# Properties

 Label 336.2.bc.f Level 336 Weight 2 Character orbit 336.bc Analytic conductor 2.683 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{5} + \beta_{8} ) q^{3} -\beta_{14} q^{5} + \beta_{10} q^{7} -\beta_{1} q^{9} +O(q^{10})$$ $$q + ( -\beta_{5} + \beta_{8} ) q^{3} -\beta_{14} q^{5} + \beta_{10} q^{7} -\beta_{1} q^{9} + ( \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{11} + ( \beta_{3} + \beta_{6} - \beta_{8} + \beta_{13} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{15} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{17} + ( \beta_{2} + \beta_{6} - \beta_{10} ) q^{19} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{21} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{23} + ( \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} ) q^{25} + ( 1 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{27} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{8} + 4 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{29} + ( 4 - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{31} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{10} + \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{33} + ( -\beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{13} + 2 \beta_{15} ) q^{35} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} ) q^{37} + ( -\beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{11} + \beta_{12} - \beta_{15} ) q^{39} + ( 2 \beta_{3} + 2 \beta_{8} ) q^{41} + ( -2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{13} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{14} + \beta_{15} ) q^{45} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{47} + ( -3 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{14} ) q^{51} + ( -2 \beta_{9} - \beta_{11} - 2 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{53} + ( -2 + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{10} - \beta_{13} ) q^{55} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{57} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{59} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{10} + \beta_{13} ) q^{61} + ( 3 - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{63} + ( 1 - \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - 6 \beta_{8} + \beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{65} + ( -\beta_{2} - 2 \beta_{4} - \beta_{10} + \beta_{13} ) q^{67} + ( 2 - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{69} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} + 3 \beta_{15} ) q^{71} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{13} ) q^{73} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + 3 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{75} + ( \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{77} + ( -4 + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{13} ) q^{79} + ( -2 