# Properties

 Label 1710.2.l.j 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

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

## 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
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −2.64575 1.00000 0 −0.500000 + 0.866025i
1261.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 2.64575 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −2.64575 1.00000 0 −0.500000 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 2.64575 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.j 4
3.b odd 2 1 570.2.i.i 4
19.c even 3 1 inner 1710.2.l.j 4
57.h odd 6 1 570.2.i.i 4

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

 $$T_{7}^{2} - 7$$ $$T_{11}^{2} - 2 T_{11} - 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$( -27 - 2 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$( 1 - T + T^{2} )^{2}$$
$29$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$
$31$ $$( -24 - 4 T + T^{2} )^{2}$$
$37$ $$( -19 - 6 T + T^{2} )^{2}$$
$41$ $$841 + 348 T + 115 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$( 36 + 6 T + T^{2} )^{2}$$
$47$ $$11664 + 432 T + 124 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$784 + 28 T^{2} + T^{4}$$
$61$ $$3844 + 124 T + 66 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$1764 - 588 T + 154 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$12996 + 2508 T + 370 T^{2} + 22 T^{3} + T^{4}$$
$73$ $$12996 + 2508 T + 370 T^{2} + 22 T^{3} + T^{4}$$
$79$ $$784 + 28 T^{2} + T^{4}$$
$83$ $$( -76 + 12 T + T^{2} )^{2}$$
$89$ $$30625 + 175 T^{2} + T^{4}$$
$97$ $$36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4}$$