# Properties

 Label 630.2.bo.b Level 630 Weight 2 Character orbit 630.bo Analytic conductor 5.031 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 630.bo (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.03057532734$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{8} ) q^{2} -\beta_{8} q^{4} + \beta_{1} q^{5} -\beta_{3} q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{8} ) q^{2} -\beta_{8} q^{4} + \beta_{1} q^{5} -\beta_{3} q^{7} - q^{8} + ( 1 + \beta_{1} - \beta_{8} - \beta_{9} ) q^{10} + ( \beta_{6} - \beta_{10} - \beta_{13} - \beta_{15} ) q^{11} + ( \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{13} -\beta_{15} q^{14} + ( -1 + \beta_{8} ) q^{16} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{17} + ( -2 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{19} + ( 1 - \beta_{8} - \beta_{9} ) q^{20} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} - \beta_{15} ) q^{22} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{23} + ( -2 + \beta_{1} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{10} - \beta_{12} - \beta_{15} ) q^{25} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{13} ) q^{26} + ( \beta_{3} - \beta_{15} ) q^{28} + ( 2 + 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{29} + ( -1 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 4 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{31} + \beta_{8} q^{32} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{34} + ( 3 + \beta_{4} - 2 \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{35} + ( \beta_{3} + 3 \beta_{5} + \beta_{10} + \beta_{13} - \beta_{15} ) q^{37} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{38} -\beta_{1} q^{40} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{13} - \beta_{15} ) q^{41} + ( -2 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} ) q^{43} + ( \beta_{3} - \beta_{5} + \beta_{13} ) q^{44} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{46} + ( -4 + \beta_{1} - \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{14} ) q^{49} + ( -3 + \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{14} - \beta_{15} ) q^{50} + ( 1 + \beta_{1} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{52} + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{53} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{55} + \beta_{3} q^{56} + ( 1 + \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{58} + ( -1 - \beta_{1} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} + ( 5 - \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{61} + ( -2 + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + 2 \beta_{13} ) q^{62} + q^{64} + ( 4 - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 6 \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{65} + ( 2 \beta_{10} + 2 \beta_{13} + 2 \beta_{15} ) q^{67} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{68} + ( 2 - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{70} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{13} + 4 \beta_{15} ) q^{71} + ( 2 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{13} + 2 \beta_{15} ) q^{73} + ( \beta_{3} + 3 \beta_{6} + \beta_{10} ) q^{74} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{76} + ( 7 