Properties

Label 1859.4.a.d
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 59 x^{7} - 12 x^{6} + 1144 x^{5} + 345 x^{4} - 7888 x^{3} - 2245 x^{2} + 9710 x - 2988\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 5 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} ) q^{5} + ( -4 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{6} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{7} + ( -5 - 3 \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{8} + ( 10 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( 5 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} ) q^{5} + ( -4 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{6} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{7} + ( -5 - 3 \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{8} + ( 10 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{9} + ( -4 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{8} ) q^{10} + 11 q^{11} + ( 20 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - 3 \beta_{8} ) q^{12} + ( 2 + 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{6} - 2 \beta_{7} + 4 \beta_{8} ) q^{14} + ( -37 + \beta_{1} - 6 \beta_{2} + 3 \beta_{4} + 2 \beta_{6} - \beta_{7} + 6 \beta_{8} ) q^{15} + ( 11 + 8 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{16} + ( 9 + 5 \beta_{1} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} + 4 \beta_{8} ) q^{17} + ( -1 - 13 \beta_{1} - 11 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{18} + ( -8 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} ) q^{19} + ( -36 + \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 11 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{20} + ( -53 + 11 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} + 3 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{21} -11 \beta_{1} q^{22} + ( 21 - 5 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 11 \beta_{6} + \beta_{7} + 5 \beta_{8} ) q^{23} + ( 17 - \beta_{1} - 4 \beta_{3} + 6 \beta_{4} - \beta_{5} + 5 \beta_{6} + \beta_{7} + 4 \beta_{8} ) q^{24} + ( 71 - 11 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} - \beta_{4} + 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 5 \beta_{8} ) q^{25} + ( 39 - 33 \beta_{1} - 8 \beta_{2} - 22 \beta_{3} + 18 \beta_{4} - \beta_{5} + 13 \beta_{6} + 6 \beta_{8} ) q^{27} + ( -25 + 5 \beta_{1} + 19 \beta_{2} + 12 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 12 \beta_{8} ) q^{28} + ( -7 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} + 14 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} ) q^{29} + ( 7 + 25 \beta_{1} + 20 \beta_{2} + 34 \beta_{3} - \beta_{4} + 14 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 16 \beta_{8} ) q^{30} + ( -68 + 12 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} + 8 \beta_{8} ) q^{31} + ( -72 + 12 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} - 31 \beta_{4} + 11 