Properties

 Label 300.5.c.c Level $300$ Weight $5$ Character orbit 300.c Analytic conductor $31.011$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 300.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$31.0109889252$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{15} + 20 x^{14} - 108 x^{13} + 492 x^{12} - 2160 x^{11} + 7360 x^{10} - 39552 x^{9} + 234752 x^{8} - 632832 x^{7} + 1884160 x^{6} - 8847360 x^{5} + 32243712 x^{4} - 113246208 x^{3} + 335544320 x^{2} - 1610612736 x + 4294967296$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{32}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{5} ) q^{4} + ( 1 + \beta_{3} ) q^{6} + ( -1 + \beta_{5} - \beta_{13} ) q^{7} + ( -12 + \beta_{2} - \beta_{8} ) q^{8} -27 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{5} ) q^{4} + ( 1 + \beta_{3} ) q^{6} + ( -1 + \beta_{5} - \beta_{13} ) q^{7} + ( -12 + \beta_{2} - \beta_{8} ) q^{8} -27 q^{9} + ( -1 + 3 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{11} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{14} - \beta_{15} ) q^{12} + ( 11 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{12} + \beta_{14} ) q^{13} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{14} + ( -22 + 5 \beta_{1} - 11 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{16} + ( 10 - \beta_{1} - 16 \beta_{2} + 3 \beta_{3} - 9 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{17} -27 \beta_{2} q^{18} + ( -5 + 15 \beta_{1} + 23 \beta_{2} - 2 \beta_{4} - 7 \beta_{5} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 4 \beta_{13} - 4 \beta_{15} ) q^{19} + ( -8 + \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{14} + 2 \beta_{15} ) q^{21} + ( -44 + 19 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{22} + ( 11 + 18 \beta_{1} - 26 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} + 5 \beta_{7} + \beta_{8} - \beta_{9} - 4 \beta_{10} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - 5 \beta_{14} - 3 \beta_{15} ) q^{23} + ( 6 + 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{9} - 3 \beta_{10} + \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{24} + ( 33 - 45 \beta_{1} + 15 \beta_{2} + 2 \beta_{3} - \beta_{5} + 4 \beta_{7} - 5 \beta_{8} + 5 \beta_{10} + 5 \beta_{13} + 2 \beta_{14} + 6 \beta_{15} ) q^{26} + 27 \beta_{1} q^{27} + ( 218 + 31 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 3 \beta_{12} + 5 \beta_{13} - 3 \beta_{14} - 3 \beta_{15} ) q^{28} + ( 137 + 4 \beta_{1} - 66 \beta_{2} + 7 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} ) q^{29} + ( -22 - 34 \beta_{1} + 57 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + \beta_{7} - 7 \beta_{8} - \beta_{9} + 7 \beta_{10} - \beta_{11} + \beta_{12} + 12 \beta_{13} - \beta_{14} - \beta_{15} ) q^{31} + ( 134 - 28 \beta_{1} - 29 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{10} - 2 \beta_{11} + \beta_{12} - 3 \beta_{13} - 10 \beta_{14} - 2 \beta_{15} ) q^{32} + ( -13 + \beta_{1} + 35 \beta_{2} + 7 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 4 \beta_{12} + 3 \beta_{13} + 5 \beta_{14} + 2 \beta_{15} ) q^{33} + ( -189 + 13 \beta_{1} + 9 \beta_{2} + \beta_{3} + \beta_{4} + 11 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - 7 \beta_{12} + 8 \beta_{13} - 2 \beta_{14} - 4 \beta_{15} ) q^{34} + ( -27 + 27 \beta_{5} ) q^{36} + ( -69 + 16 \beta_{1} - 65 \beta_{2} + 10 \beta_{3} - 6 \beta_{4} + \beta_{5} + 6 \beta_{6} - 11 \beta_{7} - \beta_{8} - 3 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} - 7 \beta_{13} - 5 \beta_{14} - 9 \beta_{15} ) q^{37} + ( -427 - 5 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} - \beta_{4} - 28 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} - \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} - 11 \beta_{13} + 3 \beta_{14} - 7 \beta_{15} ) q^{38} + ( -21 - 22 \beta_{1} + 65 \beta_{2} + \beta_{3} - 7 \beta_{4} - \beta_{5} + 6 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + 10 \beta_{12} + 6 \beta_{13} - 6 \beta_{14} ) q^{39} + ( 80 + 5 \beta_{1} - 78 \beta_{2} + \beta_{3} - 2 \beta_{4} + 27 \beta_{5} - 10 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} + 8 \beta_{9} - 5 \beta_{11} + 3 \beta_{12} + 10 \beta_{14} + 5 \beta_{15} ) q^{41} + ( 2 + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 9 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} + 8 \beta_{9} + 9 \beta_{11} - 5 \beta_{12} - 9 \beta_{13} + 6 \beta_{14} ) q^{42} + ( 24 + 7 \beta_{1} - 92 \beta_{2} - 22 \beta_{3} - 18 \beta_{4} - 6 \beta_{5} - \beta_{7} - 2 \beta_{8} + 6 \beta_{9} + \beta_{10} + 2 \beta_{11} - 17 \beta_{12} - 27 \beta_{13} + \beta_{14} - 23 \beta_{15} ) q^{43} + ( 520 - 38 \beta_{1} - 41 \beta_{2} - 22 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 16 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + 5 \beta_{10} + 4 \beta_{11} - 17 \beta_{12} + 5 \beta_{13} + 12 \beta_{14} + 4 \beta_{15} ) q^{44} + ( 352 + 29 \beta_{1} + 12 \beta_{2} - 33 \beta_{3} + 5 \beta_{4} + 30 \beta_{5} + 15 \beta_{6} - 15 \beta_{7} + 3 \beta_{8} + 11 \beta_{9} - 4 \beta_{10} - 12 \beta_{11} - 24 \beta_{12} - 7 \beta_{13} + 23 \beta_{14} - 7 \beta_{15} ) q^{46} + ( -38 - 28 \beta_{1} + 162 \beta_{2} - 54 \beta_{3} - 14 \beta_{4} - 14 \beta_{5} + 8 \beta_{7} - 12 \beta_{9} - 14 \beta_{10} + 6 \beta_{11} + 32 \beta_{12} + 4 \beta_{13} - 8 \beta_{14} + 12 \beta_{15} ) q^{47} + ( 122 + 43 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} + 7 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} + 6 \beta_{8} + 7 \beta_{9} - 6 \beta_{10} - 3 \beta_{11} - 10 \beta_{12} - 24 \beta_{13} + 12 \beta_{14} - 12 \beta_{15} ) q^{48} + ( -585 + 10 \beta_{1} - 217 \beta_{2} - 36 \beta_{3} + 20 \beta_{4} - 25 \beta_{5} + 4 \beta_{6} - 15 \beta_{7} - 27 \beta_{8} + 11 \beta_{9} - \beta_{10} - 13 \beta_{11} - 3 \beta_{12} - \beta_{13} + 17 \beta_{14} + 7 \beta_{15} ) q^{49} + ( -6 - 2 \beta_{1} - 22 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} - 3 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 9 \beta_{10} - 11 \beta_{12} - 9 \beta_{13} + 3 \beta_{14} - 9 \beta_{15} ) q^{51} + ( 179 - 30 \beta_{1} + 46 \beta_{2} + 60 \beta_{3} + 26 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 12 \beta_{7} + 20 \beta_{9} - 4 \beta_{10} + 16 \beta_{11} - 12 \beta_{12} + 2 \beta_{14} - 6 \beta_{15} ) q^{52} + ( -63 - 2 \beta_{1} + 98 \beta_{2} + 30 \beta_{3} + 23 \beta_{4} - 24 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} - 12 \beta_{8} - 16 \beta_{9} + 16 \beta_{10} - 14 \beta_{11} - 19 \beta_{12} + 16 \beta_{13} + 15 \beta_{14} + 12 \beta_{15} ) q^{53} + ( -27 - 27 \beta_{3} ) q^{54} + ( -714 - 102 \beta_{1} + 240 \beta_{2} - 22 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 12 \beta_{7} - 2 \beta_{8} + 14 \beta_{9} + \beta_{10} - 8 \beta_{11} - 11 \beta_{12} + 37 \beta_{13} + 14 \beta_{14} - 10 \beta_{15} ) q^{56} + ( 356 + 11 \beta_{1} - 13 \beta_{2} + 21 \beta_{3} - 10 \beta_{4} + 20 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} + 21 \beta_{8} - 7 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} + 10 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 6 \beta_{15} ) q^{57} + ( -1087 - 111 \beta_{1} + 165 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 53 \beta_{5} - \beta_{6} + 26 \beta_{7} + 4 \beta_{8} + 7 \beta_{10} + 7 \beta_{11} + 13 \beta_{12} + 40 \beta_{13} + 8 \beta_{14} + 10 \beta_{15} ) q^{58} + ( -98 + 26 \beta_{1} + 154 \beta_{2} + 64 \beta_{3} - 4 \beta_{4} + 54 \beta_{5} + 14 \beta_{7} - 26 \beta_{8} + 14 \beta_{9} + 14 \beta_{10} - 2 \beta_{11} + 6 \beta_{12} + 14 \beta_{13} - 14 \beta_{14} - 6 \beta_{15} ) q^{59} + ( -218 + 36 \beta_{1} - 92 \beta_{2} + 56 \beta_{3} + 32 \beta_{4} - 26 \beta_{5} + \beta_{6} + 29 \beta_{7} - 5 \beta_{8} - 31 \beta_{9} + 21 \beta_{10} - 13 \beta_{11} - 14 \beta_{12} + 21 \beta_{13} + 58 \beta_{14} + 31 \beta_{15} ) q^{61} + ( -770 + 50 \beta_{1} - 16 \beta_{2} + 32 \beta_{3} - 17 \beta_{4} - 23 \beta_{5} - 7 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} + 37 \beta_{9} - 21 \beta_{10} + 12 \beta_{11} + 18 \beta_{12} + 6 \beta_{13} - 29 \beta_{14} + 5 \beta_{15} ) q^{62} + ( 27 - 27 \beta_{5} + 27 \beta_{13} ) q^{63} + ( 612 - 166 \beta_{1} + 120 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} + 10 \beta_{7} + 14 \beta_{8} - 46 \beta_{9} + 10 \beta_{10} - 34 \beta_{11} - 22 \beta_{12} + 18 \beta_{13} + 16 \beta_{14} + 8 \beta_{15} ) q^{64} + ( 545 + 27 \beta_{1} - 23 \beta_{2} + 21 \beta_{3} + 9 \beta_{4} - 43 \beta_{5} - 3 \beta_{6} + 24 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} + 15 \beta_{10} + 3 \beta_{11} + \beta_{12} + 42 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{66} + ( 99 + 191 \beta_{1} - 143 \beta_{2} - 116 \beta_{3} + 22 \beta_{4} - 75 \beta_{5} - 38 \beta_{7} + 41 \beta_{8} - 29 \beta_{9} - 3 \beta_{11} - 22 \beta_{12} - 32 \beta_{13} + 38 \beta_{14} + 10 \beta_{15} ) q^{67} + ( 1722 + 24 \beta_{1} - 231 \beta_{2} - 22 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} - 21 \beta_{8} + 6 \beta_{9} + 17 \beta_{10} - 28 \beta_{11} - 5 \beta_{12} + 41 \beta_{13} + 2 \beta_{14} + 10 \beta_{15} ) q^{68} + ( 590 - 22 \beta_{1} + 23 \beta_{2} - \beta_{4} + 35 \beta_{5} - 18 \beta_{6} + 27 \beta_{7} + 27 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} + 10 \beta_{12} + 9 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} ) q^{69} + ( -1 - 42 \beta_{1} + 120 \beta_{2} - 141 \beta_{3} - 83 \beta_{4} - 63 \beta_{5} + 17 \beta_{7} - 9 \beta_{8} - 3 \beta_{9} - 8 \beta_{10} - 3 \beta_{12} - 8 \beta_{13} - 17 \beta_{14} - 63 \beta_{15} ) q^{71} + ( 324 - 27 \beta_{2} + 27 \beta_{8} ) q^{72} + ( 665 + 14 \beta_{1} - 101 \beta_{2} - 40 \beta_{3} - 52 \beta_{4} + 107 \beta_{5} + 34 \beta_{6} - 33 \beta_{7} + 3 \beta_{8} + 37 \beta_{9} - 23 \beta_{10} + 13 \beta_{11} + 25 \beta_{12} - 23 \beta_{13} + 33 \beta_{14} + 17 \beta_{15} ) q^{73} + ( -1096 + 316 \beta_{1} - 84 \beta_{2} - 38 \beta_{3} - 98 \beta_{4} + 84 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 18 \beta_{8} - 16 \beta_{9} + 8 \beta_{10} - 10 \beta_{11} + 22 \beta_{12} - 22 \beta_{13} - 36 \beta_{14} - 8 \beta_{15} ) q^{74} + ( -1700 - 245 \beta_{1} - 348 \beta_{2} + 46 \beta_{3} + 33 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} + 12 \beta_{7} - 27 \beta_{8} + 34 \beta_{9} + 19 \beta_{10} - 12 \beta_{11} - 55 \beta_{12} - 37 \beta_{13} + 33 \beta_{14} - 7 \beta_{15} ) q^{76} + ( 95 + 34 \beta_{1} - 108 \beta_{2} + 146 \beta_{3} - 97 \beta_{4} - 26 \beta_{5} - 24 \beta_{6} - 10 \beta_{7} + 4 \beta_{8} - 36 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} + 43 \beta_{12} - 8 \beta_{13} - 71 \beta_{14} - 24 \beta_{15} ) q^{77} + ( -1044 - 19 \beta_{1} + 20 \beta_{2} + 15 \beta_{3} + 4 \beta_{4} - 51 \beta_{5} - 36 \beta_{7} - 9 \beta_{8} + 24 \beta_{9} - 3 \beta_{10} - 18 \beta_{12} - 3 \beta_{13} + 22 \beta_{14} - 14 \beta_{15} ) q^{78} + ( 38 - 158 \beta_{1} - 36 \beta_{2} + 140 \beta_{3} + 76 \beta_{4} - 74 \beta_{5} - 46 \beta_{7} + 64 \beta_{8} + 16 \beta_{9} + 14 \beta_{10} - 32 \beta_{11} - 62 \beta_{12} - 24 \beta_{13} + 46 \beta_{14} - 18 \beta_{15} ) q^{79} + 729 q^{81} + ( -1365 + 65 \beta_{1} + 103 \beta_{2} + 45 \beta_{3} + 117 \beta_{4} + 35 \beta_{5} - 15 \beta_{6} - 40 \beta_{7} + 40 \beta_{8} + 10 \beta_{9} + 5 \beta_{10} - 25 \beta_{11} - 15 \beta_{12} - 20 \beta_{13} + 2 \beta_{14} - 8 \beta_{15} ) q^{82} + ( -68 - 334 \beta_{1} + 6 \beta_{2} - 98 \beta_{3} + 6 \beta_{4} + 164 \beta_{5} + 6 \beta_{7} - 32 \beta_{8} - 52 \beta_{9} + 60 \beta_{10} - 34 \beta_{11} - 10 \beta_{12} + 84 \beta_{13} - 6 \beta_{14} + 30 \beta_{15} ) q^{83} + ( 893 - 222 \beta_{1} + 19 \beta_{2} + 14 \beta_{3} - 28 \beta_{4} + 39 \beta_{5} - 12 \beta_{6} - 9 \beta_{8} + 18 \beta_{9} + 9 \beta_{10} + 36 \beta_{11} + 27 \beta_{12} - 27 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{84} + ( 1409 + 393 \beta_{1} - 30 \beta_{2} + 54 \beta_{3} + 7 \beta_{4} + 134 \beta_{5} - 35 \beta_{6} + 25 \beta_{7} + \beta_{8} + 71 \beta_{9} - 20 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} - 101 \beta_{13} + 3 \beta_{14} - 47 \beta_{15} ) q^{86} + ( -60 - 160 \beta_{1} + 161 \beta_{2} - 65 \beta_{3} - 13 \beta_{4} - 10 \beta_{5} - 36 \beta_{8} - 4 \beta_{9} + 9 \beta_{10} + 27 \beta_{11} + 28 \beta_{12} + 9 \beta_{13} + 18 \beta_{15} ) q^{87} + ( 3012 - 52 \beta_{1} + 508 \beta_{2} + 64 \beta_{3} + 4 \beta_{4} + 124 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} - 40 \beta_{8} + 64 \beta_{9} + 18 \beta_{10} + 48 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} + 20 \beta_{14} + 28 \beta_{15} ) q^{88} + ( -144 - 114 \beta_{1} + 346 \beta_{2} - 150 \beta_{3} - 10 \beta_{4} - 52 \beta_{5} + 20 \beta_{6} + 24 \beta_{7} + 6 \beta_{8} + 46 \beta_{9} + 