# Properties

 Label 2646.2.e.a Level $2646$ Weight $2$ Character orbit 2646.e Analytic conductor $21.128$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 882) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{8} + ( 1 - \zeta_{6} ) q^{10} -2 \zeta_{6} q^{11} + 2 \zeta_{6} q^{13} + q^{16} -7 \zeta_{6} q^{19} + ( -1 + \zeta_{6} ) q^{20} + 2 \zeta_{6} q^{22} + ( 3 - 3 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} -2 \zeta_{6} q^{26} + ( -8 + 8 \zeta_{6} ) q^{29} -4 q^{31} - q^{32} + 6 \zeta_{6} q^{37} + 7 \zeta_{6} q^{38} + ( 1 - \zeta_{6} ) q^{40} + 12 \zeta_{6} q^{41} + ( 8 - 8 \zeta_{6} ) q^{43} -2 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} -8 q^{47} -4 \zeta_{6} q^{50} + 2 \zeta_{6} q^{52} + ( 4 - 4 \zeta_{6} ) q^{53} + 2 q^{55} + ( 8 - 8 \zeta_{6} ) q^{58} + 4 q^{59} -13 q^{61} + 4 q^{62} + q^{64} -2 q^{65} -2 q^{67} + 5 q^{71} + ( -14 + 14 \zeta_{6} ) q^{73} -6 \zeta_{6} q^{74} -7 \zeta_{6} q^{76} -11 q^{79} + ( -1 + \zeta_{6} ) q^{80} -12 \zeta_{6} q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + ( -8 + 8 \zeta_{6} ) q^{86} + 2 \zeta_{6} q^{88} -14 \zeta_{6} q^{89} + ( 3 - 3 \zeta_{6} ) q^{92} + 8 q^{94} + 7 q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - q^{5} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - q^{5} - 2q^{8} + q^{10} - 2q^{11} + 2q^{13} + 2q^{16} - 7q^{19} - q^{20} + 2q^{22} + 3q^{23} + 4q^{25} - 2q^{26} - 8q^{29} - 8q^{31} - 2q^{32} + 6q^{37} + 7q^{38} + q^{40} + 12q^{41} + 8q^{43} - 2q^{44} - 3q^{46} - 16q^{47} - 4q^{50} + 2q^{52} + 4q^{53} + 4q^{55} + 8q^{58} + 8q^{59} - 26q^{61} + 8q^{62} + 2q^{64} - 4q^{65} - 4q^{67} + 10q^{71} - 14q^{73} - 6q^{74} - 7q^{76} - 22q^{79} - q^{80} - 12q^{82} - 12q^{83} - 8q^{86} + 2q^{88} - 14q^{89} + 3q^{92} + 16q^{94} + 14q^{95} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1549.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 1.00000 −0.500000 0.866025i 0 0 −1.00000 0 0.500000 + 0.866025i
2125.1 −1.00000 0 1.00000 −0.500000 + 0.866025i 0 0 −1.00000 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
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.e.a 2
3.b odd 2 1 882.2.e.j 2
7.b odd 2 1 2646.2.e.d 2
7.c even 3 1 2646.2.f.f 2
7.c even 3 1 2646.2.h.j 2
7.d odd 6 1 2646.2.f.h 2
7.d odd 6 1 2646.2.h.g 2
9.c even 3 1 2646.2.h.j 2
9.d odd 6 1 882.2.h.a 2
21.c even 2 1 882.2.e.f 2
21.g even 6 1 882.2.f.b 2
21.g even 6 1 882.2.h.d 2
21.h odd 6 1 882.2.f.c yes 2
21.h odd 6 1 882.2.h.a 2
63.g even 3 1 2646.2.f.f 2
63.h even 3 1 inner 2646.2.e.a 2
63.h even 3 1 7938.2.a.k 1
63.i even 6 1 882.2.e.f 2
63.i even 6 1 7938.2.a.ba 1
63.j odd 6 1 882.2.e.j 2
63.j odd 6 1 7938.2.a.v 1
63.k odd 6 1 2646.2.f.h 2
63.l odd 6 1 2646.2.h.g 2
63.n odd 6 1 882.2.f.c yes 2
63.o even 6 1 882.2.h.d 2
63.s even 6 1 882.2.f.b 2
63.t odd 6 1 2646.2.e.d 2
63.t odd 6 1 7938.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.f 2 21.c even 2 1
882.2.e.f 2 63.i even 6 1
882.2.e.j 2 3.b odd 2 1
882.2.e.j 2 63.j odd 6 1
882.2.f.b 2 21.g even 6 1
882.2.f.b 2 63.s even 6 1
882.2.f.c yes 2 21.h odd 6 1
882.2.f.c yes 2 63.n odd 6 1
882.2.h.a 2 9.d odd 6 1
882.2.h.a 2 21.h odd 6 1
882.2.h.d 2 21.g even 6 1
882.2.h.d 2 63.o even 6 1
2646.2.e.a 2 1.a even 1 1 trivial
2646.2.e.a 2 63.h even 3 1 inner
2646.2.e.d 2 7.b odd 2 1
2646.2.e.d 2 63.t odd 6 1
2646.2.f.f 2 7.c even 3 1
2646.2.f.f 2 63.g even 3 1
2646.2.f.h 2 7.d odd 6 1
2646.2.f.h 2 63.k odd 6 1
2646.2.h.g 2 7.d odd 6 1
2646.2.h.g 2 63.l odd 6 1
2646.2.h.j 2 7.c even 3 1
2646.2.h.j 2 9.c even 3 1
7938.2.a.f 1 63.t odd 6 1
7938.2.a.k 1 63.h even 3 1
7938.2.a.v 1 63.j odd 6 1
7938.2.a.ba 1 63.i even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13}^{2} - 2 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$64 + 8 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$36 - 6 T + T^{2}$$
$41$ $$144 - 12 T + T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$( -5 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$196 + 14 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$