Properties

Label 2013.1.bm.c
Level 2013
Weight 1
Character orbit 2013.bm
Analytic conductor 1.005
Analytic rank 0
Dimension 8
Projective image \(D_{10}\)
CM disc. -183
Inner twists 8

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2013.bm (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00461787043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{10}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{10} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{2} \) \( + \zeta_{20}^{8} q^{3} \) \( + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{4} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{6} \) \( + ( \zeta_{20} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{8} \) \( -\zeta_{20}^{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{2} \) \( + \zeta_{20}^{8} q^{3} \) \( + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{4} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{6} \) \( + ( \zeta_{20} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{8} \) \( -\zeta_{20}^{6} q^{9} \) \( -\zeta_{20}^{7} q^{11} \) \( + ( 1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{12} \) \( + ( -1 + \zeta_{20}^{2} ) q^{13} \) \( + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{16} \) \( + ( \zeta_{20} + \zeta_{20}^{7} ) q^{17} \) \( + ( \zeta_{20} + \zeta_{20}^{3} ) q^{18} \) \( + ( -1 + \zeta_{20}^{6} ) q^{19} \) \( + ( \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{22} \) \( + ( \zeta_{20} - \zeta_{20}^{9} ) q^{23} \) \( + ( \zeta_{20}^{3} + \zeta_{20}^{5} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{24} \) \( + \zeta_{20}^{8} q^{25} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{9} ) q^{26} \) \( + \zeta_{20}^{4} q^{27} \) \( + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{29} \) \( + ( -\zeta_{20} - \zeta_{20}^{3} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{32} \) \( + \zeta_{20}^{5} q^{33} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{34} \) \( + ( -1 + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{36} \) \( + ( -\zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{38} \) \( + ( -1 - \zeta_{20}^{8} ) q^{39} \) \( + ( -\zeta_{20} + \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{44} \) \( + ( \zeta_{20}^{4} + 2 \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{46} \) \( + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{48} \) \( + \zeta_{20}^{4} q^{49} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{50} \) \( + ( -\zeta_{20}^{5} + \zeta_{20}^{9} ) q^{51} \) \( + ( 1 - \zeta_{20}^{6} ) q^{52} \) \( + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{53} \) \( + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{54} \) \( + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{57} \) \( + ( 1 - \zeta_{20}^{6} - 2 \zeta_{20}^{8} ) q^{58} \) \( + ( \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{59} \) \( + \zeta_{20}^{4} q^{61} \) \( + ( -\zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{64} \) \( + ( -1 - \zeta_{20}^{2} ) q^{66} \) \( + ( -\zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{68} \) \( + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{69} \) \( + ( -\zeta_{20} - \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{72} \) \( + ( -\zeta_{20}^{6} + \zeta_{20}^{8} ) q^{73} \) \( -\zeta_{20}^{6} q^{75} \) \( + ( 2 + \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{76} \) \( + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{78} \) \( -\zeta_{20}^{2} q^{81} \) \( + ( \zeta_{20} - \zeta_{20}^{9} ) q^{87} \) \( + ( -\zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{88} \) \( + ( \zeta_{20}^{3} - \zeta_{20}^{7} ) q^{89} \) \( + ( -2 \zeta_{20} - 2 \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{92} \) \( + ( \zeta_{20} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{96} \) \( + 2 \zeta_{20}^{6} q^{97} \) \( + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{98} \) \( -\zeta_{20}^{3} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 12q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 12q^{48} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 10q^{58} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 10q^{66} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 16q^{76} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2013\mathbb{Z}\right)^\times\).

\(n\) \(1222\) \(1343\) \(1465\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
548.1
0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.587785 + 1.80902i −0.809017 + 0.587785i −2.11803 1.53884i 0 −0.587785 1.80902i 0 2.48990 1.80902i 0.309017 0.951057i 0
548.2 0.587785 1.80902i −0.809017 + 0.587785i −2.11803 1.53884i 0 0.587785 + 1.80902i 0 −2.48990 + 1.80902i 0.309017 0.951057i 0
731.1 −0.587785 1.80902i −0.809017 0.587785i −2.11803 + 1.53884i 0 −0.587785 + 1.80902i 0 2.48990 + 1.80902i 0.309017 + 0.951057i 0
731.2 0.587785 + 1.80902i −0.809017 0.587785i −2.11803 + 1.53884i 0 0.587785 1.80902i 0 −2.48990 1.80902i 0.309017 + 0.951057i 0
1280.1 −0.951057 0.690983i 0.309017 0.951057i 0.118034 + 0.363271i 0 −0.951057 + 0.690983i 0 −0.224514 + 0.690983i −0.809017 0.587785i 0
1280.2 0.951057 + 0.690983i 0.309017 0.951057i 0.118034 + 0.363271i 0 0.951057 0.690983i 0 0.224514 0.690983i −0.809017 0.587785i 0
1829.1 −0.951057 + 0.690983i 0.309017 + 0.951057i 0.118034 0.363271i 0 −0.951057 0.690983i 0 −0.224514 0.690983i −0.809017 + 0.587785i 0
1829.2 0.951057 0.690983i 0.309017 + 0.951057i 0.118034 0.363271i 0 0.951057 + 0.690983i 0 0.224514 + 0.690983i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1829.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
183.d Odd 1 CM by \(\Q(\sqrt{-183}) \) yes
3.b Odd 1 yes
11.c Even 1 yes
33.h Odd 1 yes
61.b Even 1 yes
671.x Even 1 yes
2013.bm Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} \) \(\mathstrut +\mathstrut 5 T_{2}^{6} \) \(\mathstrut +\mathstrut 10 T_{2}^{4} \) \(\mathstrut +\mathstrut 25 \) acting on \(S_{1}^{\mathrm{new}}(2013, [\chi])\).