Properties

Label 112.1.l.a
Level 112
Weight 1
Character orbit 112.l
Analytic conductor 0.056
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -7
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 112.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0558952814149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.14336.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.4917248.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} -i q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + i q^{7} -i q^{8} -i q^{9} + ( -1 - i ) q^{11} - q^{14} + q^{16} + q^{18} + ( 1 - i ) q^{22} + i q^{25} -i q^{28} + ( -1 + i ) q^{29} + i q^{32} + i q^{36} + ( -1 - i ) q^{37} + ( 1 + i ) q^{43} + ( 1 + i ) q^{44} - q^{49} - q^{50} + ( 1 + i ) q^{53} + q^{56} + ( -1 - i ) q^{58} + q^{63} - q^{64} + ( 1 - i ) q^{67} -2 i q^{71} - q^{72} + ( 1 - i ) q^{74} + ( 1 - i ) q^{77} - q^{81} + ( -1 + i ) q^{86} + ( -1 + i ) q^{88} -i q^{98} + ( -1 + i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{11} - 2q^{14} + 2q^{16} + 2q^{18} + 2q^{22} - 2q^{29} - 2q^{37} + 2q^{43} + 2q^{44} - 2q^{49} - 2q^{50} + 2q^{53} + 2q^{56} - 2q^{58} + 2q^{63} - 2q^{64} + 2q^{67} - 2q^{72} + 2q^{74} + 2q^{77} - 2q^{81} - 2q^{86} - 2q^{88} - 2q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 1.00000i 0
69.1 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 1.00000i 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 \(\Q(\sqrt{-7}) \)
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.1.l.a 2
3.b odd 2 1 1008.1.u.b 2
4.b odd 2 1 448.1.l.a 2
5.b even 2 1 2800.1.z.a 2
5.c odd 4 1 2800.1.bf.a 2
5.c odd 4 1 2800.1.bf.b 2
7.b odd 2 1 CM 112.1.l.a 2
7.c even 3 2 784.1.y.a 4
7.d odd 6 2 784.1.y.a 4
8.b even 2 1 896.1.l.b 2
8.d odd 2 1 896.1.l.a 2
16.e even 4 1 inner 112.1.l.a 2
16.e even 4 1 896.1.l.b 2
16.f odd 4 1 448.1.l.a 2
16.f odd 4 1 896.1.l.a 2
21.c even 2 1 1008.1.u.b 2
28.d even 2 1 448.1.l.a 2
28.f even 6 2 3136.1.bc.a 4
28.g odd 6 2 3136.1.bc.a 4
35.c odd 2 1 2800.1.z.a 2
35.f even 4 1 2800.1.bf.a 2
35.f even 4 1 2800.1.bf.b 2
48.i odd 4 1 1008.1.u.b 2
56.e even 2 1 896.1.l.a 2
56.h odd 2 1 896.1.l.b 2
80.i odd 4 1 2800.1.bf.a 2
80.q even 4 1 2800.1.z.a 2
80.t odd 4 1 2800.1.bf.b 2
112.j even 4 1 448.1.l.a 2
112.j even 4 1 896.1.l.a 2
112.l odd 4 1 inner 112.1.l.a 2
112.l odd 4 1 896.1.l.b 2
112.u odd 12 2 3136.1.bc.a 4
112.v even 12 2 3136.1.bc.a 4
112.w even 12 2 784.1.y.a 4
112.x odd 12 2 784.1.y.a 4
336.y even 4 1 1008.1.u.b 2
560.r even 4 1 2800.1.bf.b 2
560.bf odd 4 1 2800.1.z.a 2
560.bn even 4 1 2800.1.bf.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.1.l.a 2 1.a even 1 1 trivial
112.1.l.a 2 7.b odd 2 1 CM
112.1.l.a 2 16.e even 4 1 inner
112.1.l.a 2 112.l odd 4 1 inner
448.1.l.a 2 4.b odd 2 1
448.1.l.a 2 16.f odd 4 1
448.1.l.a 2 28.d even 2 1
448.1.l.a 2 112.j even 4 1
784.1.y.a 4 7.c even 3 2
784.1.y.a 4 7.d odd 6 2
784.1.y.a 4 112.w even 12 2
784.1.y.a 4 112.x odd 12 2
896.1.l.a 2 8.d odd 2 1
896.1.l.a 2 16.f odd 4 1
896.1.l.a 2 56.e even 2 1
896.1.l.a 2 112.j even 4 1
896.1.l.b 2 8.b even 2 1
896.1.l.b 2 16.e even 4 1
896.1.l.b 2 56.h odd 2 1
896.1.l.b 2 112.l odd 4 1
1008.1.u.b 2 3.b odd 2 1
1008.1.u.b 2 21.c even 2 1
1008.1.u.b 2 48.i odd 4 1
1008.1.u.b 2 336.y even 4 1
2800.1.z.a 2 5.b even 2 1
2800.1.z.a 2 35.c odd 2 1
2800.1.z.a 2 80.q even 4 1
2800.1.z.a 2 560.bf odd 4 1
2800.1.bf.a 2 5.c odd 4 1
2800.1.bf.a 2 35.f even 4 1
2800.1.bf.a 2 80.i odd 4 1
2800.1.bf.a 2 560.bn even 4 1
2800.1.bf.b 2 5.c odd 4 1
2800.1.bf.b 2 35.f even 4 1
2800.1.bf.b 2 80.t odd 4 1
2800.1.bf.b 2 560.r even 4 1
3136.1.bc.a 4 28.f even 6 2
3136.1.bc.a 4 28.g odd 6 2
3136.1.bc.a 4 112.u odd 12 2
3136.1.bc.a 4 112.v even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

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