# Properties

 Label 288.2.p.b Level $288$ Weight $2$ Character orbit 288.p Analytic conductor $2.300$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} - \beta_{8} ) q^{3} -\beta_{9} q^{5} -\beta_{4} q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} - \beta_{8} ) q^{3} -\beta_{9} q^{5} -\beta_{4} q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{10} ) q^{9} + ( -2 - \beta_{2} - \beta_{3} - \beta_{10} ) q^{11} -\beta_{13} q^{13} + ( -\beta_{6} - \beta_{11} + \beta_{14} ) q^{15} + ( -\beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{10} ) q^{19} + ( -2 \beta_{4} - \beta_{6} + \beta_{9} - 2 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{21} + ( -\beta_{9} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{23} + ( \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} + \beta_{8} - 3 \beta_{10} ) q^{27} + ( \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{29} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{33} + ( 3 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{35} + ( \beta_{4} + \beta_{6} + \beta_{9} + 2 \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{37} + ( -\beta_{6} + \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{39} + ( -2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{10} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + 4 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{43} + ( \beta_{4} - \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{45} + ( \beta_{4} + 2 \beta_{11} - \beta_{14} + \beta_{15} ) q^{47} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{49} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{10} ) q^{51} + ( 2 \beta_{4} + \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{53} + ( \beta_{4} + \beta_{6} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{55} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{57} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{59} + ( -2 \beta_{4} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{61} + ( -\beta_{4} - \beta_{6} + 2 \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{63} + ( 2 + 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + 4 \beta_{8} ) q^{65} + ( 5 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 3 \beta_{7} - \beta_{8} + 3 \beta_{10} ) q^{67} + ( 2 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{69} + ( \beta_{4} + \beta_{6} - \beta_{9} + \beta_{13} + \beta_{14} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{8} ) q^{73} + ( -7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 5 \beta_{7} + \beta_{8} + \beta_{10} ) q^{75} + ( \beta_{4} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{4} - \beta_{6} + 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{79} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} ) q^{81} + ( -2 - 5 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{8} + 5 \beta_{10} ) q^{83} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{85} + ( \beta_{4} + 2 \beta_{6} + \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{87} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} + \beta_{7} + \beta_{8} ) q^{89} + ( 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{7} + \beta_{8} - 8 \beta_{10} ) q^{91} + ( \beta_{4} + \beta_{6} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{93} + ( -\beta_{4} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} + ( -3 \beta_{1} + \beta_{2} - \beta_{3} - 5 \beta_{5} + 6 \beta_{7} + 5 \beta_{8} + 3 \beta_{10} ) q^{97} + ( 5 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 6q^{3} - 6q^{9} + O(q^{10})$$ $$16q + 6q^{3} - 6q^{9} - 12q^{11} + 4q^{19} - 14q^{25} + 36q^{27} + 12q^{33} - 36q^{41} - 8q^{43} + 10q^{49} - 18q^{51} + 18q^{57} - 12q^{59} - 6q^{65} + 16q^{67} - 4q^{73} - 78q^{75} - 6q^{81} - 54q^{83} + 36q^{91} + 8q^{97} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 384$$$$)/896$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 25 \nu^{13} + 38 \nu^{12} - 46 \nu^{11} + 24 \nu^{10} - 8 \nu^{9} - 68 \nu^{8} + 244 \nu^{7} - 272 \nu^{6} + 8 \nu^{5} + 128 \nu^{4} - 416 \nu^{3} + 1184 \nu^{2} - 1088 \nu + 512$$$$)/896$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{15} - 25 \nu^{14} + 33 \nu^{13} - 44 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 172 \nu^{7} - 24 \nu^{6} + 480 \nu^{5} - 720 \nu^{4} + 1024 \nu^{3} - 1088 \nu^{2} + 128 \nu - 640$$$$)/896$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - 29 \nu^{14} + 59 \nu^{13} - 102 \nu^{12} + 66 \nu^{11} + 24 \nu^{10} - 176 \nu^{9} + 492 \nu^{8} - 540 \nu^{7} + 288 \nu^{6} + 456 \nu^{5} - 1216 \nu^{4} + 2496 \nu^{3} - 2400 \nu^{2} + 1600 \nu - 384$$$$)/896$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{15} + 11 \nu^{14} - 19 \nu^{13} + 16 \nu^{12} + 2 \nu^{11} - 54 \nu^{10} + 116 \nu^{9} - 92 \nu^{8} + 4 \nu^{7} + 192 \nu^{6} - 424 \nu^{5} + 440 \nu^{4} - 352 \nu^{3} - 256 \nu^{2} + 544 \nu - 704$$$$)/448$$ $$\beta_{6}$$ $$=$$ $$($$$$6 \nu^{15} - 8 \nu^{14} + 3 \nu^{13} + 17 \nu^{12} - 39 \nu^{11} + 80 \nu^{10} - 92 \nu^{9} + 16 \nu^{8} + 272 \nu^{7} - 300 \nu^{6} + 260 \nu^{5} - 152 \nu^{4} - 304 \nu^{3} + 1072 \nu^{2} - 1312 \nu + 64$$$$)/448$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{15} + 15 \nu^{14} - 31 \nu^{13} + 4 \nu^{12} + 32 \nu^{11} - 136 \nu^{10} + 176 \nu^{9} - 100 \nu^{8} - 76 \nu^{7} + 552 \nu^{6} - 736 \nu^{5} + 880 \nu^{4} - 256 \nu^{3} - 512 \nu^{2} + 1536 \nu - 1408$$$$)/896$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} + 18 \nu^{14} - 47 \nu^{13} + 100 \nu^{12} - 117 \nu^{11} + 100 \nu^{10} + 60 \nu^{9} - 316 \nu^{8} + 592 \nu^{7} - 648 \nu^{6} + 52 \nu^{5} + 888 \nu^{4} - 2144 \nu^{3} + 2768 \nu^{2} - 3040 \nu + 1536$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$6 \nu^{15} - 15 \nu^{14} + 24 \nu^{13} - 32 \nu^{12} + 45 \nu^{11} + 24 \nu^{10} - 64 \nu^{9} + 128 \nu^{8} - 92 \nu^{7} + 176 \nu^{6} - 20 \nu^{5} - 600 \nu^{4} + 592 \nu^{3} - 944 \nu^{2} + 928 \nu - 1728$$$$)/448$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + 1184 \nu - 768$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$-19 \nu^{15} + 37 \nu^{14} - 13 \nu^{13} - 48 \nu^{12} + 232 \nu^{11} - 384 \nu^{10} + 464 \nu^{9} - 60 \nu^{8} - 740 \nu^{7} + 1608 \nu^{6} - 1696 \nu^{5} + 752 \nu^{4} + 1280 \nu^{3} - 4608 \nu^{2} + 5760 \nu - 4608$$$$)/896$$ $$\beta_{12}$$ $$=$$ $$($$$$-13 \nu^{15} + 29 \nu^{14} - 59 \nu^{13} + 116 \nu^{12} - 150 \nu^{11} + 88 \nu^{10} + 120 \nu^{9} - 324 \nu^{8} + 652 \nu^{7} - 680 \nu^{6} + 104 \nu^{5} + 1216 \nu^{4} - 2720 \nu^{3} + 3968 \nu^{2} - 4288 \nu + 3072$$$$)/448$$ $$\beta_{13}$$ $$=$$ $$($$$$-16 \nu^{15} + 40 \nu^{14} - 99 \nu^{13} + 153 \nu^{12} - 113 \nu^{11} - 36 \nu^{10} + 376 \nu^{9} - 640 \nu^{8} + 768 \nu^{7} - 124 \nu^{6} - 964 \nu^{5} + 2440 \nu^{4} - 3072 \nu^{3} + 3152 \nu^{2} - 2400 \nu - 320$$$$)/448$$ $$\beta_{14}$$ $$=$$ $$($$$$-37 \nu^{15} + 103 \nu^{14} - 183 \nu^{13} + 244 \nu^{12} - 232 \nu^{11} - 64 \nu^{10} + 712 \nu^{9} - 1172 \nu^{8} + 1300 \nu^{7} - 600 \nu^{6} - 1216 \nu^{5} + 4176 \nu^{4} - 6432 \nu^{3} + 6848 \nu^{2} - 3968 \nu + 2816$$$$)/896$$ $$\beta_{15}$$ $$=$$ $$($$$$-57 \nu^{15} + 167 \nu^{14} - 375 \nu^{13} + 528 \nu^{12} - 536 \nu^{11} + 24 \nu^{10} + 1112 \nu^{9} - 2308 \nu^{8} + 2932 \nu^{7} - 1448 \nu^{6} - 1952 \nu^{5} + 7856 \nu^{4} - 12512 \nu^{3} + 14400 \nu^{2} - 12288 \nu + 5888$$$$)/896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} - 2 \beta_{10}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{8} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 4 \beta_{1} - 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_{1}$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 3 \beta_{11} - 10 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} - \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 4 \beta_{1} - 10$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{15} - 6 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - 12 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 2$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{15} + \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 14 \beta_{6} + 2 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 18 \beta_{1} + 28$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{15} + 5 \beta_{14} - \beta_{13} + 5 \beta_{12} - \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 16 \beta_{5} - \beta_{4} + 18 \beta_{3} + 2 \beta_{2} - 16 \beta_{1} + 2$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$12 \beta_{14} + 9 \beta_{13} + 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{9} - 28 \beta_{8} - 50 \beta_{7} - 6 \beta_{6} + 18 \beta_{5} + 3 \beta_{4} + 18 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} + 10$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-\beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} + 12 \beta_{9} + 18 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + 28 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} - 58 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-3 \beta_{15} + 11 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} + 26 \beta_{10} + 2 \beta_{9} + 26 \beta_{8} + 26 \beta_{7} - 76 \beta_{5} - 21 \beta_{4} + 50 \beta_{3} + 26 \beta_{2} - 84 \beta_{1} + 34$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-38 \beta_{15} - 30 \beta_{14} + 19 \beta_{13} + 49 \beta_{12} + 39 \beta_{11} - 108 \beta_{10} - 14 \beta_{9} - 8 \beta_{8} - 54 \beta_{7} - 14 \beta_{6} + 2 \beta_{5} - 19 \beta_{4} - 2 \beta_{3} - 10 \beta_{2} + 2 \beta_{1} - 74$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-41 \beta_{15} + 5 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} - 6 \beta_{8} + 22 \beta_{6} - 6 \beta_{5} - 10 \beta_{4} - 108 \beta_{3} - 108 \beta_{2} + 130 \beta_{1} + 44$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$7 \beta_{15} - 7 \beta_{14} - 77 \beta_{13} - 7 \beta_{12} + 91 \beta_{11} - 34 \beta_{10} - 86 \beta_{9} + 102 \beta_{8} + 34 \beta_{7} + 172 \beta_{6} + 8 \beta_{5} + 91 \beta_{4} - 94 \beta_{3} - 102 \beta_{2} - 96 \beta_{1} - 190$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$12 \beta_{14} + 29 \beta_{13} - 17 \beta_{12} - 79 \beta_{11} - 130 \beta_{9} - 156 \beta_{8} - 218 \beta_{7} + 130 \beta_{6} + 202 \beta_{5} - 129 \beta_{4} + 202 \beta_{3} - 46 \beta_{2} - 326 \beta_{1} - 62$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.533474 + 1.30973i 0.867527 − 1.11687i 1.12063 − 0.862658i −0.186766 + 1.40183i −0.409484 − 1.35363i −1.37702 + 0.322193i 0.608741 + 1.27649i 1.40985 − 0.111062i −0.533474 − 1.30973i 0.867527 + 1.11687i 1.12063 + 0.862658i −0.186766 − 1.40183i −0.409484 + 1.35363i −1.37702 − 0.322193i 0.608741 − 1.27649i 1.40985 + 0.111062i
0 −0.925606 1.46399i 0 −0.895377 1.55084i 0 −2.08793 1.20546i 0 −1.28651 + 2.71015i 0
47.2 0 −0.925606 1.46399i 0 0.895377 + 1.55084i 0 2.08793 + 1.20546i 0 −1.28651 + 2.71015i 0
47.3 0 −0.418594 + 1.68071i 0 −1.60936 2.78750i 0 −1.82223 1.05206i 0 −2.64956 1.40707i 0
47.4 0 −0.418594 + 1.68071i 0 1.60936 + 2.78750i 0 1.82223 + 1.05206i 0 −2.64956 1.40707i 0
47.5 0 1.12774 1.31461i 0 −0.565188 0.978934i 0 −3.71499 2.14485i 0 −0.456412 2.96508i 0
47.6 0 1.12774 1.31461i 0 0.565188 + 0.978934i 0 3.71499 + 2.14485i 0 −0.456412 2.96508i 0
47.7 0 1.71646 + 0.231865i 0 −1.74322 3.01934i 0 1.80802 + 1.04386i 0 2.89248 + 0.795973i 0
47.8 0 1.71646 + 0.231865i 0 1.74322 + 3.01934i 0 −1.80802 1.04386i 0 2.89248 + 0.795973i 0
239.1 0 −0.925606 + 1.46399i 0 −0.895377 + 1.55084i 0 −2.08793 + 1.20546i 0 −1.28651 2.71015i 0
239.2 0 −0.925606 + 1.46399i 0 0.895377 1.55084i 0 2.08793 1.20546i 0 −1.28651 2.71015i 0
239.3 0 −0.418594 1.68071i 0 −1.60936 + 2.78750i 0 −1.82223 + 1.05206i 0 −2.64956 + 1.40707i 0
239.4 0 −0.418594 1.68071i 0 1.60936 2.78750i 0 1.82223 1.05206i 0 −2.64956 + 1.40707i 0
239.5 0 1.12774 + 1.31461i 0 −0.565188 + 0.978934i 0 −3.71499 + 2.14485i 0 −0.456412 + 2.96508i 0
239.6 0 1.12774 + 1.31461i 0 0.565188 0.978934i 0 3.71499 2.14485i 0 −0.456412 + 2.96508i 0
239.7 0 1.71646 0.231865i 0 −1.74322 + 3.01934i 0 1.80802 1.04386i 0 2.89248 0.795973i 0
239.8 0 1.71646 0.231865i 0 1.74322 3.01934i 0 −1.80802 + 1.04386i 0 2.89248 0.795973i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.p.b 16
3.b odd 2 1 864.2.p.b 16
4.b odd 2 1 72.2.l.b 16
8.b even 2 1 72.2.l.b 16
8.d odd 2 1 inner 288.2.p.b 16
9.c even 3 1 864.2.p.b 16
9.c even 3 1 2592.2.f.b 16
9.d odd 6 1 inner 288.2.p.b 16
9.d odd 6 1 2592.2.f.b 16
12.b even 2 1 216.2.l.b 16
24.f even 2 1 864.2.p.b 16
24.h odd 2 1 216.2.l.b 16
36.f odd 6 1 216.2.l.b 16
36.f odd 6 1 648.2.f.b 16
36.h even 6 1 72.2.l.b 16
36.h even 6 1 648.2.f.b 16
72.j odd 6 1 72.2.l.b 16
72.j odd 6 1 648.2.f.b 16
72.l even 6 1 inner 288.2.p.b 16
72.l even 6 1 2592.2.f.b 16
72.n even 6 1 216.2.l.b 16
72.n even 6 1 648.2.f.b 16
72.p odd 6 1 864.2.p.b 16
72.p odd 6 1 2592.2.f.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.b 16 4.b odd 2 1
72.2.l.b 16 8.b even 2 1
72.2.l.b 16 36.h even 6 1
72.2.l.b 16 72.j odd 6 1
216.2.l.b 16 12.b even 2 1
216.2.l.b 16 24.h odd 2 1