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{14} ) q^{77} + ( 1 - \beta_{1} - 4 \beta_{2} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{79} + ( -1 - \beta_{1} + \beta_{8} + \beta_{9} ) q^{80} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{13} - 2 \beta_{15} ) q^{82} + ( -2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 8 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{13} ) q^{83} + ( -\beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} ) q^{85} + ( -1 - \beta_{1} + \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{86} + ( -\beta_{6} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{88} + ( -1 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{13} + \beta_{15} ) q^{89} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 7 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{91} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{92} + ( -5 + \beta_{1} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{94} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + 3 \beta_{13} + 3 \beta_{15} ) q^{95} + ( -\beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{10} - \beta_{13} ) q^{97} + ( 1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{2} - 8q^{4} + 6q^{5} - 16q^{8} + O(q^{10})$$ $$16q + 8q^{2} - 8q^{4} + 6q^{5} - 16q^{8} + 6q^{10} - 8q^{16} + 24q^{19} + 8q^{23} - 6q^{25} + 12q^{31} + 8q^{32} - 4q^{35} + 24q^{38} - 6q^{40} - 8q^{46} - 60q^{47} - 28q^{49} - 12q^{50} - 16q^{53} + 24q^{61} + 16q^{64} + 20q^{65} - 14q^{70} + 88q^{77} + 4q^{79} - 6q^{80} + 64q^{85} - 28q^{91} - 16q^{92} - 60q^{94} + 12q^{95} - 20q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} - 6 x^{8} - 420 x^{7} + 4650 x^{6} - 21000 x^{5} + 70625 x^{4} - 168750 x^{3} + 328125 x^{2} - 468750 x + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-384 \nu^{15} + 11789 \nu^{14} - 67399 \nu^{13} + 175846 \nu^{12} - 241632 \nu^{11} + 232392 \nu^{10} - 397554 \nu^{9} - 1213884 \nu^{8} + 1002264 \nu^{7} - 726180 \nu^{6} - 12526500 \nu^{5} + 73399500 \nu^{4} - 173418750 \nu^{3} + 555896875 \nu^{2} - 834078125 \nu + 608281250$$$$)/ 568125000$$ $$\beta_{3}$$ $$=$$ $$($$$$-8097 \nu^{15} + 5902 \nu^{14} - 24132 \nu^{13} + 294133 \nu^{12} - 550416 \nu^{11} + 1092156 \nu^{10} - 99702 \nu^{9} + 7153068 \nu^{8} + 11261652 \nu^{7} + 37261770 \nu^{6} - 97970400 \nu^{5} + 98359500 \nu^{4} - 87703125 \nu^{3} + 326318750 \nu^{2} - 1186125000 \nu + 3815703125$$$$)/ 3408750000$$ $$\beta_{4}$$ $$=$$ $$($$$$6137 \nu^{15} + 20678 \nu^{14} - 42523 \nu^{13} + 210377 \nu^{12} - 519294 \nu^{11} + 1464834 \nu^{10} - 1618968 \nu^{9} - 1809558 \nu^{8} + 6197028 \nu^{7} - 1325940 \nu^{6} + 14289450 \nu^{5} + 21773250 \nu^{4} - 179800625 \nu^{3} + 473087500 \nu^{2} - 1142515625 \nu + 2128046875$$$$)/ 1704375000$$ $$\beta_{5}$$ $$=$$ $$($$$$6613 \nu^{15} + 7932 \nu^{14} - 14237 \nu^{13} + 93033 \nu^{12} + 60504 \nu^{11} + 51246 \nu^{10} + 1248438 \nu^{9} + 311568 \nu^{8} + 3451632 \nu^{7} - 1181070 \nu^{6} - 2914800 \nu^{5} - 20291250 \nu^{4} - 274375 \nu^{3} + 347887500 \nu^{2} + 48828125 \nu + 91640625$$$$)/ 1704375000$$ $$\beta_{6}$$ $$=$$ $$($$$$17122 \nu^{15} - 4902 \nu^{14} + 66757 \nu^{13} - 47583 \nu^{12} + 118266 \nu^{11} + 412344 \nu^{10} + 1336152 \nu^{9} + 2521482 \nu^{8} + 3490098 \nu^{7} + 4944330 \nu^{6} + 42895650 \nu^{5} - 62958000 \nu^{4} + 193373750 \nu^{3} + 246750000 \nu^{2} + 257328125 \nu + 1565390625$$$$)/ 1704375000$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{14} - 21 \nu^{13} + 54 \nu^{12} - 113 \nu^{11} + 168 \nu^{10} - 186 \nu^{9} + 84 \nu^{8} + 6 \nu^{7} + 420 \nu^{6} - 4650 \nu^{5} + 21000 \nu^{4} - 70625 \nu^{3} + 168750 \nu^{2} - 328125 \nu + 468750$$$$)/78125$$ $$\beta_{8}$$ $$=$$ $$($$$$-22146 \nu^{15} + 79646 \nu^{14} - 238236 \nu^{13} + 546479 \nu^{12} - 982878 \nu^{11} + 1301988 \nu^{10} - 369666 \nu^{9} - 1282116 \nu^{8} - 43854 \nu^{7} + 10333650 \nu^{6} - 65558250 \nu^{5} + 287925000 \nu^{4} - 737662500 \nu^{3} + 1557531250 \nu^{2} - 2635218750 \nu + 3913984375$$$$)/ 1704375000$$ $$\beta_{9}$$ $$=$$ $$($$$$-31084 \nu^{15} + 147184 \nu^{14} - 411169 \nu^{13} + 973141 \nu^{12} - 1435662 \nu^{11} + 2447502 \nu^{10} - 2772714 \nu^{9} + 1105386 \nu^{8} + 1076184 \nu^{7} + 27087000 \nu^{6} - 111582750 \nu^{5} + 538473750 \nu^{4} - 1441943750 \nu^{3} + 3073906250 \nu^{2} - 3831734375 \nu + 6441171875$$$$)/ 1704375000$$ $$\beta_{10}$$ $$=$$ $$($$$$29284 \nu^{15} - 117899 \nu^{14} + 303959 \nu^{13} - 699881 \nu^{12} + 1218072 \nu^{11} - 1131672 \nu^{10} + 1395684 \nu^{9} + 2003274 \nu^{8} + 3919626 \nu^{7} - 23957910 \nu^{6} + 133123800 \nu^{5} - 375636000 \nu^{4} + 984950000 \nu^{3} - 2090471875 \nu^{2} + 3278171875 \nu - 2969453125$$$$)/ 1136250000$$ $$\beta_{11}$$ $$=$$ $$($$$$-52692 \nu^{15} + 298312 \nu^{14} - 764517 \nu^{13} + 2102878 \nu^{12} - 3542736 \nu^{11} + 4898436 \nu^{10} - 2122542 \nu^{9} + 4871838 \nu^{8} + 12645312 \nu^{7} + 30074280 \nu^{6} - 179819100 \nu^{5} + 965847000 \nu^{4} - 2661213750 \nu^{3} + 6006781250 \nu^{2} - 9888984375 \nu + 14823593750$$$$)/ 1704375000$$ $$\beta_{12}$$ $$=$$ $$($$$$53838 \nu^{15} - 173323 \nu^{14} + 537918 \nu^{13} - 1115872 \nu^{12} + 1770924 \nu^{11} - 1794444 \nu^{10} + 3038328 \nu^{9} + 2591088 \nu^{8} - 12714198 \nu^{7} - 12184890 \nu^{6} + 130483800 \nu^{5} - 606171000 \nu^{4} + 1569078750 \nu^{3} - 3731140625 \nu^{2} + 5965500000 \nu - 6022343750$$$$)/ 1704375000$$ $$\beta_{13}$$ $$=$$ $$($$$$115983 \nu^{15} - 499393 \nu^{14} + 1698663 \nu^{13} - 4301152 \nu^{12} + 7691484 \nu^{11} - 9146904 \nu^{10} + 7522398 \nu^{9} + 2446458 \nu^{8} - 24049518 \nu^{7} - 62378340 \nu^{6} + 383712300 \nu^{5} - 1937064000 \nu^{4} + 5893719375 \nu^{3} - 12666678125 \nu^{2} + 22579546875 \nu - 27148437500$$$$)/ 3408750000$$ $$\beta_{14}$$ $$=$$ $$($$$$-74084 \nu^{15} + 358884 \nu^{14} - 1026644 \nu^{13} + 2840991 \nu^{12} - 4526112 \nu^{11} + 6163452 \nu^{10} - 4476264 \nu^{9} + 1398786 \nu^{8} - 4597266 \nu^{7} + 70889400 \nu^{6} - 220676100 \nu^{5} + 1302099000 \nu^{4} - 3519160000 \nu^{3} + 8129756250 \nu^{2} - 13068156250 \nu + 16796953125$$$$)/ 1704375000$$ $$\beta_{15}$$ $$=$$ $$($$$$-182946 \nu^{15} + 901171 \nu^{14} - 2551911 \nu^{13} + 6076429 \nu^{12} - 10614828 \nu^{11} + 13033788 \nu^{10} - 9796716 \nu^{9} + 6178134 \nu^{8} + 15497646 \nu^{7} + 136998150 \nu^{6} - 720224100 \nu^{5} + 2960143500 \nu^{4} - 8041421250 \nu^{3} + 17728671875 \nu^{2} - 29359171875 \nu + 36593515625$$$$)/ 3408750000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{15} - \beta_{12} - \beta_{10} + \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} - 