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Properties

Label 162.3.b.a
Level $162$
Weight $3$
Character orbit 162.b
Analytic conductor $4.414$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,3,Mod(161,162)]
 
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([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), 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: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 2 q^{4} + \beta_{2} q^{5} + (\beta_{3} - 1) q^{7} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 2 q^{4} + \beta_{2} q^{5} + (\beta_{3} - 1) q^{7} + 2 \beta_1 q^{8} + \beta_{3} q^{10} + (\beta_{2} + 3 \beta_1) q^{11} + (2 \beta_{3} + 5) q^{13} + ( - 2 \beta_{2} + \beta_1) q^{14} + 4 q^{16} + (2 \beta_{2} - 6 \beta_1) q^{17} + (2 \beta_{3} - 10) q^{19} - 2 \beta_{2} q^{20} + (\beta_{3} + 6) q^{22} + ( - \beta_{2} - 3 \beta_1) q^{23} - 2 q^{25} + ( - 4 \beta_{2} - 5 \beta_1) q^{26} + ( - 2 \beta_{3} + 2) q^{28} + (\beta_{2} - 6 \beta_1) q^{29} + ( - 3 \beta_{3} - 19) q^{31} - 4 \beta_1 q^{32} + (2 \beta_{3} - 12) q^{34} + ( - \beta_{2} + 27 \beta_1) q^{35} + ( - 2 \beta_{3} + 32) q^{37} + ( - 4 \beta_{2} + 10 \beta_1) q^{38} - 2 \beta_{3} q^{40} + (7 \beta_{2} + 18 \beta_1) q^{41} + ( - 3 \beta_{3} + 23) q^{43} + ( - 2 \beta_{2} - 6 \beta_1) q^{44} + ( - \beta_{3} - 6) q^{46} + (3 \beta_{2} - 21 \beta_1) q^{47} + ( - 2 \beta_{3} + 6) q^{49} + 2 \beta_1 q^{50} + ( - 4 \beta_{3} - 10) q^{52} + (10 \beta_{2} - 30 \beta_1) q^{53} + ( - 3 \beta_{3} - 27) q^{55} + (4 \beta_{2} - 2 \beta_1) q^{56} + (\beta_{3} - 12) q^{58} + ( - 7 \beta_{2} - 39 \beta_1) q^{59} + ( - 6 \beta_{3} - 31) q^{61} + (6 \beta_{2} + 19 \beta_1) q^{62} - 8 q^{64} + (5 \beta_{2} + 54 \beta_1) q^{65} + ( - 3 \beta_{3} + 53) q^{67} + ( - 4 \beta_{2} + 12 \beta_1) q^{68} + ( - \beta_{3} + 54) q^{70} + ( - 10 \beta_{2} - 24 \beta_1) q^{71} + ( - 6 \beta_{3} - 52) q^{73} + (4 \beta_{2} - 32 \beta_1) q^{74} + ( - 4 \beta_{3} + 20) q^{76} + (5 \beta_{2} + 24 \beta_1) q^{77} + (5 \beta_{3} - 7) q^{79} + 4 \beta_{2} q^{80} + (7 \beta_{3} + 36) q^{82} + ( - 21 \beta_{2} + 15 \beta_1) q^{83} + (6 \beta_{3} - 54) q^{85} + (6 \beta_{2} - 23 \beta_1) q^{86} + ( - 2 \beta_{3} - 12) q^{88} + ( - 10 \beta_{2} + 66 \beta_1) q^{89} + (3 \beta_{3} + 103) q^{91} + (2 \beta_{2} + 6 \beta_1) q^{92} + (3 \beta_{3} - 42) q^{94} + ( - 10 \beta_{2} + 54 \beta_1) q^{95} + (14 \beta_{3} - 7) q^{97} + (4 \beta_{2} - 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58} - 124 q^{61} - 32 q^{64} + 212 q^{67} + 216 q^{70} - 208 q^{73} + 80 q^{76} - 28 q^{79} + 144 q^{82} - 216 q^{85} - 48 q^{88} + 412 q^{91} - 168 q^{94} - 28 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 12\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.22474 + 0.707107i
1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
− 1.41421i 0 −2.00000 − 5.19615i 0 −8.34847 2.82843i 0 −7.34847
161.2 − 1.41421i 0 −2.00000 5.19615i 0 6.34847 2.82843i 0 7.34847
161.3 1.41421i 0 −2.00000 − 5.19615i 0 6.34847 − 2.82843i 0 7.34847
161.4 1.41421i 0 −2.00000 5.19615i 0 −8.34847 − 2.82843i 0 −7.34847
