# Properties

 Label 190.2.k.d Level $190$ Weight $2$ Character orbit 190.k Analytic conductor $1.517$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.k (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 24 x^{16} - 12 x^{15} + 393 x^{14} - 222 x^{13} + 3518 x^{12} - 2478 x^{11} + 22809 x^{10} - 12862 x^{9} + 77397 x^{8} - 24822 x^{7} + 178501 x^{6} - 39408 x^{5} + 132588 x^{4} + 45584 x^{3} + 50640 x^{2} + 1824 x + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{2} -\beta_{7} q^{3} -\beta_{11} q^{4} + \beta_{10} q^{5} + \beta_{5} q^{6} + \beta_{17} q^{7} + \beta_{14} q^{8} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{2} -\beta_{7} q^{3} -\beta_{11} q^{4} + \beta_{10} q^{5} + \beta_{5} q^{6} + \beta_{17} q^{7} + \beta_{14} q^{8} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} + \beta_{8} q^{10} + ( -\beta_{2} + \beta_{3} - \beta_{12} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{11} + \beta_{3} q^{12} + ( \beta_{1} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} - \beta_{16} ) q^{13} -\beta_{15} q^{14} + ( -\beta_{2} + \beta_{5} ) q^{15} + ( -\beta_{9} - \beta_{10} ) q^{16} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{16} ) q^{17} + ( -2 - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{18} + ( \beta_{2} + \beta_{3} + \beta_{6} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{19} + q^{20} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{11} + \beta_{13} + 3 \beta_{14} - \beta_{17} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} ) q^{22} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{10} - 2 \beta_{11} - \beta_{14} ) q^{23} + ( -\beta_{4} - \beta_{7} ) q^{24} + ( \beta_{8} + \beta_{11} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{14} - \beta_{17} ) q^{26} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{14} ) q^{27} + \beta_{13} q^{28} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{13} + \beta_{14} ) q^{29} -\beta_{1} q^{30} + ( -\beta_{3} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} - \beta_{17} ) q^{31} + ( -\beta_{8} - \beta_{11} ) q^{32} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + 3 \beta_{8} - \beta_{10} + 4 \beta_{11} + \beta_{13} + \beta_{14} + \beta_{17} ) q^{33} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{34} + ( -\beta_{12} - \beta_{15} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{16} ) q^{36} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{37} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{38} + ( 3 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} + \beta_{16} ) q^{39} -\beta_{9} q^{40} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} ) q^{41} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{42} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{14} ) q^{43} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{15} + \beta_{17} ) q^{44} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{45} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{14} ) q^{46} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - 3 \beta_{10} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{47} + \beta_{2} q^{48} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{8} - 4 \beta_{9} - \beta_{10} - 4 \beta_{11} + \beta_{13} - 4 \beta_{14} + \beta_{15} + \beta_{17} ) q^{49} + ( 1 - \beta_{14} ) q^{50} + ( 3 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 5 \beta_{8} + 3 \beta_{9} + \beta_{10} + 5 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{52} + ( \beta_{3} + \beta_{4} + \beta_{7} + 2 \beta_{9} - 2 \beta_{14} + \beta_{16} ) q^{53} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{14} ) q^{54} + ( \beta_{4} - \beta_{6} + \beta_{9} + \beta_{16} ) q^{55} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{56} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{12} - 3 \beta_{14} + \beta_{16} ) q^{57} + ( -\beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{58} + ( -3 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 3 \beta_{11} - \beta_{13} + 3 \beta_{14} + \beta_{17} ) q^{59} -\beta_{7} q^{60} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{61} + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{62} + ( -5 - 3 \beta_{1} - \beta_{3} - 3 \beta_{4} - 2 \beta_{7} - \beta_{8} - 5 \beta_{9} - 4 \beta_{10} - \beta_{11} + \beta_{12} + 4 \beta_{14} + \beta_{15} - \beta_{17} ) q^{63} + ( -1 + \beta_{14} ) q^{64} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{14} - \beta_{17} ) q^{65} + ( 3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - 