Properties

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

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Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 17.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.0030324701\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4669632.2
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( 1 - \beta_{2} ) q^{4} \) \( -\beta_{3} q^{5} \) \( + ( -2 \beta_{1} + \beta_{3} ) q^{6} \) \( + ( -\beta_{1} + \beta_{3} ) q^{7} \) \( + ( -1 - 7 \beta_{2} ) q^{8} \) \( + ( -13 + 6 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( 1 - \beta_{2} ) q^{4} \) \( -\beta_{3} q^{5} \) \( + ( -2 \beta_{1} + \beta_{3} ) q^{6} \) \( + ( -\beta_{1} + \beta_{3} ) q^{7} \) \( + ( -1 - 7 \beta_{2} ) q^{8} \) \( + ( -13 + 6 \beta_{2} ) q^{9} \) \( + ( -6 \beta_{1} - \beta_{3} ) q^{10} \) \( + 7 \beta_{1} q^{11} \) \( + ( 2 \beta_{1} - \beta_{3} ) q^{12} \) \( + ( 42 - 14 \beta_{2} ) q^{13} \) \( + 8 \beta_{1} q^{14} \) \( + ( -8 + 28 \beta_{2} ) q^{15} \) \( + ( -63 + 7 \beta_{2} ) q^{16} \) \( + ( 21 - 8 \beta_{1} + 14 \beta_{2} + \beta_{3} ) q^{17} \) \( + ( 61 - 13 \beta_{2} ) q^{18} \) \( -28 q^{19} \) \( + ( 6 \beta_{1} + \beta_{3} ) q^{20} \) \( + ( 48 - 34 \beta_{2} ) q^{21} \) \( + ( -14 \beta_{1} + 7 \beta_{3} ) q^{22} \) \( + ( -7 \beta_{1} - 7 \beta_{3} ) q^{23} \) \( + ( 6 \beta_{1} - 7 \beta_{3} ) q^{24} \) \( + ( -91 - 48 \beta_{2} ) q^{25} \) \( + ( -154 + 42 \beta_{2} ) q^{26} \) \( + ( 8 \beta_{1} + 6 \beta_{3} ) q^{27} \) \( -8 \beta_{1} q^{28} \) \( + 7 \beta_{3} q^{29} \) \( + ( 232 - 8 \beta_{2} ) q^{30} \) \( + ( -7 \beta_{1} - \beta_{3} ) q^{31} \) \( + ( 127 - 7 \beta_{2} ) q^{32} \) \( + ( -280 + 42 \beta_{2} ) q^{33} \) \( + ( 91 + 22 \beta_{1} + 21 \beta_{2} - 7 \beta_{3} ) q^{34} \) \( + ( 224 + 20 \beta_{2} ) q^{35} \) \( + ( -61 + 13 \beta_{2} ) q^{36} \) \( + ( -28 \beta_{1} - 7 \beta_{3} ) q^{37} \) \( + ( 28 - 28 \beta_{2} ) q^{38} \) \( + ( 56 \beta_{1} - 14 \beta_{3} ) q^{39} \) \( + ( 42 \beta_{1} + 15 \beta_{3} ) q^{40} \) \( + ( -20 \beta_{1} + 14 \beta_{3} ) q^{41} \) \( + ( -320 + 48 \beta_{2} ) q^{42} \) \( + ( -92 - 76 \beta_{2} ) q^{43} \) \( + ( 14 \beta_{1} - 7 \beta_{3} ) q^{44} \) \( + ( -36 \beta_{1} + \beta_{3} ) q^{45} \) \( + ( -28 \beta_{1} - 14 \beta_{3} ) q^{46} \) \( + ( 224 + 28 \beta_{2} ) q^{47} \) \( + ( -70 \beta_{1} + 7 \beta_{3} ) q^{48} \) \( + ( 71 + 14 \beta_{2} ) q^{49} \) \( + ( -293 - 91 \beta_{2} ) q^{50} \) \( + ( 328 + 7 \beta_{1} - 76 \beta_{2} + 14 \beta_{3} ) q^{51} \) \( + ( 154 - 42 \beta_{2} ) q^{52} \) \( + ( -98 - 92 \beta_{2} ) q^{53} \) \( + ( 20 \beta_{1} + 14 \beta_{3} ) q^{54} \) \( + ( -56 + 196 \beta_{2} ) q^{55} \) \( + ( -48 \beta_{1} - 8 \beta_{3} ) q^{56} \) \( -28 \beta_{1} q^{57} \) \( + ( 42 \beta_{1} + 7 \beta_{3} ) q^{58} \) \( + ( -364 + 140 \beta_{2} ) q^{59} \) \( + ( -232 + 8 \beta_{2} ) q^{60} \) \( + ( 64 \beta_{1} - 21 \beta_{3} ) q^{61} \) \( + ( 8 \beta_{1} - 8 \beta_{3} ) q^{62} \) \( + ( 55 \beta_{1} - 7 \beta_{3} ) q^{63} \) \( + ( 321 + 71 \beta_{2} ) q^{64} \) \( + ( 84 \beta_{1} - 14 \beta_{3} ) q^{65} \) \( + ( 616 - 280 \beta_{2} ) q^{66} \) \( + ( -28 - 64 \beta_{2} ) q^{67} \) \( + ( -91 - 22 \beta_{1} - 21 \beta_{2} + 7 \beta_{3} ) q^{68} \) \( + ( 224 + 154 \beta_{2} ) q^{69} \) \( + ( -64 + 224 \beta_{2} ) q^{70} \) \( + ( -21 \beta_{1} + 7 \beta_{3} ) q^{71} \) \( + ( -323 + 43 \beta_{2} ) q^{72} \) \( + ( -48 \beta_{1} + 56 \beta_{3} ) q^{73} \) \( + ( 14 \beta_{1} - 35 \beta_{3} ) q^{74} \) \( + ( -43 \beta_{1} - 48 \beta_{3} ) q^{75} \) \( + ( -28 + 28 \beta_{2} ) q^{76} \) \( + ( 336 - 238 \beta_{2} ) q^{77} \) \( + ( -196 \beta_{1} + 42 \beta_{3} ) q^{78} \) \( + ( 77 \beta_{1} + 21 \beta_{3} ) q^{79} \) \( + ( -42 \beta_{1} + 49 \beta_{3} ) q^{80} \) \( + ( -623 + 42 \beta_{2} ) q^{81} \) \( + ( 124 \beta_{1} - 6 \beta_{3} ) q^{82} \) \( + ( -756 + 84 \beta_{2} ) q^{83} \) \( + ( 320 - 48 \beta_{2} ) q^{84} \) \( + ( 280 - 84 \beta_{1} - 176 \beta_{2} - 49 \beta_{3} ) q^{85} \) \( + ( -516 - 92 \beta_{2} ) q^{86} \) \( + ( 56 - 196 \beta_{2} ) q^{87} \) \( + ( 42 \beta_{1} - 49 \beta_{3} ) q^{88} \) \( + ( 518 - 42 \beta_{2} ) q^{89} \) \( + ( 78 \beta_{1} - 35 \beta_{3} ) q^{90} \) \( + ( -140 \beta_{1} + 28 \beta_{3} ) q^{91} \) \( + ( 28 \beta_{1} + 14 \beta_{3} ) q^{92} \) \( + ( 272 - 14 \beta_{2} ) q^{93} \) \( + 224 \beta_{2} q^{94} \) \( + 28 \beta_{3} q^{95} \) \( + ( 134 \beta_{1} - 7 \beta_{3} ) q^{96} \) \( + ( 28 \beta_{1} - 22 \beta_{3} ) q^{97} \) \( + ( 41 + 71 \beta_{2} ) q^{98} \) \( + ( -133 \beta_{1} + 42 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut -\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut -\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut 140q^{13} \) \(\mathstrut +\mathstrut 24q^{15} \) \(\mathstrut -\mathstrut 238q^{16} \) \(\mathstrut +\mathstrut 112q^{17} \) \(\mathstrut +\mathstrut 218q^{18} \) \(\mathstrut -\mathstrut 112q^{19} \) \(\mathstrut +\mathstrut 124q^{21} \) \(\mathstrut -\mathstrut 460q^{25} \) \(\mathstrut -\mathstrut 532q^{26} \) \(\mathstrut +\mathstrut 912q^{30} \) \(\mathstrut +\mathstrut 494q^{32} \) \(\mathstrut -\mathstrut 1036q^{33} \) \(\mathstrut +\mathstrut 406q^{34} \) \(\mathstrut +\mathstrut 936q^{35} \) \(\mathstrut -\mathstrut 218q^{36} \) \(\mathstrut +\mathstrut 56q^{38} \) \(\mathstrut -\mathstrut 1184q^{42} \) \(\mathstrut -\mathstrut 520q^{43} \) \(\mathstrut +\mathstrut 952q^{47} \) \(\mathstrut +\mathstrut 312q^{49} \) \(\mathstrut -\mathstrut 1354q^{50} \) \(\mathstrut +\mathstrut 1160q^{51} \) \(\mathstrut +\mathstrut 532q^{52} \) \(\mathstrut -\mathstrut 576q^{53} \) \(\mathstrut +\mathstrut 168q^{55} \) \(\mathstrut -\mathstrut 1176q^{59} \) \(\mathstrut -\mathstrut 912q^{60} \) \(\mathstrut +\mathstrut 1426q^{64} \) \(\mathstrut +\mathstrut 1904q^{66} \) \(\mathstrut -\mathstrut 240q^{67} \) \(\mathstrut -\mathstrut 406q^{68} \) \(\mathstrut +\mathstrut 1204q^{69} \) \(\mathstrut +\mathstrut 192q^{70} \) \(\mathstrut -\mathstrut 1206q^{72} \) \(\mathstrut -\mathstrut 56q^{76} \) \(\mathstrut +\mathstrut 868q^{77} \) \(\mathstrut -\mathstrut 2408q^{81} \) \(\mathstrut -\mathstrut 2856q^{83} \) \(\mathstrut +\mathstrut 1184q^{84} \) \(\mathstrut +\mathstrut 768q^{85} \) \(\mathstrut -\mathstrut 2248q^{86} \) \(\mathstrut -\mathstrut 168q^{87} \) \(\mathstrut +\mathstrut 1988q^{89} \) \(\mathstrut +\mathstrut 1060q^{93} \) \(\mathstrut +\mathstrut 448q^{94} \) \(\mathstrut +\mathstrut 306q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(74\) \(x^{2}\mathstrut +\mathstrut \) \(1072\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 40 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 46 \nu \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(6\) \(\beta_{2}\mathstrut -\mathstrut \) \(40\)
\(\nu^{3}\)\(=\)\(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(46\) \(\beta_{1}\)

Character Values

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

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
7.36435i
7.36435i
4.44593i
4.44593i
−3.37228 7.36435i 3.37228 10.1060i 24.8347i 17.4703i 15.6060 −27.2337 34.0802i
16.2 −3.37228 7.36435i 3.37228 10.1060i 24.8347i 17.4703i 15.6060 −27.2337 34.0802i
16.3 2.37228 4.44593i −2.37228 19.4389i 10.5470i 14.9929i −24.6060 7.23369 46.1145i
16.4 2.37228 4.44593i −2.37228 19.4389i 10.5470i 14.9929i −24.6060 7.23369 46.1145i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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