# Properties

 Label 1183.2.a.r Level $1183$ Weight $2$ Character orbit 1183.a Self dual yes Analytic conductor $9.446$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{6} ) q^{3} + ( 2 - \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{4} + \beta_{7} + \beta_{11} ) q^{5} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{6} + q^{7} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + ( 3 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{6} ) q^{3} + ( 2 - \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{4} + \beta_{7} + \beta_{11} ) q^{5} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{6} + q^{7} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + ( 3 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{9} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{9} + \beta_{10} ) q^{10} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{11} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{12} + \beta_{1} q^{14} + ( 2 - \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} ) q^{15} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{16} + ( 3 + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{17} + ( -2 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{18} + ( -2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{19} + ( -3 - \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{10} ) q^{20} + ( 1 + \beta_{6} ) q^{21} + ( 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{22} + ( 2 - \beta_{1} + \beta_{3} + \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{23} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{24} + ( 4 + \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{25} + ( 2 + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{27} + ( 2 - \beta_{4} + \beta_{5} ) q^{28} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{29} + ( -3 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{30} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{31} + ( -1 + \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{32} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{33} + ( 1 + 3 \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{6} + \beta_{11} ) q^{34} + ( -1 + \beta_{4} + \beta_{7} + \beta_{11} ) q^{35} + ( 8 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{36} + ( -2 - \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{37} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{38} + ( -6 - \beta_{3} + 6 \beta_{4} - \beta_{5} + \beta_{8} - 4 \beta_{10} + 2 \beta_{11} ) q^{40} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{41} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{42} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{43} + ( 9 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{44} + ( 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{45} + ( -2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{46} + ( -1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{47} + ( 2 - 4 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} + 3 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{48} + q^{49} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{50} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{51} + ( 5 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 8 \beta_{10} ) q^{54} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{57} + ( 1 + \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 3 \beta_{10} ) q^{58} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{59} + ( 4 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{60} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} ) q^{61} + ( 5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 7 \beta_{4} - \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{62} + ( 3 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{63} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{64} + ( -5 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} ) q^{66} + ( 2 + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} + 4 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{67} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{68} + ( -1 - 3 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{69} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{9} + \beta_{10} ) q^{70} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{71} + ( -4 + 4 \beta_{1} - \beta_{2} + 7 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{72} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{73} + ( -8 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{74} + ( 3 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - \beta_{9} + 9 \beta_{10} + \beta_{11} ) q^{75} + ( 3 + 3 \beta_{1} + 3 \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{76} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{77} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{79} + ( 1 - 4 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{80} + ( 3 - 4 \beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - 4 \beta_{11} ) q^{81} + ( -3 - \beta_{1} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{82} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{11} ) q^{83} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{84} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{85} + ( 3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{86} + ( 6 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{87} + ( -3 + 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{88} + ( -1 - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{89} + ( -5 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - 7 \beta_{8} - 9 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{90} + ( 4 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - 6 \beta_{10} - 2 \beta_{11} ) q^{92} + ( -8 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{93} + ( -4 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{94} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} ) q^{95} + ( -15 + \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 5 \beta_{10} + \beta_{11} ) q^{96} + ( 3 - \beta_{1} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{97} + \beta_{1} q^{98} + ( 8 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + 6 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 3q^{2} + 8q^{3} + 15q^{4} - 4q^{5} + 2q^{6} + 12q^{7} + 12q^{8} + 26q^{9} + O(q^{10})$$ $$12q + 3q^{2} + 8q^{3} + 15q^{4} - 4q^{5} + 2q^{6} + 12q^{7} + 12q^{8} + 26q^{9} - 6q^{10} + 12q^{11} + 13q^{12} + 3q^{14} + 11q^{15} + 13q^{16} + 31q^{17} - 29q^{18} - 3q^{19} - 18q^{20} + 8q^{21} - 4q^{22} + 18q^{23} - 6q^{24} + 32q^{25} + 32q^{27} + 15q^{28} + 15q^{29} - 10q^{30} - 21q^{31} - 3q^{32} - 29q^{33} + 3q^{34} - 4q^{35} + 