# Properties

 Label 336.4.bc.e Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} - 17460 x^{7} + 57834 x^{6} - 17496 x^{5} + 59049 x^{4} - 118098 x^{3} - 531441 x^{2} - 9565938 x + 43046721$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{11}$$ Twist minimal: no (minimal twist has level 42) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} -\beta_{9} q^{5} + ( -6 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} - \beta_{15} ) q^{7} + ( -2 - 3 \beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} -\beta_{9} q^{5} + ( -6 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} - \beta_{15} ) q^{7} + ( -2 - 3 \beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{9} + ( -2 + 4 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{11} + ( -7 - \beta_{1} - 15 \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{13} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{12} + 2 \beta_{15} ) q^{15} + ( 2 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 9 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{17} + ( 33 + 17 \beta_{2} - \beta_{3} + 7 \beta_{4} + 7 \beta_{6} + 4 \beta_{10} - \beta_{11} - 4 \beta_{14} + \beta_{15} ) q^{19} + ( -31 + 8 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} + 7 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{21} + ( -8 - 4 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{8} - 4 \beta_{10} + 8 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} ) q^{23} + ( -26 - 7 \beta_{1} - 26 \beta_{2} + 11 \beta_{3} + 10 \beta_{4} + 20 \beta_{6} + 3 \beta_{10} + 4 \beta_{11} + \beta_{14} - \beta_{15} ) q^{25} + ( -11 - 3 \beta_{1} - 32 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} - 5 \beta_{12} + 8 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{27} + ( -11 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - \beta_{6} + 10 \beta_{7} + 9 \beta_{8} - \beta_{9} - 5 \beta_{10} - 5 \beta_{11} - \beta_{12} - 5 \beta_{15} ) q^{29} + ( -31 + 4 \beta_{1} + 33 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 4 \beta_{10} - \beta_{11} - 5 \beta_{14} - 5 \beta_{15} ) q^{31} + ( 117 + 10 \beta_{1} + 57 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + 5 \beta_{10} - \beta_{12} + \beta_{13} - 6 \beta_{14} - 2 \beta_{15} ) q^{33} + ( 4 - 13 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} - 5 \beta_{8} + 11 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - 7 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} - 5 \beta_{15} ) q^{35} + ( 7 + 122 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{6} + 7 \beta_{10} - 7 \beta_{11} + 7 \beta_{14} - 7 \beta_{15} ) q^{37} + ( -80 + 7 \beta_{1} - 74 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} + 12 \beta_{6} - 6 \beta_{7} + 12 \beta_{9} - \beta_{10} + 2 \beta_{11} - 6 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{39} + ( -32 + 14 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 16 \beta_{5} - 6 \beta_{6} - 16 \beta_{7} + 8 \beta_{8} + 8 \beta_{9} - 8 \beta_{10} + 8 \beta_{11} + 24 \beta_{12} - 32 \beta_{13} + 16 \beta_{14} + 8 \beta_{15} ) q^{41} + ( -93 + 7 \beta_{1} + 13 \beta_{2} - 13 \beta_{3} - 8 \beta_{4} - 4 \beta_{6} + 13 \beta_{10} - 13 \beta_{11} - 3 \beta_{14} + 10 \beta_{15} ) q^{43} + ( -98 + \beta_{1} + 103 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} - 5 \beta_{7} + 16 \beta_{8} + 3 \beta_{10} + 5 \beta_{11} - 11 \beta_{12} + 5 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{45} + ( -18 \beta_{1} + 9 \beta_{3} - 27 \beta_{4} - 7 \beta_{5} - 27 \beta_{6} + 7 \beta_{8} - 14 \beta_{9} ) q^{47} + ( 91 - 4 \beta_{1} + 70 \beta_{2} + 5 \beta_{3} - 17 \beta_{4} + 12 \beta_{6} + 3 \beta_{10} - 14 \beta_{11} + 9 \beta_{15} ) q^{49} + ( -23 + 194 \beta_{2} + 25 \beta_{3} + 4 \beta_{4} + 11 \beta_{5} - 4 \beta_{6} + 11 \beta_{8} + 11 \beta_{9} - 11 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} - 20 \beta_{13} + 9 \beta_{14} + 3 \beta_{15} ) q^{51} + ( 12 - 12 \beta_{1} + 6 \beta_{3} - 22 \beta_{5} + 12 \beta_{7} + 22 \beta_{9} - 6 \beta_{11} - 23 \beta_{12} + 12 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} ) q^{53} + ( 52 + 5 \beta_{1} + 112 \beta_{2} - 2 \beta_{3} - 12 \beta_{6} + 8 \beta_{10} + 8 \beta_{11} + 13 \beta_{14} - 5 \beta_{15} ) q^{55} + ( -171 - 40 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} - 28 \beta_{4} - 14 \beta_{6} + 9 \beta_{7} + 13 \beta_{8} + 3 \beta_{9} - 12 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 4 \beta_{14} - 16 \beta_{15} ) q^{57} + ( 2 - 12 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} - 42 \beta_{4} + 20 \beta_{5} - 2 \beta_{7} + 40 \beta_{8} + 2 \beta_{10} + \beta_{11} - 7 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{59} + ( -378 + 18 \beta_{1} - 189 \beta_{2} - 9 \beta_{3} - 13 \beta_{4} - 13 \beta_{6} + 7 \beta_{10} - 7 \beta_{14} ) q^{61} + ( 191 + 3 \beta_{1} + 136 \beta_{2} - 27 \beta_{3} + 33 \beta_{4} + 4 \beta_{5} + 12 \beta_{6} + 6 \beta_{7} + 26 \beta_{8} + \beta_{9} - 10 \beta_{10} - 3 \beta_{11} + 14 \beta_{12} - 10 \beta_{13} + 5 \beta_{14} - 13 \beta_{15} ) q^{63} + ( -4 - 2 \beta_{2} - 22 \beta_{3} - 22 \beta_{4} + 3 \beta_{5} + 22 \beta_{6} + 3 \beta_{8} + 10 \beta_{9} - 2 \beta_{10} - 16 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{65} + ( 98 + 2 \beta_{1} + 98 \beta_{2} - 2 \beta_{3} + 11 \beta_{4} + 22 \beta_{6} + 13 \beta_{10} - 13 \beta_{14} + 13 \beta_{15} ) q^{67} + ( 4 + 5 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - 38 \beta_{5} + 3 \beta_{7} - 19 \beta_{8} - 27 \beta_{9} + 11 \beta_{10} + 8 \beta_{11} + 21 \beta_{12} + 6 \beta_{13} - 5 \beta_{14} + 10 \beta_{15} ) q^{69} + ( 34 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} + 28 \beta_{4} + 14 \beta_{6} - 20 \beta_{7} + 11 \beta_{8} + 12 \beta_{9} + 10 \beta_{10} + 10 \beta_{11} + 12 \beta_{12} + 10 \beta_{15} ) q^{71} + ( 144 + 16 \beta_{1} - 134 \beta_{2} + 11 \beta_{3} - 41 \beta_{4} + 3 \beta_{10} - 5 \beta_{11} - 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( -497 - 14 \beta_{1} - 251 \beta_{2} + 12 \beta_{3} + 34 \beta_{4} + 7 \beta_{5} + 34 \beta_{6} - 7 \beta_{8} + 6 \beta_{9} + 20 \beta_{10} + 5 \beta_{11} - 20 \beta_{14} - 5 \beta_{15} ) q^{75} + ( -12 + 31 \beta_{1} + 5 \beta_{2} + 21 \beta_{3} + 50 \beta_{4} - \beta_{5} - 23 \beta_{6} - 22 \beta_{7} + 23 \beta_{8} + 29 \beta_{9} + 5 \beta_{10} + 11 \beta_{11} + 14 \beta_{12} - 12 \beta_{13} + 6 \beta_{14} + 11 \beta_{15} ) q^{77} + ( 40 - 322 \beta_{2} + 26 \beta_{3} - 26 \beta_{4} + 26 \beta_{6} - 11 \beta_{10} - 40 \beta_{11} - 11 \beta_{14} - 40 \beta_{15} ) q^{79} + ( 15 - 18 \beta_{1} + 18 \beta_{2} + 24 \beta_{3} + 21 \beta_{4} + 54 \beta_{5} + 42 \beta_{6} - 3 \beta_{7} - 6 \beta_{9} + 9 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 6 \beta_{15} ) q^{81} + ( -36 + 27 \beta_{1} - 9 \beta_{2} - 27 \beta_{3} - 52 \beta_{5} + 27 \beta_{6} - 18 \beta_{7} - 26 \beta_{8} + 19 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} + 17 \beta_{12} - 36 \beta_{13} + 18 \beta_{14} + 9 \beta_{15} ) q^{83} + ( 340 - 23 \beta_{1} + 13 \beta_{2} - 13 \beta_{3} + 46 \beta_{4} + 23 \beta_{6} + 13 \beta_{10} - 13 \beta_{11} + 13 \beta_{15} ) q^{85} + ( -59 + 43 \beta_{2} + \beta_{3} - 3 \beta_{4} - 28 \beta_{5} - 14 \beta_{7} - 56 \beta_{8} + 3 \beta_{10} + 29 \beta_{11} + 10 \beta_{12} + 14 \beta_{13} + 4 \beta_{14} + 18 \beta_{15} ) q^{87} + ( 18 - 14 \beta_{1} - 9 \beta_{2} + 7 \beta_{3} + 33 \beta_{4} - 2 \beta_{5} + 33 \beta_{6} + 36 \beta_{7} + 2 \beta_{8} - 84 \beta_{9} - 9 \beta_{10} - 18 \beta_{11} - 18 \beta_{12} + 18 \beta_{13} - 9 \beta_{14} - 18 \beta_{15} ) q^{89} + ( -305 - 20 \beta_{1} - 97 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} - 16 \beta_{6} - 6 \beta_{10} - 21 \beta_{11} + 14 \beta_{14} + 9 \beta_{15} ) q^{91} + ( -21 - 264 \beta_{2} - 18 \beta_{3} + 41 \beta_{4} - 18 \beta_{5} - 41 \beta_{6} - 18 \beta_{8} + 24 \beta_{9} - 21 \beta_{10} - 6 \beta_{11} - 21 \beta_{12} - 27 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{93} + ( 40 + 7 \beta_{1} - 27 \beta_{3} + 47 \beta_{4} + 69 \beta_{5} + 94 \beta_{6} + 40 \beta_{7} - 20 \beta_{11} - 40 \beta_{12} + 40 \beta_{13} - 20 \beta_{14} - 20 \beta_{15} ) q^{95} + ( 118 + 6 \beta_{1} + 250 \beta_{2} + 2 \beta_{3} - 7 \beta_{6} + 14 \beta_{10} + 14 \beta_{11} + 17 \beta_{14} - 3 \beta_{15} ) q^{97} + ( 243 - 48 \beta_{1} - 33 \beta_{2} + 33 \beta_{3} - 96 \beta_{4} - 48 \beta_{6} + 12 \beta_{7} - 15 \beta_{8} - 6 \beta_{9} - 33 \beta_{10} + 21 \beta_{11} - 6 \beta_{12} - 3 \beta_{14} - 36 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 80q^{7} + 18q^{9} + O(q^{10})$$ $$16q - 80q^{7} + 18q^{9} + 342q^{19} - 450q^{21} - 194q^{25} - 804q^{31} + 1332q^{33} - 962q^{37} - 594q^{39} - 1732q^{43} - 2394q^{45} + 820q^{49} - 1638q^{51} - 2664q^{57} - 4620q^{61} + 2016q^{63} + 706q^{67} + 3294q^{73} - 6174q^{75} + 2656q^{79} + 126q^{81} + 5232q^{85} - 1026q^{87} - 4098q^{91} + 2016q^{93} + 4284q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} - 17460 x^{7} + 57834 x^{6} - 17496 x^{5} + 59049 x^{4} - 118098 x^{3} - 531441 x^{2} - 9565938 x + 43046721$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$5191 \nu^{15} - 309530 \nu^{14} - 1581934 \nu^{13} - 12539069 \nu^{12} + 1431552 \nu^{11} - 2093856 \nu^{10} + 351046074 \nu^{9} - 28002824 \nu^{8} - 315020312 \nu^{7} - 6624592890 \nu^{6} + 9665820720 \nu^{5} + 16813924272 \nu^{4} - 19125837693 \nu^{3} + 82104682050 \nu^{2} + 81989654598 \nu - 4281814291149$$$$)/ 980793497376$$ $$\beta_{2}$$ $$=$$ $$($$$$-1025 \nu^{15} + 7522 \nu^{14} + 1448 \nu^{13} - 4529 \nu^{12} - 66042 \nu^{11} - 322620 \nu^{10} - 2446566 \nu^{9} - 2014304 \nu^{8} + 7719994 \nu^{7} + 24266412 \nu^{6} - 128621250 \nu^{5} - 43066404 \nu^{4} - 341572221 \nu^{3} - 683039466 \nu^{2} + 12045641706 \nu + 30596652693$$$$)/ 30024290736$$ $$\beta_{3}$$ $$=$$ $$($$$$-98467 \nu^{15} - 3277489 \nu^{14} - 15183785 \nu^{13} - 68558206 \nu^{12} - 152052840 \nu^{11} + 442320216 \nu^{10} + 1663355922 \nu^{9} + 425603006 \nu^{8} - 6212634154 \nu^{7} - 20186601156 \nu^{6} + 24902279064 \nu^{5} + 530062542648 \nu^{4} + 1100348911989 \nu^{3} + 2294813669679 \nu^{2} - 8314807374657 \nu - 17367563093094$$$$)/ 2942380492128$$ $$\beta_{4}$$ $$=$$ $$($$$$-10304 \nu^{15} + 5317 \nu^{14} - 182755 \nu^{13} + 524005 \nu^{12} - 438480 \nu^{11} + 1729920 \nu^{10} - 8920464 \nu^{9} + 14071954 \nu^{8} - 61640054 \nu^{7} + 588154338 \nu^{6} + 1339362864 \nu^{5} + 2116782720 \nu^{4} - 10184036688 \nu^{3} + 6948236781 \nu^{2} - 11769823827 \nu + 81143069085$$$$)/ 89163045216$$ $$\beta_{5}$$ $$=$$ $$($$$$99716 \nu^{15} + 525581 \nu^{14} + 4176229 \nu^{13} - 461611 \nu^{12} + 656424 \nu^{11} - 115779900 \nu^{10} + 5977428 \nu^{9} + 100103006 \nu^{8} + 2177986010 \nu^{7} - 3121868142 \nu^{6} - 5634915336 \nu^{5} + 6477453684 \nu^{4} - 27572576256 \nu^{3} - 28249454943 \nu^{2} + 1410719168997 \nu + 74485176237$$$$)/ 735595123032$$ $$\beta_{6}$$ $$=$$ $$($$$$-355015 \nu^{15} - 1366828 \nu^{14} - 1922912 \nu^{13} + 5844395 \nu^{12} + 32972292 \nu^{11} + 40769052 \nu^{10} - 195975786 \nu^{9} - 665402224 \nu^{8} - 217765000 \nu^{7} + 10270327878 \nu^{6} + 11992296564 \nu^{5} + 34438178700 \nu^{4} - 106533338643 \nu^{3} - 411473508660 \nu^{2} - 472255478712 \nu + 2323389370347$$$$)/ 980793497376$$ $$\beta_{7}$$ $$=$$ $$($$$$1475767 \nu^{15} + 15672631 \nu^{14} + 32942759 \nu^{13} + 8154460 \nu^{12} - 120991548 \nu^{11} - 989657484 \nu^{10} - 3735497922 \nu^{9} + 4281271114 \nu^{8} + 966748786 \nu^{7} - 4465722504 \nu^{6} - 318113530524 \nu^{5} - 584665940268 \nu^{4} - 1821977966745 \nu^{3} + 4559466192723 \nu^{2} + 13236460863723 \nu + 67296839372316$$$$)/ 2942380492128$$ $$\beta_{8}$$ $$=$$ $$($$$$-386047 \nu^{15} - 1698028 \nu^{14} - 7639460 \nu^{13} - 16796293 \nu^{12} + 48884112 \nu^{11} + 192629040 \nu^{10} + 26064114 \nu^{9} - 721298848 \nu^{8} - 2433981664 \nu^{7} + 3399668838 \nu^{6} + 58704418224 \nu^{5} + 122907032208 \nu^{4} + 253687212657 \nu^{3} - 929681863956 \nu^{2} - 2034388034460 \nu + 470964608523$$$$)/ 735595123032$$ $$\beta_{9}$$ $$=$$ $$($$$$-5044301 \nu^{15} - 1066547 \nu^{14} + 74264777 \nu^{13} + 103846498 \nu^{12} + 412285212 \nu^{11} + 1005805164 \nu^{10} - 9379676382 \nu^{9} - 10615473302 \nu^{8} + 26036371090 \nu^{7} + 246593916828 \nu^{6} - 51131727252 \nu^{5} - 860183867844 \nu^{4} - 4792692303585 \nu^{3} - 9626842260531 \nu^{2} - 20865967451559 \nu + 184962509874954$$$$)/ 8827141476384$$ $$\beta_{10}$$ $$=$$ $$($$$$1999363 \nu^{15} - 277637 \nu^{14} + 2684699 \nu^{13} + 34754776 \nu^{12} + 2925480 \nu^{11} + 252961716 \nu^{10} - 1826457846 \nu^{9} - 2387211554 \nu^{8} - 12485606006 \nu^{7} - 28094392284 \nu^{6} + 40613994216 \nu^{5} + 97613184564 \nu^{4} - 1329866505081 \nu^{3} + 197931007971 \nu^{2} + 2006556292143 \nu + 22962594135852$$$$)/ 2942380492128$$ $$\beta_{11}$$ $$=$$ $$($$$$-2529436 \nu^{15} + 12956441 \nu^{14} + 26278093 \nu^{13} + 45587477 \nu^{12} + 176647380 \nu^{11} + 410135172 \nu^{10} - 2382859164 \nu^{9} + 91377218 \nu^{8} - 6701661862 \nu^{7} + 62980137042 \nu^{6} - 193834271484 \nu^{5} - 224038015020 \nu^{4} - 784421157744 \nu^{3} - 4308749413767 \nu^{2} - 3303725119875 \nu + 52197572223981$$$$)/ 2942380492128$$ $$\beta_{12}$$ $$=$$ $$($$$$9038834 \nu^{15} + 13433807 \nu^{14} - 6780077 \nu^{13} - 148095511 \nu^{12} - 300478752 \nu^{11} - 81403464 \nu^{10} + 1075087608 \nu^{9} + 9396078962 \nu^{8} - 16357043710 \nu^{7} - 103630786902 \nu^{6} - 311672633328 \nu^{5} + 314707182984 \nu^{4} - 297510237738 \nu^{3} + 661815228051 \nu^{2} + 7357906401759 \nu + 18095200950033$$$$)/ 8827141476384$$ $$\beta_{13}$$ $$=$$ $$($$$$12998675 \nu^{15} - 19337332 \nu^{14} - 47538200 \nu^{13} - 375390643 \nu^{12} + 126479664 \nu^{11} - 455250396 \nu^{10} + 16091567430 \nu^{9} - 7467089716 \nu^{8} - 16357050232 \nu^{7} - 225287807490 \nu^{6} + 600940070496 \nu^{5} + 629760603060 \nu^{4} + 5903483893443 \nu^{3} - 6585663166416 \nu^{2} - 15371373974832 \nu - 180263553725439$$$$)/ 8827141476384$$ $$\beta_{14}$$ $$=$$ $$($$$$1736865 \nu^{15} - 3022417 \nu^{14} - 3556645 \nu^{13} - 24737930 \nu^{12} - 51484948 \nu^{11} - 418316724 \nu^{10} + 942508302 \nu^{9} - 449986026 \nu^{8} + 1166979326 \nu^{7} - 28086264388 \nu^{6} + 58267408860 \nu^{5} - 104928763716 \nu^{4} + 391812659325 \nu^{3} + 2109837402135 \nu^{2} + 7456720651731 \nu - 13774105728810$$$$)/ 980793497376$$ $$\beta_{15}$$ $$=$$ $$($$$$6143228 \nu^{15} - 12559 \nu^{14} - 42859679 \nu^{13} - 113257891 \nu^{12} - 84984144 \nu^{11} - 206508504 \nu^{10} + 6021465072 \nu^{9} + 4422259994 \nu^{8} - 7900452070 \nu^{7} - 143237921334 \nu^{6} + 69781506480 \nu^{5} + 597650792040 \nu^{4} + 1874434021236 \nu^{3} + 5545082851785 \nu^{2} - 2926044172935 \nu - 94337802618579$$$$)/ 2942380492128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{7} + \beta_{5} + 2 \beta_{2} + 2 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{3} - 3 \beta_{2} - 2$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 7 \beta_{8} - 8 \beta_{6} - 16 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 36 \beta_{1} + 8$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - \beta_{7} + 14 \beta_{6} + 18 \beta_{5} + 7 \beta_{4} + 8 \beta_{3} + 6 \beta_{2} - 6 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{15} + 16 \beta_{14} + 10 \beta_{13} - 74 \beta_{12} - 8 \beta_{11} + 26 \beta_{10} + 32 \beta_{9} + 79 \beta_{8} - 264 \beta_{6} + 79 \beta_{5} + 264 \beta_{4} - 24 \beta_{3} - 236 \beta_{2} + 18$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$8 \beta_{15} + 66 \beta_{14} + 56 \beta_{12} + 78 \beta_{11} - 58 \beta_{10} + 56 \beta_{9} - 26 \beta_{8} - 20 \beta_{7} + 94 \beta_{6} + 188 \beta_{4} + 58 \beta_{3} - 58 \beta_{2} - 68 \beta_{1} - 851$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$904 \beta_{15} - 18 \beta_{14} - 886 \beta_{13} + 150 \beta_{12} - 522 \beta_{11} - 504 \beta_{10} + 1472 \beta_{9} - 886 \beta_{7} - 1488 \beta_{6} + 583 \beta_{5} - 