# Properties

 Label 768.4.a.k Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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 $$\beta = 4\sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta - 4) q^{5} + (\beta + 8) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b - 4) * q^5 + (b + 8) * q^7 + 9 * q^9 $$q + 3 q^{3} + ( - \beta - 4) q^{5} + (\beta + 8) q^{7} + 9 q^{9} + ( - 4 \beta + 4) q^{11} + ( - 2 \beta + 36) q^{13} + ( - 3 \beta - 12) q^{15} + (4 \beta - 18) q^{17} + ( - 4 \beta - 68) q^{19} + (3 \beta + 24) q^{21} + ( - 2 \beta + 128) q^{23} + (8 \beta + 99) q^{25} + 27 q^{27} + (9 \beta + 76) q^{29} + (13 \beta - 40) q^{31} + ( - 12 \beta + 12) q^{33} + ( - 12 \beta - 240) q^{35} + (4 \beta + 68) q^{37} + ( - 6 \beta + 108) q^{39} + ( - 20 \beta - 218) q^{41} + (4 \beta + 356) q^{43} + ( - 9 \beta - 36) q^{45} + ( - 14 \beta - 112) q^{47} + (16 \beta - 71) q^{49} + (12 \beta - 54) q^{51} + ( - 15 \beta + 172) q^{53} + (12 \beta + 816) q^{55} + ( - 12 \beta - 204) q^{57} + 324 q^{59} + 324 q^{61} + (9 \beta + 72) q^{63} + ( - 28 \beta + 272) q^{65} + (48 \beta - 228) q^{67} + ( - 6 \beta + 384) q^{69} + (2 \beta + 1024) q^{71} + (48 \beta + 330) q^{73} + (24 \beta + 297) q^{75} + ( - 28 \beta - 800) q^{77} + ( - 59 \beta + 248) q^{79} + 81 q^{81} + (28 \beta - 388) q^{83} + (2 \beta - 760) q^{85} + (27 \beta + 228) q^{87} + (8 \beta + 266) q^{89} + (20 \beta - 128) q^{91} + (39 \beta - 120) q^{93} + (84 \beta + 1104) q^{95} + (88 \beta - 610) q^{97} + ( - 36 \beta + 36) q^{99}+O(q^{100})$$ q + 3 * q^3 + (-b - 4) * q^5 + (b + 8) * q^7 + 9 * q^9 + (-4*b + 4) * q^11 + (-2*b + 36) * q^13 + (-3*b - 12) * q^15 + (4*b - 18) * q^17 + (-4*b - 68) * q^19 + (3*b + 24) * q^21 + (-2*b + 128) * q^23 + (8*b + 99) * q^25 + 27 * q^27 + (9*b + 76) * q^29 + (13*b - 40) * q^31 + (-12*b + 12) * q^33 + (-12*b - 240) * q^35 + (4*b + 68) * q^37 + (-6*b + 108) * q^39 + (-20*b - 218) * q^41 + (4*b + 356) * q^43 + (-9*b - 36) * q^45 + (-14*b - 112) * q^47 + (16*b - 71) * q^49 + (12*b - 54) * q^51 + (-15*b + 172) * q^53 + (12*b + 816) * q^55 + (-12*b - 204) * q^57 + 324 * q^59 + 324 * q^61 + (9*b + 72) * q^63 + (-28*b + 272) * q^65 + (48*b - 228) * q^67 + (-6*b + 384) * q^69 + (2*b + 1024) * q^71 + (48*b + 330) * q^73 + (24*b + 297) * q^75 + (-28*b - 800) * q^77 + (-59*b + 248) * q^79 + 81 * q^81 + (28*b - 388) * q^83 + (2*b - 760) * q^85 + (27*b + 228) * q^87 + (8*b + 266) * q^89 + (20*b - 128) * q^91 + (39*b - 120) * q^93 + (84*b + 1104) * q^95 + (88*b - 610) * q^97 + (-36*b + 36) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 8 q^{5} + 16 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 8 * q^5 + 16 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 8 q^{5} + 16 q^{7} + 18 q^{9} + 8 q^{11} + 72 q^{13} - 24 q^{15} - 36 q^{17} - 136 q^{19} + 48 q^{21} + 256 q^{23} + 198 q^{25} + 54 q^{27} + 152 q^{29} - 80 q^{31} + 24 