Properties

 Label 2005.1.g.b Level 2005 Weight 1 Character orbit 2005.g Analytic conductor 1.001 Analytic rank 0 Dimension 8 Projective image $$D_{20}$$ RM discriminant 401 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2005 = 5 \cdot 401$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2005.g (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.00062535033$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{20}$$ Projective field Galois closure of 20.0.8181810539473601738391143798828125.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{2} + ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{4} -\zeta_{20}^{6} q^{5} + ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{8} + \zeta_{20}^{5} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{20} + \zeta_{20}^{4} ) q^{2} + ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} ) q^{4} -\zeta_{20}^{6} q^{5} + ( \zeta_{20}^{2} - \zeta_{20}^{3} ) q^{7} + ( -\zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{8} + \zeta_{20}^{5} q^{9} + ( 1 + \zeta_{20}^{7} ) q^{10} + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{11} + ( -\zeta_{20}^{3} + \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{14} + ( -1 + \zeta_{20}^{3} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{7} ) q^{16} + ( -\zeta_{20}^{6} + \zeta_{20}^{9} ) q^{18} + ( -\zeta_{20} + \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{20} + ( \zeta_{20} - \zeta_{20}^{4} + \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{22} -\zeta_{20}^{2} q^{25} + ( -1 + \zeta_{20} + \zeta_{20}^{4} - \zeta_{20}^{5} - \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{28} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{29} + ( 1 + \zeta_{20} - \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{8} ) q^{32} + ( -\zeta_{20}^{8} + \zeta_{20}^{9} ) q^{35} + ( 1 - \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{36} + ( \zeta_{20}^{2} - \zeta_{20}^{5} + \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{40} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{41} + ( \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{43} + ( -\zeta_{20} - \zeta_{20}^{2} + 2 \zeta_{20}^{5} - \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{44} + \zeta_{20} q^{45} + ( \zeta_{20} - \zeta_{20}^{4} ) q^{47} + ( \zeta_{20}^{4} - \zeta_{20}^{5} + \zeta_{20}^{6} ) q^{49} + ( \zeta_{20}^{3} - \zeta_{20}^{6} ) q^{50} + ( -\zeta_{20}^{3} - \zeta_{20}^{9} ) q^{55} + ( \zeta_{20} - \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{56} + ( -1 + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{58} + ( \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{63} + ( -\zeta_{20} - \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{5} + \zeta_{20}^{6} - \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{64} + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{70} + ( -\zeta_{20} + \zeta_{20}^{4} - \zeta_{20}^{7} - \zeta_{20}^{8} ) q^{72} + ( \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{73} + ( -1 + \zeta_{20}^{5} - \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{77} + ( 1 - \zeta_{20}^{2} - \zeta_{20}^{3} + \zeta_{20}^{6} - \zeta_{20}^{9} ) q^{80} - q^{81} + ( -1 - \zeta_{20}^{2} + \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{82} + ( 1 + \zeta_{20}^{5} ) q^{83} + ( -\zeta_{20} + \zeta_{20}^{2} - \zeta_{20}^{8} + \zeta_{20}^{9} ) q^{86} + ( -1 + \zeta_{20}^{2} + \zeta_{20}^{3} - \zeta_{20}^{5} - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{9} ) q^{88} + ( -\zeta_{20}^{2} - \zeta_{20}^{8} ) q^{89} + ( -\zeta_{20}^{2} + \zeta_{20}^{5} ) q^{90} + ( -\zeta_{20}^{2} + 2 \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{94} + ( -1 - \zeta_{20}^{5} + \zeta_{20}^{6} - \zeta_{20}^{7} + \zeta_{20}^{8} - \zeta_{20}^{9} ) q^{98} + ( \zeta_{20}^{2} + \zeta_{20}^{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 2q^{5} + 2q^{7} + O(q^{10})$$ $$8q - 2q^{2} - 2q^{5} + 2q^{7} + 8q^{10} - 12q^{16} - 2q^{18} - 2q^{25} - 12q^{28} + 8q^{32} + 2q^{35} + 8q^{36} + 2q^{43} + 2q^{47} - 2q^{50} - 6q^{58} + 2q^{63} + 10q^{70} + 2q^{73} - 10q^{77} + 8q^{80} - 8q^{81} - 10q^{82} + 8q^{83} + 4q^{86} - 10q^{88} - 2q^{90} - 8q^{98} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2005\mathbb{Z}\right)^\times$$.

 $$n$$ $$402$$ $$1206$$ $$\chi(n)$$ $$-\zeta_{20}^{5}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1202.1
 0.587785 + 0.809017i 0.951057 − 0.309017i −0.587785 + 0.809017i −0.951057 − 0.309017i 0.587785 − 0.809017i 0.951057 + 0.309017i −0.587785 − 0.809017i −0.951057 + 0.309017i
−1.39680 1.39680i 0 2.90211i −0.809017 + 0.587785i 0 0.642040 + 0.642040i 2.65688 2.65688i 1.00000i 1.95106 + 0.309017i
1202.2 −0.642040 0.642040i 0 0.175571i 0.309017 + 0.951057i 0 0.221232 + 0.221232i −0.754763 + 0.754763i 1.00000i 0.412215 0.809017i
1202.3 −0.221232 0.221232i 0 0.902113i −0.809017 0.587785i 0 −1.26007 1.26007i −0.420808 + 0.420808i 1.00000i 0.0489435 + 0.309017i
1202.4 1.26007 + 1.26007i 0 2.17557i 0.309017 0.951057i 0 1.39680 + 1.39680i −1.48131 + 1.48131i 1.00000i 1.58779 0.809017i
1603.1 −1.39680 + 1.39680i 0 2.90211i −0.809017 0.587785i 0 0.642040 0.642040i 2.65688 + 2.65688i 1.00000i 1.95106 0.309017i
1603.2 −0.642040 + 0.642040i 0 0.175571i 0.309017 0.951057i 0 0.221232 0.221232i −0.754763 0.754763i 1.00000i 0.412215 + 0.809017i
1603.3 −0.221232 + 0.221232i 0 0.902113i −0.809017 + 0.587785i 0 −1.26007 + 1.26007i −0.420808 0.420808i 1.00000i 0.0489435 0.309017i
1603.4 1.26007 1.26007i 0 2.17557i 0.309017 + 0.951057i 0 1.39680 1.39680i −1.48131 1.48131i 1.00000i 1.58779 + 0.809017i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1603.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
401.b even 2 1 RM by $$\Q(\sqrt{401})$$
5.c odd 4 1 inner
2005.g odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2005.1.g.b 8
5.c odd 4 1 inner 2005.1.g.b 8
401.b even 2 1 RM 2005.1.g.b 8
2005.g odd 4 1 inner 2005.1.g.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.1.g.b 8 1.a even 1 1 trivial
2005.1.g.b 8 5.c odd 4 1 inner
2005.1.g.b 8 401.b even 2 1 RM
2005.1.g.b 8 2005.g odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 2 T_{2}^{7} + 2 T_{2}^{6} + 11 T_{2}^{4} + 20 T_{2}^{3} + 18 T_{2}^{2} + 6 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2005, [\chi])$$.

Hecke characteristic polynomials

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