744 \beta_{4} - 608 \beta_{3} + 7310 \beta_{2} + 86 \beta_{1} + 6424$$$$)/12$$ $$\nu^{8}$$ $$=$$ $$($$$$-259 \beta_{15} + 3 \beta_{14} - 685 \beta_{13} + 1327 \beta_{12} - 259 \beta_{11} - 682 \beta_{10} - 321 \beta_{9} + 654 \beta_{8} - 450 \beta_{6} + 654 \beta_{5} + 450 \beta_{4} - 551 \beta_{3} - 10777 \beta_{2} - 426$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$1520 \beta_{15} + 3108 \beta_{14} + 740 \beta_{12} + 7732 \beta_{11} - 1588 \beta_{10} + 740 \beta_{9} - 8623 \beta_{8} - 6144 \beta_{7} + 4736 \beta_{6} + 9472 \beta_{4} + 1588 \beta_{3} - 1588 \beta_{2} + 14660 \beta_{1} + 151784$$$$)/12$$ $$\nu^{10}$$ $$=$$ $$($$$$3940 \beta_{15} + 7897 \beta_{14} - 11837 \beta_{13} + 7564 \beta_{12} + 8840 \beta_{11} + 943 \beta_{10} + 8546 \beta_{9} - 11837 \beta_{7} - 13026 \beta_{6} - 1924 \beta_{5} - 6513 \beta_{4} + 1332 \beta_{3} - 1522 \beta_{2} + 7508 \beta_{1} - 13359$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$79680 \beta_{15} + 46832 \beta_{14} - 67106 \beta_{13} - 14686 \beta_{12} + 79680 \beta_{11} - 20274 \beta_{10} + 40896 \beta_{9} + 50857 \beta_{8} + 112160 \beta_{6} + 50857 \beta_{5} - 112160 \beta_{4} - 33600 \beta_{3} - 483796 \beta_{2} - 146786$$$$)/12$$ $$\nu^{12}$$ $$=$$ $$($$$$10220 \beta_{15} - 14848 \beta_{14} - 31536 \beta_{12} - 980 \beta_{11} + 25068 \beta_{10} - 31536 \beta_{9} - 35116 \beta_{8} - 24088 \beta_{7} + 7608 \beta_{6} + 15216 \beta_{4} - 25068 \beta_{3} + 25068 \beta_{2} - 88368 \beta_{1} + 648595$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$-241968 \beta_{15} + 420802 \beta_{14} - 178834 \beta_{13} + 96850 \beta_{12} + 627458 \beta_{11} + 206656 \beta_{10} + 163968 \beta_{9} - 178834 \beta_{7} - 367232 \beta_{6} + 1162993 \beta_{5} - 183616 \beta_{4} + 686672 \beta_{3} - 6386862 \beta_{2} - 59214 \beta_{1} - 6565696$$$$)/12$$ $$\nu^{14}$$ $$=$$ $$($$$$339137 \beta_{15} - 43555 \beta_{14} + 144427 \beta_{13} - 576749 \beta_{12} + 339137 \beta_{11} + 100872 \beta_{10} + 216161 \beta_{9} + 39108 \beta_{8} - 1042408 \beta_{6} + 39108 \beta_{5} + 1042408 \beta_{4} - 47071 \beta_{3} + 4385457 \beta_{2} - 194710$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$3314576 \beta_{15} + 2956548 \beta_{14} + 5874620 \beta_{12} + 1803636 \beta_{11} + 358028 \beta_{10} + 5874620 \beta_{9} - 774953 \beta_{8} - 2161664 \beta_{7} + 1289240 \beta_{6} + 2578480 \beta_{4} - 358028 \beta_{3} + 358028 \beta_{2} - 12249284 \beta_{1} - 33554840$$$$)/12$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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
 2.99617 + 0.151487i −2.58777 + 1.51770i −1.62928 − 2.51902i 0.339489 − 2.98073i −0.0204843 + 2.99993i 2.30541 + 1.91966i 2.41164 − 1.78437i −2.81518 − 1.03671i 2.99617 − 0.151487i −2.58777 − 1.51770i −1.62928 + 2.51902i 0.339489 + 2.98073i −0.0204843 − 2.99993i 2.30541 − 1.91966i 2.41164 + 1.78437i −2.81518 + 1.03671i
0 −5.18952 0.262384i 0 −2.24534 3.88904i 0 9.71288 15.7690i 0 26.8623 + 2.72329i 0
17.2 0 −4.48216 + 2.62874i 0 −5.27257 9.13236i 0 −17.7029 + 5.44135i 0 13.1794 23.5649i 0
17.3 0 −2.82199 4.36307i 0 2.24534 + 3.88904i 0 9.71288 15.7690i 0 −11.0727 + 24.6251i 0
