# Properties

 Label 1470.2.b.c Level $1470$ Weight $2$ Character orbit 1470.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7380090971$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 16 x^{13} + 2 x^{12} + 96 x^{10} - 80 x^{9} + 2 x^{8} - 240 x^{7} + 864 x^{6} + 162 x^{4} - 3888 x^{3} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 16 x^{13} + 2 x^{12} + 96 x^{10} - 80 x^{9} + 2 x^{8} - 240 x^{7} + 864 x^{6} + 162 x^{4} - 3888 x^{3} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-86 \nu^{15} + 30 \nu^{14} - 45 \nu^{13} + 971 \nu^{12} - 814 \nu^{11} + 294 \nu^{10} - 4053 \nu^{9} + 9355 \nu^{8} - 3814 \nu^{7} + 6318 \nu^{6} - 49437 \nu^{5} + 35019 \nu^{4} - 8910 \nu^{3} + 101574 \nu^{2} - 114453 \nu + 37179$$$$)/34992$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{15} + 20 \nu^{14} + 11 \nu^{13} + 64 \nu^{12} - 271 \nu^{11} - 28 \nu^{10} - 125 \nu^{9} + 1376 \nu^{8} - 1431 \nu^{7} - 636 \nu^{6} - 3125 \nu^{5} + 12096 \nu^{4} + 2385 \nu^{3} + 4212 \nu^{2} - 28269 \nu$$$$)/3888$$ $$\beta_{3}$$ $$=$$ $$($$$$-15 \nu^{15} + 8 \nu^{14} - 33 \nu^{13} + 168 \nu^{12} - 131 \nu^{11} + 544 \nu^{10} - 597 \nu^{9} + 1392 \nu^{8} - 3055 \nu^{7} + 3232 \nu^{6} - 7809 \nu^{5} + 6696 \nu^{4} - 27459 \nu^{3} + 16200 \nu^{2} - 5589 \nu + 81648$$$$)/5832$$ $$\beta_{4}$$ $$=$$ $$($$$$185 \nu^{15} - 30 \nu^{14} - 117 \nu^{13} - 2096 \nu^{12} + 769 \nu^{11} + 354 \nu^{10} + 7347 \nu^{9} - 14656 \nu^{8} + 3769 \nu^{7} - 270 \nu^{6} + 99387 \nu^{5} - 24624 \nu^{4} - 15471 \nu^{3} - 239598 \nu^{2} + 54675 \nu - 69984$$$$)/34992$$ $$\beta_{5}$$ $$=$$ $$($$$$-97 \nu^{15} - 150 \nu^{14} - 342 \nu^{13} + 1147 \nu^{12} + 1639 \nu^{11} + 3714 \nu^{10} - 4974 \nu^{9} - 565 \nu^{8} - 5393 \nu^{7} + 27498 \nu^{6} - 42822 \nu^{5} - 75141 \nu^{4} - 192537 \nu^{3} + 125874 \nu^{2} + 258066 \nu + 527067$$$$)/34992$$ $$\beta_{6}$$ $$=$$ $$($$$$-86 \nu^{15} - 94 \nu^{14} - 165 \nu^{13} + 917 \nu^{12} + 738 \nu^{11} + 1642 \nu^{10} - 3429 \nu^{9} + 2797 \nu^{8} - 1094 \nu^{7} + 12754 \nu^{6} - 39573 \nu^{5} - 35595 \nu^{4} - 86238 \nu^{3} + 94122 \nu^{2} + 99387 \nu + 236925$$$$)/11664$$ $$\beta_{7}$$ $$=$$ $$($$$$-211 \nu^{15} - 336 \nu^{14} - 711 \nu^{13} + 2188 \nu^{12} + 3253 \nu^{11} + 6816 \nu^{10} - 9231 \nu^{9} + 716 \nu^{8} - 2675 \nu^{7} + 55008 \nu^{6} - 90423 \nu^{5} - 158004 \nu^{4} - 337851 \nu^{3} + 244944 \nu^{2} + 438129 \nu + 778572$$$$)/34992$$ $$\beta_{8}$$ $$=$$ $$($$$$-211 \nu^{15} - 462 \nu^{14} - 594 \nu^{13} + 2431 \nu^{12} + 4621 \nu^{11} + 5178 \nu^{10} - 10698 \nu^{9} - 2713 \nu^{8} + 9565 \nu^{7} + 47826 \nu^{6} - 104418 \nu^{5} - 215649 \nu^{4} - 282771 \nu^{3} + 306666 \nu^{2} + 587574 \nu + 710775$$$$)/34992$$ $$\beta_{9}$$ $$=$$ $$($$$$-70 \nu^{15} - 141 \nu^{14} - 126 \nu^{13} + 796 \nu^{12} + 1306 \nu^{11} + 1005 \nu^{10} - 3246 \nu^{9} + 2 \nu^{8} + 4714 \nu^{7} + 10803 \nu^{6} - 33534 \nu^{5} - 62424 \nu^{4} - 60102 \nu^{3} + 93069 \nu^{2} + 153090 \nu + 153090$$$$)/8748$$ $$\beta_{10}$$ $$=$$ $$($$$$122 \nu^{15} + 117 \nu^{14} + 333 \nu^{13} - 1214 \nu^{12} - 926 \nu^{11} - 3555 \nu^{10} + 4701 \nu^{9} - 3910 \nu^{8} + 6922 \nu^{7} - 26715 \nu^{6} + 51597 \nu^{5} + 42786 \nu^{4} + 185058 \nu^{3} - 123363 \nu^{2} - 140211 \nu - 470934$$$$)/11664$$ $$\beta_{11}$$ $$=$$ $$($$$$-94 \nu^{15} - 165 \nu^{14} - 459 \nu^{13} + 910 \nu^{12} + 1642 \nu^{11} + 4827 \nu^{10} - 4083 \nu^{9} - 922 \nu^{8} - 7886 \nu^{7} + 34731 \nu^{6} - 35595 \nu^{5} - 72306 \nu^{4} - 240246 \nu^{3} + 99387 \nu^{2} + 236925 \nu + 599238$$$$)/11664$$ $$\beta_{12}$$ $$=$$ $$($$$$89 \nu^{15} + 219 \nu^{14} + 426 \nu^{13} - 974 \nu^{12} - 2327 \nu^{11} - 4029 \nu^{10} + 4626 \nu^{9} + 3194 \nu^{8} + 361 \nu^{7} - 32133 \nu^{6} + 38922 \nu^{5} + 103698 \nu^{4} + 205929 \nu^{3} - 121581 \nu^{2} - 290142 \nu - 494262$$$$)/11664$$ $$\beta_{13}$$ $$=$$ $$($$$$53 \nu^{15} + 112 \nu^{14} + 117 \nu^{13} - 596 \nu^{12} - 1119 \nu^{11} - 1000 \nu^{10} + 2505 \nu^{9} + 860 \nu^{8} - 3211 \nu^{7} - 9976 \nu^{6} + 24501 \nu^{5} + 53964 \nu^{4} + 57537 \nu^{3} - 73872 \nu^{2} - 143127 \nu - 142884$$$$)/5832$$ $$\beta_{14}$$ $$=$$ $$($$$$325 \nu^{15} + 633 \nu^{14} + 1386 \nu^{13} - 3418 \nu^{12} - 6643 \nu^{11} - 13863 \nu^{10} + 15666 \nu^{9} + 6094 \nu^{8} + 8789 \nu^{7} - 106695 \nu^{6} + 137322 \nu^{5} + 313254 \nu^{4} + 699597 \nu^{3} - 415287 \nu^{2} - 919998 \nu - 1692738$$$$)/34992$$ $$\beta_{15}$$ $$=$$ $$($$$$-485 \nu^{15} - 744 \nu^{14} - 1089 \nu^{13} + 5438 \nu^{12} + 6803 \nu^{11} + 10104 \nu^{10} - 22521 \nu^{9} + 6958 \nu^{8} + 6251 \nu^{7} + 89928 \nu^{6} - 236961 \nu^{5} - 322434 \nu^{4} - 544077 \nu^{3} + 651240 \nu^{2} + 850743 \nu + 1421550$$$$)/34992$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{8} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{13} - 2 \beta_{6} - \beta_{3} + 2 \beta_{2} - 2 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{14} + 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{7} - \beta_{4} + 6$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{15} + \beta_{14} - 3 \beta_{12} + \beta_{9} + 3 \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + 3 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{15} + 8 \beta_{13} - 2 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} + \beta_{7} - 10 \beta_{6} + 10 \beta_{5} - \beta_{4} - 8 \beta_{3} + 8 \beta_{2} - 6 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{15} + 14 \beta_{14} - 7 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 20 \beta_{7} - 14 \beta_{6} + 9 \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 30$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-31 \beta_{15} - 18 \beta_{14} + 