Properties

Label 230.4.b.a
Level $230$
Weight $4$
Character orbit 230.b
Analytic conductor $13.570$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 212 x^{12} + 17560 x^{10} + 728073 x^{8} + 16036416 x^{6} + 183184060 x^{4} + 961600400 x^{2} + 1560250000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{8} q^{2} + ( \beta_{1} - \beta_{8} ) q^{3} -4 q^{4} + ( -\beta_{7} - \beta_{8} ) q^{5} + ( -2 + 2 \beta_{4} ) q^{6} + ( \beta_{1} + 6 \beta_{8} - \beta_{12} - \beta_{13} ) q^{7} + 8 \beta_{8} q^{8} + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})\) \( q -2 \beta_{8} q^{2} + ( \beta_{1} - \beta_{8} ) q^{3} -4 q^{4} + ( -\beta_{7} - \beta_{8} ) q^{5} + ( -2 + 2 \beta_{4} ) q^{6} + ( \beta_{1} + 6 \beta_{8} - \beta_{12} - \beta_{13} ) q^{7} + 8 \beta_{8} q^{8} + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{9} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{9} + 2 \beta_{11} ) q^{10} + ( -9 + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{11} + ( -4 \beta_{1} + 4 \beta_{8} ) q^{12} + ( -5 \beta_{1} + 3 \beta_{6} - 8 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - \beta_{13} ) q^{13} + ( 12 - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{14} + ( -11 - 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 6 \beta_{8} - \beta_{9} - 2 \beta_{11} + 4 \beta_{12} + \beta_{13} ) q^{15} + 16 q^{16} + ( 3 \beta_{1} + 5 \beta_{6} - 5 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{17} + ( 2 \beta_{1} + 2 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} - 2 \beta_{13} ) q^{18} + ( 12 + 4 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{19} + ( 4 \beta_{7} + 4 \beta_{8} ) q^{20} + ( -17 - 3 \beta_{2} - 2 \beta_{3} - 14 \beta_{4} - 7 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - 9 \beta_{10} - 9 \beta_{11} ) q^{21} + ( 6 \beta_{1} + 2 \beta_{6} + 4 \beta_{7} + 18 \beta_{8} - 4 \beta_{9} - 2 \beta_{12} - 2 \beta_{13} ) q^{22} + 23 \beta_{8} q^{23} + ( 8 - 8 \beta_{4} ) q^{24} + ( -15 - 5 \beta_{1} + 15 \beta_{4} + 5 \beta_{6} - \beta_{7} + 4 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 5 \beta_{12} + 5 \beta_{13} ) q^{25} + ( -16 - 6 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} - 8 \beta_{5} - 6 \beta_{10} - 6 \beta_{11} ) q^{26} + ( 8 \beta_{1} + 7 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} + 4 \beta_{12} + \beta_{13} ) q^{27} + ( -4 \beta_{1} - 24 \beta_{8} + 4 \beta_{12} + 4 \beta_{13} ) q^{28} + ( 64 + 5 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 14 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 8 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{29} + ( -12 + 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 22 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} + 8 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} ) q^{30} + ( -14 + 7 \beta_{2} + \beta_{4} + 14 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} - 9 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{31} -32 \beta_{8} q^{32} + ( -17 \beta_{1} + 6 \beta_{6} + 4 \beta_{7} - 30 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} - \beta_{12} - 8 \beta_{13} ) q^{33} + ( -10 - 10 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} - 6 \beta_{10} - 6 \beta_{11} ) q^{34} + ( -32 - 22 \beta_{1} - 5 \beta_{2} - 7 \beta_{3} - 11 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} + 14 \beta_{8} + 3 \beta_{9} - 15 \beta_{11} + \beta_{12} + 4 \beta_{13} ) q^{35} + ( 12 - 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{36} + ( -10 \beta_{1} - 3 \beta_{6} - 5 \beta_{7} + 28 \beta_{8} + 5 \beta_{9} + 12 \beta_{10} - 12 \beta_{11} + 21 \beta_{12} + 8 \beta_{13} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{6} + 6 \beta_{7} - 24 \beta_{8} - 6 \beta_{9} + 10 \beta_{10} - 10 \beta_{11} - 6 \beta_{12} - 4 \beta_{13} ) q^{38} + ( 101 - 12 \beta_{2} - 17 \beta_{3} + 6 \beta_{4} - 28 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{9} - 9 \beta_{10} - 9 \beta_{11} ) q^{39} + ( 8 - 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{4} + 8 \beta_{9} - 8 \beta_{11} ) q^{40} + ( -129 + 4 \beta_{2} + 13 \beta_{3} - 11 \beta_{4} + 20 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} + 11 \beta_{10} + 11 \beta_{11} ) q^{41} + ( 32 \beta_{1} - 8 \beta_{6} + 2 \beta_{7} + 34 \beta_{8} - 2 \beta_{9} - 20 \beta_{10} + 20 \beta_{11} + 14 \beta_{12} + 4 \beta_{13} ) q^{42} + ( -2 \beta_{1} + 14 \beta_{6} - 2 \beta_{7} - 76 \beta_{8} + 2 \beta_{9} - 17 \beta_{10} + 17 \beta_{11} + 15 \beta_{12} - 11 \beta_{13} ) q^{43} + ( 36 - 12 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{44} + ( 45 - 32 \beta_{1} + \beta_{2} - 3 \beta_{3} + 17 \beta_{4} + 5 \beta_{5} - \beta_{6} - 3 \beta_{7} + 68 \beta_{8} + \beta_{9} + 6 \beta_{10} + 18 \beta_{11} + 11 \beta_{12} - 4 \beta_{13} ) q^{45} + 46 q^{46} + ( 56 \beta_{1} + 6 \beta_{6} + 11 \beta_{7} + 8 \beta_{8} - 11 \beta_{9} - 13 \beta_{10} + 13 \beta_{11} - 18 \beta_{12} + 2 \beta_{13} ) q^{47} + ( 16 \beta_{1} - 16 \beta_{8} ) q^{48} + ( 70 + 15 \beta_{2} + 17 \beta_{3} + 9 \beta_{4} + 8 \beta_{5} + 17 \beta_{6} - 17 \beta_{7} - 17 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{49} + ( 8 - 18 \beta_{1} - 8 \beta_{2} + 10 \beta_{3} - 18 \beta_{4} - 10 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} + 30 \beta_{8} - 2 \beta_{9} - 10 \beta_{10} - 8 \beta_{11} ) q^{50} + ( -136 + 6 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 13 \beta_{10} + 13 \beta_{11} ) q^{51} + ( 20 \beta_{1} - 12 \beta_{6} + 32 \beta_{8} - 12 \beta_{10} + 12 \beta_{11} + 16 \beta_{12} + 4 \beta_{13} ) q^{52} + ( 12 \beta_{1} - 8 \beta_{6} + 18 \beta_{7} - 60 \beta_{8} - 18 \beta_{9} + \beta_{10} - \beta_{11} - 18 \beta_{12} + 21 \beta_{13} ) q^{53} + ( 18 - 20 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} + 6 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} ) q^{54} + ( 55 - 23 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 10 \beta_{4} - 16 \beta_{5} - 2 \beta_{6} + 11 \beta_{7} + 170 \beta_{8} + \beta_{9} + 11 \beta_{10} - 20 \beta_{11} - 15 \beta_{12} - 3 \beta_{13} ) q^{55} + ( -48 + 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} ) q^{56} + ( 28 \beta_{1} + 3 \beta_{6} + 6 \beta_{7} - 65 \beta_{8} - 6 \beta_{9} + 11 \beta_{10} - 11 \beta_{11} - 17 \beta_{12} - 36 \beta_{13} ) q^{57} + ( 28 \beta_{1} - 6 \beta_{6} + 16 \beta_{7} - 128 \beta_{8} - 16 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 