Properties

Label 4205.2.a.o
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 12x^{4} - 59x^{3} + 5x^{2} + 44x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + (2 \beta_{6} - \beta_{4} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{6} - 1) q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + (2 \beta_{7} + 3 \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 7 q^{3} + 7 q^{4} + 8 q^{5} + 6 q^{6} + 4 q^{7} - 6 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 7 q^{3} + 7 q^{4} + 8 q^{5} + 6 q^{6} + 4 q^{7} - 6 q^{8} + 9 q^{9} - 3 q^{10} - 8 q^{11} - 4 q^{12} + 10 q^{13} + 6 q^{14} - 7 q^{15} - 11 q^{16} - 4 q^{17} - 27 q^{18} - 3 q^{19} + 7 q^{20} - 14 q^{21} - 4 q^{22} + 5 q^{23} + 11 q^{24} + 8 q^{25} - 9 q^{26} - 28 q^{27} + q^{28} + 6 q^{30} - 5 q^{31} - 10 q^{32} - 20 q^{33} - 30 q^{34} + 4 q^{35} + 35 q^{36} - 2 q^{37} - 34 q^{38} - 10 q^{39} - 6 q^{40} - 17 q^{41} + 17 q^{42} - 33 q^{43} - 14 q^{44} + 9 q^{45} - 16 q^{46} - 17 q^{47} + 8 q^{48} - 24 q^{49} - 3 q^{50} - 11 q^{51} - 4 q^{52} + 2 q^{53} + 54 q^{54} - 8 q^{55} - 12 q^{56} + 24 q^{57} + 16 q^{59} - 4 q^{60} - 3 q^{61} + 9 q^{62} - 7 q^{63} - 4 q^{64} + 10 q^{65} + 51 q^{66} + 7 q^{67} + 20 q^{68} - 40 q^{69} + 6 q^{70} + 6 q^{71} - 38 q^{72} - 20 q^{73} + 9 q^{74} - 7 q^{75} + 31 q^{76} + 20 q^{77} - 73 q^{78} - 23 q^{79} - 11 q^{80} + 32 q^{81} + 7 q^{82} + q^{83} - 50 q^{84} - 4 q^{85} + 33 q^{86} + 29 q^{88} - 18 q^{89} - 27 q^{90} - 17 q^{91} - 19 q^{92} - 20 q^{93} + 44 q^{94} - 3 q^{95} - 5 q^{96} - 21 q^{97} + 4 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 12x^{4} - 59x^{3} + 5x^{2} + 44x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - 6\nu^{3} + 4\nu^{2} + 7\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 7\nu^{5} - 20\nu^{4} - 16\nu^{3} + 35\nu^{2} + 11\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 9\nu^{5} + 6\nu^{4} + 26\nu^{3} - 11\nu^{2} - 23\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} - 25\nu^{5} + 36\nu^{4} + 64\nu^{3} - 77\nu^{2} - 49\nu + 48 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 7\beta_{3} + 8\beta_{2} + 19\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{7} - 7\beta_{6} + 10\beta_{5} + \beta_{4} + 9\beta_{3} + 33\beta_{2} + 19\beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{7} - 8\beta_{6} + 13\beta_{5} + 10\beta_{4} + 40\beta_{3} + 54\beta_{2} + 97\beta _1 + 74 \) 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
2.45871
2.28385
1.52561
1.10443
0.435980
−1.20778
−1.62740
−1.97339
−2.45871 −3.12737 4.04526 1.00000 7.68929 2.51957 −5.02870 6.78043 −2.45871
1.2 −2.28385 1.98518 3.21595 1.00000 −4.53384 −2.69534 −2.77705 0.940935 −2.28385
1.3 −1.52561 −3.18997 0.327492 1.00000 4.86666 −1.46849 2.55160 7.17593 −1.52561
1.4 −1.10443 −0.267068 −0.780236 1.00000 0.294958 1.68257 3.07057 −2.92867 −1.10443
1.5 −0.435980 −0.0536872 −1.80992 1.00000 0.0234065 0.294570 1.66105 −2.99712 −0.435980
1.6 1.20778 −2.48499 −0.541257 1.00000 −3.00133 0.253548 −3.06929 3.17517 1.20778
1.7 1.62740 −1.12348 0.648443 1.00000 −1.82836 3.63319 −2.19953 −1.73778 1.62740
1.8 1.97339 1.26139 1.89427 1.00000 2.48922 −0.219622 −0.208650 −1.40889 1.97339
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.o 8
29.b even 2 1 4205.2.a.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.o 8 1.a even 1 1 trivial
4205.2.a.p yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{8} + 3T_{2}^{7} - 7T_{2}^{6} - 24T_{2}^{5} + 12T_{2}^{4} + 59T_{2}^{3} + 5T_{2}^{2} - 44T_{2} - 16 \) Copy content Toggle raw display
\( T_{3}^{8} + 7T_{3}^{7} + 8T_{3}^{6} - 35T_{3}^{5} - 69T_{3}^{4} + 25T_{3}^{3} + 83T_{3}^{2} + 23T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots - 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 389 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots - 102196 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots - 31361 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 2384 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 30971 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + \cdots + 109001 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 239 \) Copy content Toggle raw display
$41$ \( T^{8} + 17 T^{7} + \cdots - 83644 \) Copy content Toggle raw display
$43$ \( T^{8} + 33 T^{7} + \cdots + 153281 \) Copy content Toggle raw display
$47$ \( T^{8} + 17 T^{7} + \cdots - 79921 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots - 3539 \) Copy content Toggle raw display
$59$ \( T^{8} - 16 T^{7} + \cdots - 1378225 \) Copy content Toggle raw display
$61$ \( T^{8} + 3 T^{7} + \cdots + 1331531 \) Copy content Toggle raw display
$67$ \( T^{8} - 7 T^{7} + \cdots - 7738399 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 1110349 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + \cdots - 1284475 \) Copy content Toggle raw display
$79$ \( T^{8} + 23 T^{7} + \cdots - 1276271 \) Copy content Toggle raw display
$83$ \( T^{8} - T^{7} + \cdots + 770876 \) Copy content Toggle raw display
$89$ \( T^{8} + 18 T^{7} + \cdots - 159599 \) Copy content Toggle raw display
$97$ \( T^{8} + 21 T^{7} + \cdots + 455912669 \) Copy content Toggle raw display
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