Properties

Label 3630.2.a.e
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

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

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:

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.e 1
11.b odd 2 1 3630.2.a.q yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.e 1 1.a even 1 1 trivial
3630.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( 5 + T \)
$19$ \( -1 + T \)
$23$ \( T \)
$29$ \( 1 + T \)
$31$ \( -10 + T \)
$37$ \( 7 + T \)
$41$ \( -2 + T \)
$43$ \( 6 + T \)
$47$ \( 6 + T \)
$53$ \( -8 + T \)
$59$ \( 4 + T \)
$61$ \( 4 + T \)
$67$ \( -14 + T \)
$71$ \( 1 + T \)
$73$ \( 2 + T \)
$79$ \( 6 + T \)
$83$ \( 17 + T \)
$89$ \( T \)
$97$ \( 2 + T \)
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