Properties

Label 17.4.c.a
Level 17
Weight 4
Character orbit 17.c
Analytic conductor 1.003
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 17.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.0030324701\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} - \beta_{3} ) q^{2} \) \( + \beta_{4} q^{3} \) \( + ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{5} \) \( + ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{6} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{7} \) \( + ( -5 \beta_{1} + 13 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{8} \) \( + ( -6 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} - \beta_{3} ) q^{2} \) \( + \beta_{4} q^{3} \) \( + ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{4} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{5} \) \( + ( 3 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{6} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{7} \) \( + ( -5 \beta_{1} + 13 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{8} \) \( + ( -6 \beta_{1} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{9} \) \( + ( 10 + 4 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - \beta_{5} - \beta_{6} ) q^{10} \) \( + ( -14 + 14 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{11} \) \( + ( -23 + 9 \beta_{1} - 9 \beta_{2} - 23 \beta_{3} - 3 \beta_{4} + 5 \beta_{7} ) q^{12} \) \( + ( -12 + 6 \beta_{2} - 5 \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{13} \) \( + ( 14 + 10 \beta_{1} - 10 \beta_{2} + 14 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{14} \) \( + ( 2 \beta_{1} - 30 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{15} \) \( + ( 53 + 25 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} - \beta_{6} - \beta_{7} ) q^{16} \) \( + ( -2 - 9 \beta_{1} - 15 \beta_{2} + 25 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{17} \) \( + ( 55 - 5 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{18} \) \( + ( 10 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} \) \( + ( -38 + 4 \beta_{1} - 4 \beta_{2} - 38 \beta_{3} - 13 \beta_{4} - \beta_{7} ) q^{20} \) \( + ( -30 + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{21} \) \( + ( 3 - 17 \beta_{1} + 17 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{7} ) q^{22} \) \( + ( -3 + 9 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} + 6 \beta_{5} + \beta_{6} ) q^{23} \) \( + ( -111 - 33 \beta_{1} - 33 \beta_{2} + 111 \beta_{3} - 3 \beta_{5} + 13 \beta_{6} ) q^{24} \) \( + ( -10 \beta_{1} - 21 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} ) q^{25} \) \( + ( -34 \beta_{1} - 34 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - \beta_{6} + \beta_{7} ) q^{26} \) \( + ( 66 + 12 \beta_{1} + 12 \beta_{2} - 66 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} ) q^{27} \) \( + ( -94 + 14 \beta_{1} + 14 \beta_{2} + 94 \beta_{3} - 14 \beta_{5} - 6 \beta_{6} ) q^{28} \) \( + ( 12 + \beta_{1} - \beta_{2} + 12 \beta_{3} + 21 \beta_{4} - 25 \beta_{7} ) q^{29} \) \( + ( -12 + 4 \beta_{2} + 22 \beta_{4} + 22 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{30} \) \( + ( 73 + 31 \beta_{1} - 31 \beta_{2} + 73 \beta_{3} - 16 \beta_{4} + 13 \beta_{7} ) q^{31} \) \( + ( 