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} ) q^{32} + ( 11 - 11 \beta_{3} ) q^{33} + ( -33 + 15 \beta_{1} + 26 \beta_{2} + 26 \beta_{3} + 11 \beta_{4} + 6 \beta_{5} + 16 \beta_{6} + 3 \beta_{7} + 16 \beta_{8} ) q^{34} + ( -46 - 7 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 11 \beta_{4} - 20 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 15 \beta_{8} ) q^{35} + ( -11 - 33 \beta_{1} + 17 \beta_{2} - 38 \beta_{3} - 7 \beta_{4} + 2 \beta_{5} + 11 \beta_{6} - 4 \beta_{7} + 7 \beta_{8} ) q^{36} + ( -103 + 41 \beta_{1} - 4 \beta_{3} - 3 \beta_{4} + 6 \beta_{6} + 3 \beta_{7} + 8 \beta_{8} ) q^{37} + ( -3 - 4 \beta_{1} + 10 \beta_{2} - 17 \beta_{3} + 24 \beta_{4} - 7 \beta_{5} + 6 \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{38} + ( 62 + 3 \beta_{1} - 17 \beta_{2} + 5 \beta_{3} + 13 \beta_{4} - 2 \beta_{5} - 10 \beta_{6} - 5 \beta_{8} ) q^{40} + ( 17 - 11 \beta_{1} - 3 \beta_{2} - 4 \beta_{4} - 19 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} - 9 \beta_{8} ) q^{41} + ( -148 + 72 \beta_{1} - 32 \beta_{2} + 16 \beta_{3} - 36 \beta_{4} - 7 \beta_{5} - 11 \beta_{6} - 2 \beta_{7} - 10 \beta_{8} ) q^{42} + ( 12 + 13 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} - 6 \beta_{7} + 9 \beta_{8} ) q^{43} + ( 55 - 11 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} ) q^{44} + ( -17 + 82 \beta_{1} + 11 \beta_{2} + 73 \beta_{3} - 32 \beta_{4} - 2 \beta_{5} - 26 \beta_{6} - 7 \beta_{7} - 7 \beta_{8} ) q^{45} + ( 111 - 5 \beta_{1} - 35 \beta_{2} + 16 \beta_{3} + 11 \beta_{4} + 4 \beta_{5} - 19 \beta_{6} - \beta_{7} + 10 \beta_{8} ) q^{46} + ( 43 - 33 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} + 51 \beta_{4} - 10 \beta_{5} + 20 \beta_{6} - 7 \beta_{7} + 8 \beta_{8} ) q^{47} + ( -196 - 32 \beta_{1} + 2 \beta_{2} + 20 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 24 \beta_{8} ) q^{48} + ( 35 - 18 \beta_{1} + 17 \beta_{2} - 16 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} - 10 \beta_{6} - 5 \beta_{7} + 19 \beta_{8} ) q^{49} + ( 168 - 38 \beta_{1} - 13 \beta_{2} - 9 \beta_{3} + 43 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 14 \beta_{7} + 25 \beta_{8} ) q^{50} + ( -85 + 63 \beta_{1} - 24 \beta_{2} + 14 \beta_{3} - 35 \beta_{4} - 8 \beta_{5} - 28 \beta_{6} - 7 \beta_{7} - 10 \beta_{8} ) q^{51} + ( 128 + 33 \beta_{1} + 2 \beta_{2} + 41 \beta_{3} + 29 \beta_{4} + 18 \beta_{5} - 16 \beta_{6} + \beta_{7} + 10 \beta_{8} ) q^{53} + ( 278 - 114 \beta_{1} + 8 \beta_{2} - 34 \beta_{3} + 10 \beta_{4} - \beta_{5} + \beta_{6} - 10 \beta_{7} - 10 \beta_{8} ) q^{54} + ( -44 + 11 \beta_{1} - 11 \beta_{2} + 11 \beta_{3} - 11 \beta_{4} - 11 \beta_{8} ) q^{55} + ( -68 + 48 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + 10 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{56} + ( 26 - 61 \beta_{1} - 34 \beta_{2} - 77 \beta_{3} + 32 \beta_{4} - 7 \beta_{5} + 29 \beta_{6} - 2 \beta_{7} + 10 \beta_{8} ) q^{57} + ( -75 + 26 \beta_{1} - 51 \beta_{2} + 5 \beta_{3} - 62 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} - 31 \beta_{8} ) q^{58} + ( 41 + 70 \beta_{1} - 25 \beta_{2} + 5 \beta_{3} - 22 \beta_{4} + 6 \beta_{5} - 30 \beta_{6} + 3 \beta_{7} - 19 \beta_{8} ) q^{59} + ( 83 + 87 \beta_{1} + 6 \beta_{2} + 98 \beta_{3} - 39 \beta_{4} - 22 \beta_{6} - 11 \beta_{7} + 6 \beta_{8} ) q^{60} + ( 43 - 40 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} - 20 \beta_{5} - 6 \beta_{6} - 7 \beta_{7} - 13 \beta_{8} ) q^{61} + ( -168 + 104 \beta_{1} + 44 \beta_{2} + 18 \beta_{3} - 26 \beta_{4} + 22 \beta_{5} + 4 \beta_{6} - 14 \beta_{7} + 20 \beta_{8} ) q^{62} + ( -51 + 72 \beta_{1} + 23 \beta_{2} + 63 \beta_{3} - 25 \beta_{4} + 11 \beta_{5} - 23 \beta_{6} + 18 \beta_{7} + 9 \beta_{8} ) q^{63} + ( -124 + 71 \beta_{1} - 28 \beta_{2} + 20 \beta_{3} + 68 \beta_{4} - 20 \beta_{5} + 38 \beta_{6} + 10 \beta_{7} + 14 \beta_{8} ) q^{64} + ( -44 - 33 \beta_{1} - 11 \beta_{3} + 11 \beta_{4} - 11 \beta_{5} - 11 \beta_{8} ) q^{66} + ( 45 - 64 \beta_{1} + 71 \beta_{2} + 19 \beta_{3} - 32 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} + 9 \beta_{7} + 5 \beta_{8} ) q^{67} + ( -297 + 101 \beta_{1} + 10 \beta_{2} + 16 \beta_{3} - \beta_{4} + 12 \beta_{5} - 29 \beta_{7} + 14 \beta_{8} ) q^{68} + ( -100 + 96 \beta_{1} + 22 \beta_{2} + 48 \beta_{3} - 35 \beta_{4} - 3 \beta_{5} + 7 \beta_{6} + 21 \beta_{7} + 2 \beta_{8} ) q^{69} + ( 10 + 85 \beta_{1} + 11 \beta_{2} - 97 \beta_{3} - 31 \beta_{4} - 2 \beta_{5} + 16 \beta_{6} + 6 \beta_{7} - 57 \beta_{8} ) q^{70} + ( -211 - 27 \beta_{1} + 36 \beta_{2} + 34 \beta_{3} - \beta_{4} + 24 \beta_{5} + 68 \beta_{6} + 11 \beta_{7} + 38 \beta_{8} ) q^{71} + ( 280 + 93 \beta_{1} + 2 \beta_{2} + 20 \beta_{3} + 63 \beta_{4} - 18 \beta_{5} + 32 \beta_{6} - 2 \beta_{7} ) q^{72} + ( -417 + 37 \beta_{1} - \beta_{2} + 2 \beta_{3} - 59 \beta_{4} - 4 \beta_{5} - 44 \beta_{6} - 2 \beta_{7} + 7 \beta_{8} ) q^{73} + ( -551 + 121 \beta_{1} + 28 \beta_{2} + 72 \beta_{3} - 35 \beta_{4} - 8 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{74} + ( -88 - 23 \beta_{1} + 2 \beta_{2} - 105 \beta_{3} + 15 \beta_{4} - 18 \beta_{5} + 24 \beta_{6} + 27 \beta_{7} - 2 \beta_{8} ) q^{75} + ( -85 - 46 \beta_{1} + 49 \beta_{2} - 52 \beta_{3} - 31 \beta_{4} - 7 \beta_{5} + 24 \beta_{6} - 10 \beta_{7} + 25 \beta_{8} ) q^{76} + ( -22 + 11 \beta_{2} + 22 \beta_{3} + 11 \beta_{4} + 11 \beta_{5} + 11 \beta_{6} + 11 \beta_{8} ) q^{77} + ( -132 - 20 \beta_{1} - 76 \beta_{2} - 80 \beta_{3} - 8 \beta_{4} - 22 \beta_{5} + 36 \beta_{6} + 6 \beta_{7} - 18 \beta_{8} ) q^{79} + ( 276 - 131 \beta_{1} + 49 \beta_{2} + 31 \beta_{3} + 5 \beta_{4} + 60 \beta_{5} - 12 \beta_{7} + 3 \beta_{8} ) q^{80} + ( 267 - 84 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 15 \beta_{4} - 29 \beta_{5} - 19 \beta_{6} + \beta_{7} - 68 \beta_{8} ) q^{81} + ( 95 + 25 \beta_{1} - 7 \beta_{2} - 