# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(2013, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$11$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$13$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
$17$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$19$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
$23$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$29$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$37$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$41$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$43$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$47$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$53$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$59$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$61$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$67$ $$( 1 - T )^{4}( 1 + T )^{4}$$
$71$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$73$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$79$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$83$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$89$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$97$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$