Properties

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

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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: \(x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + 25296 x^{8} - 6656 x^{7} - 110848 x^{6} - 227328 x^{5} + 1077248 x^{4} + 589824 x^{3} - 2359296 x^{2} - 8388608 x + 16777216\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{2} + \beta_{8} ) q^{4} + ( 1 + \beta_{1} + \beta_{9} ) q^{6} + ( 1 + 3 \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{7} + ( 12 + 5 \beta_{1} + 2 \beta_{2} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{8} -27 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{1} q^{3} + ( 2 + \beta_{2} + \beta_{8} ) q^{4} + ( 1 + \beta_{1} + \beta_{9} ) q^{6} + ( 1 + 3 \beta_{2} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{7} + ( 12 + 5 \beta_{1} + 2 \beta_{2} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{8} -27 q^{9} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{11} + ( 3 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{12} + ( 17 + \beta_{1} - 9 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} - 11 \beta_{8} + 3 \beta_{10} - 5 \beta_{11} - 3 \beta_{12} + 5 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{13} + ( -47 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + 7 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} + \beta_{14} - \beta_{15} ) q^{14} + ( -10 - 5 \beta_{1} + 14 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} + 3 \beta_{11} - 4 \beta_{12} + \beta_{13} - \beta_{14} ) q^{16} + ( 5 + 3 \beta_{1} + 20 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 6 \beta_{14} + 2 \beta_{15} ) q^{17} + ( -27 - 27 \beta_{2} ) q^{18} + ( -7 - 26 \beta_{1} - 7 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} - 9 \beta_{12} + \beta_{14} ) q^{19} + ( 22 + 2 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 5 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + 6 \beta_{11} + 3 \beta_{12} - 7 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{21} + ( -24 + 16 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + \beta_{4} - 2 \beta_{5} + 11 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} + 5 \beta_{15} ) q^{22} + ( 2 - 17 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} - \beta_{5} + 6 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} + 10 \beta_{9} + 5 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{23} + ( -119 + 13 \beta_{1} + 9 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 9 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 4 \beta_{12} - 5 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} ) q^{24} + ( -55 + 10 \beta_{2} - 16 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 20 \beta_{6} - 13 \beta_{7} - 30 \beta_{8} + \beta_{9} + 14 \beta_{10} - 16 \beta_{11} - 5 \beta_{12} + 17 \beta_{13} - 7 \beta_{15} ) q^{26} -27 \beta_{1} q^{27} + ( 72 + \beta_{1} - 45 \beta_{2} + 16 \beta_{3} + 2 \beta_{4} - \beta_{5} - 9 \beta_{6} + 7 \beta_{7} + 29 \beta_{8} - 2 \beta_{9} - 15 \beta_{10} + 18 \beta_{11} + 9 \beta_{12} - 26 \beta_{13} - 7 \beta_{14} + 4 \beta_{15} ) q^{28} + ( -216 + 16 \beta_{1} - 42 \beta_{2} + 8 \beta_{3} + 10 \beta_{4} - 12 \beta_{5} + 10 \beta_{6} + 18 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 19 \beta_{13} + 2 \beta_{14} + 24 \beta_{15} ) q^{29} + ( -24 + 41 \beta_{1} - 61 \beta_{2} - 6 \beta_{3} + 11 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 11 \beta_{8} + 8 \beta_{9} + 3 \beta_{10} - 4 \beta_{11} - 8 \beta_{12} + 2 \beta_{14} ) q^{31} + ( -475 - 51 \beta_{1} + 11 \beta_{2} + 16 \beta_{3} + 11 \beta_{4} + 2 \beta_{5} - 27 \beta_{6} - 3 \beta_{7} + 33 \beta_{8} - 3 \beta_{9} - 6 \beta_{10} + 13 \beta_{11} - 7 \beta_{12} - 9 \beta_{13} + 4 \beta_{14} + 11 \beta_{15} ) q^{32} + ( -1 - 9 \beta_{1} - 9 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 23 \beta_{6} + 3 \beta_{7} + 19 \beta_{8} + 5 \beta_{9} - 5 \beta_{10} + 13 \beta_{11} - 2 \beta_{12} - 5 \beta_{13} + 6 \beta_{14} + 4 \beta_{15} ) q^{33} + ( 303 - 68 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} + 14 \beta_{4} - 12 \beta_{5} + 14 \beta_{6} + 9 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} + 10 \beta_{10} - 5 \beta_{12} - 3 \beta_{13} + 20 \beta_{14} + 17 \beta_{15} ) q^{34} + ( -54 - 27 \beta_{2} - 27 \beta_{8} ) q^{36} + ( -539 - 3 \beta_{1} + 157 \beta_{2} + 7 