Properties

Label 1728.3.q.g
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 3 - \beta_{1} + \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} + ( -7 + 5 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{13} + ( 7 + \beta_{1} + 15 \beta_{2} - \beta_{3} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( -33 - \beta_{1} - 17 \beta_{2} ) q^{23} + ( 8 - 6 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -7 + 14 \beta_{2} + 7 \beta_{3} ) q^{29} + ( 3 - 3 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{31} + ( -28 + 5 \beta_{1} - 51 \beta_{2} - 5 \beta_{3} ) q^{35} + ( 18 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{37} + ( 6 - 8 \beta_{1} - \beta_{2} ) q^{41} + ( 23 + 23 \beta_{2} ) q^{43} + ( 15 - 12 \beta_{2} + 3 \beta_{3} ) q^{47} + ( 1 - \beta_{1} + 26 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 16 - 8 \beta_{1} + 24 \beta_{2} + 8 \beta_{3} ) q^{53} + ( 24 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{55} + ( 42 - 8 \beta_{1} + 17 \beta_{2} ) q^{59} + ( -19 - 6 \beta_{1} - 22 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -25 + 32 \beta_{2} + 7 \beta_{3} ) q^{65} + ( 6 - 6 \beta_{1} + 61 \beta_{2} + 12 \beta_{3} ) q^{67} + ( -14 - 2 \beta_{1} - 30 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 23 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -93 - 17 \beta_{1} - 55 \beta_{2} ) q^{77} + ( 39 + 10 \beta_{1} + 44 \beta_{2} - 5 \beta_{3} ) q^{79} + ( -15 + 12 \beta_{2} - 3 \beta_{3} ) q^{83} + ( 6 - 6 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} ) q^{85} + ( -56 - 8 \beta_{1} - 120 \beta_{2} + 8 \beta_{3} ) q^{89} + ( 74 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 48 - 4 \beta_{1} + 22 \beta_{2} ) q^{95} + ( -99 + 4 \beta_{1} - 97 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{5} - q^{7} + O(q^{10}) \) \( 4 q + 9 q^{5} - q^{7} - 36 q^{11} - 5 q^{13} - 2 q^{19} - 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} + 126 q^{59} - 41 q^{61} - 171 q^{65} - 116 q^{67} + 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 18 q^{85} + 302 q^{91} + 144 q^{95} - 196 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} + 16 \nu - 9 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 9 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{3} + \nu^{2} + 8 \nu + 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 8 \beta_{2} + 8\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 2 \beta_{1} + 11\)\()/3\)

