# Properties

 Label 1710.2.l.f 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -1 + \zeta_{6} ) q^{5} - q^{7} - q^{8} + \zeta_{6} q^{10} -2 q^{11} + 3 \zeta_{6} q^{13} + ( -1 + \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( 5 - 2 \zeta_{6} ) q^{19} + q^{20} + ( -2 + 2 \zeta_{6} ) q^{22} -6 \zeta_{6} q^{23} -\zeta_{6} q^{25} + 3 q^{26} + \zeta_{6} q^{28} -10 \zeta_{6} q^{29} + q^{31} + \zeta_{6} q^{32} -4 \zeta_{6} q^{34} + ( 1 - \zeta_{6} ) q^{35} -5 q^{37} + ( 3 - 5 \zeta_{6} ) q^{38} + ( 1 - \zeta_{6} ) q^{40} + ( 2 - 2 \zeta_{6} ) q^{41} + ( 5 - 5 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} -6 q^{46} -6 q^{49} - q^{50} + ( 3 - 3 \zeta_{6} ) q^{52} -12 \zeta_{6} q^{53} + ( 2 - 2 \zeta_{6} ) q^{55} + q^{56} -10 q^{58} + ( -2 + 2 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + ( 1 - \zeta_{6} ) q^{62} + q^{64} -3 q^{65} + 5 \zeta_{6} q^{67} -4 q^{68} -\zeta_{6} q^{70} + ( -11 + 11 \zeta_{6} ) q^{73} + ( -5 + 5 \zeta_{6} ) q^{74} + ( -2 - 3 \zeta_{6} ) q^{76} + 2 q^{77} + ( 11 - 11 \zeta_{6} ) q^{79} -\zeta_{6} q^{80} -2 \zeta_{6} q^{82} -2 q^{83} + 4 \zeta_{6} q^{85} -5 \zeta_{6} q^{86} + 2 q^{88} -3 \zeta_{6} q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + ( -3 + 5 \zeta_{6} ) q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} + ( -6 + 6 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - q^{5} - 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} - q^{5} - 2q^{7} - 2q^{8} + q^{10} - 4q^{11} + 3q^{13} - q^{14} - q^{16} + 4q^{17} + 8q^{19} + 2q^{20} - 2q^{22} - 6q^{23} - q^{25} + 6q^{26} + q^{28} - 10q^{29} + 2q^{31} + q^{32} - 4q^{34} + q^{35} - 10q^{37} + q^{38} + q^{40} + 2q^{41} + 5q^{43} + 2q^{44} - 12q^{46} - 12q^{49} - 2q^{50} + 3q^{52} - 12q^{53} + 2q^{55} + 2q^{56} - 20q^{58} - 2q^{59} - 5q^{61} + q^{62} + 2q^{64} - 6q^{65} + 5q^{67} - 8q^{68} - q^{70} - 11q^{73} - 5q^{74} - 7q^{76} + 4q^{77} + 11q^{79} - q^{80} - 2q^{82} - 4q^{83} + 4q^{85} - 5q^{86} + 4q^{88} - 3q^{91} - 6q^{92} - q^{95} + 2q^{97} - 6q^{98} + O(q^{100})$$

## 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.f 2
3.b odd 2 1 570.2.i.b 2
19.c even 3 1 inner 1710.2.l.f 2
57.h odd 6 1 570.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.b 2 3.b odd 2 1
570.2.i.b 2 57.h odd 6 1
1710.2.l.f 2 1.a even 1 1 trivial
1710.2.l.f 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 $$S_{2}^{\mathrm{new}}(1710, [\chi])$$:

 $$T_{7} + 1$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$9 - 3 T + T^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$100 + 10 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$4 - 2 T + T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$4 + 2 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$121 - 11 T + T^{2}$$
$83$ $$( 2 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$4 - 2 T + T^{2}$$