Properties

Label 4026.2.a.k
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + (\beta - 3) q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + (\beta - 3) q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9} + ( - \beta + 3) q^{10} - q^{11} + q^{12} + 3 q^{13} - 2 q^{14} + (\beta - 3) q^{15} + q^{16} - 2 \beta q^{17} - q^{18} + (2 \beta + 4) q^{19} + (\beta - 3) q^{20} + 2 q^{21} + q^{22} + 2 \beta q^{23} - q^{24} + ( - 6 \beta + 6) q^{25} - 3 q^{26} + q^{27} + 2 q^{28} + ( - 4 \beta - 3) q^{29} + ( - \beta + 3) q^{30} + (\beta + 5) q^{31} - q^{32} - q^{33} + 2 \beta q^{34} + (2 \beta - 6) q^{35} + q^{36} + 6 \beta q^{37} + ( - 2 \beta - 4) q^{38} + 3 q^{39} + ( - \beta + 3) q^{40} + (7 \beta - 1) q^{41} - 2 q^{42} - 6 q^{43} - q^{44} + (\beta - 3) q^{45} - 2 \beta q^{46} - 6 q^{47} + q^{48} - 3 q^{49} + (6 \beta - 6) q^{50} - 2 \beta q^{51} + 3 q^{52} + (2 \beta + 8) q^{53} - q^{54} + ( - \beta + 3) q^{55} - 2 q^{56} + (2 \beta + 4) q^{57} + (4 \beta + 3) q^{58} + ( - 8 \beta + 3) q^{59} + (\beta - 3) q^{60} - q^{61} + ( - \beta - 5) q^{62} + 2 q^{63} + q^{64} + (3 \beta - 9) q^{65} + q^{66} - 6 \beta q^{67} - 2 \beta q^{68} + 2 \beta q^{69} + ( - 2 \beta + 6) q^{70} + 6 q^{71} - q^{72} + (4 \beta + 4) q^{73} - 6 \beta q^{74} + ( - 6 \beta + 6) q^{75} + (2 \beta + 4) q^{76} - 2 q^{77} - 3 q^{78} + (\beta - 3) q^{80} + q^{81} + ( - 7 \beta + 1) q^{82} + (8 \beta + 6) q^{83} + 2 q^{84} + (6 \beta - 4) q^{85} + 6 q^{86} + ( - 4 \beta - 3) q^{87} + q^{88} + (6 \beta - 3) q^{89} + ( - \beta + 3) q^{90} + 6 q^{91} + 2 \beta q^{92} + (\beta + 5) q^{93} + 6 q^{94} + ( - 2 \beta - 8) q^{95} - q^{96} + ( - 4 \beta + 5) q^{97} + 3 q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 6 q^{5} - 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 2 q^{16} - 2 q^{18} + 8 q^{19} - 6 q^{20} + 4 q^{21} + 2 q^{22} - 2 q^{24} + 12 q^{25} - 6 q^{26} + 2 q^{27} + 4 q^{28} - 6 q^{29} + 6 q^{30} + 10 q^{31} - 2 q^{32} - 2 q^{33} - 12 q^{35} + 2 q^{36} - 8 q^{38} + 6 q^{39} + 6 q^{40} - 2 q^{41} - 4 q^{42} - 12 q^{43} - 2 q^{44} - 6 q^{45} - 12 q^{47} + 2 q^{48} - 6 q^{49} - 12 q^{50} + 6 q^{52} + 16 q^{53} - 2 q^{54} + 6 q^{55} - 4 q^{56} + 8 q^{57} + 6 q^{58} + 6 q^{59} - 6 q^{60} - 2 q^{61} - 10 q^{62} + 4 q^{63} + 2 q^{64} - 18 q^{65} + 2 q^{66} + 12 q^{70} + 12 q^{71} - 2 q^{72} + 8 q^{73} + 12 q^{75} + 8 q^{76} - 4 q^{77} - 6 q^{78} - 6 q^{80} + 2 q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{84} - 8 q^{85} + 12 q^{86} - 6 q^{87} + 2 q^{88} - 6 q^{89} + 6 q^{90} + 12 q^{91} + 10 q^{93} + 12 q^{94} - 16 q^{95} - 2 q^{96} + 10 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−1.00000 1.00000 1.00000 −4.41421 −1.00000 2.00000 −1.00000 1.00000 4.41421
1.2 −1.00000 1.00000 1.00000 −1.58579 −1.00000 2.00000 −1.00000 1.00000 1.58579
\(n\): 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.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.k 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{2} + 6T_{5} + 7 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 23 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 97 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 119 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 72 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 63 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
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