# Properties

 Label 2001.1.bf.c Level $2001$ Weight $1$ Character orbit 2001.bf Analytic conductor $0.999$ Analytic rank $0$ Dimension $24$ Projective image $D_{84}$ CM discriminant -23 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2001.bf (of order $$28$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.998629090279$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$2$$ over $$\Q(\zeta_{28})$$ Coefficient field: $$\Q(\zeta_{84})$$ Defining polynomial: $$x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## Character values

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

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

## 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
 0.997204 + 0.0747301i −0.563320 + 0.826239i −0.930874 − 0.365341i 0.149042 + 0.988831i 0.997204 − 0.0747301i −0.563320 − 0.826239i −0.294755 − 0.955573i −0.680173 + 0.733052i −0.930874 + 0.365341i 0.149042 − 0.988831i −0.149042 + 0.988831i 0.930874 − 0.365341i 0.680173 − 0.733052i 0.294755 + 0.955573i 0.563320 + 0.826239i −0.997204 + 0.0747301i −0.149042 − 0.988831i 0.930874 + 0.365341i 0.563320 − 0.826239i −0.997204 − 0.0747301i
−0.940755 1.49720i 0.365341 0.930874i −0.922715 + 1.91604i 0 −1.73740 + 0.328735i 0 1.97963 0.223051i −0.733052 0.680173i 0
68.2 0.0397866 + 0.0633201i −0.988831 + 0.149042i 0.431457 0.895930i 0 −0.0487796 0.0566829i 0 0.148209 0.0166991i 0.955573 0.294755i 0
137.1 −1.23137 0.430874i 0.955573 + 0.294755i 0.548780 + 0.437637i 0 −1.04966 0.774683i 0 0.206893 + 0.329269i 0.826239 + 0.563320i 0
137.2 1.85486 + 0.649042i −0.733052 + 0.680173i 2.23740 + 1.78427i 0 −1.80117 + 0.785841i 0 1.94648 + 3.09781i 0.0747301 0.997204i 0
206.1 −0.940755 + 1.49720i 0.365341 + 0.930874i −0.922715 1.91604i 0 −1.73740 0.328735i 0 1.97963 + 0.223051i −0.733052 + 0.680173i 0
206.2 0.0397866 0.0633201i −0.988831 0.149042i 0.431457 + 0.895930i 0 −0.0487796 + 0.0566829i 0 0.148209 + 0.0166991i 0.955573 + 0.294755i 0
275.1 −1.82160 + 0.205245i 0.0747301 0.997204i 2.30117 0.525226i 0 0.0685427 + 1.83184i 0 −2.35375 + 0.823611i −0.988831 0.149042i 0
275.2 1.59908 0.180173i 0.826239 + 0.563320i 1.54966 0.353699i 0 1.42271 + 0.751927i 0 0.895403 0.313315i 0.365341 + 0.930874i 0
482.1 −1.23137 + 0.430874i 0.955573 0.294755i 0.548780 0.437637i 0 −1.04966 + 0.774683i 0 0.206893 0.329269i 0.826239 0.563320i 0
482.2 1.85486 0.649042i −0.733052 0.680173i 2.23740 1.78427i 0 −1.80117 0.785841i 0 1.94648 3.09781i 0.0747301 + 0.997204i 0
620.1 0.122805 + 0.350958i −0.733052 0.680173i 0.673741 0.537291i 0 0.148689 0.340799i 0 0.586137 + 0.368294i 0.0747301 + 0.997204i 0
620.2 0.500684 + 1.43087i 0.955573 0.294755i −1.01488 + 0.809342i 0 0.900198 + 1.21972i 0 −0.382617 0.240414i 0.826239 0.563320i 0
827.1 −0.132974 1.18017i 0.826239 + 0.563320i −0.400198 + 0.0913425i 0 0.554947 1.05001i 0 −0.231237 0.660838i 0.365341 + 0.930874i 0
827.2 −0.0895474 0.794755i 0.0747301 0.997204i 0.351311 0.0801844i 0 −0.799225 + 0.0299049i 0 −0.359338 1.02693i −0.988831 0.149042i 0
896.1 −1.69226 1.06332i −0.988831 0.149042i 1.29922 + 2.69787i 0 1.51488 + 1.30366i 0 0.446293 3.96096i 0.955573 + 0.294755i 0
896.2 0.791295 + 0.497204i 0.365341 + 0.930874i −0.0549471 0.114099i 0 −0.173741 + 0.918245i 0 0.117886 1.04627i −0.733052 + 0.680173i 0
965.1 0.122805 0.350958i −0.733052 + 0.680173i 0.673741 + 0.537291i 0 0.148689 + 0.340799i 0 0.586137 0.368294i 0.0747301 0.997204i 0
