↑

Properties

Label 9680.2.a.de
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $8$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
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: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{4} + 3) q^{9} + ( - \beta_{5} + \beta_1 - 2) q^{13} - \beta_1 q^{15} + ( - \beta_{7} - \beta_{5} - 2 \beta_{2} + \cdots - 2) q^{17}+ \cdots + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} - 12 q^{13} - q^{15} - 2 q^{17} - 6 q^{19} - 6 q^{21} - 10 q^{23} + 8 q^{25} - 13 q^{27} - 8 q^{29} - 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} + 3 q^{41} - 8 q^{43} + 19 q^{45} - 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} - 25 q^{57} - 25 q^{59} - 10 q^{61} - 64 q^{63} - 12 q^{65} + 2 q^{67} + 18 q^{69} - 25 q^{71} - 38 q^{73} - q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} - 2 q^{85} + 15 q^{87} + 12 q^{89} - 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 3\nu^{6} - 23\nu^{5} - 35\nu^{4} + 140\nu^{3} + 80\nu^{2} - 185\nu - 60 ) / 70 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + \nu^{6} + 25\nu^{5} - 70\nu^{3} - 55\nu^{2} + 55\nu + 50 ) / 35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} - 41\nu^{5} + 77\nu^{4} + 140\nu^{3} - 320\nu^{2} - 135\nu + 100 ) / 70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 5\nu^{6} + 41\nu^{5} - 77\nu^{4} - 140\nu^{3} + 390\nu^{2} + 135\nu - 520 ) / 70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 5\nu^{6} + 55\nu^{5} - 49\nu^{4} - 350\nu^{3} + 40\nu^{2} + 765\nu + 110 ) / 70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 13\nu^{6} + 73\nu^{5} - 161\nu^{4} - 280\nu^{3} + 440\nu^{2} + 155\nu - 120 ) / 70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 9\beta_{5} + 12\beta_{4} + \beta_{3} + 2\beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{7} - 10\beta_{6} + 2\beta_{5} + 16\beta_{4} - 2\beta_{3} - 15\beta_{2} + 71\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{7} + 81\beta_{5} + 123\beta_{4} + 9\beta_{3} + 5\beta_{2} + 38\beta _1 + 450 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 137\beta_{7} - 90\beta_{6} + 38\beta_{5} + 199\beta_{4} - 38\beta_{3} - 150\beta_{2} + 654\beta _1 + 300 \) 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
3.21195
3.08163
1.88088
0.713352
−0.569801
−1.64111
−2.77190
−2.90500
0 −3.21195 0 1.00000 0 −4.90937 0 7.31662 0
1.2 0 −3.08163 0 1.00000 0 0.0385298 0 6.49642 0
1.3 0 −1.88088 0 1.00000 0 3.67750 0 0.537727 0
1.4 0 −0.713352 0 1.00000 0 0.376172 0 −2.49113 0
1.5 0 0.569801 0 1.00000 0 −1.82111 0 −2.67533 0
1.6 0 1.64111 0 1.00000 0 3.37642 0 −0.306743 0
1.7 0 2.77190 0 1.00000 0 −4.55506 0 4.68342 0
1.8 0 2.90500 0 1.00000 0 −2.18309 0 5.43902 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

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 9680.2.a.de 8
4.b odd 2 1 4840.2.a.bh 8
11.b odd 2 1 9680.2.a.df 8
11.d odd 10 2 880.2.bo.k 16
44.c even 2 1 4840.2.a.bg 8
44.g even 10 2 440.2.y.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.d ✓ 16 44.g even 10 2
880.2.bo.k 16 11.d odd 10 2
4840.2.a.bg 8 44.c even 2 1
4840.2.a.bh 8 4.b odd 2 1
9680.2.a.de 8 1.a even 1 1 trivial
9680.2.a.df 8 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{8} + T_{3}^{7} - 21T_{3}^{6} - 15T_{3}^{5} + 140T_{3}^{4} + 60T_{3}^{3} - 295T_{3}^{2} - 50T_{3} + 100 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} - 21T_{7}^{6} - 151T_{7}^{5} + 55T_{7}^{4} + 954T_{7}^{3} + 709T_{7}^{2} - 444T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{8} + 12T_{13}^{7} + 17T_{13}^{6} - 261T_{13}^{5} - 871T_{13}^{4} + 1006T_{13}^{3} + 6127T_{13}^{2} + 2508T_{13} - 6064 \) Copy content Toggle raw display
\( T_{17}^{8} + 2T_{17}^{7} - 96T_{17}^{6} - 95T_{17}^{5} + 2549T_{17}^{4} - 171T_{17}^{3} - 12545T_{17}^{2} - 5794T_{17} + 3124 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + \cdots - 6064 \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 3124 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 74855 \) Copy content Toggle raw display
$23$ \( T^{8} + 10 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + \cdots + 69520 \) Copy content Toggle raw display
$31$ \( T^{8} + 19 T^{7} + \cdots - 2480 \) Copy content Toggle raw display
$37$ \( T^{8} - 12 T^{7} + \cdots - 1020420 \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} + \cdots - 126755 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + \cdots - 76780 \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{7} + \cdots + 886724 \) Copy content Toggle raw display
$53$ \( T^{8} - 28 T^{7} + \cdots + 4820 \) Copy content Toggle raw display
$59$ \( T^{8} + 25 T^{7} + \cdots + 11749 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots - 1596416 \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{7} + \cdots + 2034124 \) Copy content Toggle raw display
$71$ \( T^{8} + 25 T^{7} + \cdots + 490000 \) Copy content Toggle raw display
$73$ \( T^{8} + 38 T^{7} + \cdots - 53420 \) Copy content Toggle raw display
$79$ \( T^{8} + 38 T^{7} + \cdots + 12393280 \) Copy content Toggle raw display
$83$ \( T^{8} + 28 T^{7} + \cdots + 7869644 \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + \cdots + 2898841 \) Copy content Toggle raw display
$97$ \( T^{8} - 21 T^{7} + \cdots + 105621104 \) Copy content Toggle raw display
show more
show less