# Properties

 Label 315.2.i.e Level 315 Weight 2 Character orbit 315.i Analytic conductor 2.515 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 2 x^{10} - x^{9} - 4 x^{8} + 20 x^{7} - 38 x^{6} + 60 x^{5} - 36 x^{4} - 27 x^{3} + 162 x^{2} - 243 x + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-25 \nu^{11} + 10 \nu^{10} - 152 \nu^{9} + 301 \nu^{8} - 389 \nu^{7} + 865 \nu^{6} + 281 \nu^{5} - 2973 \nu^{4} + 45 \nu^{3} - 12906 \nu^{2} + 15228 \nu - 7533$$$$)/17496$$ $$\beta_{3}$$ $$=$$ $$($$$$13 \nu^{11} - 25 \nu^{10} - 61 \nu^{9} - 46 \nu^{8} - 49 \nu^{7} + 110 \nu^{6} - 311 \nu^{5} + 822 \nu^{4} + 1305 \nu^{3} + 351 \nu^{2} - 2349 \nu - 4860$$$$)/8748$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{11} + 17 \nu^{10} - 49 \nu^{9} + 110 \nu^{8} - 181 \nu^{7} + 68 \nu^{6} + 79 \nu^{5} - 702 \nu^{4} + 1629 \nu^{3} - 2673 \nu^{2} + 2997 \nu - 1458$$$$)/2916$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{11} + 11 \nu^{10} - 61 \nu^{9} + 116 \nu^{8} - 211 \nu^{7} + 272 \nu^{6} - 293 \nu^{5} - 672 \nu^{4} + 2115 \nu^{3} - 4833 \nu^{2} + 4455 \nu - 6318$$$$)/2916$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{11} - 77 \nu^{10} + 157 \nu^{9} - 191 \nu^{8} + 316 \nu^{7} + 13 \nu^{6} - 946 \nu^{5} + 2085 \nu^{4} - 4356 \nu^{3} + 5157 \nu^{2} - 3969 \nu - 243$$$$)/4374$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{11} + 7 \nu^{10} - 21 \nu^{9} + 51 \nu^{8} - 102 \nu^{7} + 141 \nu^{6} + 8 \nu^{5} - 199 \nu^{4} + 774 \nu^{3} - 1575 \nu^{2} + 2349 \nu - 2511$$$$)/972$$ $$\beta_{8}$$ $$=$$ $$($$$$-14 \nu^{11} + 29 \nu^{10} - 61 \nu^{9} + 197 \nu^{8} - 292 \nu^{7} + 353 \nu^{6} - 284 \nu^{5} - 1419 \nu^{4} + 2520 \nu^{3} - 3051 \nu^{2} + 7857 \nu - 7047$$$$)/4374$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} - 2 \nu^{9} + \nu^{8} + 4 \nu^{7} - 20 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + 36 \nu^{3} + 27 \nu^{2} - 162 \nu + 243$$$$)/243$$ $$\beta_{10}$$ $$=$$ $$($$$$4 \nu^{11} - 13 \nu^{10} + 35 \nu^{9} - 67 \nu^{8} + 56 \nu^{7} - 37 \nu^{6} - 134 \nu^{5} + 483 \nu^{4} - 936 \nu^{3} + 1269 \nu^{2} - 1863 \nu + 1215$$$$)/972$$ $$\beta_{11}$$ $$=$$ $$($$$$-65 \nu^{11} + 107 \nu^{10} - 217 \nu^{9} + 248 \nu^{8} - 331 \nu^{7} - 424 \nu^{6} + 1411 \nu^{5} - 3048 \nu^{4} + 2223 \nu^{3} - 5805 \nu^{2} + 7371 \nu - 12150$$$$)/8748$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{5} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + 2 \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{2} - \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{11} + 4 \beta_{9} - 3 \beta_{8} + 5 \beta_{7} - 4 \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_{1} - 6$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 5 \beta_{8} + 7 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 18 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 8 \beta_{1} + 12$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{11} - 18 \beta_{10} - 6 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 12 \beta_{5} - 27 \beta_{4} - 9 \beta_{3} - 6 \beta_{2} + \beta_{1} + 6$$ $$\nu^{8}$$ $$=$$ $$-33 \beta_{10} - 21 \beta_{9} - 