Properties

Label 1728.2.i.k
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{13} + 2 \beta_{3} q^{17} -4 q^{19} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + 4 \beta_{1} q^{25} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -5 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -1 + \beta_{3} ) q^{35} + ( -4 - 2 \beta_{3} ) q^{37} + ( -7 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 5 \beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + 3 \beta_{2} ) q^{47} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( 4 - 2 \beta_{3} ) q^{53} + ( 1 + \beta_{3} ) q^{55} + ( -7 + 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} ) q^{65} + ( 5 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 2 + 4 \beta_{3} ) q^{71} + 2 \beta_{3} q^{73} + ( -5 + 5 \beta_{1} ) q^{77} + ( -11 \beta_{1} + \beta_{2} ) q^{79} + ( 3 \beta_{1} + \beta_{2} ) q^{83} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 8 - 2 \beta_{3} ) q^{89} + ( -13 + 3 \beta_{3} ) q^{91} + ( -4 + 4 \beta_{1} ) q^{95} + ( -\beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 2q^{7} + 2q^{11} + 2q^{13} - 16q^{19} - 6q^{23} + 8q^{25} - 10q^{29} - 10q^{31} - 4q^{35} - 16q^{37} - 14q^{41} + 10q^{43} + 2q^{47} + 16q^{53} + 4q^{55} - 14q^{59} - 6q^{61} - 2q^{65} + 10q^{67} + 8q^{71} - 10q^{77} - 22q^{79} + 6q^{83} + 32q^{89} - 52q^{91} - 8q^{95} - 2q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 0.500000 0.866025i 0 −1.72474 2.98735i 0 0 0
577.2 0 0 0 0.500000 0.866025i 0 0.724745 + 1.25529i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −1.72474 + 2.98735i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 0.724745 1.25529i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.k 4
3.b odd 2 1 576.2.i.i 4
4.b odd 2 1 1728.2.i.m 4
8.b even 2 1 864.2.i.c 4
8.d odd 2 1 864.2.i.e 4
9.c even 3 1 inner 1728.2.i.k 4
9.c even 3 1 5184.2.a.bn 2
9.d odd 6 1 576.2.i.i 4
9.d odd 6 1 5184.2.a.by 2
12.b even 2 1 576.2.i.m 4
24.f even 2 1 288.2.i.c 4
24.h odd 2 1 288.2.i.e yes 4
36.f odd 6 1 1728.2.i.m 4
36.f odd 6 1 5184.2.a.bj 2
36.h even 6 1 576.2.i.m 4
36.h even 6 1 5184.2.a.bu 2
72.j odd 6 1 288.2.i.e yes 4
72.j odd 6 1 2592.2.a.n 2
72.l even 6 1 288.2.i.c 4
72.l even 6 1 2592.2.a.j 2
72.n even 6 1 864.2.i.c 4
72.n even 6 1 2592.2.a.s 2
72.p odd 6 1 864.2.i.e 4
72.p odd 6 1 2592.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 24.f even 2 1
288.2.i.c 4 72.l even 6 1
288.2.i.e yes 4 24.h odd 2 1
288.2.i.e yes 4 72.j odd 6 1
576.2.i.i 4 3.b odd 2 1
576.2.i.i 4 9.d odd 6 1
576.2.i.m 4 12.b even 2 1
576.2.i.m 4 36.h even 6 1
864.2.i.c 4 8.b even 2 1
864.2.i.c 4 72.n even 6 1
864.2.i.e 4 8.d odd 2 1
864.2.i.e 4 72.p odd 6 1
1728.2.i.k 4 1.a even 1 1 trivial
1728.2.i.k 4 9.c even 3 1 inner
1728.2.i.m 4 4.b odd 2 1
1728.2.i.m 4 36.f odd 6 1
2592.2.a.j 2 72.l even 6 1
2592.2.a.n 2 72.j odd 6 1
2592.2.a.o 2 72.p odd 6 1
2592.2.a.s 2 72.n even 6 1
5184.2.a.bj 2 36.f odd 6 1
5184.2.a.bn 2 9.c even 3 1
5184.2.a.bu 2 36.h even 6 1
5184.2.a.by 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{7}^{4} + 2 T_{7}^{3} + 9 T_{7}^{2} - 10 T_{7} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 25 - 10 T + 9 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 529 + 46 T + 27 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( ( -24 + T^{2} )^{2} \)
$19$ \( ( 4 + T )^{4} \)
$23$ \( 9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 1 + 10 T + 99 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( 361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4} \)
$37$ \( ( -8 + 8 T + T^{2} )^{2} \)
$41$ \( 625 + 350 T + 171 T^{2} + 14 T^{3} + T^{4} \)
$43$ \( 841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4} \)
$47$ \( 2809 + 106 T + 57 T^{2} - 2 T^{3} + T^{4} \)
$53$ \( ( -8 - 8 T + T^{2} )^{2} \)
$59$ \( 25 - 70 T + 201 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( 225 - 90 T + 51 T^{2} + 6 T^{3} + T^{4} \)
$67$ \( 841 + 290 T + 129 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( ( -92 - 4 T + T^{2} )^{2} \)
$73$ \( ( -24 + T^{2} )^{2} \)
$79$ \( 13225 + 2530 T + 369 T^{2} + 22 T^{3} + T^{4} \)
$83$ \( 9 - 18 T + 33 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( ( 40 - 16 T + T^{2} )^{2} \)
$97$ \( 529 - 46 T + 27 T^{2} + 2 T^{3} + T^{4} \)
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