# Properties

 Label 183.1.d.b Level $183$ Weight $1$ Character orbit 183.d Self dual yes Analytic conductor $0.091$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -183 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$183 = 3 \cdot 61$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 183.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0913288973118$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.11163.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.18385461.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} - q^{3} + q^{4} + \beta q^{6} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} - q^{3} + q^{4} + \beta q^{6} + q^{9} + \beta q^{11} - q^{12} - q^{16} + \beta q^{17} -\beta q^{18} -2 q^{22} -\beta q^{23} + q^{25} - q^{27} -\beta q^{29} + \beta q^{32} -\beta q^{33} -2 q^{34} + q^{36} + \beta q^{44} + 2 q^{46} + q^{48} + q^{49} -\beta q^{50} -\beta q^{51} + \beta q^{53} + \beta q^{54} + 2 q^{58} -\beta q^{59} - q^{61} - q^{64} + 2 q^{66} + \beta q^{68} + \beta q^{69} + \beta q^{71} -2 q^{73} - q^{75} + q^{81} + \beta q^{87} -\beta q^{89} -\beta q^{92} -\beta q^{96} -\beta q^{98} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{4} + 2q^{9} - 2q^{12} - 2q^{16} - 4q^{22} + 2q^{25} - 2q^{27} - 4q^{34} + 2q^{36} + 4q^{46} + 2q^{48} + 2q^{49} + 4q^{58} - 2q^{61} - 2q^{64} + 4q^{66} - 4q^{73} - 2q^{75} + 2q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/183\mathbb{Z}\right)^\times$$.

 $$n$$ $$62$$ $$124$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
182.1
 1.41421 −1.41421
−1.41421 −1.00000 1.00000 0 1.41421 0 0 1.00000 0
182.2 1.41421 −1.00000 1.00000 0 −1.41421 0 0 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
183.d odd 2 1 CM by $$\Q(\sqrt{-183})$$
3.b odd 2 1 inner
61.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 183.1.d.b 2
3.b odd 2 1 inner 183.1.d.b 2
4.b odd 2 1 2928.1.e.b 2
12.b even 2 1 2928.1.e.b 2
61.b even 2 1 inner 183.1.d.b 2
183.d odd 2 1 CM 183.1.d.b 2
244.c odd 2 1 2928.1.e.b 2
732.e even 2 1 2928.1.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
183.1.d.b 2 1.a even 1 1 trivial
183.1.d.b 2 3.b odd 2 1 inner
183.1.d.b 2 61.b even 2 1 inner
183.1.d.b 2 183.d odd 2 1 CM
2928.1.e.b 2 4.b odd 2 1
2928.1.e.b 2 12.b even 2 1
2928.1.e.b 2 244.c odd 2 1
2928.1.e.b 2 732.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(183, [\chi])$$.

## Hecke characteristic polynomials

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