Properties

Label 731.2.d.c
Level 731
Weight 2
Character orbit 731.d
Analytic conductor 5.837
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{13} q^{3} + ( 2 + \beta_{1} ) q^{4} + \beta_{10} q^{5} + ( -\beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{6} + \beta_{8} q^{7} + ( 1 + \beta_{4} + \beta_{15} + \beta_{16} ) q^{8} + ( -1 - \beta_{1} + \beta_{18} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} + \beta_{13} q^{3} + ( 2 + \beta_{1} ) q^{4} + \beta_{10} q^{5} + ( -\beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{6} + \beta_{8} q^{7} + ( 1 + \beta_{4} + \beta_{15} + \beta_{16} ) q^{8} + ( -1 - \beta_{1} + \beta_{18} ) q^{9} + ( \beta_{7} + \beta_{8} + \beta_{12} + \beta_{13} ) q^{10} + ( -\beta_{8} + \beta_{14} ) q^{11} + ( \beta_{2} - \beta_{3} + \beta_{10} + \beta_{11} + \beta_{12} + 4 \beta_{13} ) q^{12} + ( -\beta_{1} + \beta_{6} - \beta_{16} ) q^{13} + ( -\beta_{2} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{14} + ( 2 - \beta_{6} - \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{15} + ( \beta_{1} + \beta_{15} + \beta_{16} ) q^{16} + ( 1 - \beta_{10} - \beta_{13} - \beta_{17} ) q^{17} + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{18} + ( \beta_{5} + \beta_{15} - \beta_{18} ) q^{19} + ( -\beta_{2} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{20} + ( 2 + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{21} + ( \beta_{2} + 4 \beta_{3} - \beta_{8} + \beta_{10} - \beta_{13} ) q^{22} + ( \beta_{3} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{13} ) q^{23} + ( \beta_{3} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{24} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{25} + ( 2 - \beta_{4} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{26} + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{27} + ( -\beta_{2} + 2 \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{28} + ( -\beta_{2} - 2 \beta_{3} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{29} + ( -3 - \beta_{1} + \beta_{4} + 2 \beta_{6} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{30} + ( -\beta_{2} + \beta_{3} - \beta_{9} - \beta_{13} ) q^{31} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{32} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{33} + ( 1 - \beta_{1} + \beta_{6} - \beta_{7} - \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{19} ) q^{34} + ( -2 - \beta_{4} + \beta_{15} - \beta_{19} ) q^{35} + ( -6 - 2 \beta_{1} - 2 \beta_{15} + \beta_{17} + 4 \beta_{18} ) q^{36} + ( \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{37} + ( -1 - 3 \beta_{5} - \beta_{6} + \beta_{15} - \beta_{16} - \beta_{19} ) q^{38} + ( -\beta_{2} + \beta_{3} - \beta_{7} - 2 \beta_{8} + \beta_{11} - 3 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{39} + ( 2 \beta_{2} - \beta_{3} + \beta_{7} + 3 \beta_{8} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - 2 \beta_{14} ) q^{40} + ( -\beta_{2} + 2 \beta_{3} - \beta_{8} - \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{4} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{42} + q^{43} + ( \beta_{2} + 3 \beta_{3} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{44} + ( \beta_{2} + 2 \beta_{8} + \beta_{9} - 3 \beta_{11} + 3 \beta_{12} + \beta_{13} - \beta_{14} ) q^{45} + ( \beta_{2} - \beta_{3} - 3 \beta_{8} + \beta_{9} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{46} + ( -3 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{15} + \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{47} + ( 2 \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} - \beta_{14} ) q^{48} + ( 1 + \beta_{5} - \beta_{6} + \beta_{15} + \beta_{16} ) q^{49} + ( 2 - \beta_{1} + \beta_{5} + \beta_{15} + 2 \beta_{16} - 2 \beta_{18} - \beta_{19} ) q^{50} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} - \beta_{19} ) q^{51} + ( -4 - \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{15} - 4 \beta_{16} - \beta_{18} + \beta_{19} ) q^{52} + ( 1 + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{15} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{53} + ( -5 \beta_{3} - \beta_{7} + 5 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - \beta_{11} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{54} + ( \beta_{4} + \beta_{5} - \beta_{6} + \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{55} + ( -3 \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{12} - 2 \beta_{13} ) q^{56} + ( 3 \beta_{2} + \beta_{3} + \beta_{8} + 2 \beta_{10} + 2 \beta_{12} + 3 \beta_{13} ) q^{57} + ( -\beta_{2} - 4 \beta_{3} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{58} + ( 3 \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{59} + ( 4 + \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{15} - \beta_{16} - 3 \beta_{17} - 3 \beta_{18} + 2 \beta_{19} ) q^{60} + ( \beta_{2} - 4 \beta_{3} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{61} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - \beta_{14} ) q^{62} + ( \beta_{2} + 3 \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{63} + ( -2 - 4 \beta_{1} + \beta_{6} + 2 \beta_{15} + 3 \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{64} + ( \beta_{2} + 3 \beta_{3} + \beta_{8} + \beta_{9} - 4 \beta_{11} + 2 \beta_{12} ) q^{65} + ( 1 - \beta_{4} + \beta_{5} - 3 \beta_{15} + \beta_{16} + \beta_{17} + 4 \beta_{18} ) q^{66} + ( 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{15} - \beta_{16} - 2 \beta_{18} ) q^{67} + ( 3 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{13} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{68} + ( -5 - \beta_{1} - \beta_{4} - \beta_{6} - \beta_{15} + \beta_{16} + 2 \beta_{18} - \beta_{19} ) q^{69} + ( -2 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{15} + \beta_{16} + \beta_{17} - \beta_{19} ) q^{70} + ( -2 \beta_{2} - \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{71} + ( -3 - \beta_{1} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{15} + 2 \beta_{16} + 2 \beta_{18} ) q^{72} + ( \beta_{2} - 