Properties

Label 9680.2.a.cz
Level $9680$
Weight $2$
Character orbit 9680.a
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.507624192.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 20x^{4} + 100x^{2} - 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2420)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{4} - \beta_{3} + 3) q^{9} + (\beta_{5} + \beta_{2} + \beta_1) q^{13} - \beta_{4} q^{15} + 2 \beta_1 q^{17} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - 2 \beta_{5} - \beta_{2}) q^{21} + (\beta_{4} + \beta_{3} - 1) q^{23} + q^{25} + (4 \beta_{4} - \beta_{3} + 3) q^{27} - 2 \beta_1 q^{29} + (\beta_{4} - \beta_{3} - 5) q^{31} + ( - \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{4} + 6) q^{37} + (\beta_{5} - 5 \beta_{2} + 3 \beta_1) q^{39} + (\beta_{5} - 2 \beta_{2} + \beta_1) q^{41} + (3 \beta_{2} - \beta_1) q^{43} + ( - \beta_{4} + \beta_{3} - 3) q^{45} + 3 \beta_{4} q^{47} + ( - \beta_{4} - \beta_{3} + 3) q^{49} + (2 \beta_{5} + 2 \beta_{2}) q^{51} + (\beta_{4} + \beta_{3} - 1) q^{53} + ( - \beta_{5} + 7 \beta_{2} - 3 \beta_1) q^{57} + (3 \beta_{4} - \beta_{3} - 3) q^{59} + (\beta_{5} - \beta_1) q^{61} + ( - \beta_{5} + 9 \beta_{2} - 3 \beta_1) q^{63} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{65} + (\beta_{4} - 2 \beta_{3} + 4) q^{67} + ( - 3 \beta_{4} - \beta_{3} + 9) q^{69} + ( - 3 \beta_{4} + \beta_{3} + 3) q^{71} + 4 \beta_{2} q^{73} + \beta_{4} q^{75} + ( - 2 \beta_{5} - 4 \beta_1) q^{79} + (7 \beta_{4} - \beta_{3} + 12) q^{81} + (\beta_{5} - 5 \beta_{2} - \beta_1) q^{83} - 2 \beta_1 q^{85} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{87} + (2 \beta_{4} + 1) q^{89} + ( - 5 \beta_{4} - \beta_{3} - 1) q^{91} + ( - \beta_{4} - \beta_{3} + 3) q^{93} + (\beta_{5} - \beta_{2} - \beta_1) q^{95} + (\beta_{4} + \beta_{3} + 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 6 q^{5} + 20 q^{9} - 2 q^{15} - 4 q^{23} + 6 q^{25} + 26 q^{27} - 28 q^{31} + 32 q^{37} - 20 q^{45} + 6 q^{47} + 16 q^{49} - 4 q^{53} - 12 q^{59} + 26 q^{67} + 48 q^{69} + 12 q^{71} + 2 q^{75} + 86 q^{81} + 10 q^{89} - 16 q^{91} + 16 q^{93} + 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 20x^{4} + 100x^{2} - 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 10\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 10\nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 16\nu^{2} + 42 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 17\nu^{3} + 52\nu ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{2} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{4} + 16\beta_{3} + 70 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} + 102\beta_{2} + 118\beta_1 \) 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
2.36936
−2.36936
3.59084
−3.59084
1.22147
−1.22147
0 −2.71774 0 −1.00000 0 −4.10141 0 4.38612 0
1.2 0 −2.71774 0 −1.00000 0 4.10141 0 4.38612 0
1.3 0 0.325397 0 −1.00000 0 −1.85879 0 −2.89412 0
1.4 0 0.325397 0 −1.00000 0 1.85879 0 −2.89412 0
1.5 0 3.39234 0 −1.00000 0 −2.95353 0 8.50800 0
1.6 0 3.39234 0 −1.00000 0 2.95353 0 8.50800 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9680.2.a.cz 6
4.b odd 2 1 2420.2.a.o 6
11.b odd 2 1 inner 9680.2.a.cz 6
44.c even 2 1 2420.2.a.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2420.2.a.o 6 4.b odd 2 1
2420.2.a.o 6 44.c even 2 1
9680.2.a.cz 6 1.a even 1 1 trivial
9680.2.a.cz 6 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9680))\):

\( T_{3}^{3} - T_{3}^{2} - 9T_{3} + 3 \) Copy content Toggle raw display
\( T_{7}^{6} - 29T_{7}^{4} + 235T_{7}^{2} - 507 \) Copy content Toggle raw display
\( T_{13}^{6} - 68T_{13}^{4} + 1504T_{13}^{2} - 10800 \) Copy content Toggle raw display
\( T_{17}^{6} - 80T_{17}^{4} + 1600T_{17}^{2} - 6912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 29 T^{4} + \cdots - 507 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 68 T^{4} + \cdots - 10800 \) Copy content Toggle raw display
$17$ \( T^{6} - 80 T^{4} + \cdots - 6912 \) Copy content Toggle raw display
$19$ \( T^{6} - 104 T^{4} + \cdots - 40368 \) Copy content Toggle raw display
$23$ \( (T^{3} + 2 T^{2} - 44 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 80 T^{4} + \cdots - 6912 \) Copy content Toggle raw display
$31$ \( (T^{3} + 14 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 16 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 113 T^{4} + \cdots - 27 \) Copy content Toggle raw display
$43$ \( T^{6} - 101 T^{4} + \cdots - 6075 \) Copy content Toggle raw display
$47$ \( (T^{3} - 3 T^{2} - 81 T + 81)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} - 44 T + 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 6 T^{2} + \cdots - 540)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 89 T^{4} + \cdots - 18723 \) Copy content Toggle raw display
$67$ \( (T^{3} - 13 T^{2} + \cdots + 895)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 6 T^{2} + \cdots + 540)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} - 452 T^{4} + \cdots - 192 \) Copy content Toggle raw display
$83$ \( T^{6} - 344 T^{4} + \cdots - 73008 \) Copy content Toggle raw display
$89$ \( (T^{3} - 5 T^{2} - 29 T + 57)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 22 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
show more
show less