Properties

Label 1728.2.r.b
Level $1728$
Weight $2$
Character orbit 1728.r
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.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 4 - 2 \zeta_{12}^{2} ) q^{5} +O(q^{10})\) \( q + ( 4 - 2 \zeta_{12}^{2} ) q^{5} + 3 \zeta_{12} q^{11} + ( -4 + 2 \zeta_{12}^{2} ) q^{13} + 3 q^{17} + 7 \zeta_{12}^{3} q^{19} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} + ( 7 - 7 \zeta_{12}^{2} ) q^{25} + ( 4 + 4 \zeta_{12}^{2} ) q^{29} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} + ( 6 - 12 \zeta_{12}^{2} ) q^{37} + 3 \zeta_{12}^{2} q^{41} -5 \zeta_{12} q^{43} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + 7 \zeta_{12}^{2} q^{49} + ( 8 - 16 \zeta_{12}^{2} ) q^{53} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{55} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{59} + ( -4 - 4 \zeta_{12}^{2} ) q^{61} + ( -12 + 12 \zeta_{12}^{2} ) q^{65} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{71} -7 q^{73} + ( 10 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{79} -12 \zeta_{12} q^{83} + ( 12 - 6 \zeta_{12}^{2} ) q^{85} + 6 q^{89} + ( 14 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{95} + ( -1 + \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{5} + O(q^{10}) \) \( 4q + 12q^{5} - 12q^{13} + 12q^{17} + 14q^{25} + 24q^{29} + 6q^{41} + 14q^{49} - 24q^{61} - 24q^{65} - 28q^{73} + 36q^{85} + 24q^{89} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 3.00000 1.73205i 0 0 0 0 0
289.2 0 0 0 3.00000 1.73205i 0 0 0 0 0
1441.1 0 0 0 3.00000 + 1.73205i 0 0 0 0 0
1441.2 0 0 0 3.00000 + 1.73205i 0 0 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
4.b odd 2 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.r.b 4
3.b odd 2 1 576.2.r.a 4
4.b odd 2 1 inner 1728.2.r.b 4
8.b even 2 1 1728.2.r.a 4
8.d odd 2 1 1728.2.r.a 4
9.c even 3 1 1728.2.r.a 4
9.c even 3 1 5184.2.d.j 4
9.d odd 6 1 576.2.r.b yes 4
9.d odd 6 1 5184.2.d.c 4
12.b even 2 1 576.2.r.a 4
24.f even 2 1 576.2.r.b yes 4
24.h odd 2 1 576.2.r.b yes 4
36.f odd 6 1 1728.2.r.a 4
36.f odd 6 1 5184.2.d.j 4
36.h even 6 1 576.2.r.b yes 4
36.h even 6 1 5184.2.d.c 4
72.j odd 6 1 576.2.r.a 4
72.j odd 6 1 5184.2.d.c 4
72.l even 6 1 576.2.r.a 4
72.l even 6 1 5184.2.d.c 4
72.n even 6 1 inner 1728.2.r.b 4
72.n even 6 1 5184.2.d.j 4
72.p odd 6 1 inner 1728.2.r.b 4
72.p odd 6 1 5184.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.a 4 3.b odd 2 1
576.2.r.a 4 12.b even 2 1
576.2.r.a 4 72.j odd 6 1
576.2.r.a 4 72.l even 6 1
576.2.r.b yes 4 9.d odd 6 1
576.2.r.b yes 4 24.f even 2 1
576.2.r.b yes 4 24.h odd 2 1
576.2.r.b yes 4 36.h even 6 1
1728.2.r.a 4 8.b even 2 1
1728.2.r.a 4 8.d odd 2 1
1728.2.r.a 4 9.c even 3 1
1728.2.r.a 4 36.f odd 6 1
1728.2.r.b 4 1.a even 1 1 trivial
1728.2.r.b 4 4.b odd 2 1 inner
1728.2.r.b 4 72.n even 6 1 inner
1728.2.r.b 4 72.p odd 6 1 inner
5184.2.d.c 4 9.d odd 6 1
5184.2.d.c 4 36.h even 6 1
5184.2.d.c 4 72.j odd 6 1
5184.2.d.c 4 72.l even 6 1
5184.2.d.j 4 9.c even 3 1
5184.2.d.j 4 36.f odd 6 1
5184.2.d.j 4 72.n even 6 1
5184.2.d.j 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6 T_{5} + 12 \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 12 - 6 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 81 - 9 T^{2} + T^{4} \)
$13$ \( ( 12 + 6 T + T^{2} )^{2} \)
$17$ \( ( -3 + T )^{4} \)
$19$ \( ( 49 + T^{2} )^{2} \)
$23$ \( 144 + 12 T^{2} + T^{4} \)
$29$ \( ( 48 - 12 T + T^{2} )^{2} \)
$31$ \( 2304 + 48 T^{2} + T^{4} \)
$37$ \( ( 108 + T^{2} )^{2} \)
$41$ \( ( 9 - 3 T + T^{2} )^{2} \)
$43$ \( 625 - 25 T^{2} + T^{4} \)
$47$ \( 144 + 12 T^{2} + T^{4} \)
$53$ \( ( 192 + T^{2} )^{2} \)
$59$ \( 6561 - 81 T^{2} + T^{4} \)
$61$ \( ( 48 + 12 T + T^{2} )^{2} \)
$67$ \( 625 - 25 T^{2} + T^{4} \)
$71$ \( ( -12 + T^{2} )^{2} \)
$73$ \( ( 7 + T )^{4} \)
$79$ \( 90000 + 300 T^{2} + T^{4} \)
$83$ \( 20736 - 144 T^{2} + T^{4} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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