# Properties

 Label 1638.2.c.f Level $1638$ Weight $2$ Character orbit 1638.c Analytic conductor $13.079$ Analytic rank $0$ 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{2} - q^{4} + i q^{5} + i q^{7} + i q^{8} +O(q^{10})$$ $$q -i q^{2} - q^{4} + i q^{5} + i q^{7} + i q^{8} + q^{10} + i q^{11} + ( 3 - 2 i ) q^{13} + q^{14} + q^{16} - q^{17} + i q^{19} -i q^{20} + q^{22} -3 q^{23} + 4 q^{25} + ( -2 - 3 i ) q^{26} -i q^{28} -9 q^{29} + 4 i q^{31} -i q^{32} + i q^{34} - q^{35} + 9 i q^{37} + q^{38} - q^{40} + 8 i q^{41} + 7 q^{43} -i q^{44} + 3 i q^{46} + 8 i q^{47} - q^{49} -4 i q^{50} + ( -3 + 2 i ) q^{52} + 10 q^{53} - q^{55} - q^{56} + 9 i q^{58} -6 i q^{59} + 11 q^{61} + 4 q^{62} - q^{64} + ( 2 + 3 i ) q^{65} + 12 i q^{67} + q^{68} + i q^{70} + 6 i q^{71} -11 i q^{73} + 9 q^{74} -i q^{76} - q^{77} -12 q^{79} + i q^{80} + 8 q^{82} + 6 i q^{83} -i q^{85} -7 i q^{86} - q^{88} + 12 i q^{89} + ( 2 + 3 i ) q^{91} + 3 q^{92} + 8 q^{94} - q^{95} + 2 i q^{97} + i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{10} + 6q^{13} + 2q^{14} + 2q^{16} - 2q^{17} + 2q^{22} - 6q^{23} + 8q^{25} - 4q^{26} - 18q^{29} - 2q^{35} + 2q^{38} - 2q^{40} + 14q^{43} - 2q^{49} - 6q^{52} + 20q^{53} - 2q^{55} - 2q^{56} + 22q^{61} + 8q^{62} - 2q^{64} + 4q^{65} + 2q^{68} + 18q^{74} - 2q^{77} - 24q^{79} + 16q^{82} - 2q^{88} + 4q^{91} + 6q^{92} + 16q^{94} - 2q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
883.2 1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.f 2
3.b odd 2 1 546.2.c.a 2
12.b even 2 1 4368.2.h.k 2
13.b even 2 1 inner 1638.2.c.f 2
21.c even 2 1 3822.2.c.e 2
39.d odd 2 1 546.2.c.a 2
39.f even 4 1 7098.2.a.d 1
39.f even 4 1 7098.2.a.s 1
156.h even 2 1 4368.2.h.k 2
273.g even 2 1 3822.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 3.b odd 2 1
546.2.c.a 2 39.d odd 2 1
1638.2.c.f 2 1.a even 1 1 trivial
1638.2.c.f 2 13.b even 2 1 inner
3822.2.c.e 2 21.c even 2 1
3822.2.c.e 2 273.g even 2 1
4368.2.h.k 2 12.b even 2 1
4368.2.h.k 2 156.h even 2 1
7098.2.a.d 1 39.f even 4 1
7098.2.a.s 1 39.f even 4 1

## Hecke kernels

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

 $$T_{5}^{2} + 1$$ $$T_{11}^{2} + 1$$ $$T_{17} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$13 - 6 T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$1 + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$81 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( -7 + T )^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( -11 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$36 + T^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$144 + T^{2}$$
$97$ $$4 + T^{2}$$