# Properties

 Label 768.4.a.m 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + \beta q^{5} - 5 \beta q^{7} + 9 q^{9} +O(q^{10})$$ q + 3 * q^3 + b * q^5 - 5*b * q^7 + 9 * q^9 $$q + 3 q^{3} + \beta q^{5} - 5 \beta q^{7} + 9 q^{9} - 20 q^{11} + 14 \beta q^{13} + 3 \beta q^{15} - 34 q^{17} - 52 q^{19} - 15 \beta q^{21} - 22 \beta q^{23} - 117 q^{25} + 27 q^{27} + 71 \beta q^{29} + 39 \beta q^{31} - 60 q^{33} - 40 q^{35} - 96 \beta q^{37} + 42 \beta q^{39} - 26 q^{41} - 252 q^{43} + 9 \beta q^{45} - 122 \beta q^{47} - 143 q^{49} - 102 q^{51} - 241 \beta q^{53} - 20 \beta q^{55} - 156 q^{57} - 364 q^{59} + 260 \beta q^{61} - 45 \beta q^{63} + 112 q^{65} - 628 q^{67} - 66 \beta q^{69} + 118 \beta q^{71} + 338 q^{73} - 351 q^{75} + 100 \beta q^{77} + 279 \beta q^{79} + 81 q^{81} - 1036 q^{83} - 34 \beta q^{85} + 213 \beta q^{87} + 234 q^{89} - 560 q^{91} + 117 \beta q^{93} - 52 \beta q^{95} - 178 q^{97} - 180 q^{99} +O(q^{100})$$ q + 3 * q^3 + b * q^5 - 5*b * q^7 + 9 * q^9 - 20 * q^11 + 14*b * q^13 + 3*b * q^15 - 34 * q^17 - 52 * q^19 - 15*b * q^21 - 22*b * q^23 - 117 * q^25 + 27 * q^27 + 71*b * q^29 + 39*b * q^31 - 60 * q^33 - 40 * q^35 - 96*b * q^37 + 42*b * q^39 - 26 * q^41 - 252 * q^43 + 9*b * q^45 - 122*b * q^47 - 143 * q^49 - 102 * q^51 - 241*b * q^53 - 20*b * q^55 - 156 * q^57 - 364 * q^59 + 260*b * q^61 - 45*b * q^63 + 112 * q^65 - 628 * q^67 - 66*b * q^69 + 118*b * q^71 + 338 * q^73 - 351 * q^75 + 100*b * q^77 + 279*b * q^79 + 81 * q^81 - 1036 * q^83 - 34*b * q^85 + 213*b * q^87 + 234 * q^89 - 560 * q^91 + 117*b * q^93 - 52*b * q^95 - 178 * q^97 - 180 * 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 $$2 q + 6 q^{3} + 18 q^{9} - 40 q^{11} - 68 q^{17} - 104 q^{19} - 234 q^{25} + 54 q^{27} - 120 q^{33} - 80 q^{35} - 52 q^{41} - 504 q^{43} - 286 q^{49} - 204 q^{51} - 312 q^{57} - 728 q^{59} + 224 q^{65} - 1256 q^{67} + 676 q^{73} - 702 q^{75} + 162 q^{81} - 2072 q^{83} + 468 q^{89} - 1120 q^{91} - 356 q^{97} - 360 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 - 40 * q^11 - 68 * q^17 - 104 * q^19 - 234 * q^25 + 54 * q^27 - 120 * q^33 - 80 * q^35 - 52 * q^41 - 504 * q^43 - 286 * q^49 - 204 * q^51 - 312 * q^57 - 728 * q^59 + 224 * q^65 - 1256 * q^67 + 676 * q^73 - 702 * q^75 + 162 * q^81 - 2072 * q^83 + 468 * q^89 - 1120 * q^91 - 356 * q^97 - 360 * 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
 −1.41421 1.41421
0 3.00000 0 −2.82843 0 14.1421 0 9.00000 0
1.2 0 3.00000 0 2.82843 0 −14.1421 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.m 2
3.b odd 2 1 2304.4.a.bh 2
4.b odd 2 1 768.4.a.h 2
8.b even 2 1 768.4.a.h 2
8.d odd 2 1 inner 768.4.a.m 2
12.b even 2 1 2304.4.a.bb 2
16.e even 4 2 384.4.d.d 4
16.f odd 4 2 384.4.d.d 4
24.f even 2 1 2304.4.a.bh 2
24.h odd 2 1 2304.4.a.bb 2
48.i odd 4 2 1152.4.d.n 4
48.k even 4 2 1152.4.d.n 4

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

 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{7}^{2} - 200$$ T7^2 - 200 $$T_{11} + 20$$ T11 + 20 $$T_{19} + 52$$ T19 + 52

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 200$$
$11$ $$(T + 20)^{2}$$
$13$ $$T^{2} - 1568$$
$17$ $$(T + 34)^{2}$$
$19$ $$(T + 52)^{2}$$
$23$ $$T^{2} - 3872$$
$29$ $$T^{2} - 40328$$
$31$ $$T^{2} - 12168$$
$37$ $$T^{2} - 73728$$
$41$ $$(T + 26)^{2}$$
$43$ $$(T + 252)^{2}$$
$47$ $$T^{2} - 119072$$
$53$ $$T^{2} - 464648$$
$59$ $$(T + 364)^{2}$$
$61$ $$T^{2} - 540800$$
$67$ $$(T + 628)^{2}$$
$71$ $$T^{2} - 111392$$
$73$ $$(T - 338)^{2}$$
$79$ $$T^{2} - 622728$$
$83$ $$(T + 1036)^{2}$$
$89$ $$(T - 234)^{2}$$
$97$ $$(T + 178)^{2}$$