Properties

Label 1872.4.a.k
Level $1872$
Weight $4$
Character orbit 1872.a
Self dual yes
Analytic conductor $110.452$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.451575531\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 7 q^{5} + 13 q^{7} + O(q^{10}) \) \( q + 7 q^{5} + 13 q^{7} - 26 q^{11} + 13 q^{13} - 77 q^{17} + 126 q^{19} - 96 q^{23} - 76 q^{25} + 82 q^{29} - 196 q^{31} + 91 q^{35} - 131 q^{37} - 336 q^{41} + 201 q^{43} - 105 q^{47} - 174 q^{49} + 432 q^{53} - 182 q^{55} - 294 q^{59} - 56 q^{61} + 91 q^{65} - 478 q^{67} + 9 q^{71} + 98 q^{73} - 338 q^{77} - 1304 q^{79} - 308 q^{83} - 539 q^{85} + 1190 q^{89} + 169 q^{91} + 882 q^{95} + 70 q^{97} + O(q^{100}) \)

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 7.00000 0 13.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

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 1872.4.a.k 1
3.b odd 2 1 208.4.a.g 1
4.b odd 2 1 117.4.a.b 1
12.b even 2 1 13.4.a.a 1
24.f even 2 1 832.4.a.r 1
24.h odd 2 1 832.4.a.a 1
52.b odd 2 1 1521.4.a.a 1
60.h even 2 1 325.4.a.d 1
60.l odd 4 2 325.4.b.b 2
84.h odd 2 1 637.4.a.a 1
132.d odd 2 1 1573.4.a.a 1
156.h even 2 1 169.4.a.e 1
156.l odd 4 2 169.4.b.a 2
156.p even 6 2 169.4.c.e 2
156.r even 6 2 169.4.c.a 2
156.v odd 12 4 169.4.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 12.b even 2 1
117.4.a.b 1 4.b odd 2 1
169.4.a.e 1 156.h even 2 1
169.4.b.a 2 156.l odd 4 2
169.4.c.a 2 156.r even 6 2
169.4.c.e 2 156.p even 6 2
169.4.e.e 4 156.v odd 12 4
208.4.a.g 1 3.b odd 2 1
325.4.a.d 1 60.h even 2 1
325.4.b.b 2 60.l odd 4 2
637.4.a.a 1 84.h odd 2 1
832.4.a.a 1 24.h odd 2 1
832.4.a.r 1 24.f even 2 1
1521.4.a.a 1 52.b odd 2 1
1573.4.a.a 1 132.d odd 2 1
1872.4.a.k 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5} - 7 \)
\( T_{7} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -7 + T \)
$7$ \( -13 + T \)
$11$ \( 26 + T \)
$13$ \( -13 + T \)
$17$ \( 77 + T \)
$19$ \( -126 + T \)
$23$ \( 96 + T \)
$29$ \( -82 + T \)
$31$ \( 196 + T \)
$37$ \( 131 + T \)
$41$ \( 336 + T \)
$43$ \( -201 + T \)
$47$ \( 105 + T \)
$53$ \( -432 + T \)
$59$ \( 294 + T \)
$61$ \( 56 + T \)
$67$ \( 478 + T \)
$71$ \( -9 + T \)
$73$ \( -98 + T \)
$79$ \( 1304 + T \)
$83$ \( 308 + T \)
$89$ \( -1190 + T \)
$97$ \( -70 + T \)
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