# Properties

 Label 1600.2.s.d Level $1600$ Weight $2$ Character orbit 1600.s Analytic conductor $12.776$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.s (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ 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: $$2^{15}$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$75 \nu^{17} + 89 \nu^{16} + 248 \nu^{15} + 6 \nu^{14} - 375 \nu^{13} - 1487 \nu^{12} - 2550 \nu^{11} - 2676 \nu^{10} - 583 \nu^{9} + 4379 \nu^{8} + 10894 \nu^{7} + 15406 \nu^{6} + 13500 \nu^{5} + 848 \nu^{4} - 9920 \nu^{3} - 31296 \nu^{2} - 21696 \nu - 28416$$$$)/640$$ $$\beta_{2}$$ $$=$$ $$($$$$-191 \nu^{17} - 380 \nu^{16} - 1110 \nu^{15} - 1252 \nu^{14} - 997 \nu^{13} + 1614 \nu^{12} + 6294 \nu^{11} + 11894 \nu^{10} + 14845 \nu^{9} + 10426 \nu^{8} - 3462 \nu^{7} - 24152 \nu^{6} - 42352 \nu^{5} - 40656 \nu^{4} - 36256 \nu^{3} + 8896 \nu^{2} + 4736 \nu + 38912$$$$)/1280$$ $$\beta_{3}$$ $$=$$ $$($$$$225 \nu^{17} + 299 \nu^{16} + 798 \nu^{15} + 106 \nu^{14} - 1145 \nu^{13} - 4757 \nu^{12} - 8400 \nu^{11} - 9096 \nu^{10} - 2433 \nu^{9} + 13989 \nu^{8} + 35784 \nu^{7} + 51206 \nu^{6} + 45600 \nu^{5} + 4368 \nu^{4} - 35040 \nu^{3} - 103296 \nu^{2} - 75776 \nu - 90496$$$$)/640$$ $$\beta_{4}$$ $$=$$ $$($$$$-613 \nu^{17} - 951 \nu^{16} - 2732 \nu^{15} - 1810 \nu^{14} + 249 \nu^{13} + 9185 \nu^{12} + 20942 \nu^{11} + 29756 \nu^{10} + 24377 \nu^{9} - 5453 \nu^{8} - 56742 \nu^{7} - 108530 \nu^{6} - 126996 \nu^{5} - 64880 \nu^{4} - 1648 \nu^{3} + 165632 \nu^{2} + 115392 \nu + 196480$$$$)/640$$ $$\beta_{5}$$ $$=$$ $$($$$$-655 \nu^{17} - 817 \nu^{16} - 2314 \nu^{15} - 198 \nu^{14} + 3215 \nu^{13} + 13511 \nu^{12} + 23760 \nu^{11} + 25688 \nu^{10} + 6959 \nu^{9} - 39447 \nu^{8} - 101032 \nu^{7} - 145098 \nu^{6} - 130120 \nu^{5} - 12944 \nu^{4} + 95120 \nu^{3} + 295808 \nu^{2} + 210048 \nu + 267008$$$$)/640$$ $$\beta_{6}$$ $$=$$ $$($$$$-129 \nu^{17} - 186 \nu^{16} - 524 \nu^{15} - 232 \nu^{14} + 321 \nu^{13} + 2280 \nu^{12} + 4568 \nu^{11} + 5762 \nu^{10} + 3435 \nu^{9} - 4196 \nu^{8} - 15672 \nu^{7} - 25600 \nu^{6} - 26316 \nu^{5} - 8624 \nu^{4} + 8592 \nu^{3} + 45504 \nu^{2} + 32320 \nu + 46208$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$-1371 \nu^{17} - 1882 \nu^{16} - 5234 \nu^{15} - 1600 \nu^{14} + 4903 \nu^{13} + 25980 \nu^{12} + 48934 \nu^{11} + 57922 \nu^{10} + 26869 \nu^{9} - 60796 \nu^{8} - 185094 \nu^{7} - 284460 \nu^{6} - 275192 \nu^{5} - 63920 \nu^{4} + 137024 \nu^{3} + 537024 \nu^{2} + 383744 \nu + 513280$$$$)/1280$$ $$\beta_{8}$$ $$=$$ $$($$$$1413 \nu^{17} + 1966 \nu^{16} + 5582 \nu^{15} + 2000 \nu^{14} - 4409 \nu^{13} - 26220 \nu^{12} - 50762 \nu^{11} - 61646 \nu^{10} - 32027 \nu^{9} + 56748 \nu^{8} + 185642 \nu^{7} + 291940 \nu^{6} + 288616 \nu^{5} + 77840 \nu^{4} - 125312 \nu^{3} - 540992 \nu^{2} - 381952 \nu - 528640$$$$)/1280$$ $$\beta_{9}$$ $$=$$ $$($$$$813 \nu^{17} + 1211 \nu^{16} + 3432 \nu^{15} + 1850 \nu^{14} - 1369 \nu^{13} - 13525 \nu^{12} - 28402 \nu^{11} - 37596 \nu^{10} - 25777 \nu^{9} + 19033 \nu^{8} + 89642 \nu^{7} + 154530 \nu^{6} + 166636 \nu^{5} + 66320 \nu^{4} - 32912 \nu^{3} - 260352 \nu^{2} - 183872 \nu - 277120$$$$)/640$$ $$\beta_{10}$$ $$=$$ $$($$$$829 \nu^{17} + 1131 \nu^{16} + 3182 \nu^{15} + 1042 \nu^{14} - 2757 \nu^{13} - 15309 \nu^{12} - 29216 \nu^{11} - 34920 \nu^{10} - 17157 \nu^{9} + 34317 \nu^{8} + 108104 \nu^{7} + 167822 \nu^{6} + 164208 \nu^{5} + 40416 \nu^{4} - 75136 \nu^{3} - 313728 \nu^{2} - 220288 \nu - 303232$$$$)/640$$ $$\beta_{11}$$ $$=$$ $$($$$$1671 \nu^{17} + 2410 \nu^{16} + 6790 \nu^{15} + 3072 \nu^{14} - 3963 \nu^{13} - 29124 \nu^{12} - 58634 \nu^{11} - 74354 \nu^{10} - 45185 \nu^{9} + 51964 \nu^{8} + 198282 \nu^{7} + 326212 \nu^{6} + 337152 \nu^{5} + 113296 \nu^{4} - 103264 \nu^{3} - 575296 \nu^{2} - 405376 \nu - 588032$$$$)/1280$$ $$\beta_{12}$$ $$=$$ $$($$$$856 \nu^{17} + 1201 \nu^{16} + 3432 \nu^{15} + 1466 \nu^{14} - 2108 \nu^{13} - 14917 \nu^{12} - 29814 \nu^{11} - 37538 \nu^{10} - 22322 \nu^{9} + 27445 \nu^{8} + 102398 \nu^{7} + 167186 \nu^{6} + 172232 \nu^{5} + 56368 \nu^{4} - 54624 \nu^{3} - 299520 \nu^{2} - 209920 \nu - 306816$$$$)/640$$ $$\beta_{13}$$ $$=$$ $$($$$$1861 \nu^{17} + 2632 \nu^{16} + 7454 \nu^{15} + 3140 \nu^{14} - 4833 \nu^{13} - 32910 \nu^{12} - 65414 \nu^{11} - 81842 \nu^{10} - 47679 \nu^{9} + 62446 \nu^{8} + 226774 \nu^{7} + 368520 \nu^{6} + 376472 \nu^{5} + 119760 \nu^{4} - 127424 \nu^{3} - 661184 \nu^{2} - 466944 \nu - 670720$$$$)/1280$$ $$\beta_{14}$$ $$=$$ $$($$$$1039 \nu^{17} + 1484 \nu^{16} + 4198 \nu^{15} + 1844 \nu^{14} - 2547 \nu^{13} - 18238 \nu^{12} - 36566 \nu^{11} - 46182 \nu^{10} - 27653 \nu^{9} + 33350 \nu^{8} + 125222 \nu^{7} + 205144 \nu^{6} + 211368 \nu^{5} + 69792 \nu^{4} - 67376 \nu^{3} - 364800 \nu^{2} - 258560 \nu - 372864$$$$)/640$$ $$\beta_{15}$$ $$=$$ $$($$$$2339 \nu^{17} + 3438 \nu^{16} + 9706 \nu^{15} + 4920 \nu^{14} - 4527 \nu^{13} - 39440 \nu^{12} - 81686 \nu^{11} - 106298 \nu^{10} - 70181 \nu^{9} + 61024 \nu^{8} + 264246 \nu^{7} + 448020 \nu^{6} + 476008 \nu^{5} + 179280 \nu^{4} - 110976 \nu^{3} - 767296 \nu^{2} - 542976 \nu - 807680$$$$)/1280$$ $$\beta_{16}$$ $$=$$ $$($$$$1368 \nu^{17} + 1895 \nu^{16} + 5350 \nu^{15} + 1966 \nu^{14} - 4144 \nu^{13} - 24827 \nu^{12} - 48112 \nu^{11} - 58602 \nu^{10} - 30930 \nu^{9} + 