Properties

Label 380.2.f.a
Level $380$
Weight $2$
Character orbit 380.f
Analytic conductor $3.034$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{18} + x^{16} - 9 x^{14} + 20 x^{12} - 24 x^{10} + 80 x^{8} - 144 x^{6} + 64 x^{4} - 256 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{12} q^{2} + \beta_{10} q^{3} + \beta_{16} q^{4} - q^{5} -\beta_{14} q^{6} -\beta_{7} q^{7} -\beta_{11} q^{8} + ( 1 + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{12} q^{2} + \beta_{10} q^{3} + \beta_{16} q^{4} - q^{5} -\beta_{14} q^{6} -\beta_{7} q^{7} -\beta_{11} q^{8} + ( 1 + \beta_{3} ) q^{9} + \beta_{12} q^{10} + ( -\beta_{8} - \beta_{15} ) q^{11} + \beta_{9} q^{12} -\beta_{4} q^{13} + ( \beta_{2} - \beta_{10} ) q^{14} -\beta_{10} q^{15} + ( -\beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{16} + ( 1 + \beta_{6} - \beta_{10} + \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} + \beta_{19} ) q^{17} + ( -\beta_{2} - \beta_{6} - \beta_{12} - \beta_{19} ) q^{18} + ( \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{16} ) q^{19} -\beta_{16} q^{20} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{19} ) q^{21} + ( \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{22} + ( 1 - \beta_{6} + \beta_{8} + \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{23} + ( 2 + \beta_{3} + \beta_{7} + \beta_{15} - \beta_{18} ) q^{24} + q^{25} + ( -1 - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{26} + ( -\beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{19} ) q^{27} + ( -1 + \beta_{3} - \beta_{8} - \beta_{13} + \beta_{16} ) q^{28} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{12} + \beta_{19} ) q^{29} + \beta_{14} q^{30} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} + \beta_{12} ) q^{31} + ( \beta_{4} - \beta_{5} - 2 \beta_{10} - \beta_{11} + \beta_{19} ) q^{32} + ( -\beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{19} ) q^{33} + ( -\beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{34} + \beta_{7} q^{35} + ( -1 - \beta_{3} + 2 \beta_{7} - \beta_{8} - \beta_{13} + 2 \beta_{16} ) q^{36} + ( \beta_{2} + \beta_{5} + \beta_{11} - \beta_{12} ) q^{37} + ( \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} - \beta_{17} + 2 \beta_{19} ) q^{38} -\beta_{7} q^{39} + \beta_{11} q^{40} + ( -\beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{19} ) q^{41} + ( -3 - \beta_{3} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{16} - \beta_{19} ) q^{42} + ( 1 + \beta_{7} + \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{43} + ( 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{17} + 2 \beta_{19} ) q^{44} + ( -1 - \beta_{3} ) q^{45} + ( -\beta_{2} + 2 \beta_{5} + 2 \beta_{9} - \beta_{10} - 2 \beta_{19} ) q^{46} + ( -\beta_{6} + \beta_{8} + \beta_{10} - \beta_{15} - 2 \beta_{16} - \beta_{19} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{19} ) q^{48} + ( -\beta_{6} + \beta_{10} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{49} -\beta_{12} q^{50} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{12} + \beta_{19} ) q^{51} + ( 2 \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{52} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{53} + ( -1 - \beta_{3} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{54} + ( \beta_{8} + \beta_{15} ) q^{55} + ( \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{56} + ( -2 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{57} + ( 3 + \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{13} - \beta_{16} ) q^{58} + ( -\beta_{6} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{59} -\beta_{9} q^{60} + ( -2 - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{61} + ( -2 + \beta_{6} - \beta_{10} + 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{62} + ( 1 - \beta_{7} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{63} + ( 3 + \beta_{7} - \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} ) q^{64} + \beta_{4} q^{65} + ( 2 - \beta_{6} + 2 \beta_{8} + \beta_{10} - 2 \beta_{16} - \beta_{19} ) q^{66} + ( -\beta_{1} - \beta_{5} + \beta_{11} - 3 \beta_{12} + \beta_{19} ) q^{67} + ( -1 - \beta_{3} - 2 \beta_{7} + \beta_{8} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{68} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} - 2 \beta_{9} - \beta_{11} - 3 \beta_{12} ) q^{69} + ( -\beta_{2} + \beta_{10} ) q^{70} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{19} ) q^{71} + ( \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 3 \beta_{10} - \beta_{11} + 2 \beta_{19} ) q^{72} + ( -2 + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{19} ) q^{73} + ( -2 + \beta_{3} - \beta_{7} - \beta_{15} - \beta_{17} ) q^{74} + \beta_{10} q^{75} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} + 2 \beta_{19} ) q^{76} + ( -1 + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{13} + \beta_{14} + \beta_{15} - 3 \beta_{16} - 3 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{77} + ( \beta_{2} - \beta_{10} ) q^{78} + ( -\beta_{1} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - 3 \beta_{19} ) q^{79} + ( \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{80} + ( 4 + \beta_{3} + \beta_{6} - \beta_{10} + \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - 5 \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{81} + ( 2 - \beta_{6} + 2 \beta_{8} + \beta_{10} - 2 \beta_{16} - \beta_{19} ) q^{82} + ( -\beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{14} + \beta_{15} + 2 \beta_{16} - 2 \beta_{18} - \beta_{19} ) q^{83} + ( -2 \beta_{1} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{9} + 3 \beta_{10} + 2 \beta_{12} ) q^{84} + ( -1 - \beta_{6} + \beta_{10} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{85} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -2 - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{13} + 4 \beta_{14} + \beta_{15} - 