# Properties

 Label 768.4.a.v Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: 4.4.9792.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 2x + 7$$ x^4 - 2*x^3 - 7*x^2 + 2*x + 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + \beta_1 q^{5} + ( - \beta_{3} + \beta_1) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + b1 * q^5 + (-b3 + b1) * q^7 + 9 * q^9 $$q + 3 q^{3} + \beta_1 q^{5} + ( - \beta_{3} + \beta_1) q^{7} + 9 q^{9} + (\beta_{2} + 12) q^{11} - 3 \beta_{3} q^{13} + 3 \beta_1 q^{15} + (3 \beta_{2} + 30) q^{17} + ( - 3 \beta_{2} + 12) q^{19} + ( - 3 \beta_{3} + 3 \beta_1) q^{21} + (6 \beta_{3} + 6 \beta_1) q^{23} + ( - 6 \beta_{2} + 43) q^{25} + 27 q^{27} + (6 \beta_{3} + 3 \beta_1) q^{29} + ( - \beta_{3} + 13 \beta_1) q^{31} + (3 \beta_{2} + 36) q^{33} + (3 \beta_{2} + 120) q^{35} + (3 \beta_{3} - 18 \beta_1) q^{37} - 9 \beta_{3} q^{39} + ( - 15 \beta_{2} + 102) q^{41} + (3 \beta_{2} + 132) q^{43} + 9 \beta_1 q^{45} + (6 \beta_{3} - 30 \beta_1) q^{47} + (24 \beta_{2} + 209) q^{49} + (9 \beta_{2} + 90) q^{51} + (6 \beta_{3} + 11 \beta_1) q^{53} + ( - 8 \beta_{3} - 4 \beta_1) q^{55} + ( - 9 \beta_{2} + 36) q^{57} + ( - 16 \beta_{2} + 468) q^{59} + (15 \beta_{3} - 54 \beta_1) q^{61} + ( - 9 \beta_{3} + 9 \beta_1) q^{63} + (27 \beta_{2} - 144) q^{65} + ( - 12 \beta_{2} + 588) q^{67} + (18 \beta_{3} + 18 \beta_1) q^{69} + (18 \beta_{3} - 54 \beta_1) q^{71} + ( - 24 \beta_{2} + 242) q^{73} + ( - 18 \beta_{2} + 129) q^{75} + ( - 36 \beta_{3} + 28 \beta_1) q^{77} + ( - 9 \beta_{3} - 51 \beta_1) q^{79} + 81 q^{81} + (17 \beta_{2} + 852) q^{83} + ( - 24 \beta_{3} - 18 \beta_1) q^{85} + (18 \beta_{3} + 9 \beta_1) q^{87} + ( - 18 \beta_{2} - 918) q^{89} + (63 \beta_{2} + 1296) q^{91} + ( - 3 \beta_{3} + 39 \beta_1) q^{93} + (24 \beta_{3} + 60 \beta_1) q^{95} + ( - 6 \beta_{2} - 370) q^{97} + (9 \beta_{2} + 108) q^{99}+O(q^{100})$$ q + 3 * q^3 + b1 * q^5 + (-b3 + b1) * q^7 + 9 * q^9 + (b2 + 12) * q^11 - 3*b3 * q^13 + 3*b1 * q^15 + (3*b2 + 30) * q^17 + (-3*b2 + 12) * q^19 + (-3*b3 + 3*b1) * q^21 + (6*b3 + 6*b1) * q^23 + (-6*b2 + 43) * q^25 + 27 * q^27 + (6*b3 + 3*b1) * q^29 + (-b3 + 13*b1) * q^31 + (3*b2 + 36) * q^33 + (3*b2 + 120) * q^35 + (3*b3 - 18*b1) * q^37 - 9*b3 * q^39 + (-15*b2 + 102) * q^41 + (3*b2 + 132) * q^43 + 9*b1 * q^45 + (6*b3 - 30*b1) * q^47 + (24*b2 + 209) * q^49 + (9*b2 + 90) * q^51 + (6*b3 + 11*b1) * q^53 + (-8*b3 - 4*b1) * q^55 + (-9*b2 + 36) * q^57 + (-16*b2 + 468) * q^59 + (15*b3 - 54*b1) * q^61 + (-9*b3 + 9*b1) * q^63 + (27*b2 - 144) * q^65 + (-12*b2 + 588) * q^67 + (18*b3 + 18*b1) * q^69 + (18*b3 - 54*b1) * q^71 + (-24*b2 + 242) * q^73 + (-18*b2 + 129) * q^75 + (-36*b3 + 28*b1) * q^77 + (-9*b3 - 51*b1) * q^79 + 81 * q^81 + (17*b2 + 852) * q^83 + (-24*b3 - 18*b1) * q^85 + (18*b3 + 9*b1) * q^87 + (-18*b2 - 918) * q^89 + (63*b2 + 1296) * q^91 + (-3*b3 + 39*b1) * q^93 + (24*b3 + 60*b1) * q^95 + (-6*b2 - 370) * q^97 + (9*b2 + 108) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} + 36 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 + 36 * q^9 $$4 q + 12 q^{3} + 36 q^{9} + 48 q^{11} + 120 q^{17} + 48 q^{19} + 172 q^{25} + 108 q^{27} + 144 q^{33} + 480 q^{35} + 408 q^{41} + 528 q^{43} + 836 q^{49} + 360 q^{51} + 144 q^{57} + 1872 q^{59} - 576 q^{65} + 2352 q^{67} + 968 q^{73} + 516 q^{75} + 324 q^{81} + 3408 q^{83} - 3672 q^{89} + 5184 q^{91} - 1480 q^{97} + 432 q^{99}+O(q^{100})$$ 4 * q + 12 * q^3 + 36 * q^9 + 48 * q^11 + 120 * q^17 + 48 * q^19 + 172 * q^25 + 108 * q^27 + 144 * q^33 + 480 * q^35 + 408 * q^41 + 528 * q^43 + 836 * q^49 + 360 * q^51 + 144 * q^57 + 1872 * q^59 - 576 * q^65 + 2352 * q^67 + 968 * q^73 + 516 * q^75 + 324 * q^81 + 3408 * q^83 - 3672 * q^89 + 5184 * q^91 - 1480 * q^97 + 432 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 2x + 7$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{3} + 10\nu^{2} - 6\nu - 20$$ -2*v^3 + 10*v^2 - 6*v - 20 $$\beta_{2}$$ $$=$$ $$16\nu^{3} - 48\nu^{2} - 48\nu + 64$$ 16*v^3 - 48*v^2 - 48*v + 64 $$\beta_{3}$$ $$=$$ $$-12\nu^{3} + 44\nu^{2} + 28\nu - 80$$ -12*v^3 + 44*v^2 + 28*v - 80
 $$\nu$$ $$=$$ $$( 2\beta_{3} + \beta_{2} - 4\beta _1 + 16 ) / 32$$ (2*b3 + b2 - 4*b1 + 16) / 32 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 72 ) / 16$$ (3*b3 + 2*b2 - 2*b1 + 72) / 16 $$\nu^{3}$$ $$=$$ $$( 24\beta_{3} + 17\beta_{2} - 24\beta _1 + 352 ) / 32$$ (24*b3 + 17*b2 - 24*b1 + 352) / 32

