# Properties

 Label 368.7.f.a Level $368$ Weight $7$ Character orbit 368.f Self dual yes Analytic conductor $84.660$ Analytic rank $0$ Dimension $1$ CM discriminant -23 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,7,Mod(321,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("368.321");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 368.f (of order $$2$$, degree $$1$$, not 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: $$84.6599027721$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 23) 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)

 $$f(q)$$ $$=$$ $$q + 38 q^{3} + 715 q^{9}+O(q^{10})$$ q + 38 * q^3 + 715 * q^9 $$q + 38 q^{3} + 715 q^{9} + 1082 q^{13} + 12167 q^{23} + 15625 q^{25} - 532 q^{27} + 30746 q^{29} - 58754 q^{31} + 41116 q^{39} + 43634 q^{41} + 205342 q^{47} + 117649 q^{49} + 253942 q^{59} + 462346 q^{69} - 667154 q^{71} + 725042 q^{73} + 593750 q^{75} - 541451 q^{81} + 1168348 q^{87} - 2232652 q^{93}+O(q^{100})$$ q + 38 * q^3 + 715 * q^9 + 1082 * q^13 + 12167 * q^23 + 15625 * q^25 - 532 * q^27 + 30746 * q^29 - 58754 * q^31 + 41116 * q^39 + 43634 * q^41 + 205342 * q^47 + 117649 * q^49 + 253942 * q^59 + 462346 * q^69 - 667154 * q^71 + 725042 * q^73 + 593750 * q^75 - 541451 * q^81 + 1168348 * q^87 - 2232652 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/368\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$97$$ $$277$$ $$\chi(n)$$ $$0$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 0
0 38.0000 0 0 0 0 0 715.000 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
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.7.f.a 1
4.b odd 2 1 23.7.b.a 1
12.b even 2 1 207.7.d.a 1
23.b odd 2 1 CM 368.7.f.a 1
92.b even 2 1 23.7.b.a 1
276.h odd 2 1 207.7.d.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.7.b.a 1 4.b odd 2 1
23.7.b.a 1 92.b even 2 1
207.7.d.a 1 12.b even 2 1
207.7.d.a 1 276.h odd 2 1
368.7.f.a 1 1.a even 1 1 trivial
368.7.f.a 1 23.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 38$$ acting on $$S_{7}^{\mathrm{new}}(368, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 38$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 1082$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 12167$$
$29$ $$T - 30746$$
$31$ $$T + 58754$$
$37$ $$T$$
$41$ $$T - 43634$$
$43$ $$T$$
$47$ $$T - 205342$$
$53$ $$T$$
$59$ $$T - 253942$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T + 667154$$
$73$ $$T - 725042$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$