Properties

Label 112.4.a.e
Level $112$
Weight $4$
Character orbit 112.a
Self dual yes
Analytic conductor $6.608$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} - 12q^{5} - 7q^{7} - 23q^{9} + O(q^{10}) \) \( q + 2q^{3} - 12q^{5} - 7q^{7} - 23q^{9} - 48q^{11} + 56q^{13} - 24q^{15} - 114q^{17} - 2q^{19} - 14q^{21} + 120q^{23} + 19q^{25} - 100q^{27} - 54q^{29} - 236q^{31} - 96q^{33} + 84q^{35} + 146q^{37} + 112q^{39} + 126q^{41} + 376q^{43} + 276q^{45} + 12q^{47} + 49q^{49} - 228q^{51} + 174q^{53} + 576q^{55} - 4q^{57} - 138q^{59} + 380q^{61} + 161q^{63} - 672q^{65} + 484q^{67} + 240q^{69} - 576q^{71} - 1150q^{73} + 38q^{75} + 336q^{77} - 776q^{79} + 421q^{81} - 378q^{83} + 1368q^{85} - 108q^{87} - 390q^{89} - 392q^{91} - 472q^{93} + 24q^{95} - 1330q^{97} + 1104q^{99} + O(q^{100}) \)

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 2.00000 0 −12.0000 0 −7.00000 0 −23.0000 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.e 1
3.b odd 2 1 1008.4.a.r 1
4.b odd 2 1 14.4.a.b 1
7.b odd 2 1 784.4.a.h 1
8.b even 2 1 448.4.a.g 1
8.d odd 2 1 448.4.a.k 1
12.b even 2 1 126.4.a.d 1
20.d odd 2 1 350.4.a.f 1
20.e even 4 2 350.4.c.g 2
28.d even 2 1 98.4.a.e 1
28.f even 6 2 98.4.c.b 2
28.g odd 6 2 98.4.c.c 2
44.c even 2 1 1694.4.a.b 1
52.b odd 2 1 2366.4.a.c 1
84.h odd 2 1 882.4.a.b 1
84.j odd 6 2 882.4.g.v 2
84.n even 6 2 882.4.g.p 2
140.c even 2 1 2450.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 4.b odd 2 1
98.4.a.e 1 28.d even 2 1
98.4.c.b 2 28.f even 6 2
98.4.c.c 2 28.g odd 6 2
112.4.a.e 1 1.a even 1 1 trivial
126.4.a.d 1 12.b even 2 1
350.4.a.f 1 20.d odd 2 1
350.4.c.g 2 20.e even 4 2
448.4.a.g 1 8.b even 2 1
448.4.a.k 1 8.d odd 2 1
784.4.a.h 1 7.b odd 2 1
882.4.a.b 1 84.h odd 2 1
882.4.g.p 2 84.n even 6 2
882.4.g.v 2 84.j odd 6 2
1008.4.a.r 1 3.b odd 2 1
1694.4.a.b 1 44.c even 2 1
2366.4.a.c 1 52.b odd 2 1
2450.4.a.i 1 140.c even 2 1

Hecke kernels

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

\( T_{3} - 2 \)
\( T_{5} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( 12 + T \)
$7$ \( 7 + T \)
$11$ \( 48 + T \)
$13$ \( -56 + T \)
$17$ \( 114 + T \)
$19$ \( 2 + T \)
$23$ \( -120 + T \)
$29$ \( 54 + T \)
$31$ \( 236 + T \)
$37$ \( -146 + T \)
$41$ \( -126 + T \)
$43$ \( -376 + T \)
$47$ \( -12 + T \)
$53$ \( -174 + T \)
$59$ \( 138 + T \)
$61$ \( -380 + T \)
$67$ \( -484 + T \)
$71$ \( 576 + T \)
$73$ \( 1150 + T \)
$79$ \( 776 + T \)
$83$ \( 378 + T \)
$89$ \( 390 + T \)
$97$ \( 1330 + T \)
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