# Properties

 Label 190.2.b.b Level $190$ Weight $2$ Character orbit 190.b Analytic conductor $1.517$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} - q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{3} ) q^{6} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} - q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{3} ) q^{6} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( \beta_{2} + \beta_{5} ) q^{10} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( -\beta_{4} - \beta_{5} ) q^{12} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{15} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{18} - q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} + \beta_{2} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{23} + ( -1 - \beta_{3} ) q^{24} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{25} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{27} + ( \beta_{4} - \beta_{5} ) q^{28} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{29} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{30} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{31} -\beta_{4} q^{32} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{33} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{34} + ( 1 + \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{35} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{36} + ( -4 \beta_{4} + 2 \beta_{5} ) q^{37} + \beta_{4} q^{38} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{39} + ( -\beta_{2} - \beta_{5} ) q^{40} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{42} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{44} + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{45} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{46} + ( 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{47} + ( \beta_{4} + \beta_{5} ) q^{48} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{49} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{50} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{52} + ( \beta_{1} - \beta_{2} + 5 \beta_{4} - \beta_{5} ) q^{53} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{54} + ( 1 - 2 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( 1 - \beta_{3} ) q^{56} + ( -\beta_{4} - \beta_{5} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{58} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{60} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{62} -2 \beta_{5} q^{63} - q^{64} + ( -2 + \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{66} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{68} + ( 9 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{69} + ( -3 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( -4 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{71} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{72} + ( -3 \beta_{1} + 3 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} ) q^{73} + ( -4 + 2 \beta_{3} ) q^{74} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{75} + q^{76} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} ) q^{78} + ( 2 + 6 \beta_{3} ) q^{79} + ( -\beta_{1} - \beta_{3} ) q^{80} + ( -5 + 2 \beta_{3} ) q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{84} + ( -2 + \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 6 + \beta_{1} + \beta_{2} ) q^{86} + ( -4 \beta_{1} + 4 \beta_{2} + \beta_{4} + 9 \beta_{5} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{88} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} - 4 \beta_{4} - 3 \beta_{5} ) q^{90} + ( 1 - 3 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{92} + ( -5 \beta_{1} + 5 \beta_{2} + 4 \beta_{4} + 8 \beta_{5} ) q^{93} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( \beta_{1} + \beta_{3} ) q^{95} + ( 1 + \beta_{3} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{97} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{98} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} + 