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Properties

Label 7220.2.a.x
Level $7220$
Weight $2$
Character orbit 7220.a
Self dual yes
Analytic conductor $57.652$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 12x^{7} + 39x^{6} + 21x^{5} - 99x^{4} + 8x^{3} + 48x^{2} - 9x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (\beta_{7} + \beta_{6} - \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + (\beta_{7} - \beta_{4}) q^{7} + (\beta_{7} + \beta_{6} - \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{8} - 3 \beta_{5} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} - 9 q^{5} + 6 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 9 q^{29} - 18 q^{31} + 9 q^{33} - 30 q^{37} - 42 q^{39} - 6 q^{41} + 24 q^{43} - 6 q^{45} - 21 q^{47} + 15 q^{49} + 42 q^{51} - 9 q^{53} - 3 q^{59} - 24 q^{61} - 51 q^{63} + 3 q^{65} - 9 q^{69} - 18 q^{71} - 33 q^{73} + 3 q^{75} + 3 q^{79} + 33 q^{81} + 30 q^{83} - 3 q^{85} + 9 q^{87} + 3 q^{89} - 15 q^{91} - 15 q^{93} - 15 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 12x^{7} + 39x^{6} + 21x^{5} - 99x^{4} + 8x^{3} + 48x^{2} - 9x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 37\nu^{8} - 25\nu^{7} - 650\nu^{6} + 362\nu^{5} + 3370\nu^{4} - 1168\nu^{3} - 5409\nu^{2} + 462\nu + 1032 ) / 513 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\nu^{8} - 103\nu^{7} - 626\nu^{6} + 1382\nu^{5} + 2359\nu^{4} - 3964\nu^{3} - 3735\nu^{2} + 2991\nu + 1851 ) / 513 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{8} + 8\nu^{7} + 75\nu^{6} - 90\nu^{5} - 307\nu^{4} + 88\nu^{3} + 543\nu^{2} + 258\nu - 255 ) / 57 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -49\nu^{8} + 181\nu^{7} + 431\nu^{6} - 2231\nu^{5} + 875\nu^{4} + 4537\nu^{3} - 4095\nu^{2} - 903\nu + 921 ) / 513 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 106 \nu^{8} + 238 \nu^{7} + 1400 \nu^{6} - 3029 \nu^{5} - 3799 \nu^{4} + 6988 \nu^{3} + 2574 \nu^{2} + \cdots - 960 ) / 513 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112\nu^{8} - 316\nu^{7} - 1376\nu^{6} + 4049\nu^{5} + 2788\nu^{4} - 9784\nu^{3} - 387\nu^{2} + 4116\nu - 273 ) / 513 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{8} + 19\nu^{7} + 89\nu^{6} - 245\nu^{5} - 211\nu^{4} + 598\nu^{3} + 99\nu^{2} - 210\nu + 3 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - 4\beta_{7} - \beta_{5} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} + 12\beta_{6} - 4\beta_{5} - 7\beta_{3} + 13\beta_{2} + 7\beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{8} - 52\beta_{7} - \beta_{6} - 16\beta_{5} + 4\beta_{4} + 5\beta_{3} + 3\beta_{2} + 55\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 68\beta_{7} + 119\beta_{6} - 58\beta_{5} + 4\beta_{4} - 56\beta_{3} + 139\beta_{2} + 46\beta _1 + 303 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 119 \beta_{8} - 550 \beta_{7} - 23 \beta_{6} - 190 \beta_{5} + 59 \beta_{4} + 73 \beta_{3} + \cdots - 57 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 23 \beta_{8} + 623 \beta_{7} + 1138 \beta_{6} - 658 \beta_{5} + 71 \beta_{4} - 492 \beta_{3} + \cdots + 2865 \) 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
−3.12844
−1.58074
−0.738124
−0.179046
0.449851
0.714088
1.79921
2.55613
3.10709
0 −3.12844 0 −1.00000 0 −0.0963009 0 6.78717 0
1.2 0 −1.58074 0 −1.00000 0 −3.55916 0 −0.501249 0
1.3 0 −0.738124 0 −1.00000 0 4.65712 0 −2.45517 0
1.4 0 −0.179046 0 −1.00000 0 3.11339 0 −2.96794 0
1.5 0 0.449851 0 −1.00000 0 2.13963 0 −2.79763 0
1.6 0 0.714088 0 −1.00000 0 −2.31923 0 −2.49008 0
1.7 0 1.79921 0 −1.00000 0 0.673555 0 0.237149 0
1.8 0 2.55613 0 −1.00000 0 0.236085 0 3.53378 0
1.9 0 3.10709 0 −1.00000 0 −4.84509 0 6.65398 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.x 9
19.b odd 2 1 7220.2.a.v 9
19.f odd 18 2 380.2.u.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.a ✓ 18 19.f odd 18 2
7220.2.a.v 9 19.b odd 2 1
7220.2.a.x 9 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{9} - 3T_{3}^{8} - 12T_{3}^{7} + 39T_{3}^{6} + 21T_{3}^{5} - 99T_{3}^{4} + 8T_{3}^{3} + 48T_{3}^{2} - 9T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{9} - 39T_{7}^{7} + 10T_{7}^{6} + 429T_{7}^{5} - 234T_{7}^{4} - 1297T_{7}^{3} + 1023T_{7}^{2} - 87T_{7} - 19 \) Copy content Toggle raw display
\( T_{13}^{9} + 3 T_{13}^{8} - 57 T_{13}^{7} - 164 T_{13}^{6} + 960 T_{13}^{5} + 2730 T_{13}^{4} + \cdots + 14121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 39 T^{7} + \cdots - 19 \) Copy content Toggle raw display
$11$ \( T^{9} - 60 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{9} + 3 T^{8} + \cdots + 14121 \) Copy content Toggle raw display
$17$ \( T^{9} - 3 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 6 T^{8} + \cdots + 999 \) Copy content Toggle raw display
$29$ \( T^{9} + 9 T^{8} + \cdots + 19683 \) Copy content Toggle raw display
$31$ \( T^{9} + 18 T^{8} + \cdots + 3617 \) Copy content Toggle raw display
$37$ \( T^{9} + 30 T^{8} + \cdots + 1864711 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} + \cdots + 3935223 \) Copy content Toggle raw display
$43$ \( T^{9} - 24 T^{8} + \cdots - 14607 \) Copy content Toggle raw display
$47$ \( T^{9} + 21 T^{8} + \cdots - 11403747 \) Copy content Toggle raw display
$53$ \( T^{9} + 9 T^{8} + \cdots - 39447 \) Copy content Toggle raw display
$59$ \( T^{9} + 3 T^{8} + \cdots - 13851 \) Copy content Toggle raw display
$61$ \( T^{9} + 24 T^{8} + \cdots + 1384881 \) Copy content Toggle raw display
$67$ \( T^{9} - 363 T^{7} + \cdots + 2828523 \) Copy content Toggle raw display
$71$ \( T^{9} + 18 T^{8} + \cdots - 8830377 \) Copy content Toggle raw display
$73$ \( T^{9} + 33 T^{8} + \cdots - 4275657 \) Copy content Toggle raw display
$79$ \( T^{9} - 3 T^{8} + \cdots - 20013173 \) Copy content Toggle raw display
$83$ \( T^{9} - 30 T^{8} + \cdots + 1415691 \) Copy content Toggle raw display
$89$ \( T^{9} - 3 T^{8} + \cdots + 7145577 \) Copy content Toggle raw display
$97$ \( T^{9} + 15 T^{8} + \cdots + 1737987 \) Copy content Toggle raw display
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