Properties

Label 1872.2.a.w
Level $1872$
Weight $2$
Character orbit 1872.a
Self dual yes
Analytic conductor $14.948$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -\beta q^{7} +O(q^{10})\) \( q + \beta q^{5} -\beta q^{7} -2 q^{11} - q^{13} + ( -2 - 2 \beta ) q^{17} + \beta q^{19} -4 q^{23} + 3 q^{25} -2 q^{29} + ( 4 - \beta ) q^{31} -8 q^{35} + ( -2 - 2 \beta ) q^{37} + ( -8 + \beta ) q^{41} + ( -4 + 2 \beta ) q^{43} + ( -6 - 2 \beta ) q^{47} + q^{49} + 2 q^{53} -2 \beta q^{55} + ( 2 + 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} -\beta q^{65} + ( -4 - \beta ) q^{67} + 2 q^{71} + ( 6 - 2 \beta ) q^{73} + 2 \beta q^{77} + 4 \beta q^{79} + ( -2 + 2 \beta ) q^{83} + ( -16 - 2 \beta ) q^{85} + ( -12 - \beta ) q^{89} + \beta q^{91} + 8 q^{95} + ( -2 + 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 4q^{11} - 2q^{13} - 4q^{17} - 8q^{23} + 6q^{25} - 4q^{29} + 8q^{31} - 16q^{35} - 4q^{37} - 16q^{41} - 8q^{43} - 12q^{47} + 2q^{49} + 4q^{53} + 4q^{59} + 4q^{61} - 8q^{67} + 4q^{71} + 12q^{73} - 4q^{83} - 32q^{85} - 24q^{89} + 16q^{95} - 4q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −2.82843 0 2.82843 0 0 0
1.2 0 0 0 2.82843 0 −2.82843 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.w 2
3.b odd 2 1 624.2.a.k 2
4.b odd 2 1 117.2.a.c 2
8.b even 2 1 7488.2.a.co 2
8.d odd 2 1 7488.2.a.cl 2
12.b even 2 1 39.2.a.b 2
20.d odd 2 1 2925.2.a.v 2
20.e even 4 2 2925.2.c.u 4
24.f even 2 1 2496.2.a.bf 2
24.h odd 2 1 2496.2.a.bi 2
28.d even 2 1 5733.2.a.u 2
36.f odd 6 2 1053.2.e.e 4
36.h even 6 2 1053.2.e.m 4
39.d odd 2 1 8112.2.a.bm 2
52.b odd 2 1 1521.2.a.f 2
52.f even 4 2 1521.2.b.j 4
60.h even 2 1 975.2.a.l 2
60.l odd 4 2 975.2.c.h 4
84.h odd 2 1 1911.2.a.h 2
132.d odd 2 1 4719.2.a.p 2
156.h even 2 1 507.2.a.h 2
156.l odd 4 2 507.2.b.e 4
156.p even 6 2 507.2.e.h 4
156.r even 6 2 507.2.e.d 4
156.v odd 12 4 507.2.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 12.b even 2 1
117.2.a.c 2 4.b odd 2 1
507.2.a.h 2 156.h even 2 1
507.2.b.e 4 156.l odd 4 2
507.2.e.d 4 156.r even 6 2
507.2.e.h 4 156.p even 6 2
507.2.j.f 8 156.v odd 12 4
624.2.a.k 2 3.b odd 2 1
975.2.a.l 2 60.h even 2 1
975.2.c.h 4 60.l odd 4 2
1053.2.e.e 4 36.f odd 6 2
1053.2.e.m 4 36.h even 6 2
1521.2.a.f 2 52.b odd 2 1
1521.2.b.j 4 52.f even 4 2
1872.2.a.w 2 1.a even 1 1 trivial
1911.2.a.h 2 84.h odd 2 1
2496.2.a.bf 2 24.f even 2 1
2496.2.a.bi 2 24.h odd 2 1
2925.2.a.v 2 20.d odd 2 1
2925.2.c.u 4 20.e even 4 2
4719.2.a.p 2 132.d odd 2 1
5733.2.a.u 2 28.d even 2 1
7488.2.a.cl 2 8.d odd 2 1
7488.2.a.co 2 8.b even 2 1
8112.2.a.bm 2 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1872))\):

\( T_{5}^{2} - 8 \)
\( T_{7}^{2} - 8 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -28 + 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 8 - 8 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( 56 + 16 T + T^{2} \)
$43$ \( -16 + 8 T + T^{2} \)
$47$ \( 4 + 12 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -28 - 4 T + T^{2} \)
$61$ \( -124 - 4 T + T^{2} \)
$67$ \( 8 + 8 T + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 4 - 12 T + T^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -28 + 4 T + T^{2} \)
$89$ \( 136 + 24 T + T^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
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