Properties

Label 363.2.a.g
Level $363$
Weight $2$
Character orbit 363.a
Self dual yes
Analytic conductor $2.899$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(0\)
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: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + 3 q^{4} + 2 q^{5} - \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + 3 q^{4} + 2 q^{5} - \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} - 2 \beta q^{10} + 3 q^{12} - 10 q^{14} + 2 q^{15} - q^{16} - 2 \beta q^{17} - \beta q^{18} + 2 \beta q^{19} + 6 q^{20} + 2 \beta q^{21} - 4 q^{23} - \beta q^{24} - q^{25} + q^{27} + 6 \beta q^{28} - 2 \beta q^{29} - 2 \beta q^{30} + 3 \beta q^{32} + 10 q^{34} + 4 \beta q^{35} + 3 q^{36} + 2 q^{37} - 10 q^{38} - 2 \beta q^{40} + 2 \beta q^{41} - 10 q^{42} - 2 \beta q^{43} + 2 q^{45} + 4 \beta q^{46} + 8 q^{47} - q^{48} + 13 q^{49} + \beta q^{50} - 2 \beta q^{51} + 6 q^{53} - \beta q^{54} - 10 q^{56} + 2 \beta q^{57} + 10 q^{58} + 6 q^{60} - 4 \beta q^{61} + 2 \beta q^{63} - 13 q^{64} - 12 q^{67} - 6 \beta q^{68} - 4 q^{69} - 20 q^{70} - 8 q^{71} - \beta q^{72} + 4 \beta q^{73} - 2 \beta q^{74} - q^{75} + 6 \beta q^{76} - 6 \beta q^{79} - 2 q^{80} + q^{81} - 10 q^{82} + 4 \beta q^{83} + 6 \beta q^{84} - 4 \beta q^{85} + 10 q^{86} - 2 \beta q^{87} - 14 q^{89} - 2 \beta q^{90} - 12 q^{92} - 8 \beta q^{94} + 4 \beta q^{95} + 3 \beta q^{96} + 2 q^{97} - 13 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{9} + 6 q^{12} - 20 q^{14} + 4 q^{15} - 2 q^{16} + 12 q^{20} - 8 q^{23} - 2 q^{25} + 2 q^{27} + 20 q^{34} + 6 q^{36} + 4 q^{37} - 20 q^{38} - 20 q^{42} + 4 q^{45} + 16 q^{47} - 2 q^{48} + 26 q^{49} + 12 q^{53} - 20 q^{56} + 20 q^{58} + 12 q^{60} - 26 q^{64} - 24 q^{67} - 8 q^{69} - 40 q^{70} - 16 q^{71} - 2 q^{75} - 4 q^{80} + 2 q^{81} - 20 q^{82} + 20 q^{86} - 28 q^{89} - 24 q^{92} + 4 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 1.00000 3.00000 2.00000 −2.23607 4.47214 −2.23607 1.00000 −4.47214
1.2 2.23607 1.00000 3.00000 2.00000 2.23607 −4.47214 2.23607 1.00000 4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.a.g 2
3.b odd 2 1 1089.2.a.p 2
4.b odd 2 1 5808.2.a.bx 2
5.b even 2 1 9075.2.a.bi 2
11.b odd 2 1 inner 363.2.a.g 2
11.c even 5 2 363.2.e.a 4
11.c even 5 2 363.2.e.l 4
11.d odd 10 2 363.2.e.a 4
11.d odd 10 2 363.2.e.l 4
33.d even 2 1 1089.2.a.p 2
44.c even 2 1 5808.2.a.bx 2
55.d odd 2 1 9075.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.g 2 1.a even 1 1 trivial
363.2.a.g 2 11.b odd 2 1 inner
363.2.e.a 4 11.c even 5 2
363.2.e.a 4 11.d odd 10 2
363.2.e.l 4 11.c even 5 2
363.2.e.l 4 11.d odd 10 2
1089.2.a.p 2 3.b odd 2 1
1089.2.a.p 2 33.d even 2 1
5808.2.a.bx 2 4.b odd 2 1
5808.2.a.bx 2 44.c even 2 1
9075.2.a.bi 2 5.b even 2 1
9075.2.a.bi 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 20 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 20 \) Copy content Toggle raw display
$43$ \( T^{2} - 20 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 80 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 80 \) Copy content Toggle raw display
$79$ \( T^{2} - 180 \) Copy content Toggle raw display
$83$ \( T^{2} - 80 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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