Properties

Label 170.2.a.e
Level 170
Weight 2
Character orbit 170.a
Self dual Yes
Analytic conductor 1.357
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 170.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.35745683436\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + q^{8} - 2q^{9} - q^{10} + q^{12} - q^{13} + 2q^{14} - q^{15} + q^{16} - q^{17} - 2q^{18} - q^{19} - q^{20} + 2q^{21} - 6q^{23} + q^{24} + q^{25} - q^{26} - 5q^{27} + 2q^{28} - 3q^{29} - q^{30} + 5q^{31} + q^{32} - q^{34} - 2q^{35} - 2q^{36} + 8q^{37} - q^{38} - q^{39} - q^{40} + 6q^{41} + 2q^{42} - 10q^{43} + 2q^{45} - 6q^{46} - 3q^{47} + q^{48} - 3q^{49} + q^{50} - q^{51} - q^{52} - 3q^{53} - 5q^{54} + 2q^{56} - q^{57} - 3q^{58} + 3q^{59} - q^{60} + 11q^{61} + 5q^{62} - 4q^{63} + q^{64} + q^{65} + 2q^{67} - q^{68} - 6q^{69} - 2q^{70} + 9q^{71} - 2q^{72} + 11q^{73} + 8q^{74} + q^{75} - q^{76} - q^{78} + 8q^{79} - q^{80} + q^{81} + 6q^{82} - 12q^{83} + 2q^{84} + q^{85} - 10q^{86} - 3q^{87} + 15q^{89} + 2q^{90} - 2q^{91} - 6q^{92} + 5q^{93} - 3q^{94} + q^{95} + q^{96} - 7q^{97} - 3q^{98} + O(q^{100}) \)

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 1.00000 1.00000 −1.00000 1.00000 2.00000 1.00000 −2.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(17\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\):

\( T_{3} - 1 \)
\( T_{7} - 2 \)
\( T_{13} + 1 \)