# Properties

 Label 1782.2.e.e Level $1782$ Weight $2$ Character orbit 1782.e Analytic conductor $14.229$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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.

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

## 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 −1.00000 + 1.73205i 0 2.00000 + 3.46410i 1.00000 0 2.00000
1189.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 1.73205i 0 2.00000 3.46410i 1.00000 0 2.00000
 $$n$$: 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 1782.2.e.e 2
3.b odd 2 1 1782.2.e.v 2
9.c even 3 1 66.2.a.b 1
9.c even 3 1 inner 1782.2.e.e 2
9.d odd 6 1 198.2.a.a 1
9.d odd 6 1 1782.2.e.v 2
36.f odd 6 1 528.2.a.j 1
36.h even 6 1 1584.2.a.f 1
45.h odd 6 1 4950.2.a.bu 1
45.j even 6 1 1650.2.a.k 1
45.k odd 12 2 1650.2.c.e 2
45.l even 12 2 4950.2.c.p 2
63.l odd 6 1 3234.2.a.t 1
63.o even 6 1 9702.2.a.x 1
72.j odd 6 1 6336.2.a.bw 1
72.l even 6 1 6336.2.a.cj 1
72.n even 6 1 2112.2.a.r 1
72.p odd 6 1 2112.2.a.e 1
99.g even 6 1 2178.2.a.g 1
99.h odd 6 1 726.2.a.c 1
99.m even 15 4 726.2.e.g 4
99.o odd 30 4 726.2.e.o 4
396.k even 6 1 5808.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 9.c even 3 1
198.2.a.a 1 9.d odd 6 1
528.2.a.j 1 36.f odd 6 1
726.2.a.c 1 99.h odd 6 1
726.2.e.g 4 99.m even 15 4
726.2.e.o 4 99.o odd 30 4
1584.2.a.f 1 36.h even 6 1
1650.2.a.k 1 45.j even 6 1
1650.2.c.e 2 45.k odd 12 2
1782.2.e.e 2 1.a even 1 1 trivial
1782.2.e.e 2 9.c even 3 1 inner
1782.2.e.v 2 3.b odd 2 1
1782.2.e.v 2 9.d odd 6 1
2112.2.a.e 1 72.p odd 6 1
2112.2.a.r 1 72.n even 6 1
2178.2.a.g 1 99.g even 6 1
3234.2.a.t 1 63.l odd 6 1
4950.2.a.bu 1 45.h odd 6 1
4950.2.c.p 2 45.l even 12 2
5808.2.a.bc 1 396.k even 6 1
6336.2.a.bw 1 72.j odd 6 1
6336.2.a.cj 1 72.l even 6 1
9702.2.a.x 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 $$S_{2}^{\mathrm{new}}(1782, [\chi])$$:

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{7}^{2} - 4 T_{7} + 16$$ $$T_{13}^{2} - 6 T_{13} + 36$$ $$T_{17} - 2$$ $$T_{23}^{2} + 4 T_{23} + 16$$

## Hecke characteristic polynomials

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