Properties

Label 768.2.d.e
Level $768$
Weight $2$
Character orbit 768.d
Analytic conductor $6.133$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(385,768)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.385");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.d (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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{3} + 2 i q^{5} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + 2 i q^{5} - q^{9} - 4 i q^{11} - 2 i q^{13} + 2 q^{15} + 2 q^{17} - 4 i q^{19} + 8 q^{23} + q^{25} + i q^{27} + 6 i q^{29} + 8 q^{31} - 4 q^{33} - 6 i q^{37} - 2 q^{39} + 6 q^{41} - 4 i q^{43} - 2 i q^{45} - 7 q^{49} - 2 i q^{51} + 2 i q^{53} + 8 q^{55} - 4 q^{57} - 4 i q^{59} - 2 i q^{61} + 4 q^{65} - 4 i q^{67} - 8 i q^{69} - 8 q^{71} - 10 q^{73} - i q^{75} - 8 q^{79} + q^{81} - 4 i q^{83} + 4 i q^{85} + 6 q^{87} + 6 q^{89} - 8 i q^{93} + 8 q^{95} + 2 q^{97} + 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 4 q^{15} + 4 q^{17} + 16 q^{23} + 2 q^{25} + 16 q^{31} - 8 q^{33} - 4 q^{39} + 12 q^{41} - 14 q^{49} + 16 q^{55} - 8 q^{57} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 16 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
1.00000i
1.00000i
0 1.00000i 0 2.00000i 0 0 0 −1.00000 0
385.2 0 1.00000i 0 2.00000i 0 0 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.d.e 2
3.b odd 2 1 2304.2.d.i 2
4.b odd 2 1 768.2.d.d 2
8.b even 2 1 inner 768.2.d.e 2
8.d odd 2 1 768.2.d.d 2
12.b even 2 1 2304.2.d.k 2
16.e even 4 1 24.2.a.a 1
16.e even 4 1 192.2.a.d 1
16.f odd 4 1 48.2.a.a 1
16.f odd 4 1 192.2.a.b 1
24.f even 2 1 2304.2.d.k 2
24.h odd 2 1 2304.2.d.i 2
48.i odd 4 1 72.2.a.a 1
48.i odd 4 1 576.2.a.d 1
48.k even 4 1 144.2.a.b 1
48.k even 4 1 576.2.a.b 1
80.i odd 4 1 600.2.f.e 2
80.i odd 4 1 4800.2.f.d 2
80.j even 4 1 1200.2.f.b 2
80.j even 4 1 4800.2.f.bg 2
80.k odd 4 1 1200.2.a.d 1
80.k odd 4 1 4800.2.a.cc 1
80.q even 4 1 600.2.a.h 1
80.q even 4 1 4800.2.a.q 1
80.s even 4 1 1200.2.f.b 2
80.s even 4 1 4800.2.f.bg 2
80.t odd 4 1 600.2.f.e 2
80.t odd 4 1 4800.2.f.d 2
112.j even 4 1 2352.2.a.i 1
112.j even 4 1 9408.2.a.cc 1
112.l odd 4 1 1176.2.a.i 1
112.l odd 4 1 9408.2.a.h 1
112.u odd 12 2 2352.2.q.l 2
112.v even 12 2 2352.2.q.r 2
112.w even 12 2 1176.2.q.i 2
112.x odd 12 2 1176.2.q.a 2
144.u even 12 2 1296.2.i.e 2
144.v odd 12 2 1296.2.i.m 2
144.w odd 12 2 648.2.i.b 2
144.x even 12 2 648.2.i.g 2
176.i even 4 1 5808.2.a.s 1
176.l odd 4 1 2904.2.a.c 1
208.m odd 4 1 4056.2.c.e 2
208.o odd 4 1 8112.2.a.be 1
208.p even 4 1 4056.2.a.i 1
208.r odd 4 1 4056.2.c.e 2
240.t even 4 1 3600.2.a.v 1
240.z odd 4 1 3600.2.f.r 2
240.bb even 4 1 1800.2.f.c 2
240.bd odd 4 1 3600.2.f.r 2
240.bf even 4 1 1800.2.f.c 2
240.bm odd 4 1 1800.2.a.m 1
272.r even 4 1 6936.2.a.p 1
304.j odd 4 1 8664.2.a.j 1
336.v odd 4 1 7056.2.a.q 1
336.y even 4 1 3528.2.a.d 1
336.bo even 12 2 3528.2.s.y 2
336.bt odd 12 2 3528.2.s.j 2
528.x even 4 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 16.e even 4 1
48.2.a.a 1 16.f odd 4 1
72.2.a.a 1 48.i odd 4 1
144.2.a.b 1 48.k even 4 1
192.2.a.b 1 16.f odd 4 1
192.2.a.d 1 16.e even 4 1
576.2.a.b 1 48.k even 4 1
576.2.a.d 1 48.i odd 4 1
600.2.a.h 1 80.q even 4 1
600.2.f.e 2 80.i odd 4 1
600.2.f.e 2 80.t odd 4 1
648.2.i.b 2 144.w odd 12 2
648.2.i.g 2 144.x even 12 2
768.2.d.d 2 4.b odd 2 1
768.2.d.d 2 8.d odd 2 1
768.2.d.e 2 1.a even 1 1 trivial
768.2.d.e 2 8.b even 2 1 inner
1176.2.a.i 1 112.l odd 4 1
1176.2.q.a 2 112.x odd 12 2
1176.2.q.i 2 112.w even 12 2
1200.2.a.d 1 80.k odd 4 1
1200.2.f.b 2 80.j even 4 1
1200.2.f.b 2 80.s even 4 1
1296.2.i.e 2 144.u even 12 2
1296.2.i.m 2 144.v odd 12 2
1800.2.a.m 1 240.bm odd 4 1
1800.2.f.c 2 240.bb even 4 1
1800.2.f.c 2 240.bf even 4 1
2304.2.d.i 2 3.b odd 2 1
2304.2.d.i 2 24.h odd 2 1
2304.2.d.k 2 12.b even 2 1
2304.2.d.k 2 24.f even 2 1
2352.2.a.i 1 112.j even 4 1
2352.2.q.l 2 112.u odd 12 2
2352.2.q.r 2 112.v even 12 2
2904.2.a.c 1 176.l odd 4 1
3528.2.a.d 1 336.y even 4 1
3528.2.s.j 2 336.bt odd 12 2
3528.2.s.y 2 336.bo even 12 2
3600.2.a.v 1 240.t even 4 1
3600.2.f.r 2 240.z odd 4 1
3600.2.f.r 2 240.bd odd 4 1
4056.2.a.i 1 208.p even 4 1
4056.2.c.e 2 208.m odd 4 1
4056.2.c.e 2 208.r odd 4 1
4800.2.a.q 1 80.q even 4 1
4800.2.a.cc 1 80.k odd 4 1
4800.2.f.d 2 80.i odd 4 1
4800.2.f.d 2 80.t odd 4 1
4800.2.f.bg 2 80.j even 4 1
4800.2.f.bg 2 80.s even 4 1
5808.2.a.s 1 176.i even 4 1
6936.2.a.p 1 272.r even 4 1
7056.2.a.q 1 336.v odd 4 1
8112.2.a.be 1 208.o odd 4 1
8664.2.a.j 1 304.j odd 4 1
8712.2.a.u 1 528.x even 4 1
9408.2.a.h 1 112.l odd 4 1
9408.2.a.cc 1 112.j even 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
show more
show less