Properties

Label 1728.2.bc.a
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} + ( 3 - \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 3 - 3 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{13} + ( 4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{17} + ( 2 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{23} + ( -8 + 3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{25} + ( -2 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{31} + ( 2 - 2 \zeta_{12}^{3} ) q^{35} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + 3 \zeta_{12} q^{41} + ( -3 - 2 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{43} + ( 5 \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{47} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + ( 2 - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{55} + ( 5 + 5 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{59} + ( 6 + 6 \zeta_{12} ) q^{61} + ( 6 \zeta_{12} - 14 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{65} + ( 2 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( 4 - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( 1 - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( -3 + 3 \zeta_{12} + 7 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{77} + 12 \zeta_{12}^{2} q^{79} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{83} + ( -2 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} -2 \zeta_{12}^{3} q^{89} + ( -4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{91} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{95} + ( -\zeta_{12} - 10 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} - 6q^{7} + O(q^{10}) \) \( 4q - 4q^{5} - 6q^{7} + 8q^{11} + 14q^{13} + 16q^{17} + 6q^{19} - 6q^{23} - 24q^{25} - 6q^{29} + 8q^{31} + 8q^{35} + 12q^{37} - 2q^{43} - 2q^{47} - 6q^{49} + 16q^{53} + 12q^{59} + 24q^{61} - 28q^{65} + 14q^{67} + 2q^{77} + 24q^{79} - 2q^{83} - 16q^{85} - 20q^{91} - 20q^{97} + O(q^{100}) \)

Character values

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

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

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 −1.00000 3.73205i 0 −0.633975 + 0.366025i 0 0 0
721.1 0 0 0 −1.00000 0.267949i 0 −2.36603 1.36603i 0 0 0
1009.1 0 0 0 −1.00000 + 0.267949i 0 −2.36603 + 1.36603i 0 0 0
1585.1 0 0 0 −1.00000 + 3.73205i 0 −0.633975 0.366025i 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.a 4
3.b odd 2 1 576.2.bb.c 4
4.b odd 2 1 432.2.y.b 4
9.c even 3 1 1728.2.bc.d 4
9.d odd 6 1 576.2.bb.d 4
12.b even 2 1 144.2.x.c yes 4
16.e even 4 1 1728.2.bc.d 4
16.f odd 4 1 432.2.y.c 4
36.f odd 6 1 432.2.y.c 4
36.h even 6 1 144.2.x.b 4
48.i odd 4 1 576.2.bb.d 4
48.k even 4 1 144.2.x.b 4
144.u even 12 1 144.2.x.c yes 4
144.v odd 12 1 432.2.y.b 4
144.w odd 12 1 576.2.bb.c 4
144.x even 12 1 inner 1728.2.bc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b 4 36.h even 6 1
144.2.x.b 4 48.k even 4 1
144.2.x.c yes 4 12.b even 2 1
144.2.x.c yes 4 144.u even 12 1
432.2.y.b 4 4.b odd 2 1
432.2.y.b 4 144.v odd 12 1
432.2.y.c 4 16.f odd 4 1
432.2.y.c 4 36.f odd 6 1
576.2.bb.c 4 3.b odd 2 1
576.2.bb.c 4 144.w odd 12 1
576.2.bb.d 4 9.d odd 6 1
576.2.bb.d 4 48.i odd 4 1
1728.2.bc.a 4 1.a even 1 1 trivial
1728.2.bc.a 4 144.x even 12 1 inner
1728.2.bc.d 4 9.c even 3 1
1728.2.bc.d 4 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4 T_{5}^{3} + 20 T_{5}^{2} + 32 T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 16 + 32 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( 169 - 130 T + 41 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( 484 - 220 T + 74 T^{2} - 14 T^{3} + T^{4} \)
$17$ \( ( 13 - 8 T + T^{2} )^{2} \)
$19$ \( 9 - 18 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$31$ \( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 144 - 144 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( 81 - 9 T^{2} + T^{4} \)
$43$ \( 121 - 176 T + 65 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( 5476 - 148 T + 78 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 64 - 128 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 1521 - 234 T + 45 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 1296 - 432 T + 180 T^{2} - 24 T^{3} + T^{4} \)
$67$ \( 1369 - 592 T + 113 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( 1024 + 128 T^{2} + T^{4} \)
$73$ \( 3721 + 134 T^{2} + T^{4} \)
$79$ \( ( 144 - 12 T + T^{2} )^{2} \)
$83$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$89$ \( ( 4 + T^{2} )^{2} \)
$97$ \( 9409 + 1940 T + 303 T^{2} + 20 T^{3} + T^{4} \)
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