Properties

Label 2013.1.bm.d
Level 2013
Weight 1
Character orbit 2013.bm
Analytic conductor 1.005
Analytic rank 0
Dimension 16
Projective image \(D_{20}\)
CM disc. -183
Inner twists 8

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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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{20}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{2} \) \( -\zeta_{40}^{16} q^{3} \) \( + ( -\zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} \) \( + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{6} \) \( + ( \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} \) \( -\zeta_{40}^{12} q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{2} \) \( -\zeta_{40}^{16} q^{3} \) \( + ( -\zeta_{40}^{4} - \zeta_{40}^{10} + \zeta_{40}^{18} ) q^{4} \) \( + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{6} \) \( + ( \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{13} - \zeta_{40}^{19} ) q^{8} \) \( -\zeta_{40}^{12} q^{9} \) \( + \zeta_{40}^{19} q^{11} \) \( + ( -1 - \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{12} \) \( + ( -\zeta_{40}^{10} - \zeta_{40}^{14} ) q^{13} \) \( + ( -1 + \zeta_{40}^{2} + \zeta_{40}^{8} + \zeta_{40}^{14} - \zeta_{40}^{16} ) q^{16} \) \( + ( -\zeta_{40}^{7} - \zeta_{40}^{9} ) q^{17} \) \( + ( \zeta_{40} + \zeta_{40}^{7} ) q^{18} \) \( + ( -\zeta_{40}^{2} + \zeta_{40}^{10} ) q^{19} \) \( + ( -\zeta_{40}^{8} - \zeta_{40}^{14} ) q^{22} \) \( + ( \zeta_{40}^{7} - \zeta_{40}^{13} ) q^{23} \) \( + ( \zeta_{40} - \zeta_{40}^{3} - \zeta_{40}^{9} - \zeta_{40}^{15} ) q^{24} \) \( + \zeta_{40}^{16} q^{25} \) \( + ( \zeta_{40}^{3} + \zeta_{40}^{5} + \zeta_{40}^{9} - \zeta_{40}^{19} ) q^{26} \) \( -\zeta_{40}^{8} q^{27} \) \( + ( -\zeta_{40}^{11} + \zeta_{40}^{17} ) q^{29} \) \( + ( -\zeta_{40}^{3} + \zeta_{40}^{5} - \zeta_{40}^{9} + \zeta_{40}^{11} - \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{32} \) \( + \zeta_{40}^{15} q^{33} \) \( + ( \zeta_{40}^{2} + \zeta_{40}^{4} - \zeta_{40}^{16} - \zeta_{40}^{18} ) q^{34} \) \( + ( -\zeta_{40}^{2} + \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{36} \) \( + ( -\zeta_{40}^{5} - \zeta_{40}^{11} - \zeta_{40}^{17} + \zeta_{40}^{19} ) q^{38} \) \( + ( -\zeta_{40}^{6} - \zeta_{40}^{10} ) q^{39} \) \( + ( \zeta_{40}^{3} + \zeta_{40}^{9} - \zeta_{40}^{17} ) q^{44} \) \( + ( \zeta_{40}^{8} + \zeta_{40}^{16} ) q^{46} \) \( + ( \zeta_{40}^{4} + \zeta_{40}^{10} - \zeta_{40}^{12} + \zeta_{40}^{16} - \zeta_{40}^{18} ) q^{48} \) \( + \zeta_{40}^{8} q^{49} \) \( + ( -\zeta_{40}^{5} - \zeta_{40}^{11} ) q^{50} \) \( + ( -\zeta_{40}^{3} - \zeta_{40}^{5} ) q^{51} \) \( + ( -1 - \zeta_{40}^{4} + \zeta_{40}^{8} + \zeta_{40}^{12} + \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{52} \) \( + ( -\zeta_{40}^{9} - \zeta_{40}^{15} ) q^{53} \) \( + ( \zeta_{40}^{3} - \zeta_{40}^{17} ) q^{54} \) \( + ( \zeta_{40}^{6} + \zeta_{40}^{18} ) q^{57} \) \( + ( 1 - \zeta_{40}^{12} ) q^{58} \) \( + ( -\zeta_{40}^{13} + \zeta_{40}^{15} ) q^{59} \) \( -\zeta_{40}^{8} q^{61} \) \( + ( -1 + \zeta_{40}^{4} - \zeta_{40}^{6} + \zeta_{40}^{10} - \zeta_{40}^{12} + \zeta_{40}^{14} - \zeta_{40}^{18} ) q^{64} \) \( + ( -\zeta_{40}^{4} - \zeta_{40}^{10} ) q^{66} \) \( + ( \zeta_{40}^{5} + \zeta_{40}^{7} + \zeta_{40}^{11} + \zeta_{40}^{13} + \zeta_{40}^{17} + \zeta_{40}^{19} ) q^{68} \) \( + ( \zeta_{40}^{3} - \zeta_{40}^{9} ) q^{69} \) \( + ( \zeta_{40}^{3} + \zeta_{40}^{13} ) q^{71} \) \( + ( -\zeta_{40}^{5} - \zeta_{40}^{11} - \zeta_{40}^{17} + \zeta_{40}^{19} ) q^{72} \) \( + ( \zeta_{40}^{12} - \zeta_{40}^{16} ) q^{73} \) \( + \zeta_{40}^{12} q^{75} \) \( + ( 2 + \zeta_{40}^{6} - \zeta_{40}^{8} + \zeta_{40}^{12} - \zeta_{40}^{14} ) q^{76} \) \( + ( \zeta_{40} + \zeta_{40}^{5} - \zeta_{40}^{15} - \zeta_{40}^{19} ) q^{78} \) \( -\zeta_{40}^{4} q^{81} \) \( + ( -\zeta_{40}^{7} + \zeta_{40}^{13} ) q^{87} \) \( + ( -\zeta_{40}^{4} + \zeta_{40}^{6} + \zeta_{40}^{12} + \zeta_{40}^{18} ) q^{88} \) \( + ( \zeta_{40} - \zeta_{40}^{19} ) q^{89} \) \( + ( -\zeta_{40}^{3} - \zeta_{40}^{5} ) q^{92} \) \( + ( \zeta_{40} - \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{11} + \zeta_{40}^{13} + \zeta_{40}^{19} ) q^{96} \) \( + ( -\zeta_{40}^{3} + \zeta_{40}^{17} ) q^{98} \) \( + \zeta_{40}^{11} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 8q^{34} \) \(\mathstrut -\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 4q^{48} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 20q^{52} \) \(\mathstrut +\mathstrut 12q^{58} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 4q^{66} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 4q^{75} \) \(\mathstrut +\mathstrut 40q^{76} \) \(\mathstrut -\mathstrut 4q^{81} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(1222\) \(1343\) \(1465\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{40}^{4}\)