2 \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{87} + ( 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} + 2 \beta_{19} ) q^{88} + ( 2 \beta_{1} + \beta_{6} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{19} ) q^{89} + ( \beta_{2} + \beta_{6} + \beta_{12} + \beta_{19} ) q^{90} + ( 2 \beta_{5} - 3 \beta_{6} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{91} + ( 1 + \beta_{3} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} + 2 \beta_{19} ) q^{92} + ( -1 - 2 \beta_{3} - \beta_{7} + \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{93} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{19} ) q^{94} + ( -\beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{16} ) q^{95} + ( -6 - \beta_{3} + 3 \beta_{7} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{96} + ( \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{19} ) q^{97} + ( \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{98} + ( -1 + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{4} - 20q^{5} - 6q^{6} + 16q^{9} + O(q^{10}) \) \( 20q + 2q^{4} - 20q^{5} - 6q^{6} + 16q^{9} - 2q^{16} + 20q^{17} - 2q^{20} + 26q^{24} + 20q^{25} - 14q^{26} - 14q^{28} + 6q^{30} - 4q^{36} + 10q^{38} - 42q^{42} + 8q^{44} - 16q^{45} - 30q^{54} - 36q^{57} + 62q^{58} - 24q^{61} - 40q^{62} + 50q^{64} + 40q^{66} + 6q^{68} - 36q^{73} - 36q^{74} - 28q^{76} - 32q^{77} + 2q^{80} + 60q^{81} + 40q^{82} - 20q^{85} + 26q^{92} - 122q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - x^{18} + x^{16} - 9 x^{14} + 20 x^{12} - 24 x^{10} + 80 x^{8} - 144 x^{6} + 64 x^{4} - 256 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{19} - 3 \nu^{17} + 11 \nu^{15} - 19 \nu^{13} + 72 \nu^{11} + 144 \nu^{9} + 160 \nu^{7} - 272 \nu^{5} + 192 \nu^{3} + 1024 \nu \)\()/1536\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{18} + 3 \nu^{16} - 19 \nu^{14} + 43 \nu^{12} - 64 \nu^{10} + 104 \nu^{8} - 240 \nu^{6} + 560 \nu^{4} - 1152 \nu^{2} + 1280 \)\()/512\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{19} - 3 \nu^{17} + 19 \nu^{15} - 11 \nu^{13} + 32 \nu^{11} - 200 \nu^{9} + 336 \nu^{7} - 48 \nu^{5} + 768 \nu^{3} - 1792 \nu \)\()/1024\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{19} + 3 \nu^{17} + 5 \nu^{15} - 13 \nu^{13} + 18 \nu^{11} - 120 \nu^{9} + 16 \nu^{7} - 176 \nu^{5} + 96 \nu^{3} - 896 \nu \)\()/768\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{19} - 3 \nu^{17} + 7 \nu^{15} - 23 \nu^{13} + 66 \nu^{11} - 156 \nu^{9} + 8 \nu^{7} - 160 \nu^{5} + 768 \nu^{3} - 1024 \nu \)\()/768\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{18} + 5 \nu^{16} - 5 \nu^{14} + 13 \nu^{12} + 8 \nu^{10} + 40 \nu^{8} - 112 \nu^{6} + 144 \nu^{4} + 128 \nu^{2} - 256 \)\()/512\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{19} + 3 \nu^{17} + 6 \nu^{16} + 5 \nu^{15} + 6 \nu^{14} - 13 \nu^{13} + 18 \nu^{12} + 18 \nu^{11} - 18 \nu^{10} - 120 \nu^{9} + 84 \nu^{8} + 16 \nu^{7} + 24 \nu^{6} - 176 \nu^{5} + 528 \nu^{4} + 864 \nu^{3} + 192 \nu^{2} - 1664 \nu \)\()/768\)
