Properties

Label 4026.2.a.q
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $4$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.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: \(32.1477718538\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6809.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta_1 q^{5} + q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + \beta_1 q^{5} + q^{6} + ( - \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + q^{9} - \beta_1 q^{10} - q^{11} - q^{12} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{13} + (\beta_{2} - \beta_1 + 1) q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{17} - q^{18} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{19} + \beta_1 q^{20} + (\beta_{2} - \beta_1 + 1) q^{21} + q^{22} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{23} + q^{24} + (\beta_{2} - 2) q^{25} + (2 \beta_{3} - \beta_{2} + 1) q^{26} - q^{27} + ( - \beta_{2} + \beta_1 - 1) q^{28} + (\beta_{3} - 2 \beta_{2} - 2) q^{29} + \beta_1 q^{30} + (\beta_{2} + 3 \beta_1 - 2) q^{31} - q^{32} + q^{33} + (\beta_{3} - \beta_{2} - \beta_1) q^{34} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{35} + q^{36} + ( - 3 \beta_{2} - 4) q^{37} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{38} + (2 \beta_{3} - \beta_{2} + 1) q^{39} - \beta_1 q^{40} + ( - 2 \beta_{2} + 4 \beta_1 + 3) q^{41} + ( - \beta_{2} + \beta_1 - 1) q^{42} + ( - 4 \beta_{3} - \beta_1 - 3) q^{43} - q^{44} + \beta_1 q^{45} + (2 \beta_{3} - \beta_{2} - 4) q^{46} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{47} - q^{48} + ( - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{49} + ( - \beta_{2} + 2) q^{50} + (\beta_{3} - \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{52} + (4 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 3) q^{53} + q^{54} - \beta_1 q^{55} + (\beta_{2} - \beta_1 + 1) q^{56} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{57} + ( - \beta_{3} + 2 \beta_{2} + 2) q^{58} + ( - \beta_{2} + 3 \beta_1) q^{59} - \beta_1 q^{60} + q^{61} + ( - \beta_{2} - 3 \beta_1 + 2) q^{62} + ( - \beta_{2} + \beta_1 - 1) q^{63} + q^{64} + (\beta_{3} + \beta_1 + 3) q^{65} - q^{66} + (\beta_{3} + 6 \beta_1 - 1) q^{67} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{68} + (2 \beta_{3} - \beta_{2} - 4) q^{69} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{70} + ( - 3 \beta_{3} - 4 \beta_1 + 3) q^{71} - q^{72} + (6 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 7) q^{73} + (3 \beta_{2} + 4) q^{74} + ( - \beta_{2} + 2) q^{75} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{76} + (\beta_{2} - \beta_1 + 1) q^{77} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{78} + (5 \beta_{3} + 3 \beta_1 + 2) q^{79} + \beta_1 q^{80} + q^{81} + (2 \beta_{2} - 4 \beta_1 - 3) q^{82} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 7) q^{83} + (\beta_{2} - \beta_1 + 1) q^{84} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{85} + (4 \beta_{3} + \beta_1 + 3) q^{86} + ( - \beta_{3} + 2 \beta_{2} + 2) q^{87} + q^{88} + ( - \beta_1 - 4) q^{89} - \beta_1 q^{90} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{91} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{92} + ( - \beta_{2} - 3 \beta_1 + 2) q^{93} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{94} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 5) q^{95} + q^{96} + (4 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{97} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{12} - 4 q^{13} + 2 q^{14} + 4 q^{16} - q^{17} - 4 q^{18} + 4 q^{19} + 2 q^{21} + 4 q^{22} + 16 q^{23} + 4 q^{24} - 10 q^{25} + 4 q^{26} - 4 q^{27} - 2 q^{28} - 5 q^{29} - 10 q^{31} - 4 q^{32} + 4 q^{33} + q^{34} + 7 q^{35} + 4 q^{36} - 10 q^{37} - 4 q^{38} + 4 q^{39} + 16 q^{41} - 2 q^{42} - 8 q^{43} - 4 q^{44} - 16 q^{46} - 7 q^{47} - 4 q^{48} - 2 q^{49} + 10 q^{50} + q^{51} - 4 q^{52} + 12 q^{53} + 4 q^{54} + 2 q^{56} - 4 q^{57} + 5 q^{58} + 2 q^{59} + 4 q^{61} + 10 q^{62} - 2 q^{63} + 4 q^{64} + 11 q^{65} - 4 q^{66} - 5 q^{67} - q^{68} - 16 q^{69} - 7 q^{70} + 15 q^{71} - 4 q^{72} - 28 q^{73} + 10 q^{74} + 10 q^{75} + 4 q^{76} + 2 q^{77} - 4 q^{78} + 3 q^{79} + 4 q^{81} - 16 q^{82} + 32 q^{83} + 2 q^{84} + 17 q^{85} + 8 q^{86} + 5 q^{87} + 4 q^{88} - 16 q^{89} - 3 q^{91} + 16 q^{92} + 10 q^{93} + 7 q^{94} + 15 q^{95} + 4 q^{96} + 2 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) 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
−2.06963
−0.582772
0.361989
2.29041
−1.00000 −1.00000 1.00000 −2.06963 1.00000 −4.35299 −1.00000 1.00000 2.06963
1.2 −1.00000 −1.00000 1.00000 −0.582772 1.00000 1.07760 −1.00000 1.00000 0.582772
1.3 −1.00000 −1.00000 1.00000 0.361989 1.00000 2.23095 −1.00000 1.00000 −0.361989
1.4 −1.00000 −1.00000 1.00000 2.29041 1.00000 −0.955570 −1.00000 1.00000 −2.29041
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(11\) \(1\)
\(61\) \(-1\)

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 4026.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.q 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{4} - 5T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} - 11T_{7}^{2} - T_{7} + 10 \) Copy content Toggle raw display
\( T_{13}^{4} + 4T_{13}^{3} - 21T_{13}^{2} + 9T_{13} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} - 11 T^{2} - T + 10 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} - 21 T^{2} + 9 T + 17 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} - 25 T^{2} - 25 T + 8 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 41 T^{2} + 77 T - 8 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + 69 T^{2} + \cdots - 428 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} - 33 T^{2} - 116 T + 139 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} - 27 T^{2} + \cdots - 1195 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} - 57 T^{2} + \cdots + 1546 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} - 2 T^{2} + \cdots - 3671 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} - 51 T^{2} - 441 T - 724 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} - 68 T^{2} + \cdots + 1160 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} - 69 T^{2} + \cdots + 404 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} - 45 T^{2} + 179 T - 125 \) Copy content Toggle raw display
$61$ \( (T - 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 5 T^{3} - 152 T^{2} + \cdots - 704 \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + 4 T^{2} + 469 T - 86 \) Copy content Toggle raw display
$73$ \( T^{4} + 28 T^{3} + 105 T^{2} + \cdots - 17768 \) Copy content Toggle raw display
$79$ \( T^{4} - 3 T^{3} - 116 T^{2} + \cdots + 1136 \) Copy content Toggle raw display
$83$ \( T^{4} - 32 T^{3} + 317 T^{2} + \cdots - 1556 \) Copy content Toggle raw display
$89$ \( T^{4} + 16 T^{3} + 91 T^{2} + \cdots + 181 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} - 189 T^{2} + \cdots + 5165 \) Copy content Toggle raw display
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