Properties

Label 3969.2.a.m
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
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] = [3969,2,Mod(1,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3969.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.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: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{8} + ( - 3 \beta_1 + 1) q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11} - q^{13} + ( - \beta_{2} + 2 \beta_1) q^{16} + ( - \beta_{2} - \beta_1 + 4) q^{17} + (\beta_{2} + \beta_1 + 1) q^{19} + (\beta_{2} + 2 \beta_1 + 5) q^{20} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{22} + (2 \beta_{2} - \beta_1 + 1) q^{23} + (2 \beta_{2} - \beta_1 + 3) q^{25} + \beta_1 q^{26} + \beta_1 q^{29} + (\beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_1 - 3) q^{32} + (\beta_{2} - 2 \beta_1 + 2) q^{34} + 3 \beta_{2} q^{37} + ( - \beta_{2} - 3 \beta_1 - 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 - 7) q^{40} + ( - \beta_{2} + 7) q^{41} + ( - \beta_{2} - 4 \beta_1) q^{43} + (5 \beta_1 + 6) q^{44} + (\beta_{2} - 2 \beta_1 + 5) q^{46} + (3 \beta_{2} + 3 \beta_1 + 3) q^{47} + (\beta_{2} - 4 \beta_1 + 5) q^{50} + ( - \beta_{2} - \beta_1 - 1) q^{52} + (2 \beta_{2} - \beta_1 - 5) q^{53} + ( - \beta_{2} + 5 \beta_1) q^{55} + ( - \beta_{2} - \beta_1 - 3) q^{58} + (2 \beta_{2} - \beta_1 + 4) q^{59} + ( - 2 \beta_{2} + \beta_1 + 1) q^{61} + (2 \beta_{2} - \beta_1 + 7) q^{62} + (\beta_{2} - 2 \beta_1 - 3) q^{64} + ( - \beta_{2} - 2) q^{65} + ( - \beta_{2} - 7 \beta_1 + 2) q^{67} + (4 \beta_{2} + \beta_1 - 1) q^{68} + ( - 2 \beta_{2} + \beta_1 + 2) q^{71} + (5 \beta_{2} - \beta_1 + 1) q^{73} + ( - 3 \beta_1 + 3) q^{74} + (\beta_{2} + 4 \beta_1 + 6) q^{76} + ( - \beta_{2} + 5 \beta_1 + 3) q^{79} + (7 \beta_1 - 6) q^{80} + ( - 6 \beta_1 - 1) q^{82} + ( - \beta_{2} - \beta_1 + 4) q^{83} + (4 \beta_{2} - 2 \beta_1 + 5) q^{85} + (4 \beta_{2} + 5 \beta_1 + 11) q^{86} + ( - \beta_{2} - 7 \beta_1 - 5) q^{88} + (\beta_{2} + 6 \beta_1 - 1) q^{89} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{92} + ( - 3 \beta_{2} - 9 \beta_1 - 6) q^{94} + (\beta_{2} + 2 \beta_1 + 5) q^{95} + (2 \beta_{2} + 5 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} + 5 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} + 5 q^{5} - 6 q^{8} - 2 q^{11} - 3 q^{13} + 3 q^{16} + 12 q^{17} + 3 q^{19} + 16 q^{20} - 15 q^{22} + 6 q^{25} + q^{26} + q^{29} + 3 q^{31} - 8 q^{32} + 3 q^{34} - 3 q^{37} - 8 q^{38} - 21 q^{40} + 22 q^{41} - 3 q^{43} + 23 q^{44} + 12 q^{46} + 9 q^{47} + 10 q^{50} - 3 q^{52} - 18 q^{53} + 6 q^{55} - 9 q^{58} + 9 q^{59} + 6 q^{61} + 18 q^{62} - 12 q^{64} - 5 q^{65} - 6 q^{68} + 9 q^{71} - 3 q^{73} + 6 q^{74} + 21 q^{76} + 15 q^{79} - 11 q^{80} - 9 q^{82} + 12 q^{83} + 9 q^{85} + 34 q^{86} - 21 q^{88} + 2 q^{89} + 15 q^{92} - 24 q^{94} + 16 q^{95} - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