14 \beta_{10} - 4 \beta_{11} - 10 \beta_{12} + 14 \beta_{13} - 4 \beta_{14} + 44 \beta_{15} ) q^{89} + ( 12 + 237 \beta_{1} + 70 \beta_{2} - 250 \beta_{3} - 90 \beta_{4} + 42 \beta_{5} + 33 \beta_{7} + 36 \beta_{8} - 64 \beta_{9} - 13 \beta_{10} - 56 \beta_{11} + \beta_{12} + 21 \beta_{13} - 33 \beta_{14} - 49 \beta_{15} ) q^{91} + ( 1308 - 406 \beta_{1} + 379 \beta_{2} - 6 \beta_{3} + 52 \beta_{4} + 32 \beta_{5} + 56 \beta_{6} + 4 \beta_{7} + 7 \beta_{8} + 54 \beta_{9} + 9 \beta_{10} - 4 \beta_{11} + 3 \beta_{12} - 63 \beta_{13} + 4 \beta_{14} + 44 \beta_{15} ) q^{92} + ( -791 + 38 \beta_{1} + 190 \beta_{2} + 38 \beta_{3} + 3 \beta_{4} + 158 \beta_{5} - 15 \beta_{6} - 39 \beta_{7} - 15 \beta_{8} + 11 \beta_{9} - 21 \beta_{10} + 3 \beta_{11} - 17 \beta_{12} - 21 \beta_{13} + 25 \beta_{14} - 23 \beta_{15} ) q^{93} + ( -2748 + 702 \beta_{1} + 8 \beta_{2} + 22 \beta_{3} + 84 \beta_{4} - 214 \beta_{5} + 20 \beta_{6} - 60 \beta_{7} - 6 \beta_{8} - 44 \beta_{9} + 34 \beta_{10} - 16 \beta_{11} - 52 \beta_{12} + 30 \beta_{13} + 128 \beta_{14} + 48 \beta_{15} ) q^{94} + ( -774 - 136 \beta_{1} + 141 \beta_{2} - 6 \beta_{3} - 18 \beta_{5} + 42 \beta_{7} + 3 \beta_{8} - 12 \beta_{9} + 27 \beta_{10} - 18 \beta_{11} + 15 \beta_{12} - 45 \beta_{13} - 6 \beta_{14} + 18 \beta_{15} ) q^{96} + ( 192 + 76 \beta_{1} + 597 \beta_{2} + 266 \beta_{3} - 80 \beta_{4} - 69 \beta_{5} - 4 \beta_{6} - 19 \beta_{7} - 53 \beta_{8} - 95 \beta_{9} + 5 \beta_{10} - 29 \beta_{11} - 35 \beta_{12} + 5 \beta_{13} + 3 \beta_{14} + 11 \beta_{15} ) q^{97} + ( -3658 - 582 \beta_{1} - 456 \beta_{2} + 58 \beta_{3} + 122 \beta_{4} + 214 \beta_{5} - 22 \beta_{6} - 24 \beta_{7} - 76 \beta_{8} + 4 \beta_{9} + 78 \beta_{10} - 50 \beta_{11} - 66 \beta_{12} - 12 \beta_{13} + 20 \beta_{14} + 80 \beta_{15} ) q^{98} + ( 27 - 81 \beta_{2} + 54 \beta_{3} - 27 \beta_{5} - 27 \beta_{8} + 27 \beta_{9} + 27 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 6q^{2} + 8q^{4} + 18q^{6} - 180q^{8} - 432q^{9} + O(q^{10})$$ $$16q + 6q^{2} + 8q^{4} + 18q^{6} - 180q^{8} - 432q^{9} + 176q^{13} + 78q^{14} - 376q^{16} - 162q^{18} - 144q^{21} - 788q^{22} + 108q^{24} + 678q^{26} + 3368q^{28} + 1728q^{29} + 2016q^{32} - 2932q^{34} - 216q^{36} - 1568q^{37} - 6990q^{38} + 1248q^{41} + 162q^{42} + 8088q^{44} + 5956q^{46} + 2088q^{48} - 10720q^{49} + 3128q^{52} - 288q^{53} - 486q^{54} - 10236q^{56} + 5616q^{57} - 16164q^{58} - 3760q^{61} - 12714q^{62} + 10544q^{64} + 8100q^{66} + 26136q^{68} + 9792q^{69} + 4860q^{72} + 11040q^{73} - 17004q^{74} - 28344q^{76} + 768q^{77} - 16830q^{78} + 11664q^{81} - 21280q^{82} + 15120q^{84} + 24414q^{86} + 52840q^{88} - 768q^{89} + 23736q^{92} - 9936q^{93} - 45156q^{94} - 11088q^{96} + 7248q^{97} - 58140q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{15} + 20 x^{14} - 108 x^{13} + 492 x^{12} - 2160 x^{11} + 7360 x^{10} - 39552 x^{9} + 234752 x^{8} - 632832 x^{7} + 1884160 x^{6} - 8847360 x^{5} + 32243712 x^{4} - 113246208 x^{3} + 335544320 x^{2} - 1610612736 x + 4294967296$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$22761 \nu^{15} - 1177062 \nu^{14} + 419412 \nu^{13} + 17236980 \nu^{12} + 30915852 \nu^{11} - 386713392 \nu^{10} + 107470272 \nu^{9} + 1871005056 \nu^{8} + 10408187136 \nu^{7} - 46803376128 \nu^{6} - 247514284032 \nu^{5} + 579352461312 \nu^{4} + 5466618593280 \nu^{3} - 8277215674368 \nu^{2} - 49525033009152 \nu + 91883040669696$$$$)/ 21382226247680$$ $$\beta_{2}$$ $$=$$ $$($$$$-24219 \nu^{15} + 160488 \nu^{14} - 1269088 \nu^{13} + 2895260 \nu^{12} - 424428 \nu^{11} + 72923608 \nu^{10} - 436060768 \nu^{9} + 1029556736 \nu^{8} - 4438121984 \nu^{7} + 22265349632 \nu^{6} - 76834721792 \nu^{5} + 49264689152 \nu^{4} - 394675486720 \nu^{3} + 6387122307072 \nu^{2} - 13644691668992 \nu + 5990741180416$$$$)/ 5345556561920$$ $$\beta_{3}$$ $$=$$ $$($$$$-93651 \nu^{15} + 470862 \nu^{14} + 2835228 \nu^{13} + 8436660 \nu^{12} - 115024212 \nu^{11} + 78622752 \nu^{10} + 857582208 \nu^{9} + 3274203264 \nu^{8} - 29468779776 \nu^{7} + 17632601088 \nu^{6} + 10760036352 \nu^{5} + 1818621247488 \nu^{4} - 5337065717760 \nu^{3} - 11260853747712 \nu^{2} + 1684801585152 \nu + 338244512251904$$$$)/ 10691113123840$$ $$\beta_{4}$$ $$=$$ $$($$$$-271111 \nu^{15} + 7879162 \nu^{14} - 