Properties

 Label 290.2.b.a Level $290$ Weight $2$ Character orbit 290.b Analytic conductor $2.316$ Analytic rank $0$ Dimension $4$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [290,2,Mod(59,290)]

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([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("290.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$290 = 2 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 290.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: $$2.31566165862$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/290\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$117$$ $$\chi(n)$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i 2.61803i −1.00000 2.23607i −2.61803 0.381966i 1.00000i −3.85410 −2.23607
59.2 1.00000i 0.381966i −1.00000 2.23607i −0.381966 2.61803i 1.00000i 2.85410 2.23607
59.3 1.00000i 0.381966i −1.00000 2.23607i −0.381966 2.61803i 1.00000i 2.85410 2.23607
59.4 1.00000i 2.61803i −1.00000 2.23607i −2.61803 0.381966i 1.00000i −3.85410 −2.23607
 $$n$$: e.g. 2-40 or 990-1000 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 290.2.b.a 4
3.b odd 2 1 2610.2.e.e 4
4.b odd 2 1 2320.2.d.d 4
5.b even 2 1 inner 290.2.b.a 4
5.c odd 4 1 1450.2.a.k 2
5.c odd 4 1 1450.2.a.l 2
15.d odd 2 1 2610.2.e.e 4
20.d odd 2 1 2320.2.d.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.a 4 1.a even 1 1 trivial
290.2.b.a 4 5.b even 2 1 inner
1450.2.a.k 2 5.c odd 4 1
1450.2.a.l 2 5.c odd 4 1
2320.2.d.d 4 4.b odd 2 1
2320.2.d.d 4 20.d odd 2 1
2610.2.e.e 4 3.b odd 2 1
2610.2.e.e 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(290, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 7T^{2} + 1$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4} + 7T^{2} + 1$$
$11$ $$(T - 2)^{4}$$
$13$ $$T^{4} + 7T^{2} + 1$$
$17$ $$T^{4} + 43T^{2} + 361$$
$19$ $$(T - 2)^{4}$$
$23$ $$T^{4} + 27T^{2} + 81$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} + T - 11)^{2}$$
$37$ $$(T^{2} + 80)^{2}$$
$41$ $$(T^{2} - 2 T - 44)^{2}$$
$43$ $$T^{4} + 43T^{2} + 361$$
$47$ $$T^{4} + 112T^{2} + 256$$
$53$ $$T^{4} + 223 T^{2} + 11881$$
$59$ $$(T^{2} - T - 11)^{2}$$
$61$ $$(T^{2} + 15 T + 45)^{2}$$
$67$ $$(T^{2} + 80)^{2}$$
$71$ $$(T^{2} - 2 T - 44)^{2}$$
$73$ $$T^{4} + 243T^{2} + 6561$$
$79$ $$(T^{2} - 7 T - 89)^{2}$$
$83$ $$T^{4} + 232T^{2} + 1936$$
$89$ $$(T + 2)^{4}$$
$97$ $$T^{4} + 243T^{2} + 9801$$