# Properties

 Label 33.3.h.b Level 33 Weight 3 Character orbit 33.h Analytic conductor 0.899 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 33.h (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.899184872389$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 12 x^{14} + 180 x^{12} - 2562 x^{10} + 25179 x^{8} - 96398 x^{6} + 239275 x^{4} - 536393 x^{2} + 1771561$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{6} + \beta_{9} ) q^{3} + ( -1 - 4 \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{4} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{5} + ( 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{12} + 2 \beta_{15} ) q^{6} + ( 2 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{7} + ( -\beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{8} + ( -2 - \beta_{1} - 2 \beta_{2} - 3 \beta_{5} - \beta_{6} - \beta_{8} - 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{6} + \beta_{9} ) q^{3} + ( -1 - 4 \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{4} + ( -\beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{5} + ( 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{12} + 2 \beta_{15} ) q^{6} + ( 2 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{14} - \beta_{15} ) q^{7} + ( -\beta_{1} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{8} + ( -2 - \beta_{1} - 2 \beta_{2} - 3 \beta_{5} - \beta_{6} - \beta_{8} - 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{15} ) q^{9} + ( -1 + \beta_{1} - 3 \beta_{3} + \beta_{5} + 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{10} + ( 5 \beta_{1} - 2 \beta_{4} + 8 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{11} + ( 4 - 2 \beta_{1} - 7 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - 2 \beta_{8} + 4 \beta_{9} - \beta_{10} - 3 \beta_{11} - 6 \beta_{12} - \beta_{13} - 3 \beta_{15} ) q^{12} + ( -2 - 2 \beta_{2} - 3 \beta_{6} + 3 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} + 3 \beta_{13} ) q^{13} + ( -2 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} ) q^{14} + ( 5 + 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{15} + ( \beta_{1} + 11 \beta_{2} + 5 \beta_{3} + \beta_{5} - 11 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{16} + ( -3 \beta_{1} + 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} + \beta_{9} - 4 \beta_{11} - \beta_{14} - \beta_{15} ) q^{17} + ( -6 - 8 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} - 6 \beta_{7} + \beta_{8} - 7 \beta_{9} - \beta_{10} - 3 \beta_{11} + 7 \beta_{12} + \beta_{15} ) q^{18} + ( -14 - 5 \beta_{2} - 5 \beta_{3} + 14 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{19} + ( 3 \beta_{1} - 9 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} + 5 \beta_{11} + 7 \beta_{12} + 7 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{20} + ( 3 + 3 \beta_{1} + 9 \beta_{3} - 9 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{21} + ( 10 + 2 \beta_{1} + 6 \beta_{2} + 23 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} + \beta_{10} + 5 \beta_{12} - 5 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{22} + ( -2 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} - 5 \beta_{9} - \beta_{10} - 3 \beta_{11} + \beta_{12} + \beta_{13} + 6 \beta_{14} + 6 \beta_{15} ) q^{23} + ( 7 + 10 \beta_{1} + 7 \beta_{2} - 9 \beta_{5} - 6 \beta_{6} - 5 \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + 10 \beta_{13} + 6 \beta_{14} + \beta_{15} ) q^{24} + ( 2 - 13 \beta_{2} - 13 \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{25} + ( 4 \beta_{1} - 10 \beta_{4} + 12 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 9 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 9 \beta_{12} - 9 \beta_{13} - 6 \beta_{14} - 6 \beta_{15} ) q^{26} + ( 3 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} - 9 \beta_{4} + 21 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 12 \beta_{9} - 9 \beta_{11} - 9 \beta_{12} - 8 \beta_{13} - 12 \beta_{14} - 9 \beta_{15} ) q^{27} + ( -5 \beta_{1} - 23 \beta_{2} - 30 \beta_{3} - 5 \beta_{5} + 23 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} - 5 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} - 5 \beta_{15} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{29} + ( -8 + \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} + 8 \beta_{10} ) q^{30} + ( 15 - 5 \beta_{1} + 15 \beta_{2} - 5 \beta_{5} - 9 \beta_{6} - 10 \beta_{8} - 5 \beta_{9} + 5 \beta_{14} - 5 \beta_{15} ) q^{31} + ( -13 \beta_{4} + 19 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} + 13 \beta_{9} - 5 \beta_{10} - 5 \beta_{11} - 9 \beta_{12} - 9 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} ) q^{32} + ( -9 + 4 \beta_{1} - \beta_{2} - 13 \beta_{3} - \beta_{4} - 11 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} + \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 6 \beta_{11} + 9 \beta_{13} + 3 \beta_{14} + 6 \beta_{15} ) q^{33} + ( 5 + 3 \beta_{1} + 26 \beta_{3} + 3 \beta_{5} - 26 \beta_{6} + 