Properties

Label 224.1.h.a
Level $224$
Weight $1$
Character orbit 224.h
Self dual yes
Analytic conductor $0.112$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -7, -56, 8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 224.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.111790562830\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.1568.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} - q^{9} + O(q^{10}) \) \( q + q^{7} - q^{9} - 2q^{23} - q^{25} + q^{49} - q^{63} + 2q^{71} + 2q^{79} + q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0
0 0 0 0 0 1.00000 0 −1.00000 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 \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.1.h.a 1
3.b odd 2 1 2016.1.l.a 1
4.b odd 2 1 56.1.h.a 1
7.b odd 2 1 CM 224.1.h.a 1
7.c even 3 2 1568.1.n.a 2
7.d odd 6 2 1568.1.n.a 2
8.b even 2 1 RM 224.1.h.a 1
8.d odd 2 1 56.1.h.a 1
12.b even 2 1 504.1.l.a 1
16.e even 4 2 1792.1.c.a 1
16.f odd 4 2 1792.1.c.b 1
20.d odd 2 1 1400.1.m.a 1
20.e even 4 2 1400.1.c.a 2
21.c even 2 1 2016.1.l.a 1
24.f even 2 1 504.1.l.a 1
24.h odd 2 1 2016.1.l.a 1
28.d even 2 1 56.1.h.a 1
28.f even 6 2 392.1.j.a 2
28.g odd 6 2 392.1.j.a 2
40.e odd 2 1 1400.1.m.a 1
40.k even 4 2 1400.1.c.a 2
56.e even 2 1 56.1.h.a 1
56.h odd 2 1 CM 224.1.h.a 1
56.j odd 6 2 1568.1.n.a 2
56.k odd 6 2 392.1.j.a 2
56.m even 6 2 392.1.j.a 2
56.p even 6 2 1568.1.n.a 2
84.h odd 2 1 504.1.l.a 1
84.j odd 6 2 3528.1.bw.a 2
84.n even 6 2 3528.1.bw.a 2
112.j even 4 2 1792.1.c.b 1
112.l odd 4 2 1792.1.c.a 1
140.c even 2 1 1400.1.m.a 1
140.j odd 4 2 1400.1.c.a 2
168.e odd 2 1 504.1.l.a 1
168.i even 2 1 2016.1.l.a 1
168.v even 6 2 3528.1.bw.a 2
168.be odd 6 2 3528.1.bw.a 2
280.n even 2 1 1400.1.m.a 1
280.y odd 4 2 1400.1.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 4.b odd 2 1
56.1.h.a 1 8.d odd 2 1
56.1.h.a 1 28.d even 2 1
56.1.h.a 1 56.e even 2 1
224.1.h.a 1 1.a even 1 1 trivial
224.1.h.a 1 7.b odd 2 1 CM
224.1.h.a 1 8.b even 2 1 RM
224.1.h.a 1 56.h odd 2 1 CM
392.1.j.a 2 28.f even 6 2
392.1.j.a 2 28.g odd 6 2
392.1.j.a 2 56.k odd 6 2
392.1.j.a 2 56.m even 6 2
504.1.l.a 1 12.b even 2 1
504.1.l.a 1 24.f even 2 1
504.1.l.a 1 84.h odd 2 1
504.1.l.a 1 168.e odd 2 1
1400.1.c.a 2 20.e even 4 2
1400.1.c.a 2 40.k even 4 2
1400.1.c.a 2 140.j odd 4 2
1400.1.c.a 2 280.y odd 4 2
1400.1.m.a 1 20.d odd 2 1
1400.1.m.a 1 40.e odd 2 1
1400.1.m.a 1 140.c even 2 1
1400.1.m.a 1 280.n even 2 1
1568.1.n.a 2 7.c even 3 2
1568.1.n.a 2 7.d odd 6 2
1568.1.n.a 2 56.j odd 6 2
1568.1.n.a 2 56.p even 6 2
1792.1.c.a 1 16.e even 4 2
1792.1.c.a 1 112.l odd 4 2
1792.1.c.b 1 16.f odd 4 2
1792.1.c.b 1 112.j even 4 2
2016.1.l.a 1 3.b odd 2 1
2016.1.l.a 1 21.c even 2 1
2016.1.l.a 1 24.h odd 2 1
2016.1.l.a 1 168.i even 2 1
3528.1.bw.a 2 84.j odd 6 2
3528.1.bw.a 2 84.n even 6 2
3528.1.bw.a 2 168.v even 6 2
3528.1.bw.a 2 168.be odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(224, [\chi])\).