Properties

Label 2001.1.i.d
Level 2001
Weight 1
Character orbit 2001.i
Analytic conductor 0.999
Analytic rank 0
Dimension 4
Projective image \(D_{12}\)
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{12}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{2} \) \( -\zeta_{12}^{2} q^{3} \) \( + ( \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{4} \) \( + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{6} \) \( + ( -1 - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{8} \) \( + \zeta_{12}^{4} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{2} \) \( -\zeta_{12}^{2} q^{3} \) \( + ( \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{4} \) \( + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{6} \) \( + ( -1 - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{8} \) \( + \zeta_{12}^{4} q^{9} \) \( + ( 1 - \zeta_{12}^{4} + \zeta_{12}^{5} ) q^{12} \) \( -\zeta_{12}^{3} q^{13} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{16} \) \( + ( -1 - \zeta_{12}^{5} ) q^{18} \) \( -\zeta_{12}^{3} q^{23} \) \( + ( 1 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{24} \) \(+ q^{25}\) \( + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{26} \) \(+ q^{27}\) \( -\zeta_{12}^{5} q^{29} \) \( + ( \zeta_{12} + \zeta_{12}^{2} ) q^{31} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{32} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{36} \) \( + \zeta_{12}^{5} q^{39} \) \( + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{41} \) \( + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{46} \) \( + ( -\zeta_{12}^{4} + \zeta_{12}^{5} ) q^{47} \) \( + ( 1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{48} \) \(+ q^{49}\) \( + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{50} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{5} ) q^{52} \) \( + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{54} \) \( + ( -1 + \zeta_{12} ) q^{58} \) \( + 2 \zeta_{12}^{3} q^{59} \) \( + ( -\zeta_{12}^{2} + \zeta_{12}^{4} ) q^{62} \) \( + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} + \zeta_{12}^{5} ) q^{64} \) \( + \zeta_{12}^{5} q^{69} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{71} \) \( + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{72} \) \( + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{73} \) \( -\zeta_{12}^{2} q^{75} \) \( + ( 1 - \zeta_{12} ) q^{78} \) \( -\zeta_{12}^{2} q^{81} \) \( + ( \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{82} \) \( -\zeta_{12} q^{87} \) \( + ( -1 + \zeta_{12} - \zeta_{12}^{5} ) q^{92} \) \( + ( -\zeta_{12}^{3} - \zeta_{12}^{4} ) q^{93} \) \( + ( 2 - \zeta_{12} + \zeta_{12}^{5} ) q^{94} \) \( + ( -1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{96} \) \( + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 6q^{12} \) \(\mathstrut -\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 4q^{18} \) \(\mathstrut +\mathstrut 6q^{24} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 4q^{32} \) \(\mathstrut -\mathstrut 6q^{36} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 4q^{48} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 4q^{62} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 4q^{78} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 4q^{96} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(553\) \(668\) \(1132\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1\) \(-1\)

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
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.366025 + 0.366025i −0.500000 0.866025i 0.732051i 0 0.500000 + 0.133975i 0 −0.633975 0.633975i −0.500000 + 0.866025i 0
1172.2 1.36603 1.36603i −0.500000 + 0.866025i 2.73205i 0 0.500000 + 1.86603i 0 −2.36603 2.36603i −0.500000 0.866025i 0
1931.1 −0.366025 0.366025i −0.500000 + 0.866025i 0.732051i 0 0.500000 0.133975i 0 −0.633975 + 0.633975i −0.500000 0.866025i 0
1931.2 1.36603 + 1.36603i −0.500000 0.866025i 2.73205i 0 0.500000 1.86603i 0 −2.36603 + 2.36603i −0.500000 + 0.866025i 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 \(T_{2}^{4} \) \(\mathstrut -\mathstrut 2 T_{2}^{3} \) \(\mathstrut +\mathstrut 2 T_{2}^{2} \) \(\mathstrut +\mathstrut 2 T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).