# Properties

 Label 1152.3.m.c Level $1152$ Weight $3$ Character orbit 1152.m Analytic conductor $31.390$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.m (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{28}$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{11} q^{5} -\beta_{3} q^{7} +O(q^{10})$$ $$q + \beta_{11} q^{5} -\beta_{3} q^{7} + ( -2 - 2 \beta_{2} + \beta_{5} + \beta_{10} ) q^{11} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{14} - \beta_{15} ) q^{13} + ( \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{17} + ( -2 + 2 \beta_{2} + \beta_{6} - \beta_{9} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{19} + ( -8 + \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{23} + ( -5 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{25} + ( 2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{14} + \beta_{15} ) q^{29} + ( -8 \beta_{2} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{15} ) q^{31} + ( -6 + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{35} + ( 6 - 6 \beta_{2} - \beta_{3} - \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{37} + ( -3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{41} + ( 10 + 10 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{43} + ( -24 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( 7 - 4 \beta_{1} + 4 \beta_{3} - 5 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - 5 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{49} + ( -10 + 2 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{53} + ( 16 + 2 \beta_{1} + 2 \beta_{3} - 6 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - 6 \beta_{11} + \beta_{12} ) q^{55} + ( 8 + 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{15} ) q^{59} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 5 \beta_{10} + \beta_{14} - \beta_{15} ) q^{61} + ( 2 - 5 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{65} + ( 20 - \beta_{1} - 20 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{15} ) q^{67} + ( 32 - 2 \beta_{3} ) q^{71} + ( 6 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + 3 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{73} + ( 14 - 6 \beta_{1} + 14 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} - \beta_{15} ) q^{77} + ( 8 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} + 5 \beta_{8} - 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{12} ) q^{79} + ( 10 - 10 \beta_{2} + 2 \beta_{6} + 2 \beta_{9} - 3 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} - 2 \beta_{15} ) q^{83} + ( -10 - 8 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 2 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} + 8 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{15} ) q^{85} + ( -10 \beta_{2} + 6 \beta_{4} - 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} - 4 \beta_{15} ) q^{89} + ( -30 + 3 \beta_{1} - 30 \beta_{2} - 3 \beta_{4} + 8 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 5 \beta_{10} - 2 \beta_{14} - 5 \beta_{15} ) q^{91} + ( 40 \beta_{2} - 5 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} + 6 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 6 \beta_{15} ) q^{95} + ( 6 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 32q^{11} - 32q^{19} - 128q^{23} + 32q^{29} - 96q^{35} + 96q^{37} + 160q^{43} + 112q^{49} - 160q^{53} + 256q^{55} + 128q^{59} + 32q^{61} + 32q^{65} + 320q^{67} + 512q^{71} + 224q^{77} + 160q^{83} - 160q^{85} - 480q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5 \nu^{15} - 18 \nu^{14} - 134 \nu^{13} - 168 \nu^{12} + 170 \nu^{11} + 1156 \nu^{10} + 1848 \nu^{9} - 2928 \nu^{8} - 15600 \nu^{7} - 25632 \nu^{6} + 1792 \nu^{5} + 73472 \nu^{4} + 112128 \nu^{3} - 23552 \nu^{2} - 528384 \nu - 753664$$$$)/12288$$ $$\beta_{2}$$ $$=$$ $$($$$$-347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} - 59096 \nu^{9} - 66544 \nu^{8} + 88528 \nu^{7} + 450272 \nu^{6} + 454272 \nu^{5} - 499456 \nu^{4} - 2271744 \nu^{3} - 3177472 \nu^{2} + 3719168 \nu + 10231808$$$$)/368640$$ $$\beta_{3}$$ $$=$$ $$($$$$47 \nu^{15} + 110 \nu^{14} - 90 \nu^{13} - 528 \nu^{12} - 610 \nu^{11} + 1684 \nu^{10} + 8376 \nu^{9} + 11728 \nu^{8} - 5136 \nu^{7} - 52256 \nu^{6} - 60800 \nu^{5} + 73472 \nu^{4} + 350720 \nu^{3} + 537600 \nu^{2} - 172032 \nu - 1228800$$$$)/40960$$ $$\beta_{4}$$ $$=$$ $$($$$$-53 \nu^{15} - 89 \nu^{14} + 226 \nu^{13} + 714 \nu^{12} + 730 \nu^{11} - 2082 \nu^{10} - 9224 \nu^{9} - 8056 \nu^{8} + 19792 \nu^{7} + 75248 \nu^{6} + 67008 \nu^{5} - 110464 \nu^{4} - 377856 \nu^{3} - 429568 \nu^{2} + 837632 \nu + 1974272$$$$)/30720$$ $$\beta_{5}$$ $$=$$ $$($$$$-751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} - 156376 \nu^{9} - 19520 \nu^{8} + 641360 \nu^{7} + 1598464 \nu^{6} + 829440 \nu^{5} - 3026432 \nu^{4} - 7073280 \nu^{3} - 4517888 \nu^{2} + 22466560 \nu + 41107456$$$$)/368640$$ $$\beta_{6}$$ $$=$$ $$($$$$119 \nu^{15} + 1230 \nu^{14} + 3430 \nu^{13} + 2064 \nu^{12} - 9010 \nu^{11} - 25772 \nu^{10} - 2088 \nu^{9} + 157776 \nu^{8} + 431088 \nu^{7} + 442848 \nu^{6} - 394880 \nu^{5} - 1689856 \nu^{4} - 1328640 \nu^{3} + 4019200 \nu^{2} + 13664256 \nu + 14581760$$$$)/122880$$ $$\beta_{7}$$ $$=$$ $$($$$$197 \nu^{15} - 244 \nu^{14} - 2572 \nu^{13} - 3852 \nu^{12} + 998 \nu^{11} + 22056 \nu^{10} + 40172 \nu^{9} - 35600 \nu^{8} - 275680 \nu^{7} - 517568 \nu^{6} - 54240 \nu^{5} + 1233664 \nu^{4} + 2280960 \nu^{3} - 88064 \nu^{2} - 9620480 \nu - 12972032$$$$)/92160$$ $$\beta_{8}$$ $$=$$ $$($$$$1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + 189400 \nu^{9} + 278960 \nu^{8} - 106640 \nu^{7} - 1269472 \nu^{6} - 1748352 \nu^{5} + 775424 \nu^{4} + 6971904 \nu^{3} + 12336128 \nu^{2} - 4268032 \nu - 26214400$$$$)/368640$$ $$\beta_{9}$$ $$=$$ $$($$$$-995 \nu^{15} - 3884 \nu^{14} - 4022 \nu^{13} + 5076 \nu^{12} + 30658 \nu^{11} + 28992 \nu^{10} - 122744 \nu^{9} - 468448 \nu^{8} - 716144 \nu^{7} + 66368 \nu^{6} + 1824000 \nu^{5} + 2557952 \nu^{4} - 2188800 \nu^{3} - 16144384 \nu^{2} - 20942848 \nu - 8568832$$$$)/368640$$ $$\beta_{10}$$ $$=$$ $$($$$$134 \nu^{15} + 20 \nu^{14} - 1153 \nu^{13} - 2232 \nu^{12} - 622 \nu^{11} + 9756 \nu^{10} + 25118 \nu^{9} - 1448 \nu^{8} - 113704 \nu^{7} - 257408 \nu^{6} - 102288 \nu^{5} + 502912 \nu^{4} + 1184256 \nu^{3} + 538624 \nu^{2} - 3969536 \nu - 6232064$$$$)/46080$$ $$\beta_{11}$$ $$=$$ $$($$$$545 \nu^{15} + 1574 \nu^{14} + 302 \nu^{13} - 5256 \nu^{12} - 12838 \nu^{11} + 1188 \nu^{10} + 82544 \nu^{9} + 190768 \nu^{8} + 127664 \nu^{7} - 372128 \nu^{6} - 897600 \nu^{5} - 303872 \nu^{4} + 2511360 \nu^{3} + 7066624 \nu^{2} + 3770368 \nu - 6053888$$$$)/184320$$ $$\beta_{12}$$ $$=$$ $$($$$$-1417 \nu^{15} - 4300 \nu^{14} - 1186 \nu^{13} + 13356 \nu^{12} + 35366 \nu^{11} - 528 \nu^{10} - 219304 \nu^{9} - 508256 \nu^{8} - 401488 \nu^{7} + 933184 \nu^{6} + 2421504 \nu^{5} + 922624 \nu^{4} - 6455808 \nu^{3} - 18839552 \nu^{2} - 11743232 \nu + 13975552$$$$)/368640$$ $$\beta_{13}$$ $$=$$ $$($$$$265 \nu^{15} + 526 \nu^{14} - 578 \nu^{13} - 2856 \nu^{12} - 4406 \nu^{11} + 6228 \nu^{10} + 41008 \nu^{9} + 59984 \nu^{8} - 23888 \nu^{7} - 271264 \nu^{6} - 338496 \nu^{5} + 217088 \nu^{4} + 1423872 \nu^{3} + 2561024 \nu^{2} - 1472512 \nu - 5570560$$$$)/61440$$ $$\beta_{14}$$ $$=$$ $$($$$$1181 \nu^{15} + 2693 \nu^{14} - 2326 \nu^{13} - 13986 \nu^{12} - 22786 \nu^{11} + 22218 \nu^{10} + 190016 \nu^{9} + 302680 \nu^{8} - 52720 \nu^{7} - 1210544 \nu^{6} - 1754880 \nu^{5} + 602752 \nu^{4} + 6497280 \nu^{3} + 12133888 \nu^{2} - 3891200 \nu - 25296896$$$$)/184320$$ $$\beta_{15}$$ $$=$$ $$($$$$-2519 \nu^{15} - 4382 \nu^{14} + 9178 \nu^{13} + 33552 \nu^{12} + 37810 \nu^{11} - 85236 \nu^{10} - 427352 \nu^{9} - 462928 \nu^{8} + 698896 \nu^{7} + 3347744 \nu^{6} + 3329664 \nu^{5} - 3795712 \nu^{4} - 16501248 \nu^{3} - 21873664 \nu^{2} + 28983296 \nu + 80101376$$$$)/368640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} - 2 \beta_{6} + \beta_{4} + 4 \beta_{2} - \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_{1} + 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 4 \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 10 \beta_{2} + 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} - 4 \beta_{4} - \beta_{3} + 22 \beta_{2} - 2 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{15} - 8 \beta_{14} - 16 \beta_{13} + 2 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - 14 \beta_{8} + 4 \beta_{7} + 14 \beta_{6} - 12 \beta_{5} + 23 \beta_{4} - 12 \beta_{3} - 28 \beta_{2} + 17 \beta_{1} - 80$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$12 