# Properties

 Label 640.2.s.d Level $640$ Weight $2$ Character orbit 640.s Analytic conductor $5.110$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.s (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{13}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{7} q^{5} -\beta_{10} q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{7} q^{5} -\beta_{10} q^{7} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{13} ) q^{9} + ( \beta_{11} - \beta_{15} ) q^{11} + ( -\beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{13} + ( -1 - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{15} + ( \beta_{2} - \beta_{3} + \beta_{10} + \beta_{17} ) q^{17} + ( \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{19} + ( 1 + \beta_{2} - \beta_{4} - \beta_{8} - \beta_{13} + \beta_{17} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{17} ) q^{23} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{25} + ( 1 + \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{16} + \beta_{17} ) q^{27} + ( -1 + \beta_{2} + \beta_{8} + \beta_{17} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{31} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{33} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} + \beta_{16} - 2 \beta_{17} ) q^{35} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{15} - \beta_{17} ) q^{37} + ( 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{39} + ( 1 + \beta_{2} - \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{41} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{15} - \beta_{17} ) q^{43} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{16} ) q^{45} + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{47} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{49} + ( -\beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{10} - \beta_{13} - \beta_{14} ) q^{51} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{53} + ( -2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{12} + \beta_{16} ) q^{55} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{57} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{13} + \beta_{14} + \beta_{16} - 2 \beta_{17} ) q^{59} + ( -2 - 2 \beta_{2} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{15} + 2 \beta_{17} ) q^{61} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{17} ) q^{63} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{65} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} + 3 \beta_{17} ) q^{67} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 4 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{17} ) q^{69} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{71} + ( -\beta_{2} - \beta_{3} + \beta_{10} - \beta_{17} ) q^{73} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{15} + \beta_{16} - \beta_{17} ) q^{75} + ( 3 - 