4 \beta_{9} - \beta_{10} - 4 \beta_{14} - \beta_{15} ) q^{66} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} - 2 \beta_{13} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{68} + ( -4 + 3 \beta_{3} + 5 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} + 4 \beta_{14} - \beta_{15} + \beta_{17} ) q^{69} + ( \beta_{3} + \beta_{4} + \beta_{7} + \beta_{13} + \beta_{16} ) q^{70} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{10} + 4 \beta_{11} - 2 \beta_{13} + 4 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{71} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{72} + ( -1 - \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} + \beta_{12} + 3 \beta_{14} + \beta_{15} ) q^{73} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{14} - \beta_{16} ) q^{74} + ( -\beta_{1} - \beta_{3} ) q^{75} + ( -1 + \beta_{2} - \beta_{4} + \beta_{9} + \beta_{10} - \beta_{14} - \beta_{16} ) q^{76} + ( 1 + 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - 2 \beta_{16} ) q^{77} + ( -2 - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 3 \beta_{9} + 2 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{78} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} ) q^{79} -\beta_{11} q^{80} + ( 4 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} + \beta_{6} + 4 \beta_{7} + 4 \beta_{8} + 7 \beta_{10} - 3 \beta_{11} + \beta_{13} - 7 \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{81} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{82} + ( -2 \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} + 2 \beta_{16} ) q^{83} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{16} ) q^{84} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{85} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} ) q^{86} + ( 2 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - 4 \beta_{8} - 7 \beta_{9} - 3 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} - 3 \beta_{16} - 2 \beta_{17} ) q^{87} + ( 1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{88} + ( 4 + \beta_{1} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 5 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{89} + ( -\beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{17} ) q^{90} + ( -4 - \beta_{1} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 5 \beta_{7} + 3 \beta_{9} + 4 \beta_{10} - 7 \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{91} + ( -1 + \beta_{1} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{92} + ( 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} - \beta_{12} - \beta_{14} ) q^{93} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{94} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{15} ) q^{95} + ( \beta_{1} + \beta_{3} ) q^{96} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} + 4 \beta_{14} + \beta_{16} + \beta_{17} ) q^{97} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - \beta_{12} - \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{98} + ( 8 - 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} - 9 \beta_{9} - 8 \beta_{10} - 5 \beta_{11} + 2 \beta_{12} + \beta_{14} + \beta_{15} - 3 \beta_{16} - \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q + 9q^{8} - 18q^{9} + O(q^{10})$$ $$18q + 9q^{8} - 18q^{9} - 12q^{11} - 6q^{13} + 6q^{14} - 42q^{18} + 18q^{20} + 12q^{21} - 3q^{22} + 9q^{23} - 9q^{26} - 18q^{27} + 3q^{28} - 6q^{29} - 6q^{31} + 66q^{33} + 18q^{34} + 3q^{35} - 18q^{36} - 12q^{37} - 6q^{38} + 48q^{39} - 21q^{41} + 42q^{42} + 18q^{43} + 3q^{44} - 21q^{45} + 18q^{46} - 54q^{47} - 39q^{49} + 9q^{50} + 42q^{51} + 12q^{52} - 24q^{53} - 54q^{54} - 6q^{55} - 18q^{57} - 30q^{59} + 48q^{61} - 30q^{62} - 57q^{63} - 9q^{64} + 9q^{65} + 24q^{66} - 6q^{67} - 6q^{68} - 30q^{69} - 3q^{70} + 30q^{71} + 6q^{73} - 3q^{74} - 21q^{76} + 30q^{77} - 24q^{78} + 30q^{79} + 18q^{81} + 21q^{82} + 6q^{83} + 6q^{84} + 36q^{86} + 24q^{87} + 12q^{88} + 30q^{89} - 60q^{91} - 18q^{92} - 12q^{93} + 6q^{94} - 12q^{95} - 12q^{97} - 18q^{98} + 171q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 24 x^{16} - 12 x^{15} + 393 x^{14} - 222 x^{13} + 3518 x^{12} - 2478 x^{11} + 22809 x^{10} - 12862 x^{9} + 77397 x^{8} - 24822 x^{7} + 178501 x^{6} - 39408 x^{5} + 132588 x^{4} + 45584 x^{3} + 50640 x^{2} + 1824 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$61055476894519055 \nu^{17} - 1032210622184162986 \nu^{16} + 4915084423072614202 \nu^{15} - 18904556982892784128 \nu^{14} + 112381522660734210447 \nu^{13} - 350448690616485240916 \nu^{12} + 1484901432675530355632 \nu^{11} - 2949055427589780269094 \nu^{10} + 11426855215867581932007 \nu^{9} - 23294803089162330524192 \nu^{8} + 51608137114388154342713 \nu^{7} - 70676811003821628624804 \nu^{6} + 103073254201189644995297 \nu^{5} - 209008512165863259372458 \nu^{4} + 131930426042750651071414 \nu^{3} - 28672748737432658095620 \nu^{2} - 39173558665841641384688 \nu - 4013751372935074787872$$$$)/$$$$96\!