# Properties

 Label 768.4.a.s Level $768$ Weight $4$ Character orbit 768.a Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1436.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 11x - 12$$ x^3 - 11*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 24) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta_{2} - 3) q^{5} + (\beta_1 - 5) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b2 - 3) * q^5 + (b1 - 5) * q^7 + 9 * q^9 $$q + 3 q^{3} + ( - \beta_{2} - 3) q^{5} + (\beta_1 - 5) q^{7} + 9 q^{9} + (2 \beta_{2} - 2 \beta_1) q^{11} + (3 \beta_{2} - \beta_1 - 18) q^{13} + ( - 3 \beta_{2} - 9) q^{15} + (6 \beta_{2} + 2 \beta_1 + 6) q^{17} + ( - 6 \beta_{2} - 2 \beta_1 + 12) q^{19} + (3 \beta_1 - 15) q^{21} + ( - 2 \beta_1 - 54) q^{23} + (12 \beta_{2} - 4 \beta_1 + 15) q^{25} + 27 q^{27} + ( - 5 \beta_{2} + 2 \beta_1 - 57) q^{29} + (12 \beta_{2} - 3 \beta_1 - 109) q^{31} + (6 \beta_{2} - 6 \beta_1) q^{33} + (14 \beta_{2} + 2 \beta_1 - 36) q^{35} + (3 \beta_{2} - 11 \beta_1 - 96) q^{37} + (9 \beta_{2} - 3 \beta_1 - 54) q^{39} + ( - 6 \beta_{2} + 14 \beta_1 - 42) q^{41} + ( - 18 \beta_{2} + 10 \beta_1 - 84) q^{43} + ( - 9 \beta_{2} - 27) q^{45} + ( - 12 \beta_{2} + 6 \beta_1 - 66) q^{47} + ( - 24 \beta_{2} + 117) q^{49} + (18 \beta_{2} + 6 \beta_1 + 18) q^{51} + (11 \beta_{2} + 10 \beta_1 - 369) q^{53} + ( - 36 \beta_{2} + 4 \beta_1 - 160) q^{55} + ( - 18 \beta_{2} - 6 \beta_1 + 36) q^{57} + (8 \beta_{2} + 8 \beta_1 + 60) q^{59} + (3 \beta_{2} + 9 \beta_1 - 516) q^{61} + (9 \beta_1 - 45) q^{63} + ( - 18 \beta_{2} + 10 \beta_1 - 288) q^{65} + ( - 48 \beta_{2} - 204) q^{67} + ( - 6 \beta_1 - 162) q^{69} + (36 \beta_{2} + 6 \beta_1 + 270) q^{71} + ( - 24 \beta_{2} - 16 \beta_1 - 146) q^{73} + (36 \beta_{2} - 12 \beta_1 + 45) q^{75} + (20 \beta_{2} - 20 \beta_1 - 768) q^{77} + ( - 12 \beta_{2} - 7 \beta_1 - 1) q^{79} + 81 q^{81} + ( - 14 \beta_{2} - 2 \beta_1 + 384) q^{83} + ( - 42 \beta_{2} + 28 \beta_1 - 906) q^{85} + ( - 15 \beta_{2} + 6 \beta_1 - 171) q^{87} + ( - 12 \beta_{2} - 4 \beta_1 + 42) q^{89} + ( - 18 \beta_{2} - 38 \beta_1 - 192) q^{91} + (36 \beta_{2} - 9 \beta_1 - 327) q^{93} + (24 \beta_{2} - 28 \beta_1 + 852) q^{95} + (36 \beta_{2} - 4 \beta_1 - 418) q^{97} + (18 \beta_{2} - 18 \beta_1) q^{99}+O(q^{100})$$ q + 3 * q^3 + (-b2 - 3) * q^5 + (b1 - 5) * q^7 + 9 * q^9 + (2*b2 - 2*b1) * q^11 + (3*b2 - b1 - 18) * q^13 + (-3*b2 - 9) * q^15 + (6*b2 + 2*b1 + 6) * q^17 + (-6*b2 - 2*b1 + 12) * q^19 + (3*b1 - 15) * q^21 + (-2*b1 - 54) * q^23 + (12*b2 - 4*b1 + 15) * q^25 + 27 * q^27 + (-5*b2 + 2*b1 - 57) * q^29 + (12*b2 - 3*b1 - 