Properties

Label 8281.2.a.cg
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1279733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1183)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 5\nu - 2 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 18\beta _1 + 8 \) 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
−1.71083
2.33192
−1.54570
−0.0849355
2.10066
0.908891
−2.08281 0.0911085 2.33809 −2.63777 −0.189762 0 −0.704173 −2.99170 5.49396
1.2 −1.90785 1.08494 1.63989 −1.10591 −2.06989 0 0.687029 −1.82292 2.10992
1.3 0.312100 −1.10066 −1.90259 −1.93488 −0.343514 0 −1.21800 −1.78856 −0.603875
1.4 1.10591 −1.33192 −0.776957 1.90785 −1.47298 0 −3.07107 −1.22600 2.10992
1.5 1.93488 2.54570 1.74376 −0.312100 4.92562 0 −0.495793 3.48058 −0.603875
1.6 2.63777 2.71083 4.95781 2.08281 7.15053 0 7.80201 4.34860 5.49396
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cg 6
7.b odd 2 1 1183.2.a.o yes 6
13.b even 2 1 8281.2.a.cb 6
91.b odd 2 1 1183.2.a.n 6
91.i even 4 2 1183.2.c.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.n 6 91.b odd 2 1
1183.2.a.o yes 6 7.b odd 2 1
1183.2.c.h 12 91.i even 4 2
8281.2.a.cb 6 13.b even 2 1
8281.2.a.cg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 15T_{2}^{3} + 14T_{2}^{2} - 28T_{2} + 7 \) Copy content Toggle raw display
\( T_{3}^{6} - 4T_{3}^{5} - T_{3}^{4} + 14T_{3}^{3} - T_{3}^{2} - 11T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 2T_{5}^{5} - 8T_{5}^{4} - 15T_{5}^{3} + 14T_{5}^{2} + 28T_{5} + 7 \) Copy content Toggle raw display
\( T_{11}^{6} - 8T_{11}^{5} - 16T_{11}^{4} + 197T_{11}^{3} - 28T_{11}^{2} - 1204T_{11} + 889 \) Copy content Toggle raw display
\( T_{17}^{6} - 23T_{17}^{5} + 190T_{17}^{4} - 585T_{17}^{3} - 319T_{17}^{2} + 5190T_{17} - 7351 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} - 8 T^{4} + 15 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - T^{4} + 14 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} - 8 T^{4} - 15 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 8 T^{5} - 16 T^{4} + 197 T^{3} + \cdots + 889 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 23 T^{5} + 190 T^{4} + \cdots - 7351 \) Copy content Toggle raw display
$19$ \( T^{6} - 13 T^{5} + 47 T^{4} + \cdots + 581 \) Copy content Toggle raw display
$23$ \( T^{6} + 18 T^{5} + 97 T^{4} + \cdots - 587 \) Copy content Toggle raw display
$29$ \( T^{6} + 15 T^{5} + 9 T^{4} + \cdots + 3569 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} - 157 T^{4} + \cdots - 21463 \) Copy content Toggle raw display
$37$ \( T^{6} + 13 T^{5} + 5 T^{4} - 118 T^{3} + \cdots - 7 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} - 115 T^{4} + \cdots + 503 \) Copy content Toggle raw display
$43$ \( T^{6} + 18 T^{5} + 53 T^{4} + \cdots + 181 \) Copy content Toggle raw display
$47$ \( T^{6} - 16 T^{5} - 22 T^{4} + \cdots + 30233 \) Copy content Toggle raw display
$53$ \( T^{6} + 25 T^{5} + 111 T^{4} + \cdots + 24193 \) Copy content Toggle raw display
$59$ \( T^{6} + 18 T^{5} - 123 T^{4} + \cdots + 92911 \) Copy content Toggle raw display
$61$ \( T^{6} + 16 T^{5} - 55 T^{4} + \cdots - 12979 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} - 106 T^{4} + \cdots + 26747 \) Copy content Toggle raw display
$71$ \( T^{6} - 25 T^{5} + 24 T^{4} + \cdots + 563899 \) Copy content Toggle raw display
$73$ \( T^{6} - 5 T^{5} - 176 T^{4} + \cdots - 45367 \) Copy content Toggle raw display
$79$ \( T^{6} - 2 T^{5} - 167 T^{4} + \cdots - 10277 \) Copy content Toggle raw display
$83$ \( T^{6} - 7 T^{5} - 272 T^{4} + \cdots - 41203 \) Copy content Toggle raw display
$89$ \( T^{6} - 10 T^{5} - 242 T^{4} + \cdots - 222257 \) Copy content Toggle raw display
$97$ \( T^{6} - 5 T^{5} - 296 T^{4} + \cdots - 43931 \) Copy content Toggle raw display
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