Properties

Label 1728.2.i.e
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
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: \( k \) \(=\) \( 2 \)
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + 3 q^{17} + q^{19} -6 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} + 4 q^{37} + 9 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 12 q^{53} -3 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} + 5 \zeta_{6} q^{67} + 12 q^{71} + 11 q^{73} + 6 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -12 + 12 \zeta_{6} ) q^{83} -6 q^{89} -4 q^{91} + ( -5 + 5 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 3q^{11} + 2q^{13} + 6q^{17} + 2q^{19} - 6q^{23} + 5q^{25} - 6q^{29} + 4q^{31} + 8q^{37} + 9q^{41} - q^{43} - 6q^{47} + 3q^{49} + 24q^{53} - 3q^{59} + 8q^{61} + 5q^{67} + 24q^{71} + 22q^{73} + 6q^{77} + 4q^{79} - 12q^{83} - 12q^{89} - 8q^{91} - 5q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −1.00000 1.73205i 0 0 0
1153.1 0 0 0 0 0 −1.00000 + 1.73205i 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.e 2
3.b odd 2 1 576.2.i.g 2
4.b odd 2 1 1728.2.i.f 2
8.b even 2 1 54.2.c.a 2
8.d odd 2 1 432.2.i.b 2
9.c even 3 1 inner 1728.2.i.e 2
9.c even 3 1 5184.2.a.q 1
9.d odd 6 1 576.2.i.g 2
9.d odd 6 1 5184.2.a.r 1
12.b even 2 1 576.2.i.a 2
24.f even 2 1 144.2.i.c 2
24.h odd 2 1 18.2.c.a 2
36.f odd 6 1 1728.2.i.f 2
36.f odd 6 1 5184.2.a.p 1
36.h even 6 1 576.2.i.a 2
36.h even 6 1 5184.2.a.o 1
40.f even 2 1 1350.2.e.c 2
40.i odd 4 2 1350.2.j.a 4
56.h odd 2 1 2646.2.f.g 2
56.j odd 6 1 2646.2.e.c 2
56.j odd 6 1 2646.2.h.i 2
56.p even 6 1 2646.2.e.b 2
56.p even 6 1 2646.2.h.h 2
72.j odd 6 1 18.2.c.a 2
72.j odd 6 1 162.2.a.c 1
72.l even 6 1 144.2.i.c 2
72.l even 6 1 1296.2.a.g 1
72.n even 6 1 54.2.c.a 2
72.n even 6 1 162.2.a.b 1
72.p odd 6 1 432.2.i.b 2
72.p odd 6 1 1296.2.a.f 1
120.i odd 2 1 450.2.e.i 2
120.w even 4 2 450.2.j.e 4
168.i even 2 1 882.2.f.d 2
168.s odd 6 1 882.2.e.i 2
168.s odd 6 1 882.2.h.c 2
168.ba even 6 1 882.2.e.g 2
168.ba even 6 1 882.2.h.b 2
360.bh odd 6 1 450.2.e.i 2
360.bh odd 6 1 4050.2.a.c 1
360.bk even 6 1 1350.2.e.c 2
360.bk even 6 1 4050.2.a.v 1
360.br even 12 2 450.2.j.e 4
360.br even 12 2 4050.2.c.c 2
360.bu odd 12 2 1350.2.j.a 4
360.bu odd 12 2 4050.2.c.r 2
504.w even 6 1 2646.2.e.b 2
504.y even 6 1 882.2.e.g 2
504.bi odd 6 1 882.2.h.c 2
504.bn odd 6 1 2646.2.f.g 2
504.bn odd 6 1 7938.2.a.i 1
504.bp odd 6 1 2646.2.h.i 2
504.ca even 6 1 882.2.h.b 2
504.cc even 6 1 882.2.f.d 2
504.cc even 6 1 7938.2.a.x 1
504.cq even 6 1 2646.2.h.h 2
504.cw odd 6 1 2646.2.e.c 2
504.db odd 6 1 882.2.e.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 24.h odd 2 1
18.2.c.a 2 72.j odd 6 1
54.2.c.a 2 8.b even 2 1
54.2.c.a 2 72.n even 6 1
144.2.i.c 2 24.f even 2 1
144.2.i.c 2 72.l even 6 1
162.2.a.b 1 72.n even 6 1
162.2.a.c 1 72.j odd 6 1
432.2.i.b 2 8.d odd 2 1
432.2.i.b 2 72.p odd 6 1
450.2.e.i 2 120.i odd 2 1
450.2.e.i 2 360.bh odd 6 1
450.2.j.e 4 120.w even 4 2
450.2.j.e 4 360.br even 12 2
576.2.i.a 2 12.b even 2 1
576.2.i.a 2 36.h even 6 1
576.2.i.g 2 3.b odd 2 1
576.2.i.g 2 9.d odd 6 1
882.2.e.g 2 168.ba even 6 1
882.2.e.g 2 504.y even 6 1
882.2.e.i 2 168.s odd 6 1
882.2.e.i 2 504.db odd 6 1
882.2.f.d 2 168.i even 2 1
882.2.f.d 2 504.cc even 6 1
882.2.h.b 2 168.ba even 6 1
882.2.h.b 2 504.ca even 6 1
882.2.h.c 2 168.s odd 6 1
882.2.h.c 2 504.bi odd 6 1
1296.2.a.f 1 72.p odd 6 1
1296.2.a.g 1 72.l even 6 1
1350.2.e.c 2 40.f even 2 1
1350.2.e.c 2 360.bk even 6 1
1350.2.j.a 4 40.i odd 4 2
1350.2.j.a 4 360.bu odd 12 2
1728.2.i.e 2 1.a even 1 1 trivial
1728.2.i.e 2 9.c even 3 1 inner
1728.2.i.f 2 4.b odd 2 1
1728.2.i.f 2 36.f odd 6 1
2646.2.e.b 2 56.p even 6 1
2646.2.e.b 2 504.w even 6 1
2646.2.e.c 2 56.j odd 6 1
2646.2.e.c 2 504.cw odd 6 1
2646.2.f.g 2 56.h odd 2 1
2646.2.f.g 2 504.bn odd 6 1
2646.2.h.h 2 56.p even 6 1
2646.2.h.h 2 504.cq even 6 1
2646.2.h.i 2 56.j odd 6 1
2646.2.h.i 2 504.bp odd 6 1
4050.2.a.c 1 360.bh odd 6 1
4050.2.a.v 1 360.bk even 6 1
4050.2.c.c 2 360.br even 12 2
4050.2.c.r 2 360.bu odd 12 2
5184.2.a.o 1 36.h even 6 1
5184.2.a.p 1 36.f odd 6 1
5184.2.a.q 1 9.c even 3 1
5184.2.a.r 1 9.d odd 6 1
7938.2.a.i 1 504.bn odd 6 1
7938.2.a.x 1 504.cc even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{7}^{2} + 2 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( ( -4 + T )^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( 1 + T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 25 - 5 T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( -11 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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