# Properties

 Label 322.2.g.a Level $322$ Weight $2$ Character orbit 322.g Analytic conductor $2.571$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 18 x^{14} + 226 x^{12} + 1434 x^{10} + 6585 x^{8} + 14406 x^{6} + 22423 x^{4} + 8085 x^{2} + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$7$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( 1 - \beta_{3} - \beta_{5} ) q^{6} + \beta_{12} q^{7} + q^{8} + ( -1 + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{5} ) q^{4} + \beta_{6} q^{5} + ( 1 - \beta_{3} - \beta_{5} ) q^{6} + \beta_{12} q^{7} + q^{8} + ( -1 + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{9} -\beta_{13} q^{10} -\beta_{9} q^{11} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{12} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{13} + ( -\beta_{6} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{14} + ( \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{15} -\beta_{5} q^{16} + ( \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{17} + ( 3 - \beta_{1} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{18} + \beta_{6} q^{19} + ( -\beta_{6} + \beta_{13} ) q^{20} + ( \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{21} + ( -\beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{22} + ( \beta_{1} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{23} -\beta_{1} q^{24} + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{25} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{26} + ( -5 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + 7 \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{27} + ( \beta_{6} - \beta_{13} - \beta_{14} ) q^{28} + ( 1 - \beta_{2} + \beta_{8} ) q^{29} + ( 2 \beta_{6} - \beta_{10} + \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{30} + ( -1 - \beta_{4} - \beta_{5} - \beta_{7} ) q^{31} + ( -1 + \beta_{5} ) q^{32} + ( -2 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{33} + ( -\beta_{6} - \beta_{10} + \beta_{15} ) q^{34} + ( 3 - \beta_{1} - 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{8} ) q^{35} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{36} + ( \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{37} -\beta_{13} q^{38} + ( 5 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{39} + \beta_{6} q^{40} + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{41} + ( -\beta_{6} - \beta_{9} + \beta_{13} + \beta_{15} ) q^{42} + ( -\beta_{6} - \beta_{9} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{43} + ( \beta_{6} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{44} + ( -4 \beta_{6} + \beta_{9} + 3 \beta_{10} + \beta_{11} - 4 \beta_{12} + 5 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{45} + ( \beta_{4} - \beta_{7} + \beta_{9} ) q^{46} + ( 3 - \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{8} ) q^{47} + ( 1 - \beta_{3} - \beta_{5} ) q^{48} + ( -5 + \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} ) q^{49} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{50} + ( 3 \beta_{6} - \beta_{9} - 2 \beta_{10} + 3 \beta_{12} - 2 \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{51} + ( -\beta_{1} + \beta_{2} ) q^{52} + ( 2 \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{53} + ( 7 - 