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{10} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{81} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} - \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - 2 \beta_{15} ) q^{83} + ( 1 - \beta_{2} + 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} + 2 \beta_{13} ) q^{85} + ( -8 - 4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{87} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{89} + ( -4 + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - 2 \beta_{13} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{13} - \beta_{14} ) q^{93} + ( -2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{95} + ( -2 + \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{6} - \beta_{8} + \beta_{10} - 3 \beta_{13} ) q^{97} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{7} + 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{7} + 2q^{9} + O(q^{10})$$ $$16q - 4q^{7} + 2q^{9} - 8q^{15} + 6q^{19} + 14q^{21} - 18q^{25} + 48q^{31} - 12q^{33} - 2q^{37} + 22q^{39} - 20q^{43} - 42q^{45} - 28q^{49} - 6q^{51} - 8q^{57} + 36q^{61} + 32q^{63} - 14q^{67} + 30q^{73} - 54q^{75} - 28q^{79} + 30q^{81} + 16q^{85} - 78q^{87} - 66q^{91} + 16q^{93} - 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} - 13122 x + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} - 15 \nu^{14} + 71 \nu^{13} + 48 \nu^{12} - 110 \nu^{11} + 384 \nu^{10} - 266 \nu^{9} - 252 \nu^{8} + 854 \nu^{7} - 1272 \nu^{6} + 378 \nu^{5} + 2160 \nu^{4} - 14985 \nu^{3} + 21627 \nu^{2} + 28431 \nu - 52488$$$$)/34992$$ $$\beta_{3}$$ $$=$$ $$($$$$-11 \nu^{15} - 30 \nu^{14} + 142 \nu^{13} - 363 \nu^{12} + 662 \nu^{11} - 258 \nu^{10} - 1288 \nu^{9} + 3546 \nu^{8} - 8138 \nu^{7} + 7392 \nu^{6} + 846 \nu^{5} - 28890 \nu^{4} + 41067 \nu^{3} - 38880 \nu^{2} + 34992 \nu + 19683$$$$)/69984$$ $$\beta_{4}$$ $$=$$ $$($$$$-22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + 1483 \nu^{9} - 5787 \nu^{8} + 11651 \nu^{7} - 14727 \nu^{6} + 5787 \nu^{5} + 23193 \nu^{4} - 70227 \nu^{3} + 126846 \nu^{2} - 147987 \nu + 131220$$$$)/34992$$ $$\beta_{5}$$ $$=$$ $$($$$$-25 \nu^{15} + 111 \nu^{14} - 295 \nu^{13} + 606 \nu^{12} - 770 \nu^{11} - 66 \nu^{10} + 2656 \nu^{9} - 7812 \nu^{8} + 13268 \nu^{7} - 13818 \nu^{6} - 2826 \nu^{5} + 39366 \nu^{4} - 88047 \nu^{3} + 133893 \nu^{2} - 131949 \nu + 69984$$$$)/23328$$ $$\beta_{6}$$ $$=$$ $$($$$$-19 \nu^{15} + 95 \nu^{14} - 274 \nu^{13} + 500 \nu^{12} - 653 \nu^{11} + 145 \nu^{10} + 2131 \nu^{9} - 6443 \nu^{8} + 11435 \nu^{7} - 11899 \nu^{6} - 513 \nu^{5} + 29421 \nu^{4} - 73926 \nu^{3} + 110970 \nu^{2} - 131463 \nu + 85293$$$$)/11664$$ $$\beta_{7}$$ $$=$$ $$($$$$-125 \nu^{15} + 669 \nu^{14} - 2069 \nu^{13} + 4116 \nu^{12} - 5794 \nu^{11} + 1302 \nu^{10} + 16736 \nu^{9} - 53232 \nu^{8} + 92296 \nu^{7} - 97758 \nu^{6} + 3510 \nu^{5} + 239166 \nu^{4} - 587979 \nu^{3} + 857547 \nu^{2} - 888651 \nu + 599238$$$$)/69984$$ $$\beta_{8}$$ $$=$$ $$($$$$19 \nu^{15} - 82 \nu^{14} + 226 \nu^{13} - 433 \nu^{12} + 542 \nu^{11} - 74 \nu^{10} - 1912 \nu^{9} + 5482 \nu^{8} - 9482 \nu^{7} + 10040 \nu^{6} - 42 \nu^{5} - 24354 \nu^{4} + 60237 \nu^{3} - 95580 \nu^{2} + 105948 \nu - 75087$$$$)/7776$$ $$\beta_{9}$$ $$=$$ $$($$$$193 \nu^{15} - 753 \nu^{14} + 1975 \nu^{13} - 3624 \nu^{12} + 4292 \nu^{11} - 84 \nu^{10} - 16018 \nu^{9} + 46218 \nu^{8} - 81374 \nu^{7} + 85788 \nu^{6} + 756 \nu^{5} - 197316 \nu^{4} + 515565 \nu^{3} - 849285 \nu^{2} + 958635 \nu - 708588$$$$)/69984$$ $$\beta_{10}$$ $$=$$ $$($$$$100 \nu^{15} - 429 \nu^{14} + 1207 \nu^{13} - 2301 \nu^{12} + 2648 \nu^{11} + 510 \nu^{10} - 10624 \nu^{9} + 29190 \nu^{8} - 48824 \nu^{7} + 45918 \nu^{6} + 9288 \nu^{5} - 142074 \nu^{4} + 312012 \nu^{3} - 474093 \nu^{2} + 502281 \nu - 347733$$$$)/34992$$ $$\beta_{11}$$ $$=$$ $$($$$$-61 \nu^{15} + 302 \nu^{14} - 784 \nu^{13} + 1463 \nu^{12} - 1772 \nu^{11} + 100 \nu^{10} + 6358 \nu^{9} - 18728 \nu^{8} + 32084 \nu^{7} - 33490 \nu^{6} - 1116 \nu^{5} + 85716 \nu^{4} - 197181 \nu^{3} + 331290 \nu^{2} - 374220 \nu + 255879$$$$)/23328$$ $$\beta_{12}$$ $$=$$ $$($$$$-74 \nu^{15} + 351 \nu^{14} - 1037 \nu^{13} + 2043 \nu^{12} - 2470 \nu^{11} + 234 \nu^{10} + 8450 \nu^{9} - 24816 \nu^{8} + 42964 \nu^{7} - 43500 \nu^{6} - 3870 \nu^{5} + 111618 \nu^{4} - 271836 \nu^{3} + 402489 \nu^{2} - 438615 \nu + 321489$$$$)/23328$$ $$\beta_{13}$$ $$=$$ $$($$$$-41 \nu^{15} + 179 \nu^{14} - 482 \nu^{13} + 962 \nu^{12} - 1225 \nu^{11} + 205 \nu^{10} + 3863 \nu^{9} - 11879 \nu^{8} + 20887 \nu^{7} - 22783 \nu^{6} + 1875 \nu^{5} + 53433 \nu^{4} - 134892 \nu^{3} + 214650 \nu^{2} - 231579 \nu + 165483$$$$)/11664$$ $$\beta_{14}$$ $$=$$ $$($$$$-275 \nu^{15} + 1101 \nu^{14} - 2921 \nu^{13} + 5466 \nu^{12} - 7066 \nu^{11} + 6 \nu^{10} + 24788 \nu^{9} - 70872 \nu^{8} + 123088 \nu^{7} - 124578 \nu^{6} - 5346 \nu^{5} + 321678 \nu^{4} - 799713 \nu^{3} + 1259955 \nu^{2} - 1358127 \nu + 883548$$$$)/69984$$ $$\beta_{15}$$ $$=$$ $$($$$$-2 \nu^{15} + 8 \nu^{14} - 21 \nu^{13} + 38 \nu^{12} - 41 \nu^{11} - 17 \nu^{10} + 195 \nu^{9} - 497 \nu^{8} + 779 \nu^{7} - 661 \nu^{6} - 349 \nu^{5} + 2499 \nu^{4} - 5247 \nu^{3} + 7371 \nu^{2} - 7452 \nu + 4617$$$$)/432$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} - 6 \beta_{13} + \beta_{12} + 6 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 6 \beta_{6} + 10 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_{1} - 5$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{6} + 3 \beta_{4} - 2 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} - 6 \beta_{14} + 17 \beta_{13} - 14 \beta_{12} - 7 \beta_{11} - 16 \beta_{10} - 6 \beta_{9} + 7 \beta_{7} - \beta_{6} - 25 \beta_{5} - 15 \beta_{4} - 7 \beta_{3} - \beta_{2} - \beta_{1} - 15$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + 4 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} + 