Properties

 Label 768.4.f.b Level $768$ Weight $4$ Character orbit 768.f Analytic conductor $45.313$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(383,768)]

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([1, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.f (of order $$2$$, degree $$1$$, not minimal)

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: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + \beta_{6} q^{5} + 5 \beta_{3} q^{7} + (\beta_{5} - 21) q^{9}+O(q^{10})$$ q + b4 * q^3 + b6 * q^5 + 5*b3 * q^7 + (b5 - 21) * q^9 $$q + \beta_{4} q^{3} + \beta_{6} q^{5} + 5 \beta_{3} q^{7} + (\beta_{5} - 21) q^{9} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{11} + 13 \beta_1 q^{13} + ( - 3 \beta_{7} + 24 \beta_{3}) q^{15} + 4 \beta_{5} q^{17} - 31 \beta_{2} q^{19} + ( - 5 \beta_{6} + 15 \beta_1) q^{21} + 18 \beta_{7} q^{23} + 163 q^{25} + ( - 15 \beta_{4} - 27 \beta_{2}) q^{27} + \beta_{6} q^{29} - 9 \beta_{3} q^{31} + ( - 3 \beta_{5} + 144) q^{33} + (60 \beta_{4} - 30 \beta_{2}) q^{35} + 103 \beta_1 q^{37} + (13 \beta_{7} + 13 \beta_{3}) q^{39} + 18 \beta_{5} q^{41} + 27 \beta_{2} q^{43} + ( - 21 \beta_{6} + 144 \beta_1) q^{45} + 12 \beta_{7} q^{47} + 43 q^{49} + (24 \beta_{4} - 108 \beta_{2}) q^{51} - 3 \beta_{6} q^{53} - 144 \beta_{3} q^{55} + ( - 31 \beta_{5} - 186) q^{57} + (114 \beta_{4} - 57 \beta_{2}) q^{59} - 139 \beta_1 q^{61} + (30 \beta_{7} - 105 \beta_{3}) q^{63} + 26 \beta_{5} q^{65} + 257 \beta_{2} q^{67} + ( - 18 \beta_{6} - 432 \beta_1) q^{69} + 6 \beta_{7} q^{71} + 422 q^{73} + 163 \beta_{4} q^{75} + 30 \beta_{6} q^{77} - 193 \beta_{3} q^{79} + ( - 42 \beta_{5} + 153) q^{81} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{83} + 576 \beta_1 q^{85} + ( - 3 \beta_{7} + 24 \beta_{3}) q^{87} + 22 \beta_{5} q^{89} - 130 \beta_{2} q^{91} + (9 \beta_{6} - 27 \beta_1) q^{93} + 186 \beta_{7} q^{95} - 1070 q^{97} + (126 \beta_{4} + 81 \beta_{2}) q^{99}+O(q^{100})$$ q + b4 * q^3 + b6 * q^5 + 5*b3 * q^7 + (b5 - 21) * q^9 + (-6*b4 + 3*b2) * q^11 + 13*b1 * q^13 + (-3*b7 + 24*b3) * q^15 + 4*b5 * q^17 - 31*b2 * q^19 + (-5*b6 + 15*b1) * q^21 + 18*b7 * q^23 + 163 * q^25 + (-15*b4 - 27*b2) * q^27 + b6 * q^29 - 9*b3 * q^31 + (-3*b5 + 144) * q^33 + (60*b4 - 30*b2) * q^35 + 103*b1 * q^37 + (13*b7 + 13*b3) * q^39 + 18*b5 * q^41 + 27*b2 * q^43 + (-21*b6 + 144*b1) * q^45 + 12*b7 * q^47 + 43 * q^49 + (24*b4 - 108*b2) * q^51 - 3*b6 * q^53 - 144*b3 * q^55 + (-31*b5 - 186) * q^57 + (114*b4 - 57*b2) * q^59 - 139*b1 * q^61 + (30*b7 - 105*b3) * q^63 + 26*b5 * q^65 + 257*b2 * q^67 + (-18*b6 - 432*b1) * q^69 + 6*b7 * q^71 + 422 * q^73 + 163*b4 * q^75 + 30*b6 * q^77 - 193*b3 * q^79 + (-42*b5 + 153) * q^81 + (-6*b4 + 3*b2) * q^83 + 576*b1 * q^85 + (-3*b7 + 24*b3) * q^87 + 22*b5 * q^89 - 130*b2 * q^91 + (9*b6 - 27*b1) * q^93 + 186*b7 * q^95 - 1070 * q^97 + (126*b4 + 81*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 168 q^{9}+O(q^{10})$$ 8 * q - 168 * q^9 $$8 q - 168 q^{9} + 1304 q^{25} + 1152 q^{33} + 344 q^{49} - 1488 q^{57} + 3376 q^{73} + 1224 q^{81} - 8560 q^{97}+O(q^{100})$$ 8 * q - 168 * q^9 + 1304 * q^25 + 1152 * q^33 + 344 * q^49 - 1488 * q^57 + 3376 * q^73 + 1224 * q^81 - 8560 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{24}^{6}$$ 2*v^6 $$\beta_{2}$$ $$=$$ $$-2\zeta_{24}^{6} + 4\zeta_{24}^{2}$$ -2*v^6 + 4*v^2 $$\beta_{3}$$ $$=$$ $$4\zeta_{24}^{4} - 2$$ 4*v^4 - 2 $$\beta_{4}$$ $$=$$ $$4\zeta_{24}^{7} - \zeta_{24}^{6} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}^{2} + 2\zeta_{24}$$ 4*v^7 - v^6 + 2*v^5 - 2*v^3 + 2*v^2 + 2*v $$\beta_{5}$$ $$=$$ $$12\zeta_{24}^{5} + 12\zeta_{24}^{3} - 12\zeta_{24}$$ 12*v^5 + 12*v^3 - 12*v $$\beta_{6}$$ $$=$$ $$-12\zeta_{24}^{5} + 12\zeta_{24}^{3} + 12\zeta_{24}$$ -12*v^5 + 12*v^3 + 12*v $$\beta_{7}$$ $$=$$ $$8\zeta_{24}^{7} - 4\zeta_{24}^{5} - 4\zeta_{24}^{3} - 4\zeta_{24}$$ 8*v^7 - 4*v^5 - 4*v^3 - 4*v
 $$\zeta_{24}$$ $$=$$ $$( -3\beta_{7} + \beta_{6} - \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48$$ (-3*b7 + b6 - b5 + 6*b4 - 3*b2) / 48 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 4$$ (b2 + b1) / 4 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{6} + \beta_{5} ) / 24$$ (b6 + b5) / 24 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{3} + 2 ) / 4$$ (b3 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( -3\beta_{7} - \beta_{6} + \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48$$ (-3*b7 - b6 + b5 + 6*b4 - 3*b2) / 48 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( 3\beta_{7} + \beta_{6} + \beta_{5} + 6\beta_{4} - 3\beta_{2} ) / 48$$ (3*b7 + b6 + b5 + 6*b4 - 3*b2) / 48

