# Properties

 Label 392.3.k.l Level 392 Weight 3 Character orbit 392.k Analytic conductor 10.681 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 392.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.6812263629$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} + ( -\beta_{3} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{3} + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{4} + ( -\beta_{2} - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{5} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{6} + ( 2 \beta_{6} - 2 \beta_{11} ) q^{8} + ( 6 \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{10} q^{2} + ( -\beta_{3} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{3} + ( -1 + \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{4} + ( -\beta_{2} - \beta_{4} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{5} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{6} + ( 2 \beta_{6} - 2 \beta_{11} ) q^{8} + ( 6 \beta_{7} - \beta_{8} - 4 \beta_{9} + \beta_{10} ) q^{9} + ( 1 - \beta_{1} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{10} + ( 6 - 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{9} ) q^{11} + ( -\beta_{2} + 5 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} ) q^{12} + ( 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 5 \beta_{5} - 5 \beta_{8} - 5 \beta_{10} - 2 \beta_{11} ) q^{13} + ( 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 3 \beta_{11} ) q^{15} + ( -4 \beta_{2} - 4 \beta_{4} - 4 \beta_{7} + 8 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} ) q^{16} + ( -5 + \beta_{3} - \beta_{5} + 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{9} ) q^{17} + ( -5 + \beta_{1} + 8 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} - 5 \beta_{9} ) q^{18} + ( 16 \beta_{7} - 4 \beta_{8} + \beta_{9} + 4 \beta_{10} ) q^{19} + ( -5 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{6} + 2 \beta_{11} ) q^{20} + ( -6 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} - 6 \beta_{8} + 3 \beta_{10} + 3 \beta_{11} ) q^{22} + ( -7 \beta_{4} - 5 \beta_{8} - 5 \beta_{10} - 7 \beta_{11} ) q^{23} + ( 10 + 6 \beta_{1} + 8 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 10 \beta_{7} - 8 \beta_{9} ) q^{24} + ( -16 - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 16 \beta_{7} + 2 \beta_{9} ) q^{25} + ( -9 \beta_{2} - 4 \beta_{4} - 19 \beta_{7} + 5 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} ) q^{26} + ( -10 - 2 \beta_{3} + 2 \beta_{5} - 13 \beta_{6} - 2 \beta_{8} + 2 \beta_{10} ) q^{27} + ( 8 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{29} + ( -4 \beta_{2} - 3 \beta_{4} + 6 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{30} + ( -6 \beta_{3} - 9 \beta_{4} - 6 \beta_{5} ) q^{31} + ( 20 - 4 \beta_{1} - 8 