# Properties

 Label 64.8.a.e Level $64$ Weight $8$ Character orbit 64.a Self dual yes Analytic conductor $19.993$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## 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: $$19.9926416310$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 12 q^{3} + 210 q^{5} - 1016 q^{7} - 2043 q^{9}+O(q^{10})$$ q + 12 * q^3 + 210 * q^5 - 1016 * q^7 - 2043 * q^9 $$q + 12 q^{3} + 210 q^{5} - 1016 q^{7} - 2043 q^{9} + 1092 q^{11} - 1382 q^{13} + 2520 q^{15} + 14706 q^{17} - 39940 q^{19} - 12192 q^{21} - 68712 q^{23} - 34025 q^{25} - 50760 q^{27} + 102570 q^{29} - 227552 q^{31} + 13104 q^{33} - 213360 q^{35} - 160526 q^{37} - 16584 q^{39} + 10842 q^{41} - 630748 q^{43} - 429030 q^{45} - 472656 q^{47} + 208713 q^{49} + 176472 q^{51} + 1494018 q^{53} + 229320 q^{55} - 479280 q^{57} + 2640660 q^{59} - 827702 q^{61} + 2075688 q^{63} - 290220 q^{65} - 126004 q^{67} - 824544 q^{69} + 1414728 q^{71} + 980282 q^{73} - 408300 q^{75} - 1109472 q^{77} + 3566800 q^{79} + 3858921 q^{81} + 5672892 q^{83} + 3088260 q^{85} + 1230840 q^{87} - 11951190 q^{89} + 1404112 q^{91} - 2730624 q^{93} - 8387400 q^{95} + 8682146 q^{97} - 2230956 q^{99}+O(q^{100})$$ q + 12 * q^3 + 210 * q^5 - 1016 * q^7 - 2043 * q^9 + 1092 * q^11 - 1382 * q^13 + 2520 * q^15 + 14706 * q^17 - 39940 * q^19 - 12192 * q^21 - 68712 * q^23 - 34025 * q^25 - 50760 * q^27 + 102570 * q^29 - 227552 * q^31 + 13104 * q^33 - 213360 * q^35 - 160526 * q^37 - 16584 * q^39 + 10842 * q^41 - 630748 * q^43 - 429030 * q^45 - 472656 * q^47 + 208713 * q^49 + 176472 * q^51 + 1494018 * q^53 + 229320 * q^55 - 479280 * q^57 + 2640660 * q^59 - 827702 * q^61 + 2075688 * q^63 - 290220 * q^65 - 126004 * q^67 - 824544 * q^69 + 1414728 * q^71 + 980282 * q^73 - 408300 * q^75 - 1109472 * q^77 + 3566800 * q^79 + 3858921 * q^81 + 5672892 * q^83 + 3088260 * q^85 + 1230840 * q^87 - 11951190 * q^89 + 1404112 * q^91 - 2730624 * q^93 - 8387400 * q^95 + 8682146 * q^97 - 2230956 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 12.0000 0 210.000 0 −1016.00 0 −2043.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.a.e 1
3.b odd 2 1 576.8.a.f 1
4.b odd 2 1 64.8.a.c 1
8.b even 2 1 16.8.a.b 1
8.d odd 2 1 2.8.a.a 1
12.b even 2 1 576.8.a.g 1
16.e even 4 2 256.8.b.f 2
16.f odd 4 2 256.8.b.b 2
24.f even 2 1 18.8.a.b 1
24.h odd 2 1 144.8.a.i 1
40.e odd 2 1 50.8.a.g 1
40.f even 2 1 400.8.a.l 1
40.i odd 4 2 400.8.c.j 2
40.k even 4 2 50.8.b.c 2
56.e even 2 1 98.8.a.a 1
56.k odd 6 2 98.8.c.d 2
56.m even 6 2 98.8.c.e 2
72.l even 6 2 162.8.c.a 2
72.p odd 6 2 162.8.c.l 2
88.g even 2 1 242.8.a.e 1
104.h odd 2 1 338.8.a.d 1
104.m even 4 2 338.8.b.d 2
120.m even 2 1 450.8.a.c 1
120.q odd 4 2 450.8.c.g 2
136.e odd 2 1 578.8.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 8.d odd 2 1
16.8.a.b 1 8.b even 2 1
18.8.a.b 1 24.f even 2 1
50.8.a.g 1 40.e odd 2 1
50.8.b.c 2 40.k even 4 2
64.8.a.c 1 4.b odd 2 1
64.8.a.e 1 1.a even 1 1 trivial
98.8.a.a 1 56.e even 2 1
98.8.c.d 2 56.k odd 6 2
98.8.c.e 2 56.m even 6 2
144.8.a.i 1 24.h odd 2 1
162.8.c.a 2 72.l even 6 2
162.8.c.l 2 72.p odd 6 2
242.8.a.e 1 88.g even 2 1
256.8.b.b 2 16.f odd 4 2
256.8.b.f 2 16.e even 4 2
338.8.a.d 1 104.h odd 2 1
338.8.b.d 2 104.m even 4 2
400.8.a.l 1 40.f even 2 1
400.8.c.j 2 40.i odd 4 2
450.8.a.c 1 120.m even 2 1
450.8.c.g 2 120.q odd 4 2
576.8.a.f 1 3.b odd 2 1
576.8.a.g 1 12.b even 2 1
578.8.a.b 1 136.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 12$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 12$$
$5$ $$T - 210$$
$7$ $$T + 1016$$
$11$ $$T - 1092$$
$13$ $$T + 1382$$
$17$ $$T - 14706$$
$19$ $$T + 39940$$
$23$ $$T + 68712$$
$29$ $$T - 102570$$
$31$ $$T + 227552$$
$37$ $$T + 160526$$
$41$ $$T - 10842$$
$43$ $$T + 630748$$
$47$ $$T + 472656$$
$53$ $$T - 1494018$$
$59$ $$T - 2640660$$
$61$ $$T + 827702$$
$67$ $$T + 126004$$
$71$ $$T - 1414728$$
$73$ $$T - 980282$$
$79$ $$T - 3566800$$
$83$ $$T - 5672892$$
$89$ $$T + 11951190$$
$97$ $$T - 8682146$$