Properties

Label 2005.1.g.a
Level 2005
Weight 1
Character orbit 2005.g
Analytic conductor 1.001
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
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: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.20100125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( 1 - i ) q^{2} \) \( -i q^{4} \) \(+ q^{5}\) \( + ( -1 + i ) q^{7} \) \( + i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - i ) q^{2} \) \( -i q^{4} \) \(+ q^{5}\) \( + ( -1 + i ) q^{7} \) \( + i q^{9} \) \( + ( 1 - i ) q^{10} \) \( + 2 i q^{14} \) \(+ q^{16}\) \( + ( 1 + i ) q^{18} \) \( -i q^{20} \) \(+ q^{25}\) \( + ( 1 + i ) q^{28} \) \( -2 i q^{29} \) \( + ( 1 - i ) q^{32} \) \( + ( -1 + i ) q^{35} \) \(+ q^{36}\) \( + ( -1 - i ) q^{43} \) \( + i q^{45} \) \( + ( -1 + i ) q^{47} \) \( -i q^{49} \) \( + ( 1 - i ) q^{50} \) \( + ( -2 - 2 i ) q^{58} \) \( + ( -1 - i ) q^{63} \) \( -i q^{64} \) \( + 2 i q^{70} \) \( + ( -1 - i ) q^{73} \) \(+ q^{80}\) \(- q^{81}\) \( + ( 1 + i ) q^{83} \) \( -2 q^{86} \) \( + ( 1 + i ) q^{90} \) \( + 2 i q^{94} \) \( + ( -1 - i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(402\) \(1206\)
\(\chi(n)\) \(-i\) \(-1\)

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
1.00000i
1.00000i
1.00000 + 1.00000i 0 1.00000i 1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000 + 1.00000i
1603.1 1.00000 1.00000i 0 1.00000i 1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 T_{2} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{1}^{\mathrm{new}}(2005, [\chi])\).