Properties

Label 2001.1.bf.d
Level 2001
Weight 1
Character orbit 2001.bf
Analytic conductor 0.999
Analytic rank 0
Dimension 24
Projective image \(D_{84}\)
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: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{28})\)
Coefficient field: \(\Q(\zeta_{84})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{84}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{84} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{84}^{7} + \zeta_{84}^{20} ) q^{2} \) \( + \zeta_{84}^{25} q^{3} \) \( + ( \zeta_{84}^{14} + \zeta_{84}^{27} + \zeta_{84}^{40} ) q^{4} \) \( + ( -\zeta_{84}^{3} + \zeta_{84}^{32} ) q^{6} \) \( + ( -\zeta_{84}^{5} - \zeta_{84}^{18} + \zeta_{84}^{21} + \zeta_{84}^{34} ) q^{8} \) \( -\zeta_{84}^{8} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{84}^{7} + \zeta_{84}^{20} ) q^{2} \) \( + \zeta_{84}^{25} q^{3} \) \( + ( \zeta_{84}^{14} + \zeta_{84}^{27} + \zeta_{84}^{40} ) q^{4} \) \( + ( -\zeta_{84}^{3} + \zeta_{84}^{32} ) q^{6} \) \( + ( -\zeta_{84}^{5} - \zeta_{84}^{18} + \zeta_{84}^{21} + \zeta_{84}^{34} ) q^{8} \) \( -\zeta_{84}^{8} q^{9} \) \( + ( -\zeta_{84}^{10} - \zeta_{84}^{23} + \zeta_{84}^{39} ) q^{12} \) \( + ( \zeta_{84}^{29} + \zeta_{84}^{37} ) q^{13} \) \( + ( -\zeta_{84}^{12} - \zeta_{84}^{25} + \zeta_{84}^{28} - \zeta_{84}^{38} + \zeta_{84}^{41} ) q^{16} \) \( + ( -\zeta_{84}^{15} - \zeta_{84}^{28} ) q^{18} \) \( -\zeta_{84}^{39} q^{23} \) \( + ( \zeta_{84} - \zeta_{84}^{4} - \zeta_{84}^{17} - \zeta_{84}^{30} ) q^{24} \) \( -\zeta_{84}^{6} q^{25} \) \( + ( -\zeta_{84}^{2} - \zeta_{84}^{7} - \zeta_{84}^{15} + \zeta_{84}^{36} ) q^{26} \) \( -\zeta_{84}^{33} q^{27} \) \( -\zeta_{84}^{41} q^{29} \) \( + ( \zeta_{84}^{31} + \zeta_{84}^{38} ) q^{31} \) \( + ( \zeta_{84}^{3} - \zeta_{84}^{6} + \zeta_{84}^{16} - \zeta_{84}^{19} - \zeta_{84}^{32} + \zeta_{84}^{35} ) q^{32} \) \( + ( \zeta_{84}^{6} - \zeta_{84}^{22} - \zeta_{84}^{35} ) q^{36} \) \( + ( -\zeta_{84}^{12} - \zeta_{84}^{20} ) q^{39} \) \( + ( \zeta_{84}^{2} + \zeta_{84}^{19} ) q^{41} \) \( + ( \zeta_{84}^{4} + \zeta_{84}^{17} ) q^{46} \) \( + ( -\zeta_{84}^{11} - \zeta_{84}^{34} ) q^{47} \) \( + ( \zeta_{84}^{8} - \zeta_{84}^{11} + \zeta_{84}^{21} - \zeta_{84}^{24} - \zeta_{84}^{37} ) q^{48} \) \( -\zeta_{84}^{30} q^{49} \) \( + ( -\zeta_{84}^{13} - \zeta_{84}^{26} ) q^{50} \) \( + ( -\zeta_{84} - \zeta_{84}^{9} - \zeta_{84}^{14} - \zeta_{84}^{22} - \zeta_{84}^{27} - \zeta_{84}^{35} ) q^{52} \) \( + ( \zeta_{84}^{11} - \zeta_{84}^{40} ) q^{54} \) \( + ( \zeta_{84}^{6} + \zeta_{84}^{19} ) q^{58} \) \( + ( \zeta_{84}^{15} + \zeta_{84}^{27} ) q^{59} \) \( + ( -\zeta_{84}^{3} - \zeta_{84}^{9} - \zeta_{84}^{16} + \zeta_{84}^{38} ) q^{62} \) \( + ( -1 + \zeta_{84}^{10} - \zeta_{84}^{13} + \zeta_{84}^{23} - \zeta_{84}^{26} + \zeta_{84}^{36} - \zeta_{84}^{39} ) q^{64} \) \( + \zeta_{84}^{22} q^{69} \) \( + ( \zeta_{84} + \zeta_{84}^{23} ) q^{71} \) \( + ( 1 + \zeta_{84}^{13} + \zeta_{84}^{26} - \zeta_{84}^{29} ) q^{72} \) \( + ( \zeta_{84}^{5} + \zeta_{84}^{10} ) q^{73} \) \( -\zeta_{84}^{31} q^{75} \) \( + ( -\zeta_{84}^{19} - \zeta_{84}^{27} - \zeta_{84}^{32} - \zeta_{84}^{40} ) q^{78} \) \( + \zeta_{84}^{16} q^{81} \) \( + ( \zeta_{84}^{9} + \zeta_{84}^{22} + \zeta_{84}^{26} + \zeta_{84}^{39} ) q^{82} \) \( + \zeta_{84}^{24} q^{87} \) \( + ( \zeta_{84}^{11} + \zeta_{84}^{24} + \zeta_{84}^{37} ) q^{92} \) \( + ( -\zeta_{84}^{14} - \zeta_{84}^{21} ) q^{93} \) \( + ( \zeta_{84}^{12} - \zeta_{84}^{18} - \zeta_{84}^{31} - \zeta_{84}^{41} ) q^{94} \) \( + ( \zeta_{84}^{2} + \zeta_{84}^{15} - \zeta_{84}^{18} + \zeta_{84}^{28} - \zeta_{84}^{31} + \zeta_{84}^{41} ) q^{96} \) \( + ( \zeta_{84}^{8} - \zeta_{84}^{37} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(24q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(24q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut -\mathstrut 6q^{16} \) \(\mathstrut +\mathstrut 12q^{18} \) \(\mathstrut -\mathstrut 6q^{24} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 2q^{26} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 4q^{32} \) \(\mathstrut +\mathstrut 6q^{36} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 6q^{48} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 10q^{52} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 28q^{64} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 22q^{72} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 4q^{78} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut -\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 18q^{96} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(553\) \(668\) \(1132\)
\(\chi(n)\) \(-\zeta_{84}^{27}\) \(-1\) \(-1\)

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
−0.563320 + 0.826239i
0.997204 + 0.0747301i
0.149042 + 0.988831i
−0.930874 0.365341i
−0.563320 0.826239i
0.997204 0.0747301i
−0.680173 + 0.733052i
−0.294755 0.955573i
0.149042 0.988831i
−0.930874 + 0.365341i
0.930874 0.365341i
−0.149042 + 0.988831i
0.294755 + 0.955573i
0.680173 0.733052i
−0.997204 + 0.0747301i
0.563320 + 0.826239i
0.930874 + 0.365341i
−0.149042 0.988831i
−0.997204 0.0747301i
0.563320 0.826239i
−0.0397866 0.0633201i −0.680173 0.733052i 0.431457 0.895930i 0 −0.0193551 + 0.0722342i 0 −0.148209 + 0.0166991i −0.0747301 + 0.997204i 0