11 \beta_{8} - 6 \beta_{7} + 24 \beta_{6} - 19 \beta_{5} + 39 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} - 30 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$-28 \beta_{11} + 15 \beta_{10} + 32 \beta_{9} - 6 \beta_{8} + 25 \beta_{7} + 3 \beta_{6} + 10 \beta_{5} + 14 \beta_{4} - 78 \beta_{3} - 32 \beta_{2} + 4 \beta_{1} - 63$$ $$\nu^{10}$$ $$=$$ $$56 \beta_{11} - 17 \beta_{10} - 113 \beta_{9} + 12 \beta_{8} - 22 \beta_{7} - 55 \beta_{6} + 36 \beta_{5} - 123 \beta_{4} - 102 \beta_{3} - 82 \beta_{2} - 58 \beta_{1} + 96$$ $$\nu^{11}$$ $$=$$ $$-92 \beta_{11} - 12 \beta_{10} - 137 \beta_{9} - 18 \beta_{8} - 124 \beta_{7} - 105 \beta_{6} - 58 \beta_{5} + 142 \beta_{4} - 6 \beta_{3} + 116 \beta_{2} + 40 \beta_{1} - 339$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 1.24736 + 1.20170i 0.478182 − 1.66473i −0.764584 − 1.55416i −1.73190 − 0.0231100i 1.65373 − 0.514941i −0.382799 + 1.68922i 1.24736 − 1.20170i 0.478182 + 1.66473i −0.764584 + 1.55416i −1.73190 + 0.0231100i 1.65373 + 0.514941i −0.382799 − 1.68922i
−1.35359 + 2.34448i 1.24736 1.20170i −2.66438 4.61485i 0.500000 + 0.866025i 1.12895 + 4.55102i −0.500000 + 0.866025i 9.01155 0.111834 2.99791i −2.70717
106.2 −0.631422 + 1.09366i 0.478182 + 1.66473i 0.202612 + 0.350934i 0.500000 + 0.866025i −2.12258 0.528185i −0.500000 + 0.866025i −3.03742 −2.54268 + 1.59209i −1.26284
106.3 0.368623 0.638475i −0.764584 + 1.55416i 0.728233 + 1.26134i 0.500000 + 0.866025i 0.710448 + 1.06107i −0.500000 + 0.866025i 2.54827 −1.83082 2.37657i 0.737247
106.4 0.746337 1.29269i −1.73190 + 0.0231100i −0.114038 0.197519i 0.500000 + 0.866025i −1.26270 + 2.25606i −0.500000 + 0.866025i 2.64491 2.99893 0.0800482i 1.49267
106.5 1.09108 1.88981i 1.65373 + 0.514941i −1.38091 2.39181i 0.500000 + 0.866025i 2.77750 2.56340i −0.500000 + 0.866025i −1.66244 2.46967 + 1.70315i 2.18216
106.6 1.27897 2.21523i −0.382799 1.68922i −2.27151 3.93437i 0.500000 + 0.866025i −4.23161 1.31247i −0.500000 + 0.866025i −6.50486 −2.70693 + 1.29326i 2.55793
211.1 −1.35359 2.34448i 1.24736 + 1.20170i −2.66438 + 4.61485i 0.500000 0.866025i 1.12895 4.55102i −0.500000 0.866025i 9.01155 0.111834 + 2.99791i −2.70717
211.2 −0.631422 1.09366i 0.478182 1.66473i 0.202612 0.350934i 0.500000 0.866025i −2.12258 + 0.528185i −0.500000 0.866025i −3.03742 −2.54268 1.59209i −1.26284
211.3 0.368623 + 0.638475i −0.764584 1.55416i 0.728233 1.26134i 0.500000 0.866025i 0.710448 1.06107i −0.500000 0.866025i 2.54827 −1.83082 + 2.37657i 0.737247
211.4 0.746337 + 1.29269i −1.73190 0.0231100i −0.114038 + 0.197519i 0.500000 0.866025i −1.26270 2.25606i −0.500000 0.866025i 2.64491 2.99893 + 0.0800482i 1.49267
211.5 1.09108 + 1.88981i 1.65373 0.514941i −1.38091 + 2.39181i 0.500000 0.866025i 2.77750 + 2.56340i −0.500000 0.866025i −1.66244 2.46967 1.70315i 2.18216
211.6 1.27897 + 2.21523i −0.382799 + 1.68922i −2.27151 + 3.93437i 0.500000 0.866025i −4.23161 + 1.31247i −0.500000 0.866025i −6.50486 −2.70693 1.29326i 2.55793
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.i.e 12
3.b odd 2 1 945.2.i.e 12
9.c even 3 1 inner 315.2.i.e 12
9.c even 3 1 2835.2.a.u 6
