Properties

Label 4205.2.a.h
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1586009.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} + 2) q^{4} + q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{8} + (\beta_{4} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{4} q^{3} + ( - \beta_{4} + \beta_{3} + 2) q^{4} + q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{8} + (\beta_{4} + \beta_{2} + \beta_1) q^{9} - \beta_1 q^{10} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{12}+ \cdots + ( - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 2 q^{3} + 9 q^{4} + 5 q^{5} - 6 q^{6} - 3 q^{7} - 12 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} - 2 q^{3} + 9 q^{4} + 5 q^{5} - 6 q^{6} - 3 q^{7} - 12 q^{8} + 5 q^{9} - q^{10} - 2 q^{11} + 10 q^{12} - q^{14} - 2 q^{15} + 21 q^{16} - 19 q^{17} - 6 q^{18} + q^{19} + 9 q^{20} + 9 q^{21} + 7 q^{22} + 4 q^{23} - 25 q^{24} + 5 q^{25} - 19 q^{26} - 11 q^{27} - 18 q^{28} - 6 q^{30} - 18 q^{31} - 15 q^{32} + 8 q^{33} - 15 q^{34} - 3 q^{35} - 9 q^{36} - 2 q^{37} - 6 q^{38} - 20 q^{39} - 12 q^{40} - 2 q^{41} + 43 q^{42} - 22 q^{43} + 27 q^{44} + 5 q^{45} - 26 q^{46} + 2 q^{47} + 40 q^{48} + 14 q^{49} - q^{50} + 14 q^{51} - 32 q^{52} - 14 q^{53} - 6 q^{54} - 2 q^{55} + 23 q^{56} - 6 q^{57} + 20 q^{59} + 10 q^{60} - 5 q^{61} + 10 q^{62} - 36 q^{63} + 38 q^{64} - 19 q^{66} - 21 q^{67} - 22 q^{68} + 24 q^{69} - q^{70} - 4 q^{71} - 43 q^{72} - 28 q^{73} + 34 q^{74} - 2 q^{75} + 31 q^{76} - 13 q^{77} + 28 q^{78} - 3 q^{79} + 21 q^{80} + q^{81} + 8 q^{83} - 69 q^{84} - 19 q^{85} + 24 q^{86} + 20 q^{88} + 34 q^{89} - 6 q^{90} + 8 q^{91} + 33 q^{92} - 12 q^{93} + 5 q^{94} + q^{95} - 93 q^{96} - 9 q^{97} - 2 q^{98} - 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 8\nu^{2} + 11\nu + 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{4} + 8\beta_{3} + 2\beta_{2} + \beta _1 + 24 \) 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
2.81969
2.03739
−0.594278
−0.849666
−2.41314
−2.81969 2.21248 5.95067 1.00000 −6.23850 −3.02603 −11.1397 1.89505 −2.81969
1.2 −2.03739 −1.51980 2.15097 1.00000 3.09643 2.57663 −0.307585 −0.690207 −2.03739
1.3 0.594278 −3.18209 −1.64683 1.00000 −1.89105 −4.07314 −2.16723 7.12569 0.594278
1.4 0.849666 1.37380 −1.27807 1.00000 1.16727 3.54107 −2.78526 −1.11267 0.849666
1.5 2.41314 −0.884387 3.82326 1.00000 −2.13415 −2.01854 4.39978 −2.21786 2.41314
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.h 5
29.b even 2 1 4205.2.a.k yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.h 5 1.a even 1 1 trivial
4205.2.a.k yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{5} + T_{2}^{4} - 9T_{2}^{3} - 4T_{2}^{2} + 17T_{2} - 7 \) Copy content Toggle raw display
\( T_{3}^{5} + 2T_{3}^{4} - 8T_{3}^{3} - 11T_{3}^{2} + 12T_{3} + 13 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 9 T^{3} + \cdots - 7 \) Copy content Toggle raw display
$3$ \( T^{5} + 2 T^{4} + \cdots + 13 \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} + \cdots + 227 \) Copy content Toggle raw display
$11$ \( T^{5} + 2 T^{4} + \cdots + 67 \) Copy content Toggle raw display
$13$ \( T^{5} - 45 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{5} + 19 T^{4} + \cdots + 331 \) Copy content Toggle raw display
$19$ \( T^{5} - T^{4} + \cdots - 17 \) Copy content Toggle raw display
$23$ \( T^{5} - 4 T^{4} + \cdots - 89 \) Copy content Toggle raw display
$29$ \( T^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 18 T^{4} + \cdots + 211 \) Copy content Toggle raw display
$37$ \( T^{5} + 2 T^{4} + \cdots - 9 \) Copy content Toggle raw display
$41$ \( T^{5} + 2 T^{4} + \cdots - 567 \) Copy content Toggle raw display
$43$ \( T^{5} + 22 T^{4} + \cdots - 1053 \) Copy content Toggle raw display
$47$ \( T^{5} - 2 T^{4} + \cdots - 721 \) Copy content Toggle raw display
$53$ \( T^{5} + 14 T^{4} + \cdots - 6197 \) Copy content Toggle raw display
$59$ \( T^{5} - 20 T^{4} + \cdots + 801 \) Copy content Toggle raw display
$61$ \( T^{5} + 5 T^{4} + \cdots + 38853 \) Copy content Toggle raw display
$67$ \( T^{5} + 21 T^{4} + \cdots + 1267 \) Copy content Toggle raw display
$71$ \( T^{5} + 4 T^{4} + \cdots + 2689 \) Copy content Toggle raw display
$73$ \( T^{5} + 28 T^{4} + \cdots - 65511 \) Copy content Toggle raw display
$79$ \( T^{5} + 3 T^{4} + \cdots + 1053 \) Copy content Toggle raw display
$83$ \( T^{5} - 8 T^{4} + \cdots + 24479 \) Copy content Toggle raw display
$89$ \( T^{5} - 34 T^{4} + \cdots + 448693 \) Copy content Toggle raw display
$97$ \( T^{5} + 9 T^{4} + \cdots + 2273 \) Copy content Toggle raw display
show more
show less