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 \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00303247010\)
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: \( 2^{2} \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 36q^{4} + 14q^{5} + 22q^{6} + 2q^{7} + O(q^{10}) \) \( 8q - 36q^{4} + 14q^{5} + 22q^{6} + 2q^{7} + 78q^{10} - 108q^{11} - 174q^{12} - 88q^{13} + 108q^{14} + 420q^{16} - 10q^{17} + 428q^{18} - 306q^{20} - 260q^{21} + 30q^{22} - 22q^{23} - 862q^{24} + 540q^{27} - 764q^{28} + 46q^{29} - 120q^{30} + 610q^{31} + 816q^{33} + 1002q^{34} + 1172q^{35} - 574q^{37} - 768q^{38} - 844q^{39} - 342q^{40} - 968q^{41} + 550q^{44} - 1154q^{45} - 944q^{46} - 368q^{47} + 2494q^{48} + 468q^{50} + 296q^{51} + 2564q^{52} - 1592q^{54} - 1996q^{55} + 684q^{56} - 300q^{57} + 266q^{58} + 1258q^{61} - 2516q^{62} + 122q^{63} - 3044q^{64} + 628q^{65} + 764q^{67} + 1914q^{68} + 1812q^{69} + 1266q^{71} + 1404q^{72} - 1732q^{73} + 1538q^{74} + 1292q^{75} - 2836q^{78} + 914q^{79} + 498q^{80} + 280q^{81} - 280q^{82} - 2952q^{84} - 2498q^{85} - 4244q^{86} + 442q^{88} - 2156q^{89} + 2478q^{90} - 1632q^{91} - 1768q^{92} + 1484q^{95} + 3998q^{96} + 1836q^{97} + 6728q^{98} - 2088q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.4.c.a 8
3.b odd 2 1 153.4.f.a 8
4.b odd 2 1 272.4.o.e 8
17.c even 4 1 inner 17.4.c.a 8
17.d even 8 2 289.4.a.f 8
17.d even 8 2 289.4.b.c 8
51.f odd 4 1 153.4.f.a 8
68.f odd 4 1 272.4.o.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.c.a 8 1.a even 1 1 trivial
17.4.c.a 8 17.c even 4 1 inner
153.4.f.a 8 3.b odd 2 1
153.4.f.a 8 51.f odd 4 1
272.4.o.e 8 4.b odd 2 1
272.4.o.e 8 68.f odd 4 1
289.4.a.f 8 17.d even 8 2
289.4.b.c 8 17.d even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 14 T^{2} + 65 T^{4} - 48 T^{6} - 1152 T^{8} - 3072 T^{10} + 266240 T^{12} - 3670016 T^{14} + 16777216 T^{16} \)
$3$ \( 1 - 180 T^{3} - 124 T^{4} + 4500 T^{5} + 16200 T^{6} + 1800 T^{7} - 880154 T^{8} + 48600 T^{9} + 11809800 T^{10} + 88573500 T^{11} - 65898684 T^{12} - 2582803260 T^{13} + 282429536481 T^{16} \)
$5$ \( 1 - 14 T + 98 T^{2} - 1742 T^{3} + 12064 T^{4} + 83238 T^{5} - 830322 T^{6} + 22258486 T^{7} - 523179234 T^{8} + 2782310750 T^{9} - 12973781250 T^{10} + 162574218750 T^{11} + 2945312500000 T^{12} - 53161621093750 T^{13} + 373840332031250 T^{14} - 6675720214843750 T^{15} + 59604644775390625 T^{16} \)
$7$ \( 1 - 2 T + 2 T^{2} - 10226 T^{3} + 81296 T^{4} + 1358798 T^{5} + 49405350 T^{6} - 812471938 T^{7} - 20043782946 T^{8} - 278677874734 T^{9} + 5812490022150 T^{10} + 54832400484386 T^{11} + 1125241284292496 T^{12} - 48548564000677118 T^{13} + 3256827195820898 T^{14} - 1117091728166568014 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 108 T + 5832 T^{2} + 285984 T^{3} + 14525412 T^{4} + 631823112 T^{5} + 24418117440 T^{6} + 981926711892 T^{7} + 38344333256678 T^{8} + 1306944453528252 T^{9} + 43258184550123840 T^{10} + 1489805848060834392 T^{11} + 45586965204363734052 T^{12} + \)\(11\!\cdots\!84\)\( T^{13} + \)\(32\!\cdots\!92\)\( T^{14} + \)\(79\!\cdots\!88\)\( T^{15} + \)\(98\!\cdots\!41\)\( T^{16} \)
$13$ \( ( 1 + 44 T + 6452 T^{2} + 185644 T^{3} + 18227830 T^{4} + 407859868 T^{5} + 31142571668 T^{6} + 466597972412 T^{7} + 23298085122481 T^{8} )^{2} \)