3$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{15} + 2 \beta_{14} - 37 \beta_{13} - 11 \beta_{12} + 13 \beta_{11} - 37 \beta_{10} - 92 \beta_{9} + 51 \beta_{8} + 22 \beta_{7} - 11 \beta_{6} - 51 \beta_{5} - 11 \beta_{4} - 81 \beta_{3} + 48 \beta_{2} - 26 \beta_{1} + 22$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{14} + 12 \beta_{13} - 18 \beta_{11} + 4 \beta_{10} - 14 \beta_{9} - 4 \beta_{8} - 14 \beta_{7} + 8 \beta_{6} + 2 \beta_{5} - 12 \beta_{4} - 42 \beta_{3} - 8 \beta_{2} + 15 \beta_{1} + 12$$ $$\nu^{9}$$ $$=$$ $$($$$$-37 \beta_{15} - 10 \beta_{13} - 37 \beta_{12} - 94 \beta_{11} - 21 \beta_{10} - 299 \beta_{8} - 37 \beta_{7} - 10 \beta_{6} - 194 \beta_{4} - 102 \beta_{3} - 21 \beta_{2} + 37 \beta_{1} + 97$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$23 \beta_{15} - 28 \beta_{14} - 19 \beta_{13} - 12 \beta_{12} + 16 \beta_{11} + 19 \beta_{10} + 4 \beta_{9} - 44 \beta_{8} + 25 \beta_{6} - 44 \beta_{5} + 51 \beta_{4} - 16 \beta_{3} + 44 \beta_{2}$$ $$\nu^{11}$$ $$=$$ $$($$$$-6 \beta_{15} - 682 \beta_{14} + 179 \beta_{13} - 346 \beta_{12} - 173 \beta_{11} - 432 \beta_{10} - 794 \beta_{9} + 173 \beta_{7} + 253 \beta_{6} - 259 \beta_{5} - 349 \beta_{4} - 173 \beta_{3} + 253 \beta_{2} - 3 \beta_{1} - 349$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-44 \beta_{15} + 48 \beta_{14} + 64 \beta_{13} + 172 \beta_{12} + 24 \beta_{11} - 32 \beta_{10} + 200 \beta_{9} - 8 \beta_{8} - 172 \beta_{7} - 64 \beta_{6} + 16 \beta_{5} + 100 \beta_{3} + 32 \beta_{2} - 44 \beta_{1} - 135$$ $$\nu^{13}$$ $$=$$ $$($$$$-855 \beta_{15} + 1182 \beta_{14} - 615 \beta_{13} + 327 \beta_{12} + 855 \beta_{11} - 615 \beta_{10} + 1068 \beta_{9} - 647 \beta_{8} - 654 \beta_{7} - 681 \beta_{6} + 647 \beta_{5} - 1209 \beta_{4} + 741 \beta_{3} + 1296 \beta_{2} - 1710 \beta_{1} + 2418$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$196 \beta_{14} + 324 \beta_{13} + 372 \beta_{11} + 460 \beta_{10} - 224 \beta_{9} + 416 \beta_{8} - 20 \beta_{7} - 136 \beta_{6} - 208 \beta_{5} - 984 \beta_{4} - 468 \beta_{3} + 136 \beta_{2} - 191 \beta_{1} + 984$$ $$\nu^{15}$$ $$=$$ $$($$$$-2919 \beta_{15} + 2858 \beta_{13} + 153 \beta_{12} - 490 \beta_{11} - 1463 \beta_{10} - 9 \beta_{8} + 153 \beta_{7} + 2858 \beta_{6} - 13798 \beta_{4} - 962 \beta_{3} - 1463 \beta_{2} + 2919 \beta_{1} + 6899$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.247636 + 1.71426i −0.601642 − 1.62420i 1.60841 − 0.642670i −1.70742 + 0.291063i 1.73018 + 0.0805675i 1.22961 − 1.21986i 0.934861 + 1.45809i −0.441628 + 1.67480i 0.247636 − 1.71426i −0.601642 + 1.62420i 1.60841 + 0.642670i −1.70742 − 0.291063i 1.73018 − 0.0805675i 1.22961 + 1.21986i 0.934861 − 1.45809i −0.441628 − 1.67480i
0 −1.71426 + 0.247636i 0 −1.28955 2.23357i 0 0.203402 + 2.63792i 0 2.87735 0.849022i 0
17.2 0 −1.62420 + 0.601642i 0 0.0726693 + 0.125867i 0 −1.05451 2.42652i 0 2.27605 1.95437i 0
