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Properties

Label 1152.3.g.a
Level $1152$
Weight $3$
Character orbit 1152.g
Analytic conductor $31.390$
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] = [1152,3,Mod(127,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.g (of order \(2\), degree \(1\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2) q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} - 6) q^{13} + ( - 2 \beta_{3} + 2) q^{17} + (2 \beta_{2} - 3 \beta_1) q^{19} + (3 \beta_{2} - 5 \beta_1) q^{23} + ( - 4 \beta_{3} + 11) q^{25} + (\beta_{3} - 34) q^{29} + (4 \beta_{2} + 12 \beta_1) q^{31} - 4 \beta_{2} q^{35} + (7 \beta_{3} + 10) q^{37} + (4 \beta_{3} - 2) q^{41} + (4 \beta_{2} + 13 \beta_1) q^{43} + ( - 2 \beta_{2} + 6 \beta_1) q^{47} + ( - 8 \beta_{3} + 1) q^{49} + (\beta_{3} + 22) q^{53} + (7 \beta_{2} + 7 \beta_1) q^{55} + 11 \beta_1 q^{59} + (7 \beta_{3} + 74) q^{61} + ( - 4 \beta_{3} - 20) q^{65} + ( - 6 \beta_{2} + 9 \beta_1) q^{67} + (\beta_{2} + 9 \beta_1) q^{71} + ( - 6 \beta_{3} + 22) q^{73} + (16 \beta_{3} + 88) q^{77} + ( - 10 \beta_{2} + 22 \beta_1) q^{79} + (8 \beta_{2} + 29 \beta_1) q^{83} + (6 \beta_{3} - 68) q^{85} + ( - 10 \beta_{3} - 54) q^{89} + (12 \beta_{2} + 8 \beta_1) q^{91} + (3 \beta_{2} + 35 \beta_1) q^{95} + (10 \beta_{3} - 82) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} - 24 q^{13} + 8 q^{17} + 44 q^{25} - 136 q^{29} + 40 q^{37} - 8 q^{41} + 4 q^{49} + 88 q^{53} + 296 q^{61} - 80 q^{65} + 88 q^{73} + 352 q^{77} - 272 q^{85} - 216 q^{89} - 328 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{3} - 2\zeta_{8}^{2} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 6\zeta_{8}^{2} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} + 3\beta_1 ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 + 0.707107i
0.707107 − 0.707107i
0 0 0 −7.65685 0 − 1.65685i 0 0 0
127.2 0 0 0 −7.65685 0 1.65685i 0 0 0
127.3 0 0 0 3.65685 0 − 9.65685i 0 0 0
127.4 0 0 0 3.65685 0 9.65685i 0 0 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.g.a 4
3.b odd 2 1 128.3.c.b yes 4
4.b odd 2 1 inner 1152.3.g.a 4
8.b even 2 1 1152.3.g.b 4
8.d odd 2 1 1152.3.g.b 4
12.b even 2 1 128.3.c.b yes 4
16.e even 4 1 2304.3.b.j 4
16.e even 4 1 2304.3.b.p 4
16.f odd 4 1 2304.3.b.j 4
16.f odd 4 1 2304.3.b.p 4
24.f even 2 1 128.3.c.a ✓ 4
24.h odd 2 1 128.3.c.a ✓ 4
48.i odd 4 1 256.3.d.d 4
48.i odd 4 1 256.3.d.e 4
48.k even 4 1 256.3.d.d 4
48.k even 4 1 256.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.c.a ✓ 4 24.f even 2 1
128.3.c.a ✓ 4 24.h odd 2 1
128.3.c.b yes 4 3.b odd 2 1
128.3.c.b yes 4 12.b even 2 1
256.3.d.d 4 48.i odd 4 1
256.3.d.d 4 48.k even 4 1
256.3.d.e 4 48.i odd 4 1
256.3.d.e 4 48.k even 4 1
1152.3.g.a 4 1.a even 1 1 trivial
1152.3.g.a 4 4.b odd 2 1 inner
1152.3.g.b 4 8.b even 2 1
1152.3.g.b 4 8.d odd 2 1
2304.3.b.j 4 16.e even 4 1
2304.3.b.j 4 16.f odd 4 1
2304.3.b.p 4 16.e even 4 1
2304.3.b.p 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 4T_{5} - 28 \) Copy content Toggle raw display
\( T_{13}^{2} + 12T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} + 344T^{2} + 784 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 664 T^{2} + 99856 \) Copy content Toggle raw display
$23$ \( T^{4} + 1632 T^{2} + 565504 \) Copy content Toggle raw display
$29$ \( (T^{2} + 68 T + 1124)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2048)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20 T - 1468)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 508)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 4632 T^{2} + 5326864 \) Copy content Toggle raw display
$47$ \( T^{4} + 1408 T^{2} + 200704 \) Copy content Toggle raw display
$53$ \( (T^{2} - 44 T + 452)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2904 T^{2} + 234256 \) Copy content Toggle raw display
$61$ \( (T^{2} - 148 T + 3908)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 5976 T^{2} + 8088336 \) Copy content Toggle raw display
$71$ \( T^{4} + 1888 T^{2} + 430336 \) Copy content Toggle raw display
$73$ \( (T^{2} - 44 T - 668)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 23936 T^{2} + 93392896 \) Copy content Toggle raw display
$83$ \( T^{4} + 22104 T^{2} + 117765904 \) Copy content Toggle raw display
$89$ \( (T^{2} + 108 T - 284)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 164 T + 3524)^{2} \) Copy content Toggle raw display
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