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} + 20 \beta_{7} - 4 \beta_{9} ) q^{32} + ( -15 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} - 6 \beta_{10} ) q^{33} + ( -5 + \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{34} + ( -14 - 2 \beta_{1} + 2 \beta_{2} + 16 \beta_{3} + 4 \beta_{5} - 8 \beta_{6} + 16 \beta_{8} + 4 \beta_{10} - 10 \beta_{11} ) q^{36} + ( 9 \beta_{2} - \beta_{4} - 4 \beta_{8} - 4 \beta_{10} - \beta_{11} ) q^{37} + ( -20 + 4 \beta_{1} - 2 \beta_{3} - \beta_{4} + 15 \beta_{5} + 3 \beta_{6} - 20 \beta_{7} - 3 \beta_{9} ) q^{38} + ( 14 \beta_{1} - 17 \beta_{3} - 8 \beta_{4} - 17 \beta_{5} ) q^{39} + ( 4 \beta_{2} + 2 \beta_{4} + 4 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{40} + ( 26 + \beta_{3} - \beta_{5} - 18 \beta_{6} + \beta_{8} - \beta_{10} ) q^{41} + ( -20 - 10 \beta_{3} + 10 \beta_{5} - 10 \beta_{6} - 10 \beta_{8} + 10 \beta_{10} ) q^{43} + ( 9 \beta_{2} + 6 \beta_{4} - 21 \beta_{7} - 12 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} ) q^{44} + ( -4 \beta_{1} + 9 \beta_{3} - 2 \beta_{4} + 9 \beta_{5} ) q^{45} + ( -29 - 19 \beta_{1} - 14 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} - 29 \beta_{7} - 2 \beta_{9} ) q^{46} + ( 16 \beta_{2} - 5 \beta_{4} - 2 \beta_{8} - 2 \beta_{10} - 5 \beta_{11} ) q^{47} + ( -20 - 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 20 \beta_{5} + 4 \beta_{6} + 8 \beta_{8} + 20 \beta_{10} - 16 \beta_{11} ) q^{48} + ( 10 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 18 \beta_{5} - 4 \beta_{6} - 4 \beta_{8} - 18 \beta_{10} + 2 \beta_{11} ) q^{50} + ( -4 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} ) q^{51} + ( -10 - 6 \beta_{1} - 18 \beta_{4} - 28 \beta_{5} - 8 \beta_{6} - 10 \beta_{7} + 8 \beta_{9} ) q^{52} + ( 3 \beta_{1} + 20 \beta_{3} + 7 \beta_{4} + 20 \beta_{5} ) q^{53} + ( 2 \beta_{2} + 13 \beta_{4} - 10 \beta_{7} - 26 \beta_{8} - 15 \beta_{9} - 23 \beta_{10} + 13 \beta_{11} ) q^{54} + ( 12 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{8} + 3 \beta_{10} - 9 \beta_{11} ) q^{55} + ( 45 + 20 \beta_{3} - 20 \beta_{5} - 12 \beta_{6} + 20 \beta_{8} - 20 \beta_{10} ) q^{57} + ( -\beta_{2} - 10 \beta_{4} + 5 \beta_{7} - 12 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} - 10 \beta_{11} ) q^{58} + ( 12 - 5 \beta_{3} + 5 \beta_{5} + 9 \beta_{6} + 12 \beta_{7} - 9 \beta_{9} ) q^{59} + ( 15 - 3 \beta_{1} + 12 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 7 \beta_{6} + 15 \beta_{7} - 7 \beta_{9} ) q^{60} + ( 15 \beta_{2} + 25 \beta_{4} - 10 \beta_{8} - 10 \beta_{10} + 25 \beta_{11} ) q^{61} + ( -12 \beta_{1} + 12 \beta_{2} - 18 \beta_{3} - 9 \beta_{5} + 15 \beta_{6} - 18 \beta_{8} - 9 \beta_{10} + 9 \beta_{11} ) q^{62} + ( -24 - 8 \beta_{1} + 8 \beta_{2} + 16 \beta_{5} + 16 \beta_{6} + 16 \beta_{10} + 8 \beta_{11} ) q^{64} + ( 4 \beta_{7} + 11 \beta_{8} + 18 \beta_{9} - 11 \beta_{10} ) q^{65} + ( 30 - 6 \beta_{1} - 12 \beta_{3} - 6 \beta_{4} - 21 \beta_{5} - 12 \beta_{6} + 30 \beta_{7} + 12 \beta_{9} ) q^{66} + ( 70 - 3 \beta_{3} + 3 \beta_{5} + \beta_{6} + 70 \beta_{7} - \beta_{9} ) q^{67} + ( -7 \beta_{2} - 6 \beta_{4} + 15 \beta_{7} + 8 \beta_{8} + 5 \beta_{9} - 6 \beta_{11} ) q^{68} + ( 31 \beta_{1} - 31 \beta_{2} + 13 \beta_{3} + 13 \beta_{5} + 13 \beta_{8} + 13 \beta_{10} + 3 \beta_{11} ) q^{69} + ( -6 \beta_{1} + 6 \beta_{2} - 22 \beta_{3} - 22 \beta_{5} - 22 \beta_{8} - 22 \beta_{10} + 8 \beta_{11} ) q^{71} + ( -16 \beta_{2} + 40 \beta_{7} + 8 \beta_{8} - 24 \beta_{9} - 12 \beta_{10} ) q^{72} + ( -15 + 12 \beta_{3} - 12 \beta_{5} + 18 \beta_{6} - 15 \beta_{7} - 18 \beta_{9} ) q^{73} + ( -14 - 6 \beta_{1} - 20 \beta_{3} + 8 \beta_{4} - \beta_{5} + 6 \beta_{6} - 14 \beta_{7} - 6 \beta_{9} ) q^{74} + ( 10 \beta_{7} - 18 \beta_{8} + 24 \beta_{9} + 18 \beta_{10} ) q^{75} + ( 21 - 17 \beta_{1} + 17 \beta_{2} - 4 \beta_{3} - 18 \beta_{5} - 7 \beta_{6} - 4 \beta_{8} - 18 \beta_{10} - 6 \beta_{11} ) q^{76} + ( 35 + \beta_{1} - \beta_{2} - 44 \beta_{3} - 8 \beta_{5} + 39 \beta_{6} - 44 \beta_{8} - 8 \beta_{10} - 6 \beta_{11} ) q^{78} + ( 16 \beta_{2} + 25 \beta_{4} + 5 \beta_{8} + 5 \beta_{10} + 25 \beta_{11} ) q^{79} + ( 40 - 8 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + 40 \beta_{7} + 4 \beta_{9} ) q^{80} + ( 9 + 21 \beta_{3} - 21 \beta_{5} - 26 \beta_{6} + 9 \beta_{7} + 26 \beta_{9} ) q^{81} + ( -\beta_{2} + 18 \beta_{4} + 5 \beta_{7} - 36 \beta_{8} - 17 \beta_{9} + 8 \beta_{10} + 18 \beta_{11} ) q^{82} + ( 50 + 14 \beta_{3} - 14 \beta_{5} + 12 \beta_{6} + 14 \beta_{8} - 14 \beta_{10} ) q^{83} + ( -11 \beta_{1} + 11 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{8} - 4 \beta_{10} + 11 \beta_{11} ) q^{85} + ( 10 \beta_{2} + 10 \beta_{4} - 50 \beta_{7} - 20 \beta_{8} - 20 \beta_{9} - 30 \beta_{10} + 10 \beta_{11} ) q^{86} + ( -8 \beta_{1} - 23 \beta_{3} + 8 \beta_{4} - 23 \beta_{5} ) q^{87} + ( -30 + 6 \beta_{1} + 18 \beta_{4} - 12 \beta_{5} - 30 \beta_{7} ) q^{88} + ( 13 \beta_{7} + 36 \beta_{8} + 4 \beta_{9} - 36 \beta_{10} ) q^{89} + ( -31 - 13 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - 11 \beta_{6} + 4 \beta_{8} - 2 \beta_{10} + 6 \beta_{11} ) q^{90} + ( 35 - 7 \beta_{1} + 7 \beta_{2} + 28 \beta_{3} - 34 \beta_{5} - 3 \beta_{6} + 28 \beta_{8} - 34 \beta_{10} + 24 \beta_{11} ) q^{92} + ( -15 \beta_{2} - 21 \beta_{4} + 54 \beta_{8} + 54 \beta_{10} - 21 \beta_{11} ) q^{93} + ( -16 - 12 \beta_{1} - 42 \beta_{3} + 11 \beta_{4} - 5 \beta_{5} + 19 \beta_{6} - 16 \beta_{7} - 19 \beta_{9} ) q^{94} + ( 10 \beta_{1} + 11 \beta_{3} + \beta_{4} + 11 \beta_{5} ) q^{95} + ( -12 \beta_{2} - 16 \beta_{4} - 20 \beta_{7} + 48 \beta_{8} - 36 \beta_{9} - 