\(\beta_{9}\)\(=\)\((\)\( -5 \nu^{19} - 15 \nu^{17} + 31 \nu^{15} - 119 \nu^{13} + 96 \nu^{11} - 168 \nu^{9} + 656 \nu^{7} - 880 \nu^{5} + 4608 \nu^{3} - 4864 \nu \)\()/3072\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{19} + 9 \nu^{17} - 41 \nu^{15} + 49 \nu^{13} - 72 \nu^{11} + 264 \nu^{9} - 400 \nu^{7} + 1232 \nu^{5} - 1536 \nu^{3} + 1280 \nu \)\()/3072\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{19} - 3 \nu^{17} + 3 \nu^{15} + 5 \nu^{13} + 16 \nu^{11} - 56 \nu^{9} + 16 \nu^{7} - 176 \nu^{5} + 512 \nu^{3} \)\()/512\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{19} + \nu^{17} - \nu^{15} + 9 \nu^{13} - 20 \nu^{11} + 24 \nu^{9} - 80 \nu^{7} + 144 \nu^{5} - 64 \nu^{3} + 256 \nu \)\()/512\)
\(\beta_{13}\)\(=\)\((\)\( \nu^{19} - 3 \nu^{17} - 6 \nu^{16} - 5 \nu^{15} - 6 \nu^{14} + 13 \nu^{13} + 30 \nu^{12} - 18 \nu^{11} - 30 \nu^{10} + 120 \nu^{9} + 156 \nu^{8} - 16 \nu^{7} - 264 \nu^{6} + 176 \nu^{5} - 144 \nu^{4} - 864 \nu^{3} - 1152 \nu^{2} + 1664 \nu + 2304 \)\()/768\)
\(\beta_{14}\)\(=\)\((\)\( -5 \nu^{18} + 9 \nu^{16} - 41 \nu^{14} + 49 \nu^{12} - 72 \nu^{10} + 264 \nu^{8} - 400 \nu^{6} + 1232 \nu^{4} - 1536 \nu^{2} + 1280 \)\()/1536\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{19} - 2 \nu^{18} - 3 \nu^{17} - 5 \nu^{15} + 4 \nu^{14} + 13 \nu^{13} + 4 \nu^{12} - 18 \nu^{11} + 6 \nu^{10} + 120 \nu^{9} - 84 \nu^{8} - 16 \nu^{7} + 152 \nu^{6} + 176 \nu^{5} - 496 \nu^{4} - 864 \nu^{3} + 192 \nu^{2} + 1664 \nu - 1024 \)\()/768\)
\(\beta_{16}\)\(=\)\((\)\( -\nu^{18} + \nu^{16} - \nu^{14} + 9 \nu^{12} - 20 \nu^{10} + 24 \nu^{8} - 80 \nu^{6} + 144 \nu^{4} - 64 \nu^{2} + 256 \)\()/256\)
\(\beta_{17}\)\(=\)\((\)\( -\nu^{19} + 2 \nu^{18} + 3 \nu^{17} - 12 \nu^{16} + 5 \nu^{15} + 8 \nu^{14} - 13 \nu^{13} - 16 \nu^{12} + 18 \nu^{11} + 102 \nu^{10} - 120 \nu^{9} - 156 \nu^{8} + 16 \nu^{7} + 136 \nu^{6} - 176 \nu^{5} - 464 \nu^{4} + 864 \nu^{3} - 1664 \nu + 256 \)\()/768\)
\(\beta_{18}\)\(=\)\((\)\( 2 \nu^{19} - 9 \nu^{18} - 6 \nu^{17} + 9 \nu^{16} - 10 \nu^{15} - 33 \nu^{14} + 26 \nu^{13} + 57 \nu^{12} - 36 \nu^{11} - 60 \nu^{10} + 240 \nu^{9} + 96 \nu^{8} - 32 \nu^{7} - 96 \nu^{6} + 352 \nu^{5} + 240 \nu^{4} - 1728 \nu^{3} + 1920 \nu^{2} + 3328 \nu - 768 \)\()/1536\)
\(\beta_{19}\)\(=\)\((\)\( -17 \nu^{19} + 45 \nu^{17} - 29 \nu^{15} + 37 \nu^{13} - 192 \nu^{11} - 72 \nu^{9} - 304 \nu^{7} + 464 \nu^{5} + 2304 \nu^{3} - 7936 \nu \)\()/3072\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{18} - \beta_{15} - \beta_{14}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{19} - \beta_{10} + \beta_{6} - 2 \beta_{5} + \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{19} + \beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{6} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{18} - 4 \beta_{16} + 3 \beta_{15} + 3 \beta_{14} + 4 \beta_{8} - 