49q^{36} + 5q^{37} + 45q^{38} - 20q^{40} - 16q^{41} + 2q^{42} - 22q^{43} + 35q^{44} + 5q^{45} + 2q^{46} - 4q^{47} + 11q^{48} + 12q^{49} + 13q^{50} + 18q^{51} + 53q^{53} - 5q^{54} - 26q^{55} + 12q^{56} - 8q^{57} + 32q^{58} - 26q^{59} + 38q^{60} + 22q^{61} + 19q^{62} + 26q^{63} + 2q^{64} - 34q^{66} + 12q^{67} + 34q^{68} + 3q^{69} - 6q^{70} + 21q^{71} - 4q^{72} - 15q^{73} - 40q^{74} + 15q^{75} + 43q^{76} + 12q^{77} + 2q^{79} + 13q^{80} + 36q^{81} - 32q^{82} - 9q^{83} + 13q^{84} - 39q^{85} + 44q^{86} + 27q^{87} - 48q^{88} - 22q^{89} - 26q^{90} + 52q^{92} - 53q^{93} - 44q^{94} + 29q^{95} - 114q^{96} + 9q^{97} + 3q^{98} + 37q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 77279 \nu^{3} + 193411 \nu^{2} + 16187 \nu + 25085$$$$)/14629$$ $$\beta_{3}$$ $$=$$ $$($$$$202 \nu^{11} + 1816 \nu^{10} - 5010 \nu^{9} - 29197 \nu^{8} + 39197 \nu^{7} + 161452 \nu^{6} - 110863 \nu^{5} - 346540 \nu^{4} + 91908 \nu^{3} + 193411 \nu^{2} - 56958 \nu + 10456$$$$)/14629$$ $$\beta_{4}$$ $$=$$ $$($$$$266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 165757 \nu^{2} - 263443 \nu - 14765$$$$)/14629$$ $$\beta_{5}$$ $$=$$ $$($$$$266 \nu^{11} + 1812 \nu^{10} - 9639 \nu^{9} - 26136 \nu^{8} + 98110 \nu^{7} + 125845 \nu^{6} - 384542 \nu^{5} - 239507 \nu^{4} + 573947 \nu^{3} + 180386 \nu^{2} - 263443 \nu - 73281$$$$)/14629$$ $$\beta_{6}$$ $$=$$ $$($$$$721 \nu^{11} - 2788 \nu^{10} - 8033 \nu^{9} + 41570 \nu^{8} + 13387 \nu^{7} - 212101 \nu^{6} + 95671 \nu^{5} + 447600 \nu^{4} - 310269 \nu^{3} - 374170 \nu^{2} + 204863 \nu + 63537$$$$)/14629$$ $$\beta_{7}$$ $$=$$ $$($$$$1025 \nu^{11} - 2807 \nu^{10} - 19049 \nu^{9} + 45138 \nu^{8} + 135962 \nu^{7} - 245916 \nu^{6} - 469197 \nu^{5} + 506165 \nu^{4} + 761552 \nu^{3} - 264148 \nu^{2} - 384615 \nu - 16033$$$$)/14629$$ $$\beta_{8}$$ $$=$$ $$($$$$1181 \nu^{11} - 6474 \nu^{10} - 9303 \nu^{9} + 90086 \nu^{8} - 23075 \nu^{7} - 408596 \nu^{6} + 301924 \nu^{5} + 705799 \nu^{4} - 620810 \nu^{3} - 446758 \nu^{2} + 361943 \nu + 60552$$$$)/14629$$ $$\beta_{9}$$ $$=$$ $$($$$$1429 \nu^{11} + 825 \nu^{10} - 29069 \nu^{9} - 13256 \nu^{8} + 214356 \nu^{7} + 76988 \nu^{6} - 690923 \nu^{5} - 201544 \nu^{4} + 930739 \nu^{3} + 225077 \nu^{2} - 425386 \nu - 68266$$$$)/14629$$ $$\beta_{10}$$ $$=$$ $$($$$$2354 \nu^{11} - 5633 \nu^{10} - 34485 \nu^{9} + 79215 \nu^{8} + 175064 \nu^{7} - 364728 \nu^{6} - 391458 \nu^{5} + 632025 \nu^{4} + 415204 \nu^{3} - 345129 \nu^{2} - 144501 \nu + 5396$$$$)/14629$$ $$\beta_{11}$$ $$=$$ $$($$$$-4628 \nu^{11} + 11261 \nu^{10} + 66841 \nu^{9} - 158261 \nu^{8} - 324031 \nu^{7} + 725177 \nu^{6} + 627337 \nu^{5} - 1236319 \nu^{4} - 443637 \nu^{3} + 648376 \nu^{2} + 34553 \nu + 5226$$$$)/14629$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + \beta_{7} + 7 \beta_{5} - 7 \beta_{4} + \beta_{3} + \beta_{2} + 22$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{4} + 8 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 7$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{11} - 3 \beta_{10} - 10 \beta_{9} - \beta_{8} + 11 \beta_{7} - 2 \beta_{6} + 45 \beta_{5} - 48 \beta_{4} + 10 \beta_{3} + 11 \beta_{2} - \beta_{1} + 131$$ $$\nu^{7}$$ $$=$$ $$-23 \beta_{11} - 9 \beta_{10} - 37 \beta_{9} - 25 \beta_{8} + \beta_{7} - 13 \beta_{6} + 10 \beta_{4} + 57 \beta_{3} - 43 \beta_{2} + 166 \beta_{1} + 43$$ $$\nu^{8}$$ $$=$$ $$-27 \beta_{11} - 