Properties

Label 1856.2.e.h
Level $1856$
Weight $2$
Character orbit 1856.e
Analytic conductor $14.820$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 928)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + 2 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} - q^{5} + \beta_{3} q^{7} + 2 q^{9} + 3 \beta_{1} q^{11} + q^{13} -\beta_{1} q^{15} + \beta_{2} q^{17} -4 \beta_{1} q^{19} + \beta_{2} q^{21} + \beta_{3} q^{23} -4 q^{25} + 5 \beta_{1} q^{27} + ( 3 - \beta_{2} ) q^{29} -5 \beta_{1} q^{31} -3 q^{33} -\beta_{3} q^{35} + \beta_{1} q^{39} + \beta_{2} q^{41} -\beta_{1} q^{43} -2 q^{45} -3 \beta_{1} q^{47} + 13 q^{49} -\beta_{3} q^{51} + 9 q^{53} -3 \beta_{1} q^{55} + 4 q^{57} -\beta_{3} q^{59} + 3 \beta_{2} q^{61} + 2 \beta_{3} q^{63} - q^{65} -2 \beta_{3} q^{67} + \beta_{2} q^{69} -2 \beta_{3} q^{71} -2 \beta_{2} q^{73} -4 \beta_{1} q^{75} + 3 \beta_{2} q^{77} -15 \beta_{1} q^{79} + q^{81} + \beta_{3} q^{83} -\beta_{2} q^{85} + ( 3 \beta_{1} + \beta_{3} ) q^{87} + 3 \beta_{2} q^{89} + \beta_{3} q^{91} + 5 q^{93} + 4 \beta_{1} q^{95} + 3 \beta_{2} q^{97} + 6 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 8 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{5} + 8 q^{9} + 4 q^{13} - 16 q^{25} + 12 q^{29} - 12 q^{33} - 8 q^{45} + 52 q^{49} + 36 q^{53} + 16 q^{57} - 4 q^{65} + 4 q^{81} + 20 q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 4 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
1.61803i
0.618034i
1.61803i
0.618034i
0 1.00000i 0 −1.00000 0 −4.47214 0 2.00000 0
1217.2 0 1.00000i 0 −1.00000 0 4.47214 0 2.00000 0
1217.3 0 1.00000i 0 −1.00000 0 −4.47214 0 2.00000 0
1217.4 0 1.00000i 0 −1.00000 0 4.47214 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
29.b even 2 1 inner
116.d 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.e.h 4
4.b odd 2 1 inner 1856.2.e.h 4
8.b even 2 1 928.2.e.b 4
8.d odd 2 1 928.2.e.b 4
29.b even 2 1 inner 1856.2.e.h 4
116.d odd 2 1 inner 1856.2.e.h 4
232.b odd 2 1 928.2.e.b 4
232.g even 2 1 928.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.2.e.b 4 8.b even 2 1
928.2.e.b 4 8.d odd 2 1
928.2.e.b 4 232.b odd 2 1
928.2.e.b 4 232.g even 2 1
1856.2.e.h 4 1.a even 1 1 trivial
1856.2.e.h 4 4.b odd 2 1 inner
1856.2.e.h 4 29.b even 2 1 inner
1856.2.e.h 4 116.d odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( -20 + T^{2} )^{2} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -20 + T^{2} )^{2} \)
$29$ \( ( 29 - 6 T + T^{2} )^{2} \)
$31$ \( ( 25 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( ( -9 + T )^{4} \)
$59$ \( ( -20 + T^{2} )^{2} \)
$61$ \( ( 180 + T^{2} )^{2} \)
$67$ \( ( -80 + T^{2} )^{2} \)
$71$ \( ( -80 + T^{2} )^{2} \)
$73$ \( ( 80 + T^{2} )^{2} \)
$79$ \( ( 225 + T^{2} )^{2} \)
$83$ \( ( -20 + T^{2} )^{2} \)
$89$ \( ( 180 + T^{2} )^{2} \)
$97$ \( ( 180 + T^{2} )^{2} \)
show more
show less