↑

Properties

Label 147.6.a.n
Level $147$
Weight $6$
Character orbit 147.a
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $6$
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] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \cdots - 28) q^{8}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \cdots - 28) q^{8}+ \cdots + ( - 567 \beta_{5} - 648 \beta_{4} + \cdots + 8667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28\nu^{5} - 35\nu^{4} - 546\nu^{3} + 742\nu^{2} - 7063\nu + 11802 ) / 1941 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -87\nu^{5} - 53\nu^{4} + 3961\nu^{3} + 2547\nu^{2} - 42269\nu - 27289 ) / 1941 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 158\nu^{5} - 521\nu^{4} - 6316\nu^{3} + 21656\nu^{2} + 33579\nu - 143678 ) / 1941 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108\nu^{5} + 512\nu^{4} - 5988\nu^{3} - 17195\nu^{2} + 81453\nu + 71402 ) / 647 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -774\nu^{5} - 1297\nu^{4} + 39032\nu^{3} + 55188\nu^{2} - 447553\nu - 433643 ) / 1941 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - 12\beta_{2} - 4\beta _1 + 5 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} - 4\beta _1 + 567 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29\beta_{5} + 4\beta_{4} - 28\beta_{3} - 302\beta_{2} - 25\beta _1 - 129 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 99\beta_{5} + 176\beta_{4} + 155\beta_{3} + 80\beta_{2} + 74\beta _1 + 14843 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 862\beta_{5} + 192\beta_{4} - 737\beta_{3} - 8710\beta_{2} + 643\beta _1 - 9528 ) / 28 \) 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
−5.61145
−0.910122
4.27213
−5.10089
3.75353
5.59680
−11.1881 −9.00000 93.1729 62.3150 100.693 0 −684.407 81.0000 −697.185
1.2 −5.31815 −9.00000 −3.71724 −103.471 47.8634 0 189.950 81.0000 550.272
1.3 −3.09163 −9.00000 −22.4418 −13.7926 27.8246 0 168.314 81.0000 42.6416
1.4 3.38033 −9.00000 −20.5734 54.5253 −30.4230 0 −177.715 81.0000 184.313
1.5 8.20863 −9.00000 35.3816 −29.2259 −73.8777 0 27.7583 81.0000 −239.905
1.6 10.0089 −9.00000 68.1779 −70.3512 −90.0800 0 362.101 81.0000 −704.138
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)

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 147.6.a.n ✓ 6
3.b odd 2 1 441.6.a.bb 6
7.b odd 2 1 147.6.a.o yes 6
7.c even 3 2 147.6.e.q 12
7.d odd 6 2 147.6.e.p 12
21.c even 2 1 441.6.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n ✓ 6 1.a even 1 1 trivial
147.6.a.o yes 6 7.b odd 2 1
147.6.e.p 12 7.d odd 6 2
147.6.e.q 12 7.c even 3 2
441.6.a.ba 6 21.c even 2 1
441.6.a.bb 6 3.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2}^{6} - 2T_{2}^{5} - 169T_{2}^{4} + 336T_{2}^{3} + 6472T_{2}^{2} - 4256T_{2} - 51088 \) Copy content Toggle raw display
\( T_{5}^{6} + 100T_{5}^{5} - 6778T_{5}^{4} - 651312T_{5}^{3} + 9669292T_{5}^{2} + 959211664T_{5} + 9969962312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots - 51088 \) Copy content Toggle raw display
$3$ \( (T + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 9969962312 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 55273527989696 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 88\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 77\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 37\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 45\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 32\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 79\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 28\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 93\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 34\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 18\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 24\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 29\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
show more
show less