# Properties

 Label 464.1.h.b Level $464$ Weight $1$ Character orbit 464.h Self dual yes Analytic conductor $0.232$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -116 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 464.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.231566165862$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.53824.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + 2 q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + 2 * q^9 $$q - \beta q^{3} - q^{5} + 2 q^{9} + \beta q^{11} + q^{13} + \beta q^{15} - \beta q^{27} - q^{29} + \beta q^{31} - 3 q^{33} - \beta q^{39} + \beta q^{43} - 2 q^{45} - \beta q^{47} + q^{49} + q^{53} - \beta q^{55} - q^{65} - \beta q^{79} + q^{81} + \beta q^{87} - 3 q^{93} + 2 \beta q^{99} +O(q^{100})$$ q - b * q^3 - q^5 + 2 * q^9 + b * q^11 + q^13 + b * q^15 - b * q^27 - q^29 + b * q^31 - 3 * q^33 - b * q^39 + b * q^43 - 2 * q^45 - b * q^47 + q^49 + q^53 - b * q^55 - q^65 - b * q^79 + q^81 + b * q^87 - 3 * q^93 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 4 * q^9 $$2 q - 2 q^{5} + 4 q^{9} + 2 q^{13} - 2 q^{29} - 6 q^{33} - 4 q^{45} + 2 q^{49} + 2 q^{53} - 2 q^{65} + 2 q^{81} - 6 q^{93}+O(q^{100})$$ 2 * q - 2 * q^5 + 4 * q^9 + 2 * q^13 - 2 * q^29 - 6 * q^33 - 4 * q^45 + 2 * q^49 + 2 * q^53 - 2 * q^65 + 2 * q^81 - 6 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$$.

 $$n$$ $$117$$ $$175$$ $$321$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
463.1
 1.73205 −1.73205
0 −1.73205 0 −1.00000 0 0 0 2.00000 0
463.2 0 1.73205 0 −1.00000 0 0 0 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
116.d odd 2 1 CM by $$\Q(\sqrt{-29})$$
4.b odd 2 1 inner
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.1.h.b 2
4.b odd 2 1 inner 464.1.h.b 2
8.b even 2 1 1856.1.h.d 2
8.d odd 2 1 1856.1.h.d 2
29.b even 2 1 inner 464.1.h.b 2
116.d odd 2 1 CM 464.1.h.b 2
232.b odd 2 1 1856.1.h.d 2
232.g even 2 1 1856.1.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.1.h.b 2 1.a even 1 1 trivial
464.1.h.b 2 4.b odd 2 1 inner
464.1.h.b 2 29.b even 2 1 inner
464.1.h.b 2 116.d odd 2 1 CM
1856.1.h.d 2 8.b even 2 1
1856.1.h.d 2 8.d odd 2 1
1856.1.h.d 2 232.b odd 2 1
1856.1.h.d 2 232.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(464, [\chi])$$.

## Hecke characteristic polynomials

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