# Properties

 Label 740.2.a.d Level $740$ Weight $2$ Character orbit 740.a Self dual yes Analytic conductor $5.909$ 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] = [740,2,Mod(1,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("740.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + q^{5} + (\beta + 3) q^{7} + ( - 2 \beta + 1) q^{9}+O(q^{10})$$ q + (b - 1) * q^3 + q^5 + (b + 3) * q^7 + (-2*b + 1) * q^9 $$q + (\beta - 1) q^{3} + q^{5} + (\beta + 3) q^{7} + ( - 2 \beta + 1) q^{9} + (2 \beta - 2) q^{11} + 4 q^{13} + (\beta - 1) q^{15} - 4 q^{17} + ( - 3 \beta + 1) q^{19} + 2 \beta q^{21} + (4 \beta - 2) q^{23} + q^{25} - 4 q^{27} + ( - 2 \beta + 4) q^{29} + ( - 3 \beta + 5) q^{31} + ( - 4 \beta + 8) q^{33} + (\beta + 3) q^{35} + q^{37} + (4 \beta - 4) q^{39} + ( - 4 \beta - 2) q^{41} + 6 q^{43} + ( - 2 \beta + 1) q^{45} + (\beta + 3) q^{47} + (6 \beta + 5) q^{49} + ( - 4 \beta + 4) q^{51} + ( - 4 \beta + 6) q^{53} + (2 \beta - 2) q^{55} + (4 \beta - 10) q^{57} + ( - 5 \beta - 5) q^{59} + 14 q^{61} + ( - 5 \beta - 3) q^{63} + 4 q^{65} + ( - \beta - 7) q^{67} + ( - 6 \beta + 14) q^{69} + 8 q^{71} + 6 \beta q^{73} + (\beta - 1) q^{75} + 4 \beta q^{77} + ( - 3 \beta + 1) q^{79} + (2 \beta + 1) q^{81} + ( - 5 \beta - 7) q^{83} - 4 q^{85} + (6 \beta - 10) q^{87} - 2 q^{89} + (4 \beta + 12) q^{91} + (8 \beta - 14) q^{93} + ( - 3 \beta + 1) q^{95} - 14 q^{97} + (6 \beta - 14) q^{99}+O(q^{100})$$ q + (b - 1) * q^3 + q^5 + (b + 3) * q^7 + (-2*b + 1) * q^9 + (2*b - 2) * q^11 + 4 * q^13 + (b - 1) * q^15 - 4 * q^17 + (-3*b + 1) * q^19 + 2*b * q^21 + (4*b - 2) * q^23 + q^25 - 4 * q^27 + (-2*b + 4) * q^29 + (-3*b + 5) * q^31 + (-4*b + 8) * q^33 + (b + 3) * q^35 + q^37 + (4*b - 4) * q^39 + (-4*b - 2) * q^41 + 6 * q^43 + (-2*b + 1) * q^45 + (b + 3) * q^47 + (6*b + 5) * q^49 + (-4*b + 4) * q^51 + (-4*b + 6) * q^53 + (2*b - 2) * q^55 + (4*b - 10) * q^57 + (-5*b - 5) * q^59 + 14 * q^61 + (-5*b - 3) * q^63 + 4 * q^65 + (-b - 7) * q^67 + (-6*b + 14) * q^69 + 8 * q^71 + 6*b * q^73 + (b - 1) * q^75 + 4*b * q^77 + (-3*b + 1) * q^79 + (2*b + 1) * q^81 + (-5*b - 7) * q^83 - 4 * q^85 + (6*b - 10) * q^87 - 2 * q^89 + (4*b + 12) * q^91 + (8*b - 14) * q^93 + (-3*b + 1) * q^95 - 14 * q^97 + (6*b - 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} - 4 q^{11} + 8 q^{13} - 2 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} + 8 q^{29} + 10 q^{31} + 16 q^{33} + 6 q^{35} + 2 q^{37} - 8 q^{39} - 4 q^{41} + 12 q^{43} + 2 q^{45} + 6 q^{47} + 10 q^{49} + 8 q^{51} + 12 q^{53} - 4 q^{55} - 20 q^{57} - 10 q^{59} + 28 q^{61} - 6 q^{63} + 8 q^{65} - 14 q^{67} + 28 q^{69} + 16 q^{71} - 2 q^{75} + 2 q^{79} + 2 q^{81} - 14 q^{83} - 8 q^{85} - 20 q^{87} - 4 q^{89} + 24 q^{91} - 28 q^{93} + 2 q^{95} - 28 q^{97} - 28 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 - 4 * q^11 + 8 * q^13 - 2 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 + 8 * q^29 + 10 * q^31 + 16 * q^33 + 6 * q^35 + 2 * q^37 - 8 * q^39 - 4 * q^41 + 12 * q^43 + 2 * q^45 + 6 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 - 4 * q^55 - 20 * q^57 - 10 * q^59 + 28 * q^61 - 6 * q^63 + 8 * q^65 - 14 * q^67 + 28 * q^69 + 16 * q^71 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 14 * q^83 - 8 * q^85 - 20 * q^87 - 4 * q^89 + 24 * q^91 - 28 * q^93 + 2 * q^95 - 28 * q^97 - 28 * 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
 −1.73205 1.73205
0 −2.73205 0 1.00000 0 1.26795 0 4.46410 0
1.2 0 0.732051 0 1.00000 0 4.73205 0 −2.46410 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$-1$$

## 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 740.2.a.d 2
3.b odd 2 1 6660.2.a.i 2
4.b odd 2 1 2960.2.a.p 2
5.b even 2 1 3700.2.a.h 2
5.c odd 4 2 3700.2.d.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.d 2 1.a even 1 1 trivial
2960.2.a.p 2 4.b odd 2 1
3700.2.a.h 2 5.b even 2 1
3700.2.d.g 4 5.c odd 4 2
6660.2.a.i 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(740))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 6T + 6$$
$11$ $$T^{2} + 4T - 8$$
$13$ $$(T - 4)^{2}$$
$17$ $$(T + 4)^{2}$$
$19$ $$T^{2} - 2T - 26$$
$23$ $$T^{2} + 4T - 44$$
$29$ $$T^{2} - 8T + 4$$
$31$ $$T^{2} - 10T - 2$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 4T - 44$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 6T + 6$$
$53$ $$T^{2} - 12T - 12$$
$59$ $$T^{2} + 10T - 50$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 14T + 46$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} - 108$$
$79$ $$T^{2} - 2T - 26$$
$83$ $$T^{2} + 14T - 26$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T + 14)^{2}$$