Properties

Label 1872.2.a.f
Level $1872$
Weight $2$
Character orbit 1872.a
Self dual yes
Analytic conductor $14.948$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.9479952584\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} + 2 q^{7} - 2 q^{11} - q^{13} - 6 q^{17} + 6 q^{19} + 8 q^{23} - q^{25} - 2 q^{29} - 10 q^{31} - 4 q^{35} - 6 q^{37} + 6 q^{41} - 4 q^{43} - 2 q^{47} - 3 q^{49} - 6 q^{53} + 4 q^{55} - 10 q^{59} - 2 q^{61} + 2 q^{65} - 10 q^{67} + 10 q^{71} + 2 q^{73} - 4 q^{77} + 4 q^{79} - 6 q^{83} + 12 q^{85} + 6 q^{89} - 2 q^{91} - 12 q^{95} + 2 q^{97}+O(q^{100}) \) 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 0 0 −2.00000 0 2.00000 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 1872.2.a.f 1
3.b odd 2 1 208.2.a.c 1
4.b odd 2 1 468.2.a.b 1
8.b even 2 1 7488.2.a.bw 1
8.d odd 2 1 7488.2.a.bn 1
12.b even 2 1 52.2.a.a 1
15.d odd 2 1 5200.2.a.q 1
24.f even 2 1 832.2.a.e 1
24.h odd 2 1 832.2.a.f 1
36.f odd 6 2 4212.2.i.i 2
36.h even 6 2 4212.2.i.d 2
39.d odd 2 1 2704.2.a.g 1
39.f even 4 2 2704.2.f.f 2
48.i odd 4 2 3328.2.b.e 2
48.k even 4 2 3328.2.b.q 2
52.b odd 2 1 6084.2.a.m 1
52.f even 4 2 6084.2.b.m 2
60.h even 2 1 1300.2.a.d 1
60.l odd 4 2 1300.2.c.c 2
84.h odd 2 1 2548.2.a.e 1
84.j odd 6 2 2548.2.j.f 2
84.n even 6 2 2548.2.j.e 2
132.d odd 2 1 6292.2.a.g 1
156.h even 2 1 676.2.a.c 1
156.l odd 4 2 676.2.d.c 2
156.p even 6 2 676.2.e.c 2
156.r even 6 2 676.2.e.b 2
156.v odd 12 4 676.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 12.b even 2 1
208.2.a.c 1 3.b odd 2 1
468.2.a.b 1 4.b odd 2 1
676.2.a.c 1 156.h even 2 1
676.2.d.c 2 156.l odd 4 2
676.2.e.b 2 156.r even 6 2
676.2.e.c 2 156.p even 6 2
676.2.h.c 4 156.v odd 12 4
832.2.a.e 1 24.f even 2 1
832.2.a.f 1 24.h odd 2 1
1300.2.a.d 1 60.h even 2 1
1300.2.c.c 2 60.l odd 4 2
1872.2.a.f 1 1.a even 1 1 trivial
2548.2.a.e 1 84.h odd 2 1
2548.2.j.e 2 84.n even 6 2
2548.2.j.f 2 84.j odd 6 2
2704.2.a.g 1 39.d odd 2 1
2704.2.f.f 2 39.f even 4 2
3328.2.b.e 2 48.i odd 4 2
3328.2.b.q 2 48.k even 4 2
4212.2.i.d 2 36.h even 6 2
4212.2.i.i 2 36.f odd 6 2
5200.2.a.q 1 15.d odd 2 1
6084.2.a.m 1 52.b odd 2 1
6084.2.b.m 2 52.f even 4 2
6292.2.a.g 1 132.d odd 2 1
7488.2.a.bn 1 8.d odd 2 1
7488.2.a.bw 1 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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