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
383.1
 0.258819 − 0.965926i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i
0 −1.73205 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
383.2 0 −1.73205 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
383.3 0 −1.73205 + 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 16.9706i 0
383.4 0 −1.73205 + 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 16.9706i 0
383.5 0 1.73205 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 16.9706i 0
383.6 0 1.73205 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 16.9706i 0
383.7 0 1.73205 + 4.89898i 0 −16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
383.8 0 1.73205 + 4.89898i 0 16.9706 0 17.3205i 0 −21.0000 + 16.9706i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h 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.f.b 8
3.b odd 2 1 inner 768.4.f.b 8
4.b odd 2 1 inner 768.4.f.b 8
8.b even 2 1 inner 768.4.f.b 8
8.d odd 2 1 inner 768.4.f.b 8
12.b even 2 1 inner 768.4.f.b 8
16.e even 4 1 48.4.c.b 4
16.e even 4 1 192.4.c.c 4
16.f odd 4 1 48.4.c.b 4
16.f odd 4 1 192.4.c.c 4
24.f even 2 1 inner 768.4.f.b 8
24.h odd 2 1 inner 768.4.f.b 8
48.i odd 4 1 48.4.c.b 4
48.i odd 4 1 192.4.c.c 4
48.k even 4 1 48.4.c.b 4
48.k even 4 1 192.4.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.b 4 16.e even 4 1
48.4.c.b 4 16.f odd 4 1
48.4.c.b 4 48.i odd 4 1
48.4.c.b 4 48.k even 4 1
192.4.c.c 4 16.e even 4 1
192.4.c.c 4 16.f odd 4 1
192.4.c.c 4 48.i odd 4 1
192.4.c.c 4 48.k even 4 1
768.4.f.b 8 1.a even 1 1 trivial
768.4.f.b 8 3.b odd 2 1 inner
768.4.f.b 8 4.b odd 2 1 inner
768.4.f.b 8 8.b even 2 1 inner
768.4.f.b 8 8.d odd 2 1 inner
768.4.f.b 8 12.b even 2 1 inner
768.4.f.b 8 24.f even 2 1 inner
768.4.f.b 8 24.h odd 2 1 inner

Hecke kernels

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

 $$T_{5}^{2} - 288$$ T5^2 - 288 $$T_{19}^{2} - 11532$$ T19^2 - 11532

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 42 T^{2} + 729)^{2}$$
$5$ $$(T^{2} - 288)^{4}$$
$7$ $$(T^{2} + 300)^{4}$$
$11$ $$(T^{2} + 864)^{4}$$
$13$ $$(T^{2} + 676)^{4}$$
$17$ $$(T^{2} + 4608)^{4}$$
$19$ $$(T^{2} - 11532)^{4}$$
$23$ $$(T^{2} - 31104)^{4}$$
$29$ $$(T^{2} - 288)^{4}$$
$31$ $$(T^{2} + 972)^{4}$$
$37$ $$(T^{2} + 42436)^{4}$$
$41$ $$(T^{2} + 93312)^{4}$$
$43$ $$(T^{2} - 8748)^{4}$$
$47$ $$(T^{2} - 13824)^{4}$$
$53$ $$(T^{2} - 2592)^{4}$$
$59$ $$(T^{2} + 311904)^{4}$$
$61$ $$(T^{2} + 77284)^{4}$$
$67$ $$(T^{2} - 792588)^{4}$$
$71$ $$(T^{2} - 3456)^{4}$$
$73$ $$(T - 422)^{8}$$
$79$ $$(T^{2} + 446988)^{4}$$
$83$ $$(T^{2} + 864)^{4}$$
$89$ $$(T^{2} + 139392)^{4}$$
$97$ $$(T + 1070)^{8}$$