4 \beta_{7} + 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{19} - 8 \beta_{12} + 3 \beta_{10} - 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 3 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(2 \beta_{19} + 2 \beta_{18} - 3 \beta_{17} - 7 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 7 \beta_{13} - 2 \beta_{10} + 5 \beta_{8} - \beta_{7} + 2 \beta_{6} - 4 \beta_{3} - 9\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(2 \beta_{19} + 2 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - 6 \beta_{5} - 2 \beta_{4} + 8 \beta_{2} - 5 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-8 \beta_{19} - 3 \beta_{18} + 10 \beta_{17} - 6 \beta_{16} + \beta_{15} + 9 \beta_{14} + 2 \beta_{13} + 8 \beta_{10} + 6 \beta_{8} + 14 \beta_{7} - 8 \beta_{6} - 4 \beta_{3} + 6\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(5 \beta_{19} - 12 \beta_{12} + 4 \beta_{11} + 15 \beta_{10} - 16 \beta_{9} + 17 \beta_{6} - 6 \beta_{5} + 4 \beta_{4} + 24 \beta_{2} - 3 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-2 \beta_{19} + 12 \beta_{18} + 17 \beta_{17} + \beta_{16} + 19 \beta_{15} - 19 \beta_{14} + 7 \beta_{13} + 2 \beta_{10} + 25 \beta_{8} + 7 \beta_{7} - 2 \beta_{6} + 16 \beta_{3} - 17\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(20 \beta_{19} - 14 \beta_{12} + 30 \beta_{11} - 18 \beta_{10} - 48 \beta_{9} + 6 \beta_{6} - 42 \beta_{5} + 18 \beta_{4} + 16 \beta_{2} - 15 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-24 \beta_{19} - 35 \beta_{18} + 4 \beta_{17} + 48 \beta_{16} + 19 \beta_{15} - 21 \beta_{14} + 12 \beta_{13} + 24 \beta_{10} + 40 \beta_{8} + 8 \beta_{7} - 24 \beta_{6} - 16 \beta_{3} - 16\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-3 \beta_{19} + 160 \beta_{12} - 40 \beta_{11} - 129 \beta_{10} - 32 \beta_{9} + 17 \beta_{6} - 2 \beta_{5} + 28 \beta_{4} + 64 \beta_{2} - 29 \beta_{1}\)\()/2\)
\(\nu^{16}\)\(=\)\((\)\(26 \beta_{19} - 74 \beta_{18} - 71 \beta_{17} - 43 \beta_{16} + 77 \beta_{15} - 5 \beta_{14} - 37 \beta_{13} - 26 \beta_{10} - 15 \beta_{8} + 131 \beta_{7} + 26 \beta_{6} + 12 \beta_{3} + 283\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(150 \beta_{19} - 70 \beta_{12} - 90 \beta_{11} - 128 \beta_{10} - 128 \beta_{9} + 76 \beta_{6} - 70 \beta_{5} + 22 \beta_{4} + 104 \beta_{2} + 131 \beta_{1}\)\()/2\)
\(\nu^{18}\)\(=\)\((\)\(-48 \beta_{19} + 33 \beta_{18} - 50 \beta_{17} - 178 \beta_{16} - 183 \beta_{15} - 295 \beta_{14} - 2 \beta_{13} + 48 \beta_{10} - 6 \beta_{8} + 58 \beta_{7} - 48 \beta_{6} + 156 \beta_{3} - 142\)\()/2\)
\(\nu^{19}\)\(=\)\((\)\(137 \beta_{19} - 164 \beta_{12} - 196 \beta_{11} - 445 \beta_{10} - 208 \beta_{9} + 237 \beta_{6} - 470 \beta_{5} - 4 \beta_{4} - 104 \beta_{2} + 61 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.40065 0.195405i
1.40065 + 0.195405i
1.31446 0.521712i
1.31446 + 0.521712i
0.976481 1.02298i
0.976481 + 1.02298i
0.611950 1.27496i
