# Properties

 Label 290.2.a.b Level $290$ Weight $2$ Character orbit 290.a Self dual yes Analytic conductor $2.316$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta + 1) q^{3} + q^{4} - q^{5} + (\beta - 1) q^{6} + (\beta + 2) q^{7} - q^{8} + ( - \beta + 1) q^{9}+O(q^{10})$$ q - q^2 + (-b + 1) * q^3 + q^4 - q^5 + (b - 1) * q^6 + (b + 2) * q^7 - q^8 + (-b + 1) * q^9 $$q - q^{2} + ( - \beta + 1) q^{3} + q^{4} - q^{5} + (\beta - 1) q^{6} + (\beta + 2) q^{7} - q^{8} + ( - \beta + 1) q^{9} + q^{10} - 2 \beta q^{11} + ( - \beta + 1) q^{12} + ( - \beta + 5) q^{13} + ( - \beta - 2) q^{14} + (\beta - 1) q^{15} + q^{16} + 3 \beta q^{17} + (\beta - 1) q^{18} + (2 \beta + 2) q^{19} - q^{20} + ( - 2 \beta - 1) q^{21} + 2 \beta q^{22} + (\beta + 3) q^{23} + (\beta - 1) q^{24} + q^{25} + (\beta - 5) q^{26} + (2 \beta + 1) q^{27} + (\beta + 2) q^{28} + q^{29} + ( - \beta + 1) q^{30} + (3 \beta - 4) q^{31} - q^{32} + 6 q^{33} - 3 \beta q^{34} + ( - \beta - 2) q^{35} + ( - \beta + 1) q^{36} + ( - 6 \beta + 2) q^{37} + ( - 2 \beta - 2) q^{38} + ( - 5 \beta + 8) q^{39} + q^{40} + (2 \beta - 6) q^{41} + (2 \beta + 1) q^{42} + ( - \beta + 2) q^{43} - 2 \beta q^{44} + (\beta - 1) q^{45} + ( - \beta - 3) q^{46} + ( - \beta + 1) q^{48} + 5 \beta q^{49} - q^{50} - 9 q^{51} + ( - \beta + 5) q^{52} + 3 \beta q^{53} + ( - 2 \beta - 1) q^{54} + 2 \beta q^{55} + ( - \beta - 2) q^{56} + ( - 2 \beta - 4) q^{57} - q^{58} + (3 \beta + 3) q^{59} + (\beta - 1) q^{60} + ( - 3 \beta - 7) q^{61} + ( - 3 \beta + 4) q^{62} + ( - 2 \beta - 1) q^{63} + q^{64} + (\beta - 5) q^{65} - 6 q^{66} + (4 \beta - 4) q^{67} + 3 \beta q^{68} - 3 \beta q^{69} + (\beta + 2) q^{70} + (\beta - 1) q^{72} + ( - 5 \beta - 4) q^{73} + (6 \beta - 2) q^{74} + ( - \beta + 1) q^{75} + (2 \beta + 2) q^{76} + ( - 6 \beta - 6) q^{77} + (5 \beta - 8) q^{78} + (3 \beta - 1) q^{79} - q^{80} + (2 \beta - 8) q^{81} + ( - 2 \beta + 6) q^{82} + (2 \beta - 6) q^{83} + ( - 2 \beta - 1) q^{84} - 3 \beta q^{85} + (\beta - 2) q^{86} + ( - \beta + 1) q^{87} + 2 \beta q^{88} + (2 \beta - 12) q^{89} + ( - \beta + 1) q^{90} + (2 \beta + 7) q^{91} + (\beta + 3) q^{92} + (4 \beta - 13) q^{93} + ( - 2 \beta - 2) q^{95} + (\beta - 1) q^{96} + (3 \beta + 5) q^{97} - 5 \beta q^{98} + 6 q^{99} +O(q^{100})$$ q - q^2 + (-b + 1) * q^3 + q^4 - q^5 + (b - 1) * q^6 + (b + 2) * q^7 - q^8 + (-b + 1) * q^9 + q^10 - 2*b * q^11 + (-b + 1) * q^12 + (-b + 5) * q^13 + (-b - 