# Properties

 Label 140.2.u.a Level $140$ Weight $2$ Character orbit 140.u Analytic conductor $1.118$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.u (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} - 12338 x^{3} + 5322 x^{2} - 1484 x + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{2} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{3} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{7} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{2} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{3} + ( -2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{7} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{11} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{13} + ( -2 + 5 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{15} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{15} ) q^{17} + ( -2 \beta_{2} + 4 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{19} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{21} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{23} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{25} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{27} + ( 5 \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{29} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} ) q^{31} + ( -2 + 2 \beta_{1} - \beta_{4} + \beta_{5} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{33} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{35} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{39} + ( -1 + \beta_{1} + \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{11} - 4 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{43} + ( -6 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} - 5 \beta_{13} - \beta_{14} - 3 \beta_{15} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{47} + ( -4 \beta_{1} - 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{13} ) q^{49} + ( 4 + 4 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{51} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{53} + ( -1 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{55} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{12} + 4 \beta_{13} + \beta_{14} ) q^{57} + ( -5 \beta_{1} - \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{59} + ( 3 + \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{61} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 3 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{63} + ( 5 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 5 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{13} - \beta_{14} + 5 \beta_{15} ) q^{65} + ( -\beta_{7} - \beta_{8} - \beta_{13} - \beta_{15} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{69} + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( 6 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 6 \beta_{9} + \beta_{10} + 2 \beta_{13} + 3 \beta_{15} ) q^{73} + ( 4 + 8 \beta_{1} + 5 \beta_{2} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} + \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{75} + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{14} ) q^{77} + ( 2 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{79} + ( 4 \beta_{1} - \beta_{2} + 3 \beta_{7} + \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{81} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{83} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{85} + ( 2 - 7 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - 4 \beta_{13} - \beta_{14} + \beta_{15} ) q^{87} + ( 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{89} + ( -2 - 7 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 3 \beta_{10} - \beta_{11} - 5 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + 4 \beta_{15} ) q^{91} + ( -7 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 7 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{14} - 7 \beta_{15} ) q^{93} + ( 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} + 3 \beta_{12} + 4 \beta_{14} - \beta_{15} ) q^{95} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{97} + ( -5 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - 3 \beta_{13} - 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 6q^{5} + 2q^{7} + O(q^{10})$$ $$16q + 6q^{5} + 2q^{7} - 20q^{15} + 18q^{17} - 4q^{21} - 16q^{23} + 6q^{25} - 12q^{31} - 42q^{33} - 40q^{35} - 14q^{37} + 28q^{43} - 66q^{45} - 6q^{47} + 20q^{51} - 10q^{53} + 44q^{57} + 60q^{61} + 48q^{63} + 34q^{65} + 8q^{67} - 8q^{71} + 78q^{73} + 96q^{75} + 10q^{77} + 24q^{81} + 30q^{87} - 64q^{91} - 62q^{93} + 2q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} - 12338 x^{3} + 5322 x^{2} - 1484 x + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$25 \nu^{14} - 175 \nu^{13} + 964 \nu^{12} - 3509 \nu^{11} + 10031 \nu^{10} - 22160 \nu^{9} + 38459 \nu^{8} - 52325 \nu^{7} + 56098 \nu^{6} - 47137 \nu^{5} + 37029 \nu^{4} - 27730 \nu^{3} + 18082 \nu^{2} - 7652 \nu - 4676$$$$)/2156$$ $$\beta_{2}$$ $$=$$ $$($$$$-79 \nu^{14} + 553 \nu^{13} - 3422 \nu^{12} + 13343 \nu^{11} - 44289 \nu^{10} + 112314 \nu^{9} - 241133 \nu^{8} + 414827 \nu^{7} - 596008 \nu^{6} + 684799 \nu^{5} - 633503 \nu^{4} + 446416 \nu^{3} - 229422 \nu^{2} + 75604 \nu - 11564$$$$)/308$$ $$\beta_{3}$$ $$=$$ $$($$$$4834 \nu^{15} - 101205 \nu^{14} + 704128 \nu^{13} - 3926292 \nu^{12} + 15242328 \nu^{11} - 49407202 \nu^{10} + 126021005 \nu^{9} - 270743513 \nu^{8} + 472491670 \nu^{7} - 687499840 \nu^{6} + 806413714 \nu^{5} - 760368460 \nu^{4} + 548558607 \nu^{3} - 287680476 \nu^{2} + 95935530 \nu - 14479640$$$$)/466774$$ $$\beta_{4}$$ $$=$$ $$($$$$4834 \nu^{15} + 28695 \nu^{14} - 205172 \nu^{13} + 1782813 \nu^{12} - 7191402 \nu^{11} + 26285961 \nu^{10} - 68473935 \nu^{9} + 155008067 \nu^{8} - 273164640 \nu^{7} + 408003583 \nu^{6} - 479416158 \nu^{5} + 457303315 \nu^{4} - 328361653 \nu^{3} + 172224412 \nu^{2} - 58183892 \nu + 8834812$$$$)/466774$$ $$\beta_{5}$$ $$=$$ $$($$$$-17435 \nu^{15} + 211950 \nu^{14} - 1405979 \nu^{13} + 7058965 \nu^{12} - 26329116 \nu^{11} + 81229569 \nu^{10} - 201376883 \nu^{9} + 419520312 \nu^{8} - 717436363 \nu^{7} + 1019733523 \nu^{6} - 1176603774 \nu^{5} + 1083102693 \nu^{4} - 768004306 \nu^{3} + 389408520 \nu^{2} - 127561820 \nu + 18788784$$$$)/933548$$ $$\beta_{6}$$ $$=$$ $$($$$$-17435 \nu^{15} + 35286 \nu^{14} - 169331 \nu^{13} - 612063 \nu^{12} + 3620628 \nu^{11} - 18296347 \nu^{10} + 51186821 \nu^{9} - 121599788 \nu^{8} + 210583093 \nu^{7} - 299442545 \nu^{6} + 317228906 \nu^{5} - 253190731 \nu^{4} + 130352918 \nu^{3} - 26798008 \nu^{2} - 11351548 \nu + 8216656$$$$)/933548$$ $$\beta_{7}$$ $$=$$ $$($$$$2249 \nu^{15} - 27909 \nu^{14} + 180284 \nu^{13} - 887769 \nu^{12} + 3199847 \nu^{11} - 9537142 \nu^{10} + 22582291 \nu^{9} - 44855531 \nu^{8} + 72251158 \nu^{7} - 96645993 \nu^{6} + 103373217 \nu^{5} - 88306832 \nu^{4} + 56756774 \nu^{3} - 26082844 \nu^{2} + 7251748 \nu - 796740$$$$)/84868$$ $$\beta_{8}$$ $$=$$ $$($$$$24739 \nu^{15} - 207842 \nu^{14} + 1289025 \nu^{13} - 5512533 \nu^{12} + 18704048 \nu^{11} - 50802181 \nu^{10} + 112528069 \nu^{9} - 206198044 \nu^{8} + 308096285 \nu^{7} - 379227455 \nu^{6} + 369778726 \nu^{5} - 286728109 \nu^{4} + 162205264 \nu^{3} - 66777548 \nu^{2} + 15550132 \nu - 1768592$$$$)/933548$$ $$\beta_{9}$$ $$=$$ $$($$$$24739 \nu^{15} - 163243 \nu^{14} + 976832 \nu^{13} - 3587415 \nu^{12} + 11211849 \nu^{11} - 26017694 \nu^{10} + 49843525 \nu^{9} - 71941631 \nu^{8} + 77519022 \nu^{7} - 47126847 \nu^{6} - 13756653 \nu^{5} + 75656952 \nu^{4} - 101419858 \nu^{3} + 80642498 \nu^{2} - 39139500 \nu + 9046016$$$$)/933548$$ $$\beta_{10}$$ $$=$$ $$($$$$-34102 \nu^{15} + 371809 \nu^{14} - 2442791 \nu^{13} + 11725816 \nu^{12} - 42750227 \nu^{11} + 126934095 \nu^{10} - 304446794 \nu^{9} + 608150313 \nu^{8} - 995122009 \nu^{7} + 1346250742 \nu^{6} - 1469029887 \nu^{5} + 1277089185 \nu^{4} - 848228814 \nu^{3} + 407263392 \nu^{2} - 128511972 \nu + 21672924$$$$)/933548$$ $$\beta_{11}$$ $$=$$ $$($$$$-34102 \nu^{15} + 139721 \nu^{14} - 818175 \nu^{13} + 1712258 \nu^{12} - 3788887 \nu^{11} - 1719731 \nu^{10} + 20396734 \nu^{9} - 83878947 \nu^{8} + 186131435 \nu^{7} - 329316368 \nu^{6} + 428748493 \nu^{5} - 438556405 \nu^{4} + 327627718 \nu^{3} - 173528168 \nu^{2} + 54103180 \nu - 8891680$$$$)/933548$$ $$\beta_{12}$$ $$=$$ $$($$$$-25839 \nu^{15} + 131224 \nu^{14} - 795654 \nu^{13} + 2382058 \nu^{12} - 7119904 \nu^{11} + 12803881 \nu^{10} - 20842601 \nu^{9} + 14762885 \nu^{8} + 3093328 \nu^{7} - 49735550 \nu^{6} + 88974362 \nu^{5} - 113993809 \nu^{4} + 93361728 \nu^{3} - 54426665 \nu^{2} + 18307754 \nu - 2395204$$$$)/466774$$ $$\beta_{13}$$ $$=$$ $$($$$$6947 \nu^{15} - 52752 \nu^{14} + 331837 \nu^{13} - 1358817 \nu^{12} + 4632502 \nu^{11} - 12259977 \nu^{10} + 27244153 \nu^{9} - 49053578 \nu^{8} + 73748437 \nu^{7} - 89811387 \nu^{6} + 89051544 \nu^{5} - 69038281 \nu^{4} + 40873468 \nu^{3} - 16983728 \nu^{2} + 4309144 \nu - 431788$$$$)/84868$$ $$\beta_{14}$$ $$=$$ $$($$$$-76661 \nu^{15} + 455233 \nu^{14} - 2856600 \nu^{13} + 10109135 \nu^{12} - 33483101 \nu^{11} + 