16 \beta_{11} ) q^{96} + ( -15 \beta_{3} + 15 \beta_{5} + 8 \beta_{6} - 15 \beta_{8} + 15 \beta_{10} ) q^{97} + ( 24 - 21 \beta_{3} + 21 \beta_{5} + 12 \beta_{6} - 21 \beta_{8} + 21 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 2q^{2} + 6q^{3} - 4q^{4} + 56q^{6} + 8q^{8} - 40q^{9} + O(q^{10})$$ $$12q + 2q^{2} + 6q^{3} - 4q^{4} + 56q^{6} + 8q^{8} - 40q^{9} + 6q^{10} + 30q^{11} - 32q^{12} + 16q^{16} - 30q^{17} - 16q^{18} - 78q^{19} - 48q^{20} + 24q^{22} + 76q^{24} - 92q^{25} + 128q^{26} - 156q^{27} - 16q^{30} + 112q^{32} + 78q^{33} - 76q^{34} - 248q^{36} - 80q^{38} - 44q^{40} + 232q^{41} - 200q^{43} + 132q^{44} - 156q^{46} - 176q^{48} + 48q^{50} + 10q^{51} - 132q^{52} + 36q^{54} + 332q^{57} + 4q^{58} + 110q^{59} + 84q^{60} + 96q^{62} - 160q^{64} - 32q^{65} + 138q^{66} + 434q^{67} - 96q^{68} - 328q^{72} - 102q^{73} - 34q^{74} + 60q^{75} + 168q^{76} + 720q^{78} + 256q^{80} - 82q^{81} + 24q^{82} + 536q^{83} + 240q^{86} - 204q^{88} - 214q^{89} - 440q^{90} + 160q^{92} + 16q^{94} - 48q^{96} + 152q^{97} + 504q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2848753 \nu^{11} - 128409 \nu^{10} + 36949178 \nu^{9} + 33927641 \nu^{8} - 280693408 \nu^{7} - 324275527 \nu^{6} + 906643427 \nu^{5} - 266302456 \nu^{4} + 492489836 \nu^{3} - 675410750 \nu^{2} + 298885728 \nu - 114476732$$$$)/20513668$$ $$\beta_{2}$$ $$=$$ $$($$$$908144 \nu^{11} - 6494043 \nu^{10} + 3252536 \nu^{9} + 28087343 \nu^{8} + 91567337 \nu^{7} - 469374709 \nu^{6} + 376525973 \nu^{5} - 267919561 \nu^{4} + 416913921 \nu^{3} - 270088448 \nu^{2} + 111915639 \nu - 26684466$$$$)/5128417$$ $$\beta_{3}$$ $$=$$ $$($$$$-2016555 \nu^{11} + 4608465 \nu^{10} + 11867940 \nu^{9} + 1459025 \nu^{8} - 175954938 \nu^{7} + 201471321 \nu^{6} - 122846749 \nu^{5} + 215038026 \nu^{4} - 177078348 \nu^{3} + 48635510 \nu^{2} - 46482276 \nu + 13204764$$$$)/10256834$$ $$\beta_{4}$$ $$=$$ $$($$$$894552 \nu^{11} - 880921 \nu^{10} - 8410783 \nu^{9} - 6680758 \nu^{8} + 80688819 \nu^{7} + 13990590 \nu^{6} - 103354841 \nu^{5} + 1783409 \nu^{4} - 46480180 \nu^{3} + 87465566 \nu^{2} - 40145506 \nu + 16276324$$$$)/2930524$$ $$\beta_{5}$$ $$=$$ $$($$$$-3736366 \nu^{11} + 5070151 \nu^{10} + 28320593 \nu^{9} + 24171184 \nu^{8} - 310515677 \nu^{7} + 91183726 \nu^{6} - 2408183 \nu^{5} + 130000577 \nu^{4} + 4804172 \nu^{3} - 68273940 \nu^{2} + 70705150 \nu - 32192540$$$$)/10256834$$ $$\beta_{6}$$ $$=$$ $$($$$$-8673537 \nu^{11} + 20159299 \nu^{10} + 48705482 \nu^{9} + 5607853 \nu^{8} - 742433540 \nu^{7} + 912321093 \nu^{6} - 701452097 \nu^{5} + 889048300 \nu^{4} - 667643040 \nu^{3} + 426711138 \nu^{2} - 154926652 \nu + 29235872$$$$)/20513668$$ $$\beta_{7}$$ $$=$$ $$($$$$-8864783 \nu^{11} + 20455402 \nu^{10} + 48834261 \nu^{9} + 8785933 \nu^{8} - 751559539 \nu^{7} + 927115697 \nu^{6} - 797312818 \nu^{5} + 981317799 \nu^{4} - 