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 disc. 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 \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00062535033\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{20}\)
Projective field Galois closure of 20.0.8181810539473601738391143798828125.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
401.b Even 1 RM by \(\Q(\sqrt{401}) \) yes
5.c Odd 1 yes
2005.g Odd 1 yes

Hecke kernels

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