3 \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{73} + ( -\beta_{2} - \beta_{3} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{74} + ( -\beta_{2} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{75} + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{15} - \beta_{16} - 4 \beta_{18} + \beta_{19} ) q^{76} + ( 3 - \beta_{4} - 2 \beta_{5} + \beta_{15} + \beta_{18} - 2 \beta_{19} ) q^{77} + ( 2 \beta_{2} + 2 \beta_{3} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{78} + ( 2 \beta_{3} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{79} + ( -\beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{80} + ( 3 + \beta_{1} - \beta_{5} - \beta_{6} + 4 \beta_{15} - 2 \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{81} + ( \beta_{2} + \beta_{3} - \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{82} + ( 2 + 3 \beta_{1} + 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{83} + ( 5 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{84} + ( 4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{85} + \beta_{6} q^{86} + ( 1 - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{87} + ( 2 \beta_{2} + 8 \beta_{3} + \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - 4 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{88} + ( -1 + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{15} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{89} + ( -4 \beta_{2} - 2 \beta_{3} - \beta_{7} - 4 \beta_{8} - \beta_{9} + \beta_{10} + 4 \beta_{11} - 4 \beta_{12} - 6 \beta_{13} + 2 \beta_{14} ) q^{90} + ( -\beta_{2} + 3 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{91} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} - \beta_{12} + 4 \beta_{13} + \beta_{14} ) q^{92} + ( 3 + 2 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{93} + ( -3 - 2 \beta_{1} - \beta_{4} - \beta_{15} + \beta_{16} + 4 \beta_{17} + 3 \beta_{18} - 2 \beta_{19} ) q^{94} + ( -2 \beta_{2} + 4 \beta_{3} - \beta_{7} - 3 \beta_{8} - \beta_{9} - 3 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{95} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{12} + 4 \beta_{13} - \beta_{14} ) q^{96} + ( 3 \beta_{2} + 3 \beta_{3} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{97} + ( -4 - \beta_{1} - 2 \beta_{5} + \beta_{15} + 2 \beta_{16} + 2 \beta_{18} - \beta_{19} ) q^{98} + ( -2 \beta_{2} - 4 \beta_{3} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 2q^{2} + 42q^{4} + 18q^{8} - 18q^{9} + O(q^{10}) \) \( 20q + 2q^{2} + 42q^{4} + 18q^{8} - 18q^{9} - 4q^{13} + 26q^{15} + 6q^{16} + 16q^{17} - 22q^{18} - 4q^{19} + 20q^{21} - 2q^{25} + 22q^{26} - 72q^{30} + 38q^{32} - 12q^{33} + 12q^{34} - 30q^{35} - 104q^{36} - 22q^{38} + 26q^{42} + 20q^{43} - 34q^{47} + 22q^{49} + 42q^{50} + 52q^{51} - 110q^{52} + 14q^{53} + 12q^{55} + 20q^{59} + 42q^{60} - 22q^{64} + 50q^{66} - 12q^{67} + 50q^{68} - 82q^{69} - 30q^{70} - 50q^{72} + 2q^{76} + 78q^{77} + 44q^{81} + 20q^{83} + 62q^{84} + 76q^{85} + 2q^{86} + 12q^{87} - 46q^{89} + 58q^{93} - 18q^{94} - 62q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 29 x^{18} + 358 x^{16} + 2458 x^{14} + 10298 x^{12} + 27188 x^{10} + 45053 x^{8} + 44980 x^{6} + 24400 x^{4} + 5448 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{19} + 32 \nu^{17} + 301 \nu^{15} + 454 \nu^{13} - 8842 \nu^{11} - 60538 \nu^{9} - 170527 \nu^{7} - 239549 \nu^{5} - 159512 \nu^{3} - 37650 \nu \)\()/612\)