5 \beta_{7} + \beta_{8} - 3 \beta_{10} + \beta_{12} - \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{34} + ( -7 \beta_{1} - 6 \beta_{5} - 5 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{35} + ( -7 - 4 \beta_{1} + 16 \beta_{2} + 16 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} + 7 \beta_{6} + 9 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{36} + ( 16 + 9 \beta_{1} + 21 \beta_{2} + 16 \beta_{3} + 9 \beta_{5} + 9 \beta_{7} + 9 \beta_{8} - 9 \beta_{10} - 9 \beta_{14} + 9 \beta_{15} ) q^{37} + ( 7 \beta_{1} + 7 \beta_{4} - 20 \beta_{5} + 7 \beta_{7} + 7 \beta_{8} - 13 \beta_{9} + 20 \beta_{11} + 18 \beta_{12} + 18 \beta_{13} + 13 \beta_{14} + 13 \beta_{15} ) q^{38} + ( 15 \beta_{2} + 10 \beta_{3} + \beta_{4} - 10 \beta_{5} - 15 \beta_{6} - 9 \beta_{9} + 9 \beta_{11} + 10 \beta_{12} + 12 \beta_{13} + 9 \beta_{14} + \beta_{15} ) q^{39} + ( -27 - 7 \beta_{2} - 27 \beta_{3} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} - 3 \beta_{13} ) q^{40} + ( -31 \beta_{1} + 10 \beta_{4} - 31 \beta_{5} - 19 \beta_{7} - 19 \beta_{8} - 18 \beta_{9} + 18 \beta_{10} ) q^{41} + ( -3 - 6 \beta_{1} - 3 \beta_{2} + 30 \beta_{5} - 21 \beta_{6} + 6 \beta_{8} + 15 \beta_{9} - 12 \beta_{10} - 21 \beta_{11} - 9 \beta_{12} - 15 \beta_{13} - 9 \beta_{14} - 3 \beta_{15} ) q^{42} + ( -9 - 5 \beta_{1} + 6 \beta_{3} - 5 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} + 2 \beta_{8} + 5 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} - 5 \beta_{15} ) q^{43} + ( -16 \beta_{1} - 6 \beta_{4} + 27 \beta_{5} - 7 \beta_{7} - 7 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} - 17 \beta_{11} - 17 \beta_{12} - 17 \beta_{13} - 10 \beta_{14} - 10 \beta_{15} ) q^{44} + ( -12 - 8 \beta_{1} - 14 \beta_{3} - 8 \beta_{4} - \beta_{5} + 14 \beta_{6} - 3 \beta_{7} + \beta_{8} + 8 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} - 7 \beta_{12} - 7 \beta_{13} - 6 \beta_{14} - 10 \beta_{15} ) q^{45} + ( -11 - 11 \beta_{1} - 11 \beta_{2} - 11 \beta_{5} + 44 \beta_{6} - 22 \beta_{8} + 11 \beta_{10} - 11 \beta_{12} + 11 \beta_{13} + 11 \beta_{14} - 11 \beta_{15} ) q^{46} + ( 35 \beta_{1} - 23 \beta_{4} + 35 \beta_{5} + 14 \beta_{7} + 14 \beta_{8} + 17 \beta_{9} - 17 \beta_{10} ) q^{47} + ( -12 - 45 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} - 3 \beta_{7} - 9 \beta_{9} + 3 \beta_{11} + 9 \beta_{12} + 6 \beta_{13} - 6 \beta_{14} + 6 \beta_{15} ) q^{48} + ( 5 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} + 5 \beta_{5} - 12 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 4 \beta_{12} - 4 \beta_{13} - 5 \beta_{14} + 5 \beta_{15} ) q^{49} + ( \beta_{1} + 9 \beta_{4} - 6 \beta_{5} + \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{11} - 7 \beta_{12} - 7 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{50} + ( 14 + 12 \beta_{1} + 13 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} + 18 \beta_{11} + 5 \beta_{12} + 5 \beta_{13} + 3 \beta_{14} + 8 \beta_{15} ) q^{51} + ( 23 - 2 \beta_{2} - 2 \beta_{3} - 23 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} ) q^{52} + ( 16 \beta_{1} - 3 \beta_{5} + 3 \beta_{9} - 9 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + 6 \beta_{13} + 6 \beta_{14} + 6 \beta_{15} ) q^{53} + ( 65 + 7 \beta_{1} - 3 \beta_{3} - 9 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{10} + 6 \beta_{11} - 2 \beta_{12} + 8 \beta_{13} + 12 \beta_{14} + \beta_{15} ) q^{54} + ( -26 + 8 \beta_{1} - 31 \beta_{2} - 7 \beta_{3} + 8 \beta_{5} + 32 \beta_{6} + 14 \beta_{7} + 2 \beta_{8} + \beta_{9} - 7 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 8 \beta_{14} + 8 \beta_{15} ) q^{55} + ( 38 \beta_{1} + 9 \beta_{4} - 26 \beta_{5} + 14 \beta_{7} + 14 \beta_{8} - 9 \beta_{9} - \beta_{10} + 37 \beta_{11} + 23 \beta_{12} + 23 \beta_{13} + 10 \beta_{14} + 10 \beta_{15} ) q^{56} + ( 28 - 4 \beta_{1} + 28 \beta_{2} + 3 \beta_{5} - 13 \beta_{6} - 4 \beta_{8} - 9 \beta_{9} + \beta_{10} - 6 \beta_{11} - 7 \beta_{12} - 12 \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{57} + ( 2 + 18 \beta_{2} + 18 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{58} + ( -27 \beta_{1} + 6 \beta_{4} + 23 \beta_{5} + \beta_{7} + \beta_{8} + 22 \beta_{9} - \beta_{10} - 39 \beta_{11} - 22 \beta_{12} - 22 \beta_{13} - 21 \beta_{14} - 21 \beta_{15} ) q^{59} + ( -12 \beta_{1} - 15 \beta_{2} + 23 \beta_{3} - 7 \beta_{4} + \beta_{5} + 15 \beta_{6} - 12 \beta_{7} - 12 \beta_{8} - 6 \beta_{9} - 6 \beta_{11} + 2 \beta_{12} - 12 \beta_{13} + 6 \beta_{14} - 10 \beta_{15} ) q^{60} + ( 2 \beta_{1} + 17 \beta_{2} + 15 \beta_{3} + 2 \beta_{5} - 17 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 11 \beta_{12} - 11 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{61} + ( -19 \beta_{1} + 15 \beta_{4} - 36 \beta_{5} - 20 \beta_{7} - 20 \beta_{8} - 16 \beta_{9} + 20 \beta_{10} - 9 \beta_{11} + 16 \beta_{12} + 16 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{62} + ( -33 + 6 \beta_{1} - 33 \beta_{2} - 