Properties

Label 768.4.d.f
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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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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 384)
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.

\(f(q)\) \(=\) \( q - 3 i q^{3} + 4 i q^{5} - 10 q^{7} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 i q^{3} + 4 i q^{5} - 10 q^{7} - 9 q^{9} + 4 i q^{11} - 26 i q^{13} + 12 q^{15} + 14 q^{17} + 8 i q^{19} + 30 i q^{21} - 148 q^{23} + 109 q^{25} + 27 i q^{27} - 72 i q^{29} + 18 q^{31} + 12 q^{33} - 40 i q^{35} + 262 i q^{37} - 78 q^{39} + 378 q^{41} + 432 i q^{43} - 36 i q^{45} + 148 q^{47} - 243 q^{49} - 42 i q^{51} + 360 i q^{53} - 16 q^{55} + 24 q^{57} + 428 i q^{59} + 442 i q^{61} + 90 q^{63} + 104 q^{65} - 692 i q^{67} + 444 i q^{69} - 540 q^{71} + 1018 q^{73} - 327 i q^{75} - 40 i q^{77} + 386 q^{79} + 81 q^{81} + 108 i q^{83} + 56 i q^{85} - 216 q^{87} + 382 q^{89} + 260 i q^{91} - 54 i q^{93} - 32 q^{95} + 298 q^{97} - 36 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 20 q^{7} - 18 q^{9} + 24 q^{15} + 28 q^{17} - 296 q^{23} + 218 q^{25} + 36 q^{31} + 24 q^{33} - 156 q^{39} + 756 q^{41} + 296 q^{47} - 486 q^{49} - 32 q^{55} + 48 q^{57} + 180 q^{63} + 208 q^{65} - 1080 q^{71} + 2036 q^{73} + 772 q^{79} + 162 q^{81} - 432 q^{87} + 764 q^{89} - 64 q^{95} + 596 q^{97}+O(q^{100}) \) 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 4.00000i 0 −10.0000 0 −9.00000 0
385.2 0 3.00000i 0 4.00000i 0 −10.0000 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.f 2
4.b odd 2 1 768.4.d.k 2
8.b even 2 1 inner 768.4.d.f 2
8.d odd 2 1 768.4.d.k 2
16.e even 4 1 384.4.a.b 1
16.e even 4 1 384.4.a.g yes 1
16.f odd 4 1 384.4.a.c yes 1
16.f odd 4 1 384.4.a.f yes 1
48.i odd 4 1 1152.4.a.f 1
48.i odd 4 1 1152.4.a.h 1
48.k even 4 1 1152.4.a.e 1
48.k even 4 1 1152.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.a.b 1 16.e even 4 1
384.4.a.c yes 1 16.f odd 4 1
384.4.a.f yes 1 16.f odd 4 1
384.4.a.g yes 1 16.e even 4 1
768.4.d.f 2 1.a even 1 1 trivial
768.4.d.f 2 8.b even 2 1 inner
768.4.d.k 2 4.b odd 2 1
768.4.d.k 2 8.d odd 2 1
1152.4.a.e 1 48.k even 4 1
1152.4.a.f 1 48.i odd 4 1
1152.4.a.g 1 48.k even 4 1
1152.4.a.h 1 48.i odd 4 1

Hecke kernels

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

\( T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7} + 10 \) 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
$5$ \( T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T + 10)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 676 \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T + 148)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 5184 \) Copy content Toggle raw display
$31$ \( (T - 18)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 68644 \) Copy content Toggle raw display
$41$ \( (T - 378)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 186624 \) Copy content Toggle raw display
$47$ \( (T - 148)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 129600 \) Copy content Toggle raw display
$59$ \( T^{2} + 183184 \) Copy content Toggle raw display
$61$ \( T^{2} + 195364 \) Copy content Toggle raw display
$67$ \( T^{2} + 478864 \) Copy content Toggle raw display
$71$ \( (T + 540)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1018)^{2} \) Copy content Toggle raw display
$79$ \( (T - 386)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11664 \) Copy content Toggle raw display
$89$ \( (T - 382)^{2} \) Copy content Toggle raw display
$97$ \( (T - 298)^{2} \) Copy content Toggle raw display
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