# Properties

 Label 2880.2.t.c Level $2880$ Weight $2$ Character orbit 2880.t Analytic conductor $22.997$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.t (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ 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: $$2^{12}$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} - 632 \nu^{9} + 3284 \nu^{8} + 6101 \nu^{7} - 1962 \nu^{6} - 10212 \nu^{5} - 4496 \nu^{4} + 4544 \nu^{3} + 16768 \nu^{2} - 6656 \nu - 7296$$$$)/2688$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{15} - 80 \nu^{14} + 152 \nu^{13} + 168 \nu^{12} + 65 \nu^{11} - 740 \nu^{10} - 1008 \nu^{9} + 1260 \nu^{8} + 2359 \nu^{7} + 1444 \nu^{6} - 5092 \nu^{5} - 4728 \nu^{4} + 2944 \nu^{3} + 5856 \nu^{2} + 7168 \nu - 12288$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$41 \nu^{15} + 80 \nu^{14} - 264 \nu^{13} - 312 \nu^{12} + 47 \nu^{11} + 1380 \nu^{10} + 1312 \nu^{9} - 2204 \nu^{8} - 4071 \nu^{7} - 1316 \nu^{6} + 8052 \nu^{5} + 6376 \nu^{4} - 4208 \nu^{3} - 9952 \nu^{2} - 7680 \nu + 15488$$$$)/384$$ $$\beta_{4}$$ $$=$$ $$($$$$303 \nu^{15} - 1892 \nu^{14} + 1724 \nu^{13} + 1784 \nu^{12} + 3497 \nu^{11} - 6864 \nu^{10} - 19332 \nu^{9} + 11948 \nu^{8} + 27583 \nu^{7} + 39328 \nu^{6} - 63120 \nu^{5} - 81576 \nu^{4} + 32528 \nu^{3} + 58720 \nu^{2} + 166592 \nu - 209536$$$$)/2688$$ $$\beta_{5}$$ $$=$$ $$($$$$397 \nu^{15} - 502 \nu^{14} - 772 \nu^{13} - 664 \nu^{12} + 2523 \nu^{11} + 5226 \nu^{10} - 3292 \nu^{9} - 9932 \nu^{8} - 10139 \nu^{7} + 16854 \nu^{6} + 24480 \nu^{5} - 4720 \nu^{4} - 21776 \nu^{3} - 40576 \nu^{2} + 42176 \nu + 8832$$$$)/2688$$ $$\beta_{6}$$ $$=$$ $$($$$$-655 \nu^{15} + 60 \nu^{14} + 2516 \nu^{13} + 2480 \nu^{12} - 3641 \nu^{11} - 14664 \nu^{10} - 3308 \nu^{9} + 27372 \nu^{8} + 36625 \nu^{7} - 15880 \nu^{6} - 84768 \nu^{5} - 33296 \nu^{4} + 64416 \nu^{3} + 114624 \nu^{2} - 6592 \nu - 123008$$$$)/2688$$ $$\beta_{7}$$ $$=$$ $$($$$$-715 \nu^{15} + 2724 \nu^{14} - 1240 \nu^{13} - 1648 \nu^{12} - 7037 \nu^{11} + 2592 \nu^{10} + 27040 \nu^{9} - 2292 \nu^{8} - 22475 \nu^{7} - 67264 \nu^{6} + 45252 \nu^{5} + 102304 \nu^{4} - 7248 \nu^{3} - 26112 \nu^{2} - 239872 \nu + 227968$$$$)/2688$$ $$\beta_{8}$$ $$=$$ $$($$$$396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} - 11700 \nu^{9} - 2228 \nu^{8} + 6320 \nu^{7} + 32015 \nu^{6} - 9900 \nu^{5} - 42864 \nu^{4} - 5840 \nu^{3} - 1504 \nu^{2} + 108736 \nu - 83072$$$$)/1344$$ $$\beta_{9}$$ $$=$$ $$($$$$-137 \nu^{15} + 466 \nu^{14} - 144 \nu^{13} - 264 \nu^{12} - 1303 \nu^{11} + 106 \nu^{10} + 4672 \nu^{9} + 492 \nu^{8} - 3289 \nu^{7} - 12506 \nu^{6} + 4972 \nu^{5} + 17552 \nu^{4} + 1864 \nu^{3} - 1184 \nu^{2} - 41984 \nu + 32000$$$$)/448$$ $$\beta_{10}$$ $$=$$ $$($$$$-935 \nu^{15} + 3056 \nu^{14} - 676 \nu^{13} - 1552 \nu^{12} - 8961 \nu^{11} - 636 \nu^{10} + 30236 \nu^{9} + 5644 \nu^{8} - 17975 \nu^{7} - 81876 \nu^{6} + 25272 \nu^{5} + 108608 \nu^{4} + 16480 \nu^{3} + 1280 \nu^{2} - 270016 \nu + 203136$$$$)/2688$$ $$\beta_{11}$$ $$=$$ $$($$$$-1011 \nu^{15} + 2684 \nu^{14} - 164 \nu^{13} - 632 \nu^{12} - 8357 \nu^{11} - 3480 \nu^{10} + 25260 \nu^{9} + 9028 \nu^{8} - 7939 \nu^{7} - 73048 \nu^{6} + 8664 \nu^{5} + 88680 \nu^{4} + 19696 \nu^{3} + 21344 \nu^{2} - 240704 \nu + 166912$$$$)/2688$$ $$\beta_{12}$$ $$=$$ $$($$$$-1269 \nu^{15} + 3292 \nu^{14} + 236 \nu^{13} - 664 \nu^{12} - 11155 \nu^{11} - 6240 \nu^{10} + 31596 \nu^{9} + 17228 \nu^{8} - 5477 \nu^{7} - 95888 \nu^{6} - 4752 \nu^{5} + 107784 \nu^{4} + 43856 \nu^{3} + 44320 \nu^{2} - 296512 \nu + 176384$$$$)/2688$$ $$\beta_{13}$$ $$=$$ $$($$$$-1583 \nu^{15} + 4222 \nu^{14} - 120 \nu^{13} - 1032 \nu^{12} - 13337 \nu^{11} - 5850 \nu^{10} + 40424 \nu^{9} + 15908 \nu^{8} - 12087 \nu^{7} - 117622 \nu^{6} + 9804 \nu^{5} + 141728 \nu^{4} + 39344 \nu^{3} + 38848 \nu^{2} - 383232 \nu + 252160$$$$)/2688$$ $$\beta_{14}$$ $$=$$ $$($$$$239 \nu^{15} - 554 \nu^{14} - 56 \nu^{13} + 16 \nu^{12} + 1833 \nu^{11} + 1254 \nu^{10} - 5096 \nu^{9} - 2764 \nu^{8} + 311 \nu^{7} + 15690 \nu^{6} + 1044 \nu^{5} - 17384 \nu^{4} - 6592 \nu^{3} - 8672 \nu^{2} + 50560 \nu - 29952$$$$)/384$$ $$\beta_{15}$$ $$=$$ $$($$$$-356 \nu^{15} + 1000 \nu^{14} - 139 \nu^{13} - 312 \nu^{12} - 3008 \nu^{11} - 856 \nu^{10} + 9619 \nu^{9} + 2572 \nu^{8} - 4104 \nu^{7} - 27068 \nu^{6} + 5701 \nu^{5} + 34644 \nu^{4} + 6408 \nu^{3} + 4944 \nu^{2} - 90864 \nu + 65248$$$$)/224$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_{1} + 3$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} - \beta_{13} + \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{6} + \beta_{2} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} + 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 12 \beta_{1} - 13$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \beta_{2} - 2 \beta_{1} - 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 19 \beta_{8} + 9 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 12 \beta_{1} - 19$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-2 \beta_{15} + 13 \beta_{14} + 3 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} - 17 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 14 \beta_{6} - 11 \beta_{5} + 2 \beta_{4} + 11 \beta_{3} + 14 \beta_{2} + 11 \beta_{1} - 32$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-9 \beta_{15} - 6 \beta_{14} + 23 \beta_{13} - 16 \beta_{12} + 6 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} - 2 \beta_{5} + 10 \beta_{4} - \beta_{3} - 3 \beta_{2} - 17 \beta_{1} - 13$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-4 \beta_{15} - 13 \beta_{14} + 17 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + 3 \beta_{9} + 36 \beta_{8} - 10 \beta_{7} + 10 \beta_{6} + 7 \beta_{5} - 30 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 51 \beta_{1} + 8$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-31 \beta_{15} + 4 \beta_{14} + 84 \beta_{13} - 43 \beta_{12} + 30 \beta_{11} - 56 \beta_{10} + 35 \beta_{9} - 41 \beta_{8} + 69 \beta_{7} - 26 \beta_{6} - \beta_{5} + 37 \beta_{4} + 9 \beta_{3} + 31 \beta_{2} - 20 \beta_{1} + 23$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-46 \beta_{15} - 44 \beta_{14} + 49 \beta_{13} - 52 \beta_{12} + 26 \beta_{11} - 14 \beta_{10} + 24 \beta_{9} - 88 \beta_{8} + \beta_{7} + 33 \beta_{6} + 6 \beta_{5} + \beta_{4} - 7 \beta_{3} - 28 \beta_{2} - 47 \beta_{1} + 23$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-5 \beta_{15} - 38 \beta_{14} + 76 \beta_{13} - 17 \beta_{12} - 84 \beta_{11} + 88 \beta_{10} - 3 \beta_{9} + 139 \beta_{8} - 75 \beta_{7} + 36 \beta_{6} + 55 \beta_{5} - 75 \beta_{4} - 37 \beta_{3} - 33 \beta_{2} - 6 \beta_{1} + 139$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-32 \beta_{15} + 9 \beta_{14} - 23 \beta_{13} - 81 \beta_{12} + 27 \beta_{11} - 9 \beta_{10} + 95 \beta_{9} - 178 \beta_{8} + 50 \beta_{7} - 18 \beta_{6} - 61 \beta_{5} - 14 \beta_{4} - 33 \beta_{3} + 66 \beta_{2} - 131 \beta_{1} + 68$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-45 \beta_{15} + 4 \beta_{14} + 82 \beta_{13} - 85 \beta_{12} - 58 \beta_{11} + 11 \beta_{10} + 77 \beta_{9} - 108 \beta_{8} + 125 \beta_{6} + 24 \beta_{5} - 4 \beta_{4} - 32 \beta_{3} - 89 \beta_{2} + 32 \beta_{1} + 63$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 −0.966675 − 1.03225i 1.38652 − 0.278517i −1.39563 + 0.228522i 1.26868 − 0.624862i −0.530822 + 1.31081i −0.296075 − 1.38287i 1.32070 + 0.505727i 1.21331 − 0.726558i 1.26868 + 0.624862i −1.39563 − 0.228522i 1.38652 + 0.278517i −0.966675 + 1.03225i 1.21331 + 0.726558i 1.32070 − 0.505727i −0.296075 + 1.38287i −0.530822 − 1.31081i
0 0 0 −0.707107 + 0.707107i 0 1.73696i 0 0 0
721.2 0 0 0 −0.707107 + 0.707107i 0 0.982011i 0 0 0
721.3 0 0 0 −0.707107 + 0.707107i 0 0.690576i 0 0 0
721.4 0 0 0 −0.707107 + 0.707107i 0 4.02840i 0 0 0
721.5 0 0 0 0.707107 0.707107i 0 2.73482i 0 0 0
721.6 0 0 0 0.707107 0.707107i 0 2.66881i 0 0 0
721.7 0 0 0 0.707107 0.707107i 0 2.89402i 0 0 0
721.8 0 0 0 0.707107 0.707107i 0 4.50961i 0 0 0
2161.1 0 0 0 −0.707107 0.707107i 0 4.02840i 0 0 0
2161.2 0 0 0 −0.707107 0.707107i 0 0.690576i 0 0 0
2161.3 0 0 0 −0.707107 0.707107i 0 0.982011i 0 0 0
2161.4 0 0 0 −0.707107 0.707107i 0 1.73696i 0 0 0
2161.5 0 0 0 0.707107 + 0.707107i 0 4.50961i 0 0 0
2161.6 0 0 0 0.707107 + 0.707107i 0 2.89402i 0 0 0
2161.7 0 0 0 0.707107 + 0.707107i 0 2.66881i 0 0 0
2161.8 0 0 0 0.707107 + 0.707107i 0 2.73482i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2161.8 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.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.t.c 16
3.b odd 2 1 320.2.l.a 16
4.b odd 2 1 720.2.t.c 16
12.b even 2 1 80.2.l.a 16
15.d odd 2 1 1600.2.l.i 16
15.e even 4 1 1600.2.q.g 16
15.e even 4 1 1600.2.q.h 16
16.e even 4 1 inner 2880.2.t.c 16
16.f odd 4 1 720.2.t.c 16
24.f even 2 1 640.2.l.b 16