2.46050
0.239123
−1.69963
−2.46050 0 4.05408 2.59358 0 0 −5.05408 0 −6.38151
1.2 −0.239123 0 −1.94282 −1.18194 0 0 0.942820 0 0.282630
1.3 1.69963 0 0.888736 3.58836 0 0 −1.88874 0 6.09888
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3969.2.a.m 3
3.b odd 2 1 3969.2.a.p 3
7.b odd 2 1 567.2.a.d 3
9.c even 3 2 441.2.f.d 6
9.d odd 6 2 1323.2.f.c 6
21.c even 2 1 567.2.a.g 3
28.d even 2 1 9072.2.a.bq 3
63.g even 3 2 441.2.g.d 6
63.h even 3 2 441.2.h.b 6
63.i even 6 2 1323.2.h.d 6
63.j odd 6 2 1323.2.h.e 6
63.k odd 6 2 441.2.g.e 6
63.l odd 6 2 63.2.f.b 6
63.n odd 6 2 1323.2.g.b 6
63.o even 6 2 189.2.f.a 6
63.s even 6 2 1323.2.g.c 6
63.t odd 6 2 441.2.h.c 6
84.h odd 2 1 9072.2.a.cd 3
252.s odd 6 2 3024.2.r.g 6
252.bi even 6 2 1008.2.r.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 63.l odd 6 2
189.2.f.a 6 63.o even 6 2
441.2.f.d 6 9.c even 3 2
441.2.g.d 6 63.g even 3 2
441.2.g.e 6 63.k odd 6 2
441.2.h.b 6 63.h even 3 2
441.2.h.c 6 63.t odd 6 2
567.2.a.d 3 7.b odd 2 1
567.2.a.g 3 21.c even 2 1
1008.2.r.k 6 252.bi even 6 2
1323.2.f.c 6 9.d odd 6 2
1323.2.g.b 6 63.n odd 6 2
1323.2.g.c 6 63.s even 6 2
1323.2.h.d 6 63.i even 6 2
1323.2.h.e 6 63.j odd 6 2
3024.2.r.g 6 252.s odd 6 2
3969.2.a.m 3 1.a even 1 1 trivial
3969.2.a.p 3 3.b odd 2 1
9072.2.a.bq 3 28.d even 2 1
9072.2.a.cd 3 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 5T_{5}^{2} + 2T_{5} + 11 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 19T_{11} - 47 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + 2 T + 11 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 19 T - 47 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 12 T^{2} + 39 T - 27 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 6 T + 7 \) Copy content Toggle raw display
$23$ \( T^{3} - 33T - 9 \) Copy content Toggle raw display
$29$ \( T^{3} - T^{2} - 4T + 1 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} - 24 T - 27 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 54 T + 81 \) Copy content Toggle raw display
$41$ \( T^{3} - 22 T^{2} + 155 T - 353 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$47$ \( T^{3} - 9 T^{2} - 54 T + 189 \) Copy content Toggle raw display
$53$ \( T^{3} + 18 T^{2} + 75 T + 9 \) Copy content Toggle raw display
$59$ \( T^{3} - 9 T^{2} - 6 T + 63 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} - 21 T + 67 \) Copy content Toggle raw display
$67$ \( T^{3} - 207T + 683 \) Copy content Toggle raw display
$71$ \( T^{3} - 9 T^{2} - 6 T + 81 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} - 168 T + 243 \) Copy content Toggle raw display
$79$ \( T^{3} - 15 T^{2} - 48 T + 769 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + 39 T - 27 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 151 T - 379 \) Copy content Toggle raw display
$97$ \( T^{3} + 3 T^{2} - 114 T - 603 \) Copy content Toggle raw display
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