Properties

Label 1856.2.a.z
Level $1856$
Weight $2$
Character orbit 1856.a
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 2 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{11} + ( 4 + \beta_{3} ) q^{13} + ( \beta_{1} - 4 \beta_{2} ) q^{15} + ( -3 + \beta_{3} ) q^{17} + 2 \beta_{2} q^{19} + ( 5 - \beta_{3} ) q^{21} + ( -\beta_{1} + \beta_{2} ) q^{23} + ( 4 - 4 \beta_{3} ) q^{25} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{27} - q^{29} + \beta_{1} q^{31} + ( -5 + 4 \beta_{3} ) q^{33} + ( \beta_{1} - 3 \beta_{2} ) q^{35} + ( 2 + 2 \beta_{3} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} ) q^{39} + ( -5 + \beta_{3} ) q^{41} + ( 4 \beta_{1} - \beta_{2} ) q^{43} + ( 14 - 6 \beta_{3} ) q^{45} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + 2 \beta_{3} ) q^{49} + ( -\beta_{1} + 5 \beta_{2} ) q^{51} + ( 6 + \beta_{3} ) q^{53} + ( -\beta_{1} + 6 \beta_{2} ) q^{55} + ( -10 + 4 \beta_{3} ) q^{57} + ( -5 \beta_{1} + \beta_{2} ) q^{59} + ( 5 + \beta_{3} ) q^{61} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{63} + ( 3 - 2 \beta_{3} ) q^{65} -4 \beta_{2} q^{67} + ( -5 + 3 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + 4 \beta_{3} ) q^{73} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{75} + ( 5 + 7 \beta_{3} ) q^{77} -3 \beta_{1} q^{79} + ( 9 - 2 \beta_{3} ) q^{81} + ( -\beta_{1} + 5 \beta_{2} ) q^{83} + ( -11 + 5 \beta_{3} ) q^{85} + \beta_{2} q^{87} + ( -5 - 3 \beta_{3} ) q^{89} + ( -7 \beta_{1} - 3 \beta_{2} ) q^{91} -\beta_{3} q^{93} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{95} + ( 3 - 3 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 8 q^{9} + O(q^{10}) \) \( 4 q + 8 q^{5} + 8 q^{9} + 16 q^{13} - 12 q^{17} + 20 q^{21} + 16 q^{25} - 4 q^{29} - 20 q^{33} + 8 q^{37} - 20 q^{41} + 56 q^{45} + 12 q^{49} + 24 q^{53} - 40 q^{57} + 20 q^{61} + 12 q^{65} - 20 q^{69} + 16 q^{73} + 20 q^{77} + 36 q^{81} - 44 q^{85} - 20 q^{89} + 12 q^{97} + O(q^{100}) \)

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
1.17557
1.90211
−1.90211
−1.17557
0 −3.07768 0 4.23607 0 −2.35114 0 6.47214 0
1.2 0 −0.726543 0 −0.236068 0 −3.80423 0 −2.47214 0
1.3 0 0.726543 0 −0.236068 0 3.80423 0 −2.47214 0
1.4 0 3.07768 0 4.23607 0 2.35114 0 6.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.a.z 4
4.b odd 2 1 inner 1856.2.a.z 4
8.b even 2 1 928.2.a.f 4
8.d odd 2 1 928.2.a.f 4
24.f even 2 1 8352.2.a.ba 4
24.h odd 2 1 8352.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.a.f 4 8.b even 2 1
928.2.a.f 4 8.d odd 2 1
1856.2.a.z 4 1.a even 1 1 trivial
1856.2.a.z 4 4.b odd 2 1 inner
8352.2.a.ba 4 24.f even 2 1
8352.2.a.ba 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{4} - 10 T_{3}^{2} + 5 \)
\( T_{5}^{2} - 4 T_{5} - 1 \)
\( T_{17}^{2} + 6 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 5 - 10 T^{2} + T^{4} \)
$5$ \( ( -1 - 4 T + T^{2} )^{2} \)
$7$ \( 80 - 20 T^{2} + T^{4} \)
$11$ \( 605 - 50 T^{2} + T^{4} \)
$13$ \( ( 11 - 8 T + T^{2} )^{2} \)
$17$ \( ( 4 + 6 T + T^{2} )^{2} \)
$19$ \( 80 - 40 T^{2} + T^{4} \)
$23$ \( 80 - 20 T^{2} + T^{4} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( 5 - 10 T^{2} + T^{4} \)
$37$ \( ( -16 - 4 T + T^{2} )^{2} \)
$41$ \( ( 20 + 10 T + T^{2} )^{2} \)
$43$ \( 4805 - 170 T^{2} + T^{4} \)
$47$ \( 605 - 50 T^{2} + T^{4} \)
$53$ \( ( 31 - 12 T + T^{2} )^{2} \)
$59$ \( 9680 - 260 T^{2} + T^{4} \)
$61$ \( ( 20 - 10 T + T^{2} )^{2} \)
$67$ \( 1280 - 160 T^{2} + T^{4} \)
$71$ \( 2000 - 200 T^{2} + T^{4} \)
$73$ \( ( -64 - 8 T + T^{2} )^{2} \)
$79$ \( 405 - 90 T^{2} + T^{4} \)
$83$ \( 80 - 260 T^{2} + T^{4} \)
$89$ \( ( -20 + 10 T + T^{2} )^{2} \)
$97$ \( ( -36 - 6 T + T^{2} )^{2} \)
show more
show less