# Properties

 Label 1840.2.a.n Level $1840$ Weight $2$ Character orbit 1840.a Self dual yes Analytic conductor $14.692$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} + (\beta - 1) q^{7} + (\beta + 2) q^{9}+O(q^{10})$$ q + b * q^3 - q^5 + (b - 1) * q^7 + (b + 2) * q^9 $$q + \beta q^{3} - q^{5} + (\beta - 1) q^{7} + (\beta + 2) q^{9} + (\beta - 2) q^{11} + (\beta + 3) q^{13} - \beta q^{15} + (\beta - 2) q^{17} + ( - \beta - 3) q^{19} + 5 q^{21} - q^{23} + q^{25} + 5 q^{27} + (2 \beta + 2) q^{29} + (3 \beta - 5) q^{31} + ( - \beta + 5) q^{33} + ( - \beta + 1) q^{35} - 4 q^{37} + (4 \beta + 5) q^{39} + ( - \beta - 4) q^{41} - 4 \beta q^{43} + ( - \beta - 2) q^{45} + ( - 2 \beta + 10) q^{47} + ( - \beta - 1) q^{49} + ( - \beta + 5) q^{51} + 6 q^{53} + ( - \beta + 2) q^{55} + ( - 4 \beta - 5) q^{57} + (2 \beta + 8) q^{59} + (3 \beta + 2) q^{61} + (2 \beta + 3) q^{63} + ( - \beta - 3) q^{65} - 4 \beta q^{67} - \beta q^{69} - 3 \beta q^{71} + (6 \beta - 4) q^{73} + \beta q^{75} + ( - 2 \beta + 7) q^{77} - 8 q^{79} + (2 \beta - 6) q^{81} + 6 q^{83} + ( - \beta + 2) q^{85} + (4 \beta + 10) q^{87} + (4 \beta + 4) q^{89} + (3 \beta + 2) q^{91} + ( - 2 \beta + 15) q^{93} + (\beta + 3) q^{95} + ( - 5 \beta + 6) q^{97} + (\beta + 1) q^{99} +O(q^{100})$$ q + b * q^3 - q^5 + (b - 1) * q^7 + (b + 2) * q^9 + (b - 2) * q^11 + (b + 3) * q^13 - b * q^15 + (b - 2) * q^17 + (-b - 3) * q^19 + 5 * q^21 - q^23 + q^25 + 5 * q^27 + (2*b + 2) * q^29 + (3*b - 5) * q^31 + (-b + 5) * q^33 + (-b + 1) * q^35 - 4 * q^37 + (4*b + 5) * q^39 + (-b - 4) * q^41 - 4*b * q^43 + (-b - 2) * q^45 + (-2*b + 10) * q^47 + (-b - 1) * q^49 + (-b + 5) * q^51 + 6 * q^53 + (-b + 2) * q^55 + (-4*b - 5) * q^57 + (2*b + 8) * q^59 + (3*b + 2) * q^61 + (2*b + 3) * q^63 + (-b - 3) * q^65 - 4*b * q^67 - b * q^69 - 3*b * q^71 + (6*b - 4) * q^73 + b * q^75 + (-2*b + 7) * q^77 - 8 * q^79 + (2*b - 6) * q^81 + 6 * q^83 + (-b + 2) * q^85 + (4*b + 10) * q^87 + (4*b + 4) * q^89 + (3*b + 2) * q^91 + (-2*b + 15) * q^93 + (b + 3) * q^95 + (-5*b + 6) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} - q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 - q^7 + 5 * q^9 $$2 q + q^{3} - 2 q^{5} - q^{7} + 5 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} - 3 q^{17} - 7 q^{19} + 10 q^{21} - 2 q^{23} + 2 q^{25} + 10 q^{27} + 6 q^{29} - 7 q^{31} + 9 q^{33} + q^{35} - 8 q^{37} + 14 q^{39} - 9 q^{41} - 4 q^{43} - 5 q^{45} + 18 q^{47} - 3 q^{49} + 9 q^{51} + 12 q^{53} + 3 q^{55} - 14 q^{57} + 18 q^{59} + 7 q^{61} + 8 q^{63} - 7 q^{65} - 4 q^{67} - q^{69} - 3 q^{71} - 2 q^{73} + q^{75} + 12 q^{77} - 16 q^{79} - 10 q^{81} + 12 q^{83} + 3 q^{85} + 24 q^{87} + 12 q^{89} + 7 q^{91} + 28 q^{93} + 7 q^{95} + 7 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 - q^7 + 5 * q^9 - 3 * q^11 + 7 * q^13 - q^15 - 3 * q^17 - 7 * q^19 + 10 * q^21 - 2 * q^23 + 2 * q^25 + 10 * q^27 + 6 * q^29 - 7 * q^31 + 9 * q^33 + q^35 - 8 * q^37 + 14 * q^39 - 9 * q^41 - 4 * q^43 - 5 * q^45 + 18 * q^47 - 3 * q^49 + 9 * q^51 + 12 * q^53 + 3 * q^55 - 14 * q^57 + 18 * q^59 + 7 * q^61 + 8 * q^63 - 7 * q^65 - 4 * q^67 - q^69 - 3 * q^71 - 2 * q^73 + q^75 + 12 * q^77 - 16 * q^79 - 10 * q^81 + 12 * q^83 + 3 * q^85 + 24 * q^87 + 12 * q^89 + 7 * q^91 + 28 * q^93 + 7 * q^95 + 7 * q^97 + 3 * 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
 −1.79129 2.79129
0 −1.79129 0 −1.00000 0 −2.79129 0 0.208712 0
1.2 0 2.79129 0 −1.00000 0 1.79129 0 4.79129 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$+1$$
$$23$$ $$+1$$

## 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 1840.2.a.n 2
4.b odd 2 1 230.2.a.a 2
5.b even 2 1 9200.2.a.bs 2
8.b even 2 1 7360.2.a.bk 2
8.d odd 2 1 7360.2.a.bq 2
12.b even 2 1 2070.2.a.x 2
20.d odd 2 1 1150.2.a.o 2
20.e even 4 2 1150.2.b.g 4
92.b even 2 1 5290.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 4.b odd 2 1
1150.2.a.o 2 20.d odd 2 1
1150.2.b.g 4 20.e even 4 2
1840.2.a.n 2 1.a even 1 1 trivial
2070.2.a.x 2 12.b even 2 1
5290.2.a.e 2 92.b even 2 1
7360.2.a.bk 2 8.b even 2 1
7360.2.a.bq 2 8.d odd 2 1
9200.2.a.bs 2 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{7}^{2} + T_{7} - 5$$ T7^2 + T7 - 5 $$T_{11}^{2} + 3T_{11} - 3$$ T11^2 + 3*T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 5$$
$11$ $$T^{2} + 3T - 3$$
$13$ $$T^{2} - 7T + 7$$
$17$ $$T^{2} + 3T - 3$$
$19$ $$T^{2} + 7T + 7$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 6T - 12$$
$31$ $$T^{2} + 7T - 35$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 9T + 15$$
$43$ $$T^{2} + 4T - 80$$
$47$ $$T^{2} - 18T + 60$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 18T + 60$$
$61$ $$T^{2} - 7T - 35$$
$67$ $$T^{2} + 4T - 80$$
$71$ $$T^{2} + 3T - 45$$
$73$ $$T^{2} + 2T - 188$$
$79$ $$(T + 8)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 12T - 48$$
$97$ $$T^{2} - 7T - 119$$