# Properties

 Label 300.5.c.b 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{1} q^{3} + \beta_{7} q^{4} + ( 1 + \beta_{5} ) q^{6} + ( \beta_{7} - \beta_{12} ) q^{7} + ( 12 - \beta_{2} + \beta_{3} ) q^{8} -27 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{1} q^{3} + \beta_{7} q^{4} + ( 1 + \beta_{5} ) q^{6} + ( \beta_{7} - \beta_{12} ) q^{7} + ( 12 - \beta_{2} + \beta_{3} ) q^{8} -27 q^{9} + ( 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{11} - \beta_{14} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{8} + \beta_{10} ) q^{12} + ( -10 - 3 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{13} + ( 3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{14} + ( -22 + 5 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{16} + ( -1 + \beta_{1} + 16 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 9 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{17} + 27 \beta_{2} q^{18} + ( -12 + 15 \beta_{1} + 23 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 7 \beta_{7} - 2 \beta_{10} + \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{19} + ( -11 + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{21} + ( 50 - 19 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 6 \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} + 2 \beta_{14} + 3 \beta_{15} ) q^{22} + ( -16 - 18 \beta_{1} + 26 \beta_{2} - \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - \beta_{6} + 5 \beta_{7} - 5 \beta_{8} + 7 \beta_{10} + \beta_{11} + 2 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 5 \beta_{15} ) q^{23} + ( 8 + 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 6 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{24} + ( 32 - 45 \beta_{1} + 15 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - 5 \beta_{12} - 5 \beta_{13} - 4 \beta_{15} ) q^{26} -27 \beta_{1} q^{27} + ( -212 - 31 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} - \beta_{10} + 2 \beta_{11} + 5 \beta_{12} + \beta_{13} + 12 \beta_{15} ) q^{28} + ( 135 + 4 \beta_{1} - 66 \beta_{2} + 6 \beta_{3} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 7 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} ) q^{29} + ( -18 - 34 \beta_{1} + 57 \beta_{2} - 7 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} - 12 \beta_{12} - 7 \beta_{13} - \beta_{14} - \beta_{15} ) q^{31} + ( -144 + 28 \beta_{1} + 29 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 10 \beta_{7} - 10 \beta_{8} - 8 \beta_{9} - 4 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{32} + ( 15 - \beta_{1} - 35 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 6 \beta_{10} + 5 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + 6 \beta_{15} ) q^{33} + ( -178 + 13 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} - 7 \beta_{6} - 11 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 8 \beta_{12} + 3 \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{34} -27 \beta_{7} q^{36} + ( 68 - 16 \beta_{1} + 65 \beta_{2} + \beta_{3} + 9 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} + \beta_{7} - 5 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} - 7 \beta_{12} - 7 \beta_{13} - 3 \beta_{14} - 11 \beta_{15} ) q^{37} + ( 455 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 7 \beta_{4} + 12 \beta_{5} + 2 \beta_{6} - 28 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} + \beta_{10} - 3 \beta_{11} - 