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Properties

Label 1152.4.d.i
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{5} + (\beta_{3} - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{5} + (\beta_{3} - 8) q^{7} + (4 \beta_{2} + \beta_1) q^{11} + (2 \beta_{2} + 9 \beta_1) q^{13} + (4 \beta_{3} + 18) q^{17} + (4 \beta_{2} - 17 \beta_1) q^{19} + (2 \beta_{3} + 128) q^{23} + (8 \beta_{3} - 99) q^{25} + (9 \beta_{2} - 19 \beta_1) q^{29} + ( - 13 \beta_{3} - 40) q^{31} + ( - 12 \beta_{2} + 60 \beta_1) q^{35} + (4 \beta_{2} - 17 \beta_1) q^{37} + (20 \beta_{3} - 218) q^{41} + (4 \beta_{2} - 89 \beta_1) q^{43} + ( - 14 \beta_{3} + 112) q^{47} + ( - 16 \beta_{3} - 71) q^{49} + (15 \beta_{2} + 43 \beta_1) q^{53} + (12 \beta_{3} - 816) q^{55} + 81 \beta_1 q^{59} + 81 \beta_1 q^{61} + ( - 28 \beta_{3} - 272) q^{65} + ( - 48 \beta_{2} - 57 \beta_1) q^{67} + ( - 2 \beta_{3} + 1024) q^{71} + (48 \beta_{3} - 330) q^{73} + ( - 28 \beta_{2} + 200 \beta_1) q^{77} + (59 \beta_{3} + 248) q^{79} + (28 \beta_{2} + 97 \beta_1) q^{83} + (2 \beta_{2} + 190 \beta_1) q^{85} + ( - 8 \beta_{3} + 266) q^{89} + (20 \beta_{2} + 32 \beta_1) q^{91} + (84 \beta_{3} - 1104) q^{95} + ( - 88 \beta_{3} - 610) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{7} + 72 q^{17} + 512 q^{23} - 396 q^{25} - 160 q^{31} - 872 q^{41} + 448 q^{47} - 284 q^{49} - 3264 q^{55} - 1088 q^{65} + 4096 q^{71} - 1320 q^{73} + 992 q^{79} + 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{3} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 40\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} + 28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 28 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 5\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
− 2.30278i
− 1.30278i
1.30278i
2.30278i
0 0 0 − 18.4222i 0 −22.4222 0 0 0
577.2 0 0 0 − 10.4222i 0 6.42221 0 0 0
577.3 0 0 0 10.4222i 0 6.42221 0 0 0
577.4 0 0 0 18.4222i 0 −22.4222 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.i 4
3.b odd 2 1 384.4.d.c ✓ 4
4.b odd 2 1 1152.4.d.o 4
8.b even 2 1 inner 1152.4.d.i 4
8.d odd 2 1 1152.4.d.o 4
12.b even 2 1 384.4.d.e yes 4
16.e even 4 1 2304.4.a.t 2
16.e even 4 1 2304.4.a.bq 2
16.f odd 4 1 2304.4.a.s 2
16.f odd 4 1 2304.4.a.bp 2
24.f even 2 1 384.4.d.e yes 4
24.h odd 2 1 384.4.d.c ✓ 4
48.i odd 4 1 768.4.a.j 2
48.i odd 4 1 768.4.a.k 2
48.k even 4 1 768.4.a.e 2
48.k even 4 1 768.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c ✓ 4 3.b odd 2 1
384.4.d.c ✓ 4 24.h odd 2 1
384.4.d.e yes 4 12.b even 2 1
384.4.d.e yes 4 24.f even 2 1
768.4.a.e 2 48.k even 4 1
768.4.a.j 2 48.i odd 4 1
768.4.a.k 2 48.i odd 4 1
768.4.a.p 2 48.k even 4 1
1152.4.d.i 4 1.a even 1 1 trivial
1152.4.d.i 4 8.b even 2 1 inner
1152.4.d.o 4 4.b odd 2 1
1152.4.d.o 4 8.d odd 2 1
2304.4.a.s 2 16.f odd 4 1
2304.4.a.t 2 16.e even 4 1
2304.4.a.bp 2 16.f odd 4 1
2304.4.a.bq 2 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} + 448T_{5}^{2} + 36864 \) Copy content Toggle raw display
\( T_{7}^{2} + 16T_{7} - 144 \) Copy content Toggle raw display
\( T_{17}^{2} - 36T_{17} - 3004 \) 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
$5$ \( T^{4} + 448 T^{2} + 36864 \) Copy content Toggle raw display
$7$ \( (T^{2} + 16 T - 144)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6688 T^{2} + 10969344 \) Copy content Toggle raw display
$13$ \( T^{4} + 4256 T^{2} + 215296 \) Copy content Toggle raw display
$17$ \( (T^{2} - 36 T - 3004)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 15904 T^{2} + 1679616 \) Copy content Toggle raw display
$23$ \( (T^{2} - 256 T + 15552)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 45248 T^{2} + 122589184 \) Copy content Toggle raw display
$31$ \( (T^{2} + 80 T - 33552)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 15904 T^{2} + 1679616 \) Copy content Toggle raw display
$41$ \( (T^{2} + 436 T - 35676)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 15229534464 \) Copy content Toggle raw display
$47$ \( (T^{2} - 224 T - 28224)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 152768 T^{2} + 296390656 \) Copy content Toggle raw display
$59$ \( (T^{2} + 104976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 104976)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 182540853504 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2048 T + 1047744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 660 T - 370332)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 496 T - 662544)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 627232 T^{2} + 156950784 \) Copy content Toggle raw display
$89$ \( (T^{2} - 532 T + 57444)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1220 T - 1238652)^{2} \) Copy content Toggle raw display
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