Properties

Label 363.2.a.d
Level $363$
Weight $2$
Character orbit 363.a
Self dual yes
Analytic conductor $2.899$
Analytic rank $1$
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] = [363,2,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(2.89856959337\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - q^{3} + 3 \beta q^{4} + ( - \beta + 1) q^{5} + (\beta + 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} + q^{9} + \beta q^{10} - 3 \beta q^{12} + (2 \beta - 3) q^{13} + (\beta + 1) q^{14} + \cdots + (6 \beta + 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} + q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} + q^{10} - 3 q^{12} - 4 q^{13} + 3 q^{14} - q^{15} + 13 q^{16} - 9 q^{17} - 3 q^{18} + 5 q^{19} - 6 q^{20} + 2 q^{21}+ \cdots + 18 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−2.61803 −1.00000 4.85410 −0.618034 2.61803 −1.00000 −7.47214 1.00000 1.61803
1.2 −0.381966 −1.00000 −1.85410 1.61803 0.381966 −1.00000 1.47214 1.00000 −0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( +1 \)

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 363.2.a.d 2
3.b odd 2 1 1089.2.a.t 2
4.b odd 2 1 5808.2.a.cj 2
5.b even 2 1 9075.2.a.cb 2
11.b odd 2 1 363.2.a.i 2
11.c even 5 2 33.2.e.b 4
11.c even 5 2 363.2.e.k 4
11.d odd 10 2 363.2.e.b 4
11.d odd 10 2 363.2.e.f 4
33.d even 2 1 1089.2.a.l 2
33.h odd 10 2 99.2.f.a 4
44.c even 2 1 5808.2.a.ci 2
44.h odd 10 2 528.2.y.b 4
55.d odd 2 1 9075.2.a.u 2
55.j even 10 2 825.2.n.c 4
55.k odd 20 4 825.2.bx.d 8
99.m even 15 4 891.2.n.c 8
99.n odd 30 4 891.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 11.c even 5 2
99.2.f.a 4 33.h odd 10 2
363.2.a.d 2 1.a even 1 1 trivial
363.2.a.i 2 11.b odd 2 1
363.2.e.b 4 11.d odd 10 2
363.2.e.f 4 11.d odd 10 2
363.2.e.k 4 11.c even 5 2
528.2.y.b 4 44.h odd 10 2
825.2.n.c 4 55.j even 10 2
825.2.bx.d 8 55.k odd 20 4
891.2.n.b 8 99.n odd 30 4
891.2.n.c 8 99.m even 15 4
1089.2.a.l 2 33.d even 2 1
1089.2.a.t 2 3.b odd 2 1
5808.2.a.ci 2 44.c even 2 1
5808.2.a.cj 2 4.b odd 2 1
9075.2.a.u 2 55.d odd 2 1
9075.2.a.cb 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 11 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 45 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 71 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 99 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} - 5T - 55 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 11 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 31 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 9 \) Copy content Toggle raw display
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