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 discriminant -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 \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.998629090279\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

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

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$7$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$11$ \( ( 1 + T^{4} )^{2} \)
$13$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 1 + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( 1 - T^{2} + T^{4} \)
$31$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$53$ \( ( 1 + T^{2} )^{4} \)
$59$ \( ( 1 + T^{2} )^{4} \)
$61$ \( ( 1 + T^{4} )^{2} \)
$67$ \( ( 1 + T^{2} )^{4} \)
$71$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$73$ \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
$79$ \( ( 1 + T^{4} )^{2} \)
$83$ \( ( 1 + T^{2} )^{4} \)
$89$ \( ( 1 + T^{4} )^{2} \)
$97$ \( ( 1 + T^{4} )^{2} \)
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