Properties

Label 1728.3.g.f
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: \(\Q(\sqrt{-3}) \)
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.

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

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.f 2
3.b odd 2 1 1728.3.g.a 2
4.b odd 2 1 inner 1728.3.g.f 2
8.b even 2 1 108.3.d.b yes 2
8.d odd 2 1 108.3.d.b yes 2
12.b even 2 1 1728.3.g.a 2
24.f even 2 1 108.3.d.a 2
24.h odd 2 1 108.3.d.a 2
72.j odd 6 1 324.3.f.c 2
72.j odd 6 1 324.3.f.i 2
72.l even 6 1 324.3.f.c 2
72.l even 6 1 324.3.f.i 2
72.n even 6 1 324.3.f.b 2
72.n even 6 1 324.3.f.h 2
72.p odd 6 1 324.3.f.b 2
72.p odd 6 1 324.3.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.a 2 24.f even 2 1
108.3.d.a 2 24.h odd 2 1
108.3.d.b yes 2 8.b even 2 1
108.3.d.b yes 2 8.d odd 2 1
324.3.f.b 2 72.n even 6 1
324.3.f.b 2 72.p odd 6 1
324.3.f.c 2 72.j odd 6 1
324.3.f.c 2 72.l even 6 1
324.3.f.h 2 72.n even 6 1
324.3.f.h 2 72.p odd 6 1
324.3.f.i 2 72.j odd 6 1
324.3.f.i 2 72.l even 6 1
1728.3.g.a 2 3.b odd 2 1
1728.3.g.a 2 12.b even 2 1
1728.3.g.f 2 1.a even 1 1 trivial
1728.3.g.f 2 4.b 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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 7 T + 25 T^{2} )^{2} \)
$7$ \( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \)
$11$ \( 1 - 167 T^{2} + 14641 T^{4} \)
$13$ \( ( 1 + 20 T + 169 T^{2} )^{2} \)
$17$ \( ( 1 - 8 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 614 T^{2} + 130321 T^{4} \)
$23$ \( 1 - 1046 T^{2} + 279841 T^{4} \)
$29$ \( ( 1 - 10 T + 841 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \)
$37$ \( ( 1 - 10 T + 1369 T^{2} )^{2} \)
$41$ \( ( 1 - 50 T + 1681 T^{2} )^{2} \)
$43$ \( 1 - 3398 T^{2} + 3418801 T^{4} \)
$47$ \( 1 + 3082 T^{2} + 4879681 T^{4} \)
$53$ \( ( 1 + 47 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 5762 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 - 64 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 1478 T^{2} + 20151121 T^{4} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 + 55 T + 5329 T^{2} )^{2} \)
$79$ \( 1 - 12434 T^{2} + 38950081 T^{4} \)
$83$ \( 1 - 12911 T^{2} + 47458321 T^{4} \)
$89$ \( ( 1 + 10 T + 7921 T^{2} )^{2} \)
$97$ \( ( 1 + 25 T + 9409 T^{2} )^{2} \)
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