# Properties

 Label 91.2.h.b Level $91$ Weight $2$ Character orbit 91.h Analytic conductor $0.727$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 7 x^{10} - 2 x^{9} + 33 x^{8} - 11 x^{7} + 55 x^{6} + 17 x^{5} + 47 x^{4} + x^{3} + 8 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 7 x^{10} - 2 x^{9} + 33 x^{8} - 11 x^{7} + 55 x^{6} + 17 x^{5} + 47 x^{4} + x^{3} + 8 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-29696 \nu^{11} - 478424 \nu^{10} + 682506 \nu^{9} - 3846008 \nu^{8} + 2684563 \nu^{7} - 16878368 \nu^{6} + 16008568 \nu^{5} - 31119861 \nu^{4} + 8363982 \nu^{3} - 14058754 \nu^{2} + 5624108 \nu - 2119374$$$$)/3318773$$ $$\beta_{3}$$ $$=$$ $$($$$$-73788 \nu^{11} - 498559 \nu^{10} + 495146 \nu^{9} - 4188508 \nu^{8} + 1631143 \nu^{7} - 18206928 \nu^{6} + 16328192 \nu^{5} - 34289666 \nu^{4} + 8704710 \nu^{3} - 14803002 \nu^{2} + 21668998 \nu - 2229034$$$$)/3318773$$ $$\beta_{4}$$ $$=$$ $$($$$$-109660 \nu^{11} + 153752 \nu^{10} - 747485 \nu^{9} + 406680 \nu^{8} - 3276280 \nu^{7} + 2259680 \nu^{6} - 4702740 \nu^{5} - 2183844 \nu^{4} - 1984215 \nu^{3} - 450388 \nu^{2} - 133032 \nu - 6198231$$$$)/3318773$$ $$\beta_{5}$$ $$=$$ $$($$$$439315 \nu^{11} - 329655 \nu^{10} + 2921453 \nu^{9} - 131145 \nu^{8} + 14090715 \nu^{7} - 1556185 \nu^{6} + 21902645 \nu^{5} + 12171095 \nu^{4} + 22831649 \nu^{3} + 2423530 \nu^{2} + 646135 \nu + 572347$$$$)/3318773$$ $$\beta_{6}$$ $$=$$ $$($$$$566698 \nu^{11} - 1732988 \nu^{10} + 5617249 \nu^{9} - 9944902 \nu^{8} + 24340355 \nu^{7} - 46353032 \nu^{6} + 58565408 \nu^{5} - 63065800 \nu^{4} + 27901335 \nu^{3} - 44235433 \nu^{2} + 12588213 \nu - 6707921$$$$)/3318773$$ $$\beta_{7}$$ $$=$$ $$($$$$-572347 \nu^{11} + 1011662 \nu^{10} - 4336084 \nu^{9} + 4066147 \nu^{8} - 19018596 \nu^{7} + 20386532 \nu^{6} - 33035270 \nu^{5} + 12172746 \nu^{4} - 14729214 \nu^{3} + 22259302 \nu^{2} - 2155246 \nu + 3392561$$$$)/3318773$$ $$\beta_{8}$$ $$=$$ $$($$$$-1035034 \nu^{11} + 1869572 \nu^{10} - 7924683 \nu^{9} + 7725614 \nu^{8} - 34760912 \nu^{7} + 38513384 \nu^{6} - 61367800 \nu^{5} + 26529336 \nu^{4} - 27474213 \nu^{3} + 41650219 \nu^{2} - 4177460 \nu + 6345807$$$$)/3318773$$ $$\beta_{9}$$ $$=$$ $$($$$$1166290 \nu^{11} - 1650363 \nu^{10} + 8811506 \nu^{9} - 5639321 \nu^{8} + 40119354 \nu^{7} - 27397018 \nu^{6} + 72699666 \nu^{5} - 1266529 \nu^{4} + 44802131 \nu^{3} - 8054629 \nu^{2} + 7274619 \nu + 566698$$$$)/3318773$$ $$\beta_{10}$$ $$=$$ $$($$$$-2686072 \nu^{11} + 3882058 \nu^{10} - 19974443 \nu^{9} + 13501144 \nu^{8} - 90433689 \nu^{7} + 66981583 \nu^{6} - 158252610 \nu^{5} + 11027874 \nu^{4} - 96392052 \nu^{3} + 33752077 \nu^{2} - 15484451 \nu - 1035561$$$$)/3318773$$ $$\beta_{11}$$ $$=$$ $$\nu^{11} - \nu^{10} + 7 \nu^{9} - 2 \nu^{8} + 33 \nu^{7} - 11 \nu^{6} + 55 \nu^{5} + 17 \nu^{4} + 47 \nu^{3} + \nu^{2} + 8 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} + 2 \beta_{7} - \beta_{4} - 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} + 