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} - \beta_{6} + 14 \beta_{7} + 33 \beta_{8} - 12 \beta_{9} - 9 \beta_{10} + \beta_{11} + 11 \beta_{12} - 33 \beta_{13} + 11 \beta_{14} - 2 \beta_{15} ) q^{37} + ( 63 - 61 \beta_{1} - 26 \beta_{2} - 12 \beta_{3} - 23 \beta_{4} + \beta_{5} + 8 \beta_{6} - 14 \beta_{7} - 39 \beta_{8} - 27 \beta_{9} - \beta_{10} - 20 \beta_{11} - 29 \beta_{12} + 6 \beta_{13} + 9 \beta_{14} + 2 \beta_{15} ) q^{38} + ( 7 + 18 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + 17 \beta_{6} - 7 \beta_{7} - 6 \beta_{8} - 12 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{12} + 6 \beta_{14} ) q^{39} + ( 140 - 30 \beta_{1} + 210 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} + 16 \beta_{5} - 2 \beta_{6} - 24 \beta_{7} - 10 \beta_{8} - 32 \beta_{9} - 2 \beta_{10} - 10 \beta_{11} + 18 \beta_{12} + 16 \beta_{13} + 6 \beta_{14} + 12 \beta_{15} ) q^{41} + ( 21 - 42 \beta_{1} + 18 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} + 10 \beta_{5} - 14 \beta_{6} + 19 \beta_{7} + 24 \beta_{8} + 7 \beta_{9} + 8 \beta_{11} - \beta_{12} - 33 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} ) q^{42} + ( -12 + 64 \beta_{1} - 136 \beta_{2} + 4 \beta_{3} - 12 \beta_{4} + 10 \beta_{5} + 16 \beta_{6} + 2 \beta_{7} + 18 \beta_{8} - 12 \beta_{9} - 20 \beta_{10} + 2 \beta_{11} + 22 \beta_{12} + 12 \beta_{14} ) q^{43} + ( -416 + 3 \beta_{1} - 19 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + 9 \beta_{5} + 57 \beta_{6} + 9 \beta_{7} - 41 \beta_{8} - 14 \beta_{9} + 23 \beta_{10} - 22 \beta_{11} - 37 \beta_{12} - 34 \beta_{13} + 23 \beta_{14} ) q^{44} + ( -79 + 79 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} - 34 \beta_{6} - 12 \beta_{7} + 21 \beta_{8} - 9 \beta_{9} + 7 \beta_{10} + 20 \beta_{11} - 21 \beta_{12} + 6 \beta_{13} - 3 \beta_{14} - 10 \beta_{15} ) q^{46} + ( -34 - 119 \beta_{1} - 197 \beta_{2} + 4 \beta_{3} + 29 \beta_{4} + 17 \beta_{5} + 16 \beta_{6} - 23 \beta_{7} + 24 \beta_{8} - 16 \beta_{9} + 5 \beta_{10} + 3 \beta_{11} + 11 \beta_{12} + 20 \beta_{14} ) q^{47} + ( 227 - 11 \beta_{1} - 129 \beta_{2} - 6 \beta_{3} + 19 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} + 17 \beta_{7} + 3 \beta_{8} + \beta_{9} + 7 \beta_{11} - 5 \beta_{12} + 9 \beta_{13} + 12 \beta_{14} + 3 \beta_{15} ) q^{48} + ( -251 + 12 \beta_{1} - 96 \beta_{2} + 44 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} - 40 \beta_{6} + 60 \beta_{7} + 108 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} + 24 \beta_{11} - 116 \beta_{13} + 24 \beta_{14} + 24 \beta_{15} ) q^{49} + ( -20 - 25 \beta_{1} - 135 \beta_{2} + 18 \beta_{3} + 25 \beta_{4} + 11 \beta_{5} - 7 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 9 \beta_{9} - 2 \beta_{11} + 43 \beta_{12} + 9 \beta_{14} ) q^{51} + ( -792 + 55 \beta_{1} - 41 \beta_{2} - 48 \beta_{3} - 18 \beta_{4} - 39 \beta_{5} + 81 \beta_{6} - 27 \beta_{7} - 35 \beta_{8} - 2 \beta_{9} + 55 \beta_{10} - 34 \beta_{11} + 3 \beta_{12} + 22 \beta_{13} - 9 \beta_{14} - 16 \beta_{15} ) q^{52} + ( 326 + 30 \beta_{1} - 114 \beta_{2} + 14 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} + 34 \beta_{6} + 48 \beta_{7} + 86 \beta_{8} - 68 \beta_{9} + 34 \beta_{10} - 34 \beta_{11} + 30 \beta_{12} - 103 \beta_{13} - 18 \beta_{14} + 12 \beta_{15} ) q^{53} + ( -27 - 27 \beta_{1} - 27 \beta_{9} ) q^{54} + ( -138 - 130 \beta_{1} + 96 \beta_{2} + 42 \beta_{3} + 8 \beta_{4} + 14 \beta_{5} - 4 \beta_{6} + 64 \beta_{7} + 12 \beta_{8} + 20 \beta_{9} - 14 \beta_{10} + 36 \beta_{11} + 12 \beta_{12} - 88 \beta_{13} - 44 \beta_{14} + 10 \beta_{15} ) q^{56} + ( 732 - 8 \beta_{1} + 69 \beta_{2} + 3 \beta_{3} + 17 \beta_{4} + \beta_{5} - 5 \beta_{6} - 2 \beta_{7} - 18 \beta_{8} - 11 \beta_{9} + 6 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + 30 \beta_{13} + 18 \beta_{14} + 42 \beta_{15} ) q^{57} + ( -838 + 47 \beta_{1} - 168 \beta_{2} - 24 \beta_{3} + 13 \beta_{4} - 7 \beta_{5} + 76 \beta_{6} + 91 \beta_{7} - 97 \beta_{8} - 44 \beta_{9} + 33 \beta_{10} - 32 \beta_{12} + \beta_{13} + 49 \beta_{14} + 9 \beta_{15} ) q^{58} + ( 37 - 81 \beta_{1} + 6 \beta_{2} - 34 \beta_{3} - 65 \beta_{4} - 18 \beta_{5} + 39 \beta_{6} + 4 \beta_{7} + 76 \beta_{8} - 36 \beta_{9} - 34 \beta_{10} + 22 \beta_{11} - 14 \beta_{12} + 4 \beta_{14} ) q^{59} + ( -216 - 30 \beta_{1} + 166 \beta_{2} - 30 \beta_{3} - 28 \beta_{4} + 16 \beta_{5} + 10 \beta_{6} - 28 \beta_{7} + 42 \beta_{8} - 