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Properties

Label 3969.2.a.l
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{8} + (2 \beta_{2} - \beta_1) q^{10} + ( - \beta_{2} + \beta_1 - 2) q^{11} + (4 \beta_{2} - 2 \beta_1 + 1) q^{13} + (3 \beta_{2} - 3 \beta_1 + 1) q^{16} + ( - \beta_{2} + 2) q^{17} + ( - \beta_{2} + 2 \beta_1 + 1) q^{19} + ( - \beta_{2} + \beta_1 - 2) q^{20} + (2 \beta_{2} - 4 \beta_1 + 3) q^{22} + ( - 2 \beta_{2} - \beta_1 - 4) q^{23} + ( - 2 \beta_{2} + \beta_1 - 2) q^{25} + ( - 6 \beta_{2} + 7 \beta_1 - 1) q^{26} + (4 \beta_{2} - 5 \beta_1 - 3) q^{29} + ( - 3 \beta_{2} - 3 \beta_1 + 1) q^{31} + 3 \beta_1 q^{32} + (\beta_{2} + \beta_1 - 3) q^{34} + (3 \beta_{2} + 3 \beta_1 - 1) q^{37} + (3 \beta_{2} - 2 \beta_1 + 2) q^{38} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{40} + (\beta_{2} + \beta_1) q^{41} + ( - \beta_{2} - \beta_1 - 1) q^{43} + ( - 4 \beta_{2} + 7 \beta_1 - 5) q^{44} + (\beta_{2} - 5 \beta_1) q^{46} + (3 \beta_{2} + 2 \beta_1 + 1) q^{47} + (3 \beta_{2} - 5 \beta_1 + 2) q^{50} + (5 \beta_{2} - 10 \beta_1 + 7) q^{52} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{53} + (\beta_{2} - 2 \beta_1) q^{55} + ( - 9 \beta_{2} + 6 \beta_1 - 3) q^{58} + ( - 5 \beta_1 - 1) q^{59} + ( - 3 \beta_1 - 2) q^{61} + (\beta_1 - 10) q^{62} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{64} + (5 \beta_{2} + \beta_1 - 5) q^{65} + (3 \beta_{2} - 4) q^{67} + (2 \beta_{2} - 3 \beta_1 + 2) q^{68} + (6 \beta_{2} - 3 \beta_1 - 3) q^{71} + ( - \beta_{2} - 4 \beta_1 + 7) q^{73} + ( - \beta_1 + 10) q^{74} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{76} + (3 \beta_{2} - 7) q^{79} + (2 \beta_{2} + \beta_1 - 5) q^{80} + 3 q^{82} + ( - \beta_{2} - 4 \beta_1 - 6) q^{83} + ( - 4 \beta_{2} + 2 \beta_1 + 3) q^{85} + ( - \beta_1 - 2) q^{86} + (7 \beta_{2} - 8 \beta_1 + 9) q^{88} + (7 \beta_{2} - 3 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} + 8 \beta_1 - 1) q^{92} + ( - \beta_{2} + 2 \beta_1 + 6) q^{94} + ( - \beta_{2} + \beta_1 + 4) q^{95} + ( - 8 \beta_{2} + 7 \beta_1 + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38} + 9 q^{40} - 3 q^{43} - 15 q^{44} + 3 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} - 9 q^{58} - 3 q^{59} - 6 q^{61} - 30 q^{62} + 12 q^{64} - 15 q^{65} - 12 q^{67} + 6 q^{68} - 9 q^{71} + 21 q^{73} + 30 q^{74} - 15 q^{76} - 21 q^{79} - 15 q^{80} + 9 q^{82} - 18 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 12 q^{89} - 3 q^{92} + 18 q^{94} + 12 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.53209
−0.347296
1.87939
−2.53209 0 4.41147 −0.879385 0 0 −6.10607 0 2.22668
1.2 −1.34730 0 −0.184793 2.53209 0 0 2.94356 0 −3.41147
1.3 0.879385 0 −1.22668 1.34730 0 0 −2.83750 0 1.18479
\(n\): 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.l 3
3.b odd 2 1 3969.2.a.q 3
7.b odd 2 1 567.2.a.c 3
9.c even 3 2 1323.2.f.d 6
9.d odd 6 2 441.2.f.c 6
21.c even 2 1 567.2.a.h 3
28.d even 2 1 9072.2.a.bs 3
63.g even 3 2 1323.2.g.e 6
63.h even 3 2 1323.2.h.b 6
63.i even 6 2 441.2.h.d 6
63.j odd 6 2 441.2.h.e 6
63.k odd 6 2 1323.2.g.d 6
63.l odd 6 2 189.2.f.b 6
63.n odd 6 2 441.2.g.b 6
63.o even 6 2 63.2.f.a ✓ 6
63.s even 6 2 441.2.g.c 6
63.t odd 6 2 1323.2.h.c 6
84.h odd 2 1 9072.2.a.ca 3
252.s odd 6 2 1008.2.r.h 6
252.bi even 6 2 3024.2.r.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a ✓ 6 63.o even 6 2
189.2.f.b 6 63.l odd 6 2
441.2.f.c 6 9.d odd 6 2
441.2.g.b 6 63.n odd 6 2
441.2.g.c 6 63.s even 6 2
441.2.h.d 6 63.i even 6 2
441.2.h.e 6 63.j odd 6 2
567.2.a.c 3 7.b odd 2 1
567.2.a.h 3 21.c even 2 1
1008.2.r.h 6 252.s odd 6 2
1323.2.f.d 6 9.c even 3 2
1323.2.g.d 6 63.k odd 6 2
1323.2.g.e 6 63.g even 3 2
1323.2.h.b 6 63.h even 3 2
1323.2.h.c 6 63.t odd 6 2
3024.2.r.k 6 252.bi even 6 2
3969.2.a.l 3 1.a even 1 1 trivial
3969.2.a.q 3 3.b odd 2 1
9072.2.a.bs 3 28.d even 2 1
9072.2.a.ca 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{3} + 3T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} + 9T_{11} + 3 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 33T_{13} + 107 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 3T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} + \cdots + 107 \) Copy content Toggle raw display
$17$ \( T^{3} - 6 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} + \cdots + 17 \) Copy content Toggle raw display
$23$ \( T^{3} + 12 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$29$ \( T^{3} + 9 T^{2} + \cdots - 333 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} + \cdots + 323 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} + \cdots - 323 \) Copy content Toggle raw display
$41$ \( T^{3} - 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{3} - 3 T^{2} + \cdots - 51 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$59$ \( T^{3} + 3 T^{2} + \cdots + 51 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} + \cdots - 19 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + \cdots - 17 \) Copy content Toggle raw display
$71$ \( T^{3} + 9 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$73$ \( T^{3} - 21 T^{2} + \cdots + 269 \) Copy content Toggle raw display
$79$ \( T^{3} + 21 T^{2} + \cdots + 181 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$89$ \( T^{3} - 12 T^{2} + \cdots + 813 \) Copy content Toggle raw display
$97$ \( T^{3} - 3 T^{2} + \cdots + 323 \) Copy content Toggle raw display
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