Properties

Label 4100.2.a.d
Level $4100$
Weight $2$
Character orbit 4100.a
Self dual yes
Analytic conductor $32.739$
Analytic rank $1$
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] = [4100,2,Mod(1,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, 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("4100.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.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: \(32.7386648287\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 820)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_1 q^{7} + (\beta_{3} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_1 q^{7} + (\beta_{3} - \beta_1 + 1) q^{9} + \beta_{2} q^{11} + ( - \beta_1 - 2) q^{13} + (\beta_{2} - 2) q^{17} + ( - 2 \beta_{3} + \beta_1) q^{19} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{21} + (\beta_{3} - \beta_1 - 2) q^{23} + ( - \beta_{3} - \beta_1) q^{27} + ( - \beta_{3} - \beta_{2} + 2) q^{29} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{31} + ( - \beta_{3} + \beta_1 - 4) q^{33} + ( - \beta_{3} + \beta_{2} - 2) q^{37} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{39} + q^{41} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{43} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} + \beta_1 + 1) q^{49} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{51} + ( - \beta_{3} + \beta_{2} + \beta_1 - 6) q^{53} + (3 \beta_{3} + 3 \beta_{2}) q^{57} + ( - \beta_{3} - \beta_{2}) q^{59} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{61} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{63} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{67} + ( - \beta_{3} - \beta_1) q^{69} + ( - \beta_{2} - 2 \beta_1) q^{71} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{73} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{77} + (3 \beta_{3} + 2 \beta_1) q^{79} - 3 q^{81} + (3 \beta_{2} - \beta_1 - 6) q^{83} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{87} + (3 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{89} + (3 \beta_{2} - 3 \beta_1 - 8) q^{91} + ( - 4 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{93} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{97} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{7} + 6 q^{9} - 7 q^{13} - 8 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} + 7 q^{29} + 3 q^{31} - 18 q^{33} - 9 q^{37} + 3 q^{39} + 4 q^{41} - 5 q^{43} - 3 q^{47} + 3 q^{49} - 18 q^{51} - 26 q^{53} + 3 q^{57} - q^{59} - 8 q^{61} - 18 q^{63} - q^{67} + 2 q^{71} - 9 q^{73} + 3 q^{77} + q^{79} - 12 q^{81} - 23 q^{83} + 21 q^{87} + 10 q^{89} - 29 q^{91} - 21 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 2\nu^{2} + 5\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} + 9\beta_{2} - 5\beta _1 + 8 ) / 2 \) 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
3.04374
−0.548230
−1.82405
0.328543
0 −2.71519 0 0 0 0.176865 0 4.37228 0
1.2 0 −1.27582 0 0 0 −1.60298 0 −1.37228 0
1.3 0 1.27582 0 0 0 3.97526 0 −1.37228 0
1.4 0 2.71519 0 0 0 −3.54915 0 4.37228 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(41\) \(-1\)

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 4100.2.a.d 4
5.b even 2 1 820.2.a.d 4
5.c odd 4 2 4100.2.d.e 8
15.d odd 2 1 7380.2.a.t 4
20.d odd 2 1 3280.2.a.be 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.a.d 4 5.b even 2 1
3280.2.a.be 4 20.d odd 2 1
4100.2.a.d 4 1.a even 1 1 trivial
4100.2.d.e 8 5.c odd 4 2
7380.2.a.t 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9T_{3}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4100))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 9T^{2} + 12 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 15 T^{2} - 20 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + 3 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 15 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} - 45 T^{2} - 108 T + 108 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 7 T^{3} - 9 T^{2} + 128 T - 164 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} - 57 T^{2} - 72 T + 48 \) Copy content Toggle raw display
$37$ \( T^{4} + 9 T^{3} + 9 T^{2} - 96 T - 204 \) Copy content Toggle raw display
$41$ \( (T - 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} - 45 T^{2} + 8 T + 124 \) Copy content Toggle raw display
$47$ \( T^{4} + 3 T^{3} - 87 T^{2} + 324 T - 348 \) Copy content Toggle raw display
$53$ \( T^{4} + 26 T^{3} + 228 T^{2} + \cdots + 928 \) Copy content Toggle raw display
$59$ \( T^{4} + T^{3} - 27 T^{2} + 40 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 45 T^{2} - 244 T - 68 \) Copy content Toggle raw display
$67$ \( T^{4} + T^{3} - 123 T^{2} - 404 T - 332 \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} - 75 T^{2} - 56 T + 388 \) Copy content Toggle raw display
$73$ \( T^{4} + 9 T^{3} - 279 T^{2} + \cdots + 20964 \) Copy content Toggle raw display
$79$ \( T^{4} - T^{3} - 285 T^{2} + \cdots + 16204 \) Copy content Toggle raw display
$83$ \( T^{4} + 23 T^{3} + 111 T^{2} + \cdots - 164 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} - 171 T^{2} + \cdots - 4652 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} - 228 T^{2} + \cdots + 3936 \) Copy content Toggle raw display
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