44 \beta_{9} - 2 \beta_{10} - 50 \beta_{11} + 14 \beta_{12} + 82 \beta_{13} - 10 \beta_{14} - 92 \beta_{15} ) q^{61} + ( 953 + 145 \beta_{1} - 102 \beta_{2} - 28 \beta_{3} - 37 \beta_{4} - 13 \beta_{5} - 24 \beta_{6} + 2 \beta_{7} - 85 \beta_{8} + 59 \beta_{9} + 13 \beta_{10} - 20 \beta_{11} - 27 \beta_{12} - 10 \beta_{13} + 3 \beta_{14} + 18 \beta_{15} ) q^{62} + ( -27 - 81 \beta_{2} - 27 \beta_{6} + 27 \beta_{8} - 27 \beta_{10} ) q^{63} + ( -804 - 75 \beta_{1} - 414 \beta_{2} + 49 \beta_{3} - 9 \beta_{4} + 10 \beta_{5} - 26 \beta_{6} + 62 \beta_{7} + 40 \beta_{8} - 55 \beta_{9} - 42 \beta_{10} + 39 \beta_{11} + 22 \beta_{12} - 7 \beta_{13} + 9 \beta_{14} + 64 \beta_{15} ) q^{64} + ( -138 - 73 \beta_{1} + 48 \beta_{2} + 76 \beta_{3} + 23 \beta_{4} + 13 \beta_{5} - 98 \beta_{6} + 21 \beta_{7} + 79 \beta_{8} - 6 \beta_{9} - 71 \beta_{10} + 64 \beta_{11} + 40 \beta_{12} + \beta_{13} + 9 \beta_{14} + 37 \beta_{15} ) q^{66} + ( -28 + 268 \beta_{1} + 44 \beta_{2} - 28 \beta_{3} - 32 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 26 \beta_{7} - 44 \beta_{8} - 102 \beta_{9} - 42 \beta_{10} + 6 \beta_{11} - 88 \beta_{12} + 2 \beta_{14} ) q^{67} + ( 892 - 245 \beta_{1} + 271 \beta_{2} - 12 \beta_{3} + 14 \beta_{4} - 43 \beta_{5} + 57 \beta_{6} + 77 \beta_{7} - 43 \beta_{8} - 98 \beta_{9} + 27 \beta_{10} + 46 \beta_{11} - 5 \beta_{12} - 10 \beta_{13} + 47 \beta_{14} + 52 \beta_{15} ) q^{68} + ( 564 + 50 \beta_{1} - 180 \beta_{2} - 9 \beta_{3} - 8 \beta_{4} - 37 \beta_{5} + 56 \beta_{6} + 29 \beta_{7} - 7 \beta_{9} + 24 \beta_{10} - 32 \beta_{11} - 5 \beta_{12} - 15 \beta_{14} - 18 \beta_{15} ) q^{69} + ( 34 - 14 \beta_{1} + 204 \beta_{2} + 8 \beta_{3} - 34 \beta_{4} - 22 \beta_{5} - 70 \beta_{6} + 38 \beta_{7} + 34 \beta_{8} + 192 \beta_{9} - 30 \beta_{11} + 74 \beta_{12} - 52 \beta_{14} ) q^{71} + ( -324 - 135 \beta_{1} - 54 \beta_{2} - 27 \beta_{8} - 27 \beta_{11} + 27 \beta_{12} + 27 \beta_{13} - 27 \beta_{15} ) q^{72} + ( -620 - 22 \beta_{1} - 42 \beta_{2} + 82 \beta_{3} + 52 \beta_{4} - 78 \beta_{6} + 76 \beta_{7} + 194 \beta_{8} - 92 \beta_{9} + 22 \beta_{10} + 38 \beta_{11} + 54 \beta_{12} - 94 \beta_{13} + 14 \beta_{14} + 116 \beta_{15} ) q^{73} + ( 1777 - 2 \beta_{1} - 638 \beta_{2} + 8 \beta_{3} + 14 \beta_{4} - 18 \beta_{5} - 8 \beta_{6} + 41 \beta_{7} + 200 \beta_{8} - 23 \beta_{9} - 56 \beta_{10} + 32 \beta_{11} - \beta_{12} - 97 \beta_{13} + 2 \beta_{14} + 15 \beta_{15} ) q^{74} + ( -368 - 14 \beta_{1} + 110 \beta_{2} - 68 \beta_{3} - 64 \beta_{4} - 30 \beta_{5} + 98 \beta_{6} - 118 \beta_{7} - 222 \beta_{8} - 104 \beta_{9} - 18 \beta_{10} - 80 \beta_{11} - 138 \beta_{12} + 248 \beta_{13} + 34 \beta_{14} - 64 \beta_{15} ) q^{76} + ( 1556 + 80 \beta_{1} - 530 \beta_{2} + 24 \beta_{3} - 30 \beta_{4} - 20 \beta_{5} + 18 \beta_{6} + 18 \beta_{7} - 84 \beta_{8} + 156 \beta_{9} - 40 \beta_{10} + 82 \beta_{11} - 30 \beta_{12} - 104 \beta_{13} - 86 \beta_{14} - 8 \beta_{15} ) q^{77} + ( 249 - 49 \beta_{1} - 9 \beta_{2} + 50 \beta_{3} - 4 \beta_{4} + 13 \beta_{5} + 31 \beta_{6} - 10 \beta_{7} + 11 \beta_{8} - \beta_{9} - 13 \beta_{10} + 30 \beta_{11} - 79 \beta_{13} + 9 \beta_{14} - 7 \beta_{15} ) q^{78} + ( -114 - 195 \beta_{1} - 591 \beta_{2} - 22 \beta_{3} - \beta_{4} + 56 \beta_{5} + 60 \beta_{6} + 114 \beta_{7} - 105 \beta_{8} - 42 \beta_{9} - 21 \beta_{10} - 16 \beta_{11} + 98 \beta_{12} + 40 \beta_{14} ) q^{79} + 729 q^{81} + ( 3392 + 498 \beta_{1} + 46 \beta_{2} - 48 \beta_{3} - 90 \beta_{4} + 22 \beta_{5} + 64 \beta_{6} - 40 \beta_{7} + 70 \beta_{8} - 14 \beta_{9} + 26 \beta_{10} - 48 \beta_{11} - 2 \beta_{12} + 216 \beta_{13} - 18 \beta_{14} - 88 \beta_{15} ) q^{82} + ( -88 + 120 \beta_{1} - 244 \beta_{2} + 16 \beta_{3} - 4 \beta_{4} + 94 \beta_{5} + 94 \beta_{6} - 40 \beta_{7} - 34 \beta_{8} + 170 \beta_{9} + 70 \beta_{10} - 84 \beta_{11} - 2 \beta_{12} + 10 \beta_{14} ) q^{83} + ( -136 + 9 \beta_{1} + 39 \beta_{2} + 20 \beta_{3} + 2 \beta_{4} + 31 \beta_{5} - 101 \beta_{6} + 23 \beta_{7} + 89 \beta_{8} - 62 \beta_{9} - 79 \beta_{10} + 18 \beta_{11} - 15 \beta_{12} - 70 \beta_{13} + 21 \beta_{14} - 4 \beta_{15} ) q^{84} + ( 2428 + 184 \beta_{1} - 116 \beta_{2} + 56 \beta_{3} + 48 \beta_{4} - 4 \beta_{5} + 36 \beta_{6} - 28 \beta_{7} - 156 \beta_{8} + 52 \beta_{9} + 4 \beta_{10} + 48 \beta_{11} - 44 \beta_{12} - 8 \beta_{13} - 12 \beta_{14} - 16 \beta_{15} ) q^{86} + ( 35 - 267 \beta_{1} - 36 \beta_{2} + 72 \beta_{3} - \beta_{4} - 17 \beta_{5} - 65 \beta_{6} + 61 \beta_{7} - 81 \beta_{8} - 78 \beta_{9} - 78 \beta_{10} + 23 \beta_{11} + 113 \beta_{12} + 6 \beta_{14} ) q^{87} + ( 266 - 290 \beta_{1} - 544 \beta_{2} - 22 \beta_{3} - 44 \beta_{4} - 46 \beta_{5} + 52 \beta_{6} - 64 \beta_{7} - 84 \beta_{8} - 96 \beta_{9} - 50 \beta_{10} - 24 \beta_{11} - 172 \beta_{12} - 132 \beta_{13} + 104 \beta_{14} - 42 \beta_{15} ) q^{88} + ( 236 - 82 \beta_{1} - 634 \beta_{2} - 102 \beta_{3} - 64 \beta_{4} - 8 \beta_{5} - 54 \beta_{6} - 168 \beta_{7} - 286 \beta_{8} + 16 \beta_{9} + 98 \beta_{10} - 70 \beta_{11} - 130 \beta_{12} + 130 \beta_{13} + 82 \beta_{14} + 20 \beta_{15} ) q^{89} + ( 186 + 1260 \beta_{1} + 710 \beta_{2} - 56 \beta_{3} - 72 \beta_{4} - 44 \beta_{5} + 90 \beta_{6} - 92 \beta_{7} + 166 \beta_{8} + 268 \beta_{9} + 146 \beta_{10} - 20 \beta_{11} - 48 \beta_{12} - 64 \beta_{14} ) q^{91} + ( -1740 - 370 \beta_{1} + 42 \beta_{2} + 12 \beta_{3} - 60 \beta_{4} + 14 \beta_{5} - 134 \beta_{6} + 10 \beta_{7} + 162 \beta_{8} + 188 \beta_{9} - 78 \beta_{10} + 88 \beta_{11} + 114 \beta_{12} + 8 \beta_{13} - 66 \beta_{14} + 48 \beta_{15} ) q^{92} + ( -1289 + 15 \beta_{1} - 231 \beta_{2} - 37 \beta_{3} - 2 \beta_{4} - 40 \beta_{5} + 59 \beta_{6} - 30 \beta_{7} - 109 \beta_{8} - 50 \beta_{9} + 65 \beta_{10} - 55 \beta_{11} - 7 \beta_{12} + 14 \beta_{13} - 3 \beta_{14} + 62 \beta_{15} ) q^{93} + ( 3127 - 99 \beta_{1} - 242 \beta_{2} + 68 \beta_{3} - 49 \beta_{4} - \beta_{5} + 16 \beta_{6} + 52 \beta_{7} - 129 \beta_{8} - 95 \beta_{9} + \beta_{10} + 64 \beta_{11} + 63 \beta_{12} - 208 \beta_{13} - 13 \beta_{14} + 40 \beta_{15} ) q^{94} + ( 1206 - 481 \beta_{1} + 108 \beta_{2} - 27 \beta_{3} + 33 \beta_{4} - 6 \beta_{5} - 36 \beta_{6} + 36 \beta_{7} - 114 \beta_{8} + 15 \beta_{9} + 30 \beta_{10} + 21 \beta_{11} + 12 \beta_{12} - 45 \beta_{13} + 33 \beta_{14} + 54 \beta_{15} ) q^{96} + ( 726 - 60 \beta_{1} - 358 \beta_{2} - 92 \beta_{3} - 94 \beta_{4} + 4 \beta_{5} - 18 \beta_{6} - 110 \beta_{7} - 72 \beta_{8} - 20 \beta_{9} + 52 \beta_{10} - 106 \beta_{11} - 74 \beta_{12} + 280 \beta_{13} + 22 \beta_{14} - 160 \beta_{15} ) q^{97} + ( -1395 - 24 \beta_{1} - 23 \beta_{2} + 144 \beta_{3} + 96 \beta_{4} - 40 \beta_{5} - 136 \beta_{6} + 228 \beta_{7} + 192 \beta_{8} - 140 \beta_{9} - 288 \beta_{10} + 224 \beta_{11} + 52 \beta_{12} - 236 \beta_{13} + 88 \beta_{14} + 132 \beta_{15} ) q^{98} + ( -27 + 27 \beta_{1} - 54 \beta_{2} - 54 \beta_{3} + 27 \beta_{4} + 27 \beta_{6} + 54 \beta_{7} - 54 \beta_{8} + 54 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{2} + 26q^{4} + 18q^{6} + 180q^{8} - 432q^{9} + O(q^{10}) \) \( 16q + 12q^{2} + 26q^{4} + 18q^{6} + 180q^{8} - 432q^{9} + 352q^{13} - 804q^{14} - 190q^{16} - 324q^{18} + 288q^{21} - 436q^{22} - 1998q^{24} - 852q^{26} + 1156q^{28} - 3456q^{29} - 7668q^{32} + 4772q^{34} - 702q^{36} - 9376q^{37} + 1320q^{38} + 1248q^{41} + 324q^{42} - 6420q^{44} - 1112q^{46} + 4176q^{48} - 3952q^{49} - 12704q^{52} + 5184q^{53} - 486q^{54} - 2604q^{56} + 11232q^{57} - 12708q^{58} - 3808q^{61} + 16152q^{62} - 11902q^{64} - 2916q^{66} + 12312q^{68} + 9792q^{69} - 4860q^{72} - 11040q^{73} + 30516q^{74} - 5160q^{76} + 27456q^{77} + 3600q^{78} + 11664q^{81} + 54040q^{82} - 2052q^{84} + 39768q^{86} + 7220q^{88} + 7584q^{89} - 28848q^{92} - 19872q^{93} + 49776q^{94} + 18882q^{96} + 14496q^{97} - 23940q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 9 x^{14} + 18 x^{13} + 263 x^{12} - 444 x^{11} - 1732 x^{10} - 832 x^{9} + 25296 x^{8} - 6656 x^{7} - 110848 x^{6} - 227328 x^{5} + 1077248 x^{4} + 589824 x^{3} - 2359296 x^{2} - 8388608 x + 16777216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(6617 \nu^{15} - 26278 \nu^{14} - 88297 \nu^{13} - 60460 \nu^{12} + 2080027 \nu^{11} - 578762 \nu^{10} - 12589276 \nu^{9} - 31545336 \nu^{8} + 152490000 \nu^{7} + 165011552 \nu^{6} - 471993856 \nu^{5} - 2816759296 \nu^{4} + 3647557632 \nu^{3} + 10337263616 \nu^{2} + 3917545472 \nu - 64036536320\)\()/ 2575826944 \)
\(\beta_{2}\)\(=\)\((\)\(37951 \nu^{15} - 50364 \nu^{14} - 476215 \nu^{13} - 676818 \nu^{12} + 8411705 \nu^{11} + 6300796 \nu^{10} - 49013820 \nu^{9} - 178779072 \nu^{8} + 490096944 \nu^{7} + 1176020480 \nu^{6} - 1041215232 \nu^{5} - 12046526464 \nu^{4} + 6948376576 \nu^{3} + 44963463168 \nu^{2} + 46226472960 \nu - 214580068352\)\()/ 7727480832 \)