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + \beta_{10} + 4 \beta_{9} + 9 \beta_{8} - 4 \beta_{7} + 6 \beta_{6} - 21 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 12 \beta_{2} - 2 \beta_{1} - 9$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 8 \beta_{9} - 6 \beta_{8} - 7 \beta_{7} + 6 \beta_{6} - 12 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 6 \beta_{2} + 4 \beta_{1} + 4$$ $$\nu^{6}$$ $$=$$ $$24 \beta_{15} + 16 \beta_{14} - 22 \beta_{13} - 48 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 28 \beta_{8} - 28 \beta_{7} + 34 \beta_{6} - 10 \beta_{5} + 22 \beta_{4} + 14 \beta_{3} - 24 \beta_{2} - 4 \beta_{1} + 39$$ $$\nu^{7}$$ $$=$$ $$8 \beta_{15} - 28 \beta_{14} - 40 \beta_{13} - 28 \beta_{12} + 6 \beta_{11} + 8 \beta_{10} - 42 \beta_{8} - 48 \beta_{7} + 26 \beta_{6} - 16 \beta_{5} + 48 \beta_{4} + 40 \beta_{3} + 5 \beta_{1} - 2$$ $$\nu^{8}$$ $$=$$ $$55 \beta_{15} + 58 \beta_{13} - 3 \beta_{12} + 42 \beta_{11} + 113 \beta_{10} + 20 \beta_{9} - 162 \beta_{8} + 85 \beta_{7} - 48 \beta_{6} - 167 \beta_{4} + 58 \beta_{3} - 42 \beta_{2} + 23 \beta_{1} - 190$$ $$\nu^{9}$$ $$=$$ $$-33 \beta_{15} - 77 \beta_{14} - 142 \beta_{13} + 154 \beta_{12} + 45 \beta_{10} - 55 \beta_{9} + 825 \beta_{8} - 77 \beta_{7} + 189 \beta_{6} + 276 \beta_{5} - 190 \beta_{4} - 32 \beta_{3} - 87 \beta_{2} - 77 \beta_{1} - 403$$ $$\nu^{10}$$ $$=$$ $$-311 \beta_{15} - 121 \beta_{14} + 334 \beta_{13} + 121 \beta_{12} - 108 \beta_{11} + 311 \beta_{10} + 888 \beta_{9} + 2293 \beta_{8} + 408 \beta_{7} - 108 \beta_{6} + 769 \beta_{5} + 408 \beta_{4} + 334 \beta_{3} - 216 \beta_{2} - 444 \beta_{1} - 2293$$ $$\nu^{11}$$ $$=$$ $$-564 \beta_{15} + 384 \beta_{14} + 756 \beta_{13} - 192 \beta_{12} - 252 \beta_{11} + 768 \beta_{10} + 2736 \beta_{9} + 492 \beta_{8} + 17 \beta_{7} - 2076 \beta_{6} + 4152 \beta_{5} + 192 \beta_{4} + 180 \beta_{3} - 252 \beta_{2} - 2928 \beta_{1} + 1752$$ $$\nu^{12}$$ $$=$$ $$-1152 \beta_{15} + 3456 \beta_{14} + 2196 \beta_{13} + 1728 \beta_{11} + 1044 \beta_{10} - 2004 \beta_{9} - 4008 \beta_{8} - 552 \beta_{7} - 8604 \beta_{6} + 7740 \beta_{5} + 2004 \beta_{4} + 2412 \beta_{3} + 864 \beta_{2} + 552 \beta_{1} + 4945$$ $$\nu^{13}$$ $$=$$ $$-15024 \beta_{15} - 96 \beta_{14} - 2304 \beta_{13} - 96 \beta_{12} + 14220 \beta_{11} - 15024 \beta_{10} - 7620 \beta_{8} + 48 \beta_{7} + 4020 \beta_{6} - 9120 \beta_{5} - 48 \beta_{4} + 2304 \beta_{3} + 157 \beta_{1} - 7908$$ $$\nu^{14}$$ $$=$$ $$6947 \beta_{15} + 10116 \beta_{13} - 3169 \beta_{12} + 9612 \beta_{11} - 10873 \beta_{10} - 8472 \beta_{9} - 26052 \beta_{8} - 6467 \beta_{7} + 2342 \beta_{6} + 16103 \beta_{4} - 17820 \beta_{3} - 9612 \beta_{2} - 5303 \beta_{1} + 21406$$ $$\nu^{15}$$ $$=$$ $$39073 \beta_{15} - 25031 \beta_{14} - 11048 \beta_{13} + 50062 \beta_{12} - 59 \beta_{10} + 23433 \beta_{9} - 57141 \beta_{8} - 25031 \beta_{7} + 21685 \beta_{6} - 30086 \beta_{5} + 12180 \beta_{4} - 25090 \beta_{3} + 51771 \beta_{2} - 25031 \beta_{1} + 47975$$

