Properties

Label 288.8.a.m
Level $288$
Weight $8$
Character orbit 288.a
Self dual yes
Analytic conductor $89.967$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,8,Mod(1,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.a (trivial)

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: yes
Analytic conductor: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 48\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta + 14) q^{5} + ( - 7 \beta + 468) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta + 14) q^{5} + ( - 7 \beta + 468) q^{7} + (46 \beta + 1692) q^{11} + (42 \beta - 8538) q^{13} + (102 \beta + 9334) q^{17} + ( - 62 \beta + 22428) q^{19} + ( - 626 \beta - 21744) q^{23} + (84 \beta + 46487) q^{25} + (1569 \beta + 9742) q^{29} + ( - 1275 \beta + 166428) q^{31} + (1306 \beta - 283752) q^{35} + (2772 \beta - 4538) q^{37} + (738 \beta - 245890) q^{41} + (5110 \beta + 255492) q^{43} + (1210 \beta - 890712) q^{47} + ( - 6552 \beta + 72857) q^{49} + ( - 3339 \beta + 697846) q^{53} + (5720 \beta + 1931400) q^{55} + (664 \beta - 767052) q^{59} + (3240 \beta + 796094) q^{61} + ( - 25026 \beta + 1622292) q^{65} + ( - 31256 \beta - 584748) q^{67} + ( - 13402 \beta - 2858400) q^{71} + ( - 23064 \beta + 590442) q^{73} + (9684 \beta - 3659472) q^{77} + ( - 7575 \beta + 3269052) q^{79} + ( - 14378 \beta + 8402580) q^{83} + (29430 \beta + 4360820) q^{85} + (22116 \beta + 3059462) q^{89} + (79422 \beta - 8060040) q^{91} + (66416 \beta - 2257272) q^{95} + (32220 \beta + 11860434) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 28 q^{5} + 936 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 28 q^{5} + 936 q^{7} + 3384 q^{11} - 17076 q^{13} + 18668 q^{17} + 44856 q^{19} - 43488 q^{23} + 92974 q^{25} + 19484 q^{29} + 332856 q^{31} - 567504 q^{35} - 9076 q^{37} - 491780 q^{41} + 510984 q^{43} - 1781424 q^{47} + 145714 q^{49} + 1395692 q^{53} + 3862800 q^{55} - 1534104 q^{59} + 1592188 q^{61} + 3244584 q^{65} - 1169496 q^{67} - 5716800 q^{71} + 1180884 q^{73} - 7318944 q^{77} + 6538104 q^{79} + 16805160 q^{83} + 8721640 q^{85} + 6118924 q^{89} - 16120080 q^{91} - 4514544 q^{95} + 23720868 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
0 0 0 −338.727 0 1291.03 0 0 0
1.2 0 0 0 366.727 0 −355.029 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.8.a.m 2
3.b odd 2 1 96.8.a.c 2
4.b odd 2 1 288.8.a.l 2
8.b even 2 1 576.8.a.bh 2
8.d odd 2 1 576.8.a.bg 2
12.b even 2 1 96.8.a.f yes 2
24.f even 2 1 192.8.a.s 2
24.h odd 2 1 192.8.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.8.a.c 2 3.b odd 2 1
96.8.a.f yes 2 12.b even 2 1
192.8.a.s 2 24.f even 2 1
192.8.a.v 2 24.h odd 2 1
288.8.a.l 2 4.b odd 2 1
288.8.a.m 2 1.a even 1 1 trivial
576.8.a.bg 2 8.d odd 2 1
576.8.a.bh 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(288))\):

\( T_{5}^{2} - 28T_{5} - 124220 \) Copy content Toggle raw display
\( T_{7}^{2} - 936T_{7} - 458352 \) Copy content Toggle raw display
\( T_{11}^{2} - 3384T_{11} - 26388720 \) 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
$5$ \( T^{2} - 28T - 124220 \) Copy content Toggle raw display
$7$ \( T^{2} - 936T - 458352 \) Copy content Toggle raw display
$11$ \( T^{2} - 3384 T - 26388720 \) Copy content Toggle raw display
$13$ \( T^{2} + 17076 T + 48511908 \) Copy content Toggle raw display
$17$ \( T^{2} - 18668 T - 56701340 \) Copy content Toggle raw display
$19$ \( T^{2} - 44856 T + 449875728 \) Copy content Toggle raw display
$23$ \( T^{2} + 43488 T - 4944492288 \) Copy content Toggle raw display
$29$ \( T^{2} - 19484 T - 33936477500 \) Copy content Toggle raw display
$31$ \( T^{2} - 332856 T + 5225639184 \) Copy content Toggle raw display
$37$ \( T^{2} + 9076 T - 106202801372 \) Copy content Toggle raw display
$41$ \( T^{2} + 491780 T + 52932733444 \) Copy content Toggle raw display
$43$ \( T^{2} - 510984 T - 295697508336 \) Copy content Toggle raw display
$47$ \( T^{2} + 1781424 T + 773128148544 \) Copy content Toggle raw display
$53$ \( T^{2} - 1395692 T + 332866355812 \) Copy content Toggle raw display
$59$ \( T^{2} + 1534104 T + 582273824400 \) Copy content Toggle raw display
$61$ \( T^{2} - 1592188 T + 488646834436 \) Copy content Toggle raw display
$67$ \( T^{2} + 1169496 T - 13163254274160 \) Copy content Toggle raw display
$71$ \( T^{2} + 5716800 T + 5687472098304 \) Copy content Toggle raw display
$73$ \( T^{2} - 1180884 T - 7005028723740 \) Copy content Toggle raw display
$79$ \( T^{2} - 6538104 T + 9893471218704 \) Copy content Toggle raw display
$83$ \( T^{2} - 16805160 T + 67745558211984 \) Copy content Toggle raw display
$89$ \( T^{2} - 6118924 T + 2598748017700 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 126318807666756 \) Copy content Toggle raw display
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