# Properties

 Label 190.2.b.b Level $190$ Weight $2$ Character orbit 190.b Analytic conductor $1.517$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [190,2,Mod(39,190)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(190, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("190.39");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + \beta_{4} q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{9}+O(q^{10})$$ q - b4 * q^2 + (b5 + b4) * q^3 - q^4 + (-b3 - b1) * q^5 + (b3 + 1) * q^6 + (b5 - b4) * q^7 + b4 * q^8 + (-b3 - b2 - b1 - 2) * q^9 $$q - \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + \beta_{4} q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{9} + (\beta_{5} + \beta_{2}) q^{10} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{4}) q^{12} + (\beta_{5} + 3 \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + 4 q^{99}+O(q^{100})$$ q - b4 * q^2 + (b5 + b4) * q^3 - q^4 + (-b3 - b1) * q^5 + (b3 + 1) * q^6 + (b5 - b4) * q^7 + b4 * q^8 + (-b3 - b2 - b1 - 2) * q^9 + (b5 + b2) * q^10 + (-b2 - b1) * q^11 + (-b5 - b4) * q^12 + (b5 + 3*b4 + b2 - b1) * q^13 + (b3 - 1) * q^14 + (-b5 - 3*b4 - b3 - 2*b2 + b1 + 1) * q^15 + q^16 + (b5 + b4 + b2 - b1) * q^17 + (b5 + 2*b4 + b2 - b1) * q^18 - q^19 + (b3 + b1) * q^20 + (b3 - b2 - b1 - 3) * q^21 + (b2 - b1) * q^22 + (-3*b5 - b4 - 2*b2 + 2*b1) * q^23 + (-b3 - 1) * q^24 + (-b5 - b4 + b3 + 2*b2 + 2) * q^25 + (b3 + b2 + b1 + 3) * q^26 + (-b5 - b4 - 2*b2 + 2*b1) * q^27 + (-b5 + b4) * q^28 + (b3 + 3*b2 + 3*b1 + 3) * q^29 + (b5 - b4 - b3 - b2 - 2*b1 - 3) * q^30 + (2*b3 + 3*b2 + 3*b1 + 2) * q^31 - b4 * q^32 + (-2*b5 + 2*b4 - b2 + b1) * q^33 + (b3 + b2 + b1 + 1) * q^34 + (b5 - 3*b4 - b3 + b1 + 1) * q^35 + (b3 + b2 + b1 + 2) * q^36 + (2*b5 - 4*b4) * q^37 + b4 * q^38 + (-5*b3 - 2*b2 - 2*b1 - 5) * q^39 + (-b5 - b2) * q^40 + (-2*b3 + b2 + b1) * q^41 + (-b5 + 3*b4 + b2 - b1) * q^42 + (6*b4 + b2 - b1) * q^43 + (b2 + b1) * q^44 + (-2*b4 + 3*b3 + b2 + b1 + 4) * q^45 + (-3*b3 - 2*b2 - 2*b1 - 1) * q^46 + (-2*b5 + 4*b4 - 2*b2 + 2*b1) * q^47 + (b5 + b4) * q^48 + (3*b3 - b2 - b1 + 2) * q^49 + (-b5 - 2*b4 - b3 + 2*b1 - 1) * q^50 + (-3*b3 - 2*b2 - 2*b1 - 3) * q^51 + (-b5 - 3*b4 - b2 + b1) * q^52 + (-b5 + 5*b4 - b2 + b1) * q^53 + (-b3 - 2*b2 - 2*b1 - 1) * q^54 + (b5 - 3*b4 + b3 - 2*b1 + 1) * q^55 + (-b3 + 1) * q^56 + (-b5 - b4) * q^57 + (-b5 - 3*b4 - 3*b2 + 3*b1) * q^58 + (-b3 - 2*b2 - 2*b1 - 1) * q^59 + (b5 + 3*b4 + b3 + 2*b2 - b1 - 1) * q^60 + (-b2 - b1 - 10) * q^61 + (-2*b5 - 2*b4 - 3*b2 + 3*b1) * q^62 - 2*b5 * q^63 - q^64 + (-4*b5 - 4*b4 - 2*b2 + b1 - 2) * q^65 + (-2*b3 - b2 - b1 + 2) * q^66 + (-b5 - b4 + 2*b2 - 2*b1) * q^67 + (-b5 - b4 - b2 + b1) * q^68 + (5*b3 + 5*b2 + 5*b1 + 9) * q^69 + (b5 - b4 + b3 - b2 - 3) * q^70 + (-4*b3 + b2 + b1 - 4) * q^71 + (-b5 - 2*b4 - b2 + b1) * q^72 + (5*b5 + 5*b4 + 3*b2 - 3*b1) * q^73 + (2*b3 - 4) * q^74 + (4*b5 + 4*b4 - b3 + 2*b2 - 2*b1 + 7) * q^75 + q^76 + (-2*b5 + 2*b4 + b2 - b1) * q^77 + (5*b5 + 5*b4 + 2*b2 - 2*b1) * q^78 + (6*b3 + 2) * q^79 + (-b3 - b1) * q^80 + (2*b3 - 5) * q^81 + (2*b5 - b2 + b1) * q^82 + (-2*b4 + 2*b2 - 2*b1) * q^83 + (-b3 + b2 + b1 + 3) * q^84 + (-2*b5 - 4*b4 + b1 - 2) * q^85 + (b2 + b1 + 6) * q^86 + (9*b5 + b4 + 4*b2 - 4*b1) * q^87 + (-b2 + b1) * q^88 + (-2*b3 - b2 - b1 + 4) * q^89 + (-3*b5 - 4*b4 - b2 + b1 - 2) * q^90 + (-3*b3 + 1) * q^91 + (3*b5 + b4 + 2*b2 - 2*b1) * q^92 + (8*b5 + 4*b4 + 5*b2 - 5*b1) * q^93 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^94 + (b3 + b1) * q^95 + (b3 + 1) * q^96 + (4*b5 - 2*b4 + 4*b2 - 4*b1) * q^97 + (-3*b5 - 2*b4 + b2 - b1) * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 2 q^{5} + 4 q^{6} - 10 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 + 2 * q^5 + 4 * q^6 - 10 * q^9 $$6 q - 6 q^{4} + 2 q^{5} + 4 q^{6} - 10 q^{9} - 8 q^{14} + 8 q^{15} + 6 q^{16} - 6 q^{19} - 2 q^{20} - 20 q^{21} - 4 q^{24} + 10 q^{25} + 16 q^{26} + 16 q^{29} - 16 q^{30} + 8 q^{31} + 4 q^{34} + 8 q^{35} + 10 q^{36} - 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} - 4 q^{50} - 12 q^{51} - 4 q^{54} + 4 q^{55} + 8 q^{56} - 4 q^{59} - 8 q^{60} - 60 q^{61} - 6 q^{64} - 12 q^{65} + 16 q^{66} + 44 q^{69} - 20 q^{70} - 16 q^{71} - 28 q^{74} + 44 q^{75} + 6 q^{76} + 2 q^{80} - 34 q^{81} + 20 q^{84} - 12 q^{85} + 36 q^{86} + 28 q^{89} - 12 q^{90} + 12 q^{91} + 28 q^{94} - 2 q^{95} + 4 q^{96} + 24 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 + 2 * q^5 + 4 * q^6 - 10 * q^9 - 8 * q^14 + 8 * q^15 + 6 * q^16 - 6 * q^19 - 2 * q^20 - 20 * q^21 - 4 * q^24 + 10 * q^25 + 16 * q^26 + 16 * q^29 - 16 * q^30 + 8 * q^31 + 4 * q^34 + 8 * q^35 + 10 * q^36 - 20 * q^39 + 4 * q^41 + 18 * q^45 + 6 * q^49 - 4 * q^50 - 12 * q^51 - 4 * q^54 + 4 * q^55 + 8 * q^56 - 4 * q^59 - 8 * q^60 - 60 * q^61 - 6 * q^64 - 12 * q^65 + 16 * q^66 + 44 * q^69 - 20 * q^70 - 16 * q^71 - 28 * q^74 + 44 * q^75 + 6 * q^76 + 2 * q^80 - 34 * q^81 + 20 * q^84 - 12 * q^85 + 36 * q^86 + 28 * q^89 - 12 * q^90 + 12 * q^91 + 28 * q^94 - 2 * q^95 + 4 * q^96 + 24 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121$$ (-5*v^5 - 2*v^4 - 25*v^3 + 10*v^2 - 121*v + 100) / 121 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121$$ (7*v^5 + 27*v^4 + 35*v^3 - 14*v^2 + 223) / 121 $$\beta_{4}$$ $$=$$ $$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$$ (-25*v^5 - 10*v^4 - 4*v^3 + 50*v^2 - 605*v + 258) / 242 $$\beta_{5}$$ $$=$$ $$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242$$ (-65*v^5 - 26*v^4 + 38*v^3 + 372*v^2 - 1331*v + 574) / 242
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1$$ b5 - 3*b4 + b2 - b1 $$\nu^{3}$$ $$=$$ $$2\beta_{4} - 5\beta_{2} + 2$$ 2*b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15$$ 5*b3 + 7*b2 + 7*b1 - 15 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16$$ 2*b5 - 16*b4 - 2*b3 - 29*b1 + 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$T_{3}^{6} + 14T_{3}^{4} + 57T_{3}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(190, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 14 T^{4} + \cdots + 64$$
$5$ $$T^{6} - 2 T^{5} + \cdots + 125$$
$7$ $$T^{6} + 18 T^{4} + \cdots + 4$$
$11$ $$(T^{3} - 10 T - 8)^{2}$$
$13$ $$T^{6} + 38 T^{4} + \cdots + 4$$
$17$ $$T^{6} + 18 T^{4} + \cdots + 16$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 98 T^{4} + \cdots + 14884$$
$29$ $$(T^{3} - 8 T^{2} + \cdots + 410)^{2}$$
$31$ $$(T^{3} - 4 T^{2} + \cdots + 232)^{2}$$
$37$ $$T^{6} + 116 T^{4} + \cdots + 256$$
$41$ $$(T^{3} - 2 T^{2} + \cdots - 100)^{2}$$
$43$ $$T^{6} + 128 T^{4} + \cdots + 21904$$
$47$ $$T^{6} + 132 T^{4} + \cdots + 4096$$
$53$ $$T^{6} + 102 T^{4} + \cdots + 11236$$
$59$ $$(T^{3} + 2 T^{2} - 29 T - 80)^{2}$$
$61$ $$(T^{3} + 30 T^{2} + \cdots + 892)^{2}$$
$67$ $$T^{6} + 126 T^{4} + \cdots + 4096$$
$71$ $$(T^{3} + 8 T^{2} + \cdots - 1016)^{2}$$
$73$ $$T^{6} + 290 T^{4} + \cdots + 26896$$
$79$ $$(T^{3} - 228 T + 880)^{2}$$
$83$ $$T^{6} + 92 T^{4} + \cdots + 64$$
$89$ $$(T^{3} - 14 T^{2} + \cdots + 20)^{2}$$
$97$ $$T^{6} + 300 T^{4} + \cdots + 238144$$