Properties

Label 862.2.a.j
Level $862$
Weight $2$
Character orbit 862.a
Self dual yes
Analytic conductor $6.883$
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] = [862,2,Mod(1,862)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(862, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("862.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 862 = 2 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 862.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: \(6.88310465423\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.9783113.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 7x^{3} + 11x^{2} - 9x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 7x^{3} + 11x^{2} - 9x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} - \nu^{4} - 14\nu^{3} + 7\nu^{2} + 13\nu - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 7\nu^{3} + 21\nu^{2} + 9\nu - 17 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} + 2\nu^{4} + 33\nu^{3} - 14\nu^{2} - 51\nu + 18 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 3\nu^{4} + 47\nu^{3} - 16\nu^{2} - 59\nu + 17 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} - 7\beta_{4} - 2\beta_{3} + 8\beta_{2} - 6\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{4} - \beta_{3} + 17\beta_{2} + 29\beta _1 + 2 \) 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
0.599121
1.44560
−2.43574
0.135000
2.60430
−1.34828
1.00000 −3.44376 1.00000 −3.30426 −3.44376 3.92496 1.00000 8.85945 −3.30426
1.2 1.00000 −3.28440 1.00000 0.0124021 −3.28440 −3.12009 1.00000 7.78729 0.0124021
1.3 1.00000 −2.57408 1.00000 2.63087 −2.57408 −3.15558 1.00000 3.62588 2.63087
1.4 1.00000 −0.242119 1.00000 −2.55135 −0.242119 1.87756 1.00000 −2.94138 −2.55135
1.5 1.00000 −0.0869605 1.00000 −0.946431 −0.0869605 −4.37116 1.00000 −2.99244 −0.946431
1.6 1.00000 1.63132 1.00000 −3.84124 1.63132 −4.15570 1.00000 −0.338810 −3.84124
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(431\) \(-1\)

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 862.2.a.j 6
3.b odd 2 1 7758.2.a.r 6
4.b odd 2 1 6896.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
862.2.a.j 6 1.a even 1 1 trivial
6896.2.a.r 6 4.b odd 2 1
7758.2.a.r 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(862))\):

\( T_{3}^{6} + 8T_{3}^{5} + 16T_{3}^{4} - 13T_{3}^{3} - 53T_{3}^{2} - 16T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 8T_{5}^{5} + 12T_{5}^{4} - 44T_{5}^{3} - 131T_{5}^{2} - 79T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{5} + 16 T^{4} - 13 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 8 T^{5} + 12 T^{4} - 44 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 9 T^{5} + 3 T^{4} - 166 T^{3} + \cdots + 1318 \) Copy content Toggle raw display
$11$ \( T^{6} + 10 T^{5} - 12 T^{4} - 365 T^{3} + \cdots - 67 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} - 23 T^{4} + 46 T^{3} + \cdots - 58 \) Copy content Toggle raw display
$17$ \( T^{6} + 11 T^{5} - 4 T^{4} + \cdots - 2182 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} - 72 T^{4} - 189 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$23$ \( T^{6} + 20 T^{5} + 70 T^{4} + \cdots + 53341 \) Copy content Toggle raw display
$29$ \( T^{6} + 21 T^{5} + 109 T^{4} + \cdots + 5743 \) Copy content Toggle raw display
$31$ \( T^{6} + 28 T^{5} + 273 T^{4} + \cdots - 31336 \) Copy content Toggle raw display
$37$ \( T^{6} + 7 T^{5} - 50 T^{4} - 185 T^{3} + \cdots + 200 \) Copy content Toggle raw display
$41$ \( T^{6} + 11 T^{5} - 20 T^{4} + \cdots - 122 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} - 82 T^{4} + \cdots - 10618 \) Copy content Toggle raw display
$47$ \( T^{6} + 12 T^{5} - 81 T^{4} + \cdots - 3926 \) Copy content Toggle raw display
$53$ \( T^{6} - 11 T^{5} - 154 T^{4} + \cdots - 95369 \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} - 161 T^{4} - 1856 T^{3} + \cdots - 29 \) Copy content Toggle raw display
$61$ \( T^{6} + 16 T^{5} - 175 T^{4} + \cdots + 161894 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} - 343 T^{4} + \cdots - 95906 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} - 299 T^{4} + \cdots - 346250 \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + 88 T^{4} + \cdots - 8350 \) Copy content Toggle raw display
$79$ \( T^{6} + 4 T^{5} - 289 T^{4} + \cdots + 21416 \) Copy content Toggle raw display
$83$ \( T^{6} - 10 T^{5} - 162 T^{4} + \cdots - 9616 \) Copy content Toggle raw display
$89$ \( T^{6} - 23 T^{5} + 25 T^{4} + \cdots + 31850 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} - 223 T^{4} + \cdots - 49387 \) Copy content Toggle raw display
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