# Properties

 Label 1600.2.q.g Level $1600$ Weight $2$ Character orbit 1600.q Analytic conductor $12.776$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + ( -\beta_{6} + \beta_{9} + \beta_{10} + \beta_{15} ) q^{7} + ( \beta_{5} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + ( -\beta_{6} + \beta_{9} + \beta_{10} + \beta_{15} ) q^{7} + ( \beta_{5} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{9} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{11} + ( -1 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{13} + ( \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{17} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{21} + ( -2 + \beta_{3} + \beta_{6} + \beta_{11} + \beta_{12} ) q^{23} + ( -2 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} - 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{15} ) q^{27} + ( 2 + \beta_{3} - 2 \beta_{5} + \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{31} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{33} + ( -2 + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{37} + ( 2 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{39} + ( 1 - \beta_{1} - 2 \beta_{5} + 2 \beta_{8} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{43} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{47} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + \beta_{13} - 2 \beta_{15} ) q^{49} + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{51} + ( -2 - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{53} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{57} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{14} + \beta_{15} ) q^{59} + ( -\beta_{2} + \beta_{4} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{61} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{63} + ( 2 - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{14} - 2 \beta_{15} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{8} + 2 \beta_{9} - \beta_{14} + \beta_{15} ) q^{69} + ( -4 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{14} ) q^{71} + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{73} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{14} + \beta_{15} ) q^{77} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{79} + ( -1 - 2 \beta_{1} - 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{81} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{83} + ( -1 + \beta_{1} + \beta_{3} + 6 \beta_{4} - 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{87} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{89} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{9} - \beta_{14} + \beta_{15} ) q^{91} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{14} + \beta_{15} ) q^{93} + ( 1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{97} + ( -1 + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{7} + O(q^{10})$$ $$16q - 8q^{7} + 8q^{11} - 8q^{19} - 24q^{23} - 24q^{27} + 16q^{29} - 16q^{37} + 8q^{43} + 16q^{49} + 32q^{51} - 16q^{53} - 8q^{59} + 16q^{61} + 40q^{67} - 16q^{69} - 16q^{77} + 16q^{79} - 