# Properties

 Label 165.2.d.c Level $165$ Weight $2$ Character orbit 165.d Analytic conductor $1.318$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 24 x^{14} + 244 x^{12} - 1224 x^{10} + 2880 x^{8} - 2208 x^{6} + 3976 x^{4} + 432 x^{2} + 2116$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + \beta_{4} q^{3} + ( -2 - \beta_{7} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{11} - \beta_{13} ) q^{5} + \beta_{14} q^{6} + \beta_{10} q^{7} + ( \beta_{5} - \beta_{12} ) q^{8} + ( \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} + \beta_{4} q^{3} + ( -2 - \beta_{7} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{11} - \beta_{13} ) q^{5} + \beta_{14} q^{6} + \beta_{10} q^{7} + ( \beta_{5} - \beta_{12} ) q^{8} + ( \beta_{6} - \beta_{7} - \beta_{11} + \beta_{13} ) q^{9} + ( -\beta_{2} - \beta_{6} - \beta_{9} ) q^{10} + ( \beta_{8} + \beta_{11} - \beta_{13} ) q^{11} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{11} ) q^{12} -\beta_{10} q^{13} + ( 3 \beta_{6} + \beta_{11} - \beta_{13} ) q^{14} + ( \beta_{3} - \beta_{6} + \beta_{7} + \beta_{13} ) q^{15} + ( -1 + 2 \beta_{7} ) q^{16} + \beta_{12} q^{17} + ( 2 \beta_{2} + \beta_{5} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{18} + ( -\beta_{14} - \beta_{15} ) q^{19} + ( 3 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{11} + 2 \beta_{13} ) q^{20} + ( \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{13} ) q^{21} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{22} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{11} - \beta_{13} ) q^{23} + ( \beta_{6} + \beta_{8} + \beta_{9} + \beta_{15} ) q^{24} + ( 1 + 2 \beta_{7} - \beta_{11} - \beta_{13} ) q^{25} + ( -3 \beta_{6} - \beta_{11} + \beta_{13} ) q^{26} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{11} ) q^{27} + ( -2 \beta_{2} - 3 \beta_{10} - \beta_{11} + \beta_{13} ) q^{28} + ( -2 \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{29} + ( \beta_{2} - \beta_{5} - \beta_{8} + 2 \beta_{10} + \beta_{12} + \beta_{15} ) q^{30} -2 \beta_{7} q^{31} + \beta_{5} q^{32} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{33} + ( 1 - 3 \beta_{7} ) q^{34} + ( \beta_{12} + \beta_{14} - \beta_{15} ) q^{35} + ( 3 + \beta_{6} + 2 \beta_{7} + 2 \beta_{11} - 2 \beta_{13} ) q^{36} + ( -2 \beta_{3} + 2 \beta_{4} + \beta_{11} + \beta_{13} ) q^{37} + ( -2 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + \beta_{11} - \beta_{13} ) q^{38} + ( -\beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{39} + ( -\beta_{2} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{40} + ( -\beta_{14} + \beta_{15} ) q^{41} + ( -5 \beta_{1} + 2 \beta_{3} + \beta_{11} - 4 \beta_{13} ) q^{42} + ( 4 \beta_{2} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{43} + ( -\beta_{6} - \beta_{8} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{44} + ( 3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{11} + \beta_{13} ) q^{45} + ( -2 \beta_{6} - 2 \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{46} + ( \beta_{3} + \beta_{4} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{11} ) q^{48} + ( 1 + 2 \beta_{7} ) q^{49} + ( -3 \beta_{5} + 2 \beta_{8} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{50} + ( -\beta_{6} - \beta_{8} - \beta_{9} - \beta_{14} - \beta_{15} ) q^{51} + ( 2 \beta_{2} + 3 \beta_{10} + \beta_{11} - \beta_{13} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{11} + 2 \beta_{13} ) q^{53} + ( 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{15} ) q^{54} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{55} + ( \beta_{6} - 4 \beta_{11} + 4 \beta_{13} ) q^{56} + ( -2 \beta_{2} + 2 \beta_{5} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{57} + ( -3 \beta_{3} + 3 \beta_{4} - 2 \beta_{11} - 2 \beta_{13} ) q^{58} + ( -2 \beta_{6} + 2 \beta_{11} - 2 \beta_{13} ) q^{59} + ( -3 + \beta_{1} - \beta_{3} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} - 2 \beta_{13} ) q^{60} + ( 2 \beta_{6} + 2 \beta_{9} - \beta_{11} + \beta_{13} ) q^{61} + ( 2 \beta_{5} - 2 \beta_{12} ) q^{62} + ( -2 \beta_{2} + 2 \beta_{5} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{63} + ( 2 + 5 \beta_{7} ) q^{64} + ( -\beta_{12} - \beta_{14} + \beta_{15} ) q^{65} + ( -3 - 6 \beta_{6} - 4 \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} + \beta_{15} ) q^{66} + ( 5 \beta_{3} - 5 \beta_{4} + \beta_{11} + \beta_{13} ) q^{67} + ( 2 \beta_{5} - \beta_{12} ) q^{68} + ( -3 - 3 \beta_{6} + 3 \beta_{7} ) q^{69} + ( 1 + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{7} - \beta_{11} - \beta_{13} ) q^{70} + ( 8 \beta_{6} - 2 \beta_{11} + 2 \beta_{13} ) q^{71} + ( -3 \beta_{5} - 3 \beta_{10} ) q^{72} + \beta_{10} q^{73} + ( -2 \beta_{8} - \beta_{11} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{74} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 4 \beta_{6} + 2 \beta_{7} + \beta_{11} + \beta_{13} ) q^{75} + ( -2 \beta_{6} - 2 \beta_{9} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{76} + ( 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{77} + ( 5 \beta_{1} - 2 \beta_{3} - \beta_{11} + 4 \beta_{13} ) q^{78} + ( 3 \beta_{14} + 3 \beta_{15} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} - \beta_{11} + \beta_{13} ) q^{80} + ( -3 + 6 \beta_{6} ) q^{81} + ( -5 \beta_{3} + 5 \beta_{4} + \beta_{11} + \beta_{13} ) q^{82} + ( 2 \beta_{5} - 3 \beta_{12} ) q^{83} + ( -3 \beta_{6} + \beta_{8} - 3 \beta_{9} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{15} ) q^{84} + ( 2 \beta_{2} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{85} + ( -5 \beta_{6} + 7 \beta_{11} - 7 \beta_{13} ) q^{86} + ( -4 \beta_{2} - 2 \beta_{5} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{87} + ( \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{10} + \beta_{11} ) q^{88} + ( -4 \beta_{6} - \beta_{11} + \beta_{13} ) q^{89} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - 2 \beta_{15} ) q^{90} + ( -8 - 2 \beta_{7} ) q^{91} + ( 4 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{11} + 2 \beta_{13} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{11} ) q^{93} + ( \beta_{14} + \beta_{15} ) q^{94} + ( 2 \beta_{5} + 2 \beta_{8} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} -\beta_{14} q^{96} + ( 2 \beta_{3} - 2 \beta_{4} - \beta_{11} - \beta_{13} ) q^{97} + ( -3 \beta_{5} + 2 \beta_{12} ) q^{98} + ( 3 - 3 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} - \beta_{13} - 2 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 32q^{4} + O(q^{10})$$ $$16q - 32q^{4} - 16q^{16} + 16q^{25} + 16q^{34} + 48q^{36} + 48q^{45} + 16q^{49} - 16q^{55} - 48q^{60} + 32q^{64} - 48q^{66} - 48q^{69} + 16q^{70} - 48q^{81} - 128q^{91} + 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 24 x^{14} + 244 x^{12} - 1224 x^{10} + 2880 x^{8} - 2208 x^{6} + 3976 x^{4} + 432 x^{2} + 2116$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$363 \nu^{15} - 3071 \nu^{13} - 28361 \nu^{11} + 634291 \nu^{9} - 3652348 \nu^{7} + 6427908 \nu^{5} + 16071546 \nu^{3} - 6737122 \nu$$$$)/11947672$$ $$\beta_{2}$$ $$=$$ $$($$$$-9845 \nu^{15} + 219416 \nu^{13} - 2309677 \nu^{11} + 15411692 \nu^{9} - 74815508 \nu^{7} + 249472692 \nu^{5} - 466070970 \nu^{3} + 363771080 \nu$$$$)/ 155319736$$ $$\beta_{3}$$ $$=$$ $$($$$$-8971 \nu^{15} + 56394 \nu^{14} + 43388 \nu^{13} - 1403038 \nu^{12} + 2069306 \nu^{11} + 14059526 \nu^{10} - 33515798 \nu^{9} - 64779104 \nu^{8} + 206636446 \nu^{7} + 114208016 \nu^{6} - 561683060 \nu^{5} - 20587892 \nu^{4} + 446513512 \nu^{3} + 411164364 \nu^{2} - 469878556 \nu + 68700632$$$$)/ 310639472$$ $$\beta_{4}$$ $$=$$ $$($$$$-8971 \nu^{15} - 56394 \nu^{14} + 43388 \nu^{13} + 1403038 \nu^{12} + 2069306 \nu^{11} - 14059526 \nu^{10} - 33515798 \nu^{9} + 64779104 \nu^{8} + 206636446 \nu^{7} - 114208016 \nu^{6} - 561683060 \nu^{5} + 20587892 \nu^{4} + 446513512 \nu^{3} - 411164364 \nu^{2} - 469878556 \nu - 68700632$$$$)/ 310639472$$ $$\beta_{5}$$ $$=$$ $$($$$$19606 \nu^{14} - 567841 \nu^{12} + 6959163 \nu^{10} - 44547989 \nu^{8} + 149567906 \nu^{6} - 225854740 \nu^{4} + 101872258 \nu^{2} - 106793094$$$$)/77659868$$ $$\beta_{6}$$ $$=$$ $$($$$$60124 \nu^{15} - 1413405 \nu^{13} + 14012484 \nu^{11} - 67446266 \nu^{9} + 146255268 \nu^{7} - 83508246 \nu^{5} + 229564832 \nu^{3} + 198491328 \nu$$$$)/ 155319736$$ $$\beta_{7}$$ $$=$$ $$($$$$-232 \nu^{14} + 5459 \nu^{12} - 53458 \nu^{10} + 250659 \nu^{8} - 493628 \nu^{6} + 53274 \nu^{4} - 186956 \nu^{2} - 340216$$$$)/271538$$ $$\beta_{8}$$ $$=$$ $$($$$$58483 \nu^{15} - 29571 \nu^{14} - 1486262 \nu^{13} + 657772 \nu^{12} + 15956428 \nu^{11} - 6145510 \nu^{10} - 85789005 \nu^{9} + 26901852 \nu^{8} + 220440362 \nu^{7} - 49245546 \nu^{6} - 175612046 \nu^{5} + 9488192 \nu^{4} + 66746640 \nu^{3} - 17198024 \nu^{2} - 42258102 \nu - 494056560$$$$)/ 155319736$$ $$\beta_{9}$$ $$=$$ $$($$$$65291 \nu^{15} - 1609409 \nu^{13} + 17484637 \nu^{11} - 98553123 \nu^{9} + 279192184 \nu^{7} - 247073396 \nu^{5} - 68740986 \nu^{3} - 240799422 \nu$$$$)/ 155319736$$ $$\beta_{10}$$ $$=$$ $$($$$$-5315 \nu^{15} + 112061 \nu^{13} - 918706 \nu^{11} + 2610036 \nu^{9} + 4644706 \nu^{7} - 