Properties

Label 4205.2.a.g
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} - 1) q^{7} + \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} - 1) q^{7} + \beta_{3} q^{8} - q^{9} + \beta_1 q^{10} + (3 \beta_{3} + \beta_1) q^{11} + ( - \beta_{3} + \beta_1) q^{12} + ( - 2 \beta_{2} - 2) q^{13} + ( - \beta_{3} - 3 \beta_1) q^{14} + (\beta_{3} + \beta_1) q^{15} + ( - 2 \beta_{2} - 1) q^{16} + ( - 3 \beta_{3} + \beta_1) q^{17} - \beta_1 q^{18} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + \beta_{2} q^{20} - 2 \beta_1 q^{21} + (\beta_{2} - 1) q^{22} + (3 \beta_{2} - 3) q^{23} + ( - \beta_{2} + 1) q^{24} + q^{25} + ( - 2 \beta_{3} - 6 \beta_1) q^{26} + ( - 4 \beta_{3} - 4 \beta_1) q^{27} + ( - \beta_{2} - 3) q^{28} + (\beta_{2} + 1) q^{30} + ( - 3 \beta_{3} - 3 \beta_1) q^{31} + ( - 4 \beta_{3} - 5 \beta_1) q^{32} + ( - 2 \beta_{2} + 4) q^{33} + (\beta_{2} + 5) q^{34} + ( - \beta_{2} - 1) q^{35} - \beta_{2} q^{36} + ( - 3 \beta_{3} - 3 \beta_1) q^{37} + ( - 3 \beta_{2} - 3) q^{38} - 4 \beta_1 q^{39} + \beta_{3} q^{40} + (4 \beta_{3} - 2 \beta_1) q^{41} + ( - 2 \beta_{2} - 4) q^{42} + ( - 3 \beta_{3} - 3 \beta_1) q^{43} + ( - 5 \beta_{3} - \beta_1) q^{44} - q^{45} + (3 \beta_{3} + 3 \beta_1) q^{46} + ( - 3 \beta_{3} + 5 \beta_1) q^{47} + (\beta_{3} - 3 \beta_1) q^{48} + (2 \beta_{2} - 3) q^{49} + \beta_1 q^{50} + (4 \beta_{2} - 2) q^{51} + ( - 2 \beta_{2} - 6) q^{52} + ( - 4 \beta_{2} - 4) q^{54} + (3 \beta_{3} + \beta_1) q^{55} + (\beta_{3} + \beta_1) q^{56} - 6 q^{57} + 6 q^{59} + ( - \beta_{3} + \beta_1) q^{60} - 6 \beta_1 q^{61} + ( - 3 \beta_{2} - 3) q^{62} + (\beta_{2} + 1) q^{63} + ( - \beta_{2} - 4) q^{64} + ( - 2 \beta_{2} - 2) q^{65} - 2 \beta_{3} q^{66} + (7 \beta_{2} + 1) q^{67} + (7 \beta_{3} + 5 \beta_1) q^{68} - 6 \beta_{3} q^{69} + ( - \beta_{3} - 3 \beta_1) q^{70} - 6 q^{71} - \beta_{3} q^{72} + (3 \beta_{3} + 9 \beta_1) q^{73} + ( - 3 \beta_{2} - 3) q^{74} + (\beta_{3} + \beta_1) q^{75} + (3 \beta_{3} - 3 \beta_1) q^{76} + 2 \beta_{3} q^{77} + ( - 4 \beta_{2} - 8) q^{78} + (9 \beta_{3} + 3 \beta_1) q^{79} + ( - 2 \beta_{2} - 1) q^{80} - 5 q^{81} + ( - 2 \beta_{2} - 8) q^{82} + ( - 3 \beta_{2} - 3) q^{83} + ( - 2 \beta_{3} - 4 \beta_1) q^{84} + ( - 3 \beta_{3} + \beta_1) q^{85} + ( - 3 \beta_{2} - 3) q^{86} + ( - 3 \beta_{2} + 5) q^{88} + 4 \beta_1 q^{89} - \beta_1 q^{90} + (4 \beta_{2} + 8) q^{91} + ( - 3 \beta_{2} + 9) q^{92} - 6 q^{93} + (5 \beta_{2} + 13) q^{94} + ( - 3 \beta_{3} - 3 \beta_1) q^{95} + ( - \beta_{2} - 9) q^{96} + (3 \beta_{3} - 3 \beta_1) q^{97} + (2 \beta_{3} + \beta_1) q^{98} + ( - 3 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{13} - 4 q^{16} - 4 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{25} - 12 q^{28} + 4 q^{30} + 16 q^{33} + 20 q^{34} - 4 q^{35} - 12 q^{38} - 16 q^{42} - 4 q^{45} - 12 q^{49} - 8 q^{51} - 24 q^{52} - 16 q^{54} - 24 q^{57} + 24 q^{59} - 12 q^{62} + 4 q^{63} - 16 q^{64} - 8 q^{65} + 4 q^{67} - 24 q^{71} - 12 q^{74} - 32 q^{78} - 4 q^{80} - 20 q^{81} - 32 q^{82} - 12 q^{83} - 12 q^{86} + 20 q^{88} + 32 q^{91} + 36 q^{92} - 24 q^{93} + 52 q^{94} - 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−1.93185
−0.517638
0.517638
1.93185
−1.93185 −1.41421 1.73205 1.00000 2.73205 −2.73205 0.517638 −1.00000 −1.93185
1.2 −0.517638 1.41421 −1.73205 1.00000 −0.732051 0.732051 1.93185 −1.00000 −0.517638
1.3 0.517638 −1.41421 −1.73205 1.00000 −0.732051 0.732051 −1.93185 −1.00000 0.517638
1.4 1.93185 1.41421 1.73205 1.00000 2.73205 −2.73205 −0.517638 −1.00000 1.93185
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4205.2.a.g 4
29.b even 2 1 inner 4205.2.a.g 4
29.c odd 4 2 145.2.c.a 4
87.f even 4 2 1305.2.d.a 4
116.e even 4 2 2320.2.g.e 4
145.e even 4 2 725.2.d.b 8
145.f odd 4 2 725.2.c.d 4
145.j even 4 2 725.2.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.a 4 29.c odd 4 2
725.2.c.d 4 145.f odd 4 2
725.2.d.b 8 145.e even 4 2
725.2.d.b 8 145.j even 4 2
1305.2.d.a 4 87.f even 4 2
2320.2.g.e 4 116.e even 4 2
4205.2.a.g 4 1.a even 1 1 trivial
4205.2.a.g 4 29.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 28T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 52T^{2} + 484 \) Copy content Toggle raw display
$19$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 112T^{2} + 2704 \) Copy content Toggle raw display
$43$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 196T^{2} + 8836 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T - 6)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 144T^{2} + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T - 146)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 252T^{2} + 324 \) Copy content Toggle raw display
$79$ \( T^{4} - 252T^{2} + 324 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 64T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
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