# Properties

 Label 930.2.z.b Level $930$ Weight $2$ Character orbit 930.z Analytic conductor $7.426$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.z (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{15} + 18 x^{14} - 44 x^{13} + 63 x^{12} - 46 x^{11} + 110 x^{10} - 120 x^{9} - 79 x^{8} + 120 x^{7} + 110 x^{6} + 46 x^{5} + 63 x^{4} + 44 x^{3} + 18 x^{2} + 6 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 18 x^{14} - 44 x^{13} + 63 x^{12} - 46 x^{11} + 110 x^{10} - 120 x^{9} - 79 x^{8} + 120 x^{7} + 110 x^{6} + 46 x^{5} + 63 x^{4} + 44 x^{3} + 18 x^{2} + 6 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-18722 \nu^{15} + 185843 \nu^{14} - 838945 \nu^{13} + 2542453 \nu^{12} - 5698130 \nu^{11} + 8759897 \nu^{10} - 10751771 \nu^{9} + 15420868 \nu^{8} - 16079401 \nu^{7} + 3298328 \nu^{6} + 6615911 \nu^{5} - 873725 \nu^{4} - 1359926 \nu^{3} + 4650650 \nu^{2} - 359371 \nu - 42999$$$$)/782325$$ $$\beta_{3}$$ $$=$$ $$($$$$-81136 \nu^{15} + 480599 \nu^{14} - 1425885 \nu^{13} + 3474934 \nu^{12} - 4895655 \nu^{11} + 3505826 \nu^{10} - 8953028 \nu^{9} + 9525654 \nu^{8} + 6305807 \nu^{7} - 8317566 \nu^{6} - 10400187 \nu^{5} - 3759245 \nu^{4} - 5886168 \nu^{3} - 4477900 \nu^{2} - 1841688 \nu - 629332$$$$)/782325$$ $$\beta_{4}$$ $$=$$ $$($$$$197674 \nu^{15} - 1066126 \nu^{14} + 2668400 \nu^{13} - 5438846 \nu^{12} + 3613495 \nu^{11} + 7541156 \nu^{10} + 1487407 \nu^{9} + 3639454 \nu^{8} - 54539308 \nu^{7} + 45291284 \nu^{6} + 36771263 \nu^{5} + 54055 \nu^{4} + 9434107 \nu^{3} + 12844655 \nu^{2} + 626822 \nu + 513063$$$$)/782325$$ $$\beta_{5}$$ $$=$$ $$($$$$-202723 \nu^{15} + 1318367 \nu^{14} - 4308870 \nu^{13} + 11063617 \nu^{12} - 18261090 \nu^{11} + 18325103 \nu^{10} - 31215764 \nu^{9} + 39789357 \nu^{8} - 3525559 \nu^{7} - 23488953 \nu^{6} - 10526601 \nu^{5} - 3624410 \nu^{4} - 9300519 \nu^{3} - 4485370 \nu^{2} - 1975344 \nu - 1446151$$$$)/782325$$ $$\beta_{6}$$ $$=$$ $$($$$$-208618 \nu^{15} + 1270047 \nu^{14} - 3845335 \nu^{13} + 9367137 \nu^{12} - 13443350 \nu^{11} + 9398523 \nu^{10} - 21315414 \nu^{9} + 24116182 \nu^{8} + 18505491 \nu^{7} - 32680678 \nu^{6} - 17159506 \nu^{5} - 5841015 \nu^{4} - 11608004 \nu^{3} - 10678290 \nu^{2} - 1411609 \nu - 482471$$$$)/782325$$ $$\beta_{7}$$ $$=$$ $$($$$$-18068 \nu^{15} + 114030 \nu^{14} - 361140 \nu^{13} + 910990 \nu^{12} - 1436140 \nu^{11} + 1318156 \nu^{10} - 2478180 \nu^{9} + 3045030 \nu^{8} + 351590 \nu^{7} - 2077970 \nu^{6} - 1503207 \nu^{5} - 428460 \nu^{4} - 931050 \nu^{3} - 436540 \nu^{2} - 187100 \nu - 38786$$$$)/52155$$ $$\beta_{8}$$ $$=$$ $$($$$$-116478 \nu^{15} + 715962 \nu^{14} - 2204690 \nu^{13} + 5456387 \nu^{12} - 8121210 \nu^{11} + 6432533 \nu^{10} - 13289049 \nu^{9} + 14902967 \nu^{8} + 7965031 \nu^{7} - 16631718 \nu^{6} - 7564856 \nu^{5} - 5416770 \nu^{4} - 8525809 \nu^{3} - 3809630 \nu^{2} - 1544709 \nu - 878996$$$$)/260775$$ $$\beta_{9}$$ $$=$$ $$($$$$38786 \nu^{15} - 250784 \nu^{14} + 812178 \nu^{13} - 2067724 \nu^{12} + 3354508 \nu^{11} - 3220296 \nu^{10} + 5584616 \nu^{9} - 7132500 \nu^{8} - 19064 \nu^{7} + 5005910 \nu^{6} + 2188490 \nu^{5} + 280949 \nu^{4} + 2015058 \nu^{3} + 775534 \nu^{2} + 261608 \nu + 45616$$$$)/52155$$ $$\beta_{10}$$ $$=$$ $$($$$$629332 \nu^{15} - 3857128 \nu^{14} + 11808575 \nu^{13} - 29116493 \nu^{12} + 43122850 \nu^{11} - 33844927 \nu^{10} + 72732346 \nu^{9} - 84472868 \nu^{8} - 40191574 \nu^{7} + 81825647 \nu^{6} + 60908954 \nu^{5} + 18549085 \nu^{4} + 35888671 \nu^{3} + 21804440 \nu^{2} + 6850076 \nu + 1934304$$$$)/782325$$ $$\beta_{11}$$ $$=$$ $$($$$$41958 \nu^{15} - 268474 \nu^{14} + 860612 \nu^{13} - 2177219 \nu^{12} + 3468822 \nu^{11} - 3201630 \nu^{10} + 5693440 \nu^{9} - 7091813 \nu^{8} - 769120 \nu^{7} + 5763042 \nu^{6} + 2195322 \nu^{5} + 678181 \nu^{4} + 2484148 \nu^{3} + 972686 \nu^{2} + 335418 \nu + 145107$$$$)/52155$$ $$\beta_{12}$$ $$=$$ $$($$$$-227516 \nu^{15} + 1457174 \nu^{14} - 4702060 \nu^{13} + 12019889 \nu^{12} - 19527595 \nu^{11} + 19185661 \nu^{10} - 34032578 \nu^{9} + 42098584 \nu^{8} - 1142168 \nu^{7} - 24278411 \nu^{6} - 14305332 \nu^{5} - 6961040 \nu^{4} - 13185518 \nu^{3} - 5085725 \nu^{2} - 2607623 \nu - 683537$$$$)/260775$$ $$\beta_{13}$$ $$=$$ $$($$$$-45616 \nu^{15} + 312482 \nu^{14} - 1071872 \nu^{13} + 2819282 \nu^{12} - 4941532 \nu^{11} + 5452844 \nu^{10} - 8238056 \nu^{9} + 11058536 \nu^{8} - 3528836 \nu^{7} - 5492984 \nu^{6} - 11850 \nu^{5} + 90154 \nu^{4} - 2592859 \nu^{3} + 7954 \nu^{2} - 45554 \nu - 12088$$$$)/52155$$ $$\beta_{14}$$ $$=$$ $$($$$$-381541 \nu^{15} + 2389864 \nu^{14} - 7495250 \nu^{13} + 18750324 \nu^{12} - 28945995 \nu^{11} + 25100526 \nu^{10} - 48495373 \nu^{9} + 58529849 \nu^{8} + 15035337 \nu^{7} - 49999671 \nu^{6} - 29123452 \nu^{5} - 10020190 \nu^{4} - 20606473 \nu^{3} - 10607445 \nu^{2} - 4043868 \nu - 1113587$$$$)/260775$$ $$\beta_{15}$$ $$=$$ $$($$$$230968 \nu^{15} - 1481454 \nu^{14} + 4764706 \nu^{13} - 12092649 \nu^{12} + 19409306 \nu^{11} - 18267510 \nu^{10} + 32272344 \nu^{9} - 40300156 \nu^{8} - 2702376 \nu^{7} + 30385624 \nu^{6} + 12269836 \nu^{5} + 4760148 \nu^{4} + 12730418 \nu^{3} + 4948608 \nu^{2} + 1692418 \nu + 555224$$$$)/156465$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$-6 \beta_{15} - 6 \beta_{14} - 5 \beta_{13} - 6 \beta_{9} - 2 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{15} - 6 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 8 \beta_{11} - 6 \beta_{10} - 17 \beta_{9} + 6 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} + 6 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 6 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{14} + 8 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 33 \beta_{7} - 18 \beta_{6} - 20 \beta_{2} + 18 \beta_{1} + 2$$ $$\nu^{6}$$ $$=$$ $$-39 \beta_{15} - 21 \beta_{14} - 39 \beta_{13} + 72 \beta_{12} - 59 \beta_{11} + 57 \beta_{10} + 20 \beta_{9} - 59 \beta_{8} - 39 \beta_{7} - 59 \beta_{5} - 39 \beta_{4} + 20 \beta_{3} + 39 \beta_{2} + 20 \beta_{1} - 59$$ $$\nu^{7}$$ $$=$$ $$-288 \beta_{15} - 229 \beta_{14} - 229 \beta_{13} + 39 \beta_{12} - 136 \beta_{11} + 39 \beta_{10} - 132 \beta_{9} - 136 \beta_{8} - 229 \beta_{5} + 136 \beta_{3} + 249 \beta_{2} - 268$$ $$\nu^{8}$$ $$=$$ $$-653 \beta_{15} - 424 \beta_{14} - 424 \beta_{13} - 424 \beta_{12} + 93 \beta_{11} - 424 \beta_{10} - 653 \beta_{9} - 268 \beta_{7} - 424 \beta_{6} - 268 \beta_{5} + 424 \beta_{4} + 268 \beta_{3} + 156 \beta_{1} - 361$$ $$\nu^{9}$$ $$=$$ $$-268 \beta_{15} + 424 \beta_{14} - 716 \beta_{12} + 268 \beta_{11} - 984 \beta_{10} - 424 \beta_{9} - 1613 \beta_{7} - 1613 \beta_{6} + 984 \beta_{4} - 1881 \beta_{2} + 984 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$1613 \beta_{14} + 1613 \beta_{12} - 3021 \beta_{11} + 629 \beta_{10} + 3021 \beta_{9} - 3021 \beta_{8} - 3021 \beta_{7} - 1881 \beta_{6} - 1881 \beta_{5} - 2510 \beta_{2} + 1881 \beta_{1} - 1881$$ $$\nu^{11}$$ $$=$$ $$-8907 \beta_{15} - 5145 \beta_{14} - 7026 \beta_{13} + 3021 \beta_{12} - 12557 \beta_{11} + 1881 \beta_{10} + 4391 \beta_{9} - 11417 \beta_{8} - 11417 \beta_{5} + 4391 \beta_{3} + 7026 \beta_{2} - 13298$$ $$\nu^{12}$$ $$=$$ $$-32881 \beta_{15} - 21464 \beta_{14} - 21464 \beta_{13} - 21464 \beta_{12} - 13298 \beta_{11} - 21464 \beta_{10} - 15192 \beta_{9} - 13298 \beta_{8} - 13298 \beta_{6} - 21464 \beta_{5} + 21464 \beta_{4} + 13298 \beta_{3} + 8166 \beta_{2} - 28490$$ $$\nu^{13}$$ $$=$$ $$-21464 \beta_{15} + 13298 \beta_{14} - 80941 \beta_{12} + 8166 \beta_{11} - 89107 \beta_{10} - 21464 \beta_{9} - 49954 \beta_{7} - 80941 \beta_{6} + 80941 \beta_{4} - 94239 \beta_{2} + 30987 \beta_{1} - 8166$$ $$\nu^{14}$$ $$=$$ $$94239 \beta_{15} + 175180 \beta_{14} + 94239 \beta_{13} - 44285 \beta_{12} - 94239 \beta_{11} - 94239 \beta_{10} + 188478 \beta_{9} - 94239 \beta_{8} - 152359 \beta_{7} - 152359 \beta_{6} + 94239 \beta_{4} - 58120 \beta_{3} - 296552 \beta_{2} + 94239 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$152359 \beta_{14} + 152359 \beta_{12} - 668376 \beta_{11} + 58120 \beta_{10} + 668376 \beta_{9} - 574137 \beta_{8} - 354672 \beta_{5} - 58120 \beta_{2} - 412792$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 - \beta_{9} + \beta_{11} - \beta_{15}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −0.370800 − 0.0587290i 2.63087 + 0.416689i −0.0587290 + 0.370800i 0.416689 − 2.63087i −0.693851 − 1.36176i 0.297049 + 0.582991i 1.36176 − 0.693851i −0.582991 + 0.297049i −0.693851 + 1.36176i 0.297049 − 0.582991i 1.36176 + 0.693851i −0.582991 − 0.297049i −0.370800 + 0.0587290i 2.63087 − 0.416689i −0.0587290 − 0.370800i 0.416689 + 2.63087i
−0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 0.898602 + 2.04756i −1.00000 −2.04378 0.664066i 0.951057 0.309017i −0.309017 0.951057i 1.12833 1.93051i
