# Properties

 Label 768.4.a.g Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 3 \beta q^{5} + \beta q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 3*b * q^5 + b * q^7 + 9 * q^9 $$q - 3 q^{3} + 3 \beta q^{5} + \beta q^{7} + 9 q^{9} - 16 \beta q^{13} - 9 \beta q^{15} - 90 q^{17} + 116 q^{19} - 3 \beta q^{21} + 30 \beta q^{23} - 17 q^{25} - 27 q^{27} - 75 \beta q^{29} - 87 \beta q^{31} + 36 q^{35} + 10 \beta q^{37} + 48 \beta q^{39} + 54 q^{41} + 20 q^{43} + 27 \beta q^{45} + 114 \beta q^{47} - 331 q^{49} + 270 q^{51} + 141 \beta q^{53} - 348 q^{57} + 324 q^{59} + 166 \beta q^{61} + 9 \beta q^{63} - 576 q^{65} - 116 q^{67} - 90 \beta q^{69} - 318 \beta q^{71} - 1106 q^{73} + 51 q^{75} - 43 \beta q^{79} + 81 q^{81} - 1152 q^{83} - 270 \beta q^{85} + 225 \beta q^{87} - 918 q^{89} - 192 q^{91} + 261 \beta q^{93} + 348 \beta q^{95} + 190 q^{97} +O(q^{100})$$ q - 3 * q^3 + 3*b * q^5 + b * q^7 + 9 * q^9 - 16*b * q^13 - 9*b * q^15 - 90 * q^17 + 116 * q^19 - 3*b * q^21 + 30*b * q^23 - 17 * q^25 - 27 * q^27 - 75*b * q^29 - 87*b * q^31 + 36 * q^35 + 10*b * q^37 + 48*b * q^39 + 54 * q^41 + 20 * q^43 + 27*b * q^45 + 114*b * q^47 - 331 * q^49 + 270 * q^51 + 141*b * q^53 - 348 * q^57 + 324 * q^59 + 166*b * q^61 + 9*b * q^63 - 576 * q^65 - 116 * q^67 - 90*b * q^69 - 318*b * q^71 - 1106 * q^73 + 51 * q^75 - 43*b * q^79 + 81 * q^81 - 1152 * q^83 - 270*b * q^85 + 225*b * q^87 - 918 * q^89 - 192 * q^91 + 261*b * q^93 + 348*b * q^95 + 190 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 18 * q^9 $$2 q - 6 q^{3} + 18 q^{9} - 180 q^{17} + 232 q^{19} - 34 q^{25} - 54 q^{27} + 72 q^{35} + 108 q^{41} + 40 q^{43} - 662 q^{49} + 540 q^{51} - 696 q^{57} + 648 q^{59} - 1152 q^{65} - 232 q^{67} - 2212 q^{73} + 102 q^{75} + 162 q^{81} - 2304 q^{83} - 1836 q^{89} - 384 q^{91} + 380 q^{97}+O(q^{100})$$ 2 * q - 6 * q^3 + 18 * q^9 - 180 * q^17 + 232 * q^19 - 34 * q^25 - 54 * q^27 + 72 * q^35 + 108 * q^41 + 40 * q^43 - 662 * q^49 + 540 * q^51 - 696 * q^57 + 648 * q^59 - 1152 * q^65 - 232 * q^67 - 2212 * q^73 + 102 * q^75 + 162 * q^81 - 2304 * q^83 - 1836 * q^89 - 384 * q^91 + 380 * q^97

## 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
 −1.73205 1.73205
0 −3.00000 0 −10.3923 0 −3.46410 0 9.00000 0
1.2 0 −3.00000 0 10.3923 0 3.46410 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.g 2
3.b odd 2 1 2304.4.a.bg 2
4.b odd 2 1 768.4.a.n 2
8.b even 2 1 768.4.a.n 2
8.d odd 2 1 inner 768.4.a.g 2
12.b even 2 1 2304.4.a.be 2
16.e even 4 2 192.4.d.a 4
16.f odd 4 2 192.4.d.a 4
24.f even 2 1 2304.4.a.bg 2
24.h odd 2 1 2304.4.a.be 2
48.i odd 4 2 576.4.d.g 4
48.k even 4 2 576.4.d.g 4

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

 $$T_{5}^{2} - 108$$ T5^2 - 108 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}$$ T11 $$T_{19} - 116$$ T19 - 116

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - 108$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 3072$$
$17$ $$(T + 90)^{2}$$
$19$ $$(T - 116)^{2}$$
$23$ $$T^{2} - 10800$$
$29$ $$T^{2} - 67500$$
$31$ $$T^{2} - 90828$$
$37$ $$T^{2} - 1200$$
$41$ $$(T - 54)^{2}$$
$43$ $$(T - 20)^{2}$$
$47$ $$T^{2} - 155952$$
$53$ $$T^{2} - 238572$$
$59$ $$(T - 324)^{2}$$
$61$ $$T^{2} - 330672$$
$67$ $$(T + 116)^{2}$$
$71$ $$T^{2} - 1213488$$
$73$ $$(T + 1106)^{2}$$
$79$ $$T^{2} - 22188$$
$83$ $$(T + 1152)^{2}$$
$89$ $$(T + 918)^{2}$$
$97$ $$(T - 190)^{2}$$