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
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 0.987688i
−0.987688 0.156434i
0.987688 0.156434i
−0.156434 0.987688i
0.156434 + 0.987688i
−0.987688 + 0.156434i
0.453990 + 0.891007i
−0.891007 + 0.453990i
0.891007 0.453990i
−0.453990 0.891007i
0.453990 0.891007i
−0.891007 0.453990i
0.891007 + 0.453990i
−0.453990 + 0.891007i
−0.550672 + 1.69480i 0.809017 0.587785i −1.76007 1.27877i 0 0.550672 + 1.69480i 0 1.69480 1.23134i 0.309017 0.951057i 0
548.2 −0.280582 + 0.863541i 0.809017 0.587785i 0.142040 + 0.103198i 0 0.280582 + 0.863541i 0 −0.863541 + 0.627399i 0.309017 0.951057i 0
548.3 0.280582 0.863541i 0.809017 0.587785i 0.142040 + 0.103198i 0 −0.280582 0.863541i 0 0.863541 0.627399i 0.309017 0.951057i 0
548.4 0.550672 1.69480i 0.809017 0.587785i −1.76007 1.27877i 0 −0.550672 1.69480i 0 −1.69480 + 1.23134i 0.309017 0.951057i 0
731.1 −0.550672 1.69480i 0.809017 + 0.587785i −1.76007 + 1.27877i 0 0.550672 1.69480i 0 1.69480 + 1.23134i 0.309017 + 0.951057i 0
731.2 −0.280582 0.863541i 0.809017 + 0.587785i 0.142040 0.103198i 0 0.280582 0.863541i 0 −0.863541 0.627399i 0.309017 + 0.951057i 0
731.3 0.280582 + 0.863541i 0.809017 + 0.587785i 0.142040 0.103198i 0 −0.280582 + 0.863541i 0 0.863541 + 0.627399i 0.309017 + 0.951057i 0
731.4 0.550672 + 1.69480i 0.809017 + 0.587785i −1.76007 + 1.27877i 0 −0.550672 + 1.69480i 0 −1.69480 1.23134i 0.309017 + 0.951057i 0
1280.1 −1.59811 1.16110i −0.309017 + 0.951057i 0.896802 + 2.76007i 0 1.59811 1.16110i 0 1.16110 3.57349i −0.809017 0.587785i 0
1280.2 −0.253116 0.183900i −0.309017 + 0.951057i −0.278768 0.857960i 0 0.253116 0.183900i 0 −0.183900 + 0.565985i −0.809017 0.587785i 0
1280.3 0.253116 + 0.183900i −0.309017 + 0.951057i −0.278768 0.857960i 0 −0.253116 + 0.183900i 0 0.183900 0.565985i −0.809017 0.587785i 0
1280.4 1.59811 + 1.16110i −0.309017 + 0.951057i 0.896802 + 2.76007i 0 −1.59811 + 1.16110i 0 −1.16110 + 3.57349i −0.809017 0.587785i 0
1829.1 −1.59811 + 1.16110i −0.309017 0.951057i 0.896802 2.76007i 0 1.59811 + 1.16110i 0 1.16110 + 3.57349i −0.809017 + 0.587785i 0
1829.2 −0.253116 + 0.183900i −0.309017 0.951057i −0.278768 + 0.857960i 0 0.253116 + 0.183900i 0 −0.183900 0.565985i −0.809017 + 0.587785i 0
1829.3 0.253116 0.183900i −0.309017 0.951057i −0.278768 + 0.857960i 0 −0.253116 0.183900i 0 0.183900 + 0.565985i −0.809017 + 0.587785i 0
1829.4 1.59811 1.16110i −0.309017 0.951057i 0.896802 2.76007i 0 −1.59811 1.16110i 0 −1.16110 3.57349i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1829.4
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 \(T_{2}^{16} + \cdots\) acting on \(S_{1}^{\mathrm{new}}(2013, [\chi])\).