56 \beta_{3} - 79 \beta_{4} + 24 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 46 \beta_{8} ) q^{82} + ( 192 + 139 \beta_{1} + 58 \beta_{2} + 17 \beta_{3} - 29 \beta_{5} - 51 \beta_{6} - 4 \beta_{7} - 20 \beta_{8} ) q^{83} + ( -201 + 89 \beta_{1} + 43 \beta_{2} + 93 \beta_{3} - 33 \beta_{4} + 5 \beta_{5} - 13 \beta_{6} + 36 \beta_{7} + 4 \beta_{8} ) q^{84} + ( 85 + 57 \beta_{1} - 6 \beta_{2} - 66 \beta_{3} + 19 \beta_{4} + 22 \beta_{5} + 80 \beta_{6} - \beta_{7} - 18 \beta_{8} ) q^{85} + ( -66 - 37 \beta_{1} + 81 \beta_{2} + 97 \beta_{3} + 21 \beta_{4} + 26 \beta_{5} + 24 \beta_{6} + 18 \beta_{7} + 49 \beta_{8} ) q^{86} + ( 202 + 50 \beta_{1} + 126 \beta_{2} + 140 \beta_{3} + 58 \beta_{5} - 48 \beta_{6} + 8 \beta_{7} - 4 \beta_{8} ) q^{87} + ( -55 - 33 \beta_{1} + 11 \beta_{2} - 33 \beta_{4} + 11 \beta_{5} - 22 \beta_{6} - 11 \beta_{8} ) q^{88} + ( -156 - 62 \beta_{1} - 26 \beta_{2} - 52 \beta_{3} - 28 \beta_{4} + 12 \beta_{5} - 28 \beta_{6} - 26 \beta_{7} - 46 \beta_{8} ) q^{89} + ( -601 + 206 \beta_{1} + 81 \beta_{2} + 135 \beta_{3} - 36 \beta_{4} + 60 \beta_{5} + 14 \beta_{6} + 21 \beta_{7} + 87 \beta_{8} ) q^{90} + ( 90 - 150 \beta_{1} - 51 \beta_{2} + 82 \beta_{3} - 41 \beta_{4} + 60 \beta_{5} + 3 \beta_{6} + 16 \beta_{7} ) q^{92} + ( -196 + 6 \beta_{1} + 12 \beta_{2} + 118 \beta_{3} - 38 \beta_{4} - 20 \beta_{5} - 14 \beta_{6} + 16 \beta_{7} + 2 \beta_{8} ) q^{93} + ( 229 - 155 \beta_{1} + 142 \beta_{2} - 48 \beta_{3} - 65 \beta_{4} + 68 \beta_{5} + 8 \beta_{6} - 39 \beta_{7} + 34 \beta_{8} ) q^{94} + ( 202 + 188 \beta_{1} + 76 \beta_{2} + 106 \beta_{3} - 72 \beta_{4} - 2 \beta_{5} - 70 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} ) q^{95} + ( 368 + 284 \beta_{1} + 5 \beta_{2} + 103 \beta_{3} - 43 \beta_{4} + 61 \beta_{5} - 37 \beta_{6} - 13 \beta_{7} + 28 \beta_{8} ) q^{96} + ( -456 - 64 \beta_{1} - 18 \beta_{2} - 54 \beta_{3} + 92 \beta_{4} - 30 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 40 \beta_{8} ) q^{97} + ( 229 - 38 \beta_{1} + 3 \beta_{2} + 18 \beta_{3} + 119 \beta_{4} - 15 \beta_{5} + 10 \beta_{6} - 13 \beta_{7} + 42 \beta_{8} ) q^{98} + ( 110 + 22 \beta_{2} - 11 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} - 11 \beta_{6} - 11 \beta_{7} - 22 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 8 q^{3} + 46 q^{4} - 30 q^{5} - 34 q^{6} - 25 q^{7} - 36 q^{8} + 91 q^{9} + O(q^{10}) \) \( 9 q + 8 q^{3} + 46 q^{4} - 30 q^{5} - 34 q^{6} - 25 q^{7} - 36 q^{8} + 91 q^{9} - 22 q^{10} + 99 q^{11} + 181 q^{12} - 351 q^{15} + 130 q^{16} + 53 q^{17} - 33 q^{18} - 69 q^{19} - 282 q^{20} - 463 q^{21} + 216 q^{23} + 121 q^{24} + 617 q^{25} + 275 q^{27} - 279 q^{28} - 91 q^{29} + 29 q^{30} - 636 q^{31} - 663 q^{32} + 88 q^{33} - 423 q^{34} - 358 q^{35} - 252 q^{36} - 967 q^{37} - 101 q^{38} + 652 q^{40} + 226 q^{41} - 1186 q^{42} + 42 q^{43} + 506 q^{44} - 5 q^{45} + 1127 q^{46} + 269 q^{47} - 1820 q^{48} + 