# Properties

 Label 1470.2.m.f Level $1470$ Weight $2$ Character orbit 1470.m Analytic conductor $11.738$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1470.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + 1250 x^{4} - 100000 x^{3} + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + 1250 x^{4} - 100000 x^{3} + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$23 \nu^{15} + 151945 \nu^{14} - 513400 \nu^{13} - 30736 \nu^{12} - 3356569 \nu^{11} + 10867065 \nu^{10} - 183704 \nu^{9} + 19988016 \nu^{8} - 99781289 \nu^{7} + 133354345 \nu^{6} - 9130200 \nu^{5} + 988282000 \nu^{4} - 2959145625 \nu^{3} - 176434375 \nu^{2} - 5904875000 \nu + 17286250000$$$$)/ 263500000$$ $$\beta_{2}$$ $$=$$ $$($$$$-12176 \nu^{15} - 711160 \nu^{14} + 2464225 \nu^{13} + 334507 \nu^{12} + 13822768 \nu^{11} - 53602520 \nu^{10} + 2515873 \nu^{9} - 71108757 \nu^{8} + 489526928 \nu^{7} - 655014680 \nu^{6} - 20484575 \nu^{5} - 4050419125 \nu^{4} + 14571370000 \nu^{3} + 743925000 \nu^{2} + 21019390625 \nu - 88433203125$$$$)/ 1317500000$$ $$\beta_{3}$$ $$=$$ $$($$$$-7153 \nu^{15} + 234160 \nu^{14} - 230600 \nu^{13} + 200771 \nu^{12} - 4866801 \nu^{11} + 4897520 \nu^{10} - 1954056 \nu^{9} + 29256259 \nu^{8} - 86582961 \nu^{7} + 61131440 \nu^{6} - 80373000 \nu^{5} + 1382043875 \nu^{4} - 1294480625 \nu^{3} + 436350000 \nu^{2} - 7983125000 \nu + 7283671875$$$$)/ 263500000$$ $$\beta_{4}$$ $$=$$ $$($$$$-52781 \nu^{15} - 277595 \nu^{14} + 417475 \nu^{13} + 1210867 \nu^{12} + 5695603 \nu^{11} - 6617515 \nu^{10} - 6615837 \nu^{9} - 21020637 \nu^{8} + 97051123 \nu^{7} - 100536475 \nu^{6} - 290692925 \nu^{5} - 1656298125 \nu^{4} + 1662561875 \nu^{3} + 2208303125 \nu^{2} + 9931171875 \nu - 6183828125$$$$)/ 1317500000$$ $$\beta_{5}$$ $$=$$ $$($$$$10532 \nu^{15} - 94195 \nu^{14} + 91100 \nu^{13} - 196524 \nu^{12} + 2002804 \nu^{11} - 2187965 \nu^{10} + 1205964 \nu^{9} - 14275106 \nu^{8} + 36913924 \nu^{7} - 25413395 \nu^{6} + 64163100 \nu^{5} - 565159000 \nu^{4} + 555072500 \nu^{3} - 278928125 \nu^{2} + 3362187500 \nu - 3498281250$$$$)/ 164687500$$ $$\beta_{6}$$ $$=$$ $$($$$$5499 \nu^{15} - 36198 \nu^{14} + 69810 \nu^{13} - 110818 \nu^{12} + 799959 \nu^{11} - 1500066 \nu^{10} + 759218 \nu^{9} - 6093858 \nu^{8} + 17642267 \nu^{7} - 19619686 \nu^{6} + 35441490 \nu^{5} - 226977250 \nu^{4} + 401156375 \nu^{3} - 182746250 \nu^{2} + 1367131250 \nu - 2325781250$$$$)/65875000$$ $$\beta_{7}$$ $$=$$ $$($$$$-124784 \nu^{15} + 289765 \nu^{14} + 543850 \nu^{13} + 2971213 \nu^{12} - 6082048 \nu^{11} - 9726795 \nu^{10} - 17542518 \nu^{9} + 63233197 \nu^{8} - 18736688 \nu^{7} - 125413435 \nu^{6} - 834999350 \nu^{5} + 1529546125 \nu^{4} + 2922380000 \nu^{3} + 5544903125 \nu^{2} - 8980093750 \nu - 16529921875$$$$)/ 1317500000$$ $$\beta_{8}$$ $$=$$ $$($$$$128033 \nu^{15} - 35880 \nu^{14} + 411075 \nu^{13} - 