36782678 \nu^{5} + 16688360 \nu^{3} - 27517724 \nu$$$$)/11947672$$ $$\beta_{11}$$ $$=$$ $$($$$$-58483 \nu^{15} + 68783 \nu^{14} + 1486262 \nu^{13} - 1793454 \nu^{12} - 15956428 \nu^{11} + 20063836 \nu^{10} + 85789005 \nu^{9} - 115997830 \nu^{8} - 220440362 \nu^{7} + 348381358 \nu^{6} + 175612046 \nu^{5} - 461197672 \nu^{4} - 66746640 \nu^{3} + 376262276 \nu^{2} + 42258102 \nu - 185488836$$$$)/ 155319736$$ $$\beta_{12}$$ $$=$$ $$($$$$83727 \nu^{14} - 1962503 \nu^{12} + 19864914 \nu^{10} - 101806378 \nu^{8} + 261543794 \nu^{6} - 280985294 \nu^{4} + 423380040 \nu^{2} - 75127752$$$$)/77659868$$ $$\beta_{13}$$ $$=$$ $$($$$$58483 \nu^{15} + 68783 \nu^{14} - 1486262 \nu^{13} - 1793454 \nu^{12} + 15956428 \nu^{11} + 20063836 \nu^{10} - 85789005 \nu^{9} - 115997830 \nu^{8} + 220440362 \nu^{7} + 348381358 \nu^{6} - 175612046 \nu^{5} - 461197672 \nu^{4} + 66746640 \nu^{3} + 376262276 \nu^{2} - 42258102 \nu - 185488836$$$$)/ 155319736$$ $$\beta_{14}$$ $$=$$ $$($$$$210925 \nu^{15} - 303128 \nu^{14} - 4995848 \nu^{13} + 7315240 \nu^{12} + 49904426 \nu^{11} - 74847530 \nu^{10} - 242883212 \nu^{9} + 369656844 \nu^{8} + 535775526 \nu^{7} - 773919996 \nu^{6} - 324544792 \nu^{5} + 32762176 \nu^{4} + 745741568 \nu^{3} - 310754124 \nu^{2} + 610548224 \nu - 190436872$$$$)/ 310639472$$ $$\beta_{15}$$ $$=$$ $$($$$$210925 \nu^{15} + 303128 \nu^{14} - 4995848 \nu^{13} - 7315240 \nu^{12} + 49904426 \nu^{11} + 74847530 \nu^{10} - 242883212 \nu^{9} - 369656844 \nu^{8} + 535775526 \nu^{7} + 773919996 \nu^{6} - 324544792 \nu^{5} - 32762176 \nu^{4} + 745741568 \nu^{3} + 310754124 \nu^{2} + 610548224 \nu + 190436872$$$$)/ 310639472$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{6} - \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{8} - \beta_{5} + 3$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{11} + 4 \beta_{10} + 10 \beta_{6} - 5 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{13} - 2 \beta_{12} + 8 \beta_{11} + 4 \beta_{8} - \beta_{7} - 10 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 11$$ $$\nu^{5}$$ $$=$$ $$($$$$-30 \beta_{15} - 30 \beta_{14} + 9 \beta_{13} - 9 \beta_{11} + 8 \beta_{10} + 94 \beta_{6} - 12 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-\beta_{15} + \beta_{14} + 50 \beta_{13} - 19 \beta_{12} + 48 \beta_{11} - 2 \beta_{8} - 3 \beta_{7} - 76 \beta_{5} + 13 \beta_{4} - 13 \beta_{3} - 9$$ $$\nu^{7}$$ $$=$$ $$-105 \beta_{15} - 105 \beta_{14} + 5 \beta_{13} - 5 \beta_{11} - 47 \beta_{10} - 7 \beta_{9} + 325 \beta_{6} + 56 \beta_{4} + 56 \beta_{3} + 16 \beta_{2} - 23 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$404 \beta_{13} - 132 \beta_{12} + 180 \beta_{11} - 224 \beta_{8} + 38 \beta_{7} - 432 \beta_{5} + 48 \beta_{4} - 48 \beta_{3} - 668$$ $$\nu^{9}$$ $$=$$ $$-528 \beta_{15} - 528 \beta_{14} - 149 \beta_{13} + 149 \beta_{11} - 755 \beta_{10} - 84 \beta_{9} + 1595 \beta_{6} + 981 \beta_{4} + 981 \beta_{3} + 394 \beta_{2} - 234 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$122 \beta_{15} - 122 \beta_{14} + 2428 \beta_{13} - 622 \beta_{12} - 230 \beta_{11} - 2658 \beta_{8} + 790 \beta_{7} - 1482 \beta_{5} - 6 \beta_{4} + 6 \beta_{3} - 7632$$ $$\nu^{11}$$ $$=$$ $$-1133 \beta_{15} - 1133 \beta_{14} - 2134 \beta_{13} + 2134 \beta_{11} - 6698 \beta_{10} - 506 \beta_{9} + 3152 \beta_{6} + 9175 \beta_{4} + 9175 \beta_{3} + 4314 \beta_{2} - 1252 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$1784 \beta_{15} - 1784 \beta_{14} + 9348 \beta_{13} - 384 \beta_{12} - 11972 \beta_{11} - 21320 \beta_{8} + 8598 \beta_{7} + 3084 \beta_{5} - 1660 \beta_{4} + 1660 \beta_{3} - 59230$$ $$\nu^{13}$$ $$=$$ $$13130 \beta_{15} + 13130 \beta_{14} - 19969 \beta_{13} + 19969 \beta_{11} - 43310 \beta_{10} - 260 \beta_{9} - 41620 \beta_{6} + 62798 \beta_{4} + 62798 \beta_{3} + 33870 \beta_{2} - 2142 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$15994 \beta_{15} - 15994 \beta_{14} - 8480 \beta_{13} + 31638 \beta_{12} - 135328 \beta_{11} - 126848 \beta_{8} + 66318 \beta_{7} + 107524 \beta_{5} - 13018 \beta_{4} + 13018 \beta_{3} - 339042$$ $$\nu^{15}$$ $$=$$ $$220728 \beta_{15} + 220728 \beta_{14} - 148962 \beta_{13} + 148962 \beta_{11} - 193454 \beta_{10} + 34614 \beta_{9} - 667226 \beta_{6} + 305054 \beta_{4} + 305054 \beta_{3} + 193516 \beta_{2} + 32458 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$67$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
164.1
 2.48916 − 0.707107i −0.896709 + 0.707107i −2.48916 + 0.707107i 0.896709 − 0.707107i 0.473307 + 0.707107i 2.60307 − 0.707107i −0.473307 − 0.707107i −2.60307 + 0.707107i 2.60307 + 0.707107i 0.473307 − 0.707107i −2.60307 − 0.707107i −0.473307 + 0.707107i −0.896709 − 0.707107i 2.48916 + 0.707107i 0.896709 + 0.707107i −2.48916 − 0.707107i
2.39417i −0.796225 1.53819i −3.73205 −2.17533 0.517638i −3.68269 + 1.90630i 3.38587 4.14682i −1.73205 + 2.44949i −1.23931 + 5.20810i
164.2 2.39417i −0.796225 + 1.53819i −3.73205 −2.17533 + 0.517638i 3.68269 + 1.90630i −3.38587 4.14682i −1.73205 2.44949i 1.23931 + 5.20810i
164.3 2.39417i 0.796225 1.53819i −3.73205 2.17533 + 0.517638i −3.68269 1.90630i −3.38587 4.14682i −1.73205 2.44949i 1.23931 5.20810i
164.4 2.39417i 0.796225 + 1.53819i −3.73205 2.17533 0.517638i 3.68269 1.90630i 3.38587 4.14682i −1.73205 + 2.44949i −1.23931 5.20810i
164.5 1.50597i −1.53819 0.796225i −0.267949 1.12603 1.93185i −1.19909 + 2.31647i −2.12976 2.60842i 1.73205 + 2.44949i −2.90931 1.69577i
164.6 1.50597i −1.53819 + 0.796225i −0.267949 1.12603 + 1.93185i 1.19909 + 2.31647i 2.12976 2.60842i 1.73205 2.44949i 2.90931 1.69577i
164.7 1.50597i 1.53819 0.796225i −0.267949 −1.12603 + 1.93185i −1.19909 2.31647i 2.12976 2.60842i 1.73205 2.44949i 2.90931 + 1.69577i
164.8 1.50597i 1.53819 + 0.796225i −0.267949 −1.12603 1.93185i 1.19909 2.31647i −2.12976 2.60842i 1.73205 + 2.44949i −2.90931 + 1.69577i
164.9 1.50597i −1.53819 0.796225i −0.267949 1.12603 1.93185i 1.19909 2.31647i 2.12976 2.60842i 1.73205 + 2.44949i 2.90931 + 1.69577i
164.10 1.50597i −1.53819 + 0.796225i −0.267949 1.12603 + 1.93185i −1.19909 2.31647i −2.12976 2.60842i 1.73205 2.44949i −2.90931 + 1.69577i
