# Properties

 Label 471.1.d.b Level $471$ Weight $1$ Character orbit 471.d Self dual yes Analytic conductor $0.235$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -471 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 471.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.235059620950$$ 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 4.2.73947.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.313461333.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + q^{3} + q^{4} + \beta q^{5} -\beta q^{6} + q^{9} +O(q^{10})$$ $$q -\beta q^{2} + q^{3} + q^{4} + \beta q^{5} -\beta q^{6} + q^{9} -2 q^{10} + q^{12} -2 q^{13} + \beta q^{15} - q^{16} -\beta q^{18} + \beta q^{20} -\beta q^{23} + q^{25} + 2 \beta q^{26} + q^{27} -\beta q^{29} -2 q^{30} + \beta q^{32} + q^{36} -2 q^{39} + \beta q^{41} + \beta q^{45} + 2 q^{46} - q^{48} + q^{49} -\beta q^{50} -2 q^{52} + \beta q^{53} -\beta q^{54} + 2 q^{58} -\beta q^{59} + \beta q^{60} - q^{64} -2 \beta q^{65} -2 q^{67} -\beta q^{69} + q^{75} + 2 \beta q^{78} -\beta q^{80} + q^{81} -2 q^{82} -\beta q^{83} -\beta q^{87} -2 q^{90} -\beta q^{92} + \beta q^{96} -\beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} - 4 q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{27} - 4 q^{30} + 2 q^{36} - 4 q^{39} + 4 q^{46} - 2 q^{48} + 2 q^{49} - 4 q^{52} + 4 q^{58} - 2 q^{64} - 4 q^{67} + 2 q^{75} + 2 q^{81} - 4 q^{82} - 4 q^{90} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$$.

 $$n$$ $$158$$ $$319$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
470.1
 1.41421 −1.41421
−1.41421 1.00000 1.00000 1.41421 −1.41421 0 0 1.00000 −2.00000
470.2 1.41421 1.00000 1.00000 −1.41421 1.41421 0 0 1.00000 −2.00000
 $$n$$: 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
471.d odd 2 1 CM by $$\Q(\sqrt{-471})$$
3.b odd 2 1 inner
157.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.1.d.b 2
3.b odd 2 1 inner 471.1.d.b 2
157.b even 2 1 inner 471.1.d.b 2
471.d odd 2 1 CM 471.1.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.1.d.b 2 1.a even 1 1 trivial
471.1.d.b 2 3.b odd 2 1 inner
471.1.d.b 2 157.b even 2 1 inner
471.1.d.b 2 471.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(471, [\chi])$$.

## Hecke characteristic polynomials

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