# Properties

 Label 882.2.l.a Level $882$ Weight $2$ Character orbit 882.l Analytic conductor $7.043$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{6} q^{3} - q^{4} + ( -\beta_{1} + \beta_{6} + \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{5} + ( -\beta_{12} - \beta_{13} ) q^{6} -\beta_{4} q^{8} + ( -\beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + \beta_{6} q^{3} - q^{4} + ( -\beta_{1} + \beta_{6} + \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{5} + ( -\beta_{12} - \beta_{13} ) q^{6} -\beta_{4} q^{8} + ( -\beta_{2} + \beta_{3} ) q^{9} + ( \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{13} ) q^{10} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} + \beta_{10} ) q^{11} -\beta_{6} q^{12} + ( \beta_{6} + \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{13} + ( -\beta_{2} + \beta_{4} - 2 \beta_{8} + \beta_{9} + \beta_{15} ) q^{15} + q^{16} + ( -\beta_{1} + \beta_{6} + \beta_{7} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{17} + ( -\beta_{4} + \beta_{9} - \beta_{15} ) q^{18} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{19} + ( \beta_{1} - \beta_{6} - \beta_{7} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{20} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} + \beta_{15} ) q^{22} + ( 4 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{23} + ( \beta_{12} + \beta_{13} ) q^{24} + ( -\beta_{2} - 4 \beta_{4} + 2 \beta_{8} + \beta_{15} ) q^{25} + ( \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{26} + ( -\beta_{1} - \beta_{6} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{27} + ( -1 - \beta_{2} + 2 \beta_{4} - 2 \beta_{8} + \beta_{10} ) q^{29} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{30} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{31} + \beta_{4} q^{32} + ( \beta_{1} + \beta_{6} - 3 \beta_{7} + 3 \beta_{11} - \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{33} + ( -\beta_{1} - \beta_{5} + \beta_{11} - \beta_{13} ) q^{34} + ( \beta_{2} - \beta_{3} ) q^{36} + ( 2 - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{15} ) q^{37} + ( -\beta_{6} + \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{38} + ( -\beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{8} + 2 \beta_{9} + \beta_{15} ) q^{39} + ( -\beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{13} ) q^{40} + ( -\beta_{1} - \beta_{5} + \beta_{7} - \beta_{11} + 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{41} + ( -1 - \beta_{3} - 2 \beta_{4} + \beta_{9} - \beta_{15} ) q^{43} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{44} + ( -\beta_{1} - 3 \beta_{5} - \beta_{6} - 3 \beta_{11} ) q^{45} + ( -1 - \beta_{3} + \beta_{4} + 3 \beta_{8} - \beta_{9} - \beta_{15} ) q^{46} + ( 3 \beta_{1} - 4 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} - \beta_{11} - \beta_{12} ) q^{47} + \beta_{6} q^{48} + ( 4 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{50} + ( -\beta_{2} + 3 \beta_{4} + 