17.4 0 −0.588012 + 5.16277i 0 4.27911 + 7.41164i 0 6.41772 + 17.3728i 0 −26.3085 6.07155i 0
17.5 0 0.0354799 5.19603i 0 5.27257 + 9.13236i 0 −17.7029 + 5.44135i 0 −26.9975 0.368709i 0
17.6 0 3.99309 + 3.32495i 0 9.90442 + 17.1550i 0 −18.4277 1.84901i 0 4.88947 + 26.5536i 0
17.7 0 4.17709 3.09062i 0 −4.27911 7.41164i 0 6.41772 + 17.3728i 0 7.89612 25.8196i 0
17.8 0 4.87603 + 1.79564i 0 −9.90442 17.1550i 0 −18.4277 1.84901i 0 20.5514 + 17.5112i 0
257.1 0 −5.18952 + 0.262384i 0 −2.24534 + 3.88904i 0 9.71288 + 15.7690i 0 26.8623 2.72329i 0
257.2 0 −4.48216 2.62874i 0 −5.27257 + 9.13236i 0 −17.7029 5.44135i 0 13.1794 + 23.5649i 0
257.3 0 −2.82199 + 4.36307i 0 2.24534 3.88904i 0 9.71288 + 15.7690i 0 −11.0727 24.6251i 0
257.4 0 −0.588012 5.16277i 0 4.27911 7.41164i 0 6.41772 17.3728i 0 −26.3085 + 6.07155i 0
257.5 0 0.0354799 + 5.19603i 0 5.27257 9.13236i 0 −17.7029 5.44135i 0 −26.9975 + 0.368709i 0
257.6 0 3.99309 3.32495i 0 9.90442 17.1550i 0 −18.4277 + 1.84901i 0 4.88947 26.5536i 0
257.7 0 4.17709 + 3.09062i 0 −4.27911 + 7.41164i 0 6.41772 17.3728i 0 7.89612 + 25.8196i 0
257.8 0 4.87603 1.79564i 0 −9.90442 + 17.1550i 0 −18.4277 + 1.84901i 0 20.5514 17.5112i 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 336.4.bc.e 16
3.b odd 2 1 inner 336.4.bc.e 16
4.b odd 2 1 42.4.f.a 16
7.d odd 6 1 inner 336.4.bc.e 16
12.b even 2 1 42.4.f.a 16
21.g even 6 1 inner 336.4.bc.e 16
28.d even 2 1 294.4.f.a 16
28.f even 6 1 42.4.f.a 16
28.f even 6 1 294.4.d.a 16
28.g odd 6 1 294.4.d.a 16
28.g odd 6 1 294.4.f.a 16
84.h odd 2 1 294.4.f.a 16
84.j odd 6 1 42.4.f.a 16
84.j odd 6 1 294.4.d.a 16
84.n even 6 1 294.4.d.a 16
84.n even 6 1 294.4.f.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.f.a 16 4.b odd 2 1
42.4.f.a 16 12.b even 2 1
42.4.f.a 16 28.f even 6 1
42.4.f.a 16 84.j odd 6 1
294.4.d.a 16 28.f even 6 1
294.4.d.a 16 28.g odd 6 1
294.4.d.a 16 84.j odd 6 1
294.4.d.a 16 84.n even 6 1
294.4.f.a 16 28.d even 2 1
294.4.f.a 16 28.g odd 6 1
294.4.f.a 16 84.h odd 2 1
294.4.f.a 16 84.n even 6 1
336.4.bc.e 16 1.a even 1 1 trivial
336.4.bc.e 16 3.b odd 2 1 inner
336.4.bc.e 16 7.d odd 6 1 inner
336.4.bc.e 16 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$17\!\cdots\!41$$$$T_{5}^{4} +$$$$31\!\cdots\!40$$$$T_{5}^{2} +$$$$41\!\cdots\!56$$">$$T_{5}^{16} + \cdots$$ $$T_{13}^{8} + 4551 T_{13}^{6} + 2979108 T_{13}^{4} + 570419712 T_{13}^{2} + 16498888704$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$282429536481 - 3486784401 T^{2} + 4782969 T^{4} + 89282088 T^{5} + 12951414 T^{6} - 244944 T^{7} - 442422 T^{8} - 9072 T^{9} + 17766 T^{10} + 4536 T^{11} + 9 T^{12} - 9 T^{14} + T^{16}$$
$5$ $$4153645759078656 + 310619615073840 T^{2} + 17289861638841 T^{4} + 367182336789 T^{6} + 5550035922 T^{8} + 45374877 T^{10} + 264258 T^{12} + 597 T^{14} + T^{16}$$
$7$ $$( 13841287201 + 1614144280 T + 70001155 T^{2} + 5858440 T^{3} + 498232 T^{4} + 17080 T^{5} + 595 T^{6} + 40 T^{7} + T^{8} )^{2}$$
$11$ $$24\!\cdots\!16$$$$-$$$$10\!\cdots\!76$$$$T^{2} + 4392246020177179521 T^{4} - 13464145104313743 T^{6} + 29836951734258 T^{8} - 30424674303 T^{10} + 22614066 T^{12} - 5391 T^{14} + T^{16}$$
$13$ $$( 16498888704 + 570419712 T^{2} + 2979108 T^{4} + 4551 T^{6} + T^{8} )^{2}$$