\cdots\!48$$ $$\beta_{3}$$ $$=$$ $$($$$$-1635525082119517 \nu^{17} + 8492194184113948 \nu^{16} - 47792711775522575 \nu^{15} + 217650206396461316 \nu^{14} - 943040222741486265 \nu^{13} + 3671677898458245414 \nu^{12} - 10792835135312213281 \nu^{11} + 33637103678766889280 \nu^{10} - 85117577577455380283 \nu^{9} + 217013214072968345884 \nu^{8} - 402842204138034112340 \nu^{7} + 685066214798428861084 \nu^{6} - 1016909551599789245164 \nu^{5} + 1436505471983684001382 \nu^{4} - 1946539230130393642719 \nu^{3} + 681534385747795628040 \nu^{2} - 922419069697801510100 \nu - 33245472400761016104$$$$)/$$$$94\!\cdots\!24$$ $$\beta_{4}$$ $$=$$ $$($$$$-484588919345981797 \nu^{17} + 1575121051092986129 \nu^{16} - 9231168018118130678 \nu^{15} + 43796023402305942728 \nu^{14} - 168551952326384017361 \nu^{13} + 660135544909627110215 \nu^{12} - 1520751922853346424722 \nu^{11} + 5723373133855547047852 \nu^{10} - 12176681046920356722183 \nu^{9} + 29911238133086072202907 \nu^{8} - 47162212142358743450181 \nu^{7} + 82059566489390723167467 \nu^{6} - 139217061034678719720477 \nu^{5} + 130159014359546996203917 \nu^{4} - 175059908224639016547602 \nu^{3} + 132376946470993401358768 \nu^{2} - 33369020667095817117064 \nu + 2772471793864217551328$$$$)/$$$$96\!\cdots\!48$$ $$\beta_{5}$$ $$=$$ $$($$$$-5661108507277223089 \nu^{17} - 4938169803754683634 \nu^{16} - 133487416594392925596 \nu^{15} - 32830764928543916644 \nu^{14} - 2082011278615893963761 \nu^{13} - 318497834185614045780 \nu^{12} - 17688882567599612210342 \nu^{11} + 1665355671942828966634 \nu^{10} - 106643915322626171007461 \nu^{9} - 4768857048822705961004 \nu^{8} - 333131497615781151769293 \nu^{7} - 77028336764435244193452 \nu^{6} - 749894406880936988520413 \nu^{5} - 310172864630406771852114 \nu^{4} - 453225373012109516715344 \nu^{3} - 526814998545193841166088 \nu^{2} - 238709092089344630833744 \nu - 5849427054831566382272$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{6}$$ $$=$$ $$($$$$7061265118326284208 \nu^{17} - 20833553879539973263 \nu^{16} + 184947509852821672932 \nu^{15} - 575980991439557555464 \nu^{14} + 3393850177833466542684 \nu^{13} - 9738228420375474446487 \nu^{12} + 35349795625809934833718 \nu^{11} - 90921419739998124188058 \nu^{10} + 263573055319895541667746 \nu^{9} - 575412151238504149823551 \nu^{8} + 1127795925739463857267382 \nu^{7} - 1799505239708367848883843 \nu^{6} + 2740238653231538074321206 \nu^{5} - 3780284333044025208230387 \nu^{4} + 3751066226442860586590316 \nu^{3} - 1801940266177000524862724 \nu^{2} - 64945999769484800222216 \nu - 355065319936750139466336$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{7}$$ $$=$$ $$($$$$7120232064478913845 \nu^{17} - 4420670090951377916 \nu^{16} + 170096713436186769408 \nu^{15} - 176518448897114086940 \nu^{14} + 2868563022147134292157 \nu^{13} - 2951932328229000646866 \nu^{12} + 26212098261657925295878 \nu^{11} - 28101931376822263136126 \nu^{10} + 176757439950055842379637 \nu^{9} - 152936054047063759261762 \nu^{8} + 614538124237944153099593 \nu^{7} - 340630891724770170933642 \nu^{6} + 1442161398472860279229561 \nu^{5} - 668098226145442356951020 \nu^{4} + 1055761357623463776830468 \nu^{3} + 63550914205853885646816 \nu^{2} + 222012288383157382339584 \nu + 2790843091943609249792$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{8}$$ $$=$$ $$($$$$34032432672477862398 \nu^{17} + 5783219461066261199 \nu^{16} + 819652132698855055214 \nu^{15} - 265071606629196194776 \nu^{14} + 13369767691246558270802 \nu^{13} - 5248425729352723067701 \nu^{12} + 119343698594729763480112 \nu^{11} - 63673682729449470100638 \nu^{10} + 768682290299425173931160 \nu^{9} - 308227523279048931291601 \nu^{8} + 2592187442422267160930626 \nu^{7} - 408405271951688039988437 \nu^{6} + 6010497979226762902849242 \nu^{5} - 385109191473691322869377 \nu^{4} + 4404448023477175072733222 \nu^{3} + 2268420636039841698408604 \nu^{2} + 2192871891604607476809568 \nu + 222437131952260969078320$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{9}$$ $$=$$ $$($$$$43606923311618894528 \nu^{17} - 7120232064478913845 \nu^{16} + 