Properties

Label 7220.2.a.p
Level $7220$
Weight $2$
Character orbit 7220.a
Self dual yes
Analytic conductor $57.652$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.133593.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 3x + 2 \) 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 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} - \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} - \beta_{3} q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{3} - 2 \beta_1 + 1) q^{11} + (\beta_{3} + \beta_1 + 2) q^{13} - \beta_1 q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} + 2) q^{21} + (\beta_{3} + \beta_{2}) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{3} + \beta_1 + 1) q^{29} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{31} + (\beta_{2} + \beta_1 + 6) q^{33} - \beta_{3} q^{35} + ( - \beta_{3} + 2 \beta_1 + 6) q^{37} + ( - 2 \beta_{2} - 3 \beta_1 - 6) q^{39} + ( - \beta_{3} - 2) q^{41} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{43} + (\beta_{2} + \beta_1 + 1) q^{45} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{47} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{49} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{51} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{53} + (\beta_{3} - 2 \beta_1 + 1) q^{55} + ( - 2 \beta_{3} - \beta_1 + 3) q^{59} + (2 \beta_{3} - 3) q^{61} + (2 \beta_{3} - 5 \beta_1 + 2) q^{63} + (\beta_{3} + \beta_1 + 2) q^{65} + 2 \beta_{2} q^{67} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{69} + ( - \beta_{2} + 2 \beta_1 + 3) q^{71} + ( - \beta_{3} + \beta_{2} + 4 \beta_1) q^{73} - \beta_1 q^{75} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 6) q^{77} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{79} + (\beta_{3} + 4 \beta_1 + 5) q^{81} + ( - \beta_{2} - 3 \beta_1 + 2) q^{83} + ( - \beta_{2} - \beta_1) q^{85} + (\beta_{2} - 2 \beta_1) q^{87} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{89} + ( - 3 \beta_{3} + 2 \beta_1 - 12) q^{91} + ( - \beta_{3} + \beta_{2} - 4 \beta_1 + 8) q^{93} + ( - 2 \beta_{2} - \beta_1) q^{97} + ( - 4 \beta_{3} - \beta_{2} - 4 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47} + 10 q^{49} + 12 q^{51} + 5 q^{53} + 2 q^{55} + 11 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} - 3 q^{69} + 14 q^{71} + 4 q^{73} - q^{75} - 22 q^{77} + 13 q^{79} + 24 q^{81} + 5 q^{83} - q^{85} - 2 q^{87} + 5 q^{89} - 46 q^{91} + 28 q^{93} - q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + 3x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 7\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 8\beta _1 + 2 \) 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
3.16505
0.709218
−0.353450
−2.52082
0 −3.16505 0 1.00000 0 −1.53315 0 7.01755 0
1.2 0 −0.709218 0 1.00000 0 3.11079 0 −2.49701 0
1.3 0 0.353450 0 1.00000 0 −4.30507 0 −2.87507 0
1.4 0 2.52082 0 1.00000 0 2.72743 0 3.35453 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.p 4
19.b odd 2 1 7220.2.a.r 4
19.d odd 6 2 380.2.i.c 8
57.f even 6 2 3420.2.t.w 8
76.f even 6 2 1520.2.q.m 8
95.h odd 6 2 1900.2.i.d 8
95.l even 12 4 1900.2.s.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c 8 19.d odd 6 2
1520.2.q.m 8 76.f even 6 2
1900.2.i.d 8 95.h odd 6 2
1900.2.s.d 16 95.l even 12 4
3420.2.t.w 8 57.f even 6 2
7220.2.a.p 4 1.a even 1 1 trivial
7220.2.a.r 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7220))\):

\( T_{3}^{4} + T_{3}^{3} - 8T_{3}^{2} - 3T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 19T_{7}^{2} + 11T_{7} + 56 \) Copy content Toggle raw display
\( T_{13}^{4} - 9T_{13}^{3} - 5T_{13}^{2} + 129T_{13} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 8 T^{2} - 3 T + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 19 T^{2} + 11 T + 56 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267 \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} - 5 T^{2} + 129 T + 52 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} - 34 T^{2} + 9 T + 192 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 39 T^{2} + 99 T - 54 \) Copy content Toggle raw display
$29$ \( T^{4} - 5 T^{3} - 59 T^{2} + 318 T - 273 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} - 29 T^{2} + \cdots - 277 \) Copy content Toggle raw display
$37$ \( T^{4} - 26 T^{3} + 217 T^{2} + \cdots + 414 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + 5 T^{2} - 33 T + 18 \) Copy content Toggle raw display
$43$ \( T^{4} + 7 T^{3} - 32 T^{2} - 177 T - 178 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} - 25 T^{2} + \cdots - 1176 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} - 118 T^{2} + \cdots + 1218 \) Copy content Toggle raw display
$59$ \( T^{4} - 11 T^{3} - 55 T^{2} + \cdots + 1587 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} - 22 T^{2} - 436 T + 29 \) Copy content Toggle raw display
$67$ \( T^{4} - 124 T^{2} - 56 T + 3296 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} - T^{2} + 669 T - 2247 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} - 137 T^{2} + \cdots + 558 \) Copy content Toggle raw display
$79$ \( T^{4} - 13 T^{3} - 128 T^{2} + \cdots - 3016 \) Copy content Toggle raw display
$83$ \( T^{4} - 5 T^{3} - 82 T^{2} + \cdots - 1302 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} - 122 T^{2} + \cdots + 1176 \) Copy content Toggle raw display
$97$ \( T^{4} + T^{3} - 122 T^{2} - 51 T + 3654 \) Copy content Toggle raw display
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