8 \beta_{13} - 22 \beta_{12} - 18 \beta_{11} - 2 \beta_{10} + 24 \beta_{9} + 18 \beta_{8} - 8 \beta_{7} - 18 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} + 29 \beta_{2} - 2 \beta_{1} + 42$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$8 \beta_{15} - \beta_{14} + 32 \beta_{13} + 11 \beta_{12} - \beta_{11} - 11 \beta_{10} + 16 \beta_{9} + 21 \beta_{8} + 8 \beta_{7} - 21 \beta_{6} + 17 \beta_{5} - 6 \beta_{4} - 16 \beta_{3} + 2 \beta_{2} - 17 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$($$$$29 \beta_{15} + 74 \beta_{14} - 24 \beta_{13} - 26 \beta_{12} - 22 \beta_{11} + 2 \beta_{10} - 40 \beta_{9} + 120 \beta_{7} - 22 \beta_{6} - 26 \beta_{5} + 20 \beta_{4} + 56 \beta_{3} + 23 \beta_{2} - 74 \beta_{1} + 126$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-142 \beta_{15} - 78 \beta_{14} + 95 \beta_{13} - 78 \beta_{12} + 30 \beta_{11} + 114 \beta_{10} + 192 \beta_{9} + 30 \beta_{8} + 12 \beta_{7} + 32 \beta_{6} - 114 \beta_{5} - 96 \beta_{4} + 49 \beta_{3} + 110 \beta_{2} - 32 \beta_{1} + 66$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-12 \beta_{15} + 160 \beta_{14} + 264 \beta_{13} - 138 \beta_{12} - 160 \beta_{10} + 72 \beta_{9} + 174 \beta_{8} + 37 \beta_{7} - 138 \beta_{6} + 162 \beta_{5} - 219 \beta_{4} - 216 \beta_{3} - 72 \beta_{2} - 174 \beta_{1} - 96$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$19 \beta_{15} + 160 \beta_{14} - 32 \beta_{13} - 128 \beta_{12} - 111 \beta_{11} + 51 \beta_{10} + 47 \beta_{9} - 128 \beta_{8} + 305 \beta_{7} - 51 \beta_{6} + 160 \beta_{5} + 95 \beta_{4} + 160 \beta_{3} + 131 \beta_{2} - 111 \beta_{1} + 207$$ $$\nu^{13}$$ $$=$$ $$($$$$-800 \beta_{15} - 282 \beta_{14} + 224 \beta_{13} - 198 \beta_{12} + 198 \beta_{11} + 128 \beta_{10} + 1312 \beta_{9} - 128 \beta_{8} - 25 \beta_{7} - 544 \beta_{6} - 544 \beta_{5} - 563 \beta_{4} + 640 \beta_{3} + 32 \beta_{2} + 306$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-530 \beta_{15} - 288 \beta_{14} + 1031 \beta_{13} - 736 \beta_{12} - 1312 \beta_{11} - 1312 \beta_{10} - 192 \beta_{9} + 786 \beta_{8} + 320 \beta_{7} - 288 \beta_{6} + 786 \beta_{5} - 992 \beta_{4} - 889 \beta_{3} - 590 \beta_{2} - 736 \beta_{1} + 1632$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$843 \beta_{15} + 608 \beta_{14} + 1152 \beta_{13} - 1408 \beta_{11} + 928 \beta_{10} + 800 \beta_{9} + 608 \beta_{8} + 2912 \beta_{7} - 726 \beta_{6} + 1408 \beta_{5} + 1632 \beta_{4} + 2784 \beta_{3} + 587 \beta_{2} - 928 \beta_{1} + 864$$$$)/2$$

## Character values

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

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −0.793941 + 1.53937i 1.71739 + 0.224843i −1.11192 − 1.32802i −0.642116 − 1.60863i 1.73191 − 0.0223575i −0.519068 + 1.65244i −1.50062 + 0.864947i 1.11836 − 1.32260i −0.793941 − 1.53937i 1.71739 − 0.224843i −1.11192 + 1.32802i −0.642116 + 1.60863i 1.73191 + 0.0223575i −0.519068 − 1.65244i −1.50062 − 0.864947i 1.11836 + 1.32260i