1050986829569804846588 \nu^{15} - 693379793175613503744 \nu^{14} + 17314039310363339636444 \nu^{13} - 12549299997326528877373 \nu^{12} + 156361088538504271596370 \nu^{11} - 134270054227849545936262 \nu^{10} + 1022732245191537628425278 \nu^{9} - 737629687584098063798773 \nu^{8} + 3527981097596431339045378 \nu^{7} - 1696949174678948353073609 \nu^{6} + 8124510309772054463076170 \nu^{5} - 3160623032337137674788985 \nu^{4} + 6449852974186368344629484 \nu^{3} + 932016634613371911333884 \nu^{2} + 2144703682294526933251104 \nu - 142473260262764518720512$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{10}$$ $$=$$ $$($$$$86639743558256798479 \nu^{17} + 969177838691963594 \nu^{16} + 2076203603295977191238 \nu^{15} - 1021214586662845320392 \nu^{14} + 33961827171590309916791 \nu^{13} - 18896919165280241227616 \nu^{12} + 303478346748128162828692 \nu^{11} - 211651780691653653781518 \nu^{10} + 1964719164552568222411807 \nu^{9} - 1090007019552458228592532 \nu^{8} + 6645833755912229287473349 \nu^{7} - 2056247290318332764945376 \nu^{6} + 15301161731913615338965045 \nu^{5} - 3135864892074426475019478 \nu^{4} + 11227072290183058404325818 \nu^{3} + 4299505886808855934961940 \nu^{2} + 4122682720848137472259024 \nu + 224768933584452034659824$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{11}$$ $$=$$ $$($$$$91397297731743224723 \nu^{17} - 5661108507277223089 \nu^{16} + 2188596975758082709718 \nu^{15} - 1230254989375311622272 \nu^{14} + 35886307243646543399495 \nu^{13} - 22372211375062889852267 \nu^{12} + 321217195586087050529734 \nu^{11} - 244171386346859323073936 \nu^{10} + 2086346319635274041673541 \nu^{9} - 1282195958748307527394687 \nu^{8} + 7069107795494907657925027 \nu^{7} - 2601795221913111475843599 \nu^{6} + 16237480705649462112086771 \nu^{5} - 4351679115893473988404397 \nu^{4} + 11808012047025963907721010 \nu^{3} + 3713029046791673639057888 \nu^{2} + 4101544158590283058806632 \nu - 72000421026644988938992$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{12}$$ $$=$$ $$($$$$193672599939521177853 \nu^{17} - 5732135503686535546 \nu^{16} + 4605238891985950874212 \nu^{15} - 2484012542884763395636 \nu^{14} + 75166040667040660085229 \nu^{13} - 45094180832392128248256 \nu^{12} + 666784181932829688456702 \nu^{11} - 495156481581381846753170 \nu^{10} + 4297343729671549054262025 \nu^{9} - 2535845560255705164827176 \nu^{8} + 14267065099980243521149289 \nu^{7} - 4758623882188580338713104 \nu^{6} + 32332620281699746077039593 \nu^{5} - 7255886801588970805548322 \nu^{4} + 21458301331246703013753752 \nu^{3} + 10715920057874028174216912 \nu^{2} + 8144725479928444094141072 \nu - 513933057752306138600608$$$$)/$$$$38\!\cdots\!92$$ $$\beta_{13}$$ $$=$$ $$($$$$-196879019602708047121 \nu^{17} + 41797874796424201674 \nu^{16} - 4713042519230969592112 \nu^{15} + 3325727502525616837540 \nu^{14} - 77687986565802063558833 \nu^{13} + 58905874877127585165912 \nu^{12} - 698878921204621736144290 \nu^{11} + 616027846141178933962418 \nu^{10} - 4574099732833245797511669 \nu^{9} + 3312354529905557627095512 \nu^{8} - 15629755679129483461804601 \nu^{7} + 7186801520235321916088000 \nu^{6} - 35867074634072018735552729 \nu^{5} + 12348013541628593469174234 \nu^{4} - 26950980763744225590493364 \nu^{3} - 8310250267956919690057008 \nu^{2} - 8058021630556541263813456 \nu - 816875741882952030207008$$$$)/$$$$38\!\cdots\!92$$ $$\beta_{14}$$ $$=$$ $$($$$$-4155684050095127013 \nu^{17} + 13084200656956136 \nu^{16} - 99804354755755959896 \nu^{15} + 50250550295345704756 \nu^{14} - 1634925033338556606637 \nu^{13} + 930106180903050087006 \nu^{12} - 14649069911422322795046 \nu^{11} + 10384127757218222444462 \nu^{10} - 95056094328049887153757 \nu^{9} + 54131348872943166683470 \nu^{8} - 323373584137796292192233 \nu^{7} + 106375127124565515615406 \nu^{6} - 747274288344417697836185 \nu^{5} + 171902473458947079289616 \nu^{4} - 562485880609882172410700 \nu^{3} - 173860387898493120618840 \nu^{2} - 215896115382799596962640 \nu - 200615149791099590912$$$$)/$$$$75\!