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{51} + ( -\beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{52} + ( -\beta_{1} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{12} + \beta_{13} ) q^{54} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{11} - 3 \beta_{12} ) q^{55} + ( 3 - \beta_{2} - 4 \beta_{4} + 4 \beta_{8} - \beta_{15} ) q^{57} + ( 1 - \beta_{2} - 2 \beta_{9} - \beta_{10} ) q^{58} + ( 3 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{59} + ( \beta_{2} - \beta_{4} + 2 \beta_{8} - \beta_{9} - \beta_{15} ) q^{60} + ( \beta_{1} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{61} + ( -\beta_{1} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{62} - q^{64} + ( -2 - \beta_{2} + \beta_{3} - \beta_{8} + 6 \beta_{9} + \beta_{15} ) q^{65} + ( \beta_{1} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{66} + ( 4 + \beta_{2} + 5 \beta_{4} - 10 \beta_{8} - \beta_{10} + \beta_{15} ) q^{67} + ( \beta_{1} - \beta_{6} - \beta_{7} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{68} + ( 2 \beta_{1} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{69} + ( 1 - \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{71} + ( \beta_{4} - \beta_{9} + \beta_{15} ) q^{72} + ( -\beta_{1} + \beta_{6} - \beta_{7} - \beta_{11} - 3 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} ) q^{73} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{15} ) q^{74} + ( -3 \beta_{5} - 2 \beta_{6} - 3 \beta_{11} + \beta_{12} + 5 \beta_{13} ) q^{75} + ( \beta_{1} - \beta_{5} - \beta_{6} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{76} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{78} + ( 2 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{8} - \beta_{10} - \beta_{15} ) q^{79} + ( -\beta_{1} + \beta_{6} + \beta_{7} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{80} + ( \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 3 \beta_{9} + \beta_{10} + \beta_{15} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{12} - \beta_{14} ) q^{82} + ( \beta_{1} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{11} - 7 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} ) q^{83} + ( -2 + \beta_{3} + \beta_{4} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{15} ) q^{85} + ( 2 + \beta_{2} - \beta_{3} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{86} + ( \beta_{1} - 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{87} + ( -1 + \beta_{2} - 2 \beta_{4} + \beta_{8} + \beta_{9} - \beta_{15} ) q^{88} + ( 4 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{89} + ( 3 \beta_{7} - 3 \beta_{11} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} ) q^{90} + ( -4 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{15} ) q^{92} + ( 6 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 5 \beta_{8} - 4 \beta_{9} + \beta_{10} + \beta_{15} ) q^{93} + ( -\beta_{1} + \beta_{7} - \beta_{11} + \beta_{12} + 4 \beta_{13} - 4 \beta_{14} ) q^{94} + ( 2 - \beta_{2} - 2 \beta_{4} - 2 \beta_{9} - \beta_{10} + \beta_{15} ) q^{95} + ( -\beta_{12} - \beta_{13} ) q^{96} + ( \beta_{1} - \beta_{6} + \beta_{7} + 5 \beta_{11} - \beta_{12} - 6 \beta_{13} - \beta_{14} ) q^{97} + ( -\beta_{2} + 2 \beta_{3} + 7 \beta_{4} - 6 \beta_{9} - \beta_{10} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} - 12q^{9} + O(q^{10})$$ $$16q - 16q^{4} - 12q^{9} + 12q^{11} + 16q^{16} + 12q^{18} + 48q^{23} - 8q^{25} - 12q^{29} + 12q^{30} + 12q^{36} + 4q^{37} + 4q^{43} - 12q^{44} - 12q^{46} + 60q^{50} + 24q^{51} + 48q^{57} - 12q^{58} - 16q^{64} + 56q^{67} - 12q^{72} - 36q^{74} - 24q^{78} + 8q^{79} - 12q^{85} + 24q^{86} - 48q^{92} + 84q^{93} - 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 72 \nu^{6} + 234 \nu^{4} + 729 \nu^{2} - 243$$$$)/1944$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{14} - \nu^{12} + 6 \nu^{10} - 36 \nu^{8} + 180 \nu^{6} + 396 \nu^{4} - 972 \nu^{2} + 4131$$$$)/1944$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{12} - 9 \nu^{10} + 81 \nu^{8} - 126 \nu^{6} - 135 \nu^{4} + 1458 \nu^{2} - 2187$$$$)/1458$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{15} - 12 \nu^{13} - 144 \nu^{11} + 432 \nu^{9} - 468 \nu^{7} - 2754 \nu^{5} + 9477 \nu^{3} - 13122 \nu$$$$)/17496$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{15} + 21 \nu^{13} - 18 \nu^{11} - 108 \nu^{9} + 576 \nu^{7} - 648 \nu^{5} - 972 \nu^{3} + 9477 \nu$$$$)/5832$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + 6 \nu^{13} - 9 \nu^{11} - 54 \nu^{9} + 288 \nu^{7} - 486 \nu^{5} - 729 \nu^{3} + 4374 \nu$$$$)/2187$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{14} + 18 \nu^{12} - 216 \nu^{8} + 792 \nu^{6} + 54 \nu^{4} - 4617 \nu^{2} + 8748$$$$)/5832$$ $$\beta_{9}$$ $$=$$ $$($$$$-2 \nu^{14} + 21 \nu^{12} - 18 \nu^{10} - 108 \nu^{8} + 576 \nu^{6} - 648 \nu^{4} - 972 \nu^{2} + 9477$$$$)/5832$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{14} - 21 \nu^{12} + 126 \nu^{10} - 612 \nu^{6} + 2862 \nu^{4} - 2673 \nu^{2} + 729$$$$)/5832$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{15} - 3 \nu^{13} + 27 \nu^{9} - 45 \nu^{7} + 108 \nu^{5} + 324 \nu^{3}$$$$)/1458$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{13} + 9 \nu^{11} - 81 \nu^{9} + 126 \nu^{7} + 135 \nu^{5} - 1458 \nu^{3} + 2187 \nu$$$$)/1458$$ $$\beta_{13}$$ $$=$$ $$($$$$-5 \nu^{15} + 18 \nu^{13} - 216 \nu^{9} + 792 \nu^{7} + 54 \nu^{5} - 4617 \nu^{3} + 8748 \nu$$$$)/5832$$ $$\beta_{14}$$ $$=$$ $$($$$$-8 \nu^{15} + 21 \nu^{13} + 18 \nu^{11} - 324 \nu^{9} + 684 \nu^{7} - 4050 \nu^{3} + 3645 \nu$$$$)/5832$$ $$\beta_{15}$$ $$=$$ $$($$$$-3 \nu^{14} + 13 \nu^{12} + 12 \nu^{10} - 180 \nu^{8} + 594 \nu^{6} - 180 \nu^{4} - 3321 \nu^{2} + 8505$$$$)/972$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{14} - 2 \beta_{13} - \beta_{12} + 3 \beta_{11} + \beta_{6} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{15} + \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{14} + 