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - \beta_{7} - \beta_{8} ) q^{54} + ( 4 - \beta_{1} - 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{7} ) q^{55} + \beta_{12} q^{56} + ( \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{57} + ( 2 \beta_{2} - \beta_{5} + \beta_{8} ) q^{58} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{59} + ( -2 \beta_{6} - \beta_{9} + \beta_{13} ) q^{60} + ( -\beta_{6} - \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{61} + ( -3 + \beta_{1} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{62} + ( 5 \beta_{6} - \beta_{9} - 3 \beta_{10} + 3 \beta_{12} - 6 \beta_{13} - 3 \beta_{14} ) q^{63} + q^{64} + ( 5 \beta_{6} - \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{65} + ( -2 \beta_{6} + \beta_{10} + \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{66} + ( 2 \beta_{6} - \beta_{9} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{67} + ( \beta_{6} - \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{68} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{69} + ( -2 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} ) q^{70} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{71} + ( -1 + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{72} + ( -1 - \beta_{4} - \beta_{5} - \beta_{7} ) q^{73} + ( -2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{74} + ( -10 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + \beta_{7} ) q^{75} + ( -\beta_{6} + \beta_{13} ) q^{76} + ( -6 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + 5 \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{77} + ( -4 + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{78} + ( -\beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{79} -\beta_{13} q^{80} + ( -10 + 4 \beta_{1} + \beta_{2} + 8 \beta_{3} - \beta_{4} + 10 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{81} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{82} + ( -5 \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 4 \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{83} + ( \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{84} + ( 2 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{8} ) q^{85} + ( \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{86} + ( 1 - 3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{87} -\beta_{9} q^{88} + ( -3 \beta_{6} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{89} + ( 5 \beta_{6} + 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - 4 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{90} + ( -3 \beta_{6} - \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{91} + ( -\beta_{1} - \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{92} + ( 5 + 3 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 7 \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{94} + ( -6 + \beta_{1} + 2 \beta_{3} + 6 \beta_{5} ) q^{95} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{96} + ( 3 \beta_{6} - 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - 4 \beta_{14} ) q^{97} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{8} ) q^{98} + ( -2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 5 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{2} + 6q^{3} - 8q^{4} + 16q^{8} + 10q^{9} + O(q^{10})$$ $$16q - 8q^{2} + 6q^{3} - 8q^{4} + 16q^{8} + 10q^{9} - 6q^{12} - 8q^{16} + 10q^{18} - 4q^{23} + 6q^{24} + 2q^{25} - 6q^{26} + 16q^{29} - 24q^{31} - 8q^{32} + 4q^{35} - 20q^{36} + 22q^{39} - 4q^{46} + 30q^{47} - 58q^{49} - 4q^{50} + 6q^{52} + 54q^{54} - 8q^{58} + 36q^{59} + 16q^{64} - 32q^{70} - 12q^{71} + 10q^{72} - 24q^{73} - 96q^{75} - 38q^{77} - 44q^{78} - 36q^{81} - 6q^{82} + 24q^{85} + 42q^{87} + 8q^{92} - 38q^{93} - 30q^{94} - 38q^{95} - 6q^{96} + 56q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 18 x^{14} + 226 x^{12} + 1434 x^{10} + 6585 x^{8} + 14406 x^{6} + 22423 x^{4} + 8085 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-16016843 \nu^{14} - 357616089 \nu^{12} - 4992038792 \nu^{10} - 39156407581 \nu^{8} - 206205174673 \nu^{6} - 598846393691 \nu^{4} - 992457622834 \nu^{2} - 650423475226$$$$)/ 409935420069$$ $$\beta_{2}$$ $$=$$ $$($$$$-25899908 \nu^{14} - 838784719 \nu^{12} - 12366108886 \nu^{10} - 116812942191 \nu^{8} - 645698199869 \nu^{6} - 2357725853427 \nu^{4} - 3791385151776 \nu^{2} - 2481148141444$$$$)/ 409935420069$$ $$\beta_{3}$$ $$=$$ $$($$$$31483524 \nu^{14} + 647630005 \nu^{12} + 8339521600 \nu^{10} + 58435186409 \nu^{8} + 266212664966 \nu^{6} + 629143386508 \nu^{4} + 679011213031 \nu^{2} + 362977184262$$$$)/ 409935420069$$ $$\beta_{4}$$ $$=$$ $$($$$$46792861 \nu^{14} + 853307341 \nu^{12} + 11030105099 \nu^{10} + 71855612665 \nu^{8} + 346946458867 \nu^{6} + 723058716380 \nu^{4} + 1037405997123 \nu^{2} - 1235543183245$$$$)/ 409935420069$$ $$\beta_{5}$$ $$=$$ $$($$$$68184274 \nu^{14} + 1196540914 \nu^{12} + 14913954672 \nu^{10} + 91738182609 \nu^{8} + 416294239206 \nu^{6} + 841521367050 \nu^{4} + 1404683653213 \nu^{2} + 506321481001$$$$)/ 409935420069$$ $$\beta_{6}$$ $$=$$ $$($$$$-6975738 \nu^{15} - 112789235 \nu^{13} - 1368597087 \nu^{11} - 7411650444 \nu^{9} - 30826320447 \nu^{7} - 28154149749 \nu^{5} - 10188065349 \nu^{3} + 234153598717 \nu$$$$)/ 58562202867$$ $$\beta_{7}$$ $$=$$ $$($$$$-294128509 \nu^{14} - 5129397229 \nu^{12} - 63539668261 \nu^{10} - 386835300380 \nu^{8} - 1734524737163 \nu^{6} - 3484548118870 \nu^{4} - 5576076848873 \nu^{2} - 1717473487905$$$$)/ 409935420069$$ $$\beta_{8}$$ $$=$$ $$($$$$-379875222 \nu^{14} - 6640569708 \nu^{12} - 81813898297 \nu^{10} - 492908101323 \nu^{8} - 2133415874648 \nu^{6} - 3786373832724 \nu^{4} - 4308366097873 \nu^{2} + 671866451907$$$$)/ 409935420069$$ $$\beta_{9}$$ $$=$$ $$($$$$500045446 \nu^{15} + 8793640344 \nu^{13} + 109886589690 \nu^{11} + 679206852725 \nu^{9} + 3122871260604 \nu^{7} + 6496688880199 \nu^{5} + 11555957875297 \nu^{3} - 2682037157653 \nu$$$$)/ 2869547940483$$ $$\beta_{10}$$ $$=$$ $$($$$$159118374 \nu^{15} + 2937898622 \nu^{13} + 36966138724 \nu^{11} + 239063115702 \nu^{9} + 1081154919617 \nu^{7} + 2329068000147 \nu^{5} + 2680850200333 \nu^{3} - 421218231742 \nu$$$$)/ 409935420069$$ $$\beta_{11}$$ $$=$$ $$($$$$1482542225 \nu^{15} + 24724308082 \nu^{13} + 301479751327 \nu^{11} + 1716042596475 \nu^{9} + 7363397425490 \nu^{7} + 11201140500018 \nu^{5} + 17033080757013 \nu^{3} - 9325440613244 \nu$$$$)/ 2869547940483$$ $$\beta_{12}$$ $$=$$ $$($$$$1812976145 \nu^{15} + 33070488812 \nu^{13} + 418326213629 \nu^{11} + 2705049202442 \nu^{9} + 12617397123280 \nu^{7} + 28821327863734 \nu^{5} + 