# Properties

 Label 1755.2.i.b Level $1755$ Weight $2$ Character orbit 1755.i Analytic conductor $14.014$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1755 = 3^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1755.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0137455547$$ Analytic rank: $$1$$ 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: no (minimal twist has level 585) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
586.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 + 1.73205i −3.00000 0 1.00000
1171.1 −0.500000 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 1.73205i −3.00000 0 1.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1755.2.i.b 2
3.b odd 2 1 585.2.i.b 2
9.c even 3 1 inner 1755.2.i.b 2
9.c even 3 1 5265.2.a.m 1
9.d odd 6 1 585.2.i.b 2
9.d odd 6 1 5265.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.b 2 3.b odd 2 1
585.2.i.b 2 9.d odd 6 1
1755.2.i.b 2 1.a even 1 1 trivial
1755.2.i.b 2 9.c even 3 1 inner
5265.2.a.e 1 9.d odd 6 1
5265.2.a.m 1 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1755, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T + 3)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 5T + 25$$
$31$ $$T^{2} - T + 1$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$(T + 14)^{2}$$
$59$ $$T^{2} + 9T + 81$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$T^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 12T + 144$$
$83$ $$T^{2} + 10T + 100$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + T + 1$$