# Properties

 Label 7.5.b.a Level $7$ Weight $5$ Character orbit 7.b Self dual yes Analytic conductor $0.724$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,5,Mod(6,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("7.6");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 7.b (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: $$0.723589741587$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 15 q^{4} + 49 q^{7} - 31 q^{8} + 81 q^{9}+O(q^{10})$$ q + q^2 - 15 * q^4 + 49 * q^7 - 31 * q^8 + 81 * q^9 $$q + q^{2} - 15 q^{4} + 49 q^{7} - 31 q^{8} + 81 q^{9} - 206 q^{11} + 49 q^{14} + 209 q^{16} + 81 q^{18} - 206 q^{22} - 734 q^{23} + 625 q^{25} - 735 q^{28} + 1234 q^{29} + 705 q^{32} - 1215 q^{36} - 1294 q^{37} - 334 q^{43} + 3090 q^{44} - 734 q^{46} + 2401 q^{49} + 625 q^{50} - 5582 q^{53} - 1519 q^{56} + 1234 q^{58} + 3969 q^{63} - 2639 q^{64} + 4946 q^{67} + 2914 q^{71} - 2511 q^{72} - 1294 q^{74} - 10094 q^{77} - 3646 q^{79} + 6561 q^{81} - 334 q^{86} + 6386 q^{88} + 11010 q^{92} + 2401 q^{98} - 16686 q^{99}+O(q^{100})$$ q + q^2 - 15 * q^4 + 49 * q^7 - 31 * q^8 + 81 * q^9 - 206 * q^11 + 49 * q^14 + 209 * q^16 + 81 * q^18 - 206 * q^22 - 734 * q^23 + 625 * q^25 - 735 * q^28 + 1234 * q^29 + 705 * q^32 - 1215 * q^36 - 1294 * q^37 - 334 * q^43 + 3090 * q^44 - 734 * q^46 + 2401 * q^49 + 625 * q^50 - 5582 * q^53 - 1519 * q^56 + 1234 * q^58 + 3969 * q^63 - 2639 * q^64 + 4946 * q^67 + 2914 * q^71 - 2511 * q^72 - 1294 * q^74 - 10094 * q^77 - 3646 * q^79 + 6561 * q^81 - 334 * q^86 + 6386 * q^88 + 11010 * q^92 + 2401 * q^98 - 16686 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0
1.00000 0 −15.0000 0 0 49.0000 −31.0000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.5.b.a 1
3.b odd 2 1 63.5.d.a 1
4.b odd 2 1 112.5.c.a 1
5.b even 2 1 175.5.d.a 1
5.c odd 4 2 175.5.c.a 2
7.b odd 2 1 CM 7.5.b.a 1
7.c even 3 2 49.5.d.a 2
7.d odd 6 2 49.5.d.a 2
8.b even 2 1 448.5.c.b 1
8.d odd 2 1 448.5.c.a 1
12.b even 2 1 1008.5.f.a 1
21.c even 2 1 63.5.d.a 1
28.d even 2 1 112.5.c.a 1
35.c odd 2 1 175.5.d.a 1
35.f even 4 2 175.5.c.a 2
56.e even 2 1 448.5.c.a 1
56.h odd 2 1 448.5.c.b 1
84.h odd 2 1 1008.5.f.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.b.a 1 1.a even 1 1 trivial
7.5.b.a 1 7.b odd 2 1 CM
49.5.d.a 2 7.c even 3 2
49.5.d.a 2 7.d odd 6 2
63.5.d.a 1 3.b odd 2 1
63.5.d.a 1 21.c even 2 1
112.5.c.a 1 4.b odd 2 1
112.5.c.a 1 28.d even 2 1
175.5.c.a 2 5.c odd 4 2
175.5.c.a 2 35.f even 4 2
175.5.d.a 1 5.b even 2 1
175.5.d.a 1 35.c odd 2 1
448.5.c.a 1 8.d odd 2 1
448.5.c.a 1 56.e even 2 1
448.5.c.b 1 8.b even 2 1
448.5.c.b 1 56.h odd 2 1
1008.5.f.a 1 12.b even 2 1
1008.5.f.a 1 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T + 206$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 734$$
$29$ $$T - 1234$$
$31$ $$T$$
$37$ $$T + 1294$$
$41$ $$T$$
$43$ $$T + 334$$
$47$ $$T$$
$53$ $$T + 5582$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 4946$$
$71$ $$T - 2914$$
$73$ $$T$$
$79$ $$T + 3646$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$