Properties

Label 9702.2.a.dd
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
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 \(\beta = 2\sqrt{3}\). 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 \( q + q^{2} + q^{4} + \beta q^{5} + q^{8} + \beta q^{10} - q^{11} - 2 q^{13} + q^{16} - \beta q^{17} + (\beta - 2) q^{19} + \beta q^{20} - q^{22} + 2 \beta q^{23} + 7 q^{25} - 2 q^{26} + 6 q^{29} + (\beta - 2) q^{31} + q^{32} - \beta q^{34} + (2 \beta + 2) q^{37} + (\beta - 2) q^{38} + \beta q^{40} - \beta q^{41} + (2 \beta - 4) q^{43} - q^{44} + 2 \beta q^{46} + ( - \beta + 6) q^{47} + 7 q^{50} - 2 q^{52} + ( - 2 \beta - 6) q^{53} - \beta q^{55} + 6 q^{58} - 2 \beta q^{59} - 2 q^{61} + (\beta - 2) q^{62} + q^{64} - 2 \beta q^{65} + ( - 2 \beta + 8) q^{67} - \beta q^{68} + 12 q^{71} + (\beta + 4) q^{73} + (2 \beta + 2) q^{74} + (\beta - 2) q^{76} + (2 \beta - 4) q^{79} + \beta q^{80} - \beta q^{82} + ( - 3 \beta - 6) q^{83} - 12 q^{85} + (2 \beta - 4) q^{86} - q^{88} + (2 \beta + 6) q^{89} + 2 \beta q^{92} + ( - \beta + 6) q^{94} + ( - 2 \beta + 12) q^{95} - 2 q^{97} +O(q^{100}) \) 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 \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 4 q^{19} - 2 q^{22} + 14 q^{25} - 4 q^{26} + 12 q^{29} - 4 q^{31} + 2 q^{32} + 4 q^{37} - 4 q^{38} - 8 q^{43} - 2 q^{44} + 12 q^{47} + 14 q^{50} - 4 q^{52} - 12 q^{53} + 12 q^{58} - 4 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{67} + 24 q^{71} + 8 q^{73} + 4 q^{74} - 4 q^{76} - 8 q^{79} - 12 q^{83} - 24 q^{85} - 8 q^{86} - 2 q^{88} + 12 q^{89} + 12 q^{94} + 24 q^{95} - 4 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
1.00000 0 1.00000 −3.46410 0 0 1.00000 0 −3.46410
1.2 1.00000 0 1.00000 3.46410 0 0 1.00000 0 3.46410
\(n\): 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

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 9702.2.a.dd 2
3.b odd 2 1 3234.2.a.x 2
7.b odd 2 1 1386.2.a.p 2
21.c even 2 1 462.2.a.h 2
84.h odd 2 1 3696.2.a.bc 2
231.h odd 2 1 5082.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.h 2 21.c even 2 1
1386.2.a.p 2 7.b odd 2 1
3234.2.a.x 2 3.b odd 2 1
3696.2.a.bc 2 84.h odd 2 1
5082.2.a.bu 2 231.h odd 2 1
9702.2.a.dd 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} - 8 \) Copy content Toggle raw display
\( T_{23}^{2} - 48 \) 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
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 48 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$41$ \( T^{2} - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 72 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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