Properties

Label 162.6.a.j
Level $162$
Weight $6$
Character orbit 162.a
Self dual yes
Analytic conductor $25.982$
Analytic rank $0$
Dimension $3$
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] = [162,6,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(25.9821788097\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.125628.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 63x + 159 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + ( - \beta_1 + 18) q^{5} + ( - \beta_{2} + 44) q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + ( - \beta_1 + 18) q^{5} + ( - \beta_{2} + 44) q^{7} + 64 q^{8} + ( - 4 \beta_1 + 72) q^{10} + (\beta_{2} - 5 \beta_1 + 105) q^{11} + (2 \beta_{2} + 11 \beta_1 + 248) q^{13} + ( - 4 \beta_{2} + 176) q^{14} + 256 q^{16} + (4 \beta_{2} + 9 \beta_1 + 483) q^{17} + (6 \beta_{2} - 5 \beta_1 + 377) q^{19} + ( - 16 \beta_1 + 288) q^{20} + (4 \beta_{2} - 20 \beta_1 + 420) q^{22} + ( - \beta_{2} + 14 \beta_1 + 1056) q^{23} + ( - 10 \beta_{2} - 5 \beta_1 + 961) q^{25} + (8 \beta_{2} + 44 \beta_1 + 992) q^{26} + ( - 16 \beta_{2} + 704) q^{28} + ( - 6 \beta_{2} + 59 \beta_1 + 1716) q^{29} + ( - 17 \beta_{2} + 44 \beta_1 + 2870) q^{31} + 1024 q^{32} + (16 \beta_{2} + 36 \beta_1 + 1932) q^{34} + ( - 49 \beta_{2} - 136 \beta_1 + 450) q^{35} + (10 \beta_{2} + 40 \beta_1 + 6656) q^{37} + (24 \beta_{2} - 20 \beta_1 + 1508) q^{38} + ( - 64 \beta_1 + 1152) q^{40} + (70 \beta_{2} - 80 \beta_1 - 1683) q^{41} + (39 \beta_{2} - 117 \beta_1 + 10463) q^{43} + (16 \beta_{2} - 80 \beta_1 + 1680) q^{44} + ( - 4 \beta_{2} + 56 \beta_1 + 4224) q^{46} + ( - 49 \beta_{2} + 40 \beta_1 - 4308) q^{47} + (4 \beta_{2} - 251 \beta_1 + 17619) q^{49} + ( - 40 \beta_{2} - 20 \beta_1 + 3844) q^{50} + (32 \beta_{2} + 176 \beta_1 + 3968) q^{52} + (54 \beta_{2} + 116 \beta_1 - 16008) q^{53} + ( - \beta_{2} + 52 \beta_1 + 21042) q^{55} + ( - 64 \beta_{2} + 2816) q^{56} + ( - 24 \beta_{2} + 236 \beta_1 + 6864) q^{58} + ( - 71 \beta_{2} + 255 \beta_1 - 20985) q^{59} + ( - 82 \beta_{2} + 37 \beta_1 + 25322) q^{61} + ( - 68 \beta_{2} + 176 \beta_1 + 11480) q^{62} + 4096 q^{64} + (208 \beta_{2} - 207 \beta_1 - 36234) q^{65} + (15 \beta_{2} + 519 \beta_1 + 10997) q^{67} + (64 \beta_{2} + 144 \beta_1 + 7728) q^{68} + ( - 196 \beta_{2} - 544 \beta_1 + 1800) q^{70} + (130 \beta_{2} + 354 \beta_1 - 21612) q^{71} + ( - 68 \beta_{2} - 901 \beta_1 - 1411) q^{73} + (40 \beta_{2} + 160 \beta_1 + 26624) q^{74} + (96 \beta_{2} - 80 \beta_1 + 6032) q^{76} + ( - 308 \beta_{2} - 429 \beta_1 - 29580) q^{77} + ( - 15 \beta_{2} + 724 \beta_1 - 29734) q^{79} + ( - 256 \beta_1 + 4608) q^{80} + (280 \beta_{2} - 320 \beta_1 - 6732) q^{82} + ( - 211 \beta_{2} - 476 \beta_1 - 10878) q^{83} + (286 \beta_{2} - 232 \beta_1 - 23796) q^{85} + (156 \beta_{2} - 468 \beta_1 + 41852) q^{86} + (64 \beta_{2} - 320 \beta_1 + 6720) q^{88} + ( - 98 \beta_{2} + 294 \beta_1 + 11022) q^{89} + ( - 3 \beta_{2} + 1998 \beta_1 - 50306) q^{91} + ( - 16 \beta_{2} + 224 \beta_1 + 16896) q^{92} + ( - 196 \beta_{2} + 160 \beta_1 - 17232) q^{94} + (244 \beta_{2} + 240 \beta_1 + 27648) q^{95} + (386 \beta_{2} - 16 \beta_1 - 15415) q^{97} + (16 \beta_{2} - 1004 \beta_1 + 70476) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{2} + 48 q^{4} + 54 q^{5} + 132 q^{7} + 192 q^{8} + 216 q^{10} + 315 q^{11} + 744 q^{13} + 528 q^{14} + 768 q^{16} + 1449 q^{17} + 1131 q^{19} + 864 q^{20} + 1260 q^{22} + 3168 q^{23} + 2883 q^{25} + 2976 q^{26} + 2112 q^{28} + 5148 q^{29} + 8610 q^{31} + 3072 q^{32} + 5796 q^{34} + 1350 q^{35} + 19968 q^{37} + 4524 q^{38} + 3456 q^{40} - 5049 q^{41} + 31389 q^{43} + 5040 q^{44} + 12672 q^{46} - 