Properties

Label 8281.2.a.bi
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
2.17009
0.311108
−1.48119
−1.17009 −0.539189 −0.630898 −0.460811 0.630898 0 3.07838 −2.70928 0.539189
1.2 0.688892 2.21432 −1.52543 −3.21432 1.52543 0 −2.42864 1.90321 −2.21432
1.3 2.48119 −1.67513 4.15633 0.675131 −4.15633 0 5.35026 −0.193937 1.67513
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

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 8281.2.a.bi 3
7.b odd 2 1 1183.2.a.j 3
13.b even 2 1 8281.2.a.be 3
13.d odd 4 2 637.2.c.d 6
91.b odd 2 1 1183.2.a.h 3
91.i even 4 2 91.2.c.a 6
91.z odd 12 4 637.2.r.d 12
91.bb even 12 4 637.2.r.e 12
273.o odd 4 2 819.2.c.b 6
364.p odd 4 2 1456.2.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 91.i even 4 2
637.2.c.d 6 13.d odd 4 2
637.2.r.d 12 91.z odd 12 4
637.2.r.e 12 91.bb even 12 4
819.2.c.b 6 273.o odd 4 2
1183.2.a.h 3 91.b odd 2 1
1183.2.a.j 3 7.b odd 2 1
1456.2.k.c 6 364.p odd 4 2
8281.2.a.be 3 13.b even 2 1
8281.2.a.bi 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{3} - 2T_{2}^{2} - 2T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{3} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} + 3T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 8T_{11}^{2} + 18T_{11} - 10 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 8T_{17} - 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} - 2 T + 2 \) Copy content Toggle raw display
$3$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} + 3T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + 18 T - 10 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 8 T - 34 \) Copy content Toggle raw display
$19$ \( T^{3} + T^{2} - 59 T - 193 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 25 T - 79 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} - 21 T + 5 \) Copy content Toggle raw display
$31$ \( T^{3} + 11 T^{2} + 19 T - 65 \) Copy content Toggle raw display
$37$ \( T^{3} - 12 T^{2} + 18 T + 54 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 52 T - 40 \) Copy content Toggle raw display
$43$ \( T^{3} - 13 T^{2} + 35 T + 17 \) Copy content Toggle raw display
$47$ \( T^{3} + 19 T^{2} + 105 T + 137 \) Copy content Toggle raw display
$53$ \( T^{3} - T^{2} - 9T + 13 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} - 32 T + 52 \) Copy content Toggle raw display
$61$ \( T^{3} + 14 T^{2} + 28 T - 152 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + 32 T - 16 \) Copy content Toggle raw display
$71$ \( T^{3} - 14 T^{2} - 54 T + 890 \) Copy content Toggle raw display
$73$ \( T^{3} - 9 T^{2} - 91 T - 31 \) Copy content Toggle raw display
$79$ \( T^{3} - 13 T^{2} - 37 T + 185 \) Copy content Toggle raw display
$83$ \( T^{3} + T^{2} - 113 T - 163 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} - 55 T - 227 \) Copy content Toggle raw display
$97$ \( T^{3} + 15 T^{2} - 175 T - 2183 \) Copy content Toggle raw display
show more
show less