17.3 0 −0.642670 1.60841i 0 1.28955 + 2.23357i 0 0.203402 + 2.63792i 0 −2.17395 + 2.06735i 0
17.4 0 −0.291063 1.70742i 0 −0.0726693 0.125867i 0 −1.05451 2.42652i 0 −2.83056 + 0.993934i 0
17.5 0 −0.0805675 + 1.73018i 0 1.90017 + 3.29119i 0 −2.23495 + 1.41598i 0 −2.98702 0.278792i 0
17.6 0 1.21986 + 1.22961i 0 −1.40397 2.43175i 0 2.08606 1.62738i 0 −0.0238727 + 2.99991i 0
17.7 0 1.45809 0.934861i 0 −1.90017 3.29119i 0 −2.23495 + 1.41598i 0 1.25207 2.72623i 0
17.8 0 1.67480 + 0.441628i 0 1.40397 + 2.43175i 0 2.08606 1.62738i 0 2.60993 + 1.47928i 0
257.1 0 −1.71426 0.247636i 0 −1.28955 + 2.23357i 0 0.203402 2.63792i 0 2.87735 + 0.849022i 0
257.2 0 −1.62420 0.601642i 0 0.0726693 0.125867i 0 −1.05451 + 2.42652i 0 2.27605 + 1.95437i 0
257.3 0 −0.642670 + 1.60841i 0 1.28955 2.23357i 0 0.203402 2.63792i 0 −2.17395 2.06735i 0
257.4 0 −0.291063 + 1.70742i 0 −0.0726693 + 0.125867i 0 −1.05451 + 2.42652i 0 −2.83056 0.993934i 0
257.5 0 −0.0805675 1.73018i 0 1.90017 3.29119i 0 −2.23495 1.41598i 0 −2.98702 + 0.278792i 0
257.6 0 1.21986 1.22961i 0 −1.40397 + 2.43175i 0 2.08606 + 1.62738i 0 −0.0238727 2.99991i 0
257.7 0 1.45809 + 0.934861i 0 −1.90017 + 3.29119i 0 −2.23495 1.41598i 0 1.25207 + 2.72623i 0
257.8 0 1.67480 0.441628i 0 1.40397 2.43175i 0 2.08606 + 1.62738i 0 2.60993 1.47928i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bc.f 16
3.b odd 2 1 inner 336.2.bc.f 16
4.b odd 2 1 168.2.u.a 16
7.c even 3 1 2352.2.k.i 16
7.d odd 6 1 inner 336.2.bc.f 16
7.d odd 6 1 2352.2.k.i 16
12.b even 2 1 168.2.u.a 16
21.g even 6 1 inner 336.2.bc.f 16
21.g even 6 1 2352.2.k.i 16
21.h odd 6 1 2352.2.k.i 16
28.d even 2 1 1176.2.u.b 16
28.f even 6 1 168.2.u.a 16
28.f even 6 1 1176.2.k.a 16
28.g odd 6 1 1176.2.k.a 16
28.g odd 6 1 1176.2.u.b 16
84.h odd 2 1 1176.2.u.b 16
84.j odd 6 1 168.2.u.a 16
84.j odd 6 1 1176.2.k.a 16
84.n even 6 1 1176.2.k.a 16
84.n even 6 1 1176.2.u.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.u.a 16 4.b odd 2 1
168.2.u.a 16 12.b even 2 1
168.2.u.a 16 28.f even 6 1
168.2.u.a 16 84.j odd 6 1
336.2.bc.f 16 1.a even 1 1 trivial
336.2.bc.f 16 3.b odd 2 1 inner
336.2.bc.f 16 7.d odd 6 1 inner
336.2.bc.f 16 21.g even 6 1 inner
1176.2.k.a 16 28.f even 6 1
1176.2.k.a 16 28.g odd 6 1
1176.2.k.a 16 84.j odd 6 1
1176.2.k.a 16 84.n even 6 1
1176.2.u.b 16 28.d even 2 1
1176.2.u.b 16 28.g odd 6 1
1176.2.u.b 16 84.h odd 2 1
1176.2.u.b 16 84.n even 6 1
2352.2.k.i 16 7.c even 3 1
2352.2.k.i 16 7.d odd 6 1
2352.2.k.i 16 21.g even 6 1
2352.2.k.i 16 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{16} + \cdots$$ $$T_{13}^{8} + 55 T_{13}^{6} + 836 T_{13}^{4} + 3584 T_{13}^{2} + 4096$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - T^{2} - 7 T^{4} + 24 T^{5} + 22 T^{6} - 48 T^{7} + 10 T^{8} - 144 T^{9} + 198 T^{10} + 648 T^{11} - 567 T^{12} - 729 T^{14} + 6561 T^{16}$$