2683056 \nu^{12} + 1413601 \nu^{11} - 7551160 \nu^{10} + 15219891 \nu^{9} - 45725264 \nu^{8} + 46597281 \nu^{7} - 118375880 \nu^{6} + 765575875 \nu^{5} - 356214000 \nu^{4} + 2128850625 \nu^{3} - 4479975000 \nu^{2} + 3063921875 \nu - 10391250000$$$$)/ 1317500000$$ $$\beta_{9}$$ $$=$$ $$($$$$-221264 \nu^{15} + 115 \nu^{14} + 759725 \nu^{13} + 4513448 \nu^{12} - 596208 \nu^{11} - 16782845 \nu^{10} - 23549603 \nu^{9} + 62805512 \nu^{8} + 99497552 \nu^{7} - 180286285 \nu^{6} - 1280351475 \nu^{5} - 45651000 \nu^{4} + 4664830000 \nu^{3} + 7330671875 \nu^{2} + 435328125 \nu - 29524375000$$$$)/ 1317500000$$ $$\beta_{10}$$ $$=$$ $$($$$$-72414 \nu^{15} + 116510 \nu^{14} - 145225 \nu^{13} + 1495498 \nu^{12} - 2394398 \nu^{11} + 3173970 \nu^{10} - 8656053 \nu^{9} + 31497002 \nu^{8} - 46480158 \nu^{7} + 46384230 \nu^{6} - 430608425 \nu^{5} + 654882250 \nu^{4} - 823538750 \nu^{3} + 2448231250 \nu^{2} - 3672203125 \nu + 4653906250$$$$)/ 329375000$$ $$\beta_{11}$$ $$=$$ $$($$$$-2029 \nu^{15} - 230 \nu^{14} + 12455 \nu^{13} + 40928 \nu^{12} + 1427 \nu^{11} - 265270 \nu^{10} - 205673 \nu^{9} + 515392 \nu^{8} + 1584467 \nu^{7} - 3089990 \nu^{6} - 11581465 \nu^{5} - 2887200 \nu^{4} + 73925875 \nu^{3} + 65686250 \nu^{2} + 8384375 \nu - 448000000$$$$)/8500000$$ $$\beta_{12}$$ $$=$$ $$($$$$374136 \nu^{15} + 550015 \nu^{14} - 2860075 \nu^{13} - 7558352 \nu^{12} - 10487208 \nu^{11} + 63213055 \nu^{10} + 36024597 \nu^{9} - 32747888 \nu^{8} - 508697448 \nu^{7} + 726564415 \nu^{6} + 1997199925 \nu^{5} + 3498498000 \nu^{4} - 17288645000 \nu^{3} - 12093690625 \nu^{2} - 19156796875 \nu + 107188750000$$$$)/ 1317500000$$ $$\beta_{13}$$ $$=$$ $$($$$$-372809 \nu^{15} + 616665 \nu^{14} + 2116200 \nu^{13} + 7167888 \nu^{12} - 13513273 \nu^{11} - 43000695 \nu^{10} - 34237368 \nu^{9} + 168261072 \nu^{8} + 85472887 \nu^{7} - 527876935 \nu^{6} - 2104753400 \nu^{5} + 3333950000 \nu^{4} + 12378994375 \nu^{3} + 11838115625 \nu^{2} - 20797375000 \nu - 73231250000$$$$)/ 1317500000$$ $$\beta_{14}$$ $$=$$ $$($$$$377912 \nu^{15} - 1106320 \nu^{14} + 575 \nu^{13} - 8294559 \nu^{12} + 23323064 \nu^{11} - 2981040 \nu^{10} + 49110799 \nu^{9} - 226586671 \nu^{8} + 314783384 \nu^{7} - 46705520 \nu^{6} + 2424194175 \nu^{5} - 6401757375 \nu^{4} + 244135000 \nu^{3} - 14467050000 \nu^{2} + 36653359375 \nu - 4410859375$$$$)/ 1317500000$$ $$\beta_{15}$$ $$=$$ $$($$$$-95903 \nu^{15} + 201256 \nu^{14} - 10640 \nu^{13} + 2025921 \nu^{12} - 4520623 \nu^{11} + 677992 \nu^{10} - 11953936 \nu^{9} + 50671601 \nu^{8} - 55705439 \nu^{7} + 10534952 \nu^{6} - 590045520 \nu^{5} + 1207441825 \nu^{4} - 26643375 \nu^{3} + 3471705000 \nu^{2} - 7407050000 \nu + 559140625$$$$)/ 263500000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + 2 \beta_{9} + \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} - 2 \beta_{11} - 