Properties

Label 1728.4.a.bh
Level $1728$
Weight $4$
Character orbit 1728.a
Self dual yes
Analytic conductor $101.955$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 - \beta ) q^{5} + ( 12 + \beta ) q^{7} +O(q^{10})\) \( q + ( -4 - \beta ) q^{5} + ( 12 + \beta ) q^{7} - q^{11} + ( -16 + 4 \beta ) q^{13} + ( -28 - 4 \beta ) q^{17} + ( -92 - 2 \beta ) q^{19} + ( 46 - 4 \beta ) q^{23} + ( 188 + 8 \beta ) q^{25} + ( 168 - 2 \beta ) q^{29} + ( 188 - 5 \beta ) q^{31} + ( -345 - 16 \beta ) q^{35} + ( -174 + 4 \beta ) q^{37} + ( -156 + 10 \beta ) q^{41} + ( -40 + 14 \beta ) q^{43} + ( -114 - 8 \beta ) q^{47} + ( 98 + 24 \beta ) q^{49} + ( -76 + 13 \beta ) q^{53} + ( 4 + \beta ) q^{55} + ( -340 + 24 \beta ) q^{59} + ( 56 + 32 \beta ) q^{61} -1124 q^{65} + ( -176 - 34 \beta ) q^{67} + ( 908 - 12 \beta ) q^{71} -287 q^{73} + ( -12 - \beta ) q^{77} + ( -680 + 32 \beta ) q^{79} + ( 391 - 32 \beta ) q^{83} + ( 1300 + 44 \beta ) q^{85} + ( 120 + 18 \beta ) q^{89} + ( 996 + 32 \beta ) q^{91} + ( 962 + 100 \beta ) q^{95} + ( -169 + 8 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} + 24q^{7} + O(q^{10}) \) \( 2q - 8q^{5} + 24q^{7} - 2q^{11} - 32q^{13} - 56q^{17} - 184q^{19} + 92q^{23} + 376q^{25} + 336q^{29} + 376q^{31} - 690q^{35} - 348q^{37} - 312q^{41} - 80q^{43} - 228q^{47} + 196q^{49} - 152q^{53} + 8q^{55} - 680q^{59} + 112q^{61} - 2248q^{65} - 352q^{67} + 1816q^{71} - 574q^{73} - 24q^{77} - 1360q^{79} + 782q^{83} + 2600q^{85} + 240q^{89} + 1992q^{91} + 1924q^{95} - 338q^{97} + O(q^{100}) \)

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
3.37228
−2.37228
0 0 0 −21.2337 0 29.2337 0 0 0
1.2 0 0 0 13.2337 0 −5.23369 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.bh 2
3.b odd 2 1 1728.4.a.bt 2
4.b odd 2 1 1728.4.a.bg 2
8.b even 2 1 216.4.a.h yes 2
8.d odd 2 1 432.4.a.s 2
12.b even 2 1 1728.4.a.bs 2
24.f even 2 1 432.4.a.o 2
24.h odd 2 1 216.4.a.e 2
72.j odd 6 2 648.4.i.s 4
72.n even 6 2 648.4.i.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.e 2 24.h odd 2 1
216.4.a.h yes 2 8.b even 2 1
432.4.a.o 2 24.f even 2 1
432.4.a.s 2 8.d odd 2 1
648.4.i.m 4 72.n even 6 2
648.4.i.s 4 72.j odd 6 2
1728.4.a.bg 2 4.b odd 2 1
1728.4.a.bh 2 1.a even 1 1 trivial
1728.4.a.bs 2 12.b even 2 1
1728.4.a.bt 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 8 T_{5} - 281 \)
\( T_{7}^{2} - 24 T_{7} - 153 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -281 + 8 T + T^{2} \)
$7$ \( -153 - 24 T + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -4496 + 32 T + T^{2} \)
$17$ \( -3968 + 56 T + T^{2} \)
$19$ \( 7276 + 184 T + T^{2} \)
$23$ \( -2636 - 92 T + T^{2} \)
$29$ \( 27036 - 336 T + T^{2} \)
$31$ \( 27919 - 376 T + T^{2} \)
$37$ \( 25524 + 348 T + T^{2} \)
$41$ \( -5364 + 312 T + T^{2} \)
$43$ \( -56612 + 80 T + T^{2} \)
$47$ \( -6012 + 228 T + T^{2} \)
$53$ \( -44417 + 152 T + T^{2} \)
$59$ \( -55472 + 680 T + T^{2} \)
$61$ \( -300992 - 112 T + T^{2} \)
$67$ \( -312356 + 352 T + T^{2} \)
$71$ \( 781696 - 1816 T + T^{2} \)
$73$ \( ( 287 + T )^{2} \)
$79$ \( 158272 + 1360 T + T^{2} \)
$83$ \( -151247 - 782 T + T^{2} \)
$89$ \( -81828 - 240 T + T^{2} \)
$97$ \( 9553 + 338 T + T^{2} \)
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