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 9 \beta_{12} + 2 \beta_{11} + 9 \beta_{10} + 5 \beta_{9} + 8 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} - 21 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} - 102$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-15 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} - 3 \beta_{12} - 47 \beta_{11} - \beta_{10} - 10 \beta_{9} + 13 \beta_{8} + 14 \beta_{7} + 29 \beta_{6} - 33 \beta_{5} + 16 \beta_{4} + 11 \beta_{3} + 90 \beta_{2} - 5 \beta_{1} - 182$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$13 \beta_{15} + 21 \beta_{14} + 9 \beta_{13} + 26 \beta_{12} + 23 \beta_{11} - 24 \beta_{10} + 12 \beta_{9} + 9 \beta_{8} + 17 \beta_{7} + 8 \beta_{6} + 7 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 16 \beta_{2} + 24 \beta_{1} - 146$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-14 \beta_{15} + 24 \beta_{14} + 32 \beta_{13} - 84 \beta_{12} - 120 \beta_{11} + 66 \beta_{10} + 44 \beta_{9} + 24 \beta_{8} - 60 \beta_{7} + 28 \beta_{6} + 106 \beta_{5} + 81 \beta_{4} - 70 \beta_{3} + 40 \beta_{2} + 189 \beta_{1} - 276$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$88 \beta_{15} + 38 \beta_{14} + 80 \beta_{13} + 5 \beta_{12} - 166 \beta_{11} - 55 \beta_{10} + 59 \beta_{9} + 56 \beta_{8} - 33 \beta_{7} - 48 \beta_{6} + 86 \beta_{5} + 51 \beta_{4} + 128 \beta_{3} - 1018 \beta_{2} - 109 \beta_{1} - 786$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-167 \beta_{15} + 48 \beta_{14} + 36 \beta_{13} + 143 \beta_{12} - 105 \beta_{11} + 27 \beta_{10} - 70 \beta_{9} + 59 \beta_{8} - 74 \beta_{7} + 3 \beta_{6} + 27 \beta_{5} + 161 \beta_{4} + 199 \beta_{3} + 1006 \beta_{2} + 62 \beta_{1} + 1210$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$127 \beta_{15} + 165 \beta_{14} + 227 \beta_{13} - 175 \beta_{12} - 131 \beta_{11} - 207 \beta_{10} - 21 \beta_{9} + 17 \beta_{8} + 30 \beta_{7} - 350 \beta_{6} + 511 \beta_{5} - 84 \beta_{4} - 71 \beta_{3} - 238 \beta_{2} + 174 \beta_{1} + 8$$ $$\nu^{13}$$ $$=$$ $$($$$$-520 \beta_{15} - 904 \beta_{14} + 32 \beta_{13} - 322 \beta_{12} + 174 \beta_{11} + 296 \beta_{10} + 532 \beta_{9} - 18 \beta_{8} - 452 \beta_{7} + 114 \beta_{6} + 348 \beta_{5} - 31 \beta_{4} - 20 \beta_{3} - 2788 \beta_{2} - 281 \beta_{1} + 6448$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$204 \beta_{15} + 484 \beta_{14} - 426 \beta_{13} - 25 \beta_{12} - 210 \beta_{11} - 113 \beta_{10} - 285 \beta_{9} - 888 \beta_{8} - 395 \beta_{7} - 670 \beta_{6} + 458 \beta_{5} + 97 \beta_{4} + 692 \beta_{3} - 4926 \beta_{2} - 153 \beta_{1} + 7174$$ $$\nu^{15}$$ $$=$$ $$-1033 \beta_{15} - 1244 \beta_{14} + 920 \beta_{13} - 565 \beta_{12} + 1631 \beta_{11} + 1233 \beta_{10} - 654 \beta_{9} - 1245 \beta_{8} - 86 \beta_{7} - 1069 \beta_{6} - 695 \beta_{5} - 1836 \beta_{4} + 605 \beta_{3} + 14390 \beta_{2} - 3255 \beta_{1} + 4502$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
