# Properties

 Label 1710.2.l.n 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: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{3} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{3} ) q^{7} - q^{8} -\beta_{1} q^{10} + 3 q^{11} -2 \beta_{2} q^{13} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( -1 + \beta_{1} ) q^{16} + ( -4 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} - q^{20} + ( 3 - 3 \beta_{1} ) q^{22} + ( -\beta_{1} - 2 \beta_{2} ) q^{23} -\beta_{1} q^{25} -2 \beta_{3} q^{26} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( 4 \beta_{1} - \beta_{2} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( -5 - 2 \beta_{3} ) q^{37} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{38} + ( -1 + \beta_{1} ) q^{40} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{43} -3 \beta_{1} q^{44} + ( -1 - 2 \beta_{3} ) q^{46} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{47} -2 \beta_{3} q^{49} - q^{50} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( -5 \beta_{1} - \beta_{2} ) q^{53} + ( 3 - 3 \beta_{1} ) q^{55} + ( -1 + \beta_{3} ) q^{56} + ( 2 + 3 \beta_{3} ) q^{58} + ( -6 + 6 \beta_{1} ) q^{59} + ( -10 \beta_{1} - \beta_{2} ) q^{61} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{62} + q^{64} -2 \beta_{3} q^{65} + 3 \beta_{2} q^{67} + ( 4 - \beta_{3} ) q^{68} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( 4 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -8 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( -5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{76} + ( 3 - 3 \beta_{3} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{79} + \beta_{1} q^{80} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -2 - 4 \beta_{3} ) q^{83} + ( 4 \beta_{1} - \beta_{2} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{86} -3 q^{88} + ( -3 \beta_{1} - \beta_{2} ) q^{89} + ( 12 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -6 + 2 \beta_{3} ) q^{94} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{95} + ( 16 - 16 \beta_{1} - \beta_{2} + \beta_{3} ) q^{97} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + 2q^{5} + 4q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + 2q^{5} + 4q^{7} - 4q^{8} - 2q^{10} + 12q^{11} + 2q^{14} - 2q^{16} - 8q^{17} - 2q^{19} - 4q^{20} + 6q^{22} - 2q^{23} - 2q^{25} - 2q^{28} + 4q^{29} + 16q^{31} + 2q^{32} + 8q^{34} + 2q^{35} - 20q^{37} + 2q^{38} - 2q^{40} - 6q^{41} + 4q^{43} - 6q^{44} - 4q^{46} - 12q^{47} - 4q^{50} - 10q^{53} + 6q^{55} - 4q^{56} + 8q^{58} - 12q^{59} - 20q^{61} + 8q^{62} + 4q^{64} + 16q^{68} - 2q^{70} + 8q^{71} - 16q^{73} - 10q^{74} + 4q^{76} + 12q^{77} - 4q^{79} + 2q^{80} + 6q^{82} - 8q^{83} + 8q^{85} - 4q^{86} - 12q^{88} - 6q^{89} + 24q^{91} - 2q^{92} - 24q^{94} + 2q^{95} + 32q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1261.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.44949 −1.00000 0 −0.500000 + 0.866025i
1261.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 3.44949 −1.00000 0 −0.500000 + 0.866025i
1531.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −1.44949 −1.00000 0 −0.500000 0.866025i
1531.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 3.44949 −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.n 4
3.b odd 2 1 570.2.i.f 4
19.c even 3 1 inner 1710.2.l.n 4
57.h odd 6 1 570.2.i.f 4

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

 $$T_{7}^{2} - 2 T_{7} - 5$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( -5 - 2 T + T^{2} )^{2}$$
$11$ $$( -3 + T )^{4}$$
$13$ $$576 + 24 T^{2} + T^{4}$$
$17$ $$100 + 80 T + 54 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$361 + 38 T - 15 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$529 - 46 T + 27 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$2500 + 200 T + 66 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( -8 - 8 T + T^{2} )^{2}$$
$37$ $$( 1 + 10 T + T^{2} )^{2}$$
$41$ $$2025 - 270 T + 81 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4}$$
$59$ $$( 36 + 6 T + T^{2} )^{2}$$
$61$ $$8836 + 1880 T + 306 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$2916 + 54 T^{2} + T^{4}$$
$71$ $$1444 + 304 T + 102 T^{2} - 8 T^{3} + T^{4}$$
$73$ $$3364 + 928 T + 198 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$( -92 + 4 T + T^{2} )^{2}$$
$89$ $$9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$62500 - 8000 T + 774 T^{2} - 32 T^{3} + T^{4}$$