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.

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

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

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

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 \(T_{2}^{3} \) \(\mathstrut -\mathstrut T_{2}^{2} \) \(\mathstrut -\mathstrut 24 T_{2} \) \(\mathstrut +\mathstrut 32 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\).