# Properties

 Label 430.2.b.b Level 430 Weight 2 Character orbit 430.b Analytic conductor 3.434 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 38 x^{14} + 525 x^{12} + 3518 x^{10} + 12216 x^{8} + 20990 x^{6} + 15229 x^{4} + 4754 x^{2} + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{14} + 25 \nu^{12} + 95 \nu^{10} - 1182 \nu^{8} - 10113 \nu^{6} - 24981 \nu^{4} - 17383 \nu^{2} - 3262$$$$)/100$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} + 45 \nu^{12} + 750 \nu^{10} + 5838 \nu^{8} + 22052 \nu^{6} + 36294 \nu^{4} + 16597 \nu^{2} + 2023$$$$)/100$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{14} - 70 \nu^{12} - 850 \nu^{10} - 4811 \nu^{8} - 13439 \nu^{6} - 17188 \nu^{4} - 7809 \nu^{2} - 1031$$$$)/50$$ $$\beta_{5}$$ $$=$$ $$($$$$102 \nu^{15} + 69 \nu^{14} + 3830 \nu^{13} + 2645 \nu^{12} + 51940 \nu^{11} + 36915 \nu^{10} + 339056 \nu^{9} + 248492 \nu^{8} + 1128824 \nu^{7} + 850793 \nu^{6} + 1778408 \nu^{5} + 1365211 \nu^{4} + 999334 \nu^{3} + 761323 \nu^{2} + 163506 \nu + 133952$$$$)/4600$$ $$\beta_{6}$$ $$=$$ $$($$$$102 \nu^{15} - 69 \nu^{14} + 3830 \nu^{13} - 2645 \nu^{12} + 51940 \nu^{11} - 36915 \nu^{10} + 339056 \nu^{9} - 248492 \nu^{8} + 1128824 \nu^{7} - 850793 \nu^{6} + 1778408 \nu^{5} - 1365211 \nu^{4} + 999334 \nu^{3} - 761323 \nu^{2} + 163506 \nu - 133952$$$$)/4600$$ $$\beta_{7}$$ $$=$$ $$($$$$-47 \nu^{15} - 1740 \nu^{13} - 23065 \nu^{11} - 145796 \nu^{9} - 463499 \nu^{7} - 677433 \nu^{5} - 320439 \nu^{3} - 43831 \nu$$$$)/1150$$ $$\beta_{8}$$ $$=$$ $$($$$$-109 \nu^{15} - 4165 \nu^{13} - 57800 \nu^{11} - 385532 \nu^{9} - 1301368 \nu^{7} - 2036336 \nu^{5} - 1061823 \nu^{3} - 174267 \nu$$$$)/2300$$ $$\beta_{9}$$ $$=$$ $$($$$$18 \nu^{15} + 276 \nu^{14} + 845 \nu^{13} + 10810 \nu^{12} + 14625 \nu^{11} + 154790 \nu^{10} + 116569 \nu^{9} + 1063198 \nu^{8} + 442666 \nu^{7} + 3677792 \nu^{6} + 699797 \nu^{5} + 5861274 \nu^{4} + 223591 \nu^{3} + 3080942 \nu^{2} + 2289 \nu + 484518$$$$)/4600$$ $$\beta_{10}$$ $$=$$ $$($$$$84 \nu^{15} + 391 \nu^{14} + 2985 \nu^{13} + 14605 \nu^{12} + 37315 \nu^{11} + 196075 \nu^{10} + 222487 \nu^{9} + 1257318 \nu^{8} + 686158 \nu^{7} + 4063387 \nu^{6} + 1078611 \nu^{5} + 6077359 \nu^{4} + 775743 \nu^{3} + 3042647 \nu^{2} + 161217 \nu + 468418$$$$)/4600$$ $$\beta_{11}$$ $$=$$ $$($$$$131 \nu^{15} - 368 \nu^{14} + 4840 \nu^{13} - 13570 \nu^{12} + 64060 \nu^{11} - 178940 \nu^{10} + 406003 \nu^{9} - 1126034 \nu^{8} + 1309277 \nu^{7} - 3575166 \nu^{6} + 2007664 \nu^{5} - 5266172 \nu^{4} + 1138732 \nu^{3} - 2602726 \nu^{2} + 207693 \nu - 388424$$$$)/4600$$ $$\beta_{12}$$ $$=$$ $$($$$$131 \nu^{15} + 368 \nu^{14} + 4840 \nu^{13} + 13570 \nu^{12} + 64060 \nu^{11} + 178940 \nu^{10} + 406003 \nu^{9} + 1126034 \nu^{8} + 1309277 \nu^{7} + 3575166 \nu^{6} + 2007664 \nu^{5} + 5266172 \nu^{4} + 1138732 \nu^{3} + 2602726 \nu^{2} + 207693 \nu + 388424$$$$)/4600$$ $$\beta_{13}$$ $$=$$ $$($$$$-135 \nu^{15} - 437 \nu^{14} - 4900 \nu^{13} - 16560 \nu^{12} - 62940 \nu^{11} - 227010 \nu^{10} - 380975 \nu^{9} - 1491481 \nu^{8} - 1137065 \nu^{7} - 4945989 \nu^{6} - 1480100 \nu^{5} - 7575878 \nu^{4} - 464660 \nu^{3} - 3824164 \nu^{2} - 17225 \nu - 585281$$$$)/4600$$ $$\beta_{14}$$ $$=$$ $$($$$$564 \nu^{15} - 69 \nu^{14} + 21340 \nu^{13} - 2990 \nu^{12} + 291960 \nu^{11} - 48070 \nu^{10} + 1915152 \nu^{9} - 365447 \nu^{8} + 6351808 \nu^{7} - 1370823 \nu^{6} + 9785196 \nu^{5} - 2309706 \nu^{4} + 5082668 \nu^{3} - 1221438 \nu^{2} + 796912 \nu - 196857$$$$)/4600$$ $$\beta_{15}$$ $$=$$ $$($$$$564 \nu^{15} + 69 \nu^{14} + 21340 \nu^{13} + 2990 \nu^{12} + 291960 \nu^{11} + 48070 \nu^{10} + 1915152 \nu^{9} + 365447 \nu^{8} + 6351808 \nu^{7} + 1370823 \nu^{6} + 9785196 \nu^{5} + 2309706 \nu^{4} + 5082668 \nu^{3} + 1221438 \nu^{2} + 796912 \nu + 196857$$$$)/4600$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{15} + 2 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} - 2 \beta_{5} - 2 \beta_{2} - 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{15} - 8 \beta_{14} + \beta_{12} - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 18 \beta_{6} - 15 \beta_{5} - 18 \beta_{4} - 6 \beta_{3} - 13 \beta_{2} + 55$$ $$\nu^{5}$$ $$=$$ $$19 \beta_{15} - 43 \beta_{14} + 62 \beta_{13} + 70 \beta_{12} + 8 \beta_{11} - 44 \beta_{10} + 44 \beta_{9} + 36 \beta_{8} - 102 \beta_{7} - 6 \beta_{6} + 38 \beta_{5} + 44 \beta_{2} + 91 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-182 \beta_{15} + 182 \beta_{14} - 27 \beta_{12} + 27 \beta_{11} + 70 \beta_{10} + 70 \beta_{9} - 317 \beta_{6} + 247 \beta_{5} + 305 \beta_{4} + 138 \beta_{3} + 199 \beta_{2} - 799$$ $$\nu^{7}$$ $$=$$ $$-328 \beta_{15} + 773 \beta_{14} - 1101 \beta_{13} - 1297 \beta_{12} - 196 \beta_{11} + 810 \beta_{10} - 810 \beta_{9} - 632 \beta_{8} + 1803 \beta_{7} + 129 \beta_{6} - 681 \beta_{5} - 810 \beta_{2} - 1336 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$3362 \beta_{15} - 3362 \beta_{14} + 531 \beta_{12} - 531 \beta_{11} - 1304 \beta_{10} - 1304 \beta_{9} + 5508 \beta_{6} - 4204 \beta_{5} - 5190 \beta_{4} - 2570 \beta_{3} - 3296 \beta_{2} + 12979$$ $$\nu^{9}$$ $$=$$ $$5615 \beta_{15} - 13413 \beta_{14} + 19028 \beta_{13} + 22776 \beta_{12} + 3748 \beta_{11} - 14224 \beta_{10} + 14224 \beta_{9} + 10978 \beta_{8} - 31098 \beta_{7} - 2319 \beta_{6} + 11905 \beta_{5} + 14224 \beta_{2} + 21833 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-59022 \beta_{15} + 59022 \beta_{14} - 9536 \beta_{12} + 9536 \beta_{11} + 22949 \beta_{10} + 22949 \beta_{9} - 95079 \beta_{6} + 72130 \beta_{5} + 88811 \beta_{4} + 45300 \beta_{3} + 56012 \beta_{2} - 219133$$ $$\nu^{11}$$ $$=$$ $$-96271 \beta_{15} + 230980 \beta_{14} - 327251 \beta_{13} - 394019 \beta_{12} - 66768 \beta_{11} + 246186 \beta_{10} - 246186 \beta_{9} - 189560 \beta_{8} + 534393 \beta_{7} + 40259 \beta_{6} - 205927 \beta_{5} - 246186 \beta_{2} - 369610 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$1021100 \beta_{15} - 1021100 \beta_{14} + 166282 \beta_{12} - 166282 \beta_{11} - 397262 \beta_{10} - 397262 \beta_{9} + 1636896 \beta_{6} - 1239634 \beta_{5} - 1523938 \beta_{4} - 785056 \beta_{3} - 959414 \beta_{2} + 3745693$$ $$\nu^{13}$$ $$=$$ $$1653014 \beta_{15} - 3971476 \beta_{14} + 5624490 \beta_{13} + 6786286 \beta_{12} + 1161796 \beta_{11} - 4241248 \beta_{10} + 4241248 \beta_{9} + 3264324 \beta_{8} - 9181660 \beta_{7} - 693600 \beta_{6} + 3547648 \beta_{5} + 4241248 \beta_{2} + 6324493 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-17587244 \beta_{15} + 17587244 \beta_{14} - 2871558 \beta_{12} + 2871558 \beta_{11} + 6843034 \beta_{10} + 6843034 \beta_{9} - 28152985 \beta_{6} + 21309951 \beta_{5} + 26179015 \beta_{4} + 13530868 \beta_{3} + 16473555 \beta_{2} - 64273497$$ $$\nu^{15}$$ $$=$$ $$-28402583 \beta_{15} + 68267666 \beta_{14} - 96670249 \beta_{13} - 116724253 \beta_{12} - 20054004 \beta_{11} + 72957994 \beta_{10} - 72957994 \beta_{9} - 56150770 \beta_{8} + 157789021 \beta_{7} + 11929028 \beta_{6} - 61028966 \beta_{5} - 72957994 \beta_{2} - 108566368 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/430\mathbb{Z}\right)^\times$$.

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1
 − 4.14603i − 2.39034i − 2.20024i − 0.689100i 0.530860i 0.639453i 1.99373i 2.26167i − 2.26167i − 1.99373i − 0.639453i − 0.530860i 0.689100i 2.20024i 2.39034i 4.14603i
1.00000i 3.14603i −1.00000 2.21633 + 0.296419i −3.14603 0.417000i 1.00000i −6.89749 0.296419 2.21633i
259.2 1.00000i 1.39034i −1.00000 −1.86897 + 1.22758i −1.39034 4.65953i 1.00000i 1.06694 1.22758 + 1.86897i
259.3 1.00000i 1.20024i −1.00000 1.77243 1.36327i −1.20024 4.33187i 1.00000i 1.55943 −1.36327 1.77243i
259.4 1.00000i 0.310900i −1.00000 −0.205955 2.22656i 0.310900 2.43727i 1.00000i 2.90334 −2.22656 + 0.205955i
259.5 1.00000i 1.53086i −1.00000 −2.23604 + 0.0111723i 1.53086 2.29871i 1.00000i 0.656468 0.0111723 + 2.23604i
259.6 1.00000i 1.63945i −1.00000 0.156651 + 2.23057i 1.63945 0.479890i 1.00000i 0.312193 2.23057 0.156651i
259.7 1.00000i 2.99373i −1.00000 0.260869 2.22080i 2.99373 4.62603i 1.00000i −5.96240 −2.22080 0.260869i
259.8 1.00000i 3.26167i −1.00000 0.904683 + 2.04488i 3.26167 1.22270i 1.00000i −7.63849 2.04488 0.904683i
259.9 1.00000i 3.26167i −1.00000 0.904683 2.04488i 3.26167 1.22270i 1.00000i −7.63849 2.04488 + 0.904683i
259.10 1.00000i 2.99373i −1.00000 0.260869 + 2.22080i 2.99373 4.62603i 1.00000i −5.96240 −2.22080 + 0.260869i
259.11 1.00000i 1.63945i −1.00000 0.156651 2.23057i 1.63945 0.479890i 1.00000i 0.312193 2.23057 + 0.156651i
259.12 1.00000i 1.53086i −1.00000 −2.23604 0.0111723i 1.53086 2.29871i 1.00000i 0.656468 0.0111723 2.23604i
259.13 1.00000i 0.310900i −1.00000 −0.205955 + 2.22656i 0.310900 2.43727i 1.00000i 2.90334 −2.22656 0.205955i
