Properties

 Label 8.7.d.a Level $8$ Weight $7$ Character orbit 8.d Self dual yes Analytic conductor $1.840$ Analytic rank $0$ Dimension $1$ CM discriminant -8 Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,7,Mod(3,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, 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("8.3");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 8.d (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: $$1.84043266896$$ 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{2} + 46 q^{3} + 64 q^{4} - 368 q^{6} - 512 q^{8} + 1387 q^{9}+O(q^{10})$$ q - 8 * q^2 + 46 * q^3 + 64 * q^4 - 368 * q^6 - 512 * q^8 + 1387 * q^9 $$q - 8 q^{2} + 46 q^{3} + 64 q^{4} - 368 q^{6} - 512 q^{8} + 1387 q^{9} - 2338 q^{11} + 2944 q^{12} + 4096 q^{16} - 1726 q^{17} - 11096 q^{18} - 2482 q^{19} + 18704 q^{22} - 23552 q^{24} + 15625 q^{25} + 30268 q^{27} - 32768 q^{32} - 107548 q^{33} + 13808 q^{34} + 88768 q^{36} + 19856 q^{38} + 134642 q^{41} - 74914 q^{43} - 149632 q^{44} + 188416 q^{48} + 117649 q^{49} - 125000 q^{50} - 79396 q^{51} - 242144 q^{54} - 114172 q^{57} + 304958 q^{59} + 262144 q^{64} + 860384 q^{66} - 596626 q^{67} - 110464 q^{68} - 710144 q^{72} - 593134 q^{73} + 718750 q^{75} - 158848 q^{76} + 381205 q^{81} - 1077136 q^{82} + 678926 q^{83} + 599312 q^{86} + 1197056 q^{88} - 357262 q^{89} - 1507328 q^{96} + 1822754 q^{97} - 941192 q^{98} - 3242806 q^{99}+O(q^{100})$$ q - 8 * q^2 + 46 * q^3 + 64 * q^4 - 368 * q^6 - 512 * q^8 + 1387 * q^9 - 2338 * q^11 + 2944 * q^12 + 4096 * q^16 - 1726 * q^17 - 11096 * q^18 - 2482 * q^19 + 18704 * q^22 - 23552 * q^24 + 15625 * q^25 + 30268 * q^27 - 32768 * q^32 - 107548 * q^33 + 13808 * q^34 + 88768 * q^36 + 19856 * q^38 + 134642 * q^41 - 74914 * q^43 - 149632 * q^44 + 188416 * q^48 + 117649 * q^49 - 125000 * q^50 - 79396 * q^51 - 242144 * q^54 - 114172 * q^57 + 304958 * q^59 + 262144 * q^64 + 860384 * q^66 - 596626 * q^67 - 110464 * q^68 - 710144 * q^72 - 593134 * q^73 + 718750 * q^75 - 158848 * q^76 + 381205 * q^81 - 1077136 * q^82 + 678926 * q^83 + 599312 * q^86 + 1197056 * q^88 - 357262 * q^89 - 1507328 * q^96 + 1822754 * q^97 - 941192 * q^98 - 3242806 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0
−8.00000 46.0000 64.0000 0 −368.000 0 −512.000 1387.00 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.7.d.a 1
3.b odd 2 1 72.7.b.a 1
4.b odd 2 1 32.7.d.a 1
8.b even 2 1 32.7.d.a 1
8.d odd 2 1 CM 8.7.d.a 1
12.b even 2 1 288.7.b.a 1
16.e even 4 2 256.7.c.d 2
16.f odd 4 2 256.7.c.d 2
24.f even 2 1 72.7.b.a 1
24.h odd 2 1 288.7.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 1.a even 1 1 trivial
8.7.d.a 1 8.d odd 2 1 CM
32.7.d.a 1 4.b odd 2 1
32.7.d.a 1 8.b even 2 1
72.7.b.a 1 3.b odd 2 1
72.7.b.a 1 24.f even 2 1
256.7.c.d 2 16.e even 4 2
256.7.c.d 2 16.f odd 4 2
288.7.b.a 1 12.b even 2 1
288.7.b.a 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 46$$ acting on $$S_{7}^{\mathrm{new}}(8, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 8$$
$3$ $$T - 46$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 2338$$
$13$ $$T$$
$17$ $$T + 1726$$
$19$ $$T + 2482$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 134642$$
$43$ $$T + 74914$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T - 304958$$
$61$ $$T$$
$67$ $$T + 596626$$
$71$ $$T$$
$73$ $$T + 593134$$
$79$ $$T$$
$83$ $$T - 678926$$
$89$ $$T + 357262$$
$97$ $$T - 1822754$$