Properties

Label 1400.2.a.d
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: no (minimal twist has level 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} + q^{7} - 2q^{9} - q^{11} + q^{13} + 3q^{17} - 4q^{19} - q^{21} - 2q^{23} + 5q^{27} - q^{29} - 6q^{31} + q^{33} - 2q^{37} - q^{39} - 10q^{41} - 9q^{47} + q^{49} - 3q^{51} + 14q^{53} + 4q^{57} + 6q^{59} - 4q^{61} - 2q^{63} - 10q^{67} + 2q^{69} - 16q^{71} - 10q^{73} - q^{77} - 11q^{79} + q^{81} - 4q^{83} + q^{87} + 12q^{89} + q^{91} + 6q^{93} + 19q^{97} + 2q^{99} + O(q^{100}) \)

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
0 −1.00000 0 0 0 1.00000 0 −2.00000 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.d 1
4.b odd 2 1 2800.2.a.u 1
5.b even 2 1 1400.2.a.j 1
5.c odd 4 2 280.2.g.a 2
7.b odd 2 1 9800.2.a.bb 1
15.e even 4 2 2520.2.t.a 2
20.d odd 2 1 2800.2.a.k 1
20.e even 4 2 560.2.g.d 2
35.c odd 2 1 9800.2.a.p 1
35.f even 4 2 1960.2.g.a 2
40.i odd 4 2 2240.2.g.b 2
40.k even 4 2 2240.2.g.a 2
60.l odd 4 2 5040.2.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.a 2 5.c odd 4 2
560.2.g.d 2 20.e even 4 2
1400.2.a.d 1 1.a even 1 1 trivial
1400.2.a.j 1 5.b even 2 1
1960.2.g.a 2 35.f even 4 2
2240.2.g.a 2 40.k even 4 2
2240.2.g.b 2 40.i odd 4 2
2520.2.t.a 2 15.e even 4 2
2800.2.a.k 1 20.d odd 2 1
2800.2.a.u 1 4.b odd 2 1
5040.2.t.a 2 60.l odd 4 2
9800.2.a.p 1 35.c odd 2 1
9800.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1400))\):

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

Hecke characteristic polynomials

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