Properties

Label 1728.2.c.a
Level $1728$
Weight $2$
Character orbit 1728.c
Analytic conductor $13.798$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 432)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{7} -7 q^{13} + ( -5 + 10 \zeta_{6} ) q^{19} + 5 q^{25} + ( -6 + 12 \zeta_{6} ) q^{31} + q^{37} + ( -6 + 12 \zeta_{6} ) q^{43} + 4 q^{49} + 13 q^{61} + ( 7 - 14 \zeta_{6} ) q^{67} -17 q^{73} + ( -7 + 14 \zeta_{6} ) q^{79} + ( -7 + 14 \zeta_{6} ) q^{91} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 14q^{13} + 10q^{25} + 2q^{37} + 8q^{49} + 26q^{61} - 34q^{73} - 10q^{97} + O(q^{100}) \)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 7 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 75 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -13 + T )^{2} \)
$67$ \( 147 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 17 + T )^{2} \)
$79$ \( 147 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 5 + T )^{2} \)
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