\(\beta_{3}\)\(=\)\((\)\( 31 \nu^{19} + 839 \nu^{17} + 9484 \nu^{15} + 57832 \nu^{13} + 204482 \nu^{11} + 416180 \nu^{9} + 443207 \nu^{7} + 166912 \nu^{5} - 56522 \nu^{3} - 37092 \nu \)\()/1224\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{18} + 205 \nu^{16} + 2187 \nu^{14} + 12574 \nu^{12} + 42144 \nu^{10} + 83190 \nu^{8} + 92956 \nu^{6} + 52837 \nu^{4} + 11586 \nu^{2} + 6 \)\()/68\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{18} - 96 \nu^{16} - 1243 \nu^{14} - 8502 \nu^{12} - 33450 \nu^{10} - 76922 \nu^{8} - 99807 \nu^{6} - 65597 \nu^{4} - 16300 \nu^{2} - 134 \)\()/68\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{18} + 190 \nu^{16} + 2141 \nu^{14} + 12970 \nu^{12} + 45750 \nu^{10} + 95210 \nu^{8} + 112931 \nu^{6} + 69023 \nu^{4} + 16636 \nu^{2} + 290 \)\()/68\)
\(\beta_{7}\)\(=\)\((\)\( -131 \nu^{19} - 3733 \nu^{17} - 44786 \nu^{15} - 293564 \nu^{13} - 1140370 \nu^{11} - 2658040 \nu^{9} - 3569371 \nu^{7} - 2448146 \nu^{5} - 597674 \nu^{3} + 29112 \nu \)\()/1224\)
\(\beta_{8}\)\(=\)\((\)\( 12 \nu^{19} + 333 \nu^{17} + 3867 \nu^{15} + 24386 \nu^{13} + 90688 \nu^{11} + 202118 \nu^{9} + 261596 \nu^{7} + 178425 \nu^{5} + 49390 \nu^{3} + 1182 \nu \)\()/68\)
\(\beta_{9}\)\(=\)\((\)\( -113 \nu^{19} - 3157 \nu^{17} - 36920 \nu^{15} - 234596 \nu^{13} - 879694 \nu^{11} - 1977820 \nu^{9} - 2579461 \nu^{7} - 1761824 \nu^{5} - 477434 \nu^{3} - 9660 \nu \)\()/612\)
\(\beta_{10}\)\(=\)\((\)\( -257 \nu^{19} - 7153 \nu^{17} - 83324 \nu^{15} - 527024 \nu^{13} - 1963870 \nu^{11} - 4371820 \nu^{9} - 5602129 \nu^{7} - 3690560 \nu^{5} - 897122 \nu^{3} + 25116 \nu \)\()/1224\)
\(\beta_{11}\)\(=\)\((\)\( -95 \nu^{19} - 2581 \nu^{17} - 29360 \nu^{15} - 181442 \nu^{13} - 661246 \nu^{11} - 1442644 \nu^{9} - 1820275 \nu^{7} - 1193924 \nu^{5} - 300278 \nu^{3} + 2058 \nu \)\()/306\)
\(\beta_{12}\)\(=\)\((\)\( 49 \nu^{19} + 1347 \nu^{17} + 15514 \nu^{15} + 97148 \nu^{13} + 359078 \nu^{11} + 795432 \nu^{9} + 1020969 \nu^{7} + 685274 \nu^{5} + 182038 \nu^{3} + 2472 \nu \)\()/136\)
\(\beta_{13}\)\(=\)\((\)\( 49 \nu^{19} + 1347 \nu^{17} + 15514 \nu^{15} + 97148 \nu^{13} + 359078 \nu^{11} + 795432 \nu^{9} + 1020969 \nu^{7} + 685410 \nu^{5} + 182990 \nu^{3} + 3696 \nu \)\()/136\)
\(\beta_{14}\)\(=\)\((\)\( 299 \nu^{19} + 8191 \nu^{17} + 93824 \nu^{15} + 582812 \nu^{13} + 2129842 \nu^{11} + 4643812 \nu^{9} + 5824387 \nu^{7} + 3754124 \nu^{5} + 886166 \nu^{3} - 29292 \nu \)\()/612\)
\(\beta_{15}\)\(=\)\((\)\( 37 \nu^{18} + 1014 \nu^{16} + 11647 \nu^{14} + 72762 \nu^{12} + 268390 \nu^{10} + 593314 \nu^{8} + 759305 \nu^{6} + 506305 \nu^{4} + 131628 \nu^{2} + 950 \)\()/68\)
