# Properties

 Label 1805.2.b.d Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-19})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + (\beta - 1) q^{5} + (2 \beta - 1) q^{7} + 3 q^{9}+O(q^{10})$$ q + 2 * q^4 + (b - 1) * q^5 + (2*b - 1) * q^7 + 3 * q^9 $$q + 2 q^{4} + (\beta - 1) q^{5} + (2 \beta - 1) q^{7} + 3 q^{9} + 5 q^{11} + 4 q^{16} + (2 \beta - 1) q^{17} + (2 \beta - 2) q^{20} + ( - 4 \beta + 2) q^{23} + ( - \beta - 4) q^{25} + (4 \beta - 2) q^{28} + ( - \beta - 9) q^{35} + 6 q^{36} + ( - 6 \beta + 3) q^{43} + 10 q^{44} + (3 \beta - 3) q^{45} + (2 \beta - 1) q^{47} - 12 q^{49} + (5 \beta - 5) q^{55} - 15 q^{61} + (6 \beta - 3) q^{63} + 8 q^{64} + (4 \beta - 2) q^{68} + ( - 6 \beta + 3) q^{73} + (10 \beta - 5) q^{77} + (4 \beta - 4) q^{80} + 9 q^{81} + (4 \beta - 2) q^{83} + ( - \beta - 9) q^{85} + ( - 8 \beta + 4) q^{92} + 15 q^{99}+O(q^{100})$$ q + 2 * q^4 + (b - 1) * q^5 + (2*b - 1) * q^7 + 3 * q^9 + 5 * q^11 + 4 * q^16 + (2*b - 1) * q^17 + (2*b - 2) * q^20 + (-4*b + 2) * q^23 + (-b - 4) * q^25 + (4*b - 2) * q^28 + (-b - 9) * q^35 + 6 * q^36 + (-6*b + 3) * q^43 + 10 * q^44 + (3*b - 3) * q^45 + (2*b - 1) * q^47 - 12 * q^49 + (5*b - 5) * q^55 - 15 * q^61 + (6*b - 3) * q^63 + 8 * q^64 + (4*b - 2) * q^68 + (-6*b + 3) * q^73 + (10*b - 5) * q^77 + (4*b - 4) * q^80 + 9 * q^81 + (4*b - 2) * q^83 + (-b - 9) * q^85 + (-8*b + 4) * q^92 + 15 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - q^5 + 6 * q^9 $$2 q + 4 q^{4} - q^{5} + 6 q^{9} + 10 q^{11} + 8 q^{16} - 2 q^{20} - 9 q^{25} - 19 q^{35} + 12 q^{36} + 20 q^{44} - 3 q^{45} - 24 q^{49} - 5 q^{55} - 30 q^{61} + 16 q^{64} - 4 q^{80} + 18 q^{81} - 19 q^{85} + 30 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - q^5 + 6 * q^9 + 10 * q^11 + 8 * q^16 - 2 * q^20 - 9 * q^25 - 19 * q^35 + 12 * q^36 + 20 * q^44 - 3 * q^45 - 24 * q^49 - 5 * q^55 - 30 * q^61 + 16 * q^64 - 4 * q^80 + 18 * q^81 - 19 * q^85 + 30 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1084.1
 0.5 − 2.17945i 0.5 + 2.17945i
0 0 2.00000 −0.500000 2.17945i 0 4.35890i 0 3.00000 0
1084.2 0 0 2.00000 −0.500000 + 2.17945i 0 4.35890i 0 3.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.d 2
5.b even 2 1 inner 1805.2.b.d 2
5.c odd 4 2 9025.2.a.p 2
19.b odd 2 1 CM 1805.2.b.d 2
95.d odd 2 1 inner 1805.2.b.d 2
95.g even 4 2 9025.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.d 2 1.a even 1 1 trivial
1805.2.b.d 2 5.b even 2 1 inner
1805.2.b.d 2 19.b odd 2 1 CM
1805.2.b.d 2 95.d odd 2 1 inner
9025.2.a.p 2 5.c odd 4 2
9025.2.a.p 2 95.g even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1805, [\chi])$$:

 $$T_{2}$$ T2 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 5$$
$7$ $$T^{2} + 19$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 19$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 76$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 171$$
$47$ $$T^{2} + 19$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 15)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 171$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 76$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$