Properties

Label 2001.1.bf.a
Level 2001
Weight 1
Character orbit 2001.bf
Analytic conductor 0.999
Analytic rank 0
Dimension 12
Projective image \(D_{28}\)
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.bf (of order \(28\) and degree \(12\))

Newform invariants

Self dual: No
Analytic conductor: \(0.998629090279\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{28}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{28}^{7} - \zeta_{28}^{10} ) q^{2} \) \( -\zeta_{28}^{9} q^{3} \) \( + ( -1 - \zeta_{28}^{3} - \zeta_{28}^{6} ) q^{4} \) \( + ( -\zeta_{28}^{2} - \zeta_{28}^{5} ) q^{6} \) \( + ( -\zeta_{28}^{2} + \zeta_{28}^{7} + \zeta_{28}^{10} + \zeta_{28}^{13} ) q^{8} \) \( -\zeta_{28}^{4} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{28}^{7} - \zeta_{28}^{10} ) q^{2} \) \( -\zeta_{28}^{9} q^{3} \) \( + ( -1 - \zeta_{28}^{3} - \zeta_{28}^{6} ) q^{4} \) \( + ( -\zeta_{28}^{2} - \zeta_{28}^{5} ) q^{6} \) \( + ( -\zeta_{28}^{2} + \zeta_{28}^{7} + \zeta_{28}^{10} + \zeta_{28}^{13} ) q^{8} \) \( -\zeta_{28}^{4} q^{9} \) \( + ( -\zeta_{28} + \zeta_{28}^{9} + \zeta_{28}^{12} ) q^{12} \) \( + ( -\zeta_{28} + \zeta_{28}^{11} ) q^{13} \) \( + ( 1 + \zeta_{28}^{3} + \zeta_{28}^{6} + \zeta_{28}^{9} + \zeta_{28}^{12} ) q^{16} \) \( + ( -1 + \zeta_{28}^{11} ) q^{18} \) \( -\zeta_{28}^{9} q^{23} \) \( + ( \zeta_{28}^{2} + \zeta_{28}^{5} + \zeta_{28}^{8} + \zeta_{28}^{11} ) q^{24} \) \( -\zeta_{28}^{10} q^{25} \) \( + ( \zeta_{28}^{4} + \zeta_{28}^{7} + \zeta_{28}^{8} + \zeta_{28}^{11} ) q^{26} \) \( + \zeta_{28}^{13} q^{27} \) \( -\zeta_{28}^{3} q^{29} \) \( + ( \zeta_{28}^{5} - \zeta_{28}^{12} ) q^{31} \) \( + ( \zeta_{28}^{2} + \zeta_{28}^{5} - \zeta_{28}^{7} + \zeta_{28}^{8} - \zeta_{28}^{10} - \zeta_{28}^{13} ) q^{32} \) \( + ( \zeta_{28}^{4} + \zeta_{28}^{7} + \zeta_{28}^{10} ) q^{36} \) \( + ( \zeta_{28}^{6} + \zeta_{28}^{10} ) q^{39} \) \( + ( -\zeta_{28}^{8} + \zeta_{28}^{13} ) q^{41} \) \( + ( -\zeta_{28}^{2} - \zeta_{28}^{5} ) q^{46} \) \( + ( -\zeta_{28}^{9} - \zeta_{28}^{10} ) q^{47} \) \( + ( \zeta_{28} + \zeta_{28}^{4} + \zeta_{28}^{7} - \zeta_{28}^{9} - \zeta_{28}^{12} ) q^{48} \) \( + \zeta_{28}^{8} q^{49} \) \( + ( -\zeta_{28}^{3} - \zeta_{28}^{6} ) q^{50} \) \( + ( 1 + \zeta_{28} + \zeta_{28}^{3} + \zeta_{28}^{4} + \zeta_{28}^{7} - \zeta_{28}^{11} ) q^{52} \) \( + ( \zeta_{28}^{6} + \zeta_{28}^{9} ) q^{54} \) \( + ( \zeta_{28}^{10} + \zeta_{28}^{13} ) q^{58} \) \( + ( -\zeta_{28}^{3} - \zeta_{28}^{11} ) q^{59} \) \( + ( \zeta_{28} - \zeta_{28}^{5} - \zeta_{28}^{8} - \zeta_{28}^{12} ) q^{62} \) \( + ( -1 + \zeta_{28} - \zeta_{28}^{3} + \zeta_{28}^{4} - \zeta_{28}^{6} - \zeta_{28}^{9} - \zeta_{28}^{12} ) q^{64} \) \( -\zeta_{28}^{4} q^{69} \) \( + ( \zeta_{28} + \zeta_{28}^{11} ) q^{71} \) \( + ( 1 + \zeta_{28}^{3} + \zeta_{28}^{6} - \zeta_{28}^{11} ) q^{72} \) \( + ( -\zeta_{28}^{12} - \zeta_{28}^{13} ) q^{73} \) \( -\zeta_{28}^{5} q^{75} \) \( + ( \zeta_{28}^{2} + \zeta_{28}^{3} + \zeta_{28}^{6} - \zeta_{28}^{13} ) q^{78} \) \( + \zeta_{28}^{8} q^{81} \) \( + ( -\zeta_{28} - \zeta_{28}^{4} + \zeta_{28}^{6} + \zeta_{28}^{9} ) q^{82} \) \( + \zeta_{28}^{12} q^{87} \) \( + ( -\zeta_{28} + \zeta_{28}^{9} + \zeta_{28}^{12} ) q^{92} \) \( + ( 