Properties

Label 3969.2.a.i
Level $3969$
Weight $2$
Character orbit 3969.a
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 81)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} + \beta q^{5} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{4} + \beta q^{5} - \beta q^{8} + 3 q^{10} - 2 \beta q^{11} + q^{13} - 5 q^{16} - 3 \beta q^{17} - 2 q^{19} + \beta q^{20} - 6 q^{22} + 2 \beta q^{23} - 2 q^{25} + \beta q^{26} + \beta q^{29} - 8 q^{31} - 3 \beta q^{32} - 9 q^{34} - 7 q^{37} - 2 \beta q^{38} - 3 q^{40} + 4 \beta q^{41} + 2 q^{43} - 2 \beta q^{44} + 6 q^{46} - 4 \beta q^{47} - 2 \beta q^{50} + q^{52} - 6 q^{55} + 3 q^{58} - 8 \beta q^{59} + 7 q^{61} - 8 \beta q^{62} + q^{64} + \beta q^{65} - 10 q^{67} - 3 \beta q^{68} - 6 \beta q^{71} + 7 q^{73} - 7 \beta q^{74} - 2 q^{76} + 2 q^{79} - 5 \beta q^{80} + 12 q^{82} + 8 \beta q^{83} - 9 q^{85} + 2 \beta q^{86} + 6 q^{88} + 3 \beta q^{89} + 2 \beta q^{92} - 12 q^{94} - 2 \beta q^{95} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{10} + 2 q^{13} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 4 q^{25} - 16 q^{31} - 18 q^{34} - 14 q^{37} - 6 q^{40} + 4 q^{43} + 12 q^{46} + 2 q^{52} - 12 q^{55} + 6 q^{58} + 14 q^{61} + 2 q^{64} - 20 q^{67} + 14 q^{73} - 4 q^{76} + 4 q^{79} + 24 q^{82} - 18 q^{85} + 12 q^{88} - 24 q^{94} - 4 q^{97}+O(q^{100}) \) 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
−1.73205 0 1.00000 −1.73205 0 0 1.73205 0 3.00000
1.2 1.73205 0 1.00000 1.73205 0 0 −1.73205 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.2.a.i 2
3.b odd 2 1 inner 3969.2.a.i 2
7.b odd 2 1 81.2.a.a 2
21.c even 2 1 81.2.a.a 2
28.d even 2 1 1296.2.a.o 2
35.c odd 2 1 2025.2.a.j 2
35.f even 4 2 2025.2.b.k 4
56.e even 2 1 5184.2.a.bq 2
56.h odd 2 1 5184.2.a.br 2
63.l odd 6 2 81.2.c.b 4
63.o even 6 2 81.2.c.b 4
77.b even 2 1 9801.2.a.v 2
84.h odd 2 1 1296.2.a.o 2
105.g even 2 1 2025.2.a.j 2
105.k odd 4 2 2025.2.b.k 4
168.e odd 2 1 5184.2.a.bq 2
168.i even 2 1 5184.2.a.br 2
189.y odd 18 6 729.2.e.o 12
189.be even 18 6 729.2.e.o 12
231.h odd 2 1 9801.2.a.v 2
252.s odd 6 2 1296.2.i.s 4
252.bi even 6 2 1296.2.i.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 7.b odd 2 1
81.2.a.a 2 21.c even 2 1
81.2.c.b 4 63.l odd 6 2
81.2.c.b 4 63.o even 6 2
729.2.e.o 12 189.y odd 18 6
729.2.e.o 12 189.be even 18 6
1296.2.a.o 2 28.d even 2 1
1296.2.a.o 2 84.h odd 2 1
1296.2.i.s 4 252.s odd 6 2
1296.2.i.s 4 252.bi even 6 2
2025.2.a.j 2 35.c odd 2 1
2025.2.a.j 2 105.g even 2 1
2025.2.b.k 4 35.f even 4 2
2025.2.b.k 4 105.k odd 4 2
3969.2.a.i 2 1.a even 1 1 trivial
3969.2.a.i 2 3.b odd 2 1 inner
5184.2.a.bq 2 56.e even 2 1
5184.2.a.bq 2 168.e odd 2 1
5184.2.a.br 2 56.h odd 2 1
5184.2.a.br 2 168.i even 2 1
9801.2.a.v 2 77.b even 2 1
9801.2.a.v 2 231.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 27 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} - 3 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 48 \) Copy content Toggle raw display
$43$ \( (T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 192 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 108 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 27 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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