Properties

Label 363.4.d.a
Level $363$
Weight $4$
Character orbit 363.d
Analytic conductor $21.418$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(362,363)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.362");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.d (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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 11 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - 8 q^{4} + (\beta_{3} - 2 \beta_{2}) q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - 8 q^{4} + (\beta_{3} - 2 \beta_{2}) q^{7} + 27 q^{9} - 8 \beta_1 q^{12} + ( - 7 \beta_{3} - \beta_{2}) q^{13} + 64 q^{16} + (7 \beta_{3} - 8 \beta_{2}) q^{19} + (\beta_{3} + 11 \beta_{2}) q^{21} + 125 q^{25} + 27 \beta_1 q^{27} + ( - 8 \beta_{3} + 16 \beta_{2}) q^{28} + 30 \beta_1 q^{31} - 216 q^{36} + 84 \beta_1 q^{37} + ( - 37 \beta_{3} - 2 \beta_{2}) q^{39} + ( - 37 \beta_{3} - 16 \beta_{2}) q^{43} + 64 \beta_1 q^{48} + (120 \beta_1 - 343) q^{49} + (56 \beta_{3} + 8 \beta_{2}) q^{52} + (19 \beta_{3} + 47 \beta_{2}) q^{57} + ( - 49 \beta_{3} + 41 \beta_{2}) q^{61} + (27 \beta_{3} - 54 \beta_{2}) q^{63} - 512 q^{64} - 126 \beta_1 q^{67} + (77 \beta_{3} + 41 \beta_{2}) q^{73} + 125 \beta_1 q^{75} + ( - 56 \beta_{3} + 64 \beta_{2}) q^{76} + ( - 107 \beta_{3} - 2 \beta_{2}) q^{79} + 729 q^{81} + ( - 8 \beta_{3} - 88 \beta_{2}) q^{84} + (90 \beta_1 + 272) q^{91} + 810 q^{93} + 1330 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{4} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{4} + 108 q^{9} + 256 q^{16} + 500 q^{25} - 864 q^{36} - 1372 q^{49} - 2048 q^{64} + 2916 q^{81} + 1088 q^{91} + 3240 q^{93} + 5320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 7\nu^{3} + 27\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{3} - 7\beta_{2} ) / 66 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 9\beta_{2} ) / 22 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1
1.93185i
1.93185i
0.517638i
0.517638i
0 −5.19615 −8.00000 0 0 36.1875i 0 27.0000 0
362.2 0 −5.19615 −8.00000 0 0 36.1875i 0 27.0000 0
362.3 0 5.19615 −8.00000 0 0 7.90327i 0 27.0000 0
362.4 0 5.19615 −8.00000 0 0 7.90327i 0 27.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 363.4.d.a 4
3.b odd 2 1 CM 363.4.d.a 4
11.b odd 2 1 inner 363.4.d.a 4
33.d even 2 1 inner 363.4.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.d.a 4 1.a even 1 1 trivial
363.4.d.a 4 3.b odd 2 1 CM
363.4.d.a 4 11.b odd 2 1 inner
363.4.d.a 4 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1372 T^{2} + 81796 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 8788 T^{2} + 256036 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 27436 T^{2} + \cdots + 111978724 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24300)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 190512)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 318028 T^{2} + \cdots + 12406840996 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 907924 T^{2} + \cdots + 177104622244 \) Copy content Toggle raw display
$67$ \( (T^{2} - 428652)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 1556068 T^{2} + \cdots + 407128220356 \) Copy content Toggle raw display
$79$ \( T^{4} + 1972156 T^{2} + \cdots + 41870162884 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 1330)^{4} \) Copy content Toggle raw display
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