# Properties

 Label 2646.2.d.a Level $2646$ Weight $2$ Character orbit 2646.d Analytic conductor $21.128$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} - q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} -\zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{2} - q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} -\zeta_{12}^{3} q^{8} + ( -1 + 2 \zeta_{12}^{2} ) q^{10} + q^{16} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{20} -6 \zeta_{12}^{3} q^{23} -2 q^{25} -9 \zeta_{12}^{3} q^{29} + ( -2 + 4 \zeta_{12}^{2} ) q^{31} + \zeta_{12}^{3} q^{32} + ( 2 - 4 \zeta_{12}^{2} ) q^{34} + 4 q^{37} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{38} + ( 1 - 2 \zeta_{12}^{2} ) q^{40} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} -8 q^{43} + 6 q^{46} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -2 \zeta_{12}^{3} q^{50} + 3 \zeta_{12}^{3} q^{53} + 9 q^{58} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} ) q^{61} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{62} - q^{64} + 14 q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + 6 \zeta_{12}^{3} q^{71} + ( 7 - 14 \zeta_{12}^{2} ) q^{73} + 4 \zeta_{12}^{3} q^{74} + ( -4 + 8 \zeta_{12}^{2} ) q^{76} + 11 q^{79} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{80} + ( -2 + 4 \zeta_{12}^{2} ) q^{82} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{83} -6 q^{85} -8 \zeta_{12}^{3} q^{86} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{89} + 6 \zeta_{12}^{3} q^{92} + ( 2 - 4 \zeta_{12}^{2} ) q^{94} -12 \zeta_{12}^{3} q^{95} + ( 4 - 8 \zeta_{12}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{16} - 8q^{25} + 16q^{37} - 32q^{43} + 24q^{46} + 36q^{58} - 4q^{64} + 56q^{67} + 44q^{79} - 24q^{85} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2645.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.2 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 1.73205i
2645.3 1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.4 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 1.73205i
 $$n$$: 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 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.a 4
3.b odd 2 1 inner 2646.2.d.a 4
7.b odd 2 1 inner 2646.2.d.a 4
7.c even 3 1 378.2.k.c 4
7.d odd 6 1 378.2.k.c 4
21.c even 2 1 inner 2646.2.d.a 4
21.g even 6 1 378.2.k.c 4
21.h odd 6 1 378.2.k.c 4
63.g even 3 1 1134.2.t.c 4
63.h even 3 1 1134.2.l.b 4
63.i even 6 1 1134.2.l.b 4
63.j odd 6 1 1134.2.l.b 4
63.k odd 6 1 1134.2.t.c 4
63.n odd 6 1 1134.2.t.c 4
63.s even 6 1 1134.2.t.c 4
63.t odd 6 1 1134.2.l.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.c 4 7.c even 3 1
378.2.k.c 4 7.d odd 6 1
378.2.k.c 4 21.g even 6 1
378.2.k.c 4 21.h odd 6 1
1134.2.l.b 4 63.h even 3 1
1134.2.l.b 4 63.i even 6 1
1134.2.l.b 4 63.j odd 6 1
1134.2.l.b 4 63.t odd 6 1
1134.2.t.c 4 63.g even 3 1
1134.2.t.c 4 63.k odd 6 1
1134.2.t.c 4 63.n odd 6 1
1134.2.t.c 4 63.s even 6 1
2646.2.d.a 4 1.a even 1 1 trivial
2646.2.d.a 4 3.b odd 2 1 inner
2646.2.d.a 4 7.b odd 2 1 inner
2646.2.d.a 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -3 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 48 + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 81 + T^{2} )^{2}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$( 8 + T )^{4}$$
$47$ $$( -12 + T^{2} )^{2}$$
$53$ $$( 9 + T^{2} )^{2}$$
$59$ $$( -147 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( -14 + T )^{4}$$
$71$ $$( 36 + T^{2} )^{2}$$
$73$ $$( 147 + T^{2} )^{2}$$
$79$ $$( -11 + T )^{4}$$
$83$ $$( -300 + T^{2} )^{2}$$
$89$ $$( -108 + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$