Properties

Label 8010.2.a.bo
Level $8010$
Weight $2$
Character orbit 8010.a
Self dual yes
Analytic conductor $63.960$
Analytic rank $0$
Dimension $8$
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] = [8010,2,Mod(1,8010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8010, 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("8010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8010.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: \(63.9601720190\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} - 22x^{5} + 59x^{4} + 103x^{3} + 27x^{2} - 18x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} + ( - \beta_{5} - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{5} + ( - \beta_{5} - 1) q^{7} - q^{8} - q^{10} + \beta_{7} q^{11} + (\beta_{7} - \beta_{4} - 1) q^{13} + (\beta_{5} + 1) q^{14} + q^{16} + ( - \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{19} + q^{20} - \beta_{7} q^{22} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{23} + q^{25} + ( - \beta_{7} + \beta_{4} + 1) q^{26} + ( - \beta_{5} - 1) q^{28} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{2}) q^{29} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 1) q^{31} - q^{32} + (\beta_{2} + \beta_1 - 1) q^{34} + ( - \beta_{5} - 1) q^{35} + ( - 2 \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 3) q^{37} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{38} - q^{40} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_1 + 2) q^{41} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_1 - 2) q^{43} + \beta_{7} q^{44} + (\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{46} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_{2} + 4) q^{47} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 3) q^{49} - q^{50} + (\beta_{7} - \beta_{4} - 1) q^{52} + ( - 2 \beta_{5} + 2 \beta_1 + 2) q^{53} + \beta_{7} q^{55} + (\beta_{5} + 1) q^{56} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2}) q^{58} + (\beta_{6} - \beta_{5} - \beta_{4} + 3) q^{59} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 1) q^{61} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 1) q^{62} + q^{64} + (\beta_{7} - \beta_{4} - 1) q^{65} + (2 \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_1 - 1) q^{67} + ( - \beta_{2} - \beta_1 + 1) q^{68} + (\beta_{5} + 1) q^{70} + (2 \beta_{7} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{71} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_1 - 4) q^{73} + (2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{74} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{76} + ( - 2 \beta_{7} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{77} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{79} + q^{80} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_1 - 2) q^{82} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_1 + 2) q^{83} + ( - \beta_{2} - \beta_1 + 1) q^{85} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_1 + 2) q^{86} - \beta_{7} q^{88} - q^{89} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} + 4 \beta_1) q^{91} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{92} + (\beta_{7} - \beta_{6} - 2 \beta_{4} + 3 \beta_{2} - 4) q^{94} + (\beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{95} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 4) q^{97} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{2} - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 8 q^{5} - 5 q^{7} - 8 q^{8} - 8 q^{10} + 3 q^{11} - 2 q^{13} + 5 q^{14} + 8 q^{16} + 9 q^{17} + 5 q^{19} + 8 q^{20} - 3 q^{22} + 8 q^{23} + 8 q^{25} + 2 q^{26} - 5 q^{28} + 10 q^{29} + 2 q^{31} - 8 q^{32} - 9 q^{34} - 5 q^{35} - 16 q^{37} - 5 q^{38} - 8 q^{40} + 19 q^{41} - 4 q^{43} + 3 q^{44} - 8 q^{46} + 17 q^{47} + 11 q^{49} - 8 q^{50} - 2 q^{52} + 16 q^{53} + 3 q^{55} + 5 q^{56} - 10 q^{58} + 30 q^{59} - 15 q^{61} - 2 q^{62} + 8 q^{64} - 2 q^{65} - 3 q^{67} + 9 q^{68} + 5 q^{70} + 32 q^{71} - 20 q^{73} + 16 q^{74} + 5 q^{76} - 2 q^{77} - q^{79} + 8 q^{80} - 19 q^{82} + 13 q^{83} + 9 q^{85} + 4 q^{86} - 3 q^{88} - 8 q^{89} - 8 q^{91} + 8 q^{92} - 17 q^{94} + 5 q^{95} - 40 q^{97} - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 21x^{6} - 22x^{5} + 59x^{4} + 103x^{3} + 27x^{2} - 18x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 11\nu^{6} - 21\nu^{5} - 74\nu^{4} + 120\nu^{3} + 91\nu^{2} - 70\nu + 4 ) / 17 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6\nu^{7} + 15\nu^{6} + 95\nu^{5} + 15\nu^{4} - 249\nu^{3} - 304\nu^{2} - 80\nu + 41 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} - 4\nu^{6} - 116\nu^{5} - 89\nu^{4} + 369\nu^{3} + 395\nu^{2} + 44\nu - 37 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{7} + 47\nu^{6} + 139\nu^{5} - 225\nu^{4} - 362\nu^{3} + 123\nu^{2} + 78\nu - 3 