# Properties

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

# Learn more about

## 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} - 81 x^{3} + 567 x^{2} - 729 x + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{11} - 1173 \nu^{10} + 1963 \nu^{9} - 2091 \nu^{8} - 5201 \nu^{7} + 1905 \nu^{6} - 10452 \nu^{5} + 5769 \nu^{4} - 42354 \nu^{3} - 53541 \nu^{2} + 165564 \nu - 282366$$$$)/11421$$ $$\beta_{3}$$ $$=$$ $$($$$$-12 \nu^{11} + 221 \nu^{10} - 252 \nu^{9} + 161 \nu^{8} + 828 \nu^{7} - 316 \nu^{6} + 1677 \nu^{5} - 762 \nu^{4} + 5031 \nu^{3} + 12276 \nu^{2} - 24462 \nu + 34992$$$$)/3807$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} + 7 \nu^{9} - 3 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 21 \nu^{5} + 72 \nu^{4} - 18 \nu^{3} - 81 \nu^{2} + 567 \nu - 729$$$$)/243$$ $$\beta_{5}$$ $$=$$ $$($$$$-46 \nu^{11} - 132 \nu^{10} + 209 \nu^{9} - 456 \nu^{8} - 727 \nu^{7} - 240 \nu^{6} - 1068 \nu^{5} - 477 \nu^{4} - 8703 \nu^{3} - 2997 \nu^{2} + 14094 \nu - 37179$$$$)/3807$$ $$\beta_{6}$$ $$=$$ $$($$$$51 \nu^{11} - 199 \nu^{10} + 225 \nu^{9} + 56 \nu^{8} - 558 \nu^{7} + 497 \nu^{6} - 1311 \nu^{5} + 2604 \nu^{4} - 1395 \nu^{3} - 12834 \nu^{2} + 25920 \nu - 23085$$$$)/3807$$ $$\beta_{7}$$ $$=$$ $$($$$$196 \nu^{11} - 633 \nu^{10} + 967 \nu^{9} + 582 \nu^{8} - 1445 \nu^{7} + 2310 \nu^{6} - 3468 \nu^{5} + 10143 \nu^{4} - 1098 \nu^{3} - 39204 \nu^{2} + 97524 \nu - 69012$$$$)/11421$$ $$\beta_{8}$$ $$=$$ $$($$$$-95 \nu^{11} + 132 \nu^{10} - 68 \nu^{9} - 390 \nu^{8} + 22 \nu^{7} - 606 \nu^{6} + 504 \nu^{5} - 2907 \nu^{4} - 6102 \nu^{3} + 11880 \nu^{2} - 15363 \nu - 8505$$$$)/3807$$ $$\beta_{9}$$ $$=$$ $$($$$$-365 \nu^{11} + 1227 \nu^{10} - 1367 \nu^{9} - 222 \nu^{8} + 3565 \nu^{7} - 2922 \nu^{6} + 8556 \nu^{5} - 12015 \nu^{4} + 12132 \nu^{3} + 82512 \nu^{2} - 141912 \nu + 150660$$$$)/11421$$ $$\beta_{10}$$ $$=$$ $$($$$$151 \nu^{11} + 43 \nu^{10} - 119 \nu^{9} + 1174 \nu^{8} + 1096 \nu^{7} + 1219 \nu^{6} + 2574 \nu^{5} + 5523 \nu^{4} + 18720 \nu^{3} - 2475 \nu^{2} + 5157 \nu + 66015$$$$)/3807$$ $$\beta_{11}$$ $$=$$ $$($$$$-713 \nu^{11} + 1620 \nu^{10} - 2471 \nu^{9} - 1710 \nu^{8} + 3325 \nu^{7} - 8514 \nu^{6} + 7557 \nu^{5} - 30447 \nu^{4} - 8631 \nu^{3} + 114075 \nu^{2} - 249804 \nu + 183222$$$$)/11421$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} - 2 \beta_{8} - \beta_{6} - \beta_{4} + \beta_{2} - \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + \beta_{8} - \beta_{7} + 6 \beta_{6} + 2 \beta_{3} - 3 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{11} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - 4 \beta_{6} + 7 \beta_{5} - 4 \beta_{4} - 2 \beta_{2} + 4 \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$-\beta_{11} + 2 \beta_{10} + 10 \beta_{8} - 8 \beta_{7} + 6 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} - 11 \beta_{3} - \beta_{2} - 2 \beta_{1} + 14$$ $$\nu^{7}$$ $$=$$ $$-6 \beta_{11} - 12 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 12 \beta_{7} - 10 \beta_{6} + 17 \beta_{5} + 17 \beta_{4} - 3 \beta_{3} - 20 \beta_{2} + 24 \beta_{1} - 11$$ $$\nu^{8}$$ $$=$$ $$10 \beta_{11} + 7 \beta_{10} - 10 \beta_{9} + 25 \beta_{8} + 18 \beta_{7} + 6 \beta_{6} - 19 \beta_{5} + \beta_{4} - 9 \beta_{3} + \beta_{2} - 13 \beta_{1} - 24$$ $$\nu^{9}$$ $$=$$ $$-9 \beta_{11} - 15 \beta_{10} + 13 \beta_{9} - 41 \beta_{8} + 45 \beta_{7} - 67 \beta_{6} + 21 \beta_{5} - 31 \beta_{4} + 21 \beta_{3} + 4 \beta_{2} - 37 \beta_{1} + 25$$ $$\nu^{10}$$ $$=$$ $$-27 \beta_{11} - 27 \beta_{10} + 28 \beta_{9} + \beta_{8} - \beta_{7} + 33 \beta_{6} - 36 \beta_{5} - 90 \beta_{4} + 56 \beta_{3} + 18 \beta_{2} + 24 \beta_{1} + 37$$ $$\nu^{11}$$ $$=$$ $$42 \beta_{11} + 54 \beta_{10} - 143 \beta_{9} + 34 \beta_{8} - 39 \beta_{7} - 112 \beta_{6} - 20 \beta_{5} + 14 \beta_{4} + 108 \beta_{3} + 70 \beta_{2} + 22 \beta_{1} + 22$$

