# Properties

 Label 1782.2.e.s Level $1782$ Weight $2$ Character orbit 1782.e Analytic conductor $14.229$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1782 = 2 \cdot 3^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1782.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.2293416402$$ 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 66) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times$$.

 $$n$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
595.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.00000 1.73205i −1.00000 0 0
1189.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −1.00000 + 1.73205i −1.00000 0 0
 $$n$$: 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 1782.2.e.s 2
3.b odd 2 1 1782.2.e.f 2
9.c even 3 1 66.2.a.a 1
9.c even 3 1 inner 1782.2.e.s 2
9.d odd 6 1 198.2.a.e 1
9.d odd 6 1 1782.2.e.f 2
36.f odd 6 1 528.2.a.d 1
36.h even 6 1 1584.2.a.h 1
45.h odd 6 1 4950.2.a.g 1
45.j even 6 1 1650.2.a.m 1
45.k odd 12 2 1650.2.c.d 2
45.l even 12 2 4950.2.c.r 2
63.l odd 6 1 3234.2.a.d 1
63.o even 6 1 9702.2.a.bu 1
72.j odd 6 1 6336.2.a.bj 1
72.l even 6 1 6336.2.a.bf 1
72.n even 6 1 2112.2.a.i 1
72.p odd 6 1 2112.2.a.v 1
99.g even 6 1 2178.2.a.b 1
99.h odd 6 1 726.2.a.i 1
99.m even 15 4 726.2.e.k 4
99.o odd 30 4 726.2.e.b 4
396.k even 6 1 5808.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.a 1 9.c even 3 1
198.2.a.e 1 9.d odd 6 1
528.2.a.d 1 36.f odd 6 1
726.2.a.i 1 99.h odd 6 1
726.2.e.b 4 99.o odd 30 4
726.2.e.k 4 99.m even 15 4
1584.2.a.h 1 36.h even 6 1
1650.2.a.m 1 45.j even 6 1
1650.2.c.d 2 45.k odd 12 2
1782.2.e.f 2 3.b odd 2 1
1782.2.e.f 2 9.d odd 6 1
1782.2.e.s 2 1.a even 1 1 trivial
1782.2.e.s 2 9.c even 3 1 inner
2112.2.a.i 1 72.n even 6 1
2112.2.a.v 1 72.p odd 6 1
2178.2.a.b 1 99.g even 6 1
3234.2.a.d 1 63.l odd 6 1
4950.2.a.g 1 45.h odd 6 1
4950.2.c.r 2 45.l even 12 2
5808.2.a.l 1 396.k even 6 1
6336.2.a.bf 1 72.l even 6 1
6336.2.a.bj 1 72.j odd 6 1
9702.2.a.bu 1 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1782, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{13}^{2} - 4T_{13} + 16$$ T13^2 - 4*T13 + 16 $$T_{17} + 6$$ T17 + 6 $$T_{23}^{2} + 6T_{23} + 36$$ T23^2 + 6*T23 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 14T + 196$$