28 \beta_{12} + 10 \beta_{13} ) q^{58} + ( 200 + 12 \beta_{2} + 11 \beta_{3} - 50 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 16 \beta_{10} - 16 \beta_{11} ) q^{59} + ( 44 + 20 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} - 16 \beta_{6} + 16 \beta_{7} + 24 \beta_{8} + 4 \beta_{9} + 8 \beta_{11} - 16 \beta_{12} - 4 \beta_{13} ) q^{60} + ( -255 + 3 \beta_{2} - 27 \beta_{3} + 39 \beta_{4} - 5 \beta_{5} + 15 \beta_{6} - 15 \beta_{7} - 15 \beta_{9} + 10 \beta_{10} + 10 \beta_{11} ) q^{61} + ( 34 \beta_{1} - 4 \beta_{6} + 18 \beta_{7} + 28 \beta_{8} - 18 \beta_{9} - 28 \beta_{12} ) q^{62} + ( -52 \beta_{1} + 3 \beta_{6} - 16 \beta_{7} - 97 \beta_{8} + 16 \beta_{9} - \beta_{10} + \beta_{11} + 34 \beta_{12} + 26 \beta_{13} ) q^{63} -64 q^{64} + ( 119 + 19 \beta_{1} + 9 \beta_{2} - 25 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 15 \beta_{6} + 6 \beta_{7} + 227 \beta_{8} + \beta_{9} - 18 \beta_{10} + 8 \beta_{11} - 11 \beta_{12} - 5 \beta_{13} ) q^{65} + ( -60 - 20 \beta_{2} - 16 \beta_{3} - 50 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} + 8 \beta_{9} - 14 \beta_{10} - 14 \beta_{11} ) q^{66} + ( 34 \beta_{1} + 9 \beta_{6} + 13 \beta_{7} + 42 \beta_{8} - 13 \beta_{9} - 18 \beta_{10} + 18 \beta_{11} - 27 \beta_{12} - 22 \beta_{13} ) q^{67} + ( -12 \beta_{1} - 20 \beta_{6} + 20 \beta_{8} - 12 \beta_{10} + 12 \beta_{11} + 8 \beta_{12} - 12 \beta_{13} ) q^{68} + ( 23 - 23 \beta_{4} ) q^{69} + ( 28 + 22 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} - 32 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 64 \beta_{8} - 6 \beta_{9} - 24 \beta_{10} + 6 \beta_{11} + 8 \beta_{12} + 14 \beta_{13} ) q^{70} + ( -273 + 31 \beta_{3} + 9 \beta_{4} - 54 \beta_{5} - 17 \beta_{6} + 17 \beta_{7} + 17 \beta_{9} - 52 \beta_{10} - 52 \beta_{11} ) q^{71} + ( -8 \beta_{1} - 8 \beta_{7} - 24 \beta_{8} + 8 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} + 8 \beta_{13} ) q^{72} + ( -56 \beta_{1} + 6 \beta_{6} - 5 \beta_{7} - 280 \beta_{8} + 5 \beta_{9} + 26 \beta_{10} - 26 \beta_{11} - 26 \beta_{12} - 14 \beta_{13} ) q^{73} + ( 56 + 16 \beta_{2} + 16 \beta_{3} + 42 \beta_{5} + 10 \beta_{6} - 10 \beta_{7} - 10 \beta_{9} - 14 \beta_{10} - 14 \beta_{11} ) q^{74} + ( 204 + 8 \beta_{2} + 2 \beta_{3} + 39 \beta_{4} + 4 \beta_{5} + 9 \beta_{6} + \beta_{7} + 359 \beta_{8} - 11 \beta_{9} + 15 \beta_{10} + 58 \beta_{11} + 19 \beta_{12} - 19 \beta_{13} ) q^{75} + ( -48 - 16 \beta_{2} - 8 \beta_{3} - 20 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} + 12 \beta_{9} - 32 \beta_{10} - 32 \beta_{11} ) q^{76} + ( -18 \beta_{1} - 5 \beta_{6} - 22 \beta_{7} - 197 \beta_{8} + 22 \beta_{9} + 22 \beta_{10} - 22 \beta_{11} + 6 \beta_{12} + 41 \beta_{13} ) q^{77} + ( -44 \beta_{1} - 8 \beta_{6} - 16 \beta_{7} - 202 \beta_{8} + 16 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 56 \beta_{12} + 34 \beta_{13} ) q^{78} + ( -110 + 66 \beta_{2} - 7 \beta_{3} + 6 \beta_{4} - 17 \beta_{5} + 30 \beta_{6} - 30 \beta_{7} - 30 \beta_{9} + 35 \beta_{10} + 35 \beta_{11} ) q^{79} + ( -16 \beta_{7} - 16 \beta_{8} ) q^{80} + ( -310 + 11 \beta_{2} + 19 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 19 \beta_{10} - 19 \beta_{11} ) q^{81} + ( 