# Properties

 Label 740.2.m.b Level $740$ Weight $2$ Character orbit 740.m Analytic conductor $5.909$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(147,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("740.147");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.m (of order $$4$$, degree $$2$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{2} - 2 i q^{4} + (i + 2) q^{5} + ( - 2 i - 2) q^{8} - 3 i q^{9} +O(q^{10})$$ q + (-i + 1) * q^2 - 2*i * q^4 + (i + 2) * q^5 + (-2*i - 2) * q^8 - 3*i * q^9 $$q + ( - i + 1) q^{2} - 2 i q^{4} + (i + 2) q^{5} + ( - 2 i - 2) q^{8} - 3 i q^{9} + ( - i + 3) q^{10} + ( - i - 1) q^{13} - 4 q^{16} + ( - 3 i + 3) q^{17} + ( - 3 i - 3) q^{18} + ( - 4 i + 2) q^{20} + (4 i + 3) q^{25} - 2 q^{26} + 10 q^{29} + (4 i - 4) q^{32} - 6 i q^{34} - 6 q^{36} + ( - 6 i + 1) q^{37} + ( - 6 i - 2) q^{40} - 8 q^{41} + ( - 6 i + 3) q^{45} + 7 i q^{49} + (i + 7) q^{50} + (2 i - 2) q^{52} + (5 i - 5) q^{53} + ( - 10 i + 10) q^{58} + 10 i q^{61} + 8 i q^{64} + ( - 3 i - 1) q^{65} + ( - 6 i - 6) q^{68} + (6 i - 6) q^{72} + (5 i - 5) q^{73} + ( - 7 i - 5) q^{74} + ( - 4 i - 8) q^{80} - 9 q^{81} + (8 i - 8) q^{82} + ( - 3 i + 9) q^{85} + 10 q^{89} + ( - 9 i - 3) q^{90} + (13 i - 13) q^{97} + (7 i + 7) q^{98} +O(q^{100})$$ q + (-i + 1) * q^2 - 2*i * q^4 + (i + 2) * q^5 + (-2*i - 2) * q^8 - 3*i * q^9 + (-i + 3) * q^10 + (-i - 1) * q^13 - 4 * q^16 + (-3*i + 3) * q^17 + (-3*i - 3) * q^18 + (-4*i + 2) * q^20 + (4*i + 3) * q^25 - 2 * q^26 + 10 * q^29 + (4*i - 4) * q^32 - 6*i * q^34 - 6 * q^36 + (-6*i + 1) * q^37 + (-6*i - 2) * q^40 - 8 * q^41 + (-6*i + 3) * q^45 + 7*i * q^49 + (i + 7) * q^50 + (2*i - 2) * q^52 + (5*i - 5) * q^53 + (-10*i + 10) * q^58 + 10*i * q^61 + 8*i * q^64 + (-3*i - 1) * q^65 + (-6*i - 6) * q^68 + (6*i - 6) * q^72 + (5*i - 5) * q^73 + (-7*i - 5) * q^74 + (-4*i - 8) * q^80 - 9 * q^81 + (8*i - 8) * q^82 + (-3*i + 9) * q^85 + 10 * q^89 + (-9*i - 3) * q^90 + (13*i - 13) * q^97 + (7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{5} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^5 - 4 * q^8 $$2 q + 2 q^{2} + 4 q^{5} - 4 q^{8} + 6 q^{10} - 2 q^{13} - 8 q^{16} + 6 q^{17} - 6 q^{18} + 4 q^{20} + 6 q^{25} - 4 q^{26} + 20 q^{29} - 8 q^{32} - 12 q^{36} + 2 q^{37} - 4 q^{40} - 16 q^{41} + 6 q^{45} + 14 q^{50} - 4 q^{52} - 10 q^{53} + 20 q^{58} - 2 q^{65} - 12 q^{68} - 12 q^{72} - 10 q^{73} - 10 q^{74} - 16 q^{80} - 18 q^{81} - 16 q^{82} + 18 q^{85} + 20 q^{89} - 6 q^{90} - 26 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^5 - 4 * q^8 + 6 * q^10 - 2 * q^13 - 8 * q^16 + 6 * q^17 - 6 * q^18 + 4 * q^20 + 6 * q^25 - 4 * q^26 + 20 * q^29 - 8 * q^32 - 12 * q^36 + 2 * q^37 - 4 * q^40 - 16 * q^41 + 6 * q^45 + 14 * q^50 - 4 * q^52 - 10 * q^53 + 20 * q^58 - 2 * q^65 - 12 * q^68 - 12 * q^72 - 10 * q^73 - 10 * q^74 - 16 * q^80 - 18 * q^81 - 16 * q^82 + 18 * q^85 + 20 * q^89 - 6 * q^90 - 26 * q^97 + 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-1$$ $$i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
147.1
 1.00000i − 1.00000i
1.00000 1.00000i 0 2.00000i 2.00000 + 1.00000i 0 0 −2.00000 2.00000i 3.00000i 3.00000 1.00000i
443.1 1.00000 + 1.00000i 0 2.00000i 2.00000 1.00000i 0 0 −2.00000 + 2.00000i 3.00000i 3.00000 + 1.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.h odd 4 1 inner
740.m even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.m.b yes 2
4.b odd 2 1 CM 740.2.m.b yes 2
5.c odd 4 1 740.2.m.a 2
20.e even 4 1 740.2.m.a 2
37.b even 2 1 740.2.m.a 2
148.b odd 2 1 740.2.m.a 2
185.h odd 4 1 inner 740.2.m.b yes 2
740.m even 4 1 inner 740.2.m.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.m.a 2 5.c odd 4 1
740.2.m.a 2 20.e even 4 1
740.2.m.a 2 37.b even 2 1
740.2.m.a 2 148.b odd 2 1
740.2.m.b yes 2 1.a even 1 1 trivial
740.2.m.b yes 2 4.b odd 2 1 CM
740.2.m.b yes 2 185.h odd 4 1 inner
740.2.m.b yes 2 740.m even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13}^{2} + 2T_{13} + 2$$ T13^2 + 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T + 2$$
$17$ $$T^{2} - 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 2T + 37$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 100$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 10T + 50$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 26T + 338$$