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.b.a 4
3.b odd 2 1 inner 162.3.b.a 4
4.b odd 2 1 1296.3.e.g 4
9.c even 3 1 18.3.d.a ✓ 4
9.c even 3 1 54.3.d.a 4
9.d odd 6 1 18.3.d.a ✓ 4
9.d odd 6 1 54.3.d.a 4
12.b even 2 1 1296.3.e.g 4
36.f odd 6 1 144.3.q.c 4
36.f odd 6 1 432.3.q.d 4
36.h even 6 1 144.3.q.c 4
36.h even 6 1 432.3.q.d 4
45.h odd 6 1 450.3.i.b 4
45.h odd 6 1 1350.3.i.b 4
45.j even 6 1 450.3.i.b 4
45.j even 6 1 1350.3.i.b 4
45.k odd 12 2 450.3.k.a 8
45.k odd 12 2 1350.3.k.a 8
45.l even 12 2 450.3.k.a 8
45.l even 12 2 1350.3.k.a 8
72.j odd 6 1 576.3.q.f 4
72.j odd 6 1 1728.3.q.d 4
72.l even 6 1 576.3.q.e 4
72.l even 6 1 1728.3.q.c 4
72.n even 6 1 576.3.q.f 4
72.n even 6 1 1728.3.q.d 4
72.p odd 6 1 576.3.q.e 4
72.p odd 6 1 1728.3.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.d.a ✓ 4 9.c even 3 1
18.3.d.a ✓ 4 9.d odd 6 1
54.3.d.a 4 9.c even 3 1
54.3.d.a 4 9.d odd 6 1
144.3.q.c 4 36.f odd 6 1
144.3.q.c 4 36.h even 6 1
162.3.b.a 4 1.a even 1 1 trivial
162.3.b.a 4 3.b odd 2 1 inner
432.3.q.d 4 36.f odd 6 1
432.3.q.d 4 36.h even 6 1
450.3.i.b 4 45.h odd 6 1
450.3.i.b 4 45.j even 6 1
450.3.k.a 8 45.k odd 12 2
450.3.k.a 8 45.l even 12 2
576.3.q.e 4 72.l even 6 1
576.3.q.e 4 72.p odd 6 1
576.3.q.f 4 72.j odd 6 1
576.3.q.f 4 72.n even 6 1
1296.3.e.g 4 4.b odd 2 1
1296.3.e.g 4 12.b even 2 1
1350.3.i.b 4 45.h odd 6 1
1350.3.i.b 4 45.j even 6 1
1350.3.k.a 8 45.k odd 12 2
1350.3.k.a 8 45.l even 12 2
1728.3.q.c 4 72.l even 6 1
1728.3.q.c 4 72.p odd 6 1
1728.3.q.d 4 72.j odd 6 1
1728.3.q.d 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 27 \) acting on \(S_{3}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 53)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 90T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 10 T - 191)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 360T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 20 T - 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 90T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 198T^{2} + 2025 \) Copy content Toggle raw display
$31$ \( (T^{2} + 38 T - 125)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 64 T + 808)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 3942 T^{2} + 455625 \) Copy content Toggle raw display
$43$ \( (T^{2} - 46 T + 43)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2250 T^{2} + 408321 \) Copy content Toggle raw display
$53$ \( T^{4} + 9000 T^{2} + 810000 \) Copy content Toggle raw display
$59$ \( T^{4} + 8730 T^{2} + 2954961 \) Copy content Toggle raw display
$61$ \( (T^{2} + 62 T - 983)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 106 T + 2323)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 7704 T^{2} + 2396304 \) Copy content Toggle raw display
$73$ \( (T^{2} + 104 T + 760)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T - 1301)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 24714 T^{2} + 131262849 \) Copy content Toggle raw display
$89$ \( T^{4} + 22824 T^{2} + 36144144 \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T - 10535)^{2} \) Copy content Toggle raw display
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