Properties

Label 483.2.d.a
Level $483$
Weight $2$
Character orbit 483.d
Analytic conductor $3.857$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(461,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.461");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_{2} - 1) q^{6} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_{3} + \beta_{2} - 1) q^{6} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{9} + (\beta_{3} - 2 \beta_1) q^{10} + ( - \beta_{3} - 2 \beta_1) q^{11} + (\beta_{3} - \beta_{2} - 2) q^{12} + ( - 2 \beta_{3} + 3 \beta_1) q^{13} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{14} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{15} + 3 \beta_{2} q^{16} + (4 \beta_{2} + 5) q^{17} + (2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{18} + ( - \beta_{3} - 4 \beta_1) q^{19} - \beta_{2} q^{20} + (2 \beta_{3} + 4 \beta_1 + 1) q^{21} + ( - \beta_{2} + 2) q^{22} + \beta_{3} q^{23} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{24} + ( - 3 \beta_{2} - 3) q^{25} + (5 \beta_{2} - 3) q^{26} + (4 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{27} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{28} + ( - 5 \beta_{3} - 2 \beta_1) q^{29} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{30} + (7 \beta_{3} + 4 \beta_1) q^{31} + (5 \beta_{3} + \beta_1) q^{32} + (3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{33} + (4 \beta_{3} + \beta_1) q^{34} + (\beta_{3} + 3 \beta_{2} - 2) q^{35} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{36} + (2 \beta_{2} + 9) q^{37} + ( - 3 \beta_{2} + 4) q^{38} + ( - \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 3) q^{39} + (\beta_{3} - 3 \beta_1) q^{40} + (2 \beta_{2} + 3) q^{41} + (2 \beta_{2} + \beta_1 - 4) q^{42} + ( - \beta_{2} + 6) q^{43} + ( - 3 \beta_{3} - \beta_1) q^{44} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{45} - \beta_{2} q^{46} + 8 q^{47} + (3 \beta_{3} - 3 \beta_1 - 3) q^{48} + ( - 2 \beta_{3} - 6 \beta_{2} - 3) q^{49} - 3 \beta_{3} q^{50} + (4 \beta_{3} - 5 \beta_{2} + \beta_1 - 9) q^{51} + (\beta_{3} - 2 \beta_1) q^{52} + (\beta_{3} + 9 \beta_1) q^{53} + (\beta_{3} - 3 \beta_{2} - 4 \beta_1 - 1) q^{54} + ( - \beta_{3} + 3 \beta_1) q^{55} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 4) q^{56} + (5 \beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{57} + (3 \beta_{2} + 2) q^{58} + ( - 9 \beta_{2} - 7) q^{59} + ( - \beta_{3} + \beta_1 + 1) q^{60} + ( - 4 \beta_{3} - 3 \beta_1) q^{61} + ( - 3 \beta_{2} - 4) q^{62} + ( - 6 \beta_{3} + \beta_{2} - \beta_1 - 5) q^{63} + (2 \beta_{2} - 1) q^{64} + (5 \beta_{3} - 8 \beta_1) q^{65} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{66} + (\beta_{2} - 9) q^{67} + (5 \beta_{2} + 9) q^{68} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{69} + (3 \beta_{3} - \beta_{2} - 5 \beta_1) q^{70} + ( - 6 \beta_{3} - 7 \beta_1) q^{71} + (5 \beta_{3} + 4 \beta_{2} + 2) q^{72} + ( - 5 \beta_{3} + 2 \beta_1) q^{73} + (2 \beta_{3} + 7 \beta_1) q^{74} + ( - 3 \beta_{3} + 3 \beta_{2} + 6) q^{75} + ( - 5 \beta_{3} - \beta_1) q^{76} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1 - 4) q^{77} + (5 \beta_{3} + 3 \beta_{2} - 8 \beta_1 - 2) q^{78} + ( - 2 \beta_{2} - 11) q^{79} + ( - 