# Properties

 Label 1652.1.g.b Level 1652 Weight 1 Character orbit 1652.g Self dual yes Analytic conductor 0.824 Analytic rank 0 Dimension 2 Projective image $$D_{5}$$ CM discriminant -1652 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1652 = 2^{2} \cdot 7 \cdot 59$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 1652.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.824455400870$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.2729104.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( 1 - \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( 1 - \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} + ( 1 - \beta ) q^{11} + ( 1 - \beta ) q^{12} -\beta q^{13} + q^{14} + q^{16} + ( -1 + \beta ) q^{18} + \beta q^{19} + ( -1 + \beta ) q^{21} + ( -1 + \beta ) q^{22} + \beta q^{23} + ( -1 + \beta ) q^{24} + q^{25} + \beta q^{26} + q^{27} - q^{28} + ( -1 + \beta ) q^{29} - q^{32} + ( 2 - \beta ) q^{33} + ( 1 - \beta ) q^{36} -\beta q^{38} + q^{39} + ( 1 - \beta ) q^{42} + \beta q^{43} + ( 1 - \beta ) q^{44} -\beta q^{46} + ( 1 - \beta ) q^{48} + q^{49} - q^{50} -\beta q^{52} -\beta q^{53} - q^{54} + q^{56} - q^{57} + ( 1 - \beta ) q^{58} - q^{59} + 2 q^{61} + ( -1 + \beta ) q^{63} + q^{64} + ( -2 + \beta ) q^{66} + ( 1 - \beta ) q^{67} - q^{69} + ( -1 + \beta ) q^{72} + ( -1 + \beta ) q^{73} + ( 1 - \beta ) q^{75} + \beta q^{76} + ( -1 + \beta ) q^{77} - q^{78} + ( -1 + \beta ) q^{84} -\beta q^{86} + ( -2 + \beta ) q^{87} + ( -1 + \beta ) q^{88} + ( -1 + \beta ) q^{89} + \beta q^{91} + \beta q^{92} + ( -1 + \beta ) q^{96} + ( -1 + \beta ) q^{97} - q^{98} + ( 2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} - 2q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - 2q^{2} + q^{3} + 2q^{4} - q^{6} - 2q^{7} - 2q^{8} + q^{9} + q^{11} + q^{12} - q^{13} + 2q^{14} + 2q^{16} - q^{18} + q^{19} - q^{21} - q^{22} + q^{23} - q^{24} + 2q^{25} + q^{26} + 2q^{27} - 2q^{28} - q^{29} - 2q^{32} + 3q^{33} + q^{36} - q^{38} + 2q^{39} + q^{42} + q^{43} + q^{44} - q^{46} + q^{48} + 2q^{49} - 2q^{50} - q^{52} - q^{53} - 2q^{54} + 2q^{56} - 2q^{57} + q^{58} - 2q^{59} + 4q^{61} - q^{63} + 2q^{64} - 3q^{66} + q^{67} - 2q^{69} - q^{72} - q^{73} + q^{75} + q^{76} - q^{77} - 2q^{78} - q^{84} - q^{86} - 3q^{87} - q^{88} - q^{89} + q^{91} + q^{92} - q^{96} - q^{97} - 2q^{98} + 3q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1652\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$827$$ $$1417$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1651.1
 1.61803 −0.618034
−1.00000 −0.618034 1.00000 0 0.618034 −1.00000 −1.00000 −0.618034 0
1651.2 −1.00000 1.61803 1.00000 0 −1.61803 −1.00000 −1.00000 1.61803 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
1652.g odd 2 1 CM by $$\Q(\sqrt{-413})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1652.1.g.b yes 2
4.b odd 2 1 1652.1.g.c yes 2
7.b odd 2 1 1652.1.g.a 2
28.d even 2 1 1652.1.g.d yes 2
59.b odd 2 1 1652.1.g.d yes 2
236.c even 2 1 1652.1.g.a 2
413.b even 2 1 1652.1.g.c yes 2
1652.g odd 2 1 CM 1652.1.g.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1652.1.g.a 2 7.b odd 2 1
1652.1.g.a 2 236.c even 2 1
1652.1.g.b yes 2 1.a even 1 1 trivial
1652.1.g.b yes 2 1652.g odd 2 1 CM
1652.1.g.c yes 2 4.b odd 2 1
1652.1.g.c yes 2 413.b even 2 1
1652.1.g.d yes 2 28.d even 2 1
1652.1.g.d yes 2 59.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1652, [\chi])$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{11}^{2} - T_{11} - 1$$

## Hecke Characteristic Polynomials

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