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{16} - \beta_{17} ) q^{77} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{13} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{79} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{15} + \beta_{17} ) q^{81} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{16} + \beta_{17} ) q^{83} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{85} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{13} - \beta_{14} + \beta_{17} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{10} + 2 \beta_{11} + \beta_{13} ) q^{89} + ( -\beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{91} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{17} ) q^{93} + ( 4 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{95} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{16} + \beta_{17} ) q^{97} + ( 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} - \beta_{11} + 3 \beta_{12} + 3 \beta_{13} - \beta_{14} + 3 \beta_{15} - 2 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - 2 q^{5} + 2 q^{7} + 10 q^{9} + O(q^{10})$$ $$18 q - 2 q^{5} + 2 q^{7} + 10 q^{9} + 2 q^{11} - 20 q^{15} - 6 q^{17} + 2 q^{19} + 16 q^{21} - 2 q^{23} - 6 q^{25} + 24 q^{27} - 14 q^{29} - 8 q^{33} - 2 q^{35} + 14 q^{45} + 38 q^{47} - 8 q^{51} - 12 q^{53} - 6 q^{55} - 24 q^{57} - 10 q^{59} - 14 q^{61} - 6 q^{63} + 32 q^{69} + 24 q^{71} - 14 q^{73} - 16 q^{75} + 44 q^{77} - 16 q^{79} + 2 q^{81} - 40 q^{83} - 14 q^{85} + 24 q^{87} + 12 q^{89} + 34 q^{95} + 18 q^{97} - 22 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$75 \nu^{17} + 89 \nu^{16} + 248 \nu^{15} + 6 \nu^{14} - 375 \nu^{13} - 1487 \nu^{12} - 2550 \nu^{11} - 2676 \nu^{10} - 583 \nu^{9} + 4379 \nu^{8} + 10894 \nu^{7} + 15406 \nu^{6} + 13500 \nu^{5} + 848 \nu^{4} - 9920 \nu^{3} - 31296 \nu^{2} - 21696 \nu - 28416$$$$)/640$$ $$\beta_{2}$$ $$=$$ $$($$$$-100 \nu^{17} - 111 \nu^{16} - 342 \nu^{15} - 14 \nu^{14} + 460 \nu^{13} + 1963 \nu^{12} + 3440 \nu^{11} + 3714 \nu^{10} + 1062 \nu^{9} - 5591 \nu^{8} - 14616 \nu^{7} - 21274 \nu^{6} - 19260 \nu^{5} - 2112 \nu^{4} + 12880 \nu^{3} + 44224 \nu^{2} + 30784 \nu + 41344$$$$)/640$$ $$\beta_{3}$$ $$=$$ $$($$$$-191 \nu^{17} - 380 \nu^{16} - 1110 \nu^{15} - 1252 \nu^{14} - 997 \nu^{13} + 1614 \nu^{12} + 6294 \nu^{11} + 11894 \nu^{10} + 14845 \nu^{9} + 10426 \nu^{8} - 3462 \nu^{7} - 24152 \nu^{6} - 42352 \nu^{5} - 40656 \nu^{4} - 36256 \nu^{3} + 8896 \nu^{2} + 4736 \nu + 38912$$$$)/1280$$ $$\beta_{4}$$ $$=$$ $$($$$$-182 \nu^{17} - 271 \nu^{16} - 762 \nu^{15} - 418 \nu^{14} + 286 \nu^{13} + 2911 \nu^{12} + 6148 \nu^{11} + 8162 \nu^{10} + 5692 \nu^{9} - 3819 \nu^{8} - 18820 \nu^{7} - 32678 \nu^{6} - 35404 \nu^{5} - 14264 \nu^{4} + 5568 \nu^{3} + 54816 \nu^{2} + 37056 \nu + 58368$$$$)/640$$ $$\beta_{5}$$ $$=$$ $$($$$$-330 \nu^{17} - 407 \nu^{16} - 1174 \nu^{15} - 78 \nu^{14} + 1610 \nu^{13} + 6791 \nu^{12} + 11920 \nu^{11} + 12878 \nu^{10} + 3464 \nu^{9} - 19867 \nu^{8} - 50632 \nu^{7} - 72618 \nu^{6} - 65260 \nu^{5} - 6464 \nu^{4} + 47200 \nu^{3} + 148288 \nu^{2} + 103488 \nu + 135168$$$$)/640$$ $$\beta_{6}$$ $$=$$ $$($$$$-357 \nu^{17} - 493 \nu^{16} - 1406 \nu^{15} - 506 \nu^{14} + 1101 \nu^{13} + 6607 \nu^{12} + 12748 \nu^{11} + 15520 \nu^{10} + 8081 \nu^{9} - 14271 \nu^{8} - 46732 \nu^{7} - 73606 \nu^{6} - 73024 \nu^{5} - 19848 \nu^{4} + 31408 \nu^{3} + 136864 \nu^{2} + 97664 \nu + 134656$$$$)/640$$ $$\beta_{7}$$ $$=$$ $$($$$$228 \nu^{17} + 359 \nu^{16} + 1013 \nu^{15} + 672 \nu^{14} - 134 \nu^{13} - 3459 \nu^{12} - 7827 \nu^{11} - 11038 \nu^{10} - 8848 \nu^{9} + 2381 \nu^{8} + 21455 \nu^{7} + 40462 \nu^{6} + 46806 \nu^{5} + 23236 \nu^{4} - 752 \nu^{3} - 61744 \nu^{2} - 43104 \nu - 71232$$$$)/320$$ $$\beta_{8}$$ $$=$$ $$($$$$129 \nu^{17} + 186 \nu^{16} + 524 \nu^{15} + 232 \nu^{14} - 321 \nu^{13} - 2280 \nu^{12} - 4568 \nu^{11} - 5762 \nu^{10} - 3435 \nu^{9} + 4196 \nu^{8} + 15672 \nu^{7} + 25600 \nu^{6} + 26316 \nu^{5} + 8624 \nu^{4} - 8592 \nu^{3} - 45504 \nu^{2} - 32320 \nu - 46208$$$$)/128$$ $$\beta_{9}$$ $$=$$ $$($$$$1413 \nu^{17} + 1966 \nu^{16} + 5582 \nu^{15} + 2000 \nu^{14} - 4409 \nu^{13} - 26220 \nu^{12} - 50762 \nu^{11} - 61646 \nu^{10} - 32027 \nu^{9} + 56748 \nu^{8} + 185642 \nu^{7} + 291940 \nu^{6} + 288616 \nu^{5} + 77840 \nu^{4} - 125312 \nu^{3} - 540992 \nu^{2} - 381952 \nu - 528640$$$$)/1280$$ $$\beta_{10}$$ $$=$$ $$($$$$-1671 \nu^{17} - 2410 \nu^{16} - 6790 \nu^{15} - 3072 \nu^{14} + 3963 \nu^{13} + 29124 \nu^{12} + 58634 \nu^{11} + 74354 \nu^{10} + 45185 \nu^{9} - 51964 \nu^{8} - 198282 \nu^{7} - 326212 \nu^{6} - 337152 \nu^{5} - 113296 \nu^{4} + 103264 \nu^{3} + 575296 \nu^{2} + 405376 \nu + 588032$$$$)/1280$$ $$\beta_{11}$$ $$=$$ $$($$$$-1861 \nu^{17} - 2632 \nu^{16} - 7454 \nu^{15} - 3140 \nu^{14} + 4833 \nu^{13} + 32910 \nu^{12} + 65414 \nu^{11} + 81842 \nu^{10} + 47679 \nu^{9} - 62446 \nu^{8} - 226774 \nu^{7} - 368520 \nu^{6} - 376472 \nu^{5} - 119760 \nu^{4} + 127424 \nu^{3} + 661184 \nu^{2} + 466944 \nu + 670720$$$$)/1280$$ $$\beta_{12}$$ $$=$$ $$($$$$1039 \nu^{17} + 1484 \nu^{16} + 4198 \nu^{15} + 1844 \nu^{14} - 2547 \nu^{13} - 18238 \nu^{12} - 36566 \nu^{11} - 46182 \nu^{10} - 27653 \nu^{9} + 33350 \nu^{8} + 125222 \nu^{7} + 205144 \nu^{6} + 211368 \nu^{5} + 69792 \nu^{4} - 67376 \nu^{3} - 364800 \nu^{2} - 258560 \nu - 372864$$$$)/640$$ $$\beta_{13}$$ $$=$$ $$($$$$-593 \nu^{17} - 812 \nu^{16} - 2294 \nu^{15} - 774 \nu^{14} + 1929 \nu^{13} + 10958 \nu^{12} + 20982 \nu^{11} + 25220 \nu^{10} + 12619 \nu^{9} - 24294 \nu^{8} - 77418 \nu^{7} - 120714 \nu^{6} - 118616 \nu^{5} - 30132 \nu^{4} + 53272 \nu^{3} + 225296 \nu^{2} + 158976 \nu + 218944$$$$)/320$$ $$\beta_{14}$$ $$=$$ $$($$$$-1177 \nu^{17} - 1672 \nu^{16} - 4754 \nu^{15} - 2092 \nu^{14} + 2901 \nu^{13} + 20614 \nu^{12} + 41338 \nu^{11} + 52186 \nu^{10} + 31259 \nu^{9} - 37710 \nu^{8} - 141626 \nu^{7} - 231512 \nu^{6} - 238424 \nu^{5} - 78416 \nu^{4} + 76048 \nu^{3} + 412160 \nu^{2} + 290560 \nu + 418432$$$$)/640$$ $$\beta_{15}$$ $$=$$ $$($$$$-2339 \nu^{17} - 3438 \nu^{16} - 9706 \nu^{15} - 4920 \nu^{14} + 4527 \nu^{13} + 39440 \nu^{12} + 81686 \nu^{11} + 106298 \nu^{10} + 70181 \nu^{9} - 61024 \nu^{8} - 264246 \nu^{7} - 448020 \nu^{6} - 476008 \nu^{5} - 179280 \nu^{4} + 110976 \nu^{3} + 767296 \nu^{2} + 542976 \nu + 807680$$$$)/1280$$ $$\beta_{16}$$ $$=$$ $$($$$$-3419 \nu^{17} - 4780 \nu^{16} - 13550 \nu^{15} - 5268 \nu^{14} + 9767 \nu^{13} + 61766 \nu^{12} + 121006 \nu^{11} + 149166 \nu^{10} + 82065 \nu^{9} - 125726 \nu^{8} - 431518 \nu^{7} - 689368 \nu^{6} - 692848 \nu^{5} - 203344 \nu^{4} + 268896 \nu^{3} + 1258944 \nu^{2} + 893824 \nu + 1253888$$$$)/1280$$ $$\beta_{17}$$ $$=$$ $$($$$$-1899 \nu^{17} - 2719 \nu^{16} - 7688 \nu^{15} - 3394 \nu^{14} + 4727 \nu^{13} + 33453 \nu^{12} + 66986 \nu^{11} + 84512 \nu^{10} + 50363 \nu^{9} - 61705 \nu^{8} - 230002 \nu^{7} - 375754 \nu^{6} - 385988 \nu^{5} - 125952 \nu^{4} + 125776 \nu^{3} + 669120 \nu^{2} + 473920 \nu + 677504$$$$)/640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{16} - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{16} - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 3 \beta_{1} - 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{12} + \beta_{10} - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{17} - \beta_{16} + \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{11} - \beta_{10} - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + \beta_{1} + 6$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{17} + \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} - \beta_{11} + 4 \beta_{10} - \beta_{9} + \beta_{8} + 5 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{1} + 5$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{17} - \beta_{16} - \beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 5 \beta_{1} + 2$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{17} + 3 \beta_{16} - 2 \beta_{15} - 6 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} + 7 \beta_{8} - \beta_{7} - 7 \beta_{6} - 4 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 14 \beta_{1} - 1$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-2 \beta_{17} - \beta_{16} - 5 \beta_{15} - \beta_{14} + \beta_{12} - 3 \beta_{11} - \beta_{10} - 2 \beta_{9} - 22 \beta_{8} - 8 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 16 \beta_{2} - 3 \beta_{1} + 4$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$13 \beta_{17} - 11 \beta_{16} + \beta_{15} + 5 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} - \beta_{11} + 14 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} - 4 \beta_{6} + \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 10 \beta_{2} + 4 \beta_{1} + 20$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$12 \beta_{17} + 15 \beta_{16} - 25 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} - 5 \beta_{10} + 12 \beta_{9} + 28 \beta_{8} + 10 \beta_{7} - 26 \beta_{6} + 3 \beta_{5} + 14 \beta_{4} + 18 \beta_{3} - 28 \beta_{2} + 3 \beta_{1} + 30$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-50 \beta_{17} + 19 \beta_{16} + 14 \beta_{15} + 10 \beta_{14} + 36 \beta_{13} + 2 \beta_{12} - \beta_{11} + 6 \beta_{10} - 9 \beta_{9} - 13 \beta_{8} + 21 \beta_{7} - 19 \beta_{6} - 24 \beta_{5} + 6 \beta_{4} + 29 \beta_{3} - 8 \beta_{2} + 52 \beta_{1} - 25$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$10 \beta_{17} - 30 \beta_{16} - 19 \beta_{15} + 13 \beta_{14} - 9 \beta_{13} + 7 \beta_{12} + 12 \beta_{11} + 50 \beta_{10} + 7 \beta_{9} - 21 \beta_{8} + 13 \beta_{7} + 22 \beta_{6} - 12 \beta_{5} - 20 \beta_{4} - 10 \beta_{3} + 30 \beta_{2} + 10 \beta_{1} + 13$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$80 \beta_{17} + 13 \beta_{16} - 40 \beta_{15} - 28 \beta_{13} + 64 \beta_{12} + 67 \beta_{11} - 44 \beta_{10} + 53 \beta_{9} - 45 \beta_{8} + 9 \beta_{7} - 133 \beta_{6} + 36 \beta_{5} - 16 \beta_{4} + \beta_{3} + 4 \beta_{2} + 36 \beta_{1} + 163$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$76 \beta_{17} - 3 \beta_{16} - 39 \beta_{15} + 3 \beta_{14} - 84 \beta_{13} - 79 \beta_{12} - 81 \beta_{11} + 59 \beta_{10} + 32 \beta_{9} - 52 \beta_{8} + 58 \beta_{7} - 172 \beta_{6} + 57 \beta_{5} - 14 \beta_{4} - 8 \beta_{3} + 68 \beta_{2} - 15 \beta_{1} - 114$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-4 \beta_{17} - 66 \beta_{16} - 30 \beta_{15} + 56 \beta_{14} + 32 \beta_{13} - 101 \beta_{12} + 10 \beta_{11} + 81 \beta_{10} - 16 \beta_{9} + 104 \beta_{8} + 101 \beta_{7} + 20 \beta_{6} - 84 \beta_{5} + 13 \beta_{4} + \beta_{3} - 41 \beta_{2} + 83 \beta_{1} + 86$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$-126 \beta_{17} + 145 \beta_{16} - 25 \beta_{15} + 99 \beta_{14} + 118 \beta_{13} + 213 \beta_{12} + 23 \beta_{11} + 93 \beta_{10} + 36 \beta_{9} + 128 \beta_{8} - 66 \beta_{7} + 30 \beta_{6} - 53 \beta_{5} - 206 \beta_{4} + 164 \beta_{3} + 95 \beta_{1} + 258$$$$)/4$$ $$\nu^{17}$$ $$=$$ $$($$$$162 \beta_{17} - 113 \beta_{16} - 198 \beta_{15} - 70 \beta_{14} - 108 \beta_{13} + 84 \beta_{12} + 193 \beta_{11} - 16 \beta_{10} + 85 \beta_{9} - 685 \beta_{8} + 159 \beta_{7} - 353 \beta_{6} + 142 \beta_{5} - 10 \beta_{4} + 159 \beta_{3} + 176 \beta_{2} + 200 \beta_{1} + 39$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-\beta_{8}$$ $$-\beta_{8}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1
 −0.480367 − 1.33013i 1.41303 − 0.0578659i 1.41323 + 0.0526497i −0.635486 + 1.26339i −1.37691 + 0.322680i −1.08900 − 0.902261i 0.482716 − 1.32928i 0.0376504 − 1.41371i 0.235136 + 1.39453i −0.480367 + 1.33013i 1.41303 + 0.0578659i 1.41323 − 0.0526497i −0.635486 − 1.26339i −1.37691 − 0.322680i −1.08900 + 0.902261i 0.482716 + 1.32928i 0.0376504 + 1.41371i 0.235136 − 1.39453i
