Properties

Label 1380.4.a.a
Level $1380$
Weight $4$
Character orbit 1380.a
Self dual yes
Analytic conductor $81.423$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(81.4226358079\)
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

\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9} + 30 q^{11} - 28 q^{13} + 15 q^{15} + 78 q^{17} - 52 q^{19} + 48 q^{21} - 23 q^{23} + 25 q^{25} - 27 q^{27} - 252 q^{29} + 200 q^{31} - 90 q^{33} + 80 q^{35} + 146 q^{37} + 84 q^{39} + 438 q^{41} - 46 q^{43} - 45 q^{45} + 588 q^{47} - 87 q^{49} - 234 q^{51} + 438 q^{53} - 150 q^{55} + 156 q^{57} - 168 q^{59} - 586 q^{61} - 144 q^{63} + 140 q^{65} - 94 q^{67} + 69 q^{69} - 30 q^{71} - 466 q^{73} - 75 q^{75} - 480 q^{77} - 520 q^{79} + 81 q^{81} + 708 q^{83} - 390 q^{85} + 756 q^{87} - 600 q^{89} + 448 q^{91} - 600 q^{93} + 260 q^{95} - 340 q^{97} + 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.00000 0 −5.00000 0 −16.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(23\) \(1\)

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 1380.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.4.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 16 \) Copy content Toggle raw display
$11$ \( T - 30 \) Copy content Toggle raw display
$13$ \( T + 28 \) Copy content Toggle raw display
$17$ \( T - 78 \) Copy content Toggle raw display
$19$ \( T + 52 \) Copy content Toggle raw display
$23$ \( T + 23 \) Copy content Toggle raw display
$29$ \( T + 252 \) Copy content Toggle raw display
$31$ \( T - 200 \) Copy content Toggle raw display
$37$ \( T - 146 \) Copy content Toggle raw display
$41$ \( T - 438 \) Copy content Toggle raw display
$43$ \( T + 46 \) Copy content Toggle raw display
$47$ \( T - 588 \) Copy content Toggle raw display
$53$ \( T - 438 \) Copy content Toggle raw display
$59$ \( T + 168 \) Copy content Toggle raw display
$61$ \( T + 586 \) Copy content Toggle raw display
$67$ \( T + 94 \) Copy content Toggle raw display
$71$ \( T + 30 \) Copy content Toggle raw display
$73$ \( T + 466 \) Copy content Toggle raw display
$79$ \( T + 520 \) Copy content Toggle raw display
$83$ \( T - 708 \) Copy content Toggle raw display
$89$ \( T + 600 \) Copy content Toggle raw display
$97$ \( T + 340 \) Copy content Toggle raw display
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