259.14 1.00000i 1.20024i −1.00000 1.77243 + 1.36327i −1.20024 4.33187i 1.00000i 1.55943 −1.36327 + 1.77243i
259.15 1.00000i 1.39034i −1.00000 −1.86897 1.22758i −1.39034 4.65953i 1.00000i 1.06694 1.22758 1.86897i
259.16 1.00000i 3.14603i −1.00000 2.21633 0.296419i −3.14603 0.417000i 1.00000i −6.89749 0.296419 + 2.21633i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 259.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.b.b 16
5.b even 2 1 inner 430.2.b.b 16
5.c odd 4 1 2150.2.a.bg 8
5.c odd 4 1 2150.2.a.bh 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.b.b 16 1.a even 1 1 trivial
430.2.b.b 16 5.b even 2 1 inner
2150.2.a.bg 8 5.c odd 4 1
2150.2.a.bh 8 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$1 - 10 T^{2} + 51 T^{4} - 176 T^{6} + 667 T^{8} - 3040 T^{10} + 11605 T^{12} - 33410 T^{14} + 91072 T^{16} - 300690 T^{18} + 940005 T^{20} - 2216160 T^{22} + 4376187 T^{24} - 10392624 T^{26} + 27103491 T^{28} - 47829690 T^{30} + 43046721 T^{32}$$
$5$ $$1 - 2 T + 7 T^{2} - 37 T^{4} + 98 T^{5} - 159 T^{6} - 176 T^{7} + 728 T^{8} - 880 T^{9} - 3975 T^{10} + 12250 T^{11} - 23125 T^{12} + 109375 T^{14} - 156250 T^{15} + 390625 T^{16}$$
$7$ $$1 - 37 T^{2} + 669 T^{4} - 8530 T^{6} + 93904 T^{8} - 954556 T^{10} + 8661035 T^{12} - 68886013 T^{14} + 498263182 T^{16} - 3375414637 T^{18} + 20795145035 T^{20} - 112302558844 T^{22} + 541337873104 T^{24} - 2409513873970 T^{26} + 9259821137469 T^{28} - 25094253695413 T^{30} + 33232930569601 T^{32}$$
$11$ $$( 1 - 2 T + 28 T^{2} - 88 T^{3} + 612 T^{4} - 1532 T^{5} + 9668 T^{6} - 22990 T^{7} + 112406 T^{8} - 252890 T^{9} + 1169828 T^{10} - 2039092 T^{11} + 8960292 T^{12} - 14172488 T^{13} + 49603708 T^{14} - 38974342 T^{15} + 214358881 T^{16} )^{2}$$
$13$ $$1 - 57 T^{2} + 2221 T^{4} - 63010 T^{6} + 1470096 T^{8} - 29164612 T^{10} + 504883051 T^{12} - 7776883625 T^{14} + 106613731662 T^{16} - 1314293332625 T^{18} + 14419964819611 T^{20} - 140772011683108 T^{22} + 1199202470019216 T^{24} - 8686463571405490 T^{26} + 51745047057030301 T^{28} - 224430453984859473 T^{30} + 665416609183179841 T^{32}$$
$17$ $$1 - 82 T^{2} + 3475 T^{4} - 100800 T^{6} + 2208355 T^{8} - 36519332 T^{10} + 437841909 T^{12} - 3747068150 T^{14} + 36785708160 T^{16} - 1082902695350 T^{18} + 36568994081589 T^{20} - 881487895983908 T^{22} + 15404948823619555 T^{24} - 203212185165259200 T^{26} + 2024612274373419475 T^{28} - 13806981777870876178 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 + 15 T + 200 T^{2} + 1733 T^{3} + 13726 T^{4} + 86371 T^{5} + 508275 T^{6} + 2529942 T^{7} + 11896286 T^{8} + 48068898 T^{9} + 183487275 T^{10} + 592418689 T^{11} + 1788786046 T^{12} + 4291079567 T^{13} + 9409176200 T^{14} + 13408076085 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 - 214 T^{2} + 22679 T^{4} - 1596744 T^{6} + 84184463 T^{8} - 3537581064 T^{10} + 122756878141 T^{12} - 3589213546070 T^{14} + 89334432012112 T^{16} - 1898693965871030 T^{18} + 34352407535855581 T^{20} - 523688957718805896 T^{22} + 6592568242881889103 T^{24} - 66147533221326758856 T^{26} +$$$$49\!