Properties

Label 1856.1.h.d
Level $1856$
Weight $1$
Character orbit 1856.h
Self dual yes
Analytic conductor $0.926$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -116
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.926264663447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 464)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.53824.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} + q^{5} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{3} + q^{5} + 2 q^{9} + \beta q^{11} - q^{13} -\beta q^{15} -\beta q^{27} + q^{29} -\beta q^{31} -3 q^{33} + \beta q^{39} + \beta q^{43} + 2 q^{45} + \beta q^{47} + q^{49} - q^{53} + \beta q^{55} - q^{65} + \beta q^{79} + q^{81} -\beta q^{87} + 3 q^{93} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{5} + 4 q^{9} - 2 q^{13} + 2 q^{29} - 6 q^{33} + 4 q^{45} + 2 q^{49} - 2 q^{53} - 2 q^{65} + 2 q^{81} + 6 q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1855.1
1.73205
−1.73205
0 −1.73205 0 1.00000 0 0 0 2.00000 0
1855.2 0 1.73205 0 1.00000 0 0 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
116.d odd 2 1 CM by \(\Q(\sqrt{-29}) \)
4.b odd 2 1 inner
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.1.h.d 2
4.b odd 2 1 inner 1856.1.h.d 2
8.b even 2 1 464.1.h.b 2
8.d odd 2 1 464.1.h.b 2
29.b even 2 1 inner 1856.1.h.d 2
116.d odd 2 1 CM 1856.1.h.d 2
232.b odd 2 1 464.1.h.b 2
232.g even 2 1 464.1.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.1.h.b 2 8.b even 2 1
464.1.h.b 2 8.d odd 2 1
464.1.h.b 2 232.b odd 2 1
464.1.h.b 2 232.g even 2 1
1856.1.h.d 2 1.a even 1 1 trivial
1856.1.h.d 2 4.b odd 2 1 inner
1856.1.h.d 2 29.b even 2 1 inner
1856.1.h.d 2 116.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(1856, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -3 + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -3 + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( -3 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( -3 + T^{2} \)
$47$ \( -3 + T^{2} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( -3 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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