# Properties

 Label 768.4.a.l Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $0$ 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] = [768,4,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: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 192) 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 $$\beta = 6\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + b * q^5 + b * q^7 + 9 * q^9 $$q + 3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} - 48 q^{11} + 4 \beta q^{13} + 3 \beta q^{15} - 42 q^{17} + 92 q^{19} + 3 \beta q^{21} - 2 \beta q^{23} + 271 q^{25} + 27 q^{27} - \beta q^{29} - 7 \beta q^{31} - 144 q^{33} + 396 q^{35} + 10 \beta q^{37} + 12 \beta q^{39} + 6 q^{41} + 92 q^{43} + 9 \beta q^{45} + 2 \beta q^{47} + 53 q^{49} - 126 q^{51} - 25 \beta q^{53} - 48 \beta q^{55} + 276 q^{57} - 516 q^{59} - 18 \beta q^{61} + 9 \beta q^{63} + 1584 q^{65} - 524 q^{67} - 6 \beta q^{69} + 50 \beta q^{71} + 430 q^{73} + 813 q^{75} - 48 \beta q^{77} - 59 \beta q^{79} + 81 q^{81} - 432 q^{83} - 42 \beta q^{85} - 3 \beta q^{87} - 630 q^{89} + 1584 q^{91} - 21 \beta q^{93} + 92 \beta q^{95} + 862 q^{97} - 432 q^{99} +O(q^{100})$$ q + 3 * q^3 + b * q^5 + b * q^7 + 9 * q^9 - 48 * q^11 + 4*b * q^13 + 3*b * q^15 - 42 * q^17 + 92 * q^19 + 3*b * q^21 - 2*b * q^23 + 271 * q^25 + 27 * q^27 - b * q^29 - 7*b * q^31 - 144 * q^33 + 396 * q^35 + 10*b * q^37 + 12*b * q^39 + 6 * q^41 + 92 * q^43 + 9*b * q^45 + 2*b * q^47 + 53 * q^49 - 126 * q^51 - 25*b * q^53 - 48*b * q^55 + 276 * q^57 - 516 * q^59 - 18*b * q^61 + 9*b * q^63 + 1584 * q^65 - 524 * q^67 - 6*b * q^69 + 50*b * q^71 + 430 * q^73 + 813 * q^75 - 48*b * q^77 - 59*b * q^79 + 81 * q^81 - 432 * q^83 - 42*b * q^85 - 3*b * q^87 - 630 * q^89 + 1584 * q^91 - 21*b * q^93 + 92*b * q^95 + 862 * q^97 - 432 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 18 * q^9 $$2 q + 6 q^{3} + 18 q^{9} - 96 q^{11} - 84 q^{17} + 184 q^{19} + 542 q^{25} + 54 q^{27} - 288 q^{33} + 792 q^{35} + 12 q^{41} + 184 q^{43} + 106 q^{49} - 252 q^{51} + 552 q^{57} - 1032 q^{59} + 3168 q^{65} - 1048 q^{67} + 860 q^{73} + 1626 q^{75} + 162 q^{81} - 864 q^{83} - 1260 q^{89} + 3168 q^{91} + 1724 q^{97} - 864 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 - 96 * q^11 - 84 * q^17 + 184 * q^19 + 542 * q^25 + 54 * q^27 - 288 * q^33 + 792 * q^35 + 12 * q^41 + 184 * q^43 + 106 * q^49 - 252 * q^51 + 552 * q^57 - 1032 * q^59 + 3168 * q^65 - 1048 * q^67 + 860 * q^73 + 1626 * q^75 + 162 * q^81 - 864 * q^83 - 1260 * q^89 + 3168 * q^91 + 1724 * q^97 - 864 * 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
 −3.31662 3.31662
0 3.00000 0 −19.8997 0 −19.8997 0 9.00000 0
1.2 0 3.00000 0 19.8997 0 19.8997 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.l 2
3.b odd 2 1 2304.4.a.bl 2
4.b odd 2 1 768.4.a.i 2
8.b even 2 1 768.4.a.i 2
8.d odd 2 1 inner 768.4.a.l 2
12.b even 2 1 2304.4.a.y 2
16.e even 4 2 192.4.d.b 4
16.f odd 4 2 192.4.d.b 4
24.f even 2 1 2304.4.a.bl 2
24.h odd 2 1 2304.4.a.y 2
48.i odd 4 2 576.4.d.f 4
48.k even 4 2 576.4.d.f 4

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

 $$T_{5}^{2} - 396$$ T5^2 - 396 $$T_{7}^{2} - 396$$ T7^2 - 396 $$T_{11} + 48$$ T11 + 48 $$T_{19} - 92$$ T19 - 92

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 396$$
$7$ $$T^{2} - 396$$
$11$ $$(T + 48)^{2}$$
$13$ $$T^{2} - 6336$$
$17$ $$(T + 42)^{2}$$
$19$ $$(T - 92)^{2}$$
$23$ $$T^{2} - 1584$$
$29$ $$T^{2} - 396$$
$31$ $$T^{2} - 19404$$
$37$ $$T^{2} - 39600$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 92)^{2}$$
$47$ $$T^{2} - 1584$$
$53$ $$T^{2} - 247500$$
$59$ $$(T + 516)^{2}$$
$61$ $$T^{2} - 128304$$
$67$ $$(T + 524)^{2}$$
$71$ $$T^{2} - 990000$$
$73$ $$(T - 430)^{2}$$
$79$ $$T^{2} - 1378476$$
$83$ $$(T + 432)^{2}$$
$89$ $$(T + 630)^{2}$$
$97$ $$(T - 862)^{2}$$