↑

Properties

Label 7220.2.a.w
Level $7220$
Weight $2$
Character orbit 7220.a
Self dual yes
Analytic conductor $57.652$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \( x^{9} - 3x^{8} - 12x^{7} + 33x^{6} + 51x^{5} - 117x^{4} - 86x^{3} + 162x^{2} + 45x - 73 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + (\beta_{8} - \beta_{5} - \beta_{2}) q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + (\beta_{8} - \beta_{5} - \beta_{2}) q^{7} + (\beta_{2} + \beta_1) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + \cdots + 5) q^{99}+O(q^{100}) \) 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 \( 9 q - 3 q^{3} + 9 q^{5} + 6 q^{9} - 15 q^{13} - 3 q^{15} + 3 q^{17} - 12 q^{21} + 6 q^{23} + 9 q^{25} - 18 q^{27} - 15 q^{29} - 6 q^{31} - 9 q^{33} - 18 q^{37} - 6 q^{39} - 18 q^{41} - 6 q^{43} + 6 q^{45} - 15 q^{47} + 3 q^{49} + 18 q^{51} - 33 q^{53} + 3 q^{59} - 15 q^{63} - 15 q^{65} - 12 q^{67} - 27 q^{69} - 6 q^{71} + 3 q^{73} - 3 q^{75} - 18 q^{77} - 15 q^{79} - 15 q^{81} + 6 q^{83} + 3 q^{85} - 21 q^{87} - 3 q^{89} - 33 q^{91} - 9 q^{93} - 45 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 12x^{7} + 33x^{6} + 51x^{5} - 117x^{4} - 86x^{3} + 162x^{2} + 45x - 73 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{8} + 8\nu^{7} + 110\nu^{6} - 101\nu^{5} - 684\nu^{4} + 376\nu^{3} + 1525\nu^{2} - 416\nu - 877 ) / 111 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -10\nu^{8} + 38\nu^{7} + 97\nu^{6} - 378\nu^{5} - 326\nu^{4} + 1009\nu^{3} + 556\nu^{2} - 607\nu - 438 ) / 111 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\nu^{8} - 65\nu^{7} - 200\nu^{6} + 557\nu^{5} + 618\nu^{4} - 1168\nu^{3} - 694\nu^{2} + 494\nu + 313 ) / 111 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{8} + 43\nu^{7} + 323\nu^{6} - 381\nu^{5} - 1660\nu^{4} + 800\nu^{3} + 3410\nu^{2} - 275\nu - 2184 ) / 111 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 7\nu^{6} + 17\nu^{5} - 63\nu^{4} - 40\nu^{3} + 152\nu^{2} + 23\nu - 95 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -30\nu^{8} + 77\nu^{7} + 365\nu^{6} - 764\nu^{5} - 1533\nu^{4} + 2176\nu^{3} + 2371\nu^{2} - 1895\nu - 1018 ) / 111 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 3\beta_{6} - 3\beta_{5} + \beta_{3} + 9\beta_{2} + 11\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{8} + 2\beta_{7} - 14\beta_{6} - 16\beta_{5} - 8\beta_{4} + 21\beta_{3} + 23\beta_{2} + 45\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{8} + 14\beta_{7} - 45\beta_{6} - 47\beta_{5} + 28\beta_{3} + 80\beta_{2} + 107\beta _1 + 115 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 31 \beta_{8} + 33 \beta_{7} - 162 \beta_{6} - 186 \beta_{5} - 48 \beta_{4} + 205 \beta_{3} + \cdots + 212 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{8} + 153 \beta_{7} - 526 \beta_{6} - 561 \beta_{5} + 8 \beta_{4} + 426 \beta_{3} + \cdots + 943 \) 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
3.11553
2.74855
1.58645
1.09394
0.886461
−0.957108
−1.50873
−1.67739
−2.28771
0 −3.11553 0 1.00000 0 −1.28752 0 6.70655 0
1.2 0 −2.74855 0 1.00000 0 2.47557 0 4.55453 0
1.3 0 −1.58645 0 1.00000 0 −1.55763 0 −0.483175 0
1.4 0 −1.09394 0 1.00000 0 3.62818 0 −1.80329 0
1.5 0 −0.886461 0 1.00000 0 −0.920861 0 −2.21419 0
1.6 0 0.957108 0 1.00000 0 4.56133 0 −2.08394 0
1.7 0 1.50873 0 1.00000 0 −3.35099 0 −0.723741 0
1.8 0 1.67739 0 1.00000 0 −0.461272 0 −0.186377 0
1.9 0 2.28771 0 1.00000 0 −3.08680 0 2.23364 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.w 9
19.b odd 2 1 7220.2.a.y 9
19.e even 9 2 380.2.u.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.b ✓ 18 19.e even 9 2
7220.2.a.w 9 1.a even 1 1 trivial
7220.2.a.y 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7220))\):

