# Properties

 Label 288.2.r.b Level $288$ Weight $2$ Character orbit 288.r 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.r (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: $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{12} q^{3} + ( -\beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{9} - \beta_{14} ) q^{9} +O(q^{10})$$ $$q -\beta_{12} q^{3} + ( -\beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{9} - \beta_{14} ) q^{9} -\beta_{13} q^{11} + ( -\beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{13} + ( 2 - 2 \beta_{3} - \beta_{5} + \beta_{9} - \beta_{14} - \beta_{15} ) q^{15} + ( -1 - \beta_{14} - \beta_{15} ) q^{17} + ( -\beta_{4} + \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{11} ) q^{19} + ( -\beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{21} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{14} ) q^{23} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{14} - \beta_{15} ) q^{25} + ( -\beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{27} + ( 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{12} - \beta_{13} ) q^{29} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{14} ) q^{31} + ( -\beta_{3} - 2 \beta_{5} + \beta_{14} + \beta_{15} ) q^{33} + ( \beta_{7} - \beta_{10} - \beta_{12} ) q^{35} + ( \beta_{4} - \beta_{6} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{37} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{9} + \beta_{15} ) q^{39} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{9} + \beta_{14} + 2 \beta_{15} ) q^{41} + ( \beta_{6} + \beta_{7} ) q^{43} + ( 2 \beta_{4} - \beta_{8} - \beta_{10} + 3 \beta_{11} + \beta_{13} ) q^{45} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{9} ) q^{47} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{9} + \beta_{14} + \beta_{15} ) q^{49} + ( 2 \beta_{4} - \beta_{6} + 2 \beta_{8} - \beta_{10} + 2 \beta_{12} + \beta_{13} ) q^{51} + ( -\beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{12} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} + 2 \beta_{14} + \beta_{15} ) q^{55} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{14} - \beta_{15} ) q^{57} + ( \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{11} - \beta_{12} ) q^{59} + ( -2 \beta_{4} + \beta_{8} + 2 \beta_{12} + \beta_{13} ) q^{61} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{9} + \beta_{14} + \beta_{15} ) q^{63} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{9} ) q^{65} + ( 2 \beta_{4} + 2 \beta_{7} - 4 \beta_{8} + \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{67} + ( -\beta_{4} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{69} + ( -6 - \beta_{2} + \beta_{3} - \beta_{9} + 2 \beta_{14} + \beta_{15} ) q^{71} + ( -3 - 2 \beta_{1} + \beta_{14} + \beta_{15} ) q^{73} + ( 2 \beta_{4} + \beta_{6} + 3 \beta_{7} - 4 \beta_{8} - \beta_{10} + \beta_{12} + \beta_{13} ) q^{75} + ( \beta_{4} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{11} + 3 \beta_{12} ) q^{77} + ( 3 + 3 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{79} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + \beta_{9} - \beta_{15} ) q^{81} + ( -\beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{12} + \beta_{13} ) q^{83} + ( 3 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{85} + ( -5 + \beta_{1} - \beta_{2} + 5 \beta_{3} - 3 \beta_{5} - \beta_{14} ) q^{87} + ( 4 - \beta_{2} + \beta_{3} - \beta_{9} - \beta_{15} ) q^{89} + ( 3 \beta_{4} - 3 \beta_{6} + \beta_{7} + 2 \beta_{12} ) q^{91} + ( \beta_{4} - \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{93} + ( -6 \beta_{3} + 2 \beta_{9} - 2 \beta_{15} ) q^{95} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{9} - \beta_{14} + \beta_{15} ) q^{97} + ( -2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 6q^{7} + 2q^{9} + O(q^{10})$$ $$16q - 6q^{7} + 2q^{9} + 10q^{15} - 28q^{17} + 10q^{23} + 2q^{25} + 10q^{31} - 2q^{39} - 8q^{41} - 6q^{47} + 18q^{49} + 4q^{55} + 10q^{57} - 2q^{63} - 14q^{65} - 72q^{71} - 44q^{73} + 30q^{79} + 10q^{81} - 42q^{87} + 64q^{89} - 44q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{14} - \nu^{11} - 8 \nu^{8} - 12 \nu^{7} + 12 \nu^{6} + 4 \nu^{5} - 24 \nu^{3} + 16 \nu^{2} + 96 \nu + 96$$$$)/96$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 3 \nu^{13} - 6 \nu^{12} + 10 \nu^{11} - 4 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + 20 \nu^{7} - 40 \nu^{6} + 80 \nu^{5} - 176 \nu^{4} - 224 \nu^{3} - 384 \nu^{2} + 256 \nu - 512$$$$)/384$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 3 \nu^{13} - 4 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} - 8 \nu^{9} - 4 \nu^{8} + 12 \nu^{7} + 24 \nu^{6} + 32 \nu^{5} + 72 \nu^{4} + 96 \nu^{3} + 128 \nu^{2} + 96 \nu + 256$$$$)/192$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 24 \nu^{4} + 80 \nu^{3} + 32 \nu^{2} + 32 \nu + 64$$$$)/192$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 832 \nu + 512$$$$)/384$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{15} + \nu^{14} - 9 \nu^{13} - 6 \nu^{12} - 26 \nu^{11} - 12 \nu^{10} - 16 \nu^{9} + 16 \nu^{8} - 44 \nu^{7} + 40 \nu^{6} + 32 \nu^{5} + 240 \nu^{4} + 64 \nu^{3} + 256 \nu^{2} + 128 \nu + 896$$$$)/384$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{15} + 3 \nu^{14} + 13 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 32 \nu^{10} + 32 \nu^{9} + 24 \nu^{8} + 12 \nu^{7} - 56 \nu^{6} - 160 \nu^{5} - 320 \nu^{4} - 320 \nu^{3} - 576 \nu^{2} - 448 \nu - 768$$$$)/384$$ $$\beta_{8}$$ $$=$$ $$($$$$-7 \nu^{15} - \nu^{14} - 11 \nu^{13} - 12 \nu^{12} - 26 \nu^{11} - 4 \nu^{10} - 24 \nu^{9} + 56 \nu^{8} + 4 \nu^{7} + 112 \nu^{6} + 80 \nu^{5} + 352 \nu^{4} + 160 \nu^{3} + 320 \nu^{2} + 64 \nu + 896$$$$)/384$$ $$\beta_{9}$$ $$=$$ $$($$$$4 \nu^{15} + \nu^{14} + 5 \nu^{13} + 5 \nu^{12} + 24 \nu^{11} + 8 \nu^{10} + 24 \nu^{9} - 12 \nu^{8} + 24 \nu^{7} - 44 \nu^{6} - 72 \nu^{5} - 296 \nu^{4} - 256 \nu^{3} - 160 \nu^{2} - 32 \nu - 768$$$$)/192$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{15} - 3 \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + 18 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} + 24 \nu^{8} + 56 \nu^{7} - 4 \nu^{6} - 72 \nu^{5} - 160 \nu^{4} - 208 \nu^{3} - 288 \nu^{2} - 448 \nu - 