# Properties

 Label 1441.1.u.a Level $1441$ Weight $1$ Character orbit 1441.u Analytic conductor $0.719$ Analytic rank $0$ Dimension $20$ Projective image $D_{25}$ CM discriminant -131 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1441 = 11 \cdot 131$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1441.u (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.719152683204$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{50})$$ Defining polynomial: $$x^{20} - x^{15} + x^{10} - x^{5} + 1$$ x^20 - x^15 + x^10 - x^5 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{25}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{25} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} - \zeta_{50}^{17} - \zeta_{50}^{5}) q^{9}+O(q^{10})$$ q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9 $$q + ( - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{3} + \zeta_{50}^{10} q^{4} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{5} + ( - \zeta_{50}^{23} + \zeta_{50}^{22}) q^{7} + (\zeta_{50}^{18} - \zeta_{50}^{17} - \zeta_{50}^{5}) q^{9} + \zeta_{50}^{4} q^{11} + ( - \zeta_{50}^{19} + \zeta_{50}^{6}) q^{12} + ( - \zeta_{50}^{21} + \zeta_{50}^{14}) q^{13} + (\zeta_{50}^{24} - \zeta_{50}^{21} + \zeta_{50}^{12} + \zeta_{50}^{8}) q^{15} + \zeta_{50}^{20} q^{16} + (\zeta_{50}^{22} - \zeta_{50}^{13}) q^{20} + ( - \zeta_{50}^{19} + \zeta_{50}^{18} - \zeta_{50}^{7} + \zeta_{50}^{6}) q^{21} + (\zeta_{50}^{24} - \zeta_{50}^{15} + \zeta_{50}^{6}) q^{25} + (\zeta_{50}^{14} - \zeta_{50}^{13} + \zeta_{50}^{2} + \zeta_{50}) q^{27} + (\zeta_{50}^{8} - \zeta_{50}^{7}) q^{28} + ( - \zeta_{50}^{13} + 1) q^{33} + (\zeta_{50}^{10} - \zeta_{50}^{9} - \zeta_{50} + 1) q^{35} + ( - \zeta_{50}^{15} - \zeta_{50}^{3} + \zeta_{50}^{2}) q^{36} + ( - \zeta_{50}^{23} - \zeta_{50}^{17} + \zeta_{50}^{10} - \zeta_{50}^{5}) q^{39} + (\zeta_{50}^{4} - \zeta_{50}) q^{41} + ( - \zeta_{50}^{23} + \zeta_{50}^{2}) q^{43} + \zeta_{50}^{14} q^{44} + ( - \zeta_{50}^{21} + \zeta_{50}^{20} - \zeta_{50}^{17} + \zeta_{50}^{8} - \zeta_{50}^{5} + \zeta_{50}^{4}) q^{45} + (\zeta_{50}^{16} + \zeta_{50}^{4}) q^{48} + ( - \zeta_{50}^{21} + \zeta_{50}^{20} - \zeta_{50}^{19}) q^{49} + (\zeta_{50}^{24} + \zeta_{50}^{6}) q^{52} + (\zeta_{50}^{20} - \zeta_{50}^{15}) q^{53} + (\zeta_{50}^{16} - \zeta_{50}^{7}) q^{55} + ( - \zeta_{50}^{13} - \zeta_{50}^{7}) q^{59} + (\zeta_{50}^{22} + \zeta_{50}^{18} - \zeta_{50}^{9} + \zeta_{50}^{6}) q^{60} + (\zeta_{50}^{12} - \zeta_{50}^{3}) q^{61} + (\zeta_{50}^{16} - 2 \zeta_{50}^{15} + \zeta_{50}^{14} - \zeta_{50}^{3} + \zeta_{50}^{2}) q^{63} - \zeta_{50}^{5} q^{64} + (\zeta_{50}^{24} - \zeta_{50}^{17} + \zeta_{50}^{8} - \zeta_{50}) q^{65} + (\zeta_{50}^{24} + \zeta_{50}^{20} - \zeta_{50}^{15} - \zeta_{50}^{11} + \zeta_{50}^{8} + \zeta_{50}^{2}) q^{75} + (\zeta_{50}^{2} - \zeta_{50}) q^{77} + ( - \zeta_{50}^{23} - \zeta_{50}^{7}) q^{80} + ( - \zeta_{50}^{23} + \zeta_{50}^{22} - \zeta_{50}^{11} + \zeta_{50}^{10} - \zeta_{50}^{9}) q^{81} + ( - \zeta_{50}^{17} + \zeta_{50}^{16} + \zeta_{50}^{4} - \zeta_{50}^{3}) q^{84} + (\zeta_{50}^{20} - \zeta_{50}^{5}) q^{89} + ( - \zeta_{50}^{19} + \zeta_{50}^{18} + \zeta_{50}^{12} - \zeta_{50}^{11}) q^{91} + (\zeta_{50}^{22} - \zeta_{50}^{21} - \zeta_{50}^{9}) q^{99} +O(q^{100})$$ q + (-z^21 - z^9) * q^3 + z^10 * q^4 + (z^12 - z^3) * q^5 + (-z^23 + z^22) * q^7 + (z^18 - z^17 - z^5) * q^9 + z^4 * q^11 + (-z^19 + z^6) * q^12 + (-z^21 + z^14) * q^13 + (z^24 - z^21 + z^12 + z^8) * q^15 + z^20 * q^16 + (z^22 - z^13) * q^20 + (-z^19 + z^18 - z^7 + z^6) * q^21 + (z^24 - z^15 + z^6) * q^25 + (z^14 - z^13 + z^2 + z) * q^27 + (z^8 - z^7) * q^28 + (-z^13 + 1) * q^33 + (z^10 - z^9 - z + 1) * q^35 + (-z^15 - z^3 + z^2) * q^36 + (-z^23 - z^17 + z^10 - z^5) * q^39 + (z^4 - z) * q^41 + (-z^23 + z^2) * q^43 + z^14 * q^44 + (-z^21 + z^20 - z^17 + z^8 - z^5 + z^4) * q^45 + (z^16 + z^4) * q^48 + (-z^21 + z^20 - z^19) * q^49 + (z^24 + z^6) * q^52 + (z^20 - z^15) * q^53 + (z^16 - z^7) * q^55 + (-z^13 - z^7) * q^59 + (z^22 + z^18 - z^9 + z^6) * q^60 + (z^12 - z^3) * q^61 + (z^16 - 2*z^15 + z^14 - z^3 + z^2) * q^63 - z^5 * q^64 + (z^24 - z^17 + z^8 - z) * q^65 + (z^24 + z^20 - z^15 - z^11 + z^8 + z^2) * q^75 + (z^2 - z) * q^77 + (-z^23 - z^7) * q^80 + (-z^23 + z^22 - z^11 + z^10 - z^9) * q^81 + (-z^17 + z^16 + z^4 - z^3) * q^84 + (z^20 - z^5) * q^89 + (-z^19 + z^18 + z^12 - z^11) * q^91 + (z^22 - z^21 - z^9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 5 q^{4} - 5 q^{9}+O(q^{10})$$ 20 * q - 5 * q^4 - 5 * q^9 $$20 q - 5 q^{4} - 5 q^{9} - 5 q^{16} - 5 q^{25} + 20 q^{33} + 15 q^{35} - 5 q^{36} - 10 q^{39} - 10 q^{45} - 5 q^{49} - 10 q^{53} - 10 q^{63} - 5 q^{64} - 10 q^{75} - 5 q^{81} - 10 q^{89}+O(q^{100})$$ 20 * q - 5 * q^4 - 5 * q^9 - 5 * q^16 - 5 * q^25 + 20 * q^33 + 15 * q^35 - 5 * q^36 - 10 * q^39 - 10 * q^45 - 5 * q^49 - 10 * q^53 - 10 * q^63 - 5 * q^64 - 10 * q^75 - 5 * q^81 - 10 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1441\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$1311$$ $$\chi(n)$$ $$-1$$ $$\zeta_{50}^{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
130.1
 −0.535827 − 0.844328i 0.637424 − 0.770513i −0.968583 + 0.248690i 0.929776 + 0.368125i −0.0627905 + 0.998027i −0.535827 + 0.844328i 0.637424 + 0.770513i −0.968583 − 0.248690i 0.929776 − 0.368125i −0.0627905 − 0.998027i −0.876307 + 0.481754i 0.187381 + 0.982287i −0.728969 − 0.684547i 0.992115 + 0.125333i 0.425779 − 0.904827i −0.876307 − 0.481754i 0.187381 − 0.982287i −0.728969 + 0.684547i 0.992115 − 0.125333i 0.425779 + 0.904827i
0 −1.56720 + 1.13864i −0.809017 0.587785i −0.115808 0.356420i 0 −1.41789 1.03016i 0 0.850604 2.61789i 0
130.2 0 −0.866986 + 0.629902i −0.809017 0.587785i 0.450527 + 1.38658i 0 0.688925 + 0.500534i 0 0.0458709 0.141176i 0
130.3 0 −0.101597 + 0.0738147i −0.809017 0.587785i −0.263146 0.809880i 0 1.60528 + 1.16630i 0 −0.304144 + 0.936058i 0
130.4 0 1.03137 0.749337i −0.809017 0.587785i −0.613161 1.88711i 0 0.303189 + 0.220280i 0 0.193209 0.594636i 0
