Properties

Label 2013.1.bm.a
Level $2013$
Weight $1$
Character orbit 2013.bm
Analytic conductor $1.005$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -183
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.00461787043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.490312449.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{10} ) q^{2} + \zeta_{10}^{4} q^{3} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{4} + ( -1 - \zeta_{10}^{4} ) q^{6} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{10} ) q^{2} + \zeta_{10}^{4} q^{3} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{4} + ( -1 - \zeta_{10}^{4} ) q^{6} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{8} -\zeta_{10}^{3} q^{9} + \zeta_{10} q^{11} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{12} + ( 1 - \zeta_{10} ) q^{13} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{16} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{17} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{18} + ( 1 - \zeta_{10}^{3} ) q^{19} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{22} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + ( -1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{4} ) q^{24} + \zeta_{10}^{4} q^{25} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{26} + \zeta_{10}^{2} q^{27} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{29} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{32} - q^{33} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{34} + ( 1 - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{36} + ( -1 + \zeta_{10} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{38} + ( 1 + \zeta_{10}^{4} ) q^{39} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{44} + ( \zeta_{10}^{2} - 2 \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{46} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{48} + \zeta_{10}^{2} q^{49} + ( -1 - \zeta_{10}^{4} ) q^{50} + ( -1 - \zeta_{10}^{2} ) q^{51} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{52} + ( -1 + \zeta_{10} ) q^{53} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{54} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{57} + ( 1 - \zeta_{10}^{3} + 2 \zeta_{10}^{4} ) q^{58} + ( -1 - \zeta_{10}^{2} ) q^{59} + \zeta_{10}^{2} q^{61} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{64} + ( 1 - \zeta_{10} ) q^{66} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{68} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{69} -2 \zeta_{10}^{2} q^{71} + ( -1 + \zeta_{10} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{72} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{73} -\zeta_{10}^{3} q^{75} + ( 2 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{76} + ( -2 + \zeta_{10} - \zeta_{10}^{4} ) q^{78} -\zeta_{10} q^{81} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{87} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{88} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{89} + ( -1 - \zeta_{10}^{2} + 2 \zeta_{10}^{3} - 2 \zeta_{10}^{4} ) q^{92} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} - 2 \zeta_{10}^{4} ) q^{96} -2 \zeta_{10}^{3} q^{97} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{98} -\zeta_{10}^{4} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} - q^{3} + 2q^{4} - 3q^{6} - q^{8} - q^{9} + O(q^{10}) \) \( 4q - 3q^{2} - q^{3} + 2q^{4} - 3q^{6} - q^{8} - q^{9} + q^{11} + 2q^{12} + 3q^{13} + 2q^{17} + 2q^{18} + 3q^{19} - 2q^{22} + 2q^{23} - q^{24} - q^{25} - q^{26} - q^{27} + 2q^{29} - 4q^{32} - 4q^{33} - 4q^{34} + 2q^{36} - q^{38} + 3q^{39} + 3q^{44} - 4q^{46} - q^{49} - 3q^{50} - 3q^{51} - q^{52} - 3q^{53} + 2q^{54} - 2q^{57} + q^{58} - 3q^{59} - q^{61} + 3q^{64} + 3q^{66} + q^{68} + 2q^{69} + 2q^{71} - q^{72} - 2q^{73} - q^{75} + 4q^{76} - 6q^{78} - q^{81} + 2q^{87} - 4q^{88} + 2q^{89} + q^{92} + q^{96} - 2q^{97} + 2q^{98} + q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(1222\) \(1343\) \(1465\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{10}\)

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
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.190983 + 0.587785i −0.809017 + 0.587785i 0.500000 + 0.363271i 0 −0.190983 0.587785i 0 −0.809017 + 0.587785i 0.309017 0.951057i 0
731.1 −0.190983 0.587785i −0.809017 0.587785i 0.500000 0.363271i 0 −0.190983 + 0.587785i 0 −0.809017 0.587785i 0.309017 + 0.951057i 0
1280.1 −1.30902 0.951057i 0.309017 0.951057i 0.500000 + 1.53884i 0 −1.30902 + 0.951057i 0 0.309017 0.951057i −0.809017 0.587785i 0
1829.1 −1.30902 + 0.951057i 0.309017 + 0.951057i 0.500000 1.53884i 0 −1.30902 0.951057i 0 0.309017 + 0.951057i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
183.d odd 2 1 CM by \(\Q(\sqrt{-183}) \)
11.c even 5 1 inner
2013.bm odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2013.1.bm.a 4
3.b odd 2 1 2013.1.bm.b yes 4
11.c even 5 1 inner 2013.1.bm.a 4
33.h odd 10 1 2013.1.bm.b yes 4
61.b even 2 1 2013.1.bm.b yes 4
183.d odd 2 1 CM 2013.1.bm.a 4
671.x even 10 1 2013.1.bm.b yes 4
2013.bm odd 10 1 inner 2013.1.bm.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2013.1.bm.a 4 1.a even 1 1 trivial
2013.1.bm.a 4 11.c even 5 1 inner
2013.1.bm.a 4 183.d odd 2 1 CM
2013.1.bm.a 4 2013.bm odd 10 1 inner
2013.1.bm.b yes 4 3.b odd 2 1
2013.1.bm.b yes 4 33.h odd 10 1
2013.1.bm.b yes 4 61.b even 2 1
2013.1.bm.b yes 4 671.x even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{3} + 4 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2013, [\chi])\).