33 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 33 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - 6 \beta_{10} ) q^{63} + ( 14 + 14 \beta_{1} + 14 \beta_{2} + 14 \beta_{5} + \beta_{6} + 28 \beta_{8} + 10 \beta_{9} - 4 \beta_{10} + 4 \beta_{12} - 4 \beta_{13} - 14 \beta_{14} + 14 \beta_{15} ) q^{64} + ( 10 \beta_{1} + 16 \beta_{4} - 7 \beta_{5} - \beta_{7} - \beta_{8} - 16 \beta_{9} + 4 \beta_{10} + 14 \beta_{11} + 15 \beta_{12} + 15 \beta_{13} + 12 \beta_{14} + 12 \beta_{15} ) q^{65} + ( -39 + 9 \beta_{1} - 41 \beta_{2} - 49 \beta_{3} - 16 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} + 9 \beta_{8} - 3 \beta_{9} - 21 \beta_{10} + 9 \beta_{11} + 14 \beta_{12} - 13 \beta_{13} + 3 \beta_{14} + 5 \beta_{15} ) q^{66} + ( 38 - 12 \beta_{1} + 21 \beta_{3} - 12 \beta_{5} - 21 \beta_{6} - 9 \beta_{7} - 15 \beta_{8} + 12 \beta_{10} - 15 \beta_{12} + 15 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} ) q^{67} + ( -16 \beta_{1} + 25 \beta_{5} + 10 \beta_{9} - 7 \beta_{10} - 22 \beta_{11} - 15 \beta_{12} - 15 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{68} + ( 22 - 23 \beta_{1} + 11 \beta_{2} + 11 \beta_{3} + 17 \beta_{4} - 23 \beta_{5} - 22 \beta_{6} - 18 \beta_{7} + 4 \beta_{8} - 8 \beta_{9} + 8 \beta_{10} ) q^{69} + ( -11 - 3 \beta_{1} + 44 \beta_{2} - 11 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} - 8 \beta_{8} + 5 \beta_{9} + 8 \beta_{10} - 5 \beta_{12} + 5 \beta_{13} + 3 \beta_{14} - 3 \beta_{15} ) q^{70} + ( -5 \beta_{1} + 3 \beta_{4} - 13 \beta_{5} - 5 \beta_{7} - 5 \beta_{8} - 10 \beta_{9} + 5 \beta_{11} + \beta_{12} + \beta_{13} + 10 \beta_{14} + 10 \beta_{15} ) q^{71} + ( -3 \beta_{1} + 32 \beta_{2} + 80 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} - 32 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + 3 \beta_{11} + 2 \beta_{12} + \beta_{13} + 6 \beta_{14} + 17 \beta_{15} ) q^{72} + ( -18 - 2 \beta_{1} - 56 \beta_{2} - 18 \beta_{3} - 2 \beta_{5} - 5 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{73} + ( 52 \beta_{1} - 39 \beta_{4} + 52 \beta_{5} + 27 \beta_{7} + 27 \beta_{8} + 36 \beta_{9} - 36 \beta_{10} ) q^{74} + ( -15 \beta_{1} - 6 \beta_{5} + 9 \beta_{6} - 15 \beta_{8} + 12 \beta_{9} + 12 \beta_{10} + 3 \beta_{11} - 9 \beta_{12} + 18 \beta_{13} + 3 \beta_{14} - 12 \beta_{15} ) q^{75} + ( -53 + 11 \beta_{1} + 6 \beta_{3} + 11 \beta_{5} - 6 \beta_{6} + 17 \beta_{7} + 5 \beta_{8} - 11 \beta_{10} + 5 \beta_{12} - 5 \beta_{13} - 11 \beta_{14} + 11 \beta_{15} ) q^{76} + ( -9 \beta_{1} + 18 \beta_{4} - 29 \beta_{5} - 4 \beta_{7} - 4 \beta_{8} - 8 \beta_{9} + 18 \beta_{10} + 5 \beta_{11} + 7 \beta_{12} + 7 \beta_{13} - 10 \beta_{14} - 10 \beta_{15} ) q^{77} + ( -51 - 7 \beta_{1} - 43 \beta_{3} - \beta_{4} + 13 \beta_{5} + 43 \beta_{6} + 18 \beta_{7} + 2 \beta_{8} + \beta_{9} + 4 \beta_{10} - 3 \beta_{11} + \beta_{12} - 5 \beta_{13} - 27 \beta_{14} - 5 \beta_{15} ) q^{78} + ( -48 + 8 \beta_{1} - 48 \beta_{2} + 8 \beta_{5} + 49 \beta_{6} + 16 \beta_{8} - 9 \beta_{9} - 17 \beta_{10} + 17 \beta_{12} - 17 \beta_{13} - 8 \beta_{14} + 8 \beta_{15} ) q^{79} + ( -31 \beta_{1} + 21 \beta_{4} - 31 \beta_{5} - 6 \beta_{7} - 6 \beta_{8} - 23 \beta_{9} + 23 \beta_{10} ) q^{80} + ( -34 + 19 \beta_{1} - 15 \beta_{2} - 34 \beta_{3} - \beta_{4} + 19 \beta_{5} + 27 \beta_{7} + 10 \beta_{8} + 9 \beta_{9} - 10 \beta_{10} + 9 \beta_{11} - 9 \beta_{12} + 8 \beta_{13} + 12 \beta_{14} + 18 \beta_{15} ) q^{81} + ( -24 \beta_{1} + 5 \beta_{2} - 37 \beta_{3} - 24 \beta_{5} - 5 \beta_{6} - 24 \beta_{7} - 24 \beta_{8} - 24 \beta_{9} - 10 \beta_{12} + 10 \beta_{13} + 24 \beta_{14} - 24 \beta_{15} ) q^{82} + ( 13 \beta_{1} - 12 \beta_{4} + 20 \beta_{5} + 13 \beta_{7} + 13 \beta_{8} + 8 \beta_{9} + 5 \beta_{11} - 5 \beta_{12} - 5 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} ) q^{83} + ( 57 - 15 \beta_{1} + 93 \beta_{2} + 57 \beta_{3} - 3 \beta_{4} + 18 \beta_{5} + 15 \beta_{7} - 15 \beta_{8} + 33 \beta_{9} + 15 \beta_{10} - 15 \beta_{11} - 33 \beta_{12} - 3 \beta_{13} - 18 \beta_{15} ) q^{84} + ( 35 + 7 \beta_{2} + 7 \beta_{3} - 35 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} - 8 \beta_{9} - 8 \beta_{10} ) q^{85} + ( -24 \beta_{1} + 11 \beta_{5} - 11 \beta_{9} + 22 \beta_{10} - 22 \beta_{12} - 22 \beta_{13} - 11 \beta_{14} - 11 \beta_{15} ) q^{86} + ( 6 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} - 2 \beta_{15} ) q^{87} + ( 102 - 17 \beta_{1} + 81 \beta_{2} + 30 \beta_{3} - 17 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} - 29 \beta_{8} - 9 \beta_{9} + 8 \beta_{10} - 15 \beta_{12} + 15 \beta_{13} + 17 \beta_{14} - 17 \beta_{15} ) q^{88} + ( \beta_{1} - 9 \beta_{4} + 29 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 9 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} - 5 \beta_{14} - 5 \beta_{15} ) q^{89} + ( 43 + 9 \beta_{1} + 43 \beta_{2} - 21 \beta_{5} - 95 \beta_{6} + 18 \beta_{8} - 13 \beta_{9} + 6 \beta_{10} + 18 \beta_{11} + 12 \beta_{12} + 