3 \beta_{13} + 6 \beta_{11} - 6 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{15} - 6 \beta_{9} + 3 \beta_{8} + 12 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 9$$ $$\nu^{7}$$ $$=$$ $$9 \beta_{14} + 3 \beta_{13} - 12 \beta_{12} + 18 \beta_{11} + 9 \beta_{7} - 3 \beta_{6} - 9 \beta_{5} - 12 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$3 \beta_{15} + 12 \beta_{10} - 3 \beta_{9} + 30 \beta_{8} + 42 \beta_{4} - 3 \beta_{3} - 12 \beta_{2}$$ $$\nu^{9}$$ $$=$$ $$-36 \beta_{14} + 45 \beta_{13} - 45 \beta_{12} - 45 \beta_{11} + 9 \beta_{7} - 18 \beta_{6} - 45 \beta_{5}$$ $$\nu^{10}$$ $$=$$ $$45 \beta_{15} + 45 \beta_{10} + 27 \beta_{9} - 135 \beta_{8} + 63 \beta_{4} - 18 \beta_{3} - 108$$ $$\nu^{11}$$ $$=$$ $$-90 \beta_{13} - 126 \beta_{12} - 54 \beta_{11} + 135 \beta_{7} - 36 \beta_{6} - 270 \beta_{5} - 90 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-90 \beta_{15} + 126 \beta_{10} + 648 \beta_{9} - 126 \beta_{4} - 36 \beta_{3} - 90 \beta_{2} - 405$$ $$\nu^{13}$$ $$=$$ $$-270 \beta_{14} + 54 \beta_{12} - 378 \beta_{11} - 270 \beta_{7} + 486 \beta_{6} - 108 \beta_{5} - 243 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-54 \beta_{10} + 621 \beta_{9} - 999 \beta_{8} - 378 \beta_{4} + 486 \beta_{3} - 243 \beta_{2} - 1134$$ $$\nu^{15}$$ $$=$$ $$-729 \beta_{14} - 756 \beta_{13} + 1161 \beta_{12} + 729 \beta_{11} + 1161 \beta_{6} + 162 \beta_{5} - 1917 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$1 - \beta_{9}$$ $$1 - \beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
227.1
 −0.0967785 − 1.72934i −1.69547 + 0.354107i 1.69547 − 0.354107i 0.0967785 + 1.72934i −1.62181 − 0.608059i 1.40917 − 1.00709i −1.40917 + 1.00709i 1.62181 + 0.608059i −1.62181 + 0.608059i 1.40917 + 1.00709i −1.40917 − 1.00709i 1.62181 − 0.608059i −0.0967785 + 1.72934i −1.69547 − 0.354107i 1.69547 + 0.354107i 0.0967785 − 1.72934i
1.00000i −1.54605 0.780860i −1.00000 0.183299 0.317483i −0.780860 + 1.54605i 0 1.00000i 1.78052 + 2.41449i −0.317483 0.183299i
227.2 1.00000i −0.541068 + 1.64537i −1.00000 −0.895175 + 1.55049i 1.64537 + 0.541068i 0 1.00000i −2.41449 1.78052i 1.55049 + 0.895175i
227.3 1.00000i 0.541068 1.64537i −1.00000 0.895175 1.55049i −1.64537 0.541068i 0 1.00000i −2.41449 1.78052i −1.55049 0.895175i
227.4 1.00000i 1.54605 + 0.780860i −1.00000 −0.183299 + 0.317483i 0.780860 1.54605i 0 1.00000i 1.78052 + 2.41449i 0.317483 + 0.183299i
227.5 1.00000i −1.33750 + 1.10050i −1.00000 −1.94556 + 3.36980i −1.10050 1.33750i 0 1.00000i 0.577806 2.94383i −3.36980 1.94556i
227.6 1.00000i −0.167584 1.72392i −1.00000 −1.17468 + 2.03460i 1.72392 0.167584i 0 1.00000i −2.94383 + 0.577806i −2.03460 1.17468i
227.7 1.00000i 0.167584 + 1.72392i −1.00000 1.17468 2.03460i −1.72392 + 0.167584i 0 1.00000i −2.94383 + 0.577806i 2.03460 + 1.17468i
227.8 1.00000i 1.33750 1.10050i −1.00000 1.94556 3.36980i 1.10050 + 1.33750i 0 1.00000i 0.577806 2.94383i 3.36980 + 1.94556i
