Properties

Label 1800.2.a.d
Level $1800$
Weight $2$
Character orbit 1800.a
Self dual yes
Analytic conductor $14.373$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3730723638\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{7} + O(q^{10}) \) \( q - 4q^{7} + 4q^{11} + 4q^{13} - 6q^{17} - 4q^{19} - 4q^{23} + 4q^{29} + 4q^{37} - 8q^{41} - 12q^{47} + 9q^{49} - 2q^{53} - 12q^{59} + 2q^{61} + 8q^{67} - 8q^{71} - 16q^{73} - 16q^{77} - 8q^{79} - 8q^{83} - 16q^{91} - 8q^{97} + O(q^{100}) \)

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 0 0 0 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1800.2.a.d 1
3.b odd 2 1 1800.2.a.b 1
4.b odd 2 1 3600.2.a.bn 1
5.b even 2 1 1800.2.a.w 1
5.c odd 4 2 360.2.f.b 2
12.b even 2 1 3600.2.a.bp 1
15.d odd 2 1 1800.2.a.u 1
15.e even 4 2 360.2.f.d yes 2
20.d odd 2 1 3600.2.a.c 1
20.e even 4 2 720.2.f.a 2
40.i odd 4 2 2880.2.f.q 2
40.k even 4 2 2880.2.f.u 2
60.h even 2 1 3600.2.a.g 1
60.l odd 4 2 720.2.f.g 2
120.q odd 4 2 2880.2.f.b 2
120.w even 4 2 2880.2.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.f.b 2 5.c odd 4 2
360.2.f.d yes 2 15.e even 4 2
720.2.f.a 2 20.e even 4 2
720.2.f.g 2 60.l odd 4 2
1800.2.a.b 1 3.b odd 2 1
1800.2.a.d 1 1.a even 1 1 trivial
1800.2.a.u 1 15.d odd 2 1
1800.2.a.w 1 5.b even 2 1
2880.2.f.b 2 120.q odd 4 2
2880.2.f.f 2 120.w even 4 2
2880.2.f.q 2 40.i odd 4 2
2880.2.f.u 2 40.k even 4 2
3600.2.a.c 1 20.d odd 2 1
3600.2.a.g 1 60.h even 2 1
3600.2.a.bn 1 4.b odd 2 1
3600.2.a.bp 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\):

\( T_{7} + 4 \)
\( T_{11} - 4 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

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