Properties

Label 7220.2.a.n
Level $7220$
Weight $2$
Character orbit 7220.a
Self dual yes
Analytic conductor $57.652$
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] = [7220,2,Mod(1,7220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7220, 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("7220.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7220.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: \(57.6519902594\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 380)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{2} + 2) q^{11} - q^{13} - \beta_{2} q^{15} + (\beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_1 + 5) q^{21} + (\beta_{2} + \beta_1 - 4) q^{23} + q^{25} + (3 \beta_{2} + \beta_1 - 2) q^{27} + ( - \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - \beta_{2} - \beta_1 + 3) q^{31} + (\beta_{2} - \beta_1 + 5) q^{33} + ( - \beta_{2} - 1) q^{35} + (\beta_{2} + 1) q^{37} - \beta_{2} q^{39} + (3 \beta_{2} + 3) q^{41} + ( - 2 \beta_{2} + \beta_1 - 2) q^{43} + (\beta_{2} + \beta_1 - 2) q^{45} + (\beta_{2} + 2 \beta_1 - 1) q^{47} + (\beta_{2} - \beta_1 - 1) q^{49} + (\beta_{2} - \beta_1 + 8) q^{51} + ( - \beta_{2} - \beta_1 - 5) q^{53} + ( - \beta_{2} - 2) q^{55} + (\beta_{2} + 2 \beta_1 - 1) q^{59} + (2 \beta_{2} + 3) q^{61} + 5 \beta_{2} q^{63} + q^{65} + (2 \beta_1 + 2) q^{67} + ( - 8 \beta_{2} - \beta_1 + 2) q^{69} + (2 \beta_{2} + \beta_1 - 2) q^{71} + ( - \beta_{2} + \beta_1 - 6) q^{73} + \beta_{2} q^{75} + (2 \beta_{2} - \beta_1 + 7) q^{77} + ( - 2 \beta_{2} + 10) q^{79} + ( - 5 \beta_{2} + 6) q^{81} + ( - \beta_{2} - \beta_1 - 11) q^{83} + ( - \beta_{2} + \beta_1 + 1) q^{85} + (8 \beta_{2} + \beta_1 + 1) q^{87} + (4 \beta_{2} + 2) q^{89} + ( - \beta_{2} - 1) q^{91} + (7 \beta_{2} + \beta_1 - 2) q^{93} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{97} + (4 \beta_{2} - \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9} + 5 q^{11} - 3 q^{13} + q^{15} - 3 q^{17} + 16 q^{21} - 14 q^{23} + 3 q^{25} - 10 q^{27} + 6 q^{29} + 11 q^{31} + 15 q^{33} - 2 q^{35} + 2 q^{37} + q^{39} + 6 q^{41} - 5 q^{43} - 8 q^{45} - 6 q^{47} - 3 q^{49} + 24 q^{51} - 13 q^{53} - 5 q^{55} - 6 q^{59} + 7 q^{61} - 5 q^{63} + 3 q^{65} + 4 q^{67} + 15 q^{69} - 9 q^{71} - 18 q^{73} - q^{75} + 20 q^{77} + 32 q^{79} + 23 q^{81} - 31 q^{83} + 3 q^{85} - 6 q^{87} + 2 q^{89} - 2 q^{91} - 14 q^{93} + 11 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 10 ) / 3 \) 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
0.713538
2.19869
−1.91223
0 −3.20440 0 −1.00000 0 −2.20440 0 7.26819 0
1.2 0 −0.364448 0 −1.00000 0 0.635552 0 −2.86718 0
1.3 0 2.56885 0 −1.00000 0 3.56885 0 3.59899 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 7220.2.a.n 3
19.b odd 2 1 7220.2.a.o 3
19.c even 3 2 380.2.i.b 6
57.h odd 6 2 3420.2.t.v 6
76.g odd 6 2 1520.2.q.i 6
95.i even 6 2 1900.2.i.c 6
95.m odd 12 4 1900.2.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.b 6 19.c even 3 2
1520.2.q.i 6 76.g odd 6 2
1900.2.i.c 6 95.i even 6 2
1900.2.s.c 12 95.m odd 12 4
3420.2.t.v 6 57.h odd 6 2
7220.2.a.n 3 1.a even 1 1 trivial
7220.2.a.o 3 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7220))\):

\( T_{3}^{3} + T_{3}^{2} - 8T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 7T_{7} + 5 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 8T - 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 7 T + 5 \) Copy content Toggle raw display
$11$ \( T^{3} - 5T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} - 36 T - 81 \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + 39 T - 45 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 81 T + 513 \) Copy content Toggle raw display
$31$ \( T^{3} - 11 T^{2} + 14 T + 71 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} - 7 T + 5 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} - 63 T + 135 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} - 62 T + 105 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 81 T - 513 \) Copy content Toggle raw display
$53$ \( T^{3} + 13 T^{2} + 30 T - 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} - 81 T - 513 \) Copy content Toggle raw display
$61$ \( T^{3} - 7 T^{2} - 17 T + 63 \) Copy content Toggle raw display
$67$ \( T^{3} - 4 T^{2} - 92 T - 168 \) Copy content Toggle raw display
$71$ \( T^{3} + 9 T^{2} - 18 T - 27 \) Copy content Toggle raw display
$73$ \( T^{3} + 18 T^{2} + 69 T + 25 \) Copy content Toggle raw display
$79$ \( T^{3} - 32 T^{2} + 308 T - 856 \) Copy content Toggle raw display
$83$ \( T^{3} + 31 T^{2} + 294 T + 855 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 132 T + 72 \) Copy content Toggle raw display
$97$ \( T^{3} - 11 T^{2} - 94 T + 305 \) Copy content Toggle raw display
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