# Properties

 Label 1045.1.w.f Level $1045$ Weight $1$ Character orbit 1045.w Analytic conductor $0.522$ Analytic rank $0$ Dimension $16$ Projective image $D_{20}$ CM discriminant -95 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,1,Mod(284,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 6, 5]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.284");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.521522938201$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{40})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ x^16 - x^12 + x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} + \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{13} + \zeta_{40}^{11}) q^{3} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{4}+ \cdots + ( - \zeta_{40}^{6} + \cdots - \zeta_{40}^{2}) q^{9}+O(q^{10})$$ q + (z^7 + z) * q^2 + (z^13 + z^11) * q^3 + (z^14 + z^8 + z^2) * q^4 - z^16 * q^5 + (z^18 + z^14 + z^12 - 1) * q^6 + (z^15 + z^9 + z^3 - z) * q^8 + (-z^6 - z^4 - z^2) * q^9 $$q + (\zeta_{40}^{7} + \zeta_{40}) q^{2} + (\zeta_{40}^{13} + \zeta_{40}^{11}) q^{3} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{4}+ \cdots + (\zeta_{40}^{18} + \zeta_{40}^{16} - 1) q^{99}+O(q^{100})$$ q + (z^7 + z) * q^2 + (z^13 + z^11) * q^3 + (z^14 + z^8 + z^2) * q^4 - z^16 * q^5 + (z^18 + z^14 + z^12 - 1) * q^6 + (z^15 + z^9 + z^3 - z) * q^8 + (-z^6 - z^4 - z^2) * q^9 + (-z^17 + z^3) * q^10 - z^14 * q^11 + (z^19 + z^15 + z^13 - z^7 - z^5 - z) * q^12 + (z^17 - z^11) * q^13 + (z^9 + z^7) * q^15 + (z^16 + z^10 - z^8 + z^4 - z^2) * q^16 + (-z^13 - z^11 - z^9 - z^7 - z^5 - z^3) * q^18 + z^12 * q^19 + (-z^18 + z^10 + z^4) * q^20 + (-z^15 + z) * q^22 + (z^16 - z^12 - z^8 - z^6 - z^2 - 1) * q^24 - z^12 * q^25 + (-z^12 - z^4) * q^26 + (-z^19 - z^17 - z^15 - z^13) * q^27 + (z^16 + z^14 + z^10 + z^8) * q^30 + (z^17 - z^15 + z^11 - z^9 + z^5 - z^3) * q^32 + (z^7 + z^5) * q^33 + (-z^18 - z^16 - z^14 - z^12 - z^10 - z^8 - z^6 - z^4 + 1) * q^36 + (z^19 - z^17) * q^37 + (z^19 + z^13) * q^38 + (-z^10 - z^8 + z^4 + z^2) * q^39 + (-z^19 + z^17 + z^11 + z^5) * q^40 + (-z^16 + z^8 + z^2) * q^44 + (z^18 - z^2 - 1) * q^45 + (-z^19 + z^17 - z^13 - z^9 - z^7 - z^3) * q^48 + z^16 * q^49 + (-z^19 - z^13) * q^50 + (-z^13 - z^11) * q^52 + (z^19 - z^9) * q^53 + (-z^18 - z^16 - z^14 + z^6 + z^4 + z^2 + 2) * q^54 - z^10 * q^55 + (-z^5 - z^3) * q^57 + (z^17 + z^15 + z^11 + z^9 - z^3 - z) * q^60 + (z^10 - z^2) * q^61 + (z^18 - z^16 + z^12 - z^10 + z^6 - z^4 + z^2) * q^64 + (z^13 - z^7) * q^65 + (z^14 + z^12 + z^8 + z^6) * q^66 + (-z^11 + z^9) * q^67 + (-z^19 - z^17 - z^15 - z^13 - z^11 - z^9 + z^3 + z) * q^72 + (-z^18 - z^6 + z^4 - 1) * q^74 + (z^5 + z^3) * q^75 + (z^14 - z^6 - 1) * q^76 + (-z^17 - z^15 + z^5 + z^3) * q^78 + (z^18 + z^12 + z^6 - z^4 + 1) * q^80 + (z^12 + z^10 + z^8 + z^6 + z^4) * q^81 + (-z^17 + z^15 + z^9 + z^3) * q^88 + (z^19 - z^9 - z^7 - z^5 - z^3 - z) * q^90 + z^8 * q^95 + (z^18 - z^14 - z^10 + z^6 - z^4 + 1) * q^96 + (-z^7 - z) * q^97 + (z^17 - z^3) * q^98 + (z^18 + z^16 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{4} + 4 q^{5} - 12 q^{6} - 4 q^{9}+O(q^{10})$$ 16 * q - 4 * q^4 + 4 * q^5 - 12 * q^6 - 4 * q^9 $$16 q - 4 q^{4} + 4 q^{5} - 12 q^{6} - 4 q^{9} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 20 q^{24} - 4 q^{25} - 8 q^{26} - 8 q^{30} + 16 q^{36} + 8 q^{39} - 16 q^{45} - 4 q^{49} + 40 q^{54} + 4 q^{64} - 12 q^{74} - 16 q^{76} + 16 q^{80} + 4 q^{81} - 4 q^{95} + 12 q^{96} - 20 q^{99}+O(q^{100})$$ 16 * q - 4 * q^4 + 4 * q^5 - 12 * q^6 - 4 * q^9 + 4 * q^16 + 4 * q^19 + 4 * q^20 - 20 * q^24 - 4 * q^25 - 8 * q^26 - 8 * q^30 + 16 * q^36 + 8 * q^39 - 16 * q^45 - 4 * q^49 + 40 * q^54 + 4 * q^64 - 12 * q^74 - 16 * q^76 + 16 * q^80 + 4 * q^81 - 4 * q^95 + 12 * q^96 - 20 * q^99

