# Properties

 Label 552.1.h.c Level $552$ Weight $1$ Character orbit 552.h Analytic conductor $0.275$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -23 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: no Analytic conductor: $$0.275483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.7312896.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6}^{2} q^{12} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} -\zeta_{6} q^{18} - q^{23} + \zeta_{6} q^{24} + q^{25} + ( -1 - \zeta_{6} ) q^{26} - q^{27} + q^{29} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} + q^{36} + ( -1 + \zeta_{6}^{2} ) q^{39} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{41} -\zeta_{6}^{2} q^{46} + q^{47} - q^{48} - q^{49} + \zeta_{6}^{2} q^{50} + ( 1 - \zeta_{6}^{2} ) q^{52} -\zeta_{6}^{2} q^{54} + \zeta_{6}^{2} q^{58} + ( 1 + \zeta_{6} ) q^{62} + q^{64} -\zeta_{6} q^{69} - q^{71} + \zeta_{6}^{2} q^{72} + q^{73} + \zeta_{6} q^{75} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{78} -\zeta_{6} q^{81} + ( 1 + \zeta_{6} ) q^{82} + \zeta_{6} q^{87} + \zeta_{6} q^{92} + ( 1 - \zeta_{6}^{2} ) q^{93} + \zeta_{6}^{2} q^{94} -\zeta_{6}^{2} q^{96} -\zeta_{6}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} - 2q^{6} + 2q^{8} - q^{9} + q^{12} - q^{16} - q^{18} - 2q^{23} + q^{24} + 2q^{25} - 3q^{26} - 2q^{27} + 2q^{29} - q^{32} + 2q^{36} - 3q^{39} + q^{46} + 2q^{47} - 2q^{48} - 2q^{49} - q^{50} + 3q^{52} + q^{54} - q^{58} + 3q^{62} + 2q^{64} - q^{69} - 2q^{71} - q^{72} + 2q^{73} + q^{75} - q^{81} + 3q^{82} + q^{87} + q^{92} + 3q^{93} - q^{94} + q^{96} + q^{98} + O(q^{100})$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 0 1.00000 −0.500000 0.866025i 0
275.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 0 1.00000 −0.500000 + 0.866025i 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 inner
552.h odd 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{13}^{2} + 3$$ $$T_{29} - 1$$

## Hecke characteristic polynomials

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