Properties

Label 3630.2.a.x
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

Downloads

Learn more about

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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 5q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 5q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 5q^{13} - 5q^{14} + q^{15} + q^{16} + 3q^{17} + q^{18} + q^{19} + q^{20} - 5q^{21} - 6q^{23} + q^{24} + q^{25} - 5q^{26} + q^{27} - 5q^{28} - 9q^{29} + q^{30} - 4q^{31} + q^{32} + 3q^{34} - 5q^{35} + q^{36} + 5q^{37} + q^{38} - 5q^{39} + q^{40} - 5q^{42} - 8q^{43} + q^{45} - 6q^{46} - 6q^{47} + q^{48} + 18q^{49} + q^{50} + 3q^{51} - 5q^{52} + 6q^{53} + q^{54} - 5q^{56} + q^{57} - 9q^{58} - 6q^{59} + q^{60} - 14q^{61} - 4q^{62} - 5q^{63} + q^{64} - 5q^{65} - 10q^{67} + 3q^{68} - 6q^{69} - 5q^{70} + 3q^{71} + q^{72} + 16q^{73} + 5q^{74} + q^{75} + q^{76} - 5q^{78} + 10q^{79} + q^{80} + q^{81} - 3q^{83} - 5q^{84} + 3q^{85} - 8q^{86} - 9q^{87} - 12q^{89} + q^{90} + 25q^{91} - 6q^{92} - 4q^{93} - 6q^{94} + q^{95} + q^{96} - 4q^{97} + 18q^{98} + O(q^{100}) \)

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 −5.00000 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.x yes 1
11.b odd 2 1 3630.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.m 1 11.b odd 2 1
3630.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7} + 5 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( 5 + T \)
$11$ \( T \)
$13$ \( 5 + T \)
$17$ \( -3 + T \)
$19$ \( -1 + T \)
$23$ \( 6 + T \)
$29$ \( 9 + T \)
$31$ \( 4 + T \)
$37$ \( -5 + T \)
$41$ \( T \)
$43$ \( 8 + T \)
$47$ \( 6 + T \)
$53$ \( -6 + T \)
$59$ \( 6 + T \)
$61$ \( 14 + T \)
$67$ \( 10 + T \)
$71$ \( -3 + T \)
$73$ \( -16 + T \)
$79$ \( -10 + T \)
$83$ \( 3 + T \)
$89$ \( 12 + T \)
$97$ \( 4 + T \)
show more
show less