16q^{81} - 40q^{83} - 32q^{91} - 48q^{93} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$65 \nu^{15} - 604 \nu^{14} + 534 \nu^{13} + 720 \nu^{12} + 1271 \nu^{11} - 2160 \nu^{10} - 6494 \nu^{9} + 2932 \nu^{8} + 9201 \nu^{7} + 13624 \nu^{6} - 17886 \nu^{5} - 26360 \nu^{4} + 7384 \nu^{3} + 18176 \nu^{2} + 52128 \nu - 60352$$$$)/1344$$ $$\beta_{2}$$ $$=$$ $$($$$$-39 \nu^{15} + 74 \nu^{14} + 66 \nu^{13} + 16 \nu^{12} - 337 \nu^{11} - 454 \nu^{10} + 654 \nu^{9} + 1108 \nu^{8} + 673 \nu^{7} - 2482 \nu^{6} - 2378 \nu^{5} + 1536 \nu^{4} + 2872 \nu^{3} + 3968 \nu^{2} - 5920 \nu - 256$$$$)/448$$ $$\beta_{3}$$ $$=$$ $$($$$$-55 \nu^{15} - 38 \nu^{14} + 232 \nu^{13} + 304 \nu^{12} - 129 \nu^{11} - 1398 \nu^{10} - 968 \nu^{9} + 2156 \nu^{8} + 4001 \nu^{7} + 390 \nu^{6} - 7812 \nu^{5} - 5336 \nu^{4} + 4256 \nu^{3} + 10912 \nu^{2} + 3712 \nu - 12672$$$$)/384$$ $$\beta_{4}$$ $$=$$ $$($$$$-20 \nu^{15} + 20 \nu^{14} + 41 \nu^{13} + 44 \nu^{12} - 108 \nu^{11} - 276 \nu^{10} + 95 \nu^{9} + 472 \nu^{8} + 580 \nu^{7} - 672 \nu^{6} - 1239 \nu^{5} + 8 \nu^{4} + 940 \nu^{3} + 2048 \nu^{2} - 1744 \nu - 576$$$$)/96$$ $$\beta_{5}$$ $$=$$ $$($$$$-396 \nu^{15} + 1201 \nu^{14} - 256 \nu^{13} - 460 \nu^{12} - 3508 \nu^{11} - 489 \nu^{10} + 11700 \nu^{9} + 2228 \nu^{8} - 6320 \nu^{7} - 32015 \nu^{6} + 9900 \nu^{5} + 42864 \nu^{4} + 5840 \nu^{3} + 1504 \nu^{2} - 108736 \nu + 83072$$$$)/1344$$ $$\beta_{6}$$ $$=$$ $$($$$$556 \nu^{15} - 1613 \nu^{14} + 234 \nu^{13} + 492 \nu^{12} + 4780 \nu^{11} + 1317 \nu^{10} - 15166 \nu^{9} - 3820 \nu^{8} + 6168 \nu^{7} + 42059 \nu^{6} - 10386 \nu^{5} - 53512 \nu^{4} - 7264 \nu^{3} - 7232 \nu^{2} + 141024 \nu - 109568$$$$)/1344$$ $$\beta_{7}$$ $$=$$ $$($$$$-526 \nu^{15} + 281 \nu^{14} + 1644 \nu^{13} + 1572 \nu^{12} - 3082 \nu^{11} - 9945 \nu^{10} - 8 \nu^{9} + 18652 \nu^{8} + 23382 \nu^{7} - 16367 \nu^{6} - 54624 \nu^{5} - 14288 \nu^{4} + 43096 \nu^{3} + 77600 \nu^{2} - 24384 \nu - 65920$$$$)/1344$$ $$\beta_{8}$$ $$=$$ $$($$$$-1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + 31596 \nu^{9} + 17228 \nu^{8} - 5477 \nu^{7} - 95888 \nu^{6} - 4752 \nu^{5} + 107784 \nu^{4} + 43856 \nu^{3} + 44320 \nu^{2} - 296512 \nu + 176384$$$$)/2688$$ $$\beta_{9}$$ $$=$$ $$($$$$222 \nu^{15} - 593 \nu^{14} - 16 \nu^{13} + 148 \nu^{12} + 1914 \nu^{11} + 977 \nu^{10} - 5588 \nu^{9} - 2596 \nu^{8} + 1306 \nu^{7} + 16455 \nu^{6} - 132 \nu^{5} - 18888 \nu^{4} - 6376 \nu^{3} - 6976 \nu^{2} + 52032 \nu - 32384$$$$)/448$$ $$\beta_{10}$$ $$=$$ $$($$$$1367 \nu^{15} - 3586 \nu^{14} - 12 \nu^{13} + 720 \nu^{12} + 11729 \nu^{11} + 6366 \nu^{10} - 34004 \nu^{9} - 16892 \nu^{8} + 5967 \nu^{7} + 102370 \nu^{6} + 3072 \nu^{5} - 115064 \nu^{4} - 44192 \nu^{3} - 51040 \nu^{2} + 324288 \nu - 196096$$$$)/2688$$ $$\beta_{11}$$ $$=$$ $$($$$$-355 \nu^{15} + 1041 \nu^{14} - 124 \nu^{13} - 400 \nu^{12} - 3209 \nu^{11} - 837 \nu^{10} + 10264 \nu^{9} + 3156 \nu^{8} - 4547 \nu^{7} - 28915 \nu^{6} + 5004 \nu^{5} + 36556 \nu^{4} + 8700 \nu^{3} + 5520 \nu^{2} - 94144 \nu + 64864$$$$)/672$$ $$\beta_{12}$$ $$=$$ $$($$$$-106 \nu^{15} + 397 \nu^{14} - 200 \nu^{13} - 260 \nu^{12} - 1014 \nu^{11} + 483 \nu^{10} + 4060 \nu^{9} - 508 \nu^{8} - 3694 \nu^{7} - 10011 \nu^{6} + 7116 \nu^{5} + 15784 \nu^{4} - 1120 \nu^{3} - 4448 \nu^{2} - 36032 \nu + 33600$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$-996 \nu^{15} + 1703 \nu^{14} + 1258 \nu^{13} + 988 \nu^{12} - 7004 \nu^{11} - 9999 \nu^{10} + 13746 \nu^{9} + 19972 \nu^{8} + 14480 \nu^{7} - 53377 \nu^{6} - 39234 \nu^{5} + 37560 \nu^{4} + 46600 \nu^{3} + 73664 \nu^{2} - 153824 \nu + 36160$$$$)/1344$$ $$\beta_{14}$$ $$=$$ $$($$$$-1167 \nu^{15} + 4247 \nu^{14} - 1916 \nu^{13} - 2612 \nu^{12} - 11105 \nu^{11} + 3909 \nu^{10} + 42912 \nu^{9} - 2696 \nu^{8} - 36019 \nu^{7} - 108253 \nu^{6} + 67332 \nu^{5} + 164544 \nu^{4} - 3704 \nu^{3} - 35584 \nu^{2} - 386240 \nu + 342400$$$$)/1344$$ $$\beta_{15}$$ $$=$$ $$($$$$-1175 \nu^{15} + 3065 \nu^{14} + 8 \nu^{13} - 676 \nu^{12} - 9945 \nu^{11} - 5133 \nu^{10} + 29156 \nu^{9} + 13576 \nu^{8} - 6659 \nu^{7} - 87051 \nu^{6} + 192 \nu^{5} + 100472 \nu^{4} + 36088 \nu^{3} + 37088 \nu^{2} - 279808 \nu + 168960$$$$)/1344$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} + \beta_{5} + 2 \beta_{4} + 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 3 \beta_{1} + 3$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - 3 \beta_{13} + \beta_{12} - \beta_{11} + 5 \beta_{10} + 5 \beta_{8} - \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 6 \beta_{3} + 6 \beta_{2} - 1$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} - 3 \beta_{13} - 4 \beta_{12} - 3 \beta_{11} + \beta_{9} + 3 \beta_{8} - \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - \beta_{2} + 4 \beta_{1} - 11$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{15} - 3 \beta_{13} - \beta_{12} + 4 \beta_{10} + 8 \beta_{9} - 6 \beta_{8} + 4 \beta_{7} - 4 \beta_{6} + 12 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 12 \beta_{2} - \beta_{1} - 1$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$8 \beta_{15} - \beta_{14} - 3 \beta_{13} + 7 \beta_{11} + 10 \beta_{10} + 9 \beta_{9} + 3 \beta_{8} - \beta_{7} - 12 \beta_{6} - 13 \beta_{5} - 7 \beta_{4} + 7 \beta_{3} + 5 \beta_{2} + 4 \beta_{1} - 15$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-2 \beta_{14} + 6 \beta_{13} - 8 \beta_{12} + \beta_{11} - 13 \beta_{10} + 20 \beta_{9} - \beta_{8} - 11 \beta_{7} - 11 \beta_{6} + 3 \beta_{5} + 24 \beta_{4} - 16 \beta_{3} + 10 \beta_{2} - 5 \beta_{1} - 10$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$2 \beta_{15} - 20 \beta_{14} - \beta_{13} + 7 \beta_{12} + 28 \beta_{11} - 16 \beta_{10} + 20 \beta_{9} - 36 \beta_{8} + 6 \beta_{7} - 28 \beta_{6} + 20 \beta_{5} + 4 \beta_{4} - 22 \beta_{3} - 8 \beta_{2} + 7 \beta_{1} - 33$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-34 \beta_{15} + 30 \beta_{14} + 30 \beta_{13} - 8 \beta_{12} + 13 \beta_{11} - 7 \beta_{10} + 10 \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + 13 \beta_{6} - 49 \beta_{5} - 14 \beta_{4} - 4 \beta_{3} + 20 \beta_{2} - 13 \beta_{1} + 