836256216 \nu^{3} + 526622432 \nu^{2} - 236999870 \nu + 49862132$$$$)/20513668$$ $$\beta_{8}$$ $$=$$ $$($$$$-2781796 \nu^{11} + 6284192 \nu^{10} + 15510516 \nu^{9} + 3410783 \nu^{8} - 234558069 \nu^{7} + 282488251 \nu^{6} - 243549767 \nu^{5} + 278083421 \nu^{4} - 254949459 \nu^{3} + 160381312 \nu^{2} - 43781960 \nu + 15160294$$$$)/5128417$$ $$\beta_{9}$$ $$=$$ $$($$$$15221487 \nu^{11} - 28755150 \nu^{10} - 95322325 \nu^{9} - 55362465 \nu^{8} + 1263522623 \nu^{7} - 1071722261 \nu^{6} + 959373046 \nu^{5} - 1267262003 \nu^{4} + 975668172 \nu^{3} - 603936480 \nu^{2} + 194264578 \nu - 55637788$$$$)/20513668$$ $$\beta_{10}$$ $$=$$ $$($$$$-8729167 \nu^{11} + 23461665 \nu^{10} + 41996941 \nu^{9} - 12733522 \nu^{8} - 752734228 \nu^{7} + 1189138614 \nu^{6} - 1002633735 \nu^{5} + 1141965083 \nu^{4} - 1067833785 \nu^{3} + 678019552 \nu^{2} - 292014698 \nu + 65016998$$$$)/10256834$$ $$\beta_{11}$$ $$=$$ $$($$$$-2679889 \nu^{11} + 7514003 \nu^{10} + 13528514 \nu^{9} - 8114019 \nu^{8} - 240378660 \nu^{7} + 386668629 \nu^{6} - 225704689 \nu^{5} + 291501740 \nu^{4} - 311221520 \nu^{3} + 142102466 \nu^{2} - 51802364 \nu + 6095104$$$$)/2930524$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{7} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{9} + 5 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{11} - 6 \beta_{8} - 3 \beta_{6} - 6 \beta_{3} + 8 \beta_{2} - 8 \beta_{1} + 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$19 \beta_{11} - 20 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} + 37 \beta_{7} + 19 \beta_{4} + 17 \beta_{2}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$45 \beta_{9} + 106 \beta_{7} - 45 \beta_{6} + 20 \beta_{5} - 61 \beta_{4} - 60 \beta_{3} - 54 \beta_{1} + 106$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$231 \beta_{11} - 160 \beta_{10} + 10 \beta_{8} + 96 \beta_{6} - 160 \beta_{5} + 10 \beta_{3} + 205 \beta_{2} - 205 \beta_{1} - 225$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-311 \beta_{11} - 340 \beta_{10} + 497 \beta_{9} + 542 \beta_{8} + 1168 \beta_{7} - 311 \beta_{4} - 276 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-356 \beta_{9} - 837 \beta_{7} + 356 \beta_{6} - 1084 \beta_{5} - 2367 \beta_{4} - 452 \beta_{3} - 2101 \beta_{1} - 837$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-401 \beta_{11} - 4112 \beta_{10} + 4372 \beta_{8} + 4779 \beta_{6} - 4112 \beta_{5} + 4372 \beta_{3} - 356 \beta_{2} + 356 \beta_{1} - 11236$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-21471 \beta_{11} + 5536 \beta_{10} + 1610 \beta_{9} + 8394 \beta_{8} + 3785 \beta_{7} - 21471 \beta_{4} - 19059 \beta_{2}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-40931 \beta_{9} - 96238 \beta_{7} + 40931 \beta_{6} - 42140 \beta_{5} - 17901 \beta_{4} + 30526 \beta_{3} - 15890 \beta_{1} - 96238$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.121721 + 0.507075i −0.407369 + 0.812545i 0.378279 + 0.358951i −2.29733 + 1.90372i 2.79733 − 1.03769i 0.907369 + 0.0534805i 0.121721 − 0.507075i −0.407369 − 0.812545i 0.378279 − 0.358951i −2.29733 − 1.90372i 2.79733 + 1.03769i 0.907369 − 0.0534805i
