Properties

Label 17.4.a.b
Level 17
Weight 4
Character orbit 17.a
Self dual Yes
Analytic conductor 1.003
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 17.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.0030324701\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{5} + ( -24 + 6 \beta_{1} + 2 \beta_{2} ) q^{6} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{7} + ( -16 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + ( 19 - 8 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{4} + ( -2 + 2 \beta_{2} ) q^{5} + ( -24 + 6 \beta_{1} + 2 \beta_{2} ) q^{6} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{7} + ( -16 + 5 \beta_{1} - 9 \beta_{2} ) q^{8} + ( 19 - 8 \beta_{1} - 2 \beta_{2} ) q^{9} + ( -16 + 8 \beta_{2} ) q^{10} + ( -10 + 11 \beta_{1} - 2 \beta_{2} ) q^{11} + ( 16 - 26 \beta_{1} + 26 \beta_{2} ) q^{12} + ( 12 + 8 \beta_{1} + 6 \beta_{2} ) q^{13} + ( 24 + 4 \beta_{1} - 20 \beta_{2} ) q^{14} + ( 32 + 2 \beta_{1} - 12 \beta_{2} ) q^{15} + ( 48 - 11 \beta_{1} + 7 \beta_{2} ) q^{16} -17 q^{17} + ( -48 + 33 \beta_{1} - 41 \beta_{2} ) q^{18} + ( 24 + 22 \beta_{1} - 8 \beta_{2} ) q^{19} + ( -48 - 8 \beta_{1} + 24 \beta_{2} ) q^{20} + ( -54 - 12 \beta_{1} + 30 \beta_{2} ) q^{21} + ( 104 - 34 \beta_{1} + 26 \beta_{2} ) q^{22} + ( 46 - 39 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -224 + 46 \beta_{1} - 6 \beta_{2} ) q^{24} + ( -81 + 4 \beta_{1} - 20 \beta_{2} ) q^{25} + ( 16 + 2 \beta_{1} + 22 \beta_{2} ) q^{26} + ( -4 - 40 \beta_{1} + 8 \beta_{2} ) q^{27} + ( 144 + 4 \beta_{1} - 44 \beta_{2} ) q^{28} + ( -142 - 16 \beta_{1} + 30 \beta_{2} ) q^{29} + ( 112 + 16 \beta_{1} - 64 \beta_{2} ) q^{30} + ( 82 + 39 \beta_{1} + 16 \beta_{2} ) q^{31} + ( -16 + 37 \beta_{1} + 23 \beta_{2} ) q^{32} + ( -122 + 76 \beta_{1} - 34 \beta_{2} ) q^{33} + ( -17 \beta_{1} + 17 \beta_{2} ) q^{34} + ( -96 - 10 \beta_{1} + 44 \beta_{2} ) q^{35} + ( 440 - 91 \beta_{1} + 7 \beta_{2} ) q^{36} + ( 102 - 28 \beta_{1} - 50 \beta_{2} ) q^{37} + ( 240 - 28 \beta_{1} - 4 \beta_{2} ) q^{38} + ( 84 + 36 \beta_{1} - 16 \beta_{2} ) q^{39} + ( -128 - 8 \beta_{1} + 40 \beta_{2} ) q^{40} + ( -118 - 52 \beta_{1} - 60 \beta_{2} ) q^{41} + ( -336 + 120 \beta_{2} ) q^{42} + ( 204 + 2 \beta_{1} + 56 \beta_{2} ) q^{43} + ( -400 + 110 \beta_{1} - 78 \beta_{2} ) q^{44} + ( -110 - 20 \beta_{1} + 54 \beta_{2} ) q^{45} + ( -280 + 120 \beta_{1} - 136 \beta_{2} ) q^{46} + ( 228 - 48 \beta_{1} + 44 \beta_{2} ) q^{47} + ( 288 - 114 \beta_{1} + 90 \beta_{2} ) q^{48} + ( -121 + 20 \beta_{1} - 94 \beta_{2} ) q^{49} + ( 192 - 109 \beta_{1} + 29 \beta_{2} ) q^{50} + ( -34 + 17 \beta_{1} - 34 \beta_{2} ) q^{51} + ( -256 - 30 \beta_{1} + 6 \beta_{2} ) q^{52} + ( 98 + 116 \beta_{1} - 8 \beta_{2} ) q^{53} + ( -384 + 84 \beta_{1} - 52 \beta_{2} ) q^{54} + ( 24 + 18 \beta_{1} - 4 \beta_{2} ) q^{55} + ( 192 + 60 \beta_{1} - 108 \beta_{2} ) q^{56} + ( -228 + 108 \beta_{1} + 36 \beta_{2} ) q^{57} + ( -368 - 80 \beta_{1} + 200 \beta_{2} ) q^{58} + ( 212 - 130 \beta_{1} ) q^{59} + ( 384 - 176 \beta_{2} ) q^{60} + ( -2 - 64 \beta_{1} + 78 \beta_{2} ) q^{61} + ( 184 + 20 \beta_{1} + 44 \beta_{2} ) q^{62} + ( 342 + 9 \beta_{1} - 96 \beta_{2} ) q^{63} + ( -272 + 21 \beta_{1} + 103 \beta_{2} ) q^{64} + ( 128 + 28 \beta_{1} - 24 \beta_{2} ) q^{65} + ( 880 - 308 \beta_{1} + 172 \beta_{2} ) q^{66} + ( 292 + 24 \beta_{1} - 132 \beta_{2} ) q^{67} + ( -136 + 51 \beta_{1} + 17 \beta_{2} ) q^{68} + ( 254 - 280 \beta_{1} + 186 \beta_{2} ) q^{69} + ( -432 - 32 \beta_{1} + 208 \beta_{2} ) q^{70} + ( -110 + 185 \beta_{1} + 72 \beta_{2} ) q^{71} + ( -400 + 365 \beta_{1} - 273 \beta_{2} ) q^{72} + ( 274 + 16 \beta_{1} - 16 \beta_{2} ) q^{73} + ( 176 + 108 \beta_{1} - 308 \beta_{2} ) q^{74} + ( -546 + 105 \beta_{1} - 90 \beta_{2} ) q^{75} + ( -384 + 116 \beta_{1} - 244 \beta_{2} ) q^{76} + ( -174 - 18 \beta_{2} ) q^{77} + ( 416 - 4 \beta_{1} - 60 \beta_{2} ) q^{78} + ( -138 - 267 \beta_{1} + 180 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 40 \beta_{2} ) q^{80} + ( -137 - 20 \beta_{1} + 94 \beta_{2} ) q^{81} + ( 64 - 74 \beta_{1} - 166 \beta_{2} ) q^{82} + ( -756 + 82 \beta_{1} + 128 \beta_{2} ) q^{83} + ( -528 - 120 \beta_{1} + 456 \beta_{2} ) q^{84} + ( 34 - 34 \beta_{2} ) q^{85} + ( -432 + 256 \beta_{1} - 32 \beta_{2} ) q^{86} + ( 352 + 46 \beta_{1} - 372 \beta_{2} ) q^{87} + ( 672 - 426 \beta_{1} + 178 \beta_{2} ) q^{88} + ( -20 - 276 \beta_{1} + 110 \beta_{2} ) q^{89} + ( -592 - 16 \beta_{1} + 232 \beta_{2} ) q^{90} + ( -324 - 64 \beta_{1} + 44 \beta_{2} ) q^{91} + ( 1680 - 344 \beta_{1} + 144 \beta_{2} ) q^{92} + ( 218 + 152 \beta_{1} + 22 \beta_{2} ) q^{93} + ( -736 + 368 \beta_{1} - 192 \beta_{2} ) q^{94} + ( -120 + 28 \beta_{1} + 112 \beta_{2} ) q^{95} + ( 160 + 238 \beta_{1} - 198 \beta_{2} ) q^{96} + ( -50 + 140 \beta_{1} + 120 \beta_{2} ) q^{97} + ( 912 - 255 \beta_{1} - 121 \beta_{2} ) q^{98} + ( -1042 + 281 \beta_{1} - 206 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 4q^{3} + 25q^{4} - 8q^{5} - 74q^{6} + 22q^{7} - 39q^{8} + 59q^{9} + O(q^{10}) \) \( 3q + q^{2} + 4q^{3} + 25q^{4} - 8q^{5} - 74q^{6} + 22q^{7} - 39q^{8} + 59q^{9} - 56q^{10} - 28q^{11} + 22q^{12} + 30q^{13} + 92q^{14} + 108q^{15} + 137q^{16} - 51q^{17} - 103q^{18} + 80q^{19} - 168q^{20} - 192q^{21} + 286q^{22} + 142q^{23} - 666q^{24} - 223q^{25} + 26q^{26} - 20q^{27} + 476q^{28} - 456q^{29} + 400q^{30} + 230q^{31} - 71q^{32} - 332q^{33} - 17q^{34} - 332q^{35} + 1313q^{36} + 356q^{37} + 724q^{38} + 268q^{39} - 424q^{40} - 294q^{41} - 1128q^{42} + 556q^{43} - 1122q^{44} - 384q^{45} - 704q^{46} + 640q^{47} + 774q^{48} - 269q^{49} + 547q^{50} - 68q^{51} - 774q^{52} + 302q^{53} - 1100q^{54} + 76q^{55} + 684q^{56} - 720q^{57} - 1304q^{58} + 636q^{59} + 1328q^{60} - 84q^{61} + 508q^{62} + 1122q^{63} - 919q^{64} + 408q^{65} + 2468q^{66} + 1008q^{67} - 425q^{68} + 576q^{69} - 1504q^{70} - 402q^{71} - 927q^{72} + 838q^{73} + 836q^{74} - 1548q^{75} - 908q^{76} - 504q^{77} + 1308q^{78} - 594q^{79} - 40q^{80} - 505q^{81} + 358q^{82} - 2396q^{83} - 2040q^{84} + 136q^{85} - 1264q^{86} + 1428q^{87} + 1838q^{88} - 170q^{89} - 2008q^{90} - 1016q^{91} + 4896q^{92} + 632q^{93} - 2016q^{94} - 472q^{95} + 678q^{96} - 270q^{97} + 2857q^{98} - 2920q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 14 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + 10\)

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
−3.58966
3.87707
−0.287410
−5.03251 8.47535 17.3261 0.885690 −42.6523 3.81828 −46.9339 44.8316 −4.45724
1.2 1.36122 3.15463 −6.14708 3.03171 4.29415 −7.94049 −19.2573 −17.0483 4.12682
1.3 4.67129 −7.62999 13.8209 −11.9174 −35.6419 26.1222 27.1912 31.2167 −55.6696
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 24 T_{2} + 32 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\).