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 discriminant -23
Inner twists 4

Related objects

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.5048523.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)
87.f even 4 1 inner
2001.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.1.i.b yes 2
3.b odd 2 1 2001.1.i.a 2
23.b odd 2 1 CM 2001.1.i.b yes 2
29.c odd 4 1 2001.1.i.a 2
69.c even 2 1 2001.1.i.a 2
87.f even 4 1 inner 2001.1.i.b yes 2
667.f even 4 1 2001.1.i.a 2
2001.i odd 4 1 inner 2001.1.i.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.1.i.a 2 3.b odd 2 1
2001.1.i.a 2 29.c odd 4 1
2001.1.i.a 2 69.c even 2 1
2001.1.i.a 2 667.f even 4 1
2001.1.i.b yes 2 1.a even 1 1 trivial
2001.1.i.b yes 2 23.b odd 2 1 CM
2001.1.i.b yes 2 87.f even 4 1 inner
2001.1.i.b yes 2 2001.i odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).