# Properties

 Label 19.3.b.b Level $19$ Weight $3$ Character orbit 19.b Analytic conductor $0.518$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 19.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.517712502285$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -\beta q^{3} -9 q^{4} + 4 q^{5} + 13 q^{6} -5 q^{7} -5 \beta q^{8} -4 q^{9} +O(q^{10})$$ $$q + \beta q^{2} -\beta q^{3} -9 q^{4} + 4 q^{5} + 13 q^{6} -5 q^{7} -5 \beta q^{8} -4 q^{9} + 4 \beta q^{10} -10 q^{11} + 9 \beta q^{12} + \beta q^{13} -5 \beta q^{14} -4 \beta q^{15} + 29 q^{16} + 15 q^{17} -4 \beta q^{18} + ( -6 + 5 \beta ) q^{19} -36 q^{20} + 5 \beta q^{21} -10 \beta q^{22} + 35 q^{23} -65 q^{24} -9 q^{25} -13 q^{26} -5 \beta q^{27} + 45 q^{28} + 5 \beta q^{29} + 52 q^{30} -10 \beta q^{31} + 9 \beta q^{32} + 10 \beta q^{33} + 15 \beta q^{34} -20 q^{35} + 36 q^{36} -6 \beta q^{37} + ( -65 - 6 \beta ) q^{38} + 13 q^{39} -20 \beta q^{40} + 10 \beta q^{41} -65 q^{42} -20 q^{43} + 90 q^{44} -16 q^{45} + 35 \beta q^{46} + 10 q^{47} -29 \beta q^{48} -24 q^{49} -9 \beta q^{50} -15 \beta q^{51} -9 \beta q^{52} -21 \beta q^{53} + 65 q^{54} -40 q^{55} + 25 \beta q^{56} + ( 65 + 6 \beta ) q^{57} -65 q^{58} + 5 \beta q^{59} + 36 \beta q^{60} -40 q^{61} + 130 q^{62} + 20 q^{63} - q^{64} + 4 \beta q^{65} -130 q^{66} + 11 \beta q^{67} -135 q^{68} -35 \beta q^{69} -20 \beta q^{70} + 30 \beta q^{71} + 20 \beta q^{72} + 105 q^{73} + 78 q^{74} + 9 \beta q^{75} + ( 54 - 45 \beta ) q^{76} + 50 q^{77} + 13 \beta q^{78} -10 \beta q^{79} + 116 q^{80} -101 q^{81} -130 q^{82} -40 q^{83} -45 \beta q^{84} + 60 q^{85} -20 \beta q^{86} + 65 q^{87} + 50 \beta q^{88} -16 \beta q^{90} -5 \beta q^{91} -315 q^{92} -130 q^{93} + 10 \beta q^{94} + ( -24 + 20 \beta ) q^{95} + 117 q^{96} + 34 \beta q^{97} -24 \beta q^{98} + 40 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{4} + 8q^{5} + 26q^{6} - 10q^{7} - 8q^{9} + O(q^{10})$$ $$2q - 18q^{4} + 8q^{5} + 26q^{6} - 10q^{7} - 8q^{9} - 20q^{11} + 58q^{16} + 30q^{17} - 12q^{19} - 72q^{20} + 70q^{23} - 130q^{24} - 18q^{25} - 26q^{26} + 90q^{28} + 104q^{30} - 40q^{35} + 72q^{36} - 130q^{38} + 26q^{39} - 130q^{42} - 40q^{43} + 180q^{44} - 32q^{45} + 20q^{47} - 48q^{49} + 130q^{54} - 80q^{55} + 130q^{57} - 130q^{58} - 80q^{61} + 260q^{62} + 40q^{63} - 2q^{64} - 260q^{66} - 270q^{68} + 210q^{73} + 156q^{74} + 108q^{76} + 100q^{77} + 232q^{80} - 202q^{81} - 260q^{82} - 80q^{83} + 120q^{85} + 130q^{87} - 630q^{92} - 260q^{93} - 48q^{95} + 234q^{96} + 80q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/19\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 − 3.60555i 3.60555i
3.60555i 3.60555i −9.00000 4.00000 13.0000 −5.00000 18.0278i −4.00000 14.4222i
18.2 3.60555i 3.60555i −9.00000 4.00000 13.0000 −5.00000 18.0278i −4.00000 14.4222i
 $$n$$: 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.3.b.b 2
3.b odd 2 1 171.3.c.b 2
4.b odd 2 1 304.3.e.d 2
5.b even 2 1 475.3.c.b 2
5.c odd 4 2 475.3.d.b 4
8.b even 2 1 1216.3.e.g 2
8.d odd 2 1 1216.3.e.h 2
12.b even 2 1 2736.3.o.d 2
19.b odd 2 1 inner 19.3.b.b 2
19.c even 3 2 361.3.d.b 4
19.d odd 6 2 361.3.d.b 4
19.e even 9 6 361.3.f.d 12
19.f odd 18 6 361.3.f.d 12
57.d even 2 1 171.3.c.b 2
76.d even 2 1 304.3.e.d 2
95.d odd 2 1 475.3.c.b 2
95.g even 4 2 475.3.d.b 4
152.b even 2 1 1216.3.e.h 2
152.g odd 2 1 1216.3.e.g 2
228.b odd 2 1 2736.3.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 1.a even 1 1 trivial
19.3.b.b 2 19.b odd 2 1 inner
171.3.c.b 2 3.b odd 2 1
171.3.c.b 2 57.d even 2 1
304.3.e.d 2 4.b odd 2 1
304.3.e.d 2 76.d even 2 1
361.3.d.b 4 19.c even 3 2
361.3.d.b 4 19.d odd 6 2
361.3.f.d 12 19.e even 9 6
361.3.f.d 12 19.f odd 18 6
475.3.c.b 2 5.b even 2 1
475.3.c.b 2 95.d odd 2 1
475.3.d.b 4 5.c odd 4 2
475.3.d.b 4 95.g even 4 2
1216.3.e.g 2 8.b even 2 1
1216.3.e.g 2 152.g odd 2 1
1216.3.e.h 2 8.d odd 2 1
1216.3.e.h 2 152.b even 2 1
2736.3.o.d 2 12.b even 2 1
2736.3.o.d 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 13$$ acting on $$S_{3}^{\mathrm{new}}(19, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$13 + T^{2}$$
$3$ $$13 + T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$13 + T^{2}$$
$17$ $$( -15 + T )^{2}$$
$19$ $$361 + 12 T + T^{2}$$
$23$ $$( -35 + T )^{2}$$
$29$ $$325 + T^{2}$$
$31$ $$1300 + T^{2}$$
$37$ $$468 + T^{2}$$
$41$ $$1300 + T^{2}$$
$43$ $$( 20 + T )^{2}$$
$47$ $$( -10 + T )^{2}$$
$53$ $$5733 + T^{2}$$
$59$ $$325 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$1573 + T^{2}$$
$71$ $$11700 + T^{2}$$
$73$ $$( -105 + T )^{2}$$
$79$ $$1300 + T^{2}$$
$83$ $$( 40 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$15028 + T^{2}$$