Properties

Label 1728.2.bc.b
Level $1728$
Weight $2$
Character orbit 1728.bc
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
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.bc (of order \(12\), degree \(4\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 4 - \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( 4 - \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12} ) q^{11} + ( 2 - 2 \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{13} -4 q^{17} + ( 3 - 3 \zeta_{12}^{3} ) q^{19} + ( 2 - 3 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{23} + ( -2 - 3 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{25} + ( 1 + 2 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{31} + ( -2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} + ( -5 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{37} + ( -3 - 6 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{41} + ( 1 + 5 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{43} + ( 3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + ( 6 - 4 \zeta_{12} - 6 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{49} + ( -5 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{53} + ( 1 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{55} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( 7 + 8 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{61} + ( 2 \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{65} + ( -5 + 5 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{67} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{71} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( 5 - 5 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{77} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{79} + ( 1 - 9 \zeta_{12} - 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{83} + ( 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + ( 8 - 16 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{89} + ( 4 - 9 \zeta_{12} - 9 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{91} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} -\zeta_{12}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + 12q^{7} + O(q^{10}) \) \( 4q - 2q^{5} + 12q^{7} + 4q^{11} + 2q^{13} - 16q^{17} + 12q^{19} + 12q^{23} - 6q^{25} + 6q^{29} + 8q^{31} - 14q^{35} - 24q^{37} - 18q^{41} - 8q^{43} + 8q^{47} + 12q^{49} - 16q^{53} + 12q^{59} + 30q^{61} - 2q^{65} - 16q^{67} + 16q^{77} - 16q^{83} + 8q^{85} - 2q^{91} + 6q^{95} - 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0 0 −0.500000 1.86603i 0 3.86603 2.23205i 0 0 0
721.1 0 0 0 −0.500000 0.133975i 0 2.13397 + 1.23205i 0 0 0
1009.1 0 0 0 −0.500000 + 0.133975i 0 2.13397 1.23205i 0 0 0
1585.1 0 0 0 −0.500000 + 1.86603i 0 3.86603 + 2.23205i 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
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.bc.b 4
3.b odd 2 1 576.2.bb.b 4
4.b odd 2 1 432.2.y.a 4
9.c even 3 1 1728.2.bc.c 4
9.d odd 6 1 576.2.bb.a 4
12.b even 2 1 144.2.x.d yes 4
16.e even 4 1 1728.2.bc.c 4
16.f odd 4 1 432.2.y.d 4
36.f odd 6 1 432.2.y.d 4
36.h even 6 1 144.2.x.a 4
48.i odd 4 1 576.2.bb.a 4
48.k even 4 1 144.2.x.a 4
144.u even 12 1 144.2.x.d yes 4
144.v odd 12 1 432.2.y.a 4
144.w odd 12 1 576.2.bb.b 4
144.x even 12 1 inner 1728.2.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.a 4 36.h even 6 1
144.2.x.a 4 48.k even 4 1
144.2.x.d yes 4 12.b even 2 1
144.2.x.d yes 4 144.u even 12 1
432.2.y.a 4 4.b odd 2 1
432.2.y.a 4 144.v odd 12 1
432.2.y.d 4 16.f odd 4 1
432.2.y.d 4 36.f odd 6 1
576.2.bb.a 4 9.d odd 6 1
576.2.bb.a 4 48.i odd 4 1
576.2.bb.b 4 3.b odd 2 1
576.2.bb.b 4 144.w odd 12 1
1728.2.bc.b 4 1.a even 1 1 trivial
1728.2.bc.b 4 144.x even 12 1 inner
1728.2.bc.c 4 9.c even 3 1
1728.2.bc.c 4 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2 T_{5}^{3} + 5 T_{5}^{2} + 4 T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 + 4 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( 121 - 132 T + 59 T^{2} - 12 T^{3} + T^{4} \)
$11$ \( 1 - 2 T + 5 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 121 - 88 T + 17 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( ( 4 + T )^{4} \)
$19$ \( ( 18 - 6 T + T^{2} )^{2} \)
$23$ \( 9 - 36 T + 51 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( 9 + 9 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 4356 + 1584 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$41$ \( 81 - 162 T + 99 T^{2} + 18 T^{3} + T^{4} \)
$43$ \( 3481 + 826 T + 65 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( 676 + 416 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$59$ \( 81 - 54 T + 45 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 1089 - 396 T + 261 T^{2} - 30 T^{3} + T^{4} \)
$67$ \( 121 - 22 T + 65 T^{2} + 16 T^{3} + T^{4} \)
$71$ \( 1024 + 128 T^{2} + T^{4} \)
$73$ \( 16 + 56 T^{2} + T^{4} \)
$79$ \( 9 + 3 T^{2} + T^{4} \)
$83$ \( 32041 + 3938 T + 185 T^{2} + 16 T^{3} + T^{4} \)
$89$ \( 35344 + 392 T^{2} + T^{4} \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
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