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 $4$

Related objects

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 $$q + q^{7} - q^{9} - 2 q^{23} - q^{25} + q^{49} - q^{63} + 2 q^{71} + 2 q^{79} + q^{81}+O(q^{100})$$ q + q^7 - q^9 - 2 * q^23 - q^25 + q^49 - q^63 + 2 * q^71 + 2 * q^79 + q^81

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{1}^{\mathrm{new}}(224, [\chi])$$.

Hecke characteristic polynomials

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