# Properties

 Label 735.2.a.d Level $735$ Weight $2$ Character orbit 735.a Self dual yes Analytic conductor $5.869$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(1,735)]

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([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("735.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.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: $$5.86900454856$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} + q^{5} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 + q^5 + q^9 $$q - q^{3} - 2 q^{4} + q^{5} + q^{9} + 2 q^{12} + q^{13} - q^{15} + 4 q^{16} - 6 q^{17} - 5 q^{19} - 2 q^{20} + 6 q^{23} + q^{25} - q^{27} - 6 q^{29} - 5 q^{31} - 2 q^{36} - 7 q^{37} - q^{39} - 12 q^{41} - q^{43} + q^{45} - 6 q^{47} - 4 q^{48} + 6 q^{51} - 2 q^{52} + 5 q^{57} + 6 q^{59} + 2 q^{60} - 2 q^{61} - 8 q^{64} + q^{65} - 7 q^{67} + 12 q^{68} - 6 q^{69} + 12 q^{71} - 11 q^{73} - q^{75} + 10 q^{76} - 13 q^{79} + 4 q^{80} + q^{81} + 12 q^{83} - 6 q^{85} + 6 q^{87} - 6 q^{89} - 12 q^{92} + 5 q^{93} - 5 q^{95} + 10 q^{97}+O(q^{100})$$ q - q^3 - 2 * q^4 + q^5 + q^9 + 2 * q^12 + q^13 - q^15 + 4 * q^16 - 6 * q^17 - 5 * q^19 - 2 * q^20 + 6 * q^23 + q^25 - q^27 - 6 * q^29 - 5 * q^31 - 2 * q^36 - 7 * q^37 - q^39 - 12 * q^41 - q^43 + q^45 - 6 * q^47 - 4 * q^48 + 6 * q^51 - 2 * q^52 + 5 * q^57 + 6 * q^59 + 2 * q^60 - 2 * q^61 - 8 * q^64 + q^65 - 7 * q^67 + 12 * q^68 - 6 * q^69 + 12 * q^71 - 11 * q^73 - q^75 + 10 * q^76 - 13 * q^79 + 4 * q^80 + q^81 + 12 * q^83 - 6 * q^85 + 6 * q^87 - 6 * q^89 - 12 * q^92 + 5 * q^93 - 5 * q^95 + 10 * q^97

## 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
 0
0 −1.00000 −2.00000 1.00000 0 0 0 1.00000 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
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 735.2.a.d 1
3.b odd 2 1 2205.2.a.d 1
5.b even 2 1 3675.2.a.i 1
7.b odd 2 1 735.2.a.e 1
7.c even 3 2 735.2.i.c 2
7.d odd 6 2 105.2.i.a 2
21.c even 2 1 2205.2.a.f 1
21.g even 6 2 315.2.j.b 2
28.f even 6 2 1680.2.bg.m 2
35.c odd 2 1 3675.2.a.h 1
35.i odd 6 2 525.2.i.c 2
35.k even 12 4 525.2.r.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.a 2 7.d odd 6 2
315.2.j.b 2 21.g even 6 2
525.2.i.c 2 35.i odd 6 2
525.2.r.b 4 35.k even 12 4
735.2.a.d 1 1.a even 1 1 trivial
735.2.a.e 1 7.b odd 2 1
735.2.i.c 2 7.c even 3 2
1680.2.bg.m 2 28.f even 6 2
2205.2.a.d 1 3.b odd 2 1
2205.2.a.f 1 21.c even 2 1
3675.2.a.h 1 35.c odd 2 1
3675.2.a.i 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(735))$$:

 $$T_{2}$$ T2 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T + 6$$
$19$ $$T + 5$$
$23$ $$T - 6$$
$29$ $$T + 6$$
$31$ $$T + 5$$
$37$ $$T + 7$$
$41$ $$T + 12$$
$43$ $$T + 1$$
$47$ $$T + 6$$
$53$ $$T$$
$59$ $$T - 6$$
$61$ $$T + 2$$
$67$ $$T + 7$$
$71$ $$T - 12$$
$73$ $$T + 11$$
$79$ $$T + 13$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T - 10$$
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