# Properties

 Label 944.1.h.a Level $944$ Weight $1$ Character orbit 944.h Self dual yes Analytic conductor $0.471$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -59 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$944 = 2^{4} \cdot 59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 944.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.471117371926$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 59) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.59.1 Artin image $D_6$ Artin field Galois closure of 6.0.222784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} + O(q^{10})$$ $$q + q^{3} - q^{5} + q^{7} - q^{15} + 2q^{17} + q^{19} + q^{21} - q^{27} - q^{29} - q^{35} - q^{41} + 2q^{51} - q^{53} + q^{57} - q^{59} - 2q^{71} + q^{79} - q^{81} - 2q^{85} - q^{87} - q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/944\mathbb{Z}\right)^\times$$.

 $$n$$ $$591$$ $$709$$ $$769$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1
 0
0 1.00000 0 −1.00000 0 1.00000 0 0 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
59.b odd 2 1 CM by $$\Q(\sqrt{-59})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 944.1.h.a 1
4.b odd 2 1 59.1.b.a 1
8.b even 2 1 3776.1.h.a 1
8.d odd 2 1 3776.1.h.b 1
12.b even 2 1 531.1.c.a 1
20.d odd 2 1 1475.1.c.b 1
20.e even 4 2 1475.1.d.a 2
28.d even 2 1 2891.1.c.e 1
28.f even 6 2 2891.1.g.b 2
28.g odd 6 2 2891.1.g.d 2
59.b odd 2 1 CM 944.1.h.a 1
236.c even 2 1 59.1.b.a 1
236.g even 58 28 3481.1.d.a 28
236.h odd 58 28 3481.1.d.a 28
472.c odd 2 1 3776.1.h.a 1
472.f even 2 1 3776.1.h.b 1
708.b odd 2 1 531.1.c.a 1
1180.h even 2 1 1475.1.c.b 1
1180.l odd 4 2 1475.1.d.a 2
1652.g odd 2 1 2891.1.c.e 1
1652.k odd 6 2 2891.1.g.b 2
1652.m even 6 2 2891.1.g.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 4.b odd 2 1
59.1.b.a 1 236.c even 2 1
531.1.c.a 1 12.b even 2 1
531.1.c.a 1 708.b odd 2 1
944.1.h.a 1 1.a even 1 1 trivial
944.1.h.a 1 59.b odd 2 1 CM
1475.1.c.b 1 20.d odd 2 1
1475.1.c.b 1 1180.h even 2 1
1475.1.d.a 2 20.e even 4 2
1475.1.d.a 2 1180.l odd 4 2
2891.1.c.e 1 28.d even 2 1
2891.1.c.e 1 1652.g odd 2 1
2891.1.g.b 2 28.f even 6 2
2891.1.g.b 2 1652.k odd 6 2
2891.1.g.d 2 28.g odd 6 2
2891.1.g.d 2 1652.m even 6 2
3481.1.d.a 28 236.g even 58 28
3481.1.d.a 28 236.h odd 58 28
3776.1.h.a 1 8.b even 2 1
3776.1.h.a 1 472.c odd 2 1
3776.1.h.b 1 8.d odd 2 1
3776.1.h.b 1 472.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{1}^{\mathrm{new}}(944, [\chi])$$.

## Hecke characteristic polynomials

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