8 \beta_{13} + 3 \beta_{14} + 21 \beta_{15} ) q^{90} + ( 35 + 7 \beta_{2} + 7 \beta_{3} - 35 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} ) q^{91} + ( 15 \beta_{1} + 13 \beta_{4} - 62 \beta_{5} - 17 \beta_{7} - 17 \beta_{8} - 45 \beta_{9} + 17 \beta_{10} + 30 \beta_{11} + 45 \beta_{12} + 45 \beta_{13} + 28 \beta_{14} + 28 \beta_{15} ) q^{92} + ( 15 \beta_{1} - 19 \beta_{2} + 45 \beta_{3} + 30 \beta_{5} + 19 \beta_{6} + 15 \beta_{7} + 15 \beta_{8} + 30 \beta_{9} - 15 \beta_{11} - 15 \beta_{12} - 11 \beta_{13} - 30 \beta_{14} ) q^{93} + ( 13 \beta_{1} - 82 \beta_{2} - 95 \beta_{3} + 13 \beta_{5} + 82 \beta_{6} + 13 \beta_{7} + 13 \beta_{8} + 13 \beta_{9} - 12 \beta_{12} + 12 \beta_{13} - 13 \beta_{14} + 13 \beta_{15} ) q^{94} + ( 41 \beta_{1} - 41 \beta_{4} + 46 \beta_{5} + 23 \beta_{7} + 23 \beta_{8} + 23 \beta_{9} - 23 \beta_{10} + 14 \beta_{11} - 23 \beta_{12} - 23 \beta_{13} ) q^{95} + ( -51 + 8 \beta_{1} - 41 \beta_{2} - 41 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} + 51 \beta_{6} + 24 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 11 \beta_{10} ) q^{96} + ( -5 - 8 \beta_{1} - 5 \beta_{2} - 8 \beta_{5} + 46 \beta_{6} - 16 \beta_{8} + 2 \beta_{9} + 10 \beta_{10} - 10 \beta_{12} + 10 \beta_{13} + 8 \beta_{14} - 8 \beta_{15} ) q^{97} + ( -22 \beta_{1} - 24 \beta_{4} + 30 \beta_{5} - 7 \beta_{7} - 7 \beta_{8} + 24 \beta_{9} - 16 \beta_{10} - 38 \beta_{11} - 31 \beta_{12} - 31 \beta_{13} - 8 \beta_{14} - 8 \beta_{15} ) q^{98} + ( 2 - 16 \beta_{1} + 32 \beta_{2} + 42 \beta_{3} + 15 \beta_{4} - 12 \beta_{5} - 60 \beta_{6} - 18 \beta_{7} - 34 \beta_{8} - 7 \beta_{9} + 13 \beta_{10} - 15 \beta_{11} + 5 \beta_{12} + 14 \beta_{13} + 12 \beta_{14} - 7 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 10q^{3} + 8q^{4} - 33q^{6} + 6q^{7} - 28q^{9} + O(q^{10})$$ $$16q - 10q^{3} + 8q^{4} - 33q^{6} + 6q^{7} - 28q^{9} - 12q^{10} + 106q^{12} - 42q^{13} + 82q^{15} - 88q^{16} - 43q^{18} - 134q^{19} - 12q^{21} + 78q^{22} + 41q^{24} + 134q^{25} + 80q^{27} + 264q^{28} - 120q^{30} + 124q^{31} - 79q^{33} - 132q^{34} - 219q^{36} + 90q^{37} - 174q^{39} - 284q^{40} - 102q^{42} - 156q^{43} - 72q^{45} - 22q^{46} + 30q^{48} - 30q^{49} + 111q^{51} + 326q^{52} + 1046q^{54} - 172q^{55} + 281q^{57} - 116q^{58} + 54q^{60} - 126q^{61} - 138q^{63} + 236q^{64} - 236q^{66} + 368q^{67} + 198q^{69} - 322q^{70} - 562q^{72} + 24q^{73} - 21q^{75} - 900q^{76} - 492q^{78} - 314q^{79} - 388q^{81} + 270q^{84} + 318q^{85} + 132q^{87} + 1064q^{88} + 176q^{90} + 374q^{91} - 10q^{93} + 990q^{94} - 332q^{96} + 72q^{97} - 530q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12 x^{14} + 180 x^{12} - 2562 x^{10} + 25179 x^{8} - 96398 x^{6} + 239275 x^{4} - 536393 x^{2} + 1771561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-127407047787804 \nu^{14} - 322126035973969 \nu^{12} - 40525263758459540 \nu^{10} + 344558428769805180 \nu^{8} - 4303642725224589076 \nu^{6} + 44698619213114763312 \nu^{4} - 744861315136512930500 \nu^{2} + 374807322561273688856$$$$)/$$$$21\!\cdots\!19$$ $$\beta_{3}$$ $$=$$ $$($$$$234959048766210 \nu^{14} + 11921441073701127 \nu^{12} - 85177596039692545 \nu^{10} + 1520211191539599580 \nu^{8} - 23265517903780782960 \nu^{6} + 247688922335353991145 \nu^{4} - 208024183527244179835 \nu^{2} - 77838063410538248499$$$$)/$$$$21\!\cdots\!19$$ $$\beta_{4}$$ $$=$$ $$($$$$127407047787804 \nu^{15} + 322126035973969 \nu^{13} + 40525263758459540 \nu^{11} - 344558428769805180 \nu^{9} + 4303642725224589076 \nu^{7} - 44698619213114763312 \nu^{5} + 744861315136512930500 \nu^{3} - 374807322561273688856 \nu$$$$)/$$$$21\!\cdots\!19$$ $$\beta_{5}$$ $$=$$ $$($$$$234959048766210 \nu^{15} + 11921441073701127 \nu^{13} - 85177596039692545 \nu^{11} + 1520211191539599580 \nu^{9} - 23265517903780782960 \nu^{7} + 247688922335353991145 \nu^{5} - 208024183527244179835 \nu^{3} - 77838063410538248499 \nu$$$$)/$$$$21\!\cdots\!19$$ $$\beta_{6}$$ $$=$$ $$($$$$4438276679147244 \nu^{14} - 38143852521781896 \nu^{12} + 631568334130072500 \nu^{10} - 8766582639358512750 \nu^{8} + 76663977785573994959 \nu^{6} - 74027125217335407442 \nu^{4} - 82997600383267449850 \nu^{2} + 207581537941818757970$$$$)/$$$$21\!\cdots\!19$$ $$\beta_{7}$$ $$=$$ $$($$$$-2949643553271501 \nu^{15} - 107478072222941431 \nu^{14} + 31035928814102023 \nu^{13} + 742877598465063798 \nu^{12} - 1694622520719673097 \nu^{11} - 11853231551733086559 \nu^{10} + 18173418136790321913 \nu^{9} + 184172957827569504013 \nu^{8} - 247225326229442937739 \nu^{7} - 1221194376663844125048 \nu^{6} + 2596346362793508526818 \nu^{5} - 3066965541789538832905 \nu^{4} - 21420285449145050092157 \nu^{3} + 17059214158261006460234 \nu^{2} + 21898844749963893603650 \nu + 38060055284176600406699$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{8}$$ $$=$$ $$($$$$-2949643553271501 \nu^{15} + 107478072222941431 \nu^{14} + 31035928814102023 \nu^{13} - 742877598465063798 \nu^{12} - 1694622520719673097 \nu^{11} + 11853231551733086559 \nu^{10} + 18173418136790321913 \nu^{9} - 184172957827569504013 \nu^{8} - 247225326229442937739 \nu^{7} + 1221194376663844125048 \nu^{6} + 2596346362793508526818 \nu^{5} + 3066965541789538832905 \nu^{4} - 21420285449145050092157 \nu^{3} - 17059214158261006460234 \nu^{2} + 21898844749963893603650 \nu - 38060055284176600406699$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{9}$$ $$=$$ $$($$$$5530938613687128 \nu^{15} + 166012584300402708 \nu^{14} - 33630130225371761 \nu^{13} - 2117280218216450276 \nu^{12} + 2005487528427821201 \nu^{11} + 31045509051468975822 \nu^{10} - 18276913998033521354 \nu^{9} - 453165291912414021743 \nu^{8} + 234870744257407535442 \nu^{7} + 4437259853751563534641 \nu^{6} - 2448733533544500813279 \nu^{5} - 18835563517351609927800 \nu^{4} + 17382546759243455724562 \nu^{3} + 48992445765202921573548 \nu^{2} - 20574790310578950257680 \nu - 91659310975868063351165$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{10}$$ $$=$$ $$($$$$-5530938613687128 \nu^{15} + 166012584300402708 \nu^{14} + 33630130225371761 \nu^{13} - 2117280218216450276 \nu^{12} - 2005487528427821201 \nu^{11} + 31045509051468975822 \nu^{10} + 18276913998033521354 \nu^{9} - 453165291912414021743 \nu^{8} - 234870744257407535442 \nu^{7} + 4437259853751563534641 \nu^{6} + 2448733533544500813279 \nu^{5} - 18835563517351609927800 \nu^{4} - 17382546759243455724562 \nu^{3} + 48992445765202921573548 \nu^{2} + 20574790310578950257680 \nu - 91659310975868063351165$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{11}$$ $$=$$ $$($$$$30054413983928844 \nu^{15} - 201987708092060081 \nu^{13} + 4502519471732524742 \nu^{11} - 60964388315221160139 \nu^{9} + 517159654487638123533 \nu^{7} - 1439208800755732819107 \nu^{5} + 15876462984469736097197 \nu^{3} - 55056646013161149255646 \nu$$$$)/$$$$23\!\cdots\!09$$ $$\beta_{12}$$ $$=$$ $$($$$$-72618660380064154 \nu^{15} + 49907241408562835 \nu^{14} + 904789569200787956 \nu^{13} - 1271613859106383353 \nu^{12} - 13142255289684674994 \nu^{11} + 14689100732554831748 \nu^{10} + 193257387972027534305 \nu^{9} - 221948869246166066510 \nu^{8} - 1861216681368690697007 \nu^{7} + 2581424220326366958345 \nu^{6} + 7651271435780790370144 \nu^{5} - 15244472422698155686654 \nu^{4} - 19622760602151650003666 \nu^{3} + 22215491198933656595906 \nu^{2} + 38836150383671216498131 \nu + 2527463022467253365682$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{13}$$ $$=$$ $$($$$$-72618660380064154 \nu^{15} - 49907241408562835 \nu^{14} + 904789569200787956 \nu^{13} + 1271613859106383353 \nu^{12} - 13142255289684674994 \nu^{11} - 14689100732554831748 \nu^{10} + 193257387972027534305 \nu^{9} + 221948869246166066510 \nu^{8} - 1861216681368690697007 \nu^{7} - 2581424220326366958345 \nu^{6} + 7651271435780790370144 \nu^{5} + 15244472422698155686654 \nu^{4} - 19622760602151650003666 \nu^{3} - 22215491198933656595906 \nu^{2} + 38836150383671216498131 \nu - 2527463022467253365682$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{14}$$ $$=$$ $$($$$$99135684000027241 \nu^{15} - 20810066889315561 \nu^{14} - 862706736638173675 \nu^{13} - 476043899992241295 \nu^{12} + 14023945388217859866 \nu^{11} + 1701287476880316347 \nu^{10} - 195384258337122023097 \nu^{9} - 55882176278914975803 \nu^{8} + 1663164806667153890606 \nu^{7} + 920555929379994081361 \nu^{6} - 1429591891788951514170 \nu^{5} - 9875787159646622730582 \nu^{4} - 5780946714034994365846 \nu^{3} + 20270682318663355761963 \nu^{2} + 18115509591203040974177 \nu - 84217831418990427698342$$$$)/$$$$47\!\cdots\!18$$ $$\beta_{15}$$ $$=$$ $$($$$$94334933420026495 \nu^{15} + 186822651189718269 \nu^{14} - 1153420167662430754 \nu^{13} - 1641236318224208981 \nu^{12} + 17281610014102620849 \nu^{11} + 29344221574588659475 \nu^{10} - 246898826826540336329 \nu^{9} - 397283115633499045940 \nu^{8} + 2434586108751809455762 \nu^{7} + 3516703924371569453280 \nu^{6} - 9622707375209255559717 \nu^{5} - 8959776357704987197218 \nu^{4} + 24253609462611022050276 \nu^{3} + 28721763446539565811585 \nu^{2} - 51394738255679059357083 \nu - 7441479556877635652823$$$$)/$$$$47\!\cdots\!18$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - 8 \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - \beta_{5} + 10 \beta_{4} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{14} - 15 \beta_{13} + 15 \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 23 \beta_{6} + \beta_{5} + 85 \beta_{3} + 23 \beta_{2} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{15} - 8 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} - 85 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} - 44 \beta_{8} - 44 \beta_{7} + 99 \beta_{5} + 3 \beta_{4} - 96 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-190 \beta_{15} + 190 \beta_{14} + 220 \beta_{13} - 220 \beta_{12} + 220 \beta_{10} + 30 \beta_{9} - 380 \beta_{8} + 633 \beta_{6} - 190 \beta_{5} + 378 \beta_{2} - 190 \beta_{1} + 378$$ $$\nu^{7}$$ $$=$$ $$603 \beta_{15} + 603 \beta_{14} + 