Properties

Label 17.2.a.a
Level 17
Weight 2
Character orbit 17.a
Self dual yes
Analytic conductor 0.136
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 17.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.135745683436\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + q^{17} + 3q^{18} - 4q^{19} + 2q^{20} + 4q^{23} - q^{25} + 2q^{26} - 4q^{28} + 6q^{29} + 4q^{31} - 5q^{32} - q^{34} - 8q^{35} + 3q^{36} - 2q^{37} + 4q^{38} - 6q^{40} - 6q^{41} + 4q^{43} + 6q^{45} - 4q^{46} + 9q^{49} + q^{50} + 2q^{52} + 6q^{53} + 12q^{56} - 6q^{58} - 12q^{59} - 10q^{61} - 4q^{62} - 12q^{63} + 7q^{64} + 4q^{65} + 4q^{67} - q^{68} + 8q^{70} - 4q^{71} - 9q^{72} - 6q^{73} + 2q^{74} + 4q^{76} + 12q^{79} + 2q^{80} + 9q^{81} + 6q^{82} - 4q^{83} - 2q^{85} - 4q^{86} + 10q^{89} - 6q^{90} - 8q^{91} - 4q^{92} + 8q^{95} + 2q^{97} - 9q^{98} + O(q^{100}) \)

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
−1.00000 0 −1.00000 −2.00000 0 4.00000 3.00000 −3.00000 2.00000
\(n\): 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 17.2.a.a 1
3.b odd 2 1 153.2.a.c 1
4.b odd 2 1 272.2.a.b 1
5.b even 2 1 425.2.a.d 1
5.c odd 4 2 425.2.b.b 2
7.b odd 2 1 833.2.a.a 1
7.c even 3 2 833.2.e.b 2
7.d odd 6 2 833.2.e.a 2
8.b even 2 1 1088.2.a.i 1
8.d odd 2 1 1088.2.a.h 1
11.b odd 2 1 2057.2.a.e 1
12.b even 2 1 2448.2.a.o 1
13.b even 2 1 2873.2.a.c 1
15.d odd 2 1 3825.2.a.d 1
17.b even 2 1 289.2.a.a 1
17.c even 4 2 289.2.b.a 2
17.d even 8 4 289.2.c.a 4
17.e odd 16 8 289.2.d.d 8
19.b odd 2 1 6137.2.a.b 1
20.d odd 2 1 6800.2.a.n 1
21.c even 2 1 7497.2.a.l 1
23.b odd 2 1 8993.2.a.a 1
24.f even 2 1 9792.2.a.i 1
24.h odd 2 1 9792.2.a.n 1
51.c odd 2 1 2601.2.a.g 1
68.d odd 2 1 4624.2.a.d 1
85.c even 2 1 7225.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 1.a even 1 1 trivial
153.2.a.c 1 3.b odd 2 1
272.2.a.b 1 4.b odd 2 1
289.2.a.a 1 17.b even 2 1
289.2.b.a 2 17.c even 4 2
289.2.c.a 4 17.d even 8 4
289.2.d.d 8 17.e odd 16 8
425.2.a.d 1 5.b even 2 1
425.2.b.b 2 5.c odd 4 2
833.2.a.a 1 7.b odd 2 1
833.2.e.a 2 7.d odd 6 2
833.2.e.b 2 7.c even 3 2
1088.2.a.h 1 8.d odd 2 1
1088.2.a.i 1 8.b even 2 1
2057.2.a.e 1 11.b odd 2 1
2448.2.a.o 1 12.b even 2 1
2601.2.a.g 1 51.c odd 2 1
2873.2.a.c 1 13.b even 2 1
3825.2.a.d 1 15.d odd 2 1
4624.2.a.d 1 68.d odd 2 1
6137.2.a.b 1 19.b odd 2 1
6800.2.a.n 1 20.d odd 2 1
7225.2.a.g 1 85.c even 2 1
7497.2.a.l 1 21.c even 2 1
8993.2.a.a 1 23.b odd 2 1
9792.2.a.i 1 24.f even 2 1
9792.2.a.n 1 24.h odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - T \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 - 12 T + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 - 10 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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