# Properties

 Label 630.2.i.h Level 630 Weight 2 Character orbit 630.i Analytic conductor 5.031 Analytic rank 0 Dimension 12 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.i (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + 104 x^{3} + 197 x^{2} + 166 x + 79$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 18 \nu^{8} + 18 \nu^{7} - 6 \nu^{6} + 27 \nu^{5} + 69 \nu^{4} + 183 \nu^{3} + 287 \nu^{2} + 484 \nu + 407$$$$)/243$$ $$\beta_{2}$$ $$=$$ $$($$$$35 \nu^{11} + 82 \nu^{10} - 1108 \nu^{9} + 4209 \nu^{8} - 10470 \nu^{7} + 18003 \nu^{6} - 18903 \nu^{5} + 22194 \nu^{4} + 210 \nu^{3} + 18844 \nu^{2} + 10010 \nu + 20317$$$$)/5589$$ $$\beta_{3}$$ $$=$$ $$($$$$20 \nu^{11} - 124 \nu^{10} + 323 \nu^{9} - 506 \nu^{8} + 628 \nu^{7} - 608 \nu^{6} + 1063 \nu^{5} - 1121 \nu^{4} + 373 \nu^{3} - 640 \nu^{2} + 338 \nu - 2095$$$$)/1863$$ $$\beta_{4}$$ $$=$$ $$($$$$-21 \nu^{11} + 167 \nu^{10} - 591 \nu^{9} + 1610 \nu^{8} - 3769 \nu^{7} + 6692 \nu^{6} - 9625 \nu^{5} + 10484 \nu^{4} - 4036 \nu^{3} + 5939 \nu^{2} + 5034 \nu + 10796$$$$)/1863$$ $$\beta_{5}$$ $$=$$ $$($$$$-116 \nu^{11} + 641 \nu^{10} - 1763 \nu^{9} + 3174 \nu^{8} - 3693 \nu^{7} + 2151 \nu^{6} - 4215 \nu^{5} + 2532 \nu^{4} - 17532 \nu^{3} + 3344 \nu^{2} - 13304 \nu - 14230$$$$)/5589$$ $$\beta_{6}$$ $$=$$ $$($$$$-163 \nu^{11} + 1144 \nu^{10} - 4285 \nu^{9} + 11247 \nu^{8} - 21531 \nu^{7} + 29073 \nu^{6} - 30441 \nu^{5} + 14103 \nu^{4} - 9051 \nu^{3} - 3593 \nu^{2} - 13291 \nu + 7834$$$$)/5589$$ $$\beta_{7}$$ $$=$$ $$($$$$-67 \nu^{11} + 397 \nu^{10} - 1327 \nu^{9} + 3450 \nu^{8} - 6828 \nu^{7} + 9774 \nu^{6} - 12408 \nu^{5} + 6666 \nu^{4} - 6750 \nu^{3} - 317 \nu^{2} - 6121 \nu + 8266$$$$)/1863$$ $$\beta_{8}$$ $$=$$ $$($$$$-83 \nu^{11} + 441 \nu^{10} - 1199 \nu^{9} + 2116 \nu^{8} - 1976 \nu^{7} - 683 \nu^{6} + 1894 \nu^{5} - 7286 \nu^{4} - 3212 \nu^{3} - 6636 \nu^{2} - 9386 \nu - 9792$$$$)/1863$$ $$\beta_{9}$$ $$=$$ $$($$$$305 \nu^{11} - 1799 \nu^{10} + 6047 \nu^{9} - 15249 \nu^{8} + 28230 \nu^{7} - 37608 \nu^{6} + 46473 \nu^{5} - 20160 \nu^{4} + 31431 \nu^{3} + 11860 \nu^{2} + 29477 \nu - 16556$$$$)/5589$$ $$\beta_{10}$$ $$=$$ $$($$$$50 \nu^{11} - 310 \nu^{10} + 1026 \nu^{9} - 2323 \nu^{8} + 3594 \nu^{7} - 3452 \nu^{6} + 2876 \nu^{5} + 1050 \nu^{4} + 2830 \nu^{3} + 1620 \nu^{2} + 6250 \nu + 1720$$$$)/621$$ $$\beta_{11}$$ $$=$$ $$($$$$-670 \nu^{11} + 3349 \nu^{10} - 8923 \nu^{9} + 16284 \nu^{8} - 16530 \nu^{7} - 1827 \nu^{6} + 8607 \nu^{5} - 62922 \nu^{4} - 34794 \nu^{3} - 73550 \nu^{2} - 95158 \nu - 92048$$$$)/5589$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + 2 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 3$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_{2} + 4 \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{11} + 16 \beta_{10} - 9 \beta_{9} - \beta_{8} - 9 \beta_{6} + 5 \beta_{5} - 16 \beta_{4} - 17 \beta_{3} + 40 \beta_{2} + 17 \beta_{1} + 30$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$40 \beta_{11} + 35 \beta_{10} - 24 \beta_{9} - 38 \beta_{8} - 9 \beta_{7} - 27 \beta_{6} + 13 \beta_{5} - 38 \beta_{4} - 55 \beta_{3} + 104 \beta_{2} + 64 \beta_{1} + 75$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$37 \beta_{11} + 19 \beta_{10} - 26 \beta_{9} - 53 \beta_{8} - 16 \beta_{7} - 34 \beta_{6} + 2 \beta_{5} - 26 \beta_{4} - 56 \beta_{3} + 81 \beta_{2} + 58 \beta_{1} + 21$$ $$\nu^{7}$$ $$=$$ $$($$$$305 \beta_{11} + 46 \beta_{10} - 216 \beta_{9} - 502 \beta_{8} - 180 \beta_{7} - 402 \beta_{6} - 121 \beta_{5} - 70 \beta_{4} - 410 \beta_{3} + 598 \beta_{2} + 350 \beta_{1} - 144$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$820 \beta_{11} - 133 \beta_{10} - 444 \beta_{9} - 1442 \beta_{8} - 534 \beta_{7} - 1272 \beta_{6} - 638 \beta_{5} + 256 \beta_{4} - 796 \beta_{3} + 1400 \beta_{2} + 514 \beta_{1} - 822$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$649 \beta_{11} - 327 \beta_{10} - 210 \beta_{9} - 1281 \beta_{8} - 468 \beta_{7} - 1090 \beta_{6} - 717 \beta_{5} + 565 \beta_{4} - 403 \beta_{3} + 855 \beta_{2} + 87 \beta_{1} - 963$$ $$\nu^{10}$$ $$=$$ $$($$$$3701 \beta_{11} - 4097 \beta_{10} - 288 \beta_{9} - 9100 \beta_{8} - 3483 \beta_{7} - 7119 \beta_{6} - 6058 \beta_{5} + 6395 \beta_{4} - 806 \beta_{3} + 2332 \beta_{2} - 2101 \beta_{1} - 9297$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$4489 \beta_{11} - 13663 \beta_{10} + 2394 \beta_{9} - 17963 \beta_{8} - 7719 \beta_{7} - 13011 \beta_{6} - 15353 \beta_{5} + 19726 \beta_{4} + 4469 \beta_{3} - 6388 \beta_{2} - 12404 \beta_{1} - 27441$$$$)/3$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.142686 + 1.50500i −0.659665 − 0.495491i −0.633121 + 0.576989i 0.429413 − 1.63537i 2.48293 + 0.894932i 0.737753 − 1.71208i 0.142686 − 1.50500i −0.659665 + 0.495491i −0.633121 − 0.576989i 0.429413 + 1.63537i 2.48293 − 0.894932i 0.737753 + 1.71208i
1.00000 −1.35158 1.08315i 1.00000 0.500000 + 0.866025i −1.35158 1.08315i −0.710533 2.54856i 1.00000 0.653555 + 2.92795i 0.500000 + 0.866025i
121.2 1.00000 −1.13098 1.31183i 1.00000 0.500000 + 0.866025i −1.13098 1.31183i 0.0665372 + 2.64491i 1.00000 −0.441782 + 2.96729i 0.500000 + 0.866025i
121.3 1.00000 0.0335666 + 1.73173i 1.00000 0.500000 + 0.866025i 0.0335666 + 1.73173i 2.64400 0.0963576i 1.00000 −2.99775 + 0.116256i 0.500000 + 0.866025i
121.4 1.00000 0.400725 1.68506i 1.00000 0.500000 + 0.866025i 0.400725 1.68506i 0.0665372 + 2.64491i 1.00000 −2.67884 1.35049i 0.500000 + 0.866025i
121.5 1.00000 1.31625 1.12583i 1.00000 0.500000 + 0.866025i 1.31625 1.12583i 2.64400 0.0963576i 1.00000 0.465015 2.96374i 0.500000 + 0.866025i
121.6 1.00000 1.73202 + 0.0100417i 1.00000 0.500000 + 0.866025i 1.73202 + 0.0100417i −0.710533 2.54856i 1.00000 2.99980 + 0.0347849i 0.500000 + 0.866025i
151.1 1.00000 −1.35158 + 1.08315i 1.00000 0.500000 0.866025i −1.35158 + 1.08315i −0.710533 + 2.54856i 1.00000 0.653555 2.92795i 0.500000 0.866025i
151.2 1.00000 −1.13098 + 1.31183i 1.00000 0.500000 0.866025i −1.13098 + 1.31183i 0.0665372 2.64491i 1.00000 −0.441782 2.96729i 0.500000 0.866025i
151.3 1.00000 0.0335666 1.73173i 1.00000 0.500000 0.866025i 0.0335666 1.73173i 2.64400 + 0.0963576i 1.00000 −2.99775 0.116256i 0.500000 0.866025i
151.4 1.00000 0.400725 + 1.68506i 1.00000 0.500000 0.866025i 0.400725 + 1.68506i 0.0665372 2.64491i 1.00000 −2.67884 + 1.35049i 0.500000 0.866025i
151.5 1.00000 1.31625 + 1.12583i 1.00000 0.500000 0.866025i 1.31625 + 1.12583i 2.64400 + 0.0963576i 1.00000 0.465015 + 2.96374i 0.500000 0.866025i
151.6 1.00000 1.73202 0.0100417i 1.00000 0.500000 0.866025i 1.73202 0.0100417i −0.710533 + 2.54856i 1.00000 2.99980 0.0347849i 0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
63.h Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{12} + \cdots$$ $$T_{13}^{12} - \cdots$$