Properties

Label 17.8.a.a
Level 17
Weight 8
Character orbit 17.a
Self dual yes
Analytic conductor 5.311
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 17.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.31054543323\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 18q^{3} - 124q^{4} - 10q^{5} - 36q^{6} - 902q^{7} + 504q^{8} - 1863q^{9} + O(q^{10}) \) \( q - 2q^{2} + 18q^{3} - 124q^{4} - 10q^{5} - 36q^{6} - 902q^{7} + 504q^{8} - 1863q^{9} + 20q^{10} - 8634q^{11} - 2232q^{12} + 10858q^{13} + 1804q^{14} - 180q^{15} + 14864q^{16} + 4913q^{17} + 3726q^{18} - 784q^{19} + 1240q^{20} - 16236q^{21} + 17268q^{22} + 77330q^{23} + 9072q^{24} - 78025q^{25} - 21716q^{26} - 72900q^{27} + 111848q^{28} - 18210q^{29} + 360q^{30} - 237002q^{31} - 94240q^{32} - 155412q^{33} - 9826q^{34} + 9020q^{35} + 231012q^{36} + 230878q^{37} + 1568q^{38} + 195444q^{39} - 5040q^{40} - 304182q^{41} + 32472q^{42} - 525032q^{43} + 1070616q^{44} + 18630q^{45} - 154660q^{46} + 802752q^{47} + 267552q^{48} - 9939q^{49} + 156050q^{50} + 88434q^{51} - 1346392q^{52} + 152862q^{53} + 145800q^{54} + 86340q^{55} - 454608q^{56} - 14112q^{57} + 36420q^{58} - 1602408q^{59} + 22320q^{60} - 2601610q^{61} + 474004q^{62} + 1680426q^{63} - 1714112q^{64} - 108580q^{65} + 310824q^{66} + 1074604q^{67} - 609212q^{68} + 1391940q^{69} - 18040q^{70} - 502298q^{71} - 938952q^{72} + 3648258q^{73} - 461756q^{74} - 1404450q^{75} + 97216q^{76} + 7787868q^{77} - 390888q^{78} - 2892174q^{79} - 148640q^{80} + 2762181q^{81} + 608364q^{82} + 728104q^{83} + 2013264q^{84} - 49130q^{85} + 1050064q^{86} - 327780q^{87} - 4351536q^{88} + 7931846q^{89} - 37260q^{90} - 9793916q^{91} - 9588920q^{92} - 4266036q^{93} - 1605504q^{94} + 7840q^{95} - 1696320q^{96} - 6551038q^{97} + 19878q^{98} + 16085142q^{99} + O(q^{100}) \)

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
0
−2.00000 18.0000 −124.000 −10.0000 −36.0000 −902.000 504.000 −1863.00 20.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.8.a.a 1
3.b odd 2 1 153.8.a.a 1
4.b odd 2 1 272.8.a.b 1
5.b even 2 1 425.8.a.a 1
17.b even 2 1 289.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.8.a.a 1 1.a even 1 1 trivial
153.8.a.a 1 3.b odd 2 1
272.8.a.b 1 4.b odd 2 1
289.8.a.a 1 17.b even 2 1
425.8.a.a 1 5.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(17))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 128 T^{2} \)
$3$ \( 1 - 18 T + 2187 T^{2} \)
$5$ \( 1 + 10 T + 78125 T^{2} \)
$7$ \( 1 + 902 T + 823543 T^{2} \)
$11$ \( 1 + 8634 T + 19487171 T^{2} \)
$13$ \( 1 - 10858 T + 62748517 T^{2} \)
$17$ \( 1 - 4913 T \)
$19$ \( 1 + 784 T + 893871739 T^{2} \)
$23$ \( 1 - 77330 T + 3404825447 T^{2} \)
$29$ \( 1 + 18210 T + 17249876309 T^{2} \)
$31$ \( 1 + 237002 T + 27512614111 T^{2} \)
$37$ \( 1 - 230878 T + 94931877133 T^{2} \)
$41$ \( 1 + 304182 T + 194754273881 T^{2} \)
$43$ \( 1 + 525032 T + 271818611107 T^{2} \)
$47$ \( 1 - 802752 T + 506623120463 T^{2} \)
$53$ \( 1 - 152862 T + 1174711139837 T^{2} \)
$59$ \( 1 + 1602408 T + 2488651484819 T^{2} \)
$61$ \( 1 + 2601610 T + 3142742836021 T^{2} \)
$67$ \( 1 - 1074604 T + 6060711605323 T^{2} \)
$71$ \( 1 + 502298 T + 9095120158391 T^{2} \)
$73$ \( 1 - 3648258 T + 11047398519097 T^{2} \)
$79$ \( 1 + 2892174 T + 19203908986159 T^{2} \)
$83$ \( 1 - 728104 T + 27136050989627 T^{2} \)
$89$ \( 1 - 7931846 T + 44231334895529 T^{2} \)
$97$ \( 1 + 6551038 T + 80798284478113 T^{2} \)
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