Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 60 \beta_1 q^{5} + 2065 \beta_{2} q^{7} + 128 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + 60 \beta_1 q^{5} + 2065 \beta_{2} q^{7} + 128 \beta_{3} q^{8} - 7680 q^{10} + (588 \beta_{3} - 588 \beta_1) q^{11} + (8063 \beta_{2} - 8063) q^{13} - 2065 \beta_1 q^{14} - 16384 \beta_{2} q^{16} + 1908 \beta_{3} q^{17} - 226609 q^{19} + ( - 7680 \beta_{3} + 7680 \beta_1) q^{20} + ( - 75264 \beta_{2} + 75264) q^{22} - 32556 \beta_1 q^{23} + 70175 \beta_{2} q^{25} - 8063 \beta_{3} q^{26} + 264320 q^{28} + ( - 82824 \beta_{3} + 82824 \beta_1) q^{29} + (826370 \beta_{2} - 826370) q^{31} + 16384 \beta_1 q^{32} - 244224 \beta_{2} q^{34} + 123900 \beta_{3} q^{35} + 1344575 q^{37} + ( - 226609 \beta_{3} + 226609 \beta_1) q^{38} + (983040 \beta_{2} - 983040) q^{40} - 458904 \beta_1 q^{41} + 6147742 \beta_{2} q^{43} + 75264 \beta_{3} q^{44} + 4167168 q^{46} + (522444 \beta_{3} - 522444 \beta_1) q^{47} + ( - 1500576 \beta_{2} + 1500576) q^{49} - 70175 \beta_1 q^{50} + 1032064 \beta_{2} q^{52} + 67896 \beta_{3} q^{53} - 4515840 q^{55} + (264320 \beta_{3} - 264320 \beta_1) q^{56} + (10601472 \beta_{2} - 10601472) q^{58} + 41892 \beta_1 q^{59} + 14985697 \beta_{2} q^{61} - 826370 \beta_{3} q^{62} - 2097152 q^{64} + (483780 \beta_{3} - 483780 \beta_1) q^{65} + ( - 10023697 \beta_{2} + 10023697) q^{67} + 244224 \beta_1 q^{68} - 15859200 \beta_{2} q^{70} - 4020336 \beta_{3} q^{71} - 23261569 q^{73} + (1344575 \beta_{3} - 1344575 \beta_1) q^{74} + (29005952 \beta_{2} - 29005952) q^{76} - 1214220 \beta_1 q^{77} - 14267183 \beta_{2} q^{79} - 983040 \beta_{3} q^{80} + 58739712 q^{82} + (3198936 \beta_{3} - 3198936 \beta_1) q^{83} + (14653440 \beta_{2} - 14653440) q^{85} - 6147742 \beta_1 q^{86} - 9633792 \beta_{2} q^{88} - 10172412 \beta_{3} q^{89} - 16650095 q^{91} + (4167168 \beta_{3} - 4167168 \beta_1) q^{92} + ( - 66872832 \beta_{2} + 66872832) q^{94} - 13596540 \beta_1 q^{95} + 40571617 \beta_{2} q^{97} + 1500576 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 256 q^{4} + 4130 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 256 q^{4} + 4130 q^{7} - 30720 q^{10} - 16126 q^{13} - 32768 q^{16} - 906436 q^{19} + 150528 q^{22} + 140350 q^{25} + 1057280 q^{28} - 1652740 q^{31} - 488448 q^{34} + 5378300 q^{37} - 1966080 q^{40} + 12295484 q^{43} + 16668672 q^{46} + 3001152 q^{49} + 2064128 q^{52} - 18063360 q^{55} - 21202944 q^{58} + 29971394 q^{61} - 8388608 q^{64} + 20047394 q^{67} - 31718400 q^{70} - 93046276 q^{73} - 58011904 q^{76} - 28534366 q^{79} + 234958848 q^{82} - 29306880 q^{85} - 19267584 q^{88} - 66600380 q^{91} + 133745664 q^{94} + 81143234 q^{97}+O(q^{100}) \) 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 587.878 + 339.411i 0 1032.50 + 1788.34i 1448.15i 0 −7680.00
53.2 9.79796 5.65685i 0 64.0000 110.851i −587.878 339.411i 0 1032.50 + 1788.34i 1448.15i 0 −7680.00
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i 587.878 339.411i 0 1032.50 1788.34i 1448.15i 0 −7680.00
107.2 9.79796 + 5.65685i 0 64.0000 + 110.851i −587.878 + 339.411i 0 1032.50 1788.34i 1448.15i 0 −7680.00
\(n\): 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.c 4
3.b odd 2 1 inner 162.9.d.c 4
9.c even 3 1 54.9.b.a 2
9.c even 3 1 inner 162.9.d.c 4
9.d odd 6 1 54.9.b.a 2
9.d odd 6 1 inner 162.9.d.c 4
36.f odd 6 1 432.9.e.g 2
36.h even 6 1 432.9.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.a 2 9.c even 3 1
54.9.b.a 2 9.d odd 6 1
162.9.d.c 4 1.a even 1 1 trivial
162.9.d.c 4 3.b odd 2 1 inner
162.9.d.c 4 9.c even 3 1 inner
162.9.d.c 4 9.d odd 6 1 inner
432.9.e.g 2 36.f odd 6 1
432.9.e.g 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 460800T_{5}^{2} + 212336640000 \) 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
$5$ \( T^{4} - 460800 T^{2} + \cdots + 212336640000 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2065 T + 4264225)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 44255232 T^{2} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8063 T + 65011969)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 465979392)^{2} \) Copy content Toggle raw display
$19$ \( (T + 226609)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 135666321408 T^{2} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{4} - 878056316928 T^{2} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{2} + 826370 T + 682887376900)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1344575)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 26955888795648 T^{2} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6147742 T + 37794731698564)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 34937309841408 T^{2} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{2} + 590062952448)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 224632276992 T^{2} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 14985697 T + 224571114575809)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10023697 T + 100474501547809)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 20\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T + 23261569)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14267183 T + 203552510755489)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + 13\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 40571617 T + 16\!\cdots\!89)^{2} \) Copy content Toggle raw display
show more
show less