\cdots\!92$$ $$\beta_{15}$$ $$=$$ $$($$$$110093466354707483329 \nu^{17} + 10008211751375647830 \nu^{16} + 2654952287017311307004 \nu^{15} - 1067025603899319954224 \nu^{14} + 43490512848531584937865 \nu^{13} - 20324507300104335553752 \nu^{12} + 390642832923762531587086 \nu^{11} - 235904148869076628375654 \nu^{10} + 2538140580344124077554525 \nu^{9} - 1185937822770602685414904 \nu^{8} + 8712743369675705111656909 \nu^{7} - 1991405539095669544498228 \nu^{6} + 20525886315052473031489621 \nu^{5} - 2687710055767633239051438 \nu^{4} + 16455672097237555401464960 \nu^{3} + 5546166004746450996157332 \nu^{2} + 7489829416403767424029208 \nu + 410114541020255785394768$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{16}$$ $$=$$ $$($$$$-139103439183612193370 \nu^{17} + 3569615837665383383 \nu^{16} - 3341164204081251539494 \nu^{15} + 1774311505425595941180 \nu^{14} - 54752703948710247578774 \nu^{13} + 32806607694558362168483 \nu^{12} - 490961877766393435114708 \nu^{11} + 365463633294627112537294 \nu^{10} - 3188586085671514423360464 \nu^{9} + 1943576513197228318661575 \nu^{8} - 10871087134867573888073106 \nu^{7} + 4147066791257029612362559 \nu^{6} - 25131730767251814404780330 \nu^{5} + 7356693191530839316897827 \nu^{4} - 19032353644027934743901478 \nu^{3} - 3796235744326268436909968 \nu^{2} - 7377010696928142057337552 \nu + 135164459785755475025824$$$$)/$$$$19\!\cdots\!96$$ $$\beta_{17}$$ $$=$$ $$($$$$-228822961465924017141 \nu^{17} - 17069457929318511484 \nu^{16} - 5491645746440817469064 \nu^{15} + 2345435923324846735080 \nu^{14} - 89633278539035744001325 \nu^{13} + 44459001009776990959914 \nu^{12} - 799294910113502776880286 \nu^{11} + 512085311367039951941786 \nu^{10} - 5150625931349929178523373 \nu^{9} + 2603118085785120372196338 \nu^{8} - 17299262371659382705624745 \nu^{7} + 4573282646355888572083006 \nu^{6} - 39519550598257032163614057 \nu^{5} + 6754863310787280622560224 \nu^{4} - 27628239760151592570775604 \nu^{3} - 11990310892724571422529252 \nu^{2} - 10254485610108298199954736 \nu - 369321165030132937154528$$$$)/$$$$19\!\cdots\!96$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{17} - \beta_{16} + \beta_{15} + 5 \beta_{14} - \beta_{13} - \beta_{11} + \beta_{9} - \beta_{4} - \beta_{3} - \beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{11} - 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 8 \beta_{3} - \beta_{2} - 8 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$11 \beta_{17} + 10 \beta_{16} - 9 \beta_{15} - 43 \beta_{14} + \beta_{13} - 10 \beta_{12} + 4 \beta_{11} + 16 \beta_{10} + 4 \beta_{9} - 12 \beta_{8} - 11 \beta_{6} + \beta_{5} + 23 \beta_{4} + 10 \beta_{3} + \beta_{2} + 2 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{17} - 2 \beta_{15} + 38 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 24 \beta_{11} + 34 \beta_{10} + 58 \beta_{9} + 34 \beta_{8} - 12 \beta_{7} + 31 \beta_{5} - 31 \beta_{4} + 76 \beta_{3} - 19 \beta_{2} - 38$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{16} - 16 \beta_{15} + 86 \beta_{13} + 86 \beta_{12} + 147 \beta_{11} - 147 \beta_{10} - 194 \beta_{9} + 194 \beta_{8} - 52 \beta_{7} + 115 \beta_{6} - 36 \beta_{5} - 155 \beta_{4} - 57 \beta_{3} + 139 \beta_{2} - 41 \beta_{1} + 420$$ $$\nu^{7}$$ $$=$$ $$32 \beta_{17} - 36 \beta_{16} - 4 \beta_{15} - 534 \beta_{14} - 40 \beta_{13} + 36 \beta_{12} - 460 \beta_{11} + 264 \beta_{10} - 460 \beta_{9} - 724 \beta_{8} + 310 \beta_{7} - 32 \beta_{6} - 141 \beta_{5} + 233 \beta_{4} - 36 \beta_{3} + 378 \beta_{2} + 783 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1194 \beta_{17} - 856 \beta_{16} + 1037 \beta_{15} + 4366 \beta_{14} - 1037 \beta_{13} + 181 \beta_{12} - 2195 \beta_{11} - 388 \beta_{10} + 1807 \beta_{9} - 388 \beta_{8} + 733 \beta_{7} + 143 \beta_{5} - 999 \beta_{4} - 226 \beta_{3} - 1732 \beta_{2} - 4366$$ $$\nu^{9}$$ $$=$$ $$576 \beta_{16} + 576 \beta_{15} + 78 \beta_{13} + 78 \beta_{12} + 8696 \beta_{11} - 8696 \beta_{10} - 2938 \beta_{9} + 2938 \beta_{8} - 2516 \beta_{7} + 402 \beta_{6} - 3092 \beta_{5} + 1385 \beta_{4} - 7810 \beta_{3} - 809 \beta_{2} - 8386 \beta_{1} + 6786$$ $$\nu^{10}$$ $$=$$ $$12424 \beta_{17} + 10484 \beta_{16} - 8697 \beta_{15} - 46925 \beta_{14} + 1787 \beta_{13} - 10484 \beta_{12} + 2087 \beta_{11} + 24278 \beta_{10} + 2087 \beta_{9} - 22191 \beta_{8} - 63 \beta_{7} - 12424 \beta_{6} + 5621 \beta_{5} + 30406 \beta_{4} + 10484 \beta_{3} + 1877 \beta_{2} + 8649 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-4752 \beta_{17} - 1088 \beta_{16} - 6320 \beta_{15} + 82676 \beta_{14} + 6320 \beta_{13} - 7408 \beta_{12} - 33398 \beta_{11} + 69692 \beta_{10} + 103090 \beta_{9} + 69692 \beta_{8} - 5073 \beta_{7} + 47837 \beta_{5} - 48925 \beta_{4} + 90561 \beta_{3} - 43852 \beta_{2} - 82676$$ $$\nu^{12}$$ $$=$$ $$-16145 \beta_{16} - 16145 \beta_{15} + 89314 \beta_{13} + 89314 \beta_{12} + 271532 \beta_{11} - 271532 \beta_{10} - 266693 \beta_{9} + 266693 \beta_{8} - 116497 \beta_{7} + 129743 \beta_{6} - 100352 \beta_{5} - 220571 \beta_{4} - 128704 \beta_{3} + 204426 \beta_{2} - 112559 \beta_{1} + 513982$$ $$\nu^{13}$$ $$=$$ $$55406 \beta_{17} - 77236 \beta_{16} - 13592 \beta_{15} - 988598 \beta_{14} - 90828 \beta_{13} + 77236 \beta_{12} - 830256 \beta_{11} + 385724 \beta_{10} - 830256 \beta_{9} - 1215980 \beta_{8} + 395051 \beta_{7} - 55406 \beta_{6} - 42082 \beta_{5} + 463781 \beta_{4} - 77236 \beta_{3} + 527693 \beta_{2} + 1013533 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-1359838 \beta_{17} - 922973 \beta_{16} + 1055883 \beta_{15} + 5695827 \beta_{14} - 1055883 \beta_{13} + 132910 \beta_{12} - 2928019 \beta_{11} + 378813 \beta_{10} + 3306832 \beta_{9} + 378813 \beta_{8} + 1393103 \beta_{7} + 350034 \beta_{5} - 1273007 \beta_{4} + 500916 \beta_{3} - 2666110 \beta_{2} - 5695827$$ $$\nu^{15}$$ $$=$$ $$1089148 \beta_{16} + 1089148 \beta_{15} + 163696 \beta_{13} + 163696 \beta_{12} + 14312704 \beta_{11} - 14312704 \beta_{10} - 4498926 \beta_{9} + 4498926 \beta_{8} - 4870980 \beta_{7} + 646640 \beta_{6} - 5960128 \beta_{5} + 296250 \beta_{4} - 10207883 \beta_{3} + 792898 \beta_{2} - 11297031 \beta_{1} + 11723112$$ $$\nu^{16}$$ $$=$$ $$14303403 \beta_{17} + 10521571 \beta_{16} - 9578681 \beta_{15} - 63623749 \beta_{14} + 942890 \beta_{13} - 10521571 \beta_{12} - 7861438 \beta_{11} + 32218663 \beta_{10} - 7861438 \beta_{9} - 40080101 \beta_{8} + 903788 \beta_{7} - 14303403 \beta_{6} + 11974636 \beta_{5} + 41485230 \beta_{4} + 10521571 \beta_{3} + 4685620 \beta_{2} + 17714382 \beta_{1}$$ $$\nu^{17}$$ $$=$$ $$-7586424 \beta_{17} - 1957914 \beta_{16} - 10959612 \beta_{15} + 138514320 \beta_{14} + 10959612 \beta_{13} - 12917526 \beta_{12} - 52775282 \beta_{11} + 115532576 \beta_{10} + 168307858 \beta_{9} + 115532576 \beta_{8} + 14000123 \beta_{7} + 64164314 \beta_{5} - 66122228 \beta_{4} + 124704382 \beta_{3} - 80122351 \beta_{2} - 138514320$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/190\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$-\beta_{11}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 1.60792 + 2.78500i −0.0180720 − 0.0313015i −1.58985 − 2.75370i 1.60792 − 2.78500i −0.0180720 + 0.0313015i −1.58985 + 2.75370i −0.897451 − 1.55443i −0.288205 − 0.499186i 1.18566 + 2.05362i −0.897451 + 1.55443i −0.288205 + 0.499186i 1.18566 − 2.05362i 1.15990 − 2.00901i 0.554587 − 0.960572i −1.71449 + 2.96958i 1.15990 + 2.00901i 0.554587 + 0.960572i −1.71449 − 2.96958i
0.939693 + 0.342020i −0.558424 3.16698i 0.766044 + 0.642788i 0.766044 0.642788i 0.558424 3.16698i 0.0116976 0.0202608i 0.500000 + 0.866025i −6.89886 + 2.51098i 0.939693 0.342020i
61.2 0.939693 + 0.342020i 0.00627632 + 0.0355948i 0.766044 + 0.642788i 0.766044 0.642788i −0.00627632 + 0.0355948i −0.918706 + 1.59124i 0.500000 + 0.866025i 2.81785 1.02561i 0.939693 0.342020i
61.3 0.939693 + 0.342020i 0.552148 + 3.13139i 0.766044 + 0.642788i 0.766044 0.642788i −0.552148 + 3.13139i 1.67305 2.89781i 0.500000 + 0.866025i −6.68164 + 2.43192i 0.939693 0.342020i
81.1 0.939693 0.342020i −0.558424 + 3.16698i 0.766044 0.642788i 0.766044 + 0.642788i 0.558424 + 3.16698i 0.0116976 + 0.0202608i 0.500000 0.866025i −6.89886 2.51098i 0.939693 + 0.342020i
81.2 0.939693 0.342020i 0.00627632 0.0355948i 0.766044 0.642788i 0.766044 + 0.642788i −0.00627632 0.0355948i −0.918706 1.59124i 0.500000 0.866025i 2.81785 + 1.02561i 0.939693 + 0.342020i
81.3 0.939693 0.342020i 0.552148 3.13139i 0.766044 0.642788i 0.766044 + 0.642788i −0.552148 3.13139i 1.67305 + 2.89781i 0.500000 0.866025i −6.68164 2.43192i 0.939693 + 0.342020i
101.1 −0.766044 + 0.642788i −1.68666 0.613893i 0.173648 0.984808i 0.173648 + 0.984808i 1.68666 0.613893i −0.680736 + 1.17907i 0.500000 + 0.866025i 0.169813 + 0.142490i −0.766044 0.642788i
101.2 −0.766044 + 0.642788i −0.541649 0.197144i 0.173648 0.984808i 0.173648 + 0.984808i 0.541649 0.197144i 2.43209 4.21251i 0.500000 + 0.866025i −2.04362 1.71480i −0.766044 0.642788i
101.3 −0.766044 + 0.642788i 2.22831 + 0.811037i 0.173648 0.984808i 0.173648 + 0.984808i −2.22831 + 0.811037i −1.57771 + 2.73267i 0.500000 + 0.866025i 2.00943 + 1.68611i −0.766044 0.642788i
111.1 −0.766044 0.642788i −1.68666 + 0.613893i 0.173648 + 0.984808i 0.173648 0.984808i 1.68666 + 0.613893i −0.680736 1.17907i 0.500000 0.866025i 0.169813 0.142490i −0.766044 + 0.642788i
111.2 −0.766044 0.642788i −0.541649 + 0.197144i 0.173648 + 0.984808i 0.173648 0.984808i 0.541649 + 0.197144i 2.43209 + 4.21251i 0.500000 0.866025i −2.04362 + 1.71480i −0.766044 + 0.642788i
