# Properties

 Label 1680.2.f.k Level $1680$ Weight $2$ Character orbit 1680.f Analytic conductor $13.415$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} - 210 x^{7} + 639 x^{6} - 864 x^{5} + 810 x^{4} - 972 x^{3} + 729 x^{2} - 4374 x + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 840) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - q^{5} -\beta_{9} q^{7} + \beta_{7} q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} - q^{5} -\beta_{9} q^{7} + \beta_{7} q^{9} + \beta_{8} q^{11} + ( \beta_{1} + \beta_{8} - \beta_{9} + \beta_{15} ) q^{13} -\beta_{3} q^{15} + ( -\beta_{5} + \beta_{6} - \beta_{13} ) q^{17} + ( -\beta_{1} - \beta_{7} - \beta_{8} ) q^{19} + ( \beta_{1} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{14} ) q^{21} + ( -\beta_{7} + \beta_{10} + \beta_{14} ) q^{23} + q^{25} + ( \beta_{4} + \beta_{5} + \beta_{10} + \beta_{13} ) q^{27} + ( \beta_{1} - \beta_{4} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{29} + ( \beta_{2} + \beta_{3} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{14} ) q^{31} + ( -\beta_{2} - \beta_{4} - \beta_{10} + \beta_{12} ) q^{33} + \beta_{9} q^{35} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{37} + ( \beta_{2} + \beta_{5} - \beta_{6} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{39} + ( -2 - \beta_{10} + \beta_{11} ) q^{41} + ( -2 - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{13} ) q^{43} -\beta_{7} q^{45} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{13} ) q^{47} + ( \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{11} - \beta_{13} ) q^{49} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{53} -\beta_{8} q^{55} + ( -2 \beta_{2} - \beta_{5} - \beta_{12} - \beta_{13} ) q^{57} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} - 2 \beta_{12} ) q^{59} + ( \beta_{4} - \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{14} ) q^{61} + ( \beta_{2} + \beta_{4} + \beta_{8} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{63} + ( -\beta_{1} - \beta_{8} + \beta_{9} - \beta_{15} ) q^{65} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{67} + ( -1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{15} ) q^{69} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} + \beta_{14} ) q^{71} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{10} + 2 \beta_{14} - \beta_{15} ) q^{73} + \beta_{3} q^{75} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{13} - \beta_{15} ) q^{77} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{79} + ( -1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{81} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} - 2 \beta_{9} + 2 \beta_{12} ) q^{83} + ( \beta_{5} - \beta_{6} + \beta_{13} ) q^{85} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{87} + ( 2 - 2 \beta_{1} + 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{89} + ( -2 + \beta_{1} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{91} + ( -2 + 3 \beta_{4} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{93} + ( \beta_{1} + \beta_{7} + \beta_{8} ) q^{95} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{15} ) q^{97} + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{5} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$16q - 16q^{5} - 2q^{7} - 2q^{9} + 2q^{21} + 16q^{25} + 6q^{27} - 6q^{33} + 2q^{35} + 12q^{37} - 6q^{39} - 32q^{41} - 32q^{43} + 2q^{45} + 4q^{47} - 4q^{49} - 6q^{51} - 24q^{59} + 4q^{63} - 8q^{69} + 32q^{77} + 4q^{79} - 6q^{81} + 20q^{83} + 6q^{87} + 24q^{89} - 20q^{91} - 32q^{93} + 58q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} - 210 x^{7} + 639 x^{6} - 864 x^{5} + 810 x^{4} - 972 x^{3} + 729 x^{2} - 4374 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$13 \nu^{15} + 55 \nu^{14} - 50 \nu^{13} + 74 \nu^{12} - 176 \nu^{11} + 160 \nu^{10} - 517 \nu^{9} + 1025 \nu^{8} - 2863 \nu^{7} + 627 \nu^{6} - 1296 \nu^{5} + 8640 \nu^{4} + 5346 \nu^{3} + 23814 \nu^{2} - 12393 \nu - 37179$$$$)/46656$$ $$\beta_{3}$$ $$=$$ $$($$$$17 \nu^{15} - 17 \nu^{14} + 22 \nu^{13} + 114 \nu^{12} - 48 \nu^{11} - 152 \nu^{10} + 135 \nu^{9} + 497 \nu^{8} - 1483 \nu^{7} + 763 \nu^{6} - 1296 \nu^{5} - 1944 \nu^{4} - 4806 \nu^{3} + 23166 \nu^{2} + 11907 \nu + 9477$$$$)/46656$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{15} - 19 \nu^{14} + 14 \nu^{13} - 30 \nu^{12} + 84 \nu^{11} - 220 \nu^{10} + 333 \nu^{9} - 473 \nu^{8} + 67 \nu^{7} - 799 \nu^{6} + 1092 \nu^{5} - 3372 \nu^{4} + 3474 \nu^{3} + 4158 \nu^{2} - 10287 \nu + 8019$$$$)/15552$$ $$\beta_{5}$$ $$=$$ $$($$$$-17 \nu^{15} - 5 \nu^{14} - 182 \nu^{13} + 218 \nu^{12} - 392 \nu^{11} + 1072 \nu^{10} - 1687 \nu^{9} + 2741 \nu^{8} - 4333 \nu^{7} + 12159 \nu^{6} - 12744 \nu^{5} + 18576 \nu^{4} - 39690 \nu^{3} + 486 \nu^{2} - 83835 \nu + 111537$$$$)/34992$$ $$\beta_{6}$$ $$=$$ $$($$$$19 \nu^{15} + 45 \nu^{14} - 126 \nu^{13} + 10 \nu^{12} - 184 \nu^{11} - 276 \nu^{10} - 683 \nu^{9} + 2127 \nu^{8} - 1977 \nu^{7} + 1393 \nu^{6} + 4104 \nu^{5} + 9756 \nu^{4} - 10098 \nu^{3} + 10854 \nu^{2} - 5103 \nu - 102789$$$$)/23328$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{15} - \nu^{14} - 4 \nu^{13} - 5 \nu^{12} + 8 \nu^{11} - 34 \nu^{10} + 46 \nu^{9} + 73 \nu^{8} - 62 \nu^{7} - 198 \nu^{6} + 648 \nu^{5} + 189 \nu^{4} - 972 \nu^{3} + 486 \nu^{2} - 1458 \nu - 6561$$$$)/2187$$ $$\beta_{8}$$ $$=$$ $$($$$$71 \nu^{15} + 5 \nu^{14} + 