250368 T^{3} + 33011164416 T^{4} - 28263109632 T^{5} + 8803598336 T^{6} - 1033139760 T^{7} - 61118156 T^{8} + 22852908 T^{9} + 334368 T^{10} - 411972 T^{11} + 16180 T^{12} + 2910 T^{13} - 119 T^{14} - 15 T^{15} + T^{16} \)
$61$ \( 899128754176 + 1580477005824 T + 712427715776 T^{2} - 375497964384 T^{3} - 18813834368 T^{4} + 21949041672 T^{5} + 459130896 T^{6} - 849480150 T^{7} + 18517765 T^{8} + 18904626 T^{9} - 646868 T^{10} - 295806 T^{11} + 17373 T^{12} + 2304 T^{13} - 144 T^{14} - 12 T^{15} + T^{16} \)
$67$ \( 1893990400 - 116647526400 T + 2398480547840 T^{2} - 232523120640 T^{3} - 232338328576 T^{4} + 23556708864 T^{5} + 18546956288 T^{6} + 450238080 T^{7} - 501703296 T^{8} - 26247648 T^{9} + 10827792 T^{10} + 1321632 T^{11} + 3288 T^{12} - 5856 T^{13} - 52 T^{14} + 24 T^{15} + T^{16} \)
$71$ \( 451477504 + 2659909632 T + 13650264064 T^{2} + 13041332224 T^{3} + 12639290368 T^{4} + 409718784 T^{5} + 1706118208 T^{6} + 144080160 T^{7} + 140061424 T^{8} + 656912 T^{9} + 3757984 T^{10} + 198232 T^{11} + 56012 T^{12} + 1096 T^{13} + 268 T^{14} + 4 T^{15} + T^{16} \)
$73$ \( 10676686950400 + 9373196180480 T^{2} + 1377356153344 T^{4} + 71152240512 T^{6} + 1752551248 T^{8} + 23233516 T^{10} + 170184 T^{12} + 648 T^{14} + T^{16} \)
$79$ \( 5065884565504 - 100257497088 T - 705894678528 T^{2} + 13983252480 T^{3} + 69162463232 T^{4} - 627176448 T^{5} - 3174048000 T^{6} + 20331264 T^{7} + 103724096 T^{8} - 424896 T^{9} - 2031888 T^{10} + 5088 T^{11} + 28576 T^{12} - 204 T^{14} + T^{16} \)
$83$ \( 1363677184 - 32926777344 T + 275151003648 T^{2} - 244810874880 T^{3} + 55236196608 T^{4} + 19256559360 T^{5} - 3573728512 T^{6} - 1090003008 T^{7} + 279080080 T^{8} + 1886832 T^{9} - 4210944 T^{10} + 54528 T^{11} + 48868 T^{12} - 1560 T^{13} - 248 T^{14} + 6 T^{15} + T^{16} \)
$89$ \( 5611687210000 - 16239520170000 T + 17900141082400 T^{2} - 6468085234800 T^{3} + 489043348856 T^{4} + 207232731012 T^{5} - 26599576529 T^{6} - 5400321249 T^{7} + 864443211 T^{8} + 75868470 T^{9} - 9426321 T^{10} - 662343 T^{11} + 78537 T^{12} + 3150 T^{13} - 323 T^{14} - 9 T^{15} + T^{16} \)
$97$ \( ( 3678976 + 1843200 T - 596944 T^{2} - 501530 T^{3} - 102459 T^{4} - 7016 T^{5} + 138 T^{6} + 34 T^{7} + T^{8} )^{2} \)
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