Properties

Label 272.2.b.a
Level $272$
Weight $2$
Character orbit 272.b
Analytic conductor $2.172$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [272,2,Mod(33,272)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 272.b (of order \(2\), degree \(1\), not 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: \(2.17193093498\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 34)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + \beta q^{5} - 5 q^{9} - \beta q^{11} + 2 q^{13} - 8 q^{15} + ( - \beta - 3) q^{17} + 4 q^{19} + 2 \beta q^{23} - 3 q^{25} - 2 \beta q^{27} + \beta q^{29} + 8 q^{33} + 3 \beta q^{37} + 2 \beta q^{39} - 2 \beta q^{41} + 4 q^{43} - 5 \beta q^{45} + 7 q^{49} + ( - 3 \beta + 8) q^{51} + 6 q^{53} + 8 q^{55} + 4 \beta q^{57} - 12 q^{59} - 3 \beta q^{61} + 2 \beta q^{65} + 4 q^{67} - 16 q^{69} + 2 \beta q^{71} - 3 \beta q^{75} - 6 \beta q^{79} + q^{81} + 12 q^{83} + ( - 3 \beta + 8) q^{85} - 8 q^{87} + 6 q^{89} + 4 \beta q^{95} - 6 \beta q^{97} + 5 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{9} + 4 q^{13} - 16 q^{15} - 6 q^{17} + 8 q^{19} - 6 q^{25} + 16 q^{33} + 8 q^{43} + 14 q^{49} + 16 q^{51} + 12 q^{53} + 16 q^{55} - 24 q^{59} + 8 q^{67} - 32 q^{69} + 2 q^{81} + 24 q^{83} + 16 q^{85} - 16 q^{87} + 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.41421i
1.41421i
0 2.82843i 0 2.82843i 0 0 0 −5.00000 0
33.2 0 2.82843i 0 2.82843i 0 0 0 −5.00000 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
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.2.b.a 2
3.b odd 2 1 2448.2.c.n 2
4.b odd 2 1 34.2.b.a 2
8.b even 2 1 1088.2.b.a 2
8.d odd 2 1 1088.2.b.b 2
12.b even 2 1 306.2.b.d 2
17.b even 2 1 inner 272.2.b.a 2
17.c even 4 2 4624.2.a.s 2
20.d odd 2 1 850.2.b.f 2
20.e even 4 2 850.2.d.i 4
28.d even 2 1 1666.2.b.c 2
51.c odd 2 1 2448.2.c.n 2
68.d odd 2 1 34.2.b.a 2
68.f odd 4 2 578.2.a.d 2
68.g odd 8 2 578.2.c.a 2
68.g odd 8 2 578.2.c.d 2
68.i even 16 8 578.2.d.g 8
136.e odd 2 1 1088.2.b.b 2
136.h even 2 1 1088.2.b.a 2
204.h even 2 1 306.2.b.d 2
204.l even 4 2 5202.2.a.u 2
340.d odd 2 1 850.2.b.f 2
340.r even 4 2 850.2.d.i 4
476.e even 2 1 1666.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 4.b odd 2 1
34.2.b.a 2 68.d odd 2 1
272.2.b.a 2 1.a even 1 1 trivial
272.2.b.a 2 17.b even 2 1 inner
306.2.b.d 2 12.b even 2 1
306.2.b.d 2 204.h even 2 1
578.2.a.d 2 68.f odd 4 2
578.2.c.a 2 68.g odd 8 2
578.2.c.d 2 68.g odd 8 2
578.2.d.g 8 68.i even 16 8
850.2.b.f 2 20.d odd 2 1
850.2.b.f 2 340.d odd 2 1
850.2.d.i 4 20.e even 4 2
850.2.d.i 4 340.r even 4 2
1088.2.b.a 2 8.b even 2 1
1088.2.b.a 2 136.h even 2 1
1088.2.b.b 2 8.d odd 2 1
1088.2.b.b 2 136.e odd 2 1
1666.2.b.c 2 28.d even 2 1
1666.2.b.c 2 476.e even 2 1
2448.2.c.n 2 3.b odd 2 1
2448.2.c.n 2 51.c odd 2 1
4624.2.a.s 2 17.c even 4 2
5202.2.a.u 2 204.l even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 17 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 8 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 72 \) Copy content Toggle raw display
$41$ \( T^{2} + 32 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 72 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 32 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 288 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 288 \) Copy content Toggle raw display
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