1.00000i −1.72602 0.144414i −1.00000 −1.00000 −0.144414 + 1.72602i 0 1.00000i 2.95829 + 0.498524i 1.00000i
881.2 1.00000i −1.67271 + 0.449490i −1.00000 −1.00000 0.449490 + 1.67271i 0 1.00000i 2.59592 1.50373i 1.00000i
881.3 1.00000i −1.53549 0.801418i −1.00000 −1.00000 −0.801418 + 1.53549i 0 1.00000i 1.71546 + 2.46114i 1.00000i
881.4 1.00000i −1.24045 + 1.20883i −1.00000 −1.00000 1.20883 + 1.24045i 0 1.00000i 0.0774422 2.99900i 1.00000i
881.5 1.00000i −0.683428 1.59152i −1.00000 −1.00000 −1.59152 + 0.683428i 0 1.00000i −2.06585 + 2.17537i 1.00000i
881.6 1.00000i 0.152807 + 1.72530i −1.00000 −1.00000 1.72530 0.152807i 0 1.00000i −2.95330 + 0.527274i 1.00000i
881.7 1.00000i 1.05539 1.37337i −1.00000 −1.00000 −1.37337 1.05539i 0 1.00000i −0.772290 2.89889i 1.00000i
881.8 1.00000i 1.64990 + 0.527098i −1.00000 −1.00000 0.527098 1.64990i 0 1.00000i 2.44434 + 1.73932i 1.00000i
881.9 1.00000i −1.72602 + 0.144414i −1.00000 −1.00000 −0.144414 1.72602i 0 1.00000i 2.95829 0.498524i 1.00000i
881.10 1.00000i −1.67271 0.449490i −1.00000 −1.00000 0.449490 1.67271i 0 1.00000i 2.59592 + 1.50373i 1.00000i
881.11 1.00000i −1.53549 + 0.801418i −1.00000 −1.00000 −0.801418 1.53549i 0 1.00000i 1.71546 2.46114i 1.00000i
881.12 1.00000i −1.24045 1.20883i −1.00000 −1.00000 1.20883 1.24045i 0 1.00000i 0.0774422 + 2.99900i 1.00000i
881.13 1.00000i −0.683428 + 1.59152i −1.00000 −1.00000 −1.59152 0.683428i 0 1.00000i −2.06585 2.17537i 1.00000i
881.14 1.00000i 0.152807 1.72530i −1.00000 −1.00000 1.72530 + 0.152807i 0 1.00000i −2.95330 0.527274i 1.00000i
881.15 1.00000i 1.05539 + 1.37337i −1.00000 −1.00000 −1.37337 + 1.05539i 0 1.00000i −0.772290 + 2.89889i 1.00000i
881.16 1.00000i 1.64990 0.527098i −1.00000 −1.00000 0.527098 + 1.64990i 0 1.00000i 2.44434 1.73932i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.b.c 16
3.b odd 2 1 1470.2.b.d yes 16
7.b odd 2 1 1470.2.b.d yes 16
21.c even 2 1 inner 1470.2.b.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.b.c 16 1.a even 1 1 trivial
1470.2.b.c 16 21.c even 2 1 inner
1470.2.b.d yes 16 3.b odd 2 1
1470.2.b.d yes 16 7.b odd 2 1

## Hecke kernels

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

 $$T_{11}^{16} + \cdots$$ $$T_{17}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$6561 + 17496 T + 20412 T^{2} + 13608 T^{3} + 5670 T^{4} + 1512 T^{5} + 252 T^{6} + 24 T^{7} + 2 T^{8} + 8 T^{9} + 28 T^{10} + 56 T^{11} + 70 T^{12} + 56 T^{13} + 28 T^{14} + 8 T^{15} + T^{16}$$
$5$ $$( 1 + T )^{16}$$
$7$ $$T^{16}$$
$11$ $$64 + 2688 T^{2} + 26016 T^{4} + 83808 T^{6} + 74484 T^{8} + 20144 T^{10} + 2092 T^{12} + 80 T^{14} + T^{16}$$