5 \beta_{5} + \beta_{3} - \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{8} - 8 \beta_{7} + \beta_{6} - \beta_{2} - \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$5 \beta_{11} - \beta_{10} - 7 \beta_{9} + \beta_{8} - \beta_{7} - 24 \beta_{5} + \beta_{4} - 24 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} - 7 \beta_{10} - 9 \beta_{9} - 7 \beta_{6} - 11 \beta_{5} + 24 \beta_{4} - \beta_{3} + 9 \beta_{2} + 36$$ $$\nu^{7}$$ $$=$$ $$-11 \beta_{8} + 12 \beta_{7} - 9 \beta_{6} - 24 \beta_{3} + 40 \beta_{2} + 117 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-11 \beta_{11} + 40 \beta_{10} + 60 \beta_{9} - 117 \beta_{8} + 170 \beta_{7} + 85 \beta_{5} - 117 \beta_{4} + 85 \beta_{1} - 170$$ $$\nu^{9}$$ $$=$$ $$-117 \beta_{11} + 60 \beta_{10} + 217 \beta_{9} + 60 \beta_{6} + 581 \beta_{5} - 85 \beta_{4} + 117 \beta_{3} - 217 \beta_{2} - 99$$ $$\nu^{10}$$ $$=$$ $$581 \beta_{8} - 828 \beta_{7} + 217 \beta_{6} + 85 \beta_{3} - 362 \beta_{2} - 571 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$581 \beta_{11} - 362 \beta_{10} - 1160 \beta_{9} + 571 \beta_{8} - 695 \beta_{7} - 2933 \beta_{5} + 571 \beta_{4} - 2933 \beta_{1} + 695$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-1 + \beta_{7}$$ $$-1 + \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −0.181721 − 0.314749i 1.16700 + 2.02131i 0.756174 + 1.30973i −0.437442 − 0.757672i −1.02197 − 1.77010i 0.217953 + 0.377506i −0.181721 + 0.314749i 1.16700 − 2.02131i 0.756174 − 1.30973i −0.437442 + 0.757672i −1.02197 + 1.77010i 0.217953 − 0.377506i
−2.38804 1.37574 2.38285i 3.70272 −0.491140 + 0.850679i −3.28532 + 5.69033i −0.911766 2.48368i −4.06616 −2.28532 3.95828i 1.17286 2.03145i
16.2 −1.90556 −0.214224 + 0.371047i 1.63116 0.736565 1.27577i 0.408216 0.707051i 1.58334 + 2.11968i 0.702849 1.40822 + 2.43910i −1.40357 + 2.43105i
16.3 −0.851125 −0.330612 + 0.572636i −1.27559 −1.72074 + 2.98041i 0.281392 0.487385i −2.57273 0.617304i 2.78793 1.28139 + 2.21944i 1.46456 2.53670i
16.4 −0.268125 0.571504 0.989875i −1.92811 1.28088 2.21854i −0.153235 + 0.265410i 1.80416 1.93520i 1.05323 0.846765 + 1.46664i −0.343436 + 0.594848i
16.5 1.55469 0.244626 0.423704i 0.417051 0.595756 1.03188i 0.380316 0.658727i −2.44127 + 1.01990i −2.46099 1.38032 + 2.39078i 0.926214 1.60425i
16.6 1.85816 −1.14703 + 1.98672i 1.45276 0.0986811 0.170921i −2.13137 + 3.69165i 1.03826 2.43352i −1.01686 −1.13137 1.95960i 0.183365 0.317598i
74.1 −2.38804 1.37574 + 2.38285i 3.70272 −0.491140 0.850679i −3.28532 5.69033i −0.911766 + 2.48368i −4.06616 −2.28532 + 3.95828i 1.17286 + 2.03145i
74.2 −1.90556 −0.214224 0.371047i 1.63116 0.736565 + 1.27577i 0.408216 + 0.707051i 1.58334 2.11968i 0.702849 1.40822 2.43910i −1.40357 2.43105i
74.3 −0.851125 −0.330612 0.572636i −1.27559 −1.72074 2.98041i 0.281392 + 0.487385i −2.57273 + 0.617304i 2.78793 1.28139 2.21944i 1.46456 + 2.53670i