0.611950 + 1.27496i
0.482046 1.32952i
0.482046 + 1.32952i
−0.482046 1.32952i
−0.482046 + 1.32952i
−0.611950 1.27496i
−0.611950 + 1.27496i
−0.976481 1.02298i
−0.976481 + 1.02298i
−1.31446 0.521712i
−1.31446 + 0.521712i
−1.40065 0.195405i
−1.40065 + 0.195405i
−1.40065 0.195405i 1.89649 1.92363 + 0.547388i −1.00000 −2.65631 0.370584i 2.89109i −2.58737 1.14259i 0.596662 1.40065 + 0.195405i
151.2 −1.40065 + 0.195405i 1.89649 1.92363 0.547388i −1.00000 −2.65631 + 0.370584i 2.89109i −2.58737 + 1.14259i 0.596662 1.40065 0.195405i
151.3 −1.31446 0.521712i −2.15859 1.45563 + 1.37154i −1.00000 2.83739 + 1.12616i 1.66163i −1.19783 2.56227i 1.65950 1.31446 + 0.521712i
151.4 −1.31446 + 0.521712i −2.15859 1.45563 1.37154i −1.00000 2.83739 1.12616i 1.66163i −1.19783 + 2.56227i 1.65950 1.31446 0.521712i
151.5 −0.976481 1.02298i −0.502080 −0.0929701 + 1.99784i −1.00000 0.490271 + 0.513617i 3.57756i 2.13453 1.85574i −2.74792 0.976481 + 1.02298i
151.6 −0.976481 + 1.02298i −0.502080 −0.0929701 1.99784i −1.00000 0.490271 0.513617i 3.57756i 2.13453 + 1.85574i −2.74792 0.976481 1.02298i
151.7 −0.611950 1.27496i 3.21061 −1.25103 + 1.56042i −1.00000 −1.96474 4.09340i 3.02772i 2.75504 + 0.640115i 7.30804 0.611950 + 1.27496i
151.8 −0.611950 + 1.27496i 3.21061 −1.25103 1.56042i −1.00000 −1.96474 + 4.09340i 3.02772i 2.75504 0.640115i 7.30804 0.611950 1.27496i
151.9 −0.482046 1.32952i 0.428612 −1.53526 + 1.28178i −1.00000 −0.206611 0.569850i 1.38367i 2.44423 + 1.42329i −2.81629 0.482046 + 1.32952i
151.10 −0.482046 + 1.32952i 0.428612 −1.53526 1.28178i −1.00000 −0.206611 + 0.569850i 1.38367i 2.44423 1.42329i −2.81629 0.482046 1.32952i
151.11 0.482046 1.32952i −0.428612 −1.53526 1.28178i −1.00000 −0.206611 + 0.569850i 1.38367i −2.44423 + 1.42329i −2.81629 −0.482046 + 1.32952i
151.12 0.482046 + 1.32952i −0.428612 −1.53526 + 1.28178i −1.00000 −0.206611 0.569850i 1.38367i −2.44423 1.42329i −2.81629 −0.482046 1.32952i
151.13 0.611950 1.27496i −3.21061 −1.25103 1.56042i −1.00000 −1.96474 + 4.09340i 3.02772i −2.75504 + 0.640115i 7.30804 −0.611950 + 1.27496i
151.14 0.611950 + 1.27496i −3.21061 −1.25103 + 1.56042i −1.00000 −1.96474 4.09340i 3.02772i −2.75504 0.640115i 7.30804 −0.611950 1.27496i
151.15 0.976481 1.02298i 0.502080 −0.0929701 1.99784i −1.00000 0.490271 0.513617i 3.57756i −2.13453 1.85574i −2.74792 −0.976481 + 1.02298i
151.16 0.976481 + 1.02298i 0.502080 −0.0929701 + 1.99784i −1.00000 0.490271 + 0.513617i 3.57756i −2.13453 + 1.85574i −2.74792 −0.976481 1.02298i
151.17 1.31446 0.521712i 2.15859 1.45563 1.37154i −1.00000 2.83739 1.12616i 1.66163i 1.19783 2.56227i 1.65950 −1.31446 + 0.521712i
151.18 1.31446 + 0.521712i 2.15859 1.45563 + 1.37154i −1.00000 2.83739 + 1.12616i 1.66163i 1.19783 + 2.56227i 1.65950 −1.31446 0.521712i
151.19 1.40065 0.195405i −1.89649 1.92363 0.547388i −1.00000 −2.65631 + 0.370584i 2.89109i 2.58737 1.14259i 0.596662 −1.40065 + 0.195405i