45480525704329 \nu^{3} + 15022740886300 \nu$$$$)/ 2869547940483$$ $$\beta_{13}$$ $$=$$ $$($$$$1849992808 \nu^{15} + 33041863886 \nu^{13} + 413665642185 \nu^{11} + 2602270360005 \nu^{9} + 11908073199276 \nu^{7} + 25745509497624 \nu^{5} + 40441071229375 \nu^{3} + 14580373128671 \nu$$$$)/ 2869547940483$$ $$\beta_{14}$$ $$=$$ $$($$$$-2619126371 \nu^{15} - 45707452382 \nu^{13} - 565731629381 \nu^{11} - 3426523980851 \nu^{9} - 15140293448971 \nu^{7} - 28110655811896 \nu^{5} - 37522204462150 \nu^{3} + 9891611938649 \nu$$$$)/ 2869547940483$$ $$\beta_{15}$$ $$=$$ $$($$$$-3550950679 \nu^{15} - 62041171232 \nu^{13} - 769038087803 \nu^{11} - 4673818962915 \nu^{9} - 20759601211693 \nu^{7} - 39285915639747 \nu^{5} - 55464603304326 \nu^{3} + 2315919560941 \nu$$$$)/ 2869547940483$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - 2 \beta_{14} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{9}$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + 5 \beta_{5} - \beta_{4} - 5$$ $$\nu^{3}$$ $$=$$ $$($$$$-16 \beta_{15} + 31 \beta_{14} - 7 \beta_{12} + 8 \beta_{11} + 15 \beta_{10} + 17 \beta_{9}$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} - 9 \beta_{7} - 36 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 11 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$($$$$213 \beta_{15} - 253 \beta_{14} - 140 \beta_{13} + 182 \beta_{12} + 142 \beta_{11} - 40 \beta_{10} - 162 \beta_{9} + 182 \beta_{6}$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$13 \beta_{8} + 33 \beta_{5} + 77 \beta_{4} + 33 \beta_{3} - 13 \beta_{2} + 143 \beta_{1} + 289$$ $$\nu^{7}$$ $$=$$ $$($$$$-659 \beta_{15} - 366 \beta_{14} + 952 \beta_{13} - 1001 \beta_{12} - 1928 \beta_{11} - 976 \beta_{10} + 635 \beta_{9} - 1120 \beta_{6}$$$$)/7$$ $$\nu^{8}$$ $$=$$ $$-272 \beta_{8} + 669 \beta_{7} + 2052 \beta_{5} - 669 \beta_{4} - 796 \beta_{3} - 136 \beta_{2} - 398 \beta_{1} - 2052$$ $$\nu^{9}$$ $$=$$ $$($$$$-11468 \beta_{15} + 28780 \beta_{14} + 10192 \beta_{13} - 10255 \beta_{12} + 6175 \beta_{11} + 15548 \beta_{10} + 8270 \beta_{9} - 11074 \beta_{6}$$$$)/7$$ $$\nu^{10}$$ $$=$$ $$1339 \beta_{8} - 5932 \beta_{7} - 21473 \beta_{5} + 4260 \beta_{3} + 2678 \beta_{2} - 10192 \beta_{1} - 4260$$ $$\nu^{11}$$ $$=$$ $$($$$$162599 \beta_{15} - 249145 \beta_{14} - 199997 \beta_{13} + 202328 \beta_{12} + 104708 \beta_{11} - 75472 \beta_{10} - 147981 \beta_{9} + 202328 \beta_{6}$$$$)/7$$ $$\nu^{12}$$ $$=$$ $$12870 \beta_{8} + 43023 \beta_{5} + 53478 \beta_{4} + 43023 \beta_{3} - 12870 \beta_{2} + 139524 \beta_{1} + 192335$$ $$\nu^{13}$$ $$=$$ $$($$$$-542144 \beta_{15} - 228601 \beta_{14} + 855687 \beta_{13} - 976668 \beta_{12} - 1505451 \beta_{11} - 649764 \beta_{10} + 748067 \beta_{9} - 717801 \beta_{6}$$$$)/7$$ $$\nu^{14}$$ $$=$$ $$-244482 \beta_{8} + 487880 \beta_{7} + 1326778 \beta_{5} - 487880 \beta_{4} - 841974 \beta_{3} - 122241 \beta_{2} - 420987 \beta_{1} - 1326778$$ $$\nu^{15}$$ $$=$$ $$($$$$-8908181 \beta_{15} + 24524552 \beta_{14} + 10931949 \beta_{13} - 9308978 \beta_{12} + 5071858 \beta_{11} + 13145301 \beta_{10} + 5906596 \beta_{9} - 12167484 \beta_{6}$$$$)/7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$\beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 1.52769 + 2.64604i −1.52769 − 2.64604i −0.768626 − 1.33130i 0.768626 + 1.33130i −0.306090 − 0.530164i 0.306090 + 0.530164i 1.21724 + 2.10833i −1.21724 − 2.10833i 1.52769 − 2.64604i −1.52769 + 2.64604i −0.768626 + 1.33130i 0.768626 − 1.33130i −0.306090 + 0.530164i 0.306090 − 0.530164i 1.21724 − 2.10833i −1.21724 + 2.10833i