52407 \nu^{8} + 173656 \nu^{7} + 274106 \nu^{6} + 272636 \nu^{5} + 74528 \nu^{4} - 112112 \nu^{3} - 505408 \nu^{2} - 355008 \nu - 496256$$$$)/640$$ $$\beta_{17}$$ $$=$$ $$($$$$-3419 \nu^{17} - 4780 \nu^{16} - 13550 \nu^{15} - 5268 \nu^{14} + 9767 \nu^{13} + 61766 \nu^{12} + 121006 \nu^{11} + 149166 \nu^{10} + 82065 \nu^{9} - 125726 \nu^{8} - 431518 \nu^{7} - 689368 \nu^{6} - 692848 \nu^{5} - 203344 \nu^{4} + 268896 \nu^{3} + 1258944 \nu^{2} + 893824 \nu + 1253888$$$$)/1280$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{17} + \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{2} - 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{16} + \beta_{15} - \beta_{13} + \beta_{10} + 2 \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{1} - 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{2} + 2 \beta_{1} + 3$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{16} + \beta_{15} - 4 \beta_{14} + \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{1} + 4$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} + \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_{1} + 6$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{17} + 3 \beta_{16} - \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 7$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{17} - \beta_{16} + 5 \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + 5 \beta_{12} - 5 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} - 4 \beta_{6} - \beta_{4} + 3 \beta_{2} + 14 \beta_{1} - 4$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$7 \beta_{17} - 7 \beta_{16} + 16 \beta_{15} - 8 \beta_{13} + 9 \beta_{12} + 9 \beta_{11} - 7 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} - 11 \beta_{7} + 33 \beta_{6} - 9 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 7$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$5 \beta_{17} - 3 \beta_{16} - 11 \beta_{15} + 18 \beta_{14} + 11 \beta_{13} + 17 \beta_{12} - \beta_{11} - 2 \beta_{10} + 7 \beta_{9} - 5 \beta_{8} + 9 \beta_{7} - 13 \beta_{6} + 8 \beta_{5} + 27 \beta_{4} + 2 \beta_{3} - 14 \beta_{2} + 2 \beta_{1} + 49$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$7 \beta_{17} - 3 \beta_{16} + 2 \beta_{15} - 28 \beta_{14} + 34 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 11 \beta_{8} + 23 \beta_{7} - 51 \beta_{6} - 13 \beta_{5} - 8 \beta_{4} + 3 \beta_{3} + 18 \beta_{2} + 19 \beta_{1} + 53$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$7 \beta_{17} + 16 \beta_{16} + 14 \beta_{15} + 12 \beta_{14} - 27 \beta_{13} - 22 \beta_{12} - 18 \beta_{11} - 24 \beta_{10} + 33 \beta_{9} + 19 \beta_{8} - 28 \beta_{7} + 41 \beta_{6} - 