182 \nu^{13} - 218 \nu^{12} + 284 \nu^{11} - 1936 \nu^{10} + 1957 \nu^{9} - 3605 \nu^{8} + 5251 \nu^{7} - 7839 \nu^{6} + 24300 \nu^{5} - 9936 \nu^{4} + 65610 \nu^{3} - 31590 \nu^{2} + 12393 \nu - 181521$$$$)/69984$$ $$\beta_{9}$$ $$=$$ $$($$$$75 \nu^{15} - 47 \nu^{14} + 22 \nu^{13} - 62 \nu^{12} + 68 \nu^{11} - 1892 \nu^{10} + 1849 \nu^{9} - 421 \nu^{8} - 1129 \nu^{7} - 3835 \nu^{6} + 29940 \nu^{5} - 16596 \nu^{4} + 25866 \nu^{3} + 33534 \nu^{2} - 8019 \nu - 280665$$$$)/46656$$ $$\beta_{10}$$ $$=$$ $$($$$$-131 \nu^{15} + 19 \nu^{14} - 140 \nu^{13} + 704 \nu^{12} - 914 \nu^{11} + 2446 \nu^{10} - 1291 \nu^{9} + 5975 \nu^{8} - 9199 \nu^{7} + 19959 \nu^{6} - 36666 \nu^{5} - 1242 \nu^{4} - 60588 \nu^{3} + 71928 \nu^{2} - 86751 \nu + 194643$$$$)/69984$$ $$\beta_{11}$$ $$=$$ $$($$$$15 \nu^{15} - 13 \nu^{14} + 24 \nu^{13} - 72 \nu^{12} - 34 \nu^{11} - 362 \nu^{10} + 651 \nu^{9} - 577 \nu^{8} + 483 \nu^{7} - 1305 \nu^{6} + 4982 \nu^{5} - 3570 \nu^{4} + 12096 \nu^{3} + 2592 \nu^{2} - 4617 \nu - 60021$$$$)/7776$$ $$\beta_{12}$$ $$=$$ $$($$$$6 \nu^{15} - 4 \nu^{14} - \nu^{13} - 16 \nu^{12} + 28 \nu^{11} - 184 \nu^{10} + 215 \nu^{9} - 68 \nu^{8} + 217 \nu^{7} - 884 \nu^{6} + 2748 \nu^{5} - 2520 \nu^{4} + 1917 \nu^{3} - 1296 \nu^{2} + 486 \nu - 26244$$$$)/2916$$ $$\beta_{13}$$ $$=$$ $$($$$$57 \nu^{15} + 7 \nu^{14} + 70 \nu^{13} - 302 \nu^{12} + 440 \nu^{11} - 2048 \nu^{10} + 1951 \nu^{9} - 2839 \nu^{8} + 5717 \nu^{7} - 17941 \nu^{6} + 32376 \nu^{5} - 16992 \nu^{4} + 51786 \nu^{3} - 28674 \nu^{2} + 124659 \nu - 305451$$$$)/23328$$ $$\beta_{14}$$ $$=$$ $$($$$$35 \nu^{15} - 13 \nu^{14} + 11 \nu^{13} - 209 \nu^{12} + 185 \nu^{11} - 775 \nu^{10} + 940 \nu^{9} - 824 \nu^{8} + 2371 \nu^{7} - 6489 \nu^{6} + 13437 \nu^{5} - 3699 \nu^{4} + 20817 \nu^{3} - 23571 \nu^{2} + 20412 \nu - 131220$$$$)/11664$$ $$\beta_{15}$$ $$=$$ $$($$$$427 \nu^{15} - 491 \nu^{14} - 2 \nu^{13} - 1102 \nu^{12} + 2332 \nu^{11} - 9980 \nu^{10} + 15569 \nu^{9} - 10273 \nu^{8} + 7943 \nu^{7} - 38103 \nu^{6} + 156204 \nu^{5} - 111564 \nu^{4} + 124578 \nu^{3} - 118098 \nu^{2} + 44469 \nu - 1047573$$$$)/139968$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} - 2 \beta_{12} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 14 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 7 \beta_{6} + 2 \beta_{5} - 13 \beta_{4} - 13 \beta_{3} + \beta_{2} + 2 \beta_{1} + 32$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{15} + 8 \beta_{14} + 21 \beta_{12} - 2 \beta_{11} + 6 \beta_{10} - 34 \beta_{9} + 41 \beta_{8} - \beta_{7} - 5 \beta_{6} + 23 \beta_{5} - 25 \beta_{4} + 29 \beta_{3} - 25 \beta_{2} + 66 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{15} + 