Properties

Label 1856.2.a.a
Level $1856$
Weight $2$
Character orbit 1856.a
Self dual yes
Analytic conductor $14.820$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} - 3 q^{5} - 4 q^{7} + 6 q^{9} + O(q^{10}) \) \( q - 3 q^{3} - 3 q^{5} - 4 q^{7} + 6 q^{9} - q^{11} + 3 q^{13} + 9 q^{15} + 2 q^{17} + 4 q^{19} + 12 q^{21} + 6 q^{23} + 4 q^{25} - 9 q^{27} + q^{29} - 9 q^{31} + 3 q^{33} + 12 q^{35} + 8 q^{37} - 9 q^{39} - 8 q^{41} - 5 q^{43} - 18 q^{45} + 7 q^{47} + 9 q^{49} - 6 q^{51} + 5 q^{53} + 3 q^{55} - 12 q^{57} - 10 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{65} + 8 q^{67} - 18 q^{69} + 2 q^{71} - 12 q^{75} + 4 q^{77} + q^{79} + 9 q^{81} + 6 q^{83} - 6 q^{85} - 3 q^{87} + 12 q^{89} - 12 q^{91} + 27 q^{93} - 12 q^{95} - 6 q^{99} + O(q^{100}) \)

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
0
0 −3.00000 0 −3.00000 0 −4.00000 0 6.00000 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.a 1
4.b odd 2 1 1856.2.a.o 1
8.b even 2 1 464.2.a.g 1
8.d odd 2 1 116.2.a.a 1
24.f even 2 1 1044.2.a.c 1
24.h odd 2 1 4176.2.a.c 1
40.e odd 2 1 2900.2.a.e 1
40.k even 4 2 2900.2.c.a 2
56.e even 2 1 5684.2.a.k 1
232.b odd 2 1 3364.2.a.c 1
232.k even 4 2 3364.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.2.a.a 1 8.d odd 2 1
464.2.a.g 1 8.b even 2 1
1044.2.a.c 1 24.f even 2 1
1856.2.a.a 1 1.a even 1 1 trivial
1856.2.a.o 1 4.b odd 2 1
2900.2.a.e 1 40.e odd 2 1
2900.2.c.a 2 40.k even 4 2
3364.2.a.c 1 232.b odd 2 1
3364.2.c.b 2 232.k even 4 2
4176.2.a.c 1 24.h odd 2 1
5684.2.a.k 1 56.e even 2 1

Hecke kernels

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

\( T_{3} + 3 \)
\( T_{5} + 3 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 3 + T \)
$7$ \( 4 + T \)
$11$ \( 1 + T \)
$13$ \( -3 + T \)
$17$ \( -2 + T \)
$19$ \( -4 + T \)
$23$ \( -6 + T \)
$29$ \( -1 + T \)
$31$ \( 9 + T \)
$37$ \( -8 + T \)
$41$ \( 8 + T \)
$43$ \( 5 + T \)
$47$ \( -7 + T \)
$53$ \( -5 + T \)
$59$ \( 10 + T \)
$61$ \( 10 + T \)
$67$ \( -8 + T \)
$71$ \( -2 + T \)
$73$ \( T \)
$79$ \( -1 + T \)
$83$ \( -6 + T \)
$89$ \( -12 + T \)
$97$ \( T \)
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