36 \beta_{10} - 84 \beta_{9} - 19 \beta_{8} + 93 \beta_{7} - 24 \beta_{6} + 289 \beta_{5} - 330 \beta_{4} + 80 \beta_{3} + 94 \beta_{2} - 14 \beta_{1} + 812$$ $$\nu^{9}$$ $$=$$ $$-198 \beta_{11} - 61 \beta_{10} - 331 \beta_{9} - 231 \beta_{8} + 14 \beta_{7} - 120 \beta_{6} - \beta_{5} + 67 \beta_{4} + 396 \beta_{3} - 259 \beta_{2} + 1021 \beta_{1} + 265$$ $$\nu^{10}$$ $$=$$ $$-257 \beta_{11} - 306 \beta_{10} - 658 \beta_{9} - 224 \beta_{8} + 710 \beta_{7} - 212 \beta_{6} + 1874 \beta_{5} - 2277 \beta_{4} + 593 \beta_{3} + 734 \beta_{2} - 132 \beta_{1} + 5167$$ $$\nu^{11}$$ $$=$$ $$-1539 \beta_{11} - 370 \beta_{10} - 2625 \beta_{9} - 1909 \beta_{8} + 136 \beta_{7} - 967 \beta_{6} - 16 \beta_{5} + 356 \beta_{4} + 2724 \beta_{3} - 1555 \beta_{2} + 6448 \beta_{1} + 1686$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.58557 −2.07140 −1.35819 −1.10989 −0.961590 −0.149660 0.842530 0.983820 2.06743 2.23724 2.47725 2.62803
−2.58557 2.93994 4.68518 1.26031 −7.60143 1.00000 −6.94272 5.64327 −3.25863
1.2 −2.07140 −3.01646 2.29068 −2.69245 6.24828 1.00000 −0.602118 6.09903 5.57713
1.3 −1.35819 3.39737 −0.155322 −0.772491 −4.61428 1.00000 2.92733 8.54215 1.04919
1.4 −1.10989 0.955760 −0.768150 −3.55862 −1.06079 1.00000 3.07233 −2.08652 3.94966
1.5 −0.961590 −1.98737 −1.07534 3.39320 1.91103 1.00000 2.95722 0.949635 −3.26287
1.6 −0.149660 2.76031 −1.97760 4.13443 −0.413107 1.00000 0.595288 4.61930 −0.618759
1.7 0.842530 0.161973 −1.29014 −3.72786 0.136467 1.00000 −2.77204 −2.97376 −3.14083
1.8 0.983820 −1.57171 −1.03210 0.398447 −1.54628 1.00000 −2.98304 −0.529731 0.392000
1.9 2.06743 2.11889 2.27425 2.43928 4.38065 1.00000 0.566992 1.48970 5.04303
1.10 2.23724 3.02592 3.00523 −3.28547 6.76971 1.00000 2.24893 6.15622 −7.35037
1.11 2.47725 0.982981 4.13677 1.35413 2.43509 1.00000 5.29330 −2.03375 3.35452
1.12 2.62803 −1.76762 4.90656 −2.94291 −4.64535 1.00000 7.63852 0.124466 −7.73406
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.r yes 12
7.b odd 2 1 8281.2.a.cq 12
13.b even 2 1 1183.2.a.q 12
13.d odd 4 2 1183.2.c.j 24
91.b odd 2 1 8281.2.a.cn 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.q 12 13.b even 2 1
1183.2.a.r yes 12 1.a even 1 1 trivial
1183.2.c.j 24 13.d odd 4 2
8281.2.a.cn 12 91.b odd 2 1
8281.2.a.cq 12 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1183))$$:

 $$T_{2}^{12} - \cdots$$ $$T_{11}^{12} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$29 + 183 T - 157 T^{2} - 542 T^{3} + 262 T^{4} + 562 T^{5} - 199 T^{6} - 246 T^{7} + 80 T^{8} + 46 T^{9} - 15 T^{10} - 3 T^{11} + T^{12}$$
$3$ $$448 - 3584 T + 4928 T^{2} + 1616 T^{3} - 5492 T^{4} + 920 T^{5} + 1973 T^{6} - 691 T^{7} - 249 T^{8} + 136 T^{9} + T^{10} - 8 T^{11} + T^{12}$$
$5$ $$6208 - 13216 T - 16848 T^{2} + 25848 T^{3} + 8916 T^{4} - 11870 T^{5} - 2897 T^{6} + 2122 T^{7} + 500 T^{8} - 157 T^{9} - 38 T^{10} + 4 T^{11} + T^{12}$$
$7$ $$( -1 + T )^{12}$$
$11$ $$-2701133 + 900676 T + 1617506 T^{2} - 505685 T^{3} - 339321 T^{4} + 112394 T^{5} + 30684 T^{6} - 12005 T^{7} - 994 T^{8} + 613 T^{9} - 13 T^{10} - 12 T^{11} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$110272 - 472544 T + 675872 T^{2} - 210520 T^{3} - 402728 T^{4} + 483052 T^{5} - 228079 T^{6} + 48450 T^{7} - 861 T^{8} - 1749 T^{9} + 364 T^{10} - 31 T^{11} + T^{12}$$
