# Properties

 Label 1859.2.a.g Level $1859$ Weight $2$ Character orbit 1859.a Self dual yes Analytic conductor $14.844$ Analytic rank $1$ 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] = [1859,2,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, 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("1859.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.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: $$14.8441897358$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.361.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x + 7$$ x^3 - x^2 - 6*x + 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ 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
 2.28514 −2.50702 1.22188
−2.50702 2.28514 4.28514 1.22188 −5.72889 −0.778124 −5.72889 2.22188 −3.06327
1.2 1.22188 −2.50702 −0.507019 2.28514 −3.06327 0.285142 −3.06327 3.28514 2.79216
1.3 2.28514 1.22188 3.22188 −2.50702 2.79216 −4.50702 2.79216 −1.50702 −5.72889
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$+1$$
$$13$$ $$+1$$

## 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 1859.2.a.g 3
13.b even 2 1 1859.2.a.f 3
13.e even 6 2 143.2.e.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.b 6 13.e even 6 2
1859.2.a.f 3 13.b even 2 1
1859.2.a.g 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1859))$$:

 $$T_{2}^{3} - T_{2}^{2} - 6T_{2} + 7$$ T2^3 - T2^2 - 6*T2 + 7 $$T_{7}^{3} + 5T_{7}^{2} + 2T_{7} - 1$$ T7^3 + 5*T7^2 + 2*T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 6T + 7$$
$3$ $$T^{3} - T^{2} - 6T + 7$$
$5$ $$T^{3} - T^{2} - 6T + 7$$
$7$ $$T^{3} + 5 T^{2} + \cdots - 1$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 7 T^{2} + \cdots - 7$$
$19$ $$T^{3} + 2 T^{2} + \cdots + 77$$
$23$ $$T^{3} - 3 T^{2} + \cdots - 1$$
$29$ $$T^{3} + 4 T^{2} + \cdots - 49$$
$31$ $$T^{3} + T^{2} + \cdots + 31$$
$37$ $$T^{3} + 12 T^{2} + \cdots - 31$$
$41$ $$T^{3} + T^{2} + \cdots + 31$$
$43$ $$T^{3} + 4 T^{2} + \cdots - 581$$
$47$ $$T^{3} - 8 T^{2} + \cdots + 259$$
$53$ $$T^{3} + 9 T^{2} + \cdots - 49$$
$59$ $$T^{3} + 30 T^{2} + \cdots + 791$$
$61$ $$T^{3} + 18 T^{2} + \cdots + 7$$
$67$ $$T^{3} + 15 T^{2} + \cdots + 11$$
$71$ $$T^{3} - 11 T^{2} + \cdots - 31$$
$73$ $$(T - 2)^{3}$$
$79$ $$T^{3} - 12 T^{2} + \cdots + 88$$
$83$ $$T^{3} - 14 T^{2} + \cdots + 341$$
$89$ $$T^{3} - 17 T^{2} + \cdots - 151$$
$97$ $$T^{3} + 11 T^{2} + \cdots - 7$$