# Properties

 Label 2775.1.bb.a Level $2775$ Weight $1$ Character orbit 2775.bb Analytic conductor $1.385$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2775 = 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2775.bb (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.38490541006$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 111) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.624095613.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{5} q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{5} q^{7} -\zeta_{12}^{4} q^{9} + \zeta_{12} q^{12} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{13} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{4} q^{21} + \zeta_{12}^{3} q^{27} -\zeta_{12} q^{28} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{31} - q^{36} + \zeta_{12}^{3} q^{37} + ( -1 - \zeta_{12}^{2} ) q^{39} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{43} -\zeta_{12}^{3} q^{48} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{52} -\zeta_{12}^{3} q^{63} + q^{64} -\zeta_{12}^{5} q^{67} -\zeta_{12}^{3} q^{73} + ( 1 - \zeta_{12}^{4} ) q^{79} -\zeta_{12}^{2} q^{81} + q^{84} + ( 1 + \zeta_{12}^{2} ) q^{91} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{93} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{4} + 2 q^{9} - 2 q^{16} - 2 q^{21} - 4 q^{36} - 6 q^{39} + 4 q^{64} + 6 q^{79} - 2 q^{81} + 4 q^{84} + 6 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times$$.

 $$n$$ $$76$$ $$926$$ $$1777$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
899.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 0.500000i −0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0
899.2 0 0.866025 + 0.500000i −0.500000 + 0.866025i 0 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0
2099.1 0 −0.866025 + 0.500000i −0.500000 0.866025i 0 0 0.866025 0.500000i 0 0.500000 0.866025i 0
2099.2 0 0.866025 0.500000i −0.500000 0.866025i 0 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner
37.e even 6 1 inner
111.h odd 6 1 inner
185.l even 6 1 inner
555.ba odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.bb.a 4
3.b odd 2 1 CM 2775.1.bb.a 4
5.b even 2 1 inner 2775.1.bb.a 4
5.c odd 4 1 111.1.h.a 2
5.c odd 4 1 2775.1.w.a 2
15.d odd 2 1 inner 2775.1.bb.a 4
15.e even 4 1 111.1.h.a 2
15.e even 4 1 2775.1.w.a 2
20.e even 4 1 1776.1.bq.a 2
37.e even 6 1 inner 2775.1.bb.a 4
45.k odd 12 1 2997.1.o.a 2
45.k odd 12 1 2997.1.v.a 2
45.l even 12 1 2997.1.o.a 2
45.l even 12 1 2997.1.v.a 2
60.l odd 4 1 1776.1.bq.a 2
111.h odd 6 1 inner 2775.1.bb.a 4
185.l even 6 1 inner 2775.1.bb.a 4
185.r odd 12 1 111.1.h.a 2
185.r odd 12 1 2775.1.w.a 2
555.ba odd 6 1 inner 2775.1.bb.a 4
555.bh even 12 1 111.1.h.a 2
555.bh even 12 1 2775.1.w.a 2
740.bh even 12 1 1776.1.bq.a 2
1665.cn odd 12 1 2997.1.o.a 2
1665.cq even 12 1 2997.1.o.a 2
1665.db odd 12 1 2997.1.v.a 2
1665.dh even 12 1 2997.1.v.a 2
2220.cw odd 12 1 1776.1.bq.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.h.a 2 5.c odd 4 1
111.1.h.a 2 15.e even 4 1
111.1.h.a 2 185.r odd 12 1
111.1.h.a 2 555.bh even 12 1
1776.1.bq.a 2 20.e even 4 1
1776.1.bq.a 2 60.l odd 4 1
1776.1.bq.a 2 740.bh even 12 1
1776.1.bq.a 2 2220.cw odd 12 1
2775.1.w.a 2 5.c odd 4 1
2775.1.w.a 2 15.e even 4 1
2775.1.w.a 2 185.r odd 12 1
2775.1.w.a 2 555.bh even 12 1
2775.1.bb.a 4 1.a even 1 1 trivial
2775.1.bb.a 4 3.b odd 2 1 CM
2775.1.bb.a 4 5.b even 2 1 inner
2775.1.bb.a 4 15.d odd 2 1 inner
2775.1.bb.a 4 37.e even 6 1 inner
2775.1.bb.a 4 111.h odd 6 1 inner
2775.1.bb.a 4 185.l even 6 1 inner
2775.1.bb.a 4 555.ba odd 6 1 inner
2997.1.o.a 2 45.k odd 12 1
2997.1.o.a 2 45.l even 12 1
2997.1.o.a 2 1665.cn odd 12 1
2997.1.o.a 2 1665.cq even 12 1
2997.1.v.a 2 45.k odd 12 1
2997.1.v.a 2 45.l even 12 1
2997.1.v.a 2 1665.db odd 12 1
2997.1.v.a 2 1665.dh even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2775, [\chi])$$.

## Hecke characteristic polynomials

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