50 \beta_{1} - 6 \beta_{6} + 14 \beta_{7} + 258 \beta_{8} - 14 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} - 40 \beta_{12} - 26 \beta_{13} ) q^{82} + ( 70 \beta_{1} - 21 \beta_{6} - 5 \beta_{7} - 224 \beta_{8} + 5 \beta_{9} - 39 \beta_{10} + 39 \beta_{11} - 6 \beta_{12} - 13 \beta_{13} ) q^{83} + ( 68 + 12 \beta_{2} + 8 \beta_{3} + 56 \beta_{4} + 28 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} + 36 \beta_{10} + 36 \beta_{11} ) q^{84} + ( 224 - 10 \beta_{1} + 50 \beta_{2} + 25 \beta_{3} + 45 \beta_{4} + 36 \beta_{5} + 15 \beta_{6} - 48 \beta_{7} + 225 \beta_{8} + 20 \beta_{9} + 6 \beta_{10} + 7 \beta_{11} + 27 \beta_{12} - 5 \beta_{13} ) q^{85} + ( -152 - 24 \beta_{2} - 22 \beta_{3} + 4 \beta_{4} + 30 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{9} + 38 \beta_{10} + 38 \beta_{11} ) q^{86} + ( 16 \beta_{1} + 24 \beta_{6} - 45 \beta_{7} - 150 \beta_{8} + 45 \beta_{9} + 23 \beta_{10} - 23 \beta_{11} + 14 \beta_{12} + 12 \beta_{13} ) q^{87} + ( -24 \beta_{1} - 8 \beta_{6} - 16 \beta_{7} - 72 \beta_{8} + 16 \beta_{9} + 8 \beta_{12} + 8 \beta_{13} ) q^{88} + ( 106 - 52 \beta_{2} + 25 \beta_{3} - 14 \beta_{4} - 65 \beta_{5} - 40 \beta_{6} + 40 \beta_{7} + 40 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{89} + ( 136 - 30 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} - 56 \beta_{4} + 22 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 90 \beta_{8} - 6 \beta_{9} + 38 \beta_{10} - 6 \beta_{11} - 10 \beta_{12} + 6 \beta_{13} ) q^{90} + ( -231 + 51 \beta_{2} + 9 \beta_{3} + 191 \beta_{4} + 15 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} - 8 \beta_{9} + 42 \beta_{10} + 42 \beta_{11} ) q^{91} -92 \beta_{8} q^{92} + ( -90 \beta_{1} + 37 \beta_{6} - 33 \beta_{7} - 137 \beta_{8} + 33 \beta_{9} + 46 \beta_{10} - 46 \beta_{11} - 2 \beta_{12} - 5 \beta_{13} ) q^{93} + ( 16 - 34 \beta_{2} + 4 \beta_{3} + 68 \beta_{4} - 36 \beta_{5} - 22 \beta_{6} + 22 \beta_{7} + 22 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{94} + ( 82 + 18 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} + 5 \beta_{4} - 36 \beta_{5} + 37 \beta_{6} + 200 \beta_{8} - 36 \beta_{9} + 22 \beta_{10} - 84 \beta_{11} - 94 \beta_{12} - 22 \beta_{13} ) q^{95} + ( -32 + 32 \beta_{4} ) q^{96} + ( 9 \beta_{1} - 17 \beta_{6} - 64 \beta_{7} + 105 \beta_{8} + 64 \beta_{9} + 33 \beta_{10} - 33 \beta_{11} - 17 \beta_{12} + 5 \beta_{13} ) q^{97} + ( 50 \beta_{1} - 4 \beta_{6} + 34 \beta_{7} - 140 \beta_{8} - 34 \beta_{9} - 46 \beta_{10} + 46 \beta_{11} - 16 \beta_{12} - 34 \beta_{13} ) q^{98} + ( 208 + 8 \beta_{2} - 49 \beta_{3} - 21 \beta_{4} - 10 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + 19 \beta_{10} + 19 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 56q^{4} - 6q^{5} - 36q^{6} - 68q^{9} + O(q^{10}) \) \( 14q - 56q^{4} - 6q^{5} - 36q^{6} - 68q^{9} - 40q^{10} - 146q^{11} + 176q^{14} - 206q^{15} + 224q^{16} + 154q^{19} + 24q^{20} - 220q^{21} + 144q^{24} - 286q^{25} - 180q^{26} + 790q^{29} - 232q^{30} - 320q^{31} - 200q^{34} - 426q^{35} + 272q^{36} + 1616q^{39} + 160q^{40} - 1904q^{41} + 584q^{44} + 622q^{45} + 644q^{46} + 610q^{49} + 200q^{50} - 1834q^{51} + 192q^{54} + 854q^{55} - 704q^{56} + 2814q^{59} + 