2 \beta_{8} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - 3 \beta_{11} + 3 \beta_{8} + 2 \beta_{7} - 8 \beta_{4} - 6 \beta_{3} + 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{14} + \beta_{12} - 8 \beta_{11} + 8 \beta_{10} + 7 \beta_{9} + 8 \beta_{8} - 4 \beta_{7} - \beta_{5} + 7 \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{13} - 22 \beta_{12} - 41 \beta_{11} - 2 \beta_{9} - 31 \beta_{8} + 10 \beta_{6} - 48 \beta_{5} - 2 \beta_{4} - 22 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-25 \beta_{15} - 15 \beta_{14} - \beta_{13} + 2 \beta_{11} - \beta_{10} + 2 \beta_{8} + 20 \beta_{7} - 55 \beta_{6} - 20 \beta_{4} - 25 \beta_{3} + \beta_{2} - 15 \beta_{1} + 40$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{15} + 130 \beta_{14} + 2 \beta_{13} + 38 \beta_{12} - 75 \beta_{11} + 160 \beta_{10} + 130 \beta_{9} + 77 \beta_{8} + 120 \beta_{7} - 48 \beta_{5} + 8 \beta_{4} + 38 \beta_{3} + 90 \beta_{2} + 280$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-23 \beta_{15} + \beta_{14} - 15 \beta_{13} + 15 \beta_{12} - 80 \beta_{11} - 72 \beta_{10} + 121 \beta_{9} - 260 \beta_{8} + 4 \beta_{7} + 95 \beta_{6} - 240 \beta_{5} - 28 \beta_{4} + \beta_{3} - 23 \beta_{2} - 121 \beta_{1} + 1$$ $$\nu^{9}$$ $$=$$ $$($$$$-510 \beta_{15} - 386 \beta_{14} + 386 \beta_{13} + 54 \beta_{12} - 279 \beta_{11} - 64 \beta_{10} - 545 \beta_{8} + 616 \beta_{7} + 150 \beta_{6} + 176 \beta_{5} - 360 \beta_{4} - 438 \beta_{3} + 54 \beta_{2} + 438 \beta_{1} + 824$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$281 \beta_{15} + 399 \beta_{14} + 32 \beta_{13} + 281 \beta_{12} - 686 \beta_{11} + 696 \beta_{10} + 631 \beta_{9} + 718 \beta_{8} + 876 \beta_{7} + 135 \beta_{6} - 416 \beta_{5} - 684 \beta_{4} - 631 \beta_{3} + 399 \beta_{2} + 32 \beta_{1} - 696$$ $$\nu^{11}$$ $$=$$ $$($$$$-768 \beta_{15} - 768 \beta_{14} - 130 \beta_{13} + 954 \beta_{12} - 2725 \beta_{11} + 3680 \beta_{10} - 130 \beta_{9} - 3715 \beta_{8} - 2042 \beta_{7} + 2810 \beta_{6} - 4944 \beta_{5} - 1088 \beta_{4} + 2630 \beta_{2} - 2630 \beta_{1} - 1088$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-2537 \beta_{15} - 2336 \beta_{14} - 1905 \beta_{13} - 2336 \beta_{12} - 456 \beta_{11} - 544 \beta_{10} - 544 \beta_{9} - 1124 \beta_{8} + 4256 \beta_{7} + 2385 \beta_{6} - 4239 \beta_{5} - 2212 \beta_{4} - 1905 \beta_{3} + 544 \beta_{2} + 2537 \beta_{1} + 4239$$ $$\nu^{13}$$ $$=$$ $$($$$$-12288 \beta_{15} + 5378 \beta_{14} + 7424 \beta_{12} - 1727 \beta_{11} + 10816 \beta_{10} + 12288 \beta_{9} + 7167 \beta_{8} + 19992 \beta_{7} - 23360 \beta_{6} - 12864 \beta_{5} + 6762 \beta_{4} - 8234 \beta_{3} + 8234 \beta_{2} - 7424 \beta_{1} - 19992$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$-6432 \beta_{15} + 14112 \beta_{14} + 736 \beta_{13} + 5879 \beta_{12} - 242 \beta_{11} + 