228 q^{49} + 1374 q^{50} - 589 q^{51} + 1227 q^{53} + 2438 q^{54} - 330 q^{55} - 659 q^{56} + 71 q^{57} - 471 q^{58} + 613 q^{59} + 859 q^{60} + 427 q^{61} - 1714 q^{62} - 305 q^{63} - 1194 q^{64} - 374 q^{66} + 271 q^{67} - 2835 q^{68} - 846 q^{69} + 102 q^{70} - 2279 q^{71} + 2400 q^{72} - 3602 q^{73} - 4955 q^{74} - 883 q^{75} - 1126 q^{76} - 275 q^{77} - 1182 q^{79} + 2360 q^{80} + 2697 q^{81} + 1007 q^{82} + 1877 q^{83} - 1618 q^{84} + 441 q^{85} - 830 q^{86} + 1942 q^{87} - 396 q^{88} - 1258 q^{89} - 5669 q^{90} + 1046 q^{92} - 1556 q^{93} + 1439 q^{94} + 2032 q^{95} + 3417 q^{96} - 4002 q^{97} + 1855 q^{98} + 1001 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 59 x^{7} - 12 x^{6} + 1144 x^{5} + 345 x^{4} - 7888 x^{3} - 2245 x^{2} + 9710 x - 2988\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{8} + 357 \nu^{7} - 350 \nu^{6} - 14878 \nu^{5} + 9822 \nu^{4} + 154641 \nu^{3} - 82789 \nu^{2} - 171682 \nu + 322364 \)\()/37760\)
\(\beta_{3}\)\(=\)\((\)\( -87 \nu^{8} - 621 \nu^{7} + 6990 \nu^{6} + 29774 \nu^{5} - 168206 \nu^{4} - 418873 \nu^{3} + 1290477 \nu^{2} + 1596466 \nu - 945052 \)\()/75520\)
\(\beta_{4}\)\(=\)\((\)\( -89 \nu^{8} + 93 \nu^{7} + 6290 \nu^{6} + 18 \nu^{5} - 148562 \nu^{4} - 109591 \nu^{3} + 1200419 \nu^{2} + 1253102 \nu - 1282084 \)\()/75520\)
\(\beta_{5}\)\(=\)\((\)\( -53 \nu^{8} - 119 \nu^{7} + 4810 \nu^{6} + 6346 \nu^{5} - 133194 \nu^{4} - 131707 \nu^{3} + 1212663 \nu^{2} + 1011254 \nu - 1429268 \)\()/37760\)
\(\beta_{6}\)\(=\)\((\)\( -283 \nu^{8} + 551 \nu^{7} + 14550 \nu^{6} - 21674 \nu^{5} - 237974 \nu^{4} + 240843 \nu^{3} + 1287513 \nu^{2} - 549206 \nu - 612268 \)\()/75520\)
\(\beta_{7}\)\(=\)\((\)\( 311 \nu^{8} + 653 \nu^{7} - 16270 \nu^{6} - 35982 \nu^{5} + 263118 \nu^{4} + 525529 \nu^{3} - 1433101 \nu^{2} - 1848178 \nu + 1310876 \)\()/37760\)
\(\beta_{8}\)\(=\)\((\)\( 145 \nu^{8} - 181 \nu^{7} - 7810 \nu^{6} + 5246 \nu^{5} + 134978 \nu^{4} - 6305 \nu^{3} - 783307 \nu^{2} - 483326 \nu + 495876 \)\()/15104\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} - \beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 19 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(-2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 28 \beta_{4} - 21 \beta_{3} - 29 \beta_{2} + 8 \beta_{1} + 259\)
\(\nu^{5}\)\(=\)\(33 \beta_{8} - \beta_{7} + 60 \beta_{6} - 43 \beta_{5} + 127 \beta_{4} - 4 \beta_{3} - 37 \beta_{2} + 404 \beta_{1} + 232\)
\(\nu^{6}\)\(=\)\(-66 \beta_{8} + 90 \beta_{7} - 2 \beta_{6} - 220 \beta_{5} + 804 \beta_{4} - 436 \beta_{3} - 804 \beta_{2} + 391 \beta_{1} + 5756\)
\(\nu^{7}\)\(=\)\(928 \beta_{8} + 1662 \beta_{6} - 1458 \beta_{5} + 4309 \beta_{4} - 275 \beta_{3} - 1287 \beta_{2} + 9292 \beta_{1} + 8519\)
\(\nu^{8}\)\(=\)\(-1581 \beta_{8} + 3022 \beta_{7} + 814 \beta_{6} - 7503 \beta_{5} + 23557 \beta_{4} - 9536 \beta_{3} - 22023 \beta_{2} + 14755 \beta_{1} + 138197\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.41479
