# Properties

 Label 735.1.o.d Level $735$ Weight $1$ Character orbit 735.o Analytic conductor $0.367$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -15 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,1,Mod(569,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 2]))

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

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

chi := DirichletCharacter("735.569");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 735.o (of order $$6$$, degree $$2$$, not 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: $$0.366812784285$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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 4.2.15435.1 Artin image: $C_3\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q - b1 * q^2 - b2 * q^3 + b2 * q^4 + (-b2 - 1) * q^5 + b3 * q^6 + (-b2 - 1) * q^9 $$q - \beta_1 q^{2} - \beta_{2} q^{3} + \beta_{2} q^{4} + ( - \beta_{2} - 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{2} - 1) q^{9} + (\beta_{3} + \beta_1) q^{10} + (\beta_{2} + 1) q^{12} - q^{15} + (\beta_{2} + 1) q^{16} + (\beta_{3} + \beta_1) q^{18} - \beta_1 q^{19} + q^{20} + \beta_1 q^{23} + \beta_{2} q^{25} - q^{27} + \beta_1 q^{30} + ( - \beta_{3} - \beta_1) q^{31} + ( - \beta_{3} - \beta_1) q^{32} + q^{36} + 2 \beta_{2} q^{38} + \beta_{2} q^{45} - 2 \beta_{2} q^{46} + q^{48} - \beta_{3} q^{50} + ( - \beta_{3} - \beta_1) q^{53} + \beta_1 q^{54} + \beta_{3} q^{57} - \beta_{2} q^{60} + \beta_1 q^{61} - 2 q^{62} - q^{64} - \beta_{3} q^{69} + (\beta_{2} + 1) q^{75} - \beta_{3} q^{76} - \beta_{2} q^{80} + \beta_{2} q^{81} - \beta_{3} q^{90} + \beta_{3} q^{92} - \beta_1 q^{93} + (\beta_{3} + \beta_1) q^{95} - \beta_1 q^{96}+O(q^{100})$$ q - b1 * q^2 - b2 * q^3 + b2 * q^4 + (-b2 - 1) * q^5 + b3 * q^6 + (-b2 - 1) * q^9 + (b3 + b1) * q^10 + (b2 + 1) * q^12 - q^15 + (b2 + 1) * q^16 + (b3 + b1) * q^18 - b1 * q^19 + q^20 + b1 * q^23 + b2 * q^25 - q^27 + b1 * q^30 + (-b3 - b1) * q^31 + (-b3 - b1) * q^32 + q^36 + 2*b2 * q^38 + b2 * q^45 - 2*b2 * q^46 + q^48 - b3 * q^50 + (-b3 - b1) * q^53 + b1 * q^54 + b3 * q^57 - b2 * q^60 + b1 * q^61 - 2 * q^62 - q^64 - b3 * q^69 + (b2 + 1) * q^75 - b3 * q^76 - b2 * q^80 + b2 * q^81 - b3 * q^90 + b3 * q^92 - b1 * q^93 + (b3 + b1) * q^95 - b1 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9 $$4 q + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{9} + 2 q^{12} - 4 q^{15} + 2 q^{16} + 4 q^{20} - 2 q^{25} - 4 q^{27} + 4 q^{36} - 4 q^{38} - 2 q^{45} + 4 q^{46} + 4 q^{48} + 2 q^{60} - 8 q^{62} - 4 q^{64} + 2 q^{75} + 2 q^{80} - 2 q^{81}+O(q^{100})$$ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^5 - 2 * q^9 + 2 * q^12 - 4 * q^15 + 2 * q^16 + 4 * q^20 - 2 * q^25 - 4 * q^27 + 4 * q^36 - 4 * q^38 - 2 * q^45 + 4 * q^46 + 4 * q^48 + 2 * q^60 - 8 * q^62 - 4 * q^64 + 2 * q^75 + 2 * q^80 - 2 * q^81

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$346$$ $$442$$ $$491$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
−0.707107 + 1.22474i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.41421 0 0 −0.500000 + 0.866025i −0.707107 1.22474i
569.2 0.707107 1.22474i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.41421 0 0 −0.500000 + 0.866025i 0.707107 + 1.22474i
704.1 −0.707107 1.22474i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.41421 0 0 −0.500000 0.866025i −0.707107 + 1.22474i
704.2 0.707107 + 1.22474i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.41421 0 0 −0.500000 0.866025i 0.707107 1.22474i
 $$n$$: 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 $$\Q(\sqrt{-15})$$
7.c even 3 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
105.o odd 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{2} + 4$$ T2^4 + 2*T2^2 + 4 $$T_{17}$$ T17 $$T_{167} - 2$$ T167 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4} + 2T^{2} + 4$$
$23$ $$T^{4} + 2T^{2} + 4$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 2T^{2} + 4$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 2T^{2} + 4$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 2T^{2} + 4$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$