79191796 \nu^{10} - 170125277 \nu^{9} + 280002713 \nu^{8} - 413606074 \nu^{7} + 477906771 \nu^{6} - 490328295 \nu^{5} + 390623638 \nu^{4} - 266295000 \nu^{3} + 132858206 \nu^{2} - 47592276 \nu + 10482976$$$$)/933548$$ $$\beta_{15}$$ $$=$$ $$($$$$852 \nu^{15} - 6390 \nu^{14} + 40676 \nu^{13} - 167479 \nu^{12} + 580296 \nu^{11} - 1562572 \nu^{10} + 3560985 \nu^{9} - 6611661 \nu^{8} + 10337168 \nu^{7} - 13218775 \nu^{6} + 13918336 \nu^{5} - 11642080 \nu^{4} + 7579665 \nu^{3} - 3582677 \nu^{2} + 1115142 \nu - 175506$$$$)/9526$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 4 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 9 \beta_{11} + 7 \beta_{10} - 7 \beta_{9} - 7 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 12 \beta_{1} - 9$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{15} - 4 \beta_{14} + 27 \beta_{13} + 23 \beta_{12} - 21 \beta_{11} + 17 \beta_{10} - 17 \beta_{9} - 13 \beta_{8} - \beta_{7} + 6 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} + 44 \beta_{1} + 17$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$20 \beta_{15} + 14 \beta_{14} + 5 \beta_{13} + 25 \beta_{12} + 49 \beta_{11} - 35 \beta_{10} + 31 \beta_{9} + 41 \beta_{8} - 69 \beta_{7} - 10 \beta_{6} - 20 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 32 \beta_{2} - 52 \beta_{1} + 71$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$70 \beta_{15} + 52 \beta_{14} - 207 \beta_{13} - 137 \beta_{12} + 221 \beta_{11} - 169 \beta_{10} + 171 \beta_{9} + 121 \beta_{8} - 51 \beta_{7} - 70 \beta_{6} - 40 \beta_{5} + 42 \beta_{4} + 38 \beta_{3} - 108 \beta_{2} - 394 \beta_{1} - 33$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-116 \beta_{15} - 60 \beta_{14} - 247 \beta_{13} - 351 \beta_{12} - 157 \beta_{11} + 83 \beta_{10} - 15 \beta_{9} - 225 \beta_{8} + 479 \beta_{7} - 10 \beta_{6} + 130 \beta_{5} - 20 \beta_{4} - 34 \beta_{3} + 170 \beta_{2} + 14 \beta_{1} - 519$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-800 \beta_{15} - 492 \beta_{14} + 1383 \beta_{13} + 631 \beta_{12} - 1893 \beta_{11} + 1345 \beta_{10} - 1317 \beta_{9} - 1077 \beta_{8} + 811 \beta_{7} + 560 \beta_{6} + 452 \beta_{5} - 324 \beta_{4} - 284 \beta_{3} + 1056 \beta_{2} + 3036 \beta_{1} - 331$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$156 \beta_{15} - 10 \beta_{14} + 3339 \beta_{13} + 3381 \beta_{12} - 619 \beta_{11} + 705 \beta_{10} - 1463 \beta_{9} + 919 \beta_{8} - 2981 \beta_{7} + 738 \beta_{6} - 630 \beta_{5} - 246 \beta_{4} + 18 \beta_{3} - 454 \beta_{2} + 2804 \beta_{1} + 3427$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$7290 \beta_{15} + 4024 \beta_{14} - 7549 \beta_{13} - 1189 \beta_{12} + 14353 \beta_{11} - 9429 \beta_{10} + 8371 \beta_{9} + 9061 \beta_{8} - 9055 \beta_{7} - 3630 \beta_{6} - 4230 \beta_{5} + 2044 \beta_{4} + 1976 \beta_{3} - 9050 \beta_{2} - 20748 \beta_{1} + 6173$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$6610 \beta_{15} + 4150 \beta_{14} - 33959 \beta_{13} - 27511 \beta_{12} + 18835 \beta_{11} - 14333 \beta_{10} + 20679 \beta_{9} + 219 \beta_{8} + 15001 \beta_{7} - 9822 \beta_{6} + 1068 \beta_{5} + 4182 \beta_{4} + 1234 \beta_{3} - 4462 \beta_{2} - 41532 \beta_{1} - 19277$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-55336 \beta_{15} - 28692 \beta_{14} + 25577 \beta_{13} - 19709 \beta_{12} - 95753 \beta_{11} + 58349 \beta_{10} - 41825 \beta_{9} - 70969 \beta_{8} + 85635 \beta_{7} + 18282 \beta_{6} + 34904 \beta_{5} - 10190 \beta_{4} - 13458 \beta_{3} + 69674 \beta_{2} + 121966 \beta_{1} - 67705$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-115110 \beta_{15} - 62912 \beta_{14} + 294485 \beta_{13} + 195181 \beta_{12} - 243247 \beta_{11} + 165385 \beta_{10} - 205555 \beta_{9} - 63231 \beta_{8} - 37163 \beta_{7} + 97066 \beta_{6} + 24786 \beta_{5} - 42520 \beta_{4} - 18574 \beta_{3} + 100980 \beta_{2} + 441786 \beta_{1} + 77217$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$341246 \beta_{15} + 169732 \beta_{14} + 91907 \beta_{13} + 356907 \beta_{12} + 526209 \beta_{11} - 294961 \beta_{10} + 107171 \beta_{9} + 506405 \beta_{8} - 716103 \beta_{7} - 44304 \beta_{6} - 255624 \beta_{5} + 28968 \beta_{4} + 88256 \beta_{3} - 471362 \beta_{2} - 534052 \beta_{1} + 606325$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$1314998 \beta_{15} + 682938 \beta_{14} - 2246473 \beta_{13} - 1174327 \beta_{12} + 2458635 \beta_{11} - 1568209 \beta_{10} + 1719825 \beta_{9} + 961305 \beta_{8} - 398363 \beta_{7} - 816338 \beta_{6} - 445518 \beta_{5} + 352466 \beta_{4} + 209474 \beta_{3} - 1268392 \beta_{2} - 3999172 \beta_{1} + 26483$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-\beta_{8} - \beta_{9}$$ $$1$$ $$1 + \beta_{15}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 1.78727i 0.5 + 0.105864i 0.5 − 1.61777i 0.5 − 2.27536i 0.5 − 1.78727i 0.5 − 0.105864i 0.5 + 1.61777i 0.5 + 2.27536i 0.5 + 2.78727i 0.5 + 1.10586i 0.5 − 0.617773i 0.5 − 1.27536i 0.5 − 2.78727i 0.5 − 1.10586i 0.5 + 0.617773i 0.5 + 1.27536i
0 −3.12447 + 0.837199i 0 1.01073 1.99460i 0 0.870132 2.49857i 0 6.46333 3.73161i 0
17.2 0 −0.827625 + 0.221762i 0 0.543268 + 2.16907i 0 2.22829 + 1.42643i 0 −1.96229 + 1.13293i 0
17.3 0 1.52691 0.409133i 0 2.14461 0.632955i 0 −2.59572 0.512081i 0 −0.434025 + 0.250584i 0
17.4 0 2.42519 0.649827i 0 −2.19862 0.407542i 0 2.59537 0.513853i 0 2.86119 1.65191i 0
33.1 0 −3.12447 0.837199i 0 1.01073 + 1.99460i 0 0.870132 + 2.49857i 0 6.46333 + 3.73161i 0
33.2 0 −0.827625 0.221762i 0 0.543268 2.16907i 0 2.22829 1.42643i 0 −1.96229 1.13293i 0
33.3 0 1.52691 + 0.409133i 0 2.14461 + 0.632955i 0 −2.59572 + 0.512081i 0 −0.434025 0.250584i 0
33.4 0 2.42519 + 0.649827i 0 −2.19862 + 0.407542i 0 2.59537 + 0.513853i 0 2.86119 + 1.65191i 0
73.1 0 −0.837199 3.12447i 0 2.23274 + 0.121977i 0 −2.49857 0.870132i 0 −6.46333 + 3.73161i 0
73.2 0 −0.221762 0.827625i 0 −1.60683 1.55502i 0 1.42643 2.22829i 0 1.96229 1.13293i 0
73.3 0 0.409133 + 1.52691i 0 1.62046 1.54081i 0 −0.512081 + 2.59572i 0 0.434025 0.250584i 0
