Properties

Label 363.3.b.i
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,3,Mod(122,363)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.122");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), 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: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - 1) q^{3} - 3 q^{4} + \beta_{3} q^{5} + ( - \beta_{2} - \beta_1) q^{6} + \beta_1 q^{7} + \beta_{2} q^{8} + ( - 2 \beta_{3} - 7) q^{9} - \beta_1 q^{10} + ( - 3 \beta_{3} + 3) q^{12}+ \cdots + 7 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 12 q^{4} - 28 q^{9} + 12 q^{12} - 32 q^{15} - 76 q^{16} + 68 q^{25} + 92 q^{27} + 120 q^{31} - 224 q^{34} + 84 q^{36} - 40 q^{37} - 224 q^{42} + 64 q^{45} + 76 q^{48} + 28 q^{49} - 224 q^{58}+ \cdots + 296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} - 2\nu + 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} - 6\nu^{2} + 32\nu - 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} - 6\nu^{2} + 34\nu - 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{3} + 14\beta_{2} + 3\beta _1 - 22 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
0.500000 0.0913379i
0.500000 + 2.73709i
0.500000 2.73709i
0.500000 + 0.0913379i
2.64575i −1.00000 2.82843i −3.00000 2.82843i −7.48331 + 2.64575i 7.48331 2.64575i −7.00000 + 5.65685i −7.48331
122.2 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i 7.48331 + 2.64575i −7.48331 2.64575i −7.00000 5.65685i 7.48331
122.3 2.64575i −1.00000 2.82843i −3.00000 2.82843i 7.48331 2.64575i −7.48331 2.64575i −7.00000 + 5.65685i 7.48331
122.4 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i −7.48331 2.64575i 7.48331 2.64575i −7.00000 5.65685i −7.48331
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.i 4
3.b odd 2 1 inner 363.3.b.i 4
11.b odd 2 1 inner 363.3.b.i 4
11.c even 5 4 363.3.h.k 16
11.d odd 10 4 363.3.h.k 16
33.d even 2 1 inner 363.3.b.i 4
33.f even 10 4 363.3.h.k 16
33.h odd 10 4 363.3.h.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.i 4 1.a even 1 1 trivial
363.3.b.i 4 3.b odd 2 1 inner
363.3.b.i 4 11.b odd 2 1 inner
363.3.b.i 4 33.d even 2 1 inner
363.3.h.k 16 11.c even 5 4
363.3.h.k 16 11.d odd 10 4
363.3.h.k 16 33.f even 10 4
363.3.h.k 16 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{2} + 7 \) Copy content Toggle raw display
\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 56)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 504)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 448)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 224)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 968)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 448)^{2} \) Copy content Toggle raw display
$31$ \( (T - 30)^{4} \) Copy content Toggle raw display
$37$ \( (T + 10)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1792)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 9464)^{2} \) Copy content Toggle raw display
$67$ \( (T + 42)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 5600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 504)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 448)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3872)^{2} \) Copy content Toggle raw display
$97$ \( (T - 74)^{4} \) Copy content Toggle raw display
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