68271852 \nu^{13} + 203941940 \nu^{12} - 13607732 \nu^{11} + 1474709712 \nu^{10} - 16511118912 \nu^{9} + 62294400384 \nu^{8} - 151751375616 \nu^{7} + 1016993835008 \nu^{6} - 4022721527808 \nu^{5} + 2358687367168 \nu^{4} - 6162194104320 \nu^{3} + 212936266088448 \nu^{2} - 1126410840178688 \nu + 1490119844429824$$$$)/ 21382226247680$$ $$\beta_{5}$$ $$=$$ $$($$$$-388661 \nu^{15} + 2607502 \nu^{14} - 10397572 \nu^{13} + 97707420 \nu^{12} - 238874012 \nu^{11} + 239626992 \nu^{10} - 4774779072 \nu^{9} + 33900036224 \nu^{8} - 106722345216 \nu^{7} + 230810195968 \nu^{6} - 1350750158848 \nu^{5} + 5954721742848 \nu^{4} - 4409030082560 \nu^{3} + 28179649527808 \nu^{2} - 394858791436288 \nu + 1092980324696064$$$$)/ 21382226247680$$ $$\beta_{6}$$ $$=$$ $$($$$$126344 \nu^{15} + 1615977 \nu^{14} - 5445222 \nu^{13} + 15810740 \nu^{12} - 30606572 \nu^{11} - 119832628 \nu^{10} - 2139636272 \nu^{9} + 2387578304 \nu^{8} - 15753722496 \nu^{7} + 115544835328 \nu^{6} - 190629173248 \nu^{5} + 808000012288 \nu^{4} + 2233226035200 \nu^{3} + 10601398796288 \nu^{2} - 135566330953728 \nu + 19241185050624$$$$)/ 2672778280960$$ $$\beta_{7}$$ $$=$$ $$($$$$1099561 \nu^{15} + 8087058 \nu^{14} - 105916668 \nu^{13} + 174958420 \nu^{12} - 574807188 \nu^{11} + 2287688048 \nu^{10} - 13309116608 \nu^{9} + 108606838656 \nu^{8} - 246748559104 \nu^{7} + 1799585513472 \nu^{6} - 6317352845312 \nu^{5} + 6133141143552 \nu^{4} - 36449035223040 \nu^{3} + 88571832696832 \nu^{2} - 1409648229875712 \nu + 4944227033153536$$$$)/ 21382226247680$$ $$\beta_{8}$$ $$=$$ $$($$$$-342839 \nu^{15} + 160488 \nu^{14} - 1269088 \nu^{13} + 29659340 \nu^{12} - 52678108 \nu^{11} + 371151928 \nu^{10} - 1159965408 \nu^{9} + 10572862976 \nu^{8} - 41143145984 \nu^{7} - 23248880128 \nu^{6} - 664115105792 \nu^{5} + 2492351086592 \nu^{4} - 3359788892160 \nu^{3} + 25931813486592 \nu^{2} - 68436646428672 \nu + 390870813638656$$$$)/ 5345556561920$$ $$\beta_{9}$$ $$=$$ $$($$$$934251 \nu^{15} - 4975102 \nu^{14} + 18380292 \nu^{13} - 114934100 \nu^{12} + 362319092 \nu^{11} - 969300512 \nu^{10} + 5604213632 \nu^{9} - 36862565504 \nu^{8} + 132148219136 \nu^{7} - 355931440128 \nu^{6} + 1356358770688 \nu^{5} - 5863435010048 \nu^{4} + 14951217889280 \nu^{3} - 61237648949248 \nu^{2} + 237241041420288 \nu - 908044401639424$$$$)/ 10691113123840$$ $$\beta_{10}$$ $$=$$ $$($$$$-60250 \nu^{15} - 12401 \nu^{14} + 2285878 \nu^{13} - 2891052 \nu^{12} + 14443588 \nu^{11} - 22257372 \nu^{10} + 76150128 \nu^{9} - 2364632000 \nu^{8} + 4360913536 \nu^{7} - 36504254720 \nu^{6} + 150017800192 \nu^{5} - 16336666624 \nu^{4} + 1369636012032 \nu^{3} - 794802520064 \nu^{2} + 25242302939136 \nu - 158365611196416$$$$)/ 534555656192$$ $$\beta_{11}$$ $$=$$ $$($$$$-2504571 \nu^{15} + 23111562 \nu^{14} - 58647692 \nu^{13} + 68416260 \nu^{12} - 201651012 \nu^{11} + 4692827312 \nu^{10} - 24665202112 \nu^{9} + 56439634304 \nu^{8} - 304698391296 \nu^{7} + 1531978579968 \nu^{6} - 1619036438528 \nu^{5} + 1126111838208 \nu^{4} - 27695381217280 \nu^{3} + 430468736483328 \nu^{2} - 649518139834368 \nu - 1221605736841216$$$$)/ 21382226247680$$ $$\beta_{12}$$ $$=$$ $$($$$$-1641477 \nu^{15} + 2165514 \nu^{14} + 18313556 \nu^{13} + 72315660 \nu^{12} - 585809964 \nu^{11} + 378949824 \nu^{10} + 381751296 \nu^{9} + 24826958208 \nu^{8} - 110640826112 \nu^{7} - 203862047744 \nu^{6} + 335682248704 \nu^{5} + 9096506638336 \nu^{4} - 13326666956800 \nu^{3} - 45213799153664 \nu^{2} - 114095571337216 \nu + 435642022494208$$$$)/ 10691113123840$$ $$\beta_{13}$$ $$=$$ $$($$$$-2417571 \nu^{15} + 8485722 \nu^{14} - 14524012 \nu^{13} + 88834660 \nu^{12} - 425868452 \nu^{11} + 2776946352 \nu^{10} - 8291721152 \nu^{9} + 47132237184 \nu^{8} - 256282923776 \nu^{7} + 344508489728 \nu^{6} - 1541216534528 \nu^{5} + 6197389754368 \nu^{4} - 35056715038720 \nu^{3} + 141682391646208 \nu^{2} - 245437550297088 \nu + 868181065334784$$$$)/ 10691113123840$$ $$\beta_{14}$$ $$=$$ $$($$$$-5049151 \nu^{15} + 27147522 \nu^{14} - 33507612 \nu^{13} + 176169780 \nu^{12} - 2204571252 \nu^{11} + 6372800752 \nu^{10} - 20793500352 \nu^{9} + 161846930304 \nu^{8} - 687398633216 \nu^{7} + 1630195089408 \nu^{6} - 2617104498688 \nu^{5} + 29018916323328 \nu^{4} - 129584467804160 \nu^{3} + 166544128606208 \nu^{2} - 