# 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: $$\Q(\sqrt{-3})$$ 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.

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

## 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 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 $$S_{2}^{\mathrm{new}}(1782, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ 1
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$1 - T + T^{2}$$
$13$ $$1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 10 T + 37 T^{2} )^{2}$$
$41$ $$1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4}$$
$83$ $$1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$