Properties

Label 768.4.d.j
Level $768$
Weight $4$
Character orbit 768.d
Analytic conductor $45.313$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.385");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(45.3134668844\)
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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 3 i q^{3} + 18 i q^{5} + 8 q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 i q^{3} + 18 i q^{5} + 8 q^{7} - 9 q^{9} + 36 i q^{11} - 10 i q^{13} + 54 q^{15} + 18 q^{17} + 100 i q^{19} - 24 i q^{21} + 72 q^{23} - 199 q^{25} + 27 i q^{27} - 234 i q^{29} + 16 q^{31} + 108 q^{33} + 144 i q^{35} + 226 i q^{37} - 30 q^{39} - 90 q^{41} + 452 i q^{43} - 162 i q^{45} - 432 q^{47} - 279 q^{49} - 54 i q^{51} - 414 i q^{53} - 648 q^{55} + 300 q^{57} - 684 i q^{59} + 422 i q^{61} - 72 q^{63} + 180 q^{65} - 332 i q^{67} - 216 i q^{69} - 360 q^{71} - 26 q^{73} + 597 i q^{75} + 288 i q^{77} - 512 q^{79} + 81 q^{81} + 1188 i q^{83} + 324 i q^{85} - 702 q^{87} + 630 q^{89} - 80 i q^{91} - 48 i q^{93} - 1800 q^{95} - 1054 q^{97} - 324 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} - 18 q^{9} + 108 q^{15} + 36 q^{17} + 144 q^{23} - 398 q^{25} + 32 q^{31} + 216 q^{33} - 60 q^{39} - 180 q^{41} - 864 q^{47} - 558 q^{49} - 1296 q^{55} + 600 q^{57} - 144 q^{63} + 360 q^{65} - 720 q^{71} - 52 q^{73} - 1024 q^{79} + 162 q^{81} - 1404 q^{87} + 1260 q^{89} - 3600 q^{95} - 2108 q^{97}+O(q^{100}) \) 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 3.00000i 0 18.0000i 0 8.00000 0 −9.00000 0
385.2 0 3.00000i 0 18.0000i 0 8.00000 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
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 768.4.d.j 2
4.b odd 2 1 768.4.d.g 2
8.b even 2 1 inner 768.4.d.j 2
8.d odd 2 1 768.4.d.g 2
16.e even 4 1 48.4.a.a 1
16.e even 4 1 192.4.a.l 1
16.f odd 4 1 12.4.a.a 1
16.f odd 4 1 192.4.a.f 1
48.i odd 4 1 144.4.a.g 1
48.i odd 4 1 576.4.a.a 1
48.k even 4 1 36.4.a.a 1
48.k even 4 1 576.4.a.b 1
80.i odd 4 1 1200.4.f.d 2
80.j even 4 1 300.4.d.e 2
80.k odd 4 1 300.4.a.b 1
80.q even 4 1 1200.4.a.be 1
80.s even 4 1 300.4.d.e 2
80.t odd 4 1 1200.4.f.d 2
112.j even 4 1 588.4.a.c 1
112.l odd 4 1 2352.4.a.bk 1
112.u odd 12 2 588.4.i.d 2
112.v even 12 2 588.4.i.e 2
144.u even 12 2 324.4.e.a 2
144.v odd 12 2 324.4.e.h 2
176.i even 4 1 1452.4.a.d 1
208.l even 4 1 2028.4.b.c 2
208.o odd 4 1 2028.4.a.c 1
208.s even 4 1 2028.4.b.c 2
240.t even 4 1 900.4.a.g 1
240.z odd 4 1 900.4.d.c 2
240.bd odd 4 1 900.4.d.c 2
336.v odd 4 1 1764.4.a.b 1
336.br odd 12 2 1764.4.k.o 2
336.bu even 12 2 1764.4.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 16.f odd 4 1
36.4.a.a 1 48.k even 4 1
48.4.a.a 1 16.e even 4 1
144.4.a.g 1 48.i odd 4 1
192.4.a.f 1 16.f odd 4 1
192.4.a.l 1 16.e even 4 1
300.4.a.b 1 80.k odd 4 1
300.4.d.e 2 80.j even 4 1
300.4.d.e 2 80.s even 4 1
324.4.e.a 2 144.u even 12 2
324.4.e.h 2 144.v odd 12 2
576.4.a.a 1 48.i odd 4 1
576.4.a.b 1 48.k even 4 1
588.4.a.c 1 112.j even 4 1
588.4.i.d 2 112.u odd 12 2
588.4.i.e 2 112.v even 12 2
768.4.d.g 2 4.b odd 2 1
768.4.d.g 2 8.d odd 2 1
768.4.d.j 2 1.a even 1 1 trivial
768.4.d.j 2 8.b even 2 1 inner
900.4.a.g 1 240.t even 4 1
900.4.d.c 2 240.z odd 4 1
900.4.d.c 2 240.bd odd 4 1
1200.4.a.be 1 80.q even 4 1
1200.4.f.d 2 80.i odd 4 1
1200.4.f.d 2 80.t odd 4 1
1452.4.a.d 1 176.i even 4 1
1764.4.a.b 1 336.v odd 4 1
1764.4.k.b 2 336.bu even 12 2
1764.4.k.o 2 336.br odd 12 2
2028.4.a.c 1 208.o odd 4 1
2028.4.b.c 2 208.l even 4 1
2028.4.b.c 2 208.s even 4 1
2352.4.a.bk 1 112.l odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{7} - 8 \) 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
$5$ \( T^{2} + 324 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{2} + 100 \) Copy content Toggle raw display
$17$ \( (T - 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10000 \) Copy content Toggle raw display
$23$ \( (T - 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 54756 \) Copy content Toggle raw display
$31$ \( (T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51076 \) Copy content Toggle raw display
$41$ \( (T + 90)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 204304 \) Copy content Toggle raw display
$47$ \( (T + 432)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 171396 \) Copy content Toggle raw display
$59$ \( T^{2} + 467856 \) Copy content Toggle raw display
$61$ \( T^{2} + 178084 \) Copy content Toggle raw display
$67$ \( T^{2} + 110224 \) Copy content Toggle raw display
$71$ \( (T + 360)^{2} \) Copy content Toggle raw display
$73$ \( (T + 26)^{2} \) Copy content Toggle raw display
$79$ \( (T + 512)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1411344 \) Copy content Toggle raw display
$89$ \( (T - 630)^{2} \) Copy content Toggle raw display
$97$ \( (T + 1054)^{2} \) Copy content Toggle raw display
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