Properties

Label 768.3.b.a
Level $768$
Weight $3$
Character orbit 768.b
Analytic conductor $20.926$
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] = [768,3,Mod(127,768)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.b (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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 2 \beta_{2}) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 2 \beta_{2}) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7} + 3 q^{9} + ( - 4 \beta_1 - 8) q^{11} + (5 \beta_{3} + 4 \beta_{2}) q^{13} + ( - 6 \beta_{3} + \beta_{2}) q^{15} + ( - 8 \beta_1 - 6) q^{17} + ( - 4 \beta_1 - 24) q^{19} + (6 \beta_{3} - 2 \beta_{2}) q^{21} - 4 \beta_{3} q^{23} + (16 \beta_1 - 27) q^{25} + 3 \beta_1 q^{27} + ( - 5 \beta_{3} - 2 \beta_{2}) q^{29} + (2 \beta_{3} + 10 \beta_{2}) q^{31} + ( - 8 \beta_1 - 12) q^{33} + ( - 24 \beta_1 + 56) q^{35} + ( - 9 \beta_{3} + 8 \beta_{2}) q^{37} + (12 \beta_{3} + 5 \beta_{2}) q^{39} + ( - 8 \beta_1 + 22) q^{41} + ( - 20 \beta_1 + 40) q^{43} + (3 \beta_{3} - 6 \beta_{2}) q^{45} + (28 \beta_{3} + 4 \beta_{2}) q^{47} + (32 \beta_1 - 15) q^{49} + ( - 6 \beta_1 - 24) q^{51} + ( - 3 \beta_{3} + 2 \beta_{2}) q^{53} + (16 \beta_{3} + 12 \beta_{2}) q^{55} + ( - 24 \beta_1 - 12) q^{57} + ( - 44 \beta_1 - 32) q^{59} - 7 \beta_{3} q^{61} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{63} + (24 \beta_1 + 76) q^{65} + ( - 28 \beta_1 + 32) q^{67} - 4 \beta_{2} q^{69} + ( - 20 \beta_{3} + 24 \beta_{2}) q^{71} + ( - 32 \beta_1 + 30) q^{73} + ( - 27 \beta_1 + 48) q^{75} + ( - 8 \beta_{3} - 8 \beta_{2}) q^{77} + ( - 38 \beta_{3} - 6 \beta_{2}) q^{79} + 9 q^{81} + ( - 4 \beta_1 - 56) q^{83} + (42 \beta_{3} + 4 \beta_{2}) q^{85} + ( - 6 \beta_{3} - 5 \beta_{2}) q^{87} + ( - 16 \beta_1 + 78) q^{89} + ( - 8 \beta_1 - 56) q^{91} + (30 \beta_{3} + 2 \beta_{2}) q^{93} + 44 \beta_{2} q^{95} + ( - 48 \beta_1 - 62) q^{97} + ( - 12 \beta_1 - 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 32 q^{11} - 24 q^{17} - 96 q^{19} - 108 q^{25} - 48 q^{33} + 224 q^{35} + 88 q^{41} + 160 q^{43} - 60 q^{49} - 96 q^{51} - 48 q^{57} - 128 q^{59} + 304 q^{65} + 128 q^{67} + 120 q^{73} + 192 q^{75} + 36 q^{81} - 224 q^{83} + 312 q^{89} - 224 q^{91} - 248 q^{97} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −1.73205 0 8.92820i 0 10.9282i 0 3.00000 0
127.2 0 −1.73205 0 8.92820i 0 10.9282i 0 3.00000 0
127.3 0 1.73205 0 4.92820i 0 2.92820i 0 3.00000 0
127.4 0 1.73205 0 4.92820i 0 2.92820i 0 3.00000 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.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.3.b.a 4
3.b odd 2 1 2304.3.b.o 4
4.b odd 2 1 768.3.b.d 4
8.b even 2 1 768.3.b.d 4
8.d odd 2 1 inner 768.3.b.a 4
12.b even 2 1 2304.3.b.k 4
16.e even 4 1 96.3.g.a 4
16.e even 4 1 192.3.g.c 4
16.f odd 4 1 96.3.g.a 4
16.f odd 4 1 192.3.g.c 4
24.f even 2 1 2304.3.b.o 4
24.h odd 2 1 2304.3.b.k 4
48.i odd 4 1 288.3.g.d 4
48.i odd 4 1 576.3.g.j 4
48.k even 4 1 288.3.g.d 4
48.k even 4 1 576.3.g.j 4
80.i odd 4 1 2400.3.j.a 4
80.j even 4 1 2400.3.j.a 4
80.k odd 4 1 2400.3.e.a 4
80.q even 4 1 2400.3.e.a 4
80.s even 4 1 2400.3.j.b 4
80.t odd 4 1 2400.3.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a 4 16.e even 4 1
96.3.g.a 4 16.f odd 4 1
192.3.g.c 4 16.e even 4 1
192.3.g.c 4 16.f odd 4 1
288.3.g.d 4 48.i odd 4 1
288.3.g.d 4 48.k even 4 1
576.3.g.j 4 48.i odd 4 1
576.3.g.j 4 48.k even 4 1
768.3.b.a 4 1.a even 1 1 trivial
768.3.b.a 4 8.d odd 2 1 inner
768.3.b.d 4 4.b odd 2 1
768.3.b.d 4 8.b even 2 1
2304.3.b.k 4 12.b even 2 1
2304.3.b.k 4 24.h odd 2 1
2304.3.b.o 4 3.b odd 2 1
2304.3.b.o 4 24.f even 2 1
2400.3.e.a 4 80.k odd 4 1
2400.3.e.a 4 80.q even 4 1
2400.3.j.a 4 80.i odd 4 1
2400.3.j.a 4 80.j even 4 1
2400.3.j.b 4 80.s even 4 1
2400.3.j.b 4 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{4} + 104T_{5}^{2} + 1936 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$7$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 584T^{2} + 8464 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T - 156)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48 T + 528)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 296T^{2} + 2704 \) Copy content Toggle raw display
$31$ \( T^{4} + 2432 T^{2} + \cdots + 1401856 \) Copy content Toggle raw display
$37$ \( T^{4} + 2184 T^{2} + 197136 \) Copy content Toggle raw display
$41$ \( (T^{2} - 44 T + 292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80 T + 400)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6656 T^{2} + \cdots + 8667136 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 64 T - 4784)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 64 T - 1328)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 17024 T^{2} + \cdots + 28217344 \) Copy content Toggle raw display
$73$ \( (T^{2} - 60 T - 2172)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 12416 T^{2} + \cdots + 28558336 \) Copy content Toggle raw display
$83$ \( (T^{2} + 112 T + 3088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 156 T + 5316)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 124 T - 3068)^{2} \) Copy content Toggle raw display
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