Properties

Label 272.4.a.d
Level $272$
Weight $4$
Character orbit 272.a
Self dual yes
Analytic conductor $16.049$
Analytic rank $0$
Dimension $1$
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] = [272,4,Mod(1,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.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: \(16.0485195216\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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)
 
\(f(q)\) \(=\) \( q + 8 q^{3} + 6 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{3} + 6 q^{5} + 28 q^{7} + 37 q^{9} + 24 q^{11} - 58 q^{13} + 48 q^{15} + 17 q^{17} - 116 q^{19} + 224 q^{21} + 60 q^{23} - 89 q^{25} + 80 q^{27} + 30 q^{29} + 172 q^{31} + 192 q^{33} + 168 q^{35} - 58 q^{37} - 464 q^{39} - 342 q^{41} + 148 q^{43} + 222 q^{45} - 288 q^{47} + 441 q^{49} + 136 q^{51} + 318 q^{53} + 144 q^{55} - 928 q^{57} - 252 q^{59} + 110 q^{61} + 1036 q^{63} - 348 q^{65} + 484 q^{67} + 480 q^{69} + 708 q^{71} + 362 q^{73} - 712 q^{75} + 672 q^{77} + 484 q^{79} - 359 q^{81} - 756 q^{83} + 102 q^{85} + 240 q^{87} - 774 q^{89} - 1624 q^{91} + 1376 q^{93} - 696 q^{95} - 382 q^{97} + 888 q^{99}+O(q^{100}) \) 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
0
0 8.00000 0 6.00000 0 28.0000 0 37.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( -1 \)

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 272.4.a.d 1
3.b odd 2 1 2448.4.a.f 1
4.b odd 2 1 17.4.a.a 1
8.b even 2 1 1088.4.a.a 1
8.d odd 2 1 1088.4.a.l 1
12.b even 2 1 153.4.a.d 1
20.d odd 2 1 425.4.a.d 1
20.e even 4 2 425.4.b.c 2
28.d even 2 1 833.4.a.a 1
44.c even 2 1 2057.4.a.d 1
68.d odd 2 1 289.4.a.a 1
68.f odd 4 2 289.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 4.b odd 2 1
153.4.a.d 1 12.b even 2 1
272.4.a.d 1 1.a even 1 1 trivial
289.4.a.a 1 68.d odd 2 1
289.4.b.a 2 68.f odd 4 2
425.4.a.d 1 20.d odd 2 1
425.4.b.c 2 20.e even 4 2
833.4.a.a 1 28.d even 2 1
1088.4.a.a 1 8.b even 2 1
1088.4.a.l 1 8.d odd 2 1
2057.4.a.d 1 44.c even 2 1
2448.4.a.f 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(272))\):

\( T_{3} - 8 \) Copy content Toggle raw display
\( T_{5} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 8 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 28 \) Copy content Toggle raw display
$11$ \( T - 24 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T - 17 \) Copy content Toggle raw display
$19$ \( T + 116 \) Copy content Toggle raw display
$23$ \( T - 60 \) Copy content Toggle raw display
$29$ \( T - 30 \) Copy content Toggle raw display
$31$ \( T - 172 \) Copy content Toggle raw display
$37$ \( T + 58 \) Copy content Toggle raw display
$41$ \( T + 342 \) Copy content Toggle raw display
$43$ \( T - 148 \) Copy content Toggle raw display
$47$ \( T + 288 \) Copy content Toggle raw display
$53$ \( T - 318 \) Copy content Toggle raw display
$59$ \( T + 252 \) Copy content Toggle raw display
$61$ \( T - 110 \) Copy content Toggle raw display
$67$ \( T - 484 \) Copy content Toggle raw display
$71$ \( T - 708 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 484 \) Copy content Toggle raw display
$83$ \( T + 756 \) Copy content Toggle raw display
$89$ \( T + 774 \) Copy content Toggle raw display
$97$ \( T + 382 \) Copy content Toggle raw display
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