# Properties

 Label 1710.2.l.l Level $1710$ Weight $2$ Character orbit 1710.l Analytic conductor $13.654$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{73})$$ Defining polynomial: $$x^{4} - x^{3} + 19 x^{2} + 18 x + 324$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + q^{8} + \beta_{2} q^{10} + ( 1 - \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 1 - \beta_{1} ) q^{19} - q^{20} + ( -1 + \beta_{1} + \beta_{3} ) q^{22} + \beta_{1} q^{23} -\beta_{2} q^{25} -\beta_{3} q^{26} + \beta_{2} q^{28} -6 \beta_{2} q^{29} + ( -2 + \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( -1 + \beta_{2} ) q^{35} - q^{37} + ( \beta_{2} - \beta_{3} ) q^{38} + ( 1 - \beta_{2} ) q^{40} + ( 1 - \beta_{1} - \beta_{3} ) q^{41} + ( -4 - \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{43} -\beta_{1} q^{44} + ( -1 + \beta_{3} ) q^{46} -6 q^{49} + q^{50} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -\beta_{1} + 6 \beta_{2} ) q^{53} + ( 1 - \beta_{1} - \beta_{3} ) q^{55} - q^{56} + 6 q^{58} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{59} + ( \beta_{1} + \beta_{2} ) q^{61} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{62} + q^{64} + \beta_{3} q^{65} + ( -\beta_{1} + 7 \beta_{2} ) q^{67} -\beta_{2} q^{70} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 8 - \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{73} + ( 1 - \beta_{2} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -1 + \beta_{3} ) q^{77} + ( 4 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{79} + \beta_{2} q^{80} + \beta_{1} q^{82} -6 q^{83} + ( \beta_{1} - 5 \beta_{2} ) q^{86} + ( 1 - \beta_{3} ) q^{88} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( 1 - \beta_{1} - \beta_{3} ) q^{92} + ( -\beta_{2} + \beta_{3} ) q^{95} + ( 10 - 10 \beta_{2} ) q^{97} + ( 6 - 6 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} + 2q^{5} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} + 2q^{5} - 4q^{7} + 4q^{8} + 2q^{10} + 2q^{11} + q^{13} + 2q^{14} - 2q^{16} + 3q^{19} - 4q^{20} - q^{22} + q^{23} - 2q^{25} - 2q^{26} + 2q^{28} - 12q^{29} - 6q^{31} - 2q^{32} - 2q^{35} - 4q^{37} + 2q^{40} + q^{41} - 9q^{43} - q^{44} - 2q^{46} - 24q^{49} + 4q^{50} + q^{52} + 11q^{53} + q^{55} - 4q^{56} + 24q^{58} - 10q^{59} + 3q^{61} + 3q^{62} + 4q^{64} + 2q^{65} + 13q^{67} - 2q^{70} + 2q^{71} + 15q^{73} + 2q^{74} - 3q^{76} - 2q^{77} + 5q^{79} + 2q^{80} + q^{82} - 24q^{83} - 9q^{86} + 2q^{88} + 9q^{89} - q^{91} + q^{92} + 20q^{97} + 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 19 x^{2} + 18 x + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 19 \nu^{2} - 19 \nu + 324$$$$)/342$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 37$$$$)/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 18 \beta_{2} + \beta_{1} - 19$$ $$\nu^{3}$$ $$=$$ $$19 \beta_{3} - 37$$

## Character values

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

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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
 −1.88600 + 3.26665i 2.38600 − 4.13267i −1.88600 − 3.26665i 2.38600 + 4.13267i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 1.00000 0 0.500000 0.866025i
1261.2 −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
1531.2 −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.l 4
3.b odd 2 1 570.2.i.g 4
19.c even 3 1 inner 1710.2.l.l 4
57.h odd 6 1 570.2.i.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.g 4 3.b odd 2 1
570.2.i.g 4 57.h odd 6 1
1710.2.l.l 4 1.a even 1 1 trivial
1710.2.l.l 4 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 $$S_{2}^{\mathrm{new}}(1710, [\chi])$$:

 $$T_{7} + 1$$ $$T_{11}^{2} - T_{11} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$( -18 - T + T^{2} )^{2}$$
$13$ $$324 + 18 T + 19 T^{2} - T^{3} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$361 - 57 T + 22 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$324 + 18 T + 19 T^{2} - T^{3} + T^{4}$$
$29$ $$( 36 + 6 T + T^{2} )^{2}$$
$31$ $$( -16 + 3 T + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$324 + 18 T + 19 T^{2} - T^{3} + T^{4}$$
$43$ $$4 + 18 T + 79 T^{2} + 9 T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$144 - 132 T + 109 T^{2} - 11 T^{3} + T^{4}$$
$59$ $$2304 - 480 T + 148 T^{2} + 10 T^{3} + T^{4}$$
$61$ $$256 + 48 T + 25 T^{2} - 3 T^{3} + T^{4}$$
$67$ $$576 - 312 T + 145 T^{2} - 13 T^{3} + T^{4}$$
$71$ $$5184 + 144 T + 76 T^{2} - 2 T^{3} + T^{4}$$
$73$ $$1444 - 570 T + 187 T^{2} - 15 T^{3} + T^{4}$$
$79$ $$24964 + 790 T + 183 T^{2} - 5 T^{3} + T^{4}$$
$83$ $$( 6 + T )^{4}$$
$89$ $$20736 + 1296 T + 225 T^{2} - 9 T^{3} + T^{4}$$
$97$ $$( 100 - 10 T + T^{2} )^{2}$$