4.08298
3.71870
0.765277
0.388321
−1.62159
−3.76323
−4.11495
−4.87031
−5.41479 −1.18631 21.3199 −5.73789 6.42360 −3.58372 −72.1247 −25.5927 31.0694
1.2 −4.08298 7.19985 8.67073 7.90460 −29.3968 −23.1330 −2.73856 24.8378 −32.2743
1.3 −3.71870 7.61710 5.82875 −15.9808 −28.3257 21.9580 8.07423 31.0202 59.4279
1.4 −0.765277 −9.54214 −7.41435 17.1562 7.30238 4.60754 11.7962 64.0525 −13.1292
1.5 −0.388321 3.09988 −7.84921 −16.8933 −1.20375 −26.1569 6.15457 −17.3907 6.56001
1.6 1.62159 −2.83913 −5.37046 8.40999 −4.60389 9.04976 −21.6813 −18.9393 13.6375
1.7 3.76323 −4.70664 6.16188 −20.6048 −17.7122 28.6315 −6.91727 −4.84752 −77.5406
1.8 4.11495 9.51427 8.93279 −14.5196 39.1507 −21.0410 3.83839 63.5214 −59.7475
1.9 4.87031 −1.15688 15.7199 10.2656 −5.63435 −15.3321 37.5984 −25.6616 49.9968
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.4.a.d 9
13.b even 2 1 143.4.a.c 9
39.d odd 2 1 1287.4.a.k 9
52.b odd 2 1 2288.4.a.r 9
143.d odd 2 1 1573.4.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.c 9 13.b even 2 1
1287.4.a.k 9 39.d odd 2 1
1573.4.a.e 9 143.d odd 2 1
1859.4.a.d 9 1.a even 1 1 trivial
2288.4.a.r 9 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{9} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2988 + 9710 T + 2245 T^{2} - 7888 T^{3} - 345 T^{4} + 1144 T^{5} + 12 T^{6} - 59 T^{7} + T^{9} \)
$3$ \( 283048 + 475226 T + 145675 T^{2} - 83114 T^{3} - 27709 T^{4} + 4859 T^{5} + 1015 T^{6} - 135 T^{7} - 8 T^{8} + T^{9} \)
$5$ \( 5425892224 + 66954624 T - 238114176 T^{2} - 484320 T^{3} + 3171776 T^{4} + 38148 T^{5} - 17070 T^{6} - 421 T^{7} + 30 T^{8} + T^{9} \)
$7$ \( 18338418984 + 1196462545 T - 1339235989 T^{2} - 24653381 T^{3} + 14267352 T^{4} + 462108 T^{5} - 36052 T^{6} - 1345 T^{7} + 25 T^{8} + T^{9} \)
$11$ \( ( -11 + T )^{9} \)
$13$ \( T^{9} \)
$17$ \( 6386620512459776 + 1011834889297664 T + 12088009420992 T^{2} - 963969465568 T^{3} - 7249591648 T^{4} + 265532384 T^{5} + 1129936 T^{6} - 27946 T^{7} - 53 T^{8} + T^{9} \)
$19$ \( -1483175094148800 - 404801113461840 T - 24839285116452 T^{2} - 85405045419 T^{3} + 13489287813 T^{4} + 135681434 T^{5} - 1857827 T^{6} - 22970 T^{7} + 69 T^{8} + T^{9} \)
$23$ \( 225784040085535744 + 37879556564506128 T + 558184347978767 T^{2} - 11040212833240 T^{3} - 156666082689 T^{4} + 1089597101 T^{5} + 10759163 T^{6} - 53643 T^{7} - 216 T^{8} + T^{9} \)
$29$ \( 27221149695162175488 + 564255102670468992 T - 9110741575389696 T^{2} - 139084897122816 T^{3} + 899274593664 T^{4} + 9185210808 T^{5} - 17366128 T^{6} - 174158 T^{7} + 91 T^{8} + T^{9} \)
$31$ \( -5656313714465636352 + 133521957522855936 T + 7263520911850496 T^{2} + 57673310843648 T^{3} - 477100859168 T^{4} - 8281576528 T^{5} - 28612128 T^{6} + 75556 T^{7} + 636 T^{8} + T^{9} \)
$37$ \( 7221890531726770176 - 164994221839429632 T - 7672354452867072 T^{2} + 187648286596608 T^{3} + 137596233472 T^{4} - 12436000368 T^{5} - 22130516 T^{6} + 243580 T^{7} + 967 T^{8} + T^{9} \)
