# Properties

 Label 1710.2.l.k Level $1710$ Weight $2$ Character orbit 1710.l Analytic conductor $13.654$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## 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: $$\Q(\sqrt{-3}, \sqrt{19})$$ Defining polynomial: $$x^{4} + 19 x^{2} + 361$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} + \beta_{2} q^{10} + 3 q^{11} -4 \beta_{2} q^{13} + \beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{19} + q^{20} + ( -3 - 3 \beta_{2} ) q^{22} + 3 \beta_{2} q^{23} + \beta_{2} q^{25} -4 q^{26} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + \beta_{1} q^{35} -7 q^{37} + ( \beta_{1} + \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{40} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{41} + ( 10 + 10 \beta_{2} ) q^{43} + 3 \beta_{2} q^{44} + 3 q^{46} + 6 \beta_{2} q^{47} + 12 q^{49} + q^{50} + ( 4 + 4 \beta_{2} ) q^{52} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{53} + ( -3 - 3 \beta_{2} ) q^{55} + \beta_{3} q^{56} + ( -5 + \beta_{3} ) q^{58} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{59} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{62} + q^{64} -4 q^{65} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{67} + ( 1 + \beta_{3} ) q^{68} + ( -\beta_{1} - \beta_{3} ) q^{70} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -9 + \beta_{1} - 9 \beta_{2} ) q^{73} + ( 7 + 7 \beta_{2} ) q^{74} -\beta_{3} q^{76} + 3 \beta_{3} q^{77} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{82} + 6 q^{83} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{85} -10 \beta_{2} q^{86} + 3 q^{88} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{89} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{91} + ( -3 - 3 \beta_{2} ) q^{92} + 6 q^{94} + ( \beta_{1} + \beta_{3} ) q^{95} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{97} + ( -12 - 12 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 2q^{10} + 12q^{11} + 8q^{13} - 2q^{16} - 2q^{17} + 4q^{20} - 6q^{22} - 6q^{23} - 2q^{25} - 16q^{26} + 10q^{29} - 8q^{31} - 2q^{32} - 2q^{34} - 28q^{37} - 2q^{40} - 4q^{41} + 20q^{43} - 6q^{44} + 12q^{46} - 12q^{47} + 48q^{49} + 4q^{50} + 8q^{52} - 6q^{55} - 20q^{58} + 8q^{59} - 2q^{61} + 4q^{62} + 4q^{64} - 16q^{65} - 14q^{67} + 4q^{68} + 6q^{71} - 18q^{73} + 14q^{74} - 8q^{79} - 2q^{80} - 4q^{82} + 24q^{83} - 2q^{85} + 20q^{86} + 12q^{88} - 8q^{89} - 6q^{92} + 24q^{94} + 2q^{97} - 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 19 x^{2} + 361$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/19$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$19 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$19 \beta_{3}$$

## 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
 2.17945 + 3.77492i −2.17945 − 3.77492i 2.17945 − 3.77492i −2.17945 + 3.77492i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −4.35890 1.00000 0 −0.500000 + 0.866025i
1261.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 4.35890 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −4.35890 1.00000 0 −0.500000 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 4.35890 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.k 4
3.b odd 2 1 570.2.i.h 4
19.c even 3 1 inner 1710.2.l.k 4
57.h odd 6 1 570.2.i.h 4

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

 $$T_{7}^{2} - 19$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( -19 + T^{2} )^{2}$$
$11$ $$( -3 + T )^{4}$$
$13$ $$( 16 - 4 T + T^{2} )^{2}$$
$17$ $$324 - 36 T + 22 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$361 + 19 T^{2} + T^{4}$$
$23$ $$( 9 + 3 T + T^{2} )^{2}$$
$29$ $$36 - 60 T + 94 T^{2} - 10 T^{3} + T^{4}$$
$31$ $$( -72 + 4 T + T^{2} )^{2}$$
$37$ $$( 7 + T )^{4}$$
$41$ $$225 - 60 T + 31 T^{2} + 4 T^{3} + T^{4}$$
$43$ $$( 100 - 10 T + T^{2} )^{2}$$
$47$ $$( 36 + 6 T + T^{2} )^{2}$$
$53$ $$29241 + 171 T^{2} + T^{4}$$
$59$ $$3600 + 480 T + 124 T^{2} - 8 T^{3} + T^{4}$$
$61$ $$324 - 36 T + 22 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$900 + 420 T + 166 T^{2} + 14 T^{3} + T^{4}$$
$71$ $$26244 + 972 T + 198 T^{2} - 6 T^{3} + T^{4}$$
$73$ $$3844 + 1116 T + 262 T^{2} + 18 T^{3} + T^{4}$$
$79$ $$3600 - 480 T + 124 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$( -6 + T )^{4}$$
$89$ $$9 - 24 T + 67 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$28900 + 340 T + 174 T^{2} - 2 T^{3} + T^{4}$$
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