415.1
 −1.87459 + 0.697079i 1.84258 − 0.777752i −0.455024 − 1.94755i −1.25564 − 1.55672i 1.78012 − 0.911682i −1.96679 − 0.362960i 0.125358 + 1.99607i 1.80398 + 0.863518i −1.87459 − 0.697079i 1.84258 + 0.777752i −0.455024 + 1.94755i −1.25564 + 1.55672i 1.78012 + 0.911682i −1.96679 + 0.362960i 0.125358 − 1.99607i 1.80398 − 0.863518i
0 0 0 −5.24354 5.24354i 0 5.32796 0 0 0
415.2 0 0 0 −4.78830 4.78830i 0 10.3302 0 0 0
415.3 0 0 0 −3.40572 3.40572i 0 −12.1303 0 0 0
415.4 0 0 0 0.909023 + 0.909023i 0 0.654713 0 0 0
415.5 0 0 0 1.00772 + 1.00772i 0 −10.0236 0 0 0
415.6 0 0 0 1.69930 + 1.69930i 0 5.74280 0 0 0
415.7 0 0 0 3.32679 + 3.32679i 0 4.04088 0 0 0
415.8 0 0 0 6.49473 + 6.49473i 0 −3.94273 0 0 0
991.1 0 0 0 −5.24354 + 5.24354i 0 5.32796 0 0 0
991.2 0 0 0 −4.78830 + 4.78830i 0 10.3302 0 0 0
991.3 0 0 0 −3.40572 + 3.40572i 0 −12.1303 0 0 0
991.4 0 0 0 0.909023 0.909023i 0 0.654713 0 0 0
991.5 0 0 0 1.00772 1.00772i 0 −10.0236 0 0 0
991.6 0 0 0 1.69930 1.69930i 0 5.74280 0 0 0
991.7 0 0 0 3.32679 3.32679i 0 4.04088 0 0 0
991.8 0 0 0 6.49473 6.49473i 0 −3.94273 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.m.c 16
3.b odd 2 1 384.3.l.b 16
4.b odd 2 1 1152.3.m.f 16
8.b even 2 1 576.3.m.c 16
8.d odd 2 1 144.3.m.c 16
12.b even 2 1 384.3.l.a 16
16.e even 4 1 144.3.m.c 16
16.e even 4 1 1152.3.m.f 16
16.f odd 4 1 576.3.m.c 16
16.f odd 4 1 inner 1152.3.m.c 16
24.f even 2 1 48.3.l.a 16
24.h odd 2 1 192.3.l.a 16
48.i odd 4 1 48.3.l.a 16
48.i odd 4 1 384.3.l.a 16
48.k even 4 1 192.3.l.a 16
48.k even 4 1 384.3.l.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a 16 24.f even 2 1
48.3.l.a 16 48.i odd 4 1
144.3.m.c 16 8.d odd 2 1
144.3.m.c 16 16.e even 4 1
192.3.l.a 16 24.h odd 2 1
192.3.l.a 16 48.k even 4 1
384.3.l.a 16 12.b even 2 1
384.3.l.a 16 48.i odd 4 1
384.3.l.b 16 3.b odd 2 1
384.3.l.b 16 48.k even 4 1
576.3.m.c 16 8.b even 2 1
576.3.m.c 16 16.f odd 4 1
1152.3.m.c 16 1.a even 1 1 trivial
1152.3.m.c 16 16.f odd 4 1 inner
1152.3.m.f 16 4.b odd 2 1
1152.3.m.f 16 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{16} + \cdots$$ $$T_{7}^{8} - 224 T_{7}^{6} + 448 T_{7}^{5} + 13704 T_{7}^{4} - 53248 T_{7}^{3} - 136576 T_{7}^{2} + 720640 T_{7} - 400880$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$2117472256 - 5171462144 T + 6315081728 T^{2} - 4258177024 T^{3} + 1663893504 T^{4} - 283883520 T^{5} + 12517376 T^{6} - 4026368 T^{7} + 6221952 T^{8} - 518400 T^{9} + 512 T^{10} + 8064 T^{11} + 6656 T^{12} + 32 T^{13} + T^{16}$$
$7$ $$( -400880 + 720640 T - 136576 T^{2} - 53248 T^{3} + 13704 T^{4} + 448 T^{5} - 224 T^{6} + T^{8} )^{2}$$