6 \beta_{2} + 3) q^{80} + ( - 4 \beta_{3} - 8 \beta_1 + 1) q^{81} + (2 \beta_{3} + \beta_1) q^{82} + ( - 6 \beta_{2} + 1) q^{83} + (6 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{84} + ( - 3 \beta_{2} - 1) q^{85} + ( - \beta_{3} + 7 \beta_1) q^{86} + (7 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 2) q^{87} + 5 q^{88} + 5 \beta_{2} q^{89} + ( - 3 \beta_{3} - 4 \beta_{2} + \cdots + 2) q^{90}+ \cdots + ( - 5 \beta_{3} - 4 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} + 6 q^{7} - 6 q^{12} + 2 q^{14} - 2 q^{15} - 6 q^{16} + 12 q^{17} - 4 q^{18} + 2 q^{20} + 4 q^{21} + 10 q^{22} - 10 q^{24} - 6 q^{25} - 22 q^{26} - 14 q^{27} - 2 q^{28} + 14 q^{30} + 10 q^{33} - 14 q^{35} + 10 q^{36} + 32 q^{37} + 22 q^{38} - 22 q^{39} + 8 q^{41} - 20 q^{42} + 26 q^{43} + 10 q^{45} + 2 q^{46} + 32 q^{47} - 12 q^{48} - 26 q^{51} + 2 q^{54} + 10 q^{56} + 22 q^{57} + 2 q^{58} - 10 q^{59} + 4 q^{60} - 10 q^{62} - 22 q^{63} - 8 q^{64} - 38 q^{67} + 26 q^{68} + 2 q^{69} + 2 q^{70} + 18 q^{75} - 10 q^{77} - 14 q^{78} - 40 q^{79} + 24 q^{80} + 4 q^{81} + 16 q^{83} + 2 q^{84} + 2 q^{85} + 2 q^{87} + 20 q^{88} - 10 q^{89} + 16 q^{90} - 6 q^{91} - 10 q^{93} + 4 q^{96} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 0.618034 1.61803i −0.618034 −2.61803 −2.61803 1.00000i 2.61803 0.381966i 2.23607i −2.23607 2.00000i 4.23607i
461.2 0.618034i −1.61803 0.618034i 1.61803 −0.381966 −0.381966 + 1.00000i 0.381966 + 2.61803i 2.23607i 2.23607 + 2.00000i 0.236068i
461.3 0.618034i −1.61803 + 0.618034i 1.61803 −0.381966 −0.381966 1.00000i 0.381966 2.61803i 2.23607i 2.23607 2.00000i 0.236068i
461.4 1.61803i 0.618034 + 1.61803i −0.618034 −2.61803 −2.61803 + 1.00000i 2.61803 + 0.381966i 2.23607i −2.23607 + 2.00000i 4.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.d.a 4
3.b odd 2 1 483.2.d.b yes 4
7.b odd 2 1 483.2.d.b yes 4
21.c even 2 1 inner 483.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.d.a 4 1.a even 1 1 trivial
483.2.d.a 4 21.c even 2 1 inner
483.2.d.b yes 4 3.b odd 2 1
483.2.d.b yes 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 47T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 11)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 42T^{2} + 361 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 42T^{2} + 121 \) Copy content Toggle raw display
$31$ \( T^{4} + 90T^{2} + 25 \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T + 59)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 13 T + 41)^{2} \) Copy content Toggle raw display
$47$ \( (T - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 227T^{2} + 7921 \) Copy content Toggle raw display
$59$ \( (T^{2} + 5 T - 95)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$67$ \( (T^{2} + 19 T + 89)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 135T^{2} + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} + 82T^{2} + 961 \) Copy content Toggle raw display
$79$ \( (T^{2} + 20 T + 95)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 29)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5 T - 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 258T^{2} + 961 \) Copy content Toggle raw display
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