Properties

Label 609.2.a.a
Level $609$
Weight $2$
Character orbit 609.a
Self dual yes
Analytic conductor $4.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 609 = 3 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 609.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.86288948310\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} - 2q^{13} - q^{14} + 2q^{15} - q^{16} + 2q^{17} - q^{18} - 4q^{19} + 2q^{20} - q^{21} - 4q^{22} - 3q^{24} - q^{25} + 2q^{26} - q^{27} - q^{28} + q^{29} - 2q^{30} - 8q^{31} - 5q^{32} - 4q^{33} - 2q^{34} - 2q^{35} - q^{36} - 10q^{37} + 4q^{38} + 2q^{39} - 6q^{40} - 6q^{41} + q^{42} + 12q^{43} - 4q^{44} - 2q^{45} - 8q^{47} + q^{48} + q^{49} + q^{50} - 2q^{51} + 2q^{52} + 6q^{53} + q^{54} - 8q^{55} + 3q^{56} + 4q^{57} - q^{58} + 12q^{59} - 2q^{60} - 10q^{61} + 8q^{62} + q^{63} + 7q^{64} + 4q^{65} + 4q^{66} - 12q^{67} - 2q^{68} + 2q^{70} - 16q^{71} + 3q^{72} + 2q^{73} + 10q^{74} + q^{75} + 4q^{76} + 4q^{77} - 2q^{78} + 2q^{80} + q^{81} + 6q^{82} + 4q^{83} + q^{84} - 4q^{85} - 12q^{86} - q^{87} + 12q^{88} - 6q^{89} + 2q^{90} - 2q^{91} + 8q^{93} + 8q^{94} + 8q^{95} + 5q^{96} - 6q^{97} - q^{98} + 4q^{99} + O(q^{100}) \)

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
−1.00000 −1.00000 −1.00000 −2.00000 1.00000 1.00000 3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(29\) \(-1\)

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 609.2.a.a 1
3.b odd 2 1 1827.2.a.d 1
4.b odd 2 1 9744.2.a.l 1
7.b odd 2 1 4263.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
609.2.a.a 1 1.a even 1 1 trivial
1827.2.a.d 1 3.b odd 2 1
4263.2.a.c 1 7.b odd 2 1
9744.2.a.l 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(609))\).