# Properties

 Label 1782.2.e.n 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: $$\Q(\sqrt{-3})$$ 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.

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

## 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 −2.00000 + 3.46410i 0 1.00000 + 1.73205i −1.00000 0 −4.00000
1189.1 0.500000 0.866025i 0 −0.500000 0.866025i −2.00000 3.46410i 0 1.00000 1.73205i −1.00000 0 −4.00000
 $$n$$: 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.n 2
3.b odd 2 1 1782.2.e.l 2
9.c even 3 1 198.2.a.c 1
9.c even 3 1 inner 1782.2.e.n 2
9.d odd 6 1 66.2.a.c 1
9.d odd 6 1 1782.2.e.l 2
36.f odd 6 1 1584.2.a.s 1
36.h even 6 1 528.2.a.a 1
45.h odd 6 1 1650.2.a.c 1
45.j even 6 1 4950.2.a.bo 1
45.k odd 12 2 4950.2.c.d 2
45.l even 12 2 1650.2.c.m 2
63.l odd 6 1 9702.2.a.a 1
63.o even 6 1 3234.2.a.s 1
72.j odd 6 1 2112.2.a.n 1
72.l even 6 1 2112.2.a.bd 1
72.n even 6 1 6336.2.a.c 1
72.p odd 6 1 6336.2.a.d 1
99.g even 6 1 726.2.a.d 1
99.h odd 6 1 2178.2.a.m 1
99.n odd 30 4 726.2.e.e 4
99.p even 30 4 726.2.e.m 4
396.o odd 6 1 5808.2.a.b 1

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

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 1
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$1 - T + T^{2}$$
$13$ $$1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 - 10 T + 71 T^{2} - 290 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 2 T - 37 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 4 T + 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 2 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{2}$$
$79$ $$1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 4 T - 67 T^{2} - 332 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$