73.4 0 0.649827 + 2.42519i 0 −0.746366 + 2.10783i 0 −0.513853 2.59537i 0 −2.86119 + 1.65191i 0
117.1 0 −0.837199 + 3.12447i 0 2.23274 0.121977i 0 −2.49857 + 0.870132i 0 −6.46333 3.73161i 0
117.2 0 −0.221762 + 0.827625i 0 −1.60683 + 1.55502i 0 1.42643 + 2.22829i 0 1.96229 + 1.13293i 0
117.3 0 0.409133 1.52691i 0 1.62046 + 1.54081i 0 −0.512081 2.59572i 0 0.434025 + 0.250584i 0
117.4 0 0.649827 2.42519i 0 −0.746366 2.10783i 0 −0.513853 + 2.59537i 0 −2.86119 1.65191i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 117.4 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.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.u.a 16
3.b odd 2 1 1260.2.dq.a 16
4.b odd 2 1 560.2.ci.d 16
5.b even 2 1 700.2.bc.b 16
5.c odd 4 1 inner 140.2.u.a 16
5.c odd 4 1 700.2.bc.b 16
7.b odd 2 1 980.2.v.a 16
7.c even 3 1 980.2.m.a 16
7.c even 3 1 980.2.v.a 16
7.d odd 6 1 inner 140.2.u.a 16
7.d odd 6 1 980.2.m.a 16
15.e even 4 1 1260.2.dq.a 16
20.e even 4 1 560.2.ci.d 16
21.g even 6 1 1260.2.dq.a 16
28.f even 6 1 560.2.ci.d 16
35.f even 4 1 980.2.v.a 16
35.i odd 6 1 700.2.bc.b 16
35.k even 12 1 inner 140.2.u.a 16
35.k even 12 1 700.2.bc.b 16
35.k even 12 1 980.2.m.a 16
35.l odd 12 1 980.2.m.a 16
35.l odd 12 1 980.2.v.a 16
105.w odd 12 1 1260.2.dq.a 16
140.x odd 12 1 560.2.ci.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.u.a 16 1.a even 1 1 trivial
140.2.u.a 16 5.c odd 4 1 inner
140.2.u.a 16 7.d odd 6 1 inner
140.2.u.a 16 35.k even 12 1 inner
560.2.ci.d 16 4.b odd 2 1
560.2.ci.d 16 20.e even 4 1
560.2.ci.d 16 28.f even 6 1
560.2.ci.d 16 140.x odd 12 1
700.2.bc.b 16 5.b even 2 1
700.2.bc.b 16 5.c odd 4 1
700.2.bc.b 16 35.i odd 6 1
700.2.bc.b 16 35.k even 12 1
980.2.m.a 16 7.c even 3 1
980.2.m.a 16 7.d odd 6 1
980.2.m.a 16 35.k even 12 1
980.2.m.a 16 35.l odd 12 1
980.2.v.a 16 7.b odd 2 1
980.2.v.a 16 7.c even 3 1
980.2.v.a 16 35.f even 4 1
980.2.v.a 16 35.l odd 12 1
1260.2.dq.a 16 3.b odd 2 1
1260.2.dq.a 16 15.e even 4 1
1260.2.dq.a 16 21.g even 6 1
1260.2.dq.a 16 105.w odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$14641 + 15972 T + 8712 T^{2} + 3168 T^{3} - 14142 T^{4} - 1428 T^{5} + 7200 T^{6} - 9360 T^{7} + 6443 T^{8} - 1068 T^{9} + 120 T^{11} - 78 T^{12} + T^{16}$$
$5$ $$390625 - 468750 T + 234375 T^{2} - 56250 T^{3} - 1875 T^{4} + 4500 T^{5} - 1950 T^{6} + 1680 T^{7} - 1006 T^{8} + 336 T^{9} - 78 T^{10} + 36 T^{11} - 3 T^{12} - 18 T^{13} + 15 T^{14} - 6 T^{15} + T^{16}$$
$7$ $$5764801 - 1647086 T + 235298 T^{2} + 268912 T^{3} - 199283 T^{4} + 67228 T^{5} - 4802 T^{6} - 10290 T^{7} + 6468 T^{8} - 1470 T^{9} - 98 T^{10} + 196 T^{11} - 83 T^{12} + 16 T^{13} + 2 T^{14} - 2 T^{15} + T^{16}$$
$11$ $$( 100 + 60 T + 166 T^{2} - 78 T^{3} + 159 T^{4} - 12 T^{5} + 13 T^{6} + T^{8} )^{2}$$
$13$ $$268435456 + 58589184 T^{4} + 633872 T^{8} + 1800 T^{12} + T^{16}$$
$17$ $$256 - 2304 T + 10368 T^{2} - 31104 T^{3} + 48240 T^{4} - 6912 T^{5} - 1944 T^{6} + 7740 T^{7} - 12679 T^{8} + 3546 T^{9} + 810 T^{10} - 2880 T^{11} + 3231 T^{12} - 972 T^{13} + 162 T^{14} - 18 T^{15} + T^{16}$$
