# Properties

 Label 740.1.g.c Level $740$ Weight $1$ Character orbit 740.g Self dual yes Analytic conductor $0.369$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -740 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,1,Mod(739,740)]

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([1, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("740.739");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 740.g (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: yes Analytic conductor: $$0.369308109348$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.136900.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.1620896000.2

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta q^{3} + q^{4} + q^{5} + \beta q^{6} + \beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ q - q^2 - b * q^3 + q^4 + q^5 + b * q^6 + b * q^7 - q^8 + q^9 $$q - q^{2} - \beta q^{3} + q^{4} + q^{5} + \beta q^{6} + \beta q^{7} - q^{8} + q^{9} - q^{10} - \beta q^{12} - \beta q^{14} - \beta q^{15} + q^{16} - q^{18} - \beta q^{19} + q^{20} - 2 q^{21} + \beta q^{24} + q^{25} + \beta q^{28} + \beta q^{30} + \beta q^{31} - q^{32} + \beta q^{35} + q^{36} + q^{37} + \beta q^{38} - q^{40} - 2 q^{41} + 2 q^{42} + q^{45} + \beta q^{47} - \beta q^{48} + q^{49} - q^{50} - \beta q^{56} + 2 q^{57} + \beta q^{59} - \beta q^{60} - \beta q^{62} + \beta q^{63} + q^{64} - \beta q^{67} - \beta q^{70} - q^{72} - q^{74} - \beta q^{75} - \beta q^{76} + \beta q^{79} + q^{80} - q^{81} + 2 q^{82} - \beta q^{83} - 2 q^{84} - q^{90} - 2 q^{93} - \beta q^{94} - \beta q^{95} + \beta q^{96} - 2 q^{97} - q^{98} +O(q^{100})$$ q - q^2 - b * q^3 + q^4 + q^5 + b * q^6 + b * q^7 - q^8 + q^9 - q^10 - b * q^12 - b * q^14 - b * q^15 + q^16 - q^18 - b * q^19 + q^20 - 2 * q^21 + b * q^24 + q^25 + b * q^28 + b * q^30 + b * q^31 - q^32 + b * q^35 + q^36 + q^37 + b * q^38 - q^40 - 2 * q^41 + 2 * q^42 + q^45 + b * q^47 - b * q^48 + q^49 - q^50 - b * q^56 + 2 * q^57 + b * q^59 - b * q^60 - b * q^62 + b * q^63 + q^64 - b * q^67 - b * q^70 - q^72 - q^74 - b * q^75 - b * q^76 + b * q^79 + q^80 - q^81 + 2 * q^82 - b * q^83 - 2 * q^84 - q^90 - 2 * q^93 - b * q^94 - b * q^95 + b * q^96 - 2 * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 4 q^{21} + 2 q^{25} - 2 q^{32} + 2 q^{36} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 4 q^{42} + 2 q^{45} + 2 q^{49} - 2 q^{50} + 4 q^{57} + 2 q^{64} - 2 q^{72} - 2 q^{74} + 2 q^{80} - 2 q^{81} + 4 q^{82} - 4 q^{84} - 2 q^{90} - 4 q^{93} - 4 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^16 - 2 * q^18 + 2 * q^20 - 4 * q^21 + 2 * q^25 - 2 * q^32 + 2 * q^36 + 2 * q^37 - 2 * q^40 - 4 * q^41 + 4 * q^42 + 2 * q^45 + 2 * q^49 - 2 * q^50 + 4 * q^57 + 2 * q^64 - 2 * q^72 - 2 * q^74 + 2 * q^80 - 2 * q^81 + 4 * q^82 - 4 * q^84 - 2 * q^90 - 4 * q^93 - 4 * q^97 - 2 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 1.41421 −1.41421
−1.00000 −1.41421 1.00000 1.00000 1.41421 1.41421 −1.00000 1.00000 −1.00000
739.2 −1.00000 1.41421 1.00000 1.00000 −1.41421 −1.41421 −1.00000 1.00000 −1.00000
 $$n$$: 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
740.g odd 2 1 CM by $$\Q(\sqrt{-185})$$
4.b odd 2 1 inner
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.g.c 2
4.b odd 2 1 inner 740.1.g.c 2
5.b even 2 1 740.1.g.d yes 2
5.c odd 4 2 3700.1.b.f 4
20.d odd 2 1 740.1.g.d yes 2
20.e even 4 2 3700.1.b.f 4
37.b even 2 1 740.1.g.d yes 2
148.b odd 2 1 740.1.g.d yes 2
185.d even 2 1 inner 740.1.g.c 2
185.h odd 4 2 3700.1.b.f 4
740.g odd 2 1 CM 740.1.g.c 2
740.m even 4 2 3700.1.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.g.c 2 1.a even 1 1 trivial
740.1.g.c 2 4.b odd 2 1 inner
740.1.g.c 2 185.d even 2 1 inner
740.1.g.c 2 740.g odd 2 1 CM
740.1.g.d yes 2 5.b even 2 1
740.1.g.d yes 2 20.d odd 2 1
740.1.g.d yes 2 37.b even 2 1
740.1.g.d yes 2 148.b odd 2 1
3700.1.b.f 4 5.c odd 4 2
3700.1.b.f 4 20.e even 4 2
3700.1.b.f 4 185.h odd 4 2
3700.1.b.f 4 740.m even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(740, [\chi])$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{13}$$ T13 $$T_{97} + 2$$ T97 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 2$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 2$$
$37$ $$(T - 1)^{2}$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 2$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 2$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 2$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} - 2$$
$83$ $$T^{2} - 2$$
$89$ $$T^{2}$$
$97$ $$(T + 2)^{2}$$