Properties

 Label 7.15.b.a Level $7$ Weight $15$ Character orbit 7.b Self dual yes Analytic conductor $8.703$ 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] = [7,15,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, 15, names="a")

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

chi := DirichletCharacter("7.6");

S:= CuspForms(chi, 15);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$15$$ 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: $$8.70302777063$$ 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)

 $$f(q)$$ $$=$$ $$q - 87 q^{2} - 8815 q^{4} - 823543 q^{7} + 2192313 q^{8} + 4782969 q^{9}+O(q^{10})$$ q - 87 * q^2 - 8815 * q^4 - 823543 * q^7 + 2192313 * q^8 + 4782969 * q^9 $$q - 87 q^{2} - 8815 q^{4} - 823543 q^{7} + 2192313 q^{8} + 4782969 q^{9} + 36437514 q^{11} + 71648241 q^{14} - 46306271 q^{16} - 416118303 q^{18} - 3170063718 q^{22} - 2188914318 q^{23} + 6103515625 q^{25} + 7259531545 q^{28} + 29824366266 q^{29} - 31890210615 q^{32} - 42161871735 q^{36} - 112367216342 q^{37} + 484972531402 q^{43} - 321196685910 q^{44} + 190435545666 q^{46} + 678223072849 q^{49} - 531005859375 q^{50} + 907194972426 q^{53} - 1805464024959 q^{56} - 2594719865142 q^{58} - 3938980639167 q^{63} + 3533130267569 q^{64} + 11528240589818 q^{67} - 4338861915246 q^{71} + 10485765117297 q^{72} + 9775947821754 q^{74} - 30007859592102 q^{77} - 37193960502814 q^{79} + 22876792454961 q^{81} - 42192610231974 q^{86} + 79882435629882 q^{88} + 19295279713170 q^{92} - 59005407337863 q^{98} + 174279499899066 q^{99}+O(q^{100})$$ q - 87 * q^2 - 8815 * q^4 - 823543 * q^7 + 2192313 * q^8 + 4782969 * q^9 + 36437514 * q^11 + 71648241 * q^14 - 46306271 * q^16 - 416118303 * q^18 - 3170063718 * q^22 - 2188914318 * q^23 + 6103515625 * q^25 + 7259531545 * q^28 + 29824366266 * q^29 - 31890210615 * q^32 - 42161871735 * q^36 - 112367216342 * q^37 + 484972531402 * q^43 - 321196685910 * q^44 + 190435545666 * q^46 + 678223072849 * q^49 - 531005859375 * q^50 + 907194972426 * q^53 - 1805464024959 * q^56 - 2594719865142 * q^58 - 3938980639167 * q^63 + 3533130267569 * q^64 + 11528240589818 * q^67 - 4338861915246 * q^71 + 10485765117297 * q^72 + 9775947821754 * q^74 - 30007859592102 * q^77 - 37193960502814 * q^79 + 22876792454961 * q^81 - 42192610231974 * q^86 + 79882435629882 * q^88 + 19295279713170 * q^92 - 59005407337863 * q^98 + 174279499899066 * 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
−87.0000 0 −8815.00 0 0 −823543. 2.19231e6 4.78297e6 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 7.15.b.a 1
3.b odd 2 1 63.15.d.a 1
4.b odd 2 1 112.15.c.a 1
7.b odd 2 1 CM 7.15.b.a 1
7.c even 3 2 49.15.d.a 2
7.d odd 6 2 49.15.d.a 2
21.c even 2 1 63.15.d.a 1
28.d even 2 1 112.15.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.15.b.a 1 1.a even 1 1 trivial
7.15.b.a 1 7.b odd 2 1 CM
49.15.d.a 2 7.c even 3 2
49.15.d.a 2 7.d odd 6 2
63.15.d.a 1 3.b odd 2 1
63.15.d.a 1 21.c even 2 1
112.15.c.a 1 4.b odd 2 1
112.15.c.a 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 87$$ acting on $$S_{15}^{\mathrm{new}}(7, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 87$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 823543$$
$11$ $$T - 36437514$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 2188914318$$
$29$ $$T - 29824366266$$
$31$ $$T$$
$37$ $$T + 112367216342$$
$41$ $$T$$
$43$ $$T - 484972531402$$
$47$ $$T$$
$53$ $$T - 907194972426$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 11528240589818$$
$71$ $$T + 4338861915246$$
$73$ $$T$$
$79$ $$T + 37193960502814$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$