24.h odd 2 1 640.2.l.a 16
48.i odd 4 1 320.2.l.a 16
48.i odd 4 1 640.2.l.a 16
48.k even 4 1 80.2.l.a 16
48.k even 4 1 640.2.l.b 16
60.h even 2 1 400.2.l.h 16
60.l odd 4 1 400.2.q.g 16
60.l odd 4 1 400.2.q.h 16
96.o even 8 1 5120.2.a.s 8
96.o even 8 1 5120.2.a.v 8
96.p odd 8 1 5120.2.a.t 8
96.p odd 8 1 5120.2.a.u 8
240.t even 4 1 400.2.l.h 16
240.z odd 4 1 400.2.q.g 16
240.bb even 4 1 1600.2.q.h 16
240.bd odd 4 1 400.2.q.h 16
240.bf even 4 1 1600.2.q.g 16
240.bm odd 4 1 1600.2.l.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a 16 12.b even 2 1
80.2.l.a 16 48.k even 4 1
320.2.l.a 16 3.b odd 2 1
320.2.l.a 16 48.i odd 4 1
400.2.l.h 16 60.h even 2 1
400.2.l.h 16 240.t even 4 1
400.2.q.g 16 60.l odd 4 1
400.2.q.g 16 240.z odd 4 1
400.2.q.h 16 60.l odd 4 1
400.2.q.h 16 240.bd odd 4 1
640.2.l.a 16 24.h odd 2 1
640.2.l.a 16 48.i odd 4 1
640.2.l.b 16 24.f even 2 1
640.2.l.b 16 48.k even 4 1
720.2.t.c 16 4.b odd 2 1
720.2.t.c 16 16.f odd 4 1
1600.2.l.i 16 15.d odd 2 1
1600.2.l.i 16 240.bm odd 4 1
1600.2.q.g 16 15.e even 4 1
1600.2.q.g 16 240.bf even 4 1
1600.2.q.h 16 15.e even 4 1
1600.2.q.h 16 240.bb even 4 1
2880.2.t.c 16 1.a even 1 1 trivial
2880.2.t.c 16 16.e even 4 1 inner
5120.2.a.s 8 96.o even 8 1
5120.2.a.t 8 96.p odd 8 1
5120.2.a.u 8 96.p odd 8 1
5120.2.a.v 8 96.o even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(2880, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 1 + T^{4} )^{4}$$
$7$ $$204304 + 811008 T^{2} + 1033536 T^{4} + 549632 T^{6} + 145224 T^{8} + 20736 T^{10} + 1616 T^{12} + 64 T^{14} + T^{16}$$
$11$ $$1290496 + 799744 T + 247808 T^{2} + 848384 T^{3} + 3958016 T^{4} + 3673856 T^{5} + 1795584 T^{6} + 446848 T^{7} + 139616 T^{8} + 82368 T^{9} + 40320 T^{10} + 9568 T^{11} + 1232 T^{12} + 80 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$13$ $$20647936 + 46530560 T + 52428800 T^{2} + 27869184 T^{3} + 9146368 T^{4} + 3137536 T^{5} + 2654208 T^{6} + 1400832 T^{7} + 415872 T^{8} + 49152 T^{9} + 8192 T^{10} + 4352 T^{11} + 1600 T^{12} + 128 T^{13} + T^{16}$$
$17$ $$( 13888 + 5120 T - 7744 T^{2} - 1536 T^{3} + 1408 T^{4} + 64 T^{5} - 72 T^{6} + T^{8} )^{2}$$
$19$ $$614656 - 4164608 T + 14108672 T^{2} - 26513920 T^{3} + 30308608 T^{4} - 19398912 T^{5} + 7595520 T^{6} - 1921408 T^{7} + 731744 T^{8} - 363072 T^{9} + 132480 T^{10} - 27488 T^{11} + 3216 T^{12} - 176 T^{13} + 32 T^{14} - 8 T^{15} + T^{16}$$
$23$ $$1731856 + 5740288 T^{2} + 5719232 T^{4} + 2620928 T^{6} + 622088 T^{8} + 77504 T^{10} + 4784 T^{12} + 128 T^{14} + T^{16}$$
$29$ $$3017085184 + 4042700800 T + 2708480000 T^{2} + 456489984 T^{3} - 5714688 T^{4} - 1816064 T^{5} + 37230592 T^{6} + 2768128 T^{7} - 199840 T^{8} - 351616 T^{9} + 198144 T^{10} - 18624 T^{11} + 1104 T^{12} - 288 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$31$ $$( -20224 - 58368 T - 26112 T^{2} + 4096 T^{3} + 2848 T^{4} - 64 T^{5} - 96 T^{6} + T^{8} )^{2}$$
