# Properties

 Label 2535.2.a.d 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} + q^{9} - 2 q^{10} - 5 q^{11} + 2 q^{12} - 2 q^{14} + q^{15} - 4 q^{16} - 7 q^{17} - 2 q^{18} + 6 q^{19} + 2 q^{20} + q^{21} + 10 q^{22} + 3 q^{23} + q^{25} + q^{27} + 2 q^{28} + 2 q^{29} - 2 q^{30} - 2 q^{31} + 8 q^{32} - 5 q^{33} + 14 q^{34} + q^{35} + 2 q^{36} - 7 q^{37} - 12 q^{38} - 9 q^{41} - 2 q^{42} - 8 q^{43} - 10 q^{44} + q^{45} - 6 q^{46} - 10 q^{47} - 4 q^{48} - 6 q^{49} - 2 q^{50} - 7 q^{51} + 5 q^{53} - 2 q^{54} - 5 q^{55} + 6 q^{57} - 4 q^{58} + 2 q^{60} + 5 q^{61} + 4 q^{62} + q^{63} - 8 q^{64} + 10 q^{66} + 4 q^{67} - 14 q^{68} + 3 q^{69} - 2 q^{70} - 9 q^{71} + 6 q^{73} + 14 q^{74} + q^{75} + 12 q^{76} - 5 q^{77} - 3 q^{79} - 4 q^{80} + q^{81} + 18 q^{82} + 4 q^{83} + 2 q^{84} - 7 q^{85} + 16 q^{86} + 2 q^{87} - 11 q^{89} - 2 q^{90} + 6 q^{92} - 2 q^{93} + 20 q^{94} + 6 q^{95} + 8 q^{96} + 11 q^{97} + 12 q^{98} - 5 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 + q^9 - 2 * q^10 - 5 * q^11 + 2 * q^12 - 2 * q^14 + q^15 - 4 * q^16 - 7 * q^17 - 2 * q^18 + 6 * q^19 + 2 * q^20 + q^21 + 10 * q^22 + 3 * q^23 + q^25 + q^27 + 2 * q^28 + 2 * q^29 - 2 * q^30 - 2 * q^31 + 8 * q^32 - 5 * q^33 + 14 * q^34 + q^35 + 2 * q^36 - 7 * q^37 - 12 * q^38 - 9 * q^41 - 2 * q^42 - 8 * q^43 - 10 * q^44 + q^45 - 6 * q^46 - 10 * q^47 - 4 * q^48 - 6 * q^49 - 2 * q^50 - 7 * q^51 + 5 * q^53 - 2 * q^54 - 5 * q^55 + 6 * q^57 - 4 * q^58 + 2 * q^60 + 5 * q^61 + 4 * q^62 + q^63 - 8 * q^64 + 10 * q^66 + 4 * q^67 - 14 * q^68 + 3 * q^69 - 2 * q^70 - 9 * q^71 + 6 * q^73 + 14 * q^74 + q^75 + 12 * q^76 - 5 * q^77 - 3 * q^79 - 4 * q^80 + q^81 + 18 * q^82 + 4 * q^83 + 2 * q^84 - 7 * q^85 + 16 * q^86 + 2 * q^87 - 11 * q^89 - 2 * q^90 + 6 * q^92 - 2 * q^93 + 20 * q^94 + 6 * q^95 + 8 * q^96 + 11 * q^97 + 12 * q^98 - 5 * 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 1.00000 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.d 1
3.b odd 2 1 7605.2.a.t 1
13.b even 2 1 195.2.a.c 1
39.d odd 2 1 585.2.a.c 1
52.b odd 2 1 3120.2.a.d 1
65.d even 2 1 975.2.a.a 1
65.h odd 4 2 975.2.c.c 2
91.b odd 2 1 9555.2.a.u 1
156.h even 2 1 9360.2.a.bv 1
195.e odd 2 1 2925.2.a.s 1
195.s even 4 2 2925.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.c 1 13.b even 2 1
585.2.a.c 1 39.d odd 2 1
975.2.a.a 1 65.d even 2 1
975.2.c.c 2 65.h odd 4 2
2535.2.a.d 1 1.a even 1 1 trivial
2925.2.a.s 1 195.e odd 2 1
2925.2.c.a 2 195.s even 4 2
3120.2.a.d 1 52.b odd 2 1
7605.2.a.t 1 3.b odd 2 1
9360.2.a.bv 1 156.h even 2 1
9555.2.a.u 1 91.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 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 5$$ T11 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T + 7$$
$19$ $$T - 6$$
$23$ $$T - 3$$
$29$ $$T - 2$$
$31$ $$T + 2$$
$37$ $$T + 7$$
$41$ $$T + 9$$
$43$ $$T + 8$$
$47$ $$T + 10$$
$53$ $$T - 5$$
$59$ $$T$$
$61$ $$T - 5$$
$67$ $$T - 4$$
$71$ $$T + 9$$
$73$ $$T - 6$$
$79$ $$T + 3$$
$83$ $$T - 4$$
$89$ $$T + 11$$
$97$ $$T - 11$$