Properties

Label 3570.2.a.w
Level 3570
Weight 2
Character orbit 3570.a
Self dual Yes
Analytic conductor 28.507
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3570.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(28.5065935216\)
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

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

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 1.00000 1.00000 1.00000 −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
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(17\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3570))\):

\( T_{11} \)
\( T_{13} - 2 \)
\( T_{19} + 4 \)
\( T_{37} - 2 \)