Properties

Label 840.2.a.f
Level $840$
Weight $2$
Character orbit 840.a
Self dual yes
Analytic conductor $6.707$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.70743376979\)
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

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{5} + q^{7} + q^{9} + 4q^{11} - 2q^{13} - q^{15} - 6q^{17} + 4q^{19} - q^{21} + 8q^{23} + q^{25} - q^{27} - 2q^{29} - 4q^{33} + q^{35} - 2q^{37} + 2q^{39} + 10q^{41} + 4q^{43} + q^{45} + q^{49} + 6q^{51} + 14q^{53} + 4q^{55} - 4q^{57} + 12q^{59} - 2q^{61} + q^{63} - 2q^{65} - 4q^{67} - 8q^{69} + 2q^{73} - q^{75} + 4q^{77} - 8q^{79} + q^{81} - 4q^{83} - 6q^{85} + 2q^{87} - 6q^{89} - 2q^{91} + 4q^{95} - 6q^{97} + 4q^{99} + O(q^{100}) \)

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
0 −1.00000 0 1.00000 0 1.00000 0 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 840.2.a.f 1
3.b odd 2 1 2520.2.a.g 1
4.b odd 2 1 1680.2.a.p 1
5.b even 2 1 4200.2.a.z 1
5.c odd 4 2 4200.2.t.q 2
7.b odd 2 1 5880.2.a.z 1
8.b even 2 1 6720.2.a.bn 1
8.d odd 2 1 6720.2.a.h 1
12.b even 2 1 5040.2.a.j 1
20.d odd 2 1 8400.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.f 1 1.a even 1 1 trivial
1680.2.a.p 1 4.b odd 2 1
2520.2.a.g 1 3.b odd 2 1
4200.2.a.z 1 5.b even 2 1
4200.2.t.q 2 5.c odd 4 2
5040.2.a.j 1 12.b even 2 1
5880.2.a.z 1 7.b odd 2 1
6720.2.a.h 1 8.d odd 2 1
6720.2.a.bn 1 8.b even 2 1
8400.2.a.q 1 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(840))\):

\( T_{11} - 4 \)
\( T_{13} + 2 \)
\( T_{17} + 6 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

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