Properties

Label 1728.3.b.e
Level $1728$
Weight $3$
Character orbit 1728.b
Analytic conductor $47.085$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 4 \zeta_{12}^{2} ) q^{5} -\zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( 2 - 4 \zeta_{12}^{2} ) q^{5} -\zeta_{12}^{3} q^{7} + ( 20 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{11} + ( -1 + 2 \zeta_{12}^{2} ) q^{13} + 6 q^{17} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{19} + 30 \zeta_{12}^{3} q^{23} + 13 q^{25} + ( -12 + 24 \zeta_{12}^{2} ) q^{29} -14 \zeta_{12}^{3} q^{31} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{35} + ( -11 + 22 \zeta_{12}^{2} ) q^{37} -48 q^{41} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{43} -66 \zeta_{12}^{3} q^{47} + 48 q^{49} + ( -28 + 56 \zeta_{12}^{2} ) q^{53} -60 \zeta_{12}^{3} q^{55} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{59} + ( -25 + 50 \zeta_{12}^{2} ) q^{61} + 6 q^{65} + ( 70 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{67} + 48 \zeta_{12}^{3} q^{71} + 49 q^{73} + ( 10 - 20 \zeta_{12}^{2} ) q^{77} -83 \zeta_{12}^{3} q^{79} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} + ( 12 - 24 \zeta_{12}^{2} ) q^{85} + 66 q^{89} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{91} -6 \zeta_{12}^{3} q^{95} + 107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 24 q^{17} + 52 q^{25} - 192 q^{41} + 192 q^{49} + 24 q^{65} + 196 q^{73} + 264 q^{89} + 428 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 3.46410i 0 1.00000i 0 0 0
1567.2 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.3 0 0 0 3.46410i 0 1.00000i 0 0 0
1567.4 0 0 0 3.46410i 0 1.00000i 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.b.e yes 4
3.b odd 2 1 1728.3.b.b 4
4.b odd 2 1 inner 1728.3.b.e yes 4
8.b even 2 1 inner 1728.3.b.e yes 4
8.d odd 2 1 inner 1728.3.b.e yes 4
12.b even 2 1 1728.3.b.b 4
24.f even 2 1 1728.3.b.b 4
24.h odd 2 1 1728.3.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.b.b 4 3.b odd 2 1
1728.3.b.b 4 12.b even 2 1
1728.3.b.b 4 24.f even 2 1
1728.3.b.b 4 24.h odd 2 1
1728.3.b.e yes 4 1.a even 1 1 trivial
1728.3.b.e yes 4 4.b odd 2 1 inner
1728.3.b.e yes 4 8.b even 2 1 inner
1728.3.b.e yes 4 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{7}^{2} + 1 \)
\( T_{11}^{2} - 300 \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 12 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -300 + T^{2} )^{2} \)
$13$ \( ( 3 + T^{2} )^{2} \)
$17$ \( ( -6 + T )^{4} \)
$19$ \( ( -3 + T^{2} )^{2} \)
$23$ \( ( 900 + T^{2} )^{2} \)
$29$ \( ( 432 + T^{2} )^{2} \)
$31$ \( ( 196 + T^{2} )^{2} \)
$37$ \( ( 363 + T^{2} )^{2} \)
$41$ \( ( 48 + T )^{4} \)
$43$ \( ( -432 + T^{2} )^{2} \)
$47$ \( ( 4356 + T^{2} )^{2} \)
$53$ \( ( 2352 + T^{2} )^{2} \)
$59$ \( ( -972 + T^{2} )^{2} \)
$61$ \( ( 1875 + T^{2} )^{2} \)
$67$ \( ( -3675 + T^{2} )^{2} \)
$71$ \( ( 2304 + T^{2} )^{2} \)
$73$ \( ( -49 + T )^{4} \)
$79$ \( ( 6889 + T^{2} )^{2} \)
$83$ \( ( -192 + T^{2} )^{2} \)
$89$ \( ( -66 + T )^{4} \)
$97$ \( ( -107 + T )^{4} \)
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