# Properties

 Label 1710.2.l.q Level $1710$ Weight $2$ Character orbit 1710.l Analytic conductor $13.654$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.29654208.1 Defining polynomial: $$x^{6} + 14 x^{4} + 49 x^{2} + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} - q^{8} -\beta_{1} q^{10} + ( -1 - \beta_{2} ) q^{11} + ( -\beta_{2} - \beta_{4} ) q^{13} + \beta_{5} q^{14} + ( -1 + \beta_{1} ) q^{16} + ( -1 + \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{19} - q^{20} + ( -1 + \beta_{1} + \beta_{4} ) q^{22} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{23} -\beta_{1} q^{25} -\beta_{2} q^{26} + ( \beta_{3} + \beta_{5} ) q^{28} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{29} + \beta_{2} q^{31} + \beta_{1} q^{32} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{34} + \beta_{5} q^{35} - q^{37} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{38} + ( -1 + \beta_{1} ) q^{40} + ( 2 - 2 \beta_{1} + \beta_{4} + 3 \beta_{5} ) q^{41} + ( -4 + 4 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{44} + ( -1 - \beta_{2} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 6 + 2 \beta_{2} + 2 \beta_{3} ) q^{49} - q^{50} + \beta_{4} q^{52} + ( 6 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} + ( -1 + \beta_{1} + \beta_{4} ) q^{55} + \beta_{3} q^{56} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{58} + ( 6 - 6 \beta_{1} ) q^{59} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{61} -\beta_{4} q^{62} + q^{64} -\beta_{2} q^{65} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{67} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{68} + ( \beta_{3} + \beta_{5} ) q^{70} + ( -1 + \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{71} + ( 7 - 7 \beta_{1} + \beta_{4} + \beta_{5} ) q^{73} + ( -1 + \beta_{1} ) q^{74} + ( -\beta_{3} + \beta_{4} ) q^{76} + ( -4 - 3 \beta_{2} + \beta_{3} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{4} ) q^{79} + \beta_{1} q^{80} + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{82} + ( -2 + 4 \beta_{2} ) q^{83} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{85} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{86} + ( 1 + \beta_{2} ) q^{88} + ( -6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{89} + ( -4 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} ) q^{91} + ( -1 + \beta_{1} + \beta_{4} ) q^{92} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{95} + ( -13 + 13 \beta_{1} + \beta_{5} ) q^{97} + ( 6 - 6 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} - 3q^{4} + 3q^{5} + 2q^{7} - 6q^{8} + O(q^{10})$$ $$6q + 3q^{2} - 3q^{4} + 3q^{5} + 2q^{7} - 6q^{8} - 3q^{10} - 8q^{11} - q^{13} + q^{14} - 3q^{16} - 4q^{17} - 2q^{19} - 6q^{20} - 4q^{22} - 4q^{23} - 3q^{25} - 2q^{26} - q^{28} + 6q^{29} + 2q^{31} + 3q^{32} + 4q^{34} + q^{35} - 6q^{37} - q^{38} - 3q^{40} + 8q^{41} - 11q^{43} + 4q^{44} - 8q^{46} + 36q^{49} - 6q^{50} - q^{52} + 18q^{53} - 4q^{55} - 2q^{56} + 12q^{58} + 18q^{59} + 3q^{61} + q^{62} + 6q^{64} - 2q^{65} - 15q^{67} + 8q^{68} - q^{70} - 4q^{71} + 21q^{73} - 3q^{74} + q^{76} - 32q^{77} + 7q^{79} + 3q^{80} - 8q^{82} - 4q^{83} + 4q^{85} + 11q^{86} + 8q^{88} - 18q^{89} - 15q^{91} - 4q^{92} - q^{95} - 38q^{97} + 18q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 14 x^{4} + 49 x^{2} + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 7 \nu + 2$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} - 9 \nu^{2} - 10$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 12 \nu^{3} - 2 \nu^{2} + 35 \nu - 10$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 12 \nu^{3} + 9 \nu^{2} - 29 \nu + 10$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$($$$$-14 \beta_{5} - 14 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} + 12 \beta_{1} - 6$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{3} - 9 \beta_{2} + 35$$ $$\nu^{5}$$ $$=$$ $$($$$$98 \beta_{5} + 110 \beta_{4} + 49 \beta_{3} + 55 \beta_{2} - 144 \beta_{1} + 72$$$$)/3$$

