Properties

Label 768.3.h.a
Level $768$
Weight $3$
Character orbit 768.h
Analytic conductor $20.926$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(641,768)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.641");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.h (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: \(20.9264843029\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 i q^{3} - 2 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - 2 q^{7} - 9 q^{9} + 22 i q^{13} - 26 i q^{19} - 6 i q^{21} - 25 q^{25} - 27 i q^{27} - 46 q^{31} + 26 i q^{37} - 66 q^{39} - 22 i q^{43} - 45 q^{49} + 78 q^{57} - 74 i q^{61} + 18 q^{63} - 122 i q^{67} + 46 q^{73} - 75 i q^{75} - 142 q^{79} + 81 q^{81} - 44 i q^{91} - 138 i q^{93} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 18 q^{9} - 50 q^{25} - 92 q^{31} - 132 q^{39} - 90 q^{49} + 156 q^{57} + 36 q^{63} + 92 q^{73} - 284 q^{79} + 162 q^{81} + 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.00000i
1.00000i
0 3.00000i 0 0 0 −2.00000 0 −9.00000 0
641.2 0 3.00000i 0 0 0 −2.00000 0 −9.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
8.b 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 768.3.h.a 2
3.b odd 2 1 CM 768.3.h.a 2
4.b odd 2 1 768.3.h.b 2
8.b even 2 1 inner 768.3.h.a 2
8.d odd 2 1 768.3.h.b 2
12.b even 2 1 768.3.h.b 2
16.e even 4 1 12.3.c.a 1
16.e even 4 1 192.3.e.b 1
16.f odd 4 1 48.3.e.a 1
16.f odd 4 1 192.3.e.a 1
24.f even 2 1 768.3.h.b 2
24.h odd 2 1 inner 768.3.h.a 2
48.i odd 4 1 12.3.c.a 1
48.i odd 4 1 192.3.e.b 1
48.k even 4 1 48.3.e.a 1
48.k even 4 1 192.3.e.a 1
80.i odd 4 1 300.3.b.a 2
80.j even 4 1 1200.3.c.c 2
80.k odd 4 1 1200.3.l.b 1
80.q even 4 1 300.3.g.b 1
80.s even 4 1 1200.3.c.c 2
80.t odd 4 1 300.3.b.a 2
112.l odd 4 1 588.3.c.c 1
112.w even 12 2 588.3.p.c 2
112.x odd 12 2 588.3.p.b 2
144.u even 12 2 1296.3.q.b 2
144.v odd 12 2 1296.3.q.b 2
144.w odd 12 2 324.3.g.b 2
144.x even 12 2 324.3.g.b 2
176.l odd 4 1 1452.3.e.b 1
240.t even 4 1 1200.3.l.b 1
240.z odd 4 1 1200.3.c.c 2
240.bb even 4 1 300.3.b.a 2
240.bd odd 4 1 1200.3.c.c 2
240.bf even 4 1 300.3.b.a 2
240.bm odd 4 1 300.3.g.b 1
336.y even 4 1 588.3.c.c 1
336.bo even 12 2 588.3.p.b 2
336.bt odd 12 2 588.3.p.c 2
528.x even 4 1 1452.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 16.e even 4 1
12.3.c.a 1 48.i odd 4 1
48.3.e.a 1 16.f odd 4 1
48.3.e.a 1 48.k even 4 1
192.3.e.a 1 16.f odd 4 1
192.3.e.a 1 48.k even 4 1
192.3.e.b 1 16.e even 4 1
192.3.e.b 1 48.i odd 4 1
300.3.b.a 2 80.i odd 4 1
300.3.b.a 2 80.t odd 4 1
300.3.b.a 2 240.bb even 4 1
300.3.b.a 2 240.bf even 4 1
300.3.g.b 1 80.q even 4 1
300.3.g.b 1 240.bm odd 4 1
324.3.g.b 2 144.w odd 12 2
324.3.g.b 2 144.x even 12 2
588.3.c.c 1 112.l odd 4 1
588.3.c.c 1 336.y even 4 1
588.3.p.b 2 112.x odd 12 2
588.3.p.b 2 336.bo even 12 2
588.3.p.c 2 112.w even 12 2
588.3.p.c 2 336.bt odd 12 2
768.3.h.a 2 1.a even 1 1 trivial
768.3.h.a 2 3.b odd 2 1 CM
768.3.h.a 2 8.b even 2 1 inner
768.3.h.a 2 24.h odd 2 1 inner
768.3.h.b 2 4.b odd 2 1
768.3.h.b 2 8.d odd 2 1
768.3.h.b 2 12.b even 2 1
768.3.h.b 2 24.f even 2 1
1200.3.c.c 2 80.j even 4 1
1200.3.c.c 2 80.s even 4 1
1200.3.c.c 2 240.z odd 4 1
1200.3.c.c 2 240.bd odd 4 1
1200.3.l.b 1 80.k odd 4 1
1200.3.l.b 1 240.t even 4 1
1296.3.q.b 2 144.u even 12 2
1296.3.q.b 2 144.v odd 12 2
1452.3.e.b 1 176.l odd 4 1
1452.3.e.b 1 528.x even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 484 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 676 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 46)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 676 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 484 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 5476 \) Copy content Toggle raw display
$67$ \( T^{2} + 14884 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 46)^{2} \) Copy content Toggle raw display
$79$ \( (T + 142)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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