Properties

Label 162.9.d.a
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,9,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(65.9953348299\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 51 \beta_1 q^{5} - 2786 \beta_{2} q^{7} + 128 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 51 \beta_1 q^{5} - 2786 \beta_{2} q^{7} + 128 \beta_{3} q^{8} - 6528 q^{10} + (1983 \beta_{3} - 1983 \beta_1) q^{11} + ( - 13150 \beta_{2} + 13150) q^{13} + 2786 \beta_1 q^{14} - 16384 \beta_{2} q^{16} + 5868 \beta_{3} q^{17} + 144002 q^{19} + ( - 6528 \beta_{3} + 6528 \beta_1) q^{20} + ( - 253824 \beta_{2} + 253824) q^{22} + 4362 \beta_1 q^{23} - 57697 \beta_{2} q^{25} + 13150 \beta_{3} q^{26} - 356608 q^{28} + ( - 55455 \beta_{3} + 55455 \beta_1) q^{29} + (728738 \beta_{2} - 728738) q^{31} + 16384 \beta_1 q^{32} - 751104 \beta_{2} q^{34} - 142086 \beta_{3} q^{35} - 1964446 q^{37} + (144002 \beta_{3} - 144002 \beta_1) q^{38} + (835584 \beta_{2} - 835584) q^{40} + 87162 \beta_1 q^{41} + 78142 \beta_{2} q^{43} + 253824 \beta_{3} q^{44} - 558336 q^{46} + ( - 311100 \beta_{3} + 311100 \beta_1) q^{47} + (1996995 \beta_{2} - 1996995) q^{49} + 57697 \beta_1 q^{50} - 1683200 \beta_{2} q^{52} - 46143 \beta_{3} q^{53} - 12945024 q^{55} + ( - 356608 \beta_{3} + 356608 \beta_1) q^{56} + (7098240 \beta_{2} - 7098240) q^{58} - 442317 \beta_1 q^{59} - 17578274 \beta_{2} q^{61} - 728738 \beta_{3} q^{62} - 2097152 q^{64} + ( - 670650 \beta_{3} + 670650 \beta_1) q^{65} + ( - 17136766 \beta_{2} + 17136766) q^{67} + 751104 \beta_1 q^{68} + 18187008 \beta_{2} q^{70} - 2288430 \beta_{3} q^{71} + 28139330 q^{73} + ( - 1964446 \beta_{3} + 1964446 \beta_1) q^{74} + ( - 18432256 \beta_{2} + 18432256) q^{76} + 5524638 \beta_1 q^{77} - 9182498 \beta_{2} q^{79} - 835584 \beta_{3} q^{80} - 11156736 q^{82} + ( - 7699371 \beta_{3} + 7699371 \beta_1) q^{83} + (38306304 \beta_{2} - 38306304) q^{85} - 78142 \beta_1 q^{86} - 32489472 \beta_{2} q^{88} + 7181802 \beta_{3} q^{89} - 36635900 q^{91} + ( - 558336 \beta_{3} + 558336 \beta_1) q^{92} + (39820800 \beta_{2} - 39820800) q^{94} + 7344102 \beta_1 q^{95} + 128722558 \beta_{2} q^{97} - 1996995 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 256 q^{4} - 5572 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 256 q^{4} - 5572 q^{7} - 26112 q^{10} + 26300 q^{13} - 32768 q^{16} + 576008 q^{19} + 507648 q^{22} - 115394 q^{25} - 1426432 q^{28} - 1457476 q^{31} - 1502208 q^{34} - 7857784 q^{37} - 1671168 q^{40} + 156284 q^{43} - 2233344 q^{46} - 3993990 q^{49} - 3366400 q^{52} - 51780096 q^{55} - 14196480 q^{58} - 35156548 q^{61} - 8388608 q^{64} + 34273532 q^{67} + 36374016 q^{70} + 112557320 q^{73} + 36864512 q^{76} - 18364996 q^{79} - 44626944 q^{82} - 76612608 q^{85} - 64978944 q^{88} - 146543600 q^{91} - 79641600 q^{94} + 257445116 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 8\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−9.79796 + 5.65685i 0 64.0000 110.851i 499.696 + 288.500i 0 −1393.00 2412.75i 1448.15i 0 −6528.00
53.2 9.79796 5.65685i 0 64.0000 110.851i −499.696 288.500i 0 −1393.00 2412.75i 1448.15i 0 −6528.00
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i 499.696 288.500i 0 −1393.00 + 2412.75i 1448.15i 0 −6528.00
107.2 9.79796 + 5.65685i 0 64.0000 + 110.851i −499.696 + 288.500i 0 −1393.00 + 2412.75i 1448.15i 0 −6528.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.d.a 4
3.b odd 2 1 inner 162.9.d.a 4
9.c even 3 1 6.9.b.a 2
9.c even 3 1 inner 162.9.d.a 4
9.d odd 6 1 6.9.b.a 2
9.d odd 6 1 inner 162.9.d.a 4
36.f odd 6 1 48.9.e.d 2
36.h even 6 1 48.9.e.d 2
45.h odd 6 1 150.9.d.a 2
45.j even 6 1 150.9.d.a 2
45.k odd 12 2 150.9.b.a 4
45.l even 12 2 150.9.b.a 4
72.j odd 6 1 192.9.e.h 2
72.l even 6 1 192.9.e.c 2
72.n even 6 1 192.9.e.h 2
72.p odd 6 1 192.9.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.9.b.a 2 9.c even 3 1
6.9.b.a 2 9.d odd 6 1
48.9.e.d 2 36.f odd 6 1
48.9.e.d 2 36.h even 6 1
150.9.b.a 4 45.k odd 12 2
150.9.b.a 4 45.l even 12 2
150.9.d.a 2 45.h odd 6 1
150.9.d.a 2 45.j even 6 1
162.9.d.a 4 1.a even 1 1 trivial
162.9.d.a 4 3.b odd 2 1 inner
162.9.d.a 4 9.c even 3 1 inner
162.9.d.a 4 9.d odd 6 1 inner
192.9.e.c 2 72.l even 6 1
192.9.e.c 2 72.p odd 6 1
192.9.e.h 2 72.j odd 6 1
192.9.e.h 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 332928T_{5}^{2} + 110841053184 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 332928 T^{2} + \cdots + 110841053184 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2786 T + 7761796)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 503332992 T^{2} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13150 T + 172922500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4407478272)^{2} \) Copy content Toggle raw display
$19$ \( (T - 144002)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 2435461632 T^{2} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} - 393632899200 T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + 728738 T + 531059072644)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1964446)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 972443423232 T^{2} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{2} - 78142 T + 6106172164)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 12388250880000 T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} + 272534585472)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 25042474046592 T^{2} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} + 17578274 T + 308995716819076)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 17136766 T + 293668748938756)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 670324718707200)^{2} \) Copy content Toggle raw display
$73$ \( (T - 28139330)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9182498 T + 84318269520004)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 57\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{2} + 66\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 128722558 T + 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
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