# Properties

 Label 768.2.a.f Level $768$ Weight $2$ Character orbit 768.a Self dual yes Analytic conductor $6.133$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 4 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 4 * q^7 + q^9 $$q + q^{3} - 4 q^{7} + q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 8 q^{23} - 5 q^{25} + q^{27} + 8 q^{29} - 4 q^{31} - 4 q^{33} + 4 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 8 q^{47} + 9 q^{49} - 2 q^{51} + 8 q^{53} + 4 q^{57} + 12 q^{59} - 12 q^{61} - 4 q^{63} - 12 q^{67} - 8 q^{69} + 8 q^{71} - 6 q^{73} - 5 q^{75} + 16 q^{77} - 4 q^{79} + q^{81} + 4 q^{83} + 8 q^{87} - 6 q^{89} + 16 q^{91} - 4 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 - 4 * q^7 + q^9 - 4 * q^11 - 4 * q^13 - 2 * q^17 + 4 * q^19 - 4 * q^21 - 8 * q^23 - 5 * q^25 + q^27 + 8 * q^29 - 4 * q^31 - 4 * q^33 + 4 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 8 * q^47 + 9 * q^49 - 2 * q^51 + 8 * q^53 + 4 * q^57 + 12 * q^59 - 12 * q^61 - 4 * q^63 - 12 * q^67 - 8 * q^69 + 8 * q^71 - 6 * q^73 - 5 * q^75 + 16 * q^77 - 4 * q^79 + q^81 + 4 * q^83 + 8 * q^87 - 6 * q^89 + 16 * q^91 - 4 * q^93 - 2 * q^97 - 4 * 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 1.00000 0 0 0 −4.00000 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

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 768.2.a.f 1
3.b odd 2 1 2304.2.a.g 1
4.b odd 2 1 768.2.a.c 1
8.b even 2 1 768.2.a.b 1
8.d odd 2 1 768.2.a.g 1
12.b even 2 1 2304.2.a.j 1
16.e even 4 2 384.2.d.b yes 2
16.f odd 4 2 384.2.d.a 2
24.f even 2 1 2304.2.a.k 1
24.h odd 2 1 2304.2.a.f 1
48.i odd 4 2 1152.2.d.f 2
48.k even 4 2 1152.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.a 2 16.f odd 4 2
384.2.d.b yes 2 16.e even 4 2
768.2.a.b 1 8.b even 2 1
768.2.a.c 1 4.b odd 2 1
768.2.a.f 1 1.a even 1 1 trivial
768.2.a.g 1 8.d odd 2 1
1152.2.d.a 2 48.k even 4 2
1152.2.d.f 2 48.i odd 4 2
2304.2.a.f 1 24.h odd 2 1
2304.2.a.g 1 3.b odd 2 1
2304.2.a.j 1 12.b even 2 1
2304.2.a.k 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$:

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

## Hecke characteristic polynomials

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