2) * q^14 + (b - 1) * q^15 + q^16 + 3*b * q^17 + (b - 1) * q^18 + (2*b + 2) * q^19 - q^20 + (-2*b - 1) * q^21 + 2*b * q^22 + (b + 3) * q^23 + (b - 1) * q^24 + q^25 + (b - 5) * q^26 + (2*b + 1) * q^27 + (b + 2) * q^28 + q^29 + (-b + 1) * q^30 + (3*b - 4) * q^31 - q^32 + 6 * q^33 - 3*b * q^34 + (-b - 2) * q^35 + (-b + 1) * q^36 + (-6*b + 2) * q^37 + (-2*b - 2) * q^38 + (-5*b + 8) * q^39 + q^40 + (2*b - 6) * q^41 + (2*b + 1) * q^42 + (-b + 2) * q^43 - 2*b * q^44 + (b - 1) * q^45 + (-b - 3) * q^46 + (-b + 1) * q^48 + 5*b * q^49 - q^50 - 9 * q^51 + (-b + 5) * q^52 + 3*b * q^53 + (-2*b - 1) * q^54 + 2*b * q^55 + (-b - 2) * q^56 + (-2*b - 4) * q^57 - q^58 + (3*b + 3) * q^59 + (b - 1) * q^60 + (-3*b - 7) * q^61 + (-3*b + 4) * q^62 + (-2*b - 1) * q^63 + q^64 + (b - 5) * q^65 - 6 * q^66 + (4*b - 4) * q^67 + 3*b * q^68 - 3*b * q^69 + (b + 2) * q^70 + (b - 1) * q^72 + (-5*b - 4) * q^73 + (6*b - 2) * q^74 + (-b + 1) * q^75 + (2*b + 2) * q^76 + (-6*b - 6) * q^77 + (5*b - 8) * q^78 + (3*b - 1) * q^79 - q^80 + (2*b - 8) * q^81 + (-2*b + 6) * q^82 + (2*b - 6) * q^83 + (-2*b - 1) * q^84 - 3*b * q^85 + (b - 2) * q^86 + (-b + 1) * q^87 + 2*b * q^88 + (2*b - 12) * q^89 + (-b + 1) * q^90 + (2*b + 7) * q^91 + (b + 3) * q^92 + (4*b - 13) * q^93 + (-2*b - 2) * q^95 + (b - 1) * q^96 + (3*b + 5) * q^97 - 5*b * q^98 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 5 q^{7} - 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 5 * q^7 - 2 * q^8 + q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 5 q^{7} - 2 q^{8} + q^{9} + 2 q^{10} - 2 q^{11} + q^{12} + 9 q^{13} - 5 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} - q^{18} + 6 q^{19} - 2 q^{20} - 4 q^{21} + 2 q^{22} + 7 q^{23} - q^{24} + 2 q^{25} - 9 q^{26} + 4 q^{27} + 5 q^{28} + 2 q^{29} + q^{30} - 5 q^{31} - 2 q^{32} + 12 q^{33} - 3 q^{34} - 5 q^{35} + q^{36} - 2 q^{37} - 6 q^{38} + 11 q^{39} + 2 q^{40} - 10 q^{41} + 4 q^{42} + 3 q^{43} - 2 q^{44} - q^{45} - 7 q^{46} + q^{48} + 5 q^{49} - 2 q^{50} - 18 q^{51} + 9 q^{52} + 3 q^{53} - 4 q^{54} + 2 q^{55} - 5 q^{56} - 10 q^{57} - 2 q^{58} + 9 q^{59} - q^{60} - 17 q^{61} + 5 q^{62} - 4 q^{63} + 2 q^{64} - 9 q^{65} - 12 q^{66} - 4 q^{67} + 3 q^{68} - 3 q^{69} + 5 q^{70} - q^{72} - 13 q^{73} + 2 q^{74} + q^{75} + 6 q^{76} - 18 q^{77} - 11 q^{78} + q^{79} - 2 q^{80} - 14 q^{81} + 10 q^{82} - 10 q^{83} - 4 q^{84} - 3 q^{85} - 3 q^{86} + q^{87} + 2 q^{88} - 22 q^{89} + q^{90} + 16 q^{91} + 7 