$11$ $$25620118503424 - 29481536323584 T + 16962470019072 T^{2} + 22391525736448 T^{3} + 10536909537280 T^{4} + 2206640635904 T^{5} + 269438418944 T^{6} + 21404090368 T^{7} + 3292530688 T^{8} + 584835072 T^{9} + 68780032 T^{10} + 4242432 T^{11} + 151552 T^{12} + 4608 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$13$ $$27957043852960000 + 3777806164172800 T + 255245502513152 T^{2} - 75155778920448 T^{3} + 18426489989376 T^{4} + 1347500722176 T^{5} + 114873139200 T^{6} - 37570686976 T^{7} + 5634806368 T^{8} - 8847872 T^{9} + 5120000 T^{10} - 1717760 T^{11} + 271248 T^{12} - 3200 T^{13} + T^{16}$$
$17$ $$( 816881920 - 9390080 T - 53986304 T^{2} - 1044480 T^{3} + 508448 T^{4} + 2944 T^{5} - 1344 T^{6} + T^{8} )^{2}$$
$19$ $$10598900522979229696 + 1221045122401566720 T + 70335181828915200 T^{2} - 89104253114777600 T^{3} + 23939487244337152 T^{4} - 1508224010715136 T^{5} + 41927575470080 T^{6} + 1219751600128 T^{7} + 149480318464 T^{8} - 9271351296 T^{9} + 294903808 T^{10} + 15014400 T^{11} + 559552 T^{12} - 14208 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$23$ $$( -35037900800 + 425492480 T + 777799680 T^{2} + 42348544 T^{3} - 1089152 T^{4} - 109056 T^{5} - 736 T^{6} + 64 T^{7} + T^{8} )^{2}$$
$29$ $$56446323002698240000 - 10320873959271219200 T + 943555165480583168 T^{2} + 1809064899469205504 T^{3} + 1336868519252525056 T^{4} + 34131461984319488 T^{5} + 401836469485568 T^{6} - 22033099003904 T^{7} + 4665816875136 T^{8} + 92505709824 T^{9} + 1050931712 T^{10} - 64121984 T^{11} + 3957248 T^{12} + 45280 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$31$ $$41\!\cdots\!00$$$$+$$$$88\!\cdots\!60$$$$T^{2} + 7691970571770562816 T^{4} + 26353243415873536 T^{6} + 46734989650528 T^{8} + 46823575040 T^{10} + 26696080 T^{12} + 8064 T^{14} + T^{16}$$
$37$ $$38\!\cdots\!04$$$$+$$$$73\!\cdots\!12$$$$T + 70869372871016480768 T^{2} - 2884347249537718272 T^{3} + 703119961976168704 T^{4} + 97908324877949952 T^{5} + 6893443090702336 T^{6} - 6367638655488 T^{7} - 968313092000 T^{8} + 28675020544 T^{9} + 14234605568 T^{10} - 224961664 T^{11} + 1620496 T^{12} - 14528 T^{13} + 4608 T^{14} - 96 T^{15} + T^{16}$$
$41$ $$94\!\cdots\!00$$$$+$$$$70\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!56$$$$T^{4} + 365640212557922304 T^{6} + 356212041131520 T^{8} + 207527260160 T^{10} + 70874688 T^{12} + 13056 T^{14} + T^{16}$$
$43$ $$92\!\cdots\!00$$$$-$$$$28\!\cdots\!80$$$$T +$$$$44\!\cdots\!68$$$$T^{2} -$$$$32\!\cdots\!48$$$$T^{3} +$$$$14\!\cdots\!24$$$$T^{4} - 5239225685693136896 T^{5} + 245950489383796736 T^{6} - 12945845326176256 T^{7} + 537795837003264 T^{8} - 15508493912064 T^{9} + 306484011008 T^{10} - 4149187072 T^{11} + 46122944 T^{12} - 682624 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16}$$
$47$ $$11\!\cdots\!00$$$$+$$$$37\!\cdots\!36$$$$T^{2} + 4369756732867477504 T^{4} + 22735051785764864 T^{6} + 57150256209920 T^{8} + 69921902592 T^{10} + 41678592 T^{12} + 11200 T^{14} + T^{16}$$
