Properties

Label 112.3.c.a
Level 112
Weight 3
Character orbit 112.c
Self dual yes
Analytic conductor 3.052
Analytic rank 0
Dimension 1
CM discriminant -7
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(3.05177896084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 7q^{7} + 9q^{9} + 6q^{11} - 18q^{23} + 25q^{25} - 54q^{29} - 38q^{37} - 58q^{43} + 49q^{49} - 6q^{53} + 63q^{63} + 118q^{67} - 114q^{71} + 42q^{77} + 94q^{79} + 81q^{81} + 54q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0
0 0 0 0 0 7.00000 0 9.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.c.a 1
3.b odd 2 1 1008.3.f.a 1
4.b odd 2 1 7.3.b.a 1
7.b odd 2 1 CM 112.3.c.a 1
7.c even 3 2 784.3.s.a 2
7.d odd 6 2 784.3.s.a 2
8.b even 2 1 448.3.c.b 1
8.d odd 2 1 448.3.c.a 1
12.b even 2 1 63.3.d.a 1
20.d odd 2 1 175.3.d.a 1
20.e even 4 2 175.3.c.a 2
21.c even 2 1 1008.3.f.a 1
28.d even 2 1 7.3.b.a 1
28.f even 6 2 49.3.d.a 2
28.g odd 6 2 49.3.d.a 2
56.e even 2 1 448.3.c.a 1
56.h odd 2 1 448.3.c.b 1
84.h odd 2 1 63.3.d.a 1
84.j odd 6 2 441.3.m.a 2
84.n even 6 2 441.3.m.a 2
140.c even 2 1 175.3.d.a 1
140.j odd 4 2 175.3.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 4.b odd 2 1
7.3.b.a 1 28.d even 2 1
49.3.d.a 2 28.f even 6 2
49.3.d.a 2 28.g odd 6 2
63.3.d.a 1 12.b even 2 1
63.3.d.a 1 84.h odd 2 1
112.3.c.a 1 1.a even 1 1 trivial
112.3.c.a 1 7.b odd 2 1 CM
175.3.c.a 2 20.e even 4 2
175.3.c.a 2 140.j odd 4 2
175.3.d.a 1 20.d odd 2 1
175.3.d.a 1 140.c even 2 1
441.3.m.a 2 84.j odd 6 2
441.3.m.a 2 84.n even 6 2
448.3.c.a 1 8.d odd 2 1
448.3.c.a 1 56.e even 2 1
448.3.c.b 1 8.b even 2 1
448.3.c.b 1 56.h odd 2 1
784.3.s.a 2 7.c even 3 2
784.3.s.a 2 7.d odd 6 2
1008.3.f.a 1 3.b odd 2 1
1008.3.f.a 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( 1 - 7 T \)
$11$ \( 1 - 6 T + 121 T^{2} \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( 1 + 18 T + 529 T^{2} \)
$29$ \( 1 + 54 T + 841 T^{2} \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( 1 + 38 T + 1369 T^{2} \)
$41$ \( ( 1 - 41 T )( 1 + 41 T ) \)
$43$ \( 1 + 58 T + 1849 T^{2} \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( 1 + 6 T + 2809 T^{2} \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( 1 - 118 T + 4489 T^{2} \)
$71$ \( 1 + 114 T + 5041 T^{2} \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( 1 - 94 T + 6241 T^{2} \)
$83$ \( ( 1 - 83 T )( 1 + 83 T ) \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( ( 1 - 97 T )( 1 + 97 T ) \)
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