Properties

Label 1728.4.a.g
Level 1728
Weight 4
Character orbit 1728.a
Self dual yes
Analytic conductor 101.955
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(101.955300490\)
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)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 9q^{5} - q^{7} + O(q^{10}) \) \( q - 9q^{5} - q^{7} - 63q^{11} + 28q^{13} + 72q^{17} - 98q^{19} + 126q^{23} - 44q^{25} + 126q^{29} - 259q^{31} + 9q^{35} - 386q^{37} - 450q^{41} + 34q^{43} - 54q^{47} - 342q^{49} + 693q^{53} + 567q^{55} - 180q^{59} + 280q^{61} - 252q^{65} + 586q^{67} + 504q^{71} + 161q^{73} + 63q^{77} + 440q^{79} - 999q^{83} - 648q^{85} + 882q^{89} - 28q^{91} + 882q^{95} - 721q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −9.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.a.g 1
3.b odd 2 1 1728.4.a.y 1
4.b odd 2 1 1728.4.a.h 1
8.b even 2 1 108.4.a.d yes 1
8.d odd 2 1 432.4.a.l 1
12.b even 2 1 1728.4.a.z 1
24.f even 2 1 432.4.a.c 1
24.h odd 2 1 108.4.a.a 1
72.j odd 6 2 324.4.e.g 2
72.n even 6 2 324.4.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 24.h odd 2 1
108.4.a.d yes 1 8.b even 2 1
324.4.e.b 2 72.n even 6 2
324.4.e.g 2 72.j odd 6 2
432.4.a.c 1 24.f even 2 1
432.4.a.l 1 8.d odd 2 1
1728.4.a.g 1 1.a even 1 1 trivial
1728.4.a.h 1 4.b odd 2 1
1728.4.a.y 1 3.b odd 2 1
1728.4.a.z 1 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} + 9 \)
\( T_{7} + 1 \)
\( T_{11} + 63 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 9 T + 125 T^{2} \)
$7$ \( 1 + T + 343 T^{2} \)
$11$ \( 1 + 63 T + 1331 T^{2} \)
$13$ \( 1 - 28 T + 2197 T^{2} \)
$17$ \( 1 - 72 T + 4913 T^{2} \)
$19$ \( 1 + 98 T + 6859 T^{2} \)
$23$ \( 1 - 126 T + 12167 T^{2} \)
$29$ \( 1 - 126 T + 24389 T^{2} \)
$31$ \( 1 + 259 T + 29791 T^{2} \)
$37$ \( 1 + 386 T + 50653 T^{2} \)
$41$ \( 1 + 450 T + 68921 T^{2} \)
$43$ \( 1 - 34 T + 79507 T^{2} \)
$47$ \( 1 + 54 T + 103823 T^{2} \)
$53$ \( 1 - 693 T + 148877 T^{2} \)
$59$ \( 1 + 180 T + 205379 T^{2} \)
$61$ \( 1 - 280 T + 226981 T^{2} \)
$67$ \( 1 - 586 T + 300763 T^{2} \)
$71$ \( 1 - 504 T + 357911 T^{2} \)
$73$ \( 1 - 161 T + 389017 T^{2} \)
$79$ \( 1 - 440 T + 493039 T^{2} \)
$83$ \( 1 + 999 T + 571787 T^{2} \)
$89$ \( 1 - 882 T + 704969 T^{2} \)
$97$ \( 1 + 721 T + 912673 T^{2} \)
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