q^{92} - 22 q^{93} - 6 q^{95} - q^{96} + 13 q^{97} - 5 q^{98} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 - q^6 + 5 * q^7 - 2 * q^8 + q^9 + 2 * q^10 - 2 * q^11 + q^12 + 9 * q^13 - 5 * q^14 - q^15 + 2 * q^16 + 3 * q^17 - q^18 + 6 * q^19 - 2 * q^20 - 4 * q^21 + 2 * q^22 + 7 * q^23 - q^24 + 2 * q^25 - 9 * q^26 + 4 * q^27 + 5 * q^28 + 2 * q^29 + q^30 - 5 * q^31 - 2 * q^32 + 12 * q^33 - 3 * q^34 - 5 * q^35 + q^36 - 2 * q^37 - 6 * q^38 + 11 * q^39 + 2 * q^40 - 10 * q^41 + 4 * q^42 + 3 * q^43 - 2 * q^44 - q^45 - 7 * q^46 + q^48 + 5 * q^49 - 2 * q^50 - 18 * q^51 + 9 * q^52 + 3 * q^53 - 4 * q^54 + 2 * q^55 - 5 * q^56 - 10 * q^57 - 2 * q^58 + 9 * q^59 - q^60 - 17 * q^61 + 5 * q^62 - 4 * q^63 + 2 * q^64 - 9 * q^65 - 12 * q^66 - 4 * q^67 + 3 * q^68 - 3 * q^69 + 5 * q^70 - q^72 - 13 * q^73 + 2 * q^74 + q^75 + 6 * q^76 - 18 * q^77 - 11 * q^78 + q^79 - 2 * q^80 - 14 * q^81 + 10 * q^82 - 10 * q^83 - 4 * q^84 - 3 * q^85 - 3 * q^86 + q^87 + 2 * q^88 - 22 * q^89 + q^90 + 16 * q^91 + 7 * q^92 - 22 * q^93 - 6 * q^95 - q^96 + 13 * 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
 2.30278 −1.30278
−1.00000 −1.30278 1.00000 −1.00000 1.30278 4.30278 −1.00000 −1.30278 1.00000
1.2 −1.00000 2.30278 1.00000 −1.00000 −2.30278 0.697224 −1.00000 2.30278 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.b 2
3.b odd 2 1 2610.2.a.v 2
4.b odd 2 1 2320.2.a.i 2
5.b even 2 1 1450.2.a.m 2
5.c odd 4 2 1450.2.b.g 4
8.b even 2 1 9280.2.a.z 2
8.d odd 2 1 9280.2.a.bc 2
29.b even 2 1 8410.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.b 2 1.a even 1 1 trivial
1450.2.a.m 2 5.b even 2 1
1450.2.b.g 4 5.c odd 4 2
2320.2.a.i 2 4.b odd 2 1
2610.2.a.v 2 3.b odd 2 1
8410.2.a.r 2 29.b even 2 1
9280.2.a.z 2 8.b even 2 1
9280.2.a.bc 2 8.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T - 3$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 5T + 3$$
$11$ $$T^{2} + 2T - 12$$
$13$ $$T^{2} - 9T + 17$$
$17$ $$T^{2} - 3T - 27$$
$19$ $$T^{2} - 6T - 4$$
$23$ $$T^{2} - 7T + 9$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 5T - 23$$
$37$ $$T^{2} + 2T - 116$$
$41$ $$T^{2} + 10T + 12$$
$43$ $$T^{2} - 3T - 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 3T - 27$$
$59$ $$T^{2} - 9T - 9$$
$61$ $$T^{2} + 17T + 43$$
$67$ $$T^{2} + 4T - 48$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 13T - 39$$
$79$ $$T^{2} - T - 29$$
$83$ $$T^{2} + 10T + 12$$
$89$ $$T^{2} + 22T + 108$$
$97$ $$T^{2} - 13T + 13$$