Properties

 Label 115.5.c.b Level $115$ Weight $5$ Character orbit 115.c Self dual yes Analytic conductor $11.888$ 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] = [115,5,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, 5, names="a")

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

chi := DirichletCharacter("115.114");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$5$$ 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: $$11.8875457546$$ 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)

 $$f(q)$$ $$=$$ $$q + 16 q^{4} + 25 q^{5} - 17 q^{7} + 81 q^{9}+O(q^{10})$$ q + 16 * q^4 + 25 * q^5 - 17 * q^7 + 81 * q^9 $$q + 16 q^{4} + 25 q^{5} - 17 q^{7} + 81 q^{9} + 256 q^{16} - 457 q^{17} + 400 q^{20} + 529 q^{23} + 625 q^{25} - 272 q^{28} + 1567 q^{29} + 887 q^{31} - 425 q^{35} + 1296 q^{36} - 137 q^{37} - 2273 q^{41} - 3662 q^{43} + 2025 q^{45} - 2112 q^{49} + 4583 q^{53} - 6953 q^{59} - 1377 q^{63} + 4096 q^{64} + 3343 q^{67} - 7312 q^{68} - 9353 q^{71} + 6400 q^{80} + 6561 q^{81} - 12097 q^{83} - 11425 q^{85} + 8464 q^{92} + 11458 q^{97}+O(q^{100})$$ q + 16 * q^4 + 25 * q^5 - 17 * q^7 + 81 * q^9 + 256 * q^16 - 457 * q^17 + 400 * q^20 + 529 * q^23 + 625 * q^25 - 272 * q^28 + 1567 * q^29 + 887 * q^31 - 425 * q^35 + 1296 * q^36 - 137 * q^37 - 2273 * q^41 - 3662 * q^43 + 2025 * q^45 - 2112 * q^49 + 4583 * q^53 - 6953 * q^59 - 1377 * q^63 + 4096 * q^64 + 3343 * q^67 - 7312 * q^68 - 9353 * q^71 + 6400 * q^80 + 6561 * q^81 - 12097 * q^83 - 11425 * q^85 + 8464 * q^92 + 11458 * 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 16.0000 25.0000 0 −17.0000 0 81.0000 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 115.5.c.b yes 1
5.b even 2 1 115.5.c.a 1
23.b odd 2 1 115.5.c.a 1
115.c odd 2 1 CM 115.5.c.b yes 1

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(115, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} + 17$$ T7 + 17

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 25$$
$7$ $$T + 17$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 457$$
$19$ $$T$$
$23$ $$T - 529$$
$29$ $$T - 1567$$
$31$ $$T - 887$$
$37$ $$T + 137$$
$41$ $$T + 2273$$
$43$ $$T + 3662$$
$47$ $$T$$
$53$ $$T - 4583$$
$59$ $$T + 6953$$
$61$ $$T$$
$67$ $$T - 3343$$
$71$ $$T + 9353$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T + 12097$$
$89$ $$T$$
$97$ $$T - 11458$$