1143 \beta_{13} + 1143 \beta_{12} + 1073 \beta_{11} + 540 \beta_{10} - 1143 \beta_{9} - 540 \beta_{8} - 540 \beta_{7} - 1683 \beta_{5} - 118 \beta_{4} + 721 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2341 \beta_{10} - 2341 \beta_{9} + 588 \beta_{8} - 588 \beta_{7} + 6969 \beta_{6} - 12506 \beta_{3} - 12506 \beta_{2} - 6969$$ $$\nu^{9}$$ $$=$$ $$6381 \beta_{15} + 6381 \beta_{14} + 12816 \beta_{13} + 12816 \beta_{12} + 12827 \beta_{11} - 6381 \beta_{9} + 6446 \beta_{8} + 6446 \beta_{7} - 22393 \beta_{5} + 16012 \beta_{4} + 6446 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$28893 \beta_{15} - 28893 \beta_{14} - 38546 \beta_{13} + 38546 \beta_{12} - 28893 \beta_{10} + 38546 \beta_{8} + 19240 \beta_{7} - 77049 \beta_{6} + 28893 \beta_{5} + 77049 \beta_{3} + 28893 \beta_{1} - 80060$$ $$\nu^{11}$$ $$=$$ $$-48156 \beta_{15} - 48156 \beta_{14} - 134901 \beta_{13} - 134901 \beta_{12} - 211927 \beta_{11} - 77026 \beta_{10} + 125182 \beta_{9} + 260083 \beta_{5} - 205308 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-144620 \beta_{15} + 144620 \beta_{14} + 359360 \beta_{13} - 359360 \beta_{12} + 503980 \beta_{10} + 359360 \beta_{9} - 503980 \beta_{8} + 214740 \beta_{7} - 144620 \beta_{5} + 1043432 \beta_{3} + 1989005 \beta_{2} - 144620 \beta_{1} + 1043432$$ $$\nu^{13}$$ $$=$$ $$219120 \beta_{10} - 219120 \beta_{9} - 1152580 \beta_{8} - 1152580 \beta_{7} + 464952 \beta_{5} - 2418485 \beta_{4} + 464952 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-2055772 \beta_{15} + 2055772 \beta_{14} + 4504525 \beta_{13} - 4504525 \beta_{12} - 2055772 \beta_{9} - 2055772 \beta_{8} - 2055772 \beta_{7} + 13903349 \beta_{6} - 2055772 \beta_{5} - 25299632 \beta_{3} - 13903349 \beta_{2} - 2055772 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$9398824 \beta_{15} + 9398824 \beta_{14} + 24575190 \beta_{13} + 24575190 \beta_{12} + 36032993 \beta_{11} + 3718563 \beta_{10} - 13117387 \beta_{9} + 11457803 \beta_{8} + 11457803 \beta_{7} - 47033088 \beta_{5} + 13117387 \beta_{4} + 32314430 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −2.91048 + 0.945671i −1.90610 + 0.619331i 1.90610 − 0.619331i 2.91048 − 0.945671i −2.10855 − 2.90217i −0.974642 − 1.34148i 0.974642 + 1.34148i 2.10855 + 2.90217i −2.91048 − 0.945671i −1.90610 − 0.619331i 1.90610 + 0.619331i 2.91048 + 0.945671i −2.10855 + 2.90217i −0.974642 + 1.34148i 0.974642 − 1.34148i 2.10855 − 2.90217i
−2.91048 + 0.945671i 1.65950 2.49921i 4.34051 3.15356i 6.31437 + 2.05166i −2.46650 + 8.84324i 2.47800 1.80037i −2.45561 + 3.37986i −3.49213 8.29488i −20.3180
5.2 −1.90610 + 0.619331i −2.89787 0.776113i 0.0135968 0.00987866i −5.21596 1.69477i 6.00431 0.315387i −4.52308 + 3.28621i 4.69235 6.45847i 7.79530 + 4.49815i 10.9918
5.3 1.90610 0.619331i −1.63362 2.51621i 0.0135968 0.00987866i 5.21596 + 1.69477i −4.67221 3.78440i −4.52308 + 3.28621i −4.69235 + 6.45847i −3.66258 + 8.22104i 10.9918
5.4 2.91048 0.945671i −1.86408 + 2.35058i 4.34051 3.15356i −6.31437 2.05166i −3.20248 + 8.60410i 2.47800 1.80037i 2.45561 3.37986i −2.05042 8.76332i −20.3180
14.1 −2.10855 2.90217i 0.307087 2.98424i −2.74053 + 8.43448i −1.22635 + 1.68793i −9.30827 + 5.40120i 2.73883 8.42924i 16.6100 5.39692i −8.81140 1.83284i 7.48447
14.2 −0.974642 1.34148i 2.52902 + 1.61371i 0.386428 1.18930i 0.410570 0.565101i −0.300138 4.96542i 0.806259 2.48141i −8.28007 + 2.69036i 3.79191 + 8.16220i −1.15823
14.3 0.974642 + 1.34148i −1.09751 + 2.79204i 0.386428 1.18930i −0.410570 + 0.565101i −4.81514 + 1.24895i 0.806259 2.48141i 8.28007 2.69036i −6.59095 6.12857i −1.15823
14.4 2.10855 + 2.90217i −2.00253 2.23380i −2.74053 + 8.43448i 1.22635 1.68793i 2.26043 10.5218i 2.73883 8.42924i −16.6100 + 5.39692i −0.979734 + 8.94651i 7.48447
20.1 −2.91048 0.945671i 1.65950 + 2.49921i 4.34051 + 3.15356i 6.31437 2.05166i −2.46650 8.84324i 2.47800 + 1.80037i −2.45561 3.37986i −3.49213 + 8.29488i −20.3180
20.2 −1.90610 0.619331i −2.89787 + 0.776113i 0.0135968 + 0.00987866i −5.21596 + 1.69477i 6.00431 + 0.315387i −4.52308 3.28621i 4.69235 + 6.45847i 7.79530 4.49815i 10.9918
20.3 1.90610 + 0.619331i −1.63362 + 2.51621i 0.0135968 + 0.00987866i 5.21596 1.69477i −4.67221 + 3.78440i −4.52308 3.28621i −4.69235 6.45847i −3.66258 8.22104i 10.9918
20.4 2.91048 + 0.945671i −1.86408 2.35058i 4.34051 + 3.15356i −6.31437 + 2.05166i −3.20248 8.60410i 2.47800 + 1.80037i 2.45561 + 3.37986i −2.05042 + 8.76332i −20.3180
26.1 −2.10855 + 2.90217i 0.307087 + 2.98424i −2.74053 8.43448i −1.22635 1.68793i −9.30827 5.40120i 2.73883 + 8.42924i 16.6100 + 5.39692i −8.81140 + 1.83284i 7.48447
26.2 −0.974642 + 1.34148i 2.52902 1.61371i 0.386428 + 1.18930i 0.410570 + 0.565101i −0.300138 + 4.96542i 0.806259 + 2.48141i −8.28007 2.69036i 3.79191 8.16220i −1.15823
26.3 0.974642 1.34148i −1.09751 2.79204i 0.386428 + 1.18930i −0.410570 0.565101i −4.81514 1.24895i 0.806259 + 2.48141i 8.28007 + 2.69036i −6.59095 + 6.12857i −1.15823