\(\beta_{3}\)\(=\)\((\)\(29379 \nu^{15} - 383840 \nu^{14} + 72085 \nu^{13} + 4548650 \nu^{12} + 9905741 \nu^{11} - 58901568 \nu^{10} - 108148156 \nu^{9} + 342876944 \nu^{8} + 1512868080 \nu^{7} - 2116764224 \nu^{6} - 11278901504 \nu^{5} + 1819569152 \nu^{4} + 80125595648 \nu^{3} + 52997210112 \nu^{2} - 267076632576 \nu - 341751889920\)\()/ 7727480832 \)
\(\beta_{4}\)\(=\)\((\)\(51995 \nu^{15} + 135890 \nu^{14} - 460923 \nu^{13} - 3279624 \nu^{12} + 4859177 \nu^{11} + 30696894 \nu^{10} + 11125468 \nu^{9} - 311872888 \nu^{8} - 175212880 \nu^{7} + 1830949984 \nu^{6} + 4270099456 \nu^{5} - 14716093952 \nu^{4} - 18887659520 \nu^{3} - 7608918016 \nu^{2} + 184980930560 \nu - 71406452736\)\()/ 7727480832 \)
\(\beta_{5}\)\(=\)\((\)\(-57031 \nu^{15} - 43890 \nu^{14} + 338471 \nu^{13} + 2457920 \nu^{12} - 6417117 \nu^{11} - 20562398 \nu^{10} - 4124364 \nu^{9} + 261849912 \nu^{8} - 59670000 \nu^{7} - 1486005600 \nu^{6} - 3747883520 \nu^{5} + 11246310912 \nu^{4} + 16530591744 \nu^{3} - 20047634432 \nu^{2} - 135674003456 \nu + 21705523200\)\()/ 3863740416 \)
\(\beta_{6}\)\(=\)\((\)\(18263 \nu^{15} - 54973 \nu^{14} - 350339 \nu^{13} + 167015 \nu^{12} + 5995975 \nu^{11} - 538875 \nu^{10} - 52716712 \nu^{9} - 73912668 \nu^{8} + 468398416 \nu^{7} + 729702192 \nu^{6} - 2317975680 \nu^{5} - 8375254784 \nu^{4} + 12463949824 \nu^{3} + 43117391872 \nu^{2} - 12147032064 \nu - 215238705152\)\()/ 965935104 \)
\(\beta_{7}\)\(=\)\((\)\(-39201 \nu^{15} + 33015 \nu^{14} + 585765 \nu^{13} + 1230339 \nu^{12} - 10050521 \nu^{11} - 11545103 \nu^{10} + 55733320 \nu^{9} + 263598804 \nu^{8} - 588040880 \nu^{7} - 1636635280 \nu^{6} + 436851072 \nu^{5} + 17791716608 \nu^{4} - 8723943424 \nu^{3} - 62583123968 \nu^{2} - 87854055424 \nu + 374335602688\)\()/ 1931870208 \)
\(\beta_{8}\)\(=\)\((\)\(158297 \nu^{15} - 321188 \nu^{14} - 2321441 \nu^{13} - 1304702 \nu^{12} + 40098543 \nu^{11} + 18285284 \nu^{10} - 287175460 \nu^{9} - 710621760 \nu^{8} + 2661758544 \nu^{7} + 5196978688 \nu^{6} - 8932489472 \nu^{5} - 56814786560 \nu^{4} + 50861223936 \nu^{3} + 239692152832 \nu^{2} + 101691949056 \nu - 1177881673728\)\()/ 7727480832 \)
\(\beta_{9}\)\(=\)\((\)\(-28268 \nu^{15} + 40905 \nu^{14} + 260496 \nu^{13} + 675047 \nu^{12} - 5667738 \nu^{11} - 3456561 \nu^{10} + 24303468 \nu^{9} + 137186940 \nu^{8} - 318838944 \nu^{7} - 662545328 \nu^{6} + 86605440 \nu^{5} + 8489875200 \nu^{4} - 4521709568 \nu^{3} - 24414425088 \nu^{2} - 39889698816 \nu + 150021603328\)\()/ 1287913472 \)
\(\beta_{10}\)\(=\)\((\)\(88787 \nu^{15} - 422658 \nu^{14} - 1045059 \nu^{13} + 934396 \nu^{12} + 24980777 \nu^{11} - 21192526 \nu^{10} - 152976308 \nu^{9} - 292339752 \nu^{8} + 1922873264 \nu^{7} + 1317499424 \nu^{6} - 6319023616 \nu^{5} - 29438280192 \nu^{4} + 51661000704 \nu^{3} + 101470560256 \nu^{2} + 27957198848 \nu - 684009455616\)\()/ 3863740416 \)
\(\beta_{11}\)\(=\)\((\)\(-66849 \nu^{15} - 37184 \nu^{14} + 713289 \nu^{13} + 2048850 \nu^{12} - 9185855 \nu^{11} - 21292032 \nu^{10} + 41846436 \nu^{9} + 285719952 \nu^{8} - 349792272 \nu^{7} - 1759529280 \nu^{6} - 573712896 \nu^{5} + 14194457600 \nu^{4} + 3961020416 \nu^{3} - 44740165632 \nu^{2} - 91454242816 \nu + 163918249984\)\()/ 1931870208 \)
\(\beta_{12}\)\(=\)\((\)\(39828 \nu^{15} - 39001 \nu^{14} - 561192 \nu^{13} - 949223 \nu^{12} + 8771394 \nu^{11} + 10298721 \nu^{10} - 50310748 \nu^{9} - 213246844 \nu^{8} + 474350048 \nu^{7} + 1429650608 \nu^{6} - 694801792 \nu^{5} - 13919221504 \nu^{4} + 4471156736 \nu^{3} + 52419858432 \nu^{2} + 56009785344 \nu - 246523756544\)\()/ 965935104 \)
\(\beta_{13}\)\(=\)\((\)\(9755 \nu^{15} - 13120 \nu^{14} - 127395 \nu^{13} - 144550 \nu^{12} + 2087125 \nu^{11} + 1558880 \nu^{10} - 13170940 \nu^{9} - 40784240 \nu^{8} + 118379760 \nu^{7} + 283405760 \nu^{6} - 317326080 \nu^{5} - 2821985280 \nu^{4} + 1776168960 \nu^{3} + 11276861440 \nu^{2} + 7274496000 \nu - 48431104000\)\()/ 227278848 \)
\(\beta_{14}\)\(=\)\((\)\(-359623 \nu^{15} + 138358 \nu^{14} + 4425511 \nu^{13} + 9142168 \nu^{12} - 68656173 \nu^{11} - 87846726 \nu^{10} + 371186484 \nu^{9} + 1681049624 \nu^{8} - 3426388080 \nu^{7} - 10656051424 \nu^{6} + 4253151232 \nu^{5} + 99965980160 \nu^{4} - 38297874432 \nu^{3} - 324489093120 \nu^{2} - 337664081920 \nu + 1531047313408\)\()/ 7727480832 \)