## Character Values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$ $$\beta_{6}$$

## 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
 0.628063 − 1.61417i 1.22323 − 1.22626i 1.39898 + 1.02120i −1.67391 − 0.444996i 0.702501 + 1.58319i −0.778860 + 1.54705i 0.628063 + 1.61417i 1.22323 + 1.22626i 1.39898 − 1.02120i −1.67391 + 0.444996i 0.702501 − 1.58319i −0.778860 − 1.54705i
1.00000 −1.71194 0.263165i 1.00000 −0.500000 0.866025i −1.71194 0.263165i −2.25729 + 1.38008i 1.00000 2.86149 + 0.901046i −0.500000 0.866025i
121.2 1.00000 −1.67359 + 0.446216i 1.00000 −0.500000 0.866025i −1.67359 + 0.446216i 1.40545 + 2.24159i 1.00000 2.60178 1.49356i −0.500000 0.866025i
121.3 1.00000 0.184900 + 1.72215i 1.00000 −0.500000 0.866025i 0.184900 + 1.72215i −2.25729 + 1.38008i 1.00000 −2.93162 + 0.636851i −0.500000 0.866025i
121.4 1.00000 0.451577 1.67215i 1.00000 −0.500000 0.866025i 0.451577 1.67215i 1.85185 1.88962i 1.00000 −2.59216 1.51021i −0.500000 0.866025i
121.5 1.00000 1.01983 + 1.39998i 1.00000 −0.500000 0.866025i 1.01983 + 1.39998i 1.85185 1.88962i 1.00000 −0.919882 + 2.85549i −0.500000 0.866025i
121.6 1.00000 1.72922 + 0.0990147i 1.00000 −0.500000 0.866025i 1.72922 + 0.0990147i 1.40545 + 2.24159i 1.00000 2.98039 + 0.342436i −0.500000 0.866025i
151.1 1.00000 −1.71194 + 0.263165i 1.00000 −0.500000 + 0.866025i −1.71194 + 0.263165i −2.25729 1.38008i 1.00000 2.86149 0.901046i −0.500000 + 0.866025i
151.2 1.00000 −1.67359 0.446216i 1.00000 −0.500000 + 0.866025i −1.67359 0.446216i 1.40545 2.24159i 1.00000 2.60178 + 1.49356i −0.500000 + 0.866025i
151.3 1.00000 0.184900 1.72215i 1.00000 −0.500000 + 0.866025i 0.184900 1.72215i −2.25729 1.38008i 1.00000 −2.93162 0.636851i −0.500000 + 0.866025i
151.4 1.00000 0.451577 + 1.67215i 1.00000 −0.500000 + 0.866025i 0.451577 + 1.67215i 1.85185 + 1.88962i 1.00000 −2.59216 + 1.51021i −0.500000 + 0.866025i
151.5 1.00000 1.01983 1.39998i 1.00000 −0.500000 + 0.866025i 1.01983 1.39998i 1.85185 + 1.88962i 1.00000 −0.919882 2.85549i −0.500000 + 0.866025i
151.6 1.00000 1.72922 0.0990147i 1.00000 −0.500000 + 0.866025i 1.72922 0.0990147i 1.40545 2.24159i 1.00000 2.98039 0.342436i −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 $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

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