\(\beta_{16}\)\(=\)\((\)\( -37 \nu^{18} - 1014 \nu^{16} - 11647 \nu^{14} - 72762 \nu^{12} - 268390 \nu^{10} - 593314 \nu^{8} - 759305 \nu^{6} - 506237 \nu^{4} - 131152 \nu^{2} - 406 \)\()/68\)
\(\beta_{17}\)\(=\)\((\)\( 49 \nu^{18} + 1347 \nu^{16} + 15514 \nu^{14} + 97148 \nu^{12} + 359078 \nu^{10} + 795364 \nu^{8} + 1020153 \nu^{6} + 682146 \nu^{4} + 177890 \nu^{2} + 1112 \)\()/68\)
\(\beta_{18}\)\(=\)\((\)\( 49 \nu^{18} + 1347 \nu^{16} + 15514 \nu^{14} + 97148 \nu^{12} + 359078 \nu^{10} + 795432 \nu^{8} + 1020901 \nu^{6} + 684662 \nu^{4} + 180542 \nu^{2} + 1520 \)\()/68\)
\(\beta_{19}\)\(=\)\((\)\( 2 \nu^{18} + 55 \nu^{16} + 633 \nu^{14} + 3956 \nu^{12} + 14574 \nu^{10} + 32138 \nu^{8} + 41018 \nu^{6} + 27323 \nu^{4} + 7132 \nu^{2} + 52 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(2 \beta_{14} + 2 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{3} - 2 \beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{16} + \beta_{15} - 7 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-11 \beta_{14} + \beta_{13} - \beta_{12} - 11 \beta_{11} + 4 \beta_{10} - 4 \beta_{9} + 11 \beta_{8} + 4 \beta_{3} + 11 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-\beta_{19} + \beta_{18} + \beta_{17} - 8 \beta_{16} - 9 \beta_{15} + \beta_{6} + 42 \beta_{1} - 65\)
\(\nu^{7}\)\(=\)\(60 \beta_{14} - 9 \beta_{13} + 11 \beta_{12} + 61 \beta_{11} - 18 \beta_{10} + 19 \beta_{9} - 61 \beta_{8} - 20 \beta_{3} - 59 \beta_{2}\)
\(\nu^{8}\)\(=\)\(11 \beta_{19} - 10 \beta_{18} - 12 \beta_{17} + 51 \beta_{16} + 62 \beta_{15} - 11 \beta_{6} - 242 \beta_{1} + 345\)
\(\nu^{9}\)\(=\)\(-330 \beta_{14} + 61 \beta_{13} - 86 \beta_{12} - 342 \beta_{11} + 85 \beta_{10} - 99 \beta_{9} + 342 \beta_{8} + \beta_{7} + 113 \beta_{3} + 316 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-84 \beta_{19} + 72 \beta_{18} + 99 \beta_{17} - 302 \beta_{16} - 390 \beta_{15} + 83 \beta_{6} + \beta_{4} + 1375 \beta_{1} - 1878\)
\(\nu^{11}\)\(=\)\(1832 \beta_{14} - 375 \beta_{13} + 590 \beta_{12} + 1932 \beta_{11} - 408 \beta_{10} + 542 \beta_{9} - 1928 \beta_{8} - 16 \beta_{7} - 671 \beta_{3} - 1701 \beta_{2}\)
\(\nu^{12}\)\(=\)\(554 \beta_{19} - 458 \beta_{18} - 700 \beta_{17} + 1732 \beta_{16} + 2353 \beta_{15} - 534 \beta_{6} + 3 \beta_{5} - 20 \beta_{4} - 7769 \beta_{1} + 10341\)
\(\nu^{13}\)\(=\)\(-10245 \beta_{14} + 2204 \beta_{13} - 3796 \beta_{12} - 10968 \beta_{11} + 1958 \beta_{10} - 3050 \beta_{9} + 10892 \beta_{8} + 166 \beta_{7} + 4042 \beta_{3} + 9210 \beta_{2}\)
\(\nu^{14}\)\(=\)\(-3391 \beta_{19} + 2744 \beta_{18} + 4569 \beta_{17} - 9773 \beta_{16} - 13878 \beta_{15} + 3145 \beta_{6} - 59 \beta_{5} + 243 \beta_{4} + 43790 \beta_{1} - 57274\)
\(\nu^{15}\)\(=\)\(57593 \beta_{14} - 12641 \beta_{13} + 23561 \beta_{12} + 62463 \beta_{11} - 9295 \beta_{10} + 17437 \beta_{9} - 61572 \beta_{8} - 1424 \beta_{7} - 24361 \beta_{3} - 50124 \beta_{2}\)