\(\beta_{15}\)\(=\)\((\)\(218390 \nu^{15} - 146617 \nu^{14} - 3190762 \nu^{13} - 5199211 \nu^{12} + 50791088 \nu^{11} + 55186273 \nu^{10} - 301642548 \nu^{9} - 1199276764 \nu^{8} + 2825156288 \nu^{7} + 7751164208 \nu^{6} - 4732092544 \nu^{5} - 79129202432 \nu^{4} + 34983653376 \nu^{3} + 290853163008 \nu^{2} + 312699027456 \nu - 1460390592512\)\()/ 3863740416 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} - 9 \beta_{12} - 6 \beta_{11} + \beta_{10} + 6 \beta_{9} - 11 \beta_{8} - 11 \beta_{7} - \beta_{6} + 5 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 87 \beta_{2} + 9 \beta_{1} + 74\)\()/240\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} + 9 \beta_{14} + 13 \beta_{13} + 4 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} - 7 \beta_{9} - 17 \beta_{8} - 6 \beta_{7} + \beta_{6} + \beta_{5} - 10 \beta_{4} + 4 \beta_{3} - 39 \beta_{2} - \beta_{1} + 251\)\()/120\)
\(\nu^{3}\)\(=\)\((\)\(-16 \beta_{15} - 11 \beta_{14} + 24 \beta_{13} - 9 \beta_{12} + 6 \beta_{11} - \beta_{10} + 10 \beta_{9} - 11 \beta_{8} - 45 \beta_{7} - 11 \beta_{6} + 13 \beta_{5} + 10 \beta_{3} + 5 \beta_{2} - 65 \beta_{1} + 582\)\()/80\)
\(\nu^{4}\)\(=\)\((\)\(-118 \beta_{15} - 69 \beta_{14} + 146 \beta_{13} - 217 \beta_{12} - 70 \beta_{11} + 29 \beta_{10} + 164 \beta_{9} - 217 \beta_{8} - 273 \beta_{7} - 13 \beta_{6} + 11 \beta_{5} - 320 \beta_{4} - 118 \beta_{3} + 1683 \beta_{2} + 117 \beta_{1} - 4900\)\()/240\)
\(\nu^{5}\)\(=\)\((\)\(-9 \beta_{15} - 18 \beta_{14} + 39 \beta_{13} - 49 \beta_{12} - 4 \beta_{11} + 36 \beta_{10} + 21 \beta_{9} - 71 \beta_{7} + 22 \beta_{6} - 62 \beta_{5} - 52 \beta_{4} - 30 \beta_{3} - 498 \beta_{2} - 136 \beta_{1} + 227\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-420 \beta_{15} - 315 \beta_{14} - 140 \beta_{13} - 101 \beta_{12} + 54 \beta_{11} + 215 \beta_{10} + 390 \beta_{9} - 723 \beta_{8} - 329 \beta_{7} + 461 \beta_{6} - 259 \beta_{5} - 136 \beta_{4} - 270 \beta_{3} + 5085 \beta_{2} - 1049 \beta_{1} - 4246\)\()/80\)
\(\nu^{7}\)\(=\)\((\)\(-222 \beta_{15} - 2811 \beta_{14} - 8790 \beta_{13} + 61 \beta_{12} - 2906 \beta_{11} + 2571 \beta_{10} + 5776 \beta_{9} + 6369 \beta_{8} + 749 \beta_{7} + 509 \beta_{6} - 4555 \beta_{5} - 4496 \beta_{4} - 1122 \beta_{3} + 7197 \beta_{2} + 11939 \beta_{1} + 14624\)\()/240\)
\(\nu^{8}\)\(=\)\((\)\(457 \beta_{15} + 513 \beta_{14} + 481 \beta_{13} + 1078 \beta_{12} + 406 \beta_{11} + 505 \beta_{10} + 8561 \beta_{9} + 511 \beta_{8} - 312 \beta_{7} + 97 \beta_{6} - 6203 \beta_{5} + 4940 \beta_{4} + 2668 \beta_{3} - 18663 \beta_{2} + 23893 \beta_{1} - 59353\)\()/120\)
\(\nu^{9}\)\(=\)\((\)\(5696 \beta_{15} - 5839 \beta_{14} - 22904 \beta_{13} + 23899 \beta_{12} + 6654 \beta_{11} - 989 \beta_{10} + 11530 \beta_{9} - 2479 \beta_{8} + 16575 \beta_{7} - 8199 \beta_{6} - 23 \beta_{5} + 3880 \beta_{4} + 15490 \beta_{3} + 70985 \beta_{2} + 7075 \beta_{1} - 102602\)\()/80\)
\(\nu^{10}\)\(=\)\((\)\(3642 \beta_{15} - 13365 \beta_{14} - 25662 \beta_{13} + 22519 \beta_{12} + 15418 \beta_{11} - 16707 \beta_{10} + 23652 \beta_{9} + 98583 \beta_{8} + 7727 \beta_{7} - 51373 \beta_{6} + 3611 \beta_{5} + 9568 \beta_{4} + 32106 \beta_{3} - 71757 \beta_{2} + 42645 \beta_{1} - 79748\)\()/48\)
\(\nu^{11}\)\(=\)\((\)\(147503 \beta_{15} - 29118 \beta_{14} + 39215 \beta_{13} + 37203 \beta_{12} + 216180 \beta_{11} - 114160 \beta_{10} + 99669 \beta_{9} + 30212 \beta_{8} + 141341 \beta_{7} - 199958 \beta_{6} - 19418 \beta_{5} + 78236 \beta_{4} + 168842 \beta_{3} + 643458 \beta_{2} + 310116 \beta_{1} - 4252301\)\()/120\)
\(\nu^{12}\)\(=\)\((\)\(263580 \beta_{15} - 90575 \beta_{14} - 472940 \beta_{13} + 342399 \beta_{12} + 305054 \beta_{11} - 25765 \beta_{10} - 82370 \beta_{9} + 827737 \beta_{8} + 410251 \beta_{7} - 377319 \beta_{6} + 40201 \beta_{5} + 110424 \beta_{4} + 351690 \beta_{3} - 30615 \beta_{2} - 2501589 \beta_{1} - 765646\)\()/80\)
\(\nu^{13}\)\(=\)\((\)\(-6054 \beta_{15} - 269607 \beta_{14} - 1512030 \beta_{13} - 376863 \beta_{12} + 2571198 \beta_{11} - 3339273 \beta_{10} - 3135488 \beta_{9} + 8588613 \beta_{8} + 1637553 \beta_{7} - 2726607 \beta_{6} + 209385 \beta_{5} + 1302768 \beta_{4} + 1938966 \beta_{3} + 13660689 \beta_{2} - 2403537 \beta_{1} - 10880432\)\()/240\)
\(\nu^{14}\)\(=\)\((\)\(4832173 \beta_{15} + 313737 \beta_{14} - 3552011 \beta_{13} - 1924598 \beta_{12} + 376774 \beta_{11} + 274105 \beta_{10} - 1283411 \beta_{9} - 3224801 \beta_{8} + 4235412 \beta_{7} + 2487913 \beta_{6} - 1889267 \beta_{5} + 167420 \beta_{4} + 820372 \beta_{3} + 10976673 \beta_{2} - 11375643 \beta_{1} - 5184677\)\()/120\)