Character values

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

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

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
1.68614 + 0.396143i
−1.18614 1.26217i
1.68614 0.396143i
−1.18614 + 1.26217i
0 0 0 −2.05842 1.18843i 0 4.05842 + 7.02939i 0 0 0
449.2 0 0 0 6.55842 + 3.78651i 0 −4.55842 7.89542i 0 0 0
1601.1 0 0 0 −2.05842 + 1.18843i 0 4.05842 7.02939i 0 0 0
1601.2 0 0 0 6.55842 3.78651i 0 −4.55842 + 7.89542i 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.g 4
3.b odd 2 1 576.3.q.d 4
4.b odd 2 1 1728.3.q.h 4
8.b even 2 1 108.3.g.a 4
8.d odd 2 1 432.3.q.b 4
9.c even 3 1 576.3.q.d 4
9.d odd 6 1 inner 1728.3.q.g 4
12.b even 2 1 576.3.q.g 4
24.f even 2 1 144.3.q.b 4
24.h odd 2 1 36.3.g.a 4
36.f odd 6 1 576.3.q.g 4
36.h even 6 1 1728.3.q.h 4
40.f even 2 1 2700.3.p.b 4
40.i odd 4 2 2700.3.u.b 8
72.j odd 6 1 108.3.g.a 4
72.j odd 6 1 324.3.c.b 4
72.l even 6 1 432.3.q.b 4
72.l even 6 1 1296.3.e.e 4
72.n even 6 1 36.3.g.a 4
72.n even 6 1 324.3.c.b 4
72.p odd 6 1 144.3.q.b 4
72.p odd 6 1 1296.3.e.e 4
120.i odd 2 1 900.3.p.a 4
120.w even 4 2 900.3.u.a 8
360.bh odd 6 1 2700.3.p.b 4
360.bk even 6 1 900.3.p.a 4
360.br even 12 2 2700.3.u.b 8
360.bu odd 12 2 900.3.u.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 24.h odd 2 1
36.3.g.a 4 72.n even 6 1
108.3.g.a 4 8.b even 2 1
108.3.g.a 4 72.j odd 6 1
144.3.q.b 4 24.f even 2 1
144.3.q.b 4 72.p odd 6 1
324.3.c.b 4 72.j odd 6 1
324.3.c.b 4 72.n even 6 1
432.3.q.b 4 8.d odd 2 1
432.3.q.b 4 72.l even 6 1
576.3.q.d 4 3.b odd 2 1
576.3.q.d 4 9.c even 3 1
576.3.q.g 4 12.b even 2 1
576.3.q.g 4 36.f odd 6 1
900.3.p.a 4 120.i odd 2 1
900.3.p.a 4 360.bk even 6 1
900.3.u.a 8 120.w even 4 2
900.3.u.a 8 360.bu odd 12 2
1296.3.e.e 4 72.l even 6 1
1296.3.e.e 4 72.p odd 6 1
1728.3.q.g 4 1.a even 1 1 trivial
1728.3.q.g 4 9.d odd 6 1 inner
1728.3.q.h 4 4.b odd 2 1
1728.3.q.h 4 36.h even 6 1
2700.3.p.b 4 40.f even 2 1
2700.3.p.b 4 360.bh odd 6 1
2700.3.u.b 8 40.i odd 4 2
2700.3.u.b 8 360.br even 12 2

Hecke kernels

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

\( T_{5}^{4} - 9 T_{5}^{3} + 9 T_{5}^{2} + 162 T_{5} + 324 \)
\( T_{7}^{4} + T_{7}^{3} + 75 T_{7}^{2} - 74 T_{7} + 5476 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 + 162 T + 9 T^{2} - 9 T^{3} + T^{4} \)
$7$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
$11$ \( 81 + 324 T + 441 T^{2} + 36 T^{3} + T^{4} \)
$13$ \( 4624 - 340 T + 93 T^{2} + 5 T^{3} + T^{4} \)
$17$ \( 20736 + 387 T^{2} + T^{4} \)
$19$ \( ( -74 + T + T^{2} )^{2} \)
$23$ \( 627264 + 78408 T + 4059 T^{2} + 99 T^{3} + T^{4} \)
$29$ \( 777924 - 55566 T + 441 T^{2} + 63 T^{3} + T^{4} \)
$31$ \( 430336 - 4592 T + 705 T^{2} + 7 T^{3} + T^{4} \)
$37$ \( ( -932 - 32 T + T^{2} )^{2} \)
$41$ \( 2424249 + 28026 T - 1449 T^{2} - 18 T^{3} + T^{4} \)
$43$ \( ( 529 - 23 T + T^{2} )^{2} \)
$47$ \( 104976 - 26244 T + 2511 T^{2} - 81 T^{3} + T^{4} \)
$53$ \( 1327104 + 4032 T^{2} + T^{4} \)
$59$ \( 68121 + 32886 T + 5031 T^{2} - 126 T^{3} + T^{4} \)
$61$ \( 61504 - 10168 T + 1929 T^{2} + 41 T^{3} + T^{4} \)
$67$ \( 477481 + 80156 T + 12765 T^{2} + 116 T^{3} + T^{4} \)
$71$ \( 331776 + 1548 T^{2} + T^{4} \)
$73$ \( ( -206 - 43 T + T^{2} )^{2} \)
$79$ \( 17956 + 11122 T + 7023 T^{2} - 83 T^{3} + T^{4} \)
$83$ \( 104976 + 26244 T + 2511 T^{2} + 81 T^{3} + T^{4} \)
$89$ \( 84934656 + 24768 T^{2} + T^{4} \)
$97$ \( 86620249 + 1824172 T + 29109 T^{2} + 196 T^{3} + T^{4} \)
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