Properties

Label 3630.2.a.r
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
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: \(0\)
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

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

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 3.00000 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.r yes 1
11.b odd 2 1 3630.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.d 1 11.b odd 2 1
3630.2.a.r yes 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7} - 3 \)
\( T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( -3 + T \)
$11$ \( T \)
$13$ \( -3 + T \)
$17$ \( 1 + T \)
$19$ \( 1 + T \)
$23$ \( -6 + T \)
$29$ \( -5 + T \)
$31$ \( T \)
$37$ \( -3 + T \)
$41$ \( 4 + T \)
$43$ \( 8 + T \)
$47$ \( 2 + T \)
$53$ \( 10 + T \)
$59$ \( -14 + T \)
$61$ \( 2 + T \)
$67$ \( 10 + T \)
$71$ \( -7 + T \)
$73$ \( -8 + T \)
$79$ \( -10 + T \)
$83$ \( -9 + T \)
$89$ \( T \)
$97$ \( 12 + T \)
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