# Properties

 Label 1155.2.i.c Level 1155 Weight 2 Character orbit 1155.i Analytic conductor 9.223 Analytic rank 0 Dimension 16 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1155.i (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{8} q^{2}$$ $$+ \beta_{9} q^{3}$$ $$+ ( -2 - \beta_{3} ) q^{4}$$ $$-\beta_{9} q^{5}$$ $$-\beta_{14} q^{6}$$ $$+ ( -\beta_{7} + \beta_{14} ) q^{7}$$ $$+ ( -\beta_{4} + \beta_{8} ) q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{8} q^{2}$$ $$+ \beta_{9} q^{3}$$ $$+ ( -2 - \beta_{3} ) q^{4}$$ $$-\beta_{9} q^{5}$$ $$-\beta_{14} q^{6}$$ $$+ ( -\beta_{7} + \beta_{14} ) q^{7}$$ $$+ ( -\beta_{4} + \beta_{8} ) q^{8}$$ $$- q^{9}$$ $$+ \beta_{14} q^{10}$$ $$+ ( 1 + \beta_{1} - \beta_{10} ) q^{11}$$ $$+ ( \beta_{5} - 2 \beta_{9} ) q^{12}$$ $$+ ( -\beta_{11} + 2 \beta_{15} ) q^{13}$$ $$+ ( 1 - \beta_{1} - \beta_{5} + 4 \beta_{9} ) q^{14}$$ $$+ q^{15}$$ $$+ ( 2 + 2 \beta_{1} + \beta_{3} ) q^{16}$$ $$+ ( 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{17}$$ $$+ \beta_{8} q^{18}$$ $$+ ( -2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{19}$$ $$+ ( -\beta_{5} + 2 \beta_{9} ) q^{20}$$ $$+ ( -\beta_{8} - \beta_{13} ) q^{21}$$ $$+ ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{22}$$ $$+ ( 3 + \beta_{3} ) q^{23}$$ $$+ ( -\beta_{11} + \beta_{14} ) q^{24}$$ $$- q^{25}$$ $$+ ( -2 \beta_{2} - 2 \beta_{5} + 2 \beta_{9} - 4 \beta_{12} ) q^{26}$$ $$-\beta_{9} q^{27}$$ $$+ ( -\beta_{4} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} - 3 \beta_{14} ) q^{28}$$ $$+ ( -\beta_{4} + \beta_{8} - \beta_{10} ) q^{29}$$ $$-\beta_{8} q^{30}$$ $$+ ( -\beta_{2} - \beta_{5} - 2 \beta_{9} - \beta_{12} ) q^{31}$$ $$+ ( \beta_{4} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{32}$$ $$+ ( \beta_{9} - \beta_{12} - \beta_{15} ) q^{33}$$ $$+ ( -\beta_{2} - \beta_{5} + 2 \beta_{9} - 3 \beta_{12} ) q^{34}$$ $$+ ( \beta_{8} + \beta_{13} ) q^{35}$$ $$+ ( 2 + \beta_{3} ) q^{36}$$ $$+ ( -2 + \beta_{1} + \beta_{3} - \beta_{6} ) q^{37}$$ $$+ ( -\beta_{2} + \beta_{5} - 2 \beta_{9} + \beta_{12} ) q^{38}$$ $$+ ( \beta_{4} - 2 \beta_{10} ) q^{39}$$ $$+ ( \beta_{11} - \beta_{14} ) q^{40}$$ $$+ ( \beta_{11} - 2 \beta_{14} + 2 \beta_{15} ) q^{41}$$ $$+ ( -4 - \beta_{3} + \beta_{9} + \beta_{12} ) q^{42}$$ $$+ ( -3 \beta_{4} - \beta_{8} + \beta_{10} ) q^{43}$$ $$+ ( -3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{10} ) q^{44}$$ $$+ \beta_{9} q^{45}$$ $$+ ( \beta_{4} - 4 \beta_{8} ) q^{46}$$ $$+ ( -\beta_{2} - \beta_{5} + 8 \beta_{9} - \beta_{12} ) q^{47}$$ $$+ ( -\beta_{5} + 2 \beta_{9} - 2 \beta_{12} ) q^{48}$$ $$+ ( 1 + 2 \beta_{3} - 2 \beta_{9} - 2 \beta_{12} ) q^{49}$$ $$+ \beta_{8} q^{50}$$ $$+ ( -2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{51}$$ $$+ ( 4 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} - 6 \beta_{15} ) q^{52}$$ $$+ ( -1 - 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{6} ) q^{53}$$ $$+ \beta_{14} q^{54}$$ $$+ ( -\beta_{9} + \beta_{12} + \beta_{15} ) q^{55}$$ $$+ ( 1 + \beta_{3} + 3 \beta_{5} - \beta_{6} - 6 \beta_{9} + 2 \beta_{12} ) q^{56}$$ $$+ ( 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{57}$$ $$+ ( 6 + \beta_{1} + 3 \beta_{3} + \beta_{6} ) q^{58}$$ $$+ ( \beta_{2} - 3 \beta_{9} + \beta_{12} ) q^{59}$$ $$+ ( -2 - \beta_{3} ) q^{60}$$ $$+ ( -2 \beta_{11} - 2 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{61}$$ $$+ ( 2 \beta_{11} + 2 \beta_{14} - 4 \beta_{15} ) q^{62}$$ $$+ ( \beta_{7} - \beta_{14} ) q^{63}$$ $$+ ( -2 \beta_{1} - \beta_{3} + 2 \beta_{6} ) q^{64}$$ $$+ ( -\beta_{4} + 2 \beta_{10} ) q^{65}$$ $$+ ( \beta_{2} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{66}$$ $$+ ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{67}$$ $$+ ( 4 \beta_{11} + 2 \beta_{13} - 4 \beta_{15} ) q^{68}$$ $$+ ( -\beta_{5} + 3 \beta_{9} ) q^{69}$$ $$+ ( 4 + \beta_{3} - \beta_{9} - \beta_{12} ) q^{70}$$ $$+ ( -2 - 3 \beta_{1} - \beta_{3} + \beta_{6} ) q^{71}$$ $$+ ( \beta_{4} - \beta_{8} ) q^{72}$$ $$+ ( -2 \beta_{11} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{73}$$ $$+ ( 2 \beta_{4} + 2 \beta_{8} - 4 \beta_{10} ) q^{74}$$ $$-\beta_{9} q^{75}$$ $$+ ( -2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{76}$$ $$+ ( -\beta_{2} - \beta_{4} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{77}$$ $$+ ( -2 - 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} ) q^{78}$$ $$+ ( \beta_{4} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{79}$$ $$+ ( \beta_{5} - 2 \beta_{9} + 2 \beta_{12} ) q^{80}$$ $$+ q^{81}$$ $$+ ( -2 \beta_{2} + 4 \beta_{5} - 10 \beta_{9} ) q^{82}$$ $$+ ( 2 \beta_{11} + 2 \beta_{13} + \beta_{14} - 5 \beta_{15} ) q^{83}$$ $$+ ( -\beta_{4} + 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{84}$$ $$+ ( 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{85}$$ $$+ ( 2 + 7 \beta_{1} + 5 \beta_{3} - \beta_{6} ) q^{86}$$ $$+ ( -\beta_{11} + \beta_{14} - \beta_{15} ) q^{87}$$ $$+ ( -2 + 2 \beta_{1} + \beta_{3} - 3 \beta_{4} - \beta_{7} + 3 \beta_{8} + 3 \beta_{10} ) q^{88}$$ $$+ ( \beta_{2} - 3 \beta_{9} - \beta_{12} ) q^{89}$$ $$-\beta_{14} q^{90}$$ $$+ ( 2 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + \beta_{12} ) q^{91}$$ $$+ ( -12 - 2 \beta_{1} - 4 \beta_{3} ) q^{92}$$ $$+ ( 2 - \beta_{1} - \beta_{3} + \beta_{6} ) q^{93}$$ $$+ ( 2 \beta_{11} - 8 \beta_{14} - 4 \beta_{15} ) q^{94}$$ $$+ ( -2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{95}$$ $$+ ( \beta_{11} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{96}$$ $$+ ( \beta_{2} + 2 \beta_{5} + 5 \beta_{9} + \beta_{12} ) q^{97}$$ $$+ ( 2 \beta_{4} - 3 \beta_{8} + 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{98}$$ $$+ ( -1 - \beta_{1} + \beta_{10} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut -\mathstrut 16q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$16q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut -\mathstrut 16q^{9}$$ $$\mathstrut +\mathstrut 16q^{11}$$ $$\mathstrut +\mathstrut 16q^{14}$$ $$\mathstrut +\mathstrut 16q^{15}$$ $$\mathstrut +\mathstrut 24q^{16}$$ $$\mathstrut +\mathstrut 40q^{23}$$ $$\mathstrut -\mathstrut 16q^{25}$$ $$\mathstrut +\mathstrut 24q^{36}$$ $$\mathstrut -\mathstrut 40q^{37}$$ $$\mathstrut -\mathstrut 56q^{42}$$ $$\mathstrut -\mathstrut 24q^{44}$$ $$\mathstrut +\mathstrut 8q^{53}$$ $$\mathstrut +\mathstrut 8q^{56}$$ $$\mathstrut +\mathstrut 72q^{58}$$ $$\mathstrut -\mathstrut 24q^{60}$$ $$\mathstrut +\mathstrut 8q^{64}$$ $$\mathstrut +\mathstrut 80q^{67}$$ $$\mathstrut +\mathstrut 56q^{70}$$ $$\mathstrut -\mathstrut 24q^{71}$$ $$\mathstrut -\mathstrut 16q^{78}$$ $$\mathstrut +\mathstrut 16q^{81}$$ $$\mathstrut -\mathstrut 8q^{86}$$ $$\mathstrut -\mathstrut 40q^{88}$$ $$\mathstrut +\mathstrut 16q^{91}$$ $$\mathstrut -\mathstrut 160q^{92}$$ $$\mathstrut +\mathstrut 40q^{93}$$ $$\mathstrut -\mathstrut 16q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut +\mathstrut$$ $$x^{14}\mathstrut -\mathstrut$$ $$4$$ $$x^{12}\mathstrut -\mathstrut$$ $$49$$ $$x^{10}\mathstrut +\mathstrut$$ $$11$$ $$x^{8}\mathstrut +\mathstrut$$ $$395$$ $$x^{6}\mathstrut +\mathstrut$$ $$900$$ $$x^{4}\mathstrut +\mathstrut$$ $$1125$$ $$x^{2}\mathstrut +\mathstrut$$ $$625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4976 \nu^{14} + 6064 \nu^{12} - 53806 \nu^{10} + 1664 \nu^{8} + 584304 \nu^{6} + 1397920 \nu^{4} + 1900400 \nu^{2} - 33766125$$$$)/15616375$$ $$\beta_{2}$$ $$=$$ $$($$$$52907 \nu^{15} + 3769822 \nu^{13} + 7571087 \nu^{11} - 23218303 \nu^{9} - 220485308 \nu^{7} - 61141995 \nu^{5} + 2011459400 \nu^{3} + 5720425875 \nu$$$$)/$$$$858900625$$ $$\beta_{3}$$ $$=$$ $$($$$$75631 \nu^{14} + 26336 \nu^{12} - 423219 \nu^{10} - 3371514 \nu^{8} + 2776621 \nu^{6} + 34025725 \nu^{4} + 47891850 \nu^{2} + 32055875$$$$)/15616375$$ $$\beta_{4}$$ $$=$$ $$($$$$1098448 \nu^{14} - 1820672 \nu^{12} - 3507762 \nu^{10} - 38951322 \nu^{8} + 145662708 \nu^{6} + 254483540 \nu^{4} - 92662550 \nu^{2} - 101116250$$$$)/$$$$171780125$$ $$\beta_{5}$$ $$=$$ $$($$$$739864 \nu^{15} - 254886 \nu^{13} - 6858931 \nu^{11} - 26926811 \nu^{9} + 104639904 \nu^{7} + 341075305 \nu^{5} - 72330250 \nu^{3} - 1014860875 \nu$$$$)/$$$$858900625$$ $$\beta_{6}$$ $$=$$ $$($$$$64321 \nu^{14} + 44136 \nu^{12} - 365249 \nu^{10} - 