−1.78207 0.907869i −1.99052 + 3.44767i 2.35155 + 3.23577i −1.63031 + 0.941260i 6.67727 4.33687i 0 −1.25297 7.90127i −3.42430 5.93106i 3.75987 0.197284i
67.2 −1.19654 + 1.60259i 2.66613 4.61787i −1.13656 3.83513i −1.86796 + 1.07847i 4.21039 + 9.79818i 0 7.50608 + 2.76746i −9.71647 16.8294i 0.506759 4.28400i
67.3 0.104798 1.99725i −1.99052 + 3.44767i −3.97803 0.418616i 1.63031 0.941260i 6.67727 + 4.33687i 0 −1.25297 + 7.90127i −3.42430 5.93106i −1.70908 3.35478i
67.4 0.371518 + 1.96519i 0.824388 1.42788i −3.72395 + 1.46021i 3.95004 2.28056i 3.11234 + 1.08960i 0 −4.25310 6.77577i 3.14077 + 5.43997i 5.94924 + 6.91531i
67.5 1.51615 + 1.30434i 0.824388 1.42788i 0.597396 + 3.95514i −3.95004 + 2.28056i 3.11234 1.08960i 0 −4.25310 + 6.77577i 3.14077 + 5.43997i −8.96346 1.69454i
67.6 1.98615 0.234945i 2.66613 4.61787i 3.88960 0.933271i 1.86796 1.07847i 4.21039 9.79818i 0 7.50608 2.76746i −9.71647 16.8294i 3.45667 2.58086i
275.1 −1.78207 + 0.907869i −1.99052 3.44767i 2.35155 3.23577i −1.63031 0.941260i 6.67727 + 4.33687i 0 −1.25297 + 7.90127i −3.42430 + 5.93106i 3.75987 + 0.197284i
275.2 −1.19654 1.60259i 2.66613 + 4.61787i −1.13656 + 3.83513i −1.86796 1.07847i 4.21039 9.79818i 0 7.50608 2.76746i −9.71647 + 16.8294i 0.506759 + 4.28400i
275.3 0.104798 + 1.99725i −1.99052 3.44767i −3.97803 + 0.418616i 1.63031 + 0.941260i 6.67727 4.33687i 0 −1.25297 7.90127i −3.42430 + 5.93106i −1.70908 + 3.35478i
275.4 0.371518 1.96519i 0.824388 + 1.42788i −3.72395 1.46021i 3.95004 + 2.28056i 3.11234 1.08960i 0 −4.25310 + 6.77577i 3.14077 5.43997i 5.94924 6.91531i
275.5 1.51615 1.30434i 0.824388 + 1.42788i 0.597396 3.95514i −3.95004 2.28056i 3.11234 + 1.08960i 0 −4.25310 6.77577i 3.14077 5.43997i −8.96346 + 1.69454i
275.6 1.98615 + 0.234945i 2.66613 + 4.61787i 3.88960 + 0.933271i 1.86796 + 1.07847i 4.21039 + 9.79818i 0 7.50608 + 2.76746i −9.71647 + 16.8294i 3.45667 + 2.58086i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.6 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.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.l 12
7.b odd 2 1 56.3.k.d 12
7.c even 3 1 392.3.g.i 6
7.c even 3 1 inner 392.3.k.l 12
7.d odd 6 1 56.3.k.d 12
7.d odd 6 1 392.3.g.j 6
8.d odd 2 1 inner 392.3.k.l 12
28.d even 2 1 224.3.o.d 12
28.f even 6 1 224.3.o.d 12
28.f even 6 1 1568.3.g.j 6
28.g odd 6 1 1568.3.g.l 6
56.e even 2 1 56.3.k.d 12
56.h odd 2 1 224.3.o.d 12
56.j odd 6 1 224.3.o.d 12
56.j odd 6 1 1568.3.g.j 6
56.k odd 6 1 392.3.g.i 6
56.k odd 6 1 inner 392.3.k.l 12
