Properties

Label 3630.2.a.a
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: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - 2q^{13} + 4q^{14} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} - q^{20} + 4q^{21} - 4q^{23} + q^{24} + q^{25} + 2q^{26} - q^{27} - 4q^{28} - 6q^{29} - q^{30} - q^{32} + 2q^{34} + 4q^{35} + q^{36} - 10q^{37} + 4q^{38} + 2q^{39} + q^{40} + 6q^{41} - 4q^{42} + 12q^{43} - q^{45} + 4q^{46} - 4q^{47} - q^{48} + 9q^{49} - q^{50} + 2q^{51} - 2q^{52} - 6q^{53} + q^{54} + 4q^{56} + 4q^{57} + 6q^{58} - 4q^{59} + q^{60} - 10q^{61} - 4q^{63} + q^{64} + 2q^{65} - 12q^{67} - 2q^{68} + 4q^{69} - 4q^{70} - 4q^{71} - q^{72} - 10q^{73} + 10q^{74} - q^{75} - 4q^{76} - 2q^{78} - 4q^{79} - q^{80} + q^{81} - 6q^{82} - 4q^{83} + 4q^{84} + 2q^{85} - 12q^{86} + 6q^{87} + 10q^{89} + q^{90} + 8q^{91} - 4q^{92} + 4q^{94} + 4q^{95} + q^{96} + 18q^{97} - 9q^{98} + O(q^{100}) \)

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 −4.00000 −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.a 1
11.b odd 2 1 330.2.a.c 1
33.d even 2 1 990.2.a.g 1
44.c even 2 1 2640.2.a.j 1
55.d odd 2 1 1650.2.a.e 1
55.e even 4 2 1650.2.c.a 2
132.d odd 2 1 7920.2.a.t 1
165.d even 2 1 4950.2.a.x 1
165.l odd 4 2 4950.2.c.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.a.c 1 11.b odd 2 1
990.2.a.g 1 33.d even 2 1
1650.2.a.e 1 55.d odd 2 1
1650.2.c.a 2 55.e even 4 2
2640.2.a.j 1 44.c even 2 1
3630.2.a.a 1 1.a even 1 1 trivial
4950.2.a.x 1 165.d even 2 1
4950.2.c.y 2 165.l odd 4 2
7920.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7} + 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

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