Properties

Label 1728.2.c.e
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{12}^{3} q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{12}^{3} q^{5} + ( 1 - 2 \zeta_{12}^{2} ) q^{7} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + 2 q^{13} -6 \zeta_{12}^{3} q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} -4 q^{25} + 6 \zeta_{12}^{3} q^{29} + ( 3 - 6 \zeta_{12}^{2} ) q^{31} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{35} -8 q^{37} + ( -6 + 12 \zeta_{12}^{2} ) q^{43} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + 4 q^{49} -9 \zeta_{12}^{3} q^{53} + ( -9 + 18 \zeta_{12}^{2} ) q^{55} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{59} + 4 q^{61} + 6 \zeta_{12}^{3} q^{65} + ( -2 + 4 \zeta_{12}^{2} ) q^{67} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + q^{73} -9 \zeta_{12}^{3} q^{77} + ( 2 - 4 \zeta_{12}^{2} ) q^{79} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + 18 q^{85} + 6 \zeta_{12}^{3} q^{89} + ( 2 - 4 \zeta_{12}^{2} ) q^{91} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{13} - 16q^{25} - 32q^{37} + 16q^{49} + 16q^{61} + 4q^{73} + 72q^{85} - 20q^{97} + O(q^{100}) \)

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 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0 0 3.00000i 0 1.73205i 0 0 0
1727.2 0 0 0 3.00000i 0 1.73205i 0 0 0
1727.3 0 0 0 3.00000i 0 1.73205i 0 0 0
1727.4 0 0 0 3.00000i 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.e 4
3.b odd 2 1 inner 1728.2.c.e 4
4.b odd 2 1 inner 1728.2.c.e 4
8.b even 2 1 432.2.c.c 4
8.d odd 2 1 432.2.c.c 4
12.b even 2 1 inner 1728.2.c.e 4
24.f even 2 1 432.2.c.c 4
24.h odd 2 1 432.2.c.c 4
72.j odd 6 1 1296.2.s.g 4
72.j odd 6 1 1296.2.s.i 4
72.l even 6 1 1296.2.s.g 4
72.l even 6 1 1296.2.s.i 4
72.n even 6 1 1296.2.s.g 4
72.n even 6 1 1296.2.s.i 4
72.p odd 6 1 1296.2.s.g 4
72.p odd 6 1 1296.2.s.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.c 4 8.b even 2 1
432.2.c.c 4 8.d odd 2 1
432.2.c.c 4 24.f even 2 1
432.2.c.c 4 24.h odd 2 1
1296.2.s.g 4 72.j odd 6 1
1296.2.s.g 4 72.l even 6 1
1296.2.s.g 4 72.n even 6 1
1296.2.s.g 4 72.p odd 6 1
1296.2.s.i 4 72.j odd 6 1
1296.2.s.i 4 72.l even 6 1
1296.2.s.i 4 72.n even 6 1
1296.2.s.i 4 72.p odd 6 1
1728.2.c.e 4 1.a even 1 1 trivial
1728.2.c.e 4 3.b odd 2 1 inner
1728.2.c.e 4 4.b odd 2 1 inner
1728.2.c.e 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 \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 9 \)
\( T_{7}^{2} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 9 + T^{2} )^{2} \)
$7$ \( ( 3 + T^{2} )^{2} \)
$11$ \( ( -27 + T^{2} )^{2} \)
$13$ \( ( -2 + T )^{4} \)
$17$ \( ( 36 + T^{2} )^{2} \)
$19$ \( ( 48 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 36 + T^{2} )^{2} \)
$31$ \( ( 27 + T^{2} )^{2} \)
$37$ \( ( 8 + T )^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 108 + T^{2} )^{2} \)
$47$ \( ( -108 + T^{2} )^{2} \)
$53$ \( ( 81 + T^{2} )^{2} \)
$59$ \( ( -108 + T^{2} )^{2} \)
$61$ \( ( -4 + T )^{4} \)
$67$ \( ( 12 + T^{2} )^{2} \)
$71$ \( ( -108 + T^{2} )^{2} \)
$73$ \( ( -1 + T )^{4} \)
$79$ \( ( 12 + T^{2} )^{2} \)
$83$ \( ( -27 + T^{2} )^{2} \)
$89$ \( ( 36 + T^{2} )^{2} \)
$97$ \( ( 5 + T )^{4} \)
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