# Properties

 Label 2535.2.a.c 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 5 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 - 5 * q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 5 q^{7} + q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 10 q^{14} + q^{15} - 4 q^{16} + 2 q^{17} - 2 q^{18} + 2 q^{20} - 5 q^{21} + 4 q^{22} + 6 q^{23} + q^{25} + q^{27} - 10 q^{28} - 4 q^{29} - 2 q^{30} + 7 q^{31} + 8 q^{32} - 2 q^{33} - 4 q^{34} - 5 q^{35} + 2 q^{36} + 2 q^{37} - 6 q^{41} + 10 q^{42} + q^{43} - 4 q^{44} + q^{45} - 12 q^{46} + 8 q^{47} - 4 q^{48} + 18 q^{49} - 2 q^{50} + 2 q^{51} - 4 q^{53} - 2 q^{54} - 2 q^{55} + 8 q^{58} - 12 q^{59} + 2 q^{60} - 13 q^{61} - 14 q^{62} - 5 q^{63} - 8 q^{64} + 4 q^{66} + 7 q^{67} + 4 q^{68} + 6 q^{69} + 10 q^{70} - 12 q^{71} - 15 q^{73} - 4 q^{74} + q^{75} + 10 q^{77} + 3 q^{79} - 4 q^{80} + q^{81} + 12 q^{82} - 8 q^{83} - 10 q^{84} + 2 q^{85} - 2 q^{86} - 4 q^{87} - 14 q^{89} - 2 q^{90} + 12 q^{92} + 7 q^{93} - 16 q^{94} + 8 q^{96} + 5 q^{97} - 36 q^{98} - 2 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 - 5 * q^7 + q^9 - 2 * q^10 - 2 * q^11 + 2 * q^12 + 10 * q^14 + q^15 - 4 * q^16 + 2 * q^17 - 2 * q^18 + 2 * q^20 - 5 * q^21 + 4 * q^22 + 6 * q^23 + q^25 + q^27 - 10 * q^28 - 4 * q^29 - 2 * q^30 + 7 * q^31 + 8 * q^32 - 2 * q^33 - 4 * q^34 - 5 * q^35 + 2 * q^36 + 2 * q^37 - 6 * q^41 + 10 * q^42 + q^43 - 4 * q^44 + q^45 - 12 * q^46 + 8 * q^47 - 4 * q^48 + 18 * q^49 - 2 * q^50 + 2 * q^51 - 4 * q^53 - 2 * q^54 - 2 * q^55 + 8 * q^58 - 12 * q^59 + 2 * q^60 - 13 * q^61 - 14 * q^62 - 5 * q^63 - 8 * q^64 + 4 * q^66 + 7 * q^67 + 4 * q^68 + 6 * q^69 + 10 * q^70 - 12 * q^71 - 15 * q^73 - 4 * q^74 + q^75 + 10 * q^77 + 3 * q^79 - 4 * q^80 + q^81 + 12 * q^82 - 8 * q^83 - 10 * q^84 + 2 * q^85 - 2 * q^86 - 4 * q^87 - 14 * q^89 - 2 * q^90 + 12 * q^92 + 7 * q^93 - 16 * q^94 + 8 * q^96 + 5 * q^97 - 36 * q^98 - 2 * 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 −5.00000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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.c 1
3.b odd 2 1 7605.2.a.s 1
13.b even 2 1 2535.2.a.m 1
13.e even 6 2 195.2.i.a 2
39.d odd 2 1 7605.2.a.a 1
39.h odd 6 2 585.2.j.b 2
65.l even 6 2 975.2.i.i 2
65.r odd 12 4 975.2.bb.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.a 2 13.e even 6 2
585.2.j.b 2 39.h odd 6 2
975.2.i.i 2 65.l even 6 2
975.2.bb.f 4 65.r odd 12 4
2535.2.a.c 1 1.a even 1 1 trivial
2535.2.a.m 1 13.b even 2 1
7605.2.a.a 1 39.d odd 2 1
7605.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2535))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} + 5$$ T7 + 5 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 5$$
$11$ $$T + 2$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T - 6$$
$29$ $$T + 4$$
$31$ $$T - 7$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 1$$
$47$ $$T - 8$$
$53$ $$T + 4$$
$59$ $$T + 12$$
$61$ $$T + 13$$
$67$ $$T - 7$$
$71$ $$T + 12$$
$73$ $$T + 15$$
$79$ $$T - 3$$
$83$ $$T + 8$$
$89$ $$T + 14$$
$97$ $$T - 5$$