130.5 0 1.50441 1.09302i −0.809017 0.587785i 0.541587 + 1.66683i 0 −1.17950 0.856954i 0 0.759544 2.33764i 0
654.1 0 −1.56720 1.13864i −0.809017 + 0.587785i −0.115808 + 0.356420i 0 −1.41789 + 1.03016i 0 0.850604 + 2.61789i 0
654.2 0 −0.866986 0.629902i −0.809017 + 0.587785i 0.450527 1.38658i 0 0.688925 0.500534i 0 0.0458709 + 0.141176i 0
654.3 0 −0.101597 0.0738147i −0.809017 + 0.587785i −0.263146 + 0.809880i 0 1.60528 1.16630i 0 −0.304144 0.936058i 0
654.4 0 1.03137 + 0.749337i −0.809017 + 0.587785i −0.613161 + 1.88711i 0 0.303189 0.220280i 0 0.193209 + 0.594636i 0
654.5 0 1.50441 + 1.09302i −0.809017 + 0.587785i 0.541587 1.66683i 0 −1.17950 + 0.856954i 0 0.759544 + 2.33764i 0
785.1 0 −0.613161 + 1.88711i 0.309017 + 0.951057i 1.03137 0.749337i 0 0.598617 + 1.84235i 0 −2.37622 1.72642i 0
785.2 0 −0.263146 + 0.809880i 0.309017 + 0.951057i −0.101597 + 0.0738147i 0 −0.393950 1.21245i 0 0.222357 + 0.161552i 0
785.3 0 −0.115808 + 0.356420i 0.309017 + 0.951057i −1.56720 + 1.13864i 0 −0.574633 1.76854i 0 0.695393 + 0.505233i 0
785.4 0 0.450527 1.38658i 0.309017 + 0.951057i −0.866986 + 0.629902i 0 0.0388067 + 0.119435i 0 −0.910614 0.661600i 0
785.5 0 0.541587 1.66683i 0.309017 + 0.951057i 1.50441 1.09302i 0 0.331159 + 1.01920i 0 −1.67600 1.21769i 0
916.1 0 −0.613161 1.88711i 0.309017 0.951057i 1.03137 + 0.749337i 0 0.598617 1.84235i 0 −2.37622 + 1.72642i 0
916.2 0 −0.263146 0.809880i 0.309017 0.951057i −0.101597 0.0738147i 0 −0.393950 + 1.21245i 0 0.222357 0.161552i 0
916.3 0 −0.115808 0.356420i 0.309017 0.951057i −1.56720 1.13864i 0 −0.574633 + 1.76854i 0 0.695393 0.505233i 0
916.4 0 0.450527 + 1.38658i 0.309017 0.951057i −0.866986 0.629902i 0 0.0388067 0.119435i 0 −0.910614 + 0.661600i 0
916.5 0 0.541587 + 1.66683i 0.309017 0.951057i 1.50441 + 1.09302i 0 0.331159 1.01920i 0 −1.67600 + 1.21769i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 916.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
131.b odd 2 1 CM by $$\Q(\sqrt{-131})$$
11.c even 5 1 inner
1441.u odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1441.1.u.a 20
11.c even 5 1 inner 1441.1.u.a 20
131.b odd 2 1 CM 1441.1.u.a 20
1441.u odd 10 1 inner 1441.1.u.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.1.u.a 20 1.a even 1 1 trivial
1441.1.u.a 20 11.c even 5 1 inner
1441.1.u.a 20 131.b odd 2 1 CM
1441.1.u.a 20 1441.u odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$5$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$7$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$11$ $$T^{20} + T^{15} + T^{10} + T^{5} + 1$$
$13$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$43$ $$(T^{10} - 10 T^{8} + 35 T^{6} + T^{5} - 50 T^{4} + \cdots - 1)^{2}$$
$47$ $$T^{20}$$
$53$ $$(T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{5}$$
$59$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$61$ $$T^{20} + 5 T^{18} + 20 T^{16} + 2 T^{15} + \cdots + 1$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$(T^{2} + T - 1)^{10}$$
$97$ $$T^{20}$$