$53$ $$96\!\cdots\!00$$$$+$$$$20\!\cdots\!60$$$$T +$$$$21\!\cdots\!52$$$$T^{2} +$$$$19\!\cdots\!20$$$$T^{3} + 91449263140998676480 T^{4} + 5591746172921030656 T^{5} + 1685959386162102272 T^{6} + 18190042770372608 T^{7} + 100041121243264 T^{8} + 1029094838016 T^{9} + 299678376448 T^{10} + 3156153984 T^{11} + 16960000 T^{12} + 153504 T^{13} + 12800 T^{14} + 160 T^{15} + T^{16}$$
$59$ $$23\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T +$$$$69\!\cdots\!32$$$$T^{2} -$$$$16\!\cdots\!76$$$$T^{3} + 97922953659373584384 T^{4} - 714303909442617344 T^{5} + 215142153171501056 T^{6} - 32123905497890816 T^{7} + 2068957458595840 T^{8} - 72789256044544 T^{9} + 1586277384192 T^{10} - 20765343744 T^{11} + 158699520 T^{12} - 675840 T^{13} + 8192 T^{14} - 128 T^{15} + T^{16}$$
$61$ $$12\!\cdots\!00$$$$+$$$$90\!\cdots\!60$$$$T +$$$$33\!\cdots\!72$$$$T^{2} -$$$$22\!\cdots\!24$$$$T^{3} +$$$$35\!\cdots\!44$$$$T^{4} +$$$$20\!\cdots\!88$$$$T^{5} + 735866880468475904 T^{6} - 49175921370646016 T^{7} + 3370685859892320 T^{8} + 12153910400256 T^{9} + 42705459200 T^{10} - 2795215232 T^{11} + 115399952 T^{12} + 157120 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$67$ $$21\!\cdots\!96$$$$+$$$$11\!\cdots\!04$$$$T +$$$$30\!\cdots\!48$$$$T^{2} -$$$$28\!\cdots\!28$$$$T^{3} +$$$$13\!\cdots\!96$$$$T^{4} + 621625262253015040 T^{5} + 579595725034225664 T^{6} - 50262831949414400 T^{7} + 2061397582544896 T^{8} - 36863866568704 T^{9} + 475717435392 T^{10} - 9110568960 T^{11} + 255286272 T^{12} - 4611072 T^{13} + 51200 T^{14} - 320 T^{15} + T^{16}$$
$71$ $$( 290924400640 - 109021003776 T + 16299499520 T^{2} - 1284915200 T^{3} + 59283584 T^{4} - 1659392 T^{5} + 27776 T^{6} - 256 T^{7} + T^{8} )^{2}$$
$73$ $$98\!\cdots\!16$$$$+$$$$13\!\cdots\!52$$$$T^{2} +$$$$45\!\cdots\!44$$$$T^{4} + 53883999480140791808 T^{6} + 25520342188187648 T^{8} + 5899646435328 T^{10} + 709382400 T^{12} + 42496 T^{14} + T^{16}$$
$79$ $$18\!\cdots\!00$$$$+$$$$25\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!36$$$$T^{4} + 1719443636116761600 T^{6} + 4201068991559776 T^{8} + 2270484756224 T^{10} + 458363920 T^{12} + 36928 T^{14} + T^{16}$$
$83$ $$14\!\cdots\!76$$$$-$$$$42\!\cdots\!84$$$$T +$$$$63\!\cdots\!28$$$$T^{2} +$$$$13\!\cdots\!24$$$$T^{3} +$$$$12\!\cdots\!56$$$$T^{4} +$$$$39\!\cdots\!64$$$$T^{5} + 6381120662228434944 T^{6} - 5903701162754048 T^{7} + 105517061441536 T^{8} + 17298459828224 T^{9} + 474865041408 T^{10} - 3104705536 T^{11} + 3373056 T^{12} + 206336 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16}$$
$89$ $$52\!\cdots\!00$$$$+$$$$68\!\cdots\!60$$$$T^{2} +$$$$23\!\cdots\!56$$$$T^{4} + 30661655762820161536 T^{6} + 18282328190707200 T^{8} + 5304960677376 T^{10} + 743027648 T^{12} + 45728 T^{14} + T^{16}$$
$97$ $$( 409778579046400 + 2337541980160 T - 1019025981440 T^{2} - 1982349312 T^{3} + 383621120 T^{4} + 116224 T^{5} - 37056 T^{6} + T^{8} )^{2}$$