Properties

Label 112.4.a.g
Level $112$
Weight $4$
Character orbit 112.a
Self dual yes
Analytic conductor $6.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 10q^{3} - 8q^{5} + 7q^{7} + 73q^{9} + O(q^{10}) \) \( q + 10q^{3} - 8q^{5} + 7q^{7} + 73q^{9} + 40q^{11} - 12q^{13} - 80q^{15} - 58q^{17} - 26q^{19} + 70q^{21} + 64q^{23} - 61q^{25} + 460q^{27} - 62q^{29} - 252q^{31} + 400q^{33} - 56q^{35} + 26q^{37} - 120q^{39} + 6q^{41} - 416q^{43} - 584q^{45} + 396q^{47} + 49q^{49} - 580q^{51} - 450q^{53} - 320q^{55} - 260q^{57} - 274q^{59} - 576q^{61} + 511q^{63} + 96q^{65} + 476q^{67} + 640q^{69} + 448q^{71} - 158q^{73} - 610q^{75} + 280q^{77} + 936q^{79} + 2629q^{81} - 530q^{83} + 464q^{85} - 620q^{87} - 390q^{89} - 84q^{91} - 2520q^{93} + 208q^{95} + 214q^{97} + 2920q^{99} + O(q^{100}) \)

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 10.0000 0 −8.00000 0 7.00000 0 73.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 112.4.a.g 1
3.b odd 2 1 1008.4.a.o 1
4.b odd 2 1 28.4.a.a 1
7.b odd 2 1 784.4.a.a 1
8.b even 2 1 448.4.a.a 1
8.d odd 2 1 448.4.a.p 1
12.b even 2 1 252.4.a.d 1
20.d odd 2 1 700.4.a.n 1
20.e even 4 2 700.4.e.a 2
28.d even 2 1 196.4.a.d 1
28.f even 6 2 196.4.e.a 2
28.g odd 6 2 196.4.e.f 2
84.h odd 2 1 1764.4.a.c 1
84.j odd 6 2 1764.4.k.m 2
84.n even 6 2 1764.4.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 4.b odd 2 1
112.4.a.g 1 1.a even 1 1 trivial
196.4.a.d 1 28.d even 2 1
196.4.e.a 2 28.f even 6 2
196.4.e.f 2 28.g odd 6 2
252.4.a.d 1 12.b even 2 1
448.4.a.a 1 8.b even 2 1
448.4.a.p 1 8.d odd 2 1
700.4.a.n 1 20.d odd 2 1
700.4.e.a 2 20.e even 4 2
784.4.a.a 1 7.b odd 2 1
1008.4.a.o 1 3.b odd 2 1
1764.4.a.c 1 84.h odd 2 1
1764.4.k.d 2 84.n even 6 2
1764.4.k.m 2 84.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(112))\):

\( T_{3} - 10 \)
\( T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -10 + T \)
$5$ \( 8 + T \)
$7$ \( -7 + T \)
$11$ \( -40 + T \)
$13$ \( 12 + T \)
$17$ \( 58 + T \)
$19$ \( 26 + T \)
$23$ \( -64 + T \)
$29$ \( 62 + T \)
$31$ \( 252 + T \)
$37$ \( -26 + T \)
$41$ \( -6 + T \)
$43$ \( 416 + T \)
$47$ \( -396 + T \)
$53$ \( 450 + T \)
$59$ \( 274 + T \)
$61$ \( 576 + T \)
$67$ \( -476 + T \)
$71$ \( -448 + T \)
$73$ \( 158 + T \)
$79$ \( -936 + T \)
$83$ \( 530 + T \)
$89$ \( 390 + T \)
$97$ \( -214 + T \)
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