Properties

Label 1728.3.g.a
Level $1728$
Weight $3$
Character orbit 1728.g
Analytic conductor $47.085$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 108)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -7 q^{5} + ( 5 - 10 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -7 q^{5} + ( 5 - 10 \zeta_{6} ) q^{7} + ( 5 - 10 \zeta_{6} ) q^{11} -20 q^{13} -8 q^{17} + ( 6 - 12 \zeta_{6} ) q^{19} + ( 2 - 4 \zeta_{6} ) q^{23} + 24 q^{25} -10 q^{29} + ( 31 - 62 \zeta_{6} ) q^{31} + ( -35 + 70 \zeta_{6} ) q^{35} + 10 q^{37} -50 q^{41} + ( 10 - 20 \zeta_{6} ) q^{43} + ( -50 + 100 \zeta_{6} ) q^{47} -26 q^{49} + 47 q^{53} + ( -35 + 70 \zeta_{6} ) q^{55} + ( -20 + 40 \zeta_{6} ) q^{59} + 64 q^{61} + 140 q^{65} + ( 50 - 100 \zeta_{6} ) q^{67} -55 q^{73} -75 q^{77} + ( -4 + 8 \zeta_{6} ) q^{79} + ( 17 - 34 \zeta_{6} ) q^{83} + 56 q^{85} + 10 q^{89} + ( -100 + 200 \zeta_{6} ) q^{91} + ( -42 + 84 \zeta_{6} ) q^{95} -25 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5} + O(q^{10}) \) \( 2 q - 14 q^{5} - 40 q^{13} - 16 q^{17} + 48 q^{25} - 20 q^{29} + 20 q^{37} - 100 q^{41} - 52 q^{49} + 94 q^{53} + 128 q^{61} + 280 q^{65} - 110 q^{73} - 150 q^{77} + 112 q^{85} + 20 q^{89} - 50 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −7.00000 0 8.66025i 0 0 0
703.2 0 0 0 −7.00000 0 8.66025i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.g.a 2
3.b odd 2 1 1728.3.g.f 2
4.b odd 2 1 inner 1728.3.g.a 2
8.b even 2 1 108.3.d.a 2
8.d odd 2 1 108.3.d.a 2
12.b even 2 1 1728.3.g.f 2
24.f even 2 1 108.3.d.b yes 2
24.h odd 2 1 108.3.d.b yes 2
72.j odd 6 1 324.3.f.b 2
72.j odd 6 1 324.3.f.h 2
72.l even 6 1 324.3.f.b 2
72.l even 6 1 324.3.f.h 2
72.n even 6 1 324.3.f.c 2
72.n even 6 1 324.3.f.i 2
72.p odd 6 1 324.3.f.c 2
72.p odd 6 1 324.3.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 8.b even 2 1
108.3.d.a 2 8.d odd 2 1
108.3.d.b yes 2 24.f even 2 1
108.3.d.b yes 2 24.h odd 2 1
324.3.f.b 2 72.j odd 6 1
324.3.f.b 2 72.l even 6 1
324.3.f.c 2 72.n even 6 1
324.3.f.c 2 72.p odd 6 1
324.3.f.h 2 72.j odd 6 1
324.3.f.h 2 72.l even 6 1
324.3.f.i 2 72.n even 6 1
324.3.f.i 2 72.p odd 6 1
1728.3.g.a 2 1.a even 1 1 trivial
1728.3.g.a 2 4.b odd 2 1 inner
1728.3.g.f 2 3.b odd 2 1
1728.3.g.f 2 12.b even 2 1

Hecke kernels

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

\( T_{5} + 7 \)
\( T_{7}^{2} + 75 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 7 + T )^{2} \)
$7$ \( 75 + T^{2} \)
$11$ \( 75 + T^{2} \)
$13$ \( ( 20 + T )^{2} \)
$17$ \( ( 8 + T )^{2} \)
$19$ \( 108 + T^{2} \)
$23$ \( 12 + T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( 2883 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( ( 50 + T )^{2} \)
$43$ \( 300 + T^{2} \)
$47$ \( 7500 + T^{2} \)
$53$ \( ( -47 + T )^{2} \)
$59$ \( 1200 + T^{2} \)
$61$ \( ( -64 + T )^{2} \)
$67$ \( 7500 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 55 + T )^{2} \)
$79$ \( 48 + T^{2} \)
$83$ \( 867 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( 25 + T )^{2} \)
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