# Properties

 Label 1045.1.w.c Level $1045$ Weight $1$ Character orbit 1045.w Analytic conductor $0.522$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -19 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1045.w (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.87298324158753125.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{10} q^{4} - \zeta_{10}^{3} q^{5} + ( - \zeta_{10}^{2} + 1) q^{7} + \zeta_{10}^{3} q^{9} +O(q^{10})$$ q + z * q^4 - z^3 * q^5 + (-z^2 + 1) * q^7 + z^3 * q^9 $$q + \zeta_{10} q^{4} - \zeta_{10}^{3} q^{5} + ( - \zeta_{10}^{2} + 1) q^{7} + \zeta_{10}^{3} q^{9} - \zeta_{10}^{4} q^{11} + \zeta_{10}^{2} q^{16} + (\zeta_{10}^{4} - 1) q^{17} + \zeta_{10}^{4} q^{19} - \zeta_{10}^{4} q^{20} + ( - \zeta_{10}^{3} - \zeta_{10}^{2}) q^{23} - \zeta_{10} q^{25} + ( - \zeta_{10}^{3} + \zeta_{10}) q^{28} + ( - \zeta_{10}^{3} - 1) q^{35} + \zeta_{10}^{4} q^{36} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{43} + q^{44} + \zeta_{10} q^{45} + (\zeta_{10}^{2} + \zeta_{10}) q^{47} + (\zeta_{10}^{4} - \zeta_{10}^{2} + 1) q^{49} - \zeta_{10}^{2} q^{55} + ( - \zeta_{10}^{4} - 1) q^{61} + (\zeta_{10}^{3} + 1) q^{63} + \zeta_{10}^{3} q^{64} + ( - \zeta_{10} - 1) q^{68} - q^{76} + ( - \zeta_{10}^{4} - \zeta_{10}) q^{77} + q^{80} - \zeta_{10} q^{81} + (\zeta_{10}^{3} - \zeta_{10}) q^{83} + (\zeta_{10}^{3} + \zeta_{10}^{2}) q^{85} + ( - \zeta_{10}^{4} - \zeta_{10}^{3}) q^{92} + \zeta_{10}^{2} q^{95} + \zeta_{10}^{2} q^{99} +O(q^{100})$$ q + z * q^4 - z^3 * q^5 + (-z^2 + 1) * q^7 + z^3 * q^9 - z^4 * q^11 + z^2 * q^16 + (z^4 - 1) * q^17 + z^4 * q^19 - z^4 * q^20 + (-z^3 - z^2) * q^23 - z * q^25 + (-z^3 + z) * q^28 + (-z^3 - 1) * q^35 + z^4 * q^36 + (z^3 + z^2) * q^43 + q^44 + z * q^45 + (z^2 + z) * q^47 + (z^4 - z^2 + 1) * q^49 - z^2 * q^55 + (-z^4 - 1) * q^61 + (z^3 + 1) * q^63 + z^3 * q^64 + (-z - 1) * q^68 - q^76 + (-z^4 - z) * q^77 + q^80 - z * q^81 + (z^3 - z) * q^83 + (z^3 + z^2) * q^85 + (-z^4 - z^3) * q^92 + z^2 * q^95 + z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{4} - q^{5} + 5 q^{7} + q^{9}+O(q^{10})$$ 4 * q + q^4 - q^5 + 5 * q^7 + q^9 $$4 q + q^{4} - q^{5} + 5 q^{7} + q^{9} + q^{11} - q^{16} - 5 q^{17} - q^{19} + q^{20} - q^{25} - 5 q^{35} - q^{36} + 4 q^{44} + q^{45} + 4 q^{49} + q^{55} - 3 q^{61} + 5 q^{63} + q^{64} - 5 q^{68} - 4 q^{76} + 4 q^{80} - q^{81} - q^{95} - q^{99}+O(q^{100})$$ 4 * q + q^4 - q^5 + 5 * q^7 + q^9 + q^11 - q^16 - 5 * q^17 - q^19 + q^20 - q^25 - 5 * q^35 - q^36 + 4 * q^44 + q^45 + 4 * q^49 + q^55 - 3 * q^61 + 5 * q^63 + q^64 - 5 * q^68 - 4 * q^76 + 4 * q^80 - q^81 - q^95 - q^99

## Character values

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

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
284.1
 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i
0 0 0.809017 + 0.587785i 0.309017 0.951057i 0 0.690983 0.951057i 0 −0.309017 + 0.951057i 0
379.1 0 0 0.809017 0.587785i 0.309017 + 0.951057i 0 0.690983 + 0.951057i 0 −0.309017 0.951057i 0
664.1 0 0 −0.309017 0.951057i −0.809017 0.587785i 0 1.80902 0.587785i 0 0.809017 + 0.587785i 0
949.1 0 0 −0.309017 + 0.951057i −0.809017 + 0.587785i 0 1.80902 + 0.587785i 0 0.809017 0.587785i 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 $$\Q(\sqrt{-19})$$
55.j even 10 1 inner
1045.w odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.1.w.c yes 4
5.b even 2 1 1045.1.w.b 4
11.c even 5 1 1045.1.w.b 4
19.b odd 2 1 CM 1045.1.w.c yes 4
55.j even 10 1 inner 1045.1.w.c yes 4
95.d odd 2 1 1045.1.w.b 4
209.m odd 10 1 1045.1.w.b 4
1045.w odd 10 1 inner 1045.1.w.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.b 4 5.b even 2 1
1045.1.w.b 4 11.c even 5 1
1045.1.w.b 4 95.d odd 2 1
1045.1.w.b 4 209.m odd 10 1
1045.1.w.c yes 4 1.a even 1 1 trivial
1045.1.w.c yes 4 19.b odd 2 1 CM
1045.1.w.c yes 4 55.j even 10 1 inner
1045.1.w.c yes 4 1045.w odd 10 1 inner

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{7}^{4} - 5T_{7}^{3} + 10T_{7}^{2} - 10T_{7} + 5$$ T7^4 - 5*T7^3 + 10*T7^2 - 10*T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4} - 5 T^{3} + 10 T^{2} - 10 T + 5$$
$11$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$T^{4} + 5T^{2} + 5$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4} + 5T^{2} + 5$$
$47$ $$T^{4} + 5T + 5$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 5T + 5$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$