$17$ \( 1 + 10 T - 2720 T^{2} - 2890 T^{3} + 45425598 T^{4} - 14198570 T^{5} - 65654187680 T^{6} + 1185878764970 T^{7} + 582622237229761 T^{8} \)
$19$ \( 1 - 49548 T^{2} + 1108248628 T^{4} - 14581622201236 T^{6} + 123306344393137910 T^{8} - \)\(68\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!08\)\( T^{12} - \)\(51\!\cdots\!68\)\( T^{14} + \)\(48\!\cdots\!21\)\( T^{16} \)
$23$ \( 1 + 22 T + 242 T^{2} + 305150 T^{3} + 283710864 T^{4} + 1994580286 T^{5} + 21780998454 T^{6} + 34099874325206 T^{7} + 50076554997930590 T^{8} + 414893170914781402 T^{9} + 3224369469445515606 T^{10} + \)\(35\!\cdots\!18\)\( T^{11} + \)\(62\!\cdots\!44\)\( T^{12} + \)\(81\!\cdots\!50\)\( T^{13} + \)\(78\!\cdots\!98\)\( T^{14} + \)\(86\!\cdots\!06\)\( T^{15} + \)\(48\!\cdots\!41\)\( T^{16} \)
$29$ \( 1 - 46 T + 1058 T^{2} - 350894 T^{3} - 1469341632 T^{4} + 48677544854 T^{5} - 623040317010 T^{6} - 326680376209530 T^{7} + 1059567541310947486 T^{8} - 7967407695374227170 T^{9} - \)\(37\!\cdots\!10\)\( T^{10} + \)\(70\!\cdots\!26\)\( T^{11} - \)\(51\!\cdots\!12\)\( T^{12} - \)\(30\!\cdots\!06\)\( T^{13} + \)\(22\!\cdots\!38\)\( T^{14} - \)\(23\!\cdots\!34\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$31$ \( 1 - 610 T + 186050 T^{2} - 45370570 T^{3} + 7376742336 T^{4} - 319696487050 T^{5} - 148183743449850 T^{6} + 60089442929313630 T^{7} - 13665887910277786754 T^{8} + \)\(17\!\cdots\!30\)\( T^{9} - \)\(13\!\cdots\!50\)\( T^{10} - \)\(84\!\cdots\!50\)\( T^{11} + \)\(58\!\cdots\!96\)\( T^{12} - \)\(10\!\cdots\!70\)\( T^{13} + \)\(13\!\cdots\!50\)\( T^{14} - \)\(12\!\cdots\!10\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \)
$37$ \( 1 + 574 T + 164738 T^{2} + 36289990 T^{3} + 6240461504 T^{4} + 1192660509522 T^{5} + 315027672319726 T^{6} + 82684080067897738 T^{7} + 20284670290250127070 T^{8} + \)\(41\!\cdots\!14\)\( T^{9} + \)\(80\!\cdots\!34\)\( T^{10} + \)\(15\!\cdots\!94\)\( T^{11} + \)\(41\!\cdots\!24\)\( T^{12} + \)\(12\!\cdots\!70\)\( T^{13} + \)\(27\!\cdots\!02\)\( T^{14} + \)\(49\!\cdots\!38\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \)
$41$ \( 1 + 968 T + 468512 T^{2} + 198428824 T^{3} + 85385967612 T^{4} + 30688635230792 T^{5} + 9389247542584800 T^{6} + 2850342553409293272 T^{7} + \)\(81\!\cdots\!98\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} + \)\(44\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{11} + \)\(19\!\cdots\!72\)\( T^{12} + \)\(30\!\cdots\!24\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(71\!\cdots\!88\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \)
$43$ \( 1 - 524936 T^{2} + 127553384540 T^{4} - 18699551386799032 T^{6} + \)\(18\!\cdots\!98\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{10} + \)\(50\!\cdots\!40\)\( T^{12} - \)\(13\!\cdots\!64\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \)
$47$ \( ( 1 + 184 T + 303628 T^{2} + 45624920 T^{3} + 43219349926 T^{4} + 4736916069160 T^{5} + 3272871591913612 T^{6} + 205920007050909128 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \)
$53$ \( 1 - 429476 T^{2} + 132292785860 T^{4} - 29887976589939132 T^{6} + \)\(48\!\cdots\!18\)\( T^{8} - \)\(66\!\cdots\!28\)\( T^{10} + \)\(64\!\cdots\!60\)\( T^{12} - \)\(46\!\cdots\!64\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \)