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1 - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −2.11940 + 0.712845i −1.27963 + 1.83372i −0.442358 − 2.19188i 0.104634 + 2.23362i 0.948234 − 2.02506i 1.68760 + 1.46697i 1.98669 − 1.02619i 2.11423 + 0.728019i −2.11940 − 0.712845i −1.27963 − 1.83372i −0.442358 + 2.19188i 0.104634 − 2.23362i 0.948234 + 2.02506i 1.68760 − 1.46697i 1.98669 + 1.02619i 2.11423 − 0.728019i
0.500000 0.866025i 0 −0.500000 0.866025i −2.11940 + 0.712845i 0 1.63937 + 2.07665i −1.00000 0 −0.442358 + 2.19188i
89.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.27963 + 1.83372i 0 −0.732536 2.54232i −1.00000 0 0.948234 + 2.02506i
89.3 0.500000 0.866025i 0 −0.500000 0.866025i −0.442358 2.19188i 0 −1.63937 2.07665i −1.00000 0 −2.11940 0.712845i
89.4 0.500000 0.866025i 0 −0.500000 0.866025i 0.104634 + 2.23362i 0 −1.39924 + 2.24547i −1.00000 0 1.98669 + 1.02619i
89.5 0.500000 0.866025i 0 −0.500000 0.866025i 0.948234 2.02506i 0 0.732536 + 2.54232i −1.00000 0 −1.27963 1.83372i
89.6 0.500000 0.866025i 0 −0.500000 0.866025i 1.68760 + 1.46697i 0 −2.30608 + 1.29693i −1.00000 0 2.11423 0.728019i
89.7 0.500000 0.866025i 0 −0.500000 0.866025i 1.98669 1.02619i 0 1.39924 2.24547i −1.00000 0 0.104634 2.23362i
89.8 0.500000 0.866025i 0 −0.500000 0.866025i 2.11423 + 0.728019i 0 2.30608 1.29693i −1.00000 0 1.68760 1.46697i
269.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.11940 0.712845i 0 1.63937 2.07665i −1.00000 0 −0.442358 2.19188i
269.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.27963 1.83372i 0 −0.732536 + 2.54232i −1.00000 0 0.948234 2.02506i
269.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.442358 + 2.19188i 0 −1.63937 + 2.07665i −1.00000 0 −2.11940 + 0.712845i
269.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.104634 2.23362i 0 −1.39924 2.24547i −1.00000 0 1.98669 1.02619i
269.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.948234 + 2.02506i 0 0.732536 2.54232i −1.00000 0 −1.27963 + 1.83372i
269.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.68760 1.46697i 0 −2.30608 1.29693i −1.00000 0 2.11423 + 0.728019i
269.7 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.98669 + 1.02619i 0 1.39924 + 2.24547i −1.00000 0 0.104634 + 2.23362i
269.8 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.11423 0.728019i 0 2.30608 + 1.29693i −1.00000 0 1.68760 + 1.46697i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bo.b yes 16
3.b odd 2 1 630.2.bo.a 16
5.b even 2 1 630.2.bo.a 16
5.c odd 4 2 3150.2.bf.f 32
7.c even 3 1 4410.2.d.a 16
7.d odd 6 1 inner 630.2.bo.b yes 16
7.d odd 6 1 4410.2.d.a 16
15.d odd 2 1 inner 630.2.bo.b yes 16
15.e even 4 2 3150.2.bf.f 32
21.g even 6 1 630.2.bo.a 16
21.g even 6 1 4410.2.d.b 16
21.h odd 6 1 4410.2.d.b 16
35.i odd 6 1 630.2.bo.a 16
35.i odd 6 1 4410.2.d.b 16
35.j even 6 1 4410.2.d.b 16
35.k even 12 2 3150.2.bf.f 32
105.o odd 6 1 4410.2.d.a 16
105.p even 6 1 inner 630.2.bo.b yes 16
105.p even 6 1 4410.2.d.a 16
105.w odd 12 2 3150.2.bf.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a 16 3.b odd 2 1
630.2.bo.a 16 5.b even 2 1
630.2.bo.a 16 21.g even 6 1
630.2.bo.a 16 35.i odd 6 1
630.2.bo.b yes 16 1.a even 1 1 trivial
630.2.bo.b yes 16 7.d odd 6 1 inner
630.2.bo.b yes 16 15.d odd 2 1 inner
630.2.bo.b yes 16 105.p even 6 1 inner
3150.2.bf.f 32 5.c odd 4 2
3150.2.bf.f 32 15.e even 4 2
3150.2.bf.f 32 35.k even 12 2
3150.2.bf.f 32 105.w odd 12 2
4410.2.d.a 16 7.c even 3 1