965.2 0.500684 1.43087i 0.955573 + 0.294755i −1.01488 0.809342i 0 0.900198 1.21972i 0 −0.382617 + 0.240414i 0.826239 + 0.563320i 0
1034.1 −1.69226 + 1.06332i −0.988831 + 0.149042i 1.29922 2.69787i 0 1.51488 1.30366i 0 0.446293 + 3.96096i 0.955573 0.294755i 0
1034.2 0.791295 0.497204i 0.365341 0.930874i −0.0549471 + 0.114099i 0 −0.173741 0.918245i 0 0.117886 + 1.04627i −0.733052 0.680173i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1655.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
87.k even 28 1 inner
2001.bf odd 28 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.1.bf.c 24
3.b odd 2 1 2001.1.bf.d yes 24
23.b odd 2 1 CM 2001.1.bf.c 24
29.f odd 28 1 2001.1.bf.d yes 24
69.c even 2 1 2001.1.bf.d yes 24
87.k even 28 1 inner 2001.1.bf.c 24
667.o even 28 1 2001.1.bf.d yes 24
2001.bf odd 28 1 inner 2001.1.bf.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.1.bf.c 24 1.a even 1 1 trivial
2001.1.bf.c 24 23.b odd 2 1 CM
2001.1.bf.c 24 87.k even 28 1 inner
2001.1.bf.c 24 2001.bf odd 28 1 inner
2001.1.bf.d yes 24 3.b odd 2 1
2001.1.bf.d yes 24 29.f odd 28 1
2001.1.bf.d yes 24 69.c even 2 1
2001.1.bf.d yes 24 667.o even 28 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(2001, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 16 T + 212 T^{2} - 432 T^{3} + 1457 T^{4} - 378 T^{5} + 966 T^{6} + 16 T^{7} - 1167 T^{8} + 874 T^{9} + 1052 T^{10} + 420 T^{11} + 405 T^{12} - 476 T^{13} - 460 T^{14} - 48 T^{15} + 16 T^{16} + 12 T^{17} + 42 T^{18} + 42 T^{19} + T^{20} - 16 T^{21} - 5 T^{22} + 2 T^{23} + T^{24}$$
$3$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}$$
$5$ $$T^{24}$$
$7$ $$T^{24}$$
$11$ $$T^{24}$$
$13$ $$1 - 79 T^{2} + 1872 T^{4} + 3146 T^{6} + 4856 T^{8} - 3142 T^{10} + 2359 T^{12} - 888 T^{14} + 222 T^{16} - 46 T^{18} + 17 T^{20} - 2 T^{22} + T^{24}$$
$17$ $$T^{24}$$
$19$ $$T^{24}$$
$23$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$
$29$ $$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$
$31$ $$1 - 2 T + 2 T^{2} + 2 T^{3} - 13 T^{4} + 28 T^{5} - 28 T^{6} + 114 T^{7} - 103 T^{8} - 106 T^{9} + 674 T^{10} - 1456 T^{11} + 1455 T^{12} + 1456 T^{13} + 674 T^{14} + 106 T^{15} - 103 T^{16} - 114 T^{17} - 28 T^{18} - 28 T^{19} - 13 T^{20} - 2 T^{21} + 2 T^{22} + 2 T^{23} + T^{24}$$
$37$ $$T^{24}$$
$41$ $$1 + 16 T + 128 T^{2} - 436 T^{3} + 848 T^{4} + 910 T^{5} + 1064 T^{6} - 1640 T^{7} + 1948 T^{8} + 1492 T^{9} + 1346 T^{10} - 1470 T^{11} + 1070 T^{12} + 504 T^{13} + 464 T^{14} - 470 T^{15} + 268 T^{16} + 58 T^{17} + 56 T^{18} - 56 T^{19} + 29 T^{20} + 2 T^{21} + 2 T^{22} - 2 T^{23} + T^{24}$$
$43$ $$T^{24}$$
$47$ $$1 + 12 T + 205 T^{2} + 548 T^{3} - 328 T^{4} - 336 T^{5} + 2366 T^{6} - 1832 T^{7} - 152 T^{8} + 1224 T^{9} - 775 T^{10} + 546 T^{11} + 657 T^{12} + 14 T^{13} + 527 T^{14} + 414 T^{15} + 51 T^{16} + 110 T^{17} + 77 T^{18} + 15 T^{20} + 12 T^{21} + 2 T^{22} + 2 T^{23} + T^{24}$$
$53$ $$T^{24}$$
$59$ $$( 1 + 6 T^{2} + 5 T^{4} + T^{6} )^{4}$$
$61$ $$T^{24}$$
$67$ $$T^{24}$$
$71$ $$1 - 11 T^{2} + 96 T^{4} + 234 T^{6} + 1284 T^{8} + 1978 T^{10} + 1519 T^{12} + 716 T^{14} + 258 T^{16} + 38 T^{18} + 13 T^{20} + 6 T^{22} + T^{24}$$
$73$ $$1 + 12 T + 247 T^{2} + 1850 T^{3} + 7932 T^{4} + 21686 T^{5} + 39130 T^{6} + 47420 T^{7} + 38698 T^{8} + 20474 T^{9} + 5434 T^{10} - 1456 T^{11} - 2521 T^{12} - 1456 T^{13} - 390 T^{14} + 92 T^{15} + 128 T^{16} + 26 T^{17} - 28 T^{18} - 28 T^{19} - 13 T^{20} - 2 T^{21} + 2 T^{22} + 2 T^{23} + T^{24}$$
$79$ $$T^{24}$$
$83$ $$T^{24}$$
$89$ $$T^{24}$$
$97$ $$T^{24}$$