$5$ $$( 1 - 13 T^{2} + 93 T^{4} - 434 T^{6} + 1886 T^{8} - 10850 T^{10} + 58125 T^{12} - 203125 T^{14} + 390625 T^{16} )( 1 + 2 T^{2} - 39 T^{4} - 38 T^{6} + 836 T^{8} - 950 T^{10} - 24375 T^{12} + 31250 T^{14} + 390625 T^{16} )$$
$7$ $$( 1 + 2 T + 9 T^{2} + 10 T^{3} + 44 T^{4} + 70 T^{5} + 441 T^{6} + 686 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 49 T^{2} + 1300 T^{4} + 22265 T^{6} + 252641 T^{8} + 1395328 T^{10} - 12534274 T^{12} - 440430082 T^{14} - 6222779240 T^{16} - 53292039922 T^{18} - 183514305634 T^{20} + 2471908667008 T^{22} + 54155842054721 T^{24} + 577496758741265 T^{26} + 4079956889737300 T^{28} + 18607741845578809 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 3970655 T^{10} + 36500958 T^{12} - 236513641 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 42 T^{2} + 1059 T^{4} - 6894 T^{6} - 204407 T^{8} + 7559412 T^{10} - 49293810 T^{12} - 1407253056 T^{14} + 49740922386 T^{16} - 406696133184 T^{18} - 4117068305010 T^{20} + 182465828749428 T^{22} - 1425893651242487 T^{24} - 13898261949695406 T^{26} + 616996949226316899 T^{28} - 7071868715494839018 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 3 T + 62 T^{2} - 177 T^{3} + 2049 T^{4} - 6000 T^{5} + 55390 T^{6} - 152826 T^{7} + 1217108 T^{8} - 2903694 T^{9} + 19995790 T^{10} - 41154000 T^{11} + 267027729 T^{12} - 438269523 T^{13} + 2916844622 T^{14} - 2681615217 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 102 T^{2} + 6323 T^{4} + 248898 T^{6} + 6664873 T^{8} + 89132724 T^{10} - 1162854034 T^{12} - 108042764448 T^{14} - 3248548802990 T^{16} - 57154622392992 T^{18} - 325414235728594 T^{20} + 13194842036331636 T^{22} + 521932771402734313 T^{24} + 10310975788054808802 T^{26} +$$$$13\!\cdots\!83$$$$T^{28} +$$$$11\!\cdots\!18$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 247227929 T^{10} + 4672298286 T^{12} - 61266802063 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 24 T + 350 T^{2} - 3792 T^{3} + 33057 T^{4} - 243840 T^{5} + 1597150 T^{6} - 9593544 T^{7} + 54548420 T^{8} - 297399864 T^{9} + 1534861150 T^{10} - 7264237440 T^{11} + 30528833697 T^{12} - 108561740592 T^{13} + 310626288350 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 + T - 60 T^{2} + 689 T^{3} + 2741 T^{4} - 33360 T^{5} + 198298 T^{6} + 1145242 T^{7} - 8840520 T^{8} + 42373954 T^{9} + 271469962 T^{10} - 1689784080 T^{11} + 5137075301 T^{12} + 47777986373 T^{13} - 153943584540 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 3422058768 T^{10} + 79110004956 T^{12} + 1140025017840 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 + 5 T + 76 T^{2} + 341 T^{3} + 2710 T^{4} + 14663 T^{5} + 140524 T^{6} + 397535 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$1 - 158 T^{2} + 13171 T^{4} - 396034 T^{6} - 16747399 T^{8} + 2338864468 T^{10} - 73391270338 T^{12} - 1850417430616 T^{14} + 239524986298546 T^{16} - 4087572104230744 T^{18} - 358125987434202178 T^{20} + 25211123725919029972 T^{22} -$$$$39\!