Properties

Label 162.8.a.j
Level $162$
Weight $8$
Character orbit 162.a
Self dual yes
Analytic conductor $50.606$
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] = [162,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.43103376.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{8} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 64 q^{4} + (\beta_1 + 132) q^{5} + (\beta_{3} + \beta_{2} + 140) q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 64 q^{4} + (\beta_1 + 132) q^{5} + (\beta_{3} + \beta_{2} + 140) q^{7} + 512 q^{8} + (8 \beta_1 + 1056) q^{10} + (9 \beta_{3} + 4 \beta_{2} + \cdots + 540) q^{11}+ \cdots + ( - 50176 \beta_{3} + 448 \beta_{2} + \cdots + 2898792) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 256 q^{4} + 528 q^{5} + 560 q^{7} + 2048 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 256 q^{4} + 528 q^{5} + 560 q^{7} + 2048 q^{8} + 4224 q^{10} + 2160 q^{11} + 13460 q^{13} + 4480 q^{14} + 16384 q^{16} + 22560 q^{17} + 36704 q^{19} + 33792 q^{20} + 17280 q^{22} + 62640 q^{23} + 94696 q^{25} + 107680 q^{26} + 35840 q^{28} + 68400 q^{29} + 227504 q^{31} + 131072 q^{32} + 180480 q^{34} - 63024 q^{35} + 523580 q^{37} + 293632 q^{38} + 270336 q^{40} + 67200 q^{41} + 562640 q^{43} + 138240 q^{44} + 501120 q^{46} + 515328 q^{47} + 1449396 q^{49} + 757568 q^{50} + 861440 q^{52} + 2498016 q^{53} + 1322784 q^{55} + 286720 q^{56} + 547200 q^{58} + 4155840 q^{59} + 2130764 q^{61} + 1820032 q^{62} + 1048576 q^{64} + 8966496 q^{65} - 1205440 q^{67} + 1443840 q^{68} - 504192 q^{70} + 12486480 q^{71} - 4820860 q^{73} + 4188640 q^{74} + 2349056 q^{76} + 18686592 q^{77} - 11471680 q^{79} + 2162688 q^{80} + 537600 q^{82} + 16811232 q^{83} - 21006396 q^{85} + 4501120 q^{86} + 1105920 q^{88} + 16857600 q^{89} - 23200544 q^{91} + 4008960 q^{92} + 4122624 q^{94} + 14090160 q^{95} - 27078040 q^{97} + 11595168 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{3} - 13\nu^{2} - 759\nu + 1728 ) / 26 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -16\nu^{3} + 25\nu^{2} + 5871\nu - 3132 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{2} - 27\nu - 5184 ) / 26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 4\beta _1 + 54 ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 105\beta_{3} + \beta_{2} + 4\beta _1 + 20790 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 531\beta_{3} + 193\beta_{2} + 1474\beta _1 + 31158 ) / 108 \) 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
−17.5928
8.05976
−7.05976
18.5928
8.00000 0 64.0000 −280.425 0 −164.323 512.000 0 −2243.40
1.2 8.00000 0 64.0000 11.2463 0 1439.47 512.000 0 89.9705
1.3 8.00000 0 64.0000 325.500 0 −1450.45 512.000 0 2604.00
1.4 8.00000 0 64.0000 471.679 0 735.308 512.000 0 3773.43
\(n\): 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 162.8.a.j yes 4
3.b odd 2 1 162.8.a.g 4
9.c even 3 2 162.8.c.q 8
9.d odd 6 2 162.8.c.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.g 4 3.b odd 2 1
162.8.a.j yes 4 1.a even 1 1 trivial
162.8.c.q 8 9.c even 3 2
162.8.c.r 8 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 528T_{5}^{3} - 64206T_{5}^{2} + 43841520T_{5} - 484199775 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 528 T^{3} + \cdots - 484199775 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 252275377936 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 267055051594896 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 16\!\cdots\!11 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 45\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 83\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 16\!\cdots\!39 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 61\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 76\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 56\!\cdots\!11 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 15\!\cdots\!36 \) Copy content Toggle raw display
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