\(\nu^{16}\)\(=\)\(19887 \beta_{19} - 15908 \beta_{18} - 28475 \beta_{17} + 54640 \beta_{16} + 80739 \beta_{15} - 17485 \beta_{6} + 707 \beta_{5} - 2337 \beta_{4} - 246555 \beta_{1} + 318273\)
\(\nu^{17}\)\(=\)\(-324940 \beta_{14} + 71444 \beta_{13} - 143070 \beta_{12} - 356432 \beta_{11} + 43235 \beta_{10} - 100609 \beta_{9} + 348099 \beta_{8} + 10990 \beta_{7} + 146233 \beta_{3} + 273935 \beta_{2}\)
\(\nu^{18}\)\(=\)\(-113594 \beta_{19} + 90435 \beta_{18} + 172480 \beta_{17} - 303776 \beta_{16} - 465598 \beta_{15} + 93126 \beta_{6} - 6703 \beta_{5} + 19631 \beta_{4} + 1387569 \beta_{1} - 1772446\)
\(\nu^{19}\)\(=\)\(1837876 \beta_{14} - 400009 \beta_{13} + 856523 \beta_{12} + 2036263 \beta_{11} - 194715 \beta_{10} + 583532 \beta_{9} - 1967926 \beta_{8} - 79354 \beta_{7} - 873357 \beta_{3} - 1502086 \beta_{2}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\) \(173\) \(562\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
560.1
1.20836i
1.20836i
1.23843i
1.23843i
1.58374i
1.58374i
2.32169i
2.32169i
2.37417i
2.37417i
2.38643i
2.38643i
2.00110i
2.00110i
1.45814i
1.45814i
0.799077i
0.799077i
0.0825423i
0.0825423i
−2.35369 1.62169i 3.53987 0.259122i 3.81697i 1.04097i −3.62439 0.370107 0.609893i
560.2 −2.35369 1.62169i 3.53987 0.259122i 3.81697i 1.04097i −3.62439 0.370107 0.609893i
560.3 −2.33801 3.08166i 3.46629 3.53522i 7.20495i 0.740145i −3.42819 −6.49664 8.26537i
560.4 −2.33801 3.08166i 3.46629 3.53522i 7.20495i 0.740145i −3.42819 −6.49664 8.26537i
560.5 −2.11938 0.764619i 2.49176 0.0168131i 1.62052i 2.93256i −1.04222 2.41536 0.0356332i
560.6 −2.11938 0.764619i 2.49176 0.0168131i 1.62052i 2.93256i −1.04222 2.41536 0.0356332i
560.7 −1.26877 2.08497i −0.390228 3.03060i 2.64534i 3.31139i 3.03264 −1.34709 3.84513i
560.8 −1.26877 2.08497i −0.390228 3.03060i 2.64534i 3.31139i 3.03264 −1.34709 3.84513i
560.9 −1.16760 0.590876i −0.636707 3.35001i 0.689908i 0.173719i 3.07862 2.65087 3.91148i
560.10 −1.16760 0.590876i −0.636707 3.35001i 0.689908i 0.173719i 3.07862 2.65087 3.91148i
560.11 1.14234 0.300114i −0.695068 1.80213i 0.342831i 2.54629i −3.07867 2.90993 2.05864i
560.12 1.14234 0.300114i −0.695068 1.80213i 0.342831i 2.54629i −3.07867 2.90993 2.05864i
560.13 1.73078 2.30609i 0.995588 0.771574i 3.99132i 3.40847i −1.73841 −2.31803 1.33542i
560.14 1.73078 2.30609i 0.995588 0.771574i 3.99132i 3.40847i −1.73841 −2.31803 1.33542i
560.15 2.20768 1.64345i 2.87383 3.04599i 3.62820i 3.21768i 1.92914 0.299074 6.72456i
560.16 2.20768 1.64345i 2.87383 3.04599i 3.62820i 3.21768i 1.92914 0.299074 6.72456i
560.17 2.52220 3.34912i 4.36148 0.720920i 8.44715i 2.87134i 5.95611 −8.21664 1.81830i
560.18 2.52220 3.34912i 4.36148 0.720920i 8.44715i 2.87134i 5.95611 −8.21664 1.81830i
560.19 2.64446 1.50563i 4.99319 2.09450i 3.98159i 1.03626i 7.91537 0.733067 5.53882i
560.20 2.64446 1.50563i 4.99319 2.09450i 3.98159i 1.03626i 7.91537 0.733067 5.53882i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 560.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\).