Properties

Label 4205.2.a.m
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.66064384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 2) q^{6} + \beta_{2} q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{2} + 2) q^{6} + \beta_{2} q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + 2) q^{9} - \beta_1 q^{10} + (\beta_{5} - \beta_1) q^{11} + (\beta_{5} + \beta_{4} + 3 \beta_1) q^{12} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{5} + \beta_{4} + 3 \beta_1) q^{14} + (\beta_{5} - \beta_1) q^{15} + (\beta_{3} + 2 \beta_{2} + 4) q^{16} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{17} + (\beta_{4} + \beta_1) q^{18} + (\beta_{5} - \beta_{4} - \beta_1) q^{19} + ( - \beta_{2} - 1) q^{20} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{21} + ( - \beta_{2} - 2) q^{22} + ( - \beta_{3} - \beta_{2} - 1) q^{23} + (\beta_{3} + 3 \beta_{2} + 7) q^{24} + q^{25} + ( - \beta_{4} + 2 \beta_1) q^{26} - 4 \beta_{5} q^{27} + (\beta_{3} + 3 \beta_{2} + 9) q^{28} + ( - \beta_{2} - 2) q^{30} + ( - 3 \beta_{5} + \beta_{4} - \beta_1) q^{31} + (\beta_{4} + 5 \beta_1) q^{32} + ( - \beta_{3} - 5) q^{33} + (\beta_{3} + \beta_{2} - 3) q^{34} - \beta_{2} q^{35} + ( - \beta_{3} + 3 \beta_{2}) q^{36} + ( - \beta_{5} - \beta_{4} + 3 \beta_1) q^{37} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{38} + 4 \beta_{5} q^{39} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{40} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_1) q^{41} + (\beta_{3} + 4 \beta_{2} + 9) q^{42} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{43} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{44} + ( - \beta_{3} - 2) q^{45} + (\beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{46} + ( - \beta_{5} + \beta_1) q^{47} + ( - 5 \beta_{5} + 2 \beta_{4} + 9 \beta_1) q^{48} + (\beta_{3} + 2 \beta_{2} + 2) q^{49} + \beta_1 q^{50} + (\beta_{3} + 1) q^{51} + (\beta_{3} + 3) q^{52} + ( - \beta_{3} + 2 \beta_{2} + 5) q^{53} - 4 q^{54} + ( - \beta_{5} + \beta_1) q^{55} + ( - \beta_{5} + 2 \beta_{4} + 11 \beta_1) q^{56} + ( - \beta_{3} - 2 \beta_{2} - 5) q^{57} + (\beta_{3} - 3) q^{59} + ( - \beta_{5} - \beta_{4} - 3 \beta_1) q^{60} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{61} + (\beta_{3} + \beta_{2} - 5) q^{62} + ( - 2 \beta_{3} + 3 \beta_{2} - 2) q^{63} + ( - \beta_{3} + 3 \beta_{2} + 8) q^{64} + (\beta_{3} - 1) q^{65} + ( - \beta_{4} - 4 \beta_1) q^{66} + ( - \beta_{3} - \beta_{2} + 3) q^{67} + (\beta_{5} + \beta_1) q^{68} + (4 \beta_{5} - \beta_{4} - 4 \beta_1) q^{69} + (\beta_{5} - \beta_{4} - 3 \beta_1) q^{70} + (\beta_{3} + 9) q^{71} + ( - 3 \beta_{5} + 8 \beta_1) q^{72} + ( - 3 \beta_{5} - 3 \beta_1) q^{73} + ( - \beta_{3} + \beta_{2} + 7) q^{74} + ( - \beta_{5} + \beta_1) q^{75} + (\beta_{5} - 2 \beta_{4} - 9 \beta_1) q^{76} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_1) q^{77} + 4 q^{78} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{79} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{80} + (\beta_{3} - 4 \beta_{2} + 6) q^{81} + ( - \beta_{3} - 4 \beta_{2} - 9) q^{82} + ( - \beta_{2} - 4) q^{83} + ( - 8 \beta_{5} + 3 \beta_{4} + 16 \beta_1) q^{84} + (\beta_{5} - \beta_{4} + \beta_1) q^{85} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{86} + ( - \beta_{3} - 3 \beta_{2} - 7) q^{88} - 2 \beta_{4} q^{89} + ( - \beta_{4} - \beta_1) q^{90} + (2 \beta_{3} + 2) q^{91} + ( - 5 \beta_{2} - 8) q^{92} + (3 \beta_{3} - 2 \beta_{2} + 7) q^{93} + (\beta_{2} + 2) q^{94} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{95} + (7 \beta_{2} + 10) q^{96} + (\beta_{5} + \beta_1) q^{97} + ( - 2 \beta_{5} + 3 \beta_{4} + 7 \beta_1) q^{98} + (7 \beta_{5} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{4} - 6 q^{5} + 12 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{4} - 6 q^{5} + 12 q^{6} + 