Properties

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

Related objects

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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: no (minimal twist has level 330)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 6q^{13} - q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + q^{20} + q^{24} + q^{25} + 6q^{26} - q^{27} + 10q^{29} + q^{30} - q^{32} + 2q^{34} + q^{36} + 6q^{37} - 4q^{38} + 6q^{39} - q^{40} - 2q^{41} - 4q^{43} + q^{45} - 8q^{47} - q^{48} - 7q^{49} - q^{50} + 2q^{51} - 6q^{52} - 10q^{53} + q^{54} - 4q^{57} - 10q^{58} - 4q^{59} - q^{60} + 2q^{61} + q^{64} - 6q^{65} - 4q^{67} - 2q^{68} - 8q^{71} - q^{72} - 2q^{73} - 6q^{74} - q^{75} + 4q^{76} - 6q^{78} + 8q^{79} + q^{80} + q^{81} + 2q^{82} + 12q^{83} - 2q^{85} + 4q^{86} - 10q^{87} - 6q^{89} - q^{90} + 8q^{94} + 4q^{95} + q^{96} + 18q^{97} + 7q^{98} + O(q^{100}) \)

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 0 −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.f 1
11.b odd 2 1 330.2.a.d 1
33.d even 2 1 990.2.a.b 1
44.c even 2 1 2640.2.a.t 1
55.d odd 2 1 1650.2.a.h 1
55.e even 4 2 1650.2.c.g 2
132.d odd 2 1 7920.2.a.m 1
165.d even 2 1 4950.2.a.bg 1
165.l odd 4 2 4950.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.d 1 11.b odd 2 1
990.2.a.b 1 33.d even 2 1
1650.2.a.h 1 55.d odd 2 1
1650.2.c.g 2 55.e even 4 2
2640.2.a.t 1 44.c even 2 1
3630.2.a.f 1 1.a even 1 1 trivial
4950.2.a.bg 1 165.d even 2 1
4950.2.c.j 2 165.l odd 4 2
7920.2.a.m 1 132.d odd 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{13} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 6 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( -6 + T \)
$41$ \( 2 + T \)
$43$ \( 4 + T \)
$47$ \( 8 + T \)
$53$ \( 10 + T \)
$59$ \( 4 + T \)
$61$ \( -2 + T \)
$67$ \( 4 + T \)
$71$ \( 8 + T \)
$73$ \( 2 + T \)
$79$ \( -8 + T \)
$83$ \( -12 + T \)
$89$ \( 6 + T \)
$97$ \( -18 + T \)
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