Properties

Label 1152.4.d.p
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.1534132224.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{2} q^{7} + ( - \beta_{3} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{4}) q^{13} + (\beta_{5} - 30) q^{17} + (3 \beta_{3} + \beta_1) q^{19} + (\beta_{7} + 3 \beta_{2}) q^{23} + ( - 2 \beta_{5} - 43) q^{25} + ( - 2 \beta_{6} + 5 \beta_{4}) q^{29} + (\beta_{7} - 4 \beta_{2}) q^{31} + (3 \beta_{3} - 10 \beta_1) q^{35} + ( - \beta_{6} - 17 \beta_{4}) q^{37} + (5 \beta_{5} + 102) q^{41} + (3 \beta_{3} - 11 \beta_1) q^{43} + (2 \beta_{7} - 12 \beta_{2}) q^{47} + ( - 8 \beta_{5} + 209) q^{49} + (2 \beta_{6} - 13 \beta_{4}) q^{53} + (\beta_{7} + 5 \beta_{2}) q^{55} + (16 \beta_{3} + 39 \beta_1) q^{59} + (5 \beta_{6} + 49 \beta_{4}) q^{61} + (9 \beta_{5} + 144) q^{65} + (12 \beta_{3} + 49 \beta_1) q^{67} + ( - 3 \beta_{7} + 27 \beta_{2}) q^{71} + ( - 8 \beta_{5} - 242) q^{73} + (12 \beta_{6} + 16 \beta_{4}) q^{77} + ( - 5 \beta_{7} + 6 \beta_{2}) q^{79} + (17 \beta_{3} - 71 \beta_1) q^{83} + (8 \beta_{6} - 26 \beta_{4}) q^{85} + (6 \beta_{5} - 918) q^{89} + (63 \beta_{3} - 108 \beta_1) q^{91} + ( - 7 \beta_{7} - 3 \beta_{2}) q^{95} + (2 \beta_{5} - 370) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 240 q^{17} - 344 q^{25} + 816 q^{41} + 1672 q^{49} + 1152 q^{65} - 1936 q^{73} - 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu^{5} + 66\nu^{3} + 174\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{6} - 6\nu^{4} + 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{5} + 104\nu^{3} + 328\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{7} + 31\nu^{5} + 155\nu^{3} + 253\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 24\nu^{6} + 288\nu^{4} + 864\nu^{2} + 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -10\nu^{7} - 131\nu^{5} - 487\nu^{3} - 377\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 33\nu^{6} + 390\nu^{4} + 1086\nu^{2} - 195 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{6} + 10\beta_{4} - 3\beta_{3} - 4\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 4\beta_{5} - 3\beta_{2} - 864 ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{6} - 15\beta_{4} + 6\beta_{3} + 4\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{7} - 24\beta_{5} + 51\beta_{2} + 5280 ) / 192 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 74\beta_{6} + 370\beta_{4} - 177\beta_{3} - 44\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -15\beta_{7} + 19\beta_{5} - 63\beta_{2} - 4104 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -470\beta_{6} - 2302\beta_{4} + 1263\beta_{3} - 52\beta_1 ) / 96 \) 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
2.48330i
0.0690906i
2.21597i
2.63019i
2.21597i
2.63019i
2.48330i
0.0690906i
0 0 0 17.4288i 0 −2.99032 0 0 0
577.2 0 0 0 17.4288i 0 2.99032 0 0 0
577.3 0 0 0 5.67763i 0 −33.0917 0 0 0
577.4 0 0 0 5.67763i 0 33.0917 0 0 0
577.5 0 0 0 5.67763i 0 −33.0917 0 0 0
577.6 0 0 0 5.67763i 0 33.0917 0 0 0
577.7 0 0 0 17.4288i 0 −2.99032 0 0 0
577.8 0 0 0 17.4288i 0 2.99032 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
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 1152.4.d.p 8
3.b odd 2 1 384.4.d.f 8
4.b odd 2 1 inner 1152.4.d.p 8
8.b even 2 1 inner 1152.4.d.p 8
8.d odd 2 1 inner 1152.4.d.p 8
12.b even 2 1 384.4.d.f 8
16.e even 4 1 2304.4.a.by 4
16.e even 4 1 2304.4.a.cb 4
16.f odd 4 1 2304.4.a.by 4
16.f odd 4 1 2304.4.a.cb 4
24.f even 2 1 384.4.d.f 8
24.h odd 2 1 384.4.d.f 8
48.i odd 4 1 768.4.a.u 4
48.i odd 4 1 768.4.a.v 4
48.k even 4 1 768.4.a.u 4
48.k even 4 1 768.4.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f 8 3.b odd 2 1
384.4.d.f 8 12.b even 2 1
384.4.d.f 8 24.f even 2 1
384.4.d.f 8 24.h odd 2 1
768.4.a.u 4 48.i odd 4 1
768.4.a.u 4 48.k even 4 1
768.4.a.v 4 48.i odd 4 1
768.4.a.v 4 48.k even 4 1
1152.4.d.p 8 1.a even 1 1 trivial
1152.4.d.p 8 4.b odd 2 1 inner
1152.4.d.p 8 8.b even 2 1 inner
1152.4.d.p 8 8.d odd 2 1 inner
2304.4.a.by 4 16.e even 4 1
2304.4.a.by 4 16.f odd 4 1
2304.4.a.cb 4 16.e even 4 1
2304.4.a.cb 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 \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} + 336T_{5}^{2} + 9792 \) Copy content Toggle raw display
\( T_{7}^{4} - 1104T_{7}^{2} + 9792 \) Copy content Toggle raw display
\( T_{17}^{2} + 60T_{17} - 3708 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 336 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 1104 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1312 T^{2} + 135424)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8640 T^{2} + 12690432)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 60 T - 3708)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 9504 T^{2} + 19927296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 53568 T^{2} + \cdots + 621831168)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 41040 T^{2} + \cdots + 419577408)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 55248 T^{2} + \cdots + 461095488)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 107136 T^{2} + \cdots + 2487324672)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 204 T - 104796)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 44064 T^{2} + \cdots + 164249856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 302400 T^{2} + \cdots + 21332616192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 87888 T^{2} + \cdots + 806557248)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 700192 T^{2} + \cdots + 7735554304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1040256 T^{2} + \cdots + 270509248512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 838944 T^{2} + \cdots + 73992704256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1104192 T^{2} + \cdots + 297070322688)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 484 T - 236348)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1039824 T^{2} + \cdots + 1904357952)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1747744 T^{2} + \cdots + 334010020096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1836 T + 676836)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 740 T + 118468)^{4} \) Copy content Toggle raw display
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