\(\nu^{15}\)\(=\)\((\)\(-289520 \beta_{15} + 754233 \beta_{14} - 439336 \beta_{13} + 583651 \beta_{12} - 766418 \beta_{11} - 5285 \beta_{10} - 2325686 \beta_{9} + 1285625 \beta_{8} + 870599 \beta_{7} + 667009 \beta_{6} + 41649 \beta_{5} + 1022440 \beta_{4} + 689778 \beta_{3} - 4204591 \beta_{2} - 9617957 \beta_{1} - 8771146\)\()/16\)

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
−2.28990 1.66022i
−2.28990 + 1.66022i
−2.48191 + 1.35651i
−2.48191 1.35651i
−1.85197 2.13780i
−1.85197 + 2.13780i
2.77114 0.566380i
2.77114 + 0.566380i
2.70166 0.837276i
2.70166 + 0.837276i
1.85226 + 2.13755i
1.85226 2.13755i
−1.14149 2.58786i
−1.14149 + 2.58786i
2.44021 1.43016i
2.44021 + 1.43016i
−3.88613 0.947630i 5.19615i 14.2040 + 7.36522i 0 −4.92403 + 20.1929i 12.7755i −48.2191 42.0823i −27.0000 0
151.2 −3.88613 + 0.947630i 5.19615i 14.2040 7.36522i 0 −4.92403 20.1929i 12.7755i −48.2191 + 42.0823i −27.0000 0
151.3 −2.37483 3.21872i 5.19615i −4.72038 + 15.2878i 0 −16.7250 + 12.3400i 19.2859i 60.4174 21.1124i −27.0000 0
151.4 −2.37483 + 3.21872i 5.19615i −4.72038 15.2878i 0 −16.7250 12.3400i 19.2859i 60.4174 + 21.1124i −27.0000 0
151.5 −1.28004 3.78966i 5.19615i −12.7230 + 9.70180i 0 19.6916 6.65126i 89.0673i 53.0524 + 35.7972i −27.0000 0
151.6 −1.28004 + 3.78966i 5.19615i −12.7230 9.70180i 0 19.6916 + 6.65126i 89.0673i 53.0524 35.7972i −27.0000 0
151.7 −0.264404 3.99125i 5.19615i −15.8602 + 2.11060i 0 20.7392 1.37388i 29.5855i 12.6174 + 62.7439i −27.0000 0
151.8 −0.264404 + 3.99125i 5.19615i −15.8602 2.11060i 0 20.7392 + 1.37388i 29.5855i 12.6174 62.7439i −27.0000 0
151.9 3.08838 2.54203i 5.19615i 3.07620 15.7015i 0 −13.2088 16.0477i 86.5709i −30.4132 56.3120i −27.0000 0
151.10 3.08838 + 2.54203i 5.19615i 3.07620 + 15.7015i 0 −13.2088 + 16.0477i 86.5709i −30.4132 + 56.3120i −27.0000 0
151.11 3.34902 2.18726i 5.19615i 6.43181 14.6503i 0 11.3653 + 17.4020i 1.39605i −10.5038 63.1322i −27.0000 0
151.12 3.34902 + 2.18726i 5.19615i 6.43181 + 14.6503i 0 11.3653 17.4020i 1.39605i −10.5038 + 63.1322i −27.0000 0
151.13 3.40825 2.09376i 5.19615i 7.23235 14.2721i 0 −10.8795 17.7098i 61.3317i −5.23271 63.7857i −27.0000 0
151.14 3.40825 + 2.09376i 5.19615i 7.23235 + 14.2721i 0 −10.8795 + 17.7098i 61.3317i −5.23271 + 63.7857i −27.0000 0
151.15 3.95975 0.566024i 5.19615i 15.3592 4.48263i 0 2.94115 + 20.5755i 24.1355i 58.2814 26.4438i −27.0000 0
151.16 3.95975 + 0.566024i 5.19615i 15.3592 + 4.48263i 0 2.94115 20.5755i 24.1355i 58.2814 + 26.4438i −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.16
Significant digits:
Format:

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.d 16
4.b odd 2 1 inner 300.5.c.d 16
5.b even 2 1 60.5.c.a 16
5.c odd 4 2 300.5.f.b 32
15.d odd 2 1 180.5.c.c 16
20.d odd 2 1 60.5.c.a 16
20.e even 4 2 300.5.f.b 32
40.e odd 2 1 960.5.e.f 16
40.f even 2 1 960.5.e.f 16
60.h even 2 1 180.5.c.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.c.a 16 5.b even 2 1
60.5.c.a 16 20.d odd 2 1
180.5.c.c 16 15.d odd 2 1
180.5.c.c 16 60.h even 2 1
300.5.c.d 16 1.a even 1 1 trivial
300.5.c.d 16 4.b odd 2 1 inner
300.5.f.b 32 5.c odd 4 2
300.5.f.b 32 20.e even 4 2
960.5.e.f 16 40.e odd 2 1
960.5.e.f 16 40.f even 2 1

Hecke kernels

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

\(61\!\cdots\!24\)\( T_{7}^{8} + \)\(34\!\cdots\!04\)\( T_{7}^{6} + \)\(85\!\cdots\!24\)\( T_{7}^{4} + \)\(70\!\cdots\!36\)\( T_{7}^{2} + \)\(13\!\cdots\!76\)\( \)">\(T_{7}^{16} + \cdots\)
\(18\!\cdots\!60\)\( T_{13}^{2} - \)\(19\!\cdots\!24\)\( T_{13} - \)\(23\!\cdots\!24\)\( \)">\(T_{13}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4294967296 - 3221225472 T + 989855744 T^{2} - 201326592 T^{3} + 51118080 T^{4} - 7864320 T^{5} - 1347584 T^{6} + 589824 T^{7} - 103936 T^{8} + 36864 T^{9} - 5264 T^{10} - 1920 T^{11} + 780 T^{12} - 192 T^{13} + 59 T^{14} - 12 T^{15} + T^{16} \)
$3$ \( ( 27 + T^{2} )^{8} \)
$5$ \( T^{16} \)
$7$ \( \)\(13\!\cdots\!76\)\( + \)\(70\!\cdots\!36\)\( T^{2} + 85125139720803713024 T^{4} + 346323556398792704 T^{6} + 612313337626624 T^{8} + 484373856256 T^{10} + 157121024 T^{12} + 21184 T^{14} + T^{16} \)
$11$ \( \)\(15\!\cdots\!56\)\( + \)\(34\!\cdots\!64\)\( T^{2} + \)\(21\!\cdots\!68\)\( T^{4} + \)\(55\!\cdots\!60\)\( T^{6} + 6511204470995746816 T^{8} + 379916641239040 T^{10} + 11469588992 T^{12} + 171328 T^{14} + T^{16} \)
$13$ \( ( -236865200916224 - 197808834663424 T + 18366530120960 T^{2} - 457484808448 T^{3} + 1883800288 T^{4} + 19651648 T^{5} - 108400 T^{6} - 176 T^{7} + T^{8} )^{2} \)