151.20 1.40065 + 0.195405i −1.89649 1.92363 + 0.547388i −1.00000 −2.65631 0.370584i 2.89109i 2.58737 + 1.14259i 0.596662 −1.40065 0.195405i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.f.a 20
4.b odd 2 1 inner 380.2.f.a 20
19.b odd 2 1 inner 380.2.f.a 20
76.d even 2 1 inner 380.2.f.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.f.a 20 1.a even 1 1 trivial
380.2.f.a 20 4.b odd 2 1 inner
380.2.f.a 20 19.b odd 2 1 inner
380.2.f.a 20 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 19 T_{3}^{8} + 110 T_{3}^{6} - 218 T_{3}^{4} + 80 T_{3}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 - 256 T^{2} + 64 T^{4} - 144 T^{6} + 80 T^{8} - 24 T^{10} + 20 T^{12} - 9 T^{14} + T^{16} - T^{18} + T^{20} \)
$3$ \( ( -8 + 80 T^{2} - 218 T^{4} + 110 T^{6} - 19 T^{8} + T^{10} )^{2} \)
$5$ \( ( 1 + T )^{20} \)
$7$ \( ( 5184 + 6176 T^{2} + 2548 T^{4} + 448 T^{6} + 35 T^{8} + T^{10} )^{2} \)
$11$ \( ( 4096 + 19200 T^{2} + 8832 T^{4} + 1180 T^{6} + 60 T^{8} + T^{10} )^{2} \)
$13$ \( ( 648 + 2176 T^{2} + 2258 T^{4} + 766 T^{6} + 65 T^{8} + T^{10} )^{2} \)
$17$ \( ( -672 + 224 T + 116 T^{2} - 32 T^{3} - 5 T^{4} + T^{5} )^{4} \)
$19$ \( 6131066257801 + 577441143394 T^{2} + 26580922765 T^{4} + 828841560 T^{6} + 24231042 T^{8} + 194124 T^{10} + 67122 T^{12} + 6360 T^{14} + 565 T^{16} + 34 T^{18} + T^{20} \)
$23$ \( ( 9684544 + 2611360 T^{2} + 244628 T^{4} + 9576 T^{6} + 163 T^{8} + T^{10} )^{2} \)
$29$ \( ( 23011328 + 5474304 T^{2} + 429120 T^{4} + 13936 T^{6} + 197 T^{8} + T^{10} )^{2} \)
$31$ \( ( -401408 + 798208 T^{2} - 129024 T^{4} + 7208 T^{6} - 152 T^{8} + T^{10} )^{2} \)
$37$ \( ( 288 + 10312 T^{2} + 11072 T^{4} + 2530 T^{6} + 104 T^{8} + T^{10} )^{2} \)
$41$ \( ( 32768 + 38912 T^{2} + 16000 T^{4} + 2584 T^{6} + 128 T^{8} + T^{10} )^{2} \)
$43$ \( ( 2304 + 24896 T^{2} + 26096 T^{4} + 3772 T^{6} + 160 T^{8} + T^{10} )^{2} \)
$47$ \( ( 215296 + 1010496 T^{2} + 188784 T^{4} + 9244 T^{6} + 168 T^{8} + T^{10} )^{2} \)
$53$ \( ( 17310728 + 7433568 T^{2} + 648082 T^{4} + 20998 T^{6} + 265 T^{8} + T^{10} )^{2} \)
$59$ \( ( -49840128 + 8072192 T^{2} - 492920 T^{4} + 14256 T^{6} - 195 T^{8} + T^{10} )^{2} \)
$61$ \( ( 2656 - 3504 T - 1644 T^{2} - 162 T^{3} + 6 T^{4} + T^{5} )^{4} \)
$67$ \( ( -71472968 + 17879120 T^{2} - 1079498 T^{4} + 26654 T^{6} - 283 T^{8} + T^{10} )^{2} \)
$71$ \( ( -3612672 + 26840576 T^{2} - 8058880 T^{4} + 119592 T^{6} - 600 T^{8} + T^{10} )^{2} \)
$73$ \( ( 29936 + 9296 T - 1488 T^{2} - 208 T^{3} + 9 T^{4} + T^{5} )^{4} \)
$79$ \( ( -9152503808 + 697222144 T^{2} - 15680736 T^{4} + 150536 T^{6} - 644 T^{8} + T^{10} )^{2} \)
$83$ \( ( 1885643776 + 281106368 T^{2} + 9209568 T^{4} + 116236 T^{6} + 604 T^{8} + T^{10} )^{2} \)
$89$ \( ( 1763704832 + 324796416 T^{2} + 9964288 T^{4} + 116992 T^{6} + 580 T^{8} + T^{10} )^{2} \)
$97$ \( ( 691488 + 1784776 T^{2} + 610880 T^{4} + 29634 T^{6} + 344 T^{8} + T^{10} )^{2} \)
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