4410.2.d.a 16 7.d odd 6 1
4410.2.d.a 16 105.o odd 6 1
4410.2.d.a 16 105.p even 6 1
4410.2.d.b 16 21.g even 6 1
4410.2.d.b 16 21.h odd 6 1
4410.2.d.b 16 35.i odd 6 1
4410.2.d.b 16 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{8} - 42 T_{17}^{6} + 1516 T_{17}^{4} + 3024 T_{17}^{3} - 8688 T_{17}^{2} - 17856 T_{17} + 61504$$ acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{8}$$
$3$ 
$5$ $$1 - 6 T + 21 T^{2} - 54 T^{3} + 113 T^{4} - 168 T^{5} + 186 T^{6} - 84 T^{7} - 6 T^{8} - 420 T^{9} + 4650 T^{10} - 21000 T^{11} + 70625 T^{12} - 168750 T^{13} + 328125 T^{14} - 468750 T^{15} + 390625 T^{16}$$
$7$ $$1 + 14 T^{2} + 173 T^{4} + 1338 T^{6} + 10424 T^{8} + 65562 T^{10} + 415373 T^{12} + 1647086 T^{14} + 5764801 T^{16}$$
$11$ $$1 + 38 T^{2} + 930 T^{4} + 16780 T^{6} + 233189 T^{8} + 2491116 T^{10} + 21555910 T^{12} + 152846786 T^{14} + 1257598116 T^{16} + 18494461106 T^{18} + 315600078310 T^{20} + 4413163952076 T^{22} + 49986133101509 T^{24} + 435229984804780 T^{26} + 2918738390350530 T^{28} + 14430493676163158 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 + 32 T^{2} + 473 T^{4} + 8616 T^{6} + 148784 T^{8} + 1456104 T^{10} + 13509353 T^{12} + 154457888 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 26 T^{2} + 122 T^{4} - 648 T^{5} + 1104 T^{6} - 39888 T^{7} + 57407 T^{8} - 678096 T^{9} + 319056 T^{10} - 3183624 T^{11} + 10189562 T^{12} + 627576794 T^{14} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 12 T + 93 T^{2} - 540 T^{3} + 2231 T^{4} - 6864 T^{5} + 8928 T^{6} + 52848 T^{7} - 355902 T^{8} + 1004112 T^{9} + 3223008 T^{10} - 47080176 T^{11} + 290746151 T^{12} - 1337093460 T^{13} + 4375266933 T^{14} - 10726460868 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$( 1 - 4 T - 23 T^{2} - 132 T^{3} + 1031 T^{4} + 2792 T^{5} + 23064 T^{6} - 114968 T^{7} - 503054 T^{8} - 2644264 T^{9} + 12200856 T^{10} + 33970264 T^{11} + 288516071 T^{12} - 849597276 T^{13} - 3404825447 T^{14} - 13619301788 T^{15} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 110 T^{2} + 5617 T^{4} - 185030 T^{6} + 5268820 T^{8} - 155610230 T^{10} + 3972797377 T^{12} - 65430565310 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 6 T + 21 T^{2} - 54 T^{3} - 715 T^{4} + 4356 T^{5} - 5778 T^{6} - 5376 T^{7} + 109626 T^{8} - 166656 T^{9} - 5552658 T^{10} + 129769596 T^{11} - 660317515 T^{12} - 1545974154 T^{13} + 18637577301 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 + 184 T^{2} + 18359 T^{4} + 1197704 T^{6} + 54361073 T^{8} + 1562275568 T^{10} + 12022191570 T^{12} - 1443189401472 T^{14} - 83461258079714 T^{16} - 1975726290615168 T^{18} + 22531522575022770 T^{20} + 4008371682953075312 T^{22} +$$$$19\!\cdots\!33$$$$T^{24} +$$$$57\!\cdots\!96$$$$T^{26} +$$$$12\!\cdots\!79$$$$T^{28} +$$$$16\!\cdots\!76$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 + 84 T^{2} + 9077 T^{4} + 448128 T^{6} + 24944604 T^{8} + 753303168 T^{10} + 25649432597 T^{12} + 399008756244 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 196 T^{2} + 20648 T^{4} - 1438284 T^{6} + 72218798 T^{8} - 2659387116 T^{10} + 70591403048 T^{12} - 1238987157604 T^{14} + 11688200277601 T^{16} )^{2}$$
