# Properties

 Label 1045.1.k.d Level $1045$ Weight $1$ Character orbit 1045.k Analytic conductor $0.522$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ RM discriminant 209 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.26125.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + i q^{4} - i q^{5} + q^{8} - i q^{9} +O(q^{10})$$ q + (z + 1) * q^2 + z * q^4 - z * q^5 + q^8 - z * q^9 $$q + (i + 1) q^{2} + i q^{4} - i q^{5} + q^{8} - i q^{9} + ( - i + 1) q^{10} + q^{11} + (i - 1) q^{13} + q^{16} + ( - i + 1) q^{18} - i q^{19} + q^{20} + (i + 1) q^{22} + (i - 1) q^{23} - q^{25} - q^{26} + i q^{29} + (i + 1) q^{32} + q^{36} + ( - i + 1) q^{38} + i q^{44} - q^{45} - q^{46} + ( - i - 1) q^{47} + i q^{49} + ( - i - 1) q^{50} + ( - i - 1) q^{52} - i q^{55} + (2 i - 2) q^{58} + i q^{64} + (i + 1) q^{65} + q^{76} - i q^{80} - q^{81} + ( - i - 1) q^{90} + ( - i - 1) q^{92} + ( - 2 i + 1) q^{94} - q^{95} + (i - 1) q^{98} - i q^{99} +O(q^{100})$$ q + (z + 1) * q^2 + z * q^4 - z * q^5 + q^8 - z * q^9 + (-z + 1) * q^10 + q^11 + (z - 1) * q^13 + q^16 + (-z + 1) * q^18 - z * q^19 + q^20 + (z + 1) * q^22 + (z - 1) * q^23 - q^25 - q^26 + z * q^29 + (z + 1) * q^32 + q^36 + (-z + 1) * q^38 + z * q^44 - q^45 - q^46 + (-z - 1) * q^47 + z * q^49 + (-z - 1) * q^50 + (-z - 1) * q^52 - z * q^55 + (2*z - 2) * q^58 + z * q^64 + (z + 1) * q^65 + q^76 - z * q^80 - q^81 + (-z - 1) * q^90 + (-z - 1) * q^92 + (-2*z + 1) * q^94 - q^95 + (z - 1) * q^98 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2}+O(q^{10})$$ 2 * q + 2 * q^2 $$2 q + 2 q^{2} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 2 q^{16} + 2 q^{18} + 2 q^{20} + 2 q^{22} - 2 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{32} + 2 q^{36} + 2 q^{38} - 2 q^{45} - 4 q^{46} - 2 q^{47} - 2 q^{50} - 2 q^{52} - 4 q^{58} + 2 q^{65} + 2 q^{76} - 2 q^{81} - 2 q^{90} - 2 q^{92} - 2 q^{95} - 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 2 * q^16 + 2 * q^18 + 2 * q^20 + 2 * q^22 - 2 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^32 + 2 * q^36 + 2 * q^38 - 2 * q^45 - 4 * q^46 - 2 * q^47 - 2 * q^50 - 2 * q^52 - 4 * q^58 + 2 * q^65 + 2 * q^76 - 2 * q^81 - 2 * q^90 - 2 * q^92 - 2 * q^95 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$$.

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
208.1
 − 1.00000i 1.00000i
1.00000 1.00000i 0 1.00000i 1.00000i 0 0 0 1.00000i 1.00000 + 1.00000i
417.1 1.00000 + 1.00000i 0 1.00000i 1.00000i 0 0 0 1.00000i 1.00000 1.00000i
 $$n$$: 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
209.d even 2 1 RM by $$\Q(\sqrt{209})$$
5.c odd 4 1 inner
1045.k odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.k.d yes 2
5.c odd 4 1 inner 1045.1.k.d yes 2
11.b odd 2 1 1045.1.k.a 2
19.b odd 2 1 1045.1.k.a 2
55.e even 4 1 1045.1.k.a 2
95.g even 4 1 1045.1.k.a 2
209.d even 2 1 RM 1045.1.k.d yes 2
1045.k odd 4 1 inner 1045.1.k.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.k.a 2 11.b odd 2 1
1045.1.k.a 2 19.b odd 2 1
1045.1.k.a 2 55.e even 4 1
1045.1.k.a 2 95.g even 4 1
1045.1.k.d yes 2 1.a even 1 1 trivial
1045.1.k.d yes 2 5.c odd 4 1 inner
1045.1.k.d yes 2 209.d even 2 1 RM
1045.1.k.d yes 2 1045.k odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1045, [\chi])$$:

 $$T_{2}^{2} - 2T_{2} + 2$$ T2^2 - 2*T2 + 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 2T + 2$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$T^{2} + 2T + 2$$
$29$ $$T^{2} + 4$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 2T + 2$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$