216.2.l.b 16 36.f odd 6 1
216.2.l.b 16 72.n even 6 1
288.2.p.b 16 1.a even 1 1 trivial
288.2.p.b 16 8.d odd 2 1 inner
288.2.p.b 16 9.d odd 6 1 inner
288.2.p.b 16 72.l even 6 1 inner
648.2.f.b 16 36.f odd 6 1
648.2.f.b 16 36.h even 6 1
648.2.f.b 16 72.j odd 6 1
648.2.f.b 16 72.n even 6 1
864.2.p.b 16 3.b odd 2 1
864.2.p.b 16 9.c even 3 1
864.2.p.b 16 24.f even 2 1
864.2.p.b 16 72.p odd 6 1
2592.2.f.b 16 9.c even 3 1
2592.2.f.b 16 9.d odd 6 1
2592.2.f.b 16 72.l even 6 1
2592.2.f.b 16 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 81 - 81 T + 54 T^{2} - 45 T^{3} + 30 T^{4} - 15 T^{5} + 6 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$5$ $$266256 + 339012 T^{2} + 312453 T^{4} + 123903 T^{6} + 35106 T^{8} + 4923 T^{10} + 498 T^{12} + 27 T^{14} + T^{16}$$
$7$ $$4260096 - 2904048 T^{2} + 1279953 T^{4} - 340749 T^{6} + 66426 T^{8} - 8373 T^{10} + 750 T^{12} - 33 T^{14} + T^{16}$$
$11$ $$( 1 - 12 T + 44 T^{2} + 48 T^{3} - 9 T^{4} - 24 T^{5} + 8 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$13$ $$266256 - 1772460 T^{2} + 11382813 T^{4} - 2713221 T^{6} + 454938 T^{8} - 39129 T^{10} + 2442 T^{12} - 57 T^{14} + T^{16}$$
$17$ $$( 784 + 992 T^{2} + 360 T^{4} + 35 T^{6} + T^{8} )^{2}$$
$19$ $$( 16 + 8 T - 12 T^{2} - T^{3} + T^{4} )^{4}$$
$23$ $$639280656 + 454125924 T^{2} + 255771909 T^{4} + 42464691 T^{6} + 5182026 T^{8} + 225735 T^{10} + 7158 T^{12} + 99 T^{14} + T^{16}$$
$29$ $$17449353216 + 10074829824 T^{2} + 5081053545 T^{4} + 389228679 T^{6} + 20607630 T^{8} + 599547 T^{10} + 12654 T^{12} + 135 T^{14} + T^{16}$$
$31$ $$279189651456 - 45038923776 T^{2} + 4696155729 T^{4} - 290875401 T^{6} + 13147422 T^{8} - 398493 T^{10} + 8826 T^{12} - 117 T^{14} + T^{16}$$
$37$ $$( 74304 + 40176 T^{2} + 4896 T^{4} + 156 T^{6} + T^{8} )^{2}$$
$41$ $$( 7921 + 8010 T + 920 T^{2} - 1800 T^{3} - 51 T^{4} + 360 T^{5} + 128 T^{6} + 18 T^{7} + T^{8} )^{2}$$
$43$ $$( 6889 - 11786 T + 15184 T^{2} - 7856 T^{3} + 3115 T^{4} - 524 T^{5} + 76 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$47$ $$266256 + 1701252 T^{2} + 10094661 T^{4} + 4840839 T^{6} + 1892526 T^{8} + 160239 T^{10} + 10818 T^{12} + 111 T^{14} + T^{16}$$
$53$ $$( 297216 - 331344 T^{2} + 15408 T^{4} - 228 T^{6} + T^{8} )^{2}$$
$59$ $$( 528529 - 165756 T - 33562 T^{2} + 15960 T^{3} + 3717 T^{4} - 420 T^{5} - 58 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$61$ $$1192149524736 - 215024661360 T^{2} + 26804642049 T^{4} - 1747852317 T^{6} + 82050270 T^{8} - 1679649 T^{10} + 24750 T^{12} - 189 T^{14} + T^{16}$$
$67$ $$( 582169 + 209062 T + 125434 T^{2} - 5876 T^{3} + 5785 T^{4} - 20 T^{5} + 130 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$71$ $$( 74304 - 28944 T^{2} + 3744 T^{4} - 168 T^{6} + T^{8} )^{2}$$
$73$ $$( 172 - 224 T - 78 T^{2} + T^{3} + T^{4} )^{4}$$
$79$ $$74509345296 - 36110680524 T^{2} + 13340118429 T^{4} - 1880575641 T^{6} + 199135626 T^{8} - 3530925 T^{10} + 46758 T^{12} - 249 T^{14} + T^{16}$$
$83$ $$( 432964 - 487578 T + 162629 T^{2} + 22971 T^{3} - 6366 T^{4} - 837 T^{5} + 212 T^{6} + 27 T^{7} + T^{8} )^{2}$$
$89$ $$( 891136 + 689456 T^{2} + 23808 T^{4} + 272 T^{6} + T^{8} )^{2}$$
$97$ $$( 1018081 + 904064 T + 978382 T^{2} - 147832 T^{3} + 32851 T^{4} - 1096 T^{5} + 190 T^{6} - 4 T^{7} + T^{8} )^{2}$$