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Properties

Label 768.4.a.u
Level $768$
Weight $4$
Character orbit 768.a
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
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(1,768)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

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: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.9792.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - 3 q^{3} + \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + 9 q^{9} + ( - \beta_{2} - 12) q^{11} - 3 \beta_{3} q^{13} - 3 \beta_1 q^{15} + (3 \beta_{2} + 30) q^{17} + (3 \beta_{2} - 12) q^{19} + ( - 3 \beta_{3} + 3 \beta_1) q^{21} + ( - 6 \beta_{3} - 6 \beta_1) q^{23} + ( - 6 \beta_{2} + 43) q^{25} - 27 q^{27} + (6 \beta_{3} + 3 \beta_1) q^{29} + (\beta_{3} - 13 \beta_1) q^{31} + (3 \beta_{2} + 36) q^{33} + ( - 3 \beta_{2} - 120) q^{35} + (3 \beta_{3} - 18 \beta_1) q^{37} + 9 \beta_{3} q^{39} + ( - 15 \beta_{2} + 102) q^{41} + ( - 3 \beta_{2} - 132) q^{43} + 9 \beta_1 q^{45} + ( - 6 \beta_{3} + 30 \beta_1) q^{47} + (24 \beta_{2} + 209) q^{49} + ( - 9 \beta_{2} - 90) q^{51} + (6 \beta_{3} + 11 \beta_1) q^{53} + (8 \beta_{3} + 4 \beta_1) q^{55} + ( - 9 \beta_{2} + 36) q^{57} + (16 \beta_{2} - 468) q^{59} + (15 \beta_{3} - 54 \beta_1) q^{61} + (9 \beta_{3} - 9 \beta_1) q^{63} + (27 \beta_{2} - 144) q^{65} + (12 \beta_{2} - 588) q^{67} + (18 \beta_{3} + 18 \beta_1) q^{69} + ( - 18 \beta_{3} + 54 \beta_1) q^{71} + ( - 24 \beta_{2} + 242) q^{73} + (18 \beta_{2} - 129) q^{75} + ( - 36 \beta_{3} + 28 \beta_1) q^{77} + (9 \beta_{3} + 51 \beta_1) q^{79} + 81 q^{81} + ( - 17 \beta_{2} - 852) q^{83} + ( - 24 \beta_{3} - 18 \beta_1) q^{85} + ( - 18 \beta_{3} - 9 \beta_1) q^{87} + ( - 18 \beta_{2} - 918) q^{89} + ( - 63 \beta_{2} - 1296) q^{91} + ( - 3 \beta_{3} + 39 \beta_1) q^{93} + ( - 24 \beta_{3} - 60 \beta_1) q^{95} + ( - 6 \beta_{2} - 370) q^{97} + ( - 9 \beta_{2} - 108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 48 q^{11} + 120 q^{17} - 48 q^{19} + 172 q^{25} - 108 q^{27} + 144 q^{33} - 480 q^{35} + 408 q^{41} - 528 q^{43} + 836 q^{49} - 360 q^{51} + 144 q^{57} - 1872 q^{59} - 576 q^{65} - 2352 q^{67} + 968 q^{73} - 516 q^{75} + 324 q^{81} - 3408 q^{83} - 3672 q^{89} - 5184 q^{91} - 1480 q^{97} - 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{3} + 10\nu^{2} - 6\nu - 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\nu^{3} - 48\nu^{2} - 48\nu + 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -12\nu^{3} + 44\nu^{2} + 28\nu - 80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{3} + \beta_{2} - 4\beta _1 + 16 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 72 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{3} + 17\beta_{2} - 24\beta _1 + 352 ) / 32 \) Copy content Toggle raw display

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.06909
3.63019
−1.21597
−1.48330
0 −3.00000 0 −17.4288 0 2.99032 0 9.00000 0
1.2 0 −3.00000 0 −5.67763 0 33.0917 0 9.00000 0
1.3 0 −3.00000 0 5.67763 0 −33.0917 0 9.00000 0
1.4 0 −3.00000 0 17.4288 0 −2.99032 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.a.u 4
3.b odd 2 1 2304.4.a.cb 4
4.b odd 2 1 768.4.a.v 4
8.b even 2 1 768.4.a.v 4
8.d odd 2 1 inner 768.4.a.u 4
12.b even 2 1 2304.4.a.by 4
16.e even 4 2 384.4.d.f ✓ 8
16.f odd 4 2 384.4.d.f ✓ 8
24.f even 2 1 2304.4.a.cb 4
24.h odd 2 1 2304.4.a.by 4
48.i odd 4 2 1152.4.d.p 8
48.k even 4 2 1152.4.d.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f ✓ 8 16.e even 4 2
384.4.d.f ✓ 8 16.f odd 4 2
768.4.a.u 4 1.a even 1 1 trivial
768.4.a.u 4 8.d odd 2 1 inner
768.4.a.v 4 4.b odd 2 1
768.4.a.v 4 8.b even 2 1
1152.4.d.p 8 48.i odd 4 2
1152.4.d.p 8 48.k even 4 2
2304.4.a.by 4 12.b even 2 1
2304.4.a.by 4 24.h odd 2 1
2304.4.a.cb 4 3.b odd 2 1
2304.4.a.cb 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):

\( T_{5}^{4} - 336T_{5}^{2} + 9792 \) Copy content Toggle raw display
\( T_{7}^{4} - 1104T_{7}^{2} + 9792 \) Copy content Toggle raw display
\( T_{11}^{2} + 24T_{11} - 368 \) Copy content Toggle raw display
\( T_{19}^{2} + 24T_{19} - 4464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 336T^{2} + 9792 \) Copy content Toggle raw display
$7$ \( T^{4} - 1104 T^{2} + 9792 \) Copy content Toggle raw display
$11$ \( (T^{2} + 24 T - 368)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 8640 T^{2} + \cdots + 12690432 \) Copy content Toggle raw display
$17$ \( (T^{2} - 60 T - 3708)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24 T - 4464)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 53568 T^{2} + \cdots + 621831168 \) Copy content Toggle raw display
$29$ \( T^{4} - 41040 T^{2} + \cdots + 419577408 \) Copy content Toggle raw display
$31$ \( T^{4} - 55248 T^{2} + \cdots + 461095488 \) Copy content Toggle raw display
$37$ \( T^{4} - 107136 T^{2} + \cdots + 2487324672 \) Copy content Toggle raw display
$41$ \( (T^{2} - 204 T - 104796)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 264 T + 12816)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 302400 T^{2} + \cdots + 21332616192 \) Copy content Toggle raw display
$53$ \( T^{4} - 87888 T^{2} + \cdots + 806557248 \) Copy content Toggle raw display
$59$ \( (T^{2} + 936 T + 87952)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 1040256 T^{2} + \cdots + 270509248512 \) Copy content Toggle raw display
$67$ \( (T^{2} + 1176 T + 272016)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 1104192 T^{2} + \cdots + 297070322688 \) Copy content Toggle raw display
$73$ \( (T^{2} - 484 T - 236348)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1039824 T^{2} + \cdots + 1904357952 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1704 T + 577936)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1836 T + 676836)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 740 T + 118468)^{2} \) Copy content Toggle raw display
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