109) * q^31 + (6*b2 - 6*b1) * q^33 + (14*b2 + 2*b1 - 36) * q^35 + (3*b2 - 11*b1 - 96) * q^37 + (9*b2 - 3*b1 - 54) * q^39 + (-6*b2 + 14*b1 - 42) * q^41 + (-18*b2 + 10*b1 - 84) * q^43 + (-9*b2 - 27) * q^45 + (-12*b2 + 6*b1 - 66) * q^47 + (-24*b2 + 117) * q^49 + (18*b2 + 6*b1 + 18) * q^51 + (11*b2 + 10*b1 - 369) * q^53 + (-36*b2 + 4*b1 - 160) * q^55 + (-18*b2 - 6*b1 + 36) * q^57 + (8*b2 + 8*b1 + 60) * q^59 + (3*b2 + 9*b1 - 516) * q^61 + (9*b1 - 45) * q^63 + (-18*b2 + 10*b1 - 288) * q^65 + (-48*b2 - 204) * q^67 + (-6*b1 - 162) * q^69 + (36*b2 + 6*b1 + 270) * q^71 + (-24*b2 - 16*b1 - 146) * q^73 + (36*b2 - 12*b1 + 45) * q^75 + (20*b2 - 20*b1 - 768) * q^77 + (-12*b2 - 7*b1 - 1) * q^79 + 81 * q^81 + (-14*b2 - 2*b1 + 384) * q^83 + (-42*b2 + 28*b1 - 906) * q^85 + (-15*b2 + 6*b1 - 171) * q^87 + (-12*b2 - 4*b1 + 42) * q^89 + (-18*b2 - 38*b1 - 192) * q^91 + (36*b2 - 9*b1 - 327) * q^93 + (24*b2 - 28*b1 + 852) * q^95 + (36*b2 - 4*b1 - 418) * q^97 + (18*b2 - 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9 $$3 q + 9 q^{3} - 10 q^{5} - 14 q^{7} + 27 q^{9} - 52 q^{13} - 30 q^{15} + 26 q^{17} + 28 q^{19} - 42 q^{21} - 164 q^{23} + 53 q^{25} + 81 q^{27} - 174 q^{29} - 318 q^{31} - 92 q^{35} - 296 q^{37} - 156 q^{39} - 118 q^{41} - 260 q^{43} - 90 q^{45} - 204 q^{47} + 327 q^{49} + 78 q^{51} - 1086 q^{53} - 512 q^{55} + 84 q^{57} + 196 q^{59} - 1536 q^{61} - 126 q^{63} - 872 q^{65} - 660 q^{67} - 492 q^{69} + 852 q^{71} - 478 q^{73} + 159 q^{75} - 2304 q^{77} - 22 q^{79} + 243 q^{81} + 1136 q^{83} - 2732 q^{85} - 522 q^{87} + 110 q^{89} - 632 q^{91} - 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100})$$ 3 * q + 9 * q^3 - 10 * q^5 - 14 * q^7 + 27 * q^9 - 52 * q^13 - 30 * q^15 + 26 * q^17 + 28 * q^19 - 42 * q^21 - 164 * q^23 + 53 * q^25 + 81 * q^27 - 174 * q^29 - 318 * q^31 - 92 * q^35 - 296 * q^37 - 156 * q^39 - 118 * q^41 - 260 * q^43 - 90 * q^45 - 204 * q^47 + 327 * q^49 + 78 * q^51 - 1086 * q^53 - 512 * q^55 + 84 * q^57 + 196 * q^59 - 1536 * q^61 - 126 * q^63 - 872 * q^65 - 660 * q^67 - 492 * q^69 + 852 * q^71 - 478 * q^73 + 159 * q^75 - 2304 * q^77 - 22 * q^79 + 243 * q^81 + 1136 * q^83 - 2732 * q^85 - 522 * q^87 + 110 * q^89 - 632 * q^91 - 954 * q^93 + 2552 * q^95 - 1222 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 11x - 12$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu^{2} - 29$$ 4*v^2 - 29 $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 8\nu - 29$$ 4*v^2 - 8*v - 29
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 8$$ (-b2 + b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta _1 + 29 ) / 4$$ (b1 + 29) / 4

