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Properties

Label 8010.2.a.bn
Level $8010$
Weight $2$
Character orbit 8010.a
Self dual yes
Analytic conductor $63.960$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 24x^{5} + 34x^{4} + 111x^{3} - 127x^{2} - 20x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2670)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - \beta_{2} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} - \beta_{2} q^{7} + q^{8} + q^{10} + \beta_{3} q^{11} + ( - \beta_{5} - \beta_{2}) q^{13} - \beta_{2} q^{14} + q^{16} + ( - \beta_{6} + \beta_{2} + 1) q^{17} + ( - \beta_{4} + 1) q^{19} + q^{20} + \beta_{3} q^{22} + (\beta_{2} + \beta_1 + 1) q^{23} + q^{25} + ( - \beta_{5} - \beta_{2}) q^{26} - \beta_{2} q^{28} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{29}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 24x^{5} + 34x^{4} + 111x^{3} - 127x^{2} - 20x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{6} + 17\nu^{5} - 204\nu^{4} - 750\nu^{3} + 1539\nu^{2} + 4973\nu - 1820 ) / 964 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{6} + 59\nu^{5} + 256\nu^{4} - 1242\nu^{3} - 1265\nu^{2} + 5351\nu - 2744 ) / 964 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{6} - 101\nu^{5} - 716\nu^{4} + 1734\nu^{3} + 5033\nu^{2} - 7657\nu - 4044 ) / 964 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\nu^{6} - 101\nu^{5} - 716\nu^{4} + 1734\nu^{3} + 5033\nu^{2} - 5729\nu - 5008 ) / 964 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\nu^{6} - 33\nu^{5} - 1532\nu^{4} - 302\nu^{3} + 8297\nu^{2} - 297\nu - 2648 ) / 964 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 104\nu^{6} - 273\nu^{5} - 2508\nu^{4} + 2234\nu^{3} + 10996\nu^{2} - 6313\nu - 2670 ) / 482 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{2} - \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + 19\beta_{4} - 15\beta_{3} + 12\beta_{2} - 14\beta _1 + 49 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 2\beta_{5} + 38\beta_{4} - 19\beta_{3} + 55\beta_{2} - 42\beta _1 + 206 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{6} + 68\beta_{5} + 509\beta_{4} - 357\beta_{3} + 512\beta_{2} - 500\beta _1 + 1869 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34\beta_{6} + 116\beta_{5} + 1305\beta_{4} - 796\beta_{3} + 1658\beta_{2} - 1413\beta _1 + 5999 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
5.68936
−3.29882
1.14258
−2.69745
0.327888
2.21696
−0.380516
1.00000 0 1.00000 1.00000 0 −5.07548 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −3.64705 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 −0.407283 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 1.15694 1.00000 0 1.00000
1.5 1.00000 0 1.00000 1.00000 0 1.20963 1.00000 0 1.00000
1.6 1.00000 0 1.00000 1.00000 0 2.69061 1.00000 0 1.00000
1.7 1.00000 0 1.00000 1.00000 0 5.07263 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.bn 7
3.b odd 2 1 2670.2.a.t ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2670.2.a.t ✓ 7 3.b odd 2 1
8010.2.a.bn 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):

\( T_{7}^{7} - T_{7}^{6} - 37T_{7}^{5} + 46T_{7}^{4} + 286T_{7}^{3} - 526T_{7}^{2} + 96T_{7} + 144 \) Copy content Toggle raw display
\( T_{11}^{7} + T_{11}^{6} - 65T_{11}^{5} - 46T_{11}^{4} + 1120T_{11}^{3} + 220T_{11}^{2} - 4544T_{11} - 2304 \) Copy content Toggle raw display
\( T_{13}^{7} - 86T_{13}^{5} + 18T_{13}^{4} + 2364T_{13}^{3} - 1568T_{13}^{2} - 20848T_{13} + 29088 \) Copy content Toggle raw display
\( T_{17}^{7} - 9T_{17}^{6} - 53T_{17}^{5} + 616T_{17}^{4} - 20T_{17}^{3} - 9804T_{17}^{2} + 20464T_{17} - 9392 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - T^{6} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{7} + T^{6} + \cdots - 2304 \) Copy content Toggle raw display
$13$ \( T^{7} - 86 T^{5} + \cdots + 29088 \) Copy content Toggle raw display
$17$ \( T^{7} - 9 T^{6} + \cdots - 9392 \) Copy content Toggle raw display
$19$ \( T^{7} - 9 T^{6} + \cdots - 28128 \) Copy content Toggle raw display
$23$ \( T^{7} - 8 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{7} - 4 T^{6} + \cdots + 82816 \) Copy content Toggle raw display
$31$ \( T^{7} - 16 T^{6} + \cdots - 62464 \) Copy content Toggle raw display
$37$ \( T^{7} - 2 T^{6} + \cdots + 21664 \) Copy content Toggle raw display
$41$ \( T^{7} - T^{6} + \cdots + 15264 \) Copy content Toggle raw display
$43$ \( T^{7} + 10 T^{6} + \cdots - 12288 \) Copy content Toggle raw display
$47$ \( T^{7} - 13 T^{6} + \cdots - 36096 \) Copy content Toggle raw display
$53$ \( T^{7} - 24 T^{6} + \cdots - 139264 \) Copy content Toggle raw display
$59$ \( T^{7} - 6 T^{6} + \cdots + 508416 \) Copy content Toggle raw display
$61$ \( T^{7} - 23 T^{6} + \cdots - 805304 \) Copy content Toggle raw display
$67$ \( T^{7} - 5 T^{6} + \cdots - 4288 \) Copy content Toggle raw display
$71$ \( T^{7} - 8 T^{6} + \cdots - 1098752 \) Copy content Toggle raw display
$73$ \( T^{7} + 2 T^{6} + \cdots + 22336 \) Copy content Toggle raw display
$79$ \( T^{7} - 7 T^{6} + \cdots - 212736 \) Copy content Toggle raw display
$83$ \( T^{7} - 7 T^{6} + \cdots + 6912 \) Copy content Toggle raw display
$89$ \( (T + 1)^{7} \) Copy content Toggle raw display
$97$ \( T^{7} - 30 T^{6} + \cdots + 63104 \) Copy content Toggle raw display
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