Properties

Label 1656.2.a.n
Level $1656$
Weight $2$
Character orbit 1656.a
Self dual yes
Analytic conductor $13.223$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1656.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2232265747\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( 1 + \beta_{1} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} ) q^{11} + 2 q^{13} + ( -1 + \beta_{1} ) q^{17} + ( 2 - \beta_{2} ) q^{19} + q^{23} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{25} -2 \beta_{1} q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( 2 - 2 \beta_{1} ) q^{35} + ( 5 + \beta_{1} - \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} + ( 6 + 2 \beta_{1} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -8 + 4 \beta_{2} ) q^{55} + ( 4 - 2 \beta_{2} ) q^{59} + ( 7 - \beta_{1} + \beta_{2} ) q^{61} -2 \beta_{2} q^{65} + ( -2 \beta_{1} - \beta_{2} ) q^{67} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + 2 q^{73} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -9 - \beta_{1} ) q^{79} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{89} + ( 2 + 2 \beta_{1} ) q^{91} + ( 10 - 2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{7} + O(q^{10}) \) \( 3q + 2q^{7} - 4q^{11} + 6q^{13} - 4q^{17} + 6q^{19} + 3q^{23} + 17q^{25} + 2q^{29} - 4q^{31} + 8q^{35} + 14q^{37} + 2q^{41} - 2q^{43} + 16q^{47} + 7q^{49} + 4q^{53} - 24q^{55} + 12q^{59} + 22q^{61} + 2q^{67} + 8q^{71} + 6q^{73} + 16q^{77} - 26q^{79} + 4q^{83} + 8q^{85} + 8q^{89} + 4q^{91} + 32q^{95} + 18q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
−1.48119
2.17009
0.311108
0 0 0 −3.35026 0 −2.96239 0 0 0
1.2 0 0 0 −1.07838 0 4.34017 0 0 0
1.3 0 0 0 4.42864 0 0.622216 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 1656.2.a.n 3
3.b odd 2 1 552.2.a.g 3
4.b odd 2 1 3312.2.a.bf 3
12.b even 2 1 1104.2.a.o 3
24.f even 2 1 4416.2.a.bs 3
24.h odd 2 1 4416.2.a.bp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.g 3 3.b odd 2 1
1104.2.a.o 3 12.b even 2 1
1656.2.a.n 3 1.a even 1 1 trivial
3312.2.a.bf 3 4.b odd 2 1
4416.2.a.bp 3 24.h odd 2 1
4416.2.a.bs 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1656))\):

\( T_{5}^{3} - 16 T_{5} - 16 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 12 T_{7} + 8 \)
\( T_{11}^{3} + 4 T_{11}^{2} - 16 T_{11} - 32 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( -16 - 16 T + T^{3} \)
$7$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$11$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$13$ \( ( -2 + T )^{3} \)
$17$ \( -16 - 8 T + 4 T^{2} + T^{3} \)
$19$ \( 8 - 4 T - 6 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$31$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$37$ \( 152 + 28 T - 14 T^{2} + T^{3} \)
$41$ \( 104 - 84 T - 2 T^{2} + T^{3} \)
$43$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$47$ \( 128 + 32 T - 16 T^{2} + T^{3} \)
$53$ \( 592 - 144 T - 4 T^{2} + T^{3} \)
$59$ \( 64 - 16 T - 12 T^{2} + T^{3} \)
$61$ \( -200 + 124 T - 22 T^{2} + T^{3} \)
$67$ \( 184 - 52 T - 2 T^{2} + T^{3} \)
$71$ \( -256 - 128 T - 8 T^{2} + T^{3} \)
$73$ \( ( -2 + T )^{3} \)
$79$ \( 536 + 212 T + 26 T^{2} + T^{3} \)
$83$ \( 160 - 176 T - 4 T^{2} + T^{3} \)
$89$ \( 304 - 40 T - 8 T^{2} + T^{3} \)
$97$ \( 296 + 44 T - 18 T^{2} + T^{3} \)
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