0 −2.85601 0 1.71489 + 1.43498i 0 −0.458895 + 0.458895i 0 5.15678 0
223.2 0 −1.96251 0 1.42182 1.72581i 0 −1.60205 + 1.60205i 0 0.851447 0
223.3 0 −1.28110 0 −2.07160 0.841703i 0 −1.13975 + 1.13975i 0 −1.35879 0
223.4 0 −0.692712 0 0.245325 + 2.22257i 0 −0.343872 + 0.343872i 0 −2.52015 0
223.5 0 −0.614566 0 −0.832020 2.07551i 0 2.83610 2.83610i 0 −2.62231 0
223.6 0 0.496487 0 2.00635 0.987189i 0 1.55426 1.55426i 0 −2.75350 0
223.7 0 1.39319 0 −2.17104 + 0.535339i 0 −2.13436 + 2.13436i 0 −1.05903 0
223.8 0 2.55161 0 −1.49107 + 1.66635i 0 2.40368 2.40368i 0 3.51070 0
223.9 0 2.96561 0 0.177336 2.22902i 0 −0.115101 + 0.115101i 0 5.79486 0
287.1 0 −2.85601 0 1.71489 1.43498i 0 −0.458895 0.458895i 0 5.15678 0
287.2 0 −1.96251 0 1.42182 + 1.72581i 0 −1.60205 1.60205i 0 0.851447 0
287.3 0 −1.28110 0 −2.07160 + 0.841703i 0 −1.13975 1.13975i 0 −1.35879 0
287.4 0 −0.692712 0 0.245325 2.22257i 0 −0.343872 0.343872i 0 −2.52015 0
287.5 0 −0.614566 0 −0.832020 + 2.07551i 0 2.83610 + 2.83610i 0 −2.62231 0
287.6 0 0.496487 0 2.00635 + 0.987189i 0 1.55426 + 1.55426i 0 −2.75350 0
287.7 0 1.39319 0 −2.17104 0.535339i 0 −2.13436 2.13436i 0 −1.05903 0
287.8 0 2.55161 0 −1.49107 1.66635i 0 2.40368 + 2.40368i 0 3.51070 0
287.9 0 2.96561 0 0.177336 + 2.22902i 0 −0.115101 0.115101i 0 5.79486 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 287.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.s.d 18
4.b odd 2 1 640.2.s.c 18
5.c odd 4 1 640.2.j.d 18
8.b even 2 1 80.2.s.b yes 18
8.d odd 2 1 320.2.s.b 18
16.e even 4 1 320.2.j.b 18
16.e even 4 1 640.2.j.c 18
16.f odd 4 1 80.2.j.b 18
16.f odd 4 1 640.2.j.d 18
20.e even 4 1 640.2.j.c 18
24.h odd 2 1 720.2.z.g 18
40.e odd 2 1 1600.2.s.d 18
40.f even 2 1 400.2.s.d 18
40.i odd 4 1 80.2.j.b 18
40.i odd 4 1 400.2.j.d 18
40.k even 4 1 320.2.j.b 18
40.k even 4 1 1600.2.j.d 18
48.k even 4 1 720.2.bd.g 18
80.i odd 4 1 640.2.s.c 18
80.i odd 4 1 1600.2.s.d 18
80.j even 4 1 80.2.s.b yes 18
80.k odd 4 1 400.2.j.d 18
80.q even 4 1 1600.2.j.d 18
80.s even 4 1 400.2.s.d 18
80.s even 4 1 inner 640.2.s.d 18
80.t odd 4 1 320.2.s.b 18
120.w even 4 1 720.2.bd.g 18
240.bd odd 4 1 720.2.z.g 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 16.f odd 4 1
80.2.j.b 18 40.i odd 4 1
80.2.s.b yes 18 8.b even 2 1
80.2.s.b yes 18 80.j even 4 1
320.2.j.b 18 16.e even 4 1
320.2.j.b 18 40.k even 4 1
320.2.s.b 18 8.d odd 2 1
320.2.s.b 18 80.t odd 4 1
400.2.j.d 18 40.i odd 4 1
400.2.j.d 18 80.k odd 4 1
400.2.s.d 18 40.f even 2 1
400.2.s.d 18 80.s even 4 1
640.2.j.c 18 16.e even 4 1
640.2.j.c 18 20.e even 4 1
640.2.j.d 18 5.c odd 4 1
640.2.j.d 18 16.f odd 4 1
640.2.s.c 18 4.b odd 2 1
640.2.s.c 18 80.i odd 4 1
640.2.s.d 18 1.a even 1 1 trivial
640.2.s.d 18 80.s even 4 1 inner
720.2.z.g 18 24.h odd 2 1
720.2.z.g 18 240.bd odd 4 1