68.2 0.940755 + 1.49720i −0.294755 + 0.955573i −0.922715 + 1.91604i 0 −1.70798 + 0.457652i 0 −1.97963 + 0.223051i −0.826239 0.563320i 0
137.1 −1.85486 0.649042i −0.563320 0.826239i 2.23740 + 1.78427i 0 0.508614 + 1.89817i 0 −1.94648 3.09781i −0.365341 + 0.930874i 0
137.2 1.23137 + 0.430874i 0.997204 0.0747301i 0.548780 + 0.437637i 0 1.26012 + 0.337649i 0 −0.206893 0.329269i 0.988831 0.149042i 0
206.1 −0.0397866 + 0.0633201i −0.680173 + 0.733052i 0.431457 + 0.895930i 0 −0.0193551 0.0722342i 0 −0.148209 0.0166991i −0.0747301 0.997204i 0
206.2 0.940755 1.49720i −0.294755 0.955573i −0.922715 1.91604i 0 −1.70798 0.457652i 0 −1.97963 0.223051i −0.826239 + 0.563320i 0
275.1 −1.59908 + 0.180173i 0.149042 + 0.988831i 1.54966 0.353699i 0 −0.416490 1.55436i 0 −0.895403 + 0.313315i −0.955573 + 0.294755i 0
275.2 1.82160 0.205245i −0.930874 0.365341i 2.30117 0.525226i 0 −1.77066 0.474448i 0 2.35375 0.823611i 0.733052 + 0.680173i 0
482.1 −1.85486 + 0.649042i −0.563320 + 0.826239i 2.23740 1.78427i 0 0.508614 1.89817i 0 −1.94648 + 3.09781i −0.365341 0.930874i 0
482.2 1.23137 0.430874i 0.997204 + 0.0747301i 0.548780 0.437637i 0 1.26012 0.337649i 0 −0.206893 + 0.329269i 0.988831 + 0.149042i 0
620.1 −0.500684 1.43087i −0.997204 0.0747301i −1.01488 + 0.809342i 0 0.392355 + 1.46429i 0 0.382617 + 0.240414i 0.988831 + 0.149042i 0
620.2 −0.122805 0.350958i 0.563320 0.826239i 0.673741 0.537291i 0 −0.359154 0.0962349i 0 −0.586137 0.368294i −0.365341 0.930874i 0
827.1 0.0895474 + 0.794755i 0.930874 + 0.365341i 0.351311 0.0801844i 0 −0.206999 + 0.772532i 0 0.359338 + 1.02693i 0.733052 + 0.680173i 0
827.2 0.132974 + 1.18017i −0.149042 0.988831i −0.400198 + 0.0913425i 0 1.14717 0.307384i 0 0.231237 + 0.660838i −0.955573 + 0.294755i 0
896.1 −0.791295 0.497204i 0.294755 + 0.955573i −0.0549471 0.114099i 0 0.241876 0.902694i 0 −0.117886 + 1.04627i −0.826239 + 0.563320i 0
896.2 1.69226 + 1.06332i 0.680173 0.733052i 1.29922 + 2.69787i 0 1.93050 0.517276i 0 −0.446293 + 3.96096i −0.0747301 0.997204i 0
965.1 −0.500684 + 1.43087i −0.997204 + 0.0747301i −1.01488 0.809342i 0 0.392355 1.46429i 0 0.382617 0.240414i 0.988831 0.149042i 0
965.2 −0.122805 + 0.350958i 0.563320 + 0.826239i 0.673741 + 0.537291i 0 −0.359154 + 0.0962349i 0 −0.586137 + 0.368294i −0.365341 + 0.930874i 0
1034.1 −0.791295 + 0.497204i 0.294755 0.955573i −0.0549471 + 0.114099i 0 0.241876 + 0.902694i 0 −0.117886 1.04627i −0.826239 0.563320i 0
1034.2 1.69226 1.06332i 0.680173 + 0.733052i 1.29922 2.69787i 0 1.93050 + 0.517276i 0 −0.446293 3.96096i −0.0747301 + 0.997204i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1655.2
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 \(T_{2}^{24} - \cdots\) acting on \(S_{1}^{\mathrm{new}}(2001, [\chi])\).