Properties

Label 1728.2.a.bc
Level $1728$
Weight $2$
Character orbit 1728.a
Self dual yes
Analytic conductor $13.798$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7981494693\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + \beta q^{7} +O(q^{10})\) \( q -\beta q^{5} + \beta q^{7} - q^{11} -4 q^{13} + 2 \beta q^{19} -6 q^{23} + 8 q^{25} + 2 \beta q^{29} -\beta q^{31} -13 q^{35} -10 q^{37} -2 \beta q^{41} -2 \beta q^{43} -10 q^{47} + 6 q^{49} + \beta q^{53} + \beta q^{55} + 4 q^{59} + 4 \beta q^{65} -2 \beta q^{67} + 8 q^{71} -3 q^{73} -\beta q^{77} -4 \beta q^{79} -9 q^{83} + 2 \beta q^{89} -4 \beta q^{91} -26 q^{95} + 7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{11} - 8q^{13} - 12q^{23} + 16q^{25} - 26q^{35} - 20q^{37} - 20q^{47} + 12q^{49} + 8q^{59} + 16q^{71} - 6q^{73} - 18q^{83} - 52q^{95} + 14q^{97} + O(q^{100}) \)

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
2.30278
−1.30278
0 0 0 −3.60555 0 3.60555 0 0 0
1.2 0 0 0 3.60555 0 −3.60555 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.bc 2
3.b odd 2 1 1728.2.a.bd 2
4.b odd 2 1 1728.2.a.bd 2
8.b even 2 1 864.2.a.n yes 2
8.d odd 2 1 864.2.a.m 2
12.b even 2 1 inner 1728.2.a.bc 2
24.f even 2 1 864.2.a.n yes 2
24.h odd 2 1 864.2.a.m 2
72.j odd 6 2 2592.2.i.bc 4
72.l even 6 2 2592.2.i.bb 4
72.n even 6 2 2592.2.i.bb 4
72.p odd 6 2 2592.2.i.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 8.d odd 2 1
864.2.a.m 2 24.h odd 2 1
864.2.a.n yes 2 8.b even 2 1
864.2.a.n yes 2 24.f even 2 1
1728.2.a.bc 2 1.a even 1 1 trivial
1728.2.a.bc 2 12.b even 2 1 inner
1728.2.a.bd 2 3.b odd 2 1
1728.2.a.bd 2 4.b odd 2 1
2592.2.i.bb 4 72.l even 6 2
2592.2.i.bb 4 72.n even 6 2
2592.2.i.bc 4 72.j odd 6 2
2592.2.i.bc 4 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5}^{2} - 13 \)
\( T_{7}^{2} - 13 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -13 + T^{2} \)
$7$ \( -13 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( -52 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( -52 + T^{2} \)
$31$ \( -13 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( -52 + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( ( 10 + T )^{2} \)
$53$ \( -13 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( -52 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( ( 3 + T )^{2} \)
$79$ \( -208 + T^{2} \)
$83$ \( ( 9 + T )^{2} \)
$89$ \( -52 + T^{2} \)
$97$ \( ( -7 + T )^{2} \)
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