Properties

Label 9702.2.a.ch
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
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] = [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: \(1\)
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 154)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8} + ( - \beta + 2) q^{10} - q^{11} + (2 \beta + 1) q^{13} + q^{16} + (4 \beta - 2) q^{17} + ( - \beta + 2) q^{19} + (\beta - 2) q^{20} + q^{22} + (3 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} + ( - 2 \beta - 1) q^{26} + ( - 4 \beta + 3) q^{29} + 4 q^{31} - q^{32} + ( - 4 \beta + 2) q^{34} + ( - \beta - 8) q^{37} + (\beta - 2) q^{38} + ( - \beta + 2) q^{40} + ( - \beta - 4) q^{41} - 4 \beta q^{43} - q^{44} + ( - 3 \beta - 2) q^{46} + ( - 6 \beta - 2) q^{47} + (4 \beta - 1) q^{50} + (2 \beta + 1) q^{52} + ( - 7 \beta + 2) q^{53} + ( - \beta + 2) q^{55} + (4 \beta - 3) q^{58} + (\beta - 7) q^{59} + ( - 2 \beta - 9) q^{61} - 4 q^{62} + q^{64} + ( - 3 \beta + 2) q^{65} + ( - 3 \beta + 7) q^{67} + (4 \beta - 2) q^{68} + (5 \beta + 4) q^{71} + (\beta + 8) q^{73} + (\beta + 8) q^{74} + ( - \beta + 2) q^{76} + ( - 3 \beta - 9) q^{79} + (\beta - 2) q^{80} + (\beta + 4) q^{82} + ( - 10 \beta + 2) q^{83} + ( - 10 \beta + 12) q^{85} + 4 \beta q^{86} + q^{88} + (6 \beta + 4) q^{89} + (3 \beta + 2) q^{92} + (6 \beta + 2) q^{94} + (4 \beta - 6) q^{95} + (2 \beta + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} - 2 q^{26} + 6 q^{29} + 8 q^{31} - 2 q^{32} + 4 q^{34} - 16 q^{37} - 4 q^{38} + 4 q^{40} - 8 q^{41} - 2 q^{44} - 4 q^{46} - 4 q^{47} - 2 q^{50} + 2 q^{52} + 4 q^{53} + 4 q^{55} - 6 q^{58} - 14 q^{59} - 18 q^{61} - 8 q^{62} + 2 q^{64} + 4 q^{65} + 14 q^{67} - 4 q^{68} + 8 q^{71} + 16 q^{73} + 16 q^{74} + 4 q^{76} - 18 q^{79} - 4 q^{80} + 8 q^{82} + 4 q^{83} + 24 q^{85} + 2 q^{88} + 8 q^{89} + 4 q^{92} + 4 q^{94} - 12 q^{95} + 2 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
−1.00000 0 1.00000 −3.41421 0 0 −1.00000 0 3.41421
1.2 −1.00000 0 1.00000 −0.585786 0 0 −1.00000 0 0.585786
\(n\): 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.ch 2
3.b odd 2 1 1078.2.a.x 2
7.b odd 2 1 9702.2.a.cx 2
7.d odd 6 2 1386.2.k.t 4
12.b even 2 1 8624.2.a.bh 2
21.c even 2 1 1078.2.a.t 2
21.g even 6 2 154.2.e.e 4
21.h odd 6 2 1078.2.e.m 4
84.h odd 2 1 8624.2.a.cc 2
84.j odd 6 2 1232.2.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 21.g even 6 2
1078.2.a.t 2 21.c even 2 1
1078.2.a.x 2 3.b odd 2 1
1078.2.e.m 4 21.h odd 6 2
1232.2.q.f 4 84.j odd 6 2
1386.2.k.t 4 7.d odd 6 2
8624.2.a.bh 2 12.b even 2 1
8624.2.a.cc 2 84.h odd 2 1
9702.2.a.ch 2 1.a even 1 1 trivial
9702.2.a.cx 2 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{2} + 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 7 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} - 4T_{19} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} - 4T_{23} - 14 \) Copy content Toggle raw display
\( T_{29}^{2} - 6T_{29} - 23 \) 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
$5$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16T + 62 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$43$ \( T^{2} - 32 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 94 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 47 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 73 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 31 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 62 \) Copy content Toggle raw display
$79$ \( T^{2} + 18T + 63 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 196 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
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