Properties

Label 1960.2.a.r
Level $1960$
Weight $2$
Character orbit 1960.a
Self dual yes
Analytic conductor $15.651$
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] = [1960,2,Mod(1,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.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: \(15.6506787962\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} - q^{5} + (\beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} - q^{5} + (\beta + 1) q^{9} - \beta q^{11} + (3 \beta - 2) q^{13} + \beta q^{15} + (\beta - 6) q^{17} + ( - 2 \beta + 4) q^{19} - 2 \beta q^{23} + q^{25} + (\beta - 4) q^{27} + (\beta + 2) q^{29} + 4 \beta q^{31} + (\beta + 4) q^{33} + (4 \beta - 2) q^{37} + ( - \beta - 12) q^{39} + ( - 2 \beta - 2) q^{41} + (2 \beta - 4) q^{43} + ( - \beta - 1) q^{45} + ( - \beta - 4) q^{47} + (5 \beta - 4) q^{51} + (2 \beta - 10) q^{53} + \beta q^{55} + ( - 2 \beta + 8) q^{57} + 4 q^{59} + (2 \beta + 10) q^{61} + ( - 3 \beta + 2) q^{65} + ( - 4 \beta - 4) q^{67} + (2 \beta + 8) q^{69} + ( - 4 \beta - 2) q^{73} - \beta q^{75} + ( - 3 \beta - 4) q^{79} - 7 q^{81} - 12 q^{83} + ( - \beta + 6) q^{85} + ( - 3 \beta - 4) q^{87} + (2 \beta - 2) q^{89} + ( - 4 \beta - 16) q^{93} + (2 \beta - 4) q^{95} + ( - 3 \beta - 6) q^{97} + ( - 2 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{5} + 3 q^{9} - q^{11} - q^{13} + q^{15} - 11 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} + 4 q^{31} + 9 q^{33} - 25 q^{39} - 6 q^{41} - 6 q^{43} - 3 q^{45} - 9 q^{47} - 3 q^{51} - 18 q^{53} + q^{55} + 14 q^{57} + 8 q^{59} + 22 q^{61} + q^{65} - 12 q^{67} + 18 q^{69} - 8 q^{73} - q^{75} - 11 q^{79} - 14 q^{81} - 24 q^{83} + 11 q^{85} - 11 q^{87} - 2 q^{89} - 36 q^{93} - 6 q^{95} - 15 q^{97} - 10 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 −2.56155 0 −1.00000 0 0 0 3.56155 0
1.2 0 1.56155 0 −1.00000 0 0 0 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 1960.2.a.r 2
4.b odd 2 1 3920.2.a.bu 2
5.b even 2 1 9800.2.a.by 2
7.b odd 2 1 280.2.a.d 2
7.c even 3 2 1960.2.q.u 4
7.d odd 6 2 1960.2.q.s 4
21.c even 2 1 2520.2.a.w 2
28.d even 2 1 560.2.a.g 2
35.c odd 2 1 1400.2.a.p 2
35.f even 4 2 1400.2.g.k 4
56.e even 2 1 2240.2.a.bi 2
56.h odd 2 1 2240.2.a.be 2
84.h odd 2 1 5040.2.a.bq 2
140.c even 2 1 2800.2.a.bn 2
140.j odd 4 2 2800.2.g.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 7.b odd 2 1
560.2.a.g 2 28.d even 2 1
1400.2.a.p 2 35.c odd 2 1
1400.2.g.k 4 35.f even 4 2
1960.2.a.r 2 1.a even 1 1 trivial
1960.2.q.s 4 7.d odd 6 2
1960.2.q.u 4 7.c even 3 2
2240.2.a.be 2 56.h odd 2 1
2240.2.a.bi 2 56.e even 2 1
2520.2.a.w 2 21.c even 2 1
2800.2.a.bn 2 140.c even 2 1
2800.2.g.u 4 140.j odd 4 2
3920.2.a.bu 2 4.b odd 2 1
5040.2.a.bq 2 84.h odd 2 1
9800.2.a.by 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} - 38 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$17$ \( T^{2} + 11T + 26 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 68 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 22T + 104 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T - 8 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 15T + 18 \) Copy content Toggle raw display
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