# Properties

 Label 768.2.a.a Level $768$ Weight $2$ Character orbit 768.a Self dual yes Analytic conductor $6.133$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,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, 2, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$6.13251087523$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 2 * q^5 + 2 * q^7 + q^9 $$q - q^{3} - 2 q^{5} + 2 q^{7} + q^{9} - 4 q^{13} + 2 q^{15} - 2 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} - q^{25} - q^{27} - 6 q^{29} + 2 q^{31} - 4 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} + 4 q^{43} - 2 q^{45} - 12 q^{47} - 3 q^{49} + 2 q^{51} - 6 q^{53} - 4 q^{57} - 4 q^{59} + 2 q^{63} + 8 q^{65} - 12 q^{67} + 4 q^{69} - 12 q^{71} + 6 q^{73} + q^{75} + 10 q^{79} + q^{81} + 16 q^{83} + 4 q^{85} + 6 q^{87} + 10 q^{89} - 8 q^{91} - 2 q^{93} - 8 q^{95} - 2 q^{97}+O(q^{100})$$ q - q^3 - 2 * q^5 + 2 * q^7 + q^9 - 4 * q^13 + 2 * q^15 - 2 * q^17 + 4 * q^19 - 2 * q^21 - 4 * q^23 - q^25 - q^27 - 6 * q^29 + 2 * q^31 - 4 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 + 4 * q^43 - 2 * q^45 - 12 * q^47 - 3 * q^49 + 2 * q^51 - 6 * q^53 - 4 * q^57 - 4 * q^59 + 2 * q^63 + 8 * q^65 - 12 * q^67 + 4 * q^69 - 12 * q^71 + 6 * q^73 + q^75 + 10 * q^79 + q^81 + 16 * q^83 + 4 * q^85 + 6 * q^87 + 10 * q^89 - 8 * q^91 - 2 * q^93 - 8 * q^95 - 2 * 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
 0
0 −1.00000 0 −2.00000 0 2.00000 0 1.00000 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 768.2.a.a 1
3.b odd 2 1 2304.2.a.o 1
4.b odd 2 1 768.2.a.e 1
8.b even 2 1 768.2.a.h 1
8.d odd 2 1 768.2.a.d 1
12.b even 2 1 2304.2.a.l 1
16.e even 4 2 24.2.d.a 2
16.f odd 4 2 96.2.d.a 2
24.f even 2 1 2304.2.a.b 1
24.h odd 2 1 2304.2.a.e 1
48.i odd 4 2 72.2.d.b 2
48.k even 4 2 288.2.d.b 2
80.i odd 4 2 600.2.d.b 2
80.j even 4 2 2400.2.d.c 2
80.k odd 4 2 2400.2.k.a 2
80.q even 4 2 600.2.k.b 2
80.s even 4 2 2400.2.d.b 2
80.t odd 4 2 600.2.d.c 2
112.j even 4 2 4704.2.c.a 2
112.l odd 4 2 1176.2.c.a 2
144.u even 12 4 2592.2.r.g 4
144.v odd 12 4 2592.2.r.f 4
144.w odd 12 4 648.2.n.c 4
144.x even 12 4 648.2.n.k 4
240.t even 4 2 7200.2.k.d 2
240.z odd 4 2 7200.2.d.g 2
240.bb even 4 2 1800.2.d.i 2
240.bd odd 4 2 7200.2.d.d 2
240.bf even 4 2 1800.2.d.b 2
240.bm odd 4 2 1800.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.d.a 2 16.e even 4 2
72.2.d.b 2 48.i odd 4 2
96.2.d.a 2 16.f odd 4 2
288.2.d.b 2 48.k even 4 2
600.2.d.b 2 80.i odd 4 2
600.2.d.c 2 80.t odd 4 2
600.2.k.b 2 80.q even 4 2
648.2.n.c 4 144.w odd 12 4
648.2.n.k 4 144.x even 12 4
768.2.a.a 1 1.a even 1 1 trivial
768.2.a.d 1 8.d odd 2 1
768.2.a.e 1 4.b odd 2 1
768.2.a.h 1 8.b even 2 1
1176.2.c.a 2 112.l odd 4 2
1800.2.d.b 2 240.bf even 4 2
1800.2.d.i 2 240.bb even 4 2
1800.2.k.a 2 240.bm odd 4 2
2304.2.a.b 1 24.f even 2 1
2304.2.a.e 1 24.h odd 2 1
2304.2.a.l 1 12.b even 2 1
2304.2.a.o 1 3.b odd 2 1
2400.2.d.b 2 80.s even 4 2
2400.2.d.c 2 80.j even 4 2
2400.2.k.a 2 80.k odd 4 2
2592.2.r.f 4 144.v odd 12 4
2592.2.r.g 4 144.u even 12 4
4704.2.c.a 2 112.j even 4 2
7200.2.d.d 2 240.bd odd 4 2
7200.2.d.g 2 240.z odd 4 2
7200.2.k.d 2 240.t even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 2$$ T7 - 2 $$T_{11}$$ T11 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T + 2$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T + 6$$
$31$ $$T - 2$$
$37$ $$T + 8$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T + 4$$
$61$ $$T$$
$67$ $$T + 12$$
$71$ $$T + 12$$
$73$ $$T - 6$$
$79$ $$T - 10$$
$83$ $$T - 16$$
$89$ $$T - 10$$
$97$ $$T + 2$$