$13$ $$1183744 + 7696384 T^{2} + 8337920 T^{4} + 3194624 T^{6} + 585232 T^{8} + 57632 T^{10} + 3128 T^{12} + 88 T^{14} + T^{16}$$
$17$ $$( 8452 - 7456 T - 25728 T^{2} + 18832 T^{3} - 3596 T^{4} - 304 T^{5} + 192 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$19$ $$16516096 + 110871552 T^{2} + 200189056 T^{4} + 56122048 T^{6} + 6350596 T^{8} + 362288 T^{10} + 10996 T^{12} + 168 T^{14} + T^{16}$$
$23$ $$15130968064 + 13997326336 T^{2} + 4336808960 T^{4} + 561655296 T^{6} + 36398608 T^{8} + 1280352 T^{10} + 24792 T^{12} + 248 T^{14} + T^{16}$$
$29$ $$295936 + 5415936 T^{2} + 24214656 T^{4} + 13866688 T^{6} + 2903812 T^{8} + 263888 T^{10} + 10740 T^{12} + 184 T^{14} + T^{16}$$
$31$ $$8131710976 + 11097358336 T^{2} + 3766910720 T^{4} + 550850048 T^{6} + 40430212 T^{8} + 1544624 T^{10} + 29972 T^{12} + 280 T^{14} + T^{16}$$
$37$ $$( 1820704 - 188992 T - 314768 T^{2} + 1008 T^{3} + 13490 T^{4} + 96 T^{5} - 212 T^{6} + T^{8} )^{2}$$
$41$ $$( -904672 + 699328 T - 35120 T^{2} - 64112 T^{3} + 7314 T^{4} + 1536 T^{5} - 188 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$43$ $$( -28024 + 26912 T + 8376 T^{2} - 11472 T^{3} + 182 T^{4} + 1072 T^{5} - 128 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$47$ $$( -1727552 + 697984 T + 115392 T^{2} - 63040 T^{3} - 798 T^{4} + 1808 T^{5} - 68 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$53$ $$18939904 + 333971456 T^{2} + 1463631872 T^{4} + 632717312 T^{6} + 68257280 T^{8} + 2889728 T^{10} + 51520 T^{12} + 384 T^{14} + T^{16}$$
$59$ $$( -76408 - 102944 T + 143768 T^{2} - 37168 T^{3} - 3146 T^{4} + 1696 T^{5} - 56 T^{6} - 16 T^{7} + T^{8} )^{2}$$
$61$ $$870235242496 + 31063709057024 T^{2} + 3123026231296 T^{4} + 126425292800 T^{6} + 2662916352 T^{8} + 31464192 T^{10} + 208608 T^{12} + 720 T^{14} + T^{16}$$
$67$ $$( 9094688 + 524224 T - 1123440 T^{2} - 93424 T^{3} + 29586 T^{4} + 1792 T^{5} - 300 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$71$ $$204831512068096 + 62782122754048 T^{2} + 5247191334912 T^{4} + 187913871360 T^{6} + 3539389696 T^{8} + 37900032 T^{10} + 231776 T^{12} + 752 T^{14} + T^{16}$$
$73$ $$2539518146151424 + 264511679333376 T^{2} + 11790521357440 T^{4} + 293473027264 T^{6} + 4454045828 T^{8} + 42100048 T^{10} + 241076 T^{12} + 760 T^{14} + T^{16}$$
$79$ $$( 49664 + 76800 T - 151552 T^{2} + 59136 T^{3} - 592 T^{4} - 2016 T^{5} + 24 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$83$ $$( -4420672 - 26525568 T + 9985088 T^{2} - 977344 T^{3} - 39294 T^{4} + 10336 T^{5} - 252 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$89$ $$( -3104 - 15040 T - 22992 T^{2} - 10064 T^{3} + 1746 T^{4} + 1024 T^{5} - 140 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$97$ $$24123657389056 + 25502172857344 T^{2} + 4587936214144 T^{4} + 232475057344 T^{6} + 5327273988 T^{8} + 62470736 T^{10} + 373108 T^{12} + 1016 T^{14} + T^{16}$$