Properties

Label 1728.3.q.c
Level $1728$
Weight $3$
Character orbit 1728.q
Analytic conductor $47.085$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{2} - 3) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{2} - 3) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 6) q^{11} + ( - 4 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 5) q^{13} + (2 \beta_{3} - 12 \beta_{2} + 6) q^{17} + ( - 2 \beta_{3} + 4 \beta_1 - 10) q^{19} + (3 \beta_{2} + \beta_1 + 3) q^{23} + 2 \beta_{2} q^{25} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 6) q^{29} + (6 \beta_{3} + 19 \beta_{2} - 3 \beta_1 - 19) q^{31} + ( - 9 \beta_{3} + 6 \beta_{2} - 3) q^{35} + ( - 2 \beta_{3} + 4 \beta_1 - 32) q^{37} + (21 \beta_{2} + 6 \beta_1 + 21) q^{41} + (3 \beta_{3} - 23 \beta_{2} + 3 \beta_1) q^{43} + (7 \beta_{3} - 9 \beta_{2} - 7 \beta_1 + 18) q^{47} + ( - 4 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 6) q^{49} + ( - 10 \beta_{3} + 60 \beta_{2} - 30) q^{53} + ( - 3 \beta_{3} + 6 \beta_1 + 27) q^{55} + ( - 21 \beta_{2} - 13 \beta_1 - 21) q^{59} + ( - 6 \beta_{3} - 31 \beta_{2} - 6 \beta_1) q^{61} + (18 \beta_{3} + 15 \beta_{2} - 18 \beta_1 - 30) q^{65} + ( - 6 \beta_{3} + 53 \beta_{2} + 3 \beta_1 - 53) q^{67} + ( - 8 \beta_{3} - 60 \beta_{2} + 30) q^{71} + (6 \beta_{3} - 12 \beta_1 - 52) q^{73} + ( - 15 \beta_{2} - 8 \beta_1 - 15) q^{77} + (5 \beta_{3} - 7 \beta_{2} + 5 \beta_1) q^{79} + (5 \beta_{3} - 63 \beta_{2} - 5 \beta_1 + 126) q^{83} + ( - 12 \beta_{3} + 54 \beta_{2} + 6 \beta_1 - 54) q^{85} + ( - 22 \beta_{3} + 60 \beta_{2} - 30) q^{89} + ( - 3 \beta_{3} + 6 \beta_1 + 103) q^{91} + (30 \beta_{2} - 18 \beta_1 + 30) q^{95} + ( - 14 \beta_{3} + 7 \beta_{2} - 14 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{5} - 2 q^{7} - 18 q^{11} + 10 q^{13} - 40 q^{19} + 18 q^{23} + 4 q^{25} + 18 q^{29} - 38 q^{31} - 128 q^{37} + 126 q^{41} - 46 q^{43} + 54 q^{47} - 12 q^{49} + 108 q^{55} - 126 q^{59} - 62 q^{61} - 90 q^{65} - 106 q^{67} - 208 q^{73} - 90 q^{77} - 14 q^{79} + 378 q^{83} - 108 q^{85} + 412 q^{91} + 180 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 −4.50000 2.59808i 0 −4.17423 7.22999i 0 0 0
449.2 0 0 0 −4.50000 2.59808i 0 3.17423 + 5.49794i 0 0 0
1601.1 0 0 0 −4.50000 + 2.59808i 0 −4.17423 + 7.22999i 0 0 0
1601.2 0 0 0 −4.50000 + 2.59808i 0 3.17423 5.49794i 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
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.q.c 4
3.b odd 2 1 576.3.q.e 4
4.b odd 2 1 1728.3.q.d 4
8.b even 2 1 432.3.q.d 4
8.d odd 2 1 54.3.d.a 4
9.c even 3 1 576.3.q.e 4
9.d odd 6 1 inner 1728.3.q.c 4
12.b even 2 1 576.3.q.f 4
24.f even 2 1 18.3.d.a 4
24.h odd 2 1 144.3.q.c 4
36.f odd 6 1 576.3.q.f 4
36.h even 6 1 1728.3.q.d 4
40.e odd 2 1 1350.3.i.b 4
40.k even 4 2 1350.3.k.a 8
72.j odd 6 1 432.3.q.d 4
72.j odd 6 1 1296.3.e.g 4
72.l even 6 1 54.3.d.a 4
72.l even 6 1 162.3.b.a 4
72.n even 6 1 144.3.q.c 4
72.n even 6 1 1296.3.e.g 4
72.p odd 6 1 18.3.d.a 4
72.p odd 6 1 162.3.b.a 4
120.m even 2 1 450.3.i.b 4
120.q odd 4 2 450.3.k.a 8
360.z odd 6 1 450.3.i.b 4
360.bd even 6 1 1350.3.i.b 4
360.bo even 12 2 450.3.k.a 8
360.bt odd 12 2 1350.3.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a 4 24.f even 2 1
18.3.d.a 4 72.p odd 6 1
54.3.d.a 4 8.d odd 2 1
54.3.d.a 4 72.l even 6 1
144.3.q.c 4 24.h odd 2 1
144.3.q.c 4 72.n even 6 1
162.3.b.a 4 72.l even 6 1
162.3.b.a 4 72.p odd 6 1
432.3.q.d 4 8.b even 2 1
432.3.q.d 4 72.j odd 6 1
450.3.i.b 4 120.m even 2 1
450.3.i.b 4 360.z odd 6 1
450.3.k.a 8 120.q odd 4 2
450.3.k.a 8 360.bo even 12 2
576.3.q.e 4 3.b odd 2 1
576.3.q.e 4 9.c even 3 1
576.3.q.f 4 12.b even 2 1
576.3.q.f 4 36.f odd 6 1
1296.3.e.g 4 72.j odd 6 1
1296.3.e.g 4 72.n even 6 1
1350.3.i.b 4 40.e odd 2 1
1350.3.i.b 4 360.bd even 6 1
1350.3.k.a 8 40.k even 4 2
1350.3.k.a 8 360.bt odd 12 2
1728.3.q.c 4 1.a even 1 1 trivial
1728.3.q.c 4 9.d odd 6 1 inner
1728.3.q.d 4 4.b odd 2 1
1728.3.q.d 4 36.h even 6 1

Hecke kernels

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

\( T_{5}^{2} + 9T_{5} + 27 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 57T_{7}^{2} - 106T_{7} + 2809 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 57 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + 117 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} - 18 T^{3} + 63 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} + 38 T^{3} + 1569 T^{2} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( (T^{2} + 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 126 T^{3} + 5967 T^{2} + \cdots + 455625 \) Copy content Toggle raw display
$43$ \( T^{4} + 46 T^{3} + 2073 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( T^{4} - 54 T^{3} + 333 T^{2} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 126 T^{3} + 3573 T^{2} + \cdots + 2954961 \) Copy content Toggle raw display
$61$ \( T^{4} + 62 T^{3} + 4827 T^{2} + \cdots + 966289 \) Copy content Toggle raw display
$67$ \( T^{4} + 106 T^{3} + 8913 T^{2} + \cdots + 5396329 \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + \cdots + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 14 T^{3} + 1497 T^{2} + \cdots + 1692601 \) Copy content Toggle raw display
$83$ \( T^{4} - 378 T^{3} + \cdots + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + \cdots + 36144144 \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 110986225 \) Copy content Toggle raw display
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