2q^{5} + 4q^{6} - 10q^{9} + O(q^{10})$$ $$6q - 6q^{4} + 2q^{5} + 4q^{6} - 10q^{9} - 8q^{14} + 8q^{15} + 6q^{16} - 6q^{19} - 2q^{20} - 20q^{21} - 4q^{24} + 10q^{25} + 16q^{26} + 16q^{29} - 16q^{30} + 8q^{31} + 4q^{34} + 8q^{35} + 10q^{36} - 20q^{39} + 4q^{41} + 18q^{45} + 6q^{49} - 4q^{50} - 12q^{51} - 4q^{54} + 4q^{55} + 8q^{56} - 4q^{59} - 8q^{60} - 60q^{61} - 6q^{64} - 12q^{65} + 16q^{66} + 44q^{69} - 20q^{70} - 16q^{71} - 28q^{74} + 44q^{75} + 6q^{76} + 2q^{80} - 34q^{81} + 20q^{84} - 12q^{85} + 36q^{86} + 28q^{89} - 12q^{90} + 12q^{91} + 28q^{94} - 2q^{95} + 4q^{96} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100$$$$)/121$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223$$$$)/121$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258$$$$)/242$$ $$\beta_{5}$$ $$=$$ $$($$$$-65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574$$$$)/242$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} - 5 \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/190\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 1.32001 − 1.32001i −1.75233 + 1.75233i 0.432320 − 0.432320i 0.432320 + 0.432320i −1.75233 − 1.75233i 1.32001 + 1.32001i
1.00000i 2.12489i −1.00000 1.80487 + 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 1.80487i
39.2 1.00000i 1.36333i −1.00000 1.38900 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 1.38900i
39.3 1.00000i 2.76156i −1.00000 −2.19388 + 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 + 2.19388i
39.4 1.00000i 2.76156i −1.00000 −2.19388 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 2.19388i
39.5 1.00000i 1.36333i −1.00000 1.38900 + 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 + 1.38900i
39.6 1.00000i 2.12489i −1.00000 1.80487 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 + 1.80487i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.b.b 6
3.b odd 2 1 1710.2.d.d 6
4.b odd 2 1 1520.2.d.j 6
5.b even 2 1 inner 190.2.b.b 6
5.c odd 4 1 950.2.a.i 3
5.c odd 4 1 950.2.a.n 3
15.d odd 2 1 1710.2.d.d 6
15.e even 4 1 8550.2.a.ck 3
15.e even 4 1 8550.2.a.cl 3
20.d odd 2 1 1520.2.d.j 6
20.e even 4 1 7600.2.a.bi 3
20.e even 4 1 7600.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 1.a even 1 1 trivial
190.2.b.b 6 5.b even 2 1 inner
950.2.a.i 3 5.c odd 4 1
950.2.a.n 3 5.c odd 4 1
1520.2.d.j 6 4.b odd 2 1
1520.2.d.j 6 20.d odd 2 1
1710.2.d.d 6 3.b odd 2 1
1710.2.d.d 6 15.d odd 2 1
7600.2.a.bi 3 20.e even 4 1
7600.2.a.cd 3 20.e even 4 1
8550.2.a.ck 3 15.e even 4 1
8550.2.a.cl 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 14 T_{3}^{4} + 57 T_{3}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(190, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$64 + 57 T^{2} + 14 T^{4} + T^{6}$$
$5$ $$125 - 50 T - 15 T^{2} + 24 T^{3} - 3 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$4 + 17 T^{2} + 18 T^{4} + T^{6}$$
$11$ $$( -8 - 10 T + T^{3} )^{2}$$
$13$ $$4 + 201 T^{2} + 38 T^{4} + T^{6}$$
$17$ $$16 + 65 T^{2} + 18 T^{4} + T^{6}$$
$19$ $$( 1 + T )^{6}$$
$23$ $$14884 + 2401 T^{2} + 98 T^{4} + T^{6}$$
$29$ $$( 410 - 51 T - 8 T^{2} + T^{3} )^{2}$$
$31$ $$( 232 - 62 T - 4 T^{2} + T^{3} )^{2}$$
$37$ $$256 + 1152 T^{2} + 116 T^{4} + T^{6}$$
$41$ $$( -100 - 50 T - 2 T^{2} + T^{3} )^{2}$$
$43$ $$21904 + 4276 T^{2} + 128 T^{4} + T^{6}$$
$47$ $$4096 + 2816 T^{2} + 132 T^{4} + T^{6}$$
$53$ $$11236 + 2537 T^{2} + 102 T^{4} + T^{6}$$
$59$ $$( -80 - 29 T + 2 T^{2} + T^{3} )^{2}$$
$61$ $$( 892 + 290 T + 30 T^{2} + T^{3} )^{2}$$
$67$ $$4096 + 3977 T^{2} + 126 T^{4} + T^{6}$$
$71$ $$( -1016 - 122 T + 8 T^{2} + T^{3} )^{2}$$
$73$ $$26896 + 5745 T^{2} + 290 T^{4} + T^{6}$$
$79$ $$( 880 - 228 T + T^{3} )^{2}$$
$83$ $$64 + 880 T^{2} + 92 T^{4} + T^{6}$$
$89$ $$( 20 + 46 T - 14 T^{2} + T^{3} )^{2}$$
$97$ $$238144 + 19760 T^{2} + 300 T^{4} + T^{6}$$