# Properties

 Label 1710.2.l.d Level $1710$ Weight $2$ Character orbit 1710.l Analytic conductor $13.654$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - 5 q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + (-z + 1) * q^5 - 5 * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - 5 q^{7} + q^{8} + \zeta_{6} q^{10} + 2 q^{11} + \zeta_{6} q^{13} + ( - 5 \zeta_{6} + 5) q^{14} + (\zeta_{6} - 1) q^{16} + (6 \zeta_{6} - 6) q^{17} + (2 \zeta_{6} + 3) q^{19} - q^{20} + (2 \zeta_{6} - 2) q^{22} - 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} - q^{26} + 5 \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + 7 q^{31} - \zeta_{6} q^{32} - 6 \zeta_{6} q^{34} + (5 \zeta_{6} - 5) q^{35} + q^{37} + (3 \zeta_{6} - 5) q^{38} + ( - \zeta_{6} + 1) q^{40} + ( - 9 \zeta_{6} + 9) q^{43} - 2 \zeta_{6} q^{44} + 4 q^{46} + 18 q^{49} + q^{50} + ( - \zeta_{6} + 1) q^{52} - 8 \zeta_{6} q^{53} + ( - 2 \zeta_{6} + 2) q^{55} - 5 q^{56} + 6 q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 3 \zeta_{6} q^{61} + (7 \zeta_{6} - 7) q^{62} + q^{64} + q^{65} + 9 \zeta_{6} q^{67} + 6 q^{68} - 5 \zeta_{6} q^{70} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + (\zeta_{6} - 1) q^{74} + ( - 5 \zeta_{6} + 2) q^{76} - 10 q^{77} + ( - 5 \zeta_{6} + 5) q^{79} + \zeta_{6} q^{80} + 14 q^{83} + 6 \zeta_{6} q^{85} + 9 \zeta_{6} q^{86} + 2 q^{88} - 8 \zeta_{6} q^{89} - 5 \zeta_{6} q^{91} + (4 \zeta_{6} - 4) q^{92} + ( - 3 \zeta_{6} + 5) q^{95} + ( - 14 \zeta_{6} + 14) q^{97} + (18 \zeta_{6} - 18) q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (-z + 1) * q^5 - 5 * q^7 + q^8 + z * q^10 + 2 * q^11 + z * q^13 + (-5*z + 5) * q^14 + (z - 1) * q^16 + (6*z - 6) * q^17 + (2*z + 3) * q^19 - q^20 + (2*z - 2) * q^22 - 4*z * q^23 - z * q^25 - q^26 + 5*z * q^28 - 6*z * q^29 + 7 * q^31 - z * q^32 - 6*z * q^34 + (5*z - 5) * q^35 + q^37 + (3*z - 5) * q^38 + (-z + 1) * q^40 + (-9*z + 9) * q^43 - 2*z * q^44 + 4 * q^46 + 18 * q^49 + q^50 + (-z + 1) * q^52 - 8*z * q^53 + (-2*z + 2) * q^55 - 5 * q^56 + 6 * q^58 + (-12*z + 12) * q^59 + 3*z * q^61 + (7*z - 7) * q^62 + q^64 + q^65 + 9*z * q^67 + 6 * q^68 - 5*z * q^70 + (-6*z + 6) * q^71 + (-7*z + 7) * q^73 + (z - 1) * q^74 + (-5*z + 2) * q^76 - 10 * q^77 + (-5*z + 5) * q^79 + z * q^80 + 14 * q^83 + 6*z * q^85 + 9*z * q^86 + 2 * q^88 - 8*z * q^89 - 5*z * q^91 + (4*z - 4) * q^92 + (-3*z + 5) * q^95 + (-14*z + 14) * q^97 + (18*z - 18) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + q^{5} - 10 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + q^5 - 10 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + q^{5} - 10 q^{7} + 2 q^{8} + q^{10} + 4 q^{11} + q^{13} + 5 q^{14} - q^{16} - 6 q^{17} + 8 q^{19} - 2 q^{20} - 2 q^{22} - 4 q^{23} - q^{25} - 2 q^{26} + 5 q^{28} - 6 q^{29} + 14 q^{31} - q^{32} - 6 q^{34} - 5 q^{35} + 2 q^{37} - 7 q^{38} + q^{40} + 9 q^{43} - 2 q^{44} + 8 q^{46} + 36 q^{49} + 2 q^{50} + q^{52} - 8 q^{53} + 2 q^{55} - 10 q^{56} + 12 q^{58} + 12 q^{59} + 3 q^{61} - 7 q^{62} + 2 q^{64} + 2 q^{65} + 9 q^{67} + 12 q^{68} - 5 q^{70} + 6 q^{71} + 7 q^{73} - q^{74} - q^{76} - 20 q^{77} + 5 q^{79} + q^{80} + 28 q^{83} + 6 q^{85} + 9 q^{86} + 4 q^{88} - 8 q^{89} - 5 q^{91} - 4 q^{92} + 7 q^{95} + 14 q^{97} - 18 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + q^5 - 10 * q^7 + 2 * q^8 + q^10 + 4 * q^11 + q^13 + 5 * q^14 - q^16 - 6 * q^17 + 8 * q^19 - 2 * q^20 - 2 * q^22 - 4 * q^23 - q^25 - 2 * q^26 + 5 * q^28 - 6 * q^29 + 14 * q^31 - q^32 - 6 * q^34 - 5 * q^35 + 2 * q^37 - 7 * q^38 + q^40 + 9 * q^43 - 2 * q^44 + 8 * q^46 + 36 * q^49 + 2 * q^50 + q^52 - 8 * q^53 + 2 * q^55 - 10 * q^56 + 12 * q^58 + 12 * q^59 + 3 * q^61 - 7 * q^62 + 2 * q^64 + 2 * q^65 + 9 * q^67 + 12 * q^68 - 5 * q^70 + 6 * q^71 + 7 * q^73 - q^74 - q^76 - 20 * q^77 + 5 * q^79 + q^80 + 28 * q^83 + 6 * q^85 + 9 * q^86 + 4 * q^88 - 8 * q^89 - 5 * q^91 - 4 * q^92 + 7 * q^95 + 14 * q^97 - 18 * q^98

## Character values

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

 $$n$$ $$191$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −5.00000 1.00000 0 0.500000 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −5.00000 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.d 2
3.b odd 2 1 1710.2.l.e yes 2
19.c even 3 1 inner 1710.2.l.d 2
57.h odd 6 1 1710.2.l.e yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.l.d 2 1.a even 1 1 trivial
1710.2.l.d 2 19.c even 3 1 inner
1710.2.l.e yes 2 3.b odd 2 1
1710.2.l.e yes 2 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1710, [\chi])$$:

 $$T_{7} + 5$$ T7 + 5 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 5)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T - 7)^{2}$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 9T + 81$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 8T + 64$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 3T + 9$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} - 5T + 25$$
$83$ $$(T - 14)^{2}$$
$89$ $$T^{2} + 8T + 64$$
$97$ $$T^{2} - 14T + 196$$