509.1 1.00000i −1.33750 1.10050i −1.00000 −1.94556 3.36980i −1.10050 + 1.33750i 0 1.00000i 0.577806 + 2.94383i −3.36980 + 1.94556i
509.2 1.00000i −0.167584 + 1.72392i −1.00000 −1.17468 2.03460i 1.72392 + 0.167584i 0 1.00000i −2.94383 0.577806i −2.03460 + 1.17468i
509.3 1.00000i 0.167584 1.72392i −1.00000 1.17468 + 2.03460i −1.72392 0.167584i 0 1.00000i −2.94383 0.577806i 2.03460 1.17468i
509.4 1.00000i 1.33750 + 1.10050i −1.00000 1.94556 + 3.36980i 1.10050 1.33750i 0 1.00000i 0.577806 + 2.94383i 3.36980 1.94556i
509.5 1.00000i −1.54605 + 0.780860i −1.00000 0.183299 + 0.317483i −0.780860 1.54605i 0 1.00000i 1.78052 2.41449i −0.317483 + 0.183299i
509.6 1.00000i −0.541068 1.64537i −1.00000 −0.895175 1.55049i 1.64537 0.541068i 0 1.00000i −2.41449 + 1.78052i 1.55049 0.895175i
509.7 1.00000i 0.541068 + 1.64537i −1.00000 0.895175 + 1.55049i −1.64537 + 0.541068i 0 1.00000i −2.41449 + 1.78052i −1.55049 + 0.895175i
509.8 1.00000i 1.54605 0.780860i −1.00000 −0.183299 0.317483i 0.780860 + 1.54605i 0 1.00000i 1.78052 2.41449i 0.317483 0.183299i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 509.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.l.a 16
3.b odd 2 1 2646.2.l.b 16
7.b odd 2 1 inner 882.2.l.a 16
7.c even 3 1 126.2.m.a 16
7.c even 3 1 882.2.t.b 16
7.d odd 6 1 126.2.m.a 16
7.d odd 6 1 882.2.t.b 16
9.c even 3 1 2646.2.t.a 16
9.d odd 6 1 882.2.t.b 16
21.c even 2 1 2646.2.l.b 16
21.g even 6 1 378.2.m.a 16
21.g even 6 1 2646.2.t.a 16
21.h odd 6 1 378.2.m.a 16
21.h odd 6 1 2646.2.t.a 16
28.f even 6 1 1008.2.cc.b 16
28.g odd 6 1 1008.2.cc.b 16
63.g even 3 1 378.2.m.a 16
63.h even 3 1 1134.2.d.a 16
63.h even 3 1 2646.2.l.b 16
63.i even 6 1 inner 882.2.l.a 16
63.i even 6 1 1134.2.d.a 16
63.j odd 6 1 inner 882.2.l.a 16
63.j odd 6 1 1134.2.d.a 16
63.k odd 6 1 378.2.m.a 16
63.l odd 6 1 2646.2.t.a 16
63.n odd 6 1 126.2.m.a 16
63.o even 6 1 882.2.t.b 16
63.s even 6 1 126.2.m.a 16
63.t odd 6 1 1134.2.d.a 16
63.t odd 6 1 2646.2.l.b 16
84.j odd 6 1 3024.2.cc.b 16
84.n even 6 1 3024.2.cc.b 16
252.n even 6 1 3024.2.cc.b 16
252.o even 6 1 1008.2.cc.b 16
252.bl odd 6 1 3024.2.cc.b 16
252.bn odd 6 1 1008.2.cc.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 7.c even 3 1
126.2.m.a 16 7.d odd 6 1
126.2.m.a 16 63.n odd 6 1
126.2.m.a 16 63.s even 6 1
378.2.m.a 16 21.g even 6 1
378.2.m.a 16 21.h odd 6 1
378.2.m.a 16 63.g even 3 1
378.2.m.a 16 63.k odd 6 1
882.2.l.a 16 1.a even 1 1 trivial
882.2.l.a 16 7.b odd 2 1 inner
882.2.l.a 16 63.i even 6 1 inner
882.2.l.a 16 63.j odd 6 1 inner
882.2.t.b 16 7.c even 3 1
882.2.t.b 16 7.d odd 6 1
882.2.t.b 16 9.d odd 6 1
882.2.t.b 16 63.o even 6 1
1008.2.cc.b 16 28.f even 6 1
1008.2.cc.b 16 28.g odd 6 1
1008.2.cc.b 16 252.o even 6 1
1008.2.cc.b 16 252.bn odd 6 1
1134.2.d.a 16 63.h even 3 1
1134.2.d.a 16 63.i even 6 1
1134.2.d.a 16 63.j odd 6 1
1134.2.d.a 16 63.t odd 6 1
2646.2.l.b 16 3.b odd 2 1
2646.2.l.b 16 21.c even 2 1
2646.2.l.b 16 63.h even 3 1