1 - \zeta_{28}^{7} ) q^{93} \) \( + ( -\zeta_{28}^{2} - \zeta_{28}^{3} - \zeta_{28}^{5} - \zeta_{28}^{6} ) q^{94} \) \( + ( 1 - \zeta_{28}^{2} + \zeta_{28}^{3} - \zeta_{28}^{5} - \zeta_{28}^{8} - \zeta_{28}^{11} ) q^{96} \) \( + ( \zeta_{28} + \zeta_{28}^{4} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 10q^{52} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 14q^{64} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 14q^{72} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 4q^{78} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 2q^{87} \) \(\mathstrut -\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut 12q^{96} \) \(\mathstrut -\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(553\) \(668\) \(1132\)
\(\chi(n)\) \(\zeta_{28}^{3}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−0.781831 0.623490i
−0.974928 0.222521i
−0.781831 + 0.623490i
0.433884 0.900969i
−0.974928 + 0.222521i
0.974928 0.222521i
−0.433884 + 0.900969i
0.781831 0.623490i
0.974928 + 0.222521i
0.781831 + 0.623490i
0.433884 + 0.900969i
−0.433884 0.900969i
−0.900969 1.43388i 0.974928 0.222521i −0.810394 + 1.68280i 0 −1.19745 1.19745i 0 1.46028 0.164534i 0.900969 0.433884i 0
137.1 0.623490 + 0.218169i −0.433884 + 0.900969i −0.440689 0.351438i 0 −0.467085 + 0.467085i 0 −0.549531 0.874573i −0.623490 0.781831i 0
206.1 −0.900969 + 1.43388i 0.974928 + 0.222521i −0.810394 1.68280i 0 −1.19745 + 1.19745i 0 1.46028 + 0.164534i 0.900969 + 0.433884i 0
275.1 −0.222521 + 0.0250721i 0.781831 0.623490i −0.926041 + 0.211363i 0 −0.158342 + 0.158342i 0 0.412127 0.144209i 0.222521 0.974928i 0
482.1 0.623490 0.218169i −0.433884 0.900969i −0.440689 + 0.351438i 0 −0.467085 0.467085i 0 −0.549531 + 0.874573i −0.623490 + 0.781831i 0
620.1 0.623490 + 1.78183i 0.433884 + 0.900969i −2.00435 + 1.59842i 0 −1.33485 + 1.33485i 0 −2.49939 1.57047i −0.623490 + 0.781831i 0
827.1 −0.222521 1.97493i −0.781831 + 0.623490i −2.87590 + 0.656405i 0 1.40532 + 1.40532i 0 1.27989 + 3.65773i 0.222521 0.974928i 0
896.1 −0.900969 0.566116i −0.974928 0.222521i 0.0573735 + 0.119137i 0 0.752407 + 0.752407i 0 −0.103384 + 0.917554i 0.900969 + 0.433884i 0
965.1 0.623490 1.78183i 0.433884 0.900969i −2.00435 1.59842i 0 −1.33485 1.33485i 0 −2.49939 + 1.57047i −0.623490 0.781831i 0
1034.1 −0.900969 + 0.566116i −0.974928 + 0.222521i 0.0573735 0.119137i 0 0.752407 0.752407i 0 −0.103384 0.917554i 0.900969 0.433884i 0
1448.1 −0.222521 0.0250721i 0.781831 + 0.623490i −0.926041 0.211363i 0 −0.158342 0.158342i 0 0.412127 + 0.144209i 0.222521 + 0.974928i 0
1655.1 −0.222521 + 1.97493i −0.781831 0.623490i −2.87590 0.656405i 0 1.40532 1.40532i 0 1.27989 3.65773i 0.222521 + 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1655.1
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.k Even 1 yes
2001.bf Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).