# Properties

 Label 768.4.a.b Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ 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] = [768,4,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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 8 q^{5} - 12 q^{7} + 9 q^{9}+O(q^{10})$$ 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.b 1
3.b odd 2 1 2304.4.a.e 1
4.b odd 2 1 768.4.a.d 1
8.b even 2 1 768.4.a.c 1
8.d odd 2 1 768.4.a.a 1
12.b even 2 1 2304.4.a.f 1
16.e even 4 2 384.4.d.b yes 2
16.f odd 4 2 384.4.d.a 2
24.f even 2 1 2304.4.a.l 1
24.h odd 2 1 2304.4.a.k 1
48.i odd 4 2 1152.4.d.g 2
48.k even 4 2 1152.4.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 16.f odd 4 2
384.4.d.b yes 2 16.e even 4 2
768.4.a.a 1 8.d odd 2 1
768.4.a.b 1 1.a even 1 1 trivial
768.4.a.c 1 8.b even 2 1
768.4.a.d 1 4.b odd 2 1
1152.4.d.b 2 48.k even 4 2
1152.4.d.g 2 48.i odd 4 2
2304.4.a.e 1 3.b odd 2 1
2304.4.a.f 1 12.b even 2 1
2304.4.a.k 1 24.h odd 2 1
2304.4.a.l 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(768))$$:

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

## Hecke characteristic polynomials

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