22$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$6 \beta_{15} - 37 \beta_{14} + 10 \beta_{13} - 15 \beta_{12} + 53 \beta_{11} - 40 \beta_{10} + 21 \beta_{9} - 17 \beta_{8} - 61 \beta_{7} - 34 \beta_{6} + 11 \beta_{5} + 39 \beta_{4} - 5 \beta_{3} - 25 \beta_{2} + 17 \beta_{1} + 2$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-90 \beta_{15} - 2 \beta_{14} + 41 \beta_{13} + 7 \beta_{12} + 6 \beta_{11} - 78 \beta_{10} - 20 \beta_{9} - 42 \beta_{8} + 18 \beta_{7} + 46 \beta_{6} + 124 \beta_{5} - 24 \beta_{4} - 30 \beta_{3} - 28 \beta_{2} - 9 \beta_{1} - 49$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-80 \beta_{15} + 75 \beta_{14} - 2 \beta_{13} - 49 \beta_{12} + 71 \beta_{11} + 32 \beta_{10} - 85 \beta_{9} + 17 \beta_{8} + 39 \beta_{7} + 72 \beta_{6} - 55 \beta_{5} - 17 \beta_{4} + 29 \beta_{3} - 87 \beta_{2} + 7 \beta_{1} + 102$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-46 \beta_{15} + 14 \beta_{14} + 49 \beta_{13} - 63 \beta_{12} - 55 \beta_{11} + \beta_{10} - 86 \beta_{9} + 47 \beta_{8} - 113 \beta_{7} + 45 \beta_{6} + 151 \beta_{5} - 58 \beta_{4} + 96 \beta_{3} + 96 \beta_{2} + 8 \beta_{1} + 31$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-98 \beta_{15} + 8 \beta_{14} - 147 \beta_{13} - 67 \beta_{12} + 74 \beta_{11} - 26 \beta_{10} - 176 \beta_{9} + 48 \beta_{8} + 96 \beta_{7} + 154 \beta_{6} + 332 \beta_{5} - 46 \beta_{4} + 128 \beta_{3} - 274 \beta_{2} - 39 \beta_{1} - 167$$$$)/4$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.21331 + 0.726558i −1.39563 − 0.228522i −0.530822 − 1.31081i −0.966675 + 1.03225i −0.296075 + 1.38287i 1.26868 + 0.624862i 1.32070 − 0.505727i 1.38652 + 0.278517i 1.21331 − 0.726558i −1.39563 + 0.228522i −0.530822 + 1.31081i −0.966675 − 1.03225i −0.296075 − 1.38287i 1.26868 − 0.624862i 1.32070 + 0.505727i 1.38652 − 0.278517i
0 −1.82762 1.82762i 0 0 0 −4.50961 0 3.68037i 0
49.2 0 −1.42313 1.42313i 0 0 0 −0.690576 0 1.05061i 0
49.3 0 −1.37027 1.37027i 0 0 0 2.73482 0 0.755274i 0
49.4 0 −0.209571 0.209571i 0 0 0 1.73696 0 2.91216i 0
49.5 0 0.120009 + 0.120009i 0 0 0 2.66881 0 2.97120i 0
49.6 0 0.720673 + 0.720673i 0 0 0 −4.02840 0 1.96126i 0
49.7 0 1.66366 + 1.66366i 0 0 0 −2.89402 0 2.53555i 0
49.8 0 2.32624 + 2.32624i 0 0 0 0.982011 0 7.82281i 0
849.1 0 −1.82762 + 1.82762i 0 0 0 −4.50961 0 3.68037i 0
849.2 0 −1.42313 + 1.42313i 0 0 0 −0.690576 0 1.05061i 0
849.3 0 −1.37027 + 1.37027i 0 0 0 2.73482 0 0.755274i 0
849.4 0 −0.209571 + 0.209571i 0 0 0 1.73696 0 2.91216i 0
849.5 0 0.120009 0.120009i 0 0 0 2.66881 0 2.97120i 0
849.6 0 0.720673 0.720673i 0 0 0 −4.02840 0 1.96126i 0
849.7 0 1.66366 1.66366i 0 0 0 −2.89402 0 2.53555i 0
849.8 0 2.32624 2.32624i 0 0 0 0.982011 0 7.82281i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.q 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.q.g 16
4.b odd 2 1 400.2.q.h 16
5.b even 2 1 1600.2.q.h 16
5.c odd 4 1 320.2.l.a 16
5.c odd 4 1 1600.2.l.i 16
15.e even 4 1 2880.2.t.c 16
16.e even 4 1 1600.2.q.h 16
16.f odd 4 1 400.2.q.g 16
20.d odd 2 1 400.2.q.g 16
20.e even 4 1 80.2.l.a 16
20.e even 4 1 400.2.l.h 16
