# Properties

 Label 290.2.a.d Level $290$ Weight $2$ Character orbit 290.a Self dual yes Analytic conductor $2.316$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [290,2,Mod(1,290)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(290, 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("290.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$290 = 2 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 290.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: $$2.31566165862$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

## 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
 0.772866 −2.16425 2.39138
1.00000 −3.40268 1.00000 1.00000 −3.40268 −0.772866 1.00000 8.57822 1.00000
1.2 1.00000 0.683969 1.00000 1.00000 0.683969 2.16425 1.00000 −2.53219 1.00000
1.3 1.00000 1.71871 1.00000 1.00000 1.71871 −2.39138 1.00000 −0.0460370 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 290.2.a.d 3
3.b odd 2 1 2610.2.a.w 3
4.b odd 2 1 2320.2.a.q 3
5.b even 2 1 1450.2.a.r 3
5.c odd 4 2 1450.2.b.j 6
8.b even 2 1 9280.2.a.bp 3
8.d odd 2 1 9280.2.a.bn 3
29.b even 2 1 8410.2.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.d 3 1.a even 1 1 trivial
1450.2.a.r 3 5.b even 2 1
1450.2.b.j 6 5.c odd 4 2
2320.2.a.q 3 4.b odd 2 1
2610.2.a.w 3 3.b odd 2 1
8410.2.a.w 3 29.b even 2 1
9280.2.a.bn 3 8.d odd 2 1
9280.2.a.bp 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(290))$$:

 $$T_{3}^{3} + T_{3}^{2} - 7T_{3} + 4$$ T3^3 + T3^2 - 7*T3 + 4 $$T_{7}^{3} + T_{7}^{2} - 5T_{7} - 4$$ T7^3 + T7^2 - 5*T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} + T^{2} - 7T + 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} + T^{2} - 5T - 4$$
$11$ $$T^{3} - 2 T^{2} + \cdots + 32$$
$13$ $$T^{3} - 5 T^{2} + \cdots + 98$$
$17$ $$T^{3} + 5 T^{2} + \cdots - 2$$
$19$ $$T^{3} - 2 T^{2} + \cdots - 32$$
$23$ $$T^{3} - T^{2} - 7T - 4$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} + 5 T^{2} + \cdots - 268$$
$37$ $$T^{3} + 4 T^{2} + \cdots - 8$$
$41$ $$T^{3} - 8T^{2} + 56$$
$43$ $$T^{3} - 5 T^{2} + \cdots + 500$$
$47$ $$(T + 8)^{3}$$
$53$ $$T^{3} - 3 T^{2} + \cdots - 14$$
$59$ $$T^{3} - T^{2} + \cdots + 196$$
$61$ $$T^{3} - 5T^{2} + T + 14$$
$67$ $$T^{3} - 4 T^{2} + \cdots - 256$$
$71$ $$T^{3} + 4 T^{2} + \cdots + 256$$
$73$ $$T^{3} - 11 T^{2} + \cdots + 862$$
$79$ $$T^{3} - 9 T^{2} + \cdots + 2052$$
$83$ $$T^{3} + 26 T^{2} + \cdots + 448$$
$89$ $$T^{3} - 8 T^{2} + \cdots - 136$$
$97$ $$T^{3} - 21 T^{2} + \cdots - 98$$