26.4 2.10855 2.90217i −2.00253 + 2.23380i −2.74053 8.43448i 1.22635 + 1.68793i 2.26043 + 10.5218i 2.73883 + 8.42924i −16.6100 5.39692i −0.979734 8.94651i 7.48447
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.4 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
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.h.b 16
3.b odd 2 1 inner 33.3.h.b 16
11.b odd 2 1 363.3.h.j 16
11.c even 5 1 inner 33.3.h.b 16
11.c even 5 1 363.3.b.m 8
11.c even 5 2 363.3.h.o 16
11.d odd 10 1 363.3.b.l 8
11.d odd 10 1 363.3.h.j 16
11.d odd 10 2 363.3.h.n 16
33.d even 2 1 363.3.h.j 16
33.f even 10 1 363.3.b.l 8
33.f even 10 1 363.3.h.j 16
33.f even 10 2 363.3.h.n 16
33.h odd 10 1 inner 33.3.h.b 16
33.h odd 10 1 363.3.b.m 8
33.h odd 10 2 363.3.h.o 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b 16 1.a even 1 1 trivial
33.3.h.b 16 3.b odd 2 1 inner
33.3.h.b 16 11.c even 5 1 inner
33.3.h.b 16 33.h odd 10 1 inner
363.3.b.l 8 11.d odd 10 1
363.3.b.l 8 33.f even 10 1
363.3.b.m 8 11.c even 5 1
363.3.b.m 8 33.h odd 10 1
363.3.h.j 16 11.b odd 2 1
363.3.h.j 16 11.d odd 10 1
363.3.h.j 16 33.d even 2 1
363.3.h.j 16 33.f even 10 1
363.3.h.n 16 11.d odd 10 2
363.3.h.n 16 33.f even 10 2
363.3.h.o 16 11.c even 5 2
363.3.h.o 16 33.h odd 10 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T^{2} + 12 T^{4} + 46 T^{6} + 331 T^{8} + 1570 T^{10} + 5163 T^{12} + 12695 T^{14} + 94057 T^{16} + 203120 T^{18} + 1321728 T^{20} + 6430720 T^{22} + 21692416 T^{24} + 48234496 T^{26} + 201326592 T^{28} + 1073741824 T^{30} + 4294967296 T^{32}$$
$3$ $$1 + 10 T + 64 T^{2} + 280 T^{3} + 1045 T^{4} + 3510 T^{5} + 11556 T^{6} + 36720 T^{7} + 112509 T^{8} + 330480 T^{9} + 936036 T^{10} + 2558790 T^{11} + 6856245 T^{12} + 16533720 T^{13} + 34012224 T^{14} + 47829690 T^{15} + 43046721 T^{16}$$
$5$ $$1 - 17 T^{2} + 2430 T^{4} - 10832 T^{6} + 2625694 T^{8} + 7572877 T^{10} + 2297500515 T^{12} + 7383221492 T^{14} + 1688830878256 T^{16} + 4614513432500 T^{18} + 897461138671875 T^{20} + 1848846923828125 T^{22} + 400649108886718750 T^{24} - 1033020019531250000 T^{26} +$$$$14\!\cdots\!50$$$$T^{28} -$$$$63\!\cdots\!25$$$$T^{30} +$$$$23\!\cdots\!25$$$$T^{32}$$
$7$ $$( 1 - 3 T - 37 T^{2} - 358 T^{3} + 3346 T^{4} + 4635 T^{5} + 54602 T^{6} - 1007520 T^{7} + 4844597 T^{8} - 49368480 T^{9} + 131099402 T^{10} + 545303115 T^{11} + 19289024146 T^{12} - 101126139142 T^{13} - 512127626437 T^{14} - 2034669218547 T^{15} + 33232930569601 T^{16} )^{2}$$
$11$ $$1 - 72 T^{2} - 31537 T^{4} + 407286 T^{6} + 664335375 T^{8} + 5963074326 T^{10} - 6760236030097 T^{12} - 225966843123912 T^{14} + 45949729863572161 T^{16}$$
$13$ $$( 1 + 21 T + 81 T^{2} - 770 T^{3} + 7728 T^{4} + 805195 T^{5} + 9940744 T^{6} + 7821066 T^{7} - 396624107 T^{8} + 1321760154 T^{9} + 283917589384 T^{10} + 3886522472755 T^{11} + 6303967011888 T^{12} - 106151038723730 T^{13} + 1887144894920961 T^{14} + 82684904099685069 T^{15} + 665416609183179841 T^{16} )^{2}$$
$17$ $$1 + 894 T^{2} + 336120 T^{4} + 20399550 T^{6} - 26640872055 T^{8} - 10092273774498 T^{10} - 482862517622152 T^{12} + 723393848340773460 T^{14} +$$$$32\!\cdots\!25$$$$T^{16} +$$$$60\!\cdots\!60$$$$T^{18} -$$$$33\!\cdots\!32$$$$T^{20} -$$$$58\!\cdots\!78$$$$T^{22} -$$$$12\!\cdots\!55$$$$T^{24} +$$$$82\!\cdots\!50$$$$T^{26} +$$$$11\!\cdots\!20$$$$T^{28} +$$$$25\!\cdots\!54$$$$T^{30} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$( 1 + 67 T + 1578 T^{2} + 12676 T^{3} - 81938 T^{4} - 6545735 T^{5} - 212459133 T^{6} - 2352569968 T^{7} - 4202891960 T^{8} - 849277758448 T^{9} - 27687886671693 T^{10} - 307949869867535 T^{11} - 1391599188453458 T^{12} + 77717395883885476 T^{13} + 3492610942286402058 T^{14} + 53533447947453236107 T^{15} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$( 1 - 2384 T^{2} + 3158579 T^{4} - 2740573584 T^{6} + 1706195561800 T^{8} - 766924852320144 T^{10} + 247351433577875699 T^{12} - 52244464645936445264 T^{14} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$1 + 3112 T^{2} + 6365538 T^{4} + 10682263816 T^{6} + 15305712405667 T^{8} + 18918572757319240 T^{10} + 20690280210889005732 T^{12} +$$$$20\!\cdots\!92$$$$T^{14} +$$$$18\!\cdots\!65$$$$T^{16} +$$$$14\!\cdots\!52$$$$T^{18} +$$$$10\!\cdots\!52$$$$T^{20} +$$$$66\!\cdots\!40$$$$T^{22} +$$$$38\!\cdots\!07$$$$T^{24} +$$$$18\!\cdots\!16$$$$T^{26} +$$$$79\!\cdots\!78$$$$T^{28} +$$$$27\!\cdots\!32$$$$T^{30} +$$$$62\!\cdots\!41$$$$T^{32}$$
