Properties

Label 4205.2.a.l
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $6$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.5516125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{6}+ \cdots + (2 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - q^{3} + 7 q^{4} - 6 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - q^{3} + 7 q^{4} - 6 q^{5} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 9 q^{9} + q^{10} + 7 q^{11} + 10 q^{14} + q^{15} + 5 q^{16} - 7 q^{17} - 15 q^{18} + 25 q^{19} - 7 q^{20} + 34 q^{21} - 14 q^{22} + 6 q^{23} - 5 q^{24} + 6 q^{25} + 11 q^{26} + 14 q^{27} - 13 q^{28} + 2 q^{30} + q^{31} + 2 q^{32} - 19 q^{33} + 26 q^{34} - 6 q^{35} - 17 q^{36} + 24 q^{37} - 30 q^{38} + 6 q^{40} - 15 q^{41} - 43 q^{42} + 6 q^{43} + 20 q^{44} - 9 q^{45} + 32 q^{46} - 5 q^{47} - 18 q^{48} + 48 q^{49} - q^{50} + 18 q^{51} - 20 q^{52} - 18 q^{53} - 6 q^{54} - 7 q^{55} - 16 q^{56} + 4 q^{57} - 8 q^{59} + 20 q^{61} + 9 q^{62} + 25 q^{63} - 4 q^{64} - 21 q^{66} - 11 q^{67} + 8 q^{68} - 31 q^{69} - 10 q^{70} - 3 q^{71} + 6 q^{72} + 39 q^{73} + 41 q^{74} - q^{75} + 23 q^{76} - 5 q^{77} + 9 q^{78} + 21 q^{79} - 5 q^{80} - 6 q^{81} + 13 q^{82} - 25 q^{83} + 22 q^{84} + 7 q^{85} - 3 q^{86} - 25 q^{88} + 10 q^{89} + 15 q^{90} - 11 q^{91} + 27 q^{92} - 18 q^{93} - 36 q^{94} - 25 q^{95} + 59 q^{96} + 3 q^{97} - 38 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 3x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 12\nu^{3} - 12\nu^{2} + 29\nu + 12 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 6\nu^{2} + 2\nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 15\nu^{3} - 30\nu^{2} - 22\nu + 12 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} - 15\nu^{3} + 39\nu^{2} + 13\nu - 39 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{4} + 10\beta_{3} - 5\beta_{2} + 29\beta _1 + 6 \) 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
2.55945
2.03714
0.767206
−0.796555
−1.18631
−2.38093
−2.55945 −1.19450 4.55078 −1.00000 3.05727 −3.60270 −6.52860 −1.57316 2.55945
1.2 −2.03714 2.86066 2.14995 −1.00000 −5.82757 4.62407 −0.305468 5.18336 2.03714
1.3 −0.767206 −2.90172 −1.41139 −1.00000 2.22622 −3.57594 2.61724 5.41997 0.767206
1.4 0.796555 2.46196 −1.36550 −1.00000 1.96109 4.99211 −2.68081 3.06124 −0.796555
1.5 1.18631 −1.57697 −0.592657 −1.00000 −1.87079 4.18793 −3.07571 −0.513156 −1.18631
1.6 2.38093 −0.649421 3.66882 −1.00000 −1.54622 −0.625476 3.97334 −2.57825 −2.38093
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(29\) \( -1 \)

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 4205.2.a.l 6
29.b even 2 1 4205.2.a.n yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.l 6 1.a even 1 1 trivial
4205.2.a.n yes 6 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{6} + T_{2}^{5} - 9T_{2}^{4} - 6T_{2}^{3} + 20T_{2}^{2} + 3T_{2} - 9 \) Copy content Toggle raw display
\( T_{3}^{6} + T_{3}^{5} - 13T_{3}^{4} - 16T_{3}^{3} + 36T_{3}^{2} + 65T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} - 9 T^{4} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots - 779 \) Copy content Toggle raw display
$11$ \( T^{6} - 7 T^{5} + \cdots + 171 \) Copy content Toggle raw display
$13$ \( T^{6} - 32 T^{4} + \cdots - 229 \) Copy content Toggle raw display
$17$ \( T^{6} + 7 T^{5} + \cdots + 3609 \) Copy content Toggle raw display
$19$ \( T^{6} - 25 T^{5} + \cdots - 1721 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + \cdots - 9 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - T^{5} + \cdots - 331 \) Copy content Toggle raw display
$37$ \( T^{6} - 24 T^{5} + \cdots + 190445 \) Copy content Toggle raw display
$41$ \( T^{6} + 15 T^{5} + \cdots + 2349 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} + \cdots - 4019 \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{5} + \cdots + 8091 \) Copy content Toggle raw display
$53$ \( T^{6} + 18 T^{5} + \cdots + 142461 \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{5} + \cdots + 178281 \) Copy content Toggle raw display
$61$ \( T^{6} - 20 T^{5} + \cdots + 9791 \) Copy content Toggle raw display
$67$ \( T^{6} + 11 T^{5} + \cdots - 3769 \) Copy content Toggle raw display
$71$ \( T^{6} + 3 T^{5} + \cdots - 30339 \) Copy content Toggle raw display
$73$ \( T^{6} - 39 T^{5} + \cdots - 5039 \) Copy content Toggle raw display
$79$ \( T^{6} - 21 T^{5} + \cdots + 2083195 \) Copy content Toggle raw display
$83$ \( T^{6} + 25 T^{5} + \cdots + 4689 \) Copy content Toggle raw display
$89$ \( T^{6} - 10 T^{5} + \cdots - 25821 \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + \cdots - 124471 \) Copy content Toggle raw display
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