Properties

Label 1728.3.e.b
Level 1728
Weight 3
Character orbit 1728.e
Self dual yes
Analytic conductor 47.085
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 11q^{7} + O(q^{10}) \) \( q - 11q^{7} - 23q^{13} - 37q^{19} + 25q^{25} + 46q^{31} + 73q^{37} - 22q^{43} + 72q^{49} - 47q^{61} - 13q^{67} + 143q^{73} - 11q^{79} + 253q^{91} - 169q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
0
0 0 0 0 0 −11.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.e.b 1
3.b odd 2 1 CM 1728.3.e.b 1
4.b odd 2 1 1728.3.e.c 1
8.b even 2 1 432.3.e.a 1
8.d odd 2 1 108.3.c.a 1
12.b even 2 1 1728.3.e.c 1
24.f even 2 1 108.3.c.a 1
24.h odd 2 1 432.3.e.a 1
40.e odd 2 1 2700.3.g.b 1
40.k even 4 2 2700.3.b.d 2
72.j odd 6 2 1296.3.q.c 2
72.l even 6 2 324.3.g.a 2
72.n even 6 2 1296.3.q.c 2
72.p odd 6 2 324.3.g.a 2
120.m even 2 1 2700.3.g.b 1
120.q odd 4 2 2700.3.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.c.a 1 8.d odd 2 1
108.3.c.a 1 24.f even 2 1
324.3.g.a 2 72.l even 6 2
324.3.g.a 2 72.p odd 6 2
432.3.e.a 1 8.b even 2 1
432.3.e.a 1 24.h odd 2 1
1296.3.q.c 2 72.j odd 6 2
1296.3.q.c 2 72.n even 6 2
1728.3.e.b 1 1.a even 1 1 trivial
1728.3.e.b 1 3.b odd 2 1 CM
1728.3.e.c 1 4.b odd 2 1
1728.3.e.c 1 12.b even 2 1
2700.3.b.d 2 40.k even 4 2
2700.3.b.d 2 120.q odd 4 2
2700.3.g.b 1 40.e odd 2 1
2700.3.g.b 1 120.m even 2 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} + 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( 1 + 11 T + 49 T^{2} \)
$11$ \( ( 1 - 11 T )( 1 + 11 T ) \)
$13$ \( 1 + 23 T + 169 T^{2} \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( 1 + 37 T + 361 T^{2} \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( 1 - 46 T + 961 T^{2} \)
$37$ \( 1 - 73 T + 1369 T^{2} \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( 1 + 22 T + 1849 T^{2} \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 + 47 T + 3721 T^{2} \)
$67$ \( 1 + 13 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 - 143 T + 5329 T^{2} \)
$79$ \( 1 + 11 T + 6241 T^{2} \)
$83$ \( ( 1 - 83 T )( 1 + 83 T ) \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 + 169 T + 9409 T^{2} \)
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