# Properties

 Label 2015.1.h.c 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 disc. -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 $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: Yes Analytic conductor: $$1.00561600046$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{26})^+$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{26} + \zeta_{26}^{-1}$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4 \nu^{2} + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5 \nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$6$$ $$\nu^{5}$$ $$=$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{1}$$

## 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
 1.94188 1.49702 0.709210 −0.241073 −1.13613 −1.77091
−1.94188 1.49702 2.77091 1.00000 −2.90704 1.13613 −3.43891 1.24107 −1.94188
2014.2 −1.49702 −1.13613 1.24107 1.00000 1.70081 −1.94188 −0.360892 0.290790 −1.49702
2014.3 −0.709210 −1.77091 −0.497021 1.00000 1.25595 0.241073 1.06170 2.13613 −0.709210
2014.4 0.241073 0.709210 −0.941884 1.00000 0.170972 1.77091 −0.468136 −0.497021 0.241073
2014.5 1.13613 1.94188 0.290790 1.00000 2.20623 −1.49702 −0.805754 2.77091 1.13613
2014.6 1.77091 −0.241073 2.13613 1.00000 −0.426920 −0.709210 2.01199 −0.941884 1.77091
 $$n$$: 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2015.h Odd 1 CM by $$\Q(\sqrt{-2015})$$ yes

## Hecke kernels

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

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