## Character values

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

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$\zeta_{40}^{8}$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
284.1
 −0.453990 + 0.891007i 0.891007 + 0.453990i −0.891007 − 0.453990i 0.453990 − 0.891007i −0.453990 − 0.891007i 0.891007 − 0.453990i −0.891007 + 0.453990i 0.453990 + 0.891007i −0.987688 − 0.156434i 0.156434 − 0.987688i −0.156434 + 0.987688i 0.987688 + 0.156434i −0.987688 + 0.156434i 0.156434 + 0.987688i −0.156434 − 0.987688i 0.987688 − 0.156434i
−0.610425 + 1.87869i −0.734572 + 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 3.03979 2.20854i −0.0542543 + 0.166977i 1.97538
284.2 −0.0966818 + 0.297556i 1.44168 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 −0.481456 + 0.349798i 0.672288 2.06909i 0.312869
284.3 0.0966818 0.297556i −1.44168 + 1.04744i 0.729825 + 0.530249i −0.309017 0.951057i 0.172288 + 0.530249i 0 0.481456 0.349798i 0.672288 2.06909i −0.312869
284.4 0.610425 1.87869i 0.734572 0.533698i −2.34786 1.70582i −0.309017 0.951057i −0.554254 1.70582i 0 −3.03979 + 2.20854i −0.0542543 + 0.166977i −1.97538
379.1 −0.610425 1.87869i −0.734572 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 3.03979 + 2.20854i −0.0542543 0.166977i 1.97538
379.2 −0.0966818 0.297556i 1.44168 + 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 −0.481456 0.349798i 0.672288 + 2.06909i 0.312869
379.3 0.0966818 + 0.297556i −1.44168 1.04744i 0.729825 0.530249i −0.309017 + 0.951057i 0.172288 0.530249i 0 0.481456 + 0.349798i 0.672288 + 2.06909i −0.312869
379.4 0.610425 + 1.87869i 0.734572 + 0.533698i −2.34786 + 1.70582i −0.309017 + 0.951057i −0.554254 + 1.70582i 0 −3.03979 2.20854i −0.0542543 0.166977i −1.97538
664.1 −1.44168 1.04744i 0.610425 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 0.647354 1.99235i −2.34786 1.70582i −1.78201
664.2 −0.734572 0.533698i −0.0966818 + 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 −0.329843 + 1.01515i 0.729825 + 0.530249i −0.907981
664.3 0.734572 + 0.533698i 0.0966818 0.297556i −0.0542543 0.166977i 0.809017 0.587785i 0.229825 0.166977i 0 0.329843 1.01515i 0.729825 + 0.530249i 0.907981
664.4 1.44168 + 1.04744i −0.610425 + 1.87869i 0.672288 + 2.06909i 0.809017 0.587785i −2.84786 + 2.06909i 0 −0.647354 + 1.99235i −2.34786 1.70582i 1.78201
949.1 −1.44168 + 1.04744i 0.610425 + 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 0.647354 + 1.99235i −2.34786 + 1.70582i −1.78201
949.2 −0.734572 + 0.533698i −0.0966818 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 −0.329843 1.01515i 0.729825 0.530249i −0.907981
949.3 0.734572 0.533698i 0.0966818 + 0.297556i −0.0542543 + 0.166977i 0.809017 + 0.587785i 0.229825 + 0.166977i 0 0.329843 + 1.01515i 0.729825 0.530249i 0.907981
949.4 1.44168 1.04744i −0.610425 1.87869i 0.672288 2.06909i 0.809017 + 0.587785i −2.84786 2.06909i 0 −0.647354 1.99235i −2.34786 + 1.70582i 1.78201
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 284.4 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 $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
11.c even 5 1 inner
19.b odd 2 1 inner
55.j even 10 1 inner
209.m odd 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.f 16
5.b even 2 1 inner 1045.1.w.f 16
11.c even 5 1 inner 1045.1.w.f 16
19.b odd 2 1 inner 1045.1.w.f 16
55.j even 10 1 inner 1045.1.w.f 16
95.d odd 2 1 CM 1045.1.w.f 16
209.m odd 10 1 inner 1045.1.w.f 16
1045.w odd 10 1 inner 1045.1.w.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.f 16 1.a even 1 1 trivial
1045.1.w.f 16 5.b even 2 1 inner
1045.1.w.f 16 11.c even 5 1 inner
1045.1.w.f 16 19.b odd 2 1 inner
1045.1.w.f 16 55.j even 10 1 inner
1045.1.w.f 16 95.d odd 2 1 CM
1045.1.w.f 16 209.m odd 10 1 inner
1045.1.w.f 16 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 $$S_{1}^{\mathrm{new}}(1045, [\chi])$$:

 $$T_{2}^{16} + 4T_{2}^{14} + 12T_{2}^{12} + 32T_{2}^{10} + 150T_{2}^{8} - 32T_{2}^{6} + 97T_{2}^{4} + 16T_{2}^{2} + 1$$ T2^16 + 4*T2^14 + 12*T2^12 + 32*T2^10 + 150*T2^8 - 32*T2^6 + 97*T2^4 + 16*T2^2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$3$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$5$ $$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{4}$$
$7$ $$T^{16}$$
$11$ $$(T^{8} - T^{6} + T^{4} + \cdots + 1)^{2}$$
$13$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$17$ $$T^{16}$$
$19$ $$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{4}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$(T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2}$$
$59$ $$T^{16}$$
$61$ $$(T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2}$$
$67$ $$(T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2}$$
$71$ $$T^{16}$$
$73$ $$T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$T^{16} + 4 T^{14} + \cdots + 1$$