# Properties

 Label 552.1.h.a Level $552$ Weight $1$ Character orbit 552.h Self dual yes Analytic conductor $0.275$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -23, -552, 24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$552 = 2^{3} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 552.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.275483886973$$ 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 $$\Q(\sqrt{6}, \sqrt{-23})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.12696.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{12} + q^{16} - q^{18} + q^{23} + q^{24} + q^{25} - q^{27} + 2q^{29} - q^{32} + q^{36} - q^{46} + 2q^{47} - q^{48} - q^{49} - q^{50} + q^{54} - 2q^{58} + q^{64} - q^{69} - 2q^{71} - q^{72} - 2q^{73} - q^{75} + q^{81} - 2q^{87} + q^{92} - 2q^{94} + q^{96} + q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$185$$ $$277$$ $$415$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
275.1
 0
−1.00000 −1.00000 1.00000 0 1.00000 0 −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
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
24.f even 2 1 RM by $$\Q(\sqrt{6})$$
552.h odd 2 1 CM by $$\Q(\sqrt{-138})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 552.1.h.a 1
3.b odd 2 1 552.1.h.b yes 1
4.b odd 2 1 2208.1.h.a 1
8.b even 2 1 2208.1.h.b 1
8.d odd 2 1 552.1.h.b yes 1
12.b even 2 1 2208.1.h.b 1
23.b odd 2 1 CM 552.1.h.a 1
24.f even 2 1 RM 552.1.h.a 1
24.h odd 2 1 2208.1.h.a 1
69.c even 2 1 552.1.h.b yes 1
92.b even 2 1 2208.1.h.a 1
184.e odd 2 1 2208.1.h.b 1
184.h even 2 1 552.1.h.b yes 1
276.h odd 2 1 2208.1.h.b 1
552.b even 2 1 2208.1.h.a 1
552.h odd 2 1 CM 552.1.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.1.h.a 1 1.a even 1 1 trivial
552.1.h.a 1 23.b odd 2 1 CM
552.1.h.a 1 24.f even 2 1 RM
552.1.h.a 1 552.h odd 2 1 CM
552.1.h.b yes 1 3.b odd 2 1
552.1.h.b yes 1 8.d odd 2 1
552.1.h.b yes 1 69.c even 2 1
552.1.h.b yes 1 184.h even 2 1
2208.1.h.a 1 4.b odd 2 1
2208.1.h.a 1 24.h odd 2 1
2208.1.h.a 1 92.b even 2 1
2208.1.h.a 1 552.b even 2 1
2208.1.h.b 1 8.b even 2 1
2208.1.h.b 1 12.b even 2 1
2208.1.h.b 1 184.e odd 2 1
2208.1.h.b 1 276.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(552, [\chi])$$:

 $$T_{13}$$ $$T_{29} - 2$$

## Hecke characteristic polynomials

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