Properties

Label 1050.2.a.f
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - 2q^{11} + q^{12} - 7q^{13} + q^{14} + q^{16} + 7q^{17} - q^{18} + 8q^{19} - q^{21} + 2q^{22} + 5q^{23} - q^{24} + 7q^{26} + q^{27} - q^{28} + 9q^{29} + q^{31} - q^{32} - 2q^{33} - 7q^{34} + q^{36} + 2q^{37} - 8q^{38} - 7q^{39} + 11q^{41} + q^{42} - 3q^{43} - 2q^{44} - 5q^{46} - 4q^{47} + q^{48} + q^{49} + 7q^{51} - 7q^{52} - 3q^{53} - q^{54} + q^{56} + 8q^{57} - 9q^{58} + 7q^{59} - 5q^{61} - q^{62} - q^{63} + q^{64} + 2q^{66} + 12q^{67} + 7q^{68} + 5q^{69} - 4q^{71} - q^{72} - 10q^{73} - 2q^{74} + 8q^{76} + 2q^{77} + 7q^{78} - 6q^{79} + q^{81} - 11q^{82} + 9q^{83} - q^{84} + 3q^{86} + 9q^{87} + 2q^{88} - 10q^{89} + 7q^{91} + 5q^{92} + q^{93} + 4q^{94} - q^{96} - 10q^{97} - q^{98} - 2q^{99} + O(q^{100}) \)

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:

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.f 1
3.b odd 2 1 3150.2.a.bd 1
4.b odd 2 1 8400.2.a.bb 1
5.b even 2 1 1050.2.a.n yes 1
5.c odd 4 2 1050.2.g.b 2
7.b odd 2 1 7350.2.a.i 1
15.d odd 2 1 3150.2.a.r 1
15.e even 4 2 3150.2.g.p 2
20.d odd 2 1 8400.2.a.cb 1
35.c odd 2 1 7350.2.a.cm 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.a.f 1 1.a even 1 1 trivial
1050.2.a.n yes 1 5.b even 2 1
1050.2.g.b 2 5.c odd 4 2
3150.2.a.r 1 15.d odd 2 1
3150.2.a.bd 1 3.b odd 2 1
3150.2.g.p 2 15.e even 4 2
7350.2.a.i 1 7.b odd 2 1
7350.2.a.cm 1 35.c odd 2 1
8400.2.a.bb 1 4.b odd 2 1
8400.2.a.cb 1 20.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1050))\):

\( T_{11} + 2 \)
\( T_{13} + 7 \)
\( T_{17} - 7 \)
\( T_{19} - 8 \)

Hecke Characteristic Polynomials

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