2646.2.l.b 16 63.t odd 6 1
2646.2.t.a 16 9.c even 3 1
2646.2.t.a 16 21.g even 6 1
2646.2.t.a 16 21.h odd 6 1
2646.2.t.a 16 63.l odd 6 1
3024.2.cc.b 16 84.j odd 6 1
3024.2.cc.b 16 84.n even 6 1
3024.2.cc.b 16 252.n even 6 1
3024.2.cc.b 16 252.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$6561 + 4374 T^{2} + 1458 T^{4} + 648 T^{6} + 279 T^{8} + 72 T^{10} + 18 T^{12} + 6 T^{14} + T^{16}$$
$5$ $$1296 + 10368 T^{2} + 77436 T^{4} + 42336 T^{6} + 16461 T^{8} + 3096 T^{10} + 423 T^{12} + 24 T^{14} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$( 1296 + 2592 T + 972 T^{2} - 1512 T^{3} + 261 T^{4} + 126 T^{5} - 9 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$13$ $$331776 - 497664 T^{2} + 559872 T^{4} - 238464 T^{6} + 73296 T^{8} - 9936 T^{10} + 972 T^{12} - 36 T^{14} + T^{16}$$
$17$ $$331776 + 746496 T^{2} + 1404864 T^{4} + 569808 T^{6} + 172521 T^{8} + 17442 T^{10} + 1287 T^{12} + 42 T^{14} + T^{16}$$
$19$ $$2313441 - 2819934 T^{2} + 2533842 T^{4} - 937008 T^{6} + 251199 T^{8} - 28368 T^{10} + 2322 T^{12} - 54 T^{14} + T^{16}$$
$23$ $$( 443556 - 227772 T + 17010 T^{2} + 11286 T^{3} - 981 T^{4} - 792 T^{5} + 225 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$29$ $$( 20736 - 10368 T - 2592 T^{2} + 2160 T^{3} + 612 T^{4} - 180 T^{5} - 18 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$31$ $$( 746496 + 124416 T^{2} + 6804 T^{4} + 144 T^{6} + T^{8} )^{2}$$
$37$ $$( 1784896 + 245824 T + 170128 T^{2} - 13424 T^{3} + 9436 T^{4} - 164 T^{5} + 106 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$41$ $$73499483897856 + 6883786653696 T^{2} + 448658922240 T^{4} + 13938763392 T^{6} + 307258425 T^{8} + 4294314 T^{10} + 43695 T^{12} + 258 T^{14} + T^{16}$$
$43$ $$( 10816 - 15392 T + 17848 T^{2} - 6188 T^{3} + 1921 T^{4} - 218 T^{5} + 43 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$( 1218816 - 327744 T^{2} + 16308 T^{4} - 240 T^{6} + T^{8} )^{2}$$
$53$ $$T^{16}$$
$59$ $$( 36 - 80604 T^{2} + 17649 T^{4} - 294 T^{6} + T^{8} )^{2}$$
$61$ $$( 1557504 + 350208 T^{2} + 16965 T^{4} + 240 T^{6} + T^{8} )^{2}$$
$67$ $$( -908 + 1660 T - 111 T^{2} - 14 T^{3} + T^{4} )^{4}$$
$71$ $$( 82944 + 31104 T^{2} + 2745 T^{4} + 90 T^{6} + T^{8} )^{2}$$
$73$ $$2927055626496 - 422268609024 T^{2} + 40269720240 T^{4} - 2219198688 T^{6} + 89156745 T^{8} - 2185686 T^{10} + 37215 T^{12} - 222 T^{14} + T^{16}$$
$79$ $$( -1202 + 778 T - 129 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$83$ $$337116351515590656 + 10214329542377472 T^{2} + 207879529033728 T^{4} + 2256409253376 T^{6} + 17587710864 T^{8} + 88712784 T^{10} + 326268 T^{12} + 708 T^{14} + T^{16}$$
$89$ $$34828517376 + 29023764480 T^{2} + 21767823360 T^{4} + 1934917632 T^{6} + 134182656 T^{8} + 2488320 T^{10} + 33696 T^{12} + 216 T^{14} + T^{16}$$
$97$ $$4512402164941056 - 390697151362560 T^{2} + 24485300891568 T^{4} - 714581204256 T^{6} + 15192293193 T^{8} - 85999734 T^{10} + 353727 T^{12} - 702 T^{14} + T^{16}$$