16 \beta_{14} - 7 \beta_{13} - 2 \beta_{12} - 8 \beta_{11} + 13 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 7 \beta_{5} + \beta_{4} + 12 \beta_{3} - 9 \beta_{2} + 3 \beta_{1} + 27$$ $$\nu^{9}$$ $$=$$ $$($$$$72 \beta_{15} - 20 \beta_{14} - 30 \beta_{12} + 24 \beta_{11} + 58 \beta_{10} - 108 \beta_{9} + 74 \beta_{8} + 62 \beta_{7} + 16 \beta_{6} - 35 \beta_{5} - 58 \beta_{4} - 6 \beta_{3} + 16 \beta_{2} + 40 \beta_{1} + 22$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$8 \beta_{15} + 16 \beta_{14} - 16 \beta_{13} + 16 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} + 5 \beta_{9} - 28 \beta_{8} - 10 \beta_{7} - 40 \beta_{6} + 3 \beta_{5} - 69 \beta_{4} - 3 \beta_{3} + 53 \beta_{2} + 39 \beta_{1} - 8$$ $$\nu^{11}$$ $$=$$ $$($$$$165 \beta_{15} + 136 \beta_{14} - 128 \beta_{13} - 123 \beta_{12} - 280 \beta_{11} + 112 \beta_{10} + 88 \beta_{9} + 179 \beta_{8} - 85 \beta_{7} + 67 \beta_{6} - 200 \beta_{5} + 131 \beta_{4} - 189 \beta_{3} + 61 \beta_{2} + 216 \beta_{1} - 800$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-20 \beta_{15} - 16 \beta_{14} + 80 \beta_{13} + 32 \beta_{12} - 35 \beta_{11} + 65 \beta_{10} - 61 \beta_{9} + 168 \beta_{8} + 57 \beta_{7} - 100 \beta_{6} + 157 \beta_{5} + 5 \beta_{4} + 135 \beta_{3} + 67 \beta_{2} + 57 \beta_{1} - 123$$ $$\nu^{13}$$ $$=$$ $$($$$$-447 \beta_{15} + 88 \beta_{14} - 384 \beta_{13} - 161 \beta_{12} - 56 \beta_{11} - 224 \beta_{10} + 1016 \beta_{9} + 71 \beta_{8} + 479 \beta_{7} - 215 \beta_{6} - 689 \beta_{5} - 505 \beta_{4} + 313 \beta_{3} - 473 \beta_{2} - 968 \beta_{1} + 160$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$184 \beta_{15} - 256 \beta_{14} - 160 \beta_{13} + 208 \beta_{12} + 39 \beta_{11} - 169 \beta_{10} - 327 \beta_{9} + 180 \beta_{8} + 51 \beta_{7} + 160 \beta_{6} - 377 \beta_{5} - 449 \beta_{4} - 383 \beta_{3} + 369 \beta_{2} + 52 \beta_{1} + 208$$ $$\nu^{15}$$ $$=$$ $$($$$$568 \beta_{15} + 364 \beta_{14} - 1408 \beta_{13} + 2214 \beta_{12} + 230 \beta_{11} - 2300 \beta_{10} - 2150 \beta_{9} - 494 \beta_{8} - 1340 \beta_{7} - 624 \beta_{6} + 129 \beta_{5} - 3196 \beta_{4} + 4800 \beta_{3} + 2706 \beta_{2} + 1754 \beta_{1} - 558$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.348228 + 1.69668i 0.348228 − 1.69668i −1.12510 − 1.31688i −1.12510 + 1.31688i −1.49826 − 0.869033i −1.49826 + 0.869033i 1.66912 + 0.462633i 1.66912 − 0.462633i 1.71703 − 0.227581i 1.71703 + 0.227581i −1.34935 + 1.08593i −1.34935 − 1.08593i 1.14188 − 1.30235i 1.14188 + 1.30235i 0.0964469 − 1.72936i 0.0964469 + 1.72936i
0 −1.69668 0.348228i 0 −1.00000 0 2.63166 0.272689i 0 2.75748 + 1.18166i 0
