# Properties

 Label 184.1.e.b Level $184$ Weight $1$ Character orbit 184.e Analytic conductor $0.092$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -23 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$184 = 2^{3} \cdot 23$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 184.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.0918279623245$$ 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 6.2.270848.1

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/184\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$93$$ $$97$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 0 1.50000 + 0.866025i 0 −1.00000 −2.00000 0
45.2 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 0 1.50000 0.866025i 0 −1.00000 −2.00000 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 $$\Q(\sqrt{-23})$$
8.b even 2 1 inner
184.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.1.e.b 2
3.b odd 2 1 1656.1.l.c 2
4.b odd 2 1 736.1.e.b 2
8.b even 2 1 inner 184.1.e.b 2
8.d odd 2 1 736.1.e.b 2
23.b odd 2 1 CM 184.1.e.b 2
24.h odd 2 1 1656.1.l.c 2
69.c even 2 1 1656.1.l.c 2
92.b even 2 1 736.1.e.b 2
184.e odd 2 1 inner 184.1.e.b 2
184.h even 2 1 736.1.e.b 2
552.b even 2 1 1656.1.l.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.1.e.b 2 1.a even 1 1 trivial
184.1.e.b 2 8.b even 2 1 inner
184.1.e.b 2 23.b odd 2 1 CM
184.1.e.b 2 184.e odd 2 1 inner
736.1.e.b 2 4.b odd 2 1
736.1.e.b 2 8.d odd 2 1
736.1.e.b 2 92.b even 2 1
736.1.e.b 2 184.h even 2 1
1656.1.l.c 2 3.b odd 2 1
1656.1.l.c 2 24.h odd 2 1
1656.1.l.c 2 69.c even 2 1
1656.1.l.c 2 552.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3$$ acting on $$S_{1}^{\mathrm{new}}(184, [\chi])$$.

## Hecke characteristic polynomials

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