# Properties

 Label 115.3.c.a Level $115$ Weight $3$ Character orbit 115.c Self dual yes Analytic conductor $3.134$ Analytic rank $0$ Dimension $1$ CM discriminant -115 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,3,Mod(114,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.114");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 115.c (of order $$2$$, degree $$1$$, minimal)

## 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: $$3.13352304014$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 4 q^{4} - 5 q^{5} + 9 q^{7} + 9 q^{9}+O(q^{10})$$ q + 4 * q^4 - 5 * q^5 + 9 * q^7 + 9 * q^9 $$q + 4 q^{4} - 5 q^{5} + 9 q^{7} + 9 q^{9} + 16 q^{16} - 11 q^{17} - 20 q^{20} + 23 q^{23} + 25 q^{25} + 36 q^{28} - 57 q^{29} - 53 q^{31} - 45 q^{35} + 36 q^{36} - 51 q^{37} - 33 q^{41} + 6 q^{43} - 45 q^{45} + 32 q^{49} + 101 q^{53} + 3 q^{59} + 81 q^{63} + 64 q^{64} - 111 q^{67} - 44 q^{68} + 27 q^{71} - 80 q^{80} + 81 q^{81} + 41 q^{83} + 55 q^{85} + 92 q^{92} + 174 q^{97}+O(q^{100})$$ q + 4 * q^4 - 5 * q^5 + 9 * q^7 + 9 * q^9 + 16 * q^16 - 11 * q^17 - 20 * q^20 + 23 * q^23 + 25 * q^25 + 36 * q^28 - 57 * q^29 - 53 * q^31 - 45 * q^35 + 36 * q^36 - 51 * q^37 - 33 * q^41 + 6 * q^43 - 45 * q^45 + 32 * q^49 + 101 * q^53 + 3 * q^59 + 81 * q^63 + 64 * q^64 - 111 * q^67 - 44 * q^68 + 27 * q^71 - 80 * q^80 + 81 * q^81 + 41 * q^83 + 55 * q^85 + 92 * q^92 + 174 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
114.1
 0
0 0 4.00000 −5.00000 0 9.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.c odd 2 1 CM by $$\Q(\sqrt{-115})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.c.a 1
5.b even 2 1 115.3.c.b yes 1
5.c odd 4 2 575.3.d.a 2
23.b odd 2 1 115.3.c.b yes 1
115.c odd 2 1 CM 115.3.c.a 1
115.e even 4 2 575.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.c.a 1 1.a even 1 1 trivial
115.3.c.a 1 115.c odd 2 1 CM
115.3.c.b yes 1 5.b even 2 1
115.3.c.b yes 1 23.b odd 2 1
575.3.d.a 2 5.c odd 4 2
575.3.d.a 2 115.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(115, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} - 9$$ T7 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 9$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 11$$
$19$ $$T$$
$23$ $$T - 23$$
$29$ $$T + 57$$
$31$ $$T + 53$$
$37$ $$T + 51$$
$41$ $$T + 33$$
$43$ $$T - 6$$
$47$ $$T$$
$53$ $$T - 101$$
$59$ $$T - 3$$
$61$ $$T$$
$67$ $$T + 111$$
$71$ $$T - 27$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T - 41$$
$89$ $$T$$
$97$ $$T - 174$$