11 \beta_{12} - 2 \beta_{13} + 6 \beta_{14} + 5 \beta_{15} ) q^{38} + ( -22 - 22 \beta_{1} + 65 \beta_{2} - 6 \beta_{3} + \beta_{5} + 10 \beta_{6} + \beta_{7} + 6 \beta_{8} - 7 \beta_{10} + 2 \beta_{11} - 6 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} - 6 \beta_{15} ) q^{39} + ( 107 + 5 \beta_{1} - 78 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + \beta_{5} + 3 \beta_{6} - 27 \beta_{7} - 10 \beta_{8} + 10 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} - 5 \beta_{14} + 5 \beta_{15} ) q^{41} + ( -10 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + 8 \beta_{7} + 6 \beta_{8} + 9 \beta_{9} - 5 \beta_{10} - 8 \beta_{11} - 9 \beta_{12} - 9 \beta_{14} - 6 \beta_{15} ) q^{42} + ( -18 - 7 \beta_{1} + 92 \beta_{2} + 2 \beta_{3} + 23 \beta_{4} + 22 \beta_{5} + 17 \beta_{6} - 6 \beta_{7} + \beta_{8} + 18 \beta_{10} - 6 \beta_{11} - 27 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{43} + ( 524 - 38 \beta_{1} - 41 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} - 22 \beta_{5} - 17 \beta_{6} - 4 \beta_{7} - 12 \beta_{8} - 16 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} - 5 \beta_{12} - 5 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{44} + ( 382 + 29 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} - 33 \beta_{5} - 24 \beta_{6} - 30 \beta_{7} - 23 \beta_{8} - 15 \beta_{9} + 5 \beta_{10} + 11 \beta_{11} + 7 \beta_{12} + 4 \beta_{13} - 12 \beta_{14} + 15 \beta_{15} ) q^{46} + ( 52 + 28 \beta_{1} - 162 \beta_{2} - 12 \beta_{4} + 54 \beta_{5} - 32 \beta_{6} - 14 \beta_{7} - 8 \beta_{8} + 14 \beta_{10} + 12 \beta_{11} + 4 \beta_{12} - 14 \beta_{13} - 6 \beta_{14} + 8 \beta_{15} ) q^{47} + ( -126 - 43 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} + 14 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} + 12 \beta_{8} + 6 \beta_{9} - 7 \beta_{10} - 7 \beta_{11} - 24 \beta_{12} - 6 \beta_{13} + 3 \beta_{14} - 15 \beta_{15} ) q^{48} + ( -610 + 10 \beta_{1} - 217 \beta_{2} - 27 \beta_{3} + 7 \beta_{4} - 36 \beta_{5} - 3 \beta_{6} + 25 \beta_{7} - 17 \beta_{8} - 4 \beta_{9} + 20 \beta_{10} + 11 \beta_{11} + \beta_{12} + \beta_{13} - 13 \beta_{14} + 15 \beta_{15} ) q^{49} + ( 2 - 2 \beta_{1} - 22 \beta_{2} - 6 \beta_{3} - 9 \beta_{4} - 8 \beta_{5} - 11 \beta_{6} - 8 \beta_{7} - 3 \beta_{8} - 4 \beta_{10} + 2 \beta_{11} + 9 \beta_{12} - 9 \beta_{13} + 3 \beta_{15} ) q^{51} + ( -176 + 30 \beta_{1} - 46 \beta_{2} + 6 \beta_{4} - 60 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 26 \beta_{10} - 20 \beta_{11} - 4 \beta_{13} - 16 \beta_{14} - 12 \beta_{15} ) q^{52} + ( 87 + 2 \beta_{1} - 98 \beta_{2} + 12 \beta_{3} - 12 \beta_{4} - 30 \beta_{5} + 19 \beta_{6} - 24 \beta_{7} + 15 \beta_{8} - 18 \beta_{9} - 23 \beta_{10} + 16 \beta_{11} + 16 \beta_{12} + 16 \beta_{13} + 14 \beta_{14} + 18 \beta_{15} ) q^{53} + ( -27 - 27 \beta_{5} ) q^{54} + ( -720 - 102 \beta_{1} + 240 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} - 22 \beta_{5} - 11 \beta_{6} + 6 \beta_{7} - 14 \beta_{8} + 4 \beta_{9} + 10 \beta_{10} + 14 \beta_{11} - 37 \beta_{12} - \beta_{13} - 8 \beta_{14} - 12 \beta_{15} ) q^{56} + ( -376 - 11 \beta_{1} + 13 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} - 21 \beta_{5} - 10 \beta_{6} + 20 \beta_{7} - 3 \beta_{8} + 6 \beta_{9} + 10 \beta_{10} + 7 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 12 \beta_{14} + 6 \beta_{15} ) q^{57} + ( 1034 + 111 \beta_{1} - 165 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 11 \beta_{5} - 13 \beta_{6} + 53 \beta_{7} + 8 \beta_{8} - \beta_{9} - 3 \beta_{10} + 40 \beta_{12} + 7 \beta_{13} - 7 \beta_{14} + 26 \beta_{15} ) q^{58} + ( -44 + 26 \beta_{1} + 154 \beta_{2} - 26 \beta_{3} - 6 \beta_{4} + 64 \beta_{5} + 6 \beta_{6} - 54 \beta_{7} + 14 \beta_{8} - 4 \beta_{10} + 14 \beta_{11} - 14 \beta_{12} - 14 \beta_{13} - 2 \beta_{14} - 14 \beta_{15} ) q^{59} + ( -244 + 36 \beta_{1} - 92 \beta_{2} - 5 \beta_{3} + 31 \beta_{4} + 56 \beta_{5} - 14 \beta_{6} + 26 \beta_{7} - 58 \beta_{8} - \beta_{9} + 32 \beta_{10} - 31 \beta_{11} - 21 \beta_{12} - 21 \beta_{13} - 13 \beta_{14} - 29 \beta_{15} ) q^{61} + ( 793 - 50 \beta_{1} + 16 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} - 32 \beta_{5} - 18 \beta_{6} - 23 \beta_{7} - 29 \beta_{8} - 7 \beta_{9} + 17 \beta_{10} - 37 \beta_{11} + 6 \beta_{12} - 21 \beta_{13} - 12 \beta_{14} - 5 \beta_{15} ) q^{62} + ( -27 \beta_{7} + 27 \beta_{12} ) q^{63} + ( 604 - 166 \beta_{1} + 120 \beta_{2} + 14 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} - 22 \beta_{6} + 8 \beta_{7} - 16 \beta_{8} + 16 \beta_{9} + 2 \beta_{10} - 46 \beta_{11} - 18 \beta_{12} - 10 \beta_{13} - 34 \beta_{14} - 10 \beta_{15} ) q^{64} + ( 502 + 27 \beta_{1} - 23 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} + 21 \beta_{5} + \beta_{6} + 43 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} + 9 \beta_{10} + 2 \beta_{11} - 42 \beta_{12} - 15 \beta_{13} + 3 \beta_{14} - 24 \beta_{15} ) q^{66} + ( -24 - 191 \beta_{1} + 143 \beta_{2} - 41 \beta_{3} - 10 \beta_{4} + 116 \beta_{5} + 22 \beta_{6} - 75 \beta_{7} + 38 \beta_{8} - 22 \beta_{10} + 29 \beta_{11} - 32 \beta_{12} + 3 \beta_{14} - 38 \beta_{15} ) q^{67} + ( -1716 - 24 \beta_{1} + 231 \beta_{2} + 21 \beta_{3} - 10 \beta_{4} + 22 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} - 8 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + 41 \beta_{12} + 17 \beta_{13} + 28 \beta_{14} - 4 \beta_{15} ) q^{68} + ( 625 - 22 \beta_{1} + 23 \beta_{2} + 27 \beta_{3} + 3 \beta_{4} + 10 \beta_{6} - 35 \beta_{7} + 6 \beta_{8} + 18 \beta_{9} - \beta_{10} + 5 \beta_{11} - 9 \beta_{12} - 9 \beta_{13} + 9 \beta_{14} - 27 \beta_{15} ) q^{69} + ( -64 - 42 \beta_{1} + 120 \beta_{2} - 9 \beta_{3} - 63 \beta_{4} - 141 \beta_{5} - 3 \beta_{6} + 63 \beta_{7} + 17 \beta_{8} - 83 \beta_{10} - 3 \beta_{11} + 8 \beta_{12} + 8 \beta_{13} - 17 \beta_{15} ) q^{71} + ( -324 + 27 \beta_{2} - 27 \beta_{3} ) q^{72} + ( -772 - 14 \beta_{1} + 101 \beta_{2} - 3 \beta_{3} - 17 \beta_{4} + 40 \beta_{5} - 25 \beta_{6} + 107 \beta_{7} + 33 \beta_{8} + 34 \beta_{9} + 52 \beta_{10} - 37 \beta_{11} - 23 \beta_{12} - 23 \beta_{13} - 13 \beta_{14} - 33 \beta_{15} ) q^{73} + ( -1012 + 316 \beta_{1} - 84 \beta_{2} - 18 \beta_{3} - 8 \beta_{4} - 38 \beta_{5} + 22 \beta_{6} - 84 \beta_{7} + 36 \beta_{8} + 2 \beta_{9} - 98 \beta_{10} - 16 \beta_{11} + 22 \beta_{12} - 8 \beta_{13} - 10 \beta_{14} - 4 \beta_{15} ) q^{74} + ( -1692 - 245 \beta_{1} - 348 \beta_{2} - 27 \beta_{3} - 7 \beta_{4} + 46 \beta_{5} - 55 \beta_{6} - 8 \beta_{7} - 33 \beta_{8} - 8 \beta_{9} + 33 \beta_{10} + 34 \beta_{11} + 37 \beta_{12} - 19 \beta_{13} - 12 \beta_{14} - 12 \beta_{15} ) q^{76} + ( -69 - 34 \beta_{1} + 108 \beta_{2} - 4 \beta_{3} + 24 \beta_{4} - 146 \beta_{5} - 43 \beta_{6} - 26 \beta_{7} - 71 \beta_{8} - 24 \beta_{9} + 97 \beta_{10} + 36 \beta_{11} - 8 \beta_{12} - 8 \beta_{13} - 6 \beta_{14} - 10 \beta_{15} ) q^{77} + ( 1095 + 19 \beta_{1} - 20 \beta_{2} + 9 \beta_{3} + 14 \beta_{4} - 15 \beta_{5} + 18 \beta_{6} - 51 \beta_{7} + 22 \beta_{8} - 4 \beta_{10} - 24 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 36 \beta_{15} ) q^{78} + ( -36 - 158 \beta_{1} - 36 \beta_{2} + 64 \beta_{3} - 18 \beta_{4} + 140 \beta_{5} - 62 \beta_{6} + 74 \beta_{7} - 46 \beta_{8} + 76 \beta_{10} + 16 \beta_{11} + 24 \beta_{12} - 14 \beta_{13} - 32 \beta_{14} + 46 \beta_{15} ) q^{79} + 729 q^{81} + ( 1330 - 65 \beta_{1} - 103 \beta_{2} - 40 \beta_{3} + 8 \beta_{4} - 45 \beta_{5} + 15 \beta_{6} + 35 \beta_{7} + 2 \beta_{8} - 15 \beta_{9} - 117 \beta_{10} - 10 \beta_{11} - 20 \beta_{12} + 5 \beta_{13} + 25 \beta_{14} - 40 \beta_{15} ) q^{82} + ( -96 + 334 \beta_{1} - 6 \beta_{2} + 32 \beta_{3} - 30 \beta_{4} + 98 \beta_{5} + 10 \beta_{6} + 164 \beta_{7} - 6 \beta_{8} - 6 \beta_{10} + 52 \beta_{11} + 84 \beta_{12} + 60 \beta_{13} + 34 \beta_{14} + 6 \beta_{15} ) q^{83} + ( 932 - 222 \beta_{1} + 19 \beta_{2} - 9 \beta_{3} - 4 \beta_{4} + 14 \beta_{5} + 27 \beta_{6} - 39 \beta_{7} + 4 \beta_{8} + 12 \beta_{9} - 28 \beta_{10} + 18 \beta_{11} + 27 \beta_{12} - 9 \beta_{13} + 36 \beta_{14} ) q^{84} + ( 1543 + 393 \beta_{1} - 30 \beta_{2} + \beta_{3} - 47 \beta_{4} + 54 \beta_{5} + 2 \beta_{6} - 134 \beta_{7} - 3 \beta_{8} + 35 \beta_{9} + 7 \beta_{10} + 71 \beta_{11} + 101 \beta_{12} + 20 \beta_{13} + 6 \beta_{14} - 25 \beta_{15} ) q^{86} + ( 70 + 160 \beta_{1} - 161 \beta_{2} + 36 \beta_{3} - 18 \beta_{4} + 65 \beta_{5} - 28 \beta_{6} - 10 \beta_{7} + 13 \beta_{10} + 4 \beta_{11} + 9 \beta_{12} + 9 \beta_{13} - 27 \beta_{14} ) q^{87} + ( -3136 + 52 \beta_{1} - 508 \beta_{2} + 40 \beta_{3} - 28 \beta_{4} - 64 \beta_{5} - 2 \beta_{6} + 124 \beta_{7} + 20 \beta_{8} + 12 \beta_{9} - 4 \beta_{10} - 64 \beta_{11} + 6 \beta_{12} + 18 \beta_{13} - 48 \beta_{14} + 4 \beta_{15} ) q^{88} + ( -196 - 114 \beta_{1} + 346 \beta_{2} + 6 \beta_{3} + 44 \beta_{4} - 150 \beta_{5} - 10 \beta_{6} + 52 \beta_{7} + 4 \beta_{8} - 20 \beta_{9} - 10 \beta_{10} + 46 \beta_{11} - 14 \beta_{12} - 14 \beta_{13} - 4 \beta_{14} - 24 \beta_{15} ) q^{89} + ( 54 + 237 \beta_{1} + 70 \beta_{2} + 36 \beta_{3} - 49 \beta_{4} - 250 \beta_{5} + \beta_{6} - 42 \beta_{7} + 33 \beta_{8} - 90 \beta_{10} - 64 \beta_{11} - 21 \beta_{12} + 13 \beta_{13} - 56 \beta_{14} - 33 \beta_{15} ) q^{91} + ( -1340 + 406 \beta_{1} - 379 \beta_{2} - 7 \beta_{3} - 44 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 32 \beta_{7} + 4 \beta_{8} + 56 \beta_{9} - 52 \beta_{10} - 54 \beta_{11} - 63 \beta_{12} + 9 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{92} + ( 633 - 38 \beta_{1} - 190 \beta_{2} + 15 \beta_{3} + 23 \beta_{4} - 38 \beta_{5} + 17 \beta_{6} + 158 \beta_{7} + 25 \beta_{8} - 15 \beta_{9} - 3 \beta_{10} - 11 \beta_{11} - 21 \beta_{12} - 21 \beta_{13} - 3 \beta_{14} - 39 \beta_{15} ) q^{93} + ( -2962 + 702 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} + 48 \beta_{4} + 22 \beta_{5} - 52 \beta_{6} + 214 \beta_{7} - 128 \beta_{8} - 20 \beta_{9} + 84 \beta_{10} - 44 \beta_{11} - 30 \beta_{12} - 34 \beta_{13} - 16 \beta_{14} + 60 \beta_{15} ) q^{94} + ( -792 - 136 \beta_{1} + 141 \beta_{2} + 3 \beta_{3} + 