56.m even 6 1 56.3.k.d 12
56.m even 6 1 392.3.g.j 6
56.p even 6 1 1568.3.g.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 7.b odd 2 1
56.3.k.d 12 7.d odd 6 1
56.3.k.d 12 56.e even 2 1
56.3.k.d 12 56.m even 6 1
224.3.o.d 12 28.d even 2 1
224.3.o.d 12 28.f even 6 1
224.3.o.d 12 56.h odd 2 1
224.3.o.d 12 56.j odd 6 1
392.3.g.i 6 7.c even 3 1
392.3.g.i 6 56.k odd 6 1
392.3.g.j 6 7.d odd 6 1
392.3.g.j 6 56.m even 6 1
392.3.k.l 12 1.a even 1 1 trivial
392.3.k.l 12 7.c even 3 1 inner
392.3.k.l 12 8.d odd 2 1 inner
392.3.k.l 12 56.k odd 6 1 inner
1568.3.g.j 6 28.f even 6 1
1568.3.g.j 6 56.j odd 6 1
1568.3.g.l 6 28.g odd 6 1
1568.3.g.l 6 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{6} - 3 T_{3}^{5} + 28 T_{3}^{4} - 13 T_{3}^{3} + 466 T_{3}^{2} - 665 T_{3} + 1225$$ $$T_{5}^{12} - 29 T_{5}^{10} + 654 T_{5}^{8} - 4737 T_{5}^{6} + 25022 T_{5}^{4} - 64141 T_{5}^{2} + 117649$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 4 T^{2} - 8 T^{3} + 8 T^{4} - 32 T^{5} + 80 T^{6} - 128 T^{7} + 128 T^{8} - 512 T^{9} + 1024 T^{10} - 2048 T^{11} + 4096 T^{12}$$
$3$ $$( 1 - 3 T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 333 T^{7} - 5265 T^{8} + 10206 T^{9} + 6561 T^{10} - 177147 T^{11} + 531441 T^{12} )^{2}$$
$5$ $$1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 285640786875 T^{14} + 5654256640625 T^{16} + 92916992187500 T^{18} + 1217498779296875 T^{20} + 11539459228515625 T^{22} + 59604644775390625 T^{24}$$
$7$ 1
$11$ $$( 1 - 15 T - 129 T^{2} + 924 T^{3} + 35727 T^{4} - 38013 T^{5} - 5198650 T^{6} - 4599573 T^{7} + 523079007 T^{8} + 1636922364 T^{9} - 27652295649 T^{10} - 389061369015 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} )^{2}$$
$17$ $$( 1 + 15 T - 665 T^{2} - 3920 T^{3} + 405365 T^{4} + 1221665 T^{5} - 124870762 T^{6} + 353061185 T^{7} + 33856490165 T^{8} - 94619270480 T^{9} - 4638878698265 T^{10} + 30239908506735 T^{11} + 582622237229761 T^{12} )^{2}$$
$19$ $$( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 2101362589 T^{7} + 34293580187 T^{8} - 186489872284 T^{9} + 2564518019191 T^{10} + 239111584054239 T^{11} + 2213314919066161 T^{12} )^{2}$$
$23$ $$1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 7107938910998636023 T^{14} -$$$$95\!\cdots\!15$$$$T^{16} -$$$$69\!\cdots\!00$$$$T^{18} -$$$$95\!\cdots\!41$$$$T^{20} +$$$$11\!\cdots\!93$$$$T^{22} +$$$$48\!\cdots\!41$$$$T^{24}$$
$29$ $$( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} )^{2}$$
$31$ $$1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} +$$$$12\!\cdots\!55$$$$T^{14} +$$$$10\!\cdots\!13$$$$T^{16} +$$$$65\!\cdots\!04$$$$T^{18} +$$$$42\!\cdots\!99$$$$T^{20} +$$$$23\!\cdots\!61$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$