720.2.bd.g 18 48.k even 4 1
720.2.bd.g 18 120.w even 4 1
1600.2.j.d 18 40.k even 4 1
1600.2.j.d 18 80.q even 4 1
1600.2.s.d 18 40.e odd 2 1
1600.2.s.d 18 80.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{9} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(640, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$( 16 + 20 T - 72 T^{2} - 104 T^{3} + 40 T^{4} + 76 T^{5} - 4 T^{6} - 16 T^{7} + T^{9} )^{2}$$
$5$ $$1953125 + 781250 T + 390625 T^{2} + 250000 T^{3} + 112500 T^{4} + 65000 T^{5} + 38500 T^{6} + 17200 T^{7} + 7550 T^{8} + 2572 T^{9} + 1510 T^{10} + 688 T^{11} + 308 T^{12} + 104 T^{13} + 36 T^{14} + 16 T^{15} + 5 T^{16} + 2 T^{17} + T^{18}$$
$7$ $$288 + 4128 T + 29584 T^{2} + 108160 T^{3} + 239360 T^{4} + 315488 T^{5} + 244448 T^{6} + 85184 T^{7} + 17328 T^{8} + 11648 T^{9} + 14648 T^{10} + 4160 T^{11} + 352 T^{12} - 120 T^{13} + 200 T^{14} + 32 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$11$ $$5431808 + 4060672 T + 1517824 T^{2} + 4217856 T^{3} + 18015744 T^{4} + 19945472 T^{5} + 11514112 T^{6} + 2158336 T^{7} + 289472 T^{8} + 209344 T^{9} + 182240 T^{10} + 27968 T^{11} + 1376 T^{12} + 64 T^{13} + 848 T^{14} + 80 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$13$ $$67108864 + 131948800 T^{2} + 105268224 T^{4} + 44025600 T^{6} + 10452224 T^{8} + 1437280 T^{10} + 113344 T^{12} + 4976 T^{14} + 112 T^{16} + T^{18}$$
$17$ $$512 + 1536 T + 2304 T^{2} - 186368 T^{3} + 1493504 T^{4} - 5352960 T^{5} + 11139328 T^{6} - 12464640 T^{7} + 6499264 T^{8} + 984640 T^{9} + 14816 T^{10} - 36480 T^{11} + 66080 T^{12} + 19232 T^{13} + 2768 T^{14} + 32 T^{15} + 18 T^{16} + 6 T^{17} + T^{18}$$
$19$ $$4608 + 339456 T + 12503296 T^{2} + 7467008 T^{3} + 2215424 T^{4} - 335360 T^{5} + 13924096 T^{6} + 9073664 T^{7} + 2905024 T^{8} - 1873856 T^{9} + 675296 T^{10} + 48384 T^{11} + 16480 T^{12} - 10080 T^{13} + 2864 T^{14} - 64 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$23$ $$17700587552 + 17587696352 T + 8737762576 T^{2} - 758796096 T^{3} + 938524160 T^{4} + 784929440 T^{5} + 332892064 T^{6} - 13833888 T^{7} + 11039216 T^{8} + 7905920 T^{9} + 2752216 T^{10} + 54000 T^{11} + 9120 T^{12} + 7896 T^{13} + 3480 T^{14} + 24 T^{15} + 2 T^{16} + 2 T^{17} + T^{18}$$
$29$ $$82330112 + 372641280 T + 843321600 T^{2} + 687951872 T^{3} + 281020928 T^{4} + 16436736 T^{5} + 70118656 T^{6} + 45759488 T^{7} + 14859968 T^{8} + 1819968 T^{9} + 693088 T^{10} + 360192 T^{11} + 111008 T^{12} + 15584 T^{13} + 1616 T^{14} + 320 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$31$ $$16384 + 1150976 T^{2} + 16687104 T^{4} + 32610304 T^{6} + 17532416 T^{8} + 3648384 T^{10} + 327040 T^{12} + 12832 T^{14} + 196 T^{16} + T^{18}$$
$37$ $$574297214976 + 457920768256 T^{2} + 131428787200 T^{4} + 18630090496 T^{6} + 1481216256 T^{8} + 69964768 T^{10} + 1988288 T^{12} + 32944 T^{14} + 288 T^{16} + T^{18}$$
