# Properties

 Label 1648.1.c.a Level $1648$ Weight $1$ Character orbit 1648.c Self dual yes Analytic conductor $0.822$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -103 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1648 = 2^{4} \cdot 103$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1648.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.822459140819$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.10609.1 Artin image: $D_{10}$ Artin field: Galois closure of 10.0.115252102144.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + ( 1 - \beta ) q^{7} + q^{9} -\beta q^{13} + ( -1 + \beta ) q^{17} + \beta q^{19} + \beta q^{23} + q^{25} + ( -1 + \beta ) q^{29} -\beta q^{41} + ( 1 - \beta ) q^{49} + ( 1 - \beta ) q^{59} + ( -1 + \beta ) q^{61} + ( 1 - \beta ) q^{63} + \beta q^{79} + q^{81} + ( 1 - \beta ) q^{83} + q^{91} + ( -1 + \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q + q^{7} + 2 q^{9} - q^{13} - q^{17} + q^{19} + q^{23} + 2 q^{25} - q^{29} - q^{41} + q^{49} + q^{59} - q^{61} + q^{63} + q^{79} + 2 q^{81} + q^{83} + 2 q^{91} - q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1648\mathbb{Z}\right)^\times$$.

 $$n$$ $$207$$ $$417$$ $$1237$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1441.1
 1.61803 −0.618034
0 0 0 0 0 −0.618034 0 1.00000 0
1441.2 0 0 0 0 0 1.61803 0 1.00000 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
103.b odd 2 1 CM by $$\Q(\sqrt{-103})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1648.1.c.a 2
4.b odd 2 1 103.1.b.a 2
12.b even 2 1 927.1.d.b 2
20.d odd 2 1 2575.1.d.d 2
20.e even 4 2 2575.1.c.b 4
103.b odd 2 1 CM 1648.1.c.a 2
412.d even 2 1 103.1.b.a 2
1236.f odd 2 1 927.1.d.b 2
2060.e even 2 1 2575.1.d.d 2
2060.m odd 4 2 2575.1.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 4.b odd 2 1
103.1.b.a 2 412.d even 2 1
927.1.d.b 2 12.b even 2 1
927.1.d.b 2 1236.f odd 2 1
1648.1.c.a 2 1.a even 1 1 trivial
1648.1.c.a 2 103.b odd 2 1 CM
2575.1.c.b 4 20.e even 4 2
2575.1.c.b 4 2060.m odd 4 2
2575.1.d.d 2 20.d odd 2 1
2575.1.d.d 2 2060.e even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1648, [\chi])$$.

## Hecke characteristic polynomials

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