Properties

Label 1856.2.a.y
Level $1856$
Weight $2$
Character orbit 1856.a
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{11} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{13} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{15} + 2 q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + ( 2 - 2 \beta_{1} ) q^{23} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{25} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{27} - q^{29} + ( 5 - \beta_{1} ) q^{31} + ( 1 - 2 \beta_{1} + 3 \beta_{2} ) q^{33} -2 \beta_{2} q^{37} + ( 5 + 3 \beta_{1} ) q^{39} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{43} + ( 4 - 2 \beta_{1} ) q^{45} + ( -1 + 3 \beta_{1} - 2 \beta_{2} ) q^{47} -7 q^{49} + ( 2 - 2 \beta_{1} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{53} + ( 9 - 5 \beta_{1} - 4 \beta_{2} ) q^{55} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{57} + ( 2 + 2 \beta_{1} ) q^{59} + ( -2 + 4 \beta_{1} ) q^{61} + ( -1 + 4 \beta_{1} + 3 \beta_{2} ) q^{65} + ( 8 - 4 \beta_{1} ) q^{67} + ( 10 + 2 \beta_{2} ) q^{69} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 - 4 \beta_{2} ) q^{75} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{79} + ( -5 - 2 \beta_{1} ) q^{81} + ( 10 + 2 \beta_{1} ) q^{83} + ( -2 + 2 \beta_{2} ) q^{85} + ( -1 + \beta_{1} ) q^{87} + ( 4 + 4 \beta_{1} + 6 \beta_{2} ) q^{89} + ( 9 - 4 \beta_{1} + \beta_{2} ) q^{93} + ( -10 + 4 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -4 + 2 \beta_{2} ) q^{97} + ( 12 - 2 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43} + 10 q^{45} + 2 q^{47} - 21 q^{49} + 4 q^{51} + 26 q^{55} - 12 q^{57} + 8 q^{59} - 2 q^{61} - 2 q^{65} + 20 q^{67} + 28 q^{69} - 12 q^{71} + 2 q^{73} + 16 q^{75} + 30 q^{79} - 17 q^{81} + 32 q^{83} - 8 q^{85} - 2 q^{87} + 10 q^{89} + 22 q^{93} - 28 q^{95} - 14 q^{97} + 32 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

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

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
3.12489
−0.363328
−1.76156
0 −2.12489 0 −1.48486 0 0 0 1.51514 0
1.2 0 1.36333 0 −4.14134 0 0 0 −1.14134 0
1.3 0 2.76156 0 1.62620 0 0 0 4.62620 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 1856.2.a.y 3
4.b odd 2 1 1856.2.a.x 3
8.b even 2 1 464.2.a.j 3
8.d odd 2 1 232.2.a.d 3
24.f even 2 1 2088.2.a.s 3
24.h odd 2 1 4176.2.a.bu 3
40.e odd 2 1 5800.2.a.p 3
232.b odd 2 1 6728.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.a.d 3 8.d odd 2 1
464.2.a.j 3 8.b even 2 1
1856.2.a.x 3 4.b odd 2 1
1856.2.a.y 3 1.a even 1 1 trivial
2088.2.a.s 3 24.f even 2 1
4176.2.a.bu 3 24.h odd 2 1
5800.2.a.p 3 40.e odd 2 1
6728.2.a.j 3 232.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 5 T_{3} + 8 \)
\( T_{5}^{3} + 4 T_{5}^{2} - 3 T_{5} - 10 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 8 - 5 T - 2 T^{2} + T^{3} \)
$5$ \( -10 - 3 T + 4 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 80 - 29 T - 2 T^{2} + T^{3} \)
$13$ \( -2 - 19 T + 4 T^{2} + T^{3} \)
$17$ \( ( -2 + T )^{3} \)
$19$ \( -32 - 28 T + 4 T^{2} + T^{3} \)
$23$ \( 64 - 20 T - 4 T^{2} + T^{3} \)
$29$ \( ( 1 + T )^{3} \)
$31$ \( -68 + 59 T - 14 T^{2} + T^{3} \)
$37$ \( 32 - 32 T - 2 T^{2} + T^{3} \)
$41$ \( 512 - 64 T - 10 T^{2} + T^{3} \)
$43$ \( 32 - 37 T + 6 T^{2} + T^{3} \)
$47$ \( 452 - 117 T - 2 T^{2} + T^{3} \)
$53$ \( -58 - 43 T + T^{3} \)
$59$ \( 16 - 4 T - 8 T^{2} + T^{3} \)
$61$ \( -328 - 100 T + 2 T^{2} + T^{3} \)
$67$ \( 640 + 32 T - 20 T^{2} + T^{3} \)
$71$ \( -1696 - 148 T + 12 T^{2} + T^{3} \)
$73$ \( -160 - 96 T - 2 T^{2} + T^{3} \)
$79$ \( -388 + 251 T - 30 T^{2} + T^{3} \)
$83$ \( -976 + 316 T - 32 T^{2} + T^{3} \)
$89$ \( 2816 - 256 T - 10 T^{2} + T^{3} \)
$97$ \( -64 + 32 T + 14 T^{2} + T^{3} \)
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