# Properties

 Label 735.1.f.d Level $735$ Weight $1$ Character orbit 735.f Self dual yes Analytic conductor $0.367$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,1,Mod(344,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("735.344");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 735.f (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: $$0.366812784285$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.15435.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.8338372875.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + q^{4} - q^{5} - \beta q^{6} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + q^4 - q^5 - b * q^6 + q^9 $$q - \beta q^{2} + q^{3} + q^{4} - q^{5} - \beta q^{6} + q^{9} + \beta q^{10} + q^{12} - q^{15} - q^{16} - \beta q^{18} + \beta q^{19} - q^{20} + \beta q^{23} + q^{25} + q^{27} + \beta q^{30} - \beta q^{31} + \beta q^{32} + q^{36} - 2 q^{38} - q^{45} - 2 q^{46} - q^{48} - \beta q^{50} + \beta q^{53} - \beta q^{54} + \beta q^{57} - q^{60} - \beta q^{61} + 2 q^{62} - q^{64} + \beta q^{69} + q^{75} + \beta q^{76} + q^{80} + q^{81} + \beta q^{90} + \beta q^{92} - \beta q^{93} - \beta q^{95} + \beta q^{96} +O(q^{100})$$ q - b * q^2 + q^3 + q^4 - q^5 - b * q^6 + q^9 + b * q^10 + q^12 - q^15 - q^16 - b * q^18 + b * q^19 - q^20 + b * q^23 + q^25 + q^27 + b * q^30 - b * q^31 + b * q^32 + q^36 - 2 * q^38 - q^45 - 2 * q^46 - q^48 - b * q^50 + b * q^53 - b * q^54 + b * q^57 - q^60 - b * q^61 + 2 * q^62 - q^64 + b * q^69 + q^75 + b * q^76 + q^80 + q^81 + b * q^90 + b * q^92 - b * q^93 - b * q^95 + b * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{9} + 2 q^{12} - 2 q^{15} - 2 q^{16} - 2 q^{20} + 2 q^{25} + 2 q^{27} + 2 q^{36} - 4 q^{38} - 2 q^{45} - 4 q^{46} - 2 q^{48} - 2 q^{60} + 4 q^{62} - 2 q^{64} + 2 q^{75} + 2 q^{80} + 2 q^{81}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^9 + 2 * q^12 - 2 * q^15 - 2 * q^16 - 2 * q^20 + 2 * q^25 + 2 * q^27 + 2 * q^36 - 4 * q^38 - 2 * q^45 - 4 * q^46 - 2 * q^48 - 2 * q^60 + 4 * q^62 - 2 * q^64 + 2 * q^75 + 2 * q^80 + 2 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$$.

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
344.1
 1.41421 −1.41421
−1.41421 1.00000 1.00000 −1.00000 −1.41421 0 0 1.00000 1.41421
344.2 1.41421 1.00000 1.00000 −1.00000 1.41421 0 0 1.00000 −1.41421
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
21.c even 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.1.f.d yes 2
3.b odd 2 1 735.1.f.c 2
5.b even 2 1 735.1.f.c 2
5.c odd 4 2 3675.1.c.f 4
7.b odd 2 1 735.1.f.c 2
7.c even 3 2 735.1.o.c 4
7.d odd 6 2 735.1.o.d 4
15.d odd 2 1 CM 735.1.f.d yes 2
15.e even 4 2 3675.1.c.f 4
21.c even 2 1 inner 735.1.f.d yes 2
21.g even 6 2 735.1.o.c 4
21.h odd 6 2 735.1.o.d 4
35.c odd 2 1 inner 735.1.f.d yes 2
35.f even 4 2 3675.1.c.f 4
35.i odd 6 2 735.1.o.c 4
35.j even 6 2 735.1.o.d 4
35.k even 12 4 3675.1.u.f 8
35.l odd 12 4 3675.1.u.f 8
105.g even 2 1 735.1.f.c 2
105.k odd 4 2 3675.1.c.f 4
105.o odd 6 2 735.1.o.c 4
105.p even 6 2 735.1.o.d 4
105.w odd 12 4 3675.1.u.f 8
105.x even 12 4 3675.1.u.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.1.f.c 2 3.b odd 2 1
735.1.f.c 2 5.b even 2 1
735.1.f.c 2 7.b odd 2 1
735.1.f.c 2 105.g even 2 1
735.1.f.d yes 2 1.a even 1 1 trivial
735.1.f.d yes 2 15.d odd 2 1 CM
735.1.f.d yes 2 21.c even 2 1 inner
735.1.f.d yes 2 35.c odd 2 1 inner
735.1.o.c 4 7.c even 3 2
735.1.o.c 4 21.g even 6 2
735.1.o.c 4 35.i odd 6 2
735.1.o.c 4 105.o odd 6 2
735.1.o.d 4 7.d odd 6 2
735.1.o.d 4 21.h odd 6 2
735.1.o.d 4 35.j even 6 2
735.1.o.d 4 105.p even 6 2
3675.1.c.f 4 5.c odd 4 2
3675.1.c.f 4 15.e even 4 2
3675.1.c.f 4 35.f even 4 2
3675.1.c.f 4 105.k odd 4 2
3675.1.u.f 8 35.k even 12 4
3675.1.u.f 8 35.l odd 12 4
3675.1.u.f 8 105.w odd 12 4
3675.1.u.f 8 105.x even 12 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(735, [\chi])$$:

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{17}$$ T17 $$T_{167} + 2$$ T167 + 2

## Hecke characteristic polynomials

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