8 \beta_{5} - 12 \beta_{4} - 24 \beta_{3} + 29 \beta_{2} + 36 \beta_{1} - 53$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$5 \beta_{17} + \beta_{16} + 57 \beta_{15} + 40 \beta_{14} - 43 \beta_{13} + 39 \beta_{12} - 35 \beta_{11} - 2 \beta_{10} - \beta_{9} + 33 \beta_{8} - 19 \beta_{7} + 61 \beta_{6} + 22 \beta_{5} + 65 \beta_{4} - 24 \beta_{3} - 20 \beta_{2} - 26 \beta_{1} + 7$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$37 \beta_{17} - 8 \beta_{16} - 16 \beta_{15} + 64 \beta_{14} - 11 \beta_{13} + 24 \beta_{12} + 68 \beta_{11} - 56 \beta_{10} + 21 \beta_{9} - 3 \beta_{8} + 56 \beta_{7} - 11 \beta_{6} - 32 \beta_{5} + 24 \beta_{4} + 36 \beta_{3} + \beta_{2} + 104 \beta_{1} + 219$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$42 \beta_{17} - 28 \beta_{16} + 5 \beta_{15} - 76 \beta_{14} + 115 \beta_{13} + 42 \beta_{12} - 14 \beta_{11} - 25 \beta_{10} + 116 \beta_{9} - 2 \beta_{8} + 34 \beta_{7} + 18 \beta_{6} - 35 \beta_{5} + 45 \beta_{4} + 57 \beta_{3} - 8 \beta_{2} + 77 \beta_{1} - 80$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-37 \beta_{17} - 9 \beta_{16} + 51 \beta_{15} - 90 \beta_{14} - 11 \beta_{13} - 17 \beta_{12} - 67 \beta_{11} - 10 \beta_{10} + 53 \beta_{9} - 41 \beta_{8} + 9 \beta_{7} - 217 \beta_{6} - 28 \beta_{5} + 95 \beta_{4} - 168 \beta_{3} + 2 \beta_{2} + 26 \beta_{1} + 181$$$$)/4$$ $$\nu^{16}$$ $$=$$ $$($$$$158 \beta_{17} + 292 \beta_{16} + 65 \beta_{15} + 312 \beta_{14} - 63 \beta_{13} - 86 \beta_{12} - 80 \beta_{11} - 279 \beta_{10} - 170 \beta_{9} + 76 \beta_{8} - 40 \beta_{7} - 88 \beta_{6} + 91 \beta_{5} + 13 \beta_{4} - 53 \beta_{3} + 164 \beta_{2} - 49 \beta_{1} + 218$$$$)/4$$ $$\nu^{17}$$ $$=$$ $$($$$$-50 \beta_{17} - 123 \beta_{16} + 169 \beta_{15} + 14 \beta_{14} - 164 \beta_{13} + 133 \beta_{12} + 79 \beta_{11} + 22 \beta_{10} + 276 \beta_{9} + 56 \beta_{8} + 29 \beta_{7} + 656 \beta_{6} - 4 \beta_{5} + 63 \beta_{4} + 142 \beta_{3} + 159 \beta_{2} + 346 \beta_{1} + 68$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$\beta_{6}$$ $$\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
207.1
 0.235136 − 1.39453i 0.0376504 + 1.41371i 0.482716 + 1.32928i −1.08900 + 0.902261i −1.37691 − 0.322680i −0.635486 − 1.26339i 1.41323 − 0.0526497i 1.41303 + 0.0578659i −0.480367 + 1.33013i 0.235136 + 1.39453i 0.0376504 − 1.41371i 0.482716 − 1.32928i −1.08900 − 0.902261i −1.37691 + 0.322680i −0.635486 + 1.26339i 1.41323 + 0.0526497i 1.41303 − 0.0578659i −0.480367 − 1.33013i
0 −2.96561 0 0 0 −0.115101 0.115101i 0 5.79486 0
207.2 0 −2.55161 0 0 0 2.40368 + 2.40368i 0 3.51070 0
207.3 0 −1.39319 0 0 0 −2.13436 2.13436i 0 −1.05903 0
