Properties

Label 1400.2.a.c
Level $1400$
Weight $2$
Character orbit 1400.a
Self dual yes
Analytic conductor $11.179$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.1790562830\)
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

\(f(q)\) \(=\) \( q - 2q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} + q^{7} + q^{9} + q^{11} - 4q^{13} + 6q^{19} - 2q^{21} - 3q^{23} + 4q^{27} - 3q^{29} - 2q^{33} - 9q^{37} + 8q^{39} + 2q^{41} - 9q^{43} + 6q^{47} + q^{49} - 6q^{53} - 12q^{57} + 8q^{59} - 10q^{61} + q^{63} + q^{67} + 6q^{69} - 7q^{71} + 2q^{73} + q^{77} - 9q^{79} - 11q^{81} - 12q^{83} + 6q^{87} - 4q^{89} - 4q^{91} - 16q^{97} + q^{99} + O(q^{100}) \)

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 −2.00000 0 0 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1400.2.a.c 1
4.b odd 2 1 2800.2.a.bb 1
5.b even 2 1 1400.2.a.l yes 1
5.c odd 4 2 1400.2.g.c 2
7.b odd 2 1 9800.2.a.bk 1
20.d odd 2 1 2800.2.a.f 1
20.e even 4 2 2800.2.g.f 2
35.c odd 2 1 9800.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.c 1 1.a even 1 1 trivial
1400.2.a.l yes 1 5.b even 2 1
1400.2.g.c 2 5.c odd 4 2
2800.2.a.f 1 20.d odd 2 1
2800.2.a.bb 1 4.b odd 2 1
2800.2.g.f 2 20.e even 4 2
9800.2.a.h 1 35.c odd 2 1
9800.2.a.bk 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\):

\( T_{3} + 2 \)
\( T_{11} - 1 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

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