Properties

Label 768.4.c.f
Level $768$
Weight $4$
Character orbit 768.c
Analytic conductor $45.313$
Analytic rank $1$
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(767,768)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.767");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.c (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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-26}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 26 \) 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 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 \(\beta = \sqrt{-26}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + 2 \beta q^{5} + 2 \beta q^{7} + (2 \beta - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + 2 \beta q^{5} + 2 \beta q^{7} + (2 \beta - 25) q^{9} - 46 q^{11} - 44 q^{13} + (2 \beta - 52) q^{15} - 4 \beta q^{17} - 2 \beta q^{19} + (2 \beta - 52) q^{21} + 88 q^{23} + 21 q^{25} + ( - 23 \beta - 77) q^{27} - 50 \beta q^{29} - 42 \beta q^{31} + ( - 46 \beta - 46) q^{33} - 104 q^{35} - 332 q^{37} + ( - 44 \beta - 44) q^{39} + 96 \beta q^{41} + 46 \beta q^{43} + ( - 50 \beta - 104) q^{45} + 384 q^{47} + 239 q^{49} + ( - 4 \beta + 104) q^{51} + 90 \beta q^{53} - 92 \beta q^{55} + ( - 2 \beta + 52) q^{57} + 630 q^{59} - 236 q^{61} + ( - 50 \beta - 104) q^{63} - 88 \beta q^{65} + 10 \beta q^{67} + (88 \beta + 88) q^{69} - 680 q^{71} - 422 q^{73} + (21 \beta + 21) q^{75} - 92 \beta q^{77} - 146 \beta q^{79} + ( - 100 \beta + 521) q^{81} - 186 q^{83} + 208 q^{85} + ( - 50 \beta + 1300) q^{87} + 188 \beta q^{89} - 88 \beta q^{91} + ( - 42 \beta + 1092) q^{93} + 104 q^{95} - 1062 q^{97} + ( - 92 \beta + 1150) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 50 q^{9} - 92 q^{11} - 88 q^{13} - 104 q^{15} - 104 q^{21} + 176 q^{23} + 42 q^{25} - 154 q^{27} - 92 q^{33} - 208 q^{35} - 664 q^{37} - 88 q^{39} - 208 q^{45} + 768 q^{47} + 478 q^{49} + 208 q^{51} + 104 q^{57} + 1260 q^{59} - 472 q^{61} - 208 q^{63} + 176 q^{69} - 1360 q^{71} - 844 q^{73} + 42 q^{75} + 1042 q^{81} - 372 q^{83} + 416 q^{85} + 2600 q^{87} + 2184 q^{93} + 208 q^{95} - 2124 q^{97} + 2300 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
767.1
5.09902i
5.09902i
0 1.00000 5.09902i 0 10.1980i 0 10.1980i 0 −25.0000 10.1980i 0
767.2 0 1.00000 + 5.09902i 0 10.1980i 0 10.1980i 0 −25.0000 + 10.1980i 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
12.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.c.f 2
3.b odd 2 1 768.4.c.d 2
4.b odd 2 1 768.4.c.d 2
8.b even 2 1 768.4.c.e 2
8.d odd 2 1 768.4.c.g 2
12.b even 2 1 inner 768.4.c.f 2
16.e even 4 2 384.4.f.e 4
16.f odd 4 2 384.4.f.f yes 4
24.f even 2 1 768.4.c.e 2
24.h odd 2 1 768.4.c.g 2
48.i odd 4 2 384.4.f.f yes 4
48.k even 4 2 384.4.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.e 4 16.e even 4 2
384.4.f.e 4 48.k even 4 2
384.4.f.f yes 4 16.f odd 4 2
384.4.f.f yes 4 48.i odd 4 2
768.4.c.d 2 3.b odd 2 1
768.4.c.d 2 4.b odd 2 1
768.4.c.e 2 8.b even 2 1
768.4.c.e 2 24.f even 2 1
768.4.c.f 2 1.a even 1 1 trivial
768.4.c.f 2 12.b even 2 1 inner
768.4.c.g 2 8.d odd 2 1
768.4.c.g 2 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 104 \) Copy content Toggle raw display
\( T_{7}^{2} + 104 \) Copy content Toggle raw display
\( T_{11} + 46 \) Copy content Toggle raw display
\( T_{13} + 44 \) Copy content Toggle raw display
\( T_{23} - 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} + 104 \) Copy content Toggle raw display
$7$ \( T^{2} + 104 \) Copy content Toggle raw display
$11$ \( (T + 46)^{2} \) Copy content Toggle raw display
$13$ \( (T + 44)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 416 \) Copy content Toggle raw display
$19$ \( T^{2} + 104 \) Copy content Toggle raw display
$23$ \( (T - 88)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 65000 \) Copy content Toggle raw display
$31$ \( T^{2} + 45864 \) Copy content Toggle raw display
$37$ \( (T + 332)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 239616 \) Copy content Toggle raw display
$43$ \( T^{2} + 55016 \) Copy content Toggle raw display
$47$ \( (T - 384)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 210600 \) Copy content Toggle raw display
$59$ \( (T - 630)^{2} \) Copy content Toggle raw display
$61$ \( (T + 236)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2600 \) Copy content Toggle raw display
$71$ \( (T + 680)^{2} \) Copy content Toggle raw display
$73$ \( (T + 422)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 554216 \) Copy content Toggle raw display
$83$ \( (T + 186)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 918944 \) Copy content Toggle raw display
$97$ \( (T + 1062)^{2} \) Copy content Toggle raw display
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