824q^{60} - 3742q^{61} - 896q^{64} + 1730q^{65} - 612q^{66} + 414q^{69} + 348q^{70} - 3808q^{71} + 268q^{74} + 2904q^{75} - 616q^{76} - 1528q^{79} - 96q^{80} - 4618q^{81} + 880q^{84} + 2574q^{85} - 2024q^{86} + 2336q^{89} + 2092q^{90} - 3866q^{91} + 456q^{94} + 838q^{95} - 576q^{96} + 3342q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 212 x^{12} + 17560 x^{10} + 728073 x^{8} + 16036416 x^{6} + 183184060 x^{4} + 961600400 x^{2} + 1560250000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(3186868379 \nu^{12} + 534359144753 \nu^{10} + 31796999102640 \nu^{8} + 844061002553517 \nu^{6} + 11630063904988629 \nu^{4} + 130992968757557240 \nu^{2} + 862089169013562350\)\()/ 23726484615355650 \)
\(\beta_{3}\)\(=\)\((\)\(-3499703585 \nu^{12} - 585068510510 \nu^{10} - 35830636596552 \nu^{8} - 1060027581556725 \nu^{6} - 17113281199655430 \nu^{4} - 143349007643009696 \nu^{2} - 349399293999970160\)\()/ 9490593846142260 \)
\(\beta_{4}\)\(=\)\((\)\(-289581 \nu^{12} - 56463152 \nu^{10} - 4132854900 \nu^{8} - 142227122813 \nu^{6} - 2338196775636 \nu^{4} - 16286584250280 \nu^{2} - 31018745255000\)\()/ 613694015800 \)
\(\beta_{5}\)\(=\)\((\)\(-130865103661 \nu^{12} - 25582406911102 \nu^{10} - 1878146401786800 \nu^{8} - 64661289576689253 \nu^{6} - 1050972509419489986 \nu^{4} - 6998401679059936180 \nu^{2} - 11718636801942706000\)\()/ 94905938461422600 \)
\(\beta_{6}\)\(=\)\((\)\(-5109264790403 \nu^{13} - 553302086253686 \nu^{11} + 11731267396252320 \nu^{9} + 3396592715381521581 \nu^{7} + 141829915665819432102 \nu^{5} + 2140250136020411445820 \nu^{3} + 10343004883168433498800 \nu\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(9297613576608 \nu^{13} - 185139331460725 \nu^{12} + 1932103137868146 \nu^{11} - 36954775028636200 \nu^{10} + 153312401872801980 \nu^{9} - 2802395334754738500 \nu^{8} + 5840748749653311384 \nu^{7} - 101318102481756497925 \nu^{6} + 110718511065104546178 \nu^{5} - 1762031607651829994100 \nu^{4} + 1034963871490758904980 \nu^{3} - 12614487922741170151000 \nu^{2} + 4737027171360508063200 \nu - 22067604811859508475000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-78528469 \nu^{13} - 15504190478 \nu^{11} - 1155930465240 \nu^{9} - 40849681155237 \nu^{7} - 697518061615754 \nu^{5} - 5149286513241940 \nu^{3} - 11180999413181600 \nu\)\()/ 2424091362410000 \)
\(\beta_{9}\)\(=\)\((\)\(-14406878367011 \nu^{13} - 185139331460725 \nu^{12} - 2485405224121832 \nu^{11} - 36954775028636200 \nu^{10} - 141581134476549660 \nu^{9} - 2802395334754738500 \nu^{8} - 2444156034271789803 \nu^{7} - 101318102481756497925 \nu^{6} + 31111404600714885924 \nu^{5} - 1762031607651829994100 \nu^{4} + 1105286264529652540840 \nu^{3} - 12614487922741170151000 \nu^{2} + 5605977711807925435600 \nu - 22067604811859508475000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-17013028715369 \nu^{13} + 229642540842400 \nu^{12} - 3293692394542028 \nu^{11} + 46866526897616800 \nu^{10} - 246507496373665740 \nu^{9} + 3608225438031285000 \nu^{8} - 9208768990763228937 \nu^{7} + 130490760442966435200 \nu^{6} - 180978293949048314904 \nu^{5} + 2230072164412561892400 \nu^{4} - 1714372169926391895440 \nu^{3} + 15605126250757397257000 \nu^{2} - 5168044100910054506600 \nu + 