q^{33} - 480 q^{35} + 136 q^{37} + 216 q^{39} - 436 q^{41} + 712 q^{43} - 72 q^{45} - 224 q^{47} - 142 q^{49} - 108 q^{51} + 344 q^{53} + 1632 q^{55} - 408 q^{57} + 648 q^{59} + 648 q^{61} + 144 q^{63} + 544 q^{65} - 456 q^{67} + 768 q^{69} + 2048 q^{71} + 660 q^{73} + 594 q^{75} - 1600 q^{77} + 496 q^{79} + 162 q^{81} - 776 q^{83} - 1520 q^{85} + 456 q^{87} + 532 q^{89} - 256 q^{91} - 240 q^{93} + 2208 q^{95} - 1220 q^{97} + 72 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 8 * q^5 + 16 * q^7 + 18 * q^9 + 8 * q^11 + 72 * q^13 - 24 * q^15 - 36 * q^17 - 136 * q^19 + 48 * q^21 + 256 * q^23 + 198 * q^25 + 54 * q^27 + 152 * q^29 - 80 * q^31 + 24 * q^33 - 480 * q^35 + 136 * q^37 + 216 * q^39 - 436 * q^41 + 712 * q^43 - 72 * q^45 - 224 * q^47 - 142 * q^49 - 108 * q^51 + 344 * q^53 + 1632 * q^55 - 408 * q^57 + 648 * q^59 + 648 * q^61 + 144 * q^63 + 544 * q^65 - 456 * q^67 + 768 * q^69 + 2048 * q^71 + 660 * q^73 + 594 * q^75 - 1600 * q^77 + 496 * q^79 + 162 * q^81 - 776 * q^83 - 1520 * q^85 + 456 * q^87 + 532 * q^89 - 256 * q^91 - 240 * q^93 + 2208 * q^95 - 1220 * q^97 + 72 * 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
 2.30278 −1.30278
0 3.00000 0 −18.4222 0 22.4222 0 9.00000 0
1.2 0 3.00000 0 10.4222 0 −6.42221 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.k 2
3.b odd 2 1 2304.4.a.bq 2
4.b odd 2 1 768.4.a.e 2
8.b even 2 1 768.4.a.j 2
8.d odd 2 1 768.4.a.p 2
12.b even 2 1 2304.4.a.bp 2
16.e even 4 2 384.4.d.c 4
16.f odd 4 2 384.4.d.e yes 4
24.f even 2 1 2304.4.a.s 2
24.h odd 2 1 2304.4.a.t 2
48.i odd 4 2 1152.4.d.i 4
48.k even 4 2 1152.4.d.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c 4 16.e even 4 2
384.4.d.e yes 4 16.f odd 4 2
768.4.a.e 2 4.b odd 2 1
768.4.a.j 2 8.b even 2 1
768.4.a.k 2 1.a even 1 1 trivial
768.4.a.p 2 8.d odd 2 1
1152.4.d.i 4 48.i odd 4 2
1152.4.d.o 4 48.k even 4 2
2304.4.a.s 2 24.f even 2 1
2304.4.a.t 2 24.h odd 2 1
2304.4.a.bp 2 12.b even 2 1
2304.4.a.bq 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5}^{2} + 8T_{5} - 192$$ T5^2 + 8*T5 - 192 $$T_{7}^{2} - 16T_{7} - 144$$ T7^2 - 16*T7 - 144 $$T_{11}^{2} - 8T_{11} - 3312$$ T11^2 - 8*T11 - 3312 $$T_{19}^{2} + 136T_{19} + 1296$$ T19^2 + 136*T19 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 8T - 192$$
$7$ $$T^{2} - 16T - 144$$
$11$ $$T^{2} - 8T - 3312$$
$13$ $$T^{2} - 72T + 464$$
$17$ $$T^{2} + 36T - 3004$$
$19$ $$T^{2} + 136T + 1296$$
$23$ $$T^{2} - 256T + 15552$$
$29$ $$T^{2} - 152T - 11072$$
$31$ $$T^{2} + 80T - 33552$$
$37$ $$T^{2} - 136T + 1296$$
$41$ $$T^{2} + 436T - 35676$$
$43$ $$T^{2} - 712T + 123408$$
$47$ $$T^{2} + 224T - 28224$$
$53$ $$T^{2} - 344T - 17216$$
$59$ $$(T - 324)^{2}$$
$61$ $$(T - 324)^{2}$$
$67$ $$T^{2} + 456T - 427248$$
$71$ $$T^{2} - 2048 T + 1047744$$
$73$ $$T^{2} - 660T - 370332$$
$79$ $$T^{2} - 496T - 662544$$
$83$ $$T^{2} + 776T - 12528$$
$89$ $$T^{2} - 532T + 57444$$
$97$ $$T^{2} + 1220 T - 1238652$$