# Properties

 Label 119.1.d.b Level 119 Weight 1 Character orbit 119.d Self dual yes Analytic conductor 0.059 Analytic rank 0 Dimension 2 Projective image $$D_{5}$$ CM discriminant -119 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$119 = 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 119.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.0593887365033$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.14161.1 Artin image $D_{10}$ Artin field Galois closure of 10.0.1403737447.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 - \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 - \beta ) q^{4} + \beta q^{5} + ( -2 + \beta ) q^{6} - q^{7} - q^{8} + ( 1 - \beta ) q^{9} + q^{10} + ( 2 - \beta ) q^{12} + ( 1 - \beta ) q^{14} - q^{15} - q^{17} + ( -2 + \beta ) q^{18} - q^{20} + ( -1 + \beta ) q^{21} + ( -1 + \beta ) q^{24} + \beta q^{25} + q^{27} + ( -1 + \beta ) q^{28} + ( 1 - \beta ) q^{30} + \beta q^{31} + q^{32} + ( 1 - \beta ) q^{34} -\beta q^{35} + ( 2 - \beta ) q^{36} -\beta q^{40} + ( 1 - \beta ) q^{41} + ( 2 - \beta ) q^{42} + ( -1 + \beta ) q^{43} - q^{45} + q^{49} + q^{50} + ( -1 + \beta ) q^{51} -\beta q^{53} + ( -1 + \beta ) q^{54} + q^{56} + ( -1 + \beta ) q^{60} + ( 1 - \beta ) q^{61} + q^{62} + ( -1 + \beta ) q^{63} + ( -1 + \beta ) q^{64} -\beta q^{67} + ( -1 + \beta ) q^{68} - q^{70} + ( -1 + \beta ) q^{72} + ( 1 - \beta ) q^{73} - q^{75} + ( -2 + \beta ) q^{82} + ( -2 + \beta ) q^{84} -\beta q^{85} + ( 2 - \beta ) q^{86} + ( 1 - \beta ) q^{90} - q^{93} + ( 1 - \beta ) q^{96} + \beta q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} + q^{4} + q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} + q^{4} + q^{5} - 3q^{6} - 2q^{7} - 2q^{8} + q^{9} + 2q^{10} + 3q^{12} + q^{14} - 2q^{15} - 2q^{17} - 3q^{18} - 2q^{20} - q^{21} - q^{24} + q^{25} + 2q^{27} - q^{28} + q^{30} + q^{31} + 2q^{32} + q^{34} - q^{35} + 3q^{36} - q^{40} + q^{41} + 3q^{42} - q^{43} - 2q^{45} + 2q^{49} + 2q^{50} - q^{51} - q^{53} - q^{54} + 2q^{56} - q^{60} + q^{61} + 2q^{62} - q^{63} - q^{64} - q^{67} - q^{68} - 2q^{70} - q^{72} + q^{73} - 2q^{75} - 3q^{82} - 3q^{84} - q^{85} + 3q^{86} + q^{90} - 2q^{93} + q^{96} + q^{97} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/119\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$71$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
118.1
 −0.618034 1.61803
−1.61803 1.61803 1.61803 −0.618034 −2.61803 −1.00000 −1.00000 1.61803 1.00000
118.2 0.618034 −0.618034 −0.618034 1.61803 −0.381966 −1.00000 −1.00000 −0.618034 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
119.d odd 2 1 CM by $$\Q(\sqrt{-119})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 119.1.d.b yes 2
3.b odd 2 1 1071.1.h.a 2
4.b odd 2 1 1904.1.n.a 2
5.b even 2 1 2975.1.h.c 2
5.c odd 4 2 2975.1.b.a 4
7.b odd 2 1 119.1.d.a 2
7.c even 3 2 833.1.h.a 4
7.d odd 6 2 833.1.h.b 4
17.b even 2 1 119.1.d.a 2
17.c even 4 2 2023.1.c.e 4
17.d even 8 4 2023.1.f.b 8
17.e odd 16 8 2023.1.l.b 16
21.c even 2 1 1071.1.h.b 2
28.d even 2 1 1904.1.n.b 2
35.c odd 2 1 2975.1.h.d 2
35.f even 4 2 2975.1.b.b 4
51.c odd 2 1 1071.1.h.b 2
68.d odd 2 1 1904.1.n.b 2
85.c even 2 1 2975.1.h.d 2
85.g odd 4 2 2975.1.b.b 4
119.d odd 2 1 CM 119.1.d.b yes 2
119.f odd 4 2 2023.1.c.e 4
119.h odd 6 2 833.1.h.a 4
119.j even 6 2 833.1.h.b 4
119.l odd 8 4 2023.1.f.b 8
119.p even 16 8 2023.1.l.b 16
357.c even 2 1 1071.1.h.a 2
476.e even 2 1 1904.1.n.a 2
595.b odd 2 1 2975.1.h.c 2
595.p even 4 2 2975.1.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.1.d.a 2 7.b odd 2 1
119.1.d.a 2 17.b even 2 1
119.1.d.b yes 2 1.a even 1 1 trivial
119.1.d.b yes 2 119.d odd 2 1 CM
833.1.h.a 4 7.c even 3 2
833.1.h.a 4 119.h odd 6 2
833.1.h.b 4 7.d odd 6 2
833.1.h.b 4 119.j even 6 2
1071.1.h.a 2 3.b odd 2 1
1071.1.h.a 2 357.c even 2 1
1071.1.h.b 2 21.c even 2 1
1071.1.h.b 2 51.c odd 2 1
1904.1.n.a 2 4.b odd 2 1
1904.1.n.a 2 476.e even 2 1
1904.1.n.b 2 28.d even 2 1
1904.1.n.b 2 68.d odd 2 1
2023.1.c.e 4 17.c even 4 2
2023.1.c.e 4 119.f odd 4 2
2023.1.f.b 8 17.d even 8 4
2023.1.f.b 8 119.l odd 8 4
2023.1.l.b 16 17.e odd 16 8
2023.1.l.b 16 119.p even 16 8
2975.1.b.a 4 5.c odd 4 2
2975.1.b.a 4 595.p even 4 2
2975.1.b.b 4 35.f even 4 2
2975.1.b.b 4 85.g odd 4 2
2975.1.h.c 2 5.b even 2 1
2975.1.h.c 2 595.b odd 2 1
2975.1.h.d 2 35.c odd 2 1
2975.1.h.d 2 85.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 1$$ acting on $$S_{1}^{\mathrm{new}}(119, [\chi])$$.

## Hecke characteristic polynomials

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