Properties

Label 3630.2.a.j
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
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

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

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 3630.2.a.j 1
11.b odd 2 1 3630.2.a.z yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.j 1 1.a even 1 1 trivial
3630.2.a.z yes 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7} + 2 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( 2 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 2 + T \)
$19$ \( 2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( T \)
$37$ \( 6 + T \)
$41$ \( -6 + T \)
$43$ \( 2 + T \)
$47$ \( 4 + T \)
$53$ \( -2 + T \)
$59$ \( 8 + T \)
$61$ \( T \)
$67$ \( -4 + T \)
$71$ \( -12 + T \)
$73$ \( 16 + T \)
$79$ \( -2 + T \)
$83$ \( 8 + T \)
$89$ \( 6 + T \)
$97$ \( -10 + T \)
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