Properties

Label 4205.2.a.i
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.942577.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 8x^{2} + 5x - 3 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{4} - \beta_1 + 1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{4} + \beta_{3} + 2) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{4} - \beta_1 + 1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{4} + \beta_{3} + 2) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 3) q^{9} + \beta_1 q^{10} + (2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 2) q^{14}+ \cdots + ( - 2 \beta_{4} + 8 \beta_{3} + \cdots + 16) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} + 2 q^{3} + 5 q^{4} - 5 q^{5} + 10 q^{6} + q^{7} + 6 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} + 2 q^{3} + 5 q^{4} - 5 q^{5} + 10 q^{6} + q^{7} + 6 q^{8} + 13 q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - 8 q^{13} + 11 q^{14} - 2 q^{15} + 5 q^{16} + q^{17} - 4 q^{18} + 9 q^{19} - 5 q^{20} + 5 q^{21} - 13 q^{22} - 2 q^{23} + 7 q^{24} + 5 q^{25} + 17 q^{26} + 5 q^{27} - 6 q^{28} - 10 q^{30} - 2 q^{31} + 21 q^{32} + 2 q^{33} - 15 q^{34} - q^{35} - 5 q^{36} + 4 q^{37} + 42 q^{38} + 24 q^{39} - 6 q^{40} + 22 q^{41} - 17 q^{42} + 24 q^{43} - 9 q^{44} - 13 q^{45} + 6 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} - q^{50} - 10 q^{51} + 22 q^{52} + 16 q^{53} + 6 q^{54} - 2 q^{55} + 31 q^{56} + 18 q^{57} - 4 q^{59} - 2 q^{60} + 23 q^{61} - 16 q^{63} - 10 q^{64} + 8 q^{65} + 33 q^{66} + 3 q^{67} - 4 q^{68} + 16 q^{69} - 11 q^{70} - 24 q^{71} - 21 q^{72} - 32 q^{73} - 2 q^{74} + 2 q^{75} - q^{76} - 55 q^{77} - 10 q^{78} + 9 q^{79} - 5 q^{80} - 15 q^{81} + 14 q^{82} + 20 q^{83} + 67 q^{84} - q^{85} + 12 q^{86} - 64 q^{88} - 14 q^{89} + 4 q^{90} - 4 q^{91} + 53 q^{92} - 28 q^{93} - 39 q^{94} - 9 q^{95} + 31 q^{96} + 13 q^{97} - 28 q^{98} + 59 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 7x^{3} + 8x^{2} + 5x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} - 2\beta _1 + 15 \) 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
2.17577
1.71192
0.423235
−0.740289
−2.57064
−2.17577 −2.53401 2.73398 −1.00000 5.51343 −4.33096 −1.59698 3.42121 2.17577
1.2 −1.71192 1.85943 0.930675 −1.00000 −3.18320 0.899925 1.83060 0.457478 1.71192
1.3 −0.423235 −2.22702 −1.82087 −1.00000 0.942555 3.43800 1.61713 1.95964 0.423235
1.4 0.740289 3.20870 −1.45197 −1.00000 2.37536 −1.10348 −2.55546 7.29575 −0.740289
1.5 2.57064 1.69291 4.60819 −1.00000 4.35185 2.09652 6.70471 −0.134068 −2.57064
\(n\): 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.i 5
29.b even 2 1 4205.2.a.j yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.i 5 1.a even 1 1 trivial
4205.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{5} + T_{2}^{4} - 7T_{2}^{3} - 8T_{2}^{2} + 5T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{5} - 2T_{3}^{4} - 12T_{3}^{3} + 21T_{3}^{2} + 34T_{3} - 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 7 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} + \cdots - 57 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} + \cdots - 31 \) Copy content Toggle raw display
$11$ \( T^{5} - 2 T^{4} + \cdots + 1323 \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} + \cdots - 17 \) Copy content Toggle raw display
$17$ \( T^{5} - T^{4} - 7 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$19$ \( T^{5} - 9 T^{4} + \cdots - 7297 \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} + \cdots - 795 \) Copy content Toggle raw display
$29$ \( T^{5} \) Copy content Toggle raw display
$31$ \( T^{5} + 2 T^{4} + \cdots + 1835 \) Copy content Toggle raw display
$37$ \( T^{5} - 4 T^{4} + \cdots - 239 \) Copy content Toggle raw display
$41$ \( T^{5} - 22 T^{4} + \cdots - 147 \) Copy content Toggle raw display
$43$ \( T^{5} - 24 T^{4} + \cdots + 8045 \) Copy content Toggle raw display
$47$ \( T^{5} + 12 T^{4} + \cdots - 6939 \) Copy content Toggle raw display
$53$ \( T^{5} - 16 T^{4} + \cdots - 171 \) Copy content Toggle raw display
$59$ \( T^{5} + 4 T^{4} + \cdots + 5301 \) Copy content Toggle raw display
$61$ \( T^{5} - 23 T^{4} + \cdots - 13779 \) Copy content Toggle raw display
$67$ \( T^{5} - 3 T^{4} + \cdots - 307 \) Copy content Toggle raw display
$71$ \( T^{5} + 24 T^{4} + \cdots - 33075 \) Copy content Toggle raw display
$73$ \( T^{5} + 32 T^{4} + \cdots - 7929 \) Copy content Toggle raw display
$79$ \( T^{5} - 9 T^{4} + \cdots + 11129 \) Copy content Toggle raw display
$83$ \( T^{5} - 20 T^{4} + \cdots - 7167 \) Copy content Toggle raw display
$89$ \( T^{5} + 14 T^{4} + \cdots + 1773 \) Copy content Toggle raw display
$97$ \( T^{5} - 13 T^{4} + \cdots - 17401 \) Copy content Toggle raw display
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