−0.500000 0.866025i −1.86037 1.07408i −0.500000 + 0.866025i −0.264227 0.457654i 2.14817i −1.26346 2.32458i 1.00000 0.807314 + 1.39831i −0.264227 + 0.457654i
45.2 −0.500000 0.866025i −1.86037 1.07408i −0.500000 + 0.866025i 0.264227 + 0.457654i 2.14817i 1.26346 + 2.32458i 1.00000 0.807314 + 1.39831i 0.264227 0.457654i
45.3 −0.500000 0.866025i −0.508912 0.293820i −0.500000 + 0.866025i −0.863449 1.49554i 0.587641i 1.63208 + 2.08239i 1.00000 −1.32734 2.29902i −0.863449 + 1.49554i
45.4 −0.500000 0.866025i −0.508912 0.293820i −0.500000 + 0.866025i 0.863449 + 1.49554i 0.587641i −1.63208 2.08239i 1.00000 −1.32734 2.29902i 0.863449 1.49554i
45.5 −0.500000 0.866025i 1.05590 + 0.609623i −0.500000 + 0.866025i −1.23610 2.14099i 1.21925i 1.54219 2.14980i 1.00000 −0.756718 1.31067i −1.23610 + 2.14099i
45.6 −0.500000 0.866025i 1.05590 + 0.609623i −0.500000 + 0.866025i 1.23610 + 2.14099i 1.21925i −1.54219 + 2.14980i 1.00000 −0.756718 1.31067i 1.23610 2.14099i
45.7 −0.500000 0.866025i 2.81338 + 1.62431i −0.500000 + 0.866025i −1.55135 2.68702i 3.24861i 0.334108 + 2.62457i 1.00000 3.77674 + 6.54151i −1.55135 + 2.68702i
45.8 −0.500000 0.866025i 2.81338 + 1.62431i −0.500000 + 0.866025i 1.55135 + 2.68702i 3.24861i −0.334108 2.62457i 1.00000 3.77674 + 6.54151i 1.55135 2.68702i
229.1 −0.500000 + 0.866025i −1.86037 + 1.07408i −0.500000 0.866025i −0.264227 + 0.457654i 2.14817i −1.26346 + 2.32458i 1.00000 0.807314 1.39831i −0.264227 0.457654i
229.2 −0.500000 + 0.866025i −1.86037 + 1.07408i −0.500000 0.866025i 0.264227 0.457654i 2.14817i 1.26346 2.32458i 1.00000 0.807314 1.39831i 0.264227 + 0.457654i
229.3 −0.500000 + 0.866025i −0.508912 + 0.293820i −0.500000 0.866025i −0.863449 + 1.49554i 0.587641i 1.63208 2.08239i 1.00000 −1.32734 + 2.29902i −0.863449 1.49554i
229.4 −0.500000 + 0.866025i −0.508912 + 0.293820i −0.500000 0.866025i 0.863449 1.49554i 0.587641i −1.63208 + 2.08239i 1.00000 −1.32734 + 2.29902i 0.863449 + 1.49554i
229.5 −0.500000 + 0.866025i 1.05590 0.609623i −0.500000 0.866025i −1.23610 + 2.14099i 1.21925i 1.54219 + 2.14980i 1.00000 −0.756718 + 1.31067i −1.23610 2.14099i
229.6 −0.500000 + 0.866025i 1.05590 0.609623i −0.500000 0.866025i 1.23610 2.14099i 1.21925i −1.54219 2.14980i 1.00000 −0.756718 + 1.31067i 1.23610 + 2.14099i
229.7 −0.500000 + 0.866025i 2.81338 1.62431i −0.500000 0.866025i −1.55135 + 2.68702i 3.24861i 0.334108 2.62457i 1.00000 3.77674 6.54151i −1.55135 2.68702i
229.8 −0.500000 + 0.866025i 2.81338 1.62431i −0.500000 0.866025i 1.55135 2.68702i 3.24861i −0.334108 + 2.62457i 1.00000 3.77674 6.54151i 1.55135 + 2.68702i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.8 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.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.g.a 16
7.c even 3 1 2254.2.c.c 16
7.d odd 6 1 inner 322.2.g.a 16
7.d odd 6 1 2254.2.c.c 16
23.b odd 2 1 inner 322.2.g.a 16
161.f odd 6 1 2254.2.c.c 16
161.g even 6 1 inner 322.2.g.a 16
161.g even 6 1 2254.2.c.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.a 16 1.a even 1 1 trivial