$17$ $$87\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$T^{2} +$$$$68\!\cdots\!00$$$$T^{4} + 10884654228190861980 T^{6} + 15419407094864361 T^{8} + 2831427288342 T^{10} + 387463563 T^{12} + 22782 T^{14} + T^{16}$$
$19$ $$( 69773043356676 + 11621573426826 T + 548436272553 T^{2} - 16123787289 T^{3} + 46647738 T^{4} + 1981719 T^{5} - 1842 T^{6} - 171 T^{7} + T^{8} )^{2}$$
$23$ $$40\!\cdots\!96$$$$-$$$$40\!\cdots\!92$$$$T^{2} +$$$$28\!\cdots\!00$$$$T^{4} -$$$$96\!\cdots\!24$$$$T^{6} + 238601271153713481 T^{8} - 21745254238794 T^{10} + 1432520235 T^{12} - 44802 T^{14} + T^{16}$$
$29$ $$( 20034685818372096 + 10537006777344 T^{2} + 1509134976 T^{4} + 72729 T^{6} + T^{8} )^{2}$$
$31$ $$( 81628318996303401 + 2746763081118126 T + 34275384741024 T^{2} + 116636150232 T^{3} - 855373959 T^{4} - 4877064 T^{5} + 41736 T^{6} + 402 T^{7} + T^{8} )^{2}$$
$37$ $$( 81666889425188053156 + 646161179732391370 T + 4356757010847979 T^{2} + 14673347636587 T^{3} + 50423553172 T^{4} + 102777379 T^{5} + 314992 T^{6} + 481 T^{7} + T^{8} )^{2}$$
$41$ $$( 15703157914051608576 - 1898169992301312 T^{2} + 52068871440 T^{4} - 432408 T^{6} + T^{8} )^{2}$$
$43$ $$( 966156928 - 37466096 T - 85728 T^{2} + 433 T^{3} + T^{4} )^{4}$$
$47$ $$20\!\cdots\!76$$$$+$$$$39\!\cdots\!00$$$$T^{2} +$$$$63\!\cdots\!56$$$$T^{4} +$$$$23\!\cdots\!56$$$$T^{6} +$$$$60\!\cdots\!37$$$$T^{8} + 8488943280174618 T^{10} + 85463986203 T^{12} + 339978 T^{14} + T^{16}$$
$53$ $$65\!\cdots\!00$$$$-$$$$23\!\cdots\!00$$$$T^{2} +$$$$52\!\cdots\!25$$$$T^{4} -$$$$72\!\cdots\!75$$$$T^{6} +$$$$71\!\cdots\!86$$$$T^{8} - 45756035835996519 T^{10} + 212669417970 T^{12} - 569799 T^{14} + T^{16}$$
$59$ $$11\!\cdots\!36$$$$+$$$$34\!\cdots\!48$$$$T^{2} +$$$$10\!\cdots\!93$$$$T^{4} +$$$$28\!\cdots\!93$$$$T^{6} +$$$$69\!\cdots\!22$$$$T^{8} + 271081786005979821 T^{10} + 775669902066 T^{12} + 1029045 T^{14} + T^{16}$$
$61$ $$( 25760656722530386944 + 892159531070742144 T + 13303541774870928 T^{2} + 104045523589980 T^{3} + 480636786753 T^{4} + 1367323650 T^{5} + 2370615 T^{6} + 2310 T^{7} + T^{8} )^{2}$$
$67$ $$( 3036229573514088004 + 13656359405763358 T + 297604295739955 T^{2} + 167895466285 T^{3} + 19396001164 T^{4} + 32172037 T^{5} + 260152 T^{6} - 353 T^{7} + T^{8} )^{2}$$
$71$ $$( 182329765184802816 + 123867835918848 T^{2} + 20183760144 T^{4} + 678168 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$13\!\cdots\!64$$$$- 1429202978440395444 T + 3817932715234431 T^{2} + 14737310733717 T^{3} - 42941956140 T^{4} - 194808807 T^{5} + 1022484 T^{6} - 1647 T^{7} + T^{8} )^{2}$$
$79$ $$($$$$11\!\cdots\!61$$$$-$$$$51\!\cdots\!48$$$$T + 1926071802117069670 T^{2} - 2192123600069456 T^{3} + 3059389735471 T^{4} - 1841095856 T^{5} + 2621710 T^{6} - 1328 T^{7} + T^{8} )^{2}$$
$83$ $$($$$$16\!\cdots\!56$$$$- 360446437983184128 T^{2} + 2014371869040 T^{4} - 2852067 T^{6} + T^{8} )^{2}$$
$89$ $$18\!\cdots\!36$$$$+$$$$93\!\cdots\!60$$$$T^{2} +$$$$29\!\cdots\!56$$$$T^{4} +$$$$59\!\cdots\!64$$$$T^{6} +$$$$88\!\cdots\!77$$$$T^{8} + 9111941852559810822 T^{10} + 6939745175403 T^{12} + 3316302 T^{14} + T^{16}$$
$97$ $$( 10175730882067316736 + 2451916579670784 T^{2} + 84302904096 T^{4} + 832731 T^{6} + T^{8} )^{2}$$