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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5} - 7 \) Copy content Toggle raw display
\( T_{7}^{2} + 75 \) Copy content Toggle raw display

Hecke characteristic polynomials

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