9.d odd 6 1 945.2.i.e 12
9.d odd 6 1 2835.2.a.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.e 12 1.a even 1 1 trivial
315.2.i.e 12 9.c even 3 1 inner
945.2.i.e 12 3.b odd 2 1
945.2.i.e 12 9.d odd 6 1
2835.2.a.u 6 9.c even 3 1
2835.2.a.v 6 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 4 T^{2} - 9 T^{3} + 17 T^{4} - 15 T^{5} + 18 T^{6} - 24 T^{7} - 14 T^{8} + 72 T^{9} - 127 T^{10} + 258 T^{11} - 443 T^{12} + 516 T^{13} - 508 T^{14} + 576 T^{15} - 224 T^{16} - 768 T^{17} + 1152 T^{18} - 1920 T^{19} + 4352 T^{20} - 4608 T^{21} + 4096 T^{22} - 6144 T^{23} + 4096 T^{24}$$
$3$ $$1 - T + 2 T^{2} - T^{3} - 4 T^{4} + 20 T^{5} - 38 T^{6} + 60 T^{7} - 36 T^{8} - 27 T^{9} + 162 T^{10} - 243 T^{11} + 729 T^{12}$$
$5$ $$( 1 - T + T^{2} )^{6}$$
$7$ $$( 1 + T + T^{2} )^{6}$$
$11$ $$1 - 7 T + 2 T^{2} - 101 T^{3} + 1217 T^{4} - 1299 T^{5} + 3761 T^{6} - 100272 T^{7} + 184115 T^{8} + 8971 T^{9} + 4867308 T^{10} - 13424374 T^{11} - 3841008 T^{12} - 147668114 T^{13} + 588944268 T^{14} + 11940401 T^{15} + 2695627715 T^{16} - 16148905872 T^{17} + 6662840921 T^{18} - 25313835129 T^{19} + 260874758177 T^{20} - 238152716791 T^{21} + 51874849202 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 + 10 T + 13 T^{2} - 140 T^{3} - 193 T^{4} + 1289 T^{5} - 4213 T^{6} - 33844 T^{7} + 25985 T^{8} + 133822 T^{9} - 1152335 T^{10} + 1550359 T^{11} + 32289884 T^{12} + 20154667 T^{13} - 194744615 T^{14} + 294006934 T^{15} + 742157585 T^{16} - 12566040292 T^{17} - 20335346317 T^{18} + 80882838413 T^{19} - 157436029153 T^{20} - 1484629912220 T^{21} + 1792160394037 T^{22} + 17921603940370 T^{23} + 23298085122481 T^{24}$$
$17$ $$( 1 - 7 T + 51 T^{2} - 194 T^{3} + 1000 T^{4} - 2743 T^{5} + 13847 T^{6} - 46631 T^{7} + 289000 T^{8} - 953122 T^{9} + 4259571 T^{10} - 9938999 T^{11} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 15 T + 143 T^{2} - 992 T^{3} + 5604 T^{4} - 27269 T^{5} + 123824 T^{6} - 518111 T^{7} + 2023044 T^{8} - 6804128 T^{9} + 18635903 T^{10} - 37141485 T^{11} + 47045881 T^{12} )^{2}$$
$23$ $$1 - 14 T + 34 T^{2} + 44 T^{3} + 2382 T^{4} - 11350 T^{5} - 42548 T^{6} - 846 T^{7} + 1532858 T^{8} + 1867708 T^{9} - 19984074 T^{10} - 41596486 T^{11} + 327972422 T^{12} - 956719178 T^{13} - 10571575146 T^{14} + 22724403236 T^{15} + 428956515578 T^{16} - 5445146178 T^{17} - 6298631005172 T^{18} - 38644768823450 T^{19} + 186536766939342 T^{20} + 79250717104372 T^{21} + 1408501381264066 T^{22} - 13339336610794978 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 + 13 T - 59 T^{2} - 922 T^{3} + 7941 T^{4} + 68816 T^{5} - 482663 T^{6} - 2453358 T^{7} + 29192036 T^{8} + 77970550 T^{9} - 1143760647 T^{10} - 657761371 T^{11} + 39749720555 T^{12} - 19075079759 T^{13} - 961902704127 T^{14} + 1901623743950 T^{15} + 20646972414116 T^{16} - 50321191488342 T^{17} - 287099208583823 T^{18} + 1187067488080144 T^{19} + 3972456765323301 T^{20} - 13375588589751218 T^{21} - 24821726764711859 T^{22} + 158606626954175777 T^{23} + 353814783205469041 T^{24}$$
