Properties

Label 3822.2.a.bz
Level $3822$
Weight $2$
Character orbit 3822.a
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $4$
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] = [3822,2,Mod(1,3822)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3822, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3822.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.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: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.31808.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 11x^{2} + 12x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
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 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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + q^{8} + q^{9} + \beta_1 q^{10} + (\beta_{3} + 2) q^{11} - q^{12} - q^{13} - \beta_1 q^{15} + q^{16} + \beta_{3} q^{17} + q^{18} + \beta_1 q^{19} + \beta_1 q^{20} + (\beta_{3} + 2) q^{22} + (\beta_1 + 2) q^{23} - q^{24} + (2 \beta_{2} + \beta_1 + 1) q^{25} - q^{26} - q^{27} + ( - \beta_{3} - \beta_{2} + 2) q^{29} - \beta_1 q^{30} + ( - \beta_{3} - \beta_1) q^{31} + q^{32} + ( - \beta_{3} - 2) q^{33} + \beta_{3} q^{34} + q^{36} + ( - 3 \beta_{2} - \beta_1 + 2) q^{37} + \beta_1 q^{38} + q^{39} + \beta_1 q^{40} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{41} + ( - 2 \beta_{3} - \beta_1 + 2) q^{43} + (\beta_{3} + 2) q^{44} + \beta_1 q^{45} + (\beta_1 + 2) q^{46} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{47} - q^{48} + (2 \beta_{2} + \beta_1 + 1) q^{50} - \beta_{3} q^{51} - q^{52} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{53} - q^{54} + (\beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{55} - \beta_1 q^{57} + ( - \beta_{3} - \beta_{2} + 2) q^{58} + ( - \beta_{3} - \beta_1) q^{59} - \beta_1 q^{60} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{61} + ( - \beta_{3} - \beta_1) q^{62} + q^{64} - \beta_1 q^{65} + ( - \beta_{3} - 2) q^{66} + (\beta_{2} + 2 \beta_1 + 6) q^{67} + \beta_{3} q^{68} + ( - \beta_1 - 2) q^{69} + ( - \beta_{3} + 3 \beta_1 + 2) q^{71} + q^{72} + ( - \beta_1 + 4) q^{73} + ( - 3 \beta_{2} - \beta_1 + 2) q^{74} + ( - 2 \beta_{2} - \beta_1 - 1) q^{75} + \beta_1 q^{76} + q^{78} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{79} + \beta_1 q^{80} + q^{81} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{82} + (4 \beta_{2} - 2 \beta_1) q^{83} + (\beta_{3} + 4 \beta_{2} - 2) q^{85} + ( - 2 \beta_{3} - \beta_1 + 2) q^{86} + (\beta_{3} + \beta_{2} - 2) q^{87} + (\beta_{3} + 2) q^{88} + (2 \beta_{3} - 3 \beta_{2}) q^{89} + \beta_1 q^{90} + (\beta_1 + 2) q^{92} + (\beta_{3} + \beta_1) q^{93} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{94} + (2 \beta_{2} + \beta_1 + 6) q^{95} - q^{96} + ( - 2 \beta_{3} + 2 \beta_{2} - 2) q^{97} + (\beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 6 q^{11} - 4 q^{12} - 4 q^{13} - 2 q^{15} + 4 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} + 6 q^{22} + 10 q^{23} - 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 10 q^{29} - 2 q^{30} + 4 q^{32} - 6 q^{33} - 2 q^{34} + 4 q^{36} + 6 q^{37} + 2 q^{38} + 4 q^{39} + 2 q^{40} + 10 q^{43} + 6 q^{44} + 2 q^{45} + 10 q^{46} + 8 q^{47} - 4 q^{48} + 6 q^{50} + 2 q^{51} - 4 q^{52} + 16 q^{53} - 4 q^{54} - 6 q^{55} - 2 q^{57} + 10 q^{58} - 2 q^{60} - 2 q^{61} + 4 q^{64} - 2 q^{65} - 6 q^{66} + 28 q^{67} - 2 q^{68} - 10 q^{69} + 16 q^{71} + 4 q^{72} + 14 q^{73} + 6 q^{74} - 6 q^{75} + 2 q^{76} + 4 q^{78} + 8 q^{79} + 2 q^{80} + 4 q^{81} - 4 q^{83} - 10 q^{85} + 10 q^{86} - 10 q^{87} + 6 q^{88} - 4 q^{89} + 2 q^{90} + 10 q^{92} + 8 q^{94} + 26 q^{95} - 4 q^{96} - 4 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 11x^{2} + 12x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - \nu^{2} - 8\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 9\beta _1 + 6 \) 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
−2.51304
−1.34975
2.34975
3.51304
1.00000 −1.00000 1.00000 −2.51304 −1.00000 0 1.00000 1.00000 −2.51304
1.2 1.00000 −1.00000 1.00000 −1.34975 −1.00000 0 1.00000 1.00000 −1.34975
1.3 1.00000 −1.00000 1.00000 2.34975 −1.00000 0 1.00000 1.00000 2.34975
1.4 1.00000 −1.00000 1.00000 3.51304 −1.00000 0 1.00000 1.00000 3.51304
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( +1 \)

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 3822.2.a.bz 4
7.b odd 2 1 3822.2.a.ca yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3822.2.a.bz 4 1.a even 1 1 trivial
3822.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 12T_{5} + 28 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} - 9T_{11}^{2} + 80T_{11} - 64 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 21T_{17}^{2} + 4T_{17} + 28 \) Copy content Toggle raw display
\( T_{29}^{4} - 10T_{29}^{3} + 7T_{29}^{2} + 12T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 28 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 28 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 28 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$31$ \( T^{4} - 26 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots - 46 \) Copy content Toggle raw display
$41$ \( T^{4} - 26 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 1016 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} + \cdots - 376 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$59$ \( T^{4} - 26 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots - 316 \) Copy content Toggle raw display
$67$ \( T^{4} - 28 T^{3} + \cdots + 604 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + \cdots + 3896 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + \cdots + 28 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} + \cdots + 56 \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} + \cdots - 448 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots - 644 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + \cdots + 448 \) Copy content Toggle raw display
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