$19$ $$1395008 + 523872 T - 3138960 T^{2} + 181112 T^{3} + 1033252 T^{4} - 81510 T^{5} - 123297 T^{6} + 8681 T^{7} + 6082 T^{8} - 296 T^{9} - 131 T^{10} + 3 T^{11} + T^{12}$$
$23$ $$7017331 - 11388083 T + 3342704 T^{2} + 2962993 T^{3} - 1682702 T^{4} - 103895 T^{5} + 199072 T^{6} - 20950 T^{7} - 7201 T^{8} + 1447 T^{9} + 12 T^{10} - 18 T^{11} + T^{12}$$
$29$ $$177673 - 867913 T + 828198 T^{2} + 217418 T^{3} - 481608 T^{4} + 59784 T^{5} + 77202 T^{6} - 18103 T^{7} - 3607 T^{8} + 1221 T^{9} - 24 T^{10} - 15 T^{11} + T^{12}$$
$31$ $$-15261184 + 271744 T + 13600640 T^{2} + 1249464 T^{3} - 3652072 T^{4} - 539022 T^{5} + 346335 T^{6} + 67095 T^{7} - 8438 T^{8} - 2180 T^{9} + 7 T^{10} + 21 T^{11} + T^{12}$$
$37$ $$545930729 + 4008911 T - 216159072 T^{2} - 11928582 T^{3} + 23149628 T^{4} + 1293842 T^{5} - 1076680 T^{6} - 51131 T^{7} + 24107 T^{8} + 863 T^{9} - 252 T^{10} - 5 T^{11} + T^{12}$$
$41$ $$-101827648 + 127486208 T + 53137520 T^{2} - 51226496 T^{3} - 27517248 T^{4} - 1654660 T^{5} + 1222195 T^{6} + 214272 T^{7} - 6045 T^{8} - 3532 T^{9} - 135 T^{10} + 16 T^{11} + T^{12}$$
$43$ $$-9002449 - 10108897 T + 53854685 T^{2} - 513456 T^{3} - 11811616 T^{4} - 474599 T^{5} + 880317 T^{6} + 89963 T^{7} - 21642 T^{8} - 3728 T^{9} - 28 T^{10} + 22 T^{11} + T^{12}$$
$47$ $$17156608 + 5974528 T - 28506016 T^{2} - 6563096 T^{3} + 8846240 T^{4} + 510390 T^{5} - 760339 T^{6} - 1170 T^{7} + 23426 T^{8} - 457 T^{9} - 262 T^{10} + 4 T^{11} + T^{12}$$
$53$ $$62267981 + 177543116 T - 37586257 T^{2} - 92629676 T^{3} + 43097846 T^{4} - 1557851 T^{5} - 2436927 T^{6} + 473888 T^{7} - 1896 T^{8} - 8533 T^{9} + 1054 T^{10} - 53 T^{11} + T^{12}$$
$59$ $$913870784 + 980465920 T + 130935936 T^{2} - 118535488 T^{3} - 31898324 T^{4} + 3347084 T^{5} + 1779215 T^{6} + 69801 T^{7} - 32990 T^{8} - 3699 T^{9} + 65 T^{10} + 26 T^{11} + T^{12}$$
$61$ $$-69982144 - 18517536 T + 49214640 T^{2} + 6247144 T^{3} - 12613036 T^{4} + 309150 T^{5} + 1268015 T^{6} - 196445 T^{7} - 19825 T^{8} + 5344 T^{9} - 129 T^{10} - 22 T^{11} + T^{12}$$
$67$ $$1207037 - 1648589 T - 3785554 T^{2} + 685362 T^{3} + 2433927 T^{4} + 370595 T^{5} - 308286 T^{6} - 63359 T^{7} + 11336 T^{8} + 2004 T^{9} - 199 T^{10} - 12 T^{11} + T^{12}$$
$71$ $$-22571863 + 45796935 T + 10711179 T^{2} - 29155208 T^{3} - 2962125 T^{4} + 4813988 T^{5} + 281595 T^{6} - 258995 T^{7} + 109 T^{8} + 4211 T^{9} - 125 T^{10} - 21 T^{11} + T^{12}$$
$73$ $$-4111861312 - 4922870400 T + 430370544 T^{2} + 923841136 T^{3} + 61231368 T^{4} - 41485060 T^{5} - 3463011 T^{6} + 748304 T^{7} + 58607 T^{8} - 5707 T^{9} - 406 T^{10} + 15 T^{11} + T^{12}$$
$79$ $$-33746539 + 393597671 T - 336730518 T^{2} - 15641243 T^{3} + 46453724 T^{4} - 238149 T^{5} - 2240776 T^{6} - 7420 T^{7} + 44193 T^{8} + 343 T^{9} - 364 T^{10} - 2 T^{11} + T^{12}$$
$83$ $$-478100288 - 299570240 T + 91803616 T^{2} + 78899128 T^{3} - 1911072 T^{4} - 6691902 T^{5} - 420269 T^{6} + 204661 T^{7} + 19015 T^{8} - 2362 T^{9} - 248 T^{10} + 9 T^{11} + T^{12}$$
$89$ $$-639815168 - 2900248960 T - 3531001216 T^{2} - 1826781592 T^{3} - 406843464 T^{4} - 12278850 T^{5} + 9911493 T^{6} + 1335627 T^{7} - 15214 T^{8} - 11148 T^{9} - 360 T^{10} + 22 T^{11} + T^{12}$$
$97$ $$-6873362944 + 12657080576 T + 2910438944 T^{2} - 1985269000 T^{3} - 4751296 T^{4} + 74671606 T^{5} - 4261095 T^{6} - 974036 T^{7} + 78756 T^{8} + 5062 T^{9} - 488 T^{10} - 9 T^{11} + T^{12}$$