Properties

Label 17.4.a.a
Level 17
Weight 4
Character orbit 17.a
Self dual yes
Analytic conductor 1.003
Analytic rank 1
Dimension 1
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: \(1.00303247010\)
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

\(f(q)\) \(=\) \( q - 3q^{2} - 8q^{3} + q^{4} + 6q^{5} + 24q^{6} - 28q^{7} + 21q^{8} + 37q^{9} + O(q^{10}) \) \( q - 3q^{2} - 8q^{3} + q^{4} + 6q^{5} + 24q^{6} - 28q^{7} + 21q^{8} + 37q^{9} - 18q^{10} - 24q^{11} - 8q^{12} - 58q^{13} + 84q^{14} - 48q^{15} - 71q^{16} + 17q^{17} - 111q^{18} + 116q^{19} + 6q^{20} + 224q^{21} + 72q^{22} - 60q^{23} - 168q^{24} - 89q^{25} + 174q^{26} - 80q^{27} - 28q^{28} + 30q^{29} + 144q^{30} - 172q^{31} + 45q^{32} + 192q^{33} - 51q^{34} - 168q^{35} + 37q^{36} - 58q^{37} - 348q^{38} + 464q^{39} + 126q^{40} - 342q^{41} - 672q^{42} - 148q^{43} - 24q^{44} + 222q^{45} + 180q^{46} + 288q^{47} + 568q^{48} + 441q^{49} + 267q^{50} - 136q^{51} - 58q^{52} + 318q^{53} + 240q^{54} - 144q^{55} - 588q^{56} - 928q^{57} - 90q^{58} + 252q^{59} - 48q^{60} + 110q^{61} + 516q^{62} - 1036q^{63} + 433q^{64} - 348q^{65} - 576q^{66} - 484q^{67} + 17q^{68} + 480q^{69} + 504q^{70} - 708q^{71} + 777q^{72} + 362q^{73} + 174q^{74} + 712q^{75} + 116q^{76} + 672q^{77} - 1392q^{78} - 484q^{79} - 426q^{80} - 359q^{81} + 1026q^{82} + 756q^{83} + 224q^{84} + 102q^{85} + 444q^{86} - 240q^{87} - 504q^{88} - 774q^{89} - 666q^{90} + 1624q^{91} - 60q^{92} + 1376q^{93} - 864q^{94} + 696q^{95} - 360q^{96} - 382q^{97} - 1323q^{98} - 888q^{99} + O(q^{100}) \)

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
−3.00000 −8.00000 1.00000 6.00000 24.0000 −28.0000 21.0000 37.0000 −18.0000
\(n\): 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 17.4.a.a 1
3.b odd 2 1 153.4.a.d 1
4.b odd 2 1 272.4.a.d 1
5.b even 2 1 425.4.a.d 1
5.c odd 4 2 425.4.b.c 2
7.b odd 2 1 833.4.a.a 1
8.b even 2 1 1088.4.a.l 1
8.d odd 2 1 1088.4.a.a 1
11.b odd 2 1 2057.4.a.d 1
12.b even 2 1 2448.4.a.f 1
17.b even 2 1 289.4.a.a 1
17.c even 4 2 289.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 1.a even 1 1 trivial
153.4.a.d 1 3.b odd 2 1
272.4.a.d 1 4.b odd 2 1
289.4.a.a 1 17.b even 2 1
289.4.b.a 2 17.c even 4 2
425.4.a.d 1 5.b even 2 1
425.4.b.c 2 5.c odd 4 2
833.4.a.a 1 7.b odd 2 1
1088.4.a.a 1 8.d odd 2 1
1088.4.a.l 1 8.b even 2 1
2057.4.a.d 1 11.b odd 2 1
2448.4.a.f 1 12.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 \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(17))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} \)
$3$ \( 1 + 8 T + 27 T^{2} \)
$5$ \( 1 - 6 T + 125 T^{2} \)
$7$ \( 1 + 28 T + 343 T^{2} \)
$11$ \( 1 + 24 T + 1331 T^{2} \)
$13$ \( 1 + 58 T + 2197 T^{2} \)
$17$ \( 1 - 17 T \)
$19$ \( 1 - 116 T + 6859 T^{2} \)
$23$ \( 1 + 60 T + 12167 T^{2} \)
$29$ \( 1 - 30 T + 24389 T^{2} \)
$31$ \( 1 + 172 T + 29791 T^{2} \)
$37$ \( 1 + 58 T + 50653 T^{2} \)
$41$ \( 1 + 342 T + 68921 T^{2} \)
$43$ \( 1 + 148 T + 79507 T^{2} \)
$47$ \( 1 - 288 T + 103823 T^{2} \)
$53$ \( 1 - 318 T + 148877 T^{2} \)
$59$ \( 1 - 252 T + 205379 T^{2} \)
$61$ \( 1 - 110 T + 226981 T^{2} \)
$67$ \( 1 + 484 T + 300763 T^{2} \)
$71$ \( 1 + 708 T + 357911 T^{2} \)
$73$ \( 1 - 362 T + 389017 T^{2} \)
$79$ \( 1 + 484 T + 493039 T^{2} \)
$83$ \( 1 - 756 T + 571787 T^{2} \)
$89$ \( 1 + 774 T + 704969 T^{2} \)
$97$ \( 1 + 382 T + 912673 T^{2} \)
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