14 q^{9} + 4 q^{13} + 26 q^{16} - 6 q^{20} - 12 q^{22} - 8 q^{23} + 44 q^{24} + 6 q^{25} + 56 q^{28} - 12 q^{30} - 32 q^{33} - 16 q^{34} - 2 q^{36} - 20 q^{38} + 56 q^{42} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 20 q^{52} + 28 q^{53} - 24 q^{54} - 32 q^{57} - 16 q^{59} - 28 q^{62} - 16 q^{63} + 46 q^{64} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 40 q^{74} + 24 q^{78} - 26 q^{80} + 38 q^{81} - 56 q^{82} - 24 q^{83} - 4 q^{86} - 44 q^{88} + 16 q^{91} - 48 q^{92} + 48 q^{93} + 12 q^{94} + 60 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 13x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 8\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 8\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 9\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 8\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 9\beta_{4} + 41\beta_1 \) 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
−2.68667
−1.30397
−0.285442
0.285442
1.30397
2.68667
−2.68667 −2.31446 5.21819 −1.00000 6.21819 4.21819 −8.64620 2.35673 2.68667
1.2 −1.30397 −0.537080 −0.299664 −1.00000 0.700336 −1.29966 2.99869 −2.71155 1.30397
1.3 −0.285442 3.21789 −1.91852 −1.00000 −0.918523 −2.91852 1.11851 7.35482 0.285442
1.4 0.285442 −3.21789 −1.91852 −1.00000 −0.918523 −2.91852 −1.11851 7.35482 −0.285442
1.5 1.30397 0.537080 −0.299664 −1.00000 0.700336 −1.29966 −2.99869 −2.71155 −1.30397
1.6 2.68667 2.31446 5.21819 −1.00000 6.21819 4.21819 8.64620 2.35673 −2.68667
\(n\): 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

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.m 6
29.b even 2 1 inner 4205.2.a.m 6
29.c odd 4 2 145.2.c.b 6
87.f even 4 2 1305.2.d.b 6
116.e even 4 2 2320.2.g.i 6
145.e even 4 2 725.2.d.c 12
145.f odd 4 2 725.2.c.e 6
145.j even 4 2 725.2.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.b 6 29.c odd 4 2
725.2.c.e 6 145.f odd 4 2
725.2.d.c 12 145.e even 4 2
725.2.d.c 12 145.j even 4 2
1305.2.d.b 6 87.f even 4 2
2320.2.g.i 6 116.e even 4 2
4205.2.a.m 6 1.a even 1 1 trivial
4205.2.a.m 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{6} - 9T_{2}^{4} + 13T_{2}^{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 16T_{3}^{4} + 60T_{3}^{2} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 9 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 16 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 14 T - 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 16 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( (T^{3} - 2 T^{2} - 24 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 52 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( T^{6} - 56 T^{4} + \cdots - 1296 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 26 T - 96)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 152 T^{4} + \cdots - 144 \) Copy content Toggle raw display
$37$ \( T^{6} - 100 T^{4} + \cdots - 20736 \) Copy content Toggle raw display
$41$ \( T^{6} - 184 T^{4} + \cdots - 4096 \) Copy content Toggle raw display
$43$ \( T^{6} - 176 T^{4} + \cdots - 121104 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$53$ \( (T^{3} - 14 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 104 T^{4} + \cdots - 2304 \) Copy content Toggle raw display
$67$ \( (T^{3} - 8 T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 28 T^{2} + \cdots - 576)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 252 T^{4} + \cdots - 419904 \) Copy content Toggle raw display
$79$ \( T^{6} - 176 T^{4} + \cdots - 121104 \) Copy content Toggle raw display
$83$ \( (T^{3} + 12 T^{2} + \cdots + 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 160 T^{4} + \cdots - 65536 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{4} + \cdots - 576 \) Copy content Toggle raw display
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