Properties

Label 1728.2.c.c
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: \(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{2} q^{7} -\beta_{3} q^{11} -2 q^{13} -2 \beta_{1} q^{17} -4 \beta_{3} q^{23} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} -5 \beta_{3} q^{35} + 4 q^{37} + 4 \beta_{1} q^{41} -2 \beta_{2} q^{43} -2 \beta_{3} q^{47} -8 q^{49} -\beta_{1} q^{53} + \beta_{2} q^{55} -2 \beta_{3} q^{59} + 4 q^{61} + 2 \beta_{1} q^{65} + 2 \beta_{2} q^{67} -6 \beta_{3} q^{71} + 5 q^{73} + 3 \beta_{1} q^{77} + 2 \beta_{2} q^{79} -7 \beta_{3} q^{83} -10 q^{85} -2 \beta_{1} q^{89} + 2 \beta_{2} q^{91} + 11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 8q^{13} + 16q^{37} - 32q^{49} + 16q^{61} + 20q^{73} - 40q^{85} + 44q^{97} + O(q^{100}) \)

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

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

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
0.866025 + 1.11803i
−0.866025 + 1.11803i
−0.866025 1.11803i
0.866025 1.11803i
0 0 0 2.23607i 0 3.87298i 0 0 0
1727.2 0 0 0 2.23607i 0 3.87298i 0 0 0
1727.3 0 0 0 2.23607i 0 3.87298i 0 0 0
1727.4 0 0 0 2.23607i 0 3.87298i 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.c 4
3.b odd 2 1 inner 1728.2.c.c 4
4.b odd 2 1 inner 1728.2.c.c 4
8.b even 2 1 108.2.b.a 4
8.d odd 2 1 108.2.b.a 4
12.b even 2 1 inner 1728.2.c.c 4
24.f even 2 1 108.2.b.a 4
24.h odd 2 1 108.2.b.a 4
72.j odd 6 2 324.2.h.d 8
72.l even 6 2 324.2.h.d 8
72.n even 6 2 324.2.h.d 8
72.p odd 6 2 324.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.b.a 4 8.b even 2 1
108.2.b.a 4 8.d odd 2 1
108.2.b.a 4 24.f even 2 1
108.2.b.a 4 24.h odd 2 1
324.2.h.d 8 72.j odd 6 2
324.2.h.d 8 72.l even 6 2
324.2.h.d 8 72.n even 6 2
324.2.h.d 8 72.p odd 6 2
1728.2.c.c 4 1.a even 1 1 trivial
1728.2.c.c 4 3.b odd 2 1 inner
1728.2.c.c 4 4.b odd 2 1 inner
1728.2.c.c 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 \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 5 \)
\( T_{7}^{2} + 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( 15 + T^{2} )^{2} \)
$11$ \( ( -3 + T^{2} )^{2} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( -48 + T^{2} )^{2} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( 15 + T^{2} )^{2} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( ( 80 + T^{2} )^{2} \)
$43$ \( ( 60 + T^{2} )^{2} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( 5 + T^{2} )^{2} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( ( -4 + T )^{4} \)
$67$ \( ( 60 + T^{2} )^{2} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( -5 + T )^{4} \)
$79$ \( ( 60 + T^{2} )^{2} \)
$83$ \( ( -147 + T^{2} )^{2} \)
$89$ \( ( 20 + T^{2} )^{2} \)
$97$ \( ( -11 + T )^{4} \)
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