26015723462652981340000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(17013028715369 \nu^{13} + 229642540842400 \nu^{12} + 3293692394542028 \nu^{11} + 46866526897616800 \nu^{10} + 246507496373665740 \nu^{9} + 3608225438031285000 \nu^{8} + 9208768990763228937 \nu^{7} + 130490760442966435200 \nu^{6} + 180978293949048314904 \nu^{5} + 2230072164412561892400 \nu^{4} + 1714372169926391895440 \nu^{3} + 15605126250757397257000 \nu^{2} + 5168044100910054506600 \nu + 26015723462652981340000\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(3805919420404 \nu^{13} + 748011503552023 \nu^{11} + 55178102284700640 \nu^{9} + 1900228325596386492 \nu^{7} + 30371107726805052939 \nu^{5} + 186449255521962680440 \nu^{3} + 216134304422029615600 \nu\)\()/ 18743922846130963500 \)
\(\beta_{13}\)\(=\)\((\)\(-41766387715051 \nu^{13} - 8724753222259912 \nu^{11} - 689286446470969260 \nu^{9} - 25600458153130596723 \nu^{7} - 448191100548218946516 \nu^{5} - 3178705667917618266160 \nu^{3} - 5134147721185226373400 \nu\)\()/ \)\(18\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 29\)
\(\nu^{3}\)\(=\)\(4 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} - 25 \beta_{8} + 7 \beta_{6} - 52 \beta_{1}\)
\(\nu^{4}\)\(=\)\(84 \beta_{11} + 84 \beta_{10} + 73 \beta_{9} + 73 \beta_{7} - 73 \beta_{6} - 4 \beta_{5} + 123 \beta_{4} - 72 \beta_{3} - 36 \beta_{2} + 1316\)
\(\nu^{5}\)\(=\)\(-393 \beta_{13} - 384 \beta_{12} + 729 \beta_{11} - 729 \beta_{10} - 171 \beta_{9} + 2915 \beta_{8} + 171 \beta_{7} - 576 \beta_{6} + 3297 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-5919 \beta_{11} - 5919 \beta_{10} - 4836 \beta_{9} - 4836 \beta_{7} + 4836 \beta_{6} + 954 \beta_{5} - 11426 \beta_{4} + 4734 \beta_{3} + 1077 \beta_{2} - 72393\)
\(\nu^{7}\)\(=\)\(31526 \beta_{13} + 27528 \beta_{12} - 60661 \beta_{11} + 60661 \beta_{10} + 19979 \beta_{9} - 270718 \beta_{8} - 19979 \beta_{7} + 41481 \beta_{6} - 220815 \beta_{1}\)
\(\nu^{8}\)\(=\)\(409088 \beta_{11} + 409088 \beta_{10} + 317283 \beta_{9} + 317283 \beta_{7} - 317283 \beta_{6} - 103040 \beta_{5} + 941783 \beta_{4} - 316493 \beta_{3} - 16393 \beta_{2} + 4397417\)
\(\nu^{9}\)\(=\)\(-2366262 \beta_{13} - 1851602 \beta_{12} + 4677821 \beta_{11} - 4677821 \beta_{10} - 1734830 \beta_{9} + 22387525 \beta_{8} + 1734830 \beta_{7} - 2909981 \beta_{6} + 15130036 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-28340982 \beta_{11} - 28340982 \beta_{10} - 21095829 \beta_{9} - 21095829 \beta_{7} + 21095829 \beta_{6} + 8946672 \beta_{5} - 72874659 \beta_{4} + 21654816 \beta_{3} - 1340352 \beta_{2} - 282821788\)
\(\nu^{11}\)\(=\)\(172412469 \beta_{13} + 123616692 \beta_{12} - 348171957 \beta_{11} + 348171957 \beta_{10} + 135862743 \beta_{9} - 1737485295 \beta_{8} - 135862743 \beta_{7} + 203608608 \beta_{6} - 1049728261 \beta_{1}\)
\(\nu^{12}\)\(=\)\(1972644007 \beta_{11} + 1972644007 \beta_{10} + 1427099788 \beta_{9} + 1427099788 \beta_{7} - 1427099788 \beta_{6} - 710128842 \beta_{5} + 5441110618 \beta_{4} - 1505344402 \beta_{3} + 200773799 \beta_{2} + 18839621909\)
\(\nu^{13}\)\(=\)\(-12379981858 \beta_{13} - 8321998624 \beta_{12} + 25422950533 \beta_{11} - 25422950533 \beta_{10} - 10161400047 \beta_{9} + 130037778814 \beta_{8} + 10161400047 \beta_{7} - 14285407453 \beta_{6} + 73387455835 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
