# Properties

 Label 190.2.a.d Level $190$ Weight $2$ Character orbit 190.a Self dual yes Analytic conductor $1.517$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [190,2,Mod(1,190)]

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([0, 0]))

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

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

chi := DirichletCharacter("190.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$190 = 2 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 190.a (trivial)

## 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: yes Analytic conductor: $$1.51715763840$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
−1.00000 −2.56155 1.00000 1.00000 2.56155 −2.56155 −1.00000 3.56155 −1.00000
1.2 −1.00000 1.56155 1.00000 1.00000 −1.56155 1.56155 −1.00000 −0.561553 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$-1$$
$$19$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.a.d 2
3.b odd 2 1 1710.2.a.w 2
4.b odd 2 1 1520.2.a.n 2
5.b even 2 1 950.2.a.h 2
5.c odd 4 2 950.2.b.f 4
7.b odd 2 1 9310.2.a.bc 2
8.b even 2 1 6080.2.a.bh 2
8.d odd 2 1 6080.2.a.bb 2
15.d odd 2 1 8550.2.a.br 2
19.b odd 2 1 3610.2.a.t 2
20.d odd 2 1 7600.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 1.a even 1 1 trivial
950.2.a.h 2 5.b even 2 1
950.2.b.f 4 5.c odd 4 2
1520.2.a.n 2 4.b odd 2 1
1710.2.a.w 2 3.b odd 2 1
3610.2.a.t 2 19.b odd 2 1
6080.2.a.bb 2 8.d odd 2 1
6080.2.a.bh 2 8.b even 2 1
7600.2.a.y 2 20.d odd 2 1
8550.2.a.br 2 15.d odd 2 1
9310.2.a.bc 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(190))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + T - 4$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + T - 38$$
$17$ $$T^{2} - 11T + 26$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 3T - 36$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} + 2T - 16$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 8T - 52$$
$43$ $$T^{2} + 14T + 32$$
$47$ $$T^{2} + 4T - 64$$
$53$ $$T^{2} + 5T + 2$$
$59$ $$T^{2} - T - 4$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} + T - 4$$
$71$ $$T^{2} - 4T - 64$$
$73$ $$T^{2} - 9T - 18$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} - 14T + 32$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T - 6)^{2}$$