109.2 −0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 1.71943 1.42953i −1.00000 2.04378 + 0.664066i 0.951057 0.309017i −0.309017 0.951057i −2.16717 0.550794i
109.3 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 0.898602 2.04756i −1.00000 2.04378 + 0.664066i −0.951057 + 0.309017i −0.309017 0.951057i 2.18470 0.476543i
109.4 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.71943 + 1.42953i −1.00000 −2.04378 0.664066i −0.951057 + 0.309017i −0.309017 0.951057i −0.145857 + 2.23131i
349.1 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i −1.82930 + 1.28594i −1.00000 0.907165 1.24861i −0.587785 0.809017i 0.809017 0.587785i 2.13715 0.657718i
349.2 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i 2.21127 + 0.332092i −1.00000 −0.907165 + 1.24861i −0.587785 0.809017i 0.809017 0.587785i −2.00042 0.999158i
349.3 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −1.82930 1.28594i −1.00000 −0.907165 + 1.24861i 0.587785 + 0.809017i 0.809017 0.587785i −1.34239 1.78829i
349.4 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i 2.21127 0.332092i −1.00000 0.907165 1.24861i 0.587785 + 0.809017i 0.809017 0.587785i 2.20566 + 0.367482i
469.1 −0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i −1.82930 1.28594i −1.00000 0.907165 + 1.24861i −0.587785 + 0.809017i 0.809017 + 0.587785i 2.13715 + 0.657718i
469.2 −0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i 2.21127 0.332092i −1.00000 −0.907165 1.24861i −0.587785 + 0.809017i 0.809017 + 0.587785i −2.00042 + 0.999158i
469.3 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i −1.82930 + 1.28594i −1.00000 −0.907165 1.24861i 0.587785 0.809017i 0.809017 + 0.587785i −1.34239 + 1.78829i
469.4 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i 2.21127 + 0.332092i −1.00000 0.907165 + 1.24861i 0.587785 0.809017i 0.809017 + 0.587785i 2.20566 0.367482i
529.1 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 0.898602 2.04756i −1.00000 −2.04378 + 0.664066i 0.951057 + 0.309017i −0.309017 + 0.951057i 1.12833 + 1.93051i
529.2 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 1.71943 + 1.42953i −1.00000 2.04378 0.664066i 0.951057 + 0.309017i −0.309017 + 0.951057i −2.16717 + 0.550794i
529.3 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 0.898602 + 2.04756i −1.00000 2.04378 0.664066i −0.951057 0.309017i −0.309017 + 0.951057i 2.18470 + 0.476543i
529.4 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 1.71943 1.42953i −1.00000 −2.04378 + 0.664066i −0.951057 0.309017i −0.309017 + 0.951057i −0.145857 2.23131i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.z.b 16
5.b even 2 1 inner 930.2.z.b 16
31.d even 5 1 inner 930.2.z.b 16
155.n even 10 1 inner 930.2.z.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.z.b 16 1.a even 1 1 trivial
930.2.z.b 16 5.b even 2 1 inner
930.2.z.b 16 31.d even 5 1 inner
