Properties

Label 1728.3.o.b
Level 1728
Weight 3
Character orbit 1728.o
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.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 + 4 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -4 + 4 \zeta_{6} ) q^{5} + ( 4 - 2 \zeta_{6} ) q^{7} + ( 14 - 7 \zeta_{6} ) q^{11} + ( -22 + 22 \zeta_{6} ) q^{13} + 11 q^{17} + ( 9 - 18 \zeta_{6} ) q^{19} + ( 14 + 14 \zeta_{6} ) q^{23} + 9 \zeta_{6} q^{25} -34 \zeta_{6} q^{29} + ( -4 - 4 \zeta_{6} ) q^{31} + ( -8 + 16 \zeta_{6} ) q^{35} + 16 q^{37} + ( 13 - 13 \zeta_{6} ) q^{41} + ( -58 + 29 \zeta_{6} ) q^{43} + ( 4 - 2 \zeta_{6} ) q^{47} + ( -37 + 37 \zeta_{6} ) q^{49} + 52 q^{53} + ( -28 + 56 \zeta_{6} ) q^{55} + ( 31 + 31 \zeta_{6} ) q^{59} -16 \zeta_{6} q^{61} -88 \zeta_{6} q^{65} + ( 67 + 67 \zeta_{6} ) q^{67} -25 q^{73} + ( 42 - 42 \zeta_{6} ) q^{77} + ( -32 + 16 \zeta_{6} ) q^{79} + ( -40 + 20 \zeta_{6} ) q^{83} + ( -44 + 44 \zeta_{6} ) q^{85} + 2 q^{89} + ( -44 + 88 \zeta_{6} ) q^{91} + ( 36 + 36 \zeta_{6} ) q^{95} + 43 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 6q^{7} + O(q^{10}) \) \( 2q - 4q^{5} + 6q^{7} + 21q^{11} - 22q^{13} + 22q^{17} + 42q^{23} + 9q^{25} - 34q^{29} - 12q^{31} + 32q^{37} + 13q^{41} - 87q^{43} + 6q^{47} - 37q^{49} + 104q^{53} + 93q^{59} - 16q^{61} - 88q^{65} + 201q^{67} - 50q^{73} + 42q^{77} - 48q^{79} - 60q^{83} - 44q^{85} + 4q^{89} + 108q^{95} + 43q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −2.00000 3.46410i 0 3.00000 + 1.73205i 0 0 0
1279.1 0 0 0 −2.00000 + 3.46410i 0 3.00000 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
36.f 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.o.b 2
3.b odd 2 1 576.3.o.a 2
4.b odd 2 1 1728.3.o.a 2
8.b even 2 1 108.3.f.b 2
8.d odd 2 1 108.3.f.a 2
9.c even 3 1 1728.3.o.a 2
9.d odd 6 1 576.3.o.b 2
12.b even 2 1 576.3.o.b 2
24.f even 2 1 36.3.f.b yes 2
24.h odd 2 1 36.3.f.a 2
36.f odd 6 1 inner 1728.3.o.b 2
36.h even 6 1 576.3.o.a 2
72.j odd 6 1 36.3.f.b yes 2
72.j odd 6 1 324.3.d.b 2
72.l even 6 1 36.3.f.a 2
72.l even 6 1 324.3.d.b 2
72.n even 6 1 108.3.f.a 2
72.n even 6 1 324.3.d.c 2
72.p odd 6 1 108.3.f.b 2
72.p odd 6 1 324.3.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.a 2 24.h odd 2 1
36.3.f.a 2 72.l even 6 1
36.3.f.b yes 2 24.f even 2 1
36.3.f.b yes 2 72.j odd 6 1
108.3.f.a 2 8.d odd 2 1
108.3.f.a 2 72.n even 6 1
108.3.f.b 2 8.b even 2 1
108.3.f.b 2 72.p odd 6 1
324.3.d.b 2 72.j odd 6 1
324.3.d.b 2 72.l even 6 1
324.3.d.c 2 72.n even 6 1
324.3.d.c 2 72.p odd 6 1
576.3.o.a 2 3.b odd 2 1
576.3.o.a 2 36.h even 6 1
576.3.o.b 2 9.d odd 6 1
576.3.o.b 2 12.b even 2 1
1728.3.o.a 2 4.b odd 2 1
1728.3.o.a 2 9.c even 3 1
1728.3.o.b 2 1.a even 1 1 trivial
1728.3.o.b 2 36.f odd 6 1 inner

Hecke kernels

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

\( T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{7}^{2} - 6 T_{7} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 4 T - 9 T^{2} + 100 T^{3} + 625 T^{4} \)
$7$ \( 1 - 6 T + 61 T^{2} - 294 T^{3} + 2401 T^{4} \)
$11$ \( 1 - 21 T + 268 T^{2} - 2541 T^{3} + 14641 T^{4} \)
$13$ \( ( 1 - T + 169 T^{2} )( 1 + 23 T + 169 T^{2} ) \)
$17$ \( ( 1 - 11 T + 289 T^{2} )^{2} \)
$19$ \( 1 - 479 T^{2} + 130321 T^{4} \)
$23$ \( 1 - 42 T + 1117 T^{2} - 22218 T^{3} + 279841 T^{4} \)
$29$ \( 1 + 34 T + 315 T^{2} + 28594 T^{3} + 707281 T^{4} \)
$31$ \( 1 + 12 T + 1009 T^{2} + 11532 T^{3} + 923521 T^{4} \)
$37$ \( ( 1 - 16 T + 1369 T^{2} )^{2} \)
$41$ \( 1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4} \)
$43$ \( 1 + 87 T + 4372 T^{2} + 160863 T^{3} + 3418801 T^{4} \)
$47$ \( 1 - 6 T + 2221 T^{2} - 13254 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 - 52 T + 2809 T^{2} )^{2} \)
$59$ \( 1 - 93 T + 6364 T^{2} - 323733 T^{3} + 12117361 T^{4} \)
$61$ \( 1 + 16 T - 3465 T^{2} + 59536 T^{3} + 13845841 T^{4} \)
$67$ \( ( 1 - 67 T )^{2}( 1 - 67 T + 4489 T^{2} ) \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( ( 1 + 25 T + 5329 T^{2} )^{2} \)
$79$ \( 1 + 48 T + 7009 T^{2} + 299568 T^{3} + 38950081 T^{4} \)
$83$ \( 1 + 60 T + 8089 T^{2} + 413340 T^{3} + 47458321 T^{4} \)
$89$ \( ( 1 - 2 T + 7921 T^{2} )^{2} \)
$97$ \( 1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4} \)
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