\cdots\!39$$$$T^{24} -$$$$20\!\cdots\!66$$$$T^{26} +$$$$15\!\cdots\!11$$$$T^{28} -$$$$40\!\cdots\!02$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 265 T^{2} + 32968 T^{4} + 3168029 T^{6} + 288393293 T^{8} + 22350648928 T^{10} + 1460664537746 T^{12} + 90342579430370 T^{14} + 5160533915433520 T^{16} + 253772305619909330 T^{18} + 11525345782458595826 T^{20} +$$$$49\!\cdots\!12$$$$T^{22} +$$$$17\!\cdots\!73$$$$T^{24} +$$$$55\!\cdots\!21$$$$T^{26} +$$$$16\!\cdots\!88$$$$T^{28} +$$$$36\!\cdots\!85$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 187 T^{2} + 11904 T^{4} - 267479 T^{6} + 18518765 T^{8} - 3131679552 T^{10} + 215589458578 T^{12} - 8824777678534 T^{14} + 393339242196864 T^{16} - 30719051098976854 T^{18} + 2612375297384172658 T^{20} -$$$$13\!\cdots\!32$$$$T^{22} +$$$$27\!\cdots\!65$$$$T^{24} -$$$$13\!\cdots\!79$$$$T^{26} +$$$$21\!\cdots\!24$$$$T^{28} -$$$$11\!\cdots\!07$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 - 18 T + 331 T^{2} - 4014 T^{3} + 46777 T^{4} - 446148 T^{5} + 4203790 T^{6} - 34615152 T^{7} + 285617722 T^{8} - 2111524272 T^{9} + 15642302590 T^{10} - 101267119188 T^{11} + 647666904457 T^{12} - 3390209552214 T^{13} + 17053243913491 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 7 T - 200 T^{2} - 865 T^{3} + 28259 T^{4} + 67468 T^{5} - 2804524 T^{6} - 1482940 T^{7} + 222922288 T^{8} - 99356980 T^{9} - 12589508236 T^{10} + 20291878084 T^{11} + 569450528339 T^{12} - 1167858217555 T^{13} - 18091676433800 T^{14} + 42424981237261 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 4982766368 T^{10} + 539845751164 T^{12} - 28694463598304 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 - 15 T + 312 T^{2} - 3555 T^{3} + 48909 T^{4} - 547104 T^{5} + 5639826 T^{6} - 55083990 T^{7} + 457851344 T^{8} - 4021131270 T^{9} + 30054632754 T^{10} - 212832756768 T^{11} + 1388929569069 T^{12} - 7369769513115 T^{13} + 47216278602168 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 14 T - 152 T^{2} - 1448 T^{3} + 34763 T^{4} + 170492 T^{5} - 3816772 T^{6} - 3401582 T^{7} + 382142944 T^{8} - 268724978 T^{9} - 23820474052 T^{10} + 84059205188 T^{11} + 1354021665803 T^{12} - 4455577665752 T^{13} - 36949293239192 T^{14} + 268854725806226 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 12391788531 T^{10} + 1057276475238 T^{12} + 46098592645029 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 378 T^{2} + 69795 T^{4} - 8481534 T^{6} + 768803689 T^{8} - 53847736620 T^{10} + 2382276614382 T^{12} + 24847322779392 T^{14} - 11761219221673230 T^{16} + 196815643735564032 T^{18} +$$$$14\!\cdots\!62$$$$T^{20} -$$$$26\!\cdots\!20$$$$T^{22} +$$$$30\!\cdots\!09$$$$T^{24} -$$$$26\!\cdots\!34$$$$T^{26} +$$$$17\!\cdots\!95$$$$T^{28} -$$$$73\!\cdots\!98$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 107733228771 T^{10} + 7724888001498 T^{12} - 357344990114541 T^{14} + 7837433594376961 T^{16} )^{2}$$