\cdots\!59$$$$T^{28} -$$$$24\!\cdots\!26$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 3 T + 78 T^{2} - 569 T^{3} + 3594 T^{4} - 31051 T^{5} + 181717 T^{6} - 931820 T^{7} + 6832480 T^{8} - 27022780 T^{9} + 152823997 T^{10} - 757302839 T^{11} + 2541967914 T^{12} - 11670843781 T^{13} + 46396219038 T^{14} - 51749628927 T^{15} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 25 T + 452 T^{2} - 5865 T^{3} + 63046 T^{4} - 563375 T^{5} + 4370993 T^{6} - 29418450 T^{7} + 174910256 T^{8} - 911971950 T^{9} + 4200524273 T^{10} - 16783504625 T^{11} + 58224304966 T^{12} - 167909970615 T^{13} + 401151663812 T^{14} - 687815352775 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$1 - 394 T^{2} + 77383 T^{4} - 10014624 T^{6} + 953289279 T^{8} - 70658881624 T^{10} + 4214634576301 T^{12} - 206262118894570 T^{14} + 8368262409986752 T^{16} - 282372840766666330 T^{18} + 7898903752154858461 T^{20} -$$$$18\!\cdots\!16$$$$T^{22} +$$$$33\!\cdots\!59$$$$T^{24} -$$$$48\!\cdots\!76$$$$T^{26} +$$$$50\!\cdots\!23$$$$T^{28} -$$$$35\!\cdots\!66$$$$T^{30} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$( 1 - 19 T + 406 T^{2} - 5039 T^{3} + 63478 T^{4} - 587281 T^{5} + 5386299 T^{6} - 39029276 T^{7} + 278582162 T^{8} - 1600200316 T^{9} + 9054368619 T^{10} - 40475993801 T^{11} + 179373656758 T^{12} - 583799396839 T^{13} + 1928542321846 T^{14} - 3700331203739 T^{15} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + T^{2} )^{8}$$
$47$ $$1 - 482 T^{2} + 116711 T^{4} - 18746928 T^{6} + 2228078775 T^{8} - 207244774732 T^{10} + 15578128441725 T^{12} - 964122051100670 T^{14} + 49615909496598816 T^{16} - 2129745610881380030 T^{18} + 76016297372645089725 T^{20} -$$$$22\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!75$$$$T^{24} -$$$$98\!\cdots\!72$$$$T^{26} +$$$$13\!\cdots\!51$$$$T^{28} -$$$$12\!\cdots\!58$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 - 576 T^{2} + 154696 T^{4} - 25836080 T^{6} + 3031271836 T^{8} - 269432012400 T^{10} + 19302108479096 T^{12} - 1183214040569248 T^{14} + 65349852889172742 T^{16} - 3323648239959017632 T^{18} +$$$$15\!\cdots\!76$$$$T^{20} -$$$$59\!\cdots\!00$$$$T^{22} +$$$$18\!\cdots\!96$$$$T^{24} -$$$$45\!\cdots\!20$$$$T^{26} +$$$$75\!\cdots\!36$$$$T^{28} -$$$$79\!\cdots\!44$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 - 12 T + 376 T^{2} - 3324 T^{3} + 61004 T^{4} - 432892 T^{5} + 6064456 T^{6} - 36188204 T^{7} + 421065286 T^{8} - 2135104036 T^{9} + 21110371336 T^{10} - 88906926068 T^{11} + 739207490444 T^{12} - 2376408369876 T^{13} + 15859880649016 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$( 1 - 29 T + 693 T^{2} - 10726 T^{3} + 148982 T^{4} - 1635184 T^{5} + 16908739 T^{6} - 147864185 T^{7} + 1244942418 T^{8} - 9019715285 T^{9} + 62917417819 T^{10} - 371155699504 T^{11} + 2062781083862 T^{12} - 9059139924526 T^{13} + 35703619432173 T^{14} - 91139542244609 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$1 - 461 T^{2} + 124889 T^{4} - 23767026 T^{6} + 3523375680 T^{8} - 423196090940 T^{10} + 42444100674783 T^{12} - 3599926920976653 T^{14} + 260857140941399214 T^{16} - 16160071948264195317 T^{18} +$$$$85\!