$17$ \( ( 31387559961760000 - 3402695341056000 T - 101347242579200 T^{2} + 1101359124480 T^{3} + 26518288224 T^{4} - 10007040 T^{5} - 376016 T^{6} + T^{8} )^{2} \)
$19$ \( \)\(70\!\cdots\!76\)\( + \)\(11\!\cdots\!56\)\( T^{2} + \)\(80\!\cdots\!24\)\( T^{4} + \)\(20\!\cdots\!84\)\( T^{6} + \)\(26\!\cdots\!64\)\( T^{8} + 178536142820458496 T^{10} + 675039593984 T^{12} + 1308224 T^{14} + T^{16} \)
$23$ \( \)\(55\!\cdots\!56\)\( + \)\(10\!\cdots\!08\)\( T^{2} + \)\(34\!\cdots\!68\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(46\!\cdots\!56\)\( T^{8} + 361427060279459840 T^{10} + 1275082515968 T^{12} + 1931072 T^{14} + T^{16} \)
$29$ \( ( \)\(81\!\cdots\!36\)\( - \)\(45\!\cdots\!92\)\( T - 1118291516329069824 T^{2} + 2848989078524928 T^{3} + 2663603833184 T^{4} - 4231564032 T^{5} - 2669904 T^{6} + 1728 T^{7} + T^{8} )^{2} \)
$31$ \( \)\(48\!\cdots\!16\)\( + \)\(10\!\cdots\!16\)\( T^{2} + \)\(33\!\cdots\!56\)\( T^{4} + \)\(25\!\cdots\!16\)\( T^{6} + \)\(83\!\cdots\!64\)\( T^{8} + 14181599268396580864 T^{10} + 12694643475968 T^{12} + 5688640 T^{14} + T^{16} \)
$37$ \( ( -\)\(27\!\cdots\!04\)\( + \)\(21\!\cdots\!88\)\( T + 6449925012927361280 T^{2} - 2147497175572736 T^{3} - 15073362040352 T^{4} - 8307353536 T^{5} + 4335440 T^{6} + 4688 T^{7} + T^{8} )^{2} \)
$41$ \( ( \)\(12\!\cdots\!64\)\( + \)\(12\!\cdots\!44\)\( T - 37139646368459191552 T^{2} - 22118152285890816 T^{3} + 39511986540384 T^{4} + 8589432000 T^{5} - 13730512 T^{6} - 624 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(10\!\cdots\!76\)\( + \)\(73\!\cdots\!24\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{4} + \)\(26\!\cdots\!76\)\( T^{6} + \)\(20\!\cdots\!64\)\( T^{8} + \)\(86\!\cdots\!64\)\( T^{10} + 202841606319104 T^{12} + 23359616 T^{14} + T^{16} \)
$47$ \( \)\(16\!\cdots\!00\)\( + \)\(27\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!96\)\( T^{8} + \)\(13\!\cdots\!88\)\( T^{10} + 323664680027648 T^{12} + 31283648 T^{14} + T^{16} \)
$53$ \( ( \)\(86\!\cdots\!36\)\( + \)\(38\!\cdots\!88\)\( T - \)\(42\!\cdots\!40\)\( T^{2} - 1235641682182903296 T^{3} + 709905831679328 T^{4} + 108032893824 T^{5} - 46935120 T^{6} - 2592 T^{7} + T^{8} )^{2} \)
$59$ \( \)\(44\!\cdots\!36\)\( + \)\(49\!\cdots\!56\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{4} + \)\(74\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!64\)\( T^{8} + \)\(13\!\cdots\!56\)\( T^{10} + 6195985168034304 T^{12} + 128706624 T^{14} + T^{16} \)
$61$ \( ( -\)\(20\!\cdots\!24\)\( - \)\(61\!\cdots\!12\)\( T - \)\(41\!\cdots\!48\)\( T^{2} + 924285269314105600 T^{3} + 937740836323936 T^{4} - 73948573120 T^{5} - 55178512 T^{6} + 1904 T^{7} + T^{8} )^{2} \)
$67$ \( \)\(57\!\cdots\!76\)\( + \)\(33\!\cdots\!56\)\( T^{2} + \)\(82\!\cdots\!24\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{6} + \)\(86\!\cdots\!04\)\( T^{8} + \)\(40\!\cdots\!16\)\( T^{10} + 11247396539055104 T^{12} + 165284224 T^{14} + T^{16} \)
$71$ \( \)\(21\!\cdots\!00\)\( + \)\(58\!\cdots\!00\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(14\!\cdots\!56\)\( T^{8} + \)\(77\!\cdots\!12\)\( T^{10} + 19482745425010688 T^{12} + 227828992 T^{14} + T^{16} \)
$73$ \( ( -\)\(68\!\cdots\!96\)\( + \)\(30\!\cdots\!60\)\( T + \)\(48\!\cdots\!24\)\( T^{2} + 20581224701118447360 T^{3} + 1557357338282592 T^{4} - 715952815680 T^{5} - 103764752 T^{6} + 5520 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(46\!\cdots\!56\)\( + \)\(28\!\cdots\!80\)\( T^{2} + \)\(98\!\cdots\!88\)\( T^{4} + \)\(13\!\cdots\!64\)\( T^{6} + \)\(82\!\cdots\!84\)\( T^{8} + \)\(25\!\cdots\!16\)\( T^{10} + 41506580262811136 T^{12} + 327654976 T^{14} + T^{16} \)
$83$ \( \)\(61\!\cdots\!96\)\( + \)\(11\!\cdots\!20\)\( T^{2} + \)\(92\!\cdots\!68\)\( T^{4} + \)\(39\!\cdots\!96\)\( T^{6} + \)\(96\!\cdots\!64\)\( T^{8} + \)\(14\!\cdots\!04\)\( T^{10} + 120617011204729856 T^{12} + 544982144 T^{14} + T^{16} \)
$89$ \( ( -\)\(46\!\cdots\!76\)\( - \)\(21\!\cdots\!12\)\( T - \)\(10\!\cdots\!28\)\( T^{2} + 1094553074551375104 T^{3} + 33177059847968864 T^{4} + 642136991040 T^{5} - 324562320 T^{6} - 3792 T^{7} + T^{8} )^{2} \)
$97$ \( ( -\)\(12\!\cdots\!16\)\( - \)\(14\!\cdots\!36\)\( T - \)\(46\!\cdots\!28\)\( T^{2} + 5459530641178662144 T^{3} + 18322782073722720 T^{4} + 784704421440 T^{5} - 233395152 T^{6} - 7248 T^{7} + T^{8} )^{2} \)
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