19849 \beta_{10} - 5879 \beta_{9} - 29742 \beta_{8} + 16608 \beta_{7} + 18560 \beta_{6} - 26464 \beta_{5} - 6432 \beta_{4} + 6432 \beta_{3} + 736 \beta_{2} - 14112 \beta_{1} + 32672$$ $$\nu^{15}$$ $$=$$ $$($$$$-51456 \beta_{15} - 52480 \beta_{14} - 51456 \beta_{13} + 31744 \beta_{12} + 76979 \beta_{11} - 64320 \beta_{10} + 52480 \beta_{9} - 69645 \beta_{8} + 116800 \beta_{7} + 58790 \beta_{6} - 39104 \beta_{5} - 58816 \beta_{4} - 31744 \beta_{3} + 50406 \beta_{1} + 88640$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −2.01595 + 0.967451i −0.410538 + 2.19806i −1.22045 − 1.87363i 2.23272 + 0.122339i 1.39977 − 1.74375i 2.21573 + 0.300921i −1.12594 − 1.93191i −1.07534 + 1.96052i −2.01595 − 0.967451i −0.410538 − 2.19806i −1.22045 + 1.87363i 2.23272 − 0.122339i 1.39977 + 1.74375i 2.21573 − 0.300921i −1.12594 + 1.93191i −1.07534 − 1.96052i
−0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.49226 + 1.66528i 1.00000i 0 0.707107 0.707107i 1.00000i 2.23272 0.122339i
97.2 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −0.461873 2.18785i 1.00000i 0 0.707107 0.707107i 1.00000i −1.22045 + 1.87363i
97.3 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.84456 + 1.26397i 1.00000i 0 0.707107 0.707107i 1.00000i −0.410538 2.19806i
97.4 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 2.10958 0.741398i 1.00000i 0 0.707107 0.707107i 1.00000i −2.01595 0.967451i
97.5 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −2.14668 0.625913i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −1.07534 1.96052i
97.6 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.569907 + 2.16222i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −1.12594 + 1.93191i
97.7 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.35397 1.77954i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 2.21573 0.300921i
97.8 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 2.22280 + 0.243230i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 1.39977 + 1.74375i
1273.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.49226 1.66528i 1.00000i 0 0.707107 + 0.707107i 1.00000i 2.23272 + 0.122339i
1273.2 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.461873 + 2.18785i 1.00000i 0 0.707107 + 0.707107i 1.00000i −1.22045 1.87363i
1273.3 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 1.84456 1.26397i 1.00000i 0 0.707107 + 0.707107i 1.00000i −0.410538 + 2.19806i
1273.4 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 2.10958 + 0.741398i 1.00000i 0 0.707107 + 0.707107i 1.00000i −2.01595 + 0.967451i
1273.5 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i −2.14668 + 0.625913i 1.00000i 0 −0.707107 0.707107i 1.00000i −1.07534 + 1.96052i
1273.6 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0.569907 2.16222i 1.00000i 0 −0.707107 0.707107i 1.00000i −1.12594 1.93191i
