# Properties

 Label 56.1.h.a Level $56$ Weight $1$ Character orbit 56.h Self dual yes Analytic conductor $0.028$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -7, -56, 8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 56.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0279476407074$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{-7})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.392.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^7 - q^8 - q^9 $$q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + q^{14} + q^{16} + q^{18} + 2 q^{23} - q^{25} - q^{28} - q^{32} - q^{36} - 2 q^{46} + q^{49} + q^{50} + q^{56} + q^{63} + q^{64} - 2 q^{71} + q^{72} - 2 q^{79} + q^{81} + 2 q^{92} - q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^7 - q^8 - q^9 + q^14 + q^16 + q^18 + 2 * q^23 - q^25 - q^28 - q^32 - q^36 - 2 * q^46 + q^49 + q^50 + q^56 + q^63 + q^64 - 2 * q^71 + q^72 - 2 * q^79 + q^81 + 2 * q^92 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/56\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 −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 56.1.h.a 1
3.b odd 2 1 504.1.l.a 1
4.b odd 2 1 224.1.h.a 1
5.b even 2 1 1400.1.m.a 1
5.c odd 4 2 1400.1.c.a 2
7.b odd 2 1 CM 56.1.h.a 1
7.c even 3 2 392.1.j.a 2
7.d odd 6 2 392.1.j.a 2
8.b even 2 1 RM 56.1.h.a 1
8.d odd 2 1 224.1.h.a 1
12.b even 2 1 2016.1.l.a 1
16.e even 4 2 1792.1.c.b 1
16.f odd 4 2 1792.1.c.a 1
21.c even 2 1 504.1.l.a 1
21.g even 6 2 3528.1.bw.a 2
21.h odd 6 2 3528.1.bw.a 2
24.f even 2 1 2016.1.l.a 1
24.h odd 2 1 504.1.l.a 1
28.d even 2 1 224.1.h.a 1
28.f even 6 2 1568.1.n.a 2
28.g odd 6 2 1568.1.n.a 2
35.c odd 2 1 1400.1.m.a 1
35.f even 4 2 1400.1.c.a 2
40.f even 2 1 1400.1.m.a 1
40.i odd 4 2 1400.1.c.a 2
56.e even 2 1 224.1.h.a 1
56.h odd 2 1 CM 56.1.h.a 1
56.j odd 6 2 392.1.j.a 2
56.k odd 6 2 1568.1.n.a 2
56.m even 6 2 1568.1.n.a 2
56.p even 6 2 392.1.j.a 2
84.h odd 2 1 2016.1.l.a 1
112.j even 4 2 1792.1.c.a 1
112.l odd 4 2 1792.1.c.b 1
168.e odd 2 1 2016.1.l.a 1
168.i even 2 1 504.1.l.a 1
168.s odd 6 2 3528.1.bw.a 2
168.ba even 6 2 3528.1.bw.a 2
280.c odd 2 1 1400.1.m.a 1
280.s even 4 2 1400.1.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 1.a even 1 1 trivial
56.1.h.a 1 7.b odd 2 1 CM
56.1.h.a 1 8.b even 2 1 RM
56.1.h.a 1 56.h odd 2 1 CM
224.1.h.a 1 4.b odd 2 1
224.1.h.a 1 8.d odd 2 1
224.1.h.a 1 28.d even 2 1
224.1.h.a 1 56.e even 2 1
392.1.j.a 2 7.c even 3 2
392.1.j.a 2 7.d odd 6 2
392.1.j.a 2 56.j odd 6 2
392.1.j.a 2 56.p even 6 2
504.1.l.a 1 3.b odd 2 1
504.1.l.a 1 21.c even 2 1
504.1.l.a 1 24.h odd 2 1
504.1.l.a 1 168.i even 2 1
1400.1.c.a 2 5.c odd 4 2
1400.1.c.a 2 35.f even 4 2
1400.1.c.a 2 40.i odd 4 2
1400.1.c.a 2 280.s even 4 2
1400.1.m.a 1 5.b even 2 1
1400.1.m.a 1 35.c odd 2 1
1400.1.m.a 1 40.f even 2 1
1400.1.m.a 1 280.c odd 2 1
1568.1.n.a 2 28.f even 6 2
1568.1.n.a 2 28.g odd 6 2
1568.1.n.a 2 56.k odd 6 2
1568.1.n.a 2 56.m even 6 2
1792.1.c.a 1 16.f odd 4 2
1792.1.c.a 1 112.j even 4 2
1792.1.c.b 1 16.e even 4 2
1792.1.c.b 1 112.l odd 4 2
2016.1.l.a 1 12.b even 2 1
2016.1.l.a 1 24.f even 2 1
2016.1.l.a 1 84.h odd 2 1
2016.1.l.a 1 168.e odd 2 1
3528.1.bw.a 2 21.g even 6 2
3528.1.bw.a 2 21.h odd 6 2
3528.1.bw.a 2 168.s odd 6 2
3528.1.bw.a 2 168.ba even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(56, [\chi])$$.

## Hecke characteristic polynomials

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