881.2 0 −1.69668 + 0.348228i 0 −1.00000 0 2.63166 + 0.272689i 0 2.75748 1.18166i 0
881.3 0 −1.31688 1.12510i 0 −1.00000 0 −2.64497 + 0.0644212i 0 0.468322 + 2.96322i 0
881.4 0 −1.31688 + 1.12510i 0 −1.00000 0 −2.64497 0.0644212i 0 0.468322 2.96322i 0
881.5 0 −0.869033 1.49826i 0 −1.00000 0 0.807952 + 2.51937i 0 −1.48956 + 2.60407i 0
881.6 0 −0.869033 + 1.49826i 0 −1.00000 0 0.807952 2.51937i 0 −1.48956 2.60407i 0
881.7 0 −0.462633 1.66912i 0 −1.00000 0 −1.62879 2.08496i 0 −2.57194 + 1.54438i 0
881.8 0 −0.462633 + 1.66912i 0 −1.00000 0 −1.62879 + 2.08496i 0 −2.57194 1.54438i 0
881.9 0 0.227581 1.71703i 0 −1.00000 0 1.22074 + 2.34729i 0 −2.89641 0.781528i 0
881.10 0 0.227581 + 1.71703i 0 −1.00000 0 1.22074 2.34729i 0 −2.89641 + 0.781528i 0
881.11 0 1.08593 1.34935i 0 −1.00000 0 −2.53123 + 0.769995i 0 −0.641511 2.93061i 0
881.12 0 1.08593 + 1.34935i 0 −1.00000 0 −2.53123 0.769995i 0 −0.641511 + 2.93061i 0
881.13 0 1.30235 1.14188i 0 −1.00000 0 1.35345 2.27336i 0 0.392236 2.97425i 0
881.14 0 1.30235 + 1.14188i 0 −1.00000 0 1.35345 + 2.27336i 0 0.392236 + 2.97425i 0
881.15 0 1.72936 0.0964469i 0 −1.00000 0 −0.208829 + 2.63750i 0 2.98140 0.333584i 0
881.16 0 1.72936 + 0.0964469i 0 −1.00000 0 −0.208829 2.63750i 0 2.98140 + 0.333584i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.k 16
3.b odd 2 1 1680.2.f.l 16
4.b odd 2 1 840.2.f.a 16
7.b odd 2 1 1680.2.f.l 16
12.b even 2 1 840.2.f.b yes 16
21.c even 2 1 inner 1680.2.f.k 16
28.d even 2 1 840.2.f.b yes 16
84.h odd 2 1 840.2.f.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.f.a 16 4.b odd 2 1
840.2.f.a 16 84.h odd 2 1
840.2.f.b yes 16 12.b even 2 1
840.2.f.b yes 16 28.d even 2 1
1680.2.f.k 16 1.a even 1 1 trivial
1680.2.f.k 16 21.c even 2 1 inner
1680.2.f.l 16 3.b odd 2 1
1680.2.f.l 16 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{16} + \cdots$$ $$T_{17}^{8} - 79 T_{17}^{6} - 98 T_{17}^{5} + 1612 T_{17}^{4} + 4048 T_{17}^{3} - 1232 T_{17}^{2} - 4640 T_{17} + 1856$$ $$T_{41}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$6561 + 729 T^{2} - 486 T^{3} + 162 T^{4} + 54 T^{5} - 81 T^{6} - 60 T^{7} - 70 T^{8} - 20 T^{9} - 9 T^{10} + 2 T^{11} + 2 T^{12} - 2 T^{13} + T^{14} + T^{16}$$
$5$ $$( 1 + T )^{16}$$
$7$ $$5764801 + 1647086 T + 470596 T^{2} + 571438 T^{3} + 48020 T^{4} - 18522 T^{5} - 9604 T^{6} - 11130 T^{7} - 4714 T^{8} - 1590 T^{9} - 196 T^{10} - 54 T^{11} + 20 T^{12} + 34 T^{13} + 4 T^{14} + 2 T^{15} + T^{16}$$
$11$ $$4096 + 124928 T^{2} + 495360 T^{4} + 443392 T^{6} + 161008 T^{8} + 27032 T^{10} + 2169 T^{12} + 78 T^{14} + T^{16}$$
$13$ $$256000000 + 373760000 T^{2} + 171417600 T^{4} + 36520704 T^{6} + 4091920 T^{8} + 251724 T^{10} + 8481 T^{12} + 146 T^{14} + T^{16}$$
