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

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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:

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

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