960$$$$)/192$$ $$\beta_{11}$$ $$=$$ $$($$$$-2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 6 \nu^{12} - 11 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 8 \nu^{8} - 4 \nu^{7} + 44 \nu^{6} + 44 \nu^{5} + 128 \nu^{4} + 104 \nu^{3} + 176 \nu^{2} + 32 \nu + 352$$$$)/96$$ $$\beta_{12}$$ $$=$$ $$($$$$-9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + 40 \nu^{8} - 52 \nu^{7} + 48 \nu^{6} + 96 \nu^{5} + 384 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} + 64 \nu + 1664$$$$)/384$$ $$\beta_{13}$$ $$=$$ $$($$$$3 \nu^{15} - 5 \nu^{14} - \nu^{13} + 6 \nu^{12} + 14 \nu^{11} + 10 \nu^{10} + 12 \nu^{9} + 4 \nu^{8} + 44 \nu^{7} + 24 \nu^{6} - 8 \nu^{5} - 184 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 160 \nu - 960$$$$)/192$$ $$\beta_{14}$$ $$=$$ $$($$$$11 \nu^{15} - \nu^{14} + 13 \nu^{13} + 18 \nu^{12} + 62 \nu^{11} + 12 \nu^{10} + 80 \nu^{9} + 16 \nu^{8} + 92 \nu^{7} - 104 \nu^{6} - 80 \nu^{5} - 720 \nu^{4} - 352 \nu^{3} - 768 \nu^{2} - 384 \nu - 2560$$$$)/384$$ $$\beta_{15}$$ $$=$$ $$($$$$-15 \nu^{15} - 5 \nu^{14} - 15 \nu^{13} - 20 \nu^{12} - 62 \nu^{11} - 16 \nu^{10} - 24 \nu^{9} + 32 \nu^{8} - 28 \nu^{7} + 160 \nu^{6} + 80 \nu^{5} + 784 \nu^{4} + 480 \nu^{3} + 768 \nu^{2} - 128 \nu + 2560$$$$)/384$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + \beta_{8} + \beta_{5}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{12} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} - 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{13} + 3 \beta_{12} - 2 \beta_{9} - 4 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 6 \beta_{1} + 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{13} - \beta_{12} - 4 \beta_{11} + 3 \beta_{10} + \beta_{9} + 4 \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 18 \beta_{3}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-3 \beta_{15} + 3 \beta_{14} - \beta_{13} + 3 \beta_{12} + 8 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} - 3 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 7$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$12 \beta_{15} + 13 \beta_{14} - 2 \beta_{12} - 8 \beta_{11} - 3 \beta_{10} - \beta_{9} - 5 \beta_{7} + 7 \beta_{6} - 7 \beta_{4} + \beta_{3} - \beta_{2} + 6 \beta_{1} - 7$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$3 \beta_{15} - 4 \beta_{14} + 13 \beta_{13} - \beta_{12} - 16 \beta_{11} - 13 \beta_{10} - 3 \beta_{9} + 16 \beta_{8} - \beta_{7} - 13 \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + 2 \beta_{3} - 4 \beta_{2}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} - 5 \beta_{13} - 13 \beta_{12} + 4 \beta_{9} + 24 \beta_{8} - 26 \beta_{7} - 26 \beta_{6} - 2 \beta_{5} + 13 \beta_{4} - 51 \beta_{3} - 3 \beta_{2} + 2 \beta_{1} + 47$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-12 \beta_{15} + 5 \beta_{14} + 30 \beta_{12} - 24 \beta_{11} + 21 \beta_{10} - 17 \beta_{9} - 21 \beta_{7} - 9 \beta_{6} + 9 \beta_{4} + 17 \beta_{3} - 17 \beta_{2} + 30 \beta_{1} + 9$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$3 \beta_{15} - 4 \beta_{14} - 19 \beta_{13} + 15 \beta_{12} + 24 \beta_{11} + 