# Properties

 Label 2015.1.h.e Level 2015 Weight 1 Character orbit 2015.h Self dual yes Analytic conductor 1.006 Analytic rank 0 Dimension 6 Projective image $$D_{13}$$ CM discriminant -2015 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2015 = 5 \cdot 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2015.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00561600046$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{26})^+$$ Defining polynomial: $$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{13}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{13} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -3 \beta + \beta^{3} ) q^{3} + ( -1 + \beta^{2} ) q^{4} - q^{5} + ( -3 \beta^{2} + \beta^{4} ) q^{6} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( -2 \beta + \beta^{3} ) q^{8} + ( 3 \beta + 3 \beta^{2} - 4 \beta^{3} - \beta^{4} + \beta^{5} ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( -3 \beta + \beta^{3} ) q^{3} + ( -1 + \beta^{2} ) q^{4} - q^{5} + ( -3 \beta^{2} + \beta^{4} ) q^{6} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( -2 \beta + \beta^{3} ) q^{8} + ( 3 \beta + 3 \beta^{2} - 4 \beta^{3} - \beta^{4} + \beta^{5} ) q^{9} -\beta q^{10} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{11} + ( 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{12} - q^{13} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{14} + ( 3 \beta - \beta^{3} ) q^{15} + ( 1 - 3 \beta^{2} + \beta^{4} ) q^{16} + ( 2 - \beta^{2} ) q^{17} + ( 1 + 3 \beta - 3 \beta^{2} - \beta^{3} + \beta^{4} ) q^{18} + ( 1 - \beta^{2} ) q^{20} + ( -1 + 2 \beta + 3 \beta^{2} - 4 \beta^{3} - \beta^{4} + \beta^{5} ) q^{21} + ( -1 - 3 \beta + \beta^{2} + 4 \beta^{3} - \beta^{5} ) q^{22} + ( 1 - 3 \beta - 3 \beta^{2} + 4 \beta^{3} + \beta^{4} - \beta^{5} ) q^{23} + ( 1 + 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{24} + q^{25} -\beta q^{26} + ( -2 - 3 \beta + 4 \beta^{2} + \beta^{3} - \beta^{4} ) q^{27} + ( 1 - 3 \beta + 4 \beta^{3} - \beta^{5} ) q^{28} + ( 3 \beta^{2} - \beta^{4} ) q^{30} + q^{31} + ( 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{32} + ( 2 + 5 \beta - \beta^{2} - 5 \beta^{3} + \beta^{5} ) q^{33} + ( 2 \beta - \beta^{3} ) q^{34} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{35} + ( -2 \beta + \beta^{3} ) q^{36} + ( 3 \beta - \beta^{3} ) q^{39} + ( 2 \beta - \beta^{3} ) q^{40} + ( 1 + 2 \beta - 4 \beta^{2} - \beta^{3} + \beta^{4} ) q^{42} + \beta q^{43} + ( -1 + \beta + 3 \beta^{2} - \beta^{4} ) q^{44} + ( -3 \beta - 3 \beta^{2} + 4 \beta^{3} + \beta^{4} - \beta^{5} ) q^{45} + ( -1 - 2 \beta + 3 \beta^{2} + \beta^{3} - \beta^{4} ) q^{46} + ( 1 - 3 \beta - 3 \beta^{2} + 4 \beta^{3} + \beta^{4} - \beta^{5} ) q^{47} + ( 1 + \beta - 3 \beta^{2} + \beta^{4} ) q^{48} + ( 1 - 5 \beta + 5 \beta^{3} - \beta^{5} ) q^{49} + \beta q^{50} + ( -6 \beta + 5 \beta^{3} - \beta^{5} ) q^{51} + ( 1 - \beta^{2} ) q^{52} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{53} + ( -2 \beta - 3 \beta^{2} + 4 \beta^{3} + \beta^{4} - \beta^{5} ) q^{54} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{55} + ( -1 + 3 \beta^{2} - \beta^{4} ) q^{56} + ( -3 \beta + 4 \beta^{3} - \beta^{5} ) q^{60} + \beta q^{62} + ( -3 \beta + 3 \beta^{2} + \beta^{3} - \beta^{4} ) q^{63} + ( 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{64} + q^{65} + ( 1 + 5 \beta - \beta^{2} - 5 \beta^{3} + \beta^{5} ) q^{66} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{67} + ( -2 + 3 \beta^{2} - \beta^{4} ) q^{68} + ( 2 + 3 \beta - 4 \beta^{2} - \beta^{3} + \beta^{4} ) q^{69} + ( 2 \beta - 4 \beta^{3} + \beta^{5} ) q^{70} + ( -1 - 3 \beta + \beta^{2} + \beta^{3} ) q^{72} + ( -3 \beta + \beta^{3} ) q^{75} + ( -2 + \beta + 4 \beta^{2} - \beta^{4} ) q^{77} + ( 3 \beta^{2} - \beta^{4} ) q^{78} + ( -1 + 3 \beta^{2} - \beta^{4} ) q^{80} + ( 2 \beta + 3 \beta^{2} - 4 \beta^{3} - \beta^{4} + \beta^{5} ) q^{81} + ( 1 - \beta - \beta^{2} ) q^{84} + ( -2 + \beta^{2} ) q^{85} + \beta^{2} q^{86} + ( 1 + 2 \beta - \beta^{3} ) q^{88} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{89} + ( -1 - 3 \beta + 3 \beta^{2} + \beta^{3} - \beta^{4} ) q^{90} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{91} + ( -1 + 2 \beta + \beta^{2} - \beta^{3} ) q^{92} + ( -3 \beta + \beta^{3} ) q^{93} + ( -1 - 2 \beta + 3 \beta^{2} + \beta^{3} - \beta^{4} ) q^{94} + ( -1 - 2 \beta + \beta^{2} + \beta^{3} ) q^{96} + ( -3 \beta + \beta^{3} ) q^{97} + ( -1 - 2 \beta + \beta^{2} + 4 \beta^{3} - \beta^{5} ) q^{98} + ( -2 - 6 \beta + \beta^{2} + 5 \beta^{3} - \beta^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} + q^{3} + 5q^{4} - 6q^{5} - 2q^{6} + q^{7} + 2q^{8} + 5q^{9} + O(q^{10})$$ $$6q + q^{2} + q^{3} + 5q^{4} - 6q^{5} - 2q^{6} + q^{7} + 2q^{8} + 5q^{9} - q^{10} - q^{11} + 3q^{12} - 6q^{13} - 2q^{14} - q^{15} + 4q^{16} + q^{17} + 3q^{18} - 5q^{20} - 2q^{21} + 2q^{22} + q^{23} + 9q^{24} + 6q^{25} - q^{26} + 2q^{27} + 3q^{28} + 2q^{30} + 6q^{31} + 3q^{32} + 2q^{33} - 2q^{34} - q^{35} + 2q^{36} - q^{39} - 2q^{40} - 9q^{42} + q^{43} - 3q^{44} - 5q^{45} - 2q^{46} + q^{47} + 5q^{48} + 5q^{49} + q^{50} - 2q^{51} - 5q^{52} + q^{53} - 4q^{54} + q^{55} - 4q^{56} - 3q^{60} + q^{62} + 3q^{63} + 3q^{64} + 6q^{65} - 4q^{66} + q^{67} - 10q^{68} - 2q^{69} + 2q^{70} + 6q^{72} + q^{75} + 2q^{77} + 2q^{78} - 4q^{80} + 4q^{81} - 6q^{84} - q^{85} + 11q^{86} + 4q^{88} - q^{89} - 3q^{90} - q^{91} + 3q^{92} + q^{93} - 2q^{94} + 7q^{96} + q^{97} + 3q^{98} - 3q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times$$.