$37$ $$18939904 + 236191744 T + 1472724992 T^{2} + 2707357696 T^{3} + 2705047552 T^{4} + 1380728832 T^{5} + 381124608 T^{6} + 40185856 T^{7} + 22554112 T^{8} + 9370624 T^{9} + 2144256 T^{10} + 249088 T^{11} + 16320 T^{12} + 704 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$41$ $$110660014336 + 325786435584 T^{2} + 63304144128 T^{4} + 5024061440 T^{6} + 206041952 T^{8} + 4686848 T^{10} + 59088 T^{12} + 384 T^{14} + T^{16}$$
$43$ $$53640976 - 331044800 T + 1021520000 T^{2} - 1500385792 T^{3} + 1191507520 T^{4} - 280880992 T^{5} + 26364032 T^{6} + 1854880 T^{7} + 5444360 T^{8} - 950192 T^{9} + 76832 T^{10} + 18816 T^{11} + 5328 T^{12} - 440 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$47$ $$( 575044 + 693376 T + 244768 T^{2} + 584 T^{3} - 14936 T^{4} - 2392 T^{5} - 8 T^{6} + 20 T^{7} + T^{8} )^{2}$$
$53$ $$383725735936 + 196640112640 T + 50384076800 T^{2} + 10552958976 T^{3} + 26109784064 T^{4} + 13942091776 T^{5} + 3861454848 T^{6} + 498471936 T^{7} + 43316352 T^{8} + 6504448 T^{9} + 1892352 T^{10} + 236416 T^{11} + 15552 T^{12} + 448 T^{13} + 128 T^{14} + 16 T^{15} + T^{16}$$
$59$ $$12227051776 - 30451745792 T + 37920376832 T^{2} - 27384356352 T^{3} + 12226302208 T^{4} - 3038184192 T^{5} + 314344960 T^{6} + 20658560 T^{7} + 7403616 T^{8} - 2562240 T^{9} + 291200 T^{10} + 65632 T^{11} + 8464 T^{12} - 272 T^{13} + 32 T^{14} + 8 T^{15} + T^{16}$$
$61$ $$1393986371584 + 1092943347712 T + 428456542208 T^{2} - 95024578560 T^{3} + 27549499392 T^{4} + 9643622400 T^{5} + 2332164096 T^{6} - 240730112 T^{7} + 17876992 T^{8} + 4136960 T^{9} + 1425408 T^{10} - 176640 T^{11} + 11520 T^{12} + 384 T^{13} + 128 T^{14} - 16 T^{15} + T^{16}$$
$67$ $$46120451769616 + 26534918246592 T + 7633293466752 T^{2} + 2087109786752 T^{3} + 1043974444608 T^{4} + 475604544352 T^{5} + 148072661120 T^{6} + 31795927072 T^{7} + 4867387016 T^{8} + 537576688 T^{9} + 43098400 T^{10} + 2604960 T^{11} + 147664 T^{12} + 10680 T^{13} + 800 T^{14} + 40 T^{15} + T^{16}$$
$71$ $$3333516427264 + 2007385505792 T^{2} + 408856297472 T^{4} + 33695596544 T^{6} + 1144522752 T^{8} + 18817024 T^{10} + 157440 T^{12} + 640 T^{14} + T^{16}$$
$73$ $$15847788544 + 28362989568 T^{2} + 17757564928 T^{4} + 4377603072 T^{6} + 321195136 T^{8} + 9035648 T^{10} + 108992 T^{12} + 560 T^{14} + T^{16}$$
$79$ $$( 4352 - 31232 T - 61952 T^{2} + 4992 T^{3} + 5856 T^{4} - 352 T^{5} - 160 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$83$ $$2050640656 + 6041972416 T + 8900981888 T^{2} + 5315507456 T^{3} + 1230069056 T^{4} - 140621856 T^{5} + 1135671424 T^{6} + 379938272 T^{7} + 64272392 T^{8} - 9891472 T^{9} + 315168 T^{10} + 36928 T^{11} + 37520 T^{12} - 7976 T^{13} + 800 T^{14} - 40 T^{15} + T^{16}$$
$89$ $$684153962496 + 380947267584 T^{2} + 69045698560 T^{4} + 5734359040 T^{6} + 244188672 T^{8} + 5576192 T^{10} + 68032 T^{12} + 416 T^{14} + T^{16}$$
$97$ $$( -8549312 + 7621376 T - 1675968 T^{2} - 78720 T^{3} + 47936 T^{4} - 416 T^{5} - 440 T^{6} + T^{8} )^{2}$$