Properties

Label 4026.2.a.x
Level $4026$
Weight $2$
Character orbit 4026.a
Self dual yes
Analytic conductor $32.148$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.46101901.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 5x^{4} + 12x^{3} + 6x^{2} - 5x - 2 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{5} + 3\nu^{4} + 5\nu^{3} - 12\nu^{2} - 5\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{5} + 4\nu^{4} + 2\nu^{3} - 16\nu^{2} + 4\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 10\nu^{4} + 11\nu^{3} - 37\nu^{2} - 3\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 10\nu^{4} + 11\nu^{3} - 39\nu^{2} - \nu + 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -13\beta_{5} + 7\beta_{4} + 4\beta_{3} + 5\beta_{2} + 13\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -42\beta_{5} + 14\beta_{4} + 17\beta_{3} + 24\beta_{2} + 52\beta _1 + 40 \) 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
−0.654438
1.90193
3.29072
−1.86911
−0.371781
0.702675
1.00000 −1.00000 1.00000 −4.17718 −1.00000 2.87039 1.00000 1.00000 −4.17718
1.2 1.00000 −1.00000 1.00000 −4.05628 −1.00000 −1.59539 1.00000 1.00000 −4.05628
1.3 1.00000 −1.00000 1.00000 −0.656263 −1.00000 4.00423 1.00000 1.00000 −0.656263
1.4 1.00000 −1.00000 1.00000 0.199215 −1.00000 −0.938542 1.00000 1.00000 0.199215
1.5 1.00000 −1.00000 1.00000 1.28209 −1.00000 −0.306395 1.00000 1.00000 1.28209
1.6 1.00000 −1.00000 1.00000 1.40842 −1.00000 −3.03430 1.00000 1.00000 1.40842
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.x 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4026))\):

\( T_{5}^{6} + 6T_{5}^{5} - T_{5}^{4} - 33T_{5}^{3} + 17T_{5}^{2} + 18T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{5} - 18T_{7}^{4} + 76T_{7}^{2} + 75T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} - 18T_{13}^{4} + 22T_{13}^{3} + 69T_{13}^{2} + 35T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} - T^{4} - 33 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} - 18 T^{4} + 76 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} - 18 T^{4} + 22 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{6} + 13 T^{5} - T^{4} - 626 T^{3} + \cdots - 407 \) Copy content Toggle raw display
$19$ \( T^{6} - T^{5} - 94 T^{4} - 18 T^{3} + \cdots + 3544 \) Copy content Toggle raw display
$23$ \( T^{6} + 11 T^{5} - 25 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$29$ \( T^{6} + 14 T^{5} - 5 T^{4} - 501 T^{3} + \cdots - 88 \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} - 92 T^{4} - 407 T^{3} + \cdots + 152 \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} - 65 T^{4} + \cdots + 5659 \) Copy content Toggle raw display
$41$ \( T^{6} + 25 T^{5} + 125 T^{4} + \cdots + 2248 \) Copy content Toggle raw display
$43$ \( T^{6} - 19 T^{5} + 31 T^{4} + \cdots - 20684 \) Copy content Toggle raw display
$47$ \( T^{6} + 10 T^{5} - 22 T^{4} + \cdots - 1402 \) Copy content Toggle raw display
$53$ \( T^{6} + 17 T^{5} + 52 T^{4} + \cdots + 4642 \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} - 100 T^{4} + \cdots + 16246 \) Copy content Toggle raw display
$61$ \( (T - 1)^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - 12 T^{5} - 142 T^{4} + \cdots - 143100 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 147 T^{4} + \cdots - 24796 \) Copy content Toggle raw display
$73$ \( T^{6} + 29 T^{5} + 211 T^{4} + \cdots + 60412 \) Copy content Toggle raw display
$79$ \( T^{6} + 24 T^{5} - 156 T^{4} + \cdots + 1492388 \) Copy content Toggle raw display
$83$ \( T^{6} + 9 T^{5} - 501 T^{4} + \cdots - 1256272 \) Copy content Toggle raw display
$89$ \( T^{6} + 4 T^{5} - 363 T^{4} + \cdots - 958108 \) Copy content Toggle raw display
$97$ \( T^{6} + 5 T^{5} - 248 T^{4} + \cdots - 185347 \) Copy content Toggle raw display
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