Properties

Label 4205.2.a.z
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $36$
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: \(0\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 50 q^{4} - 36 q^{5} + 2 q^{6} + 14 q^{7} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 50 q^{4} - 36 q^{5} + 2 q^{6} + 14 q^{7} + 42 q^{9} + 10 q^{13} + 58 q^{16} - 50 q^{20} + 82 q^{22} - 20 q^{23} + 40 q^{24} + 36 q^{25} + 28 q^{28} - 2 q^{30} + 18 q^{33} + 72 q^{34} - 14 q^{35} + 100 q^{36} - 8 q^{38} - 14 q^{42} - 42 q^{45} + 42 q^{49} + 104 q^{51} + 8 q^{52} - 28 q^{53} + 80 q^{54} + 88 q^{57} + 86 q^{59} - 42 q^{62} - 12 q^{63} + 136 q^{64} - 10 q^{65} + 96 q^{67} + 28 q^{71} - 12 q^{74} - 24 q^{78} - 58 q^{80} + 32 q^{81} + 28 q^{82} + 10 q^{83} + 46 q^{86} + 212 q^{88} + 152 q^{91} - 22 q^{92} + 8 q^{93} + 114 q^{94} + 122 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.73278 −1.21520 5.46806 −1.00000 3.32088 −1.57483 −9.47743 −1.52328 2.73278
1.2 −2.69299 −3.06406 5.25219 −1.00000 8.25149 −2.38206 −8.75812 6.38847 2.69299
1.3 −2.67123 0.713943 5.13549 −1.00000 −1.90711 2.77457 −8.37562 −2.49029 2.67123
1.4 −2.65705 2.29699 5.05993 −1.00000 −6.10322 4.10952 −8.13038 2.27615 2.65705
1.5 −2.25867 −3.23671 3.10160 −1.00000 7.31067 1.13372 −2.48814 7.47631 2.25867
1.6 −2.19416 0.113339 2.81436 −1.00000 −0.248683 3.93157 −1.78684 −2.98715 2.19416
1.7 −2.04322 1.70781 2.17475 −1.00000 −3.48944 −3.88337 −0.357058 −0.0833797 2.04322
1.8 −2.01118 2.62919 2.04484 −1.00000 −5.28777 −3.87224 −0.0901779 3.91265 2.01118
1.9 −1.77096 −2.16305 1.13629 −1.00000 3.83068 4.83294 1.52959 1.67880 1.77096
1.10 −1.75945 3.29205 1.09567 −1.00000 −5.79221 0.522998 1.59112 7.83761 1.75945
1.11 −1.54851 −0.450009 0.397871 −1.00000 0.696842 1.74706 2.48091 −2.79749 1.54851
1.12 −1.26166 0.253469 −0.408220 −1.00000 −0.319791 −2.75032 3.03835 −2.93575 1.26166
1.13 −1.12063 1.00389 −0.744181 −1.00000 −1.12499 −0.443804 3.07522 −1.99221 1.12063
1.14 −1.00099 −2.52600 −0.998018 −1.00000 2.52851 −1.84414 3.00099 3.38070 1.00099
1.15 −0.688039 −1.31919 −1.52660 −1.00000 0.907653 −2.91509 2.42644 −1.25974 0.688039
1.16 −0.665686 1.73079 −1.55686 −1.00000 −1.15216 3.49242 2.36775 −0.00438097 0.665686
1.17 −0.623107 2.18308 −1.61174 −1.00000 −1.36029 1.60747 2.25050 1.76584 0.623107
1.18 −0.405675 −2.31455 −1.83543 −1.00000 0.938954 2.51358 1.55594 2.35713 0.405675
1.19 0.405675 2.31455 −1.83543 −1.00000 0.938954 2.51358 −1.55594 2.35713 −0.405675
1.20 0.623107 −2.18308 −1.61174 −1.00000 −1.36029 1.60747 −2.25050 1.76584 −0.623107
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
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.z 36
29.b even 2 1 inner 4205.2.a.z 36
29.f odd 28 2 145.2.m.b 36
145.o even 28 2 725.2.p.c 72
145.s odd 28 2 725.2.q.c 36
145.t even 28 2 725.2.p.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.m.b 36 29.f odd 28 2
725.2.p.c 72 145.o even 28 2
725.2.p.c 72 145.t even 28 2
725.2.q.c 36 145.s odd 28 2
4205.2.a.z 36 1.a even 1 1 trivial
4205.2.a.z 36 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{36} - 61 T_{2}^{34} + 1699 T_{2}^{32} - 28638 T_{2}^{30} + 326457 T_{2}^{28} - 2664319 T_{2}^{26} + \cdots + 707281 \) Copy content Toggle raw display
\( T_{3}^{36} - 75 T_{3}^{34} + 2548 T_{3}^{32} - 51946 T_{3}^{30} + 709680 T_{3}^{28} - 6871220 T_{3}^{26} + \cdots + 57121 \) Copy content Toggle raw display