Properties

Label 768.6.d.r
Level $768$
Weight $6$
Character orbit 768.d
Analytic conductor $123.175$
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] = [768,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.385");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(123.174773616\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - 9 i q^{3} + 34 i q^{5} + 240 q^{7} - 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 i q^{3} + 34 i q^{5} + 240 q^{7} - 81 q^{9} + 124 i q^{11} + 46 i q^{13} + 306 q^{15} + 1954 q^{17} - 1924 i q^{19} - 2160 i q^{21} - 2840 q^{23} + 1969 q^{25} + 729 i q^{27} - 8922 i q^{29} - 4648 q^{31} + 1116 q^{33} + 8160 i q^{35} + 4362 i q^{37} + 414 q^{39} + 2886 q^{41} - 11332 i q^{43} - 2754 i q^{45} + 7008 q^{47} + 40793 q^{49} - 17586 i q^{51} + 22594 i q^{53} - 4216 q^{55} - 17316 q^{57} + 28 i q^{59} - 6386 i q^{61} - 19440 q^{63} - 1564 q^{65} - 39076 i q^{67} + 25560 i q^{69} + 54872 q^{71} - 21034 q^{73} - 17721 i q^{75} + 29760 i q^{77} + 26632 q^{79} + 6561 q^{81} + 56188 i q^{83} + 66436 i q^{85} - 80298 q^{87} - 64410 q^{89} + 11040 i q^{91} + 41832 i q^{93} + 65416 q^{95} - 116158 q^{97} - 10044 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 480 q^{7} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 480 q^{7} - 162 q^{9} + 612 q^{15} + 3908 q^{17} - 5680 q^{23} + 3938 q^{25} - 9296 q^{31} + 2232 q^{33} + 828 q^{39} + 5772 q^{41} + 14016 q^{47} + 81586 q^{49} - 8432 q^{55} - 34632 q^{57} - 38880 q^{63} - 3128 q^{65} + 109744 q^{71} - 42068 q^{73} + 53264 q^{79} + 13122 q^{81} - 160596 q^{87} - 128820 q^{89} + 130832 q^{95} - 232316 q^{97}+O(q^{100}) \) 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 9.00000i 0 34.0000i 0 240.000 0 −81.0000 0
385.2 0 9.00000i 0 34.0000i 0 240.000 0 −81.0000 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 768.6.d.r 2
4.b odd 2 1 768.6.d.a 2
8.b even 2 1 inner 768.6.d.r 2
8.d odd 2 1 768.6.d.a 2
16.e even 4 1 24.6.a.a 1
16.e even 4 1 192.6.a.n 1
16.f odd 4 1 48.6.a.d 1
16.f odd 4 1 192.6.a.f 1
48.i odd 4 1 72.6.a.e 1
48.i odd 4 1 576.6.a.k 1
48.k even 4 1 144.6.a.i 1
48.k even 4 1 576.6.a.l 1
80.i odd 4 1 600.6.f.f 2
80.q even 4 1 600.6.a.i 1
80.t odd 4 1 600.6.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.a 1 16.e even 4 1
48.6.a.d 1 16.f odd 4 1
72.6.a.e 1 48.i odd 4 1
144.6.a.i 1 48.k even 4 1
192.6.a.f 1 16.f odd 4 1
192.6.a.n 1 16.e even 4 1
576.6.a.k 1 48.i odd 4 1
576.6.a.l 1 48.k even 4 1
600.6.a.i 1 80.q even 4 1
600.6.f.f 2 80.i odd 4 1
600.6.f.f 2 80.t odd 4 1
768.6.d.a 2 4.b odd 2 1
768.6.d.a 2 8.d odd 2 1
768.6.d.r 2 1.a even 1 1 trivial
768.6.d.r 2 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 1156 \) Copy content Toggle raw display
\( T_{7} - 240 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 1156 \) Copy content Toggle raw display
$7$ \( (T - 240)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 15376 \) Copy content Toggle raw display
$13$ \( T^{2} + 2116 \) Copy content Toggle raw display
$17$ \( (T - 1954)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3701776 \) Copy content Toggle raw display
$23$ \( (T + 2840)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 79602084 \) Copy content Toggle raw display
$31$ \( (T + 4648)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 19027044 \) Copy content Toggle raw display
$41$ \( (T - 2886)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 128414224 \) Copy content Toggle raw display
$47$ \( (T - 7008)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 510488836 \) Copy content Toggle raw display
$59$ \( T^{2} + 784 \) Copy content Toggle raw display
$61$ \( T^{2} + 40780996 \) Copy content Toggle raw display
$67$ \( T^{2} + 1526933776 \) Copy content Toggle raw display
$71$ \( (T - 54872)^{2} \) Copy content Toggle raw display
$73$ \( (T + 21034)^{2} \) Copy content Toggle raw display
$79$ \( (T - 26632)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3157091344 \) Copy content Toggle raw display
$89$ \( (T + 64410)^{2} \) Copy content Toggle raw display
$97$ \( (T + 116158)^{2} \) Copy content Toggle raw display
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