# Properties

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

# Related objects

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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{9} - 2 q^{10} - 6 q^{11} + 2 q^{12} - 3 q^{13} + q^{15} - 4 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 2 q^{20} + 12 q^{22} - 4 q^{23} + q^{25} + 6 q^{26} + q^{27} - 8 q^{29} - 2 q^{30} + q^{31} + 8 q^{32} - 6 q^{33} + 8 q^{34} + 2 q^{36} + 7 q^{37} - 2 q^{38} - 3 q^{39} - 6 q^{41} + q^{43} - 12 q^{44} + q^{45} + 8 q^{46} + 2 q^{47} - 4 q^{48} - 2 q^{50} - 4 q^{51} - 6 q^{52} + 4 q^{53} - 2 q^{54} - 6 q^{55} + q^{57} + 16 q^{58} - 8 q^{59} + 2 q^{60} - 14 q^{61} - 2 q^{62} - 8 q^{64} - 3 q^{65} + 12 q^{66} + 7 q^{67} - 8 q^{68} - 4 q^{69} + 6 q^{71} + q^{73} - 14 q^{74} + q^{75} + 2 q^{76} + 6 q^{78} - q^{79} - 4 q^{80} + q^{81} + 12 q^{82} + 2 q^{83} - 4 q^{85} - 2 q^{86} - 8 q^{87} - 12 q^{89} - 2 q^{90} - 8 q^{92} + q^{93} - 4 q^{94} + q^{95} + 8 q^{96} - 6 q^{97} - 6 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^9 - 2 * q^10 - 6 * q^11 + 2 * q^12 - 3 * q^13 + q^15 - 4 * q^16 - 4 * q^17 - 2 * q^18 + q^19 + 2 * q^20 + 12 * q^22 - 4 * q^23 + q^25 + 6 * q^26 + q^27 - 8 * q^29 - 2 * q^30 + q^31 + 8 * q^32 - 6 * q^33 + 8 * q^34 + 2 * q^36 + 7 * q^37 - 2 * q^38 - 3 * q^39 - 6 * q^41 + q^43 - 12 * q^44 + q^45 + 8 * q^46 + 2 * q^47 - 4 * q^48 - 2 * q^50 - 4 * q^51 - 6 * q^52 + 4 * q^53 - 2 * q^54 - 6 * q^55 + q^57 + 16 * q^58 - 8 * q^59 + 2 * q^60 - 14 * q^61 - 2 * q^62 - 8 * q^64 - 3 * q^65 + 12 * q^66 + 7 * q^67 - 8 * q^68 - 4 * q^69 + 6 * q^71 + q^73 - 14 * q^74 + q^75 + 2 * q^76 + 6 * q^78 - q^79 - 4 * q^80 + q^81 + 12 * q^82 + 2 * q^83 - 4 * q^85 - 2 * q^86 - 8 * q^87 - 12 * q^89 - 2 * q^90 - 8 * q^92 + q^93 - 4 * q^94 + q^95 + 8 * q^96 - 6 * q^97 - 6 * q^99

## 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
−2.00000 1.00000 2.00000 1.00000 −2.00000 0 0 1.00000 −2.00000
 $$n$$: 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.b 1
3.b odd 2 1 2205.2.a.k 1
5.b even 2 1 3675.2.a.o 1
7.b odd 2 1 735.2.a.a 1
7.c even 3 2 105.2.i.b 2
7.d odd 6 2 735.2.i.f 2
21.c even 2 1 2205.2.a.m 1
21.h odd 6 2 315.2.j.a 2
28.g odd 6 2 1680.2.bg.l 2
35.c odd 2 1 3675.2.a.p 1
35.j even 6 2 525.2.i.a 2
35.l odd 12 4 525.2.r.d 4

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

## Hecke kernels

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

 $$T_{2} + 2$$ T2 + 2 $$T_{13} + 3$$ T13 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 6$$
$13$ $$T + 3$$
$17$ $$T + 4$$
$19$ $$T - 1$$
$23$ $$T + 4$$
$29$ $$T + 8$$
$31$ $$T - 1$$
$37$ $$T - 7$$
$41$ $$T + 6$$
$43$ $$T - 1$$
$47$ $$T - 2$$
$53$ $$T - 4$$
$59$ $$T + 8$$
$61$ $$T + 14$$
$67$ $$T - 7$$
$71$ $$T - 6$$
$73$ $$T - 1$$
$79$ $$T + 1$$
$83$ $$T - 2$$
$89$ $$T + 12$$
$97$ $$T + 6$$