$31$ $$1 + 10 T - 32 T^{2} + 202 T^{3} + 8390 T^{4} + 1445 T^{5} - 77485 T^{6} + 2882135 T^{7} + 6359417 T^{8} - 43083374 T^{9} + 605642146 T^{10} + 2584009420 T^{11} - 10106229799 T^{12} + 80104292020 T^{13} + 582022102306 T^{14} - 1283496794834 T^{15} + 5873055147257 T^{16} + 82513078117385 T^{17} - 68768222722285 T^{18} + 39755727390395 T^{19} + 7155755804129990 T^{20} + 5340803676455542 T^{21} - 26228105183385632 T^{22} + 254084768964048310 T^{23} + 787662783788549761 T^{24}$$
$37$ $$( 1 - 21 T + 257 T^{2} - 2192 T^{3} + 15138 T^{4} - 92879 T^{5} + 560324 T^{6} - 3436523 T^{7} + 20723922 T^{8} - 111031376 T^{9} + 481659377 T^{10} - 1456223097 T^{11} + 2565726409 T^{12} )^{2}$$
$41$ $$1 + 4 T - 158 T^{2} + 14 T^{3} + 16080 T^{4} - 33799 T^{5} - 940223 T^{6} + 3880863 T^{7} + 37316861 T^{8} - 184680932 T^{9} - 912886272 T^{10} + 3736722920 T^{11} + 26275273283 T^{12} + 153205639720 T^{13} - 1534561823232 T^{14} - 12728394514372 T^{15} + 105448530456221 T^{16} + 449622043781463 T^{17} - 4466157259785743 T^{18} - 6582499702903919 T^{19} + 128397597684265680 T^{20} + 4583347081515454 T^{21} - 2120780171004079358 T^{22} + 2201316126864993764 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$1 + 13 T - 59 T^{2} - 1298 T^{3} + 1643 T^{4} + 58718 T^{5} - 206473 T^{6} - 2100910 T^{7} + 20656952 T^{8} + 88473724 T^{9} - 1065054743 T^{10} - 2017009199 T^{11} + 42128186297 T^{12} - 86731395557 T^{13} - 1969286219807 T^{14} + 7034280374068 T^{15} + 70622008154552 T^{16} - 308851507983130 T^{17} - 1305190792816177 T^{18} + 15960645206980826 T^{19} + 19203713056098443 T^{20} - 652365210294022214 T^{21} - 1275077456483770691 T^{22} + 12080818613125895191 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 - 8 T - 163 T^{2} + 1298 T^{3} + 15011 T^{4} - 102465 T^{5} - 1195399 T^{6} + 5422512 T^{7} + 87138701 T^{8} - 218512510 T^{9} - 5181134511 T^{10} + 4257232207 T^{11} + 259375228416 T^{12} + 200089913729 T^{13} - 11445126134799 T^{14} - 22686624325730 T^{15} + 425209063634381 T^{16} + 1243626052597584 T^{17} - 12885463225071271 T^{18} - 51911138038241295 T^{19} + 357431224079694371 T^{20} + 1452631354087391566 T^{21} - 8573658554440297987 T^{22} - 19777273720672098424 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 - 10 T + 194 T^{2} - 1431 T^{3} + 17912 T^{4} - 104359 T^{5} + 1085054 T^{6} - 5531027 T^{7} + 50314808 T^{8} - 213042987 T^{9} + 1530753314 T^{10} - 4181954930 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$1 - 21 T + 31 T^{2} + 1548 T^{3} + 1085 T^{4} - 95916 T^{5} - 961065 T^{6} + 11023368 T^{7} + 57284362 T^{8} - 412976394 T^{9} - 5014563901 T^{10} - 1966652877 T^{11} + 492853263145 T^{12} - 116032519743 T^{13} - 17455696939381 T^{14} - 84816678823326 T^{15} + 694135294008682 T^{16} + 7880873640019032 T^{17} - 40538234563687665 T^{18} - 238701495817899204 T^{19} + 159311024800688285 T^{20} + 13410317527277845572 T^{21} + 15844619352319883431 T^{22} -$$$$63\!\cdots\!39$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 2 T - 229 T^{2} + 106 T^{3} + 27669 T^{4} - 60786 T^{5} - 2252600 T^{6} + 7627838 T^{7} + 144871004 T^{8} - 483509158 T^{9} - 8189045917 T^{10} + 12920273306 T^{11} + 479371943129 T^{12} + 788136671666 T^{13} - 30471439857157 T^{14} - 109747392191998 T^{15} + 2005860886894364 T^{16} + 6442443759427238 T^{17} - 116054795285588600 T^{18} - 191034766030372506 T^{19} + 5304349643321767989 T^{20} + 1239579485840418946 T^{21} -$$$$16\!