$41$ $$242788765696 + 208906121472 T^{2} + 68836776960 T^{4} + 11517059840 T^{6} + 1077600512 T^{8} + 58367328 T^{10} + 1831744 T^{12} + 32176 T^{14} + 288 T^{16} + T^{18}$$
$43$ $$337207844416 + 627531510928 T^{2} + 248751738624 T^{4} + 38727202720 T^{6} + 3041915424 T^{8} + 132726008 T^{10} + 3310976 T^{12} + 46520 T^{14} + 340 T^{16} + T^{18}$$
$47$ $$16870640672 - 178795286432 T + 947437476496 T^{2} - 866389389184 T^{3} + 419056561664 T^{4} - 124183363808 T^{5} + 28973153120 T^{6} - 7620275328 T^{7} + 2411169968 T^{8} - 643585920 T^{9} + 126078328 T^{10} - 18512576 T^{11} + 2503968 T^{12} - 394472 T^{13} + 64680 T^{14} - 8272 T^{15} + 722 T^{16} - 38 T^{17} + T^{18}$$
$53$ $$( -220832 + 334608 T - 44032 T^{2} - 85984 T^{3} + 10768 T^{4} + 5720 T^{5} - 480 T^{6} - 136 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$59$ $$144166720393728 + 63266193922560 T + 13881883705600 T^{2} - 2799567403008 T^{3} + 1079531100672 T^{4} + 262683543040 T^{5} + 38509938944 T^{6} - 5387559424 T^{7} + 1811701952 T^{8} + 356104896 T^{9} + 37394528 T^{10} - 1999488 T^{11} + 826528 T^{12} + 162272 T^{13} + 16016 T^{14} - 96 T^{15} + 50 T^{16} + 10 T^{17} + T^{18}$$
$61$ $$121236758528 + 124325191680 T + 63746150400 T^{2} - 35916963840 T^{3} + 28713865216 T^{4} + 15011409920 T^{5} + 5616384000 T^{6} - 2393885696 T^{7} + 974385920 T^{8} + 231668992 T^{9} + 33596800 T^{10} - 3516032 T^{11} + 740800 T^{12} + 135872 T^{13} + 14496 T^{14} - 720 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$67$ $$555525752896 + 1362082550416 T^{2} + 742701558272 T^{4} + 154793099680 T^{6} + 14366974496 T^{8} + 616717432 T^{10} + 12278976 T^{12} + 120376 T^{14} + 564 T^{16} + T^{18}$$
$71$ $$( -27648 - 72640 T + 110336 T^{2} - 8704 T^{3} - 23136 T^{4} + 3344 T^{5} + 1408 T^{6} - 152 T^{7} - 12 T^{8} + T^{9} )^{2}$$
$73$ $$35535647232 + 3228962304 T + 146700544 T^{2} + 5453305856 T^{3} + 14687183360 T^{4} + 5125904896 T^{5} + 823567616 T^{6} - 175353344 T^{7} + 212641728 T^{8} + 49164608 T^{9} + 5990112 T^{10} - 758144 T^{11} + 628896 T^{12} + 125856 T^{13} + 13072 T^{14} - 544 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$79$ $$( 45002752 + 3950848 T - 7267840 T^{2} - 485376 T^{3} + 296064 T^{4} + 27488 T^{5} - 2976 T^{6} - 320 T^{7} + 8 T^{8} + T^{9} )^{2}$$
$83$ $$( 8413744 + 2612884 T - 2578680 T^{2} - 329896 T^{3} + 197504 T^{4} + 7292 T^{5} - 3612 T^{6} - 136 T^{7} + 20 T^{8} + T^{9} )^{2}$$
$89$ $$( -251904 + 5727232 T + 4338688 T^{2} - 1356288 T^{3} - 330368 T^{4} + 55104 T^{5} + 2752 T^{6} - 448 T^{7} - 6 T^{8} + T^{9} )^{2}$$
$97$ $$380349381734912 + 440719049317888 T + 255335343974656 T^{2} + 73928552845312 T^{3} + 10687493231104 T^{4} + 76758095360 T^{5} + 98986508544 T^{6} + 49913260032 T^{7} + 9402360768 T^{8} - 513287616 T^{9} + 21224416 T^{10} + 9460992 T^{11} + 2501024 T^{12} - 238368 T^{13} + 11856 T^{14} + 512 T^{15} + 162 T^{16} - 18 T^{17} + T^{18}$$