18 \beta_{4} - 6 \beta_{5} + 15 \beta_{6} + 18 \beta_{7} + 6 \beta_{8} - 12 \beta_{11} + 45 \beta_{12} - 27 \beta_{13} - 18 \beta_{14} - 42 \beta_{15} ) q^{96} + ( -123 - 76 \beta_{1} - 597 \beta_{2} + 53 \beta_{3} - 11 \beta_{4} - 266 \beta_{5} + 35 \beta_{6} - 69 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} + 80 \beta_{10} + 95 \beta_{11} + 5 \beta_{12} + 5 \beta_{13} + 29 \beta_{14} - 19 \beta_{15} ) q^{97} + ( 3444 + 582 \beta_{1} + 456 \beta_{2} + 76 \beta_{3} - 80 \beta_{4} - 58 \beta_{5} + 66 \beta_{6} + 214 \beta_{7} + 20 \beta_{8} - 22 \beta_{9} - 122 \beta_{10} - 4 \beta_{11} - 12 \beta_{12} + 78 \beta_{13} + 50 \beta_{14} - 24 \beta_{15} ) q^{98} + ( -81 \beta_{2} - 27 \beta_{3} + 54 \beta_{5} + 27 \beta_{7} + 27 \beta_{11} + 27 \beta_{14} ) 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 $$x^{16} - 6 x^{15} + 20 x^{14} - 108 x^{13} + 492 x^{12} - 2160 x^{11} + 7360 x^{10} - 39552 x^{9} + 234752 x^{8} - 632832 x^{7} + 1884160 x^{6} - 8847360 x^{5} + 32243712 x^{4} - 113246208 x^{3} + 335544320 x^{2} - 1610612736 x + 4294967296$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-22761 \nu^{15} + 1177062 \nu^{14} - 419412 \nu^{13} - 17236980 \nu^{12} - 30915852 \nu^{11} + 386713392 \nu^{10} - 107470272 \nu^{9} - 1871005056 \nu^{8} - 10408187136 \nu^{7} + 46803376128 \nu^{6} + 247514284032 \nu^{5} - 579352461312 \nu^{4} - 5466618593280 \nu^{3} + 8277215674368 \nu^{2} + 49525033009152 \nu - 91883040669696$$$$)/ 21382226247680$$ $$\beta_{2}$$ $$=$$ $$($$$$21677 \nu^{15} + 746 \nu^{14} - 410316 \nu^{13} - 410780 \nu^{12} + 7032284 \nu^{11} + 1251056 \nu^{10} - 57734336 \nu^{9} - 105520768 \nu^{8} + 674989312 \nu^{7} + 6049992704 \nu^{6} - 16265150464 \nu^{5} - 98598322176 \nu^{4} + 118742056960 \nu^{3} + 1190882443264 \nu^{2} - 1341573300224 \nu + 2036619804672$$$$)/ 2672778280960$$ $$\beta_{3}$$ $$=$$ $$($$$$21677 \nu^{15} + 746 \nu^{14} - 410316 \nu^{13} - 410780 \nu^{12} + 7032284 \nu^{11} + 1251056 \nu^{10} - 57734336 \nu^{9} - 105520768 \nu^{8} + 674989312 \nu^{7} + 6049992704 \nu^{6} - 16265150464 \nu^{5} - 98598322176 \nu^{4} + 2791520337920 \nu^{3} + 1190882443264 \nu^{2} - 1341573300224 \nu - 30036719566848$$$$)/ 2672778280960$$ $$\beta_{4}$$ $$=$$ $$($$$$-109161 \nu^{15} - 2497938 \nu^{14} + 17073788 \nu^{13} - 139010900 \nu^{12} + 269177748 \nu^{11} + 617126032 \nu^{10} + 4237170368 \nu^{9} - 42390216576 \nu^{8} + 73112360704 \nu^{7} - 335353413632 \nu^{6} + 1942748332032 \nu^{5} - 6616495357952 \nu^{4} + 637742612480 \nu^{3} - 22392426463232 \nu^{2} + 634581954854912 \nu - 1621026286862336$$$$)/ 10691113123840$$ $$\beta_{5}$$ $$=$$ $$($$$$-65031 \nu^{15} - 2238 \nu^{14} + 1230948 \nu^{13} + 1232340 \nu^{12} - 21096852 \nu^{11} - 3753168 \nu^{10} + 173203008 \nu^{9} + 316562304 \nu^{8} - 2024967936 \nu^{7} - 18149978112 \nu^{6} + 48795451392 \nu^{5} + 295794966528 \nu^{4} - 356226170880 \nu^{3} - 3572647329792 \nu^{2} + 20061389586432 \nu - 8782637694976$$$$)/ 2672778280960$$ $$\beta_{6}$$ $$=$$ $$($$$$699729 \nu^{15} - 4691478 \nu^{14} + 62915828 \nu^{13} - 142283500 \nu^{12} - 270249492 \nu^{11} - 3305599408 \nu^{10} + 21538701248 \nu^{9} - 29554953856 \nu^{8} + 141704772864 \nu^{7} - 966521116672 \nu^{6} + 3794299502592 \nu^{5} + 1279264882688 \nu^{4} + 2510022246400 \nu^{3} - 317222417334272 \nu^{2} + 517403670740992 \nu + 597877969649664$$$$)/ 21382226247680$$ $$\beta_{7}$$ $$=$$ $$($$$$16351 \nu^{15} - 105482 \nu^{14} + 241292 \nu^{13} - 454100 \nu^{12} + 6009172 \nu^{11} - 27159632 \nu^{10} + 93980992 \nu^{9} - 551716224 \nu^{8} + 2470986496 \nu^{7} - 7138510848 \nu^{6} + 11648237568 \nu^{5} - 72525611008 \nu^{4} + 455715061760 \nu^{3} - 1410993225728 \nu^{2} + 4618734010368 \nu - 11637750759424$$$$)/ 334097285120$$ $$\beta_{8}$$ $$=$$ $$($$$$528347 \nu^{15} - 4346834 \nu^{14} + 5696444 \nu^{13} - 156454180 \nu^{12} + 699098404 \nu^{11} + 84682736 \nu^{10} + 5667009344 \nu^{9} - 64428230528 \nu^{8} + 177359884032 \nu^{7} - 440713701376 \nu^{6} + 2144612663296 \nu^{5} - 13983082151936 \nu^{4} + 18693416550400 \nu^{3} + 29091843538944 \nu^{2} + 751684808605696 \nu - 2265573908021248$$$$)/ 10691113123840$$ $$\beta_{9}$$ $$=$$ $$($$$$-1660583 \nu^{15} - 11482214 \nu^{14} + 34112084 \nu^{13} + 59858740 \nu^{12} - 39385396 \nu^{11} + 1107584336 \nu^{10} + 13491647424 \nu^{9} + 20024460672 \nu^{8} - 65625180928 \nu^{7} - 812485687296 \nu^{6} - 456613249024 \nu^{5} + 3541552922624 \nu^{4} - 15751914455040 \nu^{3} - 113014531424256 \nu^{2} + 579146912628736 \nu + 1937016023416832$$$$)/ 21382226247680$$ $$\beta_{10}$$ $$=$$ $$($$$$-1668199 \nu^{15} + 15383658 \nu^{14} - 71945868 \nu^{13} + 66251380 \nu^{12} - 206745588 \nu^{11} + 3773057808 \nu^{10} - 15539076928 \nu^{9} + 62880326016 \nu^{8} - 250126258944 \nu^{7} + 1161470072832 \nu^{6} - 2210368438272 \nu^{5} - 1017690390528 \nu^{4} - 44969388933120 \nu^{3} + 246835725729792 \nu^{2} - 712848841900032 \nu + 1053528902598656$$$$)/ 21382226247680$$ $$\beta_{11}$$ $$=$$ $$($$$$1651199 \nu^{15} - 10261698 \nu^{14} + 29651228 \nu^{13} - 103582260 \nu^{12} + 635794548 \nu^{11} - 2576039408 \nu^{10} + 7081148608 \nu^{9} - 54975181696 \nu^{8} + 281339002624 \nu^{7} - 733386637312 \nu^{6} + 1605397184512 \nu^{5} - 8632613404672 \nu^{4} + 45681956618240 \nu^{3} - 111042229174272 \nu^{2} + 403359706120192 \nu - 1917097173057536$$$$)/ 10691113123840$$ $$\beta_{12}$$ $$=$$ $$($$$$-3400017 \nu^{15} + 7613094 \nu^{14} - 3207764 \nu^{13} + 50899500 \nu^{12} - 228275884 \nu^{11} + 3576049264 \nu^{10} - 5793879744 \nu^{9} + 25054599808 \nu^{8} - 247700366592 \nu^{7} + 1342089216 \nu^{6} - 986195705856 \nu^{5} + 1798418661376 \nu^{4} - 36538636042240 \nu^{3} + 164881567318016 \nu^{2} + 199582667505664 \nu - 80052016381952$$$$)/ 21382226247680$$ $$\beta_{13}$$ $$=$$ $$($$$$-362307 \nu^{15} + 2146026 \nu^{14} - 203980 \nu^{13} + 7231524 \nu^{12} - 18736100 \nu^{11} + 338099312 \nu^{10} - 2666809024 \nu^{9} + 2203154816 \nu^{8} - 27023719168 \nu^{7} + 78454321152 \nu^{6} + 197005836288 \nu^{5} + 780827033600 \nu^{4} - 2502974767104 \nu^{3} + 30237331030016 \nu^{2} - 47861236498432 \nu - 386897901780992$$$$)/ 2138222624768$$ $$\beta_{14}$$ $$=$$ $$($$$$-1827337 \nu^{15} + 8732214 \nu^{14} - 63863604 \nu^{13} + 291078860 \nu^{12} - 805581644 \nu^{11} + 3260490224 \nu^{10} - 14708375744 \nu^{9} + 114776696448 \nu^{8} - 416548658432 \nu^{7} + 960995076096 \nu^{6} - 5661721051136 \nu^{5} + 16930287845376 \nu^{4} - 46545584783360 \nu^{3} + 176630483910656 \nu^{2} - 1065137897209856 \nu + 4459902227447808$$$$)/ 10691113123840$$ $$\beta_{15}$$ $$=$$ $$($$$$244581 \nu^{15} - 1253922 \nu^{14} + 3714332 \nu^{13} - 16702380 \nu^{12} + 26117452 \nu^{11} - 341380192 \nu^{10} + 2041696512 \nu^{9} - 6334100864 \nu^{8} + 25508211456 \nu^{7} - 