$37$ $$1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} +$$$$11\!\cdots\!14$$$$T^{12} +$$$$16\!\cdots\!67$$$$T^{14} +$$$$22\!\cdots\!81$$$$T^{16} +$$$$25\!\cdots\!12$$$$T^{18} +$$$$22\!\cdots\!07$$$$T^{20} +$$$$13\!\cdots\!85$$$$T^{22} +$$$$43\!\cdots\!61$$$$T^{24}$$
$41$ $$( 1 - 58 T + 3139 T^{2} - 88960 T^{3} + 5276659 T^{4} - 163894138 T^{5} + 4750104241 T^{6} )^{4}$$
$43$ $$( 1 + 50 T + 4047 T^{2} + 107900 T^{3} + 7482903 T^{4} + 170940050 T^{5} + 6321363049 T^{6} )^{4}$$
$47$ $$1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} -$$$$24\!\cdots\!46$$$$T^{12} -$$$$48\!\cdots\!93$$$$T^{14} -$$$$68\!\cdots\!19$$$$T^{16} +$$$$24\!\cdots\!32$$$$T^{18} +$$$$37\!\cdots\!27$$$$T^{20} +$$$$10\!\cdots\!05$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} +$$$$12\!\cdots\!10$$$$T^{12} +$$$$59\!\cdots\!91$$$$T^{14} +$$$$30\!\cdots\!97$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{18} +$$$$23\!\cdots\!51$$$$T^{20} +$$$$32\!\cdots\!61$$$$T^{22} +$$$$24\!\cdots\!81$$$$T^{24}$$
$59$ $$( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 2082660463687 T^{7} + 619437155669327 T^{8} + 7640666224798022 T^{9} - 1081699833831032807 T^{10} - 28111421431535277055 T^{11} +$$$$17\!\cdots\!81$$$$T^{12} )^{2}$$
$61$ $$1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} -$$$$19\!\cdots\!14$$$$T^{12} -$$$$54\!\cdots\!09$$$$T^{14} -$$$$14\!\cdots\!35$$$$T^{16} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$20\!\cdots\!71$$$$T^{20} +$$$$46\!\cdots\!01$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$
$67$ $$( 1 - 217 T + 18053 T^{2} - 1666970 T^{3} + 209974835 T^{4} - 15078221277 T^{5} + 815515698066 T^{6} - 67686135312453 T^{7} + 4231228307040035 T^{8} - 150791409324257930 T^{9} + 7330739782930039973 T^{10} -$$$$39\!\cdots\!33$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$
$71$ $$( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$( 1 + 51 T - 9165 T^{2} - 579660 T^{3} + 45654729 T^{4} + 1993134921 T^{5} - 160997144218 T^{6} + 10621415994009 T^{7} + 1296513996931689 T^{8} - 87722397610681740 T^{9} - 7391206742209252365 T^{10} +$$$$21\!\cdots\!99$$$$T^{11} +$$$$22\!\cdots\!21$$$$T^{12} )^{2}$$
$79$ $$1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} -$$$$37\!\cdots\!74$$$$T^{12} -$$$$17\!\cdots\!25$$$$T^{14} -$$$$14\!\cdots\!39$$$$T^{16} +$$$$37\!\cdots\!92$$$$T^{18} +$$$$34\!\cdots\!43$$$$T^{20} +$$$$14\!\cdots\!93$$$$T^{22} +$$$$34\!\cdots\!81$$$$T^{24}$$
$83$ $$( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{4}$$
$89$ $$( 1 + 107 T + 1395 T^{2} - 497084 T^{3} - 62702935 T^{4} - 2422278015 T^{5} + 93422604838 T^{6} - 19186864156815 T^{7} - 3934122659177335 T^{8} - 247041448036057724 T^{9} + 5491541383954402995 T^{10} +$$$$33\!\cdots\!07$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} )^{2}$$
$97$ $$( 1 - 38 T + 25191 T^{2} - 597344 T^{3} + 237022119 T^{4} - 3364112678 T^{5} + 832972004929 T^{6} )^{4}$$