\( T_{3}^{9} + 3T_{3}^{8} - 12T_{3}^{7} - 33T_{3}^{6} + 51T_{3}^{5} + 117T_{3}^{4} - 86T_{3}^{3} - 162T_{3}^{2} + 45T_{3} + 73 \) Copy content Toggle raw display
\( T_{7}^{9} - 33T_{7}^{7} - 28T_{7}^{6} + 327T_{7}^{5} + 528T_{7}^{4} - 723T_{7}^{3} - 2157T_{7}^{2} - 1587T_{7} - 361 \) Copy content Toggle raw display
\( T_{13}^{9} + 15 T_{13}^{8} + 57 T_{13}^{7} - 154 T_{13}^{6} - 1488 T_{13}^{5} - 2574 T_{13}^{4} + \cdots - 971 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots + 73 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 33 T^{7} + \cdots - 361 \) Copy content Toggle raw display
$11$ \( T^{9} - 36 T^{7} + \cdots - 111 \) Copy content Toggle raw display
$13$ \( T^{9} + 15 T^{8} + \cdots - 971 \) Copy content Toggle raw display
$17$ \( T^{9} - 3 T^{8} + \cdots + 3027 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 6 T^{8} + \cdots + 8721 \) Copy content Toggle raw display
$29$ \( T^{9} + 15 T^{8} + \cdots - 3554871 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} + \cdots - 32113 \) Copy content Toggle raw display
$37$ \( T^{9} + 18 T^{8} + \cdots - 587421 \) Copy content Toggle raw display
$41$ \( T^{9} + 18 T^{8} + \cdots + 2190249 \) Copy content Toggle raw display
$43$ \( T^{9} + 6 T^{8} + \cdots - 127009 \) Copy content Toggle raw display
$47$ \( T^{9} + 15 T^{8} + \cdots - 249261 \) Copy content Toggle raw display
$53$ \( T^{9} + 33 T^{8} + \cdots + 235113 \) Copy content Toggle raw display
$59$ \( T^{9} - 3 T^{8} + \cdots + 5034423 \) Copy content Toggle raw display
$61$ \( T^{9} - 303 T^{7} + \cdots + 13764617 \) Copy content Toggle raw display
$67$ \( T^{9} + 12 T^{8} + \cdots + 44760491 \) Copy content Toggle raw display
$71$ \( T^{9} + 6 T^{8} + \cdots + 13281 \) Copy content Toggle raw display
$73$ \( T^{9} - 3 T^{8} + \cdots + 97928109 \) Copy content Toggle raw display
$79$ \( T^{9} + 15 T^{8} + \cdots - 19646371 \) Copy content Toggle raw display
$83$ \( T^{9} - 6 T^{8} + \cdots - 268167 \) Copy content Toggle raw display
$89$ \( T^{9} + 3 T^{8} + \cdots + 12502239 \) Copy content Toggle raw display
$97$ \( T^{9} + 45 T^{8} + \cdots + 6720723 \) Copy content Toggle raw display
show more
show less