Properties

Label 1728.2.c.d
Level $1728$
Weight $2$
Character orbit 1728.c
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + \beta_{2} q^{7} + \beta_{3} q^{11} + q^{13} -\beta_{1} q^{17} + 3 \beta_{2} q^{19} + \beta_{3} q^{23} -3 q^{25} -2 \beta_{1} q^{29} + 2 \beta_{2} q^{31} -\beta_{3} q^{35} + q^{37} + 2 \beta_{1} q^{41} + 2 \beta_{2} q^{43} -\beta_{3} q^{47} + 4 q^{49} -2 \beta_{1} q^{53} + 8 \beta_{2} q^{55} -\beta_{3} q^{59} -11 q^{61} + \beta_{1} q^{65} + 7 \beta_{2} q^{67} - q^{73} + 3 \beta_{1} q^{77} + \beta_{2} q^{79} -2 \beta_{3} q^{83} + 8 q^{85} -\beta_{1} q^{89} + \beta_{2} q^{91} -3 \beta_{3} q^{95} -13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} - 12q^{25} + 4q^{37} + 16q^{49} - 44q^{61} - 4q^{73} + 32q^{85} - 52q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
1.22474 0.707107i
−1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
0 0 0 2.82843i 0 1.73205i 0 0 0
1727.2 0 0 0 2.82843i 0 1.73205i 0 0 0
1727.3 0 0 0 2.82843i 0 1.73205i 0 0 0
1727.4 0 0 0 2.82843i 0 1.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
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.c.d 4
3.b odd 2 1 inner 1728.2.c.d 4
4.b odd 2 1 inner 1728.2.c.d 4
8.b even 2 1 108.2.b.b 4
8.d odd 2 1 108.2.b.b 4
12.b even 2 1 inner 1728.2.c.d 4
24.f even 2 1 108.2.b.b 4
24.h odd 2 1 108.2.b.b 4
72.j odd 6 1 324.2.h.a 4
72.j odd 6 1 324.2.h.b 4
72.l even 6 1 324.2.h.a 4
72.l even 6 1 324.2.h.b 4
72.n even 6 1 324.2.h.a 4
72.n even 6 1 324.2.h.b 4
72.p odd 6 1 324.2.h.a 4
72.p odd 6 1 324.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.b 4 8.b even 2 1
108.2.b.b 4 8.d odd 2 1
108.2.b.b 4 24.f even 2 1
108.2.b.b 4 24.h odd 2 1
324.2.h.a 4 72.j odd 6 1
324.2.h.a 4 72.l even 6 1
324.2.h.a 4 72.n even 6 1
324.2.h.a 4 72.p odd 6 1
324.2.h.b 4 72.j odd 6 1
324.2.h.b 4 72.l even 6 1
324.2.h.b 4 72.n even 6 1
324.2.h.b 4 72.p odd 6 1
1728.2.c.d 4 1.a even 1 1 trivial
1728.2.c.d 4 3.b odd 2 1 inner
1728.2.c.d 4 4.b odd 2 1 inner
1728.2.c.d 4 12.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} + 8 \)
\( T_{7}^{2} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 8 + T^{2} )^{2} \)
$7$ \( ( 3 + T^{2} )^{2} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( ( 27 + T^{2} )^{2} \)
$23$ \( ( -24 + T^{2} )^{2} \)
$29$ \( ( 32 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( -1 + T )^{4} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( ( 12 + T^{2} )^{2} \)
$47$ \( ( -24 + T^{2} )^{2} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( ( -24 + T^{2} )^{2} \)
$61$ \( ( 11 + T )^{4} \)
$67$ \( ( 147 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 1 + T )^{4} \)
$79$ \( ( 3 + T^{2} )^{2} \)
$83$ \( ( -96 + T^{2} )^{2} \)
$89$ \( ( 8 + T^{2} )^{2} \)
$97$ \( ( 13 + T )^{4} \)
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