$31$ $$( 1 - 62 T + 2136 T^{2} - 51368 T^{3} + 1877341 T^{4} + 2567854 T^{5} - 2516216556 T^{6} + 116136241856 T^{7} - 2932487147363 T^{8} + 111606928423616 T^{9} - 2323778830013676 T^{10} + 2278979877270574 T^{11} + 1601167313120524381 T^{12} - 42102665845629785768 T^{13} +$$$$16\!\cdots\!96$$$$T^{14} -$$$$46\!\cdots\!02$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 - 45 T + 645 T^{2} - 16850 T^{3} - 1356006 T^{4} + 101768095 T^{5} + 1589698690 T^{6} - 185593158390 T^{7} + 6348820327681 T^{8} - 254077033835910 T^{9} + 2979351286549090 T^{10} + 261109088935120855 T^{11} - 4762943214393599526 T^{12} - 81024646675240755650 T^{13} +$$$$42\!\cdots\!45$$$$T^{14} -$$$$40\!\cdots\!05$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$1 - 995 T^{2} + 1570671 T^{4} + 1044796720 T^{6} - 7793718391520 T^{8} + 15577888312588045 T^{10} + 8181601625193452394 T^{12} -$$$$48\!\cdots\!50$$$$T^{14} +$$$$11\!\cdots\!09$$$$T^{16} -$$$$13\!\cdots\!50$$$$T^{18} +$$$$65\!\cdots\!74$$$$T^{20} +$$$$35\!\cdots\!45$$$$T^{22} -$$$$49\!\cdots\!20$$$$T^{24} +$$$$18\!\cdots\!20$$$$T^{26} +$$$$79\!\cdots\!31$$$$T^{28} -$$$$14\!\cdots\!95$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$( 1 + 39 T + 5868 T^{2} + 143114 T^{3} + 14177853 T^{4} + 264617786 T^{5} + 20061524268 T^{6} + 246533158911 T^{7} + 11688200277601 T^{8} )^{4}$$
$47$ $$1 - 995 T^{2} + 2709666 T^{4} - 5345087660 T^{6} + 2560879190710 T^{8} + 1540723914202015 T^{10} + 68482758794882002179 T^{12} -$$$$35\!\cdots\!80$$$$T^{14} +$$$$20\!\cdots\!64$$$$T^{16} -$$$$17\!\cdots\!80$$$$T^{18} +$$$$16\!\cdots\!19$$$$T^{20} +$$$$17\!\cdots\!15$$$$T^{22} +$$$$14\!\cdots\!10$$$$T^{24} -$$$$14\!\cdots\!60$$$$T^{26} +$$$$36\!\cdots\!46$$$$T^{28} -$$$$65\!\cdots\!95$$$$T^{30} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 9862 T^{2} + 44250456 T^{4} + 83411867458 T^{6} - 136888230588719 T^{8} - 1037106678398694674 T^{10} -$$$$80\!\cdots\!96$$$$T^{12} +$$$$10\!\cdots\!24$$$$T^{14} +$$$$50\!\cdots\!97$$$$T^{16} +$$$$85\!\cdots\!44$$$$T^{18} -$$$$50\!\cdots\!56$$$$T^{20} -$$$$50\!\cdots\!34$$$$T^{22} -$$$$53\!\cdots\!99$$$$T^{24} +$$$$25\!\cdots\!58$$$$T^{26} +$$$$10\!\cdots\!36$$$$T^{28} +$$$$18\!\cdots\!82$$$$T^{30} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 - 1959 T^{2} - 22685748 T^{4} + 35157453984 T^{6} + 163809750679236 T^{8} + 484303216945908915 T^{10} -$$$$24\!\cdots\!77$$$$T^{12} -$$$$76\!\cdots\!00$$$$T^{14} +$$$$50\!\cdots\!92$$$$T^{16} -$$$$92\!\cdots\!00$$$$T^{18} -$$$$36\!\cdots\!17$$$$T^{20} +$$$$86\!\cdots\!15$$$$T^{22} +$$$$35\!\cdots\!76$$$$T^{24} +$$$$91\!\cdots\!84$$$$T^{26} -$$$$71\!\cdots\!28$$$$T^{28} -$$$$75\!\cdots\!39$$$$T^{30} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$( 1 + 63 T - 7114 T^{2} - 428758 T^{3} + 14002276 T^{4} + 1658235189 T^{5} + 137533071599 T^{6} - 2282091395694 T^{7} - 914529740532628 T^{8} - 8491662083377374 T^{9} + 1904261041601369759 T^{10} + 85432897715863589229 T^{11} +$$$$26\!\cdots\!56$$$$T^{12} -$$$$30\!\cdots\!58$$$$T^{13} -$$$$18\!\cdots\!94$$$$T^{14} +$$$$62\!\cdots\!83$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 - 92 T + 14029 T^{2} - 686396 T^{3} + 73672501 T^{4} - 3081231644 T^{5} + 282700076509 T^{6} - 8322171159548 T^{7} + 406067677556641 T^{8} )^{4}$$
$71$ $$1 + 18983 T^{2} + 212109414 T^{4} + 1794032145180 T^{6} + 12788451374810298 T^{8} + 81069844229226960885 T^{10} +$$$$47\!\cdots\!91$$$$T^{12} +$$$$25\!\cdots\!92$$$$T^{14} +$$$$13\!\cdots\!08$$$$T^{16} +$$$$64\!\cdots\!52$$$$T^{18} +$$$$30\!\cdots\!51$$$$T^{20} +$$$$13\!\cdots\!85$$$$T^{22} +$$$$53\!\cdots\!58$$$$T^{24} +$$$$19\!\cdots\!80$$$$T^{26} +$$$$57\!\cdots\!34$$$$T^{28} +$$$$12\!\cdots\!63$$$$T^{30} +$$$$17\!\cdots\!41$$$$T^{32}$$
$73$ $$( 1 - 12 T - 4158 T^{2} + 690642 T^{3} + 34555347 T^{4} + 1402256700 T^{5} - 68554055572 T^{6} + 4759965946776 T^{7} + 2851625758685025 T^{8} + 25365858530369304 T^{9} - 1946814591661048852 T^{10} +$$$$21\!\cdots\!00$$$$T^{11} +$$$$27\!\cdots\!07$$$$T^{12} +$$$$29\!\cdots\!58$$$$T^{13} -$$$$95\!\cdots\!18$$$$T^{14} -$$$$14\!\cdots\!08$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 + 157 T + 7773 T^{2} + 944146 T^{3} + 137198002 T^{4} + 10621769425 T^{5} + 1102248505962 T^{6} + 91369308370742 T^{7} + 5518958597130505 T^{8} + 570235853541800822 T^{9} + 42932668589348882922 T^{10} +$$$$25\!\cdots\!25$$$$T^{11} +$$$$20\!\cdots\!22$$$$T^{12} +$$$$89\!\cdots\!46$$$$T^{13} +$$$$45\!\cdots\!93$$$$T^{14} +$$$$57\!\cdots\!17$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$1 + 28010 T^{2} + 425035818 T^{4} + 4516595468385 T^{6} + 39017282313118833 T^{8} +$$$$29\!\cdots\!60$$$$T^{10} +$$$$20\!\cdots\!81$$$$T^{12} +$$$$13\!\cdots\!25$$$$T^{14} +$$$$89\!\cdots\!30$$$$T^{16} +$$$$63\!\cdots\!25$$$$T^{18} +$$$$46\!\cdots\!21$$$$T^{20} +$$$$31\!\cdots\!60$$$$T^{22} +$$$$19\!\cdots\!73$$$$T^{24} +$$$$10\!\cdots\!85$$$$T^{26} +$$$$48\!\cdots\!78$$$$T^{28} +$$$$15\!\cdots\!10$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$( 1 - 54839 T^{2} + 1370536886 T^{4} - 20394246717258 T^{6} + 197742529962478711 T^{8} -$$$$12\!\cdots\!78$$$$T^{10} +$$$$53\!\cdots\!66$$$$T^{12} -$$$$13\!\cdots\!19$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 36 T - 15693 T^{2} + 1693818 T^{3} + 119942457 T^{4} - 7626572010 T^{5} - 578486645527 T^{6} + 23274720379476 T^{7} + 13991351802208980 T^{8} + 218991844050489684 T^{9} - 51213006796607176087 T^{10} -$$$$63\!\cdots\!90$$$$T^{11} +$$$$94\!\cdots\!77$$$$T^{12} +$$$$12\!\cdots\!82$$$$T^{13} -$$$$10\!\cdots\!13$$$$T^{14} -$$$$23\!\cdots\!84$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$