# Properties

 Label 2015.1.bq.c Level $2015$ Weight $1$ Character orbit 2015.bq Analytic conductor $1.006$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$2015 = 5 \cdot 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2015.bq (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00561600046$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{4}$$ Projective field Galois closure of 4.0.4060225.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{4} q^{3} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{4} q^{3} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} + \zeta_{12}^{5} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{10} -\zeta_{12} q^{11} - q^{13} - q^{14} -\zeta_{12} q^{15} -\zeta_{12}^{4} q^{16} + \zeta_{12}^{2} q^{17} + \zeta_{12}^{2} q^{19} -\zeta_{12}^{3} q^{21} -\zeta_{12}^{2} q^{22} + \zeta_{12}^{4} q^{23} + \zeta_{12} q^{24} - q^{25} -\zeta_{12} q^{26} - q^{27} -\zeta_{12} q^{29} -\zeta_{12}^{2} q^{30} - q^{31} -\zeta_{12}^{5} q^{33} + \zeta_{12}^{3} q^{34} -\zeta_{12}^{2} q^{35} -\zeta_{12}^{4} q^{37} + \zeta_{12}^{3} q^{38} -\zeta_{12}^{4} q^{39} + q^{40} -\zeta_{12}^{4} q^{41} -\zeta_{12}^{4} q^{42} + \zeta_{12}^{2} q^{43} + \zeta_{12}^{5} q^{46} + \zeta_{12}^{2} q^{48} -\zeta_{12} q^{50} - q^{51} -\zeta_{12} q^{54} -\zeta_{12}^{4} q^{55} + \zeta_{12}^{2} q^{56} - q^{57} -\zeta_{12}^{2} q^{58} + \zeta_{12}^{2} q^{59} -\zeta_{12}^{5} q^{61} -\zeta_{12} q^{62} - q^{64} -\zeta_{12}^{3} q^{65} + q^{66} -\zeta_{12} q^{67} -\zeta_{12}^{2} q^{69} -\zeta_{12}^{3} q^{70} + \zeta_{12}^{2} q^{71} -\zeta_{12}^{5} q^{74} -\zeta_{12}^{4} q^{75} + q^{77} -\zeta_{12}^{5} q^{78} + 2 \zeta_{12}^{3} q^{79} + \zeta_{12} q^{80} -\zeta_{12}^{4} q^{81} -\zeta_{12}^{5} q^{82} + 2 q^{83} + \zeta_{12}^{5} q^{85} + \zeta_{12}^{3} q^{86} -\zeta_{12}^{5} q^{87} + \zeta_{12}^{4} q^{88} + \zeta_{12} q^{89} -\zeta_{12}^{5} q^{91} -\zeta_{12}^{4} q^{93} + \zeta_{12}^{5} q^{95} -\zeta_{12}^{5} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + O(q^{10})$$ $$4q - 2q^{3} - 2q^{10} - 4q^{13} - 4q^{14} + 2q^{16} + 2q^{17} + 2q^{19} - 2q^{22} - 2q^{23} - 4q^{25} - 4q^{27} - 2q^{30} - 4q^{31} - 2q^{35} + 2q^{37} + 2q^{39} + 4q^{40} + 2q^{41} + 2q^{42} + 2q^{43} + 2q^{48} - 4q^{51} + 2q^{55} + 2q^{56} - 4q^{57} - 2q^{58} + 2q^{59} - 4q^{64} + 4q^{66} - 2q^{69} + 2q^{71} + 2q^{75} + 4q^{77} + 2q^{81} + 8q^{83} - 2q^{88} + 2q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$716$$ $$807$$ $$1861$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
464.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.500000 + 0.866025i 0 1.00000i 0.866025 0.500000i 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
464.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0 1.00000i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
1394.1 −0.866025 + 0.500000i −0.500000 0.866025i 0 1.00000i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
1394.2 0.866025 0.500000i −0.500000 0.866025i 0 1.00000i −0.866025 0.500000i −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
 $$n$$: 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
13.c even 3 1 inner
155.c odd 2 1 inner
2015.bq odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.bq.c 4
5.b even 2 1 2015.1.bq.e yes 4
13.c even 3 1 inner 2015.1.bq.c 4
31.b odd 2 1 2015.1.bq.e yes 4
65.n even 6 1 2015.1.bq.e yes 4
155.c odd 2 1 inner 2015.1.bq.c 4
403.p odd 6 1 2015.1.bq.e yes 4
2015.bq odd 6 1 inner 2015.1.bq.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.c 4 1.a even 1 1 trivial
2015.1.bq.c 4 13.c even 3 1 inner
2015.1.bq.c 4 155.c odd 2 1 inner
2015.1.bq.c 4 2015.bq odd 6 1 inner
2015.1.bq.e yes 4 5.b even 2 1
2015.1.bq.e yes 4 31.b odd 2 1
2015.1.bq.e yes 4 65.n even 6 1
2015.1.bq.e yes 4 403.p odd 6 1

## Hecke kernels

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

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