Properties

Label 2001.1.i.b
Level 2001
Weight 1
Character orbit 2001.i
Analytic conductor 0.999
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM disc. -23
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2001.i (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.998629090279\)
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 4.0.116116029.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( 1 - i ) q^{2} \) \( + i q^{3} \) \( -i q^{4} \) \( + ( 1 + i ) q^{6} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + ( 1 - i ) q^{2} \) \( + i q^{3} \) \( -i q^{4} \) \( + ( 1 + i ) q^{6} \) \(- q^{9}\) \(+ q^{12}\) \( + 2 i q^{13} \) \(+ q^{16}\) \( + ( -1 + i ) q^{18} \) \( + i q^{23} \) \(+ q^{25}\) \( + ( 2 + 2 i ) q^{26} \) \( -i q^{27} \) \( -i q^{29} \) \( + ( -1 - i ) q^{31} \) \( + ( 1 - i ) q^{32} \) \( + i q^{36} \) \( -2 q^{39} \) \( + ( -1 - i ) q^{41} \) \( + ( 1 + i ) q^{46} \) \( + ( 1 + i ) q^{47} \) \( + i q^{48} \) \(+ q^{49}\) \( + ( 1 - i ) q^{50} \) \( + 2 q^{52} \) \( + ( -1 - i ) q^{54} \) \( + ( -1 - i ) q^{58} \) \( -2 i q^{59} \) \( -2 q^{62} \) \( -i q^{64} \) \(- q^{69}\) \( + ( -1 + i ) q^{73} \) \( + i q^{75} \) \( + ( -2 + 2 i ) q^{78} \) \(+ q^{81}\) \( -2 q^{82} \) \(+ q^{87}\) \(+ q^{92}\) \( + ( 1 - i ) q^{93} \) \( + 2 q^{94} \) \( + ( 1 + i ) q^{96} \) \( + ( 1 - i ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 4q^{78} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 2q^{87} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut 2q^{96} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times\).

\(n\) \(553\) \(668\) \(1132\)
\(\chi(n)\) \(i\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1172.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000i 1.00000i 0 1.00000 + 1.00000i 0 0 −1.00000 0
1931.1 1.00000 + 1.00000i 1.00000i 1.00000i 0 1.00000 1.00000i 0 0 −1.00000 0
\(n\): 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
23.b Odd 1 CM by \(\Q(\sqrt{-23}) \) yes
87.f Even 1 yes
2001.i 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 \(S_{1}^{\mathrm{new}}(2001, [\chi])\).