40.i odd 4 1 640.2.l.a 16
40.k even 4 1 640.2.l.b 16
60.l odd 4 1 720.2.t.c 16
80.i odd 4 1 640.2.l.a 16
80.i odd 4 1 1600.2.l.i 16
80.j even 4 1 80.2.l.a 16
80.k odd 4 1 400.2.q.h 16
80.q even 4 1 inner 1600.2.q.g 16
80.s even 4 1 400.2.l.h 16
80.s even 4 1 640.2.l.b 16
80.t odd 4 1 320.2.l.a 16
160.u even 8 1 5120.2.a.s 8
160.u even 8 1 5120.2.a.v 8
160.bb odd 8 1 5120.2.a.t 8
160.bb odd 8 1 5120.2.a.u 8
240.bd odd 4 1 720.2.t.c 16
240.bf even 4 1 2880.2.t.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a 16 20.e even 4 1
80.2.l.a 16 80.j even 4 1
320.2.l.a 16 5.c odd 4 1
320.2.l.a 16 80.t odd 4 1
400.2.l.h 16 20.e even 4 1
400.2.l.h 16 80.s even 4 1
400.2.q.g 16 16.f odd 4 1
400.2.q.g 16 20.d odd 2 1
400.2.q.h 16 4.b odd 2 1
400.2.q.h 16 80.k odd 4 1
640.2.l.a 16 40.i odd 4 1
640.2.l.a 16 80.i odd 4 1
640.2.l.b 16 40.k even 4 1
640.2.l.b 16 80.s even 4 1
720.2.t.c 16 60.l odd 4 1
720.2.t.c 16 240.bd odd 4 1
1600.2.l.i 16 5.c odd 4 1
1600.2.l.i 16 80.i odd 4 1
1600.2.q.g 16 1.a even 1 1 trivial
1600.2.q.g 16 80.q even 4 1 inner
1600.2.q.h 16 5.b even 2 1
1600.2.q.h 16 16.e even 4 1
2880.2.t.c 16 15.e even 4 1
2880.2.t.c 16 240.bf even 4 1
5120.2.a.s 8 160.u even 8 1
5120.2.a.t 8 160.bb odd 8 1
5120.2.a.u 8 160.bb odd 8 1
5120.2.a.v 8 160.u even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$16 - 64 T + 128 T^{2} + 1088 T^{3} + 5824 T^{4} - 2656 T^{5} + 1024 T^{6} + 2560 T^{7} + 2632 T^{8} + 176 T^{9} + 32 T^{10} + 80 T^{11} + 112 T^{12} + 8 T^{13} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 452 - 32 T - 896 T^{2} + 328 T^{3} + 232 T^{4} - 72 T^{5} - 24 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$11$ $$1290496 - 799744 T + 247808 T^{2} - 848384 T^{3} + 3958016 T^{4} - 3673856 T^{5} + 1795584 T^{6} - 446848 T^{7} + 139616 T^{8} - 82368 T^{9} + 40320 T^{10} - 9568 T^{11} + 1232 T^{12} - 80 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$13$ $$20647936 + 46530560 T + 52428800 T^{2} + 27869184 T^{3} + 9146368 T^{4} + 3137536 T^{5} + 2654208 T^{6} + 1400832 T^{7} + 415872 T^{8} + 49152 T^{9} + 8192 T^{10} + 4352 T^{11} + 1600 T^{12} + 128 T^{13} + T^{16}$$
$17$ $$192876544 + 241311744 T^{2} + 114806784 T^{4} + 26821632 T^{6} + 3321984 T^{8} + 222336 T^{10} + 8000 T^{12} + 144 T^{14} + T^{16}$$
$19$ $$614656 + 4164608 T + 14108672 T^{2} + 26513920 T^{3} + 30308608 T^{4} + 19398912 T^{5} + 7595520 T^{6} + 1921408 T^{7} + 731744 T^{8} + 363072 T^{9} + 132480 T^{10} + 27488 T^{11} + 3216 T^{12} + 176 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$23$ $$( 1316 - 2288 T - 192 T^{2} + 1320 T^{3} - 136 T^{4} - 208 T^{5} + 8 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$29$ $$3017085184 + 4042700800 T + 2708480000 T^{2} + 456489984 T^{3} - 5714688 T^{4} - 1816064 T^{5} + 37230592 T^{6} + 2768128 T^{7} - 199840 T^{8} - 351616 T^{9} + 198144 T^{10} - 18624 T^{11} + 1104 T^{12} - 288 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$31$ $$( -20224 - 58368 T - 26112 T^{2} + 4096 T^{3} + 2848 T^{4} - 64 T^{5} - 96 T^{6} + T^{8} )^{2}$$