1273.7 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 1.35397 + 1.77954i 1.00000i 0 −0.707107 0.707107i 1.00000i 2.21573 + 0.300921i
1273.8 0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 2.22280 0.243230i 1.00000i 0 −0.707107 0.707107i 1.00000i 1.39977 1.74375i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1273.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.m.f yes 16
5.c odd 4 1 1470.2.m.c 16
7.b odd 2 1 1470.2.m.c 16
35.f even 4 1 inner 1470.2.m.f yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.m.c 16 5.c odd 4 1
1470.2.m.c 16 7.b odd 2 1
1470.2.m.f yes 16 1.a even 1 1 trivial
1470.2.m.f yes 16 35.f even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1470, [\chi])$$:

 $$T_{11}^{8} - 40 T_{11}^{6} + 8 T_{11}^{5} + 382 T_{11}^{4} - 160 T_{11}^{3} - 1088 T_{11}^{2} + 512 T_{11} + 512$$ $$T_{13}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{4}$$
$3$ $$( 1 + T^{4} )^{4}$$
$5$ $$390625 - 625000 T + 437500 T^{2} - 175000 T^{3} + 43750 T^{4} - 7000 T^{5} + 700 T^{6} - 40 T^{7} + 2 T^{8} - 8 T^{9} + 28 T^{10} - 56 T^{11} + 70 T^{12} - 56 T^{13} + 28 T^{14} - 8 T^{15} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$( 512 + 512 T - 1088 T^{2} - 160 T^{3} + 382 T^{4} + 8 T^{5} - 40 T^{6} + T^{8} )^{2}$$
$13$ $$18496 + 13056 T + 4608 T^{2} - 394624 T^{3} + 1910048 T^{4} - 2597824 T^{5} + 1900160 T^{6} - 648992 T^{7} + 107444 T^{8} - 12480 T^{9} + 21504 T^{10} - 5632 T^{11} + 740 T^{12} - 16 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$17$ $$1336336 - 15092736 T + 85229568 T^{2} - 153372096 T^{3} + 151093568 T^{4} - 75580672 T^{5} + 18393216 T^{6} + 598240 T^{7} - 130504 T^{8} - 242176 T^{9} + 109952 T^{10} + 26096 T^{11} + 3152 T^{12} - 64 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$19$ $$( 31744 + 54272 T + 5248 T^{2} - 12352 T^{3} - 3708 T^{4} + 144 T^{5} + 180 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$23$ $$4194304 - 8388608 T + 8388608 T^{2} - 1572864 T^{3} - 475136 T^{4} + 49152 T^{5} + 1146880 T^{6} + 119808 T^{7} - 8176 T^{8} + 26176 T^{9} + 49280 T^{10} + 14752 T^{11} + 2184 T^{12} - 48 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$29$ $$64 + 77184 T^{2} + 620448 T^{4} + 779424 T^{6} + 351348 T^{8} + 63568 T^{10} + 4204 T^{12} + 112 T^{14} + T^{16}$$
$31$ $$303038464 + 9677307904 T^{2} + 7772106752 T^{4} + 1273638912 T^{6} + 80125444 T^{8} + 2468928 T^{10} + 39804 T^{12} + 320 T^{14} + T^{16}$$
$37$ $$1224191770624 - 1025352663040 T + 429404979200 T^{2} - 88779096064 T^{3} + 14443317248 T^{4} - 4308895744 T^{5} + 1761953792 T^{6} - 362205056 T^{7} + 40893956 T^{8} - 2436896 T^{9} + 759424 T^{10} - 162000 T^{11} + 17924 T^{12} - 272 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$41$ $$14484603904 + 22834207744 T^{2} + 10787820672 T^{4} + 1468735296 T^{6} + 89708676 T^{8} + 2790480 T^{10} + 44724 T^{12} + 344 T^{14} + T^{16}$$