## 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
 −2.48361 3.76644 −1.28282
0 3.00000 0 −18.5422 0 −9.32669 0 9.00000 0
1.2 0 3.00000 0 −0.612661 0 22.7441 0 9.00000 0
1.3 0 3.00000 0 9.15486 0 −27.4175 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.s 3
3.b odd 2 1 2304.4.a.bv 3
4.b odd 2 1 768.4.a.q 3
8.b even 2 1 768.4.a.r 3
8.d odd 2 1 768.4.a.t 3
12.b even 2 1 2304.4.a.bw 3
16.e even 4 2 24.4.d.a 6
16.f odd 4 2 96.4.d.a 6
24.f even 2 1 2304.4.a.bu 3
24.h odd 2 1 2304.4.a.bt 3
48.i odd 4 2 72.4.d.d 6
48.k even 4 2 288.4.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.d.a 6 16.e even 4 2
72.4.d.d 6 48.i odd 4 2
96.4.d.a 6 16.f odd 4 2
288.4.d.d 6 48.k even 4 2
768.4.a.q 3 4.b odd 2 1
768.4.a.r 3 8.b even 2 1
768.4.a.s 3 1.a even 1 1 trivial
768.4.a.t 3 8.d odd 2 1
2304.4.a.bt 3 24.h odd 2 1
2304.4.a.bu 3 24.f even 2 1
2304.4.a.bv 3 3.b odd 2 1
2304.4.a.bw 3 12.b even 2 1

## Hecke kernels

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

 $$T_{5}^{3} + 10T_{5}^{2} - 164T_{5} - 104$$ T5^3 + 10*T5^2 - 164*T5 - 104 $$T_{7}^{3} + 14T_{7}^{2} - 580T_{7} - 5816$$ T7^3 + 14*T7^2 - 580*T7 - 5816 $$T_{11}^{3} - 2816T_{11} + 49152$$ T11^3 - 2816*T11 + 49152 $$T_{19}^{3} - 28T_{19}^{2} - 11088T_{19} - 274752$$ T19^3 - 28*T19^2 - 11088*T19 - 274752

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 3)^{3}$$
$5$ $$T^{3} + 10 T^{2} - 164 T - 104$$
$7$ $$T^{3} + 14 T^{2} - 580 T - 5816$$
$11$ $$T^{3} - 2816T + 49152$$
$13$ $$T^{3} + 52 T^{2} - 1104 T - 55872$$
$17$ $$T^{3} - 26 T^{2} - 11124 T + 477576$$
$19$ $$T^{3} - 28 T^{2} - 11088 T - 274752$$
$23$ $$T^{3} + 164 T^{2} + 6384 T + 45504$$
$29$ $$T^{3} + 174 T^{2} + 3964 T - 61368$$
$31$ $$T^{3} + 318 T^{2} + 4476 T - 3749624$$
$37$ $$T^{3} + 296 T^{2} - 46080 T - 82944$$
$41$ $$T^{3} + 118 T^{2} + \cdots - 19985976$$
$43$ $$T^{3} + 260 T^{2} - 80976 T - 8601408$$
$47$ $$T^{3} + 204 T^{2} - 27792 T - 1964736$$
$53$ $$T^{3} + 1086 T^{2} + \cdots + 20665224$$
$59$ $$T^{3} - 196 T^{2} - 50000 T + 8523584$$
$61$ $$T^{3} + 1536 T^{2} + \cdots + 104841216$$
$67$ $$T^{3} + 660 T^{2} + \cdots - 32228928$$
$71$ $$T^{3} - 852 T^{2} + \cdots + 85084992$$
$73$ $$T^{3} + 478 T^{2} + \cdots - 120833304$$
$79$ $$T^{3} + 22 T^{2} - 71524 T - 7902616$$
$83$ $$T^{3} - 1136 T^{2} + \cdots - 37953536$$
$89$ $$T^{3} - 110 T^{2} - 41364 T - 1423656$$
$97$ $$T^{3} + 1222 T^{2} + \cdots - 74802424$$