# Properties

 Label 768.4.a.d Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $0$ 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] = [768,4,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, 4, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$45.3134668844$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 8 q^{5} + 12 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 8 * q^5 + 12 * q^7 + 9 * q^9 $$q + 3 q^{3} + 8 q^{5} + 12 q^{7} + 9 q^{9} - 12 q^{11} - 20 q^{13} + 24 q^{15} + 62 q^{17} + 108 q^{19} + 36 q^{21} - 72 q^{23} - 61 q^{25} + 27 q^{27} + 128 q^{29} + 204 q^{31} - 36 q^{33} + 96 q^{35} + 228 q^{37} - 60 q^{39} + 22 q^{41} - 204 q^{43} + 72 q^{45} + 600 q^{47} - 199 q^{49} + 186 q^{51} - 256 q^{53} - 96 q^{55} + 324 q^{57} - 828 q^{59} + 84 q^{61} + 108 q^{63} - 160 q^{65} + 348 q^{67} - 216 q^{69} + 456 q^{71} - 822 q^{73} - 183 q^{75} - 144 q^{77} + 1356 q^{79} + 81 q^{81} + 108 q^{83} + 496 q^{85} + 384 q^{87} + 938 q^{89} - 240 q^{91} + 612 q^{93} + 864 q^{95} + 1278 q^{97} - 108 q^{99}+O(q^{100})$$ q + 3 * q^3 + 8 * q^5 + 12 * q^7 + 9 * q^9 - 12 * q^11 - 20 * q^13 + 24 * q^15 + 62 * q^17 + 108 * q^19 + 36 * q^21 - 72 * q^23 - 61 * q^25 + 27 * q^27 + 128 * q^29 + 204 * q^31 - 36 * q^33 + 96 * q^35 + 228 * q^37 - 60 * q^39 + 22 * q^41 - 204 * q^43 + 72 * q^45 + 600 * q^47 - 199 * q^49 + 186 * q^51 - 256 * q^53 - 96 * q^55 + 324 * q^57 - 828 * q^59 + 84 * q^61 + 108 * q^63 - 160 * q^65 + 348 * q^67 - 216 * q^69 + 456 * q^71 - 822 * q^73 - 183 * q^75 - 144 * q^77 + 1356 * q^79 + 81 * q^81 + 108 * q^83 + 496 * q^85 + 384 * q^87 + 938 * q^89 - 240 * q^91 + 612 * q^93 + 864 * q^95 + 1278 * q^97 - 108 * 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 3.00000 0 8.00000 0 12.0000 0 9.00000 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 768.4.a.d 1
3.b odd 2 1 2304.4.a.f 1
4.b odd 2 1 768.4.a.b 1
8.b even 2 1 768.4.a.a 1
8.d odd 2 1 768.4.a.c 1
12.b even 2 1 2304.4.a.e 1
16.e even 4 2 384.4.d.a 2
16.f odd 4 2 384.4.d.b yes 2
24.f even 2 1 2304.4.a.k 1
24.h odd 2 1 2304.4.a.l 1
48.i odd 4 2 1152.4.d.b 2
48.k even 4 2 1152.4.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 16.e even 4 2
384.4.d.b yes 2 16.f odd 4 2
768.4.a.a 1 8.b even 2 1
768.4.a.b 1 4.b odd 2 1
768.4.a.c 1 8.d odd 2 1
768.4.a.d 1 1.a even 1 1 trivial
1152.4.d.b 2 48.i odd 4 2
1152.4.d.g 2 48.k even 4 2
2304.4.a.e 1 12.b even 2 1
2304.4.a.f 1 3.b odd 2 1
2304.4.a.k 1 24.f even 2 1
2304.4.a.l 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5} - 8$$ T5 - 8 $$T_{7} - 12$$ T7 - 12 $$T_{11} + 12$$ T11 + 12 $$T_{19} - 108$$ T19 - 108

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 8$$
$7$ $$T - 12$$
$11$ $$T + 12$$
$13$ $$T + 20$$
$17$ $$T - 62$$
$19$ $$T - 108$$
$23$ $$T + 72$$
$29$ $$T - 128$$
$31$ $$T - 204$$
$37$ $$T - 228$$
$41$ $$T - 22$$
$43$ $$T + 204$$
$47$ $$T - 600$$
$53$ $$T + 256$$
$59$ $$T + 828$$
$61$ $$T - 84$$
$67$ $$T - 348$$
$71$ $$T - 456$$
$73$ $$T + 822$$
$79$ $$T - 1356$$
$83$ $$T - 108$$
$89$ $$T - 938$$
$97$ $$T - 1278$$