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -15\nu^{7} + 46\nu^{6} + 212\nu^{5} - 90\nu^{4} - 563\nu^{3} - 369\nu^{2} - 13\nu + 43 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 14\nu^{6} - 185\nu^{5} - 82\nu^{4} + 620\nu^{3} + 558\nu^{2} - 152\nu - 155 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\nu^{7} - 45\nu^{6} - 302\nu^{5} + 23\nu^{4} + 934\nu^{3} + 606\nu^{2} - 202\nu - 72 ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{3} + 3\beta_{2} - \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 6\beta_{6} - 10\beta_{5} + 2\beta_{4} + 11\beta_{3} + 23\beta_{2} - 11\beta _1 + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -44\beta_{7} + 40\beta_{6} - 62\beta_{5} + 8\beta_{4} + 43\beta_{3} + 109\beta_{2} - 37\beta _1 + 176 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -208\beta_{7} + 190\beta_{6} - 322\beta_{5} + 54\beta_{4} + 249\beta_{3} + 603\beta_{2} - 225\beta _1 + 816 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1150 \beta_{7} + 1036 \beta_{6} - 1730 \beta_{5} + 268 \beta_{4} + 1247 \beta_{3} + 3113 \beta_{2} - 1085 \beta _1 + 4460 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5928 \beta_{7} + 5348 \beta_{6} - 9062 \beta_{5} + 1462 \beta_{4} + 6647 \beta_{3} + 16477 \beta_{2} - 5847 \beta _1 + 22938 ) / 2 \) 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
1.95321
−2.15922
−0.335485
−0.704235
5.24943
−1.29983
−2.12012
0.416251
−1.00000 0 1.00000 1.00000 0 −4.45998 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 −4.29170 −1.00000 0 −1.00000
1.3 −1.00000 0 1.00000 1.00000 0 −2.47766 −1.00000 0 −1.00000
1.4 −1.00000 0 1.00000 1.00000 0 −1.81337 −1.00000 0 −1.00000
1.5 −1.00000 0 1.00000 1.00000 0 0.561204 −1.00000 0 −1.00000
1.6 −1.00000 0 1.00000 1.00000 0 2.22167 −1.00000 0 −1.00000
1.7 −1.00000 0 1.00000 1.00000 0 2.33061 −1.00000 0 −1.00000
1.8 −1.00000 0 1.00000 1.00000 0 2.92921 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(89\) \(1\)

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 8010.2.a.bo 8
3.b odd 2 1 8010.2.a.bp yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8010.2.a.bo 8 1.a even 1 1 trivial
8010.2.a.bp yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\( T_{7}^{8} + 5T_{7}^{7} - 21T_{7}^{6} - 100T_{7}^{5} + 174T_{7}^{4} + 614T_{7}^{3} - 630T_{7}^{2} - 1164T_{7} + 732 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 37T_{11}^{6} + 140T_{11}^{5} - 12T_{11}^{4} - 532T_{11}^{3} + 828T_{11}^{2} - 480T_{11} + 96 \) Copy content Toggle raw display
\( T_{13}^{8} + 2T_{13}^{7} - 66T_{13}^{6} - 96T_{13}^{5} + 1142T_{13}^{4} + 1412T_{13}^{3} - 3928T_{13}^{2} - 2304T_{13} + 3968 \) Copy content Toggle raw display
\( T_{17}^{8} - 9T_{17}^{7} - 41T_{17}^{6} + 388T_{17}^{5} + 840T_{17}^{4} - 5148T_{17}^{3} - 9620T_{17}^{2} + 16320T_{17} + 26672 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 5 T^{7} - 21 T^{6} - 100 T^{5} + \cdots + 732 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} - 37 T^{6} + 140 T^{5} + \cdots + 96 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} - 66 T^{6} + \cdots + 3968 \) Copy content Toggle raw display
$17$ \( T^{8} - 9 T^{7} - 41 T^{6} + \cdots + 26672 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} - 77 T^{6} + 390 T^{5} + \cdots - 272 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} - 148 T^{6} + \cdots - 2856272 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} - 108 T^{6} + \cdots - 117872 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} - 150 T^{6} + \cdots + 74528 \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} - 26 T^{6} + \cdots + 249856 \) Copy content Toggle raw display
$41$ \( T^{8} - 19 T^{7} + 85 T^{6} + \cdots - 9792 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} - 252 T^{6} + \cdots + 151552 \) Copy content Toggle raw display
$47$ \( T^{8} - 17 T^{7} - 89 T^{6} + \cdots - 13056 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} - 84 T^{6} + \cdots - 541696 \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + 298 T^{6} + \cdots - 192 \) Copy content Toggle raw display
$61$ \( T^{8} + 15 T^{7} - 295 T^{6} + \cdots - 1436804 \) Copy content Toggle raw display
$67$ \( T^{8} + 3 T^{7} - 203 T^{6} + \cdots + 2478128 \) Copy content Toggle raw display
$71$ \( T^{8} - 32 T^{7} + 80 T^{6} + \cdots - 4435712 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} - 196 T^{6} + \cdots + 9767168 \) Copy content Toggle raw display
$79$ \( T^{8} + T^{7} - 451 T^{6} + \cdots - 5975232 \) Copy content Toggle raw display
$83$ \( T^{8} - 13 T^{7} - 87 T^{6} + \cdots - 1846784 \) Copy content Toggle raw display
$89$ \( (T + 1)^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 40 T^{7} + 380 T^{6} + \cdots + 14344192 \) Copy content Toggle raw display
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