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Properties

Label 162.3.d.a
Level $162$
Weight $3$
Character orbit 162.d
Analytic conductor $4.414$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
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: \(4.41418028264\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + (6 \beta_{3} - 6 \beta_1) q^{5} + (5 \beta_{2} - 5) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + (6 \beta_{3} - 6 \beta_1) q^{5} + (5 \beta_{2} - 5) q^{7} + 2 \beta_{3} q^{8} - 12 q^{10} - 6 \beta_1 q^{11} + \beta_{2} q^{13} + (5 \beta_{3} - 5 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + 18 \beta_{3} q^{17} + 29 q^{19} - 12 \beta_1 q^{20} - 12 \beta_{2} q^{22} + (6 \beta_{3} - 6 \beta_1) q^{23} + ( - 47 \beta_{2} + 47) q^{25} + \beta_{3} q^{26} - 10 q^{28} + 12 \beta_1 q^{29} + 10 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (36 \beta_{2} - 36) q^{34} - 30 \beta_{3} q^{35} - 25 q^{37} + 29 \beta_1 q^{38} - 24 \beta_{2} q^{40} + ( - 12 \beta_{3} + 12 \beta_1) q^{41} + (14 \beta_{2} - 14) q^{43} - 12 \beta_{3} q^{44} - 12 q^{46} - 6 \beta_1 q^{47} + 24 \beta_{2} q^{49} + ( - 47 \beta_{3} + 47 \beta_1) q^{50} + (2 \beta_{2} - 2) q^{52} - 36 \beta_{3} q^{53} + 72 q^{55} - 10 \beta_1 q^{56} + 24 \beta_{2} q^{58} + ( - 66 \beta_{3} + 66 \beta_1) q^{59} + (23 \beta_{2} - 23) q^{61} + 10 \beta_{3} q^{62} - 8 q^{64} - 6 \beta_1 q^{65} + 19 \beta_{2} q^{67} + (36 \beta_{3} - 36 \beta_1) q^{68} + ( - 60 \beta_{2} + 60) q^{70} + 72 \beta_{3} q^{71} - 97 q^{73} - 25 \beta_1 q^{74} + 58 \beta_{2} q^{76} + ( - 30 \beta_{3} + 30 \beta_1) q^{77} + (77 \beta_{2} - 77) q^{79} - 24 \beta_{3} q^{80} + 24 q^{82} + 84 \beta_1 q^{83} - 216 \beta_{2} q^{85} + (14 \beta_{3} - 14 \beta_1) q^{86} + ( - 24 \beta_{2} + 24) q^{88} - 54 \beta_{3} q^{89} - 5 q^{91} - 12 \beta_1 q^{92} - 12 \beta_{2} q^{94} + (174 \beta_{3} - 174 \beta_1) q^{95} + ( - 49 \beta_{2} + 49) q^{97} + 24 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 10 q^{7} - 48 q^{10} + 2 q^{13} - 8 q^{16} + 116 q^{19} - 24 q^{22} + 94 q^{25} - 40 q^{28} + 20 q^{31} - 72 q^{34} - 100 q^{37} - 48 q^{40} - 28 q^{43} - 48 q^{46} + 48 q^{49} - 4 q^{52} + 288 q^{55} + 48 q^{58} - 46 q^{61} - 32 q^{64} + 38 q^{67} + 120 q^{70} - 388 q^{73} + 116 q^{76} - 154 q^{79} + 96 q^{82} - 432 q^{85} + 48 q^{88} - 20 q^{91} - 24 q^{94} + 98 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

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

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
−1.22474 + 0.707107i 0 1.00000 − 1.73205i 7.34847 + 4.24264i 0 −2.50000 − 4.33013i 2.82843i 0 −12.0000
53.2 1.22474 − 0.707107i 0 1.00000 − 1.73205i −7.34847 − 4.24264i 0 −2.50000 − 4.33013i − 2.82843i 0 −12.0000
107.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 7.34847 − 4.24264i 0 −2.50000 + 4.33013i − 2.82843i 0 −12.0000
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.34847 + 4.24264i 0 −2.50000 + 4.33013i 2.82843i 0 −12.0000
\(n\): 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 162.3.d.a 4
3.b odd 2 1 inner 162.3.d.a 4
4.b odd 2 1 1296.3.q.i 4
9.c even 3 1 54.3.b.a ✓ 2
9.c even 3 1 inner 162.3.d.a 4
9.d odd 6 1 54.3.b.a ✓ 2
9.d odd 6 1 inner 162.3.d.a 4
12.b even 2 1 1296.3.q.i 4
36.f odd 6 1 432.3.e.d 2
36.f odd 6 1 1296.3.q.i 4
36.h even 6 1 432.3.e.d 2
36.h even 6 1 1296.3.q.i 4
45.h odd 6 1 1350.3.d.d 2
45.j even 6 1 1350.3.d.d 2
45.k odd 12 2 1350.3.b.b 4
45.l even 12 2 1350.3.b.b 4
72.j odd 6 1 1728.3.e.l 2
72.l even 6 1 1728.3.e.f 2
72.n even 6 1 1728.3.e.l 2
72.p odd 6 1 1728.3.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.3.b.a ✓ 2 9.c even 3 1
54.3.b.a ✓ 2 9.d odd 6 1
162.3.d.a 4 1.a even 1 1 trivial
162.3.d.a 4 3.b odd 2 1 inner
162.3.d.a 4 9.c even 3 1 inner
162.3.d.a 4 9.d odd 6 1 inner
432.3.e.d 2 36.f odd 6 1
432.3.e.d 2 36.h even 6 1
1296.3.q.i 4 4.b odd 2 1
1296.3.q.i 4 12.b even 2 1
1296.3.q.i 4 36.f odd 6 1
1296.3.q.i 4 36.h even 6 1
1350.3.b.b 4 45.k odd 12 2
1350.3.b.b 4 45.l even 12 2
1350.3.d.d 2 45.h odd 6 1
1350.3.d.d 2 45.j even 6 1
1728.3.e.f 2 72.l even 6 1
1728.3.e.f 2 72.p odd 6 1
1728.3.e.l 2 72.j odd 6 1
1728.3.e.l 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 72T_{5}^{2} + 5184 \) acting on \(S_{3}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$7$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 648)^{2} \) Copy content Toggle raw display
$19$ \( (T - 29)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( T^{4} - 288 T^{2} + 82944 \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T + 25)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 288 T^{2} + 82944 \) Copy content Toggle raw display
$43$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8712 T^{2} + 75898944 \) Copy content Toggle raw display
$61$ \( (T^{2} + 23 T + 529)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 19 T + 361)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 10368)^{2} \) Copy content Toggle raw display
$73$ \( (T + 97)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 77 T + 5929)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 14112 T^{2} + 199148544 \) Copy content Toggle raw display
$89$ \( (T^{2} + 5832)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 49 T + 2401)^{2} \) Copy content Toggle raw display
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