$59$ \( 1 - 803128 T^{2} + 306274528508 T^{4} - 82679168899041416 T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(34\!\cdots\!56\)\( T^{10} + \)\(54\!\cdots\!48\)\( T^{12} - \)\(60\!\cdots\!88\)\( T^{14} + \)\(31\!\cdots\!61\)\( T^{16} \)
$61$ \( 1 - 1258 T + 791282 T^{2} - 465239298 T^{3} + 289545560288 T^{4} - 163355761361878 T^{5} + 84613159959199710 T^{6} - 47961410201311305310 T^{7} + \)\(25\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!10\)\( T^{9} + \)\(43\!\cdots\!10\)\( T^{10} - \)\(19\!\cdots\!98\)\( T^{11} + \)\(76\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!98\)\( T^{13} + \)\(10\!\cdots\!42\)\( T^{14} - \)\(39\!\cdots\!38\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \)
$67$ \( ( 1 - 382 T + 518844 T^{2} + 127863586 T^{3} + 50809502070 T^{4} + 38456635716118 T^{5} + 46933788838092636 T^{6} - 10392896139384669754 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \)
$71$ \( 1 - 1266 T + 801378 T^{2} - 406667714 T^{3} + 257625327744 T^{4} - 188037640245138 T^{5} + 114289697458506374 T^{6} - 42774297641864818786 T^{7} + \)\(12\!\cdots\!34\)\( T^{8} - \)\(15\!\cdots\!46\)\( T^{9} + \)\(14\!\cdots\!54\)\( T^{10} - \)\(86\!\cdots\!78\)\( T^{11} + \)\(42\!\cdots\!04\)\( T^{12} - \)\(23\!\cdots\!14\)\( T^{13} + \)\(16\!\cdots\!58\)\( T^{14} - \)\(95\!\cdots\!86\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \)
$73$ \( 1 + 1732 T + 1499912 T^{2} + 1036616588 T^{3} + 594980634828 T^{4} + 355861503522212 T^{5} + 261220505412716920 T^{6} + \)\(20\!\cdots\!24\)\( T^{7} + \)\(15\!\cdots\!50\)\( T^{8} + \)\(81\!\cdots\!08\)\( T^{9} + \)\(39\!\cdots\!80\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!88\)\( T^{12} + \)\(92\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!36\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$79$ \( 1 - 914 T + 417698 T^{2} + 360102614 T^{3} + 55218179088 T^{4} - 435936491163994 T^{5} + 440218376260007590 T^{6} - 25818998319394847330 T^{7} - \)\(81\!\cdots\!34\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{9} + \)\(10\!\cdots\!90\)\( T^{10} - \)\(52\!\cdots\!86\)\( T^{11} + \)\(32\!\cdots\!08\)\( T^{12} + \)\(10\!\cdots\!86\)\( T^{13} + \)\(59\!\cdots\!78\)\( T^{14} - \)\(64\!\cdots\!06\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \)
$83$ \( 1 - 2421080 T^{2} + 2978675688956 T^{4} - 2441080253881110760 T^{6} + \)\(15\!\cdots\!06\)\( T^{8} - \)\(79\!\cdots\!40\)\( T^{10} + \)\(31\!\cdots\!16\)\( T^{12} - \)\(84\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \)
$89$ \( ( 1 + 1078 T + 2962392 T^{2} + 2155154546 T^{3} + 3159947923854 T^{4} + 1519317145139074 T^{5} + 1472253400492538712 T^{6} + \)\(37\!\cdots\!02\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \)
$97$ \( 1 - 1836 T + 1685448 T^{2} - 1048047300 T^{3} + 1167661514444 T^{4} - 1645617931539948 T^{5} + 1602523329629378616 T^{6} - \)\(45\!\cdots\!92\)\( T^{7} - \)\(36\!\cdots\!30\)\( T^{8} - \)\(41\!\cdots\!16\)\( T^{9} + \)\(13\!\cdots\!64\)\( T^{10} - \)\(12\!\cdots\!16\)\( T^{11} + \)\(81\!\cdots\!04\)\( T^{12} - \)\(66\!\cdots\!00\)\( T^{13} + \)\(97\!\cdots\!72\)\( T^{14} - \)\(96\!\cdots\!92\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \)
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