37 \beta_{1} - 301 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} ) q^{32} \) \( + ( 100 + 14 \beta_{2} - 25 \beta_{4} - 25 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{33} \) \( + ( 129 - 6 \beta_{1} + 7 \beta_{2} + 164 \beta_{3} + 12 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} - 20 \beta_{7} ) q^{34} \) \( + ( 138 - 10 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 17 \beta_{6} + 17 \beta_{7} ) q^{35} \) \( + ( 19 \beta_{1} + 37 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{36} \) \( + ( -72 - 15 \beta_{1} + 15 \beta_{2} - 72 \beta_{3} + 33 \beta_{4} + \beta_{7} ) q^{37} \) \( + ( -100 + 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{38} \) \( + ( -106 + 28 \beta_{1} - 28 \beta_{2} - 106 \beta_{3} + 26 \beta_{4} + 2 \beta_{7} ) q^{39} \) \( + ( -46 - 16 \beta_{1} - 16 \beta_{2} + 46 \beta_{3} - 55 \beta_{5} + 13 \beta_{6} ) q^{40} \) \( + ( -119 + 16 \beta_{1} + 16 \beta_{2} + 119 \beta_{3} - 8 \beta_{5} - 8 \beta_{6} ) q^{41} \) \( + ( 4 \beta_{1} - 20 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{42} \) \( + ( 36 \beta_{1} - 92 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{43} \) \( + ( 71 - 15 \beta_{1} - 15 \beta_{2} - 71 \beta_{3} + 31 \beta_{5} - 9 \beta_{6} ) q^{44} \) \( + ( -140 - 25 \beta_{1} - 25 \beta_{2} + 140 \beta_{3} + 49 \beta_{5} - 17 \beta_{6} ) q^{45} \) \( + ( -124 - 124 \beta_{3} - 36 \beta_{4} + 24 \beta_{7} ) q^{46} \) \( + ( -40 + 52 \beta_{2} - 20 \beta_{4} - 20 \beta_{5} - 12 \beta_{6} - 12 \beta_{7} ) q^{47} \) \( + ( 319 - 81 \beta_{1} + 81 \beta_{2} + 319 \beta_{3} + 51 \beta_{4} - 29 \beta_{7} ) q^{48} \) \( + ( -96 \beta_{1} - 91 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} + 31 \beta_{6} - 31 \beta_{7} ) q^{49} \) \( + ( 59 - 81 \beta_{2} - 17 \beta_{4} - 17 \beta_{5} - \beta_{6} - \beta_{7} ) q^{50} \) \( + ( 32 + 8 \beta_{1} + 2 \beta_{2} + 42 \beta_{3} - 50 \beta_{4} - 47 \beta_{5} - \beta_{6} + 21 \beta_{7} ) q^{51} \) \( + ( 334 + 26 \beta_{2} + 21 \beta_{4} + 21 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} ) q^{52} \) \( + ( -86 \beta_{1} + 134 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} - 21 \beta_{6} + 21 \beta_{7} ) q^{53} \) \( + ( -204 + 44 \beta_{1} - 44 \beta_{2} - 204 \beta_{3} - 12 \beta_{4} + 20 \beta_{7} ) q^{54} \) \( + ( -246 - 30 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{55} \) \( + ( 86 - 10 \beta_{1} + 10 \beta_{2} + 86 \beta_{3} + 30 \beta_{4} - 2 \beta_{7} ) q^{56} \) \( + ( -34 + 34 \beta_{3} + 28 \beta_{5} - 14 \beta_{6} ) q^{57} \) \( + ( 38 + 108 \beta_{1} + 108 \beta_{2} - 38 \beta_{3} + 61 \beta_{5} - 19 \beta_{6} ) q^{58} \) \( + ( 80 \beta_{1} + 44 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} ) q^{59} \) \( + ( 84 \beta_{1} - 396 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{60} \) \( + ( 160 - 25 \beta_{1} - 25 \beta_{2} - 160 \beta_{3} - 47 \beta_{5} - 11 \beta_{6} ) q^{61} \) \( + ( -334 + 18 \beta_{1} + 18 \beta_{2} + 334 \beta_{3} - 110 \beta_{5} + 78 \beta_{6} ) q^{62} \) \( + ( 1 - 53 \beta_{1} + 53 \beta_{2} + \beta_{3} - 24 \beta_{4} + 57 \beta_{7} ) q^{63} \) \( + ( -405 - 249 \beta_{2} - 25 \beta_{4} - 25 \beta_{5} + 49 \beta_{6} + 