# Properties

 Label 64.8.a.i Level $64$ Weight $8$ Character orbit 64.a Self dual yes Analytic conductor $19.993$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 15$$ x^2 - 15 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 32) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 16\sqrt{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 70 q^{5} - 18 \beta q^{7} + 1653 q^{9} +O(q^{10})$$ q + b * q^3 - 70 * q^5 - 18*b * q^7 + 1653 * q^9 $$q + \beta q^{3} - 70 q^{5} - 18 \beta q^{7} + 1653 q^{9} - 117 \beta q^{11} - 13758 q^{13} - 70 \beta q^{15} + 16994 q^{17} + 549 \beta q^{19} - 69120 q^{21} + 522 \beta q^{23} - 73225 q^{25} - 534 \beta q^{27} - 34190 q^{29} + 1944 \beta q^{31} - 449280 q^{33} + 1260 \beta q^{35} - 35206 q^{37} - 13758 \beta q^{39} - 484550 q^{41} - 10845 \beta q^{43} - 115710 q^{45} + 19476 \beta q^{47} + 420617 q^{49} + 16994 \beta q^{51} - 851702 q^{53} + 8190 \beta q^{55} + 2108160 q^{57} + 11223 \beta q^{59} - 71630 q^{61} - 29754 \beta q^{63} + 963060 q^{65} - 4959 \beta q^{67} + 2004480 q^{69} + 12222 \beta q^{71} + 3912042 q^{73} - 73225 \beta q^{75} + 8087040 q^{77} - 5076 \beta q^{79} - 5665671 q^{81} - 24795 \beta q^{83} - 1189580 q^{85} - 34190 \beta q^{87} - 2510630 q^{89} + 247644 \beta q^{91} + 7464960 q^{93} - 38430 \beta q^{95} - 50094 q^{97} - 193401 \beta q^{99} +O(q^{100})$$ q + b * q^3 - 70 * q^5 - 18*b * q^7 + 1653 * q^9 - 117*b * q^11 - 13758 * q^13 - 70*b * q^15 + 16994 * q^17 + 549*b * q^19 - 69120 * q^21 + 522*b * q^23 - 73225 * q^25 - 534*b * q^27 - 34190 * q^29 + 1944*b * q^31 - 449280 * q^33 + 1260*b * q^35 - 35206 * q^37 - 13758*b * q^39 - 484550 * q^41 - 10845*b * q^43 - 115710 * q^45 + 19476*b * q^47 + 420617 * q^49 + 16994*b * q^51 - 851702 * q^53 + 8190*b * q^55 + 2108160 * q^57 + 11223*b * q^59 - 71630 * q^61 - 29754*b * q^63 + 963060 * q^65 - 4959*b * q^67 + 2004480 * q^69 + 12222*b * q^71 + 3912042 * q^73 - 73225*b * q^75 + 8087040 * q^77 - 5076*b * q^79 - 5665671 * q^81 - 24795*b * q^83 - 1189580 * q^85 - 34190*b * q^87 - 2510630 * q^89 + 247644*b * q^91 + 7464960 * q^93 - 38430*b * q^95 - 50094 * q^97 - 193401*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 140 q^{5} + 3306 q^{9}+O(q^{10})$$ 2 * q - 140 * q^5 + 3306 * q^9 $$2 q - 140 q^{5} + 3306 q^{9} - 27516 q^{13} + 33988 q^{17} - 138240 q^{21} - 146450 q^{25} - 68380 q^{29} - 898560 q^{33} - 70412 q^{37} - 969100 q^{41} - 231420 q^{45} + 841234 q^{49} - 1703404 q^{53} + 4216320 q^{57} - 143260 q^{61} + 1926120 q^{65} + 4008960 q^{69} + 7824084 q^{73} + 16174080 q^{77} - 11331342 q^{81} - 2379160 q^{85} - 5021260 q^{89} + 14929920 q^{93} - 100188 q^{97}+O(q^{100})$$ 2 * q - 140 * q^5 + 3306 * q^9 - 27516 * q^13 + 33988 * q^17 - 138240 * q^21 - 146450 * q^25 - 68380 * q^29 - 898560 * q^33 - 70412 * q^37 - 969100 * q^41 - 231420 * q^45 + 841234 * q^49 - 1703404 * q^53 + 4216320 * q^57 - 143260 * q^61 + 1926120 * q^65 + 4008960 * q^69 + 7824084 * q^73 + 16174080 * q^77 - 11331342 * q^81 - 2379160 * q^85 - 5021260 * q^89 + 14929920 * q^93 - 100188 * q^97

## 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
 −3.87298 3.87298
0 −61.9677 0 −70.0000 0 1115.42 0 1653.00 0
1.2 0 61.9677 0 −70.0000 0 −1115.42 0 1653.00 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.a.i 2
3.b odd 2 1 576.8.a.bk 2
4.b odd 2 1 inner 64.8.a.i 2
8.b even 2 1 32.8.a.c 2
8.d odd 2 1 32.8.a.c 2
12.b even 2 1 576.8.a.bk 2
16.e even 4 2 256.8.b.i 4
16.f odd 4 2 256.8.b.i 4
24.f even 2 1 288.8.a.k 2
24.h odd 2 1 288.8.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.a.c 2 8.b even 2 1
32.8.a.c 2 8.d odd 2 1
64.8.a.i 2 1.a even 1 1 trivial
64.8.a.i 2 4.b odd 2 1 inner
256.8.b.i 4 16.e even 4 2
256.8.b.i 4 16.f odd 4 2
288.8.a.k 2 24.f even 2 1
288.8.a.k 2 24.h odd 2 1
576.8.a.bk 2 3.b odd 2 1
576.8.a.bk 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3840$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3840$$
$5$ $$(T + 70)^{2}$$
$7$ $$T^{2} - 1244160$$
$11$ $$T^{2} - 52565760$$
$13$ $$(T + 13758)^{2}$$
$17$ $$(T - 16994)^{2}$$
$19$ $$T^{2} - 1157379840$$
$23$ $$T^{2} - 1046338560$$
$29$ $$(T + 34190)^{2}$$
$31$ $$T^{2} - 14511882240$$
$37$ $$(T + 35206)^{2}$$
$41$ $$(T + 484550)^{2}$$
$43$ $$T^{2} - 451637856000$$
$47$ $$T^{2} - 1456567971840$$
$53$ $$(T + 851702)^{2}$$
$59$ $$T^{2} - 483669999360$$
$61$ $$(T + 71630)^{2}$$
$67$ $$T^{2} - 94432055040$$
$71$ $$T^{2} - 573608770560$$
$73$ $$(T - 3912042)^{2}$$
$79$ $$T^{2} - 98940579840$$
$83$ $$T^{2} - 2360801376000$$
$89$ $$(T + 2510630)^{2}$$
$97$ $$(T + 50094)^{2}$$