74.4 −0.268125 0.571504 + 0.989875i −1.92811 1.28088 + 2.21854i −0.153235 0.265410i 1.80416 + 1.93520i 1.05323 0.846765 1.46664i −0.343436 0.594848i
74.5 1.55469 0.244626 + 0.423704i 0.417051 0.595756 + 1.03188i 0.380316 + 0.658727i −2.44127 1.01990i −2.46099 1.38032 2.39078i 0.926214 + 1.60425i
74.6 1.85816 −1.14703 1.98672i 1.45276 0.0986811 + 0.170921i −2.13137 3.69165i 1.03826 + 2.43352i −1.01686 −1.13137 + 1.95960i 0.183365 + 0.317598i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 74.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.h.b yes 12
3.b odd 2 1 819.2.s.d 12
7.b odd 2 1 637.2.h.l 12
7.c even 3 1 91.2.g.b 12
7.c even 3 1 637.2.f.k 12
7.d odd 6 1 637.2.f.j 12
7.d odd 6 1 637.2.g.l 12
13.c even 3 1 91.2.g.b 12
13.c even 3 1 1183.2.e.h 12
13.e even 6 1 1183.2.e.g 12
21.h odd 6 1 819.2.n.d 12
39.i odd 6 1 819.2.n.d 12
91.g even 3 1 637.2.f.k 12
91.g even 3 1 1183.2.e.h 12
91.h even 3 1 inner 91.2.h.b yes 12
91.h even 3 1 8281.2.a.bz 6
91.k even 6 1 8281.2.a.ce 6
91.l odd 6 1 8281.2.a.cf 6
91.m odd 6 1 637.2.f.j 12
91.n odd 6 1 637.2.g.l 12
91.u even 6 1 1183.2.e.g 12
91.v odd 6 1 637.2.h.l 12
91.v odd 6 1 8281.2.a.ca 6
273.s odd 6 1 819.2.s.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 7.c even 3 1
91.2.g.b 12 13.c even 3 1
91.2.h.b yes 12 1.a even 1 1 trivial
91.2.h.b yes 12 91.h even 3 1 inner
637.2.f.j 12 7.d odd 6 1
637.2.f.j 12 91.m odd 6 1
637.2.f.k 12 7.c even 3 1
637.2.f.k 12 91.g even 3 1
637.2.g.l 12 7.d odd 6 1
637.2.g.l 12 91.n odd 6 1
637.2.h.l 12 7.b odd 2 1
637.2.h.l 12 91.v odd 6 1
819.2.n.d 12 21.h odd 6 1
819.2.n.d 12 39.i odd 6 1
819.2.s.d 12 3.b odd 2 1
819.2.s.d 12 273.s odd 6 1
1183.2.e.g 12 13.e even 6 1
1183.2.e.g 12 91.u even 6 1
1183.2.e.h 12 13.c even 3 1
1183.2.e.h 12 91.g even 3 1
8281.2.a.bz 6 91.h even 3 1
8281.2.a.ca 6 91.v odd 6 1
8281.2.a.ce 6 91.k even 6 1
8281.2.a.cf 6 91.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2 T_{2}^{5} - 6 T_{2}^{4} - 11 T_{2}^{3} + 8 T_{2}^{2} + 14 T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + 14 T + 8 T^{2} - 11 T^{3} - 6 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$3$ $$1 + T + 7 T^{2} + 2 T^{3} + 33 T^{4} + 11 T^{5} + 55 T^{6} - 17 T^{7} + 47 T^{8} - T^{9} + 8 T^{10} - T^{11} + T^{12}$$
$5$ $$9 - 51 T + 271 T^{2} - 210 T^{3} + 375 T^{4} - 269 T^{5} + 379 T^{6} - 203 T^{7} + 133 T^{8} - 25 T^{9} + 12 T^{10} - T^{11} + T^{12}$$
$7$ $$117649 + 50421 T + 14406 T^{2} + 15435 T^{3} + 7203 T^{4} + 2436 T^{5} + 1069 T^{6} + 348 T^{7} + 147 T^{8} + 45 T^{9} + 6 T^{10} + 3 T^{11} + T^{12}$$
$11$ $$6561 - 16767 T + 36288 T^{2} - 29079 T^{3} + 23994 T^{4} - 2862 T^{5} + 6811 T^{6} - 1155 T^{7} + 664 T^{8} - 68 T^{9} + 37 T^{10} - 4 T^{11} + T^{12}$$
$13$ $$4826809 + 742586 T - 456976 T^{2} + 6591 T^{3} + 102583 T^{4} + 5629 T^{5} - 5615 T^{6} + 433 T^{7} + 607 T^{8} + 3 T^{9} - 16 T^{10} + 2 T^{11} + T^{12}$$