$17$ $$( 1856 - 4640 T - 1232 T^{2} + 4048 T^{3} + 1612 T^{4} - 98 T^{5} - 79 T^{6} + T^{8} )^{2}$$
$19$ $$65536 + 1572864 T^{2} + 4210688 T^{4} + 3403776 T^{6} + 841216 T^{8} + 90624 T^{10} + 4672 T^{12} + 112 T^{14} + T^{16}$$
$23$ $$16777216 + 205520896 T^{2} + 499515392 T^{4} + 133120000 T^{6} + 12747008 T^{8} + 597760 T^{10} + 14944 T^{12} + 192 T^{14} + T^{16}$$
$29$ $$2316304384 + 5350359040 T^{2} + 3379879936 T^{4} + 501650432 T^{6} + 33617040 T^{8} + 1200988 T^{10} + 23665 T^{12} + 242 T^{14} + T^{16}$$
$31$ $$89718784 + 15636971520 T^{2} + 5015478272 T^{4} + 624229376 T^{6} + 39538176 T^{8} + 1381568 T^{10} + 26592 T^{12} + 260 T^{14} + T^{16}$$
$37$ $$( 1573888 - 103936 T - 424384 T^{2} - 22880 T^{3} + 15856 T^{4} + 784 T^{5} - 212 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$41$ $$( -80896 - 33536 T + 51520 T^{2} + 11392 T^{3} - 4688 T^{4} - 1328 T^{5} - 20 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$43$ $$( 8192 - 20480 T + 1408 T^{2} + 7744 T^{3} - 864 T^{4} - 672 T^{5} + 4 T^{6} + 16 T^{7} + T^{8} )^{2}$$
$47$ $$( 696832 + 242816 T - 151456 T^{2} - 32112 T^{3} + 11312 T^{4} + 504 T^{5} - 199 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$53$ $$2083195715584 + 980583383040 T^{2} + 165097832448 T^{4} + 12637560832 T^{6} + 485496064 T^{8} + 9645824 T^{10} + 100192 T^{12} + 512 T^{14} + T^{16}$$
$59$ $$( -495616 - 405504 T - 20992 T^{2} + 45056 T^{3} + 5424 T^{4} - 1520 T^{5} - 144 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$61$ $$1073741824 + 7113539584 T^{2} + 7919894528 T^{4} + 2723938304 T^{6} + 194650112 T^{8} + 5504000 T^{10} + 72720 T^{12} + 444 T^{14} + T^{16}$$
$67$ $$( 6326272 - 427008 T - 922240 T^{2} - 4416 T^{3} + 28768 T^{4} + 192 T^{5} - 316 T^{6} + T^{8} )^{2}$$
$71$ $$83534872576 + 1641010642944 T^{2} + 397017784320 T^{4} + 28141681664 T^{6} + 898139648 T^{8} + 14859968 T^{10} + 131168 T^{12} + 580 T^{14} + T^{16}$$
$73$ $$3761489575936 + 2344003043328 T^{2} + 487134347264 T^{4} + 41591709696 T^{6} + 1411139584 T^{8} + 22394112 T^{10} + 178064 T^{12} + 684 T^{14} + T^{16}$$
$79$ $$( 352256 + 6895616 T - 772608 T^{2} - 350720 T^{3} + 50800 T^{4} + 2088 T^{5} - 439 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$83$ $$( 1067008 - 95232 T - 212352 T^{2} - 384 T^{3} + 13248 T^{4} + 896 T^{5} - 188 T^{6} - 10 T^{7} + T^{8} )^{2}$$
$89$ $$( -2134016 + 1326080 T + 13312 T^{2} - 113728 T^{3} + 10000 T^{4} + 2368 T^{5} - 232 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$97$ $$895060875034624 + 164741580283904 T^{2} + 10413359167488 T^{4} + 321704965888 T^{6} + 5485687568 T^{8} + 53800060 T^{10} + 299169 T^{12} + 866 T^{14} + T^{16}$$