Properties

Label 216.4.a.f
Level $216$
Weight $4$
Character orbit 216.a
Self dual yes
Analytic conductor $12.744$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7444125612\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta ) q^{5} + ( -3 + 2 \beta ) q^{7} +O(q^{10})\) \( q + ( -2 - \beta ) q^{5} + ( -3 + 2 \beta ) q^{7} + ( -26 + 3 \beta ) q^{11} + ( 13 - 2 \beta ) q^{13} + ( -94 + \beta ) q^{17} + ( -37 - 8 \beta ) q^{19} + ( -74 - 5 \beta ) q^{23} + ( 59 + 4 \beta ) q^{25} + ( -144 - 8 \beta ) q^{29} + ( -124 + 8 \beta ) q^{31} + ( -354 - \beta ) q^{35} + ( 171 + 10 \beta ) q^{37} + 32 \beta q^{41} + ( -128 - 4 \beta ) q^{43} + ( -66 - \beta ) q^{47} + ( 386 - 12 \beta ) q^{49} + ( -476 - 14 \beta ) q^{53} + ( -488 + 20 \beta ) q^{55} + ( 502 - 21 \beta ) q^{59} + ( -17 - 34 \beta ) q^{61} + ( 334 - 9 \beta ) q^{65} + ( -433 - 16 \beta ) q^{67} + ( -388 + 30 \beta ) q^{71} + ( 937 - 12 \beta ) q^{73} + ( 1158 - 61 \beta ) q^{77} + ( 91 - 26 \beta ) q^{79} + ( 668 + 46 \beta ) q^{83} + ( 8 + 92 \beta ) q^{85} + ( -438 + 21 \beta ) q^{89} + ( -759 + 32 \beta ) q^{91} + ( 1514 + 53 \beta ) q^{95} + ( -19 + 88 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} - 6q^{7} + O(q^{10}) \) \( 2q - 4q^{5} - 6q^{7} - 52q^{11} + 26q^{13} - 188q^{17} - 74q^{19} - 148q^{23} + 118q^{25} - 288q^{29} - 248q^{31} - 708q^{35} + 342q^{37} - 256q^{43} - 132q^{47} + 772q^{49} - 952q^{53} - 976q^{55} + 1004q^{59} - 34q^{61} + 668q^{65} - 866q^{67} - 776q^{71} + 1874q^{73} + 2316q^{77} + 182q^{79} + 1336q^{83} + 16q^{85} - 876q^{89} - 1518q^{91} + 3028q^{95} - 38q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
0 0 0 −15.4164 0 23.8328 0 0 0
1.2 0 0 0 11.4164 0 −29.8328 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 216.4.a.f 2
3.b odd 2 1 216.4.a.g yes 2
4.b odd 2 1 432.4.a.p 2
8.b even 2 1 1728.4.a.bq 2
8.d odd 2 1 1728.4.a.br 2
9.c even 3 2 648.4.i.r 4
9.d odd 6 2 648.4.i.o 4
12.b even 2 1 432.4.a.r 2
24.f even 2 1 1728.4.a.bj 2
24.h odd 2 1 1728.4.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.a.f 2 1.a even 1 1 trivial
216.4.a.g yes 2 3.b odd 2 1
432.4.a.p 2 4.b odd 2 1
432.4.a.r 2 12.b even 2 1
648.4.i.o 4 9.d odd 6 2
648.4.i.r 4 9.c even 3 2
1728.4.a.bi 2 24.h odd 2 1
1728.4.a.bj 2 24.f even 2 1
1728.4.a.bq 2 8.b even 2 1
1728.4.a.br 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 4 T_{5} - 176 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(216))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -176 + 4 T + T^{2} \)
$7$ \( -711 + 6 T + T^{2} \)
$11$ \( -944 + 52 T + T^{2} \)
$13$ \( -551 - 26 T + T^{2} \)
$17$ \( 8656 + 188 T + T^{2} \)
$19$ \( -10151 + 74 T + T^{2} \)
$23$ \( 976 + 148 T + T^{2} \)
$29$ \( 9216 + 288 T + T^{2} \)
$31$ \( 3856 + 248 T + T^{2} \)
$37$ \( 11241 - 342 T + T^{2} \)
$41$ \( -184320 + T^{2} \)
$43$ \( 13504 + 256 T + T^{2} \)
$47$ \( 4176 + 132 T + T^{2} \)
$53$ \( 191296 + 952 T + T^{2} \)
$59$ \( 172624 - 1004 T + T^{2} \)
$61$ \( -207791 + 34 T + T^{2} \)
$67$ \( 141409 + 866 T + T^{2} \)
$71$ \( -11456 + 776 T + T^{2} \)
$73$ \( 852049 - 1874 T + T^{2} \)
$79$ \( -113399 - 182 T + T^{2} \)
$83$ \( 65344 - 1336 T + T^{2} \)
$89$ \( 112464 + 876 T + T^{2} \)
$97$ \( -1393559 + 38 T + T^{2} \)
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