19 \beta_{10} - 3 \beta_{9} - 24 \beta_{8} + 15 \beta_{7} - 61 \beta_{6} + 34 \beta_{5} - 46 \beta_{4} - 14 \beta_{3} - 4 \beta_{2}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-31 \beta_{15} + 31 \beta_{14} - 53 \beta_{13} - 5 \beta_{12} - 4 \beta_{9} + 40 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 66 \beta_{5} + 5 \beta_{4} + 125 \beta_{3} - 27 \beta_{2} + 66 \beta_{1} - 121$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-12 \beta_{15} + 29 \beta_{14} - 98 \beta_{12} - 8 \beta_{11} - 3 \beta_{10} - 41 \beta_{9} - 29 \beta_{7} + 127 \beta_{6} - 127 \beta_{4} + 41 \beta_{3} - 41 \beta_{2} - 18 \beta_{1} + 145$$$$)/2$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$

## 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
 0.820200 − 1.15207i 1.05026 + 0.947078i −1.12494 + 0.857038i 1.41411 + 0.0174668i −0.722180 − 1.21592i −0.179748 + 1.40274i −1.34532 − 0.436011i 0.587625 − 1.28635i 0.820200 + 1.15207i 1.05026 − 0.947078i −1.12494 − 0.857038i 1.41411 − 0.0174668i −0.722180 + 1.21592i −0.179748 − 1.40274i −1.34532 + 0.436011i 0.587625 + 1.28635i
0 −1.69028 + 0.378078i 0 −1.97542 1.14051i 0 0.907824 + 1.57240i 0 2.71411 1.27812i 0
49.2 0 −1.52768 0.816201i 0 −0.602794 0.348023i 0 −0.795065 1.37709i 0 1.66763 + 2.49379i 0
49.3 0 −0.986088 + 1.42395i 0 −1.19115 0.687709i 0 −1.80469 3.12581i 0 −1.05526 2.80828i 0
49.4 0 −0.294546 + 1.70682i 0 3.17262 + 1.83171i 0 0.191926 + 0.332426i 0 −2.82649 1.00547i 0
49.5 0 0.294546 1.70682i 0 −3.17262 1.83171i 0 0.191926 + 0.332426i 0 −2.82649 1.00547i 0
49.6 0 0.986088 1.42395i 0 1.19115 + 0.687709i 0 −1.80469 3.12581i 0 −1.05526 2.80828i 0
49.7 0 1.52768 + 0.816201i 0 0.602794 + 0.348023i 0 −0.795065 1.37709i 0 1.66763 + 2.49379i 0
49.8 0 1.69028 0.378078i 0 1.97542 + 1.14051i 0 0.907824 + 1.57240i 0 2.71411 1.27812i 0
241.1 0 −1.69028 0.378078i 0 −1.97542 + 1.14051i 0 0.907824 1.57240i 0 2.71411 + 1.27812i 0
241.2 0 −1.52768 + 0.816201i 0 −0.602794 + 0.348023i 0 −0.795065 + 1.37709i 0 1.66763 2.49379i 0
241.3 0 −0.986088 1.42395i 0 −1.19115 + 0.687709i 0 −1.80469 + 3.12581i 0 −1.05526 + 2.80828i 0
241.4 0 −0.294546 1.70682i 0 3.17262 1.83171i 0 0.191926 0.332426i 0 −2.82649 + 1.00547i 0
241.5 0 0.294546 + 1.70682i 0 −3.17262 + 1.83171i 0 0.191926 0.332426i 0 −2.82649 + 1.00547i 0
241.6 0 0.986088 + 1.42395i 0 1.19115 0.687709i 0 −1.80469 + 3.12581i 0 −1.05526 + 2.80828i 0
241.7 0 1.52768 0.816201i 0 0.602794 0.348023i 0 −0.795065 + 1.37709i 0 1.66763 2.49379i 0
241.8 0 1.69028 + 0.378078i 0 1.97542 1.14051i 0 0.907824 1.57240i 0 2.71411 + 1.27812i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.8 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.b even 2 1 inner
9.c even 3 1 inner
72.n 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.r.b 16
3.b odd 2 1 864.2.r.b 16
4.b odd 2 1 72.2.n.b 16
8.b even 2 1 inner 288.2.r.b 16
8.d odd 2 1 72.2.n.b 16
9.c even 3 1 inner 288.2.r.b 16
9.c even 3 1 2592.2.d.j 8
9.d odd 6 1 864.2.r.b 16
9.d odd 6 1 2592.2.d.k 8
12.b even 2 1 216.2.n.b 16
24.f even 2 1 216.2.n.b 16
24.h odd 2 1 864.2.r.b 16
36.f odd 6 1 72.2.n.b 16