$17$ $$( -9 + 8 T + 20 T^{2} - 14 T^{3} - 12 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$19$ $$762129 + 1346166 T + 1828647 T^{2} + 1163724 T^{3} + 622675 T^{4} + 128430 T^{5} + 52781 T^{6} + 7388 T^{7} + 3578 T^{8} + 158 T^{9} + 65 T^{10} + T^{11} + T^{12}$$
$23$ $$( -24387 - 440 T + 3031 T^{2} + 63 T^{3} - 106 T^{4} - T^{5} + T^{6} )^{2}$$
$29$ $$40401 + 225924 T + 1362670 T^{2} - 457168 T^{3} + 502614 T^{4} - 54205 T^{5} + 94294 T^{6} - 17192 T^{7} + 6322 T^{8} - 254 T^{9} + 87 T^{10} - 3 T^{11} + T^{12}$$
$31$ $$6135529 - 9908000 T + 10941966 T^{2} - 6706570 T^{3} + 3113614 T^{4} - 962758 T^{5} + 248171 T^{6} - 46594 T^{7} + 9262 T^{8} - 1390 T^{9} + 206 T^{10} - 16 T^{11} + T^{12}$$
$37$ $$( -13477 + 17436 T - 7753 T^{2} + 1351 T^{3} - 38 T^{4} - 13 T^{5} + T^{6} )^{2}$$
$41$ $$4173849 + 2939877 T + 2648890 T^{2} + 728671 T^{3} + 523034 T^{4} + 155456 T^{5} + 63915 T^{6} + 11805 T^{7} + 2948 T^{8} + 388 T^{9} + 85 T^{10} + 8 T^{11} + T^{12}$$
$43$ $$1369 + 59940 T + 2634945 T^{2} - 442016 T^{3} + 512108 T^{4} + 72977 T^{5} + 53295 T^{6} + 7624 T^{7} + 3212 T^{8} + 543 T^{9} + 120 T^{10} + 11 T^{11} + T^{12}$$
$47$ $$318515409 + 112846581 T + 141422677 T^{2} - 39081004 T^{3} + 28592513 T^{4} - 2756381 T^{5} + 984441 T^{6} - 2115 T^{7} + 25733 T^{8} - T^{9} + 178 T^{10} + T^{11} + T^{12}$$
$53$ $$4761 - 23046 T + 187801 T^{2} + 343402 T^{3} + 1276249 T^{4} - 138868 T^{5} + 144290 T^{6} + 14514 T^{7} + 9267 T^{8} + 172 T^{9} + 104 T^{10} + 2 T^{11} + T^{12}$$
$59$ $$( 9123 + 18461 T + 666 T^{2} - 996 T^{3} - 59 T^{4} + 13 T^{5} + T^{6} )^{2}$$
$61$ $$1055015361 + 1195333281 T + 1062894069 T^{2} + 390333384 T^{3} + 121103191 T^{4} + 6648335 T^{5} + 2541805 T^{6} + 133207 T^{7} + 36059 T^{8} + 847 T^{9} + 226 T^{10} + 5 T^{11} + T^{12}$$
$67$ $$276324129 - 183966741 T + 160212699 T^{2} - 4433604 T^{3} + 13229425 T^{4} + 145321 T^{5} + 875958 T^{6} + 55361 T^{7} + 18745 T^{8} + 612 T^{9} + 227 T^{10} + 11 T^{11} + T^{12}$$
$71$ $$530979849 - 315527799 T + 189871678 T^{2} - 50943317 T^{3} + 18814920 T^{4} - 4116692 T^{5} + 1239901 T^{6} - 175105 T^{7} + 26800 T^{8} - 1426 T^{9} + 177 T^{10} - 6 T^{11} + T^{12}$$
$73$ $$196812841 + 343233514 T + 480685440 T^{2} + 198569706 T^{3} + 67825152 T^{4} + 13334350 T^{5} + 2769075 T^{6} + 420036 T^{7} + 72578 T^{8} + 7642 T^{9} + 662 T^{10} + 30 T^{11} + T^{12}$$
$79$ $$110859841 + 199598253 T + 283043128 T^{2} + 130891313 T^{3} + 48229623 T^{4} + 7784759 T^{5} + 1322709 T^{6} + 74563 T^{7} + 16825 T^{8} + 416 T^{9} + 197 T^{10} - 7 T^{11} + T^{12}$$
$83$ $$( 2673 + 1188 T - 1797 T^{2} - 403 T^{3} + 158 T^{4} + 27 T^{5} + T^{6} )^{2}$$
$89$ $$( -304479 - 79486 T + 32872 T^{2} - 132 T^{3} - 367 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$97$ $$15202201 + 33180490 T + 68189685 T^{2} + 18481778 T^{3} + 12693220 T^{4} + 4789025 T^{5} + 2092673 T^{6} + 500330 T^{7} + 92800 T^{8} + 10403 T^{9} + 860 T^{10} + 35 T^{11} + T^{12}$$
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