↑

Properties

Label 8910.2.a.cd
Level $8910$
Weight $2$
Character orbit 8910.a
Self dual yes
Analytic conductor $71.147$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8910.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,7,0,7,7,0,5,7,0,7,7,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.1467082010\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 13x^{5} + 31x^{4} + 21x^{3} - 37x^{2} - 15x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 990)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + ( - \beta_{3} + 1) q^{7} + q^{8} + q^{10} + q^{11} + (\beta_1 + 1) q^{13} + ( - \beta_{3} + 1) q^{14} + q^{16} - \beta_{4} q^{17} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{19}+ \cdots + (\beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + 5 q^{7} + 7 q^{8} + 7 q^{10} + 7 q^{11} + 5 q^{13} + 5 q^{14} + 7 q^{16} + 3 q^{17} + 11 q^{19} + 7 q^{20} + 7 q^{22} + 3 q^{23} + 7 q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 13x^{5} + 31x^{4} + 21x^{3} - 37x^{2} - 15x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 9\nu^{5} + 4\nu^{4} + 104\nu^{3} - 132\nu^{2} - 87\nu + 43 ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{6} - 19\nu^{5} - 74\nu^{4} + 196\nu^{3} + 80\nu^{2} - 213\nu - 33 ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 17\nu^{5} + 35\nu^{4} - 178\nu^{3} + 87\nu^{2} + 135\nu - 64 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} - 21\nu^{5} - 45\nu^{4} + 224\nu^{3} - 96\nu^{2} - 192\nu + 56 ) / 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{6} - 28\nu^{5} - 108\nu^{4} + 275\nu^{3} + 85\nu^{2} - 201\nu - 37 ) / 11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{6} + 24\nu^{5} + 60\nu^{4} - 254\nu^{3} + 43\nu^{2} + 210\nu - 7 ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + 4\beta_{3} + 4\beta _1 + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{6} + \beta_{5} + 15\beta_{4} + 17\beta_{3} - 6\beta_{2} - 4\beta _1 + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 25\beta_{6} + 32\beta_{5} + 18\beta_{4} + 67\beta_{3} - 3\beta_{2} + 58\beta _1 + 186 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{6} + 52\beta_{5} + 225\beta_{4} + 287\beta_{3} - 72\beta_{2} + 17\beta _1 + 420 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 406\beta_{6} + 500\beta_{5} + 480\beta_{4} + 1162\beta_{3} - 99\beta_{2} + 811\beta _1 + 2790 ) / 3 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
4.18014
−0.888981
1.25457
−3.18331
2.10401
0.263313
−0.729749
1.00000 0 1.00000 1.00000 0 −4.64523 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −0.840248 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 −0.341930 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 0.591383 1.00000 0 1.00000
1.5 1.00000 0 1.00000 1.00000 0 2.17324 1.00000 0 1.00000
1.6 1.00000 0 1.00000 1.00000 0 3.31654 1.00000 0 1.00000
1.7 1.00000 0 1.00000 1.00000 0 4.74624 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
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(11\) \( -1 \)

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 8910.2.a.cd 7
3.b odd 2 1 8910.2.a.ca 7
9.c even 3 2 990.2.i.j ✓ 14
9.d odd 6 2 2970.2.i.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.i.j ✓ 14 9.c even 3 2
2970.2.i.k 14 9.d odd 6 2
8910.2.a.ca 7 3.b odd 2 1
8910.2.a.cd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8910))\):

\( T_{7}^{7} - 5T_{7}^{6} - 18T_{7}^{5} + 114T_{7}^{4} - 81T_{7}^{3} - 141T_{7}^{2} + 45T_{7} + 27 \) Copy content Toggle raw display
\( T_{13}^{7} - 5T_{13}^{6} - 42T_{13}^{5} + 285T_{13}^{4} - 54T_{13}^{3} - 2472T_{13}^{2} + 4968T_{13} - 2592 \) Copy content Toggle raw display
\( T_{17}^{7} - 3T_{17}^{6} - 69T_{17}^{5} + 149T_{17}^{4} + 1029T_{17}^{3} - 1779T_{17}^{2} - 383T_{17} + 24 \) Copy content Toggle raw display
\( T_{23}^{7} - 3T_{23}^{6} - 57T_{23}^{5} + 60T_{23}^{4} + 1047T_{23}^{3} + 630T_{23}^{2} - 3456T_{23} - 1944 \) Copy content Toggle raw display
\( T_{29}^{7} - 12T_{29}^{6} - 12T_{29}^{5} + 404T_{29}^{4} - 450T_{29}^{3} - 474T_{29}^{2} + 622T_{29} - 81 \) 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
$7$ \( T^{7} - 5 T^{6} + \cdots + 27 \) Copy content Toggle raw display
$11$ \( (T - 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 5 T^{6} + \cdots - 2592 \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} + \cdots + 24 \) Copy content Toggle raw display
$19$ \( T^{7} - 11 T^{6} + \cdots - 40178 \) Copy content Toggle raw display
$23$ \( T^{7} - 3 T^{6} + \cdots - 1944 \) Copy content Toggle raw display
$29$ \( T^{7} - 12 T^{6} + \cdots - 81 \) Copy content Toggle raw display
$31$ \( T^{7} - 5 T^{6} + \cdots + 4806 \) Copy content Toggle raw display
$37$ \( T^{7} - 2 T^{6} + \cdots + 4644 \) Copy content Toggle raw display
$41$ \( T^{7} - 6 T^{6} + \cdots + 9816 \) Copy content Toggle raw display
$43$ \( T^{7} - 17 T^{6} + \cdots - 1013024 \) Copy content Toggle raw display
$47$ \( T^{7} + 3 T^{6} + \cdots + 7104 \) Copy content Toggle raw display
$53$ \( T^{7} - 3 T^{6} + \cdots + 21744 \) Copy content Toggle raw display
$59$ \( T^{7} - 18 T^{6} + \cdots + 4580736 \) Copy content Toggle raw display
$61$ \( T^{7} - 11 T^{6} + \cdots - 17807 \) Copy content Toggle raw display
$67$ \( T^{7} - 20 T^{6} + \cdots + 41912443 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} + \cdots - 164064 \) Copy content Toggle raw display
$73$ \( T^{7} - 2 T^{6} + \cdots + 22888 \) Copy content Toggle raw display
$79$ \( T^{7} + 7 T^{6} + \cdots - 66944 \) Copy content Toggle raw display
$83$ \( T^{7} - 219 T^{5} + \cdots - 8424 \) Copy content Toggle raw display
$89$ \( T^{7} + 3 T^{6} + \cdots - 227448 \) Copy content Toggle raw display
$97$ \( T^{7} - 23 T^{6} + \cdots - 827712 \) Copy content Toggle raw display
show more
show less