36.f odd 6 1 648.2.d.j 8
36.h even 6 1 216.2.n.b 16
36.h even 6 1 648.2.d.k 8
72.j odd 6 1 864.2.r.b 16
72.j odd 6 1 2592.2.d.k 8
72.l even 6 1 216.2.n.b 16
72.l even 6 1 648.2.d.k 8
72.n even 6 1 inner 288.2.r.b 16
72.n even 6 1 2592.2.d.j 8
72.p odd 6 1 72.2.n.b 16
72.p odd 6 1 648.2.d.j 8

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

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$6561 - 729 T^{2} - 162 T^{4} + 81 T^{6} + 18 T^{8} + 9 T^{10} - 2 T^{12} - T^{14} + T^{16}$$
$5$ $$4096 - 11712 T^{2} + 26129 T^{4} - 18357 T^{6} + 9318 T^{8} - 2049 T^{10} + 326 T^{12} - 21 T^{14} + T^{16}$$
$7$ $$( 16 - 36 T + 101 T^{2} + 21 T^{3} + 48 T^{4} + 3 T^{5} + 14 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$11$ $$10556001 - 6822900 T^{2} + 2895966 T^{4} - 718680 T^{6} + 129907 T^{8} - 14440 T^{10} + 1134 T^{12} - 40 T^{14} + T^{16}$$
$13$ $$20736 - 119664 T^{2} + 587601 T^{4} - 578901 T^{6} + 467038 T^{8} - 36233 T^{10} + 2094 T^{12} - 53 T^{14} + T^{16}$$
$17$ $$( 36 - 48 T - 2 T^{2} + 7 T^{3} + T^{4} )^{4}$$
$19$ $$( 5184 + 7056 T^{2} + 1660 T^{4} + 83 T^{6} + T^{8} )^{2}$$
$23$ $$( 90000 - 67500 T + 40125 T^{2} - 10875 T^{3} + 2650 T^{4} - 275 T^{5} + 60 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$29$ $$4032758016 - 1453416048 T^{2} + 356481729 T^{4} - 46463373 T^{6} + 4385038 T^{8} - 241441 T^{10} + 9246 T^{12} - 109 T^{14} + T^{16}$$
$31$ $$( 92416 + 62320 T + 57529 T^{2} - 7415 T^{3} + 3322 T^{4} - 155 T^{5} + 76 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$37$ $$( 1327104 + 177264 T^{2} + 8080 T^{4} + 152 T^{6} + T^{8} )^{2}$$
$41$ $$( 335241 + 264024 T + 165090 T^{2} + 38376 T^{3} + 7879 T^{4} + 616 T^{5} + 90 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$43$ $$81 - 1080 T^{2} + 13446 T^{4} - 12360 T^{6} + 8827 T^{8} - 1880 T^{10} + 294 T^{12} - 20 T^{14} + T^{16}$$
$47$ $$( 2178576 - 225828 T + 142965 T^{2} + 3537 T^{3} + 5544 T^{4} + 63 T^{5} + 90 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$53$ $$( 451584 + 113520 T^{2} + 7744 T^{4} + 160 T^{6} + T^{8} )^{2}$$
$59$ $$131079601 - 286866144 T^{2} + 591967766 T^{4} - 75952296 T^{6} + 7079403 T^{8} - 287928 T^{10} + 8534 T^{12} - 108 T^{14} + T^{16}$$
$61$ $$176319369216 - 57650719680 T^{2} + 15331541409 T^{4} - 1011826485 T^{6} + 47134062 T^{8} - 1107945 T^{10} + 18846 T^{12} - 165 T^{14} + T^{16}$$
$67$ $$9881774573841 - 1672043075100 T^{2} + 223335161334 T^{4} - 8572738680 T^{6} + 228454587 T^{8} - 3485160 T^{10} + 38646 T^{12} - 240 T^{14} + T^{16}$$
$71$ $$( -864 - 252 T + 48 T^{2} + 18 T^{3} + T^{4} )^{4}$$
$73$ $$( 36 - 84 T - 14 T^{2} + 11 T^{3} + T^{4} )^{4}$$
$79$ $$( 3136 - 34440 T + 377609 T^{2} - 8445 T^{3} + 9402 T^{4} - 1065 T^{5} + 236 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$83$ $$31713911056 - 8048506380 T^{2} + 1398102029 T^{4} - 126163065 T^{6} + 8173602 T^{8} - 289605 T^{10} + 7406 T^{12} - 105 T^{14} + T^{16}$$
$89$ $$( -504 + 156 T + 52 T^{2} - 16 T^{3} + T^{4} )^{4}$$
$97$ $$( 1324801 + 897780 T + 454166 T^{2} + 104520 T^{3} + 19107 T^{4} + 1560 T^{5} + 134 T^{6} + T^{8} )^{2}$$