\cdots\!29$$$$T^{22} + 87027835222871677322 T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 6 T - 135 T^{2} - 1082 T^{3} + 1827 T^{4} + 35682 T^{5} + 102358 T^{6} + 2130660 T^{7} + 45193590 T^{8} + 41844472 T^{9} - 3116106003 T^{10} - 9659182248 T^{11} + 64516763713 T^{12} - 647165210616 T^{13} - 13988199847467 T^{14} + 12585268932136 T^{15} + 910701500514390 T^{16} + 2876657560480620 T^{17} + 9259139082054502 T^{18} + 216258311501135286 T^{19} + 741885646895983107 T^{20} - 29437470216791132654 T^{21} -$$$$24\!\cdots\!15$$$$T^{22} +$$$$73\!\cdots\!98$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$( 1 + 29 T + 653 T^{2} + 10236 T^{3} + 134114 T^{4} + 1436171 T^{5} + 13182683 T^{6} + 101968141 T^{7} + 676068674 T^{8} + 3663576996 T^{9} + 16593827693 T^{10} + 52322651179 T^{11} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 8 T + 205 T^{2} - 1029 T^{3} + 18060 T^{4} - 73748 T^{5} + 1316041 T^{6} - 5383604 T^{7} + 96241740 T^{8} - 400298493 T^{9} + 5821639405 T^{10} - 16584572744 T^{11} + 151334226289 T^{12} )^{2}$$
$79$ $$1 - 22 T + 12 T^{2} + 2648 T^{3} - 8141 T^{4} - 130972 T^{5} - 734750 T^{6} + 15896118 T^{7} + 81246412 T^{8} - 907461286 T^{9} - 9153348513 T^{10} - 1468497462 T^{11} + 1287301310934 T^{12} - 116011299498 T^{13} - 57126048069633 T^{14} - 447413804988154 T^{15} + 3164554328359372 T^{16} + 48913251611159082 T^{17} - 178608507944054750 T^{18} - 2515174367735216548 T^{19} - 12350782821449313101 T^{20} +$$$$31\!\cdots\!12$$$$T^{21} +$$$$11\!\cdots\!12$$$$T^{22} -$$$$16\!\cdots\!38$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 5 T - 361 T^{2} + 1484 T^{3} + 71060 T^{4} - 224895 T^{5} - 10522018 T^{6} + 24042225 T^{7} + 1282356419 T^{8} - 1800322606 T^{9} - 131803281744 T^{10} + 61097984545 T^{11} + 11701835672526 T^{12} + 5071132717235 T^{13} - 907992807934416 T^{14} - 1029401061916922 T^{15} + 60858482569312499 T^{16} + 94703301423150675 T^{17} - 3440072493515338642 T^{18} - 6102762187312164165 T^{19} +$$$$16\!\cdots\!60$$$$T^{20} +$$$$27\!\cdots\!52$$$$T^{21} -$$$$56\!\cdots\!89$$$$T^{22} -$$$$64\!\cdots\!35$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 - T + 315 T^{2} - 80 T^{3} + 52930 T^{4} - 6403 T^{5} + 5804756 T^{6} - 569867 T^{7} + 419258530 T^{8} - 56397520 T^{9} + 19763805915 T^{10} - 5584059449 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$1 + 32 T + 96 T^{2} - 4240 T^{3} + 41557 T^{4} + 1467386 T^{5} - 2429426 T^{6} - 116703246 T^{7} + 1775844382 T^{8} + 18457608542 T^{9} - 163047635337 T^{10} - 64121694798 T^{11} + 28748647747254 T^{12} - 6219804395406 T^{13} - 1534115200885833 T^{14} + 16845760960852766 T^{15} + 157214226306349342 T^{16} - 1002170482498374222 T^{17} - 2023643846046640754 T^{18} +$$$$11\!\cdots\!18$$$$T^{19} +$$$$32\!\cdots\!77$$$$T^{20} -$$$$32\!\cdots\!80$$$$T^{21} +$$$$70\!\cdots\!04$$$$T^{22} +$$$$22\!\cdots\!96$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$