Properties

Label 9800.2.a.cc
Level $9800$
Weight $2$
Character orbit 9800.a
Self dual yes
Analytic conductor $78.253$
Analytic rank $0$
Dimension $3$
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] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, 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("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.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: \(78.2533939809\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1400)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + (\beta_{2} - 2 \beta_1) q^{9} + ( - 2 \beta_{2} + \beta_1 - 2) q^{11} + ( - 4 \beta_{2} + 3 \beta_1) q^{13} + (3 \beta_{2} - 3 \beta_1 + 2) q^{17} + (\beta_{2} - 3 \beta_1 + 2) q^{19} + ( - \beta_{2} - 2 \beta_1 - 1) q^{23} - 3 \beta_{2} q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + ( - 3 \beta_{2} - \beta_1 - 1) q^{31} + (3 \beta_{2} - 5 \beta_1 + 2) q^{33} + ( - 4 \beta_{2} + 3 \beta_1 + 2) q^{37} + (7 \beta_{2} - 7 \beta_1 + 2) q^{39} + ( - \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - 3 \beta_{2} + 4 \beta_1 + 3) q^{43} + (3 \beta_{2} - 2 \beta_1 - 6) q^{47} + ( - 6 \beta_{2} + 8 \beta_1 - 5) q^{51} + (5 \beta_{2} + 4 \beta_1 - 2) q^{53} + ( - 4 \beta_{2} + 6 \beta_1 - 7) q^{57} + (3 \beta_1 - 1) q^{59} + (\beta_{2} + 5 \beta_1 + 3) q^{61} + (6 \beta_1 - 4) q^{67} + ( - \beta_{2} - 4) q^{69} + (6 \beta_{2} + 3) q^{71} + (4 \beta_{2} + 4) q^{73} + ( - \beta_{2} - 1) q^{79} + (3 \beta_1 - 3) q^{81} + ( - 3 \beta_{2} + 5 \beta_1) q^{83} + (4 \beta_1 - 7) q^{87} + ( - \beta_{2} - 6 \beta_1 + 2) q^{89} + (2 \beta_{2} - 3 \beta_1 - 4) q^{93} + (6 \beta_{2} - 3 \beta_1 + 7) q^{97} + ( - 2 \beta_{2} + 7 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 6 q^{11} + 6 q^{17} + 6 q^{19} - 3 q^{23} + 12 q^{29} - 3 q^{31} + 6 q^{33} + 6 q^{37} + 6 q^{39} + 3 q^{41} + 9 q^{43} - 18 q^{47} - 15 q^{51} - 6 q^{53} - 21 q^{57} - 3 q^{59} + 9 q^{61} - 12 q^{67} - 12 q^{69} + 9 q^{71} + 12 q^{73} - 3 q^{79} - 9 q^{81} - 21 q^{87} + 6 q^{89} - 12 q^{93} + 21 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
−1.53209
−0.347296
1.87939
0 −2.53209 0 0 0 0 0 3.41147 0
1.2 0 −1.34730 0 0 0 0 0 −1.18479 0
1.3 0 0.879385 0 0 0 0 0 −2.22668 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 9800.2.a.cc 3
5.b even 2 1 9800.2.a.ci 3
7.b odd 2 1 9800.2.a.ch 3
7.c even 3 2 1400.2.q.k yes 6
35.c odd 2 1 9800.2.a.cb 3
35.j even 6 2 1400.2.q.i 6
35.l odd 12 4 1400.2.bh.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.i 6 35.j even 6 2
1400.2.q.k yes 6 7.c even 3 2
1400.2.bh.h 12 35.l odd 12 4
9800.2.a.cb 3 35.c odd 2 1
9800.2.a.cc 3 1.a even 1 1 trivial
9800.2.a.ch 3 7.b odd 2 1
9800.2.a.ci 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3}^{3} + 3T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} + 3T_{11} - 19 \) Copy content Toggle raw display
\( T_{13}^{3} - 39T_{13} - 19 \) Copy content Toggle raw display
\( T_{19}^{3} - 6T_{19}^{2} - 9T_{19} + 17 \) Copy content Toggle raw display
\( T_{23}^{3} + 3T_{23}^{2} - 18T_{23} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + 3 T - 19 \) Copy content Toggle raw display
$13$ \( T^{3} - 39T - 19 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} - 15 T + 19 \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} - 9 T + 17 \) Copy content Toggle raw display
$23$ \( T^{3} + 3 T^{2} - 18 T + 17 \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + 39 T - 19 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 36 T - 19 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 27 T + 51 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} - 18 T + 57 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} - 12 T + 179 \) Copy content Toggle raw display
$47$ \( T^{3} + 18 T^{2} + 87 T + 107 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 171 T - 1077 \) Copy content Toggle raw display
$59$ \( T^{3} + 3 T^{2} - 24 T - 53 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} - 66 T - 37 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} - 60 T - 584 \) Copy content Toggle raw display
$71$ \( T^{3} - 9 T^{2} - 81 T + 513 \) Copy content Toggle raw display
$73$ \( T^{3} - 12T^{2} + 192 \) Copy content Toggle raw display
$79$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$83$ \( T^{3} - 57T + 163 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 117 T + 699 \) Copy content Toggle raw display
$97$ \( T^{3} - 21 T^{2} + 66 T + 467 \) Copy content Toggle raw display
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