$41$ \( \)\(58\!\cdots\!96\)\( - 19247276606096634666 T + 191703456737784097 T^{2} - 202236274629678 T^{3} - 6306359254017 T^{4} + 21609891843 T^{5} + 63750727 T^{6} - 282435 T^{7} - 226 T^{8} + T^{9} \)
$43$ \( \)\(51\!\cdots\!52\)\( + 3745159933714018560 T - 110487315152691840 T^{2} - 945106239520144 T^{3} + 2429355295720 T^{4} + 29024570884 T^{5} - 10703918 T^{6} - 302261 T^{7} - 42 T^{8} + T^{9} \)
$47$ \( \)\(51\!\cdots\!84\)\( + 51796098595737847296 T - 1275770712980367936 T^{2} - 10399685852156128 T^{3} + 2271547150288 T^{4} + 132311976056 T^{5} + 72580584 T^{6} - 606028 T^{7} - 269 T^{8} + T^{9} \)
$53$ \( -\)\(12\!\cdots\!32\)\( - \)\(16\!\cdots\!28\)\( T + 28176524689842303366 T^{2} + 17941237062097609 T^{3} - 240660019233381 T^{4} - 3472942078 T^{5} + 905955235 T^{6} - 429690 T^{7} - 1227 T^{8} + T^{9} \)
$59$ \( -\)\(15\!\cdots\!52\)\( - \)\(44\!\cdots\!44\)\( T + 4507635855927083776 T^{2} - 1000333847064864 T^{3} - 70182997140432 T^{4} + 90481814952 T^{5} + 364432436 T^{6} - 572432 T^{7} - 613 T^{8} + T^{9} \)
$61$ \( \)\(49\!\cdots\!64\)\( - 19276390395427431552 T + 124644631383491904 T^{2} + 1855682143709632 T^{3} - 15772751105488 T^{4} - 33043073776 T^{5} + 406641104 T^{6} - 624964 T^{7} - 427 T^{8} + T^{9} \)
$67$ \( -\)\(89\!\cdots\!72\)\( + \)\(10\!\cdots\!16\)\( T + 46723867168090464256 T^{2} - 133358096368324800 T^{3} - 258199815903216 T^{4} + 861271633320 T^{5} + 444504500 T^{6} - 1689290 T^{7} - 271 T^{8} + T^{9} \)
$71$ \( -\)\(15\!\cdots\!32\)\( + \)\(83\!\cdots\!92\)\( T + \)\(50\!\cdots\!68\)\( T^{2} + 1436318669482760768 T^{3} + 593150091807408 T^{4} - 2812976322272 T^{5} - 3713478448 T^{6} + 13710 T^{7} + 2279 T^{8} + T^{9} \)
$73$ \( \)\(24\!\cdots\!24\)\( + \)\(28\!\cdots\!38\)\( T + \)\(11\!\cdots\!99\)\( T^{2} - 117610659471505742 T^{3} - 1221939300094063 T^{4} - 1579613817773 T^{5} + 1215878513 T^{6} + 4382917 T^{7} + 3602 T^{8} + T^{9} \)
$79$ \( \)\(19\!\cdots\!00\)\( + \)\(31\!\cdots\!20\)\( T - 76901370480790721152 T^{2} - 374192120558897088 T^{3} + 675537196748256 T^{4} + 1303275364544 T^{5} - 1699593072 T^{6} - 1819212 T^{7} + 1182 T^{8} + T^{9} \)
$83$ \( -\)\(41\!\cdots\!88\)\( - \)\(86\!\cdots\!96\)\( T + \)\(39\!\cdots\!88\)\( T^{2} - 711519704689235001 T^{3} - 3294242954026009 T^{4} + 1880951108764 T^{5} + 4721478375 T^{6} - 2350840 T^{7} - 1877 T^{8} + T^{9} \)
$89$ \( \)\(94\!\cdots\!48\)\( - \)\(95\!\cdots\!28\)\( T - \)\(25\!\cdots\!40\)\( T^{2} + 28581002230567360 T^{3} + 1540586755214240 T^{4} + 755047072832 T^{5} - 2572230784 T^{6} - 1799516 T^{7} + 1258 T^{8} + T^{9} \)
$97$ \( -\)\(56\!\cdots\!64\)\( + \)\(84\!\cdots\!56\)\( T + \)\(44\!\cdots\!04\)\( T^{2} + 5684446199048870016 T^{3} - 3001968418497856 T^{4} - 11317665267008 T^{5} - 5733497880 T^{6} + 3458952 T^{7} + 4002 T^{8} + T^{9} \)
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