1198421000060928 \nu + 3637698787016704$$$$)/ 21382226247680$$ $$\beta_{15}$$ $$=$$ $$($$$$3451111 \nu^{15} - 17471882 \nu^{14} + 38280652 \nu^{13} - 124703220 \nu^{12} + 1026964852 \nu^{11} - 4440964752 \nu^{10} + 12858348352 \nu^{9} - 88665743744 \nu^{8} + 430771520256 \nu^{7} - 919234787328 \nu^{6} + 1545770057728 \nu^{5} - 12429149667328 \nu^{4} + 70140487270400 \nu^{3} - 169647252963328 \nu^{2} + 615710875189248 \nu - 2016819065913344$$$$)/ 10691113123840$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{12} - 2 \beta_{9} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1} + 17$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} - 3 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 8 \beta_{2} + 5 \beta_{1} - 6$$$$)/24$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{14} + 5 \beta_{12} + 4 \beta_{11} + 2 \beta_{9} + 4 \beta_{7} + 7 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 5 \beta_{1} + 83$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-15 \beta_{15} + 9 \beta_{14} - 33 \beta_{13} - 21 \beta_{12} - 3 \beta_{11} + 15 \beta_{10} - 9 \beta_{9} + 33 \beta_{8} + 3 \beta_{7} - 39 \beta_{5} + 38 \beta_{3} + 27 \beta_{2} + 108 \beta_{1} - 127$$$$)/12$$ $$\nu^{5}$$ $$=$$ $$($$$$-120 \beta_{15} - 48 \beta_{14} - 156 \beta_{13} - 3 \beta_{12} - 12 \beta_{10} + 42 \beta_{9} - 36 \beta_{8} - 72 \beta_{7} - 48 \beta_{6} + 303 \beta_{5} - 6 \beta_{4} + 18 \beta_{3} + 891 \beta_{2} - 65 \beta_{1} + 1599$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$27 \beta_{15} + 23 \beta_{14} + 13 \beta_{13} - 18 \beta_{12} + 19 \beta_{11} + 29 \beta_{10} + \beta_{9} - 77 \beta_{8} + 29 \beta_{7} - 32 \beta_{6} - 30 \beta_{5} + 12 \beta_{4} + 234 \beta_{3} + 300 \beta_{2} + 373 \beta_{1} + 1130$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$464 \beta_{15} + 620 \beta_{14} - 312 \beta_{13} + 47 \beta_{12} + 36 \beta_{11} + 264 \beta_{10} - 686 \beta_{9} + 504 \beta_{8} + 132 \beta_{7} - 192 \beta_{6} - 2219 \beta_{5} + 906 \beta_{4} - 850 \beta_{3} - 5887 \beta_{2} - 1075 \beta_{1} + 49881$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-421 \beta_{15} - 49 \beta_{14} - 879 \beta_{13} - 423 \beta_{12} + 627 \beta_{11} - 303 \beta_{10} - 171 \beta_{9} + 111 \beta_{8} - 291 \beta_{7} - 912 \beta_{6} + 5763 \beta_{5} - 952 \beta_{4} - 4338 \beta_{3} - 8255 \beta_{2} - 2512 \beta_{1} - 56145$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$-316 \beta_{15} - 892 \beta_{14} - 536 \beta_{13} + 771 \beta_{12} + 1476 \beta_{11} - 8 \beta_{10} + 694 \beta_{9} + 984 \beta_{8} + 660 \beta_{7} - 688 \beta_{6} - 2943 \beta_{5} + 1458 \beta_{4} + 6394 \beta_{3} - 17427 \beta_{2} + 3401 \beta_{1} - 123007$$ $$\nu^{10}$$ $$=$$ $$($$$$7686 \beta_{15} - 8058 \beta_{14} - 1614 \beta_{13} + 17820 \beta_{12} - 1266 \beta_{11} + 17394 \beta_{10} + 8730 \beta_{9} + 3534 \beta_{8} + 24978 \beta_{7} - 12960 \beta_{6} + 35844 \beta_{5} + 10272 \beta_{4} + 8532 \beta_{3} + 61224 \beta_{2} - 258134 \beta_{1} + 329820$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-38256 \beta_{15} + 52728 \beta_{14} - 8976 \beta_{13} - 200042 \beta_{12} - 71064 \beta_{11} + 107952 \beta_{10} - 10900 \beta_{9} - 15216 \beta_{8} + 7944 \beta_{7} - 53952 \beta_{6} + 143474 \beta_{5} + 93452 \beta_{4} + 169412 \beta_{3} - 653590 \beta_{2} + 313850 \beta_{1} - 469766$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$38652 \beta_{15} - 13220 \beta_{14} + 23684 \beta_{13} - 2732 \beta_{12} - 60020 \beta_{11} + 39620 \beta_{10} - 92860 \beta_{9} - 107652 \beta_{8} - 31500 \beta_{7} - 55808 \beta_{6} + 141596 \beta_{5} + 178656 \beta_{4} + 241160 \beta_{3} + 11284 \beta_{2} - 710544 \beta_{1} - 570116$$ $$\nu^{13}$$ $$=$$ $$($$$$2160128 \beta_{15} + 946400 \beta_{14} + 1860144 \beta_{13} + 131652 \beta_{12} + 391584 \beta_{11} + 298608 \beta_{10} + 513960 \beta_{9} - 1381680 \beta_{8} + 159936 \beta_{7} - 654528 \beta_{6} + 3040044 \beta_{5} + 1101896 \beta_{4} + 2202408 \beta_{3} - 6884132 \beta_{2} - 6503092 \beta_{1} + 47110956$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$3947688 \beta_{15} + 2188104 \beta_{14} + 4768152 \beta_{13} - 3173808 \beta_{12} + 462696 \beta_{11} + 2830104 \beta_{10} - 5646408 \beta_{9} + 4096104 \beta_{8} + 2794008 \beta_{7} + 2102016 \beta_{6} - 7696080 \beta_{5} + 4454112 \beta_{4} - 6575504 \beta_{3} - 134740512 \beta_{2} - 29994792 \beta_{1} + 110092528$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$-294976 \beta_{15} - 3295648 \beta_{14} - 5312000 \beta_{13} + 594696 \beta_{12} + 404128 \beta_{11} - 3533312 \beta_{10} + 527792 \beta_{9} + 5093376 \beta_{8} - 3642336 \beta_{7} + 20480 \beta_{6} + 35907800 \beta_{5} - 2899920 \beta_{4} - 8452848 \beta_{3} + 5799416 \beta_{2} - 102182056 \beta_{1} - 530397384$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 −1.18006 + 3.82197i −1.18006 − 3.82197i −3.55818 − 1.82740i −3.55818 + 1.82740i 2.11879 + 3.39275i 2.11879 − 3.39275i −3.04390 + 2.59512i −3.04390 − 2.59512i 3.97720 + 0.426493i 3.97720 − 0.426493i −1.06635 + 3.85524i −1.06635 − 3.85524i 3.79586 − 1.26152i 3.79586 + 1.26152i 1.95664 + 3.48878i 1.95664 − 3.48878i
−3.89995 0.889027i 5.19615i 14.4193 + 6.93433i 0 4.61952 20.2647i 44.0991i −50.0696 39.8627i −27.0000 0
151.2 −3.89995 + 0.889027i 5.19615i 14.4193 6.93433i 0 4.61952 + 20.2647i 44.0991i −50.0696 + 39.8627i −27.0000 0
151.3 −3.36166 2.16777i 5.19615i 6.60152 + 14.5746i 0 −11.2641 + 17.4677i 36.6738i 9.40241 63.3056i −27.0000 0
151.4 −3.36166 + 2.16777i 5.19615i 6.60152 14.5746i 0 −11.2641 17.4677i 36.6738i 9.40241 + 63.3056i −27.0000 0
151.5 −1.87881 3.53130i 5.19615i −8.94016 + 13.2693i 0 18.3492 9.76257i 7.86734i 63.6546 + 6.63999i −27.0000 0
151.6 −1.87881 + 3.53130i 5.19615i −8.94016 13.2693i 0 18.3492 + 9.76257i 7.86734i 63.6546 6.63999i −27.0000 0
151.7 0.725488 3.93366i 5.19615i −14.9473 5.70765i 0 −20.4399 3.76975i 36.7329i −33.2960 + 54.6569i −27.0000 0
151.8 0.725488 + 3.93366i 5.19615i −14.9473 + 5.70765i 0 −20.4399 + 3.76975i 36.7329i −33.2960 54.6569i −27.0000 0
151.9 1.61924 3.65760i 5.19615i −10.7561 11.8451i 0 19.0055 + 8.41384i 95.1090i −60.7414 + 20.1614i −27.0000 0
151.10 1.61924 + 3.65760i 5.19615i −10.7561 + 11.8451i 0 19.0055 8.41384i 95.1090i −60.7414 20.1614i −27.0000 0
151.11 2.80556 2.85111i 5.19615i −0.257663 15.9979i 0 −14.8148 14.5781i 63.6232i −46.3347 44.1485i −27.0000 0
151.12 2.80556 + 2.85111i 5.19615i −0.257663 + 15.9979i 0 −14.8148 + 14.5781i 63.6232i −46.3347 + 44.1485i −27.0000 0
151.13 2.99044 2.65655i 5.19615i 1.88545 15.8885i 0 13.8039 + 15.5388i 74.3539i −36.5704 52.5224i −27.0000 0
151.14 2.99044 + 2.65655i 5.19615i 1.88545 + 15.8885i 0 13.8039 15.5388i 74.3539i −36.5704 + 52.5224i −27.0000 0
151.15 3.99969 0.0498899i 5.19615i 15.9950 0.399088i 0 −0.259236 20.7830i 35.2842i 63.9552 2.39422i −27.0000 0
151.16 3.99969 + 0.0498899i 5.19615i 15.9950 + 0.399088i 0 −0.259236 + 20.7830i 35.2842i 63.9552 + 2.39422i −27.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.5.c.c yes 16
4.b odd 2 1 inner 300.5.c.c yes 16
5.b even 2 1 300.5.c.b 16
5.c odd 4 2 300.5.f.c 32
20.d odd 2 1 300.5.c.b 16
20.e even 4 2 300.5.f.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.5.c.b 16 5.b even 2 1
300.5.c.b 16 20.d odd 2 1
300.5.c.c yes 16 1.a even 1 1 trivial
300.5.c.c yes 16 4.b odd 2 1 inner
300.5.f.c 32 5.c odd 4 2
300.5.f.c 32 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(300, [\chi])$$:

 $$11\!\cdots\!56$$$$T_{7}^{10} +$$$$28\!\cdots\!38$$$$T_{7}^{8} +$$$$41\!\cdots\!04$$$$T_{7}^{6} +$$$$32\!\cdots\!84$$$$T_{7}^{4} +$$$$10\!\cdots\!52$$$$T_{7}^{2} +$$$$55\!\cdots\!61$$">$$T_{7}^{16} + \cdots$$ $$76\!\cdots\!48$$$$T_{13}^{2} +$$$$33\!\cdots\!08$$$$T_{13} +$$$$62\!\cdots\!21$$">$$T_{13}^{8} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4294967296 - 1610612736 T + 234881024 T^{2} + 50331648 T^{3} - 18087936 T^{4} - 393216 T^{5} + 778240 T^{6} - 129024 T^{7} - 26368 T^{8} - 8064 T^{9} + 3040 T^{10} - 96 T^{11} - 276 T^{12} + 48 T^{13} + 14 T^{14} - 6 T^{15} + T^{16}$$
$3$ $$( 27 + T^{2} )^{8}$$
$5$ $$T^{16}$$
$7$ $$55\!\cdots\!61$$$$+$$$$10\!\cdots\!52$$$$T^{2} +$$$$32\!\cdots\!84$$$$T^{4} + 4195788114349404104 T^{6} + 2891061394272838 T^{8} + 1109365466056 T^{10} + 232923164 T^{12} + 24568 T^{14} + T^{16}$$
$11$ $$23\!\cdots\!64$$$$+$$$$15\!\cdots\!72$$$$T^{2} +$$$$14\!\cdots\!80$$$$T^{4} +$$$$52\!\cdots\!84$$$$T^{6} + 947044849627010560 T^{8} + 90784698812416 T^{10} + 4590189248 T^{12} + 112000 T^{14} + T^{16}$$
