Properties

Label 9702.2.a.ec
Level $9702$
Weight $2$
Character orbit 9702.a
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. 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 \( q + q^{2} + q^{4} + \beta_{2} q^{5} + q^{8} + \beta_{2} q^{10} + q^{11} + (\beta_{3} + \beta_1) q^{13} + q^{16} + (2 \beta_{3} + \beta_{2}) q^{17} + (\beta_{3} + \beta_{2} - \beta_1) q^{19} + \beta_{2} q^{20} + q^{22} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{23} + (2 \beta_{3} + 3) q^{25} + (\beta_{3} + \beta_1) q^{26} - 2 \beta_{3} q^{29} + ( - \beta_{3} + \beta_{2} + 4) q^{31} + q^{32} + (2 \beta_{3} + \beta_{2}) q^{34} + ( - 4 \beta_{3} - \beta_1 + 2) q^{37} + (\beta_{3} + \beta_{2} - \beta_1) q^{38} + \beta_{2} q^{40} + ( - 2 \beta_{3} - \beta_{2}) q^{41} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{43} + q^{44} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{46} + (\beta_{3} - \beta_{2} + \beta_1 - 4) q^{47} + (2 \beta_{3} + 3) q^{50} + (\beta_{3} + \beta_1) q^{52} + ( - 4 \beta_{3} + \beta_1 - 2) q^{53} + \beta_{2} q^{55} - 2 \beta_{3} q^{58} + ( - 2 \beta_{3} - 8) q^{59} + ( - \beta_{3} + \beta_1 + 4) q^{61} + ( - \beta_{3} + \beta_{2} + 4) q^{62} + q^{64} + (8 \beta_{3} + \beta_1 + 4) q^{65} + (2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{67} + (2 \beta_{3} + \beta_{2}) q^{68} + ( - 2 \beta_{3} + 2 \beta_1) q^{71} + ( - 4 \beta_{3} + \beta_{2} - 2 \beta_1) q^{73} + ( - 4 \beta_{3} - \beta_1 + 2) q^{74} + (\beta_{3} + \beta_{2} - \beta_1) q^{76} + ( - 6 \beta_{3} - \beta_1 + 4) q^{79} + \beta_{2} q^{80} + ( - 2 \beta_{3} - \beta_{2}) q^{82} + ( - 3 \beta_{3} + \beta_{2}) q^{83} + (2 \beta_{3} + 2 \beta_1 + 8) q^{85} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{86} + q^{88} + (\beta_{3} - 3 \beta_1 - 4) q^{89} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{92} + (\beta_{3} - \beta_{2} + \beta_1 - 4) q^{94} + ( - 6 \beta_{3} + \beta_1 + 4) q^{95} + ( - \beta_{3} - 2 \beta_1 - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} + 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} + 12 q^{50} - 8 q^{53} - 32 q^{59} + 16 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} - 16 q^{89} + 8 q^{92} - 16 q^{94} + 16 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 8x^{2} + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta_1 \) 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
−2.32685
1.60804
−1.60804
2.32685
1.00000 0 1.00000 −3.29066 0 0 1.00000 0 −3.29066
1.2 1.00000 0 1.00000 −2.27411 0 0 1.00000 0 −2.27411
1.3 1.00000 0 1.00000 2.27411 0 0 1.00000 0 2.27411
1.4 1.00000 0 1.00000 3.29066 0 0 1.00000 0 3.29066
\(n\): 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.ec 4
3.b odd 2 1 3234.2.a.bk yes 4
7.b odd 2 1 9702.2.a.eb 4
21.c even 2 1 3234.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bj 4 21.c even 2 1
3234.2.a.bk yes 4 3.b odd 2 1
9702.2.a.eb 4 7.b odd 2 1
9702.2.a.ec 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9702))\):

\( T_{5}^{4} - 16T_{5}^{2} + 56 \) Copy content Toggle raw display
\( T_{13}^{4} - 36T_{13}^{2} - 32T_{13} + 164 \) Copy content Toggle raw display
\( T_{17}^{4} - 32T_{17}^{2} - 32T_{17} - 8 \) Copy content Toggle raw display
\( T_{19}^{4} - 36T_{19}^{2} + 80T_{19} - 4 \) Copy content Toggle raw display
\( T_{23}^{4} - 8T_{23}^{3} - 56T_{23}^{2} + 608T_{23} - 1168 \) Copy content Toggle raw display
\( T_{29}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 16T^{2} + 56 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 36 T^{2} - 32 T + 164 \) Copy content Toggle raw display
$17$ \( T^{4} - 32 T^{2} - 32 T - 8 \) Copy content Toggle raw display
$19$ \( T^{4} - 36 T^{2} + 80 T - 4 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 56 T^{2} + \cdots - 1168 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 16 T^{3} + 76 T^{2} + \cdots - 100 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} - 72 T^{2} + 480 T - 400 \) Copy content Toggle raw display
$41$ \( T^{4} - 32 T^{2} + 32 T - 8 \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} - 56 T^{2} + \cdots - 1168 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + 60 T^{2} + 48 T - 4 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} - 72 T^{2} - 224 T + 112 \) Copy content Toggle raw display
$59$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 60 T^{2} + \cdots - 284 \) Copy content Toggle raw display
$67$ \( T^{4} - 16 T^{3} - 144 T^{2} + \cdots + 3872 \) Copy content Toggle raw display
$71$ \( T^{4} - 144 T^{2} + 256 T + 2624 \) Copy content Toggle raw display
$73$ \( T^{4} - 176 T^{2} - 448 T + 184 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} - 80 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$83$ \( T^{4} - 52 T^{2} + 48 T + 92 \) Copy content Toggle raw display
$89$ \( T^{4} + 16 T^{3} - 196 T^{2} + \cdots + 12004 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} - 36 T^{2} + \cdots + 1988 \) Copy content Toggle raw display
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