Properties

Label 1728.2.i.g
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
Analytic rank $0$
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.i (of order \(3\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( -3 + 3 \zeta_{6} ) q^{7} + ( 5 - 5 \zeta_{6} ) q^{11} -5 \zeta_{6} q^{13} + 2 q^{17} -4 q^{19} + \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + ( 9 - 9 \zeta_{6} ) q^{29} -\zeta_{6} q^{31} -3 q^{35} + 6 q^{37} + 3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{47} -2 \zeta_{6} q^{49} + 2 q^{53} + 5 q^{55} + 11 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} + ( 5 - 5 \zeta_{6} ) q^{65} + \zeta_{6} q^{67} + 4 q^{71} -2 q^{73} + 15 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( 1 - \zeta_{6} ) q^{83} + 2 \zeta_{6} q^{85} + 18 q^{89} + 15 q^{91} -4 \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 3q^{7} + O(q^{10}) \) \( 2q + q^{5} - 3q^{7} + 5q^{11} - 5q^{13} + 4q^{17} - 8q^{19} + q^{23} + 4q^{25} + 9q^{29} - q^{31} - 6q^{35} + 12q^{37} + 3q^{41} - q^{43} + 3q^{47} - 2q^{49} + 4q^{53} + 10q^{55} + 11q^{59} + 7q^{61} + 5q^{65} + q^{67} + 8q^{71} - 4q^{73} + 15q^{77} + q^{79} + q^{83} + 2q^{85} + 36q^{89} + 30q^{91} - 4q^{95} + 13q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 0.866025i 0 −1.50000 2.59808i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.g 2
3.b odd 2 1 576.2.i.c 2
4.b odd 2 1 1728.2.i.h 2
8.b even 2 1 432.2.i.a 2
8.d odd 2 1 216.2.i.a 2
9.c even 3 1 inner 1728.2.i.g 2
9.c even 3 1 5184.2.a.n 1
9.d odd 6 1 576.2.i.c 2
9.d odd 6 1 5184.2.a.x 1
12.b even 2 1 576.2.i.d 2
24.f even 2 1 72.2.i.a 2
24.h odd 2 1 144.2.i.b 2
36.f odd 6 1 1728.2.i.h 2
36.f odd 6 1 5184.2.a.i 1
36.h even 6 1 576.2.i.d 2
36.h even 6 1 5184.2.a.s 1
72.j odd 6 1 144.2.i.b 2
72.j odd 6 1 1296.2.a.e 1
72.l even 6 1 72.2.i.a 2
72.l even 6 1 648.2.a.a 1
72.n even 6 1 432.2.i.a 2
72.n even 6 1 1296.2.a.i 1
72.p odd 6 1 216.2.i.a 2
72.p odd 6 1 648.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 24.f even 2 1
72.2.i.a 2 72.l even 6 1
144.2.i.b 2 24.h odd 2 1
144.2.i.b 2 72.j odd 6 1
216.2.i.a 2 8.d odd 2 1
216.2.i.a 2 72.p odd 6 1
432.2.i.a 2 8.b even 2 1
432.2.i.a 2 72.n even 6 1
576.2.i.c 2 3.b odd 2 1
576.2.i.c 2 9.d odd 6 1
576.2.i.d 2 12.b even 2 1
576.2.i.d 2 36.h even 6 1
648.2.a.a 1 72.l even 6 1
648.2.a.c 1 72.p odd 6 1
1296.2.a.e 1 72.j odd 6 1
1296.2.a.i 1 72.n even 6 1
1728.2.i.g 2 1.a even 1 1 trivial
1728.2.i.g 2 9.c even 3 1 inner
1728.2.i.h 2 4.b odd 2 1
1728.2.i.h 2 36.f odd 6 1
5184.2.a.i 1 36.f odd 6 1
5184.2.a.n 1 9.c even 3 1
5184.2.a.s 1 36.h even 6 1
5184.2.a.x 1 9.d odd 6 1

Hecke kernels

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

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{2} + 3 T_{7} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 9 + 3 T + T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 1 - T + T^{2} \)
$29$ \( 81 - 9 T + T^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 1 + T + T^{2} \)
$47$ \( 9 - 3 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( 121 - 11 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( 1 - T + T^{2} \)
$89$ \( ( -18 + T )^{2} \)
$97$ \( 169 - 13 T + T^{2} \)
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