Properties

Label 9702.2.a.dq
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} + q^{11} - 2 \beta q^{13} + q^{16} - 4 \beta q^{17} - 2 \beta q^{19} + \beta q^{20} + q^{22} + 2 q^{23} - 3 q^{25} - 2 \beta q^{26} + 6 q^{29} + \beta q^{31} + q^{32} - 4 \beta q^{34} - 2 q^{37} - 2 \beta q^{38} + \beta q^{40} + 4 \beta q^{41} + 4 q^{43} + q^{44} + 2 q^{46} + 3 \beta q^{47} - 3 q^{50} - 2 \beta q^{52} + 12 q^{53} + \beta q^{55} + 6 q^{58} + 3 \beta q^{59} + 4 \beta q^{61} + \beta q^{62} + q^{64} - 4 q^{65} - 2 q^{67} - 4 \beta q^{68} + 10 q^{71} + 10 \beta q^{73} - 2 q^{74} - 2 \beta q^{76} - 8 q^{79} + \beta q^{80} + 4 \beta q^{82} - 2 \beta q^{83} - 8 q^{85} + 4 q^{86} + q^{88} - 5 \beta q^{89} + 2 q^{92} + 3 \beta q^{94} - 4 q^{95} + 11 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} + 4 q^{23} - 6 q^{25} + 12 q^{29} + 2 q^{32} - 4 q^{37} + 8 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{50} + 24 q^{53} + 12 q^{58} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 20 q^{71} - 4 q^{74} - 16 q^{79} - 16 q^{85} + 8 q^{86} + 2 q^{88} + 4 q^{92} - 8 q^{95}+O(q^{100}) \) 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
−1.41421
1.41421
1.00000 0 1.00000 −1.41421 0 0 1.00000 0 −1.41421
1.2 1.00000 0 1.00000 1.41421 0 0 1.00000 0 1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9702.2.a.dq 2
3.b odd 2 1 1078.2.a.q 2
7.b odd 2 1 inner 9702.2.a.dq 2
12.b even 2 1 8624.2.a.bw 2
21.c even 2 1 1078.2.a.q 2
21.g even 6 2 1078.2.e.s 4
21.h odd 6 2 1078.2.e.s 4
84.h odd 2 1 8624.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.q 2 3.b odd 2 1
1078.2.a.q 2 21.c even 2 1
1078.2.e.s 4 21.g even 6 2
1078.2.e.s 4 21.h odd 6 2
8624.2.a.bw 2 12.b even 2 1
8624.2.a.bw 2 84.h odd 2 1
9702.2.a.dq 2 1.a even 1 1 trivial
9702.2.a.dq 2 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 32 \) Copy content Toggle raw display
\( T_{19}^{2} - 8 \) Copy content Toggle raw display
\( T_{23} - 2 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 32 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 32 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 18 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 18 \) Copy content Toggle raw display
$61$ \( T^{2} - 32 \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 200 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 242 \) Copy content Toggle raw display
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