## 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
 1.06909 3.63019 −1.21597 −1.48330
0 3.00000 0 −17.4288 0 −2.99032 0 9.00000 0
1.2 0 3.00000 0 −5.67763 0 −33.0917 0 9.00000 0
1.3 0 3.00000 0 5.67763 0 33.0917 0 9.00000 0
1.4 0 3.00000 0 17.4288 0 2.99032 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

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 768.4.a.v 4
3.b odd 2 1 2304.4.a.by 4
4.b odd 2 1 768.4.a.u 4
8.b even 2 1 768.4.a.u 4
8.d odd 2 1 inner 768.4.a.v 4
12.b even 2 1 2304.4.a.cb 4
16.e even 4 2 384.4.d.f 8
16.f odd 4 2 384.4.d.f 8
24.f even 2 1 2304.4.a.by 4
24.h odd 2 1 2304.4.a.cb 4
48.i odd 4 2 1152.4.d.p 8
48.k even 4 2 1152.4.d.p 8

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

 $$T_{5}^{4} - 336T_{5}^{2} + 9792$$ T5^4 - 336*T5^2 + 9792 $$T_{7}^{4} - 1104T_{7}^{2} + 9792$$ T7^4 - 1104*T7^2 + 9792 $$T_{11}^{2} - 24T_{11} - 368$$ T11^2 - 24*T11 - 368 $$T_{19}^{2} - 24T_{19} - 4464$$ T19^2 - 24*T19 - 4464

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 3)^{4}$$
$5$ $$T^{4} - 336T^{2} + 9792$$
$7$ $$T^{4} - 1104 T^{2} + 9792$$
$11$ $$(T^{2} - 24 T - 368)^{2}$$
$13$ $$T^{4} - 8640 T^{2} + \cdots + 12690432$$
$17$ $$(T^{2} - 60 T - 3708)^{2}$$
$19$ $$(T^{2} - 24 T - 4464)^{2}$$
$23$ $$T^{4} - 53568 T^{2} + \cdots + 621831168$$
$29$ $$T^{4} - 41040 T^{2} + \cdots + 419577408$$
$31$ $$T^{4} - 55248 T^{2} + \cdots + 461095488$$
$37$ $$T^{4} - 107136 T^{2} + \cdots + 2487324672$$
$41$ $$(T^{2} - 204 T - 104796)^{2}$$
$43$ $$(T^{2} - 264 T + 12816)^{2}$$
$47$ $$T^{4} - 302400 T^{2} + \cdots + 21332616192$$
$53$ $$T^{4} - 87888 T^{2} + \cdots + 806557248$$
$59$ $$(T^{2} - 936 T + 87952)^{2}$$
$61$ $$T^{4} - 1040256 T^{2} + \cdots + 270509248512$$
$67$ $$(T^{2} - 1176 T + 272016)^{2}$$
$71$ $$T^{4} - 1104192 T^{2} + \cdots + 297070322688$$
$73$ $$(T^{2} - 484 T - 236348)^{2}$$
$79$ $$T^{4} - 1039824 T^{2} + \cdots + 1904357952$$
$83$ $$(T^{2} - 1704 T + 577936)^{2}$$
$89$ $$(T^{2} + 1836 T + 676836)^{2}$$
$97$ $$(T^{2} + 740 T + 118468)^{2}$$