Properties

 Label 1045.1.w.e Level $1045$ Weight $1$ Character orbit 1045.w Analytic conductor $0.522$ Analytic rank $0$ Dimension $8$ Projective image $D_{10}$ 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.87298324158753125.1

$q$-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.

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

Character values

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

 $$n$$ $$496$$ $$761$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{2}$$ $$-1$$

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
 −0.951057 − 0.309017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i −0.587785 + 0.809017i 0.587785 − 0.809017i −0.587785 − 0.809017i 0.587785 + 0.809017i
−0.363271 + 1.11803i 1.53884 1.11803i −0.309017 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 −0.587785 + 0.427051i 0.809017 2.48990i −1.17557
284.2 0.363271 1.11803i −1.53884 + 1.11803i −0.309017 0.224514i 0.309017 + 0.951057i 0.690983 + 2.12663i 0 0.587785 0.427051i 0.809017 2.48990i 1.17557
379.1 −0.363271 1.11803i 1.53884 + 1.11803i −0.309017 + 0.224514i 0.309017 0.951057i 0.690983 2.12663i 0 −0.587785 0.427051i 0.809017 + 2.48990i −1.17557
379.2 0.363271 + 1.11803i −1.53884 1.11803i −0.309017 + 0.224514i 0.309017 0.951057i 0.690983 2.12663i 0 0.587785 + 0.427051i 0.809017 + 2.48990i 1.17557
664.1 −1.53884 1.11803i −0.363271 + 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 1.31433i 0 0.951057 2.92705i −0.309017 0.224514i 1.90211
664.2 1.53884 + 1.11803i 0.363271 1.11803i 0.809017 + 2.48990i −0.809017 + 0.587785i 1.80902 1.31433i 0 −0.951057 + 2.92705i −0.309017 0.224514i −1.90211
949.1 −1.53884 + 1.11803i −0.363271 1.11803i 0.809017 2.48990i −0.809017 0.587785i 1.80902 + 1.31433i 0 0.951057 + 2.92705i −0.309017 + 0.224514i 1.90211
949.2 1.53884 1.11803i 0.363271 + 1.11803i 0.809017 2.48990i −0.809017 0.587785i 1.80902 + 1.31433i 0 −0.951057 2.92705i −0.309017 + 0.224514i −1.90211
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 284.2 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.e 8
5.b even 2 1 inner 1045.1.w.e 8
11.c even 5 1 inner 1045.1.w.e 8
19.b odd 2 1 inner 1045.1.w.e 8
55.j even 10 1 inner 1045.1.w.e 8
95.d odd 2 1 CM 1045.1.w.e 8
209.m odd 10 1 inner 1045.1.w.e 8
1045.w odd 10 1 inner 1045.1.w.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.1.w.e 8 1.a even 1 1 trivial
1045.1.w.e 8 5.b even 2 1 inner
1045.1.w.e 8 11.c even 5 1 inner
1045.1.w.e 8 19.b odd 2 1 inner
1045.1.w.e 8 55.j even 10 1 inner
1045.1.w.e 8 95.d odd 2 1 CM
1045.1.w.e 8 209.m odd 10 1 inner
1045.1.w.e 8 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 $$S_{1}^{\mathrm{new}}(1045, [\chi])$$:

 $$T_{2}^{8} + 10T_{2}^{4} + 25T_{2}^{2} + 25$$ T2^8 + 10*T2^4 + 25*T2^2 + 25 $$T_{7}$$ T7

Hecke characteristic polynomials

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