# Properties

 Label 768.2.a.i Level $768$ Weight $2$ Character orbit 768.a Self dual yes Analytic conductor $6.133$ 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] = [768,2,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$6.13251087523$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 384) 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} - \beta q^{7} + q^{9} +O(q^{10})$$ q - q^3 + b * q^5 - b * q^7 + q^9 $$q - q^{3} + \beta q^{5} - \beta q^{7} + q^{9} - 4 q^{11} - 2 \beta q^{13} - \beta q^{15} - 2 q^{17} - 4 q^{19} + \beta q^{21} + 2 \beta q^{23} + 3 q^{25} - q^{27} - \beta q^{29} + 3 \beta q^{31} + 4 q^{33} - 8 q^{35} + 2 \beta q^{39} - 10 q^{41} - 12 q^{43} + \beta q^{45} - 2 \beta q^{47} + q^{49} + 2 q^{51} - \beta q^{53} - 4 \beta q^{55} + 4 q^{57} + 4 q^{59} + 4 \beta q^{61} - \beta q^{63} - 16 q^{65} - 4 q^{67} - 2 \beta q^{69} - 2 \beta q^{71} + 2 q^{73} - 3 q^{75} + 4 \beta q^{77} + 3 \beta q^{79} + q^{81} + 4 q^{83} - 2 \beta q^{85} + \beta q^{87} - 6 q^{89} + 16 q^{91} - 3 \beta q^{93} - 4 \beta q^{95} + 14 q^{97} - 4 q^{99} +O(q^{100})$$ q - q^3 + b * q^5 - b * q^7 + q^9 - 4 * q^11 - 2*b * q^13 - b * q^15 - 2 * q^17 - 4 * q^19 + b * q^21 + 2*b * q^23 + 3 * q^25 - q^27 - b * q^29 + 3*b * q^31 + 4 * q^33 - 8 * q^35 + 2*b * q^39 - 10 * q^41 - 12 * q^43 + b * q^45 - 2*b * q^47 + q^49 + 2 * q^51 - b * q^53 - 4*b * q^55 + 4 * q^57 + 4 * q^59 + 4*b * q^61 - b * q^63 - 16 * q^65 - 4 * q^67 - 2*b * q^69 - 2*b * q^71 + 2 * q^73 - 3 * q^75 + 4*b * q^77 + 3*b * q^79 + q^81 + 4 * q^83 - 2*b * q^85 + b * q^87 - 6 * q^89 + 16 * q^91 - 3*b * q^93 - 4*b * q^95 + 14 * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 8 q^{11} - 4 q^{17} - 8 q^{19} + 6 q^{25} - 2 q^{27} + 8 q^{33} - 16 q^{35} - 20 q^{41} - 24 q^{43} + 2 q^{49} + 4 q^{51} + 8 q^{57} + 8 q^{59} - 32 q^{65} - 8 q^{67} + 4 q^{73} - 6 q^{75} + 2 q^{81} + 8 q^{83} - 12 q^{89} + 32 q^{91} + 28 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 8 * q^11 - 4 * q^17 - 8 * q^19 + 6 * q^25 - 2 * q^27 + 8 * q^33 - 16 * q^35 - 20 * q^41 - 24 * q^43 + 2 * q^49 + 4 * q^51 + 8 * q^57 + 8 * q^59 - 32 * q^65 - 8 * q^67 + 4 * q^73 - 6 * q^75 + 2 * q^81 + 8 * q^83 - 12 * q^89 + 32 * q^91 + 28 * q^97 - 8 * 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
 −1.41421 1.41421
0 −1.00000 0 −2.82843 0 2.82843 0 1.00000 0
1.2 0 −1.00000 0 2.82843 0 −2.82843 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.a.i 2
3.b odd 2 1 2304.2.a.x 2
4.b odd 2 1 768.2.a.l 2
8.b even 2 1 768.2.a.l 2
8.d odd 2 1 inner 768.2.a.i 2
12.b even 2 1 2304.2.a.r 2
16.e even 4 2 384.2.d.c 4
16.f odd 4 2 384.2.d.c 4
24.f even 2 1 2304.2.a.x 2
24.h odd 2 1 2304.2.a.r 2
48.i odd 4 2 1152.2.d.h 4
48.k even 4 2 1152.2.d.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 16.e even 4 2
384.2.d.c 4 16.f odd 4 2
768.2.a.i 2 1.a even 1 1 trivial
768.2.a.i 2 8.d odd 2 1 inner
768.2.a.l 2 4.b odd 2 1
768.2.a.l 2 8.b even 2 1
1152.2.d.h 4 48.i odd 4 2
1152.2.d.h 4 48.k even 4 2
2304.2.a.r 2 12.b even 2 1
2304.2.a.r 2 24.h odd 2 1
2304.2.a.x 2 3.b odd 2 1
2304.2.a.x 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11} + 4$$ T11 + 4 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 8$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 32$$
$29$ $$T^{2} - 8$$
$31$ $$T^{2} - 72$$
$37$ $$T^{2}$$
$41$ $$(T + 10)^{2}$$
$43$ $$(T + 12)^{2}$$
$47$ $$T^{2} - 32$$
$53$ $$T^{2} - 8$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 128$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 32$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} - 72$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T - 14)^{2}$$