# Properties

 Label 107.1.b.a Level 107 Weight 1 Character orbit 107.b Self dual yes Analytic conductor 0.053 Analytic rank 0 Dimension 1 Projective image $$D_{3}$$ CM discriminant -107 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$107$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 107.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0533999563517$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.107.1 Artin image $S_3$ Artin field Galois closure of 3.1.107.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} + O(q^{10})$$ $$q - q^{3} + q^{4} - q^{11} - q^{12} - q^{13} + q^{16} - q^{19} - q^{23} + q^{25} + q^{27} + 2q^{29} + q^{33} - q^{37} + q^{39} - q^{41} - q^{44} + 2q^{47} - q^{48} + q^{49} - q^{52} - q^{53} + q^{57} - q^{61} + q^{64} + q^{69} - q^{75} - q^{76} - q^{79} - q^{81} + 2q^{83} - 2q^{87} - q^{89} - q^{92} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/107\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 0
0 −1.00000 1.00000 0 0 0 0 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
107.b odd 2 1 CM by $$\Q(\sqrt{-107})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 107.1.b.a 1
3.b odd 2 1 963.1.b.a 1
4.b odd 2 1 1712.1.g.a 1
5.b even 2 1 2675.1.c.a 1
5.c odd 4 2 2675.1.d.a 2
107.b odd 2 1 CM 107.1.b.a 1
321.d even 2 1 963.1.b.a 1
428.b even 2 1 1712.1.g.a 1
535.d odd 2 1 2675.1.c.a 1
535.e even 4 2 2675.1.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
107.1.b.a 1 1.a even 1 1 trivial
107.1.b.a 1 107.b odd 2 1 CM
963.1.b.a 1 3.b odd 2 1
963.1.b.a 1 321.d even 2 1
1712.1.g.a 1 4.b odd 2 1
1712.1.g.a 1 428.b even 2 1
2675.1.c.a 1 5.b even 2 1
2675.1.c.a 1 535.d odd 2 1
2675.1.d.a 2 5.c odd 4 2
2675.1.d.a 2 535.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(107, [\chi])$$.

## Hecke characteristic polynomials

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