$13$ $$( 6282398897791921 + 33216073577608 T - 7653778912948 T^{2} - 9799175336 T^{3} + 2076207574 T^{4} + 3887096 T^{5} - 85588 T^{6} - 88 T^{7} + T^{8} )^{2}$$
$17$ $$( 973504517115808000 + 1700982354984960 T - 430858738860032 T^{2} + 379598346240 T^{3} + 18224825952 T^{4} - 7595520 T^{5} - 251264 T^{6} + T^{8} )^{2}$$
$19$ $$19\!\cdots\!41$$$$+$$$$95\!\cdots\!72$$$$T^{2} +$$$$11\!\cdots\!44$$$$T^{4} +$$$$10\!\cdots\!84$$$$T^{6} +$$$$35\!\cdots\!58$$$$T^{8} + 53234150608563176 T^{10} + 360827961404 T^{12} + 1023128 T^{14} + T^{16}$$
$23$ $$87\!\cdots\!16$$$$+$$$$26\!\cdots\!40$$$$T^{2} +$$$$34\!\cdots\!32$$$$T^{4} +$$$$23\!\cdots\!64$$$$T^{6} +$$$$97\!\cdots\!76$$$$T^{8} + 2454119857823737856 T^{10} + 3684668964032 T^{12} + 2994560 T^{14} + T^{16}$$
$29$ $$( -$$$$10\!\cdots\!44$$$$+ 62334214990770610176 T + 99105602888626176 T^{2} - 1001984840673792 T^{3} + 875025320288 T^{4} + 1806056448 T^{5} - 1966464 T^{6} - 864 T^{7} + T^{8} )^{2}$$
$31$ $$75\!\cdots\!09$$$$+$$$$69\!\cdots\!84$$$$T^{2} +$$$$10\!\cdots\!92$$$$T^{4} +$$$$55\!\cdots\!60$$$$T^{6} +$$$$14\!\cdots\!18$$$$T^{8} + 21324287935279871752 T^{10} + 16476492748892 T^{12} + 6451768 T^{14} + T^{16}$$
$37$ $$($$$$86\!\cdots\!76$$$$-$$$$50\!\cdots\!36$$$$T - 11390748757819708672 T^{2} + 8386070949986048 T^{3} + 16309243794784 T^{4} - 4335816512 T^{5} - 7421392 T^{6} + 784 T^{7} + T^{8} )^{2}$$
$41$ $$($$$$51\!\cdots\!32$$$$-$$$$22\!\cdots\!36$$$$T - 73411816733676279808 T^{2} - 4417366119286272 T^{3} + 54861865758912 T^{4} + 4976285376 T^{5} - 13204624 T^{6} - 624 T^{7} + T^{8} )^{2}$$
$43$ $$44\!\cdots\!21$$$$+$$$$17\!\cdots\!28$$$$T^{2} +$$$$77\!\cdots\!04$$$$T^{4} +$$$$12\!\cdots\!36$$$$T^{6} +$$$$88\!\cdots\!38$$$$T^{8} +$$$$28\!\cdots\!04$$$$T^{10} + 441304651117244 T^{12} + 33759512 T^{14} + T^{16}$$
$47$ $$28\!\cdots\!00$$$$+$$$$20\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!16$$$$T^{4} +$$$$14\!\cdots\!72$$$$T^{6} +$$$$42\!\cdots\!36$$$$T^{8} +$$$$75\!\cdots\!88$$$$T^{10} + 780447236298176 T^{12} + 43339232 T^{14} + T^{16}$$
$53$ $$( -$$$$44\!\cdots\!44$$$$+$$$$21\!\cdots\!64$$$$T -$$$$21\!\cdots\!92$$$$T^{2} - 131443387574794752 T^{3} + 154375141456064 T^{4} + 10160870208 T^{5} - 25145232 T^{6} + 144 T^{7} + T^{8} )^{2}$$
$59$ $$62\!\cdots\!96$$$$+$$$$19\!\cdots\!72$$$$T^{2} +$$$$43\!\cdots\!24$$$$T^{4} +$$$$30\!\cdots\!64$$$$T^{6} +$$$$95\!\cdots\!28$$$$T^{8} +$$$$16\!\cdots\!36$$$$T^{10} + 1419528561476544 T^{12} + 61807968 T^{14} + T^{16}$$
$61$ $$( -$$$$22\!\cdots\!11$$$$-$$$$92\!\cdots\!96$$$$T -$$$$80\!\cdots\!60$$$$T^{2} + 3828100717148664232 T^{3} + 1653249809252950 T^{4} - 191431094008 T^{5} - 79352788 T^{6} + 1880 T^{7} + T^{8} )^{2}$$
$67$ $$16\!\cdots\!41$$$$+$$$$38\!\cdots\!92$$$$T^{2} +$$$$13\!\cdots\!24$$$$T^{4} +$$$$19\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!58$$$$T^{8} +$$$$64\!\cdots\!36$$$$T^{10} + 15444288581016764 T^{12} + 195207448 T^{14} + T^{16}$$
$71$ $$31\!\cdots\!00$$$$+$$$$33\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!16$$$$T^{4} +$$$$12\!\cdots\!88$$$$T^{6} +$$$$68\!\cdots\!36$$$$T^{8} +$$$$20\!\cdots\!72$$$$T^{10} + 33408631558645376 T^{12} + 287151328 T^{14} + T^{16}$$
$73$ $$( -$$$$23\!\cdots\!48$$$$+$$$$29\!\cdots\!60$$$$T -$$$$40\!\cdots\!96$$$$T^{2} - 24511044693386622720 T^{3} + 4188875574048096 T^{4} + 646830269760 T^{5} - 115731920 T^{6} - 5520 T^{7} + T^{8} )^{2}$$
$79$ $$22\!\cdots\!04$$$$+$$$$45\!\cdots\!92$$$$T^{2} +$$$$25\!\cdots\!08$$$$T^{4} +$$$$29\!\cdots\!48$$$$T^{6} +$$$$14\!\cdots\!72$$$$T^{8} +$$$$37\!\cdots\!20$$$$T^{10} + 51681030281796608 T^{12} + 364268416 T^{14} + T^{16}$$
$83$ $$31\!\cdots\!64$$$$+$$$$17\!\cdots\!48$$$$T^{2} +$$$$36\!\cdots\!08$$$$T^{4} +$$$$34\!\cdots\!92$$$$T^{6} +$$$$16\!\cdots\!32$$$$T^{8} +$$$$41\!\cdots\!00$$$$T^{10} + 55342713702177728 T^{12} + 372516704 T^{14} + T^{16}$$
$89$ $$( -$$$$56\!\cdots\!12$$$$+$$$$12\!\cdots\!88$$$$T -$$$$35\!\cdots\!80$$$$T^{2} - 56816085262141489152 T^{3} + 19327010090713088 T^{4} + 369241767936 T^{5} - 259302912 T^{6} + 384 T^{7} + T^{8} )^{2}$$
$97$ $$($$$$64\!\cdots\!77$$$$+$$$$33\!\cdots\!36$$$$T -$$$$66\!\cdots\!72$$$$T^{2} -$$$$51\!\cdots\!44$$$$T^{3} + 101624878423701990 T^{4} + 2464983082152 T^{5} - 550033476 T^{6} - 3624 T^{7} + T^{8} )^{2}$$