6.80605i
6.41782i
3.85861i
1.66599i
3.36381i
4.96965i
8.41501i
8.41501i
4.96965i
3.36381i
1.66599i
3.85861i
6.41782i
6.80605i
2.00000i 7.80605i −4.00000 −4.43387 10.2636i −15.6121 21.1583i 8.00000i −33.9344 −20.5271 + 8.86773i
139.2 2.00000i 7.41782i −4.00000 2.14945 10.9718i −14.8356 26.7462i 8.00000i −28.0240 −21.9436 4.29890i
139.3 2.00000i 4.85861i −4.00000 8.38650 + 7.39369i −9.71722 11.5462i 8.00000i 3.39393 14.7874 16.7730i
139.4 2.00000i 2.66599i −4.00000 −3.67145 + 10.5603i −5.33198 1.62258i 8.00000i 19.8925 21.1206 + 7.34291i
139.5 2.00000i 2.36381i −4.00000 −6.34330 9.20666i 4.72762 7.27979i 8.00000i 21.4124 −18.4133 + 12.6866i
139.6 2.00000i 3.96965i −4.00000 10.8788 2.57927i 7.93930 5.88989i 8.00000i 11.2419 −5.15854 21.7575i
139.7 2.00000i 7.41501i −4.00000 −9.96609 + 5.06726i 14.8300 26.6333i 8.00000i −27.9823 10.1345 + 19.9322i
139.8 2.00000i 7.41501i −4.00000 −9.96609 5.06726i 14.8300 26.6333i 8.00000i −27.9823 10.1345 19.9322i
139.9 2.00000i 3.96965i −4.00000 10.8788 + 2.57927i 7.93930 5.88989i 8.00000i 11.2419 −5.15854 + 21.7575i
139.10 2.00000i 2.36381i −4.00000 −6.34330 + 9.20666i 4.72762 7.27979i 8.00000i 21.4124 −18.4133 12.6866i
139.11 2.00000i 2.66599i −4.00000 −3.67145 10.5603i −5.33198 1.62258i 8.00000i 19.8925 21.1206 7.34291i
139.12 2.00000i 4.85861i −4.00000 8.38650 7.39369i −9.71722 11.5462i 8.00000i 3.39393 14.7874 + 16.7730i
139.13 2.00000i 7.41782i −4.00000 2.14945 + 10.9718i −14.8356 26.7462i 8.00000i −28.0240 −21.9436 + 4.29890i
139.14 2.00000i 7.80605i −4.00000 −4.43387 + 10.2636i −15.6121 21.1583i 8.00000i −33.9344 −20.5271 8.86773i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.b.a 14
5.b even 2 1 inner 230.4.b.a 14
5.c odd 4 1 1150.4.a.y 7
5.c odd 4 1 1150.4.a.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.b.a 14 1.a even 1 1 trivial
230.4.b.a 14 5.b even 2 1 inner
1150.4.a.y 7 5.c odd 4 1
1150.4.a.z 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 223 T_{3}^{12} + 19539 T_{3}^{10} + 852894 T_{3}^{8} + 19553159 T_{3}^{6} + 231696639 T_{3}^{4} + 1302477653 T_{3}^{2} + 2723378596 \) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T^{2} )^{7} \)
$3$ \( 2723378596 + 1302477653 T^{2} + 231696639 T^{4} + 19553159 T^{6} + 852894 T^{8} + 19539 T^{10} + 223 T^{12} + T^{14} \)
$5$ \( 476837158203125 + 22888183593750 T + 4913330078125 T^{2} - 293945312500 T^{3} + 8732421875 T^{4} - 2773593750 T^{5} + 217946875 T^{6} - 9147000 T^{7} + 1743575 T^{8} - 177510 T^{9} + 4471 T^{10} - 1204 T^{11} + 161 T^{12} + 6 T^{13} + T^{14} \)
$7$ \( 146581095556624 + 64505469416944 T^{2} + 3527972102360 T^{4} + 67486596460 T^{6} + 509973305 T^{8} + 1578060 T^{10} + 2096 T^{12} + T^{14} \)
$11$ \( ( -1520797760 + 198575596 T + 38988576 T^{2} - 455360 T^{3} - 109358 T^{4} - 631 T^{5} + 73 T^{6} + T^{7} )^{2} \)
$13$ \( \)\(10\!\cdots\!44\)\( + 34540243723768967469 T^{2} + 452490786891503211 T^{4} + 678391089469599 T^{6} + 413458452030 T^{8} + 123387971 T^{10} + 17875 T^{12} + T^{14} \)
$17$ \( \)\(33\!\cdots\!00\)\( + \)\(56\!\cdots\!56\)\( T^{2} + 11457269989496190028 T^{4} + 8422342016811800 T^{6} + 2808080791889 T^{8} + 460795336 T^{10} + 35676 T^{12} + T^{14} \)