2994144 \nu^{8} + 2128291 \nu^{6} + 28658755 \nu^{4} + 40690320 \nu^{2} + 30499350$$$$)/3123275$$ $$\beta_{7}$$ $$=$$ $$($$$$-3708119 \nu^{14} + 2186986 \nu^{12} + 13453081 \nu^{10} + 157291286 \nu^{8} - 302484579 \nu^{6} - 1049398975 \nu^{4} - 1461240500 \nu^{2} - 1320364000$$$$)/$$$$171780125$$ $$\beta_{8}$$ $$=$$ $$($$$$1073735 \nu^{14} - 529596 \nu^{12} - 3783171 \nu^{10} - 46979616 \nu^{8} + 81866729 \nu^{6} + 310606859 \nu^{4} + 477661630 \nu^{2} + 434987125$$$$)/34356025$$ $$\beta_{9}$$ $$=$$ $$($$$$2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + 1102430 \nu^{3} + 955700 \nu$$$$)/633875$$ $$\beta_{10}$$ $$=$$ $$($$$$-1446284 \nu^{14} + 355734 \nu^{12} + 4966964 \nu^{10} + 65045254 \nu^{8} - 95756546 \nu^{6} - 437603572 \nu^{4} - 798277730 \nu^{2} - 730658725$$$$)/34356025$$ $$\beta_{11}$$ $$=$$ $$($$$$515378 \nu^{15} + 520818 \nu^{13} - 2369422 \nu^{11} - 27002382 \nu^{9} + 8880998 \nu^{7} + 226234550 \nu^{5} + 509713400 \nu^{3} + 379881750 \nu$$$$)/78081875$$ $$\beta_{12}$$ $$=$$ $$($$$$599 \nu^{15} + 90 \nu^{13} - 3000 \nu^{11} - 26110 \nu^{9} + 31110 \nu^{7} + 237486 \nu^{5} + 287550 \nu^{3} + 249500 \nu$$$$)/74525$$ $$\beta_{13}$$ $$=$$ $$($$$$-1468861 \nu^{15} + 201274 \nu^{13} + 5426879 \nu^{11} + 66369199 \nu^{9} - 90901286 \nu^{7} - 466969935 \nu^{5} - 843562400 \nu^{3} - 722601875 \nu$$$$)/78081875$$ $$\beta_{14}$$ $$=$$ $$($$$$2100604 \nu^{15} - 547346 \nu^{13} - 7374841 \nu^{11} - 94005021 \nu^{9} + 141150094 \nu^{7} + 639251405 \nu^{5} + 1106504500 \nu^{3} + 977629125 \nu$$$$)/78081875$$ $$\beta_{15}$$ $$=$$ $$($$$$-2699561 \nu^{15} + 1648844 \nu^{13} + 8983874 \nu^{11} + 116269094 \nu^{9} - 219278916 \nu^{7} - 742870540 \nu^{5} - 1128269700 \nu^{3} - 1087173250 \nu$$$$)/78081875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{10}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{15}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$3$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$9$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$3$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18$$ $$\beta_{9}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$15$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$31$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$29$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$59$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$12$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$34$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$58$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$99$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$53$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$191$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$167$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$69$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$38$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$61$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$39$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$153$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$343$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$191$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$152$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$457$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$152$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$38$$ $$\beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$124$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$276$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$248$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$85$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$199$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$150$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$838$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$954$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$857$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$97$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1942$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$97$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