49 \beta_{7} ) q^{64} \) \( + ( 74 - 20 \beta_{1} + 20 \beta_{2} + 74 \beta_{3} - 72 \beta_{4} + 18 \beta_{7} ) q^{65} \) \( + ( 26 \beta_{1} - 126 \beta_{3} + 61 \beta_{4} - 61 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} ) q^{66} \) \( + ( 98 - 90 \beta_{2} + 81 \beta_{4} + 81 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{67} \) \( + ( 240 + 111 \beta_{1} + 134 \beta_{2} - 25 \beta_{3} + 16 \beta_{4} + 81 \beta_{5} - 33 \beta_{6} + 30 \beta_{7} ) q^{68} \) \( + ( 230 + 52 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} ) q^{69} \) \( + ( 40 \beta_{1} - 64 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} ) q^{70} \) \( + ( 173 - 71 \beta_{1} + 71 \beta_{2} + 173 \beta_{3} + 14 \beta_{4} - 59 \beta_{7} ) q^{71} \) \( + ( 173 + 25 \beta_{2} - 73 \beta_{4} - 73 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{72} \) \( + ( -209 + 70 \beta_{1} - 70 \beta_{2} - 209 \beta_{3} - 46 \beta_{4} - 30 \beta_{7} ) q^{73} \) \( + ( 208 - 42 \beta_{1} - 42 \beta_{2} - 208 \beta_{3} + 129 \beta_{5} - 63 \beta_{6} ) q^{74} \) \( + ( 166 + 58 \beta_{1} + 58 \beta_{2} - 166 \beta_{3} - 39 \beta_{5} - 18 \beta_{6} ) q^{75} \) \( + ( -76 \beta_{1} + 108 \beta_{3} + 24 \beta_{4} - 24 \beta_{5} - 8 \beta_{6} + 8 \beta_{7} ) q^{76} \) \( + ( 88 \beta_{1} + 386 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + \beta_{6} - \beta_{7} ) q^{77} \) \( + ( -362 - 86 \beta_{1} - 86 \beta_{2} + 362 \beta_{3} + 22 \beta_{5} + 30 \beta_{6} ) q^{78} \) \( + ( 125 - 123 \beta_{1} - 123 \beta_{2} - 125 \beta_{3} + 2 \beta_{5} - 43 \beta_{6} ) q^{79} \) \( + ( 86 - 108 \beta_{1} + 108 \beta_{2} + 86 \beta_{3} + 93 \beta_{4} - 95 \beta_{7} ) q^{80} \) \( + ( 35 + 150 \beta_{2} + 15 \beta_{4} + 15 \beta_{5} ) q^{81} \) \( + ( -41 - 103 \beta_{1} + 103 \beta_{2} - 41 \beta_{3} - 8 \beta_{4} + 24 \beta_{7} ) q^{82} \) \( + ( 24 \beta_{1} - 100 \beta_{3} - \beta_{4} + \beta_{5} - 76 \beta_{6} + 76 \beta_{7} ) q^{83} \) \( + ( -356 - 44 \beta_{2} - 18 \beta_{4} - 18 \beta_{5} - 26 \beta_{6} - 26 \beta_{7} ) q^{84} \) \( + ( -316 - 11 \beta_{1} - 109 \beta_{2} - 232 \beta_{3} + 39 \beta_{4} + 54 \beta_{5} + 46 \beta_{6} - 31 \beta_{7} ) q^{85} \) \( + ( -550 - 74 \beta_{2} - 45 \beta_{4} - 45 \beta_{5} + 39 \beta_{6} + 39 \beta_{7} ) q^{86} \) \( + ( -174 \beta_{1} + 382 \beta_{3} - 28 \beta_{4} + 28 \beta_{5} + 49 \beta_{6} - 49 \beta_{7} ) q^{87} \) \( + ( 49 - 7 \beta_{1} + 7 \beta_{2} + 49 \beta_{3} - 135 \beta_{4} + 25 \beta_{7} ) q^{88} \) \( + ( -290 - 16 \beta_{4} - 16 \beta_{5} + 41 \beta_{6} + 41 \beta_{7} ) q^{89} \) \( + ( 310 - 40 \beta_{1} + 40 \beta_{2} + 310 \beta_{3} - 97 \beta_{4} - \beta_{7} ) q^{90} \) \( + ( -220 + 106 \beta_{1} + 106 \beta_{2} + 220 \beta_{3} - 56 \beta_{5} + 64 \beta_{6} ) q^{91} \) \( + ( -232 - 160 \beta_{1} - 160 \beta_{2} + 232 \beta_{3} - 60 \beta_{5} + 44 \beta_{6} ) q^{92} \) \( + ( 184 \beta_{1} - 530 \beta_{3} + 46 \beta_{4} - 46 \beta_{5} - 57 \beta_{6} + 57 \beta_{7} ) q^{93} \) \( + ( -8 \beta_{1} - 440 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} - 32 \beta_{6} + 32 \beta_{7} ) q^{94} \) \( + ( 178 + 26 \beta_{1} + 26 \beta_{2} - 178 \beta_{3} - 24 \beta_{5} + 30 \beta_{6} ) q^{95} \) \( + ( 527 + 193 \beta_{1} + 193 \beta_{2} - 527 \beta_{3} + 291 \beta_{5} - 109 \beta_{6} ) q^{96} \) \( + ( 233 + 168 \beta_{1} - 168 \beta_{2} + 233 \beta_{3} - 24 \beta_{4} - 14 \beta_{7} ) q^{97} \) \( + ( 879 + 55 \beta_{2} + 36 \beta_{4} + 36 \beta_{5} - 76 \beta_{6} - 76 \beta_{7} ) q^{98} \) \( + ( -248 + 144 \beta_{1} - 144 \beta_{2} - 248 \beta_{3} + 189 \beta_{4} - 52 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 22q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 36q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 22q^{6} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 78q^{10} \) \(\mathstrut -\mathstrut 108q^{11} \) \(\mathstrut -\mathstrut 174q^{12} \) \(\mathstrut -\mathstrut 88q^{13} \) \(\mathstrut +\mathstrut 108q^{14} \) \(\mathstrut +\mathstrut 420q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 428q^{18} \) \(\mathstrut -\mathstrut 306q^{20} \) \(\mathstrut -\mathstrut 260q^{21} \) \(\mathstrut +\mathstrut 30q^{22} \) \(\mathstrut -\mathstrut 22q^{23} \) \(\mathstrut -\mathstrut 862q^{24} \) \(\mathstrut +\mathstrut 540q^{27} \) \(\mathstrut -\mathstrut 764q^{28} \) \(\mathstrut +\mathstrut 46q^{29} \) \(\mathstrut -\mathstrut 120q^{30} \) \(\mathstrut +\mathstrut 610q^{31} \) \(\mathstrut +\mathstrut 816q^{33} \) \(\mathstrut +\mathstrut 1002q^{34} \) \(\mathstrut +\mathstrut 1172q^{35} \) \(\mathstrut -\mathstrut 574q^{37} \) \(\mathstrut -\mathstrut 768q^{38} \) \(\mathstrut -\mathstrut 844q^{39} \) \(\mathstrut -\mathstrut 342q^{40} \) \(\mathstrut -\mathstrut 968q^{41} \) \(\mathstrut +\mathstrut 550q^{44} \) \(\mathstrut -\mathstrut 1154q^{45} \) \(\mathstrut -\mathstrut 944q^{46} \) \(\mathstrut -\mathstrut 368q^{47} \) \(\mathstrut +\mathstrut 2494q^{48} \) \(\mathstrut +\mathstrut 468q^{50} \) \(\mathstrut +\mathstrut 296q^{51} \) \(\mathstrut +\mathstrut 2564q^{52} \) \(\mathstrut -\mathstrut 1592q^{54} \) \(\mathstrut -\mathstrut 1996q^{55} \) \(\mathstrut +\mathstrut 684q^{56} \) \(\mathstrut -\mathstrut 300q^{57} \) \(\mathstrut +\mathstrut 266q^{58} \) \(\mathstrut +\mathstrut 1258q^{61} \) \(\mathstrut -\mathstrut 2516q^{62} \) \(\mathstrut +\mathstrut 122q^{63} \) \(\mathstrut -\mathstrut 3044q^{64} \) \(\mathstrut +\mathstrut 628q^{65} \) \(\mathstrut +\mathstrut 764q^{67} \) \(\mathstrut +\mathstrut 1914q^{68} \) \(\mathstrut +\mathstrut 1812q^{69} \) \(\mathstrut +\mathstrut 1266q^{71} \) \(\mathstrut +\mathstrut 1404q^{72} \) \(\mathstrut -\mathstrut 1732q^{73} \) \(\mathstrut +\mathstrut 1538q^{74} \) \(\mathstrut +\mathstrut 1292q^{75} \) \(\mathstrut -\mathstrut 2836q^{78} \) \(\mathstrut +\mathstrut 914q^{79} \) \(\mathstrut +\mathstrut 498q^{80} \) \(\mathstrut +\mathstrut 280q^{81} \) \(\mathstrut -\mathstrut 280q^{82} \) \(\mathstrut -\mathstrut 2952q^{84} \) \(\mathstrut -\mathstrut 2498q^{85} \) \(\mathstrut -\mathstrut 4244q^{86} \) \(\mathstrut +\mathstrut 442q^{88} \) \(\mathstrut -\mathstrut 2156q^{89} \) \(\mathstrut +\mathstrut 2478q^{90} \) \(\mathstrut -\mathstrut 1632q^{91} \) \(\mathstrut -\mathstrut 1768q^{92} \) \(\mathstrut +\mathstrut 1484q^{95} \) \(\mathstrut +\mathstrut 3998q^{96} \) \(\mathstrut +\mathstrut 1836q^{97} \) \(\mathstrut +\mathstrut 6728q^{98} \) \(\mathstrut -\mathstrut 2088q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(46\) \(x^{6}\mathstrut +\mathstrut \) \(561\) \(x^{4}\mathstrut +\mathstrut \) \(836\) \(x^{2}\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} - 23 \nu^{2} - 16 \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 46 \nu^{5} + 545 \nu^{3} + 468 \nu \)\()/160\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 172 \nu^{4} - 413 \nu^{3} + 1864 \nu^{2} + 396 \nu + 1120 \)\()/160\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{6} + 42 \nu^{5} + 172 \nu^{4} + 413 \nu^{3} + 1864 \nu^{2} - 396 \nu + 1120 \)\()/160\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{6} - 226 \nu^{5} + 180 \nu^{4} - 2673 \nu^{3} + 2128 \nu^{2} - 3156 \nu + 2208 \)\()/160\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{6} + 226 \nu^{5} + 180 \nu^{4} + 2673 \nu^{3} + 2128 \nu^{2} + 3156 \nu + 2208 \)\()/160\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(12\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(23\) \(\beta_{7}\mathstrut -\mathstrut \) \(23\) \(\beta_{6}\mathstrut +\mathstrut \) \(23\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(33\) \(\beta_{2}\mathstrut +\mathstrut \) \(260\)
\(\nu^{5}\)\(=\)\(-\)\(33\) \(\beta_{7}\mathstrut +\mathstrut \) \(33\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(304\) \(\beta_{3}\mathstrut +\mathstrut \) \(477\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(523\) \(\beta_{7}\mathstrut +\mathstrut \) \(523\) \(\beta_{6}\mathstrut -\mathstrut \) \(503\) \(\beta_{5}\mathstrut -\mathstrut \) \(503\) \(\beta_{4}\mathstrut +\mathstrut \) \(953\) \(\beta_{2}\mathstrut -\mathstrut \) \(5868\)
\(\nu^{7}\)\(=\)\(973\) \(\beta_{7}\mathstrut -\mathstrut \) \(973\) \(\beta_{6}\mathstrut -\mathstrut \) \(53\) \(\beta_{5}\mathstrut +\mathstrut \) \(53\) \(\beta_{4}\mathstrut -\mathstrut \) \(9464\) \(\beta_{3}\mathstrut -\mathstrut \) \(10965\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
4.93651i
0.648995i
1.11783i
4.46767i
4.46767i
1.11783i
0.648995i
4.93651i
3.93651i −0.299807 + 0.299807i −7.49613 1.37942 1.37942i 1.18019 + 1.18019i 17.9849 + 17.9849i 1.98349i 26.8202i −5.43011 5.43011i
4.2 0.351005i 2.28193 2.28193i 7.87680 −9.32676 + 9.32676i 0.800971 + 0.800971i −23.5385 23.5385i 5.57284i 16.5856i −3.27374 3.27374i
4.3 2.11783i −5.92758 + 5.92758i 3.51478 10.1567 10.1567i −12.5536 12.5536i 3.21600 + 3.21600i 24.3864i 43.2725i 21.5102 + 21.5102i
4.4 5.46767i 3.94546 3.94546i −21.8954 4.79064 4.79064i 21.5725 + 21.5725i 3.33761 + 3.33761i 75.9757i 4.13329i 26.1937 + 26.1937i
13.1 5.46767i 3.94546 + 3.94546i −21.8954 4.79064 + 4.79064i 21.5725 21.5725i 3.33761 3.33761i 75.9757i 4.13329i 26.1937 26.1937i
13.2 2.11783i −5.92758 5.92758i 3.51478 10.1567 + 10.1567i −12.5536 + 12.5536i 3.21600 3.21600i 24.3864i 43.2725i 21.5102 21.5102i
13.3 0.351005i 2.28193 + 2.28193i 7.87680 −9.32676 9.32676i 0.800971 0.800971i −23.5385 + 23.5385i 5.57284i 16.5856i −3.27374 + 3.27374i
13.4 3.93651i −0.299807 0.299807i −7.49613 1.37942 + 1.37942i 1.18019 1.18019i 17.9849 17.9849i 1.98349i 26.8202i −5.43011 + 5.43011i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(17, [\chi])\).