 $$n$$ $$716$$ $$807$$ $$1861$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2014.1
 −1.77091 −1.13613 −0.241073 0.709210 1.49702 1.94188
−1.77091 −0.241073 2.13613 −1.00000 0.426920 0.709210 −2.01199 −0.941884 1.77091
2014.2 −1.13613 1.94188 0.290790 −1.00000 −2.20623 1.49702 0.805754 2.77091 1.13613
2014.3 −0.241073 0.709210 −0.941884 −1.00000 −0.170972 −1.77091 0.468136 −0.497021 0.241073
2014.4 0.709210 −1.77091 −0.497021 −1.00000 −1.25595 −0.241073 −1.06170 2.13613 −0.709210
2014.5 1.49702 −1.13613 1.24107 −1.00000 −1.70081 1.94188 0.360892 0.290790 −1.49702
2014.6 1.94188 1.49702 2.77091 −1.00000 2.90704 −1.13613 3.43891 1.24107 −1.94188
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2014.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2015.h odd 2 1 CM by $$\Q(\sqrt{-2015})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.h.e yes 6
5.b even 2 1 2015.1.h.b 6
13.b even 2 1 2015.1.h.c yes 6
31.b odd 2 1 2015.1.h.d yes 6
65.d even 2 1 2015.1.h.d yes 6
155.c odd 2 1 2015.1.h.c yes 6
403.b odd 2 1 2015.1.h.b 6
2015.h odd 2 1 CM 2015.1.h.e yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.h.b 6 5.b even 2 1
2015.1.h.b 6 403.b odd 2 1
2015.1.h.c yes 6 13.b even 2 1
2015.1.h.c yes 6 155.c odd 2 1
2015.1.h.d yes 6 31.b odd 2 1
2015.1.h.d yes 6 65.d even 2 1
2015.1.h.e yes 6 1.a even 1 1 trivial
2015.1.h.e yes 6 2015.h odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2015, [\chi])$$:

 $$T_{2}^{6} - T_{2}^{5} - 5 T_{2}^{4} + 4 T_{2}^{3} + 6 T_{2}^{2} - 3 T_{2} - 1$$ $$T_{3}^{6} - T_{3}^{5} - 5 T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} - 3 T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$5$ $$( 1 + T )^{6}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$11$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$
$13$ $$( 1 + T )^{6}$$
$17$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$19$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$23$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$29$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$31$ $$( 1 - T )^{6}$$
$37$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$41$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$43$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$47$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$53$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$59$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$61$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$67$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$
$71$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$73$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$79$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$83$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$89$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$
$97$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12}$$