83272363008 \nu^{6} + 322552496128 \nu^{5} - 853910028288 \nu^{4} + 2757534679040 \nu^{3} - 27160124325888 \nu^{2} + 77615771680768 \nu + 1607357956096$$$$)/ 1336389140480$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + 3 \beta_{2} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{8} - 3 \beta_{7} - \beta_{4} + \beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{2} + 12$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{15} + 3 \beta_{14} - 12 \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 6 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + 13 \beta_{2} - 29 \beta_{1} - 30$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{15} + 6 \beta_{14} + 18 \beta_{13} - 18 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 12 \beta_{9} + 12 \beta_{8} - 24 \beta_{7} - 6 \beta_{6} - 12 \beta_{4} - 114 \beta_{2} + 26 \beta_{1} + 612$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{15} - 34 \beta_{14} - 10 \beta_{13} - 18 \beta_{12} - 46 \beta_{11} + 2 \beta_{10} + 16 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 22 \beta_{6} + 4 \beta_{5} + 8 \beta_{4} + 14 \beta_{3} + 120 \beta_{2} - 166 \beta_{1} + 604$$ $$\nu^{7}$$ $$=$$ $$($$$$-528 \beta_{15} - 144 \beta_{14} - 228 \beta_{13} - 516 \beta_{12} - 72 \beta_{11} + 136 \beta_{10} + 96 \beta_{9} + 152 \beta_{8} + 192 \beta_{7} + 60 \beta_{6} + 840 \beta_{5} - 56 \beta_{4} - 144 \beta_{3} - 2032 \beta_{2} + 784 \beta_{1} + 24000$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$300 \beta_{15} + 108 \beta_{14} + 360 \beta_{13} - 216 \beta_{12} + 1516 \beta_{11} + 828 \beta_{10} + 912 \beta_{9} + 40 \beta_{8} - 3304 \beta_{7} + 440 \beta_{6} + 4784 \beta_{5} - 856 \beta_{4} + 696 \beta_{3} + 17420 \beta_{2} + 2300 \beta_{1} - 61208$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$-1528 \beta_{15} - 552 \beta_{14} - 640 \beta_{13} - 1168 \beta_{12} - 2200 \beta_{11} - 1080 \beta_{10} + 688 \beta_{9} + 1760 \beta_{8} + 16 \beta_{7} + 1328 \beta_{6} - 3600 \beta_{5} - 1120 \beta_{4} + 1496 \beta_{3} - 5552 \beta_{2} - 2568 \beta_{1} - 126928$$ $$\nu^{10}$$ $$=$$ $$($$$$-16728 \beta_{15} + 4200 \beta_{14} - 1992 \beta_{13} - 21000 \beta_{12} + 3096 \beta_{11} + 4632 \beta_{10} + 12960 \beta_{9} - 192 \beta_{8} + 44448 \beta_{7} + 648 \beta_{6} - 41616 \beta_{5} + 4896 \beta_{4} + 21720 \beta_{3} - 258576 \beta_{2} + 222008 \beta_{1} + 472080$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-73152 \beta_{15} + 19584 \beta_{14} + 41664 \beta_{13} - 75264 \beta_{12} - 42816 \beta_{11} + 12096 \beta_{10} + 53952 \beta_{9} + 12480 \beta_{8} + 174240 \beta_{7} + 117216 \beta_{6} - 49280 \beta_{5} + 68160 \beta_{4} - 21456 \beta_{3} + 1220016 \beta_{2} - 347424 \beta_{1} - 1040192$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$41936 \beta_{15} - 26096 \beta_{14} + 33280 \beta_{13} + 17344 \beta_{12} - 37488 \beta_{11} + 18640 \beta_{10} + 55808 \beta_{9} + 2784 \beta_{8} + 330528 \beta_{7} - 94848 \beta_{6} + 32448 \beta_{5} - 153248 \beta_{4} - 58240 \beta_{3} - 578512 \beta_{2} + 921680 \beta_{1} - 400800$$ $$\nu^{13}$$ $$=$$ $$($$$$25248 \beta_{15} - 1057824 \beta_{14} - 101856 \beta_{13} + 1459680 \beta_{12} + 158176 \beta_{11} - 802848 \beta_{10} + 654528 \beta_{9} - 1131584 \beta_{8} - 1094272 \beta_{7} - 654688 \beta_{6} - 1595392 \beta_{5} - 1023424 \beta_{4} + 449664 \beta_{3} + 9630944 \beta_{2} + 6514976 \beta_{1} + 42598720$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$-5928288 \beta_{15} + 997920 \beta_{14} - 800736 \beta_{13} + 1137312 \beta_{12} - 277024 \beta_{11} + 726752 \beta_{10} - 2102016 \beta_{9} + 946176 \beta_{8} + 15856768 \beta_{7} + 7981024 \beta_{6} + 7811264 \beta_{5} - 1479552 \beta_{4} + 2455968 \beta_{3} + 42491648 \beta_{2} + 34407776 \beta_{1} + 29247552$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$4799232 \beta_{15} + 6359296 \beta_{14} + 4069184 \beta_{13} + 2290496 \beta_{12} + 8967296 \beta_{11} - 6544512 \beta_{10} - 20480 \beta_{9} + 2138752 \beta_{8} - 5565440 \beta_{7} + 1689408 \beta_{6} + 1307520 \beta_{5} - 7580800 \beta_{4} + 830976 \beta_{3} + 65203712 \beta_{2} + 98251520 \beta_{1} - 532596736$$

## 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
 1.95664 + 3.48878i 1.95664 − 3.48878i 3.79586 − 1.26152i 3.79586 + 1.26152i −1.06635 + 3.85524i −1.06635 − 3.85524i 3.97720 + 0.426493i 3.97720 − 0.426493i −3.04390 + 2.59512i −3.04390 − 2.59512i 2.11879 + 3.39275i 2.11879 − 3.39275i −3.55818 − 1.82740i −3.55818 + 1.82740i −1.18006 + 3.82197i −1.18006 − 3.82197i
−3.99969 0.0498899i 5.19615i 15.9950 + 0.399088i 0 −0.259236 + 20.7830i 35.2842i −63.9552 2.39422i −27.0000 0
151.2 −3.99969 + 0.0498899i 5.19615i 15.9950 0.399088i 0 −0.259236 20.7830i 35.2842i −63.9552 + 2.39422i −27.0000 0
151.3 −2.99044 2.65655i 5.19615i 1.88545 + 15.8885i 0 13.8039 15.5388i 74.3539i 36.5704 52.5224i −27.0000 0
151.4 −2.99044 + 2.65655i 5.19615i 1.88545 15.8885i 0 13.8039 + 15.5388i 74.3539i 36.5704 + 52.5224i −27.0000 0
151.5 −2.80556 2.85111i 5.19615i −0.257663 + 15.9979i 0 −14.8148 + 14.5781i 63.6232i 46.3347 44.1485i −27.0000 0
151.6 −2.80556 + 2.85111i 5.19615i −0.257663 15.9979i 0 −14.8148 14.5781i 63.6232i 46.3347 + 44.1485i −27.0000 0
151.7 −1.61924 3.65760i 5.19615i −10.7561 + 11.8451i 0 19.0055 8.41384i 95.1090i 60.7414 + 20.1614i −27.0000 0
151.8 −1.61924 + 3.65760i 5.19615i −10.7561 11.8451i 0 19.0055 + 8.41384i 95.1090i 60.7414 20.1614i −27.0000 0
151.9 −0.725488 3.93366i 5.19615i −14.9473 + 5.70765i 0 −20.4399 + 3.76975i 36.7329i 33.2960 + 54.6569i −27.0000 0
151.10 −0.725488 + 3.93366i 5.19615i −14.9473 5.70765i 0 −20.4399 3.76975i 36.7329i 33.2960 54.6569i −27.0000 0
151.11 1.87881 3.53130i 5.19615i −8.94016 13.2693i 0 18.3492 + 9.76257i 7.86734i −63.6546 + 6.63999i −27.0000 0
151.12 1.87881 + 3.53130i 5.19615i −8.94016 + 13.2693i 0 18.3492 9.76257i 7.86734i −63.6546 6.63999i −27.0000 0
151.13 3.36166 2.16777i 5.19615i 6.60152 14.5746i 0 −11.2641 17.4677i 36.6738i −9.40241 63.3056i −27.0000 0
151.14 3.36166 + 2.16777i 5.19615i 6.60152 + 14.5746i 0 −11.2641 + 17.4677i 36.6738i −9.40241 + 63.3056i −27.0000 0
151.15 3.89995 0.889027i 5.19615i 14.4193 6.93433i 0 4.61952 + 20.2647i 44.0991i 50.0696 39.8627i −27.0000 0
151.16 3.89995 + 0.889027i 5.19615i 14.4193 + 6.93433i 0 4.61952 20.2647i 44.0991i 50.0696 + 39.8627i −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.b 16
4.b odd 2 1 inner 300.5.c.b 16
5.b even 2 1 300.5.c.c yes 16
5.c odd 4 2 300.5.f.c 32
20.d odd 2 1 300.5.c.c yes 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 1.a even 1 1 trivial
300.5.c.b 16 4.b odd 2 1 inner
300.5.c.c yes 16 5.b even 2 1
300.5.c.c yes 16 20.d odd 2 1
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

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