$571$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1453$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$843$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$494$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$494$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2092$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$77$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$299$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$1104$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$2808$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1666$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$169$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$935$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$542$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3947$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$935$$ $$\beta_{2}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$1018$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$5734$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$6634$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$169$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$731$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$731$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5903$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$13386$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$1546$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$367$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$3459$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1040$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$3092$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2334$$ $$\beta_{9}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$232$$ $$386$$ $$661$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −1.86824 + 0.357358i 1.86824 − 0.357358i −0.0566033 − 1.17421i 0.0566033 + 1.17421i −0.644389 + 0.983224i 0.644389 − 0.983224i 0.917186 − 1.66637i −0.917186 + 1.66637i −0.917186 − 1.66637i 0.917186 + 1.66637i 0.644389 + 0.983224i −0.644389 − 0.983224i 0.0566033 − 1.17421i −0.0566033 + 1.17421i 1.86824 + 0.357358i −1.86824 − 0.357358i
2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 0.474903i 7.22133i −1.00000 2.60278
76.2 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 0.474903i 7.22133i −1.00000 −2.60278
76.3 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 + 1.47195i 1.83215i −1.00000 2.19849
76.4 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 + 1.47195i 1.83215i −1.00000 −2.19849
76.5 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 + 2.19849i 2.69862i −1.00000 1.47195
76.6 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 + 2.19849i 2.69862i −1.00000 −1.47195
76.7 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 2.60278i 1.79251i −1.00000 0.474903
76.8 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 2.60278i 1.79251i −1.00000 −0.474903
76.9 0.474903i 1.00000i 1.77447 1.00000i 0.474903 −0.474903 + 2.60278i 1.79251i −1.00000 −0.474903
76.10 0.474903i 1.00000i 1.77447 1.00000i −0.474903 0.474903 + 2.60278i 1.79251i −1.00000 0.474903
76.11 1.47195i 1.00000i −0.166634 1.00000i 1.47195 −1.47195 2.19849i 2.69862i −1.00000 −1.47195
76.12 1.47195i 1.00000i −0.166634 1.00000i −1.47195 1.47195 2.19849i 2.69862i −1.00000 1.47195
76.13 2.19849i 1.00000i −2.83337 1.00000i 2.19849 −2.19849 1.47195i 1.83215i −1.00000 −2.19849
76.14 2.19849i 1.00000i −2.83337 1.00000i −2.19849 2.19849 1.47195i 1.83215i −1.00000 2.19849
76.15 2.60278i 1.00000i −4.77447 1.00000i 2.60278 −2.60278 + 0.474903i 7.22133i −1.00000 −2.60278
76.16 2.60278i 1.00000i −4.77447 1.00000i −2.60278 2.60278 + 0.474903i 7.22133i −1.00000 2.60278
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
11.b Odd 1 yes
77.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{8}$$ $$\mathstrut +\mathstrut 14 T_{2}^{6}$$ $$\mathstrut +\mathstrut 61 T_{2}^{4}$$ $$\mathstrut +\mathstrut 84 T_{2}^{2}$$ $$\mathstrut +\mathstrut 16$$ acting on $$S_{2}^{\mathrm{new}}(1155, [\chi])$$.