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Properties

Label 768.4.c.o
Level $768$
Weight $4$
Character orbit 768.c
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{-14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{2} - 3 \beta_1) q^{7} + ( - 3 \beta_{2} + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{3} + ( - \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{2} - 3 \beta_1) q^{7} + ( - 3 \beta_{2} + 15) q^{9} + ( - 2 \beta_{3} - \beta_1 + 32) q^{11} + (8 \beta_{3} + 4 \beta_1 - 28) q^{13} + (4 \beta_{3} - 6 \beta_{2} - 5 \beta_1 - 24) q^{15} + ( - 2 \beta_{2} + 16 \beta_1) q^{17} + ( - 16 \beta_{2} - 7 \beta_1) q^{19} + ( - 8 \beta_{3} - 9 \beta_{2} + \cdots - 36) q^{21}+ \cdots + ( - 30 \beta_{3} - 96 \beta_{2} + \cdots + 480) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 60 q^{9} + 128 q^{11} - 112 q^{13} - 96 q^{15} - 144 q^{21} + 320 q^{23} - 108 q^{25} + 168 q^{33} - 128 q^{35} - 496 q^{37} - 672 q^{39} - 672 q^{45} - 388 q^{49} + 768 q^{51} - 336 q^{57} - 112 q^{61} + 1344 q^{63} - 672 q^{69} + 320 q^{71} + 616 q^{73} - 2688 q^{75} - 1116 q^{81} - 2688 q^{83} + 2624 q^{85} - 672 q^{87} + 336 q^{93} - 4928 q^{95} + 2856 q^{97} + 1920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{3} - 8\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + 8\nu + 10 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} - 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 0.646084i
0.646084i
3.09557i
− 3.09557i
0 −4.58258 − 2.44949i 0 − 2.31464i 0 − 29.6636i 0 15.0000 + 22.4499i 0
767.2 0 −4.58258 + 2.44949i 0 2.31464i 0 29.6636i 0 15.0000 − 22.4499i 0
767.3 0 4.58258 − 2.44949i 0 − 17.2813i 0 0.269691i 0 15.0000 − 22.4499i 0
767.4 0 4.58258 + 2.44949i 0 17.2813i 0 − 0.269691i 0 15.0000 + 22.4499i 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.o 4
3.b odd 2 1 768.4.c.m 4
4.b odd 2 1 768.4.c.m 4
8.b even 2 1 768.4.c.n 4
8.d odd 2 1 768.4.c.p 4
12.b even 2 1 inner 768.4.c.o 4
16.e even 4 2 384.4.f.g ✓ 8
16.f odd 4 2 384.4.f.h yes 8
24.f even 2 1 768.4.c.n 4
24.h odd 2 1 768.4.c.p 4
48.i odd 4 2 384.4.f.h yes 8
48.k even 4 2 384.4.f.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.f.g ✓ 8 16.e even 4 2
384.4.f.g ✓ 8 48.k even 4 2
384.4.f.h yes 8 16.f odd 4 2
384.4.f.h yes 8 48.i odd 4 2
768.4.c.m 4 3.b odd 2 1
768.4.c.m 4 4.b odd 2 1
768.4.c.n 4 8.b even 2 1
768.4.c.n 4 24.f even 2 1
768.4.c.o 4 1.a even 1 1 trivial
768.4.c.o 4 12.b even 2 1 inner
768.4.c.p 4 8.d odd 2 1
768.4.c.p 4 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 \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{4} + 304T_{5}^{2} + 1600 \) Copy content Toggle raw display
\( T_{7}^{4} + 880T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} - 64T_{11} + 940 \) Copy content Toggle raw display
\( T_{13}^{2} + 56T_{13} - 560 \) Copy content Toggle raw display
\( T_{23}^{2} - 160T_{23} + 5056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 30T^{2} + 729 \) Copy content Toggle raw display
$5$ \( T^{4} + 304T^{2} + 1600 \) Copy content Toggle raw display
$7$ \( T^{4} + 880T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} - 64 T + 940)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 56 T - 560)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 12736 T^{2} + 35046400 \) Copy content Toggle raw display
$19$ \( T^{4} + 31024 T^{2} + 173185600 \) Copy content Toggle raw display
$23$ \( (T^{2} - 160 T + 5056)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 68656 T^{2} + 621006400 \) Copy content Toggle raw display
$31$ \( T^{4} + 6384 T^{2} + 705600 \) Copy content Toggle raw display
$37$ \( (T^{2} + 248 T - 50480)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 76800 T^{2} + 681836544 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1873850944 \) Copy content Toggle raw display
$47$ \( (T^{2} - 86016)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 462000 T^{2} + 17640000 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 56 T - 387632)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19993960000 \) Copy content Toggle raw display
$71$ \( (T^{2} - 160 T - 833600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 308 T - 169820)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 734586126400 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1344 T + 336588)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 126280729600 \) Copy content Toggle raw display
$97$ \( (T^{2} - 1428 T + 423780)^{2} \) Copy content Toggle raw display
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