# Properties

 Label 1638.2.r.c Level 1638 Weight 2 Character orbit 1638.r Analytic conductor 13.079 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.r (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$1$$ 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 182) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} -\zeta_{6} q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} -\zeta_{6} q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} + ( 4 - 3 \zeta_{6} ) q^{13} + q^{14} + ( -1 + \zeta_{6} ) q^{16} + 6 \zeta_{6} q^{17} + 4 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} + ( 3 - 3 \zeta_{6} ) q^{23} + 4 q^{25} + ( -1 + 4 \zeta_{6} ) q^{26} + ( -1 + \zeta_{6} ) q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -10 q^{31} -\zeta_{6} q^{32} -6 q^{34} + 3 \zeta_{6} q^{35} + ( -8 + 8 \zeta_{6} ) q^{37} -4 q^{38} -3 q^{40} -8 \zeta_{6} q^{43} + 3 \zeta_{6} q^{46} -6 q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} + ( -3 - \zeta_{6} ) q^{52} -12 q^{53} -\zeta_{6} q^{56} + 6 \zeta_{6} q^{58} + 3 \zeta_{6} q^{59} -11 \zeta_{6} q^{61} + ( 10 - 10 \zeta_{6} ) q^{62} + q^{64} + ( -12 + 9 \zeta_{6} ) q^{65} + ( -2 + 2 \zeta_{6} ) q^{67} + ( 6 - 6 \zeta_{6} ) q^{68} -3 q^{70} -3 \zeta_{6} q^{71} + 2 q^{73} -8 \zeta_{6} q^{74} + ( 4 - 4 \zeta_{6} ) q^{76} -4 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} -18 \zeta_{6} q^{85} + 8 q^{86} + ( -6 + 6 \zeta_{6} ) q^{89} + ( -3 - \zeta_{6} ) q^{91} -3 q^{92} + ( 6 - 6 \zeta_{6} ) q^{94} -12 \zeta_{6} q^{95} -2 \zeta_{6} q^{97} -\zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 6q^{5} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 6q^{5} - q^{7} + 2q^{8} + 3q^{10} + 5q^{13} + 2q^{14} - q^{16} + 6q^{17} + 4q^{19} + 3q^{20} + 3q^{23} + 8q^{25} + 2q^{26} - q^{28} + 6q^{29} - 20q^{31} - q^{32} - 12q^{34} + 3q^{35} - 8q^{37} - 8q^{38} - 6q^{40} - 8q^{43} + 3q^{46} - 12q^{47} - q^{49} - 4q^{50} - 7q^{52} - 24q^{53} - q^{56} + 6q^{58} + 3q^{59} - 11q^{61} + 10q^{62} + 2q^{64} - 15q^{65} - 2q^{67} + 6q^{68} - 6q^{70} - 3q^{71} + 4q^{73} - 8q^{74} + 4q^{76} - 8q^{79} + 3q^{80} - 18q^{85} + 16q^{86} - 6q^{89} - 7q^{91} - 6q^{92} + 6q^{94} - 12q^{95} - 2q^{97} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −3.00000 0 −0.500000 + 0.866025i 1.00000 0 1.50000 + 2.59808i
1387.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −3.00000 0 −0.500000 0.866025i 1.00000 0 1.50000 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.r.c 2
3.b odd 2 1 182.2.g.b 2
12.b even 2 1 1456.2.s.e 2
13.c even 3 1 inner 1638.2.r.c 2
21.c even 2 1 1274.2.g.g 2
21.g even 6 1 1274.2.e.i 2
21.g even 6 1 1274.2.h.i 2
21.h odd 6 1 1274.2.e.d 2
21.h odd 6 1 1274.2.h.j 2
39.h odd 6 1 2366.2.a.n 1
39.i odd 6 1 182.2.g.b 2
39.i odd 6 1 2366.2.a.f 1
39.k even 12 2 2366.2.d.f 2
156.p even 6 1 1456.2.s.e 2
273.r even 6 1 1274.2.h.i 2
273.s odd 6 1 1274.2.h.j 2
273.bf even 6 1 1274.2.e.i 2
273.bm odd 6 1 1274.2.e.d 2
273.bn even 6 1 1274.2.g.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 3.b odd 2 1
182.2.g.b 2 39.i odd 6 1
1274.2.e.d 2 21.h odd 6 1
1274.2.e.d 2 273.bm odd 6 1
1274.2.e.i 2 21.g even 6 1
1274.2.e.i 2 273.bf even 6 1
1274.2.g.g 2 21.c even 2 1
1274.2.g.g 2 273.bn even 6 1
1274.2.h.i 2 21.g even 6 1
1274.2.h.i 2 273.r even 6 1
1274.2.h.j 2 21.h odd 6 1
1274.2.h.j 2 273.s odd 6 1
1456.2.s.e 2 12.b even 2 1
1456.2.s.e 2 156.p even 6 1
1638.2.r.c 2 1.a even 1 1 trivial
1638.2.r.c 2 13.c even 3 1 inner
2366.2.a.f 1 39.i odd 6 1
2366.2.a.n 1 39.h odd 6 1
2366.2.d.f 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5} + 3$$ $$T_{11}$$ $$T_{17}^{2} - 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$1 - 5 T + 13 T^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$1 - 6 T + 7 T^{2} - 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 10 T + 31 T^{2} )^{2}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{2}$$
$53$ $$( 1 + 12 T + 53 T^{2} )^{2}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$1 + 3 T - 62 T^{2} + 213 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$