111.3 −0.766044 0.642788i 2.22831 0.811037i 0.173648 + 0.984808i 0.173648 0.984808i −2.22831 0.811037i −1.57771 2.73267i 0.500000 0.866025i 2.00943 1.68611i −0.766044 + 0.642788i
131.1 −0.173648 + 0.984808i −1.77707 1.49114i −0.939693 0.342020i −0.939693 + 0.342020i 1.77707 1.49114i 2.45837 + 4.25802i 0.500000 0.866025i 0.413538 + 2.34529i −0.173648 0.984808i
131.2 −0.173648 + 0.984808i −0.849676 0.712963i −0.939693 0.342020i −0.939693 + 0.342020i 0.849676 0.712963i −2.46456 4.26875i 0.500000 0.866025i −0.307311 1.74285i −0.173648 0.984808i
131.3 −0.173648 + 0.984808i 2.62675 + 2.20410i −0.939693 0.342020i −0.939693 + 0.342020i −2.62675 + 2.20410i −0.933500 1.61687i 0.500000 0.866025i 1.52078 + 8.62480i −0.173648 0.984808i
161.1 −0.173648 0.984808i −1.77707 + 1.49114i −0.939693 + 0.342020i −0.939693 0.342020i 1.77707 + 1.49114i 2.45837 4.25802i 0.500000 + 0.866025i 0.413538 2.34529i −0.173648 + 0.984808i
161.2 −0.173648 0.984808i −0.849676 + 0.712963i −0.939693 + 0.342020i −0.939693 0.342020i 0.849676 + 0.712963i −2.46456 + 4.26875i 0.500000 + 0.866025i −0.307311 + 1.74285i −0.173648 + 0.984808i
161.3 −0.173648 0.984808i 2.62675 2.20410i −0.939693 + 0.342020i −0.939693 0.342020i −2.62675 2.20410i −0.933500 + 1.61687i 0.500000 + 0.866025i 1.52078 8.62480i −0.173648 + 0.984808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.k.d 18
5.b even 2 1 950.2.l.i 18
5.c odd 4 2 950.2.u.g 36
19.e even 9 1 inner 190.2.k.d 18
19.e even 9 1 3610.2.a.bi 9
19.f odd 18 1 3610.2.a.bj 9
95.p even 18 1 950.2.l.i 18
95.q odd 36 2 950.2.u.g 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.d 18 1.a even 1 1 trivial
190.2.k.d 18 19.e even 9 1 inner
950.2.l.i 18 5.b even 2 1
950.2.l.i 18 95.p even 18 1
950.2.u.g 36 5.c odd 4 2
950.2.u.g 36 95.q odd 36 2
3610.2.a.bi 9 19.e even 9 1
3610.2.a.bj 9 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(190, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{3} + T^{6} )^{3}$$
$3$ $$64 - 288 T + 46512 T^{2} + 244544 T^{3} + 501912 T^{4} + 488322 T^{5} + 206005 T^{6} - 15876 T^{7} - 36531 T^{8} - 4636 T^{9} + 2925 T^{10} + 1422 T^{11} + 788 T^{12} + 216 T^{13} + 36 T^{14} + 6 T^{15} + 9 T^{16} + T^{18}$$
$5$ $$( 1 + T^{3} + T^{6} )^{3}$$
$7$ $$18496 - 760512 T + 32545056 T^{2} + 53371104 T^{3} + 67870608 T^{4} + 47109648 T^{5} + 28601840 T^{6} + 10883520 T^{7} + 4616220 T^{8} + 1223644 T^{9} + 514728 T^{10} + 88560 T^{11} + 35149 T^{12} + 3021 T^{13} + 1797 T^{14} + 70 T^{15} + 51 T^{16} + T^{18}$$
$11$ $$5079555441 + 6166153107 T + 6896991726 T^{2} + 3820014981 T^{3} + 2392061715 T^{4} + 972386325 T^{5} + 497942785 T^{6} + 161951652 T^{7} + 62400093 T^{8} + 15500399 T^{9} + 4942728 T^{10} + 1031922 T^{11} + 257928 T^{12} + 40383 T^{13} + 7866 T^{14} + 986 T^{15} + 153 T^{16} + 12 T^{17} + T^{18}$$
$13$ $$183439936 + 24704256 T + 385593696 T^{2} - 111797504 T^{3} + 137053344 T^{4} - 62056032 T^{5} + 18852032 T^{6} - 2466576 T^{7} + 928380 T^{8} - 411632 T^{9} + 339162 T^{10} - 111276 T^{11} - 14731 T^{12} + 9024 T^{13} + 870 T^{14} - 359 T^{15} - 36 T^{16} + 6 T^{17} + T^{18}$$
$17$ $$205291584 + 600916320 T + 673297920 T^{2} + 342974664 T^{3} + 133247412 T^{4} + 140616936 T^{5} + 161581753 T^{6} + 105016740 T^{7} + 52260096 T^{8} + 16013050 T^{9} + 3458322 T^{10} + 421620 T^{11} + 97637 T^{12} - 3000 T^{13} - 966 T^{14} + 38 T^{15} - 24 T^{16} + T^{18}$$
$19$ $$322687697779 - 53632304340 T^{2} - 893871739 T^{3} + 2867322642 T^{4} - 128235864 T^{5} - 13402486 T^{6} + 42897630 T^{7} - 1878606 T^{8} - 3550597 T^{9} - 98874 T^{10} + 118830 T^{11} - 1954 T^{12} - 984 T^{13} + 1158 T^{14} - 19 T^{15} - 60 T^{16} + T^{18}$$
$23$ $$121352256 - 249842880 T + 157883904 T^{2} - 32285952 T^{3} + 74719584 T^{4} - 127044288 T^{5} + 103453200 T^{6} - 56096064 T^{7} + 26155872 T^{8} - 8763552 T^{9} + 1444230 T^{10} + 36450 T^{11} - 53469 T^{12} + 10017 T^{13} + 333 T^{14} - 414 T^{15} + 108 T^{16} - 9 T^{17} + T^{18}$$
$29$ $$95883264 + 1153419264 T + 6025300992 T^{2} + 2434950144 T^{3} + 4696035840 T^{4} + 2556008064 T^{5} + 822659392 T^{6} + 277384512 T^{7} + 73078368 T^{8} + 3908032 T^{9} - 544032 T^{10} + 636768 T^{11} + 229440 T^{12} + 5472 T^{13} - 2232 T^{14} - 344 T^{15} - 24 T^{16} + 6 T^{17} + T^{18}$$
$31$ $$533794816 + 1335226368 T + 4155024384 T^{2} - 836384768 T^{3} + 2585081088 T^{4} + 156627840 T^{5} + 767118784 T^{6} - 34805184 T^{7} + 90750240 T^{8} + 694432 T^{9} + 6103152 T^{10} + 53376 T^{11} + 259568 T^{12} + 9816 T^{13} + 6900 T^{14} + 264 T^{15} + 120 T^{16} + 6 T^{17} + T^{18}$$