$47$ $$( 1 + 30 T + 585 T^{2} + 8550 T^{3} + 104411 T^{4} + 1086708 T^{5} + 9949392 T^{6} + 80614644 T^{7} + 585105414 T^{8} + 3788888268 T^{9} + 21978206928 T^{10} + 112825284684 T^{11} + 509492372891 T^{12} + 1960899809850 T^{13} + 6305840967465 T^{14} + 15198693613890 T^{15} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 + 8 T - 104 T^{2} - 744 T^{3} + 6851 T^{4} + 22388 T^{5} - 596424 T^{6} - 31316 T^{7} + 43664392 T^{8} - 1659748 T^{9} - 1675355016 T^{10} + 3333058276 T^{11} + 54057685331 T^{12} - 311137446792 T^{13} - 2305093557416 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} )^{2}$$
$59$ $$1 - 426 T^{2} + 99991 T^{4} - 16373094 T^{6} + 2068006933 T^{8} - 212127519060 T^{10} + 18234588368746 T^{12} - 1338906778092288 T^{14} + 84873222726826858 T^{16} - 4660734494539254528 T^{18} +$$$$22\!\cdots\!06$$$$T^{20} -$$$$89\!\cdots\!60$$$$T^{22} +$$$$30\!\cdots\!93$$$$T^{24} -$$$$83\!\cdots\!94$$$$T^{26} +$$$$17\!\cdots\!71$$$$T^{28} -$$$$26\!\cdots\!86$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 12 T + 106 T^{2} - 696 T^{3} - 1526 T^{4} + 2700 T^{5} + 111184 T^{6} - 2327412 T^{7} + 40396639 T^{8} - 141972132 T^{9} + 413715664 T^{10} + 612848700 T^{11} - 21128753366 T^{12} - 587839025496 T^{13} + 5461159682266 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 + 352 T^{2} + 62036 T^{4} + 7763648 T^{6} + 806269898 T^{8} + 74285375456 T^{10} + 6186081613776 T^{12} + 467069870756640 T^{14} + 32434315696789747 T^{16} + 2096676649826556960 T^{18} +$$$$12\!\cdots\!96$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{22} +$$$$32\!\cdots\!18$$$$T^{24} +$$$$14\!\cdots\!52$$$$T^{26} +$$$$50\!\cdots\!96$$$$T^{28} +$$$$12\!\cdots\!08$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 + 52 T^{2} + 10456 T^{4} + 567388 T^{6} + 78505582 T^{8} + 2860202908 T^{10} + 265704536536 T^{12} + 6661214763892 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$1 - 312 T^{2} + 45700 T^{4} - 4488720 T^{6} + 369040906 T^{8} - 28689322824 T^{10} + 2019175463824 T^{12} - 122728472139336 T^{14} + 7818824571234835 T^{16} - 654020028030521544 T^{18} + 57341031442960733584 T^{20} -$$$$43\!\cdots\!36$$$$T^{22} +$$$$29\!\cdots\!86$$$$T^{24} -$$$$19\!\cdots\!80$$$$T^{26} +$$$$10\!\cdots\!00$$$$T^{28} -$$$$38\!\cdots\!08$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 2 T - 55 T^{2} + 1622 T^{3} - 4375 T^{4} - 95596 T^{5} + 995034 T^{6} + 1066704 T^{7} - 79120334 T^{8} + 84269616 T^{9} + 6210007194 T^{10} - 47132556244 T^{11} - 170406604375 T^{12} + 4990985479178 T^{13} - 13369810053655 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 - 250 T^{2} + 32249 T^{4} - 3769146 T^{6} + 372693188 T^{8} - 25965646794 T^{10} + 1530483393929 T^{12} - 81735093342250 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 492 T^{2} + 129352 T^{4} - 23026392 T^{6} + 3057469378 T^{8} - 313387087572 T^{10} + 25702504802560 T^{12} - 1837765181416836 T^{14} + 143644724514270163 T^{16} - 14556938002002757956 T^{18} +$$$$16\!\cdots\!60$$$$T^{20} -$$$$15\!\cdots\!92$$$$T^{22} +$$$$12\!\cdots\!18$$$$T^{24} -$$$$71\!\cdots\!92$$$$T^{26} +$$$$31\!\cdots\!92$$$$T^{28} -$$$$96\!\cdots\!72$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 + 438 T^{2} + 104849 T^{4} + 16454334 T^{6} + 1868658564 T^{8} + 154818828606 T^{10} + 9282206583569 T^{12} + 364841738158902 T^{14} + 7837433594376961 T^{16} )^{2}$$