Properties

 Label 2646.2.a.bf Level $2646$ Weight $2$ Character orbit 2646.a Self dual yes Analytic conductor $21.128$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$21.1284163748$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} + ( 1 - \beta ) q^{10} + ( -1 + \beta ) q^{11} + ( 2 - \beta ) q^{13} + q^{16} + ( 1 - \beta ) q^{17} -2 q^{19} + ( -1 + \beta ) q^{20} + ( 1 - \beta ) q^{22} + ( -4 - 2 \beta ) q^{23} + ( 3 - 2 \beta ) q^{25} + ( -2 + \beta ) q^{26} + ( -5 - \beta ) q^{29} + ( 2 - \beta ) q^{31} - q^{32} + ( -1 + \beta ) q^{34} + ( -4 + 3 \beta ) q^{37} + 2 q^{38} + ( 1 - \beta ) q^{40} + ( 3 - 3 \beta ) q^{41} + 5 q^{43} + ( -1 + \beta ) q^{44} + ( 4 + 2 \beta ) q^{46} + ( -3 - 3 \beta ) q^{47} + ( -3 + 2 \beta ) q^{50} + ( 2 - \beta ) q^{52} -6 q^{53} + ( 8 - 2 \beta ) q^{55} + ( 5 + \beta ) q^{58} + ( 11 + \beta ) q^{59} + ( -10 - \beta ) q^{61} + ( -2 + \beta ) q^{62} + q^{64} + ( -9 + 3 \beta ) q^{65} + ( 3 + 2 \beta ) q^{67} + ( 1 - \beta ) q^{68} + ( -13 + \beta ) q^{71} + 4 \beta q^{73} + ( 4 - 3 \beta ) q^{74} -2 q^{76} + ( -2 - 5 \beta ) q^{79} + ( -1 + \beta ) q^{80} + ( -3 + 3 \beta ) q^{82} + ( -8 + 2 \beta ) q^{83} + ( -8 + 2 \beta ) q^{85} -5 q^{86} + ( 1 - \beta ) q^{88} + ( -3 - 3 \beta ) q^{89} + ( -4 - 2 \beta ) q^{92} + ( 3 + 3 \beta ) q^{94} + ( 2 - 2 \beta ) q^{95} + ( -3 + 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} - 2q^{8} + 2q^{10} - 2q^{11} + 4q^{13} + 2q^{16} + 2q^{17} - 4q^{19} - 2q^{20} + 2q^{22} - 8q^{23} + 6q^{25} - 4q^{26} - 10q^{29} + 4q^{31} - 2q^{32} - 2q^{34} - 8q^{37} + 4q^{38} + 2q^{40} + 6q^{41} + 10q^{43} - 2q^{44} + 8q^{46} - 6q^{47} - 6q^{50} + 4q^{52} - 12q^{53} + 16q^{55} + 10q^{58} + 22q^{59} - 20q^{61} - 4q^{62} + 2q^{64} - 18q^{65} + 6q^{67} + 2q^{68} - 26q^{71} + 8q^{74} - 4q^{76} - 4q^{79} - 2q^{80} - 6q^{82} - 16q^{83} - 16q^{85} - 10q^{86} + 2q^{88} - 6q^{89} - 8q^{92} + 6q^{94} + 4q^{95} - 6q^{97} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
−1.00000 0 1.00000 −3.64575 0 0 −1.00000 0 3.64575
1.2 −1.00000 0 1.00000 1.64575 0 0 −1.00000 0 −1.64575
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.a.bf 2
3.b odd 2 1 2646.2.a.bo 2
7.b odd 2 1 2646.2.a.bi 2
7.d odd 6 2 378.2.g.h yes 4
21.c even 2 1 2646.2.a.bl 2
21.g even 6 2 378.2.g.g 4
63.i even 6 2 1134.2.e.t 4
63.k odd 6 2 1134.2.h.t 4
63.s even 6 2 1134.2.h.q 4
63.t odd 6 2 1134.2.e.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 21.g even 6 2
378.2.g.h yes 4 7.d odd 6 2
1134.2.e.q 4 63.t odd 6 2
1134.2.e.t 4 63.i even 6 2
1134.2.h.q 4 63.s even 6 2
1134.2.h.t 4 63.k odd 6 2
2646.2.a.bf 2 1.a even 1 1 trivial
2646.2.a.bi 2 7.b odd 2 1
2646.2.a.bl 2 21.c even 2 1
2646.2.a.bo 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2646))$$:

 $$T_{5}^{2} + 2 T_{5} - 6$$ $$T_{11}^{2} + 2 T_{11} - 6$$ $$T_{13}^{2} - 4 T_{13} - 3$$ $$T_{17}^{2} - 2 T_{17} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-6 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-6 + 2 T + T^{2}$$
$13$ $$-3 - 4 T + T^{2}$$
$17$ $$-6 - 2 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-12 + 8 T + T^{2}$$
$29$ $$18 + 10 T + T^{2}$$
$31$ $$-3 - 4 T + T^{2}$$
$37$ $$-47 + 8 T + T^{2}$$
$41$ $$-54 - 6 T + T^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$-54 + 6 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$114 - 22 T + T^{2}$$
$61$ $$93 + 20 T + T^{2}$$
$67$ $$-19 - 6 T + T^{2}$$
$71$ $$162 + 26 T + T^{2}$$
$73$ $$-112 + T^{2}$$
$79$ $$-171 + 4 T + T^{2}$$
$83$ $$36 + 16 T + T^{2}$$
$89$ $$-54 + 6 T + T^{2}$$
$97$ $$-103 + 6 T + T^{2}$$