# Properties

 Label 1045.1.w.d Level $1045$ Weight $1$ Character orbit 1045.w Analytic conductor $0.522$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -95 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.132135025.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{2} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9}+O(q^{10})$$ q + (-z^4 - z^2) * q^2 + (-z^2 + z) * q^3 + (z^4 - z^3 - z) * q^4 + z^2 * q^5 + (z^4 - z^3 - z + 1) * q^6 + (z^3 - z^2 + z + 1) * q^8 + (z^4 - z^3 + z^2) * q^9 $$q + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{2} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{3} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}) q^{4} + \zeta_{10}^{2} q^{5} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{6} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{8} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{9} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{10} - \zeta_{10}^{3} q^{11} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{12} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{13} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{15} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{16} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{18} + \zeta_{10}^{4} q^{19} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{20} + ( - \zeta_{10}^{2} - 1) q^{22} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} + 1) q^{24} + \zeta_{10}^{4} q^{25} + (\zeta_{10}^{4} - \zeta_{10}^{3} - 2 \zeta_{10}) q^{26} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10} + 1) q^{27} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{30} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{32} + ( - \zeta_{10}^{4} - 1) q^{33} + (\zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{36} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{37} + (\zeta_{10}^{3} + \zeta_{10}) q^{38} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10} + 1) q^{39} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{40} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10}) q^{44} + (\zeta_{10}^{4} - \zeta_{10} + 1) q^{45} + ( - 2 \zeta_{10}^{4} + 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{48} + \zeta_{10}^{2} q^{49} + (\zeta_{10}^{3} + \zeta_{10}) q^{50} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{52} + \zeta_{10}^{3} q^{53} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 2) q^{54} + q^{55} + (\zeta_{10} - 1) q^{57} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10} - 1) q^{60} + (\zeta_{10}^{4} + 1) q^{61} + (2 \zeta_{10}^{4} - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - \zeta_{10} - 1) q^{64} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{65} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{66} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{67} + ( - 3 \zeta_{10}^{4} + 2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{72} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{74} + (\zeta_{10} - 1) q^{75} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{76} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{78} + (\zeta_{10}^{4} - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{80} + (\zeta_{10}^{4} - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{81} + ( - \zeta_{10}^{4} + \zeta_{10}^{3} + \zeta_{10} - 1) q^{88} + ( - \zeta_{10}^{4} + 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{90} - \zeta_{10} q^{95} + (2 \zeta_{10}^{4} - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10} + 2) q^{96} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{97} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{98} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{99} +O(q^{100})$$ q + (-z^4 - z^2) * q^2 + (-z^2 + z) * q^3 + (z^4 - z^3 - z) * q^4 + z^2 * q^5 + (z^4 - z^3 - z + 1) * q^6 + (z^3 - z^2 + z + 1) * q^8 + (z^4 - z^3 + z^2) * q^9 + (-z^4 + z) * q^10 - z^3 * q^11 + (-z^4 + z^3 - z^2 + z - 2) * q^12 + (-z^4 - z^2) * q^13 + (-z^4 + z^3) * q^15 + (z^4 - z^3 + z^2 - z - 1) * q^16 + (-z^4 + z^3 - z^2 + 2*z - 1) * q^18 + z^4 * q^19 + (-z^3 - z + 1) * q^20 + (-z^2 - 1) * q^22 + (2*z^4 - 2*z^3 + 2*z^2 - z + 1) * q^24 + z^4 * q^25 + (z^4 - z^3 - 2*z) * q^26 + (-z^4 + z^3 + z + 1) * q^27 + (-z^3 + z^2 - z + 1) * q^30 + (-z^4 + z^3 - z^2 + z - 2) * q^32 + (-z^4 - 1) * q^33 + (z^4 - z^3 + 2*z^2 - 2*z + 2) * q^36 + (-z^4 + z^3) * q^37 + (z^3 + z) * q^38 + (z^4 - z^3 - z + 1) * q^39 + (-z^4 + z^3 - z^2 - 1) * q^40 + (z^4 + z^2 - z) * q^44 + (z^4 - z + 1) * q^45 + (-2*z^4 + 2*z^3 - 2*z^2 + z - 2) * q^48 + z^2 * q^49 + (z^3 + z) * q^50 + (z^3 - z^2 + z - 2) * q^52 + z^3 * q^53 + (z^4 - z^3 + z^2 - z + 2) * q^54 + q^55 + (z - 1) * q^57 + (-z^4 + z^3 - 2*z^2 + z - 1) * q^60 + (z^4 + 1) * q^61 + (2*z^4 - z^3 + 2*z^2 - z - 1) * q^64 + (-z^4 + z) * q^65 + (z^4 - z^3 + z^2 - z) * q^66 + (z^3 - z^2) * q^67 + (-3*z^4 + 2*z^3 - z^2 + 2*z - 3) * q^72 + (-z^3 + z^2 - z + 1) * q^74 + (z - 1) * q^75 + (-z^3 + z^2 + 1) * q^76 + (-z^4 + z^3 - z^2 + z - 2) * q^78 + (z^4 - z^3 + z^2 - z + 1) * q^80 + (z^4 - z^3 - z^2 + z - 1) * q^81 + (-z^4 + z^3 + z - 1) * q^88 + (-z^4 + 2*z^3 - z^2 + z - 1) * q^90 - z * q^95 + (2*z^4 - 2*z^3 + 3*z^2 - z + 2) * q^96 + (-z^4 - z^2) * q^97 + (-z^4 + z) * q^98 + (z^2 - z + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 3 q^{4} - q^{5} + q^{6} - q^{8} - 3 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 - 3 * q^4 - q^5 + q^6 - q^8 - 3 * q^9 $$4 q + 2 q^{2} + 2 q^{3} - 3 q^{4} - q^{5} + q^{6} - q^{8} - 3 q^{9} + 2 q^{10} - q^{11} - 4 q^{12} + 2 q^{13} + 2 q^{15} + q^{18} - q^{19} + 2 q^{20} - 3 q^{22} - 3 q^{24} - q^{25} - 4 q^{26} - q^{27} + q^{30} - 4 q^{32} - 3 q^{33} + q^{36} + 2 q^{37} + 2 q^{38} + q^{39} - q^{40} - 3 q^{44} + 2 q^{45} - q^{49} + 2 q^{50} - 4 q^{52} + 2 q^{53} + 2 q^{54} + 4 q^{55} - 3 q^{57} + q^{60} + 3 q^{61} - 2 q^{64} + 2 q^{65} - 4 q^{66} + 2 q^{67} - 3 q^{72} + q^{74} - 3 q^{75} + 2 q^{76} - 2 q^{78} - q^{88} + q^{90} - q^{95} - 2 q^{96} + 2 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 - 3 * q^4 - q^5 + q^6 - q^8 - 3 * q^9 + 2 * q^10 - q^11 - 4 * q^12 + 2 * q^13 + 2 * q^15 + q^18 - q^19 + 2 * q^20 - 3 * q^22 - 3 * q^24 - q^25 - 4 * q^26 - q^27 + q^30 - 4 * q^32 - 3 * q^33 + q^36 + 2 * q^37 + 2 * q^38 + q^39 - q^40 - 3 * q^44 + 2 * q^45 - q^49 + 2 * q^50 - 4 * q^52 + 2 * q^53 + 2 * q^54 + 4 * q^55 - 3 * q^57 + q^60 + 3 * q^61 - 2 * q^64 + 2 * q^65 - 4 * q^66 + 2 * q^67 - 3 * q^72 + q^74 - 3 * q^75 + 2 * q^76 - 2 * q^78 - q^88 + q^90 - q^95 - 2 * q^96 + 2 * q^97 + 2 * q^98 + 2 * 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
0.500000 1.53884i 0.500000 0.363271i −1.30902 0.951057i 0.309017 + 0.951057i −0.309017 0.951057i 0 −0.809017 + 0.587785i −0.190983 + 0.587785i 1.61803
379.1 0.500000 + 1.53884i 0.500000 + 0.363271i −1.30902 + 0.951057i 0.309017 0.951057i −0.309017 + 0.951057i 0 −0.809017 0.587785i −0.190983 0.587785i 1.61803
664.1 0.500000 + 0.363271i 0.500000 1.53884i −0.190983 0.587785i −0.809017 + 0.587785i 0.809017 0.587785i 0 0.309017 0.951057i −1.30902 0.951057i −0.618034
949.1 0.500000 0.363271i 0.500000 + 1.53884i −0.190983 + 0.587785i −0.809017 0.587785i 0.809017 + 0.587785i 0 0.309017 + 0.951057i −1.30902 + 0.951057i −0.618034
 $$n$$: 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
11.c even 5 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.d yes 4
5.b even 2 1 1045.1.w.a 4
11.c even 5 1 inner 1045.1.w.d yes 4
19.b odd 2 1 1045.1.w.a 4
55.j even 10 1 1045.1.w.a 4
95.d odd 2 1 CM 1045.1.w.d yes 4
209.m odd 10 1 1045.1.w.a 4
1045.w odd 10 1 inner 1045.1.w.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.a 4 5.b even 2 1
1045.1.w.a 4 19.b odd 2 1
1045.1.w.a 4 55.j even 10 1
1045.1.w.a 4 209.m odd 10 1
1045.1.w.d yes 4 1.a even 1 1 trivial
1045.1.w.d yes 4 11.c even 5 1 inner
1045.1.w.d yes 4 95.d odd 2 1 CM
1045.1.w.d 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 $$S_{1}^{\mathrm{new}}(1045, [\chi])$$:

 $$T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1$$ T2^4 - 2*T2^3 + 4*T2^2 - 3*T2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$3$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$5$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$13$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$17$ $$T^{4}$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1$$
$67$ $$(T^{2} - T - 1)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$