Properties

Label 112.10.a.b
Level $112$
Weight $10$
Character orbit 112.a
Self dual yes
Analytic conductor $57.684$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.6840136504\)
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

\(f(q)\) \(=\) \( q + 6q^{3} + 560q^{5} + 2401q^{7} - 19647q^{9} + O(q^{10}) \) \( q + 6q^{3} + 560q^{5} + 2401q^{7} - 19647q^{9} + 54152q^{11} - 113172q^{13} + 3360q^{15} + 6262q^{17} - 257078q^{19} + 14406q^{21} + 266000q^{23} - 1639525q^{25} - 235980q^{27} + 1574714q^{29} + 4637484q^{31} + 324912q^{33} + 1344560q^{35} - 11946238q^{37} - 679032q^{39} + 21909126q^{41} - 27520592q^{43} - 11002320q^{45} - 52927836q^{47} + 5764801q^{49} + 37572q^{51} + 16221222q^{53} + 30325120q^{55} - 1542468q^{57} + 140509618q^{59} - 202963560q^{61} - 47172447q^{63} - 63376320q^{65} - 153734572q^{67} + 1596000q^{69} - 279655936q^{71} - 404022830q^{73} - 9837150q^{75} + 130018952q^{77} + 130689816q^{79} + 385296021q^{81} - 420134014q^{83} + 3506720q^{85} + 9448284q^{87} - 469542390q^{89} - 271725972q^{91} + 27824904q^{93} - 143963680q^{95} - 872501690q^{97} - 1063924344q^{99} + O(q^{100}) \)

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 6.00000 0 560.000 0 2401.00 0 −19647.0 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 112.10.a.b 1
4.b odd 2 1 14.10.a.a 1
12.b even 2 1 126.10.a.e 1
20.d odd 2 1 350.10.a.c 1
20.e even 4 2 350.10.c.b 2
28.d even 2 1 98.10.a.a 1
28.f even 6 2 98.10.c.e 2
28.g odd 6 2 98.10.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 4.b odd 2 1
98.10.a.a 1 28.d even 2 1
98.10.c.e 2 28.f even 6 2
98.10.c.f 2 28.g odd 6 2
112.10.a.b 1 1.a even 1 1 trivial
126.10.a.e 1 12.b even 2 1
350.10.a.c 1 20.d odd 2 1
350.10.c.b 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 6 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -6 + T \)
$5$ \( -560 + T \)
$7$ \( -2401 + T \)
$11$ \( -54152 + T \)
$13$ \( 113172 + T \)
$17$ \( -6262 + T \)
$19$ \( 257078 + T \)
$23$ \( -266000 + T \)
$29$ \( -1574714 + T \)
$31$ \( -4637484 + T \)
$37$ \( 11946238 + T \)
$41$ \( -21909126 + T \)
$43$ \( 27520592 + T \)
$47$ \( 52927836 + T \)
$53$ \( -16221222 + T \)
$59$ \( -140509618 + T \)
$61$ \( 202963560 + T \)
$67$ \( 153734572 + T \)
$71$ \( 279655936 + T \)
$73$ \( 404022830 + T \)
$79$ \( -130689816 + T \)
$83$ \( 420134014 + T \)
$89$ \( 469542390 + T \)
$97$ \( 872501690 + T \)
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