Properties

 Label 1710.2.l.a Level $1710$ Weight $2$ Character orbit 1710.l Analytic conductor $13.654$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1710.l (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$13.6544187456$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{7} + q^{8} -\zeta_{6} q^{10} -6 q^{11} -5 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 5 - 2 \zeta_{6} ) q^{19} + q^{20} + ( 6 - 6 \zeta_{6} ) q^{22} + 6 \zeta_{6} q^{23} -\zeta_{6} q^{25} + 5 q^{26} + \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} + 5 q^{31} -\zeta_{6} q^{32} + ( 1 - \zeta_{6} ) q^{35} + 11 q^{37} + ( -3 + 5 \zeta_{6} ) q^{38} + ( -1 + \zeta_{6} ) q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} + ( 1 - \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{44} -6 q^{46} + 12 \zeta_{6} q^{47} -6 q^{49} + q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} -12 \zeta_{6} q^{53} + ( 6 - 6 \zeta_{6} ) q^{55} - q^{56} -6 q^{58} + ( 6 - 6 \zeta_{6} ) q^{59} + 7 \zeta_{6} q^{61} + ( -5 + 5 \zeta_{6} ) q^{62} + q^{64} + 5 q^{65} + \zeta_{6} q^{67} + \zeta_{6} q^{70} + ( 1 - \zeta_{6} ) q^{73} + ( -11 + 11 \zeta_{6} ) q^{74} + ( -2 - 3 \zeta_{6} ) q^{76} + 6 q^{77} + ( 7 - 7 \zeta_{6} ) q^{79} -\zeta_{6} q^{80} + 6 \zeta_{6} q^{82} + 6 q^{83} + \zeta_{6} q^{86} -6 q^{88} -12 \zeta_{6} q^{89} + 5 \zeta_{6} q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} -12 q^{94} + ( -3 + 5 \zeta_{6} ) q^{95} + ( -14 + 14 \zeta_{6} ) q^{97} + ( 6 - 6 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{5} - 2q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{5} - 2q^{7} + 2q^{8} - q^{10} - 12q^{11} - 5q^{13} + q^{14} - q^{16} + 8q^{19} + 2q^{20} + 6q^{22} + 6q^{23} - q^{25} + 10q^{26} + q^{28} + 6q^{29} + 10q^{31} - q^{32} + q^{35} + 22q^{37} - q^{38} - q^{40} + 6q^{41} + q^{43} + 6q^{44} - 12q^{46} + 12q^{47} - 12q^{49} + 2q^{50} - 5q^{52} - 12q^{53} + 6q^{55} - 2q^{56} - 12q^{58} + 6q^{59} + 7q^{61} - 5q^{62} + 2q^{64} + 10q^{65} + q^{67} + q^{70} + q^{73} - 11q^{74} - 7q^{76} + 12q^{77} + 7q^{79} - q^{80} + 6q^{82} + 12q^{83} + q^{86} - 12q^{88} - 12q^{89} + 5q^{91} + 6q^{92} - 24q^{94} - q^{95} - 14q^{97} + 6q^{98} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1261.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 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
19.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.l.a 2
3.b odd 2 1 570.2.i.d 2
19.c even 3 1 inner 1710.2.l.a 2
57.h odd 6 1 570.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.d 2 3.b odd 2 1
570.2.i.d 2 57.h odd 6 1
1710.2.l.a 2 1.a even 1 1 trivial
1710.2.l.a 2 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1710, [\chi])$$:

 $$T_{7} + 1$$ $$T_{11} + 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$144 - 12 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$1 - T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$49 - 7 T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$144 + 12 T + T^{2}$$
$97$ $$196 + 14 T + T^{2}$$