12924 q^{47} + 52857 q^{49} + 11532 q^{50} + 11904 q^{52} - 48024 q^{53} + 63126 q^{55} + 8448 q^{56} + 20592 q^{58} - 62955 q^{59} + 75966 q^{61} + 34440 q^{62} + 12288 q^{64} - 108702 q^{65} + 32991 q^{67} + 23184 q^{68} + 5400 q^{70} - 64836 q^{71} - 4233 q^{73} + 79872 q^{74} + 18096 q^{76} - 88740 q^{77} - 89202 q^{79} + 13824 q^{80} - 20196 q^{82} - 32634 q^{83} - 71388 q^{85} + 125556 q^{86} + 20160 q^{88} + 33066 q^{89} - 150918 q^{91} + 50688 q^{92} - 51696 q^{94} + 82944 q^{95} - 46245 q^{97} + 211428 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 63x + 159 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{2} + 24\nu - 135 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -15\nu^{2} - 12\nu + 639 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 5\beta _1 + 18 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{2} - 2\beta _1 + 1143 ) / 27 \) 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
6.81933
2.72791
−8.54724
4.00000 0 16.0000 −66.0868 0 114.190 64.0000 0 −264.347
1.2 4.00000 0 16.0000 41.6029 0 −203.321 64.0000 0 166.412
1.3 4.00000 0 16.0000 78.4839 0 221.131 64.0000 0 313.936
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

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 162.6.a.j 3
3.b odd 2 1 162.6.a.i 3
9.c even 3 2 18.6.c.b 6
9.d odd 6 2 54.6.c.b 6
36.f odd 6 2 144.6.i.b 6
36.h even 6 2 432.6.i.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b 6 9.c even 3 2
54.6.c.b 6 9.d odd 6 2
144.6.i.b 6 36.f odd 6 2
162.6.a.i 3 3.b odd 2 1
162.6.a.j 3 1.a even 1 1 trivial
432.6.i.b 6 36.h even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 54T_{5}^{2} - 4671T_{5} + 215784 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 54 T^{2} - 4671 T + 215784 \) Copy content Toggle raw display
$7$ \( T^{3} - 132 T^{2} - 42927 T + 5134078 \) Copy content Toggle raw display
$11$ \( T^{3} - 315 T^{2} + \cdots + 41768163 \) Copy content Toggle raw display
$13$ \( T^{3} - 744 T^{2} + \cdots + 384824422 \) Copy content Toggle raw display
$17$ \( T^{3} - 1449 T^{2} + \cdots + 930192444 \) Copy content Toggle raw display
$19$ \( T^{3} - 1131 T^{2} + \cdots - 352455920 \) Copy content Toggle raw display
$23$ \( T^{3} - 3168 T^{2} + \cdots - 425600514 \) Copy content Toggle raw display
$29$ \( T^{3} - 5148 T^{2} + \cdots - 6505725654 \) Copy content Toggle raw display
$31$ \( T^{3} - 8610 T^{2} + \cdots + 59316561604 \) Copy content Toggle raw display
$37$ \( T^{3} - 19968 T^{2} + \cdots - 188019064016 \) Copy content Toggle raw display
$41$ \( T^{3} + 5049 T^{2} + \cdots - 2157363401913 \) Copy content Toggle raw display
$43$ \( T^{3} - 31389 T^{2} + \cdots + 513661035961 \) Copy content Toggle raw display
$47$ \( T^{3} + 12924 T^{2} + \cdots + 84596352750 \) Copy content Toggle raw display
$53$ \( T^{3} + 48024 T^{2} + \cdots + 1764512817552 \) Copy content Toggle raw display
$59$ \( T^{3} + 62955 T^{2} + \cdots - 5778215946243 \) Copy content Toggle raw display
$61$ \( T^{3} - 75966 T^{2} + \cdots - 5359653497960 \) Copy content Toggle raw display
$67$ \( T^{3} - 32991 T^{2} + \cdots + 3034793402875 \) Copy content Toggle raw display
$71$ \( T^{3} + 64836 T^{2} + \cdots - 139951336896 \) Copy content Toggle raw display
$73$ \( T^{3} + 4233 T^{2} + \cdots + 14322358753732 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 115303078320596 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 103421607911604 \) Copy content Toggle raw display
$89$ \( T^{3} - 33066 T^{2} + \cdots + 9104584153608 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 292114819729997 \) Copy content Toggle raw display
show more
show less