\cdots\!43$$$$T^{20} -$$$$38\!\cdots\!60$$$$T^{22} +$$$$14\!\cdots\!80$$$$T^{24} -$$$$43\!\cdots\!74$$$$T^{26} +$$$$10\!\cdots\!29$$$$T^{28} -$$$$16\!\cdots\!69$$$$T^{30} +$$$$16\!\cdots\!81$$$$T^{32}$$
$71$ $$( 1 - 12 T + 350 T^{2} - 3178 T^{3} + 57608 T^{4} - 469734 T^{5} + 6497602 T^{6} - 47379416 T^{7} + 534124718 T^{8} - 3363938536 T^{9} + 32754411682 T^{10} - 168122965674 T^{11} + 1463916119048 T^{12} - 5733840877478 T^{13} + 44835099372350 T^{14} - 109141441900692 T^{15} + 645753531245761 T^{16} )^{2}$$
$73$ $$1 - 905 T^{2} + 395545 T^{4} - 111083010 T^{6} + 22471297760 T^{8} - 3475557609012 T^{10} + 425393441711119 T^{12} - 42042952532589265 T^{14} + 3391469041132093230 T^{16} -$$$$22\!\cdots\!85$$$$T^{18} +$$$$12\!\cdots\!79$$$$T^{20} -$$$$52\!\cdots\!68$$$$T^{22} +$$$$18\!\cdots\!60$$$$T^{24} -$$$$47\!\cdots\!90$$$$T^{26} +$$$$90\!\cdots\!45$$$$T^{28} -$$$$11\!\cdots\!45$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 + 5 T + 234 T^{2} + 995 T^{3} + 29332 T^{4} + 88335 T^{5} + 2715965 T^{6} + 3567410 T^{7} + 217842536 T^{8} + 281825390 T^{9} + 16950337565 T^{10} + 43552600065 T^{11} + 1142483775892 T^{12} + 3061671117005 T^{13} + 56882464591914 T^{14} + 96019544930795 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 580 T^{2} + 173632 T^{4} - 35649596 T^{6} + 5673283660 T^{8} - 749699370772 T^{10} + 85546383012736 T^{12} - 8573159301594284 T^{14} + 757692623249501222 T^{16} - 59060494428683022476 T^{18} +$$$$40\!\cdots\!56$$$$T^{20} -$$$$24\!\cdots\!68$$$$T^{22} +$$$$12\!\cdots\!60$$$$T^{24} -$$$$55\!\cdots\!04$$$$T^{26} +$$$$18\!\cdots\!52$$$$T^{28} -$$$$42\!\cdots\!20$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$( 1 - 20 T + 460 T^{2} - 5698 T^{3} + 87296 T^{4} - 929226 T^{5} + 12020020 T^{6} - 113397072 T^{7} + 1243372926 T^{8} - 10092339408 T^{9} + 95210578420 T^{10} - 655075523994 T^{11} + 5477146670336 T^{12} - 31817970740402 T^{13} + 228611393842060 T^{14} - 884626697910580 T^{15} + 3936588805702081 T^{16} )^{2}$$
$97$ $$1 - 774 T^{2} + 301691 T^{4} - 79641640 T^{6} + 16078651115 T^{8} - 2644465230904 T^{10} + 367166336275261 T^{12} - 43893175395348070 T^{14} + 4560761585591764624 T^{16} -$$$$41\!\cdots\!30$$$$T^{18} +$$$$32\!\cdots\!41$$$$T^{20} -$$$$22\!\cdots\!16$$$$T^{22} +$$$$12\!\cdots\!15$$$$T^{24} -$$$$58\!\cdots\!60$$$$T^{26} +$$$$20\!\cdots\!31$$$$T^{28} -$$$$50\!\cdots\!06$$$$T^{30} +$$$$61\!\cdots\!21$$$$T^{32}$$
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