$43$ $$3347316736 - 6635388928 T + 6576668672 T^{2} - 3299573760 T^{3} + 932091904 T^{4} - 145625088 T^{5} + 83593216 T^{6} - 41145216 T^{7} + 11314820 T^{8} - 911680 T^{9} + 25088 T^{10} - 14784 T^{11} + 9164 T^{12} - 224 T^{13} + T^{16}$$
$47$ $$1212153856 + 2353004544 T + 2283798528 T^{2} - 21445607424 T^{3} + 49173127168 T^{4} - 40742932480 T^{5} + 17973970944 T^{6} - 2530418560 T^{7} + 179488004 T^{8} - 4097984 T^{9} + 3069440 T^{10} - 402592 T^{11} + 26380 T^{12} - 96 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$53$ $$1176027464704 - 2890288160768 T + 3551688163328 T^{2} - 1667752409088 T^{3} + 373704430336 T^{4} + 20328189440 T^{5} + 3965147136 T^{6} - 1148941824 T^{7} + 176869584 T^{8} + 4147648 T^{9} + 681600 T^{10} - 178336 T^{11} + 23960 T^{12} + 208 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$59$ $$( 15023104 - 11758592 T + 2958208 T^{2} - 174912 T^{3} - 37060 T^{4} + 5440 T^{5} - 52 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$61$ $$98867482624 + 266839572480 T^{2} + 189494849280 T^{4} + 31395442176 T^{6} + 1183933316 T^{8} + 19485456 T^{10} + 160788 T^{12} + 648 T^{14} + T^{16}$$
$67$ $$4228120576 + 125778264064 T + 1870827880448 T^{2} + 2849732984832 T^{3} + 2237420570624 T^{4} + 1006404622336 T^{5} + 294240651264 T^{6} + 58689541248 T^{7} + 8252020868 T^{8} + 825235648 T^{9} + 59987456 T^{10} + 3494880 T^{11} + 222668 T^{12} + 17248 T^{13} + 1152 T^{14} + 48 T^{15} + T^{16}$$
$71$ $$( 591872 - 208896 T - 112384 T^{2} + 28544 T^{3} + 8004 T^{4} - 896 T^{5} - 228 T^{6} + T^{8} )^{2}$$
$73$ $$36991502500096 - 18860431807488 T + 4808075692032 T^{2} - 615420529152 T^{3} + 134410557248 T^{4} - 48415579776 T^{5} + 12334017024 T^{6} - 1580872000 T^{7} + 123435140 T^{8} - 11231360 T^{9} + 2850816 T^{10} - 370112 T^{11} + 24668 T^{12} - 416 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$79$ $$148243480576 + 739137028096 T^{2} + 296029519872 T^{4} + 37532991488 T^{6} + 1442709760 T^{8} + 24132352 T^{10} + 193056 T^{12} + 720 T^{14} + T^{16}$$
$83$ $$1401249857536 - 4121323110400 T + 6060769280000 T^{2} - 2513080156160 T^{3} + 540572778496 T^{4} - 30152458240 T^{5} + 4115660800 T^{6} - 1911521280 T^{7} + 509747456 T^{8} - 13629440 T^{9} + 204800 T^{10} - 97280 T^{11} + 45600 T^{12} - 640 T^{13} + T^{16}$$
$89$ $$( 15376 - 781696 T + 196112 T^{2} + 173952 T^{3} - 32566 T^{4} - 5920 T^{5} + 52 T^{6} + 32 T^{7} + T^{8} )^{2}$$
$97$ $$244234884096256 + 257312657933312 T + 135545346394112 T^{2} + 29994906685440 T^{3} + 4080338085696 T^{4} + 512357261440 T^{5} + 117151090688 T^{6} + 22710734080 T^{7} + 2926317188 T^{8} + 269183232 T^{9} + 28910080 T^{10} + 4021248 T^{11} + 480604 T^{12} + 38880 T^{13} + 2048 T^{14} + 64 T^{15} + T^{16}$$