# Properties

 Label 2535.2.a.e Level $2535$ Weight $2$ Character orbit 2535.a Self dual yes Analytic conductor $20.242$ 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] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2535 = 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2535.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: $$20.2420769124$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9} + q^{10} - q^{12} + 2 q^{14} - q^{15} - q^{16} - 2 q^{17} - q^{18} - 2 q^{19} + q^{20} - 2 q^{21} + 8 q^{23} + 3 q^{24} + q^{25} + q^{27} + 2 q^{28} + 2 q^{29} + q^{30} + 2 q^{31} - 5 q^{32} + 2 q^{34} + 2 q^{35} - q^{36} - 8 q^{37} + 2 q^{38} - 3 q^{40} + 2 q^{41} + 2 q^{42} + 4 q^{43} - q^{45} - 8 q^{46} - 4 q^{47} - q^{48} - 3 q^{49} - q^{50} - 2 q^{51} - 6 q^{53} - q^{54} - 6 q^{56} - 2 q^{57} - 2 q^{58} - 12 q^{59} + q^{60} + 10 q^{61} - 2 q^{62} - 2 q^{63} + 7 q^{64} + 6 q^{67} + 2 q^{68} + 8 q^{69} - 2 q^{70} - 8 q^{71} + 3 q^{72} + 16 q^{73} + 8 q^{74} + q^{75} + 2 q^{76} - 8 q^{79} + q^{80} + q^{81} - 2 q^{82} - 12 q^{83} + 2 q^{84} + 2 q^{85} - 4 q^{86} + 2 q^{87} + 6 q^{89} + q^{90} - 8 q^{92} + 2 q^{93} + 4 q^{94} + 2 q^{95} - 5 q^{96} - 16 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^3 - q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + q^9 + q^10 - q^12 + 2 * q^14 - q^15 - q^16 - 2 * q^17 - q^18 - 2 * q^19 + q^20 - 2 * q^21 + 8 * q^23 + 3 * q^24 + q^25 + q^27 + 2 * q^28 + 2 * q^29 + q^30 + 2 * q^31 - 5 * q^32 + 2 * q^34 + 2 * q^35 - q^36 - 8 * q^37 + 2 * q^38 - 3 * q^40 + 2 * q^41 + 2 * q^42 + 4 * q^43 - q^45 - 8 * q^46 - 4 * q^47 - q^48 - 3 * q^49 - q^50 - 2 * q^51 - 6 * q^53 - q^54 - 6 * q^56 - 2 * q^57 - 2 * q^58 - 12 * q^59 + q^60 + 10 * q^61 - 2 * q^62 - 2 * q^63 + 7 * q^64 + 6 * q^67 + 2 * q^68 + 8 * q^69 - 2 * q^70 - 8 * q^71 + 3 * q^72 + 16 * q^73 + 8 * q^74 + q^75 + 2 * q^76 - 8 * q^79 + q^80 + q^81 - 2 * q^82 - 12 * q^83 + 2 * q^84 + 2 * q^85 - 4 * q^86 + 2 * q^87 + 6 * q^89 + q^90 - 8 * q^92 + 2 * q^93 + 4 * q^94 + 2 * q^95 - 5 * q^96 - 16 * q^97 + 3 * q^98

## 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
−1.00000 1.00000 −1.00000 −1.00000 −1.00000 −2.00000 3.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## 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 2535.2.a.e 1
3.b odd 2 1 7605.2.a.p 1
13.b even 2 1 2535.2.a.l 1
13.d odd 4 2 195.2.b.b 2
39.d odd 2 1 7605.2.a.d 1
39.f even 4 2 585.2.b.a 2
52.f even 4 2 3120.2.g.a 2
65.f even 4 2 975.2.h.d 2
65.g odd 4 2 975.2.b.b 2
65.k even 4 2 975.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.b 2 13.d odd 4 2
585.2.b.a 2 39.f even 4 2
975.2.b.b 2 65.g odd 4 2
975.2.h.a 2 65.k even 4 2
975.2.h.d 2 65.f even 4 2
2535.2.a.e 1 1.a even 1 1 trivial
2535.2.a.l 1 13.b even 2 1
3120.2.g.a 2 52.f even 4 2
7605.2.a.d 1 39.d odd 2 1
7605.2.a.p 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2535))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7} + 2$$ T7 + 2 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 2$$
$19$ $$T + 2$$
$23$ $$T - 8$$
$29$ $$T - 2$$
$31$ $$T - 2$$
$37$ $$T + 8$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T - 10$$
$67$ $$T - 6$$
$71$ $$T + 8$$
$73$ $$T - 16$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T - 6$$
$97$ $$T + 16$$