Properties

Label 1050.2.a.l
Level $1050$
Weight $2$
Character orbit 1050.a
Self dual yes
Analytic conductor $8.384$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} + 6q^{11} - q^{12} + q^{13} - q^{14} + q^{16} - 3q^{17} + q^{18} - 4q^{19} + q^{21} + 6q^{22} + 3q^{23} - q^{24} + q^{26} - q^{27} - q^{28} + 3q^{29} + 5q^{31} + q^{32} - 6q^{33} - 3q^{34} + q^{36} + 10q^{37} - 4q^{38} - q^{39} + 9q^{41} + q^{42} + q^{43} + 6q^{44} + 3q^{46} - q^{48} + q^{49} + 3q^{51} + q^{52} - 9q^{53} - q^{54} - q^{56} + 4q^{57} + 3q^{58} + 9q^{59} + 11q^{61} + 5q^{62} - q^{63} + q^{64} - 6q^{66} + 4q^{67} - 3q^{68} - 3q^{69} - 12q^{71} + q^{72} + 10q^{73} + 10q^{74} - 4q^{76} - 6q^{77} - q^{78} - 10q^{79} + q^{81} + 9q^{82} - 9q^{83} + q^{84} + q^{86} - 3q^{87} + 6q^{88} - 6q^{89} - q^{91} + 3q^{92} - 5q^{93} - q^{96} - 14q^{97} + q^{98} + 6q^{99} + O(q^{100}) \)

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 0 −1.00000 −1.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

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 1050.2.a.l yes 1
3.b odd 2 1 3150.2.a.a 1
4.b odd 2 1 8400.2.a.ci 1
5.b even 2 1 1050.2.a.j 1
5.c odd 4 2 1050.2.g.e 2
7.b odd 2 1 7350.2.a.cz 1
15.d odd 2 1 3150.2.a.bg 1
15.e even 4 2 3150.2.g.a 2
20.d odd 2 1 8400.2.a.a 1
35.c odd 2 1 7350.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.j 1 5.b even 2 1
1050.2.a.l yes 1 1.a even 1 1 trivial
1050.2.g.e 2 5.c odd 4 2
3150.2.a.a 1 3.b odd 2 1
3150.2.a.bg 1 15.d odd 2 1
3150.2.g.a 2 15.e even 4 2
7350.2.a.r 1 35.c odd 2 1
7350.2.a.cz 1 7.b odd 2 1
8400.2.a.a 1 20.d odd 2 1
8400.2.a.ci 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} - 6 \)
\( T_{13} - 1 \)
\( T_{17} + 3 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

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