$19$ $$4711998736 + 1304236000 T^{2} + 233802668 T^{4} + 25047688 T^{6} + 1958965 T^{8} + 99122 T^{10} + 3623 T^{12} + 74 T^{14} + T^{16}$$
$23$ $$7890481 - 595508 T + 22472 T^{2} - 5529808 T^{3} - 3336286 T^{4} + 2059820 T^{5} + 1791744 T^{6} + 1250976 T^{7} + 1050667 T^{8} + 429492 T^{9} + 114336 T^{10} + 29720 T^{11} + 6482 T^{12} + 968 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$29$ $$( 16384 + 53760 T^{2} + 7073 T^{4} + 162 T^{6} + T^{8} )^{2}$$
$31$ $$( 414736 - 54096 T - 36932 T^{2} + 5124 T^{3} + 2909 T^{4} - 366 T^{5} - 49 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$37$ $$4096 + 8192 T + 8192 T^{2} + 16384 T^{3} + 12224 T^{4} - 9344 T^{5} - 10368 T^{6} - 15744 T^{7} - 4799 T^{8} + 9330 T^{9} + 6882 T^{10} + 9352 T^{11} + 7943 T^{12} + 1144 T^{13} + 98 T^{14} + 14 T^{15} + T^{16}$$
$41$ $$( 12544 + 18592 T^{2} + 7001 T^{4} + 170 T^{6} + T^{8} )^{2}$$
$43$ $$( 10432900 - 5335960 T + 1364552 T^{2} - 179032 T^{3} + 13021 T^{4} - 518 T^{5} + 98 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$47$ $$10000 + 96000 T + 460800 T^{2} + 1474560 T^{3} + 3376236 T^{4} + 5560632 T^{5} + 6521472 T^{6} + 5010708 T^{7} + 1764461 T^{8} - 185190 T^{9} + 3330 T^{10} - 4368 T^{11} - 1605 T^{12} + 36 T^{13} + 18 T^{14} + 6 T^{15} + T^{16}$$
$53$ $$35477982736 + 50355846464 T + 35736407168 T^{2} + 26114307744 T^{3} + 16523164364 T^{4} + 6618760456 T^{5} + 2361839200 T^{6} + 854109060 T^{7} + 133930861 T^{8} - 8357982 T^{9} - 667358 T^{10} - 166756 T^{11} - 4429 T^{12} + 1248 T^{13} + 50 T^{14} + 10 T^{15} + T^{16}$$
$59$ $$6263627420176 + 1008958164256 T^{2} + 114520335692 T^{4} + 6401255896 T^{6} + 258171733 T^{8} + 4295858 T^{10} + 51575 T^{12} + 266 T^{14} + T^{16}$$
$61$ $$( 4092529 - 1954218 T + 234178 T^{2} + 36708 T^{3} - 6193 T^{4} - 1140 T^{5} + 338 T^{6} - 30 T^{7} + T^{8} )^{2}$$
$67$ $$2401 + 48020 T + 480200 T^{2} + 9845472 T^{3} + 98439530 T^{4} - 24754604 T^{5} + 3032512 T^{6} - 811272 T^{7} - 223373 T^{8} + 81516 T^{9} - 10016 T^{10} + 2624 T^{11} + 746 T^{12} - 264 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$71$ $$( 448 - 560 T - 142 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$73$ $$3841600000000 - 8758848000000 T + 9985086720000 T^{2} - 7588665907200 T^{3} + 4106290163136 T^{4} - 1595998728576 T^{5} + 461102671008 T^{6} - 102374438376 T^{7} + 17753663537 T^{8} - 2421175254 T^{9} + 261571986 T^{10} - 22450812 T^{11} + 1517871 T^{12} - 79092 T^{13} + 3042 T^{14} - 78 T^{15} + T^{16}$$
$79$ $$3647809685776 - 617486068896 T^{2} + 73987701580 T^{4} - 4137948696 T^{6} + 166446117 T^{8} - 3670422 T^{10} + 56911 T^{12} - 270 T^{14} + T^{16}$$
$83$ $$5006411536 + 726098856 T^{4} + 25716161 T^{8} + 25542 T^{12} + T^{16}$$
$89$ $$33243864241 + 12201456680 T^{2} + 3176092682 T^{4} + 410845568 T^{6} + 38512555 T^{8} + 1180288 T^{10} + 26714 T^{12} + 184 T^{14} + T^{16}$$
$97$ $$47698139955456 + 320344015104 T^{4} + 337548384 T^{8} + 68112 T^{12} + T^{16}$$