322.2.g.a 16 7.d odd 6 1 inner
322.2.g.a 16 23.b odd 2 1 inner
322.2.g.a 16 161.g even 6 1 inner
2254.2.c.c 16 7.c even 3 1
2254.2.c.c 16 7.d odd 6 1
2254.2.c.c 16 161.f odd 6 1
2254.2.c.c 16 161.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{8}$$
$3$ $$( 25 + 45 T - 8 T^{2} - 63 T^{3} + 35 T^{4} + 21 T^{5} - 4 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$5$ $$2401 + 10045 T^{2} + 36586 T^{4} + 20893 T^{6} + 8377 T^{8} + 1699 T^{10} + 250 T^{12} + 19 T^{14} + T^{16}$$
$7$ $$5764801 + 3411821 T^{2} + 1152480 T^{4} + 264061 T^{6} + 43787 T^{8} + 5389 T^{10} + 480 T^{12} + 29 T^{14} + T^{16}$$
$11$ $$5764801 - 5577523 T^{2} + 3994145 T^{4} - 1145344 T^{6} + 236443 T^{8} - 21050 T^{10} + 1352 T^{12} - 44 T^{14} + T^{16}$$
$13$ $$( 51529 + 14828 T^{2} + 1538 T^{4} + 67 T^{6} + T^{8} )^{2}$$
$17$ $$671898241 + 845957756 T^{2} + 981746560 T^{4} + 99721334 T^{6} + 7020499 T^{8} + 259544 T^{10} + 6985 T^{12} + 101 T^{14} + T^{16}$$
$19$ $$2401 + 10045 T^{2} + 36586 T^{4} + 20893 T^{6} + 8377 T^{8} + 1699 T^{10} + 250 T^{12} + 19 T^{14} + T^{16}$$
$23$ $$78310985281 + 13619301788 T + 1776430668 T^{2} + 154472232 T^{3} + 34140602 T^{4} + 7446204 T^{5} + 2632304 T^{6} + 1339244 T^{7} + 108771 T^{8} + 58228 T^{9} + 4976 T^{10} + 612 T^{11} + 122 T^{12} + 24 T^{13} + 12 T^{14} + 4 T^{15} + T^{16}$$
$29$ $$( -335 + 416 T - 87 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$31$ $$( 561001 - 199983 T - 15185 T^{2} + 13884 T^{3} + 887 T^{4} - 624 T^{5} - 4 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$37$ $$1958569461121 - 1022921497325 T^{2} + 500073035267 T^{4} - 17083730378 T^{6} + 394761145 T^{8} - 5229778 T^{10} + 50654 T^{12} - 274 T^{14} + T^{16}$$
$41$ $$( 51529 + 14828 T^{2} + 1538 T^{4} + 67 T^{6} + T^{8} )^{2}$$
$43$ $$( 117649 + 146461 T^{2} + 16268 T^{4} + 260 T^{6} + T^{8} )^{2}$$
$47$ $$( 14161 + 18921 T + 7832 T^{2} - 795 T^{3} - 889 T^{4} + 75 T^{5} + 70 T^{6} - 15 T^{7} + T^{8} )^{2}$$
$53$ $$13731656640625 - 2902975508125 T^{2} + 512751106484 T^{4} - 19038752515 T^{6} + 494947933 T^{8} - 6906401 T^{10} + 69476 T^{12} - 311 T^{14} + T^{16}$$
$59$ $$( 2025 - 27945 T + 129897 T^{2} - 18630 T^{3} - 2781 T^{4} + 540 T^{5} + 78 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$61$ $$212558803681 + 53731562304 T^{2} + 10320177820 T^{4} + 683586798 T^{6} + 31777503 T^{8} + 849540 T^{10} + 16333 T^{12} + 153 T^{14} + T^{16}$$
$67$ $$2401 - 493822 T^{2} + 100829810 T^{4} - 151408018 T^{6} + 223311517 T^{8} - 3661214 T^{10} + 44999 T^{12} - 245 T^{14} + T^{16}$$
$71$ $$( 693 - 540 T - 132 T^{2} + 3 T^{3} + T^{4} )^{4}$$
$73$ $$( 561001 - 199983 T - 15185 T^{2} + 13884 T^{3} + 887 T^{4} - 624 T^{5} - 4 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$79$ $$4408324359201 - 1850109612372 T^{2} + 715693242240 T^{4} - 24232284162 T^{6} + 568660419 T^{8} - 7007688 T^{10} + 62865 T^{12} - 303 T^{14} + T^{16}$$
$83$ $$( 2941225 - 1752436 T^{2} + 73917 T^{4} - 520 T^{6} + T^{8} )^{2}$$
$89$ $$13539502754078161 + 653203448052337 T^{2} + 21139770445210 T^{4} + 378752661649 T^{6} + 4895590453 T^{8} + 35398627 T^{10} + 184378 T^{12} + 523 T^{14} + T^{16}$$
$97$ $$( 62236321 - 3493308 T^{2} + 65297 T^{4} - 456 T^{6} + T^{8} )^{2}$$