$37$ $$18939904 + 236191744 T + 1472724992 T^{2} + 2707357696 T^{3} + 2705047552 T^{4} + 1380728832 T^{5} + 381124608 T^{6} + 40185856 T^{7} + 22554112 T^{8} + 9370624 T^{9} + 2144256 T^{10} + 249088 T^{11} + 16320 T^{12} + 704 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$41$ $$110660014336 + 325786435584 T^{2} + 63304144128 T^{4} + 5024061440 T^{6} + 206041952 T^{8} + 4686848 T^{10} + 59088 T^{12} + 384 T^{14} + T^{16}$$
$43$ $$53640976 + 331044800 T + 1021520000 T^{2} + 1500385792 T^{3} + 1191507520 T^{4} + 280880992 T^{5} + 26364032 T^{6} - 1854880 T^{7} + 5444360 T^{8} + 950192 T^{9} + 76832 T^{10} - 18816 T^{11} + 5328 T^{12} + 440 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$47$ $$330675601936 + 199265537792 T^{2} + 41923796288 T^{4} + 4004140672 T^{6} + 195376712 T^{8} + 5016512 T^{10} + 65872 T^{12} + 416 T^{14} + T^{16}$$
$53$ $$383725735936 + 196640112640 T + 50384076800 T^{2} + 10552958976 T^{3} + 26109784064 T^{4} + 13942091776 T^{5} + 3861454848 T^{6} + 498471936 T^{7} + 43316352 T^{8} + 6504448 T^{9} + 1892352 T^{10} + 236416 T^{11} + 15552 T^{12} + 448 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$59$ $$12227051776 - 30451745792 T + 37920376832 T^{2} - 27384356352 T^{3} + 12226302208 T^{4} - 3038184192 T^{5} + 314344960 T^{6} + 20658560 T^{7} + 7403616 T^{8} - 2562240 T^{9} + 291200 T^{10} + 65632 T^{11} + 8464 T^{12} - 272 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$61$ $$1393986371584 + 1092943347712 T + 428456542208 T^{2} - 95024578560 T^{3} + 27549499392 T^{4} + 9643622400 T^{5} + 2332164096 T^{6} - 240730112 T^{7} + 17876992 T^{8} + 4136960 T^{9} + 1425408 T^{10} - 176640 T^{11} + 11520 T^{12} + 384 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$67$ $$46120451769616 - 26534918246592 T + 7633293466752 T^{2} - 2087109786752 T^{3} + 1043974444608 T^{4} - 475604544352 T^{5} + 148072661120 T^{6} - 31795927072 T^{7} + 4867387016 T^{8} - 537576688 T^{9} + 43098400 T^{10} - 2604960 T^{11} + 147664 T^{12} - 10680 T^{13} + 800 T^{14} - 40 T^{15} + T^{16}$$
$71$ $$3333516427264 + 2007385505792 T^{2} + 408856297472 T^{4} + 33695596544 T^{6} + 1144522752 T^{8} + 18817024 T^{10} + 157440 T^{12} + 640 T^{14} + T^{16}$$
$73$ $$( -125888 - 139008 T + 35904 T^{2} + 73088 T^{3} + 15296 T^{4} - 736 T^{5} - 280 T^{6} + T^{8} )^{2}$$
$79$ $$( 4352 + 31232 T - 61952 T^{2} - 4992 T^{3} + 5856 T^{4} + 352 T^{5} - 160 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$83$ $$2050640656 - 6041972416 T + 8900981888 T^{2} - 5315507456 T^{3} + 1230069056 T^{4} + 140621856 T^{5} + 1135671424 T^{6} - 379938272 T^{7} + 64272392 T^{8} + 9891472 T^{9} + 315168 T^{10} - 36928 T^{11} + 37520 T^{12} + 7976 T^{13} + 800 T^{14} + 40 T^{15} + T^{16}$$
$89$ $$684153962496 + 380947267584 T^{2} + 69045698560 T^{4} + 5734359040 T^{6} + 244188672 T^{8} + 5576192 T^{10} + 68032 T^{12} + 416 T^{14} + T^{16}$$
$97$ $$73090735673344 + 29428625465344 T^{2} + 3189138534400 T^{4} + 153010863104 T^{6} + 3690118272 T^{8} + 45708672 T^{10} + 289472 T^{12} + 880 T^{14} + T^{16}$$