## 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
 − 2.35084i 2.86514i − 0.514306i 2.35084i − 2.86514i 0.514306i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −4.59821 −1.00000 0 −0.500000 + 0.866025i
1261.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.75353 −1.00000 0 −0.500000 + 0.866025i
1261.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 3.84469 −1.00000 0 −0.500000 + 0.866025i
1531.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −4.59821 −1.00000 0 −0.500000 0.866025i
1531.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 1.75353 −1.00000 0 −0.500000 0.866025i
1531.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 3.84469 −1.00000 0 −0.500000 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1531.3 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.q 6
3.b odd 2 1 570.2.i.j 6
19.c even 3 1 inner 1710.2.l.q 6
57.h odd 6 1 570.2.i.j 6

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

 $$T_{7}^{3} - T_{7}^{2} - 19 T_{7} + 31$$ $$T_{11}^{3} + 4 T_{11}^{2} - 11 T_{11} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{3}$$
$3$ $$T^{6}$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$( 31 - 19 T - T^{2} + T^{3} )^{2}$$
$11$ $$( -6 - 11 T + 4 T^{2} + T^{3} )^{2}$$
$13$ $$64 - 128 T + 248 T^{2} - 32 T^{3} + 17 T^{4} + T^{5} + T^{6}$$
$17$ $$46656 + 10800 T + 3364 T^{2} + 232 T^{3} + 66 T^{4} + 4 T^{5} + T^{6}$$
$19$ $$6859 + 722 T + 152 T^{2} + 140 T^{3} + 8 T^{4} + 2 T^{5} + T^{6}$$
$23$ $$36 + 66 T + 145 T^{2} - 32 T^{3} + 27 T^{4} + 4 T^{5} + T^{6}$$
$29$ $$467856 - 69768 T + 14508 T^{2} - 756 T^{3} + 138 T^{4} - 6 T^{5} + T^{6}$$
$31$ $$( -8 - 16 T - T^{2} + T^{3} )^{2}$$
$37$ $$( 1 + T )^{6}$$
$41$ $$1077444 - 129750 T + 23929 T^{2} - 1076 T^{3} + 189 T^{4} - 8 T^{5} + T^{6}$$
$43$ $$1296 + 864 T + 972 T^{2} - 192 T^{3} + 145 T^{4} + 11 T^{5} + T^{6}$$
$47$ $$20736 - 12096 T + 7056 T^{2} - 288 T^{3} + 84 T^{4} + T^{6}$$
$53$ $$11664 - 9396 T + 5625 T^{2} - 1350 T^{3} + 237 T^{4} - 18 T^{5} + T^{6}$$
$59$ $$( 36 - 6 T + T^{2} )^{3}$$
$61$ $$4 - 36 T + 330 T^{2} + 50 T^{3} + 27 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$4 + 108 T + 2886 T^{2} + 806 T^{3} + 171 T^{4} + 15 T^{5} + T^{6}$$
$71$ $$46656 + 10800 T + 3364 T^{2} + 232 T^{3} + 66 T^{4} + 4 T^{5} + T^{6}$$
$73$ $$45796 - 26964 T + 11382 T^{2} - 2218 T^{3} + 315 T^{4} - 21 T^{5} + T^{6}$$
$79$ $$144 + 84 T^{2} - 24 T^{3} + 49 T^{4} - 7 T^{5} + T^{6}$$
$83$ $$( -1032 - 260 T + 2 T^{2} + T^{3} )^{2}$$
$89$ $$11664 + 9396 T + 5625 T^{2} + 1350 T^{3} + 237 T^{4} + 18 T^{5} + T^{6}$$
$97$ $$3283344 + 837144 T + 144588 T^{2} + 13932 T^{3} + 982 T^{4} + 38 T^{5} + T^{6}$$