164.11 1.50597i 1.53819 0.796225i −0.267949 −1.12603 + 1.93185i 1.19909 + 2.31647i −2.12976 2.60842i 1.73205 2.44949i −2.90931 1.69577i
164.12 1.50597i 1.53819 + 0.796225i −0.267949 −1.12603 1.93185i −1.19909 + 2.31647i 2.12976 2.60842i 1.73205 + 2.44949i 2.90931 1.69577i
164.13 2.39417i −0.796225 1.53819i −3.73205 −2.17533 0.517638i 3.68269 1.90630i −3.38587 4.14682i −1.73205 + 2.44949i 1.23931 5.20810i
164.14 2.39417i −0.796225 + 1.53819i −3.73205 −2.17533 + 0.517638i −3.68269 1.90630i 3.38587 4.14682i −1.73205 2.44949i −1.23931 5.20810i
164.15 2.39417i 0.796225 1.53819i −3.73205 2.17533 + 0.517638i 3.68269 + 1.90630i 3.38587 4.14682i −1.73205 2.44949i −1.23931 + 5.20810i
164.16 2.39417i 0.796225 + 1.53819i −3.73205 2.17533 0.517638i −3.68269 + 1.90630i −3.38587 4.14682i −1.73205 + 2.44949i 1.23931 + 5.20810i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 164.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.d.c 16
3.b odd 2 1 inner 165.2.d.c 16
5.b even 2 1 inner 165.2.d.c 16
5.c odd 4 2 825.2.f.f 16
11.b odd 2 1 inner 165.2.d.c 16
15.d odd 2 1 inner 165.2.d.c 16
15.e even 4 2 825.2.f.f 16
33.d even 2 1 inner 165.2.d.c 16
55.d odd 2 1 inner 165.2.d.c 16
55.e even 4 2 825.2.f.f 16
165.d even 2 1 inner 165.2.d.c 16
165.l odd 4 2 825.2.f.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.c 16 1.a even 1 1 trivial
165.2.d.c 16 3.b odd 2 1 inner
165.2.d.c 16 5.b even 2 1 inner
165.2.d.c 16 11.b odd 2 1 inner
165.2.d.c 16 15.d odd 2 1 inner
165.2.d.c 16 33.d even 2 1 inner
165.2.d.c 16 55.d odd 2 1 inner
165.2.d.c 16 165.d even 2 1 inner
825.2.f.f 16 5.c odd 4 2
825.2.f.f 16 15.e even 4 2
825.2.f.f 16 55.e even 4 2
825.2.f.f 16 165.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(165, [\chi])$$:

 $$T_{2}^{4} + 8 T_{2}^{2} + 13$$ $$T_{23}^{4} - 36 T_{23}^{2} + 216$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 13 + 8 T^{2} + T^{4} )^{4}$$
$3$ $$( 81 + 6 T^{4} + T^{8} )^{2}$$
$5$ $$( 625 - 100 T^{2} + 6 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$7$ $$( 52 - 16 T^{2} + T^{4} )^{4}$$
$11$ $$( 14641 - 3388 T^{2} + 390 T^{4} - 28 T^{6} + T^{8} )^{2}$$
$13$ $$( 52 - 16 T^{2} + T^{4} )^{4}$$
$17$ $$( 52 + 20 T^{2} + T^{4} )^{4}$$
$19$ $$( 312 + 36 T^{2} + T^{4} )^{4}$$
$23$ $$( 216 - 36 T^{2} + T^{4} )^{4}$$
$29$ $$( 312 - 84 T^{2} + T^{4} )^{4}$$
$31$ $$( -12 + T^{2} )^{8}$$
$37$ $$( 96 + 72 T^{2} + T^{4} )^{4}$$
$41$ $$( 312 - 60 T^{2} + T^{4} )^{4}$$
$43$ $$( 6292 - 160 T^{2} + T^{4} )^{4}$$
$47$ $$( 24 - 12 T^{2} + T^{4} )^{4}$$
$53$ $$( 1536 - 96 T^{2} + T^{4} )^{4}$$
$59$ $$( 24 + T^{2} )^{8}$$
$61$ $$( 1248 + 120 T^{2} + T^{4} )^{4}$$
$67$ $$( 26136 + 324 T^{2} + T^{4} )^{4}$$
$71$ $$( 2304 + 192 T^{2} + T^{4} )^{4}$$
$73$ $$( 52 - 16 T^{2} + T^{4} )^{4}$$
$79$ $$( 25272 + 324 T^{2} + T^{4} )^{4}$$
$83$ $$( 8788 + 188 T^{2} + T^{4} )^{4}$$
$89$ $$( 1936 + 112 T^{2} + T^{4} )^{4}$$
$97$ $$( 96 + 72 T^{2} + T^{4} )^{4}$$
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