$19$ \( ( 3897418182928 - 100904860052 T - 9306645048 T^{2} + 180870142 T^{3} + 1753538 T^{4} - 27117 T^{5} - 77 T^{6} + T^{7} )^{2} \)
$23$ \( ( 529 + T^{2} )^{7} \)
$29$ \( ( 43908716160740 + 6467467381706 T - 51881011787 T^{2} - 1494071099 T^{3} + 16326882 T^{4} - 3712 T^{5} - 395 T^{6} + T^{7} )^{2} \)
$31$ \( ( -108456344870975 - 1679247428902 T + 41479291658 T^{2} + 514231961 T^{3} - 5149851 T^{4} - 42052 T^{5} + 160 T^{6} + T^{7} )^{2} \)
$37$ \( \)\(32\!\cdots\!76\)\( + \)\(13\!\cdots\!24\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{4} + \)\(56\!\cdots\!00\)\( T^{6} + 11106103668950600 T^{8} + 111087964296 T^{10} + 539049 T^{12} + T^{14} \)
$41$ \( ( -4461048431070103 - 286383952230598 T - 5435839306486 T^{2} - 41123402959 T^{3} - 104273579 T^{4} + 144276 T^{5} + 952 T^{6} + T^{7} )^{2} \)
$43$ \( \)\(30\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + \)\(18\!\cdots\!36\)\( T^{4} + \)\(84\!\cdots\!00\)\( T^{6} + 17369480297490832 T^{8} + 166880257660 T^{10} + 694656 T^{12} + T^{14} \)
$47$ \( \)\(34\!\cdots\!24\)\( + \)\(52\!\cdots\!96\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{4} + \)\(85\!\cdots\!29\)\( T^{6} + 21249691598777608 T^{8} + 211245073374 T^{10} + 798808 T^{12} + T^{14} \)
$53$ \( \)\(17\!\cdots\!36\)\( + \)\(26\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{4} + \)\(40\!\cdots\!40\)\( T^{6} + 53783511939144944 T^{8} + 339190483192 T^{10} + 966577 T^{12} + T^{14} \)
$59$ \( ( -306214699126946240 + 1352168743838400 T + 17570394341952 T^{2} - 133963830592 T^{3} + 219942096 T^{4} + 395292 T^{5} - 1407 T^{6} + T^{7} )^{2} \)
$61$ \( ( -353194702681136200 + 3443682965165420 T + 11760074125908 T^{2} - 107616895766 T^{3} - 202966384 T^{4} + 790365 T^{5} + 1871 T^{6} + T^{7} )^{2} \)
$67$ \( \)\(73\!\cdots\!00\)\( + \)\(10\!\cdots\!40\)\( T^{2} + \)\(29\!\cdots\!84\)\( T^{4} + \)\(93\!\cdots\!68\)\( T^{6} + 107590701060929640 T^{8} + 537642031896 T^{10} + 1206121 T^{12} + T^{14} \)
$71$ \( ( 6375361089013134193 + 53088774775260400 T - 129245498346378 T^{2} - 1392412886391 T^{3} - 2294009991 T^{4} - 173482 T^{5} + 1904 T^{6} + T^{7} )^{2} \)
$73$ \( \)\(16\!\cdots\!96\)\( + \)\(61\!\cdots\!20\)\( T^{2} + \)\(47\!\cdots\!04\)\( T^{4} + \)\(71\!\cdots\!13\)\( T^{6} + 446783836058135192 T^{8} + 1337740543966 T^{10} + 1885088 T^{12} + T^{14} \)
$79$ \( ( 257926872164009632 + 40155721569516640 T + 258294797901552 T^{2} + 227665705692 T^{3} - 960259724 T^{4} - 1202882 T^{5} + 764 T^{6} + T^{7} )^{2} \)
$83$ \( \)\(42\!\cdots\!04\)\( + \)\(61\!\cdots\!56\)\( T^{2} + \)\(17\!\cdots\!96\)\( T^{4} + \)\(19\!\cdots\!24\)\( T^{6} + 988178783732744092 T^{8} + 2353718926896 T^{10} + 2532285 T^{12} + T^{14} \)
$89$ \( ( \)\(40\!\cdots\!00\)\( - 255604711463700800 T - 2359976360272496 T^{2} + 1703867755788 T^{3} + 3794613044 T^{4} - 3050274 T^{5} - 1168 T^{6} + T^{7} )^{2} \)
$97$ \( \)\(57\!\cdots\!44\)\( + \)\(78\!\cdots\!60\)\( T^{2} + \)\(42\!\cdots\!36\)\( T^{4} + \)\(11\!\cdots\!00\)\( T^{6} + 18037700754978417688 T^{8} + 14865762561577 T^{10} + 6164419 T^{12} + T^{14} \)
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