# Properties

 Label 115.7.c.b Level $115$ Weight $7$ Character orbit 115.c Self dual yes Analytic conductor $26.456$ 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] = [115,7,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, 7, names="a")

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

chi := DirichletCharacter("115.114");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$26.4562196163$$ 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)

 $$f(q)$$ $$=$$ $$q + 64 q^{4} + 125 q^{5} + 594 q^{7} + 729 q^{9}+O(q^{10})$$ q + 64 * q^4 + 125 * q^5 + 594 * q^7 + 729 * q^9 $$q + 64 q^{4} + 125 q^{5} + 594 q^{7} + 729 q^{9} + 4096 q^{16} - 8206 q^{17} + 8000 q^{20} - 12167 q^{23} + 15625 q^{25} + 38016 q^{28} - 41382 q^{29} + 3922 q^{31} + 74250 q^{35} + 46656 q^{36} - 76806 q^{37} + 130482 q^{41} + 33066 q^{43} + 91125 q^{45} + 235187 q^{49} - 179174 q^{53} - 31302 q^{59} + 433026 q^{63} + 262144 q^{64} - 127206 q^{67} - 525184 q^{68} - 388638 q^{71} + 512000 q^{80} + 531441 q^{81} + 778426 q^{83} - 1025750 q^{85} - 778688 q^{92} - 356526 q^{97}+O(q^{100})$$ q + 64 * q^4 + 125 * q^5 + 594 * q^7 + 729 * q^9 + 4096 * q^16 - 8206 * q^17 + 8000 * q^20 - 12167 * q^23 + 15625 * q^25 + 38016 * q^28 - 41382 * q^29 + 3922 * q^31 + 74250 * q^35 + 46656 * q^36 - 76806 * q^37 + 130482 * q^41 + 33066 * q^43 + 91125 * q^45 + 235187 * q^49 - 179174 * q^53 - 31302 * q^59 + 433026 * q^63 + 262144 * q^64 - 127206 * q^67 - 525184 * q^68 - 388638 * q^71 + 512000 * q^80 + 531441 * q^81 + 778426 * q^83 - 1025750 * q^85 - 778688 * q^92 - 356526 * 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 64.0000 125.000 0 594.000 0 729.000 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 115.7.c.b yes 1
5.b even 2 1 115.7.c.a 1
23.b odd 2 1 115.7.c.a 1
115.c odd 2 1 CM 115.7.c.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.7.c.a 1 5.b even 2 1
115.7.c.a 1 23.b odd 2 1
115.7.c.b yes 1 1.a even 1 1 trivial
115.7.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 $$S_{7}^{\mathrm{new}}(115, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} - 594$$ T7 - 594

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 125$$
$7$ $$T - 594$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 8206$$
$19$ $$T$$
$23$ $$T + 12167$$
$29$ $$T + 41382$$
$31$ $$T - 3922$$
$37$ $$T + 76806$$
$41$ $$T - 130482$$
$43$ $$T - 33066$$
$47$ $$T$$
$53$ $$T + 179174$$
$59$ $$T + 31302$$
$61$ $$T$$
$67$ $$T + 127206$$
$71$ $$T + 388638$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T - 778426$$
$89$ $$T$$
$97$ $$T + 356526$$