# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(731, [\chi])$$.