Properties

Label 1152.4.d.h
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(577,1152)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (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: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{5} + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{5} + 32 q^{7} + 2 \beta q^{11} - 5 \beta q^{13} + 98 q^{17} + 22 \beta q^{19} + 32 q^{23} - 19 q^{25} - 43 \beta q^{29} + 256 q^{31} + 96 \beta q^{35} - 23 \beta q^{37} + 102 q^{41} - 74 \beta q^{43} - 320 q^{47} + 681 q^{49} + 19 \beta q^{53} - 96 q^{55} - 102 \beta q^{59} + 159 \beta q^{61} + 240 q^{65} - 138 \beta q^{67} - 416 q^{71} - 138 q^{73} + 64 \beta q^{77} + 64 q^{79} + 98 \beta q^{83} + 294 \beta q^{85} - 582 q^{89} - 160 \beta q^{91} - 1056 q^{95} + 238 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{7} + 196 q^{17} + 64 q^{23} - 38 q^{25} + 512 q^{31} + 204 q^{41} - 640 q^{47} + 1362 q^{49} - 192 q^{55} + 480 q^{65} - 832 q^{71} - 276 q^{73} + 128 q^{79} - 1164 q^{89} - 2112 q^{95} + 476 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 0 0 12.0000i 0 32.0000 0 0 0
577.2 0 0 0 12.0000i 0 32.0000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.4.d.h 2
3.b odd 2 1 128.4.b.d yes 2
4.b odd 2 1 1152.4.d.a 2
8.b even 2 1 inner 1152.4.d.h 2
8.d odd 2 1 1152.4.d.a 2
12.b even 2 1 128.4.b.a 2
16.e even 4 1 2304.4.a.c 1
16.e even 4 1 2304.4.a.m 1
16.f odd 4 1 2304.4.a.d 1
16.f odd 4 1 2304.4.a.n 1
24.f even 2 1 128.4.b.a 2
24.h odd 2 1 128.4.b.d yes 2
48.i odd 4 1 256.4.a.c 1
48.i odd 4 1 256.4.a.f 1
48.k even 4 1 256.4.a.b 1
48.k even 4 1 256.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.a 2 12.b even 2 1
128.4.b.a 2 24.f even 2 1
128.4.b.d yes 2 3.b odd 2 1
128.4.b.d yes 2 24.h odd 2 1
256.4.a.b 1 48.k even 4 1
256.4.a.c 1 48.i odd 4 1
256.4.a.f 1 48.i odd 4 1
256.4.a.g 1 48.k even 4 1
1152.4.d.a 2 4.b odd 2 1
1152.4.d.a 2 8.d odd 2 1
1152.4.d.h 2 1.a even 1 1 trivial
1152.4.d.h 2 8.b even 2 1 inner
2304.4.a.c 1 16.e even 4 1
2304.4.a.d 1 16.f odd 4 1
2304.4.a.m 1 16.e even 4 1
2304.4.a.n 1 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 \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{7} - 32 \) Copy content Toggle raw display
\( T_{17} - 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 144 \) Copy content Toggle raw display
$7$ \( (T - 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( (T - 98)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 7744 \) Copy content Toggle raw display
$23$ \( (T - 32)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 29584 \) Copy content Toggle raw display
$31$ \( (T - 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8464 \) Copy content Toggle raw display
$41$ \( (T - 102)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 87616 \) Copy content Toggle raw display
$47$ \( (T + 320)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 5776 \) Copy content Toggle raw display
$59$ \( T^{2} + 166464 \) Copy content Toggle raw display
$61$ \( T^{2} + 404496 \) Copy content Toggle raw display
$67$ \( T^{2} + 304704 \) Copy content Toggle raw display
$71$ \( (T + 416)^{2} \) Copy content Toggle raw display
$73$ \( (T + 138)^{2} \) Copy content Toggle raw display
$79$ \( (T - 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 153664 \) Copy content Toggle raw display
$89$ \( (T + 582)^{2} \) Copy content Toggle raw display
$97$ \( (T - 238)^{2} \) Copy content Toggle raw display
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