930.2.z.b 16 155.n even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 6 T_{7}^{6} + 16 T_{7}^{4} - 11 T_{7}^{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$5$ $$( 625 - 750 T + 350 T^{2} - 50 T^{3} - 10 T^{4} - 10 T^{5} + 14 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$7$ $$( 121 - 11 T^{2} + 16 T^{4} - 6 T^{6} + T^{8} )^{2}$$
$11$ $$( 16 + 32 T + 72 T^{2} + 40 T^{3} - 4 T^{4} - 20 T^{5} + 18 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$13$ $$( 1 - 11 T^{2} + 46 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$17$ $$40960000 + 8192000 T^{2} + 24166400 T^{4} - 8632320 T^{6} + 1220096 T^{8} - 28416 T^{10} + 1616 T^{12} - 56 T^{14} + T^{16}$$
$19$ $$( 93025 - 42700 T + 10365 T^{2} - 200 T^{3} + 276 T^{4} - 4 T^{5} + 46 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$23$ $$( 160000 - 8000 T^{2} + 400 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$29$ $$( 13456 - 16704 T + 30968 T^{2} - 11520 T^{3} + 3016 T^{4} - 720 T^{5} + 152 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$31$ $$( 923521 - 595820 T + 193161 T^{2} - 50530 T^{3} + 10671 T^{4} - 1630 T^{5} + 201 T^{6} - 20 T^{7} + T^{8} )^{2}$$
$37$ $$( 21025 + 8510 T^{2} + 1171 T^{4} + 62 T^{6} + T^{8} )^{2}$$
$41$ $$( 400 - 2000 T + 4240 T^{2} - 2960 T^{3} + 1776 T^{4} - 536 T^{5} + 126 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$43$ $$324928500625 + 121118912000 T^{2} + 137027261900 T^{4} - 4867118670 T^{6} + 93853646 T^{8} - 1370316 T^{10} + 21041 T^{12} - 206 T^{14} + T^{16}$$
$47$ $$83181696160000 + 3023595008000 T^{2} + 818244478400 T^{4} - 25852012480 T^{6} + 384032336 T^{8} - 3482864 T^{10} + 30236 T^{12} - 224 T^{14} + T^{16}$$
$53$ $$100000000 - 164000000 T^{2} + 112760000 T^{4} - 26702400 T^{6} + 2492816 T^{8} + 55344 T^{10} + 4076 T^{12} + 44 T^{14} + T^{16}$$
$59$ $$( 839056 + 54960 T + 38096 T^{2} + 9840 T^{3} + 2416 T^{4} - 600 T^{5} + 86 T^{6} + T^{8} )^{2}$$
$61$ $$( 1525 - 470 T - 161 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$67$ $$( 201601 + 75566 T^{2} + 7791 T^{4} + 166 T^{6} + T^{8} )^{2}$$
$71$ $$( 336400 + 174000 T + 62760 T^{2} + 20800 T^{3} + 6456 T^{4} + 1036 T^{5} + 96 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$73$ $$19858796855041 - 9405110033710 T^{2} + 1680509696781 T^{4} + 18811069780 T^{6} + 1043156246 T^{8} - 165230 T^{10} + 48036 T^{12} - 340 T^{14} + T^{16}$$
$79$ $$( 36481 - 14134 T + 19542 T^{2} + 928 T^{3} + 480 T^{4} - 68 T^{5} + 7 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$83$ $$2342560000 - 1320352000 T^{2} + 287302400 T^{4} + 601920 T^{6} + 1648736 T^{8} - 30624 T^{10} + 1316 T^{12} - 44 T^{14} + T^{16}$$
$89$ $$( 16 + 80 T + 216 T^{2} + 320 T^{3} + 336 T^{4} + 60 T^{5} + 36 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$97$ $$6575592154500625 - 661206442049500 T^{2} + 27563427955900 T^{4} - 348402348430 T^{6} + 1928503566 T^{8} - 2345564 T^{10} + 107061 T^{12} + 146 T^{14} + T^{16}$$