Properties

Label 4026.2.a.u
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.9176805.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 12x^{3} + 7x^{2} + 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 12x^{3} + 7x^{2} + 30x - 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - \nu^{3} - 8\nu^{2} + \nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 10\nu^{2} + 3\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{3} - 2\nu^{2} - 7\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 2\beta_{3} + 2\beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 10\beta_{3} + 12\beta_{2} + 16\beta _1 + 26 \) 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
3.32361
1.62906
0.689091
−2.15973
−2.48204
−1.00000 1.00000 1.00000 −3.32361 −1.00000 −3.91566 −1.00000 1.00000 3.32361
1.2 −1.00000 1.00000 1.00000 −1.62906 −1.00000 −4.09482 −1.00000 1.00000 1.62906
1.3 −1.00000 1.00000 1.00000 −0.689091 −1.00000 4.91945 −1.00000 1.00000 0.689091
1.4 −1.00000 1.00000 1.00000 2.15973 −1.00000 −1.48663 −1.00000 1.00000 −2.15973
1.5 −1.00000 1.00000 1.00000 2.48204 −1.00000 1.57766 −1.00000 1.00000 −2.48204
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.u 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{5} + T_{5}^{4} - 12T_{5}^{3} - 7T_{5}^{2} + 30T_{5} + 20 \) Copy content Toggle raw display
\( T_{7}^{5} + 3T_{7}^{4} - 26T_{7}^{3} - 84T_{7}^{2} + 62T_{7} + 185 \) Copy content Toggle raw display
\( T_{13}^{5} - 2T_{13}^{4} - 34T_{13}^{3} + 56T_{13}^{2} + 193T_{13} - 339 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + T^{4} - 12 T^{3} - 7 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} - 26 T^{3} - 84 T^{2} + \cdots + 185 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 2 T^{4} - 34 T^{3} + 56 T^{2} + \cdots - 339 \) Copy content Toggle raw display
$17$ \( T^{5} - 6 T^{4} - 31 T^{3} + 201 T^{2} + \cdots + 141 \) Copy content Toggle raw display
$19$ \( T^{5} - 7 T^{4} - 64 T^{3} + \cdots - 3807 \) Copy content Toggle raw display
$23$ \( T^{5} + 12 T^{4} + 13 T^{3} - 195 T^{2} + \cdots - 44 \) Copy content Toggle raw display
$29$ \( T^{5} - 4 T^{4} - 49 T^{3} + 355 T^{2} + \cdots + 564 \) Copy content Toggle raw display
$31$ \( T^{5} + T^{4} - 22 T^{3} - 13 T^{2} + \cdots + 60 \) Copy content Toggle raw display
$37$ \( T^{5} - 3 T^{4} - 82 T^{3} + \cdots - 3375 \) Copy content Toggle raw display
$41$ \( T^{5} + 3 T^{4} - 93 T^{3} - 279 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$43$ \( T^{5} - 24 T^{4} + 185 T^{3} + \cdots + 1796 \) Copy content Toggle raw display
$47$ \( T^{5} - 18 T^{4} + 26 T^{3} + \cdots - 8823 \) Copy content Toggle raw display
$53$ \( T^{5} + 13 T^{4} + 40 T^{3} + 8 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$59$ \( T^{5} + 16 T^{4} - 108 T^{3} + \cdots + 16125 \) Copy content Toggle raw display
$61$ \( (T + 1)^{5} \) Copy content Toggle raw display
$67$ \( T^{5} - 18 T^{4} - 56 T^{3} + \cdots - 39877 \) Copy content Toggle raw display
$71$ \( T^{5} + 31 T^{4} + 332 T^{3} + \cdots - 3764 \) Copy content Toggle raw display
$73$ \( T^{5} - 8 T^{4} - 189 T^{3} + \cdots - 2820 \) Copy content Toggle raw display
$79$ \( T^{5} - 32 T^{4} + 126 T^{3} + \cdots + 3503 \) Copy content Toggle raw display
$83$ \( T^{5} - 8 T^{4} - 193 T^{3} + \cdots + 1200 \) Copy content Toggle raw display
$89$ \( T^{5} - T^{4} - 196 T^{3} + \cdots - 11604 \) Copy content Toggle raw display
$97$ \( T^{5} + 4 T^{4} - 124 T^{3} + \cdots - 7895 \) Copy content Toggle raw display
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