$37$ $$( 25992 + 53352 T - 44388 T^{2} - 21012 T^{3} + 12096 T^{4} + 2544 T^{5} - 743 T^{6} - 129 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$41$ $$14058877733289 - 18943156057203 T + 9396908734059 T^{2} - 1501674935508 T^{3} - 70401243042 T^{4} - 74492648343 T^{5} + 56684970541 T^{6} - 4930914552 T^{7} + 1623756402 T^{8} - 79573424 T^{9} - 14045745 T^{10} - 2924121 T^{11} + 33056 T^{12} + 71979 T^{13} + 21045 T^{14} + 2687 T^{15} + 261 T^{16} + 21 T^{17} + T^{18}$$
$43$ $$448422976 + 1623267456 T + 2091136416 T^{2} + 1115358976 T^{3} + 540221784 T^{4} + 442217118 T^{5} + 121125217 T^{6} + 9959424 T^{7} + 83210022 T^{8} + 39543538 T^{9} - 4675020 T^{10} - 1208094 T^{11} + 955259 T^{12} - 206400 T^{13} + 35676 T^{14} - 3594 T^{15} + 306 T^{16} - 18 T^{17} + T^{18}$$
$47$ $$1871424 - 24624000 T + 94019616 T^{2} + 1597652352 T^{3} + 4823560512 T^{4} + 5248199520 T^{5} + 4376119888 T^{6} + 2892259488 T^{7} + 1518788832 T^{8} + 632295280 T^{9} + 210109968 T^{10} + 55313652 T^{11} + 11406593 T^{12} + 1831962 T^{13} + 227388 T^{14} + 21023 T^{15} + 1356 T^{16} + 54 T^{17} + T^{18}$$
$53$ $$24208870464 + 50826305088 T + 56156589792 T^{2} + 41073673824 T^{3} + 21349221552 T^{4} + 7761104784 T^{5} + 1838061568 T^{6} + 232069824 T^{7} - 4228800 T^{8} - 8542096 T^{9} - 1562394 T^{10} + 233142 T^{11} + 238433 T^{12} + 75615 T^{13} + 15318 T^{14} + 2383 T^{15} + 291 T^{16} + 24 T^{17} + T^{18}$$
$59$ $$11195428245849 + 4786859922480 T + 11370336750990 T^{2} + 279055365417 T^{3} - 1059269897142 T^{4} - 190956654288 T^{5} + 25574578183 T^{6} + 17308939446 T^{7} + 3161319504 T^{8} - 168177754 T^{9} - 148868205 T^{10} - 18665016 T^{11} + 1383131 T^{12} + 668829 T^{13} + 96102 T^{14} + 8389 T^{15} + 573 T^{16} + 30 T^{17} + T^{18}$$
$61$ $$176773345494962176 - 85573732711231488 T + 23880532149123072 T^{2} - 4337973100116480 T^{3} + 500391124830720 T^{4} - 23942247976704 T^{5} - 2535684944320 T^{6} + 540829630464 T^{7} - 12934624320 T^{8} - 9013627328 T^{9} + 1779767328 T^{10} - 173706336 T^{11} + 10942960 T^{12} - 565920 T^{13} + 69216 T^{14} - 10736 T^{15} + 990 T^{16} - 48 T^{17} + T^{18}$$
$67$ $$261685448704 + 512452331520 T + 477865314816 T^{2} + 216702068352 T^{3} + 91323859392 T^{4} + 28340406996 T^{5} - 2785258351 T^{6} - 3783063198 T^{7} - 263789235 T^{8} + 198653998 T^{9} + 40629747 T^{10} + 3536226 T^{11} + 642460 T^{12} + 73764 T^{13} + 6414 T^{14} + 604 T^{15} + 207 T^{16} + 6 T^{17} + T^{18}$$
$71$ $$170103363047424 - 270217414526976 T + 222057452519424 T^{2} - 125458113927168 T^{3} + 49031249967360 T^{4} - 12058622660736 T^{5} + 2021706701632 T^{6} - 257518705536 T^{7} + 31811445696 T^{8} - 4996442752 T^{9} + 914800992 T^{10} - 130466496 T^{11} + 11626160 T^{12} - 627312 T^{13} + 41688 T^{14} - 3116 T^{15} + 354 T^{16} - 30 T^{17} + T^{18}$$
$73$ $$35070801984 - 10314941760 T + 119653324416 T^{2} - 130448519712 T^{3} + 32183734536 T^{4} - 14846063130 T^{5} + 15399199413 T^{6} + 5566169664 T^{7} + 854212635 T^{8} - 252057114 T^{9} - 21661164 T^{10} + 720720 T^{11} + 825103 T^{12} - 72750 T^{13} + 21495 T^{14} - 586 T^{15} - 102 T^{16} - 6 T^{17} + T^{18}$$
$79$ $$33597696507904 + 88885063974912 T + 104998599917568 T^{2} + 17211336294400 T^{3} + 6782946115584 T^{4} - 4192001900544 T^{5} - 697198567424 T^{6} - 93523814400 T^{7} + 168902624256 T^{8} - 12650972672 T^{9} + 2080400640 T^{10} - 136284672 T^{11} + 2187776 T^{12} + 520992 T^{13} - 54096 T^{14} + 1944 T^{15} + 264 T^{16} - 30 T^{17} + T^{18}$$
$83$ $$57361719357504 - 94613763879648 T + 108776361247776 T^{2} - 69085774290168 T^{3} + 33545688282648 T^{4} - 9716757545070 T^{5} + 2397764311885 T^{6} - 322424160156 T^{7} + 78865764327 T^{8} - 8302652396 T^{9} + 1556086149 T^{10} - 89458122 T^{11} + 15501830 T^{12} - 678420 T^{13} + 108579 T^{14} - 2548 T^{15} + 396 T^{16} - 6 T^{17} + T^{18}$$
$89$ $$156263975312481 - 417406153068441 T + 523568468479068 T^{2} - 354809479185675 T^{3} + 152612936571249 T^{4} - 44903929254426 T^{5} + 7051875196438 T^{6} + 253567115175 T^{7} - 26724808020 T^{8} + 5620308037 T^{9} + 569006235 T^{10} - 79162074 T^{11} + 6454254 T^{12} + 566901 T^{13} - 8748 T^{14} - 3971 T^{15} + 435 T^{16} - 30 T^{17} + T^{18}$$
$97$ $$3392829504 + 48061589760 T + 393057013968 T^{2} + 1580902117560 T^{3} + 6254595335808 T^{4} - 272469482478 T^{5} + 1303192495485 T^{6} + 101372899896 T^{7} - 28773266193 T^{8} - 6070056708 T^{9} + 519938145 T^{10} + 78662406 T^{11} + 617044 T^{12} + 103140 T^{13} + 8916 T^{14} - 3406 T^{15} - 117 T^{16} + 12 T^{17} + T^{18}$$