207.4 0 −0.496487 0 0 0 1.55426 + 1.55426i 0 −2.75350 0
207.5 0 0.614566 0 0 0 2.83610 + 2.83610i 0 −2.62231 0
207.6 0 0.692712 0 0 0 −0.343872 0.343872i 0 −2.52015 0
207.7 0 1.28110 0 0 0 −1.13975 1.13975i 0 −1.35879 0
207.8 0 1.96251 0 0 0 −1.60205 1.60205i 0 0.851447 0
207.9 0 2.85601 0 0 0 −0.458895 0.458895i 0 5.15678 0
943.1 0 −2.96561 0 0 0 −0.115101 + 0.115101i 0 5.79486 0
943.2 0 −2.55161 0 0 0 2.40368 2.40368i 0 3.51070 0
943.3 0 −1.39319 0 0 0 −2.13436 + 2.13436i 0 −1.05903 0
943.4 0 −0.496487 0 0 0 1.55426 1.55426i 0 −2.75350 0
943.5 0 0.614566 0 0 0 2.83610 2.83610i 0 −2.62231 0
943.6 0 0.692712 0 0 0 −0.343872 + 0.343872i 0 −2.52015 0
943.7 0 1.28110 0 0 0 −1.13975 + 1.13975i 0 −1.35879 0
943.8 0 1.96251 0 0 0 −1.60205 + 1.60205i 0 0.851447 0
943.9 0 2.85601 0 0 0 −0.458895 + 0.458895i 0 5.15678 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.9 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 1600.2.s.d 18
4.b odd 2 1 400.2.s.d 18
5.b even 2 1 320.2.s.b 18
5.c odd 4 1 320.2.j.b 18
5.c odd 4 1 1600.2.j.d 18
16.e even 4 1 400.2.j.d 18
16.f odd 4 1 1600.2.j.d 18
20.d odd 2 1 80.2.s.b yes 18
20.e even 4 1 80.2.j.b 18
20.e even 4 1 400.2.j.d 18
40.e odd 2 1 640.2.s.d 18
40.f even 2 1 640.2.s.c 18
40.i odd 4 1 640.2.j.c 18
40.k even 4 1 640.2.j.d 18
60.h even 2 1 720.2.z.g 18
60.l odd 4 1 720.2.bd.g 18
80.i odd 4 1 400.2.s.d 18
80.i odd 4 1 640.2.s.d 18
80.j even 4 1 320.2.s.b 18
80.k odd 4 1 320.2.j.b 18
80.k odd 4 1 640.2.j.c 18
80.q even 4 1 80.2.j.b 18
80.q even 4 1 640.2.j.d 18
80.s even 4 1 640.2.s.c 18
80.s even 4 1 inner 1600.2.s.d 18
80.t odd 4 1 80.2.s.b yes 18
240.bf even 4 1 720.2.z.g 18
240.bm odd 4 1 720.2.bd.g 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 20.e even 4 1
80.2.j.b 18 80.q even 4 1
80.2.s.b yes 18 20.d odd 2 1
80.2.s.b yes 18 80.t odd 4 1
320.2.j.b 18 5.c odd 4 1
320.2.j.b 18 80.k odd 4 1
320.2.s.b 18 5.b even 2 1
320.2.s.b 18 80.j even 4 1
400.2.j.d 18 16.e even 4 1
400.2.j.d 18 20.e even 4 1
400.2.s.d 18 4.b odd 2 1
400.2.s.d 18 80.i odd 4 1
640.2.j.c 18 40.i odd 4 1
640.2.j.c 18 80.k odd 4 1
640.2.j.d 18 40.k even 4 1
640.2.j.d 18 80.q even 4 1
640.2.s.c 18 40.f even 2 1
640.2.s.c 18 80.s even 4 1
640.2.s.d 18 40.e odd 2 1
640.2.s.d 18 80.i odd 4 1
720.2.z.g 18 60.h even 2 1
720.2.z.g 18 240.bf even 4 1
720.2.bd.g 